COSMOS-e’-GTachyon from string theory

Abstract

In this article, our prime objective is to study the inflationary paradigm in the context of the generalized tachyon (GTachyon) living on the world volume of a non-BPS string theory. The tachyon action is considered here is modified compared to the original action. One can quantify the amount of the modification via a power q instead of 1 / 2 in the effective action. Using this set-up we study inflation by various types of tachyonic potentials, using which we constrain the index q within, $1/2<q<2$, and a specific combination ($\propto \alpha ^{\prime }M^4_s/g_s$) of the Regge slope $\alpha ^{\prime }$, the string coupling constant $g_{s}$ and the mass scale of tachyon $M_s$, from the recent Planck 2015 and Planck+BICEP2/Keck Array joint data. We explicitly study the inflationary consequences from single field, assisted field and multi-field tachyon set-ups. Specifically for the single field and assisted field cases we derive the results in the quasi-de Sitter background in which we will utilize the details of cosmological perturbations and quantum fluctuations. Also we derive the expressions for all inflationary observables using any arbitrary vacuum and the Bunch–Davies vacuum. For the single field and the assisted field cases we derive the inflationary flow equations, new sets of consistency relations. Also we derive the field excursion formula for the tachyon, which shows that assisted inflation is on the safe side compared to the single field case to validate the effective field theory framework. Further we study the features of the CMB angular power spectrum from TT, TE and EE correlations from scalar fluctuations within the allowed range of q for each of the potentials from the single field set-up. We also put constraints from the temperature anisotropy and polarization spectra, which shows that our analysis is consistent with the Planck 2015 data. Finally, using the $\delta N$ formalism we derive the expressions for inflationary observables in the context of multi-field tachyons.

1 Introduction

The primordial inflationary paradigm is a very sublime thought aimed to explain various aspects of the early universe, in which perturbations occur and matter is created. For recent developments see Refs. [1–10]. The success of primordial inflation can be gauged by the current observations arising from the cosmic microwave background (CMB) radiation [11–13]. The observations from Planck have put interestingly tight bounds on a number of cosmological observables related to the perturbations, which also determines any departure from the Gaussian perturbations and the constraint on the tensor-to-scalar ratio, $r_{\star }$, which can potentially reveal the scale of New Physics within any given effective field theory set-up. But a big issue may crop up in model discrimination and also in the removal of the degeneracy of cosmological parameters obtained from CMB observations. Non-canonical interactions in the effective theory set-up [14–23] are among the possibilities through which one can address this issue.¹ The natural source for such non-canonical interactions is string theory. The most promising example is the Dirac–Bonn–Infeld (DBI) effective action for the tachyon field [48–51]. In this paper, we will only focus on tachyons in weakly coupled Type-IIA/IIB string theory [52, 53].

The phenomenon of tachyon condensation was introduced in Refs. [48, 49], where Type-IIA/IIB string theory and the tachyon instability on D-branes have been studied elaborately. The rolling tachyon [54–56] in weakly coupled Type-IIA/IIB string theory may be described in terms of an effective field theory for the tachyon condensate in Refs. [57–73]. The cosmological implications of the tachyon were studied in [74–86]. For more details see also Refs. [87–95]. In the context of cosmology, the detailed consequences have been studied in the following topics over many years:

• 1. Inflationary paradigm [75–86, 92],

• 2. Primordial non-Gaussianity and CMB aspects [96–104],

• 3. Reheating scenario and particle creation [105, 106],

• 4. Late time cosmic acceleration (dark energy) [107–110].

In this work, we introduce the most generalized version of the tachyon effective action in which we are interested to study the cosmological consequences from the non-canonical higher-dimensional effective field theory operators, originating from string theory. Technically we rename such a non-canonical field a generalized tachyon (GTachyon), which name we will frequently use throughout the rest of the paper. The prime motivation of writing this paper is the following:²

• To update the present status of non-canonical interaction for stringy tachyon field appearing within the framework of string theory after releasing the Planck 2015 data.

• To explicitly study the role of tachyon field and its generalized version, GTachyon, to explain the observed temperature anisotropy and polarization in the CMB angular power spectrum.

• To study the specific role of the most generalized version of non-canonical higher-dimensional effective field theory Wilsonian operators.

• To give a broad overview of the present constraints on the inflationary paradigm from the most generalized version of the tachyon string theory.

• To test the explicit dynamical features of various tachyonic potentials obtained from string theory and also to learn about the specific structural form of effective field theory operators for a specific type of potential.

Throughout the paper we have taken the following assumptions:

• 1. The tachyon field T is minimally coupled to the Einstein gravity sector.

• 2. The initial condition for inflation is fixed via Bunch–Davies (BD) vacuum [111]. For completeness we also present the results obtained from an arbitrary vacuum (AV). For a classical initial condition the amplitude of the primordial gravitational waves would be very tiny and practically undetectable, therefore this can be treated as the first observable evidence of quantum theory gravity, such as string theory. However, apart from the importance and applicability of the quantum version of a Bunch–Davis vacuum on its theoretical and observational ground, it is still not at all well understood from the previous works in this area whether the quantum Bunch–Davies vacuum is the only possible source generating a large value of the tensor-to-scalar ratio during the inflationary epoch or not. One of the prime possibilities coming from the deviation from the quantum Bunch–Davies vacuum and consideration of the quantum non-Bunch–Davies or an arbitrary vacuum in the present non-canonical picture may also be responsible for the generation of a large tensor-to-scalar ratio and large non-Gaussianity during inflation. Within the framework of effective field theory, such an arbitrary vacuum is commonly identified with the $\alpha$-vacuum [112–118], which has a string theoretic origin.

• 3. We would like to point out that the tachyon mode appears in the quantization of the open string struck to the non-BPS brane. The effective action of this tachyon field is constructed on the assumption that the tachyon field couples only to the graviton of the closed string sector with fixed vacuum expectation value for the dilation field. The open string tachyon condensation phenomenon fits in well with this assumption.

• 4. The sound speed is $c_{S}< 1$ in general for non-canonical interactions [14–23], which is the most promising ingredient for a tachyonic set-up, to generate simultaneously a detectable value of the tensor-to-scalar ratio and large non-Gaussianity. This fact can be more clearly visualized when we go to an all higher order expansion in slow-roll or more precisely by taking the exact solution of the mode functions for scalar and tensor fluctuations as obtained from cosmological perturbation theory by appropriately choosing the Bunch–Davies case or any arbitrary initial conditions for inflation.

• 5. UV cut-off of the effective field theory is fixed at $\Lambda _{\mathrm{UV}}\sim M_p$, where $M_p=2.43\times 10^{18}~\mathrm{GeV}$ is the reduced Planck mass. But in principle one can fix the scale between GUT scale and the reduced Planck scale, i.e., $\Lambda _{\mathrm{GUT}}<\Lambda _{\mathrm{UV}}\le M_p$. But in such a situation $\Lambda _{\mathrm{UV}}$ acts as a regulating parameter in the effective field theory [23, 119].

• 6. Within the region of $N_{\star }(=$ $N_{\mathrm{cmb}})\approx \mathcal{O}(50$–70) e-foldings, we will use the following constraints in the background of $\Lambda$CDM model for: Planck (2015)+WMAP-9+high L(TT) data: [120]

  $$\begin{aligned}&r(k_\star )\le 0.11~~ (\mathrm{within}~ 2\sigma ~C.L.),\end{aligned}$$

  (1.1)
  $$\begin{aligned}&\ln (10^{10}P_{S})=3.089\pm 0.036~~ (\mathrm{within}~ 2\sigma ~C.L.),\end{aligned}$$

  (1.2)
  $$\begin{aligned}&n_{S}=0.9569 \pm 0.0077~~ (\mathrm{within}~ 3\sigma ~C.L.),\end{aligned}$$

  (1.3)
  $$\begin{aligned}&\alpha _{S}=\mathrm{d}n_{S}/\mathrm{d}\ln k=0.011^{+ 0.014}_{-0.013}~~(\mathrm{within}~1.5\sigma ~C.L.),\nonumber \\\end{aligned}$$

  (1.4)
  $$\begin{aligned}&\kappa _{S}=\mathrm{d}^{2}n_{S}/\mathrm{d}\ln k^{2}=0.029^{+0.015}_{-0.016}~~(\mathrm{within}~1.5\sigma ~C.L.).\nonumber \\ \end{aligned}$$

  (1.5)

  Planck (2015)+BICEP2/Keck Array joint data: [121]

  $$\begin{aligned}&r(k_\star )\le 0.12~~ (\mathrm{within}~ 2\sigma ~C.L.),\end{aligned}$$

  (1.6)
  $$\begin{aligned}&\ln (10^{10}P_{S})=3.089^{+0.024}_{-0.027}~~ (\mathrm{within}~ 2\sigma ~C.L.),\end{aligned}$$

  (1.7)
  $$\begin{aligned}&n_{S}=0.9600 \pm 0.0071~~ (\mathrm{within}~ 3\sigma ~C.L.),\end{aligned}$$

  (1.8)
  $$\begin{aligned}&\alpha _{S}=\mathrm{d}n_{S}/\mathrm{d}\ln k=-0.022\pm 0.010\nonumber \\&\quad (\mathrm{within}~1.5\sigma ~C.L.),\end{aligned}$$

  (1.9)
  $$\begin{aligned}&\kappa _{S}=\mathrm{d}^{2}n_{S}/\mathrm{d}\ln k^{2}=0.020^{+0.016}_{-0.015}\nonumber \\&\quad (\mathrm{within}~1.5\sigma ~C.L.). \end{aligned}$$

  (1.10)

The plan of the paper is as follows:

• In Sect. 2, we have discussed the role of tachyon in non-BPS barnes in weakly coupled Type-IIA/IIB string theory and also introduced the GTachyon in the effective action of string theory.

• In Sect. 3, we have introduced and studied the features from variants of the tachyonic potentials inspired by non-BPS branes in string theory.

• In Sect. 4, we have studied the cosmological dynamics from GTachyons, in which we have explicitly discussed the unperturbed evolution and dynamical solution in various phases of the universe.

• In Sect. 5, we have explicitly studied inflationary paradigm from single, assisted and multi-field Gtachyons. Particularly for the single field and assisted field cases in the presence of GTachyon, we have derived the inflationary Hubble flow and potential dependent flow equations, new sets of consistency relations, which are valid in the slow-roll regime and field excursion formula for tachyon in terms of the inflationary observables.

• In Sect. 6, we have mentioned the future prospects of the present work and summarized the context of the present work.

In this paper, we have explored various cosmological consequences from the GTachyonic field. We start with the basic introduction of tachyons in the context of non-BPS string theory, where we also introduce the GTachyon field, in the presence of which the tachyon action is modified and one can quantify the amount of the modification via a superscript q instead of 1 / 2. This modification exactly mimics the role of effective field theory operators and studying the various cosmological features from this theory, one of our final objectives is to constrain the index q and a specific combination ($\propto \alpha ^{\prime }M^4_s/g_s$) of the Regge slope parameter $\alpha ^{\prime }$, the string coupling constant $g_{s}$ and mass scale of tachyon $M_s$, from the recent Planck 2015 and Planck+BICEP2/Keck Array joint data. To serve this purpose, we introduce various types of tachyonic potentials: the inverse cosh, logarithmic, exponential and inverse polynomials, using which we constrain the index q. To explore this issue in detail, we start with the characteristic features of the each potentials. Next we discuss the dynamics of GTachyon as well as usual tachyon for single, assisted and multi-field scenario. Next we have explicitly studied the inflationary paradigm from a single field, assisted field and multi-field tachyon set-up. Specifically for the single field and assisted field cases we have derived the results in the quasi-de Sitter background in which we have utilized the details of: (1) cosmological perturbations and quantum fluctuations for scalar and tensor modes, (2) the slow-roll prescription. In this context we have derived the expressions for all inflationary observables using any arbitrary vacuum and also for Bunch–Davies vacuum. For single field and assisted field case in the presence of GTachyon we have derived the inflationary Hubble flow and potential dependent flow equations, new sets of consistency relations valid in the slow-roll regime and also derived the expression for the field excursion formula for the tachyon in terms of the inflationary observables from both of the solutions obtained from arbitrary and Bunch–Davies initial conditions for inflation. Particularly the derived formula for the field excursion for GTachyon can be treated as an one of the important probes through which one can distinguish between various tachyon models and also check the validity of effective field theory prescription and compare the results obtained from assisted inflation as well. The results obtained in this context explicitly show that assisted inflation is better compared to single field inflation from the tachyon portal, provided the number of identical tachyon fields is required to be large to validate effective field theory prescription. Next using the explicit form of the tachyonic potentials we have studied the inflationary constraints and quantify the allowed range of the generalized index q for each of the potentials. Hence using each specific form of the tachyonic potentials in the context of the single field scenario, we have studied the features of CMB angular power spectrum from TT, TE and EE correlations from scalar fluctuations within the allowed range of q for each potentials. We also put the constraints from the Planck temperature anisotropy and polarization data, which shows that our analysis is consistent with the data. We have additionally studied the features of the tensor contribution in the CMB angular power spectrum from TT, BB, TE and EE correlations, which will give more interesting information in near future while the signature of primordial B-modes can be detected. Further, using the $\delta N$ formalism, we have derived the expressions for inflationary observables in the context of multi-field tachyons and demonstrated the results for inverse cosh potential for completeness.

2 GTachyon in string theory

In this section we explicitly study the world volume actions for non-BPS branes which finally govern their cosmological dynamics. For the sake of simplicity in this discussion we neglect the contribution from the fermions and concentrate only on the massless bosonic fields for the non-BPS branes. The world volume action for non-BPS branes is described by the sum of the Dirac–Born–Infeld (DBI) and the Wess–Zumino (WZ) terms in Type-IIA/IIB string theory. The effective action for DBI in a non-BPS p-brane is given by [122–124]

$$\begin{aligned} S^{(p)}_\mathrm{DBI}= & {} -\mathcal{T}_{p}\int \mathrm{d}^{p+1}\sigma ~e^{-\phi }~\sqrt{-\mathrm{det}\left( \mathcal{Z}_{\mu \nu }+\mathcal{F}_{\mu \nu }\right) }, \end{aligned}$$

(2.1)

where the metric has signature $(-, +, +, +)$ and $\mathcal{Z}_{\mu \nu }$ is defined as

$$\begin{aligned} \mathcal{Z}_{\mu \nu }= & {} G_{\mu \nu }+\alpha ^{\prime }\partial _{\mu }T\partial _{\nu }T. \end{aligned}$$

(2.2)

Here T is the dimensionless tachyon field whose properties have been discussed later in detail. Also in Eq. (2.1) $\mathcal{T}_{p}$ represents the brane tension defined as [122–124]

$$\begin{aligned} \begin{array}{lll} \displaystyle \mathcal{T}_{p} = \displaystyle \sqrt{2}(2\pi )^{-p}g^{-1}_{s} \end{array} \end{aligned}$$

(2.3)

and $\alpha ^{\prime }$ represents the Regge slope parameter in string theory. Here Type-IIA/IIB string theory contains the non-BPS Dp-branes [125], which have precisely those dimensions which BPS D-branes do not have explicitly. This implies that Type-IIA string theory has non-BPS Dp-branes for only odd p and Type-IIB string theory has non-BPS Dp-branes only for even p in the present context. Additionally, it is important to note that $g_{s}$ characterizes the string coupling constant. Also in Eq. (2.1), $G_{\mu \nu }$ is defined via the following transformation equation:

$$\begin{aligned} G_{\mu \nu }=G_{MN}\partial _{\mu }X^{M}\partial _{\nu }X^{N}. \end{aligned}$$

(2.4)

In Eq. (2.4) M, N characterize the ten-dimensional ($D=10$) indices, which run from $0,1,\ldots ,9$; $\sigma ^{\mu }(0\le \mu \le p)$ denotes the world volume coordinates of the Dp brane. Also it is important to note that, in this discussion, $G_{MN}$ represents the ten-dimensional ($D=10$) background metric for Type-IIA/IIB string theory.

It is important to mention here that, in the context of non-BPS D-brane, the stringy tachyon comes from only one specific sector of string theory and consequently it is a real scalar field using which we will explain the cosmological dynamics in this article for $p=3$. Additionally it is also important to note that Type-IIA/IIB string theories contain unstable non-BPS D-branes in their spectra. The most easiest way to define these types of D-branes in the context of IIA/IIB string theory is to start the computation with a coincident BPS Dp-$\bar{\mathrm{D}}p$-brane pair in Type-IIB/IIA string theory, and then take a specific orbifold of the string theory by $(-1)^{F_L}$, where $F_L$ signifies the specific contribution to the space-time fermion number from the left-moving sector of the world-sheet string theory. Now in this context the Ramond–Ramond (RR) fields are odd under $(-1)^{F_L}$ transformation and consequently all the Ramond–Ramond (RR) fields of Type-IIB/IIA string theory are projected out by the same amount via $(-1)^{F_L}$ projection. As a result the twisted sector stringy states then give us back the RR fields of Type-IIA/IIB string theory in the present context. Most importantly, here the $(-1)^{F_L}$ projection reverses the signature of the RR charge and consequently it transforms a BPS Dp-brane to a $\bar{\mathrm{D}}p$-brane and vice versa. This further implies that due to its operation on the open string states on a Dp–$\bar{\mathrm{D}}p$-brane stringy system will do the job of conjugate operation on the Chan–Paton factor by the action of exchange operator $\sigma _{1}$ in this context. Thus technically the modding out operation on the Dp–$\bar{\mathrm{D}}p$-brane by exactly the amount of $(-1)^{F_L}$ eliminates all the open string states which carry a Chan–Paton factor $\sigma _{2}$ and $\sigma _{3}$, as both of them anti-commute with the exchange operator $\sigma _{1}$. Additionally it is important to note that this operation finally keeps the open string states which are characterized via the Chan–Paton factors I and $\sigma _{1}$. Finally all such operations give us a non-BPS Dp-brane in the present context. Although in this discussion we are only interested in the non-BPS D-branes, the most important characteristic feature that distinguishes the physics of non-BPS D-branes from BPS D-branes is that the mass spectrum of open strings on a non-BPS D-brane contains a single mode of negative mass squared besides an infinite number of other modes of positive definite mass squared. This negative mass squired mode is identified with the tachyonic mode which is exactly equivalent to a particular linear combination of the two tachyonic modes living on the original brane–antibrane pair stringy system that survives the previously mentioned $(-1)^{F_L}$ projection and contains exactly the same mass squared contribution.

In our analysis for the sake of simplicity we have neglected the contribution from the antisymmetric Kalb–Ramond two form field from the effective action but the gauge invariance of the action requires the presence of all such antisymmetric tensor contributions in the original version of the string effective action. In the present context, it is important to note that $G_{\mu \nu }$ can be physically interpreted as the induced metric on the membrane. Additionally it is important to note that the background metric $G_{MN}$ is not at all arbitrary for the present set-up but the structural form of the metric is restricted in the specific sense that it has to satisfy the sets of background field equations in this context. Also in our discussion the transverse component of the fluctuations of the D-membrane is described by ($9-p$) scalar fields $X^{i}$, where the index i runs from $p+1\le i\le 9$, and the gauge field $A_{\mu }$ describes the fluctuations along the longitudinal direction of the membrane.

Before writing down the total effective action for non-BPS D-brane in string theory it is important to mention that the non-BPS p-brane has an extra tachyon field appearing in both of the Dirac–Born–Infeld (DBI) and the Wess–Zumino (WZ) stringy effective actions. The corresponding effective actions can be written in the non-BPS string theory set-up as [122–124]

$$\begin{aligned} S^{(p)}_\mathrm{DBI}= & {} -\int \mathrm{d}^{p+1}\sigma ~e^{-\phi }~\sqrt{-\mathrm{det}\left( \mathcal{Z}_{\mu \nu }+\mathcal{F}_{\mu \nu }\right) }~\nonumber \\&\quad \times \Theta (T,\partial _{\mu }T,D_{\mu }\partial _{\nu }T),\end{aligned}$$

(2.5)

$$\begin{aligned} S^{(p)}_\mathrm{WZ}= & {} \int \mathrm{d}^{p+1}\sigma ~C\wedge \mathrm{d}T\wedge e^{\mathcal{F}}, \end{aligned}$$

(2.6)

where the field C contains Ramond–Ramond (RR) fields and the leading term has a ($p+1$)-differential form. This also mimics the role of a source term for the membrane and its presence is explicitly required for consistency of the specific version of the field theory like anomaly cancellation within the set-up of string theory. It is important to mention here that we have the world volume action for a $\mathrm{Dp}$-brane in the $(p+1)$-dimensional case, where p characterizes the spatial dimension and 1 stands for time. The WZ term plays an important role since the BPS $\mathrm{Dp}$-brane is charged under a $(p+1)$ rank RR gauge field. Consequently the total action is therefore the DBI action together with the WZ action. For the non-BPS $\mathrm{Dp}$-brane, the brane is charged under a rank p RR gauge field. As a result the WZ action consists of a wedge product of this p form and additionally a one form $\mathrm{d}T$, where T is the tachyon field. Also it is important to note that for the non-BPS case $\mathcal{F}_{\mu \nu }$ is explicitly defined as [122–124]

$$\begin{aligned} \mathcal{F}_{\mu \nu }= & {} B_{\mu \nu }+2\pi \alpha ^{\prime }F_{\mu \nu } +\partial _{\mu }Y^{I}\partial _{\nu }Y^{I}\nonumber \\&+\left( G_{IJ}+B_{IJ}\right) \partial _{\mu }Y^{I}\partial _{\nu }Y^{J}+\left( G_{\mu I}+B_{\mu I}\right) \partial _{\nu }Y^{I}\nonumber \\&+\;\left( G_{I\nu }+B_{I\nu }\right) \partial _{\mu }Y^{I}, \end{aligned}$$

(2.7)

and T represents the dimensionless tachyon field and $\Theta$ characterizes the generalized functional in non-BPS p-membrane. In Eq. (2.7), the rank-2 field strength tensor $F_{\mu \nu }$ is defined as

$$\begin{aligned} F_{\mu \nu }=\partial _{[\mu }A_{\nu ]} \end{aligned}$$

(2.8)

and $B_{\mu \nu }$ represents a rank-2 Kalb–Ramond field and sometimes this can be interpreted as the pullback of $B_{MN}$ onto the D-brane world volume. Using this specific action, mentioned in Eq. (2.7), we can compute the source contributions and terms for various closed string fields produced by the membrane. Additionally, it is important to note that, on a non-BPS Dp-membrane world volume we have an infinite tower of massive fields, a $\mathrm{U}$(1) gauge field $A_{\mu }$ with the restriction on gauge indices, $0 \le \mu \le p$, and a set of scalar fields $Y^{I}$, one for each direction $y^{I}$ transverse besides the tachyonic field to the D-brane. Here it is important to note that $(p + 1) \le I \le D$, where D is 9 for superstring theory and 25 for bosonic string theory. Within the present set-up the tachyon field is defined in such a way that for

$$\begin{aligned} T=0,~~F=\mathcal{T}_{p}, \end{aligned}$$

(2.9)

the a constraint condition is explicitly satisfied. For a non-BPS p membrane the field content C as appearing in the Wess–Zumino (WZ) action contains the Ramond–Ramond (RR) fields but careful observation clearly indicates that the leading order contribution in C is characterized by the p form in the effective action. Also it is important to mention here that, for constant tachyon background T, the Wess–Zumino (WZ) effective action automatically vanishes in the present context as

$$\begin{aligned} \mathrm{d}T=0, \end{aligned}$$

(2.10)

and for such a specific background the generalized functional $\Theta$ can be recast in the following form:

$$\begin{aligned} \Theta (T,\partial _{\mu }T,D_{\mu }\partial _{\nu }T)= & {} V(T), \end{aligned}$$

(2.11)

where V(T) represents the effective tachyon potential within which the contribution from the membrane tension is already taken. Consequently Eq. (2.5) can be recast in the following simplified form:

$$\begin{aligned} S^{(p)}_\mathrm{DBI}= & {} -\int \mathrm{d}^{p+1}\sigma ~e^{-\phi }~V(T)~\sqrt{-\mathrm{det}\left( \mathcal{Z}_{\mu \nu }+\mathcal{F}_{\mu \nu }\right) }.\nonumber \\ \end{aligned}$$

(2.12)

For non-BPS string theory in the constant dilaton background the purely tachyonic part of the action, after inclusion of the massless fields on the Dp-membrane world volume around the tachyon vacuum, is given by

$$\begin{aligned} S^{(p)}_\mathrm{D}= & {} -\int \mathrm{d}^{p+1}\sigma ~V(T)~\sqrt{-\mathrm{det}\left( \mathcal{Z}_{\mu \nu }+F_{\mu \nu }+\partial _{\mu }Y^{I}\partial _{\nu }Y^{I}\right) }.\nonumber \\ \end{aligned}$$

(2.13)

After neglecting the contribution from the massless fields for non-BPS string theory the tachyonic part of the effective action describing the Dp-brane world volume is given by [122–124] the following simplified form:

$$\begin{aligned} S^{(p)}_\mathrm{D}= & {} -\int \mathrm{d}^{p+1}\sigma ~V(T)~\sqrt{-\mathrm{det}\left( \mathcal{Z}_{\mu \nu }\right) } \nonumber \\= & {} -\int \mathrm{d}^{p+1}\sigma ~V(T)~\sqrt{-\mathrm{det}\left( G_{\mu \nu }+\alpha ^{\prime }\partial _{\mu }T\partial _{\nu }T\right) }\nonumber \\= & {} -\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~V(T)~\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T}.\nonumber \\ \end{aligned}$$

(2.14)

In a more generalized prescription Eq. (2.14) can be modified into the following effective action:

$$\begin{aligned} S^{(pq)}_\mathrm{D}= & {} -\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~V(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q},\nonumber \\ \end{aligned}$$

(2.15)

which we identified with the most generalized Gtachyon action in string theory. Here for $p=3$, i.e., for the D3 brane, Eq. (2.14) refers to the following crucial issues:

• Here $p=3, q=1/2$ corresponds to the exact tachyonic behavior of the effective action and it is commonly used to describe the cosmological dynamics,

• Here $p=3, q=1$ corresponds to the single field behavior in cosmological dynamics where the kinetic term of the tachyon field is non-canonical. In this case the non-canonical contribution in the effective action is given by $\alpha ^{\prime }V(T)$, where V(T) is the tachyon effective potential. This situation is different from the usual single field models of inflation where the kinetic term is canonical within the framework of string theory.

• For $p=3, 1/2<q<1$ or $p=3, q<1/2$ or $p=3, q>1$ it contains various non-trivial features in cosmological dynamics. In this case the effective action is significantly different from the usual tachyon action as appearing in the context of string theory. In this case the effective action describes a huge class of effective field theories of inflationary models which can be embedded within the framework of tachyon in string theory. In the present context, the generalized factor $V(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q}$ with exponent q can be treated as Wilsonian operators as appearing in the context of effective field theory. For an example let us consider a situation where we treat the Regge slope parameter $\alpha ^{\prime }$ to be small. In this case the generalized factor $V(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q}$ takes the following structure:

  $$\begin{aligned}&V(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q} \nonumber \\&\quad =\;V(T)~\sum ^{q}_{k=0}{}^{q}C_{k}\left( \alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{k}\nonumber \\&\quad =\;V(T)\left[ 1+q\left( \alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) \right. \nonumber \\&\qquad \left. +\;\frac{q(q-1)}{2}\left( \alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^2+\cdots \right] , \end{aligned}$$

  (2.16)

  where $\cdots$ contains higher order terms which are suppressed by the powers of the Regge slope parameter $\alpha ^{\prime }$. Here for each value of $k(=0,1,2,\ldots ,q)$ the expansion factor ${}^{q}C_{k}\left( \alpha ^{\prime }\right) ^{k}$ mimics the role of Wilson coefficients.

Here we would like to point out that the tachyon mode appears in the quantization of the open string struck to the non-BPS brane. The effective action of this tachyon field is constructed on the assumption that the tachyon field couples only to the graviton of the closed string sector with fixed vacuum expectation value for dilation field. The open string tachyon condensation phenomenon fits in well with this assumption.

For a multi-tachyonic field scenario one can generalize the tachyonic part of the non-BPS action as stated in Eqs. (2.14) and (2.15):

$$\begin{aligned} S^{(p)}_\mathrm{D}= & {} -\sum ^{N}_{i=1}\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~V(T_i)~\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_i \partial _{\nu }T_i},\nonumber \\\end{aligned}$$

(2.17)

$$\begin{aligned} S^{(pq)}_\mathrm{D}= & {} -\sum ^{N}_{i=1}\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~V(T_i)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_i \partial _{\nu }T_i \right) ^{q}.\nonumber \\ \end{aligned}$$

(2.18)

In this case one can introduce a total effective potential of tachyonic fields $V_{E}(T)$, which can be expressed in terms of N component tachyonic fields as

$$\begin{aligned} V_{E}(T)=\sum ^{N}_{i=1}V(T_i), \end{aligned}$$

(2.19)

which is very useful to study the cosmological dynamics for the $p=3$ case, i.e., for the D3 brane.

Also for the assisted case one can assume all N multi-tachyonic fields are identical to each other, i.e.,

$$\begin{aligned} T_{i}=T \quad \forall \; i=1,2,3,\ldots ,N. \end{aligned}$$

(2.20)

Consequently in such a prescription Eqs. (2.17) and (2.18) can be rewritten as

$$\begin{aligned} S^{(p)}_\mathrm{D}= & {} -\sum ^{N}_{i=1}\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~V(T)~\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T \partial _{\nu }T} \nonumber \\= & {} -\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~NV(T)~\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T \partial _{\nu }T},\nonumber \\\end{aligned}$$

(2.21)

$$\begin{aligned} S^{(pq)}_\mathrm{D}= & {} -\sum ^{N}_{i=1}\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~V(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T \partial _{\nu }T_i \right) ^{q}\nonumber \\= & {} -\int \mathrm{d}^{p+1}\sigma ~\sqrt{-g}~NV(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T \right) ^{q}.\nonumber \\ \end{aligned}$$

(2.22)

In this case the total effective potential of the tachyonic fields $V_{E}(T)$ can be recast as

$$\begin{aligned} V_{E}(T)=\sum ^{N}_{i=1}V(T_i)=\sum ^{N}_{i=1}V(T)=NV(T). \end{aligned}$$

(2.23)

In the next section we will discuss the various aspects of tachyonic potential V(T) and also mention the various models of tachyonic potential that can be derived from a string theory background.

3 Variants of tachyonic models in string theory

On general string theoretic grounds, one can argued that at the specified minimum $T_{0}$ of the effective potential V(T) vanishes [122–124], i.e.,

$$\begin{aligned} V(T_{0})=0. \end{aligned}$$

(3.1)

Consequently the world volume action vanishes identically and in this situation the gauge field mimics the role of Lagrange multiplier field. Finally this imposes a constraint on the non-BPS set-up such that the gauge current also vanishes identically. This implies that all the states which are charged under this gauge field are to disappear from the spectrum. Also it is important to mention another important feature of the tachyonic potential in which it admits kink profile for the stringy tachyonic field. Also on an unstable non-BPS p-brane tachyon condensation occurs to form a kink profile and finally it forms a stable BPS ($p-1$) brane configuration. This kink profile for the tachyon is expected to give a $\delta$-function from $\mathrm{d}T$- contribution and thus to reproduce the standard WZ term in the resulting $D(p-1)$-brane. Most importantly, the kink solution effectively reduces the dimension of the world volume by one. Although finding an explicit form of the tachyonic potential is a very difficult, string theory predicts the approximated form of the tachyonic potential. Additionally, we also assume that the tachyonic potential V(T) satisfies the following properties to describe the cosmological dynamics for the non-BPS D3 brane set-up:

• 1. The tachyonic potential at $T=0$ satisfies

  $$\begin{aligned} V(T=0)=\lambda =\frac{M^4_s}{(2\pi )^3 g_s}, \end{aligned}$$

  (3.2)

  where $g_{s}$ is the string coupling constant and $M_{s}$ signifies the mass scale of the tachyonic string theory. For multi-tachyonic field and assisted case Eq. (3.2) is modified as

  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}V(T_{i}=0)=\sum ^{N}_{i=1}\lambda _{i}\nonumber \\= & {} \sum ^{N}_{i=1}\frac{M^4_s}{(2\pi )^3 g^{(i)}_s},\end{aligned}$$

  (3.3)
  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}V(T_{i}=T=0)=\sum ^{N}_{i=1}\lambda _{i}=N\lambda \nonumber \\= & {} \frac{M^4_s N}{(2\pi )^3 g_s}. \end{aligned}$$

  (3.4)

  Here $g^{(i)}_s$ represents the string coupling constant for the ith field content, which are not same for all N tachyonic degrees of freedom. For the sake of simplicity, here we also assume that the mass scale associated with N tachyons for multi-tachyonic case and assisted case is identical for all degrees of freedom. On the other hand, for the assisted case we assume all couplings are exactly identical and consequently we get an overall factor of N multiplied with the result obtained for the single tachyonic field case.

• 2. Inflation generally takes place at an energy scale:

  $$\begin{aligned} V^{1/4}_{\mathrm{inf}}=V^{1/4}(\tilde{T}_{0})\propto \lambda ^{1/4} \end{aligned}$$

  (3.5)

  with the single tachyon field fixed at $T \sim \tilde{T}_{0}$, and within the set-up of string theory $\tilde{T}_{0}$ is identified to be the mass scale of the tachyon by in the following fashion:

  $$\begin{aligned} \tilde{T}_{0} \sim M_{s}. \end{aligned}$$

  (3.6)

  For the multi-tachyonic field case Eqs. (3.5) and (3.6) are modified as

  $$\begin{aligned} V^{1/4}_{\mathrm{inf}}= & {} V^{1/4}_{E}=\left\{ \sum ^{N}_{i=1}V(\tilde{T}_{0i})\right\} ^{1/4}\propto \left\{ \sum ^{N}_{i=1}\lambda _{i}\right\} ^{1/4}\nonumber \\= & {} \left\{ \sum ^{N}_{i=1}\frac{M^4_s}{(2\pi )^3 g^{(i)}_s}\right\} ^{1/4}, \end{aligned}$$

  (3.7)
  $$\begin{aligned}&\tilde{T}_{0} \sim \sum ^{N}_{i=1} \tilde{T}_{0i}=M_{s}. \end{aligned}$$

  (3.8)

  Similarly, for the assisted case Eqs. (3.5) and (3.6) are modified as

  $$\begin{aligned} V^{1/4}_{\mathrm{inf}}= & {} V^{1/4}_{E}=\left\{ \sum ^{N}_{i=1}V(\tilde{T}_{0i})\right\} ^{1/4}\propto \left\{ \sum ^{N}_{i=1}\lambda _{i}\right\} ^{1/4}\nonumber \\= & {} (N\lambda )^{1/4}=\left\{ \frac{M^4_s N}{(2\pi )^3 g_s}\right\} ^{1/4}, \end{aligned}$$

  (3.9)
  $$\begin{aligned} \tilde{T}_{0}\sim & {} \sum ^{N}_{i=1} \tilde{T}_{0i}=\sum ^{N}_{i=1} \tilde{T}_{0}=N\tilde{T}_{0}=NM_{s}. \end{aligned}$$

  (3.10)

• 3. For $T>0$ the first derivative of the single tachyonic potential is always positive, i.e.,

  $$\begin{aligned} V^{\prime }(T>0)>0 \end{aligned}$$

  (3.11)

  where $'$ represents differentiation with respect to the tachyon field T. For the multi-tachyonic and assisted cases Eq. (3.11) can be recast as

  $$\begin{aligned} V^{\prime }_{E}(T>0)= & {} \sum ^{N}_{i=1}\frac{\mathrm{d}V(T_{i}>0)}{\mathrm{d}T_i}>0,\end{aligned}$$

  (3.12)
  $$\begin{aligned} V^{\prime }_{E}(T>0)= & {} \sum ^{N}_{i=1}\frac{\mathrm{d}V(T_{i}>0)}{\mathrm{d}T_i}=\sum ^{N}_{i=1}\frac{\mathrm{d}V(T>0)}{\mathrm{d}T}\nonumber \\= & {} N V^{\prime }(T>0)>0. \end{aligned}$$

  (3.13)

• 4. At the asymptotic case of the single tachyonic field $|T|\rightarrow \infty$ the potential should satisfy

  $$\begin{aligned} V(|T|\rightarrow \infty )\rightarrow 0. \end{aligned}$$

  (3.14)

  For the multi-tachyonic and assisted cases Eq. (3.13) can be recast thus:

  $$\begin{aligned} V_{E}(|T|\rightarrow \infty )= & {} \sum ^{N}_{i=1}V(|T_{i}|\rightarrow \infty )\rightarrow 0,\end{aligned}$$

  (3.15)
  $$\begin{aligned} V_{E}(|T|\rightarrow \infty )= & {} \sum ^{N}_{i=1}V(|T_{i}|\rightarrow \infty )\nonumber \\= & {} NV(|T|\rightarrow \infty )\rightarrow 0. \end{aligned}$$

  (3.16)

• 5. Also one can consider that the tachyonic potential contains a global maximum at $T = 0$, i.e.,

  $$\begin{aligned} V^{\prime \prime }(T=0)<0 \end{aligned}$$

  (3.17)

  for which the value of the potential is given by Eq. (3.2). Similarly for the multi-tachyonic and assisted cases Eq. (3.18) can be recast thus:

  $$\begin{aligned} V^{\prime \prime }_{E}(T=0)= & {} \sum ^{N}_{i=1}\frac{\mathrm{d}^{2}V(T_{i}>0)}{\mathrm{d}T^{2}_i}<0,\end{aligned}$$

  (3.18)
  $$\begin{aligned} V^{\prime \prime }_{E}(T=0)= & {} \sum ^{N}_{i=1}\frac{\mathrm{d}^{2}V(T_{i}>0)}{\mathrm{d}T^{2}_i}=\sum ^{N}_{i=1}\frac{\mathrm{d}^{2}V(T>0)}{\mathrm{d}T^{2}}\nonumber \\= & {} NV^{\prime \prime }(T=0)<0. \end{aligned}$$

  (3.19)

In the next subsections we mention variants of the tachyonic potentials which satisfy the various above mentioned characteristics.

3.1 Model I: inverse cosh potential

For single field case the first model of the tachyonic potential is given by [55, 56, 84, 126]

$$\begin{aligned} V(T)=\frac{\lambda }{\mathrm{cosh}\left( \frac{T}{T_{0}}\right) }, \end{aligned}$$

(3.20)

and for the multi-tachyonic and assisted cases the total effective potential is given by

$$\begin{aligned} \begin{array}{lll} \displaystyle V_{E}(T) =\left\{ \begin{array}{lll} \displaystyle \sum ^{N}_{i=1}\frac{\lambda _{i}}{\mathrm{cosh}\left( \frac{T_i}{T_{0i}}\right) }, &{} \quad \mathrm{for~~Multi~~tachyonic} ,\\ \displaystyle \sum ^{N}_{i=1}\frac{\lambda }{\mathrm{cosh}\left( \frac{T}{T_{0}}\right) }\\ \quad =\frac{N\lambda }{\mathrm{cosh}\left( \frac{T}{T_{0}}\right) }, &{} \quad \mathrm{for~~Assisted~~tachyonic}. \end{array} \right. \end{array} \end{aligned}$$

(3.21)

Here the potential satisfies the following criteria:

• At $T=0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=0)=\lambda \end{aligned}$$

  (3.22)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.23)
  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i}=N\lambda . \end{aligned}$$

  (3.24)

• At $T=T_0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=T_0)=\frac{\lambda }{\mathrm{cosh}(1)}\ne 0 \end{aligned}$$

  (3.25)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\frac{\lambda _{i}}{\mathrm{cosh}(1)}\ne 0,\end{aligned}$$

  (3.26)
  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\frac{\lambda _{i}}{\mathrm{cosh}(1)}=\frac{N\lambda }{\mathrm{cosh}(1)}\ne 0. \end{aligned}$$

  (3.27)

• For the single field tachyonic potential:

  $$\begin{aligned} V^{\prime }(T)= & {} -\frac{\lambda }{T_0}\mathrm{sech}\left( \frac{T}{T_0}\right) \mathrm{tanh}\left( \frac{T}{T_0}\right) ,\end{aligned}$$

  (3.28)
  $$\begin{aligned} V^{\prime \prime }(T)= & {} -\frac{\lambda }{T^2_0}\left[ \mathrm{sech}^3\left( \frac{T}{T_0}\right) \right. \nonumber \\&\left. -\,\mathrm{sech}\left( \frac{T}{T_0}\right) \mathrm{tanh}^2\left( \frac{T}{T_0}\right) \right] . \end{aligned}$$

  (3.29)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T)=0 \end{aligned}$$

  (3.30)

  which give rise to the following solutions for the tachyonic field:

  $$\begin{aligned} |T|=2m~ T_{0}\pi ,\quad \left( 2m+1\right) ~T_{0}\pi \end{aligned}$$

  (3.31)

  where $m\in \mathbb {Z}$. Further substituting the solutions for tachyonic field in Eq. (3.29) we get

  $$\begin{aligned}&V^{\prime \prime }(|T|=2m~ T_{0}\pi )=-\frac{\lambda }{T^2_0},\end{aligned}$$

  (3.32)
  $$\begin{aligned}&V^{\prime \prime }(|T|=\left( 2m+1\right) ~T_{0}\pi )=-\frac{\lambda }{T^2_0}, \end{aligned}$$

  (3.33)

  and at these points the value of the potential is computed as

  $$\begin{aligned}&V(|T|=2m~ T_{0}\pi )=\frac{\lambda }{\mathrm{cosh}\left( 2m~\pi \right) }=\lambda ~ \mathrm{sech}\left( 2m~\pi \right) ,\nonumber \\\end{aligned}$$

  (3.34)
  $$\begin{aligned}&V(|T|=\left( 2m+1\right) ~T_{0}\pi )=\frac{\lambda }{\mathrm{cosh}\left( \left( 2m+1\right) ~\pi \right) } \nonumber \\&\qquad = \; \lambda ~ \mathrm{sech}\left( \left( 2m+1\right) ~\pi \right) . \end{aligned}$$

  (3.35)

  It is important to note for the single tachyonic case that for $\lambda >0$, $V^{\prime \prime }(|T|=2m~ T_{0}\pi ,~~ \left( 2m+1\right) ~T_{0}\pi )>0$, i.e., we get maxima on the potential and for the assisted case the results are the same, provided the following replacement occurs:

  $$\begin{aligned} \lambda \rightarrow N\lambda , \end{aligned}$$

  (3.36)

  and finally for the multi-tachyonic case we have

  $$\begin{aligned} V^{\prime }(T_{i})= & {} -\frac{\lambda _{i}}{T_{0i}}\mathrm{sech}\left( \frac{T_{i}}{T_{0i}}\right) \mathrm{tanh}\left( \frac{T_{i}}{T_{0i}}\right) ,\end{aligned}$$

  (3.37)
  $$\begin{aligned} V^{\prime \prime }(T_{i})= & {} -\frac{\lambda _{i}}{T^2_{0i}}\left[ \mathrm{sech}^3\left( \frac{T_{i}}{T_{0i}}\right) \right. \nonumber \\&\left. -\;\mathrm{sech}\left( \frac{T_{i}}{T_{0i}}\right) \mathrm{tanh}^2\left( \frac{T_{i}}{T_{0i}}\right) \right] . \end{aligned}$$

  (3.38)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T_{j})=0 \quad \forall \; j=1,2,\ldots ,N \end{aligned}$$

  (3.39)

  which gives rise to the following solutions for the jth tachyonic field:

  $$\begin{aligned} |T_{j}|=2m~ T_{0j}\pi , \quad \left( 2m+1\right) ~T_{0j}\pi \quad \forall \; j=1,2,\ldots ,N\nonumber \\ \end{aligned}$$

  (3.40)

  where $m\in \mathbb {Z}$. Further substituting the solutions for tachyonic field in Eq. (3.38) we get

  $$\begin{aligned}&V^{\prime \prime }(|T_{j}|=2m~T_{0j}\pi )=-\frac{\lambda _{j}}{T^2_{0j}},\end{aligned}$$

  (3.41)
  $$\begin{aligned}&V^{\prime \prime }(|T_{j}|=\left( 2m+1\right) ~T_{0j}\pi )=-\frac{\lambda _{j}}{T^2_{0j}}, \end{aligned}$$

  (3.42)

  and at these points the value of the total effective potential is computed as

  $$\begin{aligned} V^{(1)}_{E}= & {} \sum ^{N}_{j=1}V(|T_{j}|=2m~ T_{0j}\pi )=\sum ^{N}_{j=1}\frac{\lambda _{j}}{\mathrm{cosh}\left( 2m~\pi \right) }\nonumber \\= & {} \mathrm{sech}\left( 2m~\pi \right) \sum ^{N}_{j=1}\lambda _{j},\end{aligned}$$

  (3.43)
  $$\begin{aligned} V^{(2)}_{E}= & {} \sum ^{N}_{j=1}V(|T_{j}|=\left( 2m+1\right) ~T_{0j}\pi )\nonumber \\= & {} \sum ^{N}_{j=1}\frac{\lambda _{j}}{\mathrm{cosh}\left( \left( 2m+1\right) ~\pi \right) }\nonumber \\= & {} \mathrm{sech}\left( \left( 2m+1\right) ~\pi \right) \sum ^{N}_{j=1}\lambda _{j}. \end{aligned}$$

  (3.44)

  It is important to note for the multi-tachyonic case that for $\lambda _{j}>0$, $V^{\prime \prime }(|T_{j}|=2m~ T_{0j}\pi ,~~ \left( 2m+1\right) ~T_{0j}\pi )>0$, i.e., we get maxima on the potential $V(T_{j})$ as well as in $V_{E}(T)$.

3.2 Model II: logarithmic potential

For single field case the second model of the tachyonic potential is given by [79]

$$\begin{aligned} V(T)=\lambda \left\{ \left( \frac{T}{T_{0}}\right) ^2\left[ \ln \left( \frac{T}{T_{0}}\right) \right] ^2+1\right\} , \end{aligned}$$

(3.45)

and for the multi-tachyonic and assisted cases the total effective potential is given by

$$\begin{aligned} \begin{array}{lll} \displaystyle V_{E}(T) \\ \quad =\left\{ \begin{array}{lll} \displaystyle \sum ^{N}_{i=1}\lambda _{i}\left\{ \left( \frac{T_i}{T_{0i}}\right) ^2 \left[ \ln \left( \frac{T_i}{T_{0i}}\right) \right] ^2+1\right\} , &{} \ \mathrm{for~~Multi~~tachyonic}, \\ \displaystyle \sum ^{N}_{i=1}\lambda \left\{ \left( \frac{T}{T_{0}}\right) ^2\left[ \ln \left( \frac{T}{T_{0}}\right) \right] ^2+1\right\} \\ \displaystyle =N\lambda \left\{ \left( \frac{T}{T_{0}}\right) ^2\left[ \ln \left( \frac{T}{T_{0}}\right) \right] ^2+1\right\} , &{} \ \mathrm{for~~Assisted~~tachyonic}. \end{array} \right. \end{array} \end{aligned}$$

(3.46)

Here the potential satisfies the following criteria:

• At $T=0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=0)=\lambda \end{aligned}$$

  (3.47)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.48)
  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i}=N\lambda . \end{aligned}$$

  (3.49)

• At $T=T_0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=T_0)=\lambda , \end{aligned}$$

  (3.50)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=T_0)= & {} \sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.51)
  $$\begin{aligned} V_{E}(T=T_0)= & {} \sum ^{N}_{i=1}\lambda _{i}=N\lambda . \end{aligned}$$

  (3.52)

• For the single field tachyonic potential:

  $$\begin{aligned} V^{\prime }(T)= & {} \frac{2 T\lambda }{T^2_0}\ln \left( \frac{T}{T_0}\right) \left[ 1+\ln \left( \frac{T}{T_0}\right) \right] ,\end{aligned}$$

  (3.53)
  $$\begin{aligned} V^{\prime \prime }(T)= & {} \frac{2\lambda }{T^2_0}\left[ 1+3\ln \left( \frac{T}{T_0}\right) +\ln ^2 \left( \frac{T}{T_0}\right) \right] . \end{aligned}$$

  (3.54)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T)=0 \end{aligned}$$

  (3.55)

  which give rise to the following solutions for the tachyonic field:

  $$\begin{aligned} T=0, \quad T_{0}, \quad \frac{T_{0}}{e}. \end{aligned}$$

  (3.56)

  Further substituting the solutions for the tachyonic field in Eq. (3.54) we get

  $$\begin{aligned}&V^{\prime \prime }(T=0)\rightarrow \infty ,\end{aligned}$$

  (3.57)
  $$\begin{aligned}&V^{\prime \prime }(T=T_{0})=\frac{2\lambda }{T^2_0},\end{aligned}$$

  (3.58)
  $$\begin{aligned}&V^{\prime \prime }\left( T=\frac{T_{0}}{e}\right) =-\frac{2\lambda }{T^2_0}, \end{aligned}$$

  (3.59)

  and at these points the value of the potential is computed as

  $$\begin{aligned}&V(T=0)=\lambda ,\end{aligned}$$

  (3.60)
  $$\begin{aligned}&V(T=T_{0})=\lambda ,\end{aligned}$$

  (3.61)
  $$\begin{aligned}&V\left( T=\frac{T_{0}}{e}\right) =\lambda \left( 1+\frac{1}{e^2}\right) . \end{aligned}$$

  (3.62)

  It is important to note for the single tachyonic case that for $\lambda >0$, $V^{\prime \prime }(T=T_{0})>0$ and $V^{\prime \prime }(T=\frac{T_{0}}{e})<0$ i.e., we get both maxima and minima on the potential. For the assisted case the results are the same, provided the following replacement occurs:

  $$\begin{aligned} \lambda \rightarrow N\lambda , \end{aligned}$$

  (3.63)

  and finally for the multi-tachyonic case we have

  $$\begin{aligned} V^{\prime }(T_{i})= & {} \frac{2 \lambda _{i}T_{i} }{T^2_{0i}}\ln \left( \frac{T_{i}}{T_{0i}}\right) \left[ 1+\ln \left( \frac{T_{i}}{T_{0i}}\right) \right] ,\end{aligned}$$

  (3.64)
  $$\begin{aligned} V^{\prime \prime }(T_{i})= & {} \frac{2\lambda _{i}}{T^2_{0i}}\left[ 1+3\ln \left( \frac{T_{i}}{T_{0i}}\right) +\ln ^2 \left( \frac{T_{i}}{T_{0i}}\right) \right] . \end{aligned}$$

  (3.65)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T_{j})=0\quad \forall j=1,2,\ldots ,N \end{aligned}$$

  (3.66)

  which give rise to the following solutions for the jth tachyonic field:

  $$\begin{aligned} T_{j}=0, \quad T_{0j}, \quad \frac{T_{0j}}{e} \quad \forall \; j=1,2,\ldots ,N. \end{aligned}$$

  (3.67)

  Further substituting the solutions for the tachyonic field in Eq. (3.65) we get

  $$\begin{aligned}&V^{\prime \prime }(T_{j}=0)\rightarrow \infty ,\end{aligned}$$

  (3.68)
  $$\begin{aligned}&V^{\prime \prime }(T_{j}=T_{0j})=\frac{2\lambda _{j}}{T^2_{0j}},\end{aligned}$$

  (3.69)
  $$\begin{aligned}&V^{\prime \prime }\left( T_{j}=\frac{T_{0j}}{e}\right) =-\frac{2\lambda _{j}}{T^2_{0j}},\quad \quad \end{aligned}$$

  (3.70)

  and at these points the value of the total effective potential is computed as

  $$\begin{aligned} V^{(1)}_{E}= & {} \sum ^{N}_{j=1}V(T_{j}=0)=\sum ^{N}_{j=1}\lambda _{j},\end{aligned}$$

  (3.71)
  $$\begin{aligned} V^{(2)}_{E}= & {} \sum ^{N}_{j=1}V(T_{j}=T_{0j})=\sum ^{N}_{j=1}\lambda _{j},\end{aligned}$$

  (3.72)
  $$\begin{aligned} V^{(3)}_{E}= & {} \sum ^{N}_{j=1}V\left( T_{j}=\frac{T_{0j}}{e}\right) =\left( 1+\frac{1}{e^2}\right) \sum ^{N}_{j=1}\lambda _{j}.\quad \quad \end{aligned}$$

  (3.73)

  It is important to note for the multi-tachyonic case that for $\lambda _{j}>0$, $V^{\prime \prime }(T=T_{0j})>0$ and $V^{\prime \prime }(T=\frac{T_{0j}}{e})<0$, i.e., we get both maxima and minima on the potential $V(T_{j})$ as well as in $V_{E}(T)$.

3.3 Model III: exponential potential Type-I

For single field case the third model of the tachyonic potential is given by [83]

$$\begin{aligned} V(T)=\lambda \exp \left( -\frac{T}{T_{0}}\right) , \end{aligned}$$

(3.74)

and for the multi-tachyonic and assisted cases the total effective potential is given by

$$\begin{aligned} \begin{array}{lll} \displaystyle V_{E}(T) =\left\{ \begin{array}{lll} \displaystyle \sum ^{N}_{i=1}\lambda _{i}\exp \left( -\frac{T_{i}}{T_{0i}}\right) ,&{} \quad \mathrm{for~~Multi~~tachyonic} , \\ \displaystyle \sum ^{N}_{i=1}\lambda \exp \left( -\frac{T}{T_{0}}\right) \\ \quad =N\lambda \exp \left( -\frac{T}{T_{0}}\right) ,&{} \quad \mathrm{for~~Assisted~~tachyonic}. \end{array} \right. \end{array} \end{aligned}$$

(3.75)

Here the potential satisfies the following criteria:

• At $T=0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=0)=\lambda \end{aligned}$$

  (3.76)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.77)
  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i}=N\lambda . \end{aligned}$$

  (3.78)

• At $T=T_0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=T_0)=\frac{\lambda }{e} \end{aligned}$$

  (3.79)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=T_0)= & {} \frac{1}{e}\sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.80)
  $$\begin{aligned} V_{E}(T=T_0)= & {} \frac{1}{e}\sum ^{N}_{i=1}\lambda _{i}=\frac{N\lambda }{e}. \end{aligned}$$

  (3.81)

• For the single field tachyonic potential:

  $$\begin{aligned} V^{\prime }(T)= & {} -\frac{\lambda }{T_0}\exp \left( -\frac{T}{T_{0}}\right) ,\end{aligned}$$

  (3.82)
  $$\begin{aligned} V^{\prime \prime }(T)= & {} \frac{\lambda }{T^2_0}\exp \left( -\frac{T}{T_{0}}\right) . \end{aligned}$$

  (3.83)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T)=0, \end{aligned}$$

  (3.84)

  which gives rise to the following solution for the tachyonic field:

  $$\begin{aligned} T\rightarrow \infty . \end{aligned}$$

  (3.85)

  Further substituting the solutions for the tachyonic field in Eq. (3.83) we get

  $$\begin{aligned} V^{\prime \prime }(T\rightarrow \infty )\rightarrow 0, \end{aligned}$$

  (3.86)

  and also at the points $T=0,~T_{0}$ we have

  $$\begin{aligned}&V^{\prime \prime }(T=0)= \frac{\lambda }{T^2_0},\end{aligned}$$

  (3.87)
  $$\begin{aligned}&V^{\prime \prime }(T=T_{0})= \frac{\lambda }{eT^2_0}, \end{aligned}$$

  (3.88)

  and at these points the value of the potential is computed as

  $$\begin{aligned}&V(T\rightarrow \infty )\rightarrow 0,\end{aligned}$$

  (3.89)
  $$\begin{aligned}&V(T=0)= \lambda ,\end{aligned}$$

  (3.90)
  $$\begin{aligned}&V(T=T_{0})= \frac{\lambda }{e}. \end{aligned}$$

  (3.91)

  It is important to note for the single tachyonic case that for $\lambda >0$, $V^{\prime \prime }(T=0,~~T_{0} )>0$, i.e., we get an asymptotic behavior of the potential. For the assisted case the results are the same, provided the following replacement occurs:

  $$\begin{aligned} \lambda \rightarrow N\lambda , \end{aligned}$$

  (3.92)

  and finally for the multi-tachyonic case we have

  $$\begin{aligned}&V^{\prime }(T_{i})=-\frac{\lambda _{i} }{T_0}\exp \left( -\frac{T_{i}}{T_{0i}}\right) ,\end{aligned}$$

  (3.93)
  $$\begin{aligned}&V^{\prime \prime }(T_{i})=\frac{\lambda _{i} }{T^2_{0i}}\exp \left( -\frac{T_{i}}{T_{0i}}\right) . \end{aligned}$$

  (3.94)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T_{j})=0 \quad \forall \; j=1,2,\ldots ,N \end{aligned}$$

  (3.95)

  which gives rise to the following solutions for the jth tachyonic field:

  $$\begin{aligned} T_{j}\rightarrow \infty . \end{aligned}$$

  (3.96)

  Further substituting the solutions for the tachyonic field in Eq. (3.94) we get

  $$\begin{aligned} V^{\prime \prime }(T_{j}\rightarrow \infty )\rightarrow & {} 0, \end{aligned}$$

  (3.97)

  and also at the points $T_{j}=0,~T_{0j}$ we have

  $$\begin{aligned}&V^{\prime \prime }(T_{j}=0)= \frac{\lambda _{j}}{T^2_{0j}},\end{aligned}$$

  (3.98)
  $$\begin{aligned}&V^{\prime \prime }(T_{j}=T_{0j})= \frac{\lambda _{j}}{eT^2_{0j}}, \end{aligned}$$

  (3.99)

  and at these points the value of the total effective potential is computed as

  $$\begin{aligned}&V^{(1)}_{E}=\sum ^{N}_{j=1}V(T_{j}\rightarrow \infty ) \rightarrow 0,\end{aligned}$$

  (3.100)
  $$\begin{aligned}&V^{(2)}_{E}=\sum ^{N}_{j=1}V(T_{j}=0) = \sum ^{N}_{j=1}\lambda _{j},\end{aligned}$$

  (3.101)
  $$\begin{aligned}&V^{(3)}_{E}= \sum ^{N}_{j=1}V(T_{j}=T_{0j}) = \frac{1}{e}\sum ^{N}_{j=1}\lambda _{j}. \end{aligned}$$

  (3.102)

  It is important to note for the multi-tachyonic case that for $\lambda _{j}>0$, $V^{\prime \prime }(T=0,~~T_{0j} )>0$, i.e., we get an asymptotic behavior of the potential $V(T_{j})$ as well as in $V_{E}(T)$.

3.4 Model IV: exponential potential Type-II (Gaussian)

For single field case the first model of the tachyonic potential is given by [127]

$$\begin{aligned} V(T)=\lambda \exp \left[ -\left( \frac{T}{T_{0}}\right) ^2\right] , \end{aligned}$$

(3.103)

and for the multi-tachyonic and assisted cases the total effective potential is given by

$$\begin{aligned} \begin{array}{lll} \displaystyle V_{E}(T) =\left\{ \begin{array}{lll} \displaystyle \sum ^{N}_{i=1}\lambda \exp \left[ -\left( \frac{T}{T_{0}}\right) ^2\right] ,&{} \quad \mathrm{for~~Multi~~tachyonic} \\ \displaystyle \sum ^{N}_{i=1}\lambda \exp \left[ -\left( \frac{T}{T_{0}}\right) ^2\right] \\ \quad =N\lambda \exp \left[ -\left( \frac{T}{T_{0}}\right) ^2\right] ,&{} \quad \mathrm{for~~Assisted~~tachyonic}. \end{array} \right. \end{array} \end{aligned}$$

(3.104)

Here the potential satisfies the following criteria:

• At $T=0$ for single field tachyonic potential:

  $$\begin{aligned} V(T=0)=\lambda \end{aligned}$$

  (3.105)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.106)
  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i}=N\lambda . \end{aligned}$$

  (3.107)

• At $T=T_0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=T_0)=\frac{\lambda }{e} \end{aligned}$$

  (3.108)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=T_0)= & {} \frac{1}{e}\sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.109)
  $$\begin{aligned} V_{E}(T=T_0)= & {} \frac{1}{e}\sum ^{N}_{i=1}\lambda _{i}=\frac{N\lambda }{e}. \end{aligned}$$

  (3.110)

• For the single field tachyonic potential:

  $$\begin{aligned} V^{\prime }(T)= & {} -\frac{2\lambda T }{T^2_0}\exp \left[ -\left( \frac{T}{T_{0}}\right) ^2\right] ,\end{aligned}$$

  (3.111)
  $$\begin{aligned} V^{\prime \prime }(T)= & {} -\frac{2\lambda }{T^2_0}\exp \left[ -\left( \frac{T}{T_{0}}\right) ^2\right] \left\{ 1-\frac{2T^2}{T^2_0}\right\} . \end{aligned}$$

  (3.112)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T)=0, \end{aligned}$$

  (3.113)

  which gives rise to the following solution for the tachyonic field:

  $$\begin{aligned} T=0, \; \infty . \end{aligned}$$

  (3.114)

  Further substituting the solutions for the tachyonic field in Eq. (3.112) we get

  $$\begin{aligned}&V^{\prime \prime }\left( T=0\right) = -\frac{2\lambda }{T^2_{0}},\end{aligned}$$

  (3.115)
  $$\begin{aligned}&V^{\prime \prime }(T\rightarrow \infty ) \rightarrow 0 \end{aligned}$$

  (3.116)

  and also additionally for $T=T_{0}$ we have

  $$\begin{aligned} V^{\prime \prime }(T=T_0)= & {} \frac{2\lambda }{eT^2_0}, \end{aligned}$$

  (3.117)

  and at these points the value of the potential is computed as

  $$\begin{aligned}&V(T\rightarrow \infty ) \rightarrow 0,\end{aligned}$$

  (3.118)
  $$\begin{aligned}&V(T=0)= \lambda ,\end{aligned}$$

  (3.119)
  $$\begin{aligned}&V(T=T_{0})= \frac{\lambda }{e}. \end{aligned}$$

  (3.120)

  It is important to note for the single tachyonic case that for $\lambda >0$, $V^{\prime \prime }(T=0,~~T_{0} )>0$, i.e., we get maxima on the potential. For the assisted case the results are the same, provided the following replacement occurs:

  $$\begin{aligned} \lambda \rightarrow N\lambda , \end{aligned}$$

  (3.121)

  and finally for the multi-tachyonic case we have

  $$\begin{aligned} V^{\prime }(T_{i})= & {} -\frac{2\lambda _{i} T_{i} }{T^2_{0i}}\exp \left[ -\left( \frac{T_{i}}{T_{0i}}\right) ^2\right] ,\end{aligned}$$

  (3.122)
  $$\begin{aligned} V^{\prime \prime }(T_{i})= & {} -\frac{2\lambda _{i} }{T^2_{0i}}\exp \left[ -\left( \frac{T_{i}}{T_{0i}}\right) ^2\right] \left\{ 1-\frac{2T^2_{i}}{T^2_{0i}}\right\} .\nonumber \\ \end{aligned}$$

  (3.123)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T_{j})=0 \quad \forall \; j=1,2,\ldots ,N \end{aligned}$$

  (3.124)

  which give rise to the following solutions for the jth tachyonic field:

  $$\begin{aligned} T_{j}=0, \; \infty . \end{aligned}$$

  (3.125)

  Further substituting the solutions for the tachyonic field in Eq. (3.123) we get

  $$\begin{aligned}&V^{\prime \prime }(T_{j}=0)= -\frac{2\lambda }{T^2_{0}},\end{aligned}$$

  (3.126)
  $$\begin{aligned}&V^{\prime \prime }(T_{j}\rightarrow \infty ) \rightarrow 0, \end{aligned}$$

  (3.127)

  Additionally for the point $T_{j}=T_{0j}$ we have

  $$\begin{aligned} V^{\prime \prime }(T_{j}=T_{0j})= & {} \frac{2\lambda _{j}}{eT^2_{0j}}, \end{aligned}$$

  (3.128)

  and at these points the value of the total effective potential is computed as

  $$\begin{aligned}&V^{(1)}_{E}=\sum ^{N}_{j=1}V(T_{j}\rightarrow \infty ) \rightarrow 0,\end{aligned}$$

  (3.129)
  $$\begin{aligned}&V^{(2)}_{E}=\sum ^{N}_{j=1}V(T_{j}=0)= \sum ^{N}_{j=1}\lambda _{j},\end{aligned}$$

  (3.130)
  $$\begin{aligned}&V^{(3)}_{E}= \sum ^{N}_{j=1}V(T_{j}=T_{0j})= \frac{1}{e}\sum ^{N}_{j=1}\lambda _{j}. \end{aligned}$$

  (3.131)

  It is important to note for the multi-tachyonic case that for $\lambda _{j}>0$, $V^{\prime \prime }(T=0,~~T_{0j} )>0$, i.e., we get maxima on the potential $V(T_{j})$ as well as in $V_{E}(T)$.

3.5 Model V: inverse power-law potential

For the single field case the first model of the tachyonic potential is given by [86, 128]

$$\begin{aligned} V(T)=\frac{\lambda }{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] }, \end{aligned}$$

(3.132)

and for the multi-tachyonic and assisted cases the total effective potential is given by

$$\begin{aligned} \begin{array}{lll} \displaystyle V_{E}(T) =\left\{ \begin{array}{lll} \displaystyle \sum ^{N}_{i=1}\frac{\lambda _{i}}{\left[ 1+\left( \frac{T_{i}}{T_{0i}}\right) ^4\right] } , &{} \quad \mathrm{for~~Multi~~tachyonic} \\ \displaystyle \sum ^{N}_{i=1}\frac{\lambda }{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] }\\ \quad =\frac{N\lambda }{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] } , &{} \quad \mathrm{for~~Assisted~~tachyonic}. \end{array} \right. \end{array} \end{aligned}$$

(3.133)

Here the potential satisfies the following criteria:

• At $T=0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=0)=\lambda , \end{aligned}$$

  (3.134)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.135)
  $$\begin{aligned} V_{E}(T=0)= & {} \sum ^{N}_{i=1}\lambda _{i}=N\lambda . \end{aligned}$$

  (3.136)

• At $T=T_0$ for the single field tachyonic potential:

  $$\begin{aligned} V(T=T_0)=\frac{\lambda }{2} \end{aligned}$$

  (3.137)

  and for the multi-tachyonic and assisted cases we have

  $$\begin{aligned} V_{E}(T=T_0)= & {} \frac{1}{2}\sum ^{N}_{i=1}\lambda _{i},\end{aligned}$$

  (3.138)
  $$\begin{aligned} V_{E}(T=T_0)= & {} \frac{1}{2}\sum ^{N}_{i=1}\lambda _{i}=\frac{N\lambda }{2}. \end{aligned}$$

  (3.139)

• For single field tachyonic potential:

  $$\begin{aligned} V^{\prime }(T)= & {} -\frac{4\lambda T^3 }{T^4_0 \left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] ^2},\end{aligned}$$

  (3.140)
  $$\begin{aligned} V^{\prime \prime }(T)= & {} \frac{32 \lambda T^6}{T^8_0 \left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] ^3}-\frac{12 \lambda T^2}{T^4_0 \left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] ^2}.\nonumber \\ \end{aligned}$$

  (3.141)

  Now to find the extrema of the potential we substitute

  $$\begin{aligned} V^{\prime }(T)=0, \end{aligned}$$

  (3.142)

  which give rise to the following solution for the tachyonic field:

  $$\begin{aligned} T=0,\; \infty . \end{aligned}$$

  (3.143)

  Further substituting the solutions for the tachyonic field in Eq. (3.141) we get

  $$\begin{aligned}&V^{\prime \prime }\left( T=0\right) = 0,\end{aligned}$$

  (3.144)
  $$\begin{aligned}&V^{\prime \prime }(T\rightarrow \infty )\rightarrow 0 \end{aligned}$$

  (3.145)

  and also additionally for $T=T_{0}$ we have

  $$\begin{aligned} V^{\prime \prime }(T=T_0)= & {} \frac{\lambda }{T^2_0}, \end{aligned}$$

  (3.146)

  and at these points the value of the potential is computed as

  $$\begin{aligned}&V(T\rightarrow \infty )\rightarrow 0,\end{aligned}$$

  (3.147)
  $$\begin{aligned}&V(T=0)= \lambda ,\end{aligned}$$

  (3.148)
  $$\begin{aligned}&V(T=T_{0})= \frac{\lambda }{2}. \end{aligned}$$

  (3.149)

  It is important to note for the single tachyonic case that for $\lambda >0$, $V^{\prime \prime }(T=0,~~T_{0} )>0$, i.e., we get maxima on the potential. For the assisted case the results are the same, provided the following replacement occurs:

  $$\begin{aligned} \lambda \rightarrow N\lambda , \end{aligned}$$

  (3.150)

  and finally for the multi-tachyonic case we have

  $$\begin{aligned} V^{\prime }(T_{i})= & {} -\frac{4\lambda _{i} T^3_{i} }{T^4_{0i} \left[ 1+\left( \frac{T_{i}}{T_{0i}}\right) ^4\right] ^2},\end{aligned}$$

  (3.151)
  $$\begin{aligned} V^{\prime \prime }(T_{i})= & {} \frac{32 \lambda _{i} T^6_{i}}{T^8_{0i} \left[ 1+\left( \frac{T_{i}}{T_{0i}}\right) ^4\right] ^3}-\frac{12 \lambda _{i} T^2_{i}}{T^4_{0i} \left[ 1+\left( \frac{T_{i}}{T_{0i}}\right) ^4\right] ^2}.\nonumber \\ \end{aligned}$$

  (3.152)

  Now to find the extrema of the potential we substitute:

  $$\begin{aligned} V^{\prime }(T_{j})=0 \quad \forall \; j=1,2,\ldots ,N \end{aligned}$$

  (3.153)

  which give rise to the following solutions for the jth tachyonic field:

  $$\begin{aligned} T_{j}=0,\; \infty . \end{aligned}$$

  (3.154)

  Further substituting the solutions for the tachyonic field in Eq. (3.152) we get

  $$\begin{aligned}&V^{\prime \prime }(T_{j}=0)=0,\end{aligned}$$

  (3.155)
  $$\begin{aligned}&V^{\prime \prime }(T_{j}\rightarrow \infty )\rightarrow 0, \end{aligned}$$

  (3.156)

  Additionally for the point $T_{j}=T_{0j}$ we have

  $$\begin{aligned} V^{\prime \prime }(T_{j}=T_{0j})= & {} \frac{\lambda _{j}}{T^2_{0j}}, \end{aligned}$$

  (3.157)

  and at these points the value of the total effective potential is computed as

  $$\begin{aligned}&V^{(1)}_{E}=\sum ^{N}_{j=1}V(T_{j}\rightarrow \infty ) \rightarrow 0,\end{aligned}$$

  (3.158)
  $$\begin{aligned}&V^{(2)}_{E}=\sum ^{N}_{j=1}V(T_{j}=0)= \sum ^{N}_{j=1}\lambda _{j},\end{aligned}$$

  (3.159)
  $$\begin{aligned}&V^{(3)}_{E}= \sum ^{N}_{j=1}V(T_{j}=T_{0j})= \frac{1}{2}\sum ^{N}_{j=1}\lambda _{j}. \end{aligned}$$

  (3.160)

  It is important to note for the multi-tachyonic case that for $\lambda _{j}>0$, $V^{\prime \prime }(T=0,~~T_{0j} )>0$, i.e., we get maxima on the potential $V(T_{j})$ as well as in $V_{E}(T)$.

4 Cosmological dynamics from GTachyon

4.1 Unperturbed evolution

For $p=3$ non-BPS branes the total tachyonic model action can be written as

$$\begin{aligned} \begin{array}{lll} \displaystyle S_{T} =\left\{ \begin{array}{lll} \displaystyle \int \mathrm{d}^{4}\sigma ~\sqrt{-g}~\left[ \frac{M^{2}_p}{2}R-~V(T)~\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T}\right] , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \int \mathrm{d}^{4}\sigma ~\sqrt{-g}~\left[ \frac{M^{2}_p}{2}R-~\sum ^{N}_{i=1}V(T_{i})~\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_{i}\partial _{\nu }T_{i}}\right] ,&{} \quad \mathrm{for~~~Multi}, \\ \displaystyle \int \mathrm{d}^{4}\sigma ~\sqrt{-g}~\left[ \frac{M^{2}_p}{2}R-~NV(T)~\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T}\right] ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.1)

and in a more generalized situation Eq. (4.1) is modified as

$$\begin{aligned} \begin{array}{lll} \displaystyle S^{(q)}_{T} =\left\{ \begin{array}{lll} \displaystyle \int \mathrm{d}^{4}\sigma ~\sqrt{-g}~\left[ \frac{M^{2}_p}{2}R-~V(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q}\right] , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \int \mathrm{d}^{4}\sigma ~\sqrt{-g}~\left[ \frac{M^{2}_p}{2}R-~\sum ^{N}_{i=1}V(T_{i})~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_{i}\partial _{\nu }T_{i}\right) ^{q}\right] ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \int \mathrm{d}^{4}\sigma ~\sqrt{-g}~\left[ \frac{M^{2}_p}{2}R-~NV(T)~\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q}\right] ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.2)

where $M_p$ is the reduced Planck mass, $M_p =2.43\times 10^{18}\mathrm{GeV}$. By varying the action as stated in Eqs. (4.1) and (4.2), with respect to the metric $g_{\mu \nu }$ we get the following equation of motion:

$$\begin{aligned} G_{\mu \nu }=\frac{T_{\mu \nu }}{M^{2}_{p}} \end{aligned}$$

(4.3)

where $G_{\mu \nu }$ is defined as

$$\begin{aligned} G_{\nu \nu }= R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R \end{aligned}$$

(4.4)

and the energy-momentum stress tensor $T_{\mu \nu }$ is defined as

$$\begin{aligned} T_{\mu \nu }=-\frac{2}{\sqrt{-g}}\frac{\delta (\sqrt{-g}\mathcal{L}_\mathrm{Tachyon})}{\delta g^{\mu \nu }} \end{aligned}$$

(4.5)

where $\mathcal{L}_\mathrm{Tachyon}$ is the tachyonic part of the Lagrangian for non-BPS set-up. Explicitly using Eq. (4.1) the energy-momentum stress tensor can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle T_{\mu \nu } =\left\{ \begin{array}{lll} \displaystyle V(T)\left[ \frac{\alpha ^{\prime }\partial _{\mu }T\partial _{\nu }T}{\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T}}-g_{\mu \nu }\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T}\right] , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V(T_{i})\left[ \frac{\alpha ^{\prime }\partial _{\mu }T_{i}\partial _{\nu }T_{i}}{\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_{i}\partial _{\nu }T_{i}}}-g_{\mu \nu }\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_{i}\partial _{\nu }T_{i}}\right] ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle NV(T)\left[ \frac{\alpha ^{\prime }\partial _{\mu }T\partial _{\nu }T}{\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T}}-g_{\mu \nu }\sqrt{1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T}\right] ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.6)

and similarly in a more generalized situation using Eq. (4.2) the energy-momentum stress tensor can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle T^{(q)}_{\mu \nu } =\left\{ \begin{array}{lll} \displaystyle V(T)\left[ \frac{2q\alpha ^{\prime }\partial _{\mu }T\partial _{\nu }T}{\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{1-q}}-g_{\mu \nu }\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q}\right] ,~ &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V(T_{i})\left[ \frac{2q\alpha ^{\prime }\partial _{\mu }T_{i}\partial _{\nu }T_{i}}{\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_{i}\partial _{\nu }T_{i}\right) ^{1-q}}-g_{\mu \nu }\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T_{i}\partial _{\nu }T_{i}\right) ^{q}\right] ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle NV(T)\left[ \frac{2q\alpha ^{\prime }\partial _{\mu }T\partial _{\nu }T}{\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{1-q}}-g_{\mu \nu }\left( 1+\alpha ^{\prime }g^{\mu \nu }\partial _{\mu }T\partial _{\nu }T\right) ^{q}\right] ,~ &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.7)

It clearly appears that, for $q=1/2$, the result is perfectly consistent with Eq. (4.6). Further using the perfect fluid assumption the energy-momentum stress tensor can be written as

$$\begin{aligned} \begin{array}{lll} \displaystyle T_{\mu \nu } =\left\{ \begin{array}{lll} \displaystyle (\rho +p)u_{\mu }u_{\nu }+ g_{\mu \nu } p ,&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}(\rho _{i} +p_{i})u^{(i)}_{\mu }u^{(i)}_{\nu }+ g_{\mu \nu } p_{i} , &{} \quad \mathrm{for~~~Multi}, \\ \displaystyle N\left[ (\rho +p)u_{\mu }u_{\nu }+ g_{\mu \nu } p\right] , &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(4.8)

where for the assisted case we assume that the density and pressure of all identical tachyonic modes are the same. Here $u_{\mu }$ signifies the four velocity of the fluid, which is defined as

$$\begin{aligned} \begin{array}{lll} \displaystyle u_{\mu } =\left\{ \begin{array}{lll} \displaystyle \frac{\partial _{\mu }T}{\sqrt{-g^{\alpha \beta }\partial _{\alpha }T\partial _{\beta }T}} , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \frac{\partial _{\mu }T_{i}}{\sqrt{-g^{\alpha \beta }\partial _{\alpha }T_{i}\partial _{\beta }T_{i}}} ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N\partial _{\mu }T}{\sqrt{-g^{\alpha \beta }\partial _{\alpha }T\partial _{\beta }T}} , &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.9)

Further comparing Eqs. (4.6) and (4.8) the density $\rho$ and the pressure p for the tachyonic field can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle \rho =\left\{ \begin{array}{lll} \displaystyle \frac{V(T)}{\sqrt{1-\alpha ^{\prime }\dot{T}^2}},&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \frac{V(T)}{\sqrt{1-\alpha ^{\prime }\dot{T}^2}},&{} \quad \mathrm{for~~~Multi}, \\ \displaystyle \frac{NV(T)}{\sqrt{1-\alpha ^{\prime }\dot{T}^2}}, &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.10)

and

$$\begin{aligned} \begin{array}{lll} \displaystyle p =\left\{ \begin{array}{lll} \displaystyle -V(T)\sqrt{1-\alpha ^{\prime }\dot{T}^2}, &{} \quad \mathrm{for~~~Single}, \\ \displaystyle -\sum ^{N}_{i=1} V(T)\sqrt{1-\alpha ^{\prime }\dot{T}^2},&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle -NV(T)\sqrt{1-\alpha ^{\prime }\dot{T}^2}, &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.11)

Similarly for the generalized situation comparing Eqs. (4.7) and (4.8) the density $\rho$ and pressure p for tachyonic field can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle \rho =\left\{ \begin{array}{lll} \displaystyle \frac{V(T)}{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{1-q}}\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] ,~&{} \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \frac{V(T_{i})}{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{1-q}}\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right] ,&{} \mathrm{for~~~Multi} ,\\ \displaystyle \frac{NV(T)}{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{1-q}}\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] ,&{} \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.12)

and

$$\begin{aligned} \begin{array}{lll} \displaystyle p =\left\{ \begin{array}{lll} \displaystyle -V(T)\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{q} ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle -\sum ^{N}_{i=1} V(T_{i})\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{q} ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle -NV(T)\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{q} ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.13)

Next using Eqs. (4.10) and (4.11) one can write down the expression for the equation of state parameter:

$$\begin{aligned} \begin{array}{lll} \displaystyle w =\left\{ \begin{array}{lll} \displaystyle \frac{p}{\rho }=\left( \alpha ^{\prime }\dot{T}^2-1\right) ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \frac{p_{i}}{\rho _{i}}=\sum ^{N}_{i=1}\left( \alpha ^{\prime }\dot{T}^2_{i}-1\right) ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{Np}{\rho }= N\left( \alpha ^{\prime }\dot{T}^2-1\right) ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.14)

and for the generalized case using Eqs. (4.12) and (4.13) one can write down the expression for the equation of state parameter:

$$\begin{aligned} \begin{array}{lll} \displaystyle w =\left\{ \begin{array}{lll} \displaystyle \frac{p}{\rho }=\frac{\left( \alpha ^{\prime }\dot{T}^2-1\right) }{\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] } ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \frac{p_{i}}{\rho _{i}}=\sum ^{N}_{i=1}\frac{\left( \alpha ^{\prime }\dot{T}^2_{i}-1\right) }{\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right] } ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{Np}{\rho }= \frac{N\left( \alpha ^{\prime }\dot{T}^2-1\right) }{\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] } ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.15)

Further using the spatially flat $k=0$ FLRW metric defined through the following line element:

$$\begin{aligned} \mathrm{d}s^2=-\mathrm{d}t^2+a^{2}(t)\mathrm{d}{} \mathbf{x}^2, \end{aligned}$$

(4.16)

the Friedmann equations can be written as

$$\begin{aligned}&H^{2}=\left( \frac{\dot{a}}{a}\right) ^2=\frac{\rho }{3M^{2}_{p}},\end{aligned}$$

(4.17)

$$\begin{aligned}&\dot{H}+ H^{2} = \frac{\ddot{a}}{a}=-\frac{(\rho +3p)}{6M^{2}_{p}}, \end{aligned}$$

(4.18)

where the density $\rho$ and pressure p is computed in Eqs. (4.10)–(4.11). Also H is the Hubble parameter, defined as

$$\begin{aligned} H=\frac{1}{a(t)}\frac{\mathrm{d}a(t)}{\mathrm{d}t}=\frac{\dot{a}}{a}. \end{aligned}$$

(4.19)

On the other hand, varying the action as stated in Eq. (4.1) with respect to the tachyon field the equation of motion can be written as

$$\begin{aligned} \begin{array}{lll} \displaystyle 0 =\left\{ \begin{array}{lll} \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V(T)\sqrt{1+g^{\alpha \beta }\partial _{\alpha }T\partial _{\beta }T}\right) =\frac{\alpha ^{\prime }\ddot{T}}{\left( 1-\alpha ^{\prime }\dot{T}^2\right) } +3H\alpha ^{\prime }\dot{T}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T} , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V(T_{i})\sqrt{1+g^{\alpha \beta }\partial _{\alpha }T_{i}\partial _{\beta }T_{i}}\right) =\frac{\alpha ^{\prime }\ddot{T}_{i}}{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) } +3H\alpha ^{\prime }\dot{T}_{i}+\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}} , &{} \quad \mathrm{for~~~Multi}, \\ \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V(T)\sqrt{1+g^{\alpha \beta }\partial _{\alpha }T\partial _{\beta }T}\right) =\frac{\alpha ^{\prime }\ddot{T}}{\left( 1-\alpha ^{\prime }\dot{T}^2\right) } +3H\alpha ^{\prime }\dot{T}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T} , &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.20)

Similarly, in the most generalized case, varying the action as stated in Eq. (4.2) with respect to the tachyon field the equation of motion can be written as

$$\begin{aligned} {}\begin{array}{lll} \displaystyle 0 =\left\{ \begin{array}{lll} \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V(T)\left( 1+g^{\alpha \beta }\partial _{\alpha }T\partial _{\beta }T\right) ^q\right) \displaystyle =\frac{2q\alpha ^{\prime }\ddot{T}}{\left( 1-\alpha ^{\prime }\dot{T}^2\right) } +6q\alpha ^{\prime }H\frac{\dot{T}}{1-\alpha ^{\prime }(1-2q)\dot{T}^2}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T} ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V(T_{i})\left( 1+g^{\alpha \beta }\partial _{\alpha }T_{i}\partial _{\beta }T_{i}\right) ^q\right) \displaystyle =\frac{2q\alpha ^{\prime }\ddot{T}_{i}}{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) } +6q\alpha ^{\prime }H\frac{\dot{T}_{i}}{1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}}+\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}} ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V(T)\left( 1+g^{\alpha \beta }\partial _{\alpha }T\partial _{\beta }T\right) ^q\right) \displaystyle =\frac{2q\alpha ^{\prime }\ddot{T}}{\left( 1-\alpha ^{\prime }\dot{T}^2\right) } +6q\alpha ^{\prime }H\frac{\dot{T}}{1-\alpha ^{\prime }(1-2q)\dot{T}^2}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T} ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(4.21)

Further using Eqs. (4.10), (4.11) and (5.328) the expression for the adiabatic sound speed $c_{A}$ turns out to be

$$\begin{aligned} {} \begin{array}{lll} \displaystyle c_{A} =\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{\dot{p}}{\dot{\rho }}}=\sqrt{-w\left[ 1+\frac{2}{3H\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] } ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sqrt{\sum ^{N}_{i=1} \frac{\dot{p}_{i}}{\dot{\rho }_{i}}}=\sqrt{\sum ^{N}_{i=1}\left\{ -w_{i}\left[ 1+\frac{2}{3H\alpha ^{\prime }\dot{T}_{i}}\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right] \right\} } ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \sqrt{\frac{N\dot{p}}{\dot{\rho }}}=\sqrt{-wN\left[ 1+\frac{2}{3H\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] } ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.22)

and for the generalized case using Eqs. (4.10), (4.11) and (4.21) the expression for the adiabatic sound speed turns out to be:

$$\begin{aligned} {} \begin{array}{lll} \displaystyle c_{A} =\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{-w\left[ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{3qH\alpha ^{\prime }\dot{T}} \frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }{\left[ \left( \frac{1}{q}-1\right) \left\{ 1-\frac{(1-2q)}{(1-q)}w\right\} \left\{ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right\} -\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }} ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sqrt{\sum ^{N}_{i=1}\left\{ \frac{-w_{i}\left[ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right) }{3qH\alpha ^{\prime }\dot{T}_{i}} \frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right] }{\left[ \left( \frac{1}{q}-1\right) \left\{ 1-\frac{(1-2q)}{(1-q)}w_{i}\right\} \left\{ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right) }{6qH\alpha ^{\prime }\dot{T}_{i}}\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right\} -\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right) }{6qH\alpha ^{\prime }\dot{T}_{i}}\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right] }\right\} } ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \sqrt{\frac{-wN\left[ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{3qH\alpha ^{\prime }\dot{T}} \frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }{\left[ \left( \frac{1}{q}-1\right) \left\{ 1-\frac{(1-2q)}{(1-q)}w\right\} \left\{ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right\} -\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }} ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(4.23)

It is important to mention here that, substituting $q=1/2$ in Eq. (4.23) one can get back the result obtained in Eq. (4.22). Similarly in the present context the effective sound speed $c_{S}$ is defined as

$$\begin{aligned} \begin{array}{lll} \displaystyle c_{S} =\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{\frac{\partial p}{\partial \dot{T}^2}}{\frac{\partial \rho }{\partial \dot{T}^2}}}=\sqrt{-w} ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sqrt{\sum ^{N}_{i=1} \frac{\frac{\partial p_{i}}{\partial \dot{T}^2_{i}}}{\frac{\partial \rho _{i}}{\partial \dot{T}^2_{i}}}} =\sqrt{-\sum ^{N}_{i=1}w_{i}} ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \sqrt{N\frac{\frac{\partial p}{\partial \dot{T}^2}}{\frac{\partial \rho }{\partial \dot{T}^2}}}=\sqrt{-wN} ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.24)

and for the generalized case the effective sound speed $c_{S}$ is defined as

$$\begin{aligned} \begin{array}{lll} \displaystyle c_{S} =\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{\frac{\partial p}{\partial \dot{T}^2}}{\frac{\partial \rho }{\partial \dot{T}^2}}}=\sqrt{\left[ 1+\frac{2(q-1)(1+w)}{1+(1-2q)(2w+1)}\right] } ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sqrt{\sum ^{N}_{i=1} \frac{\frac{\partial p_{i}}{\partial \dot{T}^2_{i}}}{\frac{\partial \rho _{i}}{\partial \dot{T}^2_{i}}}} =\sqrt{\sum ^{N}_{i=1}\left[ 1+\frac{2(q-1)(1+w_{i})}{1+(1-2q)(2w_{i}+1)}\right] } ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \sqrt{N\frac{\frac{\partial p}{\partial \dot{T}^2}}{\frac{\partial \rho }{\partial \dot{T}^2}}}=\sqrt{N\left[ 1+\frac{2(q-1)(1+w)}{1+(1-2q)(2w+1)}\right] } ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.25)

Finally comparing Eqs. (4.22), (4.23), (4.24) and (4.25) we get the following relationship between adiabatic and effective sound speed in tachyonic field theory:

$$\begin{aligned} \begin{array}{lll} \displaystyle c_{A} =\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{\dot{p}}{\dot{\rho }}}=c_{S}\sqrt{\left[ 1+\frac{2}{3H\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] } ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sqrt{\sum ^{N}_{i=1} \frac{\dot{p}_{i}}{\dot{\rho }_{i}}}=\sqrt{\sum ^{N}_{i=1}\left\{ c^{2}_{S,i}\left[ 1+\frac{2}{3H\alpha ^{\prime }\dot{T}_{i}}\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right] \right\} } ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \sqrt{\frac{N\dot{p}}{\dot{\rho }}}=c_{S}\sqrt{\left[ 1+\frac{2}{3H\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] } ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.26)

and for the generalized case we get

$$\begin{aligned} \begin{array}{lll} \displaystyle c_{A} =\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{\frac{c^{2}_{S}}{\left[ 1+\frac{(1-2q)}{(1-q)}\left( c^{2}_{S}-1\right) \right] } \left[ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{3qH\alpha ^{\prime }\dot{T}} \frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }{\left[ \left( \frac{1}{q}-1\right) \left\{ 1+\frac{\frac{(1-2q)}{(1-q)}c^{2}_{S}}{\left[ 1+\frac{(1-2q)}{(1-q)}\left( c^{2}_{S}-1\right) \right] }\right\} \left\{ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right\} -\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }} ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sqrt{\sum ^{N}_{i=1}\left\{ \frac{\frac{c^{2}_{S,i}}{\left[ 1+\frac{(1-2q)}{(1-q)}\left( c^{2}_{S,i}-1\right) \right] } \left[ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right) }{3qH\alpha ^{\prime }\dot{T}_{i}} \frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right] }{\left[ \left( \frac{1}{q}-1\right) \left\{ 1+\frac{\frac{(1-2q)}{(1-q)}c^{2}_{S,i}}{\left[ 1+\frac{(1-2q)}{(1-q)}\left( c^{2}_{S,i}-1\right) \right] }\right\} \left\{ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right) }{6qH\alpha ^{\prime }\dot{T}_{i}}\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right\} -\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right) }{6qH\alpha ^{\prime }\dot{T}_{i}}\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\right] }\right\} } ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \sqrt{\frac{\frac{c^{2}_{S}}{\left[ 1+\frac{(1-2q)}{(1-q)}\left( \frac{c^{2}_{S}}{N}-1\right) \right] } \left[ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{3qH\alpha ^{\prime }\dot{T}} \frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }{\left[ \left( \frac{1}{q}-1\right) \left\{ 1+\frac{\frac{(1-2q)}{(1-q)}c^{2}_{S}}{N\left[ 1+\frac{(1-2q)}{(1-q)}\left( \frac{c^{2}_{S}}{N}-1\right) \right] }\right\} \left\{ 1+\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right\} -\frac{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) }{6qH\alpha ^{\prime }\dot{T}}\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\right] }} ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.27)

Let us mention other crucial issues which we use throughout the analysis performed in this paper:

• 1. At early times the tachyonic field satisfies the following small field criteria to validate the effective field theory prescription within the framework of tachyonic string theory:

  $$\begin{aligned}&\frac{|T|}{M_{p}}<<1,\end{aligned}$$

  (4.28)
  $$\begin{aligned}&|\dot{T}|<<\frac{1}{\sqrt{\alpha ^{\prime }}} \end{aligned}$$

  (4.29)

  and for the generalized $q\ne 1/2$ case additionally we have to satisfy another constraint:

  $$\begin{aligned} \sqrt{(1-2q)}|\dot{T}|<<\frac{1}{\sqrt{\alpha ^{\prime }}}. \end{aligned}$$

  (4.30)

• 2. At early times using Eqs. (4.28), (4.29) and (4.30) in Eqs. (4.14) and (4.15), the equation of state parameter w can be approximated by

  $$\begin{aligned} w\approx -1. \end{aligned}$$

  (4.31)

• 3. At late times the tachyonic field satisfies the following small field criteria within the framework of tachyonic string theory:

  $$\begin{aligned}&\frac{|T|}{M_{p}}\sim 1,\end{aligned}$$

  (4.32)
  $$\begin{aligned}&\frac{|\dot{T}|}{M^{2}_{p}}\sim \frac{1}{\sqrt{\alpha ^{\prime }}} \end{aligned}$$

  (4.33)

  and for the generalized $q\ne 1/2$ case additionally we have to satisfy another constraint:

  $$\begin{aligned} \sqrt{(1-2q)}|\dot{T}|\sim \frac{1}{\sqrt{\alpha ^{\prime }}}. \end{aligned}$$

  (4.34)

• 4. At late times using Eqs. (4.32), (4.33) and (4.34) in Eqs. (4.14) and (4.15), the equation of state parameter w can be approximated by

  $$\begin{aligned} w\approx 0. \end{aligned}$$

  (4.35)

• 5. There might be another interesting possibility appear in the present context, where the tachyonic modes satisfy the large field criteria, represented by the following constraint:

  $$\begin{aligned}&\frac{|T|}{M_{p}}>>1,\end{aligned}$$

  (4.36)
  $$\begin{aligned}&|\dot{T}|>>\frac{1}{\sqrt{\alpha ^{\prime }}} \end{aligned}$$

  (4.37)

  and for the generalized $q\ne 1/2$ case additionally we have to satisfy another constraint:

  $$\begin{aligned} \sqrt{(1-2q)}|\dot{T}|>>\frac{1}{\sqrt{\alpha ^{\prime }}}. \end{aligned}$$

  (4.38)

• 6. Further using Eqs. (4.36), (4.37) and (4.38) in Eqs. (4.14) and (4.15), the equation of state parameter w can be approximated by

  $$\begin{aligned} \begin{array}{lll} \displaystyle w =\left\{ \begin{array}{lll} \displaystyle \frac{p}{\rho }=\alpha ^{\prime }\dot{T}^2 ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \frac{p_{i}}{\rho _{i}}=\sum ^{N}_{i=1}\alpha ^{\prime }\dot{T}^2_{i} ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{Np}{\rho }= N\alpha ^{\prime }\dot{T}^2 ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.39)

  and for the generalized $q\ne 1/2$ case we have

  $$\begin{aligned} \begin{array}{lll} \displaystyle w =\left\{ \begin{array}{lll} \displaystyle \frac{p}{\rho }=\frac{1}{(2q-1) } ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \frac{p_{i}}{\rho _{i}}=\sum ^{N}_{i=1}\frac{1}{(2q-1) } ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{Np}{\rho }= \frac{N}{(2q-1) } ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

  (4.40)

  Here one can control the parameter q and N to get the desired value of equation of state parameter w, which is necessarily required to explain the cosmological dynamics.

• 7. Additionally, it is important to note that, within the set-up of string theory, the tachyonic modes can form cluster on small cosmological scales. Consequently tachyonic string theory can be treated as a unified prescription to explain the inflationary paradigm and dark matter.

• 8. Reheating and creation of matter particles in a class of specific models where the minimum of the tachyon potential V(T) is at $T\rightarrow \infty$, which is a pathological issue in the present context because the tachyon field in such a type of string theories does not participate in oscillations.³ To solve this crucial pathological problem in the present context, one can think about a particular physical situation where the universe is initially dominated by a inflationary epoch and this can be explained via the energy density of the tachyon condensate as mentioned in the introduction of this article and according to this proposal the set-up will always remain dominated by the tachyons. To resolve this pathological issue it may happen that the tachyon condensation phenomenon is potentially responsible for a short period of an inflationary epoch prior, which occurs at a Planckian mass scale, $M_p\sim 2.43\times 10^{18}\,\mathrm{GeV}$ and also one needs to require a second stage of inflationary epoch just followed by the previous one which occurs at the vicinity of the GUT scale ($10^{16}\,\mathrm{GeV}$). This directly implies that the tachyon serves no crucial purpose in the post inflationary epoch till at the very later stages of its cosmological evolution on time scales. All these crucial pathological problems do not appear in the context of the well-known hybrid inflationary set-up where the complex tachyon field has a specific minimum value at the sub-Planckian ( $M_{p}$) regime given by the list of constraint equations as stated in Eqs. (4.28), (4.29), and Eq. (4.30). In this paper, we have studied the cosmological consequences from different classes of tachyonic potentials appearing in the context of string theory which have no connection to the hybrid inflationary model, but to explain CMB constraints the tachyon condensation phenomenon plays an important role. Instead of studying the tachyon condensation phenomenon, in this paper we study the cosmological perturbation theory and its physical consequences in detail in later sections.

• 9. The energy density of tachyons after inflation should be fine tuned to be sub-dominant until the very later stages of the cosmological evolution of the universe.

4.2 Dynamical solution for various phases

In this section our prime objective is to study the dynamical behavior of the tachyonic field in the background of spatially flat FLRW metric and in the presence of the Einstein–Hilbert term in the gravity sector. Below we explicitly show that the solution for the tachyonic field can explain various phases of the universe starting from inflation to the dust formation. To study the cosmological dynamics from the tachyonic string field theoretic set-up let us start with the following solution ansatz of the tachyon field:

$$\begin{aligned} \begin{array}{lll} \displaystyle \dot{T} =\left\{ \begin{array}{lll} \displaystyle \frac{1}{\sqrt{\alpha ^{\prime }}}\left( \frac{\exp \left( \frac{2t}{bT_0}\right) -1}{\exp \left( \frac{2t}{bT_0}\right) +1}\right) =\frac{1}{\sqrt{\alpha ^{\prime }}}\mathrm{tanh}\left( \frac{t}{b T_0}\right) ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \dot{T}_{i}=\sum ^{N}_{i=1}\frac{1}{\sqrt{\alpha ^{\prime }}}\left( \frac{\exp \left( \frac{2t}{bT_{0i}}\right) -1}{\exp \left( \frac{2t}{bT_{0i}}\right) +1}\right) =\sum ^{N}_{i=1}\frac{1}{\sqrt{\alpha ^{\prime }}}\mathrm{tanh}\left( \frac{t}{b T_{0i}}\right) ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N}{\sqrt{\alpha ^{\prime }}}\left( \frac{\exp \left( \frac{2t}{bT_0}\right) -1}{\exp \left( \frac{2t}{bT_0}\right) +1}\right) =\frac{N}{\sqrt{\alpha ^{\prime }}}\mathrm{tanh}\left( \frac{t}{b T_0}\right) ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.41)

which will satisfy the equation of motion of the tachyon field as stated in Eqs. (5.328) and (4.21) respectively. Here b is a new parameter of the theory which has inverse square mass dimension, i.e., $[M]^{-1}$. Consequently the argument of the hyperbolic functions, i.e., $(\frac{t}{b T_0})$ is dimensionless. From various cosmological observations it is possible to put stringent constraint on the newly introduced parameter b. Further integrating Eq. (4.41) we get the following solutions for the tachyonic field:

$$\begin{aligned} \begin{array}{lll} \displaystyle T(t) =\left\{ \begin{array}{lll} \displaystyle \frac{b T_{0}}{\sqrt{\alpha ^{\prime }}}\ln \left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}T_{i}(t)=\sum ^{N}_{i=1}\frac{b T_{0i}}{\sqrt{\alpha ^{\prime }}}\ln \left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right] ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{Nb T_{0}}{\sqrt{\alpha ^{\prime }}}\ln \left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.42)

For both of the cases we fix the boundary condition in such a way that the tachyonic field T satisfy the constraint: $T(t=0)=T(0)=0.$ Further using Eqs. (4.17) and (5.328) we get the following constraint condition for the cosmological time dependent potential V(t):

$$\begin{aligned} \begin{array}{lll} \displaystyle 0 =\left\{ \begin{array}{lll} \displaystyle \left[ \frac{1}{b T_{0}}+\frac{\dot{V}}{V}\mathrm{coth}\left( \frac{t}{b T_0}\right) +\sqrt{\frac{3V}{\mathrm{cosh} \left( \frac{t}{b T_0}\right) }}\frac{\mathrm{sinh}\left( \frac{t}{b T_0}\right) }{M_p }\right] ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \left[ \frac{1}{b T_{0i}}+\frac{\dot{V}}{V}\mathrm{coth}\left( \frac{t}{b T_{0i}}\right) +\sqrt{\frac{3V}{ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) }}\frac{\mathrm{sinh}\left( \frac{t}{b T_{0i}}\right) }{M_p}\right] ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle N\left[ \frac{1}{b T_{0}}+\frac{\dot{V}}{V}\mathrm{coth}\left( \frac{t}{b T_0}\right) +\sqrt{\frac{3V}{\mathrm{cosh}\left( \frac{t}{b T_0}\right) }}\frac{\mathrm{sinh}\left( \frac{t}{b T_0}\right) }{ M_p}\right] ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array}\end{aligned}$$

(4.43)

and for the generalized case we get

$$\begin{aligned} \begin{array}{lll} \displaystyle 0 =\left\{ \begin{array}{lll} \displaystyle \left[ \frac{1}{b T_{0}}+\frac{1}{2q}\frac{\dot{V}}{V}\mathrm{coth}\left( \frac{t}{b T_0}\right) +\frac{\sqrt{\frac{3V}{\mathrm{cosh}\left( \frac{t}{b T_0}\right) }}\mathrm{sinh} \left( \frac{t}{b T_0}\right) }{ M_p \left[ 1-(1-2q)\mathrm{tanh}^2\left( \frac{t}{b T_0}\right) \right] }\right] ,~ &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} \left[ \frac{1}{b T_{0i}}+\frac{1}{2q}\frac{\dot{V}}{V}\mathrm{coth}\left( \frac{t}{b T_{0i}}\right) +\frac{\sqrt{\frac{3V}{\mathrm{cosh} \left( \frac{t}{b T_{0i}}\right) }}\mathrm{sinh}\left( \frac{t}{b T_{0i}}\right) }{ M_p \left[ 1-(1-2q)\mathrm{tanh}^2\left( \frac{t}{b T_{0i}}\right) \right] }\right] ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle N\left[ \frac{1}{b T_{0}}+\frac{1}{2q}\frac{\dot{V}}{V}\mathrm{coth}\left( \frac{t}{b T_0}\right) +\frac{\sqrt{\frac{3V}{\mathrm{cosh}\left( \frac{t}{b T_0}\right) }}\mathrm{sinh} \left( \frac{t}{b T_0}\right) }{ M_p \left[ 1-(1-2q)\mathrm{tanh}^2\left( \frac{t}{b T_0}\right) \right] }\right] ,~ &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.44)

The solutions of Eqs. (4.43) and (4.44) are given by

$$\begin{aligned} \begin{array}{lll} \displaystyle V(t) =\left\{ \begin{array}{lll} \displaystyle \frac{\lambda }{\mathrm{cosh}\left( \frac{t}{b T_0}\right) }\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V_{i}(t)=\sum ^{N}_{i=1}\frac{\lambda _{i}}{\mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) }\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \frac{t}{b T_{0i}}-\mathrm{tanh}\left( \frac{t}{b T_{0i}}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N\lambda }{\mathrm{cosh}\left( \frac{t}{b T_0}\right) }\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.45)

and for the generalized case we get

$$\begin{aligned} \begin{array}{lll} \displaystyle V(t) =\left\{ \begin{array}{lll} \displaystyle \frac{\lambda }{\left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ^{2q}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V_{i}(t)=\sum ^{N}_{i=1}\frac{\lambda _{i}}{\left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right] ^{2q}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \frac{t}{b T_{0i}}-\mathrm{tanh}\left( \frac{t}{b T_{0i}}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N\lambda }{\left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ^{2q}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.46)

where to get the analytical solution from the generalized case we assume that the time scale of our consideration satisfies the following constraint:

$$\begin{aligned} t<<b T_{0}~\mathrm{tanh}^{-1}\left[ \sqrt{\frac{1}{2q-1}}\right] \end{aligned}$$

(4.47)

which is valid for all values of q except $q\ne 1/2$. For the usual tachyonic case and for the generalized situation we use the following normalization condition:

$$\begin{aligned} V(t=0)=V(0)=\lambda \end{aligned}$$

(4.48)

to fix the value of arbitrary integration constant. Further using the explicit solution for the tachyonic field as appearing in Eq. (4.42), we can write the time as a function of tachyonic field as

$$\begin{aligned} \begin{array}{lll} \displaystyle t =\left\{ \begin{array}{lll} \displaystyle b T_0 \left[ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right] ,~&{} \mathrm{for~~~Single} \\ \displaystyle \sum ^{N}_{i=1}b T_{0i} \left[ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0i}}\right) \right\} \right] ,&{} \mathrm{for~~~Multi} ,\\ \displaystyle Nb T_0 \left[ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right] ,&{} \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.49)

and further substituting Eq. (4.49) in Eqs. (4.45) and (4.46) we get the following expression for the potential as a function of tachyonic field:

$$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \lambda \exp \left( -\frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}\lambda _{i}\exp \left( -\frac{\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle N\lambda \exp \left( -\frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.50)

and for the generalized case we get

$$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \lambda \exp \left( -\frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}\lambda _{i}\exp \left( -\frac{2q\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle N\lambda \exp \left( -\frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.51)

Next we use the following redefinition in the tachyonic field:

$$\begin{aligned} \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\rightarrow & {} \frac{T}{T_{0}},\end{aligned}$$

(4.52)

$$\begin{aligned} \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0i}}\rightarrow & {} \frac{T}{T_{0i}}. \end{aligned}$$

(4.53)

Hence using the redefinition, the potential as stated in Eqs. (4.50) and (4.51) can be re-expressed as

$$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \lambda \exp \left( -\frac{T}{T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}\lambda _{i}\exp \left( -\frac{T_{i}}{T_{0i}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle N\lambda \exp \left( -\frac{T}{T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.54)

and for the generalized case we get

$$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \lambda \exp \left( -\frac{2qT}{T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 , &{} \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}\lambda _{i}\exp \left( -\frac{2qT_{i}}{ T_{0i}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T_{i}}{b T_{0i}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \mathrm{for~~~Multi} ,\\ \displaystyle N\lambda \exp \left( -\frac{2qT}{ T_{0}}\right) \left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} -\mathrm{tanh}\left( \mathrm{cosh}^{-1}\left\{ \exp \left( \frac{2q\sqrt{\alpha ^{\prime }}T}{b T_{0}}\right) \right\} \right) \right\} }\right] ^2 ,~ &{} \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

(4.55)

Now let us explicitly study the limiting behavior of the potential V(T) in detail, which is appended below:

• At $T<<T_{0}$ and $T_{i}<<T_{0i}$ limiting case one can use the following approximation for the usual tachyonic case:

  $$\begin{aligned}&\ln \left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right]<<1,\end{aligned}$$

  (4.56)
  $$\begin{aligned}&\ln \left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right]<<1. \end{aligned}$$

  (4.57)

  Using this approximation one can use the following expansion:

  $$\begin{aligned} \ln \left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right]\approx & {} \frac{1}{2}\left( \frac{t}{b T_{0}}\right) ^2,\end{aligned}$$

  (4.58)
  $$\begin{aligned} \ln \left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right]\approx & {} \frac{1}{2}\left( \frac{t}{b T_{0i}}\right) ^2. \end{aligned}$$

  (4.59)

• Hence using the solution obtained for the tachyonic field as stated in Eq. (4.42) we get

  $$\begin{aligned} \begin{array}{lll} \displaystyle T(t) =\left\{ \begin{array}{lll} \displaystyle \frac{T_{0}}{2}\left( \frac{t}{b T_{0}}\right) ^2 ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}T_{i}(t)=\sum ^{N}_{i=1}\frac{T_{0i}}{2}\left( \frac{t}{b T_{0i}}\right) ^2 ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{NT_{0}}{2}\left( \frac{t}{b T_{0}}\right) ^2 ,&{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.60)

  and by inverting Eq. (4.78) the associated time scale can be computed as

  $$\begin{aligned} \begin{array}{lll} \displaystyle t =\left\{ \begin{array}{lll} \displaystyle b T_{0}\sqrt{\frac{2T}{ T_{0}}} ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}b T_{0i}\sqrt{\frac{2T}{ T_{0i}}} ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle Nb T_{0}\sqrt{\frac{2T}{ T_{0}}} ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

  (4.61)

  Consequently we have

  $$\begin{aligned}&\mathrm{Single}: \quad \mathrm{tanh}\left( \frac{t}{b T_{0}}\right) \approx \mathrm{tanh}\left( \sqrt{\frac{2T}{ T_{0}}}\right) \approx \sqrt{\frac{2T}{ T_{0}}},\end{aligned}$$

  (4.62)
  $$\begin{aligned}&\mathrm{Multi}: \quad \mathrm{tanh}\left( \frac{t}{b T_{0i}}\right) \approx \mathrm{tanh}\left( \sqrt{\frac{2T}{ T_{0i}}}\right) \approx \sqrt{\frac{2T}{ T_{0i}}},\end{aligned}$$

  (4.63)
  $$\begin{aligned}&\mathrm{Assisted}: \quad \mathrm{tanh}\left( \frac{t}{b T_{0}}\right) \approx \mathrm{tanh}\left( \sqrt{\frac{2T}{ T_{0}}}\right) \approx \sqrt{\frac{2T}{ T_{0}}}. \end{aligned}$$

  (4.64)

  and finally for the usual tachyonic case the potential V(T) can be approximated by

  $$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \lambda \exp \left( -\frac{T}{T_{0}}\right) , &{} \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V(T_{i})=\sum ^{N}_{i=1}\lambda _{i}\exp \left( -\frac{T_{i}}{T_{0i}}\right) ,~ &{} \mathrm{for~~~Multi} ,\\ \displaystyle N\lambda \exp \left( -\frac{T}{T_{0}}\right) , &{} \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.65)

  and for the generalized case we have

  $$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \lambda \exp \left( -\frac{2qT}{T_{0}}\right) , &{}\quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V(T_{i})=\sum ^{N}_{i=1}\lambda _{i}\exp \left( -\frac{2qT_{i}}{T_{0i}}\right) ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle N\lambda \exp \left( -\frac{2qT}{T_{0}}\right) , &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.66)

  where the behavior of these types of potentials has been elaborated in the earlier section.

• At $T>>T_{0}$ and $T_{i}>>T_{0i}$ limiting case one can use the following approximation for the usual tachyonic case:

  $$\begin{aligned}&\ln \left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right]>>1,\end{aligned}$$

  (4.67)
  $$\begin{aligned}&\ln \left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right]>>1. \end{aligned}$$

  (4.68)

  Using this approximation one can use the following expansion:

  $$\begin{aligned} \ln \left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right]\approx & {} \ln \left[ \frac{1}{2}{\exp }\left( \frac{t}{b T_0}\right) \right] \approx \frac{t}{b T_0},\end{aligned}$$

  (4.69)
  $$\begin{aligned} \ln \left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right]\approx & {} \ln \left[ \frac{1}{2}{\exp }\left( \frac{t}{b T_{0i}}\right) \right] \approx \frac{t}{b T_{0i}}. \end{aligned}$$

  (4.70)

  Hence using the solution obtained for the tachyonic field as stated in Eq. (4.42) we get

  $$\begin{aligned} \begin{array}{lll} \displaystyle T(t) =\left\{ \begin{array}{lll} \displaystyle \frac{t}{b} ,~&{} \mathrm{for~~~Single} \\ \displaystyle \sum ^{N}_{i=1}T_{i}(t)=\sum ^{N}_{i=1} \frac{t}{2b}=\frac{Nt}{b} ,&{} \mathrm{for~~~Multi} ,\\ \displaystyle \frac{Nt}{b} ,&{} \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.71)

  and by inverting Eq. (4.71) the associated time scale can be computed as

  $$\begin{aligned} \begin{array}{lll} \displaystyle t =\left\{ \begin{array}{lll} \displaystyle b T ,~&{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}b T_{i} ,&{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle Nb T ,&{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

  (4.72)

  Consequently we have

  $$\begin{aligned}&\mathrm{Single: } \quad \mathrm{tanh}\left( \frac{t}{b T_{0}}\right) \approx 1,\end{aligned}$$

  (4.73)
  $$\begin{aligned}&\mathrm{Multi: } \quad \mathrm{tanh}\left( \frac{t}{b T_{0i}}\right) \approx 1,\end{aligned}$$

  (4.74)
  $$\begin{aligned}&\mathrm{Assisted: } \quad \mathrm{tanh}\left( \frac{t}{b T_{0}}\right) \approx 1. \end{aligned}$$

  (4.75)

  and finally for the usual tachyonic case the potential V(T) can be approximated by

  $$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \frac{4M^{2}_{p}}{3 b^2 T^{2}_{0}}\left( \frac{T_{0}}{T}\right) ^2 \exp \left( -\frac{T}{T_{0}}\right) , &{} \mathrm{for~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V(T_{i})=\sum ^{N}_{i=1}\frac{4M^{2}_{p}}{3 b^2 T^{2}_{0i}}\left( \frac{T_{0i}}{T_{i}}\right) ^2\exp \left( -\frac{T_{i}}{T_{0i}}\right) ,~ &{} \mathrm{for~~Multi} ,\\ \displaystyle \frac{4M^{2}_{p}N}{3 b^2 T^{2}_{0}}\left( \frac{T_{0}}{T}\right) ^2\exp \left( -\frac{T}{T_{0}}\right) ,~ &{} \mathrm{for~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.76)

  and for the generalized case we have

  $$\begin{aligned} \begin{array}{lll} \displaystyle V(T) =\left\{ \begin{array}{lll} \displaystyle \frac{4M^{2}_{p}}{3 b^2 T^{2}_{0}}\left( \frac{T_{0}}{T}\right) ^2\exp \left( -\frac{2qT}{T_{0}}\right) , &{} \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}V(T_{i})=\sum ^{N}_{i=1}\frac{4M^{2}_{p}}{3 b^2 T^{2}_{0i}}\left( \frac{T_{0i}}{T_{i}}\right) ^2\exp \left( -\frac{2qT_{i}}{T_{0i}}\right) ,~ &{} \mathrm{for~~~Multi} ,\\ \displaystyle \frac{4M^{2}_{p}N}{3 b^2 T^{2}_{0}}\left( \frac{T_{0}}{T}\right) ^2\exp \left( -\frac{2qT}{T_{0}}\right) ,~ &{} \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.77)

  where the scale of inflation for the usual tachyonic case is fixed by:

  $$\begin{aligned} \begin{array}{lll} \displaystyle V^{1/4}_{\mathrm{inf}}\propto \left\{ \begin{array}{lll} \displaystyle \lambda ^{1/4} , &{} \quad \mathrm{for~~~Single} \\ \displaystyle \left\{ \sum ^{N}_{i=1}\lambda _{i}\right\} ^{1/4} ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle (N\lambda )^{1/4} ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.78)

  and for the generalized case it is fixed by

  $$\begin{aligned} \begin{array}{lll} \displaystyle V^{1/4}_{\mathrm{inf}}\propto \left\{ \begin{array}{lll} \displaystyle \left( \frac{4M^{2}_{p}}{3 b^2 T^{2}_{0}}\right) ^{1/4} , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \left\{ \sum ^{N}_{i=1}\frac{4M^{2}_{p}}{3 b^2 T^{2}_{0i}}\right\} ^{1/4} ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \left( \frac{4M^{2}_{p}N}{3 b^2 T^{2}_{0}}\right) ^{1/4} ,~ &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

  (4.79)

  Here the potential should satisfy the following criteria:

  • At $T=0$ for the single field tachyonic potential:

    $$\begin{aligned} V(T=0)\rightarrow \infty , \end{aligned}$$

    (4.80)

    and for the multi-tachyonic and assisted cases we have

    $$\begin{aligned} V_{E}(T=0)\rightarrow & {} \infty . \end{aligned}$$

    (4.81)

  • At $T=T_0$ for the single field tachyonic potential:

    $$\begin{aligned} V(T=T_0)=\frac{4M^{2}_{p}}{3e b^2 T^{2}_{0}} \end{aligned}$$

    (4.82)

    and also for the multi-tachyonic and assisted cases we have

    $$\begin{aligned} V_{E}(T=T_0)= & {} \sum ^{N}_{i=1}\frac{4M^{2}_{p}}{3e b^2 T^{2}_{0i}},\end{aligned}$$

    (4.83)
    $$\begin{aligned} V_{E}(T=T_0)= & {} \frac{4M^{2}_{p}N}{3e b^2 T^{2}_{0}}. \end{aligned}$$

    (4.84)

    For the generalized case one can repeat the same computation with the following redefinition of the b parameter of tachyonic field theory:

    $$\begin{aligned} b^{-2}\exp (-2q)\rightarrow b^{-2}. \end{aligned}$$

    (4.85)

  • For single field tachyonic potential:

    $$\begin{aligned} V^{\prime }(T)= & {} -\frac{4 M^{2}_{p}}{3 b^2 T^3}\exp \left( -\frac{T}{T_{0}}\right) \left( 2 +\frac{T}{T_{0}}\right) ,\end{aligned}$$

    (4.86)
    $$\begin{aligned} V^{\prime \prime }(T)= & {} \frac{4 M^{2}_{p}}{3 b^2 T^4}\exp \left( -\frac{T}{T_{0}}\right) \nonumber \\&\times \left\{ 6+4\left( \frac{T}{T_{0}}\right) +\left( \frac{T}{T_{0}}\right) ^2\right\} . \end{aligned}$$

    (4.87)

    Now to find the extrema of the potential we substitute

    $$\begin{aligned} V^{\prime }(T)=0 \end{aligned}$$

    (4.88)

    which give rise to the following solution for the tachyonic field:

    $$\begin{aligned} T=-2T_{0}, \; \infty . \end{aligned}$$

    (4.89)

    Further substituting the solutions for the tachyonic field in Eq. (4.87) we get

    $$\begin{aligned}&V^{\prime \prime }\left( T=-2T_{0}\right) = \frac{e^2 M^{2}_{p}}{6 b^2 T^4_{0}},\end{aligned}$$

    (4.90)
    $$\begin{aligned}&V^{\prime \prime }(T\rightarrow \infty )\rightarrow 0 \end{aligned}$$

    (4.91)

    and also additionally for $T=T_{0}$ we have

    $$\begin{aligned} V^{\prime \prime }(T=T_0)= & {} \frac{44 M^{2}_{p}}{3e b^2 T^4_{0}}, \end{aligned}$$

    (4.92)

    and at these points the value of the potential is computed as

    $$\begin{aligned}&V(T\rightarrow \infty )\rightarrow 0,\end{aligned}$$

    (4.93)
    $$\begin{aligned}&V(T=-2T_{0})= \frac{e^2 M^{2}_{p}}{3 b^2 T^2_{0}},\end{aligned}$$

    (4.94)
    $$\begin{aligned}&V(T=T_{0})= \frac{4 M^{2}_{p}}{3 b^2 T^2_{0}}. \end{aligned}$$

    (4.95)

    It is important to note for the single tachyonic case that:

    • for $b^2>0$, $V^{\prime \prime }(T=-2T_{0},~~T_{0} )>0$, i.e., we get maxima on the potential.

    • for $b^2<0$, $V^{\prime \prime }(T=-2T_{0},~~T_{0} )<0$, i.e., we get minima on the potential.

    and for the assisted case the results are the same, provided the following replacement occurs:

    $$\begin{aligned} \lambda \rightarrow N\lambda , \end{aligned}$$

    (4.96)

    and finally for the multi-tachyonic case we have

    $$\begin{aligned} V^{\prime }(T_{i})= & {} -\frac{4 M^{2}_{p}}{3 b^2 T^3_{i}}\exp \left( -\frac{T_{i}}{T_{0i}}\right) \left( 2 +\frac{T_{i}}{T_{0i}}\right) ,\nonumber \\\end{aligned}$$

    (4.97)
    $$\begin{aligned} V^{\prime \prime }(T_{i})= & {} \frac{4 M^{2}_{p}}{3 b^2 T^4_{i}}\exp \left( -\frac{T_{i}}{T_{0i}}\right) \nonumber \\&\quad \times \left\{ 6+4\left( \frac{T_{i}}{T_{0i}}\right) +\left( \frac{T_{i}}{T_{0i}}\right) ^2\right\} . \end{aligned}$$

    (4.98)

    Now to find the extrema of the potential we substitute:

    $$\begin{aligned} V^{\prime }(T_{j})=0 \quad \forall \;j=1,2,\ldots ,N \end{aligned}$$

    (4.99)

    which give rise to the following solutions for the jth tachyonic field:

    $$\begin{aligned} T_{j}=-2T_{0j},~ \infty . \end{aligned}$$

    (4.100)

    Further substituting the solutions for the tachyonic field in Eq. (4.98) we get

    $$\begin{aligned}&V^{\prime \prime }\left( T_{j}=-2T_{0j}\right) = \frac{e^2 M^{2}_{p}}{6 b^2 T^4_{0j}},\end{aligned}$$

    (4.101)
    $$\begin{aligned}&V^{\prime \prime }(T_{j}\rightarrow \infty )\rightarrow 0. \end{aligned}$$

    (4.102)

    Additionally for the point $T_{j}=T_{0j}$ we have

    $$\begin{aligned} V^{\prime \prime }(T_{j}=T_{0j})= & {} \frac{44 M^{2}_{p}}{3e b^2 T^4_{0j}}, \end{aligned}$$

    (4.103)

    and at these points the value of the total effective potential is computed as

    $$\begin{aligned}&V^{(1)}_{E}=\sum ^{N}_{j=1}V(T_{j}\rightarrow \infty )\rightarrow 0,\end{aligned}$$

    (4.104)
    $$\begin{aligned}&V^{(2)}_{E}=\sum ^{N}_{j=1}V(T_{j}=-2T_{0j})= \sum ^{N}_{j=1}\frac{e^2 M^{2}_{p}}{3 b^2 T^2_{0j}},\end{aligned}$$

    (4.105)
    $$\begin{aligned}&V^{(3)}_{E}= \sum ^{N}_{j=1}V(T_{j}=T_{0j})= \frac{4N M^{2}_{p}}{3 b^2 T^2_{0}}. \end{aligned}$$

    (4.106)

    It is important to note for the multi-tachyonic case that:

    • for $b^2>0$, $V^{\prime \prime }(T=-2T_{0j},~~T_{0j} )>0$, i.e., we get maxima on the potential $V_{j}(T)$ as well in $V_{E}(T)$.

    • for $b^2<0$, $V^{\prime \prime }(T=-2T_{0j},~~T_{0j} )<0$, i.e., we get minima on the potential $V_{j}(T)$ as well in $V_{E}(T)$.

Next using Eq. (4.17) the Hubble parameter can be expressed in terms of the usual tachyonic potential V(T) as

$$\begin{aligned} \begin{array}{lll} \displaystyle H^{2}(t)= \left\{ \begin{array}{lll} \displaystyle \frac{\lambda }{3M^{2}_{p}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}H_{i}(t)=\sum ^{N}_{i=1}\frac{\lambda _{i}}{3M^{2}_{p}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \frac{t}{b T_{0i}}-\mathrm{tanh}\left( \frac{t}{b T_{0i}}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N\lambda }{3M^{2}_{p}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

(4.107)

and for the generalized case we have

$$\begin{aligned} \begin{array}{lll} \displaystyle H^{2}(t)= \left\{ \begin{array}{lll} \displaystyle \frac{\lambda }{3M^{2}_{p}\left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ^{2(2q-1)}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 , &{} \quad \mathrm{for~~~Single} \\ \displaystyle \sum ^{N}_{i=1}\frac{\lambda _{i}}{3M^{2}_{p}\left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right] ^{2(2q-1)}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0i}}{2M_{p}}\left\{ \frac{t}{b T_{0i}}-\mathrm{tanh}\left( \frac{t}{b T_{0i}}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N\lambda }{3M^{2}_{p}\left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ^{2(2q-1)}}\left[ \frac{1}{1+\frac{\sqrt{3\lambda }bT_{0}}{2M_{p}}\left\{ \frac{t}{b T_{0}}-\mathrm{tanh}\left( \frac{t}{b T_0}\right) \right\} }\right] ^2 ,~ &{} \quad \mathrm{for~~~Assisted} .\end{array} \right. \end{array} \end{aligned}$$

(4.108)

Let us study the limiting situation from the expression obtained for the Hubble parameter explicitly:

• In the $T<<T_{0}$ and $T_{j}<<T_{0j}$ limiting situation the Hubble parameter can be approximated for the usual tachyonic case as:

  $$\begin{aligned} \begin{array}{lll} \displaystyle H^{2}(t)= \left\{ \begin{array}{lll} \displaystyle \frac{\lambda }{3M^{2}_{p}}, &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}\frac{\lambda _{i}}{3M^{2}_{p}} ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N\lambda }{3M^{2}_{p}} ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.109)

  and for the generalized case we have

  $$\begin{aligned} \begin{array}{lll} \displaystyle H^{2}(t)= \left\{ \begin{array}{lll} \displaystyle \frac{\lambda }{3M^{2}_{p}\left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ^{2(2q-1)}}, &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}\frac{\lambda _{i}}{3M^{2}_{p}\left[ \mathrm{cosh}\left( \frac{t}{b T_{0i}}\right) \right] ^{2(2q-1)}} ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{N\lambda }{3M^{2}_{p}\left[ \mathrm{cosh}\left( \frac{t}{b T_0}\right) \right] ^{2(2q-1)}} ,~ &{} \quad \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

  (4.110)

  In this limiting situation the solution for the scale factor a(t) from Eq. (4.109) can be expressed as

  $$\begin{aligned} \begin{array}{lll} \displaystyle a(t)= \left\{ \begin{array}{lll} \displaystyle a_{\mathrm{inf}}\exp \left[ \frac{\sqrt{\lambda }}{\sqrt{3}M_{p}}\left( t-t_{\mathrm{inf}}\right) \right] , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}a_{\mathrm{inf}}\exp \left[ \sqrt{\frac{\lambda _{i}}{3M^{2}_{p}}}\left( t-t_{\mathrm{inf}}\right) \right] ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle a_{\mathrm{inf}}\exp \left[ \frac{\sqrt{N\lambda }}{\sqrt{3}M_{p}}\left( t-t_{\mathrm{inf}}\right) \right] ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.111)

  which is exactly the de Sitter solution required for inflation. Here at the inflationary time scale $t=t_{\mathrm{inf}}$ the scale factor is given by

  $$\begin{aligned} a_{\mathrm{inf}}=a(t=t_{\mathrm{inf}}). \end{aligned}$$

  (4.112)

• On the other hand for the generalized case we have the following solution for the scale factor a(t):

  $$\begin{aligned} \begin{array}{lll} \displaystyle a(t)= \left\{ \begin{array}{lll} \displaystyle a_{\mathrm{inf}}\exp \left[ \frac{\sqrt{\lambda }b T_{0}}{\sqrt{3}M_{p}}\left\{ \mathrm{sinh}\left( \frac{t}{b T_0}\right) \, _2F_1\left[ \frac{1}{2},q;\frac{3}{2};-\sinh ^2\left( \frac{t}{b T_0}\right) \right] \right\} ^{t}_{t_{\mathrm{inf}}}\right] , &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} a_{\mathrm{inf}}\exp \left[ \frac{\sqrt{\lambda _{i}}b T_{0i}}{\sqrt{3}M_{p}}\left\{ \mathrm{sinh}\left( \frac{t}{b T_{0i}}\right) \, _2F_1\left[ \frac{1}{2},q;\frac{3}{2};-\sinh ^2\left( \frac{t}{b T_{0i}}\right) \right] \right\} ^{t}_{t_{\mathrm{inf}}}\right] ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle a_{\mathrm{inf}}\exp \left[ \frac{\sqrt{N\lambda }b T_{0}}{\sqrt{3}M_{p}}\left\{ \mathrm{sinh}\left( \frac{t}{b T_0}\right) \, _2F_1\left[ \frac{1}{2},q;\frac{3}{2};-\sinh ^2\left( \frac{t}{b T_0}\right) \right] \right\} ^{t}_{t_{\mathrm{inf}}}\right] ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array}\quad \quad \qquad \qquad \end{aligned}$$

  (4.113)

  which replicates the behavior of quasi-de Sitter solution during inflation. For the $q=1/2$ case it exactly follows the de Sitter behavior.

• In the $T>>T_{0}$ and $T_{j}>>T_{0j}$ limiting situation the Hubble parameter can be approximated for the usual tachyonic case as:

  $$\begin{aligned} \begin{array}{lll} \displaystyle H^{2}(t)= \left\{ \begin{array}{lll} \displaystyle \frac{4}{9t^2}, &{} \quad \mathrm{for~~~Single} \\ \displaystyle \sum ^{N}_{i=1}\frac{4}{9t^2} ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle \frac{4N}{9t^2} ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.114)

  and for the generalized case we have

  $$\begin{aligned} \begin{array}{lll} \displaystyle H^{2}(t)= \left\{ \begin{array}{lll} \displaystyle \frac{4}{9t^2\left[ \frac{1}{2}\exp \left( \frac{t}{b T_0}\right) \right] ^{2(2q-1)}}, &{} \mathrm{for~~~Single} \\ \displaystyle \sum ^{N}_{i=1}\frac{4}{9t^2\left[ \frac{1}{2}\exp \left( \frac{t}{b T_{0i}}\right) \right] ^{2(2q-1)}} ,~ &{} \mathrm{for~~~Multi} ,\\ \displaystyle \frac{4N}{9t^2\left[ \frac{1}{2}\exp \left( \frac{t}{b T_0}\right) \right] ^{2(2q-1)}} ,~ &{} \mathrm{for~~~Assisted}. \end{array} \right. \end{array} \end{aligned}$$

  (4.115)

  In this limiting situation the solution for the scale factor a(t) from Eq. (4.114) can be expressed as

  $$\begin{aligned} \begin{array}{lll} \displaystyle a(t)= \left\{ \begin{array}{lll} \displaystyle a_{\mathrm{dust}}\left( \frac{t}{t_{\mathrm{dust}}}\right) ^{2/3}, &{} \quad \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1}a_{\mathrm{dust}}\left( \frac{t}{t_{\mathrm{dust}}}\right) ^{2/3} ,~ &{} \quad \mathrm{for~~~Multi} ,\\ \displaystyle a_{\mathrm{dust}}\left( \frac{t}{t_{\mathrm{dust}}}\right) ^{2/3} ,~ &{} \quad \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.116)

  which is exactly the dust like solution required for the formation of dark matter. Here at the inflationary time scale $t=t_{\mathrm{dust}}$ the scale factor is given by

  $$\begin{aligned} a_{\mathrm{dust}}=a(t=t_{\mathrm{dust}}). \end{aligned}$$

  (4.117)

  On the other hand for the generalized case we have the following solution for the scale factor a(t):

  $$\begin{aligned} \begin{array}{lll} \displaystyle a(t)= \left\{ \begin{array}{lll} \displaystyle a_{\mathrm{dust}}\exp \left[ \frac{4^q}{3} \left\{ \text {Ei}\left[ \left( \frac{t}{b T_{0}}\right) -2q\right] \right. \right. &{}\,\\ \left. \left. \,\,-\text {Ei}\left[ \left( \frac{t_{\mathrm{dust}}}{b T_{0}}\right) -2q\right] \right\} \right] , &{} \mathrm{for~~~Single}, \\ \displaystyle \sum ^{N}_{i=1} a_{\mathrm{dust}}\exp \left[ \frac{4^q}{3} \left\{ \text {Ei}\left[ \left( \frac{t}{b T_{0i}}\right) -2q\right] \right. \right. &{}\,\\ \left. \left. \,\,-\text {Ei}\left[ \left( \frac{t_{\mathrm{dust}}}{b T_{0i}}\right) -2q\right] \right\} \right] ,~ &{} \mathrm{for~~~Multi} ,\\ \displaystyle a_{\mathrm{dust}}\exp \left[ \frac{N4^q}{3} \left\{ \text {Ei}\left[ \left( \frac{t}{b T_{0}}\right) -2q\right] \right. \right. &{}\\ \left. \left. \,\,-\text {Ei}\left[ \left( \frac{t_{\mathrm{dust}}}{b T_{0}}\right) -2q\right] \right\} \right] ,~ &{} \mathrm{for~~~Assisted}, \end{array} \right. \end{array} \end{aligned}$$

  (4.118)

  which replicates the behavior of quasi-dust like solution. For the $q=1/2$ case it exactly follows the dust like behavior.

5 Inflationary paradigm from GTachyon

It is a very well known fact that during cosmological inflation, quantum fluctuations are stretched on the scales larger than the size of the horizon. Consequently, they are frozen until they re-enter the horizon at the end of inflationary phase. In the present context a single tachyonic field drives the inflationary paradigm, which finally gives rise to the large-scale perturbations with a quasi-scale invariant primordial power spectrum corresponding to the scalar and tensor modes. Deviations from the scale invariance in the primordial power spectrum can be measured in terms of the slow-roll parameters which we will explicitly discuss in the next subsections. By detailed computation we explicitly show that at lowest order of the primordial spectrum the scalar perturbations is exactly the same as that obtained for the usual single field inflationary set-up. For completeness the next to leading order corrections to the cosmological perturbations are also derived, which finally give rise to the sufficient change in the cosmological consistency relations with respect to the results obtained for the usual single field inflationary set-up. Hence we apply all the derived results for the tachyonic inflationary models as explicitly mentioned in the previous section and study the CMB constraints by applying the recent Planck 2015 data.

5.1 Computation for single field inflation

5.1.1 Condition for inflation

For single field tachyonic inflation, the prime condition for inflation is given by

$$\begin{aligned} \dot{H}+H^{2}= & {} \left( \frac{\ddot{a}}{{a}}\right) =-\frac{(\rho +3p)}{6 M^2_p}>0, \end{aligned}$$

(5.1)

which can be re-expressed in terms of the following constraint condition in the context of single field tachyonic inflation:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{V(T)}{3M^{2}_{p}\sqrt{1-\alpha ^{\prime }\dot{T}^2}}\left( 1-\frac{3}{2}\alpha ^{\prime }\dot{T}^2\right) >0. \end{array} \end{aligned}$$

(5.2)

Here Eq. (5.6) implies that, to satisfy inflationary constraints in the slow-roll regime, the following constraint always holds good:

$$\begin{aligned} \dot{T}< & {} \sqrt{\frac{2}{3\alpha ^{\prime }}},\end{aligned}$$

(5.3)

$$\begin{aligned} \ddot{T}<&3H \dot{T}<\sqrt{\frac{6}{\alpha ^{\prime }}}H. \end{aligned}$$

(5.4)

Consequently the field equations are approximated by

$$\begin{aligned} \begin{array}{lll} \displaystyle 3H\alpha ^{\prime }\dot{T}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\approx 0. \end{array} \end{aligned}$$

(5.5)

Similarly, in the most generalized case,

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{V(T)}{3M^{2}_{p}\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{1-q}}\left( 1-(1+q)\alpha ^{\prime }\dot{T}^2\right) >0. \end{array} \end{aligned}$$

(5.6)

Here Eq. (5.6) implies that to satisfy inflationary constraints in the slow-roll regime the following constraint always holds good:

$$\begin{aligned}&\dot{T}< \sqrt{\frac{1}{\alpha ^{\prime }(1+q)}},\end{aligned}$$

(5.7)

$$\begin{aligned}&\ddot{T}< 3H \dot{T}<\sqrt{\frac{9}{\alpha ^{\prime }(1+q)}}H. \end{aligned}$$

(5.8)

Consequently the field equations are approximated by

$$\begin{aligned} \begin{array}{lll} \displaystyle 6q\alpha ^{\prime }H\dot{T}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\approx 0. \end{array} \end{aligned}$$

(5.9)

Also for both cases in the slow-roll regime the Friedmann equation is modified as

$$\begin{aligned} H^{2}\approx & {} \frac{V(T)}{3M^{2}_{p}}. \end{aligned}$$

(5.10)

Further substituting Eq. (5.10) in Eqs. (5.5) and (5.9) we get

$$\begin{aligned}&\displaystyle \frac{\sqrt{3V(T)}}{M_{p}}\alpha ^{\prime }\dot{T}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\approx 0 ,\end{aligned}$$

(5.11)

$$\begin{aligned}&6q\frac{\sqrt{V(T)}}{\sqrt{3}M_{p}}\alpha ^{\prime }\dot{T}+\frac{\mathrm{d}V(T)}{V(T)\mathrm{d}T}\approx 0. \end{aligned}$$

(5.12)

Finally the general solution for both cases can be expressed in terms of the single field tachyonic potential V(T) as

$$\begin{aligned}&\displaystyle t-t_{i}\approx -\frac{\sqrt{3}\alpha ^{\prime }}{M_{p}}\int ^{T}_{T_{i}}\mathrm{d}T~\frac{V^{3/2}(T)}{V^{\prime }(T)} ,\end{aligned}$$

(5.13)

$$\begin{aligned}&t-t_{i}\approx -\frac{6q\alpha ^{\prime }}{\sqrt{3}M_{p}}\int ^{T}_{T_{i}}\mathrm{d}T~\frac{V^{3/2}(T)}{V^{\prime }(T)}. \end{aligned}$$

(5.14)

Let us now re-write Eqs. (5.332) and (5.333), in terms of the string theoretic tachyonic potentials as already mentioned in the last section. For the $q=1/2$ situation we get

$$\begin{aligned} \begin{array}{lll} \displaystyle t-t_{i}\approx \left\{ \begin{array}{lll} \displaystyle \frac{\alpha ^{\prime }T^{2}_{0}}{2M_{p}}\sqrt{3\lambda } \left\{ \ln \left[ \frac{\left( \sqrt{\text {sech}\left( \frac{T_{i}}{T_{0}}\right) }+1\right) \left( 1-\sqrt{\text {sech}\left( \frac{T}{T_{0}}\right) }\right) }{\left( \sqrt{\text {sech}\left( \frac{T}{T_{0}}\right) }+1\right) \left( 1-\sqrt{\text {sech} \left( \frac{T_{i}}{T_{0}}\right) }\right) }\right] \right. \\ \left. \displaystyle +2 \tan ^{-1}\left[ \frac{\sqrt{\text {sech}\left( \frac{T_{i}}{T_{0}}\right) }-\sqrt{\text {sech}\left( \frac{T}{T_{0}}\right) }}{1+\sqrt{\text {sech}\left( \frac{T_{i}}{T_{0}}\right) \text {sech}\left( \frac{T}{T_{0}}\right) }} \right] \right\} , &{} \quad \mathrm{for~Model~1,} \\ \displaystyle -\frac{\alpha ^{\prime }T^{2}_{0}}{2M_{p}}\sqrt{3\lambda }~ \ln \left( \frac{\ln \left( \frac{T}{T_{0}}\right) \left( \ln \left( \frac{T_{i}}{T_{0}}\right) +1\right) }{\ln \left( \frac{T_{i}}{T_{0}}\right) \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) }\right) , &{} \quad \mathrm{for~Model~2,} \\ \displaystyle -\frac{2\alpha ^{\prime } T^2_{0}}{M_{p}}\sqrt{3\lambda } \left\{ \exp \left[ -\frac{T}{2T_{0}}\right] -\exp \left[ -\frac{T_{i}}{2T_{0}}\right] \right\} , &{} \quad \mathrm{for~Model~3,} \\ \displaystyle -\frac{\alpha ^{\prime }T^{2}_{0}}{4M_{p}}\sqrt{3\lambda }\left\{ \text {Ei}\left( -\frac{T^2_{i}}{2 T^2_{0}}\right) -\text {Ei}\left( -\frac{T^2}{2 T^2_{0}}\right) \right\} , &{} \quad \mathrm{for~Model~4,} \\ \displaystyle \frac{\alpha ^{\prime }T^{2}_{0}}{8 M_{p}} \sqrt{3\lambda } \left\{ \ln \left( \frac{\sqrt{T^4_{0}+T^4}+T^2}{\sqrt{T^4_{0}+T^4_{i}}+T^2_{i}}\right) +\sqrt{1+\left( \frac{T_{0}}{T_{i}}\right) ^4}-\sqrt{1+\left( \frac{T_{0}}{T}\right) ^4}\right\} , &{} \quad \mathrm{for~Model~5,} \end{array} \right. \end{array} \end{aligned}$$

(5.15)

and for any arbitrary q we get the following generalized result:

$$\begin{aligned} \begin{array}{lll} \displaystyle t-t_{i}\approx \left\{ \begin{array}{lll} \displaystyle \frac{3q\alpha ^{\prime }T^{2}_{0}}{M_{p}}\sqrt{\frac{\lambda }{3}} \left\{ \ln \left[ \frac{\left( \sqrt{\text {sech}\left( \frac{T_{i}}{T_{0}}\right) }+1\right) \left( 1-\sqrt{\text {sech}\left( \frac{T}{T_{0}}\right) }\right) }{\left( \sqrt{\text {sech}\left( \frac{T}{T_{0}}\right) }+1\right) \left( 1-\sqrt{\text {sech}\left( \frac{T_{i}}{T_{0}}\right) }\right) }\right] \right. \\ \left. \displaystyle +2 \tan ^{-1}\left[ \frac{\sqrt{\text {sech}\left( \frac{T_{i}}{T_{0}}\right) }-\sqrt{\text {sech}\left( \frac{T}{T_{0}}\right) }}{1+\sqrt{\text {sech}\left( \frac{T_{i}}{T_{0}}\right) \text {sech}\left( \frac{T}{T_{0}}\right) }} \right] \right\} , &{} \quad \mathrm{for~Model~1,} \\ \displaystyle -\frac{3q\alpha ^{\prime }T^{2}_{0}}{M_{p}}\sqrt{\frac{\lambda }{3}}~ \ln \left( \frac{\ln \left( \frac{T}{T_{0}}\right) \left( \ln \left( \frac{T_{i}}{T_{0}}\right) +1\right) }{\ln \left( \frac{T_{i}}{T_{0}}\right) \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) }\right) , &{} \quad \mathrm{for~Model~2,} \\ \displaystyle -\frac{12q\alpha ^{\prime } T^2_{0}}{M_{p}}\sqrt{\frac{\lambda }{3}} \left\{ \exp \left[ -\frac{T}{2T_{0}}\right] -\exp \left[ -\frac{T_{i}}{2T_{0}}\right] \right\} , &{} \quad \mathrm{for~Model~3,} \\ \displaystyle -\frac{3q\alpha ^{\prime }T^{2}_{0}}{2M_{p}}\sqrt{\frac{\lambda }{3}}\left\{ \text {Ei}\left( -\frac{T^2_{i}}{2 T^2_{0}}\right) -\text {Ei}\left( -\frac{T^2}{2 T^2_{0}}\right) \right\} , &{} \quad \mathrm{for~Model~4,} \\ \displaystyle \frac{3q\alpha ^{\prime }T^{2}_{0}}{4 M_{p}} \sqrt{\frac{\lambda }{3}} \left\{ \ln \left( \frac{\sqrt{T^4_{0}+T^4}+T^2}{\sqrt{T^4_{0}+T^4_{i}}+T^2_{i}}\right) +\sqrt{1+\left( \frac{T_{0}}{T_{i}}\right) ^4}-\sqrt{1+\left( \frac{T_{0}}{T}\right) ^4}\right\} . &{} \quad \mathrm{for~Model~5.} \end{array} \right. \end{array} \end{aligned}$$

(5.16)

Further using Eqs. (5.15), (5.16) and (5.10) we get the following solution for the scale factor in terms of the tachyonic field for the usual $q=1/2$ and for a generalized value of q as:

$$\begin{aligned} \begin{array}{lll} \displaystyle a= a_{i}\times \left\{ \begin{array}{lll} \displaystyle \exp \left[ -\frac{\alpha ^{\prime }}{M^2_p}\int ^{T}_{T_{i}}\mathrm{d}T~\frac{V^{2}(T)}{V^{\prime }(T)}\right] , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \exp \left[ -\frac{2q\alpha ^{\prime }}{M^2_p}\int ^{T}_{T_{i}}\mathrm{d}T~\frac{V^{2}(T)}{V^{\prime }(T)}\right] ,&{} \quad \mathrm{for~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.17)

Finally re-writing Eq. (5.334), in terms of the string theoretic tachyonic potentials as already mentioned in the last section for $q=1/2$, we get

$$\begin{aligned} \begin{array}{lll} \displaystyle a\approx a_{i}\times \left\{ \begin{array}{lll} \displaystyle \exp \left[ \frac{\alpha ^{\prime }T^{2}_{0}\lambda }{M^{2}_{p}}\ln \left( \frac{\tanh \left( \frac{T}{2 T_{0}}\right) }{\tanh \left( \frac{T_{i}}{2 T_{0}}\right) }\right) \right] , &{} \mathrm{for~Model~1,} \\ \displaystyle \exp \left[ -\frac{\alpha ^{\prime }T^{2}_{0}\lambda }{2M^{2}_{p}}~ \ln \left( \frac{\ln \left( \frac{T}{T_{0}}\right) \left( \ln \left( \frac{T_{i}}{T_{0}}\right) +1\right) }{\ln \left( \frac{T_{i}}{T_{0}}\right) \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) }\right) \right] , &{} \mathrm{for~Model~2,} \\ \displaystyle \exp \left[ \frac{\alpha ^{\prime }T^2_{0}\lambda }{M^2_{p}} \left( \exp \left[ -\frac{T_{i}}{T_{0}}\right] -\exp \left[ -\frac{T}{T_{0}}\right] \right) \right] , &{} \mathrm{for~Model~3,} \\ \displaystyle \exp \left[ -\frac{\alpha ^{\prime }T^{2}_{0}\lambda }{4M^2_{p}}\left\{ \text {Ei}\left( -\frac{T^2_{i}}{T^2_{0}}\right) -\text {Ei}\left( -\frac{T^2}{T^2_{0}}\right) \right\} \right] , &{} \mathrm{for~Model~4,} \\ \displaystyle \exp \left[ -\frac{\alpha ^{\prime }T^4_{0}\lambda }{8M^2_{p}} \left( \frac{1}{T^2}-\frac{1}{T^2_{i}}\right) \right] , &{} \mathrm{for~Model~5,} \end{array} \right. \end{array} \end{aligned}$$

(5.18)

and for any arbitrary q we get

$$\begin{aligned} \begin{array}{lll} \displaystyle a\approx a_{i}\times \left\{ \begin{array}{lll} \displaystyle \exp \left[ \frac{2q\alpha ^{\prime }T^{2}_{0}\lambda }{3M^{2}_{p}}\ln \left( \frac{\tanh \left( \frac{T}{2 T_{0}}\right) }{\tanh \left( \frac{T_{i}}{2 T_{0}}\right) }\right) \right] , &{} \mathrm{for~Model~1,} \\ \displaystyle \exp \left[ -\frac{q\alpha ^{\prime }T^{2}_{0}\lambda }{M^{2}_{p}}~ \ln \left( \frac{\ln \left( \frac{T}{T_{0}}\right) \left( \ln \left( \frac{T_{i}}{T_{0}}\right) +1\right) }{\ln \left( \frac{T_{i}}{T_{0}}\right) \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) }\right) \right] , &{} \mathrm{for~Model~2,} \\ \displaystyle \exp \left[ \frac{2q\alpha ^{\prime }T^2_{0}\lambda }{M^2_{p}} \left( \exp \left[ -\frac{T_{i}}{T_{0}}\right] -\exp \left[ -\frac{T}{T_{0}}\right] \right) \right] , &{} \mathrm{for~Model~3,} \\ \displaystyle \exp \left[ -\frac{q\alpha ^{\prime }T^{2}_{0}\lambda }{2M^2_{p}}\left\{ \text {Ei}\left( -\frac{T^2_{i}}{T^2_{0}}\right) -\text {Ei}\left( -\frac{T^2}{T^2_{0}}\right) \right\} \right] , &{} \mathrm{for~Model~4,} \\ \displaystyle \exp \left[ -\frac{q\alpha ^{\prime }T^4_{0}\lambda }{4M^2_{p}} \left( \frac{1}{T^2}-\frac{1}{T^2_{i}}\right) \right] . &{} \mathrm{for~Model~5.} \end{array} \right. \end{array} \end{aligned}$$

(5.19)

Hence using Eqs. (5.15), (5.16), (5.18) and (5.19) one can study the parametric behavior of the scale factor a(t) with respect to time t and expected to be as like exact de Sitter or quasi-de Sitter solution during the inflationary slow-roll phase, as explicitly derived in the previous section.

5.1.2 Analysis using slow-roll formalism

Here our prime objective is to define slow-roll parameters for tachyon inflation in terms of the Hubble parameter and the single field tachyonic inflationary potential. Using the slow-roll approximation one can expand various cosmological observables in terms of small dynamical quantities derived from the appropriate derivatives of the Hubble parameter and of the inflationary potential. To start with here we use the horizon-flow parameters based on derivatives of Hubble parameter with respect to the number of e-foldings N, defined as

$$\begin{aligned} N(t)= & {} \int ^{t_{\mathrm{end}}}_{t}H(t)~\mathrm{d}t, \end{aligned}$$

(5.20)

where $t_{\mathrm{end}}$ signifies the end of inflation. Further using Eqs. (5.5), (5.9), (5.10) and (5.335) we get

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\mathrm{d}T}{\mathrm{d}N}=\frac{\dot{T}}{H}\\ \quad =\left\{ \begin{array}{lll} \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}, &{} \quad \mathrm{for}\; q=1/2, \\ \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q}\right) ,&{} \quad \mathrm{for~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.21)

where $H^{\prime }>0$ which makes always $\dot{T}>0$ during the inflationary phase. Further using Eq. (5.21) we get the following differential operator identity for tachyonic inflation:

$$\begin{aligned}&\displaystyle \frac{1}{H}\frac{\mathrm{d}}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}N}=\frac{\mathrm{d}}{\mathrm{d}\ln k}\nonumber \\&\quad =\left\{ \begin{array}{lll} \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\frac{\mathrm{d}}{\mathrm{d}T}, &{} \quad \mathrm{for }\; q=1/2, \\ \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q}\right) \frac{\mathrm{d}}{\mathrm{d}T},&{} \quad \mathrm{for~any~arbitrary~} q. \end{array} \right. \nonumber \\ \end{aligned}$$

(5.22)

Next we define the following Hubble slow-roll parameters:

$$\begin{aligned}&\epsilon _{0}=\frac{H_{\star }}{H},\end{aligned}$$

(5.23)

$$\begin{aligned}&\epsilon _{i+1}=\frac{\mathrm{d}\ln |\epsilon _{i}|}{\mathrm{d}N}, \quad i\ge 1 \end{aligned}$$

(5.24)

where $H_{\star }$ is the Hubble parameter at the pivot scale. Further using the differential operator identity as mentioned in Eq. (5.22) we get the following Hubble flow equation for tachyonic inflation for $i\ge 0$:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{H}\frac{\mathrm{d}\epsilon _{i}}{\mathrm{d}t}=\frac{\mathrm{d}\epsilon _{i}}{\mathrm{d}N}=\epsilon _{i+1}\epsilon _{i}=\left\{ \begin{array}{lll} \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\frac{\mathrm{d}\epsilon _{i}}{\mathrm{d}T}, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q}\right) \frac{\mathrm{d}\epsilon _{i}}{\mathrm{d}T},&{} \quad \mathrm{for~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.25)

For a realistic estimate from the single field tachyonic inflationary model substituting the free index i to $i=0,1,2$ in Eqs. (5.23) and (5.25) we get the contributions from the first three Hubble slow-roll parameter, which can be depicted as

$$\begin{aligned} \epsilon _{1}= & {} \frac{\mathrm{d}\ln |\epsilon _{0}|}{\mathrm{d}N}=-\frac{\dot{H}}{H^2} =\left\{ \begin{array}{lll} \displaystyle \frac{2}{3\alpha ^{\prime }}\left( \frac{H^{\prime }}{H^2}\right) ^{2}=\frac{3}{2}\alpha ^{\prime }\dot{T}^{2},~&{} \quad \mathrm{for}~~ q=1/2 \\ \displaystyle \frac{2}{3\alpha ^{\prime }}\left( \frac{H^{\prime }}{H^2}\right) ^{2}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q}\right) \\ \quad \displaystyle =\frac{3}{2}\alpha ^{\prime }\dot{T}^{2}\left( \frac{2q}{1-\alpha ^{\prime }(1-2q)\dot{T}^2}\right) ,~&{} \quad \mathrm{for~any~arbitrary~} q, \end{array}\right. \end{aligned}$$

(5.26)

$$\begin{aligned} \displaystyle \epsilon _{2}= & {} \frac{\mathrm{d}\ln |\epsilon _{1}|}{\mathrm{d}N}=\frac{\ddot{H}}{H\dot{H}}+2\epsilon _{1} =\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{2}{3\alpha ^{\prime }\epsilon _{1}}}\frac{\epsilon ^{\prime }_{1}}{H}=2\frac{\ddot{T}}{H\dot{T}},~&{} \quad \mathrm{for}~~ q=1/2\\ \displaystyle \sqrt{\frac{2}{3\alpha ^{\prime }\epsilon _{1}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q}\right) }\frac{\epsilon ^{\prime }_{1}}{H}\\ \quad \displaystyle =\frac{2\ddot{T}}{H\dot{T}\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) },~&{} \quad \mathrm{for~any~arbitrary~} q, \end{array}\right. \end{aligned}$$

(5.27)

$$\begin{aligned} \displaystyle \epsilon _{3}= & {} \frac{\mathrm{d}\ln |\epsilon _{2}|}{\mathrm{d}N}=\frac{1}{\epsilon _{2}}\left[ \frac{\dddot{H}}{H^2 \dot{H}}-3\frac{\ddot{H}}{H^3}-\frac{\ddot{H}^2}{H^2 \dot{H}^2}+4\frac{\dot{H}^2}{H^4}\right] \nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{2\epsilon _{1}}{3\alpha ^{\prime }}}\frac{\epsilon ^{\prime }_{2}}{H}=\left[ \frac{2\dddot{T}}{H^2\dot{T}\epsilon _{2}}+\epsilon _{1}-\frac{\epsilon _{2}}{2}\right] ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \sqrt{\frac{2\epsilon _{1}}{3\alpha ^{\prime }}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q}\right) }\frac{\epsilon ^{\prime }_{2}}{H}=\frac{\left[ \frac{2\dddot{T}}{H^2\dot{T}\epsilon _{2}}+\epsilon _{1}-\frac{\epsilon _{2}}{2}\right] }{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) } +\frac{\frac{4\alpha ^{\prime }(1-2q)\ddot{T}^2}{H} }{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right) ^2},~&{} \quad \mathrm{for~any~arbitrary~} q. \end{array} \right. \end{aligned}$$

(5.28)

It is important to note that:

• In the present context $\epsilon _{1}$ is characterized by the part of the total tachyonic energy density $\dot{T}^2$. Inflation occurs when $\epsilon _{1}<1$ and ends when $\epsilon _{1}=1$, which is exactly the same as the other single field slow-roll inflationary paradigm.

• The slow-roll parameter $\epsilon _{2}$ characterizes the ratio of the field acceleration relative to the frictional contribution acting on it due to the expansion.

• The third slow-roll parameter $\epsilon _{3}$ is made up of both $\epsilon _{1}$ and $\epsilon _{2}$. More precisely, $\epsilon _{3}$ is made up of $\dot{T}$, $\ddot{T}$ and $\dddot{T}$. This clearly implies that the third slow-roll parameter $\epsilon _{3}$ carries the contribution from the part of total tachyonic energy density, field acceleration relative to the frictional contribution and rate of change of field acceleration.

• The slow-roll conditions stated in Eqs. (5.3) and (5.4) are satisfied when the slow-roll parameters satisfy $\epsilon _{1}<<1$, $\epsilon _{2}<<1$ and $\epsilon _{3}<<1$. This also implies that in the slow-roll regime of the tachyonic inflation product of the two slow-roll parameters are also less than unity. For an example from Eq. (5.38) it is clearly observed that to satisfy the slow-roll condition we need to have additionally $\epsilon _{2}\epsilon _{3}<<1$.

Now for the sake of clarity, using Hamilton–Jacobi formalism, the Friedman equations and conservation equation can be rewritten as

$$\begin{aligned} \begin{array}{lll} \displaystyle 0\approx \left\{ \begin{array}{lll} \displaystyle \left[ H^{\prime }(T)\right] ^2 -\frac{9\alpha ^{\prime }}{4}H^{4}(T)+\frac{\alpha ^{\prime }}{4M^{4}_{p}}V^{2}(T), &{} \mathrm{for}~~ q=1/2, \\ \displaystyle H^{2}(T)\left[ 1-\frac{4\left[ H^{\prime }(T)\right] ^2}{9\alpha ^{\prime } H^{4}(T)}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q} \right) ^2\right] ^{1-q}\\ \displaystyle -\frac{V(T)}{3M^{2}_{p}}\left[ 1-\frac{4(1-2q)\left[ H^{\prime }(T)\right] ^2}{9\alpha ^{\prime } H^{4}(T)}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2}{2q}\right) ^2\right] ,&{} \mathrm{for~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.29)

and

$$\begin{aligned} \begin{array}{lll} \displaystyle H^{\prime }(T)\approx \left\{ \begin{array}{lll} \displaystyle -\frac{3\alpha ^{\prime }}{2}H^{2}(T)\dot{T}, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\frac{3\alpha ^{\prime }}{2}H^{2}(T)\dot{T}\left( \frac{2q}{1-\alpha ^{\prime }(1-2q)\dot{T}^2}\right) ,&{} \quad \mathrm{for~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.30)

Further using the definition of the first Hubble slow-roll parameter $\epsilon _{1}$ in Eq. (5.29) we get

$$\begin{aligned} \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle H^{2}(T)\left[ 1-\frac{2}{3}\epsilon _{1}(T)\right] ^{1/2}=\frac{V(T)}{3M^{2}_{p}}, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle H^{2}(T)\left[ 1-\frac{1}{3q}\epsilon _{1}(T)\right] ^{1-q}\approx \frac{V(T)}{3M^{2}_{p}}\left[ 1-\frac{(1-2q)}{3q}\epsilon _{1}(T)\right] ,&{} \quad \mathrm{for~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.31)

where for the arbitrary q we have used the following constraint condition:

$$\begin{aligned} 1-\underbrace{\alpha ^{\prime }(1-2q)\dot{T}^{2}}_{<<1}\approx 1. \end{aligned}$$

(5.32)

Now as in the slow-roll regime of tachyonic inflation $\epsilon _{1}(T)<<1$, consequently one can expand the exponents appearing in the left hand side of Eq. (5.31), which leads to the following simplified expression:

$$\begin{aligned} \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle H^{2}(T)\left[ 1-\frac{1}{3}\epsilon _{1}(T)\right] +\mathcal{O}(\epsilon ^2_{1} (T))=\frac{V(T)}{3M^{2}_{p}}, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle H^{2}(T)\left[ 1-\frac{1-q}{3q}\epsilon _{1}(T)\right] +\mathcal{O}(\epsilon ^2_{1} (T))\approx \frac{V(T)}{3M^{2}_{p}}\left[ 1-\frac{(1-2q)}{3q}\epsilon _{1}(T)\right] ,&{} \quad \mathrm{for~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.33)

It is important to mention here that:

• The result for $q=1/2$ implies that except for the second order correction term in slow-roll i.e., $\mathcal{O}(\epsilon ^2_{1} (T))$ the rest of the contribution exactly matches with the known result for the single field slow-roll inflationary models. But in the non-slow-roll limiting situation truncating at second order in slow-roll is not allowed and in that case for correct computation one needs to consider the full binomial series expansion.

• For $q\ne \frac{1}{2}$, i.e., for any other arbitrary q in the slow-roll regime of tachyonic inflation we have allowed the second order correction term in slow-roll i.e., $\mathcal{O}(\epsilon ^2_{1} (T))$ as appearing for $q\ne \frac{1}{2}$. But the final result implies significant deviation from the result that is well known for single field slow-roll inflationary models in the slow-roll regime. As mentioned earlier in the non-slow-roll limiting situation truncating at a certain order in slow-roll is not allowed and in that case for correct computation one needs to consider the full binomial series expansion.Also for $q\ne \frac{1}{2}$ case, the right hand side of Eq. (5.33) gets modified in the presence of slow-roll parameter $\epsilon _{1}$.

Our next objective is to express the Hubble slow-roll parameters in terms of the tachyon potential dependent slow-roll parameters. To serve this purpose let us start with writing the expression for the derivatives of the potential in terms of the Hubble slow-roll parameters. Allowing up to the second order contribution in the Hubble slow-roll parameters we get [86]

$$\begin{aligned}&M^{2}_{p}\frac{V^{\prime }(T)}{V(T)H(T)}=-\sqrt{6\epsilon _{1}}\frac{\left( 1-\frac{2}{3}\epsilon _{1}+\frac{\epsilon _{2}}{6}\right) }{\left( 1-\frac{2}{3}\epsilon _{1}\right) },\end{aligned}$$

(5.34)

$$\begin{aligned}&M^{4}_{p}\frac{V^{\prime \prime }(T)}{V(T)H^2(T)}=6\epsilon _{1}\frac{\left( 1-\frac{2}{3}\epsilon _{1}+\frac{\epsilon _{2}}{6}\right) \left( 1-\frac{2}{3}\epsilon _{1}-\frac{\epsilon _{2}}{3}\right) }{\left( 1-\frac{2}{3}\epsilon _{1}\right) ^2}\nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad +\frac{\epsilon _{2}}{2}\frac{\left( 5\epsilon _{1}-\frac{\epsilon _{2}}{3}-\epsilon _{3}\right) }{\left( 1-\frac{2}{3}\epsilon _{1}\right) } +3\left( \epsilon _{1}-\frac{\epsilon _{2}}{2}\right) , \end{aligned}$$

(5.35)

Further using Eqs. (5.36), (5.37), (5.38), (5.34) and (5.35) one can re-express the Hubble slow-roll parameters in terms of the potential dependent slow-roll parameter as

$$\begin{aligned} \displaystyle \epsilon _{1}\approx & {} \left\{ \begin{array}{lll} \displaystyle \frac{M^{2}_{p}}{2\alpha ^{\prime }}\frac{V^{'2}(T)}{V^{3}(T)}=\frac{\epsilon _{V}}{V(T)\alpha ^{\prime }}\equiv \bar{\epsilon }_{V}, &{} \quad \mathrm{for~} q=1/2, \\ \displaystyle \frac{M^{2}_{p}}{4q\alpha ^{\prime }}\frac{V^{'2}(T)}{V^{3}(T)}=\frac{\epsilon _{V}}{2qV(T)\alpha ^{\prime }}\equiv \frac{\bar{\epsilon }_{V}}{2q},~&{} \quad \mathrm{for~any~} q, \end{array}\right. \end{aligned}$$

(5.36)

$$\begin{aligned} \displaystyle \epsilon _{2}\approx & {} \left\{ \begin{array}{lll} \displaystyle \frac{M^2_{p}}{\alpha ^{\prime }}\left( 3\frac{V^{'2}(T)}{V^{3}(T)}-2\frac{V^{\prime \prime }(T)}{V^{2}(T)}\right) =\frac{2\left( 3\epsilon _{V}-\eta _{V}\right) }{V(T)\alpha ^{\prime }}=2\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ,~ &{} \quad \mathrm{for~} q=1/2, \\ \displaystyle \frac{M^2_{p}}{\sqrt{2q}\alpha ^{\prime }}\left( 3\frac{V^{'2}(T)}{V^{3}(T)}-2\frac{V^{\prime \prime }(T)}{V^{2}(T)}\right) =\frac{\sqrt{\frac{2}{q}}\left( 3\epsilon _{V}-\eta _{V}\right) }{V(T)\alpha ^{\prime }}=\sqrt{\frac{2}{q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) . &{} \quad \mathrm{for~any~} q \end{array} \right. \end{aligned}$$

(5.37)

$$\begin{aligned} \displaystyle \epsilon _{3}\epsilon _{2}\approx & {} \left\{ \begin{array}{lll} \displaystyle \frac{M^{4}_{p}}{V^{2}(T)\alpha ^{'2}}\left( 2\frac{V^{\prime \prime \prime }(T)V^{\prime }(T)}{V^{2}(T)}-10\frac{V^{\prime \prime }(T)V^{'2}(T)}{V^{3}(T)}+9\frac{V^{'4}(T)}{V^{4}(T)}\right) \\ \quad \displaystyle =\frac{\left( 2\xi ^2_{V}-5\eta _{V}\epsilon _{V}+36\epsilon ^2_{V}\right) }{V^{2}(T)\alpha ^{'2}}=\left( 2\bar{\xi }^2_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^2_{V}\right) ,~&{} \quad \mathrm{for~} q=1/2, \\ \displaystyle \frac{M^{4}_{p}}{\sqrt{2q}V^{2}(T)\alpha ^{'2}}\left( 2\frac{V^{\prime \prime \prime }(T)V^{\prime }(T)}{V^{2}(T)}-10\frac{V^{\prime \prime }(T)V^{'2}(T)}{V^{3}(T)}+9\frac{V^{'4}(T)}{V^{4}(T)}\right) \\ \quad \displaystyle =\frac{\left( 2\xi ^2_{V}-5\eta _{V}\epsilon _{V}+36\epsilon ^2_{V}\right) }{\sqrt{2q} V^{2}(T)\alpha ^{'2}}=\frac{\left( 2\bar{\xi }^2_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^2_{V}\right) }{\sqrt{2q}},~&{} \quad \mathrm{for~any~} q, \end{array} \right. \end{aligned}$$

(5.38)

where the potential dependent slow-roll parameters $\epsilon _{V}, \eta _{V}, \xi ^{2}_{V}, \sigma ^{3}_{V}$ are defined as

$$\begin{aligned}&\epsilon _{V}=\frac{M^{2}_{p}}{2}\left( \frac{V^{\prime }(T)}{V(T)}\right) ^2,\end{aligned}$$

(5.39)

$$\begin{aligned}&\eta _{V}= M^{2}_{p}\left( \frac{V^{\prime \prime }(T)}{V(T)}\right) ,\end{aligned}$$

(5.40)

$$\begin{aligned}&\xi ^{2}_{V}= M^{4}_{p}\left( \frac{V^{\prime }(T)V^{\prime \prime \prime }(T)}{V^2(T)}\right) ,\end{aligned}$$

(5.41)

$$\begin{aligned}&\sigma ^{3}_{V}= M^{6}_{p}\left( \frac{V^{'2}(T)V^{\prime \prime \prime \prime }(T)}{V^3(T)}\right) , \end{aligned}$$

(5.42)

which is exactly similar to the expression for the slow-roll parameter as appearing in the context of single field slow-roll inflationary models. However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for tachyonic inflation by rescaling with the appropriate powers of $\alpha ^{\prime }V(T)$:

$$\begin{aligned}&\bar{\epsilon }_{V}=\frac{\epsilon _{V}}{\alpha ^{\prime }V(T)}=\frac{M^{2}_{p}}{2\alpha ^{\prime }V(T)}\left( \frac{V^{\prime }(T)}{V(T)}\right) ^2,\end{aligned}$$

(5.43)

$$\begin{aligned}&\bar{\eta }_{V}= \frac{{\eta }_{V}}{\alpha ^{\prime }V(T)}=\frac{M^{2}_{p}}{\alpha ^{\prime }V(T)}\left( \frac{V^{\prime \prime }(T)}{V(T)}\right) ,\end{aligned}$$

(5.44)

$$\begin{aligned}&\bar{\xi }^{2}_{V}= \frac{\xi ^{2}_{V}}{\alpha ^{'2}V^2(T)}=\frac{M^{4}_{p}}{\alpha ^{'2}V^2(T)}\left( \frac{V^{\prime }(T)V^{\prime \prime \prime }(T)}{V^2(T)}\right) ,\end{aligned}$$

(5.45)

$$\begin{aligned}&\bar{\sigma }^{3}_{V}= \frac{\sigma ^{3}_{V}}{\alpha ^{'3}V^3(T)}=\frac{M^{6}_{p}}{\alpha ^{'3}V^3(T)}\left( \frac{V^{'2}(T)V^{\prime \prime \prime \prime }(T)}{V^3(T)}\right) . \end{aligned}$$

(5.46)

Further using Eqs. (5.39)–(5.46) we get the following operator identity for tachyonic inflation:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{H}\frac{\mathrm{d}}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}N}=\frac{\mathrm{d}}{\mathrm{d}\ln k}\approx \left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{2\bar{\epsilon }_{V}}{V(T)\alpha ^{\prime }}}M_{p}\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\frac{\mathrm{d}}{\mathrm{d}T}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{2\bar{\epsilon }_{V}}{V(T)\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}\frac{\mathrm{d}}{\mathrm{d}T},&{} \quad \mathrm{for~any~arbitrary~} q. \end{array}\right. \end{array} \end{aligned}$$

(5.47)

Finally using Eq. (5.47) we get the following sets of flow equations in the context of tachyonic inflation:

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}\epsilon _{1}}{\mathrm{d}N}=\left\{ \begin{array}{lll} \displaystyle \frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}N}=2\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q}\frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}N}=\frac{\bar{\epsilon }_{V}}{q}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4},~&{} \quad \mathrm{for~any~} q, \end{array} \right. \end{aligned}$$

(5.48)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}\epsilon _{2}}{\mathrm{d}N}=\left\{ \begin{array}{lll} \displaystyle 2\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{2}{q}}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.49)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}(\epsilon _{2}\epsilon _{3})}{\mathrm{d}N}=\left\{ \begin{array}{lll} \displaystyle \left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V}+194\bar{\epsilon }^2_{V}\bar{\eta }_{V}-10\bar{\eta }^2_{V}\bar{\epsilon }_{V} \right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4},~&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{\left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V}+194\bar{\epsilon }^2_{V}\bar{\eta }_{V}-10\bar{\eta }^2_{V}\bar{\epsilon }_{V} \right) }{\sqrt{2q}}\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.50)

where we use the following consistency conditions for the rescaled potential dependent slow-roll parameters:

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}N}=\left\{ \begin{array}{lll} \displaystyle 2\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 2\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}. &{} \quad \mathrm{for ~any~} q \end{array} \right. \end{aligned}$$

(5.51)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}\bar{\eta }_{V}}{\mathrm{d}N}=\left\{ \begin{array}{lll} \displaystyle \left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}. &{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.52)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}\bar{\xi }^2_{V}}{\mathrm{d}N}=\left\{ \begin{array}{lll} \displaystyle \left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}. &{} \quad \mathrm{for ~any~} q \end{array} \right. \end{aligned}$$

(5.53)

$$\begin{aligned} \displaystyle \frac{\mathrm{d}\bar{\sigma }^3_{V}}{\mathrm{d}N}=\left\{ \begin{array}{lll} \displaystyle \bar{\sigma }^3_{V}\left( \bar{\eta }_{V}-12\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4},~&{} \mathrm{for}~~~q=1/2, \\ \displaystyle \bar{\sigma }^3_{V}\left( \bar{\eta }_{V}-12\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4},~&{} \mathrm{for ~any~} q. \end{array} \right. \end{aligned}$$

(5.54)

In terms of the slow-roll parameters, the number of e-foldings can be re-expressed as

$$\begin{aligned} \begin{array}{lll} \displaystyle N(T)=\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{3\alpha ^{\prime }}{2}}\int ^{T_{\mathrm{end}}}_{T}\frac{H(T)}{\sqrt{\epsilon _{1}}}~\mathrm{d}T\\ \displaystyle \quad =\sqrt{\frac{3\alpha ^{\prime }}{2}}\int ^{T_{\mathrm{end}}}_{T}\frac{H(T)}{\sqrt{\bar{\epsilon }_{V}}}~\mathrm{d}T\\ \displaystyle \quad \approx \frac{\alpha ^{\prime }}{M^{2}_{p}}\int ^{T}_{T_{\mathrm{end}}} \frac{V^{2}(T)}{V^{\prime }(T)}~\mathrm{d}T, &{} \quad \mathrm{for}~~~q=1/2\\ \displaystyle 2q\sqrt{\frac{3\alpha ^{\prime }}{2}}\int ^{T_{\mathrm{end}}}_{T}\frac{H(T)}{\sqrt{\epsilon _{1}}}~\mathrm{d}T\\ \displaystyle \quad = \sqrt{3\alpha ^{\prime }q}\int ^{T_{\mathrm{end}}}_{T}\frac{H(T)}{\sqrt{\bar{\epsilon }_{V}}}~\mathrm{d}T\\ \displaystyle \quad \approx \frac{\sqrt{2q}\alpha ^{\prime }}{M^{2}_{p}}\int ^{T}_{T_{\mathrm{end}}} \frac{V^{2}(T)}{V^{\prime }(T)}~\mathrm{d}T. &{} \quad \mathrm{for ~any~} q \end{array} \right. \end{array} \end{aligned}$$

(5.55)

where $T_{\mathrm{end}}$ characterizes the tachyonic field value at the end of inflation $t=t_{\mathrm{end}}$. It is important to mention here that in the single field tachyonic inflationary paradigm the field value of the tachyon at the end of inflation is computed from the following condition:

$$\begin{aligned} \mathrm{max}_{\phi =\phi _{e}}\left[ \epsilon _{V},|\eta _{V}|,|\xi ^{2}_{V}|,|\sigma ^{3}_{V}|\right] \equiv 1. \end{aligned}$$

(5.56)

Let $T_{\star }$ denote the value of tachyonic field T at which a length scale or more precisely the modes crosses the Hubble radius during inflation, which is given by the momentum at pivot scale $k_{\star } = a_{\star }H_{\star }$. Here $a_{\star }$ and $H_{\star }$ signify the scale factor and Hubble parameter at the horizon crossing scale or at the pivot scale. Then the definition of the number of e-foldings as stated in Eqs. (5.335) and (5.55) gives

$$\begin{aligned} N_{\star }=N(T_{\star })=\ln \left( \frac{a_{\mathrm{end}}}{a_{\star }}\right) , \end{aligned}$$

(5.57)

where $a_{\mathrm{end}}$ is the scale factor at the end of inflation. Then using Eq. (5.57) the corresponding horizon crossing momentum scale or the pivot scale can be computed as

$$\begin{aligned} c_{S}k_{\star }=a_{\star }H_{\star }=a_{\mathrm{end}}H_{\star }\exp \left( -N_{\star }\right) . \end{aligned}$$

(5.58)

Now at any arbitrary momentum scale the number of e-foldings, N(k), between the Hubble exit of the relevant modes and the end of inflation can be expressed as

$$\begin{aligned} N(k)\approx & {} 71.21 -\ln \left( \frac{k}{k_{0}}\right) +\frac{1}{4}\ln \left( \frac{V(T_{\star })}{M^{4}_{p}}\right) \nonumber \\&+\frac{1}{4}\ln \left( \frac{V(T_{\star })}{\rho _{\mathrm{end}}}\right) +\frac{1-3w_{\mathrm{int}}}{12(1+w_{\mathrm{int}})} \ln \left( \frac{\rho _{\mathrm{reh}}}{\rho _{\mathrm{end}}}\right) , \end{aligned}$$

(5.59)

where $\rho _{\mathrm{end}}$ is the energy density at the end of inflation, $\rho _{\mathrm{reh}}$ is an energy scale during reheating, $c_{S}k_{0}= a_{0}H_{0}$ is the present Hubble scale, $V(T_{\star })$ corresponds to the potential energy when the relevant modes left the Hubble patch during inflation corresponding to the momentum scale $c_{S}k_{\star } = a_{\star }H_{\star }=c_{S}k_{\mathrm{cmb}}$, and $w_{\mathrm{int}}$ characterizes the effective equation of state parameter between the end of inflation and the energy scale during reheating. Further using Eq. (5.59) in Eq. (5.57) we get the following expression:

$$\begin{aligned} N_{\star }\approx & {} 71.21 -\ln \left( \frac{k_{\star }}{k_{0}}\right) +\frac{1}{4}\ln \left( \frac{V(T_{\star })}{M^{4}_{p}}\right) \nonumber \\&+\frac{1}{4}\ln \left( \frac{V(T_{\star })}{\rho _{\mathrm{end}}}\right) +\frac{1-3w_{\mathrm{int}}}{12(1+w_{\mathrm{int}})} \ln \left( \frac{\rho _{\mathrm{reh}}}{\rho _{\mathrm{end}}}\right) , \end{aligned}$$

(5.60)

which is very useful to fix the number of e-foldings within $50<N_{\star }<70$ for tachyonic inflation.

5.1.3 Basics of tachyonic perturbations

In this subsection we explicitly discuss the cosmological linear perturbation theory within the framework of tachyonic inflation. Let us clearly mention that here we have various ways of characterizing cosmological perturbations in the context of inflation, which finally depend on the choice of gauge. Let us do the computation in the longitudinal guage, where the scalar metric perturbations of the FLRW background are given by the following infinitesimal line element:

$$\begin{aligned} \mathrm{d}s^2= & {} -\left( 1+2\mathbf{\Phi }(t,\mathbf{x})\right) \mathrm{d}t^2\nonumber \\&+\,a^{2}(t)\left( 1-2\mathbf{\Psi }(t,\mathbf{x})\right) \delta _{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}, \end{aligned}$$

(5.61)

where a(t) is the scale factor, $\mathbf{\Phi }(t,\mathbf{x})$ and $\mathbf{\Psi }(t,\mathbf{x})$ characterizes the gauge invariant metric perturbations. Specifically, the perturbation of the FLRW metric leads to the perturbation in the energy-momentum stress tensor via the Einstein field equation or equivalently through the Friedmann equations. For the perturbed metric as mentioned in Eq. (5.61), the perturbed Einstein field equations can be expressed for the $q=1/2$ case of the tachyonic inflationary set-up as

$$\begin{aligned}&3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +\frac{k^{2}}{a^{2}(t)}=-\frac{1}{2M^{2}_{p}}\delta \rho , \end{aligned}$$

(5.62)

$$\begin{aligned}&\ddot{\mathbf{\Psi }}(t,\mathbf{k})+3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +H\dot{\mathbf{\Phi }}(t,\mathbf{k})+2\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+\frac{k^{2}}{3a^{2}(t)} \left( \mathbf{\Phi }(t,\mathbf{k})-\mathbf{\Psi }(t,\mathbf{k})\right) =\frac{1}{2M^{2}_{p}}\delta p,\end{aligned}$$

(5.63)

$$\begin{aligned}&\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\mathbf{\Phi }(t,\mathbf{k})=-\frac{\alpha ^{\prime }V(T)}{\sqrt{1-\alpha ^{\prime }\dot{T}^{2}}}\frac{\dot{T}}{M^{2}_{p}}\delta T,\end{aligned}$$

(5.64)

$$\begin{aligned}&{} \mathbf{\Psi }(t,\mathbf{k})-\mathbf{\Phi }(t,\mathbf{k})=0. \end{aligned}$$

(5.65)

Similarly, for any arbitrary q the perturbed Einstein field equations can be expressed as

$$\begin{aligned}&3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +\frac{k^{2}}{a^{2}(t)}=-\frac{1}{2M^{2}_{p}}\delta \rho ,\end{aligned}$$

(5.66)

$$\begin{aligned}&\ddot{\mathbf{\Psi }}(t,\mathbf{k})+3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +H\dot{\mathbf{\Phi }}(t,\mathbf{k})+2\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+\frac{k^{2}}{3a^{2}(t)} \left( \mathbf{\Phi }(t,\mathbf{k})-\mathbf{\Psi }(t,\mathbf{k})\right) =\frac{1}{2M^{2}_{p}}\delta p,\end{aligned}$$

(5.67)

$$\begin{aligned}&\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\mathbf{\Phi }(t,\mathbf{k})=-\frac{\alpha ^{\prime }V(T)\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] }{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{1-q}} \frac{\dot{T}}{M^{2}_{p}}\delta T,\nonumber \\\end{aligned}$$

(5.68)

$$\begin{aligned}&{} \mathbf{\Psi }(t,\mathbf{k})-\mathbf{\Phi }(t,\mathbf{k})=0. \end{aligned}$$

(5.69)

Here $\mathbf{\Phi }(t,\mathbf{k})$ and $\mathbf{\Psi }(t,\mathbf{k})$ are the two gauge invariant metric perturbations in the Fourier space, defined via the following transformation:

$$\begin{aligned} \mathbf{\Phi }(t,\mathbf{x})=\int \mathrm{d}^{3}k~ \mathbf{\Phi }(t,\mathbf{k})~\exp (i \mathbf{k}. \mathbf{x}),\end{aligned}$$

(5.70)

$$\begin{aligned} \mathbf{\Psi }(t,\mathbf{x})=\int \mathrm{d}^{3}k~ \mathbf{\Psi }(t,\mathbf{k})~\exp (i \mathbf{k}. \mathbf{x}). \end{aligned}$$

(5.71)

Additionally, it is important to note that in Eq. (5.65), the two gauge invariant metric perturbations $\mathbf{\Phi }(t,\mathbf{k})$ and $\mathbf{\Psi }(t,\mathbf{k})$ are equal in the context of minimally coupled tachyonic string field theoretic model with an Einstein gravity sector. In Eqs. (5.62) and (5.63) the perturbed energy density $\delta \rho$ and pressure $\delta p$ are given by

$$\begin{aligned} \begin{array}{lll} \displaystyle \delta \rho =\left\{ \begin{array}{lll} \displaystyle \frac{V^{\prime }(T)\delta T}{\sqrt{1-\alpha ^{\prime }\dot{T}^2}} +\frac{\alpha ^{\prime }V(T)\left( \dot{T}\delta \dot{T}+\dot{T}^2\mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{3/2}} ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{\left\{ V^{\prime }(T)\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] \delta T-4\alpha ^{\prime }(1-2q)V(T)\dot{T}\delta \dot{T}\right\} }{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{1-q}}\\ \quad \displaystyle +\frac{2\alpha ^{\prime }(1-q)V(T)\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] \left( \dot{T}\delta \dot{T}+\dot{T}^2\mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{2-q}} ,&{} \quad \mathrm{for~any~arbitrary}~q, \end{array} \right. \end{array} \end{aligned}$$

(5.72)

and

$$\begin{aligned} \begin{array}{lll} \displaystyle \delta p =\left\{ \begin{array}{lll} \displaystyle -V^{\prime }(T)\sqrt{1-\alpha ^{\prime }\dot{T}^2}\delta T +\frac{\alpha ^{\prime }V(T)\left( \dot{T}\delta \dot{T}+\dot{T}^2\mathbf{\Phi }(t,\mathbf{k})\right) }{\sqrt{1-\alpha ^{\prime }\dot{T}^2}} ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -V^{\prime }(T)\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{q}\delta T+\frac{2q\alpha ^{\prime }V(T)\left( \dot{T}\delta \dot{T}+\dot{T}^2\mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{1-q}} ,&{} \quad \mathrm{for ~any~arbitrary}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.73)

Similarly after the variation of the tachyoinic field equation motion we get the following expressions for the perturbed equation of motion:

$$\begin{aligned} \begin{array}{lll} \displaystyle 0 \approx \left\{ \begin{array}{lll} \displaystyle \delta \ddot{T}+3H\delta \dot{T}+\frac{2\alpha ^{\prime }\ddot{T}\left( \dot{T}\delta \dot{T}+\dot{T}^2 \mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2\right) } +\frac{M_{p}\sqrt{1-\alpha ^{\prime }\dot{T}^2}}{\alpha ^{\prime }V(T)}\left[ \left( \frac{k^2}{a^2} - 3\dot{H}\right) \mathbf{\Phi }(t,\mathbf{k})-\frac{2k^2}{a^2}\mathbf{\Psi }(t,\mathbf{k}) \right. \\ \left. \displaystyle -3\left( \ddot{\mathbf{\Psi }}(t,\mathbf{k})+4H\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\dot{\mathbf{\Phi }}(t,\mathbf{k})+\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+4H^2 \mathbf{\Phi }(t,\mathbf{k})\right) \right] \\ -\left\{ 6H\alpha ^{\prime }\dot{T}^3 - \frac{2V^{\prime }(T)}{\alpha ^{\prime }V(T)}\left( 1-\alpha ^{\prime }\dot{T}^2\right) \right\} \mathbf{\Phi }(t,\mathbf{k})-\left( \dot{\mathbf{\Psi }}(t,\mathbf{k})+3\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) \dot{T}-\frac{M_{p}\left( 1-\alpha ^{\prime }\dot{T}^2\right) }{\alpha ^{\prime }}\left( \frac{V^{\prime \prime }(T)}{V(T)}-\frac{V^{'2}(T)}{V^{2}(T)}\right) ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \delta \ddot{T}+3H\delta \dot{T}+\frac{2\alpha ^{\prime }\ddot{T}\left( \dot{T}\delta \dot{T}+\dot{T}^2 \mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{2(1-q)}} +\frac{M_{p}\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{1-q}}{\alpha ^{\prime }V(T) \left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2\right] }\Bigg [\left( \frac{k^2}{a^2} - 3\dot{H}\right) \mathbf{\Phi }(t,\mathbf{k})-\frac{2k^2}{a^2}\mathbf{\Psi }(t,\mathbf{k}) \\ \displaystyle -3\left( \ddot{\mathbf{\Psi }}(t,\mathbf{k})+4H\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\dot{\mathbf{\Phi }}(t,\mathbf{k})+\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+4H^2 \mathbf{\Phi }(t,\mathbf{k})\right) \Bigg ]-\left\{ 6H\alpha ^{\prime }\dot{T}^3 - \sqrt{\frac{2}{q}}\frac{V^{\prime }(T)}{\alpha ^{\prime }V(T)}\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{2(1-q)}\right\} \\ \times \displaystyle \mathbf{\Phi }(t,\mathbf{k})-\left( \dot{\mathbf{\Psi }}(t,\mathbf{k})+3\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) \dot{T} \displaystyle -\frac{M_{p}\left( 1-\alpha ^{\prime }\dot{T}^2\right) ^{2(1-q)}}{\sqrt{2q}\alpha ^{\prime }}\left( \frac{V^{\prime \prime }(T)}{V(T)}-\frac{V^{'2}(T)}{V^{2}(T)}\right) ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.74)

Further we will perform the following steps throughout the next part of the computation:

• First of all we decompose the scalar perturbations into two components-(1) entropic or isocurvature perturbations which can be usually treated as the orthogonal projective part to the trajectory and (2) adiabatic or curvature perturbations which can be usually treated as the parallel projective part to the trajectory.

• If inflation is governed by a single scalar field then we deal with adiabatic or curvature perturbations. On the other hand for multiple scalar fields we deal with entropic or isocurvature perturbations.

• In the present context the inflationary dynamics is governed by a single tachyonic field, which implies the surviving part of the cosmological perturbations are governed by the adiabatic contribution.

• Within the framework of first order cosmological perturbation theory we define a gauge invariant primordial curvature perturbation on the scales outside the horizon:

  $$\begin{aligned} \zeta = \mathbf{\Psi }-\frac{H}{\dot{\rho }}\delta \rho . \end{aligned}$$

  (5.75)

• Next we consider the uniform density hypersurface in which

  $$\begin{aligned} \delta \rho =0. \end{aligned}$$

  (5.76)

  Consequently the curvature perturbation is governed by

  $$\begin{aligned} \zeta =\mathbf{\Psi }. \end{aligned}$$

  (5.77)

• Further, the time evolution of the curvature perturbation can be expressed as

  $$\begin{aligned} \dot{\zeta }=H\left( \frac{\delta \bar{p}}{\rho +p}\right) , \end{aligned}$$

  (5.78)

  where $\delta \bar{p}$ characterizes the non-adiabatic or entropic contribution in the first order linearized cosmological perturbation. In the present context $\delta \bar{p}$ can be expressed as

  $$\begin{aligned} \delta \bar{p}= \Gamma \dot{p}, \end{aligned}$$

  (5.79)

  where $\Gamma$ characterizes the relative displacement between hypersurfaces of uniform pressure and density. Additionally, it is important to note that Eq. (5.375) signifies the change in the curvature perturbation on the uniform density hypersurfaces on the large scales. Also from Eq. (5.375) it is clearly observed that the contribution from the time evolution of the adiabatic or curvature perturbations are directly proportional to the non-adiabatic contribution which comes from significantly from the isocurvature part of pressure perturbation $\delta p$ and are completely independent of the specific mathematical structure of the gravitational field equations in the context of Einstein gravity framework. In a generalized prescription the pressure perturbation in arbitrary gauge can be decomposed into the following two contributions:

  $$\begin{aligned} \delta p = c^{2}_{S}\delta \rho + \delta \bar{p}, \end{aligned}$$

  (5.80)

  where $c^{2}_{S}$ is the effective sound speed, which is mentioned in the earlier section of the paper.

• Now let us consider a situation where the pressure perturbation is completely made up of adiabatic contribution from the cosmological perturbation on large cosmological scales. Consequently we get

  $$\begin{aligned} \delta \bar{p}= \Gamma \dot{p}=0 \quad \Rightarrow \quad \zeta =\mathrm{Constant}, \end{aligned}$$

  (5.81)

  which is consistent with the single field slow-roll conditions in the context of the tachyonic inflationary set-up. Finally, in the uniform density hypersurfaces, the curvature perturbation can be written in terms of the tachyonic field fluctuations on spatially flat hypersurfaces as

$$\begin{aligned} \zeta =-H\left( \frac{\delta T}{\dot{T}}\right) . \end{aligned}$$

(5.82)

5.1.4 Computation of scalar power spectrum

In this subsection our prime objective is to compute the primordial power spectra of scalar quantum fluctuations from tachyonic inflation and study the cosmological consequences from the previously mentioned string theory originating tachyonic potentials in the light of Planck 2015 data. To serve this purpose let us start with the following canonical variable $v_\mathbf{k}$, which can be quantized with the standard techniques:

$$\begin{aligned} v_\mathbf{k}\equiv z~M_{p}~\zeta _\mathbf{k}, \end{aligned}$$

(5.83)

where $\zeta _{k}$ is the curvature perturbation in the momentum space, which can be expressed in terms of the curvature perturbation in position space through the following Fourier transformation:

$$\begin{aligned} \zeta (t,\mathbf{x})=\int \mathrm{d}^{3}k~\zeta _\mathbf{k}(t)~\exp (i\mathbf{k}.\mathbf{x}). \end{aligned}$$

(5.84)

Also z is defined as

$$\begin{aligned} \begin{array}{lll} \displaystyle z =\frac{a(t)}{c_{S} H M_{p}}\sqrt{\rho +p}=\left\{ \begin{array}{lll} \displaystyle \frac{\sqrt{3\alpha ^{\prime }}a(t)\dot{T}}{\sqrt{1-\alpha ^{\prime }\dot{T}^2}}=\frac{a(t)}{c_{S}}\sqrt{2\epsilon _{1}}= \frac{a(t)}{c_{S}}\sqrt{2\bar{\epsilon }_{V}},~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{\sqrt{6q\alpha ^{\prime }}a(t)\dot{T}}{\sqrt{1-\alpha ^{\prime }\dot{T}^2}}\sqrt{\frac{1+(1-2q)\alpha ^{\prime }\dot{T}^2}{1-(1-2q)\alpha ^{\prime }\dot{T}^2}} =\frac{a(t)\sqrt{2\epsilon _{1}}}{c_{S}} =\frac{a(t)\sqrt{2\bar{\epsilon }_{V}}}{\sqrt{2q}c_{S}} ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.85)

Next we use conformal time $\eta$ instead of using the time t, which is defined via the following infinitesimal transformation:

$$\begin{aligned} \mathrm{d}t= a~\mathrm{d}\eta \end{aligned}$$

(5.86)

using which one can redefine the Hubble parameter in conformal coordinate system as

$$\begin{aligned} \mathcal{H}(\eta )=\frac{1}{a(\eta )}\frac{\mathrm{d}a(\eta )}{\mathrm{d}\eta }=a~H(t). \end{aligned}$$

(5.87)

Further we derive the equation of motion of the scalar fluctuation by extremizing the tachyonic model action as

$$\begin{aligned} \left[ \frac{\mathrm{d}^{2}}{\mathrm{d}\eta ^2}+\left( c^{2}_{S}k^{2}-\frac{1}{z}\frac{\mathrm{d}^{2}z}{\mathrm{d}\eta ^2}\right) \right] v_\mathbf{k}=0 \end{aligned}$$

(5.88)

where

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{z}\frac{\mathrm{d}^{2}z}{\mathrm{d}\eta ^2} = 2a^2H^2+\overbrace{\frac{2}{3}\frac{\left( \frac{1}{z}\frac{\mathrm{d}z}{\mathrm{d}\eta }\frac{\mathrm{d}\epsilon _{1}}{\mathrm{d}\eta }+\frac{1}{2}\frac{\mathrm{d}^2\epsilon _{1}}{\mathrm{d}\eta ^2}\right) }{\left( 1-\frac{2}{3}\epsilon _{1}\right) } +\frac{1}{9}\frac{\left( \frac{\mathrm{d}\epsilon _{1}}{\mathrm{d}\eta }\right) ^2}{\left( 1-\frac{2}{3}\epsilon _{1}\right) ^2}}^\mathrm{Higher~order~slow-roll~correction}\\ \quad \displaystyle =\left\{ \begin{array}{lll} \displaystyle 2a^2H^2+\frac{2}{3}\frac{\left( \frac{1}{z}\frac{\mathrm{d}z}{\mathrm{d}\eta }\frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}\eta }+\frac{1}{2}\frac{\mathrm{d}^2\bar{\epsilon }_{V}}{\mathrm{d}\eta ^2}\right) }{\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) } +\frac{1}{9}\frac{\left( \frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}\eta }\right) ^2}{\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^2}\\ \quad \displaystyle =a^2H^2\left[ 2+8\bar{\epsilon }_{V}-3\bar{\eta }_{V}+\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2+\bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \displaystyle + \frac{1}{2}\left( 2\bar{\xi }^2_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^2_{V}\right) \right] +\cdots ,~&{} \mathrm{for}~~ q=1/2, \\ \displaystyle 2a^2H^2+\frac{1}{3q}\frac{\left( \frac{1}{z}\frac{\mathrm{d}z}{\mathrm{d}\eta }\frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}\eta }+\frac{1}{2}\frac{\mathrm{d}^2\bar{\epsilon }_{V}}{\mathrm{d}\eta ^2}\right) }{\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) } +\frac{1}{36q^2}\frac{\left( \frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}\eta }\right) ^2}{\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^2}\\ \quad \displaystyle =a^2H^2\left[ 2+\left( \frac{9}{\sqrt{2q}}-\frac{1}{2q}\right) \bar{\epsilon }_{V}-\frac{3}{\sqrt{2q}}\bar{\eta }_{V}+\frac{1}{2q}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2 \displaystyle +\frac{1}{(2q)^{3/2}}\bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right. \\ \qquad \left. + \frac{1}{2\sqrt{2q}}\left( 2\bar{\xi }^2_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^2_{V} \right) \right] +\cdots ,&{} \mathrm{for ~any}~q, \end{array} \right. \end{array} \end{aligned}$$

(5.89)

and the factor aH can be expressed in terms of the conformal time $\eta$ as

$$\begin{aligned} \begin{array}{lll} \displaystyle \eta \approx \left\{ \begin{array}{lll} \displaystyle -\frac{1}{aH}\left( 1+\bar{\epsilon }_{V}\right) +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\frac{1}{aH}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) +\cdots ,&{} \quad \mathrm{for ~any~arbitrary}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.90)

Further replacing the factor aH in Eq. (5.89), we finally get the following simplified expression:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{z}\frac{\mathrm{d}^{2}z}{\mathrm{d}\eta ^2} \approx \left\{ \begin{array}{lll} \displaystyle \frac{1}{\eta ^2}\left[ \left( \frac{3}{2}+4\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2-\frac{1}{4}\right] +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{1}{\eta ^2}\left[ \left\{ \frac{3}{2}+\left( \frac{1}{2q}+\frac{3}{\sqrt{2q}}\right) \bar{\epsilon }_{V}-\frac{1}{\sqrt{2q}}\bar{\eta }_{V}\right\} ^2-\frac{1}{4}\right] +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.91)

Now for further simplification in the computation of scalar power spectrum we introduce a new factor $\nu$, which is defined as

$$\begin{aligned} \begin{array}{lll} \displaystyle \nu \approx \left\{ \begin{array}{lll} \displaystyle \left( \frac{3}{2}+4\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \frac{3}{2}+\left( \frac{1}{2q}+\frac{3}{\sqrt{2q}}\right) \bar{\epsilon }_{V}-\frac{1}{\sqrt{2q}}\bar{\eta }_{V}\right\} +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.92)

Hence using Eq. (5.100) in Eq. (5.88), we get the following simplified form of the equation of motion:

$$\begin{aligned} \left[ \frac{\mathrm{d}^{2}}{\mathrm{d}\eta ^2}+\left( c^{2}_{S}k^{2}-\frac{\left( \nu ^2-\frac{1}{4}\right) }{\eta ^2}\right) \right] v_\mathbf{k}(\eta )=0, \end{aligned}$$

(5.93)

and the most general solution of Eq. (5.123) is given by

$$\begin{aligned} v_\mathbf{k}(\eta )=\sqrt{-\eta }\left[ C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu }\left( -kc_{S}\eta \right) \right] , \end{aligned}$$

(5.94)

where $C_{1}$ and $C_{2}$ are two arbitrary integration constants, which can be fixed from the appropriate choice of the boundary conditions. Additionally $H^{(1)}_{\nu }$ and $H^{(2)}_{\nu }$ represent the Hankel function of the first and second kind with rank $\nu$. Now to impose the well known Bunch–Davies boundary condition at early times we have used:

$$\begin{aligned}&\lim _{kc_{S}\eta \rightarrow -\infty } H^{(1)}_{\nu }\left( -kc_{S}\eta \right) \nonumber \\&\quad = \sqrt{\frac{2}{\pi }}\frac{1}{\sqrt{-\eta }}\exp \left( ikc_{S}\eta \right) \exp \left( i\frac{\pi }{2}\left( \nu +\frac{1}{2}\right) \right) ,\end{aligned}$$

(5.95)

$$\begin{aligned}&\lim _{kc_{S}\eta \rightarrow -\infty } H^{(2)}_{\nu }\left( -kc_{S}\eta \right) \nonumber \\&\quad = \sqrt{\frac{2}{\pi }}\frac{1}{\sqrt{-\eta }}\exp \left( -ikc_{S}\eta \right) \exp \left( -i\frac{\pi }{2}\left( \nu +\frac{1}{2}\right) \right) . \end{aligned}$$

(5.96)

As a result the previously mentioned integration constants are fixed at the values $C_{1}=\sqrt{\frac{\pi }{2}},\; C_{2}= 0$. Consequently the solution of the mode function for scalar fluctuations takes the following form:

$$\begin{aligned} v_\mathbf{k}(\eta )=\sqrt{-\frac{\eta \pi }{2}}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) . \end{aligned}$$

(5.97)

On the other hand, the solution stated in Eq. (5.94) determines the future evolution of the mode including its super-horizon dynamics at $c_{S}k<<aH$ or $|k c_{S}\eta |<<1$ or $k c_{S}\eta \rightarrow 0$ and this is due to

$$\begin{aligned} \lim _{kc_{S}\eta \rightarrow 0} H^{(1)}_{\nu }\left( -kc_{S}\eta \right)= & {} \frac{i}{\pi }\Gamma (\nu )\left( \frac{-kc_{S}\eta }{2}\right) ^{-\nu }. \end{aligned}$$

(5.98)

Consequently the solution of the mode function for scalar fluctuations takes the following form:

$$\begin{aligned} v_\mathbf{k}(\eta )=\sqrt{-\frac{\eta \pi }{2}}\frac{i}{\pi }\Gamma (\nu )\left( \frac{-kc_{S}\eta }{2}\right) ^{-\nu }. \end{aligned}$$

(5.99)

Finally combining the results obtained in Eqs. (5.94), (5.97) and (5.128) we get

$$\begin{aligned} \begin{array}{lll} \displaystyle v_\mathbf{k}(\eta ) =\left\{ \begin{array}{lll} \displaystyle \sqrt{-\eta }\Big [C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) &{}\,\\ \,\,+C_{2}H^{(2)}_{\nu }\left( -kc_{S}\eta \right) \Big ],~&{} \mathrm{for~AV} ,\\ \displaystyle \sqrt{-\frac{\eta \pi }{2}}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) ,&{} \mathrm{for~BD}~ + |k c_{S}\eta |>>1 ,\\ \displaystyle \sqrt{-\frac{\eta \pi }{2}}\frac{i}{\pi }\Gamma (\nu )\left( \frac{-kc_{S}\eta }{2}\right) ^{-\nu } ,&{} \mathrm{for~BD}~ + |k c_{S}\eta |<<1, \end{array} \right. \end{array} \end{aligned}$$

(5.100)

where $\mathrm{AV}$ and $\mathrm{BD}$ signify the arbitrary vacuum and the Bunch–Davies vacuum respectively. Finally the two point function from a scalar fluctuation for both $\mathrm{AV}$ and $\mathrm{BD}$ can be expressed as

$$\begin{aligned} \langle \zeta _\mathbf{k}\zeta _\mathbf{k^{\prime }}\rangle= & {} \left( \frac{H}{\dot{T}}\right) ^{2}\langle \delta T_\mathbf{k} \delta T_\mathbf{k^{\prime }}\rangle =\frac{1}{z^{2}M^{2}_{p}}\langle v_\mathbf{k} v_\mathbf{k^{\prime }}\rangle \nonumber \\= & {} (2\pi )^{3}\delta ^{3}(\mathbf{k}+\mathbf{k^{\prime }})\frac{2\pi ^{2}}{k^{3}}\Delta _{\zeta }(k), \end{aligned}$$

(5.101)

where the primordial power spectrum for the scalar modes at any arbitrary momentum scale k can be written for both $\mathrm{AV}$ and $\mathrm{BD}$ with $q=1/2$ as

$$\begin{aligned} \Delta _{\zeta }(k)\equiv \frac{k^{3}P_{\zeta }(k)}{2\pi ^2}=\frac{k^{3}|v_{k}|^{2}}{2\pi ^{2}z^{2}M^{2}_{p}}=\left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle \frac{2^{2\nu -3}\left( -k\eta c_{S}\right) ^{3-2\nu } H^2}{8c_{S}\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})^2 \pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\nu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |<<1 , \\ \displaystyle \frac{2^{2\nu -3}c^{2-2\nu }_{S}\left( 1+\bar{\epsilon }_{V}\right) ^{1-2\nu } H^2}{8\bar{\epsilon }_{V} \pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\nu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |=1 , \\ \displaystyle \frac{\left( -k\eta c_{S}\right) ^3 H^2|H^{(1)}_{\nu }\left( -kc_{S}\eta \right) |^{2}}{8c_{S}\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})^2 \pi M^{2}_{p}},&{} \quad \mathrm{for}~~~ |k c_{S}\eta |>>1, \end{array} \right. ~~~&{} {\underline{\mathrm{for~BD}}} , \\ \displaystyle \frac{\left( -k\eta c_{S}\right) ^3 H^2\left| C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right| ^2}{4c_{S}\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})^2 \pi ^{2}M^{2}_{p}},~~~&{} {\underline{\mathrm{for~~AV}}}, \end{array} \right. \end{aligned}$$

(5.102)

and similarly the primordial power spectrum for the scalar modes at any arbitrary momentum scale k can be written for both $\mathrm{AV}$ and $\mathrm{BD}$ with any arbitrary q as

$$\begin{aligned} \Delta _{\zeta }(k)\equiv \frac{k^{3}P_{\zeta }(k)}{2\pi ^2}=\frac{k^{3}|v_{k}|^{2}}{2\pi ^{2}z^{2}M^{2}_{p}}=\left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle \frac{2^{2\nu -3}q\left( -k\eta c_{S}\right) ^{3-2\nu } H^2}{4c_{S}\bar{\epsilon }_{V}\left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^2 \pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\nu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |<<1 , \\ \displaystyle \frac{2^{2\nu -3}q ~c^{2-2\nu }_{S}\left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^{1-2\nu } H^2}{4\bar{\epsilon }_{V} \pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\nu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |=1 ,\\ \displaystyle \frac{q\left( -k\eta c_{S}\right) ^3 H^2|H^{(1)}_{\nu }\left( -kc_{S}\eta \right) |^{2}}{4c_{S}\bar{\epsilon }_{V}\left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^2 \pi M^{2}_{p}},&{} \quad \mathrm{for}~~~ |k c_{S}\eta |>>1. \end{array} \right. ~~~&{} {\underline{\mathrm{for~BD}}} \\ \displaystyle \frac{q\left( -k\eta c_{S}\right) ^3 H^2\left| C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right| ^2}{2c_{S}\bar{\epsilon }_{V}\left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^2 \pi ^{2}M^{2}_{p}},~~~&{} {\underline{\mathrm{for~~AV}}}, \end{array} \right. \end{aligned}$$

(5.103)

where the effective sound speed $c_{S}$ is given by

$$\begin{aligned} \begin{array}{lll} \displaystyle c_{S} =\left\{ \begin{array}{lll} \displaystyle \sqrt{1-\frac{2}{3} {\epsilon }_{1}}=\sqrt{1-\frac{2}{3}\bar{\epsilon }_{V}},~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \sqrt{\frac{1-\frac{2}{3} {\epsilon }_{1}}{1+\frac{2(1-2q)}{3q} {\epsilon }_{1}}}=\sqrt{\frac{1-\frac{1}{3q} \bar{\epsilon }_{V}}{1+\frac{(1-2q)}{3q^2} \bar{\epsilon }_{V}}} ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.104)

Now starting from the expression for primordial power spectrum for the scalar modes one can compute the spectral tilt at any arbitrary momentum scale k for both $\mathrm{AV}$ and $\mathrm{BD}$ with $q=1/2$ as

$$\begin{aligned} n_{\zeta }(k)-1\equiv & {} \frac{\mathrm{d}\ln \Delta _{\zeta }(k)}{\mathrm{d}\ln k} = \frac{\mathrm{d}\ln \Delta _{\zeta }(k)}{\mathrm{d} N}\nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle (3-2\nu )\left[ 1-\frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots , &{} \quad \mathrm{for}~~~ |k c_{S}\eta |<<1 \\ \displaystyle (3-2\nu )\left[ 1-\frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots ,&{} \quad \mathrm{for}~~~ |k c_{S}\eta |=1 , \\ \displaystyle \bar{\epsilon }_{V}-\left[ \frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) +2(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\right] \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \quad \displaystyle +\frac{\left( -c_{S}\eta \right) \left[ H^{(1)}_{\nu -1}\left( -kc_{S}\eta \right) -H^{(1)}_{\nu +1} \left( -kc_{S}\eta \right) \right] }{H^{(1)}_{\nu }\left( -kc_{S}\eta \right) }+\cdots , &{} \quad \mathrm{for}~~~ |k c_{S}\eta |>>1, \end{array} \right. &{} {\underline{\mathrm{for \ BD}}}, \\ \displaystyle \bar{\epsilon }_{V}-\left[ \frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) +2(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\right] \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \quad \displaystyle +\frac{\left( -c_{S}\eta \right) C_{1}\left[ H^{(1)}_{\nu -1}\left( -kc_{S}\eta \right) -H^{(1)}_{\nu +1} \left( -kc_{S}\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right] }\\ \quad +\frac{\left( -c_{S}\eta \right) C_{2}\left[ H^{(2)}_{\nu -1}\left( -kc_{S}\eta \right) -H^{(2)}_{\nu +1} \left( -kc_{S}\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right] }+\cdots , &{} {\underline{\mathrm{for~~AV}}}. \end{array} \right. \end{aligned}$$

(5.105)

and for both $\mathrm{AV}$ and $\mathrm{BD}$ with any arbitrary q as

$$\begin{aligned} n_{\zeta }(k)-1\equiv & {} \frac{\mathrm{d}\ln \Delta _{\zeta }(k)}{\mathrm{d}\ln k} = \frac{\mathrm{d}\ln \Delta _{\zeta }(k)}{\mathrm{d} N}\nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle (3-2\nu )\left[ 1-\frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots ,&{} \quad \mathrm{for}~~~ |k c_{S}\eta |<<1 , \\ \displaystyle (3-2\nu )\left[ 1-\frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots , &{} \quad \mathrm{for}~~~ |k c_{S}\eta |=1 ,\\ \displaystyle \frac{\bar{\epsilon }_{V}}{2q}-\left[ \frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) +\frac{1}{q}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\right] \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \quad \displaystyle +\frac{\left( -c_{S}\eta \right) \left[ H^{(1)}_{\nu -1}\left( -kc_{S}\eta \right) -H^{(1)}_{\nu +1} \left( -kc_{S}\eta \right) \right] }{H^{(1)}_{\nu }\left( -kc_{S}\eta \right) }+\cdots , &{} \quad \mathrm{for}~~~ |k c_{S}\eta |>>1, \end{array} \right. &{} {\underline{\mathrm{for~BD}}}, \\ \displaystyle \frac{\bar{\epsilon }_{V}}{2q}-\left[ \frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) \displaystyle +\frac{1}{q}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\right] \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \displaystyle \quad +\frac{\left( -c_{S}\eta \right) C_{1}\left[ H^{(1)}_{\nu -1}\left( -kc_{S}\eta \right) -H^{(1)}_{\nu +1} \left( -kc_{S}\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right] }\\ \displaystyle \quad +\frac{\left( -c_{S}\eta \right) C_{1}\left[ H^{(2)}_{\nu -1}\left( -kc_{S}\eta \right) -H^{(2)}_{\nu +1} \left( -kc_{S}\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right] }+\cdots , &{} {\underline{\mathrm{for~~AV}}}. \end{array} \right. \end{aligned}$$

(5.106)

One can also consider the following approximations to simplify the final derived form of the primordial scalar power spectrum for the BD vacuum with the $|k c_{S}\eta |=1$ case:

• 1. We start with the Laurent expansion of the Gamma function:

  $$\begin{aligned} \Gamma (\nu )= & {} \frac{1}{\nu }-\gamma +\frac{1}{2}\left( \gamma ^2+\frac{\pi ^2}{6}\right) \nu -\frac{1}{6}\left( \gamma ^3+\frac{\gamma \pi ^2}{2}+2\zeta (3)\right) \nu ^2 +\mathcal{O}(\nu ^3). \end{aligned}$$

  (5.107)

  where $\gamma$ being the Euler–Mascheroni constant and $\zeta (3)$ characterizing the Riemann zeta function of order 3 originating in the expansion of the gamma function.

• 2. Hence using the result of Eq. (5.107) for $q=1/2$ and for arbitrary q we can write:

  $$\begin{aligned} \begin{array}{lll} \displaystyle \Gamma (\nu ) =\left\{ \begin{array}{lll} \displaystyle \frac{1}{\left( \frac{3}{2}+4\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) }-\gamma +\frac{1}{2} \left( \gamma ^2+\frac{\pi ^2}{6}\right) \left( \frac{3}{2}+4\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \\ \quad \displaystyle -\frac{1}{6}\left( \gamma ^3+\frac{\gamma \pi ^2}{2}+2\zeta (3)\right) \left( \frac{3}{2}+4\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2 +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{1}{ \left\{ \frac{3}{2}+\left( \frac{1}{2q}+\frac{3}{\sqrt{2q}}\right) \bar{\epsilon }_{V}-\frac{1}{\sqrt{2q}}\bar{\eta }_{V}\right\} } -\gamma \\ \displaystyle ~~~+\frac{1}{2}\left( \gamma ^2+\frac{\pi ^2}{6}\right) \left\{ \frac{3}{2}+\left( \frac{1}{2q}+\frac{3}{\sqrt{2q}}\right) \bar{\epsilon }_{V}-\frac{1}{\sqrt{2q}}\bar{\eta }_{V}\right\} \\ \displaystyle ~-\frac{1}{6}\left( \gamma ^3+\frac{\gamma \pi ^2}{2}+2\zeta (3)\right) \left\{ \frac{3}{2}+\left( \frac{1}{2q}+\frac{3}{\sqrt{2q}}\right) \bar{\epsilon }_{V}-\frac{1}{\sqrt{2q}}\bar{\eta }_{V}\right\} ^2 +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

  (5.108)

• 3. In the slow-roll regime of inflation all the slow-roll parameters satisfy the following constraint:

  $$\begin{aligned}&\bar{\epsilon }_{V}<< 1,\end{aligned}$$

  (5.109)
  $$\begin{aligned}&|\bar{\eta }_{V}|<< 1,\end{aligned}$$

  (5.110)
  $$\begin{aligned}&|\bar{\xi }^{2}_{V}|<< 1,\end{aligned}$$

  (5.111)
  $$\begin{aligned}&|\bar{\sigma }^{3}_{V}|<< 1. \end{aligned}$$

  (5.112)

  Using these approximations the primordial scalar power spectrum can be expressed as

  $$\begin{aligned} \Delta _{\zeta , \star }\approx & {} \left\{ \left[ 1-(\mathcal{C}_{E}+1){\epsilon }_{1}-\frac{\mathcal{C}_{E}}{2}\epsilon _{2}\right] ^2 \frac{H^{2}}{8\pi ^2M^{2}_{p}c_{S}{\epsilon }_{1}}\right\} _{k_{\star }=a_{\star }H_{\star }}\nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}-\mathcal{C}_{E}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2M^{2}_{p}c_{S}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }},~~~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2M^{2}_{p}c_{S}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }} ,~~~&{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

  (5.113)

  where $\mathcal{C}_{E}$ is given by

  $$\begin{aligned} \mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72. \end{aligned}$$

  (5.114)

• 4. Using the slow-roll approximations one can further approximate the expression for the sound speed as

  $$\begin{aligned} \begin{array}{lll} \displaystyle c^2_{S} =\left\{ \begin{array}{lll} \displaystyle 1-\frac{2}{3}\bar{\epsilon }_{V}+\mathcal{O}(\bar{\epsilon }^2_{V})+\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle 1-\frac{(1-q)}{3q^2}\bar{\epsilon }_{V}+\mathcal{O}(\bar{\epsilon }^2_{V})+\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

  (5.115)

• 5. Hence using the result in Eq. (5.169) we get the following simplified expression for the primordial scalar power spectrum:

  $$\begin{aligned} \Delta _{\zeta , \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \bar{\epsilon }_{V}-\mathcal{C}_{E} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2M^{2}_{p}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }} ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2M^{2}_{p} \bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }} ,~&{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

  (5.116)

  where the factor $\Sigma$ is defined as

  $$\begin{aligned} \Sigma= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \frac{1}{6} ,~&{} \quad \mathrm{for}~~ q=1/2\\ \displaystyle \frac{1-q}{6q} ,~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.117)

• 6. Next one can compute the scalar spectral tilt ($n_{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} n_{\zeta , \star }-1\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -2{\epsilon }_{1}-{\epsilon }_{2}-2{\epsilon }^{2}_{1}-\left( 2\mathcal{C}_{E}+\frac{8}{3}\right) {\epsilon }_{1}{\epsilon }_{2}-\mathcal{C}_{E}{\epsilon }_{2}{\epsilon }_{3}+\cdots \\ \quad \displaystyle =2\bar{\eta }_{V}-8\bar{\epsilon }_{V}-2\bar{\epsilon }^{2}_{V}-2\left( 2\mathcal{C}_{E}+\frac{8}{3}\right) \bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \\ \qquad -\mathcal{C}_{E}\left( 2\bar{\xi }^{2}_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^{2}_{V}\right) +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle -2{\epsilon }_{1}-{\epsilon }_{2}-2{\epsilon }^{2}_{1}-\left( 2\mathcal{C}_{E}+3-2\Sigma \right) {\epsilon }_{1}{\epsilon }_{2}-\mathcal{C}_{E}{\epsilon }_{2}{\epsilon }_{3}+\cdots \\ \quad \displaystyle =\sqrt{\frac{2}{q}}\bar{\eta }_{V}-\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \bar{\epsilon }_{V}-\frac{\bar{\epsilon }^{2}_{V}}{2q^2} \displaystyle -\frac{2}{(2q)^{3/2}}\left( 2\mathcal{C}_{E}+3-2\Sigma \right) \bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \\ \qquad \displaystyle -\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 2\bar{\xi }^{2}_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^{2}_{V}\right) +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.118)

• 7. Next one can compute the running of the scalar spectral tilt ($\alpha _{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} \alpha _{\zeta , \star }= & {} \left( \frac{\mathrm{d}n_{\zeta }(k)}{\mathrm{d}\ln k}\right) _{k_{\star }=a_{\star }H_{\star }}=\left( \frac{\mathrm{d}n_{\zeta }(k)}{\mathrm{d}N}\right) _{k_{\star }=a_{\star }H_{\star }}\nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\left\{ \left[ 4\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})(\bar{\eta }_{V}-3\bar{\epsilon }_{V})+2\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] \right. \\ \left. \displaystyle -\mathcal{C}_{E}\left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V} +194\bar{\epsilon }^2_{V}\bar{\eta }_{V}-10\bar{\eta }_{V}\bar{\epsilon }_{V} \right) \displaystyle \right. \\ \left. \displaystyle -\left( 2\mathcal{C}_{E}+\frac{8}{3}\right) \left[ 2\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V} -\bar{\xi }^2_{V}\right) \displaystyle -4\bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2\right] \right\} \displaystyle \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\left\{ \left[ \sqrt{\frac{2}{q}}\frac{\bar{\epsilon }_{V}}{q}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) (\bar{\eta }_{V}-3\bar{\epsilon }_{V})+\sqrt{\frac{2}{q}}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] \right. \\ \left. \displaystyle -\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V} +194\bar{\epsilon }^2_{V}\bar{\eta }_{V}-10\bar{\eta }_{V}\bar{\epsilon }_{V} \right) \right. \\ \left. \displaystyle -\left( 2\mathcal{C}_{E}+\frac{8}{3}\right) \left[ \sqrt{\frac{2}{q}}\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V} -\bar{\xi }^2_{V}\right) \displaystyle -\frac{4}{(2q)^{3/2}}\bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2\right] \right\} \\ \quad \times \displaystyle \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \nonumber \\\end{aligned}$$

  (5.119)

• 8. Finally, one can also compute the running of the running of scalar spectral tilt ($\kappa _{S}$) of the primordial scalar power spectrum as

$$\begin{aligned} \kappa _{\zeta , \star }= & {} \left( \frac{\mathrm{d}^2n_{\zeta }(k)}{\mathrm{d}\ln k^2}\right) _{k_{\star }=a_{\star }H_{\star }} =\left( \frac{\mathrm{d}^2n_{\zeta }(k)}{\mathrm{d}N^2}\right) _{k_{\star }=a_{\star }H_{\star }}\nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\left[ 8\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2+8\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 +8\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})\left( \bar{\xi }^2_{V}-10\bar{\epsilon }_{V}\bar{\eta }_{V} +18\bar{\epsilon }^{2}_{V}\right) \right. \\ \left. \displaystyle +2\left( 20\bar{\epsilon }_{V}\bar{\eta }_{V} \left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) +10\bar{\epsilon }_{V}\left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \right. \right. \\ \left. \left. \displaystyle -72\bar{\epsilon }^2_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) - \left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \right) \right] \displaystyle \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle \displaystyle -\left[ \left( \frac{2}{q}\right) ^{3/2}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) (\bar{\eta }_{V} -3\bar{\epsilon }_{V})^2+\sqrt{\frac{2}{q}}\frac{2}{q^2}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2\right. \\ \left. \displaystyle +\sqrt{\frac{2}{q}}\frac{2}{q}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) \left( \bar{\xi }^2_{V}-10\bar{\epsilon }_{V}\bar{\eta }_{V} +18\bar{\epsilon }^{2}_{V}\right) +\sqrt{\frac{2}{q}}\left( \frac{10}{q}\bar{\epsilon }_{V}\bar{\eta }_{V} \left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) +10\bar{\epsilon }_{V}\left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \right. \right. \\ \left. \left. \displaystyle -\frac{36}{q}\bar{\epsilon }^2_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) - \left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \right) \right] \displaystyle \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

(5.120)

5.1.5 Computation of tensor power spectrum

In this subsection our prime objective is to compute the primordial power spectra of tensor quantum fluctuations from tachyonic inflation and study the cosmological consequences from the previously mentioned string theory originating tachyonic potentials in the light of Planck 2015 data. To serve this purpose let us start with the following canonical variable $u_\mathbf{k}$, which can be quantized with the standard techniques:

$$\begin{aligned} u^{\gamma }_\mathbf{k}\equiv \frac{a}{\sqrt{2}}~M_{p}~h^{\gamma }_\mathbf{k}, \end{aligned}$$

(5.121)

where $h^{\gamma }_\mathbf{k}$ is the curvature perturbation in the momentum space, which can be expressed in terms of the curvature perturbation in position space through the following Fourier transformation:

$$\begin{aligned} h^{\gamma }(t,\mathbf{x})=\int \mathrm{d}^{3}k~h^{\gamma }_\mathbf{k}(t)~\exp (i\mathbf{k}.\mathbf{x}). \end{aligned}$$

(5.122)

Here the superscript $\gamma$ stands for the helicity index for the transverse and traceless spin-2 graviton degrees of freedom. In general, the tensor modes can be written in terms of the two orthogonal polarization basis vectors.

Further we derive the equation of motion of the tensor fluctuation by extremizing the tachyonic model action as

$$\begin{aligned} \left[ \frac{\mathrm{d}^{2}}{\mathrm{d}\eta ^2}+\left( k^{2}-\frac{1}{a}\frac{\mathrm{d}^{2}a}{\mathrm{d}\eta ^2}\right) \right] u_\mathbf{k}=0, \end{aligned}$$

(5.123)

where the helicity index $\gamma$ is summed over in the Fourier modes for the tensor contribution $u_\mathbf{k}$ and

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{a}\frac{\mathrm{d}^{2}a}{\mathrm{d}\eta ^2} =\left\{ \begin{array}{lll} \displaystyle a^2H^2\left[ 2-\bar{\epsilon }_{V}\right] +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle a^2H^2\left[ 2-\frac{1}{2q}\bar{\epsilon }_{V}\right] +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.124)

Further replacing the factor aH in Eq. (5.89), we finally get the following simplified expression:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{a}\frac{\mathrm{d}^{2}a}{\mathrm{d}\eta ^2} \approx \left\{ \begin{array}{lll} \displaystyle \frac{1}{\eta ^2}\left[ \left( \frac{3}{2}+\bar{\epsilon }_{V}\right) ^2-\frac{1}{4}\right] +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{1}{\eta ^2}\left[ \left\{ \frac{3}{2}+\frac{1}{2q}\bar{\epsilon }_{V}\right\} ^2-\frac{1}{4}\right] +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.125)

Now for further simplification in the computation of tensor power spectrum we introduce a new factor $\mu$ which is defined as

$$\begin{aligned} \begin{array}{lll} \displaystyle \mu \approx \left\{ \begin{array}{lll} \displaystyle \left( \frac{3}{2}+\bar{\epsilon }_{V}\right) +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \frac{3}{2}+\frac{1}{2q}\bar{\epsilon }_{V}\right\} +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.126)

Hence using Eq. (5.126) in Eq. (5.123), we get the following simplified form of the equation of motion:

$$\begin{aligned} \left[ \frac{\mathrm{d}^{2}}{\mathrm{d}\eta ^2}+\left( k^{2}-\frac{\left( \mu ^2-\frac{1}{4}\right) }{\eta ^2}\right) \right] u_\mathbf{k}(\eta )=0, \end{aligned}$$

(5.127)

and the most general solution of Eq. (5.127) is given by

$$\begin{aligned} u_\mathbf{k}(\eta )=\sqrt{-\eta }\left[ D_{1}H^{(1)}_{\mu }\left( -k\eta \right) +D_{2}H^{(2)}_{\mu }\left( -k\eta \right) \right] , \end{aligned}$$

(5.128)

where $D_{1}$ and $D_{2}$ are two arbitrary integration constants, which can be fixed from the appropriate choice of the boundary conditions. Additionally $H^{(1)}_{\mu }$ and $H^{(2)}_{\mu }$ represent the Hankel function of the first and second kind with rank $\mu$. Now to impose the well-known Bunch–Davies boundary condition at early times we have used

$$\begin{aligned}&\lim _{k\eta \rightarrow -\infty } H^{(1)}_{\mu }\left( -k\eta \right) \nonumber \\&\quad = \sqrt{\frac{2}{\pi }}\frac{1}{\sqrt{-\eta }}\exp \left( ik\eta \right) \exp \left( i\frac{\pi }{2}\left( \mu +\frac{1}{2}\right) \right) ,\end{aligned}$$

(5.129)

$$\begin{aligned}&\lim _{k\eta \rightarrow -\infty } H^{(2)}_{\mu }\left( -k\eta \right) \nonumber \\&\quad = \sqrt{\frac{2}{\pi }}\frac{1}{\sqrt{-\eta }}\exp \left( -ik\eta \right) \exp \left( -i\frac{\pi }{2}\left( \mu +\frac{1}{2}\right) \right) . \end{aligned}$$

(5.130)

As a result the previously mentioned integration constants are fixed at the following values:

$$\begin{aligned} D_{1}= & {} \sqrt{\frac{\pi }{2}},\end{aligned}$$

(5.131)

$$\begin{aligned} D_{2}= & {} 0. \end{aligned}$$

(5.132)

Consequently the solution of the mode function for tensor fluctuations takes the following form:

$$\begin{aligned} u_\mathbf{k}(\eta )=\sqrt{-\frac{\eta \pi }{2}}H^{(1)}_{\mu }\left( -k\eta \right) . \end{aligned}$$

(5.133)

On the other hand, the solution stated in Eq. (5.128) determines the future evolution of the mode including its super-horizon dynamics at $k<<aH$ or $|k \eta |<<1$ or $k \eta \rightarrow 0$ and this is due to

$$\begin{aligned} \lim _{k\eta \rightarrow 0} H^{(1)}_{\mu }\left( -k\eta \right)= & {} \frac{i}{\pi }\Gamma (\mu )\left( \frac{-k\eta }{2}\right) ^{-\mu }. \end{aligned}$$

(5.134)

Consequently the solution of the mode function for tensor fluctuations takes the following form:

$$\begin{aligned} u_\mathbf{k}(\eta )=\sqrt{-\frac{\eta \pi }{2}}\frac{i}{\pi }\Gamma (\mu )\left( \frac{-k\eta }{2}\right) ^{-\mu }. \end{aligned}$$

(5.135)

Finally combining the results obtained in Eqs. (5.94), (5.97) and (5.128) we get

$$\begin{aligned} \begin{array}{lll} \displaystyle u_\mathbf{k}(\eta ) =\left\{ \begin{array}{lll} \displaystyle \sqrt{-\eta }\left[ D_{1}H^{(1)}_{\mu }\left( -k\eta \right) +D_{2}H^{(2)}_{\mu }\left( -k\eta \right) \right] ,~&{} \quad \mathrm{for~~AV} , \\ \displaystyle \sqrt{-\frac{\eta \pi }{2}}H^{(1)}_{\mu }\left( -k\eta \right) ,&{} \quad \mathrm{for~~BD} + |k \eta |>>1 ,\\ \displaystyle \sqrt{-\frac{\eta \pi }{2}}\frac{i}{\pi }\Gamma (\mu )\left( \frac{-k\eta }{2}\right) ^{-\mu } ,&{}\quad \mathrm{for~~BD} + |k \eta |<<1, \end{array} \right. \end{array} \end{aligned}$$

(5.136)

where $\mathrm{AV}$ and $\mathrm{BD}$ signify an arbitrary vacuum and the Bunch–Davies vacuum respectively. Finally the two point function from the tensor fluctuations for both $\mathrm{AV}$ and $\mathrm{BD}$ can be expressed as

$$\begin{aligned} \langle h_\mathbf{k}h_\mathbf{k^{\prime }}\rangle= & {} \left( \frac{4}{M^2_{p}}\right) \langle \delta \psi _\mathbf{k} \delta \psi _\mathbf{k^{\prime }}\rangle =\frac{2}{a^{2}M^{2}_{p}}\langle u_\mathbf{k} u_\mathbf{k^{\prime }}\rangle \nonumber \\= & {} (2\pi )^{3}\delta ^{3}(\mathbf{k}+\mathbf{k^{\prime }})\frac{2\pi ^{2}}{k^{3}}\Delta _{h}(k), \end{aligned}$$

(5.137)

where the primordial power spectrum for tensor modes at any arbitrary momentum scale k can be written for both $\mathrm{AV}$ and $\mathrm{BD}$ with $q=1/2$ as

$$\begin{aligned} {}\Delta _{h}(k)\equiv & {} \overbrace{2}^\mathrm{Due~to~graviton~helicity}\times \frac{k^{3}P_{h}(k)}{2\pi ^2}=\frac{k^{3}|u_{k}|^{2}}{\pi ^{2}a^{2}M^{2}_{p}}\nonumber \\ {}= & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle \frac{2^{2\mu -2}\left( -k\eta \right) ^{3-2\mu } H^2}{(1+\bar{\epsilon }_{V})^2 \pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\mu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k \eta |<<1 \\ \displaystyle \frac{2^{2\mu -2}\left( 1+\bar{\epsilon }_{V}\right) ^{1-2\mu } H^2}{\pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\mu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k \eta |=1 \\ \displaystyle \frac{\left( -k\eta \right) ^3 H^2|H^{(1)}_{\mu }\left( -k\eta \right) |^{2}}{(1+\bar{\epsilon }_{V})^2 \pi M^{2}_{p}},&{} \quad \mathrm{for}~~~ |k \eta |>>1, \end{array} \right. ~~~&{} {\underline{\mathrm{for~BD}}} , \\ \displaystyle \frac{2\left( -k\eta \right) ^3 H^2\left| D_{1}H^{(1)}_{\mu }\left( -k\eta \right) +D_{2}H^{(2)}_{\mu } \left( -k\eta \right) \right| ^2}{(1+\bar{\epsilon }_{V})^2 \pi ^{2}M^{2}_{p}},~~~&{} {\underline{\mathrm{for~~AV}}}, \end{array} \right. \end{aligned}$$

(5.138)

and similarly the primordial power spectrum for tensor modes at any arbitrary momentum scale k can be written for both $\mathrm{AV}$ and $\mathrm{BD}$ with any arbitrary q as

$$\begin{aligned} \Delta _{h}(k)\equiv & {} \overbrace{2}^\mathrm{Due~to~graviton~helicity}\times \frac{k^{3}P_{h}(k)}{2\pi ^2}=\frac{k^{3}|u_{k}|^{2}}{\pi ^{2}a^{2}M^{2}_{p}}\nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle \frac{2^{2\mu -1}q\left( -k\eta \right) ^{3-2\mu } H^2}{\left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^2 \pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\mu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k \eta |<<1 \\ \displaystyle \frac{2^{2\mu -1}q \left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^{1-2\mu } H^2}{\pi ^{2}M^{2}_{p}}\left| \frac{\Gamma (\mu )}{\Gamma \left( \frac{3}{2}\right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k \eta |=1 \\ \displaystyle \frac{2q\left( -k\eta \right) ^3 H^2|H^{(1)}_{\mu }\left( -k\eta \right) |^{2}}{\bar{\epsilon }_{V}\left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^2 \pi M^{2}_{p}},&{} \quad \mathrm{for}~~~ |k \eta |>>1. \end{array} \right. ~~~&{} \underline{\mathrm{for~BD}} \\ \displaystyle \frac{4q\left( -k\eta \right) ^3 H^2\left| C_{1}H^{(1)}_{\mu }\left( -k\eta \right) +C_{2}H^{(2)}_{\mu } \left( -k\eta \right) \right| ^2}{\left( 1+\frac{1}{2q}\bar{\epsilon }_{V}\right) ^2\pi ^{2}M^{2}_{p}},~~~&{} {\underline{\mathrm{for~~AV}}}. \end{array} \right. \end{aligned}$$

(5.139)

Now starting from the expression for the primordial power spectrum for the tensor modes one can compute the spectral tilt at any arbitrary momentum scale k for both $\mathrm{AV}$ and $\mathrm{BD}$ with $q=1/2$ as

$$\begin{aligned} n_{h}(k)\equiv & {} \frac{\mathrm{d}\ln \Delta _{h}(k)}{\mathrm{d}\ln k} = \frac{\mathrm{d}\ln \Delta _{h}(k)}{\mathrm{d} N}\nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle (3-2\mu )\left[ 1-\frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots , &{} \quad \mathrm{for}~~~ |k \eta |<<1 , \\ \displaystyle (3-2\mu )\left[ 1-\frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots ,&{} \quad \mathrm{for}~~~ |k \eta |=1 , \\ \displaystyle \bar{\epsilon }_{V}-\frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4} -2(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \displaystyle \frac{\left( -\eta \right) \left[ H^{(1)}_{\mu -1}\left( -k\eta \right) -H^{(1)}_{\mu +1} \left( -k\eta \right) \right] }{H^{(1)}_{\mu }\left( -k\eta \right) }+\cdots , &{} \quad \mathrm{for}~~~ |k \eta |>>1, \end{array} \right. &{} {\underline{\mathrm{for~BD}}} ,\\ \displaystyle \bar{\epsilon }_{V}-\frac{2}{3}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4} \displaystyle -2(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \displaystyle \frac{\left( -\eta \right) C_{1}\left[ H^{(1)}_{\mu -1}\left( -k\eta \right) -H^{(1)}_{\mu +1} \left( -k\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\mu }\left( -k\eta \right) +C_{2}H^{(2)}_{\mu } \left( -k\eta \right) \right] } \displaystyle \frac{\left( -\eta \right) C_{1}\left[ H^{(2)}_{\mu -1}\left( -k\eta \right) -H^{(2)}_{\mu +1} \left( -k\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\mu }\left( -k\eta \right) +C_{2}H^{(2)}_{\mu } \left( -k\eta \right) \right] }+\cdots , &{} {\underline{\mathrm{for~~AV}}}, \end{array} \right. \end{aligned}$$

(5.140)

and for both $\mathrm{AV}$ and $\mathrm{BD}$ with any arbitrary q as

$$\begin{aligned} n_{h}(k)\equiv & {} \frac{\mathrm{d}\ln \Delta _{h}(k)}{\mathrm{d}\ln k} = \frac{\mathrm{d}\ln \Delta _{h}(k)}{\mathrm{d} N}\nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle (3-2\mu )\left[ 1-\frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots ,&{} \quad \mathrm{for}~~~ |k \eta |<<1 , \\ \displaystyle (3-2\mu )\left[ 1-\frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}\right] +\cdots , &{} \quad \mathrm{for}~~~ |k \eta |=1 , \\ \displaystyle \frac{\bar{\epsilon }_{V}}{2q}-\frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}-\frac{1}{q}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \displaystyle \frac{\left( -\eta \right) \left[ H^{(1)}_{\mu -1}\left( -k\eta \right) -H^{(1)}_{\mu +1} \left( -k\eta \right) \right] }{H^{(1)}_{\mu }\left( -k\eta \right) }+\cdots , &{} \quad \mathrm{for}~~~ |k \eta |>>1, \end{array} \right. &{} {\underline{\mathrm{for~BD}}} ,\\ \displaystyle \frac{\bar{\epsilon }_{V}}{2q}-\frac{1}{3q}\bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}\displaystyle -\frac{1}{q}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}\\ \displaystyle \frac{\left( -\eta \right) C_{1}\left[ H^{(1)}_{\mu -1}\left( -k\eta \right) -H^{(1)}_{\mu +1} \left( -k\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\mu }\left( -k\eta \right) +C_{2}H^{(2)}_{\mu } \left( -k\eta \right) \right] }\\ \displaystyle \frac{\left( -\eta \right) C_{1}\left[ H^{(2)}_{\mu -1}\left( -k\eta \right) -H^{(2)}_{\mu +1} \left( -k\eta \right) \right] }{\left[ C_{1}H^{(1)}_{\mu }\left( -k\eta \right) +C_{2}H^{(2)}_{\mu } \left( -k\eta \right) \right] }+\cdots , &{} {\underline{\mathrm{for~~AV}}}. \end{array} \right. \end{aligned}$$

(5.141)

One can also consider the following approximations to simplify the final derived form of the primordial scalar power spectrum for the BD vacuum with $|k c_{S}\eta |=1$ case:

• 1. We start with the Laurent expansion of the Gamma function:

  $$\begin{aligned} \Gamma (\mu )= & {} \frac{1}{\mu }-\gamma +\frac{1}{2}\left( \gamma ^2+\frac{\pi ^2}{6}\right) \mu - \; \frac{1}{6}\left( \gamma ^3+\frac{\gamma \pi ^2}{2}+2\zeta (3)\right) \mu ^2 +\mathcal{O}(\mu ^3). \end{aligned}$$

  (5.142)

  where $\gamma$ being the Euler–Mascheroni constant and $\zeta (3)$ characterizing the Riemann zeta function of order 3 originating in the expansion of the gamma function.

• 2. Hence using the result of Eq. (5.142) for $q=1/2$ and for arbitrary q we can write:

  $$\begin{aligned} \begin{array}{lll} \displaystyle \Gamma (\mu ) =\left\{ \begin{array}{lll} \displaystyle \frac{1}{\left( \frac{3}{2}+\bar{\epsilon }_{V}\right) }-\gamma +\frac{1}{2} \left( \gamma ^2+\frac{\pi ^2}{6}\right) \left( \frac{3}{2}+\bar{\epsilon }_{V}\right) \\ \displaystyle -\frac{1}{6}\left( \gamma ^3+\frac{\gamma \pi ^2}{2}+2\zeta (3)\right) \left( \frac{3}{2}+\bar{\epsilon }_{V}\right) ^2 +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{1}{ \left\{ \frac{3}{2}+\frac{1}{2q}\bar{\epsilon }_{V}\right\} } -\gamma +\frac{1}{2}\left( \gamma ^2+\frac{\pi ^2}{6}\right) \left\{ \frac{3}{2}+\frac{1}{2q}\bar{\epsilon }_{V}\right\} \\ \displaystyle ~-\frac{1}{6}\left( \gamma ^3+\frac{\gamma \pi ^2}{2}+2\zeta (3)\right) \left\{ \frac{3}{2}+\frac{1}{2q}\bar{\epsilon }_{V}\right\} ^2 +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

  (5.143)

• 3. In the slow-roll regime of inflation all the slow-roll parameters satisfy the following constraint:

  $$\begin{aligned}&\bar{\epsilon }_{V}<< 1,\end{aligned}$$

  (5.144)
  $$\begin{aligned}&|\bar{\eta }_{V}|<< 1,\end{aligned}$$

  (5.145)
  $$\begin{aligned}&|\bar{\xi }^{2}_{V}|<< 1,\end{aligned}$$

  (5.146)
  $$\begin{aligned}&|\bar{\sigma }^{3}_{V}|<< 1. \end{aligned}$$

  (5.147)

  Using these approximations the primordial scalar power spectrum can be expressed as

  $$\begin{aligned} \Delta _{\zeta , \star }\approx & {} \left\{ \left[ 1-(\mathcal{C}_{E}+1){\epsilon }_{1}-\frac{\mathcal{C}_{E}}{2}\epsilon _{2}\right] ^2 \frac{H^{2}}{8\pi ^2M^{2}_{p}c_{S}{\epsilon }_{1}}\right\} _{k_{\star }=a_{\star }H_{\star }}\nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}-\mathcal{C}_{E}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2M^{2}_{p}c_{S}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }},~~~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2M^{2}_{p}c_{S}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }} ,~~~&{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

  (5.148)

  where $\mathcal{C}_{E}$ is given by

  $$\begin{aligned} \mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72. \end{aligned}$$

  (5.149)

• 4. Next one can compute the scalar spectral tilt ($n_{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} n_{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -2{\epsilon }_{1}\left[ 1+{\epsilon }_{1}+\left( \mathcal{C}_{E}+1\right) {\epsilon }_{2}\right] +\cdots \\ \displaystyle =-2\bar{\epsilon }_{V}\left[ 1+\bar{\epsilon }_{V}+2\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle -2{\epsilon }_{1}\left[ 1+{\epsilon }_{1}+\left( \mathcal{C}_{E}+1\right) {\epsilon }_{2}\right] +\cdots \\ \displaystyle =-\frac{\bar{\epsilon }_{V}}{q}\left[ 1+\frac{\bar{\epsilon }_{V}}{2q}+\sqrt{\frac{2}{q}}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.150)

• 5. Next one can compute the running of the tensor spectral tilt ($\alpha _{h}$) of the primordial scalar power spectrum as

  $$\begin{aligned} \alpha _{h, \star }= & {} \left( \frac{\mathrm{d}n_{h}(k)}{\mathrm{d}\ln k}\right) _{k_{\star }=a_{\star }H_{\star }}=\left( \frac{\mathrm{d}n_{h}(k)}{\mathrm{d}N}\right) _{k_{\star }=a_{\star }H_{\star }}\end{aligned}$$

  (5.151)
  $$\begin{aligned}\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\left[ 4\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})(\bar{\eta }_{V}-3\bar{\epsilon }_{V})+4\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) \displaystyle -8\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 \right. \\ \left. \displaystyle -2\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] _{\star }\displaystyle \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\left[ 2\frac{\bar{\epsilon }_{V}}{q}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) (\bar{\eta }_{V}-3\bar{\epsilon }_{V})+\frac{1}{q^2}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) \displaystyle -\frac{8}{(2q)^{5/2}}\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 \right. \\ \left. \displaystyle - \sqrt{\frac{2}{q}}\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] _{\star }\displaystyle \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array}\right. \end{aligned}$$

  (5.152)

• 6. Finally, one can also compute the running of the running of scalar spectral tilt ($\kappa _{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} \kappa _{h, \star }= & {} \left( \frac{\mathrm{d}^2n_{h}(k)}{\mathrm{d}\ln k^2}\right) _{k_{\star }=a_{\star }H_{\star }} =\left( \frac{\mathrm{d}^2n_{h}(k)}{\mathrm{d}N^2}\right) _{k_{\star }=a_{\star }H_{\star }}\nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \displaystyle -\left[ 8\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2+8\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 \right. \\ \left. \displaystyle -16\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}\left\{ (\bar{\eta }_{V}-3\bar{\epsilon }_{V})^3 -(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right\} \right. \\ \left. \displaystyle -4\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] _{\star }\displaystyle \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\left[ \frac{4}{q}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) (\bar{\eta }_{V} -3\bar{\epsilon }_{V})^2+\frac{2}{q^2}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 \displaystyle +\frac{2}{q}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) \left( \bar{\xi }^2_{V}-10\bar{\epsilon }_{V}\bar{\eta }_{V} +18\bar{\epsilon }^{2}_{V}\right) \right. \\ \left. \displaystyle -\frac{8}{q}\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}\left\{ (\bar{\eta }_{V}-3\bar{\epsilon }_{V})^3 -(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right\} \right] _{\star } \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.153)

5.1.6 Modified consistency relations

In this subsection we derive the new (modified) consistency relations for single tachyonic field inflation:

• 1. Let us first start with the tensor-to-scalar ratio r, which can be defined at any arbitrary momentum scale k for the $q=1/2$ case as

  $$\begin{aligned} r(k)\equiv & {} \frac{\Delta _{h}(k)}{\Delta _{\zeta }(k)}=2\frac{P_{h}(k)}{P_{\zeta }(k)}=2\frac{|u_{k}|^{2}}{|v_{k}|^{2}}\left( \frac{z}{a}\right) ^2\nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle 16\times 2^{2(\mu -\nu )}\bar{\epsilon }_{V}\left( -k\eta c_{S}\right) ^{2(\nu -\mu )}c^{2\mu -2}_{S}\left| \frac{\Gamma (\mu )}{\Gamma \left( \nu \right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |<<1 \\ \displaystyle 16\times 2^{2(\mu -\nu )} \bar{\epsilon }_{V}\left( 1+\bar{\epsilon }_{V}\right) ^{2(\nu -\mu )}c^{2\nu -2}_{S}\left| \frac{\Gamma (\mu )}{\Gamma \left( \nu \right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |=1 \\ \displaystyle \frac{8\bar{\epsilon }_{V}}{c^{2}_{S}}\times \frac{|H^{(1)}_{\mu }\left( -k\eta \right) |^{2}}{|H^{(1)}_{\nu }\left( -kc_{S}\eta \right) |^{2}},&{} \quad \mathrm{for}~~~ |k c_{S}\eta |>>1. \end{array} \right. ~~&{} {\underline{\mathrm{for~BD}}} \\ \displaystyle \frac{8\bar{\epsilon }_{V}}{c^{2}_{S}}\times \frac{\left| D_{1}H^{(1)}_{\mu }\left( -k\eta \right) +D_{2}H^{(2)}_{\mu } \left( -k\eta \right) \right| ^2}{\left| C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right| ^2},~&{} {\underline{\mathrm{for~~AV}}}. \end{array} \right. \end{aligned}$$

  (5.154)

  Similarly for arbitrary q one can write the following expression for the tensor-to-scalar ratio r at any arbitrary momentum scale:

  $$\begin{aligned} r(k)\equiv & {} \frac{\Delta _{h}(k)}{\Delta _{\zeta }(k)}=2\frac{P_{h}(k)}{P_{\zeta }(k)}=2\frac{|u_{k}|^{2}}{|v_{k}|^{2}}\left( \frac{z}{a}\right) ^2\nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \left\{ \begin{array}{lll} \displaystyle 16\times 2^{2(\mu -\nu )} \bar{\epsilon }_{V}\left( -k\eta c_{S}\right) ^{2(\nu -\mu )}c^{2\mu -2}_{S}\left| \frac{\Gamma (\mu )}{\Gamma \left( \nu \right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |<<1 \\ \displaystyle 16\times 2^{2(\mu -\nu )} \bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) ^{2(\nu -\mu )}c^{2\nu -2}_{S}\left| \frac{\Gamma (\mu )}{\Gamma \left( \nu \right) }\right| ^{2},~&{} \quad \mathrm{for}~~~ |k c_{S}\eta |=1 \\ \displaystyle \frac{8\bar{\epsilon }_{V}}{c^{2}_{S}}\times \frac{|H^{(1)}_{\mu }\left( -kc_{S}\eta \right) |^{2}}{|H^{(1)}_{\nu }\left( -kc_{S}\eta \right) |^{2}},&{} \quad \mathrm{for}~~~ |k c_{S}\eta |>>1. \end{array} \right. ~~&{} {\underline{\mathrm{for~BD}}} \\ \displaystyle \frac{8\bar{\epsilon }_{V}}{c^{2}_{S}}\times \frac{\left| D_{1}H^{(1)}_{\mu }\left( -kc_{S}\eta \right) +D_{2}H^{(2)}_{\mu } \left( -kc_{S}\eta \right) \right| ^2}{\left| C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right| ^2},~&{} {\underline{\mathrm{for~~AV}}}. \end{array} \right. \end{aligned}$$

  (5.155)

• 2. Next for the BD vacuum with $|k c_{S}\eta |=1$ case within the slow-roll regime we can approximately write the following expression for the tensor-to-scalar ratio:

  $$\begin{aligned} r_{\star }= & {} \frac{\Delta _{h}(k_{\star })}{\Delta _{\zeta }(k_{\star })}=2\frac{P_{h}(k_{\star })}{P_{\zeta }(k_{\star })}=2\frac{|u_{k_{\star }}|^{2}}{|v_{k_{\star }}|^{2}}\left( \frac{z}{a}\right) ^2_{\star }\end{aligned}$$

  (5.156)
  $$\begin{aligned}= & {} \left[ 16{\epsilon }_{1}c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1){\epsilon }_{1}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1){\epsilon }_{1}-\frac{\mathcal{C}_{E}}{2}{\epsilon }_{2}\right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }}\nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ 16\bar{\epsilon }_{V}c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}-\mathcal{C}_{E}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ 16\bar{\epsilon }_{V}\frac{\left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}\right] ^2}{\left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \bar{\epsilon }_{V}-\mathcal{C}_{E} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }},~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ \frac{8}{q}\bar{\epsilon }_{V}c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ \frac{8}{q}\bar{\epsilon }_{V}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }},~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.157)

• 3. Hence the consistency relation between the tensor-to-scalar ratio r and spectral tilt $n_{h}$ for tensor modes for the BD vacuum with the $|k c_{S}\eta |=1$ case can be written as

  $$\begin{aligned} r_{\star }= & {} -8n_{h,\star }\times \left[ c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1){\epsilon }_{1}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1) {\epsilon }_{1}-\frac{\mathcal{C}_{E}}{2}{\epsilon }_{2}\right] ^2\left[ 1+{\epsilon }_{1}+\left( \mathcal{C}_{E} +1\right) {\epsilon }_{2}\right] }\right] _{k_{\star }=a_{\star }H_{\star }}\nonumber \\\approx & {} -8n_{h,\star }\times \underbrace{\left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}-\mathcal{C}_{E}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2\left[ 1+\bar{\epsilon }_{V}+2\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ \frac{\left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}\right] ^2}{\left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \bar{\epsilon }_{V}-\mathcal{C}_{E} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2\left[ 1+\bar{\epsilon }_{V}+2\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }=a_{\star }H_{\star }},~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2\left[ 1+\frac{\bar{\epsilon }_{V}}{2q}+\sqrt{\frac{2}{q}}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ \frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2\left[ 1+\frac{\bar{\epsilon }_{V}}{2q}+\sqrt{\frac{2}{q}}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }=a_{\star }H_{\star }},~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. }_\mathrm{Correction~factor} \end{aligned}$$

  (5.158)

• 4. Next one can express the first two slow-roll parameters $\bar{\epsilon }_{V}$ and $\bar{\eta }_{V}$ in terms of the inflationary observables as

  $$\begin{aligned} \bar{\epsilon }_{V}\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \epsilon _{1}\approx -\frac{n_{h,\star }}{2}+\cdots \approx \frac{r_{\star }}{16}+\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle 2q\epsilon _{1}\approx -qn_{h,\star }+\cdots \approx \frac{qr_{\star }}{8}+\cdots , &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \nonumber \\ \bar{\eta }_{V}\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle 3\epsilon _{1}-\frac{\epsilon _{2}}{2}\approx \frac{1}{2}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{2}\right) +\cdots \approx \frac{1}{2}\left( n_{\zeta ,\star }-1-4n_{h,\star }\right) +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle 6q\epsilon _{1}-\sqrt{\frac{q}{2}}\epsilon _{2}\approx \sqrt{\frac{q}{2}}\left( n_{\zeta ,\star }-1+\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \frac{qr_{\star }}{8}\right) +\cdots \\ \quad \displaystyle \approx \sqrt{\frac{q}{2}}\left( n_{\zeta ,\star }-1-q\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) n_{h,\star }\right) +\cdots ,~~ &{} \quad \mathrm{for~~any}~q. \end{array} \right. \end{aligned}$$

  (5.159)

• 5. Then the connecting consistency relation between tensor and scalar spectral tilt and tensor-to-scalar ratio can be expressed as

  $$\begin{aligned} n_{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\frac{r_{\star }}{8c_{S}}\left[ 1-\frac{r_{\star }}{16}+\left( 1-n_{\zeta ,\star }\right) -\mathcal{C}_{E}\left\{ \frac{r_{\star }}{8} +\left( n_{\zeta ,\star }-1\right) \right\} \right] +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle -\frac{r_{\star }}{8c_{S}}\left[ 1+\left\{ \left( \frac{3q}{8}\sqrt{\frac{2}{q}} -\left( \sqrt{\frac{2}{q}}+5\right) \frac{1}{16}\right) r_{\star } +\frac{\left( 1-n_{\zeta ,\star }\right) }{\sqrt{2q}}\right\} \right. \\ \left. \displaystyle +\sqrt{\frac{2}{q}}\mathcal{C}_{E}\left( \frac{3qr_{\star }}{8}-\frac{1}{2}\left\{ n_{\zeta ,\star }-1+\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \frac{qr_{\star }}{8}\right\} \right) \right] +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.160)

  Finally, using the approximated version of the expression for $c_{S}$ in terms of slow-roll parameters, one can recast this consistency condition as

  $$\begin{aligned} n_{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\frac{r_{\star }}{8}\left[ 1-\frac{r_{\star }}{24}+\left( 1-n_{\zeta ,\star }\right) -\mathcal{C}_{E}\left\{ \frac{r_{\star }}{8} +\left( n_{\zeta ,\star }-1\right) \right\} \right] +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle -\frac{r_{\star }}{8}\left[ 1+\left\{ \left( \frac{3q}{8}\sqrt{\frac{2}{q}} -\left( \sqrt{\frac{2}{q}}+5\right) \frac{1}{16}+\frac{\Sigma }{8}\right) r_{\star } +\frac{\left( 1-n_{\zeta ,\star }\right) }{\sqrt{2q}}\right\} \right. \\ \left. \displaystyle +\sqrt{\frac{2}{q}}\mathcal{C}_{E}\left( \frac{3qr_{\star }}{8}-\frac{1}{2}\left\{ n_{\zeta ,\star }-1+\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \frac{qr_{\star }}{8}\right\} \right) \right] +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.161)

• 6. Next the running of the sound speed $c_{S}$ can be written in terms of slow-roll parameters as

  $$\begin{aligned} \begin{array}{lll} \displaystyle S=\frac{\dot{c}_{S}}{Hc_{S}}=\frac{\mathrm{d}\ln c_{S}}{\mathrm{d}N}= \frac{\mathrm{d}\ln c_{S}}{\mathrm{d}\ln k}=\left\{ \begin{array}{lll} \displaystyle -\frac{2}{3}\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\right) ^{1/4}+\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\frac{(1-q)}{3q^2}\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\bar{\epsilon }_{V}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any}~q, \end{array} \right. \end{array} \end{aligned}$$

  (5.162)

  which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inflationary observables as

  $$\begin{aligned} \begin{array}{lll} \displaystyle S=\frac{\dot{c}_{S}}{Hc_{S}}=\frac{\mathrm{d}\ln c_{S}}{\mathrm{d}N}= \frac{\mathrm{d}\ln c_{S}}{\mathrm{d}\ln k}=\left\{ \begin{array}{lll} \displaystyle -\frac{r_{\star }}{48}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\frac{(1-q)}{24q^2}\sqrt{\frac{q}{2}}r_{\star }\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array}\right. \end{array} \end{aligned}$$

  (5.163)

• 7. Further the running of the tensor spectral tilt can be written in terms of the inflationary observables as

  $$\begin{aligned} \alpha _{h, \star }= & {} \left( \frac{\mathrm{d}n_{h}(k)}{\mathrm{d}\ln k}\right) _{k_{\star }=a_{\star }H_{\star }}=\left( \frac{\mathrm{d}n_{h}(k)}{\mathrm{d}N}\right) _{k_{\star }=a_{\star }H_{\star }}\end{aligned}$$

  (5.164)
  $$\begin{aligned}\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\left[ \frac{r_{\star }}{8}\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) -\frac{r_{\star }}{8}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right. \\ \quad \left. \displaystyle -\mathcal{C}_{E}\frac{r_{\star }}{8}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \quad -\left[ \frac{r_{\star }}{4}\sqrt{\frac{q}{2}}\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right. \\ \quad \left. \displaystyle -\frac{1}{(2q)^{1/2}}\left( \mathcal{C}_{E}+1\right) \frac{r_{\star }}{8}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.165)

• 8. Next the scalar power spectrum can be expressed in terms of the other inflationary observables as

  $$\begin{aligned} \Delta _{\zeta , \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2} \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right] ^2 \frac{2H^{2}_{\star }}{\pi ^2M^{2}_{p}r_{\star }}+\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] ^2 \displaystyle \frac{2 H^{2}_{\star }}{\pi ^2M^{2}_{p} r_{\star }}+\cdots ,~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.166)

• 9. Further the tensor power spectrum can be expressed in terms of the other inflationary observables as

  $$\begin{aligned} \Delta _{h, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ 1-(\mathcal{C}_{E}+1)\frac{r_{\star }}{16}\right] ^2 \frac{2H^{2}_{\star }}{\pi ^2M^{2}_{p}}+\cdots ,~~~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1)\frac{r_{\star }}{16}\right] ^2 \frac{2H^{2}_{\star }}{\pi ^2M^{2}_{p}}+\cdots ,~~~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.167)

• 10. Next the running of the tensor-to-scalar ratio can be expressed in terms of the inflationary observables as

  $$\begin{aligned} \alpha _{r,\star }= & {} \left( \frac{\mathrm{d} r}{\mathrm{\mathrm{d}}\ln k}\right) _{\star }=\left( \frac{\mathrm{d} r}{\mathrm{\mathrm{d}} N}\right) _{\star }=-8\alpha _{h, \star }+\cdots \nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ r_{\star }\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) -r_{\star }\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right. \\ \left. \displaystyle -\mathcal{C}_{E}r_{\star }\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ 2r_{\star }\sqrt{\frac{q}{2}}\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right. \\ \left. \displaystyle -\frac{1}{(2q)^{1/2}}\left( \mathcal{C}_{E}+1\right) r_{\star }\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.168)

• 11. Finally the scale of single field tachyonic inflation can be expressed in terms of the Hubble parameter and the other inflationary observables as

  $$\begin{aligned} H_{\mathrm{inf}}= & {} H_{\star }\approx \left\{ \begin{array}{lll} \displaystyle \displaystyle \frac{\sqrt{\frac{\Delta _{\zeta , \star }r_{\star }}{2}}\pi M_{p}}{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2} \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right] } +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{\sqrt{\frac{\Delta _{\zeta , \star }r_{\star }}{2}}\pi M_{p}}{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] } +\cdots ,~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.169)

  One can recast this statement in terms of the inflationary potential as

  $$\begin{aligned} {\root 4 \of {V_{\mathrm{inf}}}}= & {} {\root 4 \of {V_{\star }}}\approx \left\{ \begin{array}{lll} \displaystyle \displaystyle \frac{\root 4 \of {\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\sqrt{\pi } M_{p}}{ \sqrt{\left[ 1-\left( \mathcal{C}_{E}+ \frac{5}{6}\right) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2} \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right] }} +\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{\root 4 \of {\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\sqrt{\pi } M_{p}}{ \sqrt{\left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] }} +\cdots , &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.170)

5.1.7 Field excursion for tachyon

In this subsection we explicitly derive the expression for the field excursion⁴ for tachyonic inflation defined as

$$\begin{aligned} |\Delta T|= & {} |T_{\mathrm{cmb}}-T_{\mathrm{end}}|=|T_{\star }-T_{\mathrm{end}}| \end{aligned}$$

(5.171)

where $T_{\mathrm{cmb}}$, $T_{\mathrm{end}}$ and $T_{\star }$ signify the tachyon field value at the time of horizon exit, at end of inflation and at pivot scale respectively. Here we perform the computation for both AV and BD vacuum. For for the sake of simplicity the pivot scale is fixed at the horizon exit scale. To compute the expression for the field excursion we perform the following steps:

• 1. We start with the operator identity for single field tachyon using which one can write expression for the tachyon field variation with respect to the momentum scale (k) or number of e-foldings (N) in terms of the inflationary observables as

  $$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{H}\frac{\mathrm{d}T}{\mathrm{d}t}=\frac{\mathrm{d}T}{\mathrm{d}N}=\frac{\mathrm{d}T}{\mathrm{d}\ln k}\approx \left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{r}{8V(T)\alpha ^{\prime }}}M_{p}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{qr}{4V(T)\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

  (5.172)

  where the tensor-to-scalar ratio r is a function of k or N.

• 2. Next using Eq. (5.172) we can write the following integral equation:

  $$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{T_{\star }}_{T_{\mathrm{end}}} \mathrm{d}T~\sqrt{V(T)} \approx \left\{ \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{r}{8\alpha ^{\prime }}}M_{p}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots \,\\ \quad \displaystyle \displaystyle =\int ^{N_{\star }}_{N_{\mathrm{end}}} \mathrm{d}N~\sqrt{\frac{r}{8\alpha ^{\prime }}}M_{p}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots \,\\ \quad \displaystyle =\int ^{N_{\star }}_{N_{\mathrm{end}}} \mathrm{d}N~\sqrt{\frac{qr}{4\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

  (5.173)

• 3. Next we parametrize the form of the tensor-to-scalar ratio for $q=1/2$ and for any arbitrary q at any arbitrary scale as

  $$\begin{aligned} \begin{array}{lll} \displaystyle r(k)\displaystyle =\left\{ \begin{array}{ll}\left\{ \begin{array}{ll} \displaystyle r_{\star } &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle r_{\star }\left( \frac{k}{k_{\star }}\right) ^{n_{h,\star }-n_{\zeta ,\star }+1} &{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle r_{\star }\left( \frac{k}{k_{*}}\right) ^{n_{h,\star }-n_{\zeta ,\star }+1+\frac{\alpha _{h,\star }-\alpha _{\zeta ,\star }}{2!}\ln \left( \frac{k}{k_{\star }}\right) +\cdots } &{} \quad \mathrm{for~~\underline{Case~~III},} \end{array} \right. &{} \quad \mathrm{for~BD} ,\\ \displaystyle \frac{qr_{\star }}{2c^{2}_{S}}\times \frac{\left| D_{1}H^{(1)}_{\mu }\left( -kc_{S}\eta \right) +D_{2}H^{(2)}_{\mu } \left( -kc_{S}\eta \right) \right| ^2}{\left| C_{1}H^{(1)}_{\nu }\left( -kc_{S}\eta \right) +C_{2}H^{(2)}_{\nu } \left( -kc_{S}\eta \right) \right| ^2}\, &{} \quad \mathrm{for~~AV}, \end{array} \right. \end{array} \end{aligned}$$

  (5.174)

  where $k_{*}$ is the pivot scale of momentum. One can also express Eq. (5.174) in terms of the number of e-foldings (N) as

  $$\begin{aligned} \begin{array}{lll} \displaystyle r(N)\displaystyle =\left\{ \begin{array}{ll}\left\{ \begin{array}{ll} \displaystyle r_{\star } &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle r_{\star }\exp \left[ (N-N_{\star })(n_{h,\star }-n_{\zeta ,\star }+1)\right] &{} \quad \mathrm{for~~\underline{Case~~II},} \\ \displaystyle r_{\star }\exp \left[ (N-N_{\star })\left\{ (n_{h,\star }-n_{\zeta ,\star }+1)\right. \right. \\ \left. \left. \displaystyle \displaystyle +\frac{\alpha _{h,\star }-\alpha _{\zeta ,\star }}{2!}(N-N_{\star })+\cdots \right\} \right] &{} \quad \mathrm{for~~\underline{Case~~III},} \end{array} \right. &{} \quad \mathrm{for~BD}, \\ \displaystyle \frac{qr_{\star }}{2c^{2}_{S}}\times \frac{\left| D_{1}H^{(1)}_{\mu }\left( -k_{\star }c_{S}\eta \exp \left[ N-N_{\star }\right] \right) +D_{2}H^{(2)}_{\mu } \left( -k_{\star }c_{S}\eta \exp \left[ N-N_{\star }\right] \right) \right| ^2}{\left| C_{1}H^{(1)}_{\nu }\left( -k_{\star }c_{S}\eta \exp \left[ N-N_{\star }\right] \right) +C_{2}H^{(2)}_{\nu } \left( -k_{\star }c_{S}\eta \exp \left[ N-N_{\star }\right] \right) \right| ^2}\, &{} \quad \mathrm{for~~AV} \end{array} \right. \end{array} \end{aligned}$$

  (5.175)

  where k and N are connected through the following expression: $\frac{k}{k_{\star }}= \exp \left[ (N-N_{\star })\right] .$ Here the three possibilities for the BD vacuum are:

  Case  I stands for a situation where the spectrum is scale invariant. This is a similar situation to the one considered in the case of the Lyth bound. This possibility also surmounts to the Harrison & Zeldovich spectrum, which is completely ruled out by Planck 2015+WMAP9 data within 5$\sigma$ C.L.

  Case  II stands for a situation where the spectrum shows a power-law feature through the spectral tilt $(n_{\zeta },n_{h})$. This possibility is also tightly constrained by the WMAP9 and Planck 2015+WMAP9 data within 2$\sigma$ C.L.

  Case  III signifies a situation where the spectrum shows deviation from power low in the presence of running of the spectral tilt $(\alpha _{\zeta },\alpha _{h})$ along with logarithmic correction in the momentum scale as appearing in the exponent. This possibility is favored by WMAP9 data and tightly constrained within 2$\sigma$ window by Planck+WMAP9 data.

• 4. For any value of q including $q=1/2$ we need to compute the following integral:

  

  (5.176)

  Similarly for AV we get the following result:

  $$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}\,\\ \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{M_{p}}{2c_{S}}\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}\ln \left( \frac{k_{\star }}{k_{\mathrm{end}}}\right) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{M_{p}\left| D\right| }{2c_{S}\left| C\right| }\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}\ln \left( \frac{k_{\star }}{k_{\mathrm{end}}}\right) &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~~AV}}}. \end{array} \end{aligned}$$

  (5.177)

  In terms of the number of e-foldings N one can re-express Eqs. (5.176) and (5.177) as

  $$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}\,\\ \approx \left\{ \begin{array}{ll} \displaystyle \sqrt{\frac{qr_{\star }}{4\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r_{\star }}{24}\right) ^{1/4}\left( N_{\star }-N_{\mathrm{end}}\right) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \frac{\sqrt{\frac{qr_{\star }}{4\alpha ^{\prime }}}\frac{M_{p}}{q}}{3(n_{h,\star }-n_{\zeta ,\star }+1)}\left[ \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \right] +2 \left( 1-\frac{r_{\star }}{24} \right) ^{1/4}\right\} \right. \\ \left. \displaystyle -e^{\frac{n_{h,\star }-n_{\zeta ,\star }+1}{2}(N_{\mathrm{end}}-N_{\star })} \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} e^{(n_{h,\star }-n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })}\right] \right. \right. \\ \left. \left. \displaystyle +2 \left( 1-\frac{r_{\star }}{24}e^{(n_{h,\star } -n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })} \right) ^{1/4}\right\} \right] &{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle \sqrt{\frac{\pi q r_{\star }}{\alpha ^{\prime }(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\frac{M_{p}}{48q} e^{-\frac{3 (n_{h,\star }-n_{\zeta ,\star }+1)^2}{4 (\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\\ \displaystyle ~~\left[ 12 e^{\frac{(n_{h,\star }-n_{\zeta ,\star }+1)^2}{ 2(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\left\{ \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}} \right) \right. \right. \\ \left. \left. \displaystyle - \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}{2}(N_{\mathrm{end}}-N_{\star }) \right) \right\} \right. \\ \left. \displaystyle -\frac{\sqrt{3}r_{\star }}{24} \left\{ \text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}\right) \right. \right. \\ \left. \left. \displaystyle -\text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{3(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}{2} (N_{\mathrm{end}}-N_{\star })\right) \right\} \right] &{} \quad \mathrm{for~~\underline{Case~~III}}, \end{array} \right.&{\underline{\mathrm{for~BD}}}, \end{array} \end{aligned}$$

  (5.178)
  $$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}\,\\ \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{M_{p}}{2c_{S}}\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{M_{p}\left| D\right| }{2c_{S}\left| C\right| }\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~~AV}}}. \end{array} \end{aligned}$$

  (5.179)

  Here the two possibilities for AV vacuum are:

  Case  I stands for a situation where the spectrum is characterized by the constraint i) $D_{1}=D_{2}=C_{1}=C_{2}\ne 0$, ii) $D_{1}=D_{2}$, $C_{1}=C_{2}=0$, iii) $D_{1}=D_{2}=0$, $C_{1}=C_{2}$.

  Case  II stands for a situation where the spectrum is characterized by the constraint i) $\mu \approx \nu$, $D_{1}=D_{2}=D\ne 0$ and $C_{1}=C_{2}=C\ne 0$, ii) $\mu \approx \nu$, $D_{1}=D\ne 0$, $D_{2}=0$ and $C_{1}=C\ne 0$, $C_{2}=0$, iii) $\mu \approx \nu$, $D_{2}=D\ne 0$, $D_{1}=0$ and $C_{2}=C\ne 0$, $C_{1}=0$.

• 5. Next we assume that the generic tachyonic potential V(T) can be expressed as

  $$\begin{aligned} V(T)= & {} V_{0}+\sum ^{\infty }_{n=1}\frac{1}{n!}\left( \frac{\mathrm{d}^{n}V(T)}{\mathrm{d}T^{n}}\right) _{T=T_{0}}\left( T-T_{0}\right) ^{n}, \end{aligned}$$

  (5.180)

  where the contribution from $V_{0}$ fix the scale of potential and the higher order Taylor expansion co-efficients characterize the shape of potential.

• 6. Further we need to compute the following integral:

  $$\begin{aligned} \displaystyle \int ^{T_{\star }}_{T_{\mathrm{end}}} \mathrm{d}T~\sqrt{V(T)}\approx & {} \int ^{T_{\star }}_{T_{\mathrm{end}}} \mathrm{d}T~\sqrt{V_{0}+\sum ^{\infty }_{n=1}\frac{1}{n!}\left( \frac{\mathrm{d}^{n}V(T)}{\mathrm{d}T^{n}}\right) _{T=T_{0}}\left( T-T_{0}\right) ^{n}}\nonumber \\\approx & {} \int ^{T_{\star }}_{T_{\mathrm{end}}} \mathrm{d}T~\sqrt{V_{0}}\left\{ 1+\sum ^{\infty }_{n=1}\frac{1}{2 n!V_{0}}\left( \frac{\mathrm{d}^{n}V(T)}{\mathrm{d}T^{n}}\right) _{T=T_{0}}\left( T-T_{0}\right) ^{n}+\cdots \right\} \nonumber \\\approx & {} \sqrt{V_{0}}\Delta T\left\{ 1+\sum ^{\infty }_{n=1}\frac{\left( \frac{\mathrm{d}^{n}V(T)}{\mathrm{d}T^{n}}\right) _{T=T_{0}}}{2(n+1) n!V_{0}}\frac{\left[ \left( T_{\star }-T_{0}\right) ^{n+1}- \left( T_{\mathrm{end}}-T_{0}\right) ^{n+1}\right] }{\Delta T}+\cdots \right\} \end{aligned}$$

  (5.181)

• 7. Next we assume that

  $$\begin{aligned} \frac{1}{V_{0}}\left( \frac{\mathrm{d}^{n}V(T)}{\mathrm{d}T^{n}}\right) _{T=T_{0}}\frac{\left[ \left( T_{\star }-T_{0}\right) ^{n+1}- \left( T_{\mathrm{end}}-T_{0}\right) ^{n+1}\right] }{\Delta T}<<1. \end{aligned}$$

  (5.182)

  Consequently from Eq. (5.181) we get

  $$\begin{aligned} \int ^{T_{\star }}_{T_{\mathrm{end}}} \mathrm{d}T~\sqrt{V(T)}\approx \sqrt{V_{0}}\Delta T \end{aligned}$$

  (5.183)

• 8. Finally using Eqs. (5.178), (5.179), (5.183) and (5.173) we get

  $$\begin{aligned}&\begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}}\, \approx \left\{ \begin{array}{ll} \displaystyle \sqrt{\frac{qr_{\star }}{4\alpha ^{\prime }V_{0}}}\frac{1}{2q}\left( 1-\frac{r_{\star }}{24}\right) ^{1/4}\left( N_{\star }-N_{\mathrm{end}}\right) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \frac{\sqrt{\frac{qr_{\star }}{4\alpha ^{\prime }V_{0}}}\frac{1}{q}}{3(n_{h,\star }-n_{\zeta ,\star }+1)}\left[ \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \right] +2 \left( 1-\frac{r_{\star }}{24} \right) ^{1/4}\right\} \right. \\ \left. \displaystyle -e^{\frac{n_{h,\star }-n_{\zeta ,\star }+1}{2}(N_{\mathrm{end}}-N_{\star })} \left\{ \,_2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} e^{(n_{h,\star }-n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })}\right] \right. \right. \\ \left. \left. \displaystyle +2 \left( 1-\frac{r_{\star }}{24}e^{(n_{h,\star } -n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })} \right) ^{1/4}\right\} \right] &{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle \sqrt{\frac{\pi q r_{\star }}{\alpha ^{\prime }(\alpha _{h,\star }-\alpha _{\zeta ,\star })V_{0}}}\frac{1}{48q} e^{-\frac{3 (n_{h,\star }-n_{\zeta ,\star }+1)^2}{4 (\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\\ \displaystyle ~~\left[ 12 e^{\frac{(n_{h,\star }-n_{\zeta ,\star }+1)^2}{ 2(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\left\{ \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}} \right) \right. \right. \\ \left. \left. \displaystyle - \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}{2}(N_{\mathrm{end}}-N_{\star }) \right) \right\} \right. \\ \left. \displaystyle -\frac{\sqrt{3}r_{\star }}{24} \left\{ \text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}\right) \right. \right. \\ \left. \left. \displaystyle -\text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{3(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}{2} (N_{\mathrm{end}}-N_{\star })\right) \right\} \right] &{} \quad \mathrm{for~~\underline{Case~~III}}, \end{array} \right. &{} {\underline{\mathrm{for~BD}}},\\ \end{array}\end{aligned}$$

  (5.184)
  $$\begin{aligned}&\begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}} \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }V_{0}}}\frac{1}{2c_{S}}\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }V_{0}}}\frac{\left| D\right| }{2c_{S}\left| C\right| }\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~~AV}}}. \end{array} \end{aligned}$$

  (5.185)

• 9. Next using Eq. (5.170) in Eqs. (5.184) and (5.185) we get

  $$\begin{aligned}&\begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}}\, \approx \left\{ \begin{array}{ll} \displaystyle \sqrt{\frac{qr_{\star }}{4\alpha ^{\prime }}}\frac{1}{2q\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\left( 1-\frac{r_{\star }}{24}\right) ^{1/4}\left( N_{\star }-N_{\mathrm{end}}\right) \\ \displaystyle \Bigg [1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 \right. \\ \left. \displaystyle +\left( 1+3\sqrt{2q}-6q\right) \frac{r_{\star }}{8}\right) \Bigg ]&{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \frac{\sqrt{\frac{qr_{\star }}{4\alpha ^{\prime }}}\frac{1}{q\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}}{3(n_{h,\star }-n_{\zeta ,\star }+1)}\left[ \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \right] +2 \left( 1-\frac{r_{\star }}{24} \right) ^{1/4}\right\} \right. \\ \left. \displaystyle -e^{\frac{n_{h,\star }-n_{\zeta ,\star }+1}{2}(N_{\mathrm{end}}-N_{\star })} \left\{ \,_2F_1\Bigg [\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} e^{(n_{h,\star }-n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })}\right] \right. \\ \left. \displaystyle +2 \left( 1-\frac{r_{\star }}{24}e^{(n_{h,\star } -n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })} \right) ^{1/4}\right\} \Bigg ]\\ \displaystyle \Bigg [1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 \right. \\ \left. \displaystyle +\left( 1+3\sqrt{2q}-6q\right) \frac{r_{\star }}{8}\right) \Bigg ]&{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle \frac{ \sqrt{\frac{\pi q r_{\star }}{\alpha ^{\prime }(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}}{48q\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}} e^{-\frac{3 (n_{h,\star }-n_{\zeta ,\star }+1)^2}{4 (\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\\ \displaystyle ~~\left[ 12 e^{\frac{(n_{h,\star }-n_{\zeta ,\star }+1)^2}{ 2(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\left\{ \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}} \right) \right. \right. \\ \left. \left. \displaystyle - \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}{2}(N_{\mathrm{end}}-N_{\star }) \right) \right\} \right. \\ \left. \displaystyle -\frac{\sqrt{3}r_{\star }}{24} \left\{ \text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}\right) \right. \right. \\ \left. \left. \displaystyle -\text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{3(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}{2} (N_{\mathrm{end}}-N_{\star })\right) \right\} \right] \\ \displaystyle \Bigg [1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 \right. \\ \left. \displaystyle +\left( 1+3\sqrt{2q}-6q\right) \frac{r_{\star }}{8}\right) \Bigg ]&{} \quad \mathrm{for~~\underline{Case~~III},} \end{array} \right.&{\underline{\mathrm{for~BD}}}, \end{array}\end{aligned}$$

  (5.186)
  $$\begin{aligned}&\begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}} \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2c_{S}\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 +\left( 1+3\sqrt{2q}-6q\right) \frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{\left| D\right| \left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2c_{S}\left| C\right| \sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 +\left( 1+3\sqrt{2q}-6q\right) \frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~AV}}}. \end{array} \end{aligned}$$

  (5.187)

Further using the approximated form of the sound speed $c_{S}$ the expression for the field excursion for AV can be rewritten as

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}} \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{\left( 1-\frac{qr_{\star }}{48\left[ 1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}\right] }\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2\sqrt{1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}}\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 +\left( 1+3\sqrt{2q}-6q\right) \frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8\alpha ^{\prime }}}\frac{\left| D\right| \left( 1-\frac{qr_{\star }}{48\left[ 1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}\right] }\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2\sqrt{1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}}\left| C\right| \sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 +\left( 1+3\sqrt{2q}-6q\right) \frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~AV}}}. \end{array} \end{aligned}$$

(5.188)

5.1.8 Semi-analytical study and cosmological parameter estimation

In this subsection our prime objective are:

• To compute various inflationary observables from variants of tachyonic single field potentials as mentioned earlier in this paper.

• To estimate the relevant cosmological parameters from the proposed models.

• Next to compare the effectiveness of all of these models in the light of recent Planck 2015 data alongwith other combined constraints.

• Finally to check the compatibility of all of these models with the CMB TT, TE and EE angular power spectra as observed by Planck 2015.

Model I: inverse cosh potential

For the single field case the first model of the tachyonic potential is given by

$$\begin{aligned} V(T)=\frac{\lambda }{\mathrm{cosh}\left( \frac{T}{T_{0}}\right) }, \end{aligned}$$

(5.189)

where $\lambda$ characterizes the scale of inflation and $T_{0}$ is the parameter of the model. In Fig. 1 we have depicted the symmetric behavior of the inverse cosh potential with respect to scaled field coordinate $T/T_{0}$ in dimensionless units around the origin fixed at $T/T_{0}=0$. In this case the tachyon field started rolling down from the top height of the potential and takes part in the inflationary dynamics.

Fig. 1

Variation of the inverse cosh potential $V(T)/\lambda$ with field $T/T_{0}$ in dimensionless units

Next using specified form of the potential the potential dependent slow-roll parameters are computed as

$$\begin{aligned}&\bar{\epsilon }_{V}=\frac{1}{2g}\frac{\sinh ^2\left( \frac{T}{T_{0}}\right) }{\cosh \left( \frac{T}{T_{0}}\right) },\end{aligned}$$

(5.190)

$$\begin{aligned}&\bar{\eta }_{V}=\frac{1}{g}\left[ \frac{\sinh ^2\left( \frac{T}{T_{0}}\right) }{\cosh \left( \frac{T}{T_{0}}\right) }-\mathrm{sech}\left( \frac{T}{T_{0}}\right) \right] ,\end{aligned}$$

(5.191)

$$\begin{aligned}&\bar{\xi }^2_{V}=\frac{1}{g^2}\frac{\sinh ^2\left( \frac{T}{T_{0}}\right) }{\cosh \left( \frac{T}{T_{0}}\right) }\left[ \frac{\sinh ^2\left( \frac{T}{T_{0}}\right) }{\cosh \left( \frac{T}{T_{0}}\right) }-5~\mathrm{sech}\left( \frac{T}{T_{0}}\right) \right] ,\end{aligned}$$

(5.192)

$$\begin{aligned}&\bar{\sigma }^3_{V}=\frac{1}{g^3}\frac{\sinh ^2\left( \frac{T}{T_{0}}\right) }{\cosh \left( \frac{T}{T_{0}}\right) }\left[ \frac{\sinh ^2\left( \frac{T}{T_{0}}\right) }{\cosh \left( \frac{T}{T_{0}}\right) }\right. \nonumber \\&\quad \times \left. \left\{ \frac{\sinh ^2\left( \frac{T}{T_{0}}\right) }{\cosh \left( \frac{T}{T_{0}}\right) }-18~\mathrm{sech}\left( \frac{T}{T_{0}}\right) \right\} +5~\mathrm{sech}^2\left( \frac{T}{T_{0}}\right) \right] , \end{aligned}$$

(5.193)

where the factor g is defined as

$$\begin{aligned} g= \frac{\alpha ^{\prime }\lambda T^2_0}{M^2_p}=\frac{M^4_s}{(2\pi )^3 g_s}\frac{\alpha ^{\prime } T^2_0}{M^2_p}. \end{aligned}$$

(5.194)

Next we compute the number of e-foldings from this model:

$$\begin{aligned} \begin{array}{lll} \displaystyle N(T)=\left\{ \begin{array}{lll} \displaystyle g~\ln \left[ \frac{\tanh \left( \frac{T_{\mathrm{end}}}{2T_{0}}\right) }{\tanh \left( \frac{T}{2T_{0}}\right) }\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{2q}~g~\ln \left[ \frac{\tanh \left( \frac{T_{\mathrm{end}}}{2T_{0}}\right) }{\tanh \left( \frac{T}{2T_{0}}\right) }\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.195)

Further using the condition to end inflation:

$$\begin{aligned} \bar{\epsilon }_{V}(T_{\mathrm{end}})=1,\end{aligned}$$

(5.196)

$$\begin{aligned} |\bar{\eta }_{V}(T_{\mathrm{end}})|=1, \end{aligned}$$

(5.197)

we get the following field value at the end of inflation:

$$\begin{aligned} T_{\mathrm{end}}=T_{0}~\mathrm{sech}^{-1}(g). \end{aligned}$$

(5.198)

Next using $N=N_{\mathrm{cmb}}=N_{\star }$ and $T=T_{\mathrm{cmb}}=T_{\star }$ at the horizon crossing we get

$$\begin{aligned} \begin{array}{lll} \displaystyle T_{\star }=2T_{0}\times \left\{ \begin{array}{lll} \displaystyle \tanh ^{-1}\left[ \exp \left( -\frac{N_{\star }}{g}\right) \tanh \left( \frac{1}{2}\mathrm{sech}^{-1}(g)\right) \right] , &{} \quad \mathrm{for}~~~q=1/2\\ \displaystyle \tanh ^{-1}\left[ \exp \left( -\frac{N_{\star }}{\sqrt{2q}g}\right) \tanh \left( \frac{1}{2}\mathrm{sech}^{-1}(g)\right) \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.199)

Consequently the field excursion can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle |\Delta T|=T_{0}\times \left\{ \begin{array}{lll} \displaystyle \left| 2\tanh ^{-1}\left[ \exp \left( -\frac{N_{\star }}{g}\right) \tanh \left( \frac{1}{2}\mathrm{sech}^{-1}(g)\right) \right] -\mathrm{sech}^{-1}(g)\right| , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left| 2\tanh ^{-1}\left[ \exp \left( -\frac{N_{\star }}{\sqrt{2q}g}\right) \tanh \left( \frac{1}{2}\mathrm{sech}^{-1}(g)\right) \right] -\mathrm{sech}^{-1}(g)\right| ,&{} \quad \mathrm{for ~any~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.200)

In the slow-roll regime of inflation the following approximations hold good:

$$\begin{aligned}&\begin{array}{lll} \displaystyle \cosh \left( \frac{T_{\star }}{T_{0}}\right) \approx \left\{ \begin{array}{lll} \displaystyle \frac{1}{\tanh \left( \frac{N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\tanh \left( \frac{N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.201)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \sinh \left( \frac{T_{\star }}{T_{0}}\right) \approx \left\{ \begin{array}{lll} \displaystyle \frac{1}{\sinh \left( \frac{N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\sinh \left( \frac{N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.202)

Using Eqs. (5.201) and (5.202) in the definition of potential dependent slow-roll parameter finally we compute the following inflationary observables:

$$\begin{aligned}&\begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{g\lambda }{12\pi ^2M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \sinh ^2\left( \frac{N_{\star }}{g}\right) , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 2q~\sinh ^2\left( \frac{N_{\star }}{\sqrt{2q}g}\right) ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.203)

$$\begin{aligned}&\begin{array}{lll} \displaystyle n_{\zeta ,\star }-1\approx -\frac{2}{g}\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{\tanh \left( \frac{N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\sqrt{2q}\tanh \left( \frac{N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.204)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \alpha _{\zeta ,\star }\approx -\frac{2}{g^2}\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{\sinh ^2\left( \frac{N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q\sinh ^2\left( \frac{N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.205)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \kappa _{\zeta ,\star }\approx -\frac{4}{g^3}\times \left\{ \begin{array}{lll} \displaystyle \frac{\cosh \left( \frac{N_{\star }}{g}\right) }{\sinh ^3\left( \frac{N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{\cosh \left( \frac{N_{\star }}{2qg}\right) }{(2q)^{3/2}\sinh ^3\left( \frac{N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.206)

$$\begin{aligned}&\begin{array}{lll} \displaystyle r_{\star }\approx \frac{16}{g}\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{\sinh \left( \frac{2N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q\sinh \left( \frac{2N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.207)

For the inverse cosh potential we get the following consistency relations:

$$\begin{aligned} \displaystyle r_{\star }\approx & {} 4(1-n_{\zeta ,\star })\nonumber \\&\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{\cosh ^2\left( \frac{N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{\sqrt{2q}}{\cosh ^2\left( \frac{N_{\star }}{2qg}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{aligned}$$

(5.208)

$$\begin{aligned} \Delta _{\zeta ,\star }\approx & {} \frac{\lambda }{12\pi ^2M^{4}_{p}(1-n_{\zeta ,\star })}\nonumber \\&\times \left\{ \begin{array}{lll} \displaystyle \sinh \left( \frac{2N_{\star }}{g}\right) , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{2q}\sinh \left( \frac{2N_{\star }}{\sqrt{2q}g}\right) ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{aligned}$$

(5.209)

$$\begin{aligned} \displaystyle \Delta _{\zeta ,\star }\approx & {} \frac{2\lambda }{3\pi ^2M^{4}_{p}r_{\star }}\nonumber \\&\times \left\{ \begin{array}{lll} \displaystyle \tanh \left( \frac{N_{\star }}{g}\right) , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \tanh \left( \frac{N_{\star }}{\sqrt{2q}g}\right) ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{aligned}$$

(5.210)

$$\begin{aligned} \displaystyle \alpha _{\zeta ,\star }\approx & {} \frac{2}{g}\times \left( n_{\zeta ,\star }-1\right) \nonumber \\&\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{ \sinh \left( \frac{2N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\sqrt{2q}\sinh \left( \frac{2N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{aligned}$$

(5.211)

$$\begin{aligned} \displaystyle \kappa _{\zeta ,\star }\approx & {} \frac{2}{g^2}\times \left( n_{\zeta ,\star }-1\right) \nonumber \\&\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{ \sinh ^2\left( \frac{N_{\star }}{g}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q\sinh ^2\left( \frac{N_{\star }}{\sqrt{2q}g}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{aligned}$$

(5.212)

Fig. 2

Behavior of the tensor-to-scalar ratio r with respect to a the scalar spectral index $n_{\zeta }$ and b the parameter g for the inverse cosh potential. The purple and blue colored line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The cyan color shaded region bounded by two vertical black colored lines in a represents the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the inverse cosh potential is favored only for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. In b, we have explicitly shown that in the r–g plane the observationally favored lower bound for the characteristic index is $q\ge 1/2$

Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor q and on the parameters appearing in the expression for the inverse cosh potential. In Fig. 2a, b, we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index $n_{\zeta }$ and the model parameter g for the inverse cosh potential respectively. In both the figures the purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in Fig. 2a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck 2015 data and Planck+ BICEP2+Keck Array joint constraint. The rest of the region is completely ruled out by the present observational constraints. From Fig. 2a, b, it is also observed that, within $50<N_{\star }<70$, the inverse cosh potential is favored only for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Also in Fig. 2a for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/g\sim 0.8$. This implies that for $50<N_{\star }<70$, the prescribed window for g from the r–$n_{\zeta }$ plot is given by $63<g<88$. In Fig. 2b, we have explicitly shown that the in r–g plane the observationally favored lower bound for the characteristic index is $q\ge 1/2$. It is additionally important to note that, for $q>>2$, the tensor-to-scalar ratio computed from the model is negligibly small for the inverse cosh potential. This implies that if the inflationary tensor mode is detected close to its present upper bound on tensor-to-scalar ratio then all $q>>2$ possibilities for tachyonic inflation can be discarded for the inverse cosh potential. On the contrary, if inflationary tensor modes are never detected by any of the future observational probes then the $q>>2$ possibilities for tachyonic inflation in the case of the inverse cosh potential are highly prominent. Also it is important to mention that, in Fig. 2b within the window $0<g<100$, if we take a smaller value of g, then the inflationary tensor-to-scalar ratio also gradually decreases. To analyze the results more clearly let us describe the cosmological features from Fig. 2a in detail. Let us first start with the $q=1/2$ situation, in which the $2\sigma$ constraint on the scalar spectral tilt is satisfied within the window of tensor-to-scalar ratio, $0.015<r<0.025$ for $50<N_{\star }<70$. Next for the $q=1$ case, the same constraint is satisfied within the window of the tensor-to-scalar ratio, $0.050<r<0.075$ for $50<N_{\star }<70$. Further for the $q=3/2$ case, the same constraint on scalar spectral tilt is satisfied within the window of the tensor-to-scalar ratio, $0.090<r<0.12$ for $50<N_{\star }<70$. Finally, for the $q=2$ situation, the value for the tensor-to-scalar ratio is $r<0.12$, which is tightly constrained from the upper bound of the spectral tilt from Planck 2015 observational data.

Fig. 3

Variation of the a scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$, b scalar power spectrum $\Delta _{\zeta }$ vs. the stringy parameter g and c scalar spectral tilt $n_{\zeta }$ vs. the stringy parameter g. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint

In Fig. 3a–c, we have depicted the behavior of the scalar power spectrum $\Delta _{\zeta }$ vs. the stringy parameter g, scalar spectral tilt $n_{\zeta }$ vs. the stringy parameter g and scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$ for the inverse cosh potential respectively. It is important to note that, for all of the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The cyan color shaded region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is the $1\sigma$ region, which is presently disfavored by the joint Planck+WMAP constraints. The rest of the region is completely ruled out by the present observational constraints. From Fig. 3a it is clearly observed that the observational constraints on the amplitude of the scalar mode fluctuations satisfy within the window $80<g<100$. For $g>100$ the corresponding amplitude falls down in a non-trivial fashion by following the exact functional form as stated in Eq. (5.203). Next using the behavior as shown in Fig. 3b, the lower bound on the stringy parameter g is constrained by $g>50$ by using the non-trivial relationship as stated in Eq. (5.204). But at this lower bound of the parameter g the amplitude of the scalar power spectrum is larger compared to the present observational constraints. This implies that, to satisfy both the constraints from the amplitude of the scalar power spectrum and its spectral tilt, within $2\sigma$ CL the constrained numerical value of the stringy parameter is lying within the window $80<g<100$. Also in Fig. 3c for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/g\sim 0.8$, which further implies that for $50<N_{\star }<70$, the prescribed window for g from $\Delta _{\zeta }$–$n_{\zeta }$ plot is given by $63<g<88$. If we additionally impose the constraint from the upper bound on tensor-to-scalar ratio then also the allowed parameter range is lying within a similar window, i.e., $88<g<100$.

In Fig. 4a, b, we have shown the behavior of the running of the scalar spectral tilt $\alpha _{\zeta }$ and running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for the inverse cosh potential with $g=88$, respectively. For both the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in both the plots represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From both of these figures, it is also observed that, within $50<N_{\star }<70$, the inverse cosh potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. From Fig. 4a, b, it is observed that within the $2\sigma$ observed range of the scalar spectral tilt $n_{\zeta }$, as the value of the characteristic parameter q increases, the value of the running $\alpha _{\zeta }$ and running of the running $\kappa _{\zeta }$ decreases for the inverse cosh potential. It is also important to note that for $1/2<q<2$, the numerical value of the running $\alpha _{\zeta }\sim \mathcal{O}(-10^{-4})$ and running of the running $\kappa _{\zeta }\sim \mathcal{O}(-10^{-6})$, which are perfectly consistent with the $1.5\sigma$ constraints on running and running of the running as obtained from Planck 2015 data.

Model  II: logarithmic potential

For single field case the second model of tachyonic potential is given by

$$\begin{aligned} V(T)=\lambda \left\{ \left( \frac{T}{T_{0}}\right) ^2\left[ \ln \left( \frac{T}{T_{0}}\right) \right] ^2+1\right\} , \end{aligned}$$

(5.213)

where $\lambda$ characterize the scale of inflation and $T_{0}$ is the parameter of the model. In Fig. 5 we have depicted the behavior of the logarithmic potential with respect to scaled field coordinate $T/T_{0}$ in dimensionless units. In this case the tachyon field started rolling down from the top height of the potential either from the left or right hand side and takes part in the inflationary dynamics.

Fig. 4

Behavior of the a running of the scalar spectral tilt $\alpha _{\zeta }$ and b running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for the inverse cosh potential with $g=88$. For both figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The cyan color shaded region bounded by two vertical black colored lines in a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the inverse cosh potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis

Fig. 5

Variation of the logarithmic potential $V(T)/\lambda$ with field $T/T_{0}$ in dimensionless units

Next using specified form of the potential the potential dependent slow-roll parameters are computed as

$$\begin{aligned} \bar{\epsilon }_{V}= & {} \frac{1}{2g}\frac{4\left( \frac{T}{T_{0}}\right) ^2\ln ^2\left( \frac{T}{T_{0}}\right) \left[ 1+\ln \left( \frac{T}{T_{0}}\right) \right] ^2}{\left[ 1 +\left( \frac{T}{T_{0}}\right) ^2\ln ^2\left( \frac{T}{T_{0}}\right) \right] ^3},\end{aligned}$$

(5.214)

$$\begin{aligned} \bar{\eta }_{V}= & {} \frac{1}{g}\frac{2\left\{ \ln \left( \frac{T}{T_{0}}\right) +\left[ 1+\ln \left( \frac{T}{T_{0}}\right) \right] ^2\right\} }{\left[ 1 +\left( \frac{T}{T_{0}}\right) ^2\ln ^2\left( \frac{T}{T_{0}}\right) \right] ^2},\end{aligned}$$

(5.215)

$$\begin{aligned} \bar{\xi }^2_{V}= & {} \frac{1}{g^2}\frac{4\ln \left( \frac{T}{T_{0}}\right) \left[ 1+\ln \left( \frac{T}{T_{0}}\right) \right] \left[ 3+2\ln \left( \frac{T}{T_{0}}\right) \right] }{\left[ 1 +\left( \frac{T}{T_{0}}\right) ^2\ln ^2\left( \frac{T}{T_{0}}\right) \right] ^4}, \end{aligned}$$

(5.216)

$$\begin{aligned} \bar{\sigma }^3_{V}= & {} -\frac{1}{g^3}\frac{4\ln ^2\left( \frac{T}{T_{0}}\right) \left[ 1+\ln \left( \frac{T}{T_{0}}\right) \right] ^2\left[ 1+2\ln \left( \frac{T}{T_{0}}\right) \right] }{\left[ 1 +\left( \frac{T}{T_{0}}\right) ^2\ln ^2\left( \frac{T}{T_{0}}\right) \right] ^6},\nonumber \\ \end{aligned}$$

(5.217)

where the factor g is defined as

$$\begin{aligned} g= \frac{\alpha ^{\prime }\lambda T^2_0}{M^2_p}=\frac{M^4_s}{(2\pi )^3 g_s}\frac{\alpha ^{\prime } T^2_0}{M^2_p}. \end{aligned}$$

(5.218)

Next we compute the number of e-foldings from this model:

$$\begin{aligned} \begin{array}{lll} \displaystyle N(T)=\left\{ \begin{array}{ll} \displaystyle \frac{g}{2}~\left\{ -\frac{2 \text {Ei}\left[ 2 \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) \right] }{e^2}-\frac{\text {Ei}\left[ 4 \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) \right] }{e^4}\right. \\ \left. \displaystyle +\left( \frac{T}{T_{0}}\right) ^4\left[ \frac{11 }{32}-\frac{3}{8} \ln \left( \frac{T}{T_{0}}\right) +\frac{1}{4} \ln ^2\left( \frac{T}{T_{0}}\right) \right] \right. \\ \left. \displaystyle +\left( \frac{T}{T_{0}}\right) ^2+\ln \left( \frac{\ln \left( \frac{T}{T_{0}}\right) }{1+\ln \left( \frac{T}{T_{0}}\right) }\right) \right\} ^{T}_{T_{\mathrm{end}}}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{2q}\frac{g}{2}~\left\{ -\frac{2 \text {Ei}\left[ 2 \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) \right] }{e^2}\right. \\ \left. \displaystyle -\frac{\text {Ei}\left[ 4 \left( \ln \left( \frac{T}{T_{0}}\right) +1\right) \right] }{e^4}\right. \\ \left. \displaystyle +\left( \frac{T}{T_{0}}\right) ^4\left[ \frac{11 }{32}-\frac{3}{8} \ln \left( \frac{T}{T_{0}}\right) +\frac{1}{4} \ln ^2\left( \frac{T}{T_{0}}\right) \right] \right. \\ \left. \displaystyle +\left( \frac{T}{T_{0}}\right) ^2+\ln \left( \frac{\ln \left( \frac{T}{T_{0}}\right) }{1+\ln \left( \frac{T}{T_{0}}\right) }\right) \right\} ^{T}_{T_{\mathrm{end}}},&{}\mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.219)

Further using the condition to end inflation:

$$\begin{aligned} \bar{\epsilon }_{V}(T_{\mathrm{end}})=1,\end{aligned}$$

(5.220)

$$\begin{aligned} |\bar{\eta }_{V}(T_{\mathrm{end}})|=1, \end{aligned}$$

(5.221)

we get the following transcendental equation:

$$\begin{aligned}&\left[ 1+\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) \right] ^2\left\{ \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\ln ^2 \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) -1\right\} \nonumber \\&\quad =\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) , \end{aligned}$$

(5.222)

from which we get the following sets of real solutions for the field value:

$$\begin{aligned} T_{\mathrm{end}}= & {} \left( 0.07~T_{0},~ 0.69~T_{0},~1.83~T_{0}\right) . \end{aligned}$$

(5.223)

Then using this result we need to numerically solve the transcendental equation of $T_{\star }$ which involves $N_{\star }$ explicitly. However, in the slow-roll regime of inflation we get the following simplified expression for the field value $T_{\star }$ in terms of $N_{\star }$, $T_{\mathrm{end}}$ and $T_{0}$:

$$\begin{aligned} \begin{array}{lll} \displaystyle T_{\star }\approx T_{0}\times \left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.224)

where we have explicitly used the fact that in the slow-roll regime the quadratic term gives the dominant contribution in $N_{\star }$.

Also the field excursion can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle |\Delta T|=T_{0}\times \left\{ \begin{array}{lll} \displaystyle \left| \sqrt{\frac{2N_{\star }}{g}+c^2}-c\right| , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left| \sqrt{\frac{2N_{\star }}{\sqrt{2q}g}+c^2}-c\right| ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.225)

where $c=0.07,0.69,1.83$. During the numerical estimation we have taken $c=0.07$ as it is compatible with the observational constraints from Planck 2015 data.

Finally using the previously mentioned definition of potential dependent slow-roll parameter we compute the following inflationary observables:

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{g\lambda }{48\pi ^2 M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \frac{\left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^4}{\left( \frac{2N_{\star }}{g} +\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{2q\left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^4}{\left( \frac{2N_{\star }}{\sqrt{2q}g} +\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}. &{} \quad \mathrm{for ~any~} q \end{array} \right. \end{array} \end{aligned}$$

(5.226)

$$\begin{aligned} \begin{array}{lll} \displaystyle n_{\zeta ,\star }-1\approx \frac{4}{g}\times \left\{ \begin{array}{lll} \displaystyle \frac{ \vartheta _{\star }}{\left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^3}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \displaystyle \frac{ \vartheta _{\star }}{\left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^3}. &{} \quad \mathrm{for ~any~} q \end{array} \right. \end{array} \end{aligned}$$

(5.227)

$$\begin{aligned} \begin{array}{lll} \displaystyle r_{\star }\approx \frac{32}{g}\times \left\{ \begin{array}{lll} \displaystyle \frac{\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}{\left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^3}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q}\frac{\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2} \ln \left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}{\left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^3},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.228)

where

$$\begin{aligned} \begin{array}{lll} \displaystyle \vartheta _{\star }\approx \left\{ \begin{array}{lll} \displaystyle \left[ 1-\frac{3}{4}\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2 \right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{g} +\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2\\ \quad \displaystyle +\frac{1}{2}\ln \left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left[ 1-\frac{G}{4}\left( \frac{N_{\star }}{qg} \right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2\\ \quad \displaystyle +\frac{P}{2}\ln \left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.229)

Here the constants G and P are defined as

$$\begin{aligned} G= & {} \frac{1}{2q}\left( 1+2\sqrt{2q}\right) ,\end{aligned}$$

(5.230)

$$\begin{aligned} P= & {} \frac{1}{\sqrt{2q}}. \end{aligned}$$

(5.231)

For the inverse cosh potential we get the following consistency relations:

$$\begin{aligned} \begin{array}{lll} \displaystyle r_{\star }\approx 8(1-n_{\zeta ,\star })\times \left\{ \begin{array}{lll} \displaystyle \frac{\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{g} +\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}{ |\vartheta _{\star }|}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}{2q |\vartheta _{\star }|},&{} \quad \mathrm{for ~any~} q \end{array} \right. \end{array} \end{aligned}$$

(5.232)

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{\lambda }{12\pi ^2 M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \frac{ |\vartheta _{\star }|}{(n_{\zeta ,\star }-1)\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2 \right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}, &{} \mathrm{for}~q=1/2, \\ \displaystyle \frac{2q |\vartheta _{\star }|}{(n_{\zeta ,\star }-1)\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2 \right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \left[ 1+\frac{1}{2}\ln \left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ^2}. &{} \mathrm{for ~any~} q \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.233)

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{2\lambda }{3\pi ^2M^{4}_{p}r_{\star }}\times \left\{ \begin{array}{lll} \displaystyle \left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left[ 1 +\frac{1}{4}\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \ln ^2\left( \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\right) \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.234)

Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor q and on the paramters appearing in the expression for the logarithmic potential. In Fig. 6a, b, we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index $n_{\zeta }$ and the model parameter g for the logarithmic potential respectively. In both the figures the purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in Fig. 6a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck 2015 data and Planck+ BICEP2+Keck Array joint constraint. The rest of the region is completely ruled out by the present observational constraints. From Fig. 6a, b, it is also observed that, within $50<N_{\star }<70$, the logarithmic potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Also in Fig. 6a for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/g\sim 0.7$. This implies that for $50<N_{\star }<70$, the prescribed window for g from r–$n_{\zeta }$ plot is givenby $71.4<g<100$. In Fig. 6b, we have explicitly shown that the in $r-g$ plane the observationally favored lower bound for the characteristic index is $q\ge 1/2$. It is additionally important to note that, for $q>>2$, the tensor-to-scalar ratio computed from the model is negligibly small for the logarithmic potential. This implies that if the inflationary tensor mode is detected close to its present upper bound on tensor-to-scalar ratio then all $q>>2$ possibilities for tachyonic inflation can be discarded for the logarithmic potential. On the

Fig. 6

Behavior of the tensor-to-scalar ratio r with respect to a the scalar spectral index $n_{\zeta }$ and b the parameter g for the logarithmic potential. The purple and blue colored line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the logarithmic potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. In a, we have explicitly shown that in the r–$n_{\zeta }$ plane the observationally favored lower bound for the characteristic index is $q\ge 1/2$

contrary, if inflationary tensor modes are never detected by any of the future observational probes then the $q>>2$ possibilities for tachyonic inflation in the case of a logarithmic potential are highly prominent. Also it is important to mention that, in Fig. 6b within the window $40<g<150$, if we increase the value of g, then the inflationary tensor-to-scalar ratio also gradually decreases. After that within $71<g<300$ the value of the tensor-to-scalar ratio slightly increases and again it falls down to lower value within the the interval $120<g<1000$.

Fig. 7

Variation of the a scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$, b scalar power spectrum $\Delta _{\zeta }$ vs. the stringy parameter g and c scalar spectral tilt $n_{\zeta }$ vs. the stringy parameter g. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint

In Fig. 7a–c, we have depicted the behavior of the scalar power spectrum $\Delta _{\zeta }$ vs. the stringy parameter g, scalar spectral tilt $n_{\zeta }$ vs. the stringy parameter g and scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$ for the logarithmic potential respectively. It is important to note that, for all of the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The cyan color shaded region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is the $1\sigma$ region, which is presently disfavored by the joint Planck+WMAP constraints. The rest of the region is completely ruled out by the present observational constraints. From Fig. 7a it is clearly observed that the observational constraints on the amplitude of the scalar mode fluctuations satisfy within the window $g=(26,90)$ for $q=1/2$, $g=(20,60)$ for $q=1$, $g=(18,48)$ for $q=3/2$ and $g=(16,38)$ for $q=2$. For all the cases outside the mentioned window for the parameter g the corresponding amplitude gradually increases in a non-trivial fashion by following the exact functional form as stated in Eq. (5.226). Next using the behavior as shown in Fig. 7b, the lower bound on the stringy parameter g is constrained by $g>45$ by using the non-trivial relationship as stated in Eq. (5.227) for $q=2$. Similarly by observing the Fig. 7b one can find the other lower bounds on g from different value of q. But at this lower bound of the parameter g the amplitude of the scalar power spectrum is very larger compared to the present observational constraints. This implies that, to satisfy both the constraints from the amplitude of the scalar power spectrum and its spectral tilt within $2\sigma$ CL the constrained numerical value of the stringy parameter is lying within the window $48<g<90$ in which $q=2$ case is slightly disfavored compared to the other lower values of q studied in the present context. Also in Fig. 7c for $q=1/2$, and $q=1$ we fix $N_{\star }/g\sim 0.7$, which further implies that for $50<N_{\star }<70$, the prescribed window for g from the $\Delta _{\zeta }$–$n_{\zeta }$ plot is given by $71.4<g<100$. If we additionally impose the constraint from the upper bound on tensor-to-scalar ratio then also the allowed parameter range is lying within a similar window, i.e., $71.4<g<90$ and combining all the constraints the allowed value of the characteristic index is lying within $1/2<q<1$. However, $q=1/2$ is tightly constrained as depicted in Fig. 7c.

Model  III:  exponential  potential Type  I

For the single field case the third model of the tachyonic potential is given by

$$\begin{aligned} V(T)=\lambda \exp \left( -\frac{T}{T_{0}}\right) , \end{aligned}$$

(5.235)

where $\lambda$ characterizes the scale of inflation and $T_{0}$ is the parameter of the model. In Fig. 8 we have depicted the behavior of the exponential potential Type-I with respect to the scaled field coordinate $T/T_{0}$ in dimensionless units. In this case the tachyon field started rolling down from the top height of the potential from the left hand side and takes part in the inflationary dynamics.

Fig. 8

Variation of the exponential potential Type-I $V(T)/\lambda$ with field $T/T_{0}$ in dimensionless units

Next using specified form of the potential the potential dependent slow-roll parameters are computed as

$$\begin{aligned} \bar{\epsilon }_{V}= & {} \frac{1}{2g}\exp \left( \frac{T}{T_{0}}\right) ,\end{aligned}$$

(5.236)

$$\begin{aligned} \bar{\eta }_{V}= & {} \frac{1}{g}\exp \left( \frac{T}{T_{0}}\right) ,\end{aligned}$$

(5.237)

$$\begin{aligned} \bar{\xi }^2_{V}= & {} \frac{1}{g^2}\exp \left( \frac{2T}{T_{0}}\right) , \end{aligned}$$

(5.238)

$$\begin{aligned} \bar{\sigma }^3_{V}= & {} \frac{1}{g^3}\exp \left( \frac{3T}{T_{0}}\right) , \end{aligned}$$

(5.239)

where the factor g is defined as

$$\begin{aligned} g= \frac{\alpha ^{\prime }\lambda T^2_0}{M^2_p}=\frac{M^4_s}{(2\pi )^3 g_s}\frac{\alpha ^{\prime } T^2_0}{M^2_p}. \end{aligned}$$

(5.240)

Next we compute the number of e-foldings from this model:

$$\begin{aligned} \begin{array}{lll} \displaystyle N(T)=\left\{ \begin{array}{lll} \displaystyle g~\left[ \exp \left( -\frac{T}{T_{0}}\right) -\exp \left( -\frac{T_{\mathrm{end}}}{T_{0}}\right) \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{2q}g~\left[ \exp \left( -\frac{T}{T_{0}}\right) -\exp \left( -\frac{T_{\mathrm{end}}}{T_{0}}\right) \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.241)

Further using the condition to end inflation:

$$\begin{aligned} \bar{\epsilon }_{V}(T_{\mathrm{end}})=1, \end{aligned}$$

(5.242)

we get the following field value at the end of inflation:

$$\begin{aligned} T_{\mathrm{end}}=T_{0}~\ln (2g). \end{aligned}$$

(5.243)

Next using $N=N_{\mathrm{cmb}}=N_{\star }$ and $T=T_{\mathrm{cmb}}=T_{\star }$ at the horizon crossing we get

$$\begin{aligned} \begin{array}{lll} \displaystyle T_{\star }=T_{0}\times \left\{ \begin{array}{lll} \displaystyle \ln \left[ \frac{g}{\frac{1}{2}+N_{\star }}\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \ln \left[ \frac{g}{\frac{1}{2}+\frac{N_{\star }}{\sqrt{2q}}}\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.244)

Also the field excursion can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle |\Delta T|=T_{0}\times \left\{ \begin{array}{lll} \displaystyle \left| \ln \left[ \frac{1}{1+2N_{\star }}\right] \right| , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left| \ln \left[ \frac{1}{1+\frac{2N_{\star }}{\sqrt{2q}}}\right] \right| ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.245)

Fig. 9

In a the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index $n_{\zeta }$ and b the parameter g for exponential potential Type-I. The purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. In a, we have explicitly shown that in $r-n_{\zeta }$ plane the observationally favored lower bound for the characteristic index is $q\ge 1/2$. Variation of the scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$ is shown in b. The purple and blue colored line represent the upper and lower bounds allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represents the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the exponential potential Type-I is favored for the characteristic index $1<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis

Finally we compute the following inflationary observables:

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{g\lambda }{12\pi ^2M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \left( \frac{N_{\star }+\frac{1}{2}}{g}\right) ^2, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 2q~ \left( \frac{\frac{N_{\star }}{\sqrt{2q}}+\frac{1}{2}}{g}\right) ^2,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.246)

$$\begin{aligned} \begin{array}{lll} \displaystyle n_{\zeta ,\star }-1\approx \left\{ \begin{array}{lll} \displaystyle -\frac{2}{N_{\star }+\frac{1}{2}}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{2}{\sqrt{2q}\left( \frac{N_{\star }}{\sqrt{2q}}+\frac{1}{2}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.247)

$$\begin{aligned} \begin{array}{lll} \displaystyle \alpha _{\zeta ,\star }\approx \left\{ \begin{array}{lll} \displaystyle -\frac{2}{\left( N_{\star }+\frac{1}{2}\right) ^2}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{2}{2q\left( \frac{N_{\star }}{\sqrt{2q}}+\frac{1}{2}\right) ^2},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.248)

$$\begin{aligned} \begin{array}{lll} \displaystyle \kappa _{\zeta ,\star }\approx \left\{ \begin{array}{lll} \displaystyle -\frac{4}{\left( N_{\star }+\frac{1}{2}\right) ^3}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{4}{(2q)^{3/2}\left( \frac{N_{\star }}{\sqrt{2q}}+\frac{1}{2}\right) ^3},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.249)

$$\begin{aligned} \begin{array}{lll} \displaystyle r_{\star }\approx \left\{ \begin{array}{lll} \displaystyle \frac{8}{\left( N_{\star }+\frac{1}{2}\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{8}{2q\left( \frac{N_{\star }}{\sqrt{2q}}+\frac{1}{2}\right) },&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.250)

For the inverse cosh potential we get the following consistency relations:

$$\begin{aligned} \begin{array}{lll} \displaystyle r_{\star }\approx 4(1-n_{\zeta ,\star })\times \left\{ \begin{array}{lll} \displaystyle 1, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\sqrt{2q}},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.251)

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{\lambda }{3g\pi ^2M^{4}_{p}(1-n_{\zeta ,\star })^2}\times \left\{ \begin{array}{lll} \displaystyle 1, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 1,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.252)

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{2\lambda }{3g\pi ^2M^{4}_{p}r_{\star }}\times \left\{ \begin{array}{lll} \displaystyle \left( N_{\star }+\frac{1}{2}\right) , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \left( \frac{N_{\star }}{\sqrt{2q}}+\frac{1}{2}\right) ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.253)

$$\begin{aligned} \begin{array}{lll} \displaystyle \alpha _{\zeta ,\star }\approx -\frac{1}{2}\left( n_{\zeta ,\star }-1\right) ^2\times \left\{ \begin{array}{lll} \displaystyle 1, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 1,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.254)

$$\begin{aligned} \begin{array}{lll} \displaystyle \kappa _{\zeta ,\star }\approx \frac{1}{2}\left( n_{\zeta ,\star }-1\right) ^3\times \left\{ \begin{array}{lll} \displaystyle 1, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 1,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.255)

Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor q and on the parameters appearing in the expression for exponential potential Type-I. In Fig. 9a, we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index $n_{\zeta }$ for exponential potential Type-I respectively. In both the figures the purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures the red, green,  brown, orange colored curve represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in Fig. 9a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck 2015 data and Planck+ BICEP2+Keck Array joint constraint. The rest of the region is completely ruled out by the present observational constraints. From Fig. 9a, it is also observed that, within $50<N_{\star }<70$, the exponential potential Type-I is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Also in Fig. 9a for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/g\sim 0.8$. This implies that for $50<N_{\star }<70$, the prescribed window for g from r–$n_{\zeta }$ plot is given by $63<g<88$. To analyze the results more clearly let us describe the cosmological features from Fig. 9a in detail. Let us first start with the $q=1/2$ situation, in which the $2\sigma$ constraint on the scalar spectral tilt is satisfied within the window of the tensor-to-scalar ratio, $0.10<r<0.12$ for $50<N_{\star }<70$. Next for the $q=1$ case, the same constraint is satisfied within the window of the tensor-to-scalar ratio, $0.07<r<0.11$ for $50<N_{\star }<70$. Further for the $q=3/2$ case, the same constraint on scalar spectral tilt is satisfied within the window of the tensor-to-scalar ratio, $0.06<r<0.085$ for $50<N_{\star }<70$. Finally, for $q=2$ situation, the value for the tensor-to-scalar ratio is $0.05<r<0.075$.

Fig. 10

Behavior of the a running of the scalar spectral tilt $\alpha _{\zeta }$ and b running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for exponential potential Type-I. For both the figures red colored curves represent for any value of q, respectively. The cyan color shaded region bounded by two vertical black colored lines in a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the exponential potential Type-I is favored for the characteristic index $1<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis

In Fig. 9b, we have depicted the behavior of the scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral tilt $n_{\zeta }$ for exponential potential Type-I respectively. It is important to note that, for all of the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The cyan color shaded region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is the $1\sigma$ region, which is presently disfavored by the joint Planck+WMAP constraints. The rest of the region is completely ruled out by the present observational constraints. Also in Fig. 9b for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }\sim 70$. But the plots can be reproduced for $50<N_{\star } <70$ also considering the present observational constraints from Planck 2015 data. It is important to mention here that if we combine the constraints obtained from Fig. 9a, b, then by comparing the behavior of the GTachyon in the r–$n_{\zeta }$ and $\Delta _{\zeta }$–$n_{\zeta }$ planes we clearly observe that the $q=1/2$ case is almost discarded as it is not consistent with both of the $2\sigma$ constraints simultaneously.

In Fig. 10a, b, we have shown the behavior of the running of the scalar spectral tilt $\alpha _{\zeta }$ and running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for exponential potential Type I respectively. For both the figures the red colored curve represent the behavior for any arbitrary values of q. The cyan color shaded region bounded by two vertical black colored lines in both the plots represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From both of these figures, it is also observed that, within $50<N_{\star }<70$, the exponential potential Type-I is favored for the characteristic index $1<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. It is also important to note that for any values of q, the numerical value of the running $\alpha _{\zeta }\sim \mathcal{O}(-10^{-4})$ and running of the running $\kappa _{\zeta }\sim \mathcal{O}(-10^{-5})$, which are perfectly consistent with the $1.5\sigma$ constraints on running and running of the running as obtained from Planck 2015 data. For $q=1/2$, $q=1$, $q=3/2$ and $q=2$ g is not explicitly appearing in the various inflationary observables except the amplitude of scalar power spectrum in this case. To produce the correct value of the amplitude of the scalar power spectra we fix the parameter $360<g<400$.

Model  IV: exponential  potential-type  II

For the single field case the first model of the tachyonic potential is given by

$$\begin{aligned} V(T)=\lambda \exp \left[ -\left( \frac{T}{T_{0}}\right) ^2\right] , \end{aligned}$$

(5.256)

where $\lambda$ characterize the scale of inflation and $T_{0}$ is the parameter of the model. In Fig. 11 we have depicted the symmetric behavior of the exponential potential Type-I with respect to the scaled field coordinate $T/T_{0}$ in dimensionless units. In this case the tachyon field started rolling down from the top height of the potential situated at the origin and takes part in the inflationary dynamics.

Fig. 11

Variation of the exponential potential Type-II $V(T)/\lambda$ with field $T/T_{0}$ in dimensionless units

Next using specified form of the potential the potential dependent slow-roll parameters are computed as

$$\begin{aligned} \bar{\epsilon }_{V}= & {} \frac{1}{2g}4\left( \frac{T}{T_{0}}\right) ^2\exp \left[ \left( \frac{T}{T_{0}}\right) ^2\right] ,\end{aligned}$$

(5.257)

$$\begin{aligned} \bar{\eta }_{V}= & {} \frac{1}{g}\left[ 2\left( \frac{T}{T_{0}}\right) ^2-1\right] \exp \left[ \left( \frac{T}{T_{0}}\right) ^2\right] ,\end{aligned}$$

(5.258)

$$\begin{aligned} \bar{\xi }^2_{V}= & {} \frac{1}{g^2}8\left( \frac{T}{T_{0}}\right) ^2\left[ 2\left( \frac{T}{T_{0}}\right) ^2-3\right] \exp \left[ 2\left( \frac{T}{T_{0}}\right) ^2\right] ,\nonumber \\\end{aligned}$$

(5.259)

$$\begin{aligned} \bar{\sigma }^3_{V}= & {} \frac{1}{g^3}16\left( \frac{T}{T_{0}}\right) ^2\left[ 4\left( \frac{T}{T_{0}}\right) ^4-12\left( \frac{T}{T_{0}}\right) ^2+3\right] \nonumber \\&\times \exp \left[ 3\left( \frac{T}{T_{0}}\right) ^2\right] , \end{aligned}$$

(5.260)

where the factor g is defined as

$$\begin{aligned} g= \frac{\alpha ^{\prime }\lambda T^2_0}{M^2_p}=\frac{M^4_s}{(2\pi )^3 g_s}\frac{\alpha ^{\prime } T^2_0}{M^2_p}. \end{aligned}$$

(5.261)

Next we compute the number of e-foldings from this model:⁵

Further using the condition to end inflation:

we get the following transcendental equation to determine the field value at the end of inflation:

which we need to solve numerically for a given value of g. For the sake of simplicity in the further computation one can consider the following possibilities:

Next using $N=N_{\mathrm{cmb}}=N_{\star }$ and $T=T_{\mathrm{cmb}}=T_{\star }$ at the horizon crossing we get

$$\begin{aligned} \begin{array}{lll} \displaystyle T_{\star }=T_{0}\times \left\{ \begin{array}{lll} \displaystyle \exp \left[ \frac{N_{\star }}{g}+\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \exp \left[ \frac{N_{\star }}{\sqrt{2q}g}\!+\!\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.267)

Also the field excursion can be computed as

$$\begin{aligned} \begin{array}{lll} \displaystyle |\Delta T|=T_{0}\times \!\left\{ \!\begin{array}{lll} \displaystyle \left| \exp \left[ \frac{N_{\star }}{g}+\ln p\right] -p\right| , &{} \quad \mathrm{for }\;q=1/2, \\ \displaystyle \left| \exp \left[ \frac{N_{\star }}{\sqrt{2q}g}\!+\!\ln p\right] -p\right| ,&{} \quad \mathrm{for ~any~arbitrary~} \;q \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.268)

where $p=1, \sqrt{\frac{g}{2}}$. But during numerical estimation we only take $p= \sqrt{\frac{g}{2}}$ because $p=1$ is disfavored by the Planck 2015 data.

Finally we compute the following inflationary observables:

$$\begin{aligned}&\begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{g\lambda }{48\pi ^2M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \frac{\exp \left( -2\exp \left[ \frac{2N_{\star }}{g}+2\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) \right] \right) }{ \exp \left[ \frac{2N_{\star }}{g}+2\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) \right] }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 2q~ \frac{\exp \left( -2\exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}+2\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) \right] \right) }{ \exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}+2\ln \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) \right] },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.269)

$$\begin{aligned}&\begin{array}{lll} \displaystyle n_{\zeta ,\star }-1\approx \left\{ \begin{array}{lll} \displaystyle -\frac{2}{g}\left\{ 1+2\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{g}\right] \right\} , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{2}{\sqrt{2q}~g}\left\{ 1+2\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}\right] \right\} ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.270)

$$\begin{aligned} \begin{array}{lll} \displaystyle \alpha _{\zeta ,\star }\approx \frac{8}{g^2}\times \left\{ \begin{array}{lll} \displaystyle \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{g}\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q}\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.271)

$$\begin{aligned} \begin{array}{lll} \displaystyle \kappa _{\zeta ,\star }\approx -\frac{16}{g^3}\times \left\{ \begin{array}{lll} \displaystyle \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{g}\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{(2q)^{3/2}}\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.272)

$$\begin{aligned} \begin{array}{lll} \displaystyle r_{\star }\approx \frac{32}{g}\times \left\{ \begin{array}{lll} \displaystyle \left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2 \exp \left[ \frac{2N_{\star }}{g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\exp \left[ \frac{2N_{\star }}{g}\right] \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q}\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2 \exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}+\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2\exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}\right] \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.273)

For the inverse cosh potential we get the following consistency relations:

$$\begin{aligned} \begin{array}{lll} \displaystyle r_{\star }\approx \frac{16}{g}\times \left\{ \begin{array}{lll} \displaystyle \left[ \frac{g}{2}\left( 1-n_{\zeta ,\star }\right) -1\right] \exp \left[ \frac{1}{2}\left( \frac{g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right) \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q}\left[ \frac{\sqrt{2q}g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right] \exp \left[ \frac{1}{2}\left( \frac{\sqrt{2q}g}{2}\left( 1-n_{\zeta ,\star }\right) -1\right) \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.274)

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{g\lambda }{24\pi ^2M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \frac{\exp \left( -\left[ \frac{g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right] \right) }{\left[ \frac{g}{2}\left( 1-n_{\zeta ,\star }\right) -1\right] }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 2q~ \frac{\exp \left( -\left[ \frac{\sqrt{2q}g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right] \right) }{\left[ \frac{\sqrt{2q}g}{2}\left( 1-n_{\zeta ,\star }\right) -1\right] },&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.275)

$$\begin{aligned} \begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{2\lambda }{3\pi ^2M^{4}_{p}r_{\star }}\times \left\{ \begin{array}{lll} \displaystyle \exp \left[ -\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{g}\right] \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \exp \left[ -\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2~ \exp \left[ \frac{2N_{\star }}{\sqrt{2q}g}\right] \right] ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.276)

$$\begin{aligned} \begin{array}{lll} \displaystyle \alpha _{\zeta ,\star }\approx \frac{4}{g^2}\times \left\{ \begin{array}{lll} \displaystyle \left[ \frac{g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q}\left[ \frac{\sqrt{2q}g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.277)

$$\begin{aligned} \begin{array}{lll} \displaystyle \kappa _{\zeta ,\star }\approx -\frac{8}{g^3}\times \left\{ \begin{array}{lll} \displaystyle \left[ \frac{g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{(2q)^{3/2}}\left[ \frac{\sqrt{2q}g}{2}\left( 1-n_{\zeta ,\star } \right) -1\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.278)

Fig. 12

Behavior of the tensor-to-scalar ratio r with respect to a the scalar spectral index $n_{\zeta }$ and b the parameter g for exponential potential Type-II. The purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the logarithmic potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis

Fig. 13

Variation of the a scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$, b scalar power spectrum $\Delta _{\zeta }$ vs. the stringy parameter g and c scalar spectral tilt $n_{\zeta }$ vs. the stringy parameter g. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint

Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor q and on the parameters appearing in the expression for exponential potential Type-II. In Fig. 12a, b, we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index $n_{\zeta }$ and the model parameter g for exponential potential Type-II respectively. In both the figures the purple and blue colored line represent the upper bound of tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures the red, green,  brown, orange colored curve represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in Fig. 12a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck 2015 data and Planck+ BICEP2+Keck Array joint constraint. The rest of the region is completely ruled out by the present observational constraints. From Fig. 12a, b, it is also observed that, within $50<N_{\star }<70$, the exponential potential Type-II is favored only for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Also in Fig. 12a for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/g\sim 0.85$. This implies that for $50<N_{\star }<70$, the prescribed window for g from r–$n_{\zeta }$ plot is given by $59<g<82.3$ considering $1/2<q<2$. It is additionally important to note that, for $q<<1/2$, the tensor-to-scalar ratio computed from the model is negligibly small for exponential potential Type-II. This implies that if the inflationary tensor mode is detected near its present upper bound on the tensor-to-scalar ratio then all $q<<1/2$ possibilities for tachyonic inflation can be discarded for exponential potential Type-II. On the contrary, if inflationary tensor modes are never detected by any of the future observational probes then $q<<1/2$ possibilities for tachyonic inflation in the case of an exponential potential Type-II is highly prominent. Also it is important to mention that, in Fig. 12b within the window $30<g<93$, if we take a smaller value of g, then the inflationary tensor-to-scalar ratio also gradually decreases. To analyze the results more clearly let us describe the cosmological features from Fig. 12a in detail. Let us first start with the $q=1/2$ situation, in which the $2\sigma$ constraint on the scalar spectral tilt is satisfied within the window of the tensor-to-scalar ratio, $0<r<0.065$ for $50<N_{\star }<70$. Next for the $q=1$ case, the same constraint is satisfied within the window of the tensor-to-scalar ratio, $0.030<r<0.12$ for $50<N_{\star }<70$. Further for the $q=3/2$ case, the same constraint on scalar spectral tilt is satisfied within the window of the tensor-to-scalar ratio, $0.045<r<0.12$ for $50<N_{\star }<70$. Finally, for the $q=2$ situation, the value for the tensor-to-scalar ratio is $0.060<r<0.12$, which is tightly constrained from the upper bound of spectral tilt from Planck 2015 observational data.

Fig. 14

Behavior of the a running of the scalar spectral tilt $\alpha _{\zeta }$ and b running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for the inverse cosh potential with $g=88$. For both figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The cyan color shaded region bounded by two vertical black colored lines in a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the inverse cosh potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis

In Fig. 13a–c, we have depicted the behavior of the scalar power spectrum $\Delta _{\zeta }$ vs. the stringy parameter g, scalar spectral tilt $n_{\zeta }$ vs. the stringy parameter g and scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$ for exponential potential, Type-II respectively. It is important to note that, for all of the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The cyan color shaded region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is the $1\sigma$ region, which is presently disfavored by the joint Planck+WMAP constraints. The rest of the region is completely ruled out by the present observational constraints. From Fig. 13a it is clearly observed that the observational constraints on the amplitude of the scalar mode fluctuations satisfy within the window $55<g<93$ considering $1/2<q<2$. For $g>93$ the corresponding amplitude falls down in a non-trivial fashion by following the exact functional form as stated in Eq. (5.269). Next using the behavior as shown in Fig. 13b, the lower bound on the stringy parameter g is constrained by $g>73$ by using the non-trivial relationship as stated in Eq. (5.270). But at this lower bound of the parameter g the amplitude of the scalar power spectrum is slightly outside to the present observational constraints. This implies that, to satisfy both the constraints from the amplitude of the scalar power spectrum and its spectral tilt, within $2\sigma$ CL the constrained numerical value of the stringy parameter is lying within the window $73<g<93$. Also in Fig. 13c for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/g\sim 0.85$, which further implies that for $50<N_{\star }<70$, the prescribed window for g from $\Delta _{\zeta }$–$n_{\zeta }$ plot is given by $59<g<82.3$. If we additionally impose the constraint from the upper bound on the tensor-to-scalar ratio then also the allowed parameter range is lying within the window, i.e., $73<g<82.3$.

In Fig. 14a, b, we have shown the behavior of the running of the scalar spectral tilt $\alpha _{\zeta }$ and running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for exponential potential Type II with $g=76$, respectively. For both the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in both plots represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From both of these figures, it is also observed that, within $50<N_{\star }<70$, the exponential potential Type-II is favored for the characteristic index $1/2<q<2$, by the Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. From Fig. 14a, b, it is observed that within the $2\sigma$ observed range of the scalar spectral tilt $n_{\zeta }$, as the value of the characteristic parameter q increases, the value of the running $\alpha _{\zeta }$ decreases and running of the running $\kappa _{\zeta }$ increases for exponential potential Type-II. It is also important to note that for $1/2<q<2$, the numerical value of the running $\alpha _{\zeta }\sim \mathcal{O}(10^{-4})$ and running of the running $\kappa _{\zeta }\sim \mathcal{O}(-10^{-6})$, which are perfectly consistent with the $1.5\sigma$ constraints on running and running of the running as obtained from the Planck 2015 data.

Model  V: inverse  power-law  potential

For single field case the first model of tachyonic potential is given by

$$\begin{aligned} V(T)=\frac{\lambda }{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] }, \end{aligned}$$

(5.279)

where $\lambda$ characterize the scale of inflation and $T_{0}$ is the parameter of the model. In Fig. 15 we have depicted the symmetric behavior of the inverse power-law potential with respect to the scaled field coordinate $T/T_{0}$ in dimensionless units. In this case the tachyon field started rolling down from the top height of the potential situated at the origin and takes part in the inflationary dynamics.

Fig. 15

Variation of the inverse power-law potential $V(T)/\lambda$ with the field $T/T_{0}$ in dimensionless units

Next using specified form of the potential the potential dependent slow-roll parameters are computed as

$$\begin{aligned}&\bar{\epsilon }_{V}=\frac{1}{2g}\frac{16\left( \frac{T}{T_{0}}\right) ^6}{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] },\end{aligned}$$

(5.280)

$$\begin{aligned}&\bar{\eta }_{V}=\frac{1}{g}\frac{4\left[ 5\left( \frac{T}{T_{0}}\right) ^4-3\right] }{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] },\end{aligned}$$

(5.281)

$$\begin{aligned}&\bar{\xi }^2_{V}=\frac{1}{g^2}\frac{96\left( \frac{T}{T_{0}}\right) ^2\left[ 5\left( \frac{T}{T_{0}}\right) ^8-10\left( \frac{T}{T_{0}}\right) ^4 +1\right] }{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] ^2}, \end{aligned}$$

(5.282)

$$\begin{aligned}&\bar{\sigma }^3_{V}=\frac{1}{g^3}\frac{384\left( \frac{T}{T_{0}}\right) ^6 \left[ 5\left( \frac{T}{T_{0}}\right) ^4\left( 7\left( \frac{T}{T_{0}}\right) ^8 -31\left( \frac{T}{T_{0}}\right) ^4 +13\right) -1\right] }{\left[ 1+\left( \frac{T}{T_{0}}\right) ^4\right] ^3}, \end{aligned}$$

(5.283)

where the factor g is defined as

$$\begin{aligned} g= \frac{\alpha ^{\prime }\lambda T^2_0}{M^2_p}=\frac{M^4_s}{(2\pi )^3 g_s}\frac{\alpha ^{\prime } T^2_0}{M^2_p}. \end{aligned}$$

(5.284)

Next we compute the number of e-foldings from this model:

$$\begin{aligned} \begin{array}{lll} \displaystyle N(T)=\left\{ \begin{array}{lll} \displaystyle \frac{g}{8}~\left[ \frac{1}{\left( \frac{T}{T_{0}}\right) ^2}-\frac{1}{\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2}\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{2q}~\frac{g}{8}~\left[ \frac{1}{\left( \frac{T}{T_{0}}\right) ^2} -\frac{1}{\left( \frac{T_{\mathrm{end}}}{T_{0}}\right) ^2}\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.285)

Further using the condition to end inflation:

$$\begin{aligned} \bar{\epsilon }_{V}(T_{\mathrm{end}})=1, \end{aligned}$$

(5.286)

we get the following field values at the end of inflation:

$$\begin{aligned} T_{\mathrm{end}}\sim \sqrt{\frac{g}{8}}~T_{0}. \end{aligned}$$

(5.287)

Next using $N=N_{\mathrm{cmb}}=N_{\star }$ and $T=T_{\mathrm{cmb}}=T_{\star }$ at the horizon crossing we get

$$\begin{aligned} \begin{array}{lll} \displaystyle T_{\star }=T_{0}\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{\sqrt{\frac{8}{g}\left( N_{\star }+1\right) }}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\sqrt{\frac{8}{g}\left( \frac{N_{\star }}{\sqrt{2q}}+1\right) }},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.288)

Also the field excursion can be computed as

$$\begin{aligned} |\Delta T|= & {} T_{0}\nonumber \\&\times \left\{ \begin{array}{l@{\quad }l} \displaystyle \left| \frac{1}{\sqrt{\frac{8}{g}\left( N_{\star }+1\right) }}-\sqrt{\frac{g}{8}}\right| , &{} \mathrm{for}~~~q=1/2, \\ \displaystyle \left| \frac{1}{\sqrt{\frac{8}{g}\left( \frac{N_{\star }}{\sqrt{2q}}+1\right) }} -\sqrt{\frac{g}{8}}\right| ,&{} \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{aligned}$$

(5.289)

Finally we compute the following inflationary observables:

$$\begin{aligned} \displaystyle \Delta _{\zeta ,\star }\approx & {} \frac{g\lambda }{12\pi ^2M^{4}_{p}}\nonumber \\&\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{16\left( \frac{8}{g}\left( N_{\star }+1\right) \right) ^3}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{2q}{16\left( \frac{8}{g}\left( \frac{N_{\star }}{\sqrt{2q}}+1\right) \right) ^3},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \nonumber \\ \end{aligned}$$

(5.290)

Fig. 16

In a the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index $n_{\zeta }$ and b the parameter g for inverse power-law potential. The purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. In a, we have explicitly shown that in r–$n_{\zeta }$ plane the observationally disfavored value of the characteristic index is $q\ge 1/2$. Variation of the scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral index $n_{\zeta }$ is shown in b. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The green dotted region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is disfavored by the Planck+WMAP constraint. In b, we have explicitly shown that in $\Delta _{\zeta }-n_{\zeta }$ plane the observationally favored value of the characteristic index is $q=1$ and $q=3/2$. From a and b, it is also observed that, within $50<N_{\star }<70$, the inverse power-law potential is favored for the characteristic index $3/2<q<1$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis

$$\begin{aligned}&\begin{array}{lll} \displaystyle n_{\zeta ,\star }-1\approx \left\{ \begin{array}{lll} \displaystyle -\frac{3}{\left( N_{\star }+1\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{3}{\left( \frac{N_{\star }}{\sqrt{2q}}+1\right) },&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\end{aligned}$$

(5.291)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \alpha _{\zeta ,\star }\approx \left\{ \begin{array}{lll} \displaystyle -\frac{3}{\left( N_{\star }+1\right) ^2}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{3}{\sqrt{2q}\left( \frac{N_{\star }}{\sqrt{2q}}+1\right) ^2},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\\end{aligned}$$

(5.292)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \kappa _{\zeta ,\star }\approx \left\{ \begin{array}{lll} \displaystyle -\frac{6}{\left( N_{\star }+1\right) ^3}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{6}{2q\left( \frac{N_{\star }}{\sqrt{2q}}+1\right) ^3},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.293)

$$\begin{aligned}&\begin{array}{lll} \displaystyle r_{\star }\approx \left\{ \begin{array}{lll} \displaystyle \frac{16}{\left( N_{\star }+1\right) }, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{16}{2q\left( \frac{N_{\star }}{\sqrt{2q}}+1\right) },&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.294)

For the inverse cosh potential we get the following consistency relations:

$$\begin{aligned}&\begin{array}{lll} \displaystyle r_{\star }\approx \frac{16}{3}(1-n_{\zeta ,\star })\times \left\{ \begin{array}{lll} \displaystyle 1, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.295)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{g^4 \lambda (1-n_{\zeta ,\star })^3}{12\pi ^2M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{221184}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{2q}{221184},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\end{aligned}$$

(5.296)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \Delta _{\zeta ,\star }\approx \frac{g^4 \lambda r^3_{\star }}{12\pi ^2M^{4}_{p}}\times \left\{ \begin{array}{lll} \displaystyle \frac{1}{33554432}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{(2q)^3}{33554432},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\\end{aligned}$$

(5.297)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \alpha _{\zeta ,\star }\approx -\frac{1}{3}\left( n_{\zeta ,\star }-1\right) ^2 \times \left\{ \begin{array}{lll} \displaystyle 1, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\sqrt{2q}},&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array}\nonumber \\\end{aligned}$$

(5.298)

$$\begin{aligned}&\begin{array}{lll} \displaystyle \kappa _{\zeta ,\star }\approx \frac{2}{9}\left( n_{\zeta ,\star }-1\right) ^3 \times \left\{ \begin{array}{lll} \displaystyle 1, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q},&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.299)

Let us now discuss the general constraints on the parameters of tachyonic string theory including the factor q and on the parameters appearing in the expression for inverse power-law potential. In Fig. 16a, we have shown the behavior of the tensor-to-scalar ratio r with respect to the scalar spectral index $n_{\zeta }$ for inverse power-law potential respectively. In both the figures the purple and blue colored line represent the upper bound of the tensor-to-scalar ratio allowed by Planck+ BICEP2+Keck Array joint constraint and only Planck 2015 data respectively. For both the figures the red, green,  brown, orange colored curve represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in Fig. 16a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck 2015 data and Planck+ BICEP2+Keck Array joint constraint. The rest of the region is completely ruled out by the present observational constraints. From Fig. 16a, it is also observed that, within $50<N_{\star }<70$, the inverse power-law potential is favored for the characteristic index $1<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis. Also in Fig. 16a for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }\sim 70$. One can draw similar characteristic curves for any value of the number of e-foldings lying within the window $50<N_{\star }<70$ also. To analyze the results more clearly let us describe the cosmological features from Fig. 16a in detail. Let us first start with the $q=1/2$ situation, in which the $2\sigma$ constraint on the scalar spectral tilt is disfavored for $50<N_{\star }<70$. Next for the $q=1$ case, the same constraint is satisfied within the window of the tensor-to-scalar ratio, $0.07<r<0.10$ for $50<N_{\star }<70$. Further for the $q=3/2$ case, the same constraint on scalar spectral tilt is satisfied within the window of the tensor-to-scalar ratio, $0.05<r<0.07$ for $50<N_{\star }<70$. Finally, for the $q=2$ situation, the value for the tensor-to-scalar ratio is $0.035<r<0.05$.

Fig. 17

Behavior of the a running of the scalar spectral tilt $\alpha _{\zeta }$ and b running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for the inverse cosh potential with $g=88$. For both figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The cyan color shaded region bounded by two vertical black colored lines in a and a represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From a and b, it is also observed that, within $50<N_{\star }<70$, the inverse cosh potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and Planck+ BICEP2+Keck Array joint analysis

In Fig. 16b, we have depicted the behavior of the scalar power spectrum $\Delta _{\zeta }$ vs. scalar spectral tilt $n_{\zeta }$ for inverse power-law potential respectively. It is important to note that, for all of the figures the red, green,  brown, orange colored curves represent $q=1/2$, $q=1$, $q=3/2$ and $q=2$ respectively. The purple and blue colored line represent the upper and lower bound allowed by WMAP+Planck 2015 data respectively. The cyan color shaded region bounded by two vertical black colored lines represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region is the $1\sigma$ region, which is presently disfavored by the joint Planck+WMAP constraints. The rest of the region is completely ruled out by the present observational constraints. Also in Fig. 16b for $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }\sim 70$. But the plots can be reproduced for $50<N_{\star } <70$ also considering the present observational constraints from Planck 2015 data. It is clearly observed from Fig. 16b that the Planck 2015 constraint on amplitude of the scalar power spectrum $\Delta _{\zeta }$ and the scalar spectral tilt $n_{\zeta }$ disfavor $q=1/2$ and $q=2$ values for inverse power-law potential.

In Fig. 17a, b, we show the behavior of the running of the scalar spectral tilt $\alpha _{\zeta }$ and running of the running of the scalar spectral tilt $\kappa _{\zeta }$ with respect to the scalar spectral index $n_{\zeta }$ for inverse power-law potential respectively. For both the figures the red, green,  brown, and orange colored curves represent $q=1/2$, $q=1$, $q=3/2$, and $q=2$, respectively. The cyan color shaded region bounded by two vertical black colored lines in both plots represent the Planck $2\sigma$ allowed region and the rest of the light gray shaded region shows the $1\sigma$ allowed range, which is at present disfavored by the Planck data and Planck+ BICEP2+Keck Array joint constraint. From both of these figures, it is also observed that, within $50<N_{\star }<70$, the inverse power-law potential is favored for the characteristic index $1/2<q<2$, by Planck 2015 data and a Planck+ BICEP2+Keck Array joint analysis. If we additionally impose the constraints from the r–$n_{\zeta }$ and the $\Delta _{\zeta }$–$n_{\zeta }$ plane then the stringent window on the characteristic index of GTachyon is lying within $1<q<3/2$. It is also important to note that for any values of q, the numerical value of the running $\alpha _{\zeta }\sim \mathcal{O}(-10^{-4})$ and running of the running $\kappa _{\zeta }\sim \mathcal{O}(-10^{-5})$, which are perfectly consistent with the $1.5\sigma$ constraints on running and running of the running as obtained from Planck 2015 data. For $q=1/2$, $q=1$, $q=3/2$ and $q=2$ g does not explicitly appear in the various inflationary observables except the amplitude of scalar power spectrum in this case. To produce the correct value of the amplitude of the scalar power spectra we fix the parameter $600<g<700$.

5.1.9 Analyzing CMB power spectrum

In this section we explicitly study the cosmological consequences from the CMB TT, TE, EE, BB angular power spectrum computed from all the proposed models of inflation.⁶ The angular power spectra or equivalently the two point correlator of the X and Y fields are defined as [10]

$$\begin{aligned} C_\ell ^{XY} \equiv \frac{1}{2\ell + 1} \sum ^{l}_{m=-l} \langle a_{X,\ell m}^* a_{Y,\ell m} \rangle , \qquad X, Y = T, E, B.\nonumber \\ \end{aligned}$$

(5.300)

Here l characterizes the CMB multipole and m signifies the magnetic quantum number, which runs from $m=-l,\ldots , +l$. In general, the field $X(\hat{n})$ and $Y(\hat{n})$ can be expressed in terms of the following harmonic expansion:

$$\begin{aligned} \begin{array}{lll} \displaystyle X(\hat{n}), Y(\hat{n})=\left\{ \begin{array}{lll} \displaystyle \sum ^{\infty }_{l=0}\sum ^{+l}_{m=-l}a_{T,lm}V_{lm}(\hat{n}), &{} \quad \mathrm{for}~~ X=T ,\\ \displaystyle \sum ^{\infty }_{l=0}\sum ^{+l}_{m=-l}a_{E,lm}V_{lm}(\hat{n}), &{} \quad \mathrm{for}~~ X=E,\\ \displaystyle \sum ^{\infty }_{l=0}\sum ^{+l}_{m=-l}a_{B,lm}V_{lm}(\hat{n}), &{} \quad \mathrm{for}~~ X=B, \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.301)

where $\hat{n}$ is the arbitrary directional unit vector in CMB map. Here $V_{lm}(\hat{n})$ are the spherical harmonics which are chosen to be the basis of the harmonic expansion of the temperature anisotropy and the E and B polarization. Further using the inflationary power spectra at any momentum scale k:

$$\begin{aligned} \Delta (k) \equiv \{ \Delta _\zeta (k), \Delta _h (k)\} \end{aligned}$$

(5.302)

the angular power spectra for CMB temperature fluctuations and polarization can be expressed as [10]

$$\begin{aligned} \displaystyle C_\ell ^{XY} = \frac{2}{\pi } \int k^2 \mathrm{d}k \underbrace{\Delta (k)}_\mathrm{Inflation}\, \underbrace{\Theta _{X \ell }(k) \Theta _{Y \ell }(k)}_\mathrm{Anisotropies}, \end{aligned}$$

(5.303)

where the anisotropic integral kernel can be written as [10]

$$\begin{aligned} \Theta _{X \ell }(k) = \int _0^{\eta _0} \mathrm{d}\eta \, \underbrace{S_X(k, \eta )}_\mathrm{Sources}\, \underbrace{P_{X\ell }(k[\eta _0-\eta ])}_\mathrm{Projection}. \end{aligned}$$

(5.304)

Fig. 18

We show the variation of CMB TT angular power spectrum with respect to the multipole, l, for scalar modes for all five tachyonic models

The integral (5.303) relates the inhomogeneities predicted by inflation, $\Delta (k)$, to the anisotropies observed in the CMB, $C_\ell ^{XY}$. The correlations between the different X and Y modes are related by the transfer functions $\Theta _{X\ell }(k)$ and $\Theta _{Y\ell }(k)$. The transfer functions may be written as the line-of-sight integral Eq. (5.304) which factorizes into physical source terms $S_X(k, \eta )$ and geometric projection factors $P_{X\ell }(k[\eta _0-\eta ])$ through combinations of Bessel functions.

Further expressing Eq. (5.303) in terms of TT, TE, EE, BB correlation we get

For curvature perturbation

$$\begin{aligned}&C_\ell ^{TT} = \frac{2}{\pi } \int k^2 \mathrm{d}k ~{\Delta _{S}(k)}\, {\Theta _{T \ell }(k) \Theta _{T \ell }(k)},\end{aligned}$$

(5.305)

$$\begin{aligned}&C_\ell ^{TE} = (4\pi )^2 \int k^2 \mathrm{d}k ~{\Delta _{S}(k)}\, {\Theta _{T \ell }(k) \Theta _{E \ell }(k)},\end{aligned}$$

(5.306)

$$\begin{aligned}&C_\ell ^{EE} = (4\pi )^2 \int k^2 \mathrm{d}k ~{\Delta _{S}(k)}\, {\Theta _{E \ell }(k) \Theta _{E \ell }(k)}, \end{aligned}$$

(5.307)

For  tensor perturbation

$$\begin{aligned}&C_\ell ^{BB} = (4\pi )^2 \int k^2 \mathrm{d}k ~{\Delta _{h}(k)}\, {\Theta _{B \ell }(k) \Theta _{B \ell }(k)}\end{aligned}$$

(5.308)

$$\begin{aligned}&C_\ell ^{TT} = \frac{2}{\pi } \int k^2 \mathrm{d}k ~{\Delta _{h}(k)}\, {\Theta _{T \ell }(k) \Theta _{T \ell }(k)},\end{aligned}$$

(5.309)

$$\begin{aligned}&C_\ell ^{TE} = (4\pi )^2 \int k^2 \mathrm{d}k ~{\Delta _{h}(k)}\, {\Theta _{T \ell }(k) \Theta _{E \ell }(k)},\end{aligned}$$

(5.310)

$$\begin{aligned}&C_\ell ^{EE} = (4\pi )^2 \int k^2 \mathrm{d}k ~{\Delta _{h}(k)}\, {\Theta _{E \ell }(k) \Theta _{E \ell }(k)}, \end{aligned}$$

(5.311)

where the inflationary power spectra $\{ \Delta _\zeta (k), \Delta _h (k) \}$ are parametrized at any arbitrary momentum scale k as⁷

$$\begin{aligned} \Delta _{\zeta }(k)= & {} \Delta _{\zeta ,\star }\left( \frac{k}{c_{S}k_{\star }}\right) ^{n_{\zeta ,\star }-1+\frac{\alpha _{\zeta ,\star }}{2}\ln \left( \frac{k}{c_{S}k_{\star }}\right) +\frac{\kappa _{\zeta ,\star }}{6}\ln ^{2}\left( \frac{k}{c_{S}k_{\star }}\right) +\cdots },\nonumber \\\end{aligned}$$

(5.312)

$$\begin{aligned} \Delta _{h}(k)= & {} r(k)\Delta _{\zeta }(k)\nonumber \\= & {} \Delta _{h,\star }\left( \frac{k}{k_{\star }}\right) ^{n_{h,\star }+\frac{\alpha _{h,\star }}{2}\ln \left( \frac{k}{k_{\star }}\right) +\frac{\kappa _{h,\star }}{6}\ln ^{2}\left( \frac{k}{k_{\star }}\right) +\cdots }. \end{aligned}$$

(5.313)

It is important to note that the cosmological significance of the E and B decomposition of CMB polarization carries the following significant features:

• Curvature (density) perturbations create only polarizing E-modes.

• Vector (vorticity) perturbations create mainly B-modes. However, it is important to note that here the contributions of vector modes decay with the expansion of the universe and are therefore sub-dominant at the epoch of recombination. For this reason we have neglected such sub-dominant effects from our rest of the analysis.

• Tensor (gravitational wave) perturbations create both E-modes and B-modes. But to quantify the exact contribution of the primordial gravitational waves, specifically in the inflationary B modes, one needs to separate all the other significant contributions in the B modes, i.e., the primordial magnetic field, gravitational lensing, non-Gaussianity, etc., But at present in the cosmological literature no such sophisticated techniques or algorithms are available using which one can separate all of these significant contributions completely.

To compute the momentum integrals numerically and to analyze the various features of CMB angular power spectra from all of the inflationary models mentioned in the last section, here we use a semi-analytical code “CAMB”. For the numerical analysis we use here the best fit model parameters corresponding to all of the five tachyonic models, which are compatible with Planck 2015 data. Additionally we take the $\Lambda$CDM background. Further we put all these inputs to “CAMB” and modify the inbuilt parameterization of the power spectrum for scalar and tensor modes accordingly. After performing all the numerical computations via “CAMB” finally from our analysis we have generated all the theoretical CMB angular power spectra from which we observed the following significant features:

• In Fig. 18, at low $\ell$ region $(2<l<49)$ the contributions from the running ($\alpha _{\zeta },\alpha _{h}$), and running of running ($\kappa _{\zeta },\kappa _{h}$) are very small. Their additional contribution to the CMB power spectrum for scalar and tensor modes becomes unity $(\mathcal{O}(1))$ within low-l region and the original power spectrum becomes unchanged. As a result the tachyonic models will be well fitted with the CMB TT spectrum at low-l region within high cosmic variance as observed by Planck except for a few outliers according to the Planck 2013 data release.⁸ But to update our analysis with the latest Planck 2015 data set we have not shown such high cosmic variance explicitly. In case of WMAP9 data for CMB TT spectrum the cosmic variance in the low-l region is not very large compared to the Planck low-l data. But our tachyonic models are also well fitted with WMAP9 low-l data, which we have not shown explicitly in the plot to update the analysis using Planck 2015 data. It is also important to mention that if we incorporate the uncertainties in the scanning multipole l for the measurement of CMB TT spectrum, then also our prescribed analysis is fairly consistent with the Planck 2015 data. Further if we move toward the high $\ell$ regime ($47<l<2500$) the contributions of running and running of running become stronger, and this will enhance the power spectrum to a permissible value such that it will accurately fit high-l data within very a small cosmic variance as observed by Planck. In this way one can easily scan over all the multipoles starting from low-l to high-l using the same momentum dependent parameterization of the tensor-to-scalar ratio.

• From Fig. 18, we see that the Sachs–Wolfe plateau obtained from our proposed tachyonic models is non-flat, confirming the appearance of running, and running of the running in the spectrum observed for low l region ($l<47$). For larger values of the multipole ($47<l<2500$), the CMB anisotropy spectrum is dominated by the baryon acoustic oscillations (BAO), giving rise to several ups and downs in the CMB TT spectrum. In the low l region due to the presence of very large cosmic variance there may be other pre-inflationary scenarios which might be able to describe the TT-power spectrum better. In our study we have considered only the possibility for which the behavior of tachyonic models is analyzed for both low and high l regions.

• From Figs. 18, 19, 20, we observe that if we include the uncertainties in multipole l as well in the observed CMB angular power spectra then the proposed tachyonic models is fairly consistent with the CMB TT, TE, EE for scalar mode from Planck 2015 data.

• In Figs. 21, 22, 23 and 24 we have explicitly shown the theoretical CMB BB, TT, TE and EE angular power spectra from the tensor mode. Most importantly, if the inflationary paradigm is responsible for the nearly de Sitter expansion of the early universe then the CMB BB spectra for tensor modes is one of the prime components through which one can detect the contribution for primordial gravitational waves via the tensor-to-scalar ratio. But till date only the contribution from the lensing B-modes are detected via South Pole Telescope and Planck 2015 +BICEP2/Keck Array joint mission. But confirming the sole inflationary origin from the detection of the de-lensed version of the signal is not sufficient to draw any final conclusion.⁹ There are other possibilities as well through which it is possible to generate CMB B-modes—those components are the primordial magnetic field, gravitational lensing, CP asymmetry in the lepton sector of particle physics, etc.

Fig. 19

We show the variation of CMB TE angular power spectrum with respect to the multipole, l, for scalar modes for all five tachyonic models

Fig. 20

We show the variation of CMB EE angular power spectrum with respect to the multipole, l, for scalar modes for all five tachyonic models

Fig. 21

We show the variation of CMB BB angular power spectrum with respect to the multipole, l, for tensor modes for all five tachyonic models

Fig. 22

We show the variation of CMB TT angular power spectrum with respect to the multipole, l, for tensor modes for all five tachyonic models

Fig. 23

We show the variation of CMB TE angular power spectrum with respect to the multipole, l, for tensor modes for all five tachyonic models

Fig. 24

We show the variation of CMB EE angular power spectrum with respect to the multipole, l, for tensor modes for all five tachyonic models

5.2 Computation for assisted inflation

In the case of assisted inflation [134–137] all the tachyons would follow a similar trajectory with a unique late time attractor.¹⁰ In more technical language one can state that

$$\begin{aligned} T_{1}\sim T_{2}\sim \cdots T_{M}\equiv T_{i}=T \quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

(5.314)

Most importantly the detailed computation of the density and tensor perturbation responsible for the anisotropy of CMB depends on this late time attractor behavior of the fields.

5.2.1 Condition for inflation

For assisted tachyonic inflation, the prime condition for inflation is given by

$$\begin{aligned} \dot{H}+H^{2}= & {} \left( \frac{\ddot{a}}{{a}}\right) =-\sum ^{M}_{i=1}\frac{(\rho _{i}+3p_{i})}{6 M^2_p}=\frac{(\rho +3p)}{6 M^2_p}>0\nonumber \\ \end{aligned}$$

(5.315)

which can be re-expressed in terms of the following constraint condition in the context of assisted tachyonic inflation:

$$\begin{aligned}&\displaystyle \sum ^{M}_{i=1}\frac{V(T_i)}{3M^{2}_{p}\sqrt{1-\alpha ^{\prime }\dot{T}^2_i}}\left( 1-\frac{3}{2}\alpha ^{\prime }\dot{T}^2_i\right) \nonumber \\&\quad = \;\frac{MV(T)}{3M^{2}_{p}\sqrt{1-\alpha ^{\prime }\dot{T}^2}}\left( 1-\frac{3}{2}\alpha ^{\prime }\dot{T}^2\right) >0. \end{aligned}$$

(5.316)

Here Eq. (5.316) implies that to satisfy inflationary constraints in the slow-roll regime the following constraint always holds good:

$$\begin{aligned} \dot{T}< & {} \sqrt{\frac{2}{3\alpha ^{\prime }}} \quad \forall \; i=1,2,\ldots ,M,\end{aligned}$$

(5.317)

$$\begin{aligned} \ddot{T}<&3H \dot{T}<\sqrt{\frac{6}{\alpha ^{\prime }}}H \quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

(5.318)

Consequently the field equations are approximated by

$$\begin{aligned} \begin{array}{lll} \displaystyle 3H\alpha ^{\prime }\dot{T}_{i}+\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\approx 0 \quad \forall \; i=1,2,\ldots ,M, \end{array} \end{aligned}$$

(5.319)

Similarly, in the most generalized case,

$$\begin{aligned}&\displaystyle \sum ^{M}_{i=1}\frac{V(T_{i})}{3M^{2}_{p}\left( 1-\alpha ^{\prime }\dot{T}^2_{i} \right) ^{1-q}}\left( 1-(1+q)\alpha ^{\prime }\dot{T}^2_{i}\right) \nonumber \\&\quad =\frac{MV(T)}{3M^{2}_{p}\left( 1-\alpha ^{\prime }\dot{T}^2 \right) ^{1-q}}\left( 1-(1+q)\alpha ^{\prime }\dot{T}^2\right) >0. \end{aligned}$$

(5.320)

Here Eq. (5.320) implies that to satisfy inflationary constraints in the slow-roll regime the following constraint always holds good:

$$\begin{aligned} \dot{T}< & {} \sqrt{\frac{1}{\alpha ^{\prime }(1+q)}} \quad \forall \; i=1,2,\ldots ,M,\end{aligned}$$

(5.321)

$$\begin{aligned} \ddot{T}<&3H \dot{T}<\sqrt{\frac{9}{\alpha ^{\prime }(1+q)}}H \quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

(5.322)

Consequently the field equations are approximated by

$$\begin{aligned} \begin{array}{lll} \displaystyle 6q\alpha ^{\prime }H\dot{T}_{i}+\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\approx 0 \quad \forall \; i=1,2,\ldots ,M. \end{array} \end{aligned}$$

(5.323)

Also for both cases in the slow-roll regime the Friedmann equation is modified as

$$\begin{aligned} H^{2}\approx & {} \sum ^{M}_{i=1}\frac{V(T_{i})}{3M^{2}_{p}}=\frac{MV(T)}{3M^2_p}. \end{aligned}$$

(5.324)

The equation of motions can be mapped to the equations of a model with a single tachyonic field using the following redefinitions:

$$\begin{aligned} \tilde{T}= & {} \sqrt{M} T,\end{aligned}$$

(5.325)

$$\begin{aligned} \tilde{V}= & {} V(\tilde{T})=M V(T)=\sum ^{M}_{i=1}V(T_{i}). \end{aligned}$$

(5.326)

Now to show the late time attractor behavior of the solution of scalar fields we keep $T_{1}$, but we replace the rest with the redefined fields:

$$\begin{aligned} \chi _{i}= & {} T_{i}-T_{1} \quad \forall \; i=2,\ldots ,M. \end{aligned}$$

(5.327)

Using Eq. (5.327), at late times the equation of motion can be recast as

$$\begin{aligned} \begin{array}{lll} \displaystyle 0 =\left\{ \begin{array}{lll} \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V_{eff}(T_{1},\chi _{i})\sqrt{1+g^{\alpha \beta }\partial _{\alpha }\chi _{i}\partial _{\beta }\chi _{i}}\right) \\ \quad \displaystyle =\frac{\alpha ^{\prime }\ddot{\chi }_{i}}{\left( 1-\alpha ^{\prime }\dot{\chi }^2_{i}\right) } +3H\alpha ^{\prime }\dot{\chi }_{i}+\frac{\mathrm{d}V_{eff}(T_{1},\chi _{i})}{V_{eff}(T_{1},\chi _{i})\mathrm{d}\chi _{i}} ,~&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}V_{eff}(T_{1},\chi _{i})\left( 1+g^{\alpha \beta }\partial _{\alpha }\chi _{i}\partial _{\beta }\chi _{i}\right) ^q\right) \\ \quad \displaystyle =\frac{2q\alpha ^{\prime }\ddot{\chi }_{i}}{\left( 1-\alpha ^{\prime }\dot{\chi }^2_{i}\right) } +6q\alpha ^{\prime }H\frac{\dot{\chi }_{i}}{1-\alpha ^{\prime }(1-2q)\dot{\chi }^2_{i}}+\frac{\mathrm{d}V_{eff}(T_{1},\chi _{i})}{V_{eff}(T_{1},\chi _{i})\mathrm{d}\chi _{i}} ,&{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{array} \end{aligned}$$

(5.328)

where $V_{eff}(T_{1},\chi _{i})$ represents the effective potential. The minimum in the $\chi _{i}$ direction is always appearing at $\chi _{i}=0$, regardless of the explicit behavior of the tachyon field $T_{1}$. Consequently the late time solution has all the $T_{i}$ equal. From this analysis it is also observed that the length of the time interval to reach this attractor behavior will depend on the initial separation, i.e., the tachyon field value $\chi _{i}$ and the extent of the frictional contribution coming from the expansion rate H.

It is important to mention that, for assisted inflation, the equation of motion for each tachyon fields follows the following simple relationship:

$$\begin{aligned} \frac{\mathrm{d}\ln T_{i}}{\mathrm{d}t}\simeq \frac{\mathrm{d}\ln T}{\mathrm{d}t} \quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

(5.329)

Further substituting Eq. (5.324) in Eqs. (5.319) and (5.323) we get

$$\begin{aligned} \displaystyle \frac{\sqrt{3MV(T)}}{M_{p}}\alpha ^{\prime }\dot{T}_{i}+\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\approx 0 ,\end{aligned}$$

(5.330)

$$\begin{aligned} 6q\frac{\sqrt{MV(T)}}{\sqrt{3}M_{p}}\alpha ^{\prime }\dot{T}_{i}+\frac{\mathrm{d}V(T_{i})}{V(T_{i})\mathrm{d}T_{i}}\approx 0. \end{aligned}$$

(5.331)

Finally the general solution for both cases can be expressed in terms of the single field tachyonic potential V(T) as

$$\begin{aligned} \displaystyle t-t_{in}\approx -\frac{\sqrt{3M}\alpha ^{\prime }}{M_{p}}\int ^{T_{i}}_{T_{in}}\mathrm{d}T_{i}~\frac{\sqrt{V(T)}V(T_{i})}{V^{\prime }(T_{i})} ,\end{aligned}$$

(5.332)

$$\begin{aligned} t-t_{in}\approx -\frac{6q\sqrt{M}\alpha ^{\prime }}{\sqrt{3}M_{p}}\int ^{T_{i}}_{T_{in}}\mathrm{d}T_{i}~\frac{\sqrt{V(T)}V(T_{i})}{V^{\prime }(T_{i})}. \end{aligned}$$

(5.333)

Further using Eqs. (5.15), (5.16) and (5.324) we get the following solution for the scale factor in terms of the tachyonic field for the usual $q=1/2$ and for a generalized value of q:

$$\begin{aligned} \begin{array}{lll} \displaystyle a= a_{in}\times \left\{ \begin{array}{lll} \displaystyle \exp \left[ -\frac{\alpha ^{\prime }M}{M^2_p}\int ^{T_{i}}_{T_{in}}\mathrm{d}T_{i}~\frac{V(T)V(T_{i})}{V^{\prime }(T_{i})}\right] , &{} \mathrm{for}~~~q=1/2, \\ \displaystyle \exp \left[ -\frac{2q\alpha ^{\prime }M}{M^2_p}\int ^{T_{i}}_{T_{in}}\mathrm{d}T_{i}~\frac{V(T)V(T_i)}{V^{\prime }(T_{i})}\right] ,&{} \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.334)

5.2.2 Analysis using slow-roll formalism

Here our prime objective is to define slow-roll parameters for tachyon inflation in terms of the Hubble parameter and the assisted tachyonic inflationary potential. Using the slow-roll approximation one can expand various cosmological observables in terms of small dynamical quantities derived from the appropriate derivatives of the Hubble parameter and of the inflationary potential. To start with here we use the horizon-flow parameters based on the derivatives of the Hubble parameter with respect to the number of e-foldings N, defined as

$$\begin{aligned} N(t)= & {} \int ^{t_{\mathrm{end}}}_{t}H(t)~\mathrm{d}t, \end{aligned}$$

(5.335)

where $t_{\mathrm{end}}$ signifies the end of inflation. Further, using Eqs. (5.5), (5.9), (5.10) and (5.335), we get

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\mathrm{d}T_i}{\mathrm{d}N}=\frac{\dot{T}_i}{H}=\left\{ \begin{array}{lll} \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2_i}{2q}\right) ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

(5.336)

where $H^{\prime }>0$, which makes always $\dot{T}_i>0$ during inflationary phase. Further using Eq. (5.336) we get the following differential operator identity for tachyonic inflation:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{H}\frac{\mathrm{d}}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}N}=\frac{\mathrm{d}}{\mathrm{d}\ln k}=\left\{ \begin{array}{lll} \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\frac{\mathrm{d}}{\mathrm{d}T_i}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2_i}{2q}\right) \frac{\mathrm{d}}{\mathrm{d}T_i},&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.337)

Further using the differential operator identity as mentioned in Eq. (5.337) we get the following Hubble flow equation for tachyonic inflation for $j\ge 0$:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{H}\frac{\mathrm{d}\epsilon _{i}}{\mathrm{d}t}=\frac{\mathrm{d}\epsilon _{j}}{\mathrm{d}N}=\epsilon _{j+1}\epsilon _{j}=\left\{ \begin{array}{lll} \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\frac{\mathrm{d}\epsilon _{j}}{\mathrm{d}T_i}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle -\frac{2H^{\prime }}{3\alpha ^{\prime }H^{3}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2_i}{2q}\right) \frac{\mathrm{d}\epsilon _{j}}{\mathrm{d}T_i},&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.338)

For a realistic estimate from the single field tachyonic inflationary model substituting the free index j to $j=0,1,2$ in Eqs. (5.23) and (5.338) we get the contributions from the first three Hubble slow-roll parameters, which can be represented as

$$\begin{aligned} \displaystyle \epsilon _{1}=\frac{\mathrm{d}\ln |\epsilon _{0}|}{\mathrm{d}N}= & {} -\frac{\dot{H}}{H^2}=\left\{ \begin{array}{lll} \displaystyle \frac{2}{3\alpha ^{\prime }}\left( \frac{H^{\prime }}{H^2}\right) ^{2}=\frac{3}{2}\alpha ^{\prime }\dot{T}^{2}_i,~&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{2}{3\alpha ^{\prime }}\left( \frac{H^{\prime }}{H^2}\right) ^{2}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2_i}{2q}\right) \\ \quad \displaystyle =\frac{3}{2}\alpha ^{\prime }\dot{T}^{2}_i \left( \frac{2q}{1-\alpha ^{\prime }(1-2q)\dot{T}^2_i}\right) ,~&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{aligned}$$

(5.339)

$$\begin{aligned} \displaystyle \epsilon _{2}=\frac{\mathrm{d}\ln |\epsilon _{1}|}{\mathrm{d}N}= & {} \frac{\ddot{H}}{H\dot{H}}+2\epsilon _{1}=\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{2}{3\alpha ^{\prime }\epsilon _{1}}}\frac{\epsilon ^{\prime }_{1}}{H}=2\frac{\ddot{T}_i}{H\dot{T}_i},~&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{2}{3\alpha ^{\prime }\epsilon _{1}}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2_i}{2q}\right) }\frac{\epsilon ^{\prime }_{1}}{H}\\ \quad \displaystyle =\frac{2\ddot{T}_i}{H\dot{T}\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_i\right) },~&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{aligned}$$

(5.340)

$$\begin{aligned} \displaystyle \epsilon _{3}=\frac{\mathrm{d}\ln |\epsilon _{2}|}{\mathrm{d}N}= & {} \frac{1}{\epsilon _{2}}\left[ \frac{\dddot{H}}{H^2 \dot{H}}-3\frac{\ddot{H}}{H^3}-\frac{\ddot{H}^2}{H^2 \dot{H}^2}+4\frac{\dot{H}^2}{H^4}\right] \nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{2\epsilon _{1}}{3\alpha ^{\prime }}}\frac{\epsilon ^{\prime }_{2}}{H}=\left[ \frac{2\dddot{T}_i}{H^2\dot{T}_i\epsilon _{2}}+\epsilon _{1}-\frac{\epsilon _{2}}{2}\right] ,~&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{2\epsilon _{1}}{3\alpha ^{\prime }}\left( \frac{1-\alpha ^{\prime }(1-2q)\dot{T}^2_i}{2q}\right) }\frac{\epsilon ^{\prime }_{2}}{H}\\ \quad \displaystyle =\frac{\left[ \frac{2\dddot{T}_i}{H^2\dot{T}_i\epsilon _{2}}+\epsilon _{1}-\frac{\epsilon _{2}}{2}\right] }{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_i\right) } +\frac{\frac{4\alpha ^{\prime }(1-2q)\ddot{T}^2_i}{H} }{\left( 1-\alpha ^{\prime }(1-2q)\dot{T}^2_i\right) ^2},~&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{aligned}$$

(5.341)

Further using Eqs. (5.339), (5.340), (5.343), (5.34) and (5.35) one can re-express the Hubble slow-roll parameters in terms of the potential dependent slow-roll parameter as

$$\begin{aligned} \displaystyle \epsilon _{1}\approx & {} \left\{ \begin{array}{lll} \displaystyle \frac{M^{2}_{p}}{2\alpha ^{\prime }}\frac{V^{'2}(T_i)}{MV(T)V^{2}(T_i)}=\frac{\epsilon _{V}(T_i)}{MV(T)\alpha ^{\prime }}\equiv \bar{\epsilon }_{V}(T_i), &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{M^{2}_{p}}{4q\alpha ^{\prime }}\frac{V^{'2}(T_i)}{MV(T)V^{2}(T_i)}=\frac{\epsilon _{V}(T_i)}{2qMV(T)\alpha ^{\prime }}\equiv \frac{\bar{\epsilon }_{V}(T_i)}{2q},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \\ \nonumber \displaystyle \epsilon _{2}\approx & {} \left\{ \begin{array}{lll} \displaystyle \frac{M^2_{p}}{\alpha ^{\prime }M}\left( 3\frac{V^{'2}(T_i)}{V(T)V^{2}(T_i)}-2\frac{V^{\prime \prime }(T_i)}{V(T)V(T_i)}\right) \\ \quad \displaystyle =\frac{2\left( 3\epsilon _{V}(T_i)-\eta _{V}(T_i)\right) }{MV(T)\alpha ^{\prime }}=2\left( 3\bar{\epsilon }_{V}(T_i)-\bar{\eta }_{V}(T_i)\right) , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{M^2_{p}}{\sqrt{2q}\alpha ^{\prime }M}\left( 3\frac{V^{'2}(T_i)}{V(T)V^{2}(T_i)}-2\frac{V^{\prime \prime }(T_i)}{V(T)V(T_i)}\right) \\ \quad \displaystyle =\frac{\sqrt{\frac{2}{q}}\left( 3\epsilon _{V}(T_i)-\eta _{V}(T_i)\right) }{MV(T)\alpha ^{\prime }}=\sqrt{\frac{2}{q}}\left( 3\bar{\epsilon }_{V}(T_i)-\bar{\eta }_{V}(T_i)\right) ,&{} \quad \mathrm{for ~any~} q \end{array} \right. \end{aligned}$$

(5.342)

$$\begin{aligned} \displaystyle \epsilon _{3}\epsilon _{2}\approx & {} \left\{ \begin{array}{lll} \displaystyle \frac{M^{4}_{p}}{M^2V^{2}(T)\alpha ^{'2}}\left( 2\frac{V^{\prime \prime \prime }(T_i)V^{\prime }(T_i)}{V^{2}(T_i)}-10\frac{V^{\prime \prime }(T_i)V^{'2}(T_i)}{V^{3}(T_i)}+9\frac{V^{'4}(T_i)}{V^{4}(T_i)}\right) \\ \quad \displaystyle =\frac{\left( 2\xi ^2_{V}(T_i)-5\eta _{V}(T_i)\epsilon _{V}(T_i)+36\epsilon ^2_{V}(T_i)\right) }{M^2V^{2}(T)\alpha ^{'2}}\\ \quad \displaystyle =\left( 2\bar{\xi }^2_{V}(T_i)-5\bar{\eta }_{V}(T_i)\bar{\epsilon }_{V}(T_i)+36\bar{\epsilon }^2_{V}(T_i)\right) ,~&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{M^{4}_{p}}{M^2\sqrt{2q}V^{2}(T)\alpha ^{'2}}\left( 2\frac{V^{\prime \prime \prime }(T_i)V^{\prime }(T_i)}{V^{2}(T_i)}-10\frac{V^{\prime \prime }(T_i)V^{'2}(T_i)}{V^{3}(T_i)}+9\frac{V^{'4}(T_i)}{V^{4}(T_i)}\right) \\ \quad \displaystyle =\frac{\left( 2\xi ^2_{V}(T_i)-5\eta _{V}(T_i)\epsilon _{V}(T_i)+36\epsilon ^2_{V}(T_i)\right) }{M^2\sqrt{2q} V^{2}(T)\alpha ^{'2}}\\ \quad \displaystyle =\frac{\left( 2\bar{\xi }^2_{V}(T_i)-5\bar{\eta }_{V}(T_i)\bar{\epsilon }_{V}(T_i)+36\bar{\epsilon }^2_{V}(T_i)\right) }{\sqrt{2q}},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.343)

where $'=\mathrm{d}/\mathrm{d}T_{i}$ and the potential dependent slow-roll parameters $\epsilon _{V}, \eta _{V}, \xi ^{2}_{V}, \sigma ^{3}_{V}$ are defined as

$$\begin{aligned} \epsilon _{V}(T_{i})= & {} \frac{M^{2}_{p}}{2}\left( \frac{V^{\prime }(T_{i})}{V(T_{i})}\right) ^2,\end{aligned}$$

(5.344)

$$\begin{aligned} \eta _{V}(T_{i})= & {} M^{2}_{p}\left( \frac{V^{\prime \prime }(T_{i})}{V(T_{i})}\right) ,\end{aligned}$$

(5.345)

$$\begin{aligned} \xi ^{2}_{V}(T_{i})= & {} M^{4}_{p}\left( \frac{V^{\prime }(T_{i})V^{\prime \prime \prime }(T_i)}{V^2(T_{i})}\right) ,\end{aligned}$$

(5.346)

$$\begin{aligned} \sigma ^{3}_{V}(T_{i})= & {} M^{6}_{p}\left( \frac{V^{'2}(T_{i})V^{\prime \prime \prime \prime }(T_{i})}{V^3(T_{i})}\right) , \end{aligned}$$

(5.347)

which is exactly similar to the expression for the slow-roll parameter as appearing in the context of single field slow-roll inflationary models. However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for tachyonic inflation by rescaling with the appropriate powers of $\alpha ^{\prime }V(T)$:

$$\begin{aligned} \bar{\epsilon }_{V}(T_{i})= & {} \frac{\epsilon _{V}(T_{i})}{M\alpha ^{\prime }V(T)}=\frac{M^{2}_{p}}{2M\alpha ^{\prime }V(T)}\left( \frac{V^{\prime }(T_{i})}{V(T_{i})}\right) ^2=\frac{\bar{\epsilon }_{V}}{M},\nonumber \\\end{aligned}$$

(5.348)

$$\begin{aligned} \bar{\eta }_{V}(T_{i})= & {} \frac{{\eta }_{V}(T_{i})}{M\alpha ^{\prime }V(T)}=\frac{M^{2}_{p}}{M\alpha ^{\prime }V(T)}\left( \frac{V^{\prime \prime }(T_{i})}{V(T_{i})}\right) =\frac{\bar{\eta }_{V}}{M},\nonumber \\\end{aligned}$$

(5.349)

$$\begin{aligned} \bar{\xi }^{2}_{V}(T_{i})= & {} \frac{\xi ^{2}_{V}(T_{i})}{M^2\alpha ^{'2}V^2(T)}=\frac{M^{4}_{p}}{M^2\alpha ^{'2}V^2(T)}\left( \frac{V^{\prime }(T_{i})V^{\prime \prime \prime }(T_{i})}{V^2(T_{i})}\right) =\frac{\bar{\xi }^{2}_{V}}{M^2},\end{aligned}$$

(5.350)

$$\begin{aligned} \bar{\sigma }^{3}_{V}(T_{i})= & {} \frac{\sigma ^{3}_{V}(T_{i})}{M^3\alpha ^{'3}V^3(T)}=\frac{M^{6}_{p}}{M^2\alpha ^{'3}V^3(T)}\left( \frac{V^{'2}(T_{i})V^{\prime \prime \prime \prime }(T_{i})}{V^3(T_{i})}\right) =\frac{\bar{\sigma }^{3}_{V}}{M^3}. \end{aligned}$$

(5.351)

where $\bar{\epsilon }_{V}$, $\bar{\eta }_{V}$, $\bar{\xi }^{2}_{V}$ and $\bar{\sigma }^{3}_{V}$ are the single field tachyonic slow-roll parameters. Further using Eqs. (5.344)–(5.470) we get the following operator identity for tachyonic inflation:

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{H}\frac{\mathrm{d}}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}N}=\frac{\mathrm{d}}{\mathrm{d}\ln k}\approx \left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{2\bar{\epsilon }_{V}(T_i)}{MV(T)\alpha ^{\prime }}}M_{p}\left( 1-\frac{2}{3}\bar{\epsilon }_{V}(T_i)\right) ^{1/4}\frac{\mathrm{d}}{\mathrm{d}T_i}, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{2\bar{\epsilon }_{V}(T_i)}{MV(T)\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{1}{3q}\bar{\epsilon }_{V}(T_i)\right) ^{1/4}\frac{\mathrm{d}}{\mathrm{d}T_i},&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.352)

Finally using Eq. (5.352) we get the following sets of flow equations in the context of tachyonic inflation:

$$\begin{aligned} \displaystyle \frac{\mathrm{d}\epsilon _{1}}{\mathrm{d}N}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{\mathrm{d}\bar{\epsilon }_{V}(T_i)}{\mathrm{d}N}=\frac{2}{M^2}\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{2q}\frac{\mathrm{d}\bar{\epsilon }_{V}(T_i)}{\mathrm{d}N}=\frac{\bar{\epsilon }_{V}}{qM^2}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.353)

$$\begin{aligned} \displaystyle \frac{\mathrm{d}\epsilon _{2}}{\mathrm{d}N}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{2}{M^2}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{\sqrt{\frac{2}{q}}}{M^2}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.354)

$$\begin{aligned} \displaystyle \frac{\mathrm{d}(\epsilon _{2}\epsilon _{3})}{\mathrm{d}N}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{1}{M^3}\left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V}\right. \\ \left. ~~\displaystyle +194\bar{\epsilon }^2_{V} \bar{\eta }_{V}-10\bar{\eta }_{V}\bar{\epsilon }_{V} \right) \displaystyle \left( 1-\frac{2}{3}\bar{\epsilon }_{V}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{M^3\sqrt{2q}}\left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V}\right. \\ \left. ~~\displaystyle +194\bar{\epsilon }^2_{V} \bar{\eta }_{V}-10\bar{\eta }_{V}\bar{\epsilon }_{V} \right) \displaystyle \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.355)

where we use the following consistency conditions for the rescaled potential dependent slow-roll parameters:

$$\begin{aligned} \displaystyle \frac{\mathrm{d}\bar{\epsilon }_{V}}{\mathrm{d}N}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{2}{M^2}\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{2}{M^2}\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.356)

$$\begin{aligned} \displaystyle \frac{\mathrm{d}\bar{\eta }_{V}}{\mathrm{d}N}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{1}{M^2}\left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{M^2}\left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.357)

$$\begin{aligned} \displaystyle \frac{\mathrm{d}\bar{\xi }^2_{V}}{\mathrm{d}N}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{1}{M^3}\left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{M^3}\left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q, \end{array} \right. \end{aligned}$$

(5.358)

$$\begin{aligned} \displaystyle \frac{\mathrm{d}\bar{\sigma }^3_{V}}{\mathrm{d}N}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{1}{M^4}\bar{\sigma }^3_{V}\left( \bar{\eta }_{V}-12\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{1}{M^4}\bar{\sigma }^3_{V}\left( \bar{\eta }_{V}-12\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4},~&{} \quad \mathrm{for ~any~} q. \end{array} \right. \end{aligned}$$

(5.359)

In terms of the slow-roll parameters, the number of e-foldings can be re-expressed as

$$\begin{aligned} \begin{array}{lll} \displaystyle N=\left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{3\alpha ^{\prime }}{2}}\int ^{T_{i,end}}_{T_i}\frac{H(T)}{\sqrt{\epsilon _{1}}}~\mathrm{d}T_i=\sqrt{\frac{3\alpha ^{\prime }}{2}}\int ^{T_{\mathrm{end}}}_{T_i}\frac{H(T)}{\sqrt{\bar{\epsilon }_{V}(T_i)}}~\mathrm{d}T \nonumber \\ \quad \displaystyle \approx \frac{\alpha ^{\prime }M}{M^{2}_{p}}\int ^{T_i}_{T_{i,end}} \frac{V(T)V(T_i)}{V^{\prime }(T_i)}~\mathrm{d}T_i, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle 2q\sqrt{\frac{3\alpha ^{\prime }}{2}}\int ^{T_{i,end}}_{T_i}\frac{H(T)}{\sqrt{\epsilon _{1}}}~\mathrm{d}T_i= \sqrt{3\alpha ^{\prime }q}\int ^{T_{i,end}}_{T_i}\frac{H(T)}{\sqrt{\bar{\epsilon }_{V}(T_i)}}~\mathrm{d}T_i \nonumber \\ \quad \displaystyle \approx \frac{\sqrt{2q}\alpha ^{\prime }M}{M^{2}_{p}}\int ^{T_i}_{T_{i,end}} \frac{V(T)V(T_i)}{V^{\prime }(T_i)}~\mathrm{d}T_i,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

where $T_{i,end}$ characterizes the tachyonic field value at the end of inflation $t=t_{\mathrm{end}}$ for all i fields participating in assisted inflation. As in the case of assisted inflation all the M fields are identical; then one can re-express the number of e-foldings as

$$\begin{aligned} \begin{array}{lll} \displaystyle N=\left\{ \begin{array}{lll} \displaystyle \frac{\alpha ^{\prime }M}{M^{2}_{p}}\int ^{T_i}_{T_{i,end}} \frac{V^2(T_i)}{V^{\prime }(T_i)}~\mathrm{d}T_i=\frac{\alpha ^{\prime }M}{M^{2}_{p}}\int ^{T}_{T_{\mathrm{end}}} \frac{V^2(T)}{V^{\prime }(T)}~\mathrm{d}T, &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{ \sqrt{2q}\alpha ^{\prime }M}{M^{2}_{p}}\int ^{T_i}_{T_{i,end}} \frac{V^2(T_i)}{V^{\prime }(T_i)}~\mathrm{d}T_i=\frac{ \sqrt{2q}\alpha ^{\prime }M}{M^{2}_{p}}\int ^{T}_{T_{\mathrm{end}}} \frac{V^2(T)}{V^{\prime }(T)}~\mathrm{d}T,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

(5.360)

5.2.3 Basics of tachyonic perturbations

In this subsection we explicitly discuss the cosmological linear perturbation theory within the framework of assisted tachyonic inflation. Let us clearly mention that here we have various ways of characterizing cosmological perturbations in the context of inflation, which finally depend on the choice of gauge. Let us do the computation in the longitudinal gauge, where the scalar metric perturbations of the FLRW background are given by the following infinitesimal line element:

$$\begin{aligned} \mathrm{d}s^2= & {} -\left( 1+2\mathbf{\Phi }(t,{x})\right) \mathrm{d}t^2+a^{2}(t)\left( 1-2\mathbf{\Psi }(t,{x})\right) \delta _{ij}\mathrm{d}x^{i}\mathrm{d}x^{j},\nonumber \\ \end{aligned}$$

(5.361)

where a(t) is the scale factor, $\mathbf{\Phi }(t,{x})$ and $\mathbf{\Psi }(t,{x})$ characterizes the gauge invariant metric perturbations. Specifically, the perturbation of the FLRW metric leads to the perturbation in the energy-momentum stress tensor via the Einstein field equation or equivalently through the Friedmann equations. For the perturbed metric as mentioned in Eq. (5.361), the perturbed Einstein field equations can be expressed for the $q=1/2$ case of the tachyonic inflationary set-up as

$$\begin{aligned}&3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +\frac{k^{2}}{a^{2}(t)}=-\frac{1}{2M^{2}_{p}}\delta \rho _{i},\nonumber \\\end{aligned}$$

(5.362)

$$\begin{aligned}&\ddot{\mathbf{\Psi }}(t,\mathbf{k})+3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +H\dot{\mathbf{\Phi }}(t,\mathbf{k})\nonumber \\&\quad +\,2\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+\frac{k^{2}}{3a^{2}(t)} \left( \mathbf{\Phi }(t,\mathbf{k})-\mathbf{\Psi }(t,\mathbf{k})\right) =\frac{1}{2M^{2}_{p}}\delta p_{i},\nonumber \\\end{aligned}$$

(5.363)

$$\begin{aligned}&\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\mathbf{\Phi }(t,\mathbf{k})=-\frac{\alpha ^{\prime }V(T_i)}{\sqrt{1-\alpha ^{\prime }\dot{T}^{2}_{i}}}\frac{\dot{T}_{i}}{M^{2}_{p}}\delta T_{i},\end{aligned}$$

(5.364)

$$\begin{aligned}&{} \mathbf{\Psi }(t,\mathbf{k})-\mathbf{\Phi }(t,\mathbf{k})=0. \end{aligned}$$

(5.365)

Similarly, for any arbitrary q the perturbed Einstein field equations can be expressed as

$$\begin{aligned}&3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +\frac{k^{2}}{a^{2}(t)}=-\frac{1}{2M^{2}_{p}}\delta \rho _{i},\end{aligned}$$

(5.366)

$$\begin{aligned}&\ddot{\mathbf{\Psi }}(t,\mathbf{k})+3H\left( H\mathbf{\Phi }(t,\mathbf{k})+\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) +H\dot{\mathbf{\Phi }}(t,\mathbf{k})\nonumber \\&\quad +2\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+\frac{k^{2}}{3a^{2}(t)} \left( \mathbf{\Phi }(t,\mathbf{k})-\mathbf{\Psi }(t,\mathbf{k})\right) =\frac{1}{2M^{2}_{p}}\delta p_{i},\end{aligned}$$

(5.367)

$$\begin{aligned}&\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\mathbf{\Phi }(t,\mathbf{k})\nonumber \\&\quad =-\frac{\alpha ^{\prime }V(T_i)\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right] }{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{1-q}} \frac{\dot{T}_{i}}{M^{2}_{p}}\delta T_{i},\end{aligned}$$

(5.368)

$$\begin{aligned}&{} \mathbf{\Psi }(t,\mathbf{k})-\mathbf{\Phi }(t,\mathbf{k})=0. \end{aligned}$$

(5.369)

Here $\mathbf{\Phi }(t,\mathbf{k})$ and $\mathbf{\Psi }(t,\mathbf{k})$ are the two gauge invariant metric perturbations in the Fourier space. Additionally, it is important to note that in Eq. (5.65), the two gauge invariant metric perturbations $\mathbf{\Phi }(t,\mathbf{k})$ and $\mathbf{\Psi }(t,\mathbf{k})$ are equal in the context of a minimally coupled tachyonic string field theoretic model with an Einstein gravity sector. In Eqs. (5.62) and (5.63) the perturbed energy density $\delta \rho$ and pressure $\delta p$ are given by

$$\begin{aligned} \displaystyle \delta \rho _{i}= & {} \left\{ \begin{array}{lll} \displaystyle \frac{V^{\prime }(T_{i})\delta T_{i}}{\sqrt{1-\alpha ^{\prime }\dot{T}^2_{i}}} +\frac{\alpha ^{\prime }V(T_{i})\left( \dot{T}_{i}\delta \dot{T}_{i}+\dot{T}^2_{i}\mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{3/2}} ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{\left\{ V^{\prime }(T_{i})\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right] \delta T_{i}-4\alpha ^{\prime }(1-2q)V(T_{i})\dot{T}_{i}\delta \dot{T}_{i}\right\} }{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{1-q}}\\ \quad \displaystyle +\frac{2\alpha ^{\prime }(1-q)V(T_{i})\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right] \left( \dot{T}_{i}\delta \dot{T}_{i}+\dot{T}^2_{i}\mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{2-q}} ,&{} \quad \mathrm{for~any~arbitrary}~q. \end{array} \right. \\ \nonumber \displaystyle \delta p_{i}= & {} \left\{ \begin{array}{lll} \displaystyle -V^{\prime }(T_{i})\sqrt{1-\alpha ^{\prime }\dot{T}^2_{i}}\delta T_{i} +\frac{\alpha ^{\prime }V(T_{i})\left( \dot{T}_{i}\delta \dot{T}_{i}+\dot{T}^2\mathbf{\Phi }(t,\mathbf{k})\right) }{\sqrt{1-\alpha ^{\prime }\dot{T}^2}} ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -V^{\prime }(T_{i})\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{q}\delta T_{i}+\frac{2q\alpha ^{\prime }V(T_{i})\left( \dot{T}_{i}\delta \dot{T}_{i}+\dot{T}^2_{i}\mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{1-q}} ,&{} \quad \mathrm{for ~any~arbitrary}~q. \end{array} \right. \end{aligned}$$

(5.370)

Similarly after the variation of the tachyonic field we get the following expressions for the perturbed equation of motion:

$$\begin{aligned} \begin{array}{lll} \displaystyle 0 \approx \left\{ \begin{array}{lll} \displaystyle \delta \ddot{T}_{i}+3H\delta \dot{T}_{i}+\frac{2\alpha ^{\prime }\ddot{T}_{i}\left( \dot{T}_{i}\delta \dot{T}_{i}+\dot{T}^2_{i} \mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) } \\ \quad \displaystyle +\frac{M_{p}\sqrt{1-\alpha ^{\prime }\dot{T}^2_{i}}}{\alpha ^{\prime }V(T_{i})}\left[ \left( \frac{k^2}{a^2} - 3\dot{H}\right) \mathbf{\Phi }(t,\mathbf{k})-\frac{2k^2}{a^2}\mathbf{\Psi }(t,\mathbf{k}) \right. \\ \quad \left. \displaystyle -3\left( \ddot{\mathbf{\Psi }}(t,\mathbf{k})+4H\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\dot{\mathbf{\Phi }}(t,\mathbf{k})+\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+4H^2 \mathbf{\Phi }(t,\mathbf{k})\right) \right] \\ \quad \displaystyle -\left\{ 6H\alpha ^{\prime }\dot{T}^3_{i} - \frac{2V^{\prime }(T_{i})}{\alpha ^{\prime }V(T_{i})}\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) \right\} \mathbf{\Phi }(t,\mathbf{k})-\left( \dot{\mathbf{\Psi }}(t,\mathbf{k})+3\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) \dot{T}_{i}\\ \quad \displaystyle -\frac{M_{p}\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) }{\alpha ^{\prime }}\left( \frac{V^{\prime \prime }(T_{i})}{V(T_{i})}-\frac{V^{'2}(T_{i})}{V^{2}(T_{i})}\right) ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \delta \ddot{T}_{i}+3H\delta \dot{T}_{i}+\frac{2\alpha ^{\prime }\ddot{T}_{i}\left( \dot{T}_{i}\delta \dot{T}_{i}+\dot{T}^2_{i} \mathbf{\Phi }(t,\mathbf{k})\right) }{\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{2(1-q)}} \\ \quad \displaystyle +\frac{M_{p}\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{1-q}}{\alpha ^{\prime }V(T_{i})\left[ 1-\alpha ^{\prime }(1-2q)\dot{T}^2_{i}\right] }\left[ \left( \frac{k^2}{a^2} - 3\dot{H}\right) \mathbf{\Phi }(t,\mathbf{k})-\frac{2k^2}{a^2}\mathbf{\Psi }(t,\mathbf{k}) \right. \\ \quad \left. \displaystyle -3\left( \ddot{\mathbf{\Psi }}(t,\mathbf{k})+4H\dot{\mathbf{\Psi }}(t,\mathbf{k})+H\dot{\mathbf{\Phi }}(t,\mathbf{k})+\dot{H}{} \mathbf{\Phi }(t,\mathbf{k})+4H^2 \mathbf{\Phi }(t,\mathbf{k})\right) \right] \\ \quad \displaystyle -\left\{ 6H\alpha ^{\prime }\dot{T}^3_{i} - \sqrt{\frac{2}{q}}\frac{V^{\prime }(T_{i})}{\alpha ^{\prime }V(T_{i})}\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{2(1-q)}\right\} \mathbf{\Phi }(t,\mathbf{k})-\left( \dot{\mathbf{\Psi }}(t,\mathbf{k})+3\dot{\mathbf{\Psi }}(t,\mathbf{k})\right) \dot{T}_{i}\\ \quad \displaystyle -\frac{M_{p}\left( 1-\alpha ^{\prime }\dot{T}^2_{i}\right) ^{2(1-q)}}{\sqrt{2q}\alpha ^{\prime }}\left( \frac{V^{\prime \prime }(T_{i})}{V(T_{i})}-\frac{V^{'2}(T_{i})}{V^{2}(T_{i})}\right) ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

(5.371)

Further we will perform the following steps throughout the next part of the computation:

• First of all we decompose the scalar perturbations into two components-(1) entropic or isocurvature perturbations which can be usually treated as the orthogonal projective part to the trajectory and (2) adiabatic or curvature perturbations which can be usually treated as the parallel projective part to the trajectory.

• Within the framework of first order cosmological perturbation theory we define a gauge invariant primordial curvature perturbation on the scales outside the horizon:

  $$\begin{aligned} \zeta =\sum ^{M}_{i=1}\left( \zeta _{r}+\zeta _{\tilde{T}_{i}}\right) = \mathbf{\Psi }-\sum ^{M}_{i=1}\frac{H}{\dot{\rho }_{i}}\delta \rho _{i}. \end{aligned}$$

  (5.372)

• Next we consider the uniform density hypersurface in which

  $$\begin{aligned} \delta \rho _{i}=0 \quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

  (5.373)

  Consequently the curvature perturbation is governed by

  $$\begin{aligned} \zeta =\sum ^{M}_{i=1}\left( \zeta _{r}+\zeta _{\tilde{T}_{i}}\right) =\mathbf{\Psi }. \end{aligned}$$

  (5.374)

• Further, the time evolution of the curvature perturbation can be expressed as

  $$\begin{aligned} \dot{\zeta }=H\sum ^{M}_{i=1}\left( \frac{\delta \bar{p}_{i}}{\rho _{i}+p_{i}}\right) , \end{aligned}$$

  (5.375)

  where $\delta \bar{p}_{i}$ characterizes the non-adiabatic or entropic contribution in the first order linearized cosmological perturbation. In the present context $\delta \bar{p}_{i}$ can be expressed as

  $$\begin{aligned} \delta \bar{p}_{i}= \Gamma \dot{p}_{i}=\left( \frac{\partial p_{i}}{\partial S}\right) _{S}\delta S, \end{aligned}$$

  (5.376)

  where $\Gamma$ characterizes the relative displacement between hypersurfaces of uniform pressure and density. In the case of the assisted or multi-component fluid dynamical system, there are two contributions to the $\delta \bar{p}$ appearing in the computation:

  $$\begin{aligned} \delta \bar{p}_{i}=\delta \bar{p}_{rel,\tilde{T}_{i}}+\delta \bar{p}_{\mathrm{int}}, \end{aligned}$$

  (5.377)

  where $\delta \bar{p}_{rel,i}$ and $\delta \bar{p}_{\mathrm{int}}$ characterize the relative and intrinsic contributions to the multi-component fluid system given by

  $$\begin{aligned} \delta \bar{p}_{rel,\tilde{T}_{i}}= & {} \frac{1}{3H\dot{\rho }}\dot{\rho }_{r}\dot{\rho }_{\tilde{T}_{i}}\left( c^2_{r}-c^{2}_{S,\tilde{T}_{i}}\right) \mathcal{S}_{r\tilde{T}_{i}}\propto \dot{\rho }_{r},\nonumber \\\end{aligned}$$

  (5.378)
  $$\begin{aligned} \delta \bar{p}_{\mathrm{int}}= & {} \delta \bar{p}_{int,\tilde{T}_{i}}+\delta \bar{p}_{int,r}=\sum _{\alpha =r,\tilde{T}_{i}}\left( \delta p_{\alpha }-c^{2}_{\alpha }\delta \rho _{\alpha }\right) .\nonumber \\ \end{aligned}$$

  (5.379)

  Here $c^{2}_{S,\tilde{T}_{i}}$ and $c^2_{r}$ are the effective adiabatic sound speed and sound speed due to radiation. Also the relative entropic perturbation is defined as

  $$\begin{aligned} \mathcal{S}_{r\tilde{T}_{i}}= & {} 3\left( \zeta _{r}-\zeta _{\tilde{T}_{i}}\right) =-3H\left( \frac{\delta \rho _{r}}{\dot{\rho }_{r}}-\frac{\delta \rho _{\tilde{T}_{i}}}{\dot{\rho }_{\tilde{T}_{i}}}\right) \nonumber \\= & {} \frac{\delta _{r}}{1+w_{r}}-\frac{\delta _{\tilde{T}_{i}}}{1+w_{\tilde{T}_{i}}}, \end{aligned}$$

  (5.380)

  where $w_{r}$ and $w_{\tilde{T}_{i}}$ signify the equation of state parameters due to radiation and tachyonic matter, $\delta _{r}$ and $\delta _{\tilde{T}_{i}}$ characterize the energy density contrasts due to radiation and tachyonic matter; M is the number of tachyonic matter contents, defined by

  $$\begin{aligned} \delta _{\alpha }= & {} \frac{\delta \rho _{\alpha }}{\rho _{\alpha }} \quad \forall \; \alpha =r, \tilde{T}_{i}(\forall i=1,2,\ldots ,M). \end{aligned}$$

  (5.381)

  In a generalized prescription the pressure perturbation in arbitrary gauge can be decomposed into the following contributions:

  $$\begin{aligned} \delta p= & {} \sum ^{M}_{i=1}\delta p_{i} = c^{2}_{S}\delta \rho + \frac{1}{3H\dot{\rho }}\dot{\rho }_{r}\sum ^{M}_{i=1}\dot{\rho }_{\tilde{T}_{i}}\left( c^2_{r}-c^{2}_{S,\tilde{T}_{i}}\right) \mathcal{S}_{r\tilde{T}_{i}} \nonumber \\&+ \; \sum ^{M}_{i=1} \sum _{\alpha =r,\tilde{T}_{i}}\left( \delta p_{\alpha }-c^{2}_{\alpha }\delta \rho _{\alpha }\right) . \end{aligned}$$

  (5.382)

  As $\delta \bar{p}_{rel,\tilde{T}_{i}}\propto \dot{\rho }_{r}$, the relative non-adiabatic pressure perturbation is heavily suppressed in the computation and one can safely ignore such contributions during the epoch of inflation. Here the relative entropic perturbation $\mathcal{S}_{r\tilde{T}_{i}}$ remains small and finite in the limit of small $\dot{\rho }_{r}$ due to the smallness of the curvature perturbations $\zeta _{r}$ and $\zeta _{\tilde{T}_{i}}$.

• Now let us consider a situation where the equation of state parameter corresponding to radiation is

  $$\begin{aligned} w_{r}=c^2_{r}=\mathrm{Constant}. \end{aligned}$$

  (5.383)

  Consequently the fluctuation in pressure due to radiation can be expressed in terms of the fluctuation of the density:

  $$\begin{aligned} \delta p_{r}=w_{r}\delta \rho _{r}=c^{2}_{r}\delta \rho _{r}=\mathrm{Constant}\times \delta \rho _{r}\propto \delta \rho _{r}.\nonumber \\ \end{aligned}$$

  (5.384)

  This clearly implies that

  $$\begin{aligned} \delta \bar{p}_{int,r}=\left( \delta p_{r}-c^{2}_{r}\delta \rho _{r}\right) =0 \end{aligned}$$

  (5.385)

  i.e., the intrinsic non-adiabatic pressure vanishes identically. However, in terms of the effective tachyonic field $\tilde{T}$ this situation is not very simple, as the equation of state parameter and the intrinsic sound speed changes with the number of component fields M. During the inflationary epoch all M fields obey the following equation of state:

  $$\begin{aligned} w_{\tilde{T}_{i}}= & {} \frac{p_{\tilde{T}_{i}}}{\rho _{\tilde{T}_{i}}}\simeq -1=c^2_{S,\tilde{T}_{i}}=\mathrm{Constant} \nonumber \\&\quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

  (5.386)

  For all M effective tachyon fields the fluctuation in the pressure can be expressed as

  $$\begin{aligned} \delta p_{\tilde{T}_{i}} \simeq w_{\tilde{T}_{i}}\delta \rho _{\tilde{T}_{i}}=c^2_{S,\tilde{T}_{i}}\delta \rho _{\tilde{T}_{i}}\simeq -\delta \rho _{\tilde{T}_{i}}. \end{aligned}$$

  (5.387)

  Consequently the intrinsic non-adiabatic pressure within each tachyonic field amongst M fields can be expressed as

  $$\begin{aligned} \delta p_{int,\tilde{T}_{i}}=\delta p_{\tilde{T}_{i}}-c^2_{S,\tilde{T}_{i}}\delta \rho _{\tilde{T}_{i}}\simeq 0. \end{aligned}$$

  (5.388)

  Now in the present context the relative contributions to the non-adiabatic pressure of M tachyonic fields in this context can be expressed as

  $$\begin{aligned} \delta p_{rel,\tilde{T}_{i}\tilde{T}_{j}}\propto \left( c^2_{S,\tilde{T}_{i}}-c^2_{S,\tilde{T}_{j}}\right) \simeq 0, \end{aligned}$$

  (5.389)

  since

  $$\begin{aligned} c^2_{S,\tilde{T}_{i}}\simeq c^2_{S,\tilde{T}_{j}}\simeq -1 \quad \forall \; i,j=1,2,\ldots , M. \end{aligned}$$

  (5.390)

  This implies that the total non-adiabatic pressure is negligible during inflation, i.e.,

  $$\begin{aligned} \delta \bar{p}_{\mathrm{int}}= & {} \delta \bar{p}_{int,\tilde{T}_{i}}+\delta \bar{p}_{int,r}=\sum _{\alpha =r,\tilde{T}_{i}}\left( \delta p_{\alpha }-c^{2}_{\alpha }\delta \rho _{\alpha }\right) =0.\nonumber \\ \end{aligned}$$

  (5.391)

  and it is completely justified to write the fluctuation in the pressure for M individual tachyonic field contents as

  $$\begin{aligned} \delta \bar{p}_{i}=\delta \bar{p}_{rel,\tilde{T}_{i}}=\frac{1}{3H\dot{\rho }}\dot{\rho }_{r}\dot{\rho }_{\tilde{T}_{i}}\left( c^2_{r}-c^{2}_{S,\tilde{T}_{i}}\right) \mathcal{S}_{r\tilde{T}_{i}}\propto \dot{\rho }_{r}.\nonumber \\ \end{aligned}$$

  (5.392)

  Using these results the total fluctuation in the pressure can be re-expressed as

  $$\begin{aligned} \delta p= & {} \sum ^{M}_{i=1}\delta p_{i} = c^{2}_{S}\delta \rho + \sum ^{M}_{i=1}\delta \bar{p}_{i}=c^{2}_{S}\delta \rho +AM\dot{\rho }_{r}\nonumber \\\simeq & {} c^{2}_{S}\delta \rho , \end{aligned}$$

  (5.393)

  where A is the proportionality constant and we have safely ignored the contribution from the isocurvatrure modes due to its smallness in the case of assisted tachyonic inflation. Consequently we get

  $$\begin{aligned} \delta \bar{p}_{i}= & {} \Gamma \dot{p}_{i}=AM\dot{\rho }_{r}\simeq 0\nonumber \\\Rightarrow & {} \rho _{r}=\mathrm{Constant}~~~~\Rightarrow ~~~~\zeta =\mathrm{Constant}, \end{aligned}$$

  (5.394)

  which is exactly similar with the usual single field slow-roll conditions in the context of the tachyonic inflationary set-up discussed earlier. Finally, in the uniform density hypersurfaces, the curvature perturbation can be written in terms of the tachyonic field fluctuations on spatially flat hypersurfaces as

  $$\begin{aligned} \zeta= & {} -H\sum ^{M}_{i=1}\left( \frac{\delta T_{i}}{\dot{T}_{i}}\right) =-HM\left( \frac{\delta T}{\dot{T}}\right) \nonumber \\= & {} -HM\left( \frac{\delta \tilde{T}}{\dot{\tilde{T}}}\right) . \end{aligned}$$

  (5.395)

• But in the case of multi-tachyonic inflation where the M number of fields are not identical with each other the situation is not simpler like assisted inflation. In that case during the short time intervals when fields decay in a very much faster rate and its corresponding equation of state changes rapidly then it would be really interesting to investigate the production of isocurvature perturbations. On the scale smaller than the Hubble radius any entropy perturbation rapidly becomes adiabatic perturbation of the same amplitude, as local pressure differences, due to the local fluctuations in the equation of state, re-distribute the energy density. However, this change is slightly less efficient during the epoch of radiation compared to the tachyonic matter dominated epoch and can only occur after the decoupling between the photons and baryons, in the specific case of baryonic isocurvature perturbation. Causality precludes this re-distribution on scales bigger than the Hubble radius, and thus any entropy perturbation on these scales remains with a constant amplitude. Also it is important to note that the entropic or isocurvature perturbations are not affected by Silk damping, which is exactly contrary to the curvature or adiabatic perturbations.

5.2.4 Computation of scalar power spectrum

In this subsection we will not derive the results for assisted inflation as it is exactly the same as derived in the case of single tachyonic inflation. But we will state the results for the BD vacuum where the changes will appear due to the presence of M identical copies of the tachyon field. One can similarly write down the detailed expressions for AV as well. The changes will appear in the expressions for the following inflationary observables at the horizon crossing:

• In the present context amplitude of scalar power spectrum can be computed as

  $$\begin{aligned} \Delta _{\zeta , \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}-\frac{\mathcal{C}_{E}}{M}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{MH^{2}}{8\pi ^2M^{2}_{p}c_{S}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }},~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}-\frac{\mathcal{C}_{E}}{\sqrt{2q}M}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{qM H^{2}}{4\pi ^2M^{2}_{p}c_{S}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }} ,~&{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

  (5.396)

  where $H^2=MV/3M^{2}_{p}$ and $\mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72.$ Using the slow-roll approximations one can further approximate the expression for the sound speed as

  $$\begin{aligned} \begin{array}{lll} \displaystyle c^2_{S} =\left\{ \begin{array}{lll} \displaystyle 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}+\mathcal{O}\left( \frac{\bar{\epsilon }^2_{V}}{M^2}\right) +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle 1-\frac{(1-q)}{3q^2 M}\bar{\epsilon }_{V}+\mathcal{O}\left( \frac{\bar{\epsilon }^2_{V}}{M^2}\right) +\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

  (5.397)

Hence using the result in Eq. (5.408) we get the following simplified expression for the primordial scalar power spectrum:

$$\begin{aligned} \Delta _{\zeta , \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \frac{\bar{\epsilon }_{V}}{M}-\frac{\mathcal{C}_{E}}{M} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{MH^{2}}{8\pi ^2M^{2}_{p}\bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }} ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2qM}-\frac{\mathcal{C}_{E}}{\sqrt{2q}M}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{qM H^{2}}{4\pi ^2M^{2}_{p} \bar{\epsilon }_{V}}\right\} _{k_{\star }=a_{\star }H_{\star }},~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

(5.398)

• Next one can compute the scalar spectral tilt ($n_{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} n_{\zeta , \star }-1\approx & {} \left\{ \begin{array}{lll} \displaystyle \frac{1}{M}\left( 2\bar{\eta }_{V}-8\bar{\epsilon }_{V}\right) -\frac{2}{M^2}\bar{\epsilon }^{2}_{V}-\frac{2}{M^2}\left( 2\mathcal{C}_{E}+\frac{8}{3}\right) \bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \\ \quad \displaystyle -\frac{\mathcal{C}_{E}}{M^2}\left( 2\bar{\xi }^{2}_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^{2}_{V}\right) +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{1}{M}\sqrt{\frac{2}{q}}\bar{\eta }_{V}-\frac{1}{M}\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \bar{\epsilon }_{V}-\frac{\bar{\epsilon }^{2}_{V}}{2q^2M^2}\\ \quad \displaystyle -\frac{2}{(2q)^{3/2}M^2}\left( 2\mathcal{C}_{E}+3-2\Sigma \right) \bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \\ \quad \displaystyle -\frac{\mathcal{C}_{E}}{\sqrt{2q}M^2}\left( 2\bar{\xi }^{2}_{V}-5\bar{\eta }_{V}\bar{\epsilon }_{V}+36\bar{\epsilon }^{2}_{V}\right) +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.399)

• Next one can compute the running of the scalar spectral tilt ($\alpha _{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} \alpha _{\zeta , \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\left\{ \left[ \frac{4}{M^2}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{M}\right) (\bar{\eta }_{V}-3\bar{\epsilon }_{V})+\frac{2}{M^2}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] \right. \\ \quad \left. \displaystyle -\frac{\mathcal{C}_{E}}{M^3}\left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V} +194\bar{\epsilon }^2_{V}\bar{\eta }_{V}-10\bar{\eta }_{V}\bar{\epsilon }_{V} \right) \displaystyle \right. \\ \quad \left. \displaystyle -\frac{1}{M^3}\left( 2\mathcal{C}_{E}+\frac{8}{3}\right) \left[ 2\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V} -\bar{\xi }^2_{V}\right) \right. \right. \\ \quad \left. \left. \displaystyle -4\bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2\right] \right\} \displaystyle \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \quad \displaystyle -\left\{ \left[ \sqrt{\frac{2}{q}}\frac{\bar{\epsilon }_{V}}{qM^2}\left( 1+\frac{\bar{\epsilon }_{V}}{2qM}\right) (\bar{\eta }_{V}-3\bar{\epsilon }_{V})+\frac{1}{M^2}\sqrt{\frac{2}{q}}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] \right. \\ \quad \left. \displaystyle -\frac{\mathcal{C}_{E}}{\sqrt{2q}M^3}\left( 2\bar{\sigma }^{3}_{V}-216\bar{\epsilon }^{3}_{V}+2\bar{\xi }^{2}_{V}\bar{\eta }_{V}-7\bar{\xi }^{2}_{V}\bar{\epsilon }_{V} +194\bar{\epsilon }^2_{V}\bar{\eta }_{V}-10\bar{\eta }_{V}\bar{\epsilon }_{V} \right) \right. \\ \quad \left. \displaystyle -\frac{1}{M^3}\left( 2\mathcal{C}_{E}+\frac{8}{3}\right) \left[ \sqrt{\frac{2}{q}}\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V}-18\bar{\epsilon }^2_{V} -\bar{\xi }^2_{V}\right) \right. \right. \\ \quad \left. \left. \displaystyle -\frac{4}{(2q)^{3/2}}\bar{\epsilon }_{V}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) ^2\right] \right\} \displaystyle \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

• Finally, one can also compute the running of the running of scalar spectral tilt ($\kappa _{S}$) of the primordial scalar power spectrum as

$$\begin{aligned} \kappa _{\zeta , \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\frac{1}{M^3}\left[ 8\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2+c\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2\right. \\ \quad \left. \displaystyle +8\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})\left( \bar{\xi }^2_{V}-10\bar{\epsilon }_{V}\bar{\eta }_{V} +18\bar{\epsilon }^{2}_{V}\right) \right. \\ \quad \left. \displaystyle +2\left( 20\bar{\epsilon }_{V}\bar{\eta }_{V} \left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) +10\bar{\epsilon }_{V}\left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \right. \right. \\ \quad \left. \left. \displaystyle -72\bar{\epsilon }^2_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) - \left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \right) \right] \displaystyle \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \quad \displaystyle \displaystyle \displaystyle -\frac{1}{M^3}\left[ \left( \frac{2}{q}\right) ^{3/2}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2qM}\right) (\bar{\eta }_{V} -3\bar{\epsilon }_{V})^2+\sqrt{\frac{2}{q}}\frac{2}{q^2M}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2\right. \\ \quad \left. \displaystyle +\sqrt{\frac{2}{q}}\frac{2}{q}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) \left( \bar{\xi }^2_{V}-10\bar{\epsilon }_{V}\bar{\eta }_{V} +18\bar{\epsilon }^{2}_{V}\right) \right. \\ \quad \left. \displaystyle +\sqrt{\frac{2}{q}}\left( \frac{10}{q}\bar{\epsilon }_{V}\bar{\eta }_{V} \left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) +10\bar{\epsilon }_{V}\left( \bar{\xi }^2_{V}-4\bar{\epsilon }_{V}\bar{\eta }_{V}\right) \right. \right. \\ \quad \left. \left. \displaystyle -\frac{36}{q}\bar{\epsilon }^2_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) - \left( \bar{\sigma }^3_{V}+\bar{\xi }^2_{V}\bar{\eta }_{V}-\bar{\xi }^2_{V}\bar{\epsilon }_{V}\right) \right) \right] \displaystyle \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

(5.400)

5.2.5 Computation of tensor power spectrum

In this subsection we will not derive the results for assisted inflation as it is exactly the same as derived in the case of single tachyonic inflation. But we will state the results for the BD vacuum where the changes will appear due to the presence of M identical copies of the tachyon field. One can similarly write down the detailed expressions for AV as well. The changes will appear in the expressions for the following inflationary observables at the horizon crossing:

• In the present context amplitude of tensor power spectrum can be computed as

  $$\begin{aligned} \Delta _{h, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}\right] ^2 \frac{2H^{2}}{\pi ^2M^{2}_{p}}\right\} _{k_{\star }=a_{\star }H_{\star }},~~~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}\right] ^2 \frac{2H^{2}}{\pi ^2M^{2}_{p}}\right\} _{k_{\star }=a_{\star }H_{\star }} ,~~~&{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

  (5.401)

  where $\mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72.$

• Next one can compute the scalar spectral tilt ($n_{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} n_{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\frac{2}{M}\bar{\epsilon }_{V}\left[ 1+\frac{\bar{\epsilon }_{V}}{M}+\frac{2}{M}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \quad \displaystyle -\frac{\bar{\epsilon }_{V}}{qM}\left[ 1+\frac{\bar{\epsilon }_{V}}{2qM}+\frac{1}{M}\sqrt{\frac{2}{q}}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.402)

• Next one can compute the running of the tensor spectral tilt ($\alpha _{h}$) of the primordial scalar power spectrum as

  $$\begin{aligned} \alpha _{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\left[ \frac{4}{M^2}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{M}\right) (\bar{\eta }_{V}-3\bar{\epsilon }_{V})+\frac{4}{M^3}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) \right. \\ \quad \left. \displaystyle -\frac{8}{M^3}\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 \right. \\ \quad \left. \displaystyle -\frac{2}{M^3}\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] _{\star }\displaystyle \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \quad \displaystyle -\left[ 2\frac{\bar{\epsilon }_{V}}{qM^2}\left( 1+\frac{\bar{\epsilon }_{V}}{2qM}\right) (\bar{\eta }_{V}-3\bar{\epsilon }_{V})+\frac{1}{q^2M^3}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V}) \right. \\ \quad \left. \displaystyle -\frac{8}{(2q)^{5/2}M^3}\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 \right. \\ \quad \left. \displaystyle - \frac{1}{M^3}\sqrt{\frac{2}{q}}\bar{\epsilon }_{V}\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] _{\star }\displaystyle \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.403)

• Finally, one can also compute the running of the running of scalar spectral tilt ($\kappa _{S}$) of the primordial scalar power spectrum as

  $$\begin{aligned} \kappa _{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \displaystyle -\left[ \frac{8}{M^3}\bar{\epsilon }_{V}(1+\bar{\epsilon }_{V})(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2+\frac{8}{M^4}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2 \right. \\ \quad \left. \displaystyle -\frac{16}{M^4}\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}\left\{ (\bar{\eta }_{V}-3\bar{\epsilon }_{V})^3 -(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right\} \right. \\ \quad \left. \displaystyle -\frac{4}{M^3}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{M}\right) \left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right] _{\star }\displaystyle \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \quad \displaystyle \displaystyle \displaystyle -\left[ \frac{4}{qM^3}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2qM}\right) (\bar{\eta }_{V} -3\bar{\epsilon }_{V})^2+\frac{2}{q^2M^4}\bar{\epsilon }^2_{V}(\bar{\eta }_{V}-3\bar{\epsilon }_{V})^2\right. \\ \quad \left. \displaystyle +\frac{2}{qM^3}\bar{\epsilon }_{V}\left( 1+\frac{\bar{\epsilon }_{V}}{2q}\right) \left( \bar{\xi }^2_{V}-10\bar{\epsilon }_{V}\bar{\eta }_{V} +18\bar{\epsilon }^{2}_{V}\right) \right. \\ \quad \left. \displaystyle -\frac{8}{qM^4}\left( \mathcal{C}_{E}+1\right) \bar{\epsilon }_{V}\left\{ (\bar{\eta }_{V}-3\bar{\epsilon }_{V})^3 -(\bar{\eta }_{V}-3\bar{\epsilon }_{V})\left( 10\bar{\epsilon }_{V}\bar{\eta }_{V} -18\bar{\epsilon }^2_{V}-\bar{\xi }^2_{V}\right) \right\} \right] _{\star }\\ \quad \displaystyle \times \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}_{\star }+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.404)

5.2.6 Modified consistency relations

In this subsection we derive the new (modified) consistency relations for single tachyonic field inflation:

• 1. Next for the BD vacuum with $|k c_{S}\eta |=1$ case within slow-roll regime we can approximately write the following expression for the tensor-to-scalar ratio:

$$\begin{aligned} r_{\star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ \frac{16}{M}\bar{\epsilon }_{V}c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}-\frac{\mathcal{C}_{E}}{M}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ \frac{16}{M}\bar{\epsilon }_{V}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}\right] ^2}{\left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \frac{\bar{\epsilon }_{V}}{M}-\frac{\mathcal{C}_{E}}{M} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }},~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ \frac{8}{qM}\bar{\epsilon }_{V}c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}-\frac{\mathcal{C}_{E}}{M\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ \frac{8}{qM}\bar{\epsilon }_{V}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2qM}-\frac{\mathcal{C}_{E}}{M\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2}\right] _{k_{\star }=a_{\star }H_{\star }},~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

(5.405)

• 2. Hence the consistency relation between the tensor-to-scalar ratio r and spectral tilt $n_{T}$ for tensor modes for BD vacuum with $|k c_{S}\eta |=1$ case can be written as

  $$\begin{aligned} r_{\star }\approx & {} -8n_{h,\star }\times \underbrace{\left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}-\frac{\mathcal{C}_{E}}{M}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \left[ 1+\frac{\bar{\epsilon }_{V}}{M}+\frac{2}{M}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ \frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{M}\right] ^2}{\left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \frac{\bar{\epsilon }_{V}}{M}-\frac{\mathcal{C}_{E}}{M} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2\left[ 1+\frac{\bar{\epsilon }_{V}}{M}+\frac{2}{M}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }}, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ c_{S}\frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}-\frac{\mathcal{C}_{E}}{M\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2\left[ 1+\frac{\bar{\epsilon }_{V}}{2qM}+\frac{1}{M}\sqrt{\frac{2}{q}}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }=a_{\star }H_{\star }}\\ \displaystyle =\left[ \frac{\left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2qM}\right] ^2}{\left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2qM}-\frac{\mathcal{C}_{E}}{M\sqrt{2q}} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2\left[ 1+\frac{\bar{\epsilon }_{V}}{2qM}+\frac{1}{M} \sqrt{\frac{2}{q}}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] }\right] _{k_{\star }}, &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. }_\mathrm{Correction~factor} \end{aligned}$$

  (5.406)

• 3. Next one can express the first two slow-roll parameters $\bar{\epsilon }_{V}$ and $\bar{\eta }_{V}$ in terms of the inflationary observables as

  $$\begin{aligned} \bar{\epsilon }_{V}\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \epsilon _{1}\approx -\frac{n_{h,\star }M}{2}+\cdots \approx \frac{r_{\star }M}{16}+\cdots ,&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle 2q\epsilon _{1}\approx -qn_{h,\star }M+\cdots \approx \frac{qr_{\star }M}{8}+\cdots , &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

  (5.407)
  $$\begin{aligned} \bar{\eta }_{V}\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle 3\epsilon _{1}-\frac{\epsilon _{2}}{2}\approx \frac{M}{2}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{2}\right) +\cdots \approx \frac{M}{2}\left( n_{\zeta ,\star }-1-4n_{h,\star }\right) +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle 6q\epsilon _{1}-\sqrt{\frac{q}{2}}\epsilon _{2}\approx M\sqrt{\frac{q}{2}}\left( n_{\zeta ,\star }-1+\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \frac{qr_{\star }}{8}\right) +\cdots \\ \displaystyle \approx M\sqrt{\frac{q}{2}}\left( n_{\zeta ,\star }-1-q\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) n_{h,\star }\right) +\cdots ,~~ &{}\mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.408)

• 4. Then the connecting consistency relation between tensor and scalar spectral tilt and tensor-to-scalar ratio can be expressed as

  $$\begin{aligned} n_{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\frac{r_{\star }}{8c_{S}}\left[ 1-\frac{r_{\star }}{16}+\left( 1-n_{\zeta ,\star }\right) -\mathcal{C}_{E}\left\{ \frac{r_{\star }}{8} +\left( n_{\zeta ,\star }-1\right) \right\} \right] +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle -\frac{r_{\star }}{8c_{S}}\left[ 1+\left\{ \left( \frac{3q}{8}\sqrt{\frac{2}{q}} -\left( \sqrt{\frac{2}{q}}+5\right) \frac{1}{16}\right) r_{\star } +\frac{\left( 1-n_{\zeta ,\star }\right) }{\sqrt{2q}}\right\} \right. \\ \left. \displaystyle +\sqrt{\frac{2}{q}}\mathcal{C}_{E}\left( \frac{3qr_{\star }}{8}-\frac{1}{2}\left\{ n_{\zeta ,\star }-1+\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \frac{qr_{\star }}{8}\right\} \right) \right] +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.409)

  Finally using the approximated version of the expression for $c_{S}$ in terms of slow-roll parameters one can recast this consistency condition as

  $$\begin{aligned} n_{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\frac{r_{\star }}{8}\left[ 1-\frac{r_{\star }}{24}+\left( 1-n_{\zeta ,\star }\right) -\mathcal{C}_{E}\left\{ \frac{r_{\star }}{8} +\left( n_{\zeta ,\star }-1\right) \right\} \right] +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle -\frac{r_{\star }}{8}\left[ 1+\left\{ \left( \frac{3q}{8}\sqrt{\frac{2}{q}} -\left( \sqrt{\frac{2}{q}}+5\right) \frac{1}{16}+\frac{\Sigma }{8}\right) r_{\star } +\frac{\left( 1-n_{\zeta ,\star }\right) }{\sqrt{2q}}\right\} \right. \\ \left. \displaystyle +\sqrt{\frac{2}{q}}\mathcal{C}_{E}\left( \frac{3qr_{\star }}{8}-\frac{1}{2}\left\{ n_{\zeta ,\star }-1+\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \frac{qr_{\star }}{8}\right\} \right) \right] +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.410)

• 5. Next the running of the sound speed $c_{S}$ can be written in terms of slow-roll parameters as

  $$\begin{aligned} \displaystyle S= & {} \frac{\dot{c}_{S}}{Hc_{S}}=\frac{\mathrm{d}\ln c_{S}}{\mathrm{d}N}= \frac{\mathrm{d}\ln c_{S}}{\mathrm{d}\ln k} \nonumber \\= & {} \left\{ \begin{array}{lll} \displaystyle -\frac{2}{3M^2}\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{2}{3}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}+\cdots ,~&{} \quad \mathrm{for}~~ q=1/2 , \\ \displaystyle -\frac{(1-q)}{3q^2M^2}\bar{\epsilon }_{V}\left( \bar{\eta }_{V}-3\bar{\epsilon }_{V}\right) \left( 1-\frac{1}{3q}\frac{\bar{\epsilon }_{V}}{M}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any}~q, \end{array} \right. \end{aligned}$$

  (5.411)

  which can be treated as another slow-roll parameter in the present context. One can also recast the slow-roll parameter S in terms of the inflationary observables as

  $$\begin{aligned} \begin{array}{lll} \displaystyle S=\left\{ \begin{array}{lll} \displaystyle -\frac{r_{\star }}{48}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots ,~&{} \quad \mathrm{for}~~ q=1/2 ,\\ \displaystyle -\frac{(1-q)}{24q^2}\sqrt{\frac{q}{2}}r_{\star }\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any}~q. \end{array} \right. \end{array} \end{aligned}$$

  (5.412)

• 6. Further the running of the tensor spectral tilt can be written in terms of the inflationary observables as

  $$\begin{aligned} \alpha _{h, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -\left[ \frac{r_{\star }}{8}\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) -\frac{r_{\star }}{8}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right. \\ \left. \displaystyle -\mathcal{C}_{E}\frac{r_{\star }}{8}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~ q=1/2 ,\\ \displaystyle -\left[ \frac{r_{\star }}{4}\sqrt{\frac{q}{2}}\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right. \\ \left. \displaystyle -\frac{1}{(2q)^{1/2}}\left( \mathcal{C}_{E}+1\right) \frac{r_{\star }}{8}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.413)

• 7. Next the scalar power spectrum can be expressed in terms of the other inflationary observables as

  $$\begin{aligned} \Delta _{\zeta , \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2} \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right] ^2 \frac{2H^{2}_{\star }}{\pi ^2M^{2}_{p}r_{\star }}+\cdots ,~&{} \quad \mathrm{for}~~ q=1/2 ,\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] ^2 \displaystyle \frac{2 H^{2}_{\star }}{\pi ^2M^{2}_{p} r_{\star }}+\cdots ,~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.414)

• 8. Further the tensor power spectrum can be expressed in terms of the other inflationary observables as

  $$\begin{aligned} \Delta _{h, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ 1-(\mathcal{C}_{E}+1)\frac{r_{\star }}{16}\right] ^2 \frac{2H^{2}_{\star }}{\pi ^2M^{2}_{p}}+\cdots ,~~~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1)\frac{r_{\star }}{16}\right] ^2 \frac{2H^{2}_{\star }}{\pi ^2M^{2}_{p}}+\cdots ,~~~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.415)

• 9. Next the running of the tensor-to-scalar ratio can be expressed in terms of the inflationary observables as

  $$\begin{aligned} \alpha _{r,\star }= & {} -8\alpha _{h, \star }+\cdots \nonumber \\\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left[ r_{\star }\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) -r_{\star }\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right. \\ \left. \displaystyle -\mathcal{C}_{E}r_{\star }\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left[ 2r_{\star }\sqrt{\frac{q}{2}}\left( 1+\frac{r_{\star }}{8}\right) \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right. \\ \left. \displaystyle -\frac{1}{(2q)^{1/2}}\left( \mathcal{C}_{E}+1\right) r_{\star }\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) ^2 \right] \displaystyle \left( 1-\frac{r_{\star }}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.416)

• 10. Finally the scale of single field tachyonic inflation can be expressed in terms of the Hubble parameter and the other inflationary observables as

  $$\begin{aligned} H_{\mathrm{inf}}= & {} H_{\star } \approx \left\{ \begin{array}{lll} \displaystyle \displaystyle \frac{\sqrt{\frac{\Delta _{\zeta , \star }r_{\star }}{2}}\pi M_{p}}{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2} \left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right] } +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{\sqrt{\frac{\Delta _{\zeta , \star }r_{\star }}{2}}\pi M_{p}}{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] } +\cdots ,~&{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.417)

  One can recast this statement in terms of the inflationary potential as

  $$\begin{aligned} \root 4 \of {V_{\mathrm{inf}}}= & {} \root 4 \of {V_{\star }}\approx \left\{ \begin{array}{lll} \displaystyle \displaystyle \frac{\root 4 \of {\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\sqrt{\pi } M_{p}}{ \sqrt{\left[ 1-\left( \mathcal{C}_{E}+ \frac{5}{6}\right) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1+\frac{r_{\star }}{8}\right) \right] }} +\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{\root 4 \of {\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\sqrt{\pi } M_{p}}{ \sqrt{\left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] }} +\cdots , &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.418)

5.2.7 Field excursion for tachyon

In this subsection we explicitly derive the expression for the field excursion for tachyonic inflation defined as

$$\begin{aligned} |\Delta T|= & {} |T_{\mathrm{cmb}}-T_{\mathrm{end}}|=|T_{\star }-T_{\mathrm{end}}| \end{aligned}$$

(5.419)

where $T_{\mathrm{cmb}}$, $T_{\mathrm{end}}$ and $T_{\star }$ signify the tachyon field value at the time of horizon exit, at end of inflation and at pivot scale respectively. Here we perform the computation for both AV and BD vacuum. For for the sake of simplicity the pivot scale is fixed at the horizon exit scale. To compute the expression for the field excursion we perform the following steps:

• 1. We start with the operator identity for single field tachyon using which one can write expression for the tachyon field variation with respect to the momentum scale (k) or number of e-foldings (N) in terms of the inflationary observables as

  $$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{H}\frac{\mathrm{d}T}{\mathrm{d}t}=\frac{\mathrm{d}T}{\mathrm{d}N}=\frac{\mathrm{d}T}{\mathrm{d}\ln k}\approx \left\{ \begin{array}{lll} \displaystyle \sqrt{\frac{r}{8MV(T)\alpha ^{\prime }}}M_{p}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sqrt{\frac{qr}{4MV(T)\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any~arbitrary~} q, \end{array} \right. \end{array} \end{aligned}$$

  (5.420)

  where the tensor-to-scalar ratio r is a function of k or N.

• 2. Next using Eq. (5.172) we can write the following integral equation:

  $$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{T_{\star }}_{T_{\mathrm{end}}} \mathrm{d}T~\sqrt{V(T)} \approx \left\{ \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{r}{8M\alpha ^{\prime }}}M_{p}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots \,\\ \displaystyle \displaystyle \quad =\int ^{N_{\star }}_{N_{\mathrm{end}}} \mathrm{d}N~\sqrt{\frac{r}{8M\alpha ^{\prime }}}M_{p}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4M\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots \,\\ \displaystyle \quad =\int ^{N_{\star }}_{N_{\mathrm{end}}} \mathrm{d}N~\sqrt{\frac{qr}{4M\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}+\cdots ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array} \end{aligned}$$

  (5.421)

• 3. Next we use the same parametrization of the tensor-to-scalar ratio for $q=1/2$ and for any arbitrary q at any arbitrary scale as mentioned earlier for the single tachyonic field case.

• 4. For any value of q including $q=1/2$ we need to compute the following integral:

  $$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4M\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}\,\\ \approx \left\{ \begin{array}{ll} \displaystyle \sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r_{\star }}{24}\right) ^{1/4}\ln \left( \frac{k_{\star }}{k_{\mathrm{end}}}\right) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \frac{\sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }}}\frac{M_{p}}{q}}{3(n_{h,\star }-n_{\zeta ,\star }+1)}\left[ \left\{ _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \right] +2 \left( 1-\frac{r_{\star }}{24} \right) ^{1/4}\right\} \right. \\ \left. \displaystyle -\left( \frac{k_{\mathrm{end}}}{k_{\star }}\right) ^{\frac{n_{h,\star }-n_{\zeta ,\star }+1}{2}} \left\{ _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \left( \frac{k_{\mathrm{end}}}{k_{\star }}\right) ^{n_{h,\star }-n_{\zeta ,\star }+1}\right] \right. \right. \\ \left. \left. \displaystyle +2 \left( 1-\frac{r_{\star }}{24}\left( \frac{k_{\mathrm{end}}}{k_{\star }}\right) ^{n_{h,\star } -n_{\zeta ,\star }+1} \right) ^{1/4}\right\} \right] &{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle \sqrt{\frac{\pi q r_{\star }}{M\alpha ^{\prime }(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\frac{M_{p}}{48q} e^{-\frac{3 (n_{h,\star }-n_{\zeta ,\star }+1)^2}{4 (\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\\ \displaystyle ~~\left[ 12 e^{\frac{(n_{h,\star }-n_{\zeta ,\star }+1)^2}{ 2(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\left\{ \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}} \right) \right. \right. \\ \left. \left. \displaystyle - \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}{2}\ln \left( \frac{k_{\mathrm{end}}}{k_{\star }}\right) \right) \right\} \right. \\ \left. \displaystyle -\frac{\sqrt{3}r_{\star }}{24} \left\{ \text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}\right) \right. \right. \\ \left. \left. \displaystyle -\text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{3(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}{2} \ln \left( \frac{k_{\mathrm{end}}}{k_{\star }}\right) \right) \right\} \right] &{} \quad \mathrm{for~~\underline{Case~~III},} \end{array} \right.&{\underline{\mathrm{for~BD}}}. \end{array} \end{aligned}$$

  (5.422)

Similarly for AV we get the following result:

$$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4M\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}\, \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{M_{p}}{2c_{S}}\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}\ln \left( \frac{k_{\star }}{k_{\mathrm{end}}}\right) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{M_{p}\left| D\right| }{2c_{S}\left| C\right| }\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}\ln \left( \frac{k_{\star }}{k_{\mathrm{end}}}\right) &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~~AV}}}. \end{array}\nonumber \\ \end{aligned}$$

(5.423)

In terms of the number of e-foldings N one can re-express Eqs. (5.422) and (5.423) as

$$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4M\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}\,\\ \approx \left\{ \begin{array}{ll} \displaystyle \sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r_{\star }}{24}\right) ^{1/4}\left( N_{\star }-N_{\mathrm{end}}\right) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \frac{\sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }}}\frac{M_{p}}{q}}{3(n_{h,\star }-n_{\zeta ,\star }+1)}\left[ \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \right] +2 \left( 1-\frac{r_{\star }}{24} \right) ^{1/4}\right\} \right. \\ \left. \displaystyle -e^{\frac{n_{h,\star }-n_{\zeta ,\star }+1}{2}(N_{\mathrm{end}}-N_{\star })} \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} e^{(n_{h,\star }-n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })}\right] \right. \right. \\ \left. \left. \displaystyle +2 \left( 1-\frac{r_{\star }}{24}e^{(n_{h,\star } -n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })} \right) ^{1/4}\right\} \right] &{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle \sqrt{\frac{\pi q r_{\star }}{M\alpha ^{\prime }(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\frac{M_{p}}{48q} e^{-\frac{3 (n_{h,\star }-n_{\zeta ,\star }+1)^2}{4 (\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\\ \displaystyle ~~\left[ 12 e^{\frac{(n_{h,\star }-n_{\zeta ,\star }+1)^2}{ 2(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\left\{ \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}} \right) \right. \right. \\ \left. \left. \displaystyle - \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}{2}(N_{\mathrm{end}}-N_{\star }) \right) \right\} \right. \\ \left. \displaystyle -\frac{\sqrt{3}r_{\star }}{24} \left\{ \text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}\right) \right. \right. \\ \left. \left. \displaystyle -\text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{3(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}{2} (N_{\mathrm{end}}-N_{\star })\right) \right\} \right] &{} \quad \mathrm{for~~\underline{Case~~III},} \end{array} \right.&{\underline{\mathrm{for~BD}}}, \end{array} \end{aligned}$$

(5.424)

$$\begin{aligned} \begin{array}{lll} \displaystyle \int ^{k_{\star }}_{k_{\mathrm{end}}} \mathrm{d}\ln k~\sqrt{\frac{qr}{4\alpha ^{\prime }}}\frac{M_{p}}{2q}\left( 1-\frac{r}{24}\right) ^{1/4}\,\approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{M_{p}}{2c_{S}}\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{M_{p}\left| D\right| }{2c_{S}\left| C\right| }\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~~AV}}}. \end{array}\nonumber \\ \end{aligned}$$

(5.425)

Here the two possibilities for AV vacuum as appearing for the assisted inflationary framework are:

Case  I stands for a situation where the spectrum is characterized by the constraint i) $D_{1}=D_{2}=C_{1}=C_{2}\ne 0$, ii) $D_{1}=D_{2}$, $C_{1}=C_{2}=0$, iii) $D_{1}=D_{2}=0$, $C_{1}=C_{2}$.

Case  II stands for a situation where the spectrum is characterized by the constraint i) $\mu \approx \nu$, $D_{1}=D_{2}=D\ne 0$ and $C_{1}=C_{2}=C\ne 0$, ii) $\mu \approx \nu$, $D_{1}=D\ne 0$, $D_{2}=0$ and $C_{1}=C\ne 0$, $C_{2}=0$, iii) $\mu \approx \nu$, $D_{2}=D\ne 0$, $D_{1}=0$ and $C_{2}=C\ne 0$, $C_{1}=0$.

• 5. Next following exactly the same steps for single field inflation and also using Eqs. (5.424), (5.425) and (5.421) we get

  $$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}}\, \approx \left\{ \begin{array}{ll} \displaystyle \sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }V_{0}}}\frac{1}{2q}\left( 1-\frac{r_{\star }}{24}\right) ^{1/4}\left( N_{\star }-N_{\mathrm{end}}\right) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \frac{\sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }V_{0}}}\frac{1}{q}}{3(n_{h,\star }-n_{\zeta ,\star }+1)}\left[ \left\{ \,_2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \right] +2 \left( 1-\frac{r_{\star }}{24} \right) ^{1/4}\right\} \right. \\ \left. \displaystyle -e^{\frac{n_{h,\star }-n_{\zeta ,\star }+1}{2}(N_{\mathrm{end}}-N_{\star })} \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} e^{(n_{h,\star }-n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })}\right] \right. \right. \\ \left. \left. \displaystyle +2 \left( 1-\frac{r_{\star }}{24}e^{(n_{h,\star } -n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })} \right) ^{1/4}\right\} \right] &{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle \sqrt{\frac{\pi q r_{\star }}{M\alpha ^{\prime }(\alpha _{h,\star }-\alpha _{\zeta ,\star })V_{0}}}\frac{1}{48q} e^{-\frac{3 (n_{h,\star }-n_{\zeta ,\star }+1)^2}{4 (\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\\ \displaystyle ~~\left[ 12 e^{\frac{(n_{h,\star }-n_{\zeta ,\star }+1)^2}{ 2(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\left\{ \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}} \right) \right. \right. \\ \left. \left. \displaystyle - \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}{2}(N_{\mathrm{end}}-N_{\star }) \right) \right\} \right. \\ \left. \displaystyle -\frac{\sqrt{3}r_{\star }}{24} \left\{ \text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}\right) \right. \right. \\ \left. \left. \displaystyle -\text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{3(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}{2} (N_{\mathrm{end}}-N_{\star })\right) \right\} \right] &{} \quad \mathrm{for~~\underline{Case~~III},} \end{array} \right.&{\underline{\mathrm{for~BD}}}, \end{array} \end{aligned}$$

  (5.426)
  $$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}} \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }V_{0}}}\frac{1}{2c_{S}}\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }V_{0}}}\frac{\left| D\right| }{2c_{S}\left| C\right| }\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}}) &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~~AV}}}. \end{array} \end{aligned}$$

  (5.427)

• 6. Next using Eq. (5.418) in Eqs. (5.426) and (5.427) we get

  $$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}}\, \approx \left\{ \begin{array}{ll} \displaystyle \sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }}}\frac{1}{2q\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\left( 1-\frac{r_{\star }}{24}\right) ^{1/4}\left( N_{\star }-N_{\mathrm{end}}\right) \\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2\sqrt{2q}}\left( n_{\zeta ,\star }-1 \displaystyle +\frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \frac{\sqrt{\frac{qr_{\star }}{4M\alpha ^{\prime }}}\frac{1}{q\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}}{3(n_{h,\star }-n_{\zeta ,\star }+1)}\left[ \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} \right] +2 \left( 1-\frac{r_{\star }}{24} \right) ^{1/4}\right\} \right. \\ \left. \displaystyle -e^{\frac{n_{h,\star }-n_{\zeta ,\star }+1}{2}(N_{\mathrm{end}}-N_{\star })} \left\{ \, _2F_1\left[ \frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{r_{\star }}{24} e^{(n_{h,\star }-n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })}\right] \right. \right. \\ \left. \left. \displaystyle +2 \left( 1-\frac{r_{\star }}{24}e^{(n_{h,\star } -n_{\zeta ,\star }+1)(N_{\mathrm{end}}-N_{\star })} \right) ^{1/4}\right\} \right] \\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 \displaystyle +\frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~II},}\\ \displaystyle \frac{ \sqrt{\frac{\pi q r_{\star }}{M\alpha ^{\prime }(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}}{48q\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}} e^{-\frac{3 (n_{h,\star }-n_{\zeta ,\star }+1)^2}{4 (\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\\ \displaystyle ~~\left[ 12 e^{\frac{(n_{h,\star }-n_{\zeta ,\star }+1)^2}{ 2(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}\left\{ \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}} \right) \right. \right. \\ \left. \left. \displaystyle - \text {erfi}\left( \frac{n_{h,\star }-n_{\zeta ,\star }+1}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}{2}(N_{\mathrm{end}}-N_{\star }) \right) \right\} \right. \\ \left. \displaystyle -\frac{\sqrt{3}r_{\star }}{24} \left\{ \text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}\right) \right. \right. \\ \left. \left. \displaystyle -\text {erfi} \left( \frac{\sqrt{3} (n_{h,\star }-n_{\zeta ,\star }+1)}{2 \sqrt{\alpha _{h,\star }-\alpha _{\zeta ,\star }}}+\frac{\sqrt{3(\alpha _{h,\star }-\alpha _{\zeta ,\star })}}{2} (N_{\mathrm{end}}-N_{\star })\right) \right\} \right] \\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 \displaystyle +\frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~III},} \end{array} \right.&{\underline{\mathrm{for~BD}}}, \end{array} \end{aligned}$$

  (5.428)
  $$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}} \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{\left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2c_{S}\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{\left| D\right| \left( 1-\frac{qr_{\star }}{48c^{2}_{S}}\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2c_{S}\left| C\right| \sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~ AV}}}. \end{array} \end{aligned}$$

  (5.429)

  Further using the approximated form of the sound speed $c_{S}$ the expression for the field excursion for AV can be rewritten as

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{\Delta T}{M_{p}} \approx \left\{ \begin{array}{ll} \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{\left( 1-\frac{qr_{\star }}{48\left[ 1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}\right] }\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2\sqrt{1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}}\sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~I},} \\ \displaystyle \displaystyle \sqrt{\frac{r_{\star }}{8M\alpha ^{\prime }}}\frac{\left| D\right| \left( 1-\frac{qr_{\star }}{48\left[ 1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}\right] }\frac{\left| D\right| ^2}{\left| C\right| ^2}\right) ^{1/4}(N_{\star }-N_{\mathrm{end}})}{2\sqrt{1-\frac{(1-q)}{3q}\frac{r_{\star }}{8}}\left| C\right| \sqrt{\frac{3\Delta _{\zeta , \star }r_{\star }}{2}}\pi M^2_{p}}\\ \displaystyle \left[ 1-(\mathcal{C}_{E}+1-\Sigma ) \frac{r_{\star }}{16}+\frac{\mathcal{C}_{E}}{2}\left( n_{\zeta ,\star }-1 +\frac{r_{\star }}{8}\right) \right] &{} \quad \mathrm{for~~\underline{Case~~II},} \end{array} \right.&{\underline{\mathrm{for~~AV}}}. \end{array}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{aligned}$$

(5.430)

This implies that for BD and AV we get roughly the following result from this analysis:

$$\begin{aligned} \left| \frac{\Delta T}{M_p}\right| _\mathrm{Assisted}= & {} \frac{1}{\sqrt{M}}\times \left| \frac{\Delta T}{M_p}\right| _\mathrm{Single}, \end{aligned}$$

(5.431)

which means that if the number of tachyonic fields participating in the assisted inflation gradually increases, then the tachyonic field excursion for assisted inflation becomes more and more sub-Planckian compared to the single field result.

5.2.8 Semi-analytical study and cosmological parameter estimation

In this subsection our prime objective are:

• To compute various inflationary observables from variants of tachyonic potentials in the presence of M identical degrees of freedom.

• To estimate the relevant cosmological parameters from the proposed models.

• To compare the effectiveness of all of these models in the light of recent Planck 2015 data along with other combined constraints.

• Finally to check the compatibility of all of these models with the CMB TT, TE and EE angular power spectra as observed by Planck 2015.

However, instead of computing everything in detail we will not do any further computation in the context of assisted inflation using all the individual five potentials for that we have already done the analysis in the context of single tachyonic field earlier in this paper. In this context the results are exactly the same for all potentials that have already done for a single field, provided for all the models the stringy parameter g, appearing almost everywhere, is rescaled by the number of identical scalar fields in the present context, i.e., for the sake of simplicity here we define a new stringy parameter $\tilde{g}$, which is given by

$$\begin{aligned} \tilde{g}=gM= \underbrace{\frac{\alpha ^{\prime }\lambda T^2_0}{M^2_p}}\times M=\underbrace{\frac{M^4_s}{(2\pi )^3 g_s}\frac{\alpha ^{\prime } T^2_0 }{M^2_p}}\times M, \end{aligned}$$

(5.432)

where the terms pointed by the $\mathbf{\underbrace{}}$ symbol signify the exact contribution from the single tachyonic field. Now from the observational constraints instead of constraining the parameter g here we need to constrain the value of $\tilde{g}$ for all five tachyonic potentials mentioned earlier. Let us mention all the constraints on the stringy parameter $\tilde{g}$ for the assisted inflationary framework:

• Model  I: inverse  cosh  potential For $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/\tilde{g}\sim 0.8$, which further implies that for $50<N_{\star }<70$, the prescribed window for $\tilde{g}$ from the $\Delta _{\zeta }+n_{\zeta }$ plot is given by $63<\tilde{g}<88$. If we additionally impose the constraint from the upper bound on the tensor-to-scalar ratio then also the allowed parameter range is lying within a similar window, i.e., $88<\tilde{g}<100$.

  Table 1 Comparison between the field excursion obtained from all the tachyonic potentials from the single field and assisted inflationary frameworks. Here we define $y_0=T_0/M_p$, where $M_p=2.43\times 10^{18}\,\mathrm{GeV}$. The numerics clearly implies that the assisted tachyonic inflationary framework pushes the field excursion value to a sub-Planckian value for large M by a large amount, compared to the value obtained from single inflationary set-up. Technically this implies the doing effective field theory (EFT) with assisted framework is safer compared to single field case, as it involves an additional parameter M

  Model  II: logarithmic  potential For $q=1/2$, and $q=1$ we fix $N_{\star }/\tilde{g}\sim 0.7$, which further implies that for $50<N_{\star }<70$, the prescribed window for $\tilde{g}$ from the $\Delta _{\zeta }+n_{\zeta }$ plot is given by $71.4<g<100$. If we additionally impose the constraint from the upper bound on the tensor-to-scalar ratio then also the allowed parameter range is lying within a similar window, i.e., $71.4<\tilde{g}<90$. Model  III: exponential  potential-type  I For $q=1/2$, $q=1$, $q=3/2$ and $q=2$ case $\tilde{g}$ is not explicitly appearing in the various inflationary observables except the amplitude of scalar power spectrum in this case. To produce the correct value of the amplitude of the scalar power spectra we fix the parameter $360<\tilde{g}<400$. Model  IV: exponential  potential-type  II For $q=1/2$, $q=1$, $q=3/2$ and $q=2$ we fix $N_{\star }/g\sim 0.85$, which further implies that for $50<N_{\star }<70$, the prescribed window for g from $\Delta _{\zeta }+n_{\zeta }$ constraints is given by $59<\tilde{g}<82.3$. If we additionally impose the constraint from the upper bound on the tensor-to-scalar ratio then also the allowed parameter range is lying within the window, i.e., $73<\tilde{g}<82.3$. Model  V: inverse  power-law  potential For the $q=1/2$, $q=1$, $q=3/2$ and $q=2$ cases $\tilde{g}$ is not explicitly appearing in the various inflationary observables except the amplitude of scalar power spectrum in this case. To produce the correct value of the amplitude of the scalar power spectra we fix the parameter $600<\tilde{g}<700$.

In Table 1 we have shown the comparison between the field excursion obtained from all the tachyonic potentials from single field and assisted inflationary framework. Here we define $y_0=T_0/M_p$, where $M_p=2.43\times 10^{18}~\mathrm{GeV}$. The numerics clearly implies that the assisted tachyonic inflationary framework pushes the field excursion value to a sub-Planckian value for large M with large amount, compared to the value obtained from the single inflationary set-up. Technically this implies that effective field theory (EFT) with assisted framework is safer compared to the single field case, as it involves an additional parameter M.

Additionally it is important to mention here that the other conclusions and the rest of the constraints are exactly the same as analyzed in the single field case. Similarly the CMB TT, TE, EE spectra for the scalar modes are exactly the same as obtained in the context of single tachyonic inflation and compatible with the observed Planck 2015 data.

5.3 Computation for the multi-field inflation

In case of multi-tachyonic inflation all the tachyons are not identical. In more technical language for the most generalized prescription one can state that

$$\begin{aligned} T_{1}\ne T_{2}\ne \cdots \ne T_{M}. \end{aligned}$$

(5.433)

In the next subsections we will explore the detailed features of multi-tachyonic inflation by computing the curvature, isocurvature and tensor perturbations and then we discuss the observational constraints and cosmological consequences from the set-up. We will give all the analytical results for M non-identical tachyonic fields and for completeness also give the results for M number of non-identical inverse cosh separable potential.

5.3.1 Condition for inflation

For assisted tachyonic inflation, the prime condition for inflation is given by

$$\begin{aligned} \dot{H}+H^{2}= & {} \left( \frac{\ddot{a}}{{a}}\right) =-\sum ^{M}_{i=1}\frac{(\rho _{i}+3p_{i})}{6 M^2_p}>0 \end{aligned}$$

(5.434)

which can be re-expressed in terms of the following constraint condition in the context of assisted tachyonic inflation:

$$\begin{aligned} \begin{array}{lll} \displaystyle \sum ^{M}_{i=1}\frac{V(T_i)}{3M^{2}_{p}\sqrt{1-\alpha ^{\prime }\dot{T}^2_i}}\left( 1-\frac{3}{2}\alpha ^{\prime }\dot{T}^2_i\right) >0. \end{array} \end{aligned}$$

(5.435)

Here Eq. (5.316) implies that to satisfy inflationary constraints in the slow-roll regime the following constraint always holds good:

$$\begin{aligned} \dot{T}_{i}< & {} \sqrt{\frac{2}{3\alpha ^{\prime }}} \quad \forall \; i=1,2,\ldots ,M,\end{aligned}$$

(5.436)

$$\begin{aligned} \ddot{T}_{i}<&3H \dot{T}_{i}<\sqrt{\frac{6}{\alpha ^{\prime }}}H \quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

(5.437)

Consequently the field equations are approximated by

$$\begin{aligned} \begin{array}{lll} \displaystyle 3H\alpha ^{\prime }\dot{T}_{i}+\left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1}\frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) \approx 0 , \end{array}\nonumber \\ \end{aligned}$$

(5.438)

Similarly, in the most generalized case,

$$\begin{aligned} \begin{array}{lll} \displaystyle \sum ^{M}_{i=1}\frac{V(T_{i})}{3M^{2}_{p}\left( 1-\alpha ^{\prime }\dot{T}^2_{i} \right) ^{1-q}}\left( 1-(1+q)\alpha ^{\prime }\dot{T}^2_{i}\right) >0. \end{array} \end{aligned}$$

(5.439)

Here Eq. (5.439) implies that to satisfy inflationary constraints in the slow-roll regime the following constraint always holds good:

$$\begin{aligned} \dot{T}_{i}< & {} \sqrt{\frac{1}{\alpha ^{\prime }(1+q)}} \quad \forall \; i=1,2,\ldots ,M,\end{aligned}$$

(5.440)

$$\begin{aligned} \ddot{T_{i}}<&3H \dot{T}_{i}<\sqrt{\frac{9}{\alpha ^{\prime }(1+q)}}H \quad \forall \; i=1,2,\ldots ,M. \end{aligned}$$

(5.441)

Consequently the field equations are approximated by

$$\begin{aligned} \begin{array}{lll} \displaystyle 6q\alpha ^{\prime }H\dot{T}_{i}+\left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1}\frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) \approx 0 \end{array}\nonumber \\ \end{aligned}$$

(5.442)

Also for both cases in the slow-roll regime the Friedmann equation is modified as

$$\begin{aligned} H^{2}\approx & {} \sum ^{M}_{i=1}\frac{V(T_{i})}{3M^{2}_{p}}. \end{aligned}$$

(5.443)

Further substituting Eq. (5.443) in Eqs. (5.438) and (5.442) we get

$$\begin{aligned}&\displaystyle \sqrt{\sum ^{M}_{j=1} V(T_{j})}\frac{\sqrt{3}\alpha ^{\prime }}{M_p}\dot{T}_{i}+\left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1}\frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) \approx 0\nonumber \\&\qquad \forall i=1,2,\ldots ,M,\end{aligned}$$

(5.444)

$$\begin{aligned}&\frac{6q}{\sqrt{3}M_{p}}\sqrt{\sum ^{M}_{j=1} V(T_{j})}\alpha ^{\prime }\dot{T}_{i}+\left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1}\frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) \approx 0\nonumber \\&\qquad \forall i=1,2,\ldots ,M. \end{aligned}$$

(5.445)

Finally the general solution for both cases can be expressed in terms of the single field tachyonic potential V(T) as

$$\begin{aligned} \displaystyle&t-t_{in,i}\approx -\frac{\sqrt{3}\alpha ^{\prime }}{M_{p}}\nonumber \\&\int ^{T_{i}}_{T_{in,i}}\mathrm{d}T_{i}~ \sqrt{\sum ^{M}_{j=1} V(T_{j})}\left[ \left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1}\frac{\partial }{\partial T_{i}} \left( \sum ^{M}_{j=1} V(T_{j})\right) \right] ^{-1} ,\nonumber \\\end{aligned}$$

(5.446)

$$\begin{aligned}&t-t_{in,i}\approx -\frac{6q\alpha ^{\prime }}{\sqrt{3}M_{p}}\nonumber \\&\int ^{T_{i}}_{T_{in,i}}\mathrm{d}T_{i}~\sqrt{\sum ^{M}_{j=1} V(T_{j})}\left[ \left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1} \frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) \right] ^{-1}.\nonumber \\ \end{aligned}$$

(5.447)

Further using Eqs. (5.446), (5.447) and (5.443) we get the following solution for the scale factor in terms of the tachyonic field for the usual $q=1/2$ and for a generalized value of q:

$$\begin{aligned} \begin{array}{lll} \displaystyle a= a_{in,i}\times \left\{ \begin{array}{lll} \displaystyle \exp \left[ -\frac{\alpha ^{\prime }M}{M^2_p}\int ^{T_{i}}_{T_{in}}\mathrm{d}T_{i}~\sum ^{M}_{j=1} V(T_{j})\left[ \left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1} \frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) \right] ^{-1}\right] ,&{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \exp \left[ -\frac{2q\alpha ^{\prime }M}{M^2_p}\int ^{T_{i}}_{T_{in}}\mathrm{d}T_{i}~\sum ^{M}_{j=1} V(T_{j})\left[ \left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1} \frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) \right] ^{-1}\right] . &{} \quad \mathrm{for ~any}~~q. \end{array} \right. \end{array}\quad \quad \quad \quad \end{aligned}$$

(5.448)

Sometimes it is convenient to identify the inflaton field direction as the direction in field space corresponding to the evolution of the background spatially homogeneous tachyonic fields during inflation. To serve this purpose one can write

$$\begin{aligned} \sigma =\int \sum ^{M}_{i=1}\hat{\sigma }_{i}\dot{T}_{i}~\mathrm{d}t, \end{aligned}$$

(5.449)

where the inflaton direction is defined as

$$\begin{aligned} \hat{\sigma }_{i}\equiv \dot{T}\times \left[ \sum ^{M}_{j=1}\dot{T}^2_{j}\right] ^{-1/2}. \end{aligned}$$

(5.450)

The M evolution equations for the homogeneous scalar fields can be written as the evolution of single scalar field in the slow-roll regime as

$$\begin{aligned}&\displaystyle 3H\dot{\sigma }+\left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1}\frac{\partial }{\partial \sigma }\left( \sum ^{M}_{j=1} V(T_{j})\right) \nonumber \\&\quad =3H\dot{\sigma }+\sum ^{M}_{i=1} \hat{\sigma }_{i}\left( \sum ^{M}_{j=1} V(T_{j})\right) ^{-1}\frac{\partial }{\partial T_{i}}\left( \sum ^{M}_{j=1} V(T_{j})\right) =0.\nonumber \\ \end{aligned}$$

(5.451)

5.3.2 Analysis using slow-roll formalism

Here our prime objective is to define slow-roll parameters for tachyon inflation in terms of the Hubble parameter and the multi-tachyonic inflationary potential, where the M tachyon fields are not identical. Using the slow-roll approximation one can expand various cosmological observables in terms of small dynamical quantities derived from the appropriate derivatives of the Hubble parameter and of the inflationary potential. To start with, in the present context the potential dependent slow-roll parameters are defined as

$$\begin{aligned}&\epsilon _{V;T_{j}T_{j}}(T_{i})=\frac{M^{2}_{p}}{2}\left( \frac{\partial _{T_{j}}V(T_{i})}{V(T_{i})}\right) ^2, \end{aligned}$$

(5.452)

$$\begin{aligned}&\eta _{V;T_{j}T_{j}}(T_{i})= M^{2}_{p}\left( \frac{\partial _{T_{j}}\partial _{T_{j}}V(T_{i})}{V(T_{i})}\right) ,\end{aligned}$$

(5.453)

$$\begin{aligned}&\Theta _{V;T_{j}T_{k}}(T_{i})=M^{2}_{p}\left( \frac{\partial _{T_{j}}\partial _{T_{k}}V(T_{i})}{V(T_{i})}\right) , \end{aligned}$$

(5.454)

$$\begin{aligned}&\Delta _{V;T_{j}T_{k}}(T_{i})= \frac{M^{2}_{p}}{2}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{k}}V(T_{i})}{V^2(T_{i})}\right) ,\end{aligned}$$

(5.455)

$$\begin{aligned}&\xi ^{2}_{V;T_{j}T_{j}T_{j}T_{j}}(T_{i})= M^{4}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{j}}V(T_i)}{V^2(T_{i})}\right) ,\nonumber \\\end{aligned}$$

(5.456)

$$\begin{aligned}&\vartheta ^{2}_{V;T_{j}T_{j}T_{k}T_{k}}(T_{i})= M^{4}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{j}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^2(T_{i})}\right) ,\nonumber \\\end{aligned}$$

(5.457)

$$\begin{aligned}&\vartheta ^{2}_{V;T_{k}T_{j}T_{k}T_{k}}(T_{i})= M^{4}_{p}\left( \frac{\partial _{T_{k}}V(T_{i})\partial _{T_{j}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^2(T_{i})}\right) ,\nonumber \\\end{aligned}$$

(5.458)

$$\begin{aligned}&\vartheta ^{2}_{V;T_{j}T_{k}T_{k}T_{k}}(T_{i})= M^{4}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{k}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^2(T_{i})}\right) ,\nonumber \\ \end{aligned}$$

(5.459)

$$\begin{aligned}&\sigma ^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{j}}V(T_{i})\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{j}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.460)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{k}T_{k}}(T_{i})\nonumber \\&\quad =M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{j}}V(T_{i})\partial _{T_{k}}\partial _{T_{k}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.461)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{k}T_{k}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{k}}V(T_{i})\partial _{T_{k}}\partial _{T_{k}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.462)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{j}T_{k}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{j}}V(T_{i})\partial _{T_{j}}\partial _{T_{k}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.463)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{j}}V(T_{i})\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.464)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{j}}V(T_{i})\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.465)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{j}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{k}}V(T_{i})\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{j}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.466)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{k}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{k}}V(T_{i})\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.467)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{k}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{k}}V(T_{i})\partial _{T_{j}}\partial _{T_{j}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) ,\end{aligned}$$

(5.468)

$$\begin{aligned}&\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{k}}(T_{i})\nonumber \\&\quad = M^{6}_{p}\left( \frac{\partial _{T_{j}}V(T_{i})\partial _{T_{k}}V(T_{i})\partial _{T_{j}}\partial _{T_{k}}\partial _{T_{k}}\partial _{T_{k}}V(T_i)}{V^3(T_{i})}\right) . \end{aligned}$$

(5.469)

However, for the sake of clarity here we introduce new sets of potential dependent slow-roll parameters for multi-tachyonic inflation by rescaling with the appropriate powers of $\alpha ^{\prime }\left( \sum ^{M}_{k=1}V(T_{k})\right)$:

$$\begin{aligned}&\bar{\epsilon }_{V;T_{j}T_{j}}(T_{i})=\frac{\epsilon _{V;T_{j}T_{j}}(T_{i})}{\alpha ^{\prime }\left( \sum ^{M}_{k=1}V(T_{k})\right) },\nonumber \\&\quad \bar{\Delta }_{V;T_{j}T_{k}}(T_{i})=\frac{\Delta _{V;T_{j}T_{k}}(T_{i})}{\alpha ^{\prime }\left( \sum ^{M}_{j=1}V(T_{j})\right) },\nonumber \\&\bar{\eta }_{V;T_{j}T_{j}}(T_{i})= \frac{{\eta }_{V;T_{j}T_{j}}(T_{i})}{\alpha ^{\prime }\left( \sum ^{M}_{j=1}V(T_{j})\right) },\nonumber \\&\quad \bar{\Theta }_{V;T_{j}T_{k}}(T_{i})= \frac{{\Theta }_{V;T_{j}T_{k}}(T_{i})}{\alpha ^{\prime }\left( \sum ^{M}_{j=1}V(T_{j})\right) },\nonumber \\&\bar{\xi }^{2}_{V;T_{j}T_{j}T_{j}T_{j}}(T_{i})= \frac{\xi ^{2}_{V;T_{j}T_{j}T_{j}T_{j}}(T_{i})}{\alpha ^{'2}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^2},\nonumber \\&\quad \bar{\vartheta }^{2}_{V;T_{j}T_{j}T_{k}T_{k}}(T_{i})= \frac{\vartheta ^{2}_{V;T_{j}T_{j}T_{k}T_{k}}(T_{i})}{\alpha ^{'2}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^2},\nonumber \\&\bar{\vartheta }^{2}_{V;T_{k}T_{j}T_{k}T_{k}}(T_{i})= \frac{\vartheta ^{2}_{V;T_{k}T_{j}T_{k}T_{k}}(T_{i})}{\alpha ^{'2}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^2},\qquad \nonumber \\&\quad \bar{\vartheta }^{2}_{V;T_{j}T_{k}T_{k}T_{k}}(T_{i})= \frac{\vartheta ^{2}_{V;T_{k}T_{k}T_{j}T_{k}}(T_{i})}{\alpha ^{'2}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^2},\end{aligned}$$

(5.470)

$$\begin{aligned}&\bar{\sigma }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i})=\frac{{\sigma }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\nonumber \\&\quad \Upsilon ^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{k}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{k}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\end{aligned}$$

(5.471)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{k}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{k}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\nonumber \\&\quad \bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{j}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{j}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\end{aligned}$$

(5.472)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\nonumber \\&\quad \bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\end{aligned}$$

(5.473)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{j}}(T_{i})=\frac{\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{j}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\nonumber \\&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\end{aligned}$$

(5.474)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3},\nonumber \\&\quad \bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{k}}(T_{i})= \frac{\Upsilon ^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{k}}(T_{i})}{\alpha ^{'3}\left( \sum ^{M}_{j=1}V(T_{j})\right) ^3}. \end{aligned}$$

(5.475)

where in all cases $i,j,k =1,2,\ldots ,M$ and $j\ne k$. For the sake of simplicity let us parametrize the flow-functions in the slow-roll regime via two angular parameters, by making use of the following transformation equations:

$$\begin{aligned} \cos \alpha _{V;T_{j}T_{j}}:= & {} \sqrt{\frac{\bar{\epsilon }_{V;T_{j}T_{j}}(T_i)}{\bar{{\epsilon }}_{V}}},\end{aligned}$$

(5.476)

$$\begin{aligned} \sin \alpha _{V;T_{k}T_{k}}:= & {} \sqrt{\frac{\bar{\epsilon }_{V;T_{k}T_{k}}(T_i)}{\bar{{\epsilon }}_{V}}},\end{aligned}$$

(5.477)

$$\begin{aligned} \cos \beta _{V;T_{k}}:= & {} M_p ~\bar{{\epsilon }}_{V;T_{k}},\end{aligned}$$

(5.478)

$$\begin{aligned} \sin \beta _{V;T_{j}}:= & {} M_p ~\bar{\epsilon }_{V;T_{k}}. \end{aligned}$$

(5.479)

For the two-field set-up this can be visualized in a better way. In that case we need to fix $j=1$, $k=2$ or $j=2$, $k=1$. and i is the free index which can take values $i=1,2$ depending on the field derivative, $\partial _{T_{1}}$ or $\partial _{T_{2}}$ acting on it. In the present context these two sets of angular parameters physically represent the angle between the adiabatic perturbation, the tangent of the first slow-roll parameter and the field contents. Here also we define the following reduced parameters for multi-tachyonic inflation:

$$\begin{aligned} \bar{\epsilon }_{V}= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\epsilon }_{V;T_{j}T_{j}}(T_{i})\nonumber \\&+\;\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\left( \bar{\Delta }_{V;T_{j}T_{k}}(T_{i})+\bar{\Delta }_{V;T_{k}T_{j}}(T_{i})\right) ,\nonumber \\ \end{aligned}$$

(5.480)

$$\begin{aligned} \bar{\eta }_{V}= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\eta }_{V;T_{j}T_{j}}(T_{i})\nonumber \\&+\;\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\left( \bar{\Theta }_{V;T_{j}T_{k}}(T_{i})\!+\!\bar{\Theta }_{V;T_{k}T_{j}}(T_{i})\right) , \end{aligned}$$

(5.481)

$$\begin{aligned} \bar{\xi }^2_{V}= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\xi }^{2}_{V;T_{j}T_{j}T_{j}T_{j}}(T_{i})\nonumber \\&+\;\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\left( \bar{\vartheta }^{2}_{V;T_{j}T_{j}T_{k}T_{k}}(T_{i})+\bar{\vartheta }^{2}_{V;T_{k}T_{j}T_{k}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\vartheta }^{2}_{V;T_{j}T_{k}T_{k}T_{k}}(T_{i})+\bar{\vartheta }^{2}_{V;T_{j}T_{k}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\vartheta }^{2}_{V;T_{j}T_{k}T_{k}T_{j}}(T_{i})+\bar{\vartheta }^{2}_{V;T_{k}T_{k}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\vartheta }^{2}_{V;T_{k}T_{k}T_{k}T_{j}}(T_{i})\right) , \end{aligned}$$

(5.482)

$$\begin{aligned} \bar{\sigma }^3_{V}= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\sigma }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i})\\&+\;\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\left( \Upsilon ^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{k}T_{k}}(T_{i})\right. \\&+\;\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{k}T_{k}}(T_{i})+\;\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})\\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{j}}(T_{i})\right. \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{k}}(T_{i})\right. \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{k}T_{k}}(T_{i})\right. \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{j}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{k}T_{j}}(T_{i})\right. \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{j}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{k}T_{j}}(T_{i})\right. \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{j}T_{k}}(T_{i})\right. \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{j}T_{j}}(T_{i})\right. \end{aligned}$$

$$\begin{aligned}&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{j}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{j}T_{j}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{j}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{j}T_{j}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{j}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{j}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{j}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{j}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{k}T_{j}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{k}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{k}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{k}T_{k}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{k}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\left. +\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{k}T_{k}T_{j}}(T_{i})+\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{k}T_{k}}(T_{i})\right) .\nonumber \\ \end{aligned}$$

(5.483)

Now simplify the results let us mention the symmetries appearing due to various permutation of the indices:

$$\begin{aligned}&\bar{\Delta }_{V;T_{j}T_{k}}(T_{i})=\bar{\Delta }_{V;T_{k}T_{j}}(T_{i}),\end{aligned}$$

(5.484)

$$\begin{aligned}&\bar{\Theta }_{V;T_{j}T_{k}}(T_{i})=\bar{\Theta }_{V;T_{k}T_{j}}(T_{i}),\end{aligned}$$

(5.485)

$$\begin{aligned}&\bar{\vartheta }^{2}_{V;T_{j}T_{j}T_{k}T_{k}}(T_{i})=\bar{\vartheta }^{2}_{V;T_{j}T_{k}T_{j}T_{k}}(T_{i})=\bar{\vartheta }^{2}_{V;T_{j}T_{k}T_{k}T_{j}}(T_{i}),\nonumber \\\end{aligned}$$

(5.486)

$$\begin{aligned}&\bar{\vartheta }^{2}_{V;T_{k}T_{j}T_{k}T_{k}}(T_{i})=\bar{\vartheta }^{2}_{V;T_{k}T_{k}T_{j}T_{k}}(T_{i})=\bar{\vartheta }^{2}_{V;T_{k}T_{k}T_{k}T_{j}}(T_{i}),\nonumber \\\end{aligned}$$

(5.487)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{k}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{k}T_{k}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{j}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{k}T_{j}}(T_{i}),\nonumber \\\end{aligned}$$

(5.488)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{j}T_{k}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{k}T_{j}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{k}T_{j}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{j}T_{k}}(T_{i}),\end{aligned}$$

(5.489)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{j}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{j}T_{j}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{k}T_{j}T_{j}T_{j}}(T_{i}),\nonumber \\\end{aligned}$$

(5.490)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{j}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i}),\end{aligned}$$

(5.491)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{j}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{j}T_{j}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{j}T_{j}}(T_{i}),\nonumber \\\end{aligned}$$

(5.492)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{k}T_{j}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{j}T_{j}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{j}T_{j}}(T_{i}),\nonumber \\\end{aligned}$$

(5.493)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{j}T_{k}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{j}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{j}}(T_{i})\nonumber \\&\qquad =\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{k}T_{j}}(T_{i}),\end{aligned}$$

(5.494)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{j}T_{k}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{j}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{k}T_{j}}(T_{i})\nonumber \\&\qquad =\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{k}T_{j}}(T_{i}),\end{aligned}$$

(5.495)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{j}T_{k}T_{k}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{j}T_{k}T_{k}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{k}T_{j}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{k}T_{j}T_{k}T_{k}T_{k}T_{j}}(T_{i}),\nonumber \\\end{aligned}$$

(5.496)

$$\begin{aligned}&\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{k}}(T_{i})=\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{j}T_{k}T_{k}}(T_{i})\nonumber \\&\qquad =\;\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{j}T_{k}}(T_{i})= \bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{k}T_{j}}(T_{i}).\nonumber \\ \end{aligned}$$

(5.497)

Using these results one can finally re-express the reduced slow-roll parameters as

$$\begin{aligned} \bar{\epsilon }_{V}= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\epsilon }_{V;T_{j}T_{j}}(T_{i})+2\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\bar{\Delta }_{V;T_{j}T_{k}}(T_{i}), \nonumber \\ \end{aligned}$$

(5.498)

$$\begin{aligned} \bar{\eta }_{V}= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\eta }_{V;T_{j}T_{j}}(T_{i})+2\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\bar{\Theta }_{V;T_{j}T_{k}}(T_{i}), \nonumber \\\end{aligned}$$

(5.499)

$$\begin{aligned} \bar{\xi }^2_{V}= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\xi }^{2}_{V;T_{j}T_{j}T_{j}T_{j}}(T_{i})\nonumber \\&+\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\left( 3\bar{\vartheta }^{2}_{V;T_{j}T_{j}T_{k}T_{k}}(T_{i})\right. \nonumber \\&\left. +\;3\bar{\vartheta }^{2}_{V;T_{k}T_{j}T_{k}T_{k}}(T_{i})+\bar{\vartheta }^{2}_{V;T_{j}T_{k}T_{k}T_{k}}(T_{i})\right) , \end{aligned}$$

(5.500)

$$\begin{aligned}&\bar{\sigma }^3_{V}=\sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\sigma }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i})\nonumber \\&\quad +\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{k}T_{k}T_{k}T_{k}}(T_{i})+\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}\left( \Upsilon ^{3}_{V;T_{j}T_{j}T_{k}T_{k}T_{k}T_{k}}(T_{i})\right. \nonumber \\&\quad \left. +\;5\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{k}T_{k}}(T_{i})+4\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{k}}(T_{i})\right. \nonumber \\&\quad +2\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{j}}(T_{i})+8\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{j}T_{k}}(T_{i})\nonumber \\&\quad +\;10\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{j}T_{k}T_{k}}(T_{i})+8\bar{\Upsilon }^{3}_{V;T_{j}T_{k}T_{j}T_{k}T_{k}T_{k}}(T_{i})\nonumber \\&\quad \left. +4\bar{\Upsilon }^{3}_{V;T_{j}T_{j}T_{j}T_{k}T_{k}T_{k}}(T_{i}) \right) . \end{aligned}$$

(5.501)

Now in our computation we take separable potentials having same structural form for all M number of non-identical tachyons. In such case, for an example if take:

$$\begin{aligned}&V(T_1)= A_{1}\exp (-T_{1}/T_{01}),\end{aligned}$$

(5.502)

$$\begin{aligned}&V(T_2)= A_{2}\exp (-T_{2}/T_{02}),\end{aligned}$$

(5.503)

$$\begin{aligned}&\cdots \cdots \cdots \quad \cdots \cdots \cdots \cdots \cdots \cdots \end{aligned}$$

(5.504)

$$\begin{aligned}&V(T_M)= A_{M}\exp (-T_{M}/T_{0M}), \end{aligned}$$

(5.505)

which implies the structural form, we get the following simplified expression:

$$\begin{aligned}&\bar{\epsilon }_{V}=\sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\epsilon }_{V;T_{j}T_{j}}(T_{i}),\end{aligned}$$

(5.506)

$$\begin{aligned}&\bar{\eta }_{V}=\sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\eta }_{V;T_{j}T_{j}}(T_{i}),\end{aligned}$$

(5.507)

$$\begin{aligned}&\bar{\xi }^2_{V}=\sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\xi }^{2}_{V;T_{j}T_{j}T_{j}T_{j}}(T_{i}),\end{aligned}$$

(5.508)

$$\begin{aligned}&\bar{\sigma }^3_{V}=\sum ^{M}_{i=1}\sum ^{M}_{j=1}\bar{\sigma }^{3}_{V;T_{j}T_{j}T_{j}T_{j}T_{j}T_{j}}(T_{i}). \end{aligned}$$

(5.509)

The cross terms only appear when the structural form of the M number of tachyons are different. For example, if we choose

$$\begin{aligned}&V(T_{1})=A_{1}\exp (-T_{1}/T_{01}),\end{aligned}$$

(5.510)

$$\begin{aligned}&V(T_{2})= B_{2}\mathrm{cosh}(T_{2}/T_{02}),\end{aligned}$$

(5.511)

$$\begin{aligned}&\cdots \cdots \cdots \quad \cdots \cdots \cdots \cdots \cdots \cdots \end{aligned}$$

(5.512)

then all the cross terms in slow-roll vanish. Also for non-separable potentials this explanation works.

5.3.3 The $\delta N$ formalism for Multi tachyons

In this section we have used the $\delta N$ formalism to compute the inflationary observables, for the multi-tachyonic field set-up. Here N signifies the number of e-foldings, which can be expressed in the multi-tachyonic set-up as

$$\begin{aligned} \begin{array}{lll} N= \int ^{t_{\mathrm{end}}}_{t} Hdt=\left\{ \begin{array}{lll} \displaystyle \frac{\alpha ^{\prime }}{M^{2}_{p}}\sum ^{M}_{i=1}\int ^{T_i}_{T_{i,end}} \frac{V^2(T_i)}{V^{\prime }(T_i)}~\mathrm{d}T_i, &{} \mathrm{for}~~~q=1/2, \\ \displaystyle \frac{\sqrt{2q}\alpha ^{\prime }}{M^{2}_{p}}\sum ^{M}_{i=1}\int ^{T_i}_{T_{i,end}} \frac{V^2(T_i)}{V^{\prime }(T_i)}~\mathrm{d}T_i,&{} \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.513)

In the non-attractor regime, the $\delta N$ formalism shows various non-trivial features which have to be taken into account during explicit calculations. Once the solution reaches the attractor behavior, the dominant contribution comes from only on the perturbations of the scalar-field trajectories with respect to the tachyon field value at the initial hypersurface, $T_{i}$, as the velocity, $\dot{T}_{i}$, is uniquely determined by $T_{i}$ where $i=1,2,\ldots ,M$. However, in the non-attractor regime of solution, both the information from the field value $T_{i}$ and also $\dot{T}_{i}$ are required to determine the trajectory. This can be understood by providing two initial conditions on $T_i$ and $\dot{T}_i$ on the initial hypersurface. During the computation of the trajectories let us assume here that the universe has already arrived at the adiabatic limit via attractor phase by this epoch, or equivalently it can be stated that a typical phase transition phenomenon appears to an attractor phase at the time $t=t_*$. More specifically, in the present context, we have assumed that the evolution of the universe is unique after the value of the scalar field arrives at $T=T_*$ where it is mimicking the role of a standard clock, irrespective of the value of its velocity $\dot{T}_*$. Let us mention that only in this case $\delta N$ is equal to the final value of the comoving curvature perturbation $\zeta$ which is conserved at $t\ge t_*$. In Figs. 25 and 26, we have shown the schematic picture of the $\delta N$ formalism and the trajectories and perturbation decompositions in the field space for the multi-field tachyon inflation.

Fig. 25

Diagrammatic representation of $\delta N$ formalism. In this schematic picture $\Sigma (t_{i})$ and $\Sigma (t_{f})$ represent the initial and final hypersurface where time arrow flows from $t_{i}\rightarrow t_{f}$

Fig. 26

Diagrammatic representation of trajectories and perturbation decompositions in the field space. Here it is explicitly shown that any perturbation including field perturbations can be decomposed into a curvature component “curve” and an isocurvature component “iso”

On sufficiently large scales for given suitable assumptions as regards the dynamical behavior which permit us to ignore time derivatives of the perturbations, we expect that each horizon volume will evolve in such a manner that it behaves as a self contained universe, and consequently the curvature perturbation can be written beyond linear order in cosmological perturbation theory as [18, 139]

$$\begin{aligned} \zeta= & {} \delta N \nonumber \\= & {} \sum ^{M}_{i=1}N_{,i}\delta T^{i}+\frac{1}{2!}\sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,ij}\delta T^{i}\delta T^{j}\nonumber \\&+ \; \frac{1}{3!}\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{k=1}N_{,ijk}\delta T^{i}\delta T^{j}\delta T^{k}+\cdots , \end{aligned}$$

(5.514)

where we use the following short hand notations:

$$\begin{aligned} N_{,i}= & {} \partial _{T_{i}}N,\end{aligned}$$

(5.515)

$$\begin{aligned} N_{,ij}= & {} \partial _{T_{i}}\partial _{T_{j}}N,\end{aligned}$$

(5.516)

$$\begin{aligned} N_{,ijk}= & {} \partial _{T_{i}}\partial _{T_{j}}\partial _{T_{k}}N. \end{aligned}$$

(5.517)

More precisely here $\delta T_{i} \; \forall \, i=1,2,\ldots ,M$ represent the deviations of the fields from their unperturbed values in some specified region of the universe.

When the potential is sum-separable, the derivatives of N can be simplified to the following expressions:

$$\begin{aligned} N_{,i}= & {} \frac{1}{\sqrt{2\bar{\epsilon }_{V}(T^{\star }_{i})}}\frac{V(T^{\star }_{i})+Z^{c}_{i}}{\left( \sum ^{M}_{j=1}V(T^{\star }_{j})\right) },\end{aligned}$$

(5.518)

$$\begin{aligned} N_{,ij}= & {} \delta _{ij}\left[ 1-\frac{\bar{\eta }_{V}(T^{\star }_{i})}{2\bar{\epsilon }_{V}(T^{\star }_{i})}\frac{V(T^{\star }_{i})+Z^{c}_{i}}{\left( \sum ^{M}_{j=1}V(T^{\star }_{j})\right) }\right] \nonumber \\&+\;\frac{1}{\sqrt{2\bar{\epsilon }_{V}(T^{\star }_{j})}\left( \sum ^{M}_{m=1}V(T^{\star }_{m})\right) }Z^{c}_{j,i}, \end{aligned}$$

(5.519)

where $Z^{c}_{i}$ and $Z^{c}_{i,j}$ is defined as

$$\begin{aligned} Z^{c}_{i}= & {} \left( \sum ^{M}_{m=1}V(T^{c}_{m})\right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}-V(T^{c}_{i}),\end{aligned}$$

(5.520)

$$\begin{aligned} Z^{c}_{i,j}= & {} -\frac{\left( \sum ^{M}_{m=1}V(T^{c}_{m})\right) ^2}{\sum ^{M}_{p=1}V(T^{\star }_{p})}\sqrt{\frac{2}{\bar{\epsilon }_{V}(T_{j})}}\nonumber \\&\times \;\left[ \sum ^{M}_{k=1}\bar{\epsilon }_{V}(T_{k})\left( \frac{\bar{\epsilon }_{V}(T_{i})}{\bar{\epsilon }_{V}}-\delta _{ik}\right) \left( \frac{\bar{\epsilon }_{V}(T_{j})}{\bar{\epsilon }_{V}}-\delta _{jk}\right) \right. \nonumber \\&\times \; \left. \left( 1-\frac{\bar{\eta }_{V}(T_{k})}{\bar{\epsilon }_{V}}\right) \right] _{c}\nonumber \\= & {} \sqrt{\frac{2}{\bar{\epsilon }_{V}(T_{j})}}\left( \sum ^{M}_{m=1}V(T^{\star }_{m})\right) \mathcal{K}_{ij}, \end{aligned}$$

(5.521)

where $\star$ indicates the horizon crossing and c denotes the constant density surface. In this context additionally we get

$$\begin{aligned} \mathrm{d}N= & {} \sum ^{M}_{j=1}\left[ \left( \frac{V(T_{j})}{\partial _{j}V(T_{i})}\right) -\sum ^{M}_{i=1}\frac{\partial T^{c}_{i}}{\partial T^{\star }_{j}}\left( \frac{V(T_{i})}{\partial _{i}V(T_{k})}\right) \right] \mathrm{d}T^{\star }_{j},\nonumber \\ \end{aligned}$$

(5.522)

where

$$\begin{aligned} \frac{\partial T^{c}_{i}}{\partial T^{\star }_{j}}= & {} -\frac{\sum ^{M}_{k=1}V(T^{c}_{k})}{\sum ^{M}_{m=1}V(T^{\star }_{m})}\sqrt{\frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }_{V}(T^{\star }_{j})}}\left( \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\sum ^{M}_{p=1}\bar{\epsilon }_{V}(T_{p})}-\delta _{ij}\right) .\nonumber \\ \end{aligned}$$

(5.523)

Further using this, one can write the following differential operator identity which is very useful to compute various inflationary observables mentioned in the next section:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}N}= & {} \sum ^{M}_{j=1}\left[ \left( \frac{V(T_{j})}{\partial _{j}V(T_{i})}\right) -\sum ^{M}_{i=1}\frac{\partial T^{c}_{i}}{\partial T^{\star }_{j}}\left( \frac{V(T_{i})}{\partial _{i}V(T_{k})}\right) \right] ^{-1}\frac{\mathrm{d}}{\mathrm{d}T^{\star }_{j}}.\nonumber \\ \end{aligned}$$

(5.524)

In the next section we will write all the inflationary observables in terms of the components of $\delta N$ by including all the effects of non-linearities in cosmological perturbations for multi-tachyons.

5.3.4 Computation of scalar power spectrum

In this subsection we start with two point function for multi-tachyonic scalar modes. Implementing the $\delta N$ procedure we get:

$$\begin{aligned} \langle \zeta _\mathbf{k} \zeta _\mathbf{k^{\prime }} \rangle= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\langle \zeta ^{i}_\mathbf{k} \zeta ^{j}_\mathbf{k^{\prime }} \rangle =(2\pi )^3 \delta ^{3}(\mathbf{k}+\mathbf{k^{\prime }})\frac{2\pi ^2}{k^3}\Delta _{\zeta }(k),\nonumber \\ \end{aligned}$$

(5.525)

where the primordial power spectrum for the scalar modes at any arbitrary momentum scale k can be expressed as

$$\begin{aligned} \Delta _{\zeta }(k)= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{'j}G^{ij}\frac{k^3 P_{\zeta }(k)}{2\pi ^2}. \end{aligned}$$

(5.526)

Sometimes it is convenient to express everything in terms of the normalized adiabatic curvature power spectrum in the following way:

$$\begin{aligned} \langle \bar{\zeta }_\mathbf{k} \bar{\zeta }_\mathbf{k^{\prime }} \rangle= & {} (2\pi )^3 \delta ^{3}(\mathbf{k}+\mathbf{k^{\prime }})\frac{2\pi ^2}{k^3}\bar{\Delta }_{\zeta }(k), \end{aligned}$$

(5.527)

where

$$\begin{aligned}&\bar{\zeta }= -\frac{H}{\dot{T}_0}\sum ^{M}_{i=1}\sum ^{M}_{j=1}\Theta _{i}\delta T_{j}G^{ij},\end{aligned}$$

(5.528)

$$\begin{aligned}&\bar{\Delta }_{\zeta }(k)= \frac{1}{2\bar{\epsilon }_{V}}\sum ^{M}_{i=1}\sum ^{M}_{j=1}\Theta _{i}\Theta _{j}G^{ij}\frac{k^3 P_{\zeta }(k)}{2\pi ^2}. \end{aligned}$$

(5.529)

Here $\Theta _{i}=\frac{\dot{T}_{i}}{\dot{T}_{0}}$ is a basis vector that projects $\delta T_{i}$ along the direction of classical background trajectory. The vector $\mathbf{\Theta }$ and a complementary set of $(M-1)$ mutually orthonormal basis vectors $\mathbf{s_{p}}$ form the kinematic basis. Applying the Gram–Schmidt orthogonalization technique one can determine $\mathbf{s_{p}}$.

On the other hand for the multi-tachyonic scenario the isocurvature perturbations $\mathcal{S}_{p}$ is the orthogonal projection along the $\mathbf{s_{K}}$ directions:

$$\begin{aligned} \mathcal{S}_{p}= & {} -\frac{H}{\dot{T}_{0}}\sum ^{M}_{j=1}\sum ^{M}_{m=1}s_{pj}G^{jm}\delta T_{m}. \end{aligned}$$

(5.530)

Using this one can write down the expression for the normalized isocurvature perturbation:

$$\begin{aligned} \langle \mathcal{S}_{\mathbf{k}} \mathcal{S}_{\mathbf{k^{\prime }}}\rangle= & {} \sum ^{M-1}_{p=1}\sum ^{M-1}_{q=1}\langle \mathcal{S}_{p\mathbf{k}} \mathcal{S}_{q\mathbf{k^{\prime }}}\rangle \nonumber \\= & {} (2\pi )^3\delta ^{3}(\mathbf{k}+\mathbf{k^{\prime }})\frac{2\pi ^2}{k^3}\bar{\Delta }_\mathcal{S}(k), \end{aligned}$$

(5.531)

where

$$\begin{aligned} \bar{\Delta }_\mathcal{S}(k)= & {} \frac{1}{2\bar{\epsilon }_{V}}\sum ^{M-1}_{p=1}\sum ^{M-1}_{q=1}\sum ^{M}_{j=1}\sum ^{M}_{m=1}s_{j}^{p}s_{m}^{q}G^{jm}\frac{k^3 P_{\zeta }(k)}{2\pi ^2},\nonumber \\ \end{aligned}$$

(5.532)

where we have explicitly shown all the summations to indicate that the isocurvature basis vectors are $(M-1)$-dimensional. For the sake of simplicity applying the Gram–Schmidt orthogonalization technique one can choose a basis where $G^{ij}$ is diagonal and given by $G^{ij}=\delta ^{ij}$.

Similarly, for completeness one can define the adiabatic-isocurvature cross spectra in the following fashion:

$$\begin{aligned} \langle \bar{\zeta }_{\mathbf{k}} \mathcal{S}_{\mathbf{k^{\prime }}}\rangle= & {} \sum ^{M-1}_{p=1}\langle \bar{\zeta }_{\mathbf{k}} \mathcal{S}_{p\mathbf{k^{\prime }}}\rangle \nonumber \\ {}= & {} (2\pi )^3\delta ^{3}(\mathbf{k}+\mathbf{k^{\prime }})\frac{2\pi ^2}{k^3}\bar{\Delta }_{\zeta \mathcal{S}}(k), \end{aligned}$$

(5.533)

where

$$\begin{aligned} \bar{\Delta }_{\zeta \mathcal{S}}(k)= & {} \frac{1}{2\bar{\epsilon }_{V}}\sum ^{M-1}_{p=1}\sum ^{M}_{j=1}\sum ^{M}_{m=1}\Theta _{j}s_{m}^{p}\left( G^{jm}+G^{mj}\right) \frac{k^3 P_{\zeta }(k)}{2\pi ^2}.\nonumber \\ \end{aligned}$$

(5.534)

Cross correlations are generically expected if the background trajectory is curved as the modes of interest leave the horizon.

Now we write down the expressions for all the inflationary observables computed from the multi-tachyonic set-up at a horizon crossing:

• In the present context the amplitude of scalar power spectrum can be computed as

$$\begin{aligned} \Delta _{\zeta , \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij} \left\{ \left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}-\mathcal{C}_{E}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{4\pi ^2c_{S}}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij}\left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{2\pi ^2c_{S}}\right\} _{\star }, &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

(5.535)

where $H^2=V/3M^{2}_{p}=\sum ^{M}_{k=1}V(T_{k})/3M^2_p$ and $\mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72.$ Using the slow-roll approximations one can further approximate the expression for the sound speed as

$$\begin{aligned} \begin{array}{lll} \displaystyle c^2_{S} =\left\{ \begin{array}{lll} \displaystyle 1-\frac{2}{3}\bar{\epsilon }_{V}+\mathcal{O}\left( \bar{\epsilon }^2_{V}\right) +\cdots ,~&{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle 1-\frac{(1-q)}{3q^2 }\bar{\epsilon }_{V}+\mathcal{O}\left( \bar{\epsilon }^2_{V}\right) +\cdots ,&{} \quad \mathrm{for ~any}~q, \end{array}\right. \end{array} \end{aligned}$$

(5.536)

where $\bar{\epsilon }_{V}$ and $\bar{\eta }_{V}$ are the cumulative or total contribution to the slow-roll parameter as defined earlier. Hence using the result in Eq. (5.535) we get the following simplified expression for the primordial scalar power spectrum:

$$\begin{aligned} \Delta _{\zeta , \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij} \left\{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \bar{\epsilon }_{V}-\mathcal{C}_{E} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{4\pi ^2}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=1/2\\ \displaystyle \sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij}\left\{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{2\pi ^2}\right\} _{\star }, &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

(5.537)

Similarly, using the normalized adiabatic curvature power spectrum, we get

$$\begin{aligned} \bar{\Delta }_{\zeta , \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \sum ^{M}_{i=1}\sum ^{M}_{j=1}\Theta _{i}\Theta _{j}G^{ij} \left\{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \bar{\epsilon }_{V}-\mathcal{C}_{E} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \sum ^{M}_{i=1}\sum ^{M}_{j=1}\Theta _{i}\Theta _{j}G^{ij}\left\{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

(5.538)

• In the present context the normalized amplitude of isocurvature power spectrum can be computed as

  $$\begin{aligned} \bar{\Delta }_{\mathcal{S}, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \sum ^{M-1}_{p,q=1}\sum ^{M}_{j,m=1}s_{j}^{p}s_{m}^{q}G^{jm} \left\{ \left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}-\mathcal{C}_{E}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2c_{S}\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \sum ^{M-1}_{p,q=1}\sum ^{M}_{j,m=1}s_{j}^{p}s_{m}^{q}G^{jm}\left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2c_{S}\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for~~any}~~ q, \end{array}\right. \end{aligned}$$

  (5.539)

  where $H^2=V/3M^{2}_{p}=\sum ^{M}_{k=1}V(T_{k})/3M^2_p$ and $\mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72.$ Further using an approximated expression for $c_{S}$ in the slow-roll regime and also using the result in Eq. (5.539) we get the following simplified expression for the power spectrum:

  $$\begin{aligned} \bar{\Delta }_{\mathcal{S}, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \sum ^{M-1}_{p,q=1}\sum ^{M}_{j,m=1}s_{j}^{p}s_{m}^{q}G^{jm} \left\{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \bar{\epsilon }_{V}-\mathcal{C}_{E} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=\frac{1}{2} \\ \displaystyle \sum ^{M-1}_{p,q=1}\sum ^{M}_{j,m=1}s_{j}^{p}s_{m}^{q}G^{jm}\left\{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \end{aligned}$$

  (5.540)

• Similarly the normalized amplitude of the adiabatic-isocurvature cross power spectrum can be computed as

  $$\begin{aligned} \bar{\Delta }_{\zeta \mathcal{S}, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \sum ^{M-1}_{p=1}\sum ^{M}_{j,m=1}\Theta _{j}s_{m}^{p}\left( G^{jm}+G^{mj}\right) \left\{ \left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}-\mathcal{C}_{E}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2c_{S}\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \sum ^{M-1}_{p=1}\sum ^{M}_{j,m=1}\Theta _{j}s_{m}^{p}\left( G^{jm}+G^{mj}\right) \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2c_{S}\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \nonumber \\ \end{aligned}$$

  (5.541)

  where $H^2=V/3M^{2}_{p}=\sum ^{M}_{k=1}V(T_{k})/3M^2_p$ and $\mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72.$ Further using an approximated expression for $c_{S}$ in the slow-roll regime and also using the result in Eq. (5.541) we get the following simplified expression for the power spectrum:

  $$\begin{aligned} \bar{\Delta }_{\zeta \mathcal{S}, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \sum ^{M-1}_{p=1}\sum ^{M}_{j,m=1}\Theta _{j}s_{m}^{p}\left( G^{jm}+G^{mj}\right) \left\{ \left[ 1-\left( \mathcal{C}_{E}+\frac{5}{6}\right) \bar{\epsilon }_{V}-\mathcal{C}_{E} \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{H^{2}}{8\pi ^2\bar{\epsilon }_{V}}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=\frac{1}{2} \\ \displaystyle \sum ^{M-1}_{p=1}\sum ^{M}_{j,m=1}\Theta _{j}s_{m}^{p}\left( G^{jm}+G^{mj}\right) \left\{ \left[ 1-(\mathcal{C}_{E}+1-\Sigma )\frac{\bar{\epsilon }_{V}}{2q}-\frac{\mathcal{C}_{E}}{\sqrt{2q}}\left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] ^2 \frac{q H^{2}}{4\pi ^2\bar{\epsilon }_{V}}\right\} _{\star }\, &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \nonumber \\ \end{aligned}$$

  (5.542)

• Next one can compute the scalar spectral tilt ($n_{\zeta }, n_\mathcal{S},n_{\zeta \mathcal{S}}$) of the primordial adiabatic, isocurvature and cross power spectrum as

  $$\begin{aligned} n_{\zeta , \star }-1\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \displaystyle \left( 2\bar{\eta }_{V}-8\bar{\epsilon }_{V}\right) -\frac{2}{\sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij}}\\ \displaystyle +\frac{2}{\sum ^{M}_{m=1}V(T_{m})}\frac{\sum ^{M}_{n=1}\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{l=1}\sum ^{M}_{k=1}V_{,ij}(T_{n})N_{,l}N_{,k}G^{il}G^{jk}}{\sum ^{M}_{i^{\prime }=1}\sum ^{M}_{j^{\prime }=1}N_{,i^{\prime }}N_{,j^{\prime }}G^{i^{\prime }j^{\prime }}}+\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle \displaystyle \sqrt{\frac{2}{q}}\bar{\eta }_{V}-\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \bar{\epsilon }_{V}-\frac{1}{q\sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij}}\\ \displaystyle + \frac{1}{q\sum ^{M}_{m=1}V(T_{m})}\frac{\sum ^{M}_{n=1}\sum ^{M}_{i=1}\sum ^{M}_{j=1}\sum ^{M}_{l=1}\sum ^{M}_{k=1}V_{,ij}(T_{n})N_{,l}N_{,k}G^{il}G^{jk}}{\sum ^{M}_{i^{\prime }=1}\sum ^{M}_{j^{\prime }=1}N_{,i^{\prime }}N_{,j^{\prime }}G^{i^{\prime }j^{\prime }}}+\cdots , &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

  (5.543)

  and additionally we have

  $$\begin{aligned} n_\mathcal{S,\star }-1\approx & {} n_{\zeta ,\star }-1\approx n_{\zeta \mathcal{S},\star }-1. \end{aligned}$$

  (5.544)

• One can also compute the expression for the running and running of the running by following the same procedure as mentioned above.

For completeness let us mention the behavior of the cosmological perturbation at later times for multi Gtachyonic inflation. To start with it is important to mention here that in a more generalized physical prescription the time dependence of adiabatic and entropy perturbations in the large cosmological scale limit can always be written in the following simplified form for multi Gtachyonic inflationary paradigm as

$$\begin{aligned} \frac{\mathrm{d}\zeta }{\mathrm{d}t}=\dot{\zeta }= & {} \Upsilon H \mathcal{S},\end{aligned}$$

(5.545)

$$\begin{aligned} \frac{\mathrm{d}\mathcal{S}}{\mathrm{d}t}=\dot{\mathcal{S}}= & {} \Sigma H \mathcal{S}, \end{aligned}$$

(5.546)

where $\Upsilon$ and $\Sigma$ physically represent the generalized time-dependent dimensionless functions in cosmological perturbation theory of multi Gtachyonic fields. Now to extract more informations form Eqs. (5.545) and (5.546), we further integrate both equations over the specified cosmological time scale and finally following this prescription one can easily compute the expression for the generalized form of the transfer matrix which relates the curvature and entropic perturbations generated during the situation where the fluctuating mode is stretched outside the expansion (Hubble) scale during the epoch of inflation to the curvature and entropic perturbations at later time via the following simplified form of the matrix equation:

$$\begin{aligned} \left( \begin{array}{c} {\zeta }(t) \\ {\mathcal{S}(t)} \end{array}\right) = \left( \begin{array}{ll} 1 &{} \hat{T}_{\zeta \mathcal{S}} \\ 0 &{} \hat{T}_{\mathcal{S}\mathcal{S}} \end{array} \right) \left( \begin{array}{c}\zeta (t_{\star }) \\ \mathcal{S}(t_{\star }) \end{array} \right) , \end{aligned}$$

(5.547)

where the transfer functions are represented by the following equations in the context of multi-tachyonic inflation from the effective GTachyon set-up:

$$\begin{aligned} \hat{T}_{\zeta \mathcal{S}}(t_{\star },t)= & {} \int ^t_{t_{\star }} \Upsilon (t^{\prime \prime }) \hat{T}_{\mathcal{S}\mathcal{S}}(t_{\star },t^{\prime \prime }) H(t^{\prime \prime }) \mathrm{d}t^{\prime \prime },\end{aligned}$$

(5.548)

$$\begin{aligned} \hat{T}_{\mathcal{S}\mathcal{S}}(t_{\star },t)= & {} \exp \left( \int ^t_{t_{\star }} \Sigma (t^{\prime }) H(t^{\prime \prime }) \mathrm{d}t^{\prime \prime } \right) . \end{aligned}$$

(5.549)

It is also important to note that the evolution in the large-scale limit is independent of the cosmological scale under consideration in the present context and consequently the derived generalized form of the transfer functions $\hat{T}_{\zeta \mathcal{S}}$ and $\hat{T}_{\mathcal{S}\mathcal{S}}$ are implicit functions of the cosmological scale due to their dependence upon the cosmic time scale $t_{\star }(k)$ as appearing in the argument of the transfer functions. The scale dependence of the transfer functions are governed by the following sets of evolution equations:

$$\begin{aligned}&\left( H_{\star }^{-1} {\partial _{t_{\star }}} +\Sigma _{\star }\right) \hat{T}_{\zeta \mathcal{S}}+\Upsilon _{\star }= 0, \end{aligned}$$

(5.550)

$$\begin{aligned}&\left( H_{\star }^{-1} \partial _{t_{\star }}+\Sigma _{\star }\right) \hat{T}_{\mathcal{S}\mathcal{S}} =0. \end{aligned}$$

(5.551)

where we introduce the notation $\partial _{t_{\star }}=\partial /\partial t_{\star }$ to define the partial differentiation in a simplified way. In the present context, the cosmological (momentum) scale dependence of the generalized transfer functions for the multi-tachyonic inflationary paradigm is explicitly determined by the two factors, $\Upsilon _{\star }$ and $\Sigma _{\star }$, which physically represent the cosmic time scale evolution of the curvature and entropic fluctuations at the horizon crossing during the epoch of multi GTachyonic inflation.

Now one can apply the above mentioned generalized transfer matrix as stated in Eq. (5.547) to the primordial scalar power spectra explicitly computed in the previous subsection. After doing the detailed analysis one can finally compute the resulting curvature and entropic primordial power spectra at the beginning point of the radiation dominated epoch:

$$\begin{aligned}&\bar{\Delta }_{\zeta } = \left( 1 + \hat{T}_{\zeta \mathcal{S}}^2 \right) \bar{\Delta }_{\zeta ,\star },\end{aligned}$$

(5.552)

$$\begin{aligned}&\bar{\Delta }_{\mathcal{S}} = \hat{T}_{\mathcal{S}\mathcal{S}}^2 \bar{\Delta }_{\zeta ,\star },\end{aligned}$$

(5.553)

$$\begin{aligned}&\bar{\Delta }_{\zeta \mathcal{S}} = \hat{T}_{\zeta \mathcal{S}} \hat{T}_{\mathcal{S}\mathcal{S}} \bar{\Delta }_{\zeta ,\star }. \end{aligned}$$

(5.554)

For the sake of simplicity let us define a dimensionless cosmological measure of the correlation function in terms of a correlation angle $\theta$ in the following simplified way:

$$\begin{aligned}&\cos \theta \equiv {\bar{\Delta }_{\zeta \mathcal{S}} \over \sqrt{\bar{\Delta }_{\zeta } \bar{\Delta }_\mathcal{S}}} = {\hat{T}_{\zeta \mathcal{S}} \over \sqrt{1+\hat{T}_{\zeta \mathcal{S}}^2}}, \end{aligned}$$

(5.555)

$$\begin{aligned}&\sin \theta \equiv \sqrt{1-\left( {\bar{\Delta }_{\zeta \mathcal{S}} \over \sqrt{{\Delta }_{\zeta } \bar{\Delta }_\mathcal{S}}}\right) ^2} = \frac{1}{\sqrt{1+\hat{T}_{\zeta \mathcal{S}}^2}} ,\end{aligned}$$

(5.556)

$$\begin{aligned}&\theta = \cot ^{-1}(\hat{T}_{\zeta \mathcal{S}}) , \end{aligned}$$

(5.557)

which are surely very useful for further computation in the present context. Further using Eq. (5.552) in Eq. (5.555), finally we get the following expression for the scalar metric perturbation at the horizon crossing which is expressed in terms of the observed curvature perturbation at later times in the cosmic time scale and also in terms of the cross-correlation angle:

$$\begin{aligned} \bar{\Delta }_{\zeta ,\star } \simeq \bar{\Delta }_{\zeta } \sin ^2\theta . \end{aligned}$$

(5.558)

Next using Eqs. (5.552–5.554), the spectral indices of the primordial power spectrum at later stage of times in cosmological time scale can be expressed by the following simplified expressions:

$$\begin{aligned} n_\zeta -1= & {} n_{\zeta ,\star }-1 + H_{\star }^{-1} \left( \partial _{t_{\star }}{\hat{T}}_{\zeta \mathcal{S}}\right) \sin 2\theta ,\end{aligned}$$

(5.559)

$$\begin{aligned} n_\mathcal{S}-1= & {} n_{\zeta ,\star }-1 + 2 H_{\star }^{-1} \left( \partial _{t_{\star }}{\hat{T}}_{\mathcal{S}\mathcal{S}}\right) ,\end{aligned}$$

(5.560)

$$\begin{aligned} n_\mathcal{C}-1= & {} n_{\zeta ,\star }-1 + H_{\star }^{-1} \left\{ \left( \partial _{t_{\star }}{\hat{T}}_{\zeta \mathcal{S}}\right) \tan \theta + \left( \partial _{t_{\star }}{\hat{T}}_{\mathcal{S}\mathcal{S}}\right) \right\} ,\nonumber \\ \end{aligned}$$

(5.561)

which are very useful to study the scale dependent behavior of the primordial power spectra in the present context. Following the same procedure one can also compute the expressions for the running and running of the running of the spectral indices. Additionally, it is important to mention here that the overall amplitude of the generalized transfer functions $\hat{T}_{\zeta \mathcal{S}}$ and $\hat{T}_{\mathcal{S}\mathcal{S}}$ are dependent on the time scale evolution after horizon crossing via the reheating phenomenon and into the radiation dominated epoch. But in spite of this important fact, the spectral tilts of the resulting primordial perturbation spectra can be finally expressed in terms of the slow-roll parameters at the horizon crossing during the epoch of inflation and also in terms of the cross-correlation angle $\theta$, which we have already introduced earlier in Eq. (5.557).

5.3.5 Computation of tensor power spectrum

In this subsection we will not derive the crucial results for multi-Gtachyonic inflation. But we will state the results for BD vacuum where the changes will appear due to the presence of M number of different tachyon field. One can similarly write down the detailed expressions for AV (including $\alpha$ vacuum) as well.

$$\begin{aligned} \langle h_\mathbf{k} h_\mathbf{k^{\prime }} \rangle= & {} \sum ^{M}_{i=1}\sum ^{M}_{j=1}\langle h^{i}_\mathbf{k} h^{j}_\mathbf{k^{\prime }} \rangle =(2\pi )^3 \delta ^{3}(\mathbf{k}+\mathbf{k^{\prime }})\frac{2\pi ^2}{k^3}\Delta _{h}(k),\nonumber \\ \end{aligned}$$

(5.562)

where the primordial power spectrum for the scalar modes at any arbitrary momentum scale k can be expressed as

$$\begin{aligned} \Delta _{h}(k)= & {} 8 \frac{\sum ^{M}_{m=1}\sum ^{M}_{n=1}N_{,m}N_{,n}G^{mn}}{\sum ^{M}_{i=1}\sum ^{M}_{j=1}N_{,i}N_{'j}G^{ij}}\frac{k^3 P_{\zeta }(k)}{2\pi ^2}=\frac{4k^3 P_{\zeta }(k)}{\pi ^2}.\nonumber \\ \end{aligned}$$

(5.563)

The changes will appear in the expressions for the following inflationary observables at the horizon crossing:

• In the present context amplitude of tensor power spectrum can be computed as

  $$\begin{aligned} \Delta _{h, \star }= & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\bar{\epsilon }_{V}\right] ^2 \frac{2H^{2}}{\pi ^2M^{2}_{p}}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \left[ 1-(\mathcal{C}_{E}+1)\frac{\bar{\epsilon }_{V}}{2q}\right] ^2 \frac{2H^{2}}{\pi ^2M^{2}_{p}}\right\} _{\star } , &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \nonumber \\ \end{aligned}$$

  (5.564)

  where $\mathcal{C}_{E}= -2 + \ln 2 + \gamma \approx -0.72.$

• Next one can compute the tensor spectral tilt ($n_{h}$) of the primordial scalar power spectrum as

  $$\begin{aligned} n_{h, \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle -2\bar{\epsilon }_{V}\left[ 1+\bar{\epsilon }_{V}+2\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] +\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle -\frac{\bar{\epsilon }_{V}}{q}\left[ 1+\frac{\bar{\epsilon }_{V}}{2q}+\sqrt{\frac{2}{q}}\left( \mathcal{C}_{E}+1\right) \left( 3\bar{\epsilon }_{V}-\bar{\eta }_{V}\right) \right] +\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{aligned}$$

  (5.565)

• Finally the tensor-to-scalar ratio for the multi-tachyonic set-up can be expressed as

  $$\begin{aligned} r_{ \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \frac{8}{\sum ^{M}_{i}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij}}+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{4}{q\sum ^{M}_{i}\sum ^{M}_{j=1}N_{,i}N_{,j}G^{ij}}+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \nonumber \\ \end{aligned}$$

  (5.566)

On the contrary, compared to the scalar (curvature and entropic) part of the cosmological perturbations, the tensor perturbations remain frozen in on the large cosmological scales and finally decoupled from the scalar (curvature and entropic) part of the cosmological perturbations at the linear order of the cosmological perturbation theory. Consequently, in the present context, the primordial cosmological perturbation spectrum for gravitational waves is given by the following simplified expression:

$$\begin{aligned} {\Delta }_h= & {} {\Delta }_{h,\star },\end{aligned}$$

(5.567)

$$\begin{aligned} n_h= & {} n_{h,\star }. \end{aligned}$$

(5.568)

Finally, it also important to note that the consistency condition for the tensor-to-scalar amplitudes of the primordial power spectrum at the Hubble-crossing can be rewritten, using Eqs. (5.558) and (5.567), as a model independent consistency relation between the tensor-to-scalar amplitudes of the primordial power spectrum at late times in cosmological scale for GTachyon as

$$\begin{aligned}&r={{\Delta }_{h}\over {\Delta }_{\zeta }} \simeq - 8 n_h \sin ^2\theta +\cdots \nonumber \\&\quad = - 8 n_{h,\star } \sin ^2\theta +\cdots = r_{\star } \sin ^2\theta +\cdots . \end{aligned}$$

(5.569)

In the present context, the scale dependence of the final scalar (curvature and entropic) power spectra depends on both on the cosmological scale dependence of the initial spectral index ($n_{\zeta ,\star }$) and on the explicit form of the generalized transfer functions $T_{\zeta \mathcal{S}}$ and $T_{\mathcal{S}\mathcal{S}}$. Here we have not explicitly discussed the cosmological parameter estimation and numerical estimations from a specific class of multi-tachyonic potentials. But to understand this more clearly, one can carry forward the results obtained in this section to test various models of the multi-tachyonic potential.

5.3.6 Analytical study for the multi-field model

Here we compute the expression for the inverse cosh potential. One can repeat the computation for the other proposed models of multi-field inflation as well. For the multi-field case the inverse cosh potential is given by

$$\begin{aligned} V(T_{j})=\frac{\lambda _{j}}{\mathrm{cosh}\left( \frac{T_{j}}{T_{0j}}\right) } \quad \forall \; j=1,2,\ldots ,M, \end{aligned}$$

(5.570)

and the total effective potential is given by

$$\begin{aligned} V=\sum ^{M}_{j=1}V(T_{j})=\sum ^{M}_{j=1}\frac{\lambda _{j}}{\mathrm{cosh}\left( \frac{T_{j}}{T_{0j}}\right) }, \end{aligned}$$

(5.571)

where $\lambda _{j}$ characterize the scale of inflation in each branch and $T_{0j}$ are j number of different parameter of the model. Next using specified form of the potential the potential dependent slow-roll parameters for i th species are computed as

$$\begin{aligned} \bar{\epsilon }_{V}(T_i)= & {} \frac{1}{\sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T_m}{T_{0m}}\right) }\left[ \frac{\lambda _{i}}{2g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] ,\nonumber \\\end{aligned}$$

(5.572)

$$\begin{aligned} \bar{\eta }_{V}(T_{i})= & {} \frac{1}{\sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T_m}{T_{0m}}\right) }\nonumber \\&\times \;\left\{ \frac{\lambda _{i}}{g_{i}}\left[ \tanh ^2\left( \frac{T_{i}}{T_{0i}}\right) -\mathrm{sech}^2\left( \frac{T_i}{T_{0i}}\right) \right] \right\} .\nonumber \\ \end{aligned}$$

(5.573)

Also the total contribution in the slow-roll is expressed through the reduced slow-roll parameters:

$$\begin{aligned} \bar{\epsilon }_{V}= & {} \frac{1}{\sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T_m}{T_{0m}}\right) }\left[ \sum ^{M}_{i=1}\frac{\lambda _{i}}{2g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] ,\nonumber \\\end{aligned}$$

(5.574)

$$\begin{aligned} \bar{\eta }_{V}= & {} \frac{1}{\sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T_m}{T_{0m}}\right) }\nonumber \\&\times \;\left\{ \sum ^{M}_{i=1}\frac{\lambda _{i}}{g_{i}}\left[ \tanh ^2\left( \frac{T_{i}}{T_{0i}}\right) -\mathrm{sech}^2\left( \frac{T_i}{T_{0i}}\right) \right] \right\} .\nonumber \\ \end{aligned}$$

(5.575)

Additionally here we introduce a new factor $g_{j}$ for each non-identical j number of multi-tachyonic fields, defined as

$$\begin{aligned} g_{j}= \frac{\alpha ^{\prime }\lambda _{j} T^2_{0j}}{M^2_p}=\frac{M^4_{sj}}{(2\pi )^3 g_{sj}}\frac{\alpha ^{\prime } T^2_{0j}}{M^2_p}. \end{aligned}$$

(5.576)

Next we compute the number of e-foldings from this model:

$$\begin{aligned} \begin{array}{lll} \displaystyle N=\left\{ \begin{array}{lll} \displaystyle \sum ^{M}_{j=1}g_j~\ln \left[ \frac{\tanh \left( \frac{T_{end,j}}{2T_{0j}}\right) }{\tanh \left( \frac{T_j}{2T_{0j}}\right) }\right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \sum ^{M}_{j=1}\sqrt{2q}~g_j~\ln \left[ \frac{\tanh \left( \frac{T_{end,j}}{2T_{0j}}\right) }{\tanh \left( \frac{T_j}{2T_{0j}}\right) }\right] ,&{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.577)

Further using the condition to end inflation:

$$\begin{aligned} \bar{\epsilon }_{V}(T_{end,i})=1,\end{aligned}$$

(5.578)

$$\begin{aligned} |\bar{\eta }_{V}(T_{end,i})|=1, \end{aligned}$$

(5.579)

we get the following field value at the end of inflation:

$$\begin{aligned} T_{end,i}=T_{0i}~\mathrm{sech}^{-1}(g_{i}). \end{aligned}$$

(5.580)

Next using $N_i=N_{\mathrm{cmb},i}=N_{\star ,i}$ and $T_{i}=T_{\mathrm{cmb},i}=T^{\star }_{i}$ at the horizon crossing we get

$$\begin{aligned} \begin{array}{lll} \displaystyle T^{\star }_{i}\approx 2T_{0,i}\times \left\{ \begin{array}{lll} \displaystyle \tanh ^{-1}\left[ \exp \left( -\frac{N_{\star ,i}}{g_i}\right) \right] , &{} \quad \mathrm{for}~~~q=1/2, \\ \displaystyle \tanh ^{-1}\left[ \exp \left( -\frac{N_{\star ,i}}{\sqrt{2q}g_i}\right) \right] . &{} \quad \mathrm{for ~any~arbitrary~} q. \end{array} \right. \end{array}\nonumber \\ \end{aligned}$$

(5.581)

Using these results one can compute

$$\begin{aligned} N_{,i}= & {} \frac{1}{\sqrt{2\bar{\epsilon }_{V}(T^{\star }_{i})}}\frac{V(T^{\star }_{i})-V(T^{c}_{i})+\left( \sum ^{M}_{m=1}V(T^{c}_{m})\right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}}{\left( \sum ^{M}_{j=1}V(T^{\star }_{j})\right) }\nonumber \\= & {} \frac{\lambda _{i}\mathrm{sech}\left( \frac{T^{\star }_i}{T_{0i}}\right) -\lambda _{i}\mathrm{sech}\left( \frac{T^{c}_i}{T_{0i}}\right) +\left( \sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T^{c}_m}{T_{0m}}\right) \right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}}{\sqrt{\left[ \frac{\lambda _{i}}{g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }}\nonumber \\ \end{aligned}$$

(5.582)

which is very useful to further compute the inflationary observables from the multi-tachyonic set-up. Here $\star$ indicates the horizon crossing and c denotes the constant density surface.

Using these results finally we compute the following inflationary observables:

$$\begin{aligned} \Delta _{\zeta , \star }= & {} \sum ^{M}_{i=1}\left( \frac{\lambda _{i}\mathrm{sech}\left( \frac{T^{\star }_i}{T_{0i}}\right) -\lambda _{i}\mathrm{sech}\left( \frac{T^{c}_i}{T_{0i}}\right) +\left( \sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T^{c}_m}{T_{0m}}\right) \right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}}{\sqrt{\left[ \frac{\lambda _{i}}{g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }}\right) ^2\nonumber \\&\times \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \frac{\left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }{12\pi ^2M^2_p}\right\} _{\star }, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \frac{q \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }{6\pi ^2M^2_p}\right\} _{\star }, &{} \quad \mathrm{for~~any}~~ q, \end{array}\right. \end{aligned}$$

(5.583)

$$\begin{aligned} \Delta _{\zeta ,c}= & {} \left[ \sum ^{M}_{i=1}\frac{\lambda _{i}}{2g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] ^{-1}_{c}\times \left\{ \begin{array}{lll} \displaystyle \displaystyle \left\{ \frac{\left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) ^2}{12\pi ^2M^2_p }\right\} _{c}, &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \left\{ \frac{q \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) ^2}{6\pi ^2M^2_p}\right\} _{c}, &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

(5.584)

$$\begin{aligned} n_{\zeta , \star }-1\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \displaystyle \left( 2\bar{\eta }_{V}-8\bar{\epsilon }_{V}\right) -\frac{2}{\sum ^{M}_{i=1}\left( \frac{\lambda _{i}\mathrm{sech}\left( \frac{T^{\star }_i}{T_{0i}}\right) -\lambda _{i}\mathrm{sech}\left( \frac{T^{c}_i}{T_{0i}}\right) +\left( \sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T^{c}_m}{T_{0m}}\right) \right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}}{\sqrt{\left[ \frac{\lambda _{i}}{g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }}\right) ^2}+\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle \displaystyle \sqrt{\frac{2}{q}}\bar{\eta }_{V}-\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \bar{\epsilon }_{V}\\ \displaystyle -\frac{1}{q\sum ^{M}_{i=1}\left( \frac{\lambda _{i}\mathrm{sech}\left( \frac{T^{\star }_i}{T_{0i}}\right) -\lambda _{i}\mathrm{sech}\left( \frac{T^{c}_i}{T_{0i}}\right) +\left( \sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T^{c}_m}{T_{0m}}\right) \right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}}{\sqrt{\left[ \frac{\lambda _{i}}{g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }}\right) ^2}+\cdots , &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

(5.585)

$$\begin{aligned} n_{\zeta ,c}-1\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \displaystyle -6\bar{\epsilon }_{V}+2\frac{\sum ^{M}_{i=1}\bar{\epsilon }_{V}(T_i)\bar{\eta }_{V}(T_i)}{\bar{\epsilon }_{V}}+\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle \displaystyle -3\sqrt{\frac{2}{q}}\bar{\epsilon }_{V}+\frac{\sum ^{M}_{i=1}\bar{\epsilon }_{V}(T_i)\bar{\eta }_{V}(T_i)}{q\bar{\epsilon }_{V}}\cdots , &{} \quad \mathrm{for~~any}~~ q, \end{array} \right. \end{aligned}$$

(5.586)

$$\begin{aligned} r_{ \star }\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \frac{8}{\sum ^{M}_{i=1}\left( \frac{\lambda _{i}\mathrm{sech}\left( \frac{T^{\star }_i}{T_{0i}}\right) -\lambda _{i}\mathrm{sech}\left( \frac{T^{c}_i}{T_{0i}}\right) +\left( \sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T^{c}_m}{T_{0m}}\right) \right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}}{\sqrt{\left[ \frac{\lambda _{i}}{g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }}\right) ^2}+\cdots ,~~ &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \frac{4}{q\sum ^{M}_{i=1}\left( \frac{\lambda _{i}\mathrm{sech}\left( \frac{T^{\star }_i}{T_{0i}}\right) -\lambda _{i}\mathrm{sech}\left( \frac{T^{c}_i}{T_{0i}}\right) +\left( \sum ^{M}_{m=1}\lambda _{m}\mathrm{sech}\left( \frac{T^{c}_m}{T_{0m}}\right) \right) \frac{\bar{\epsilon }_{V}(T^{c}_{i})}{\bar{\epsilon }^{c}_{V}}}{\sqrt{\left[ \frac{\lambda _{i}}{g_{i}}\frac{\sinh ^2\left( \frac{T_{i}}{T_{0i}}\right) }{\cosh \left( \frac{T_i}{T_{0i}}\right) }\right] \left( \sum ^{M}_{j=1}\lambda _{j}\mathrm{sech}\left( \frac{T^{\star }_j}{T_{0j}}\right) \right) }}\right) ^2}+\cdots ,~~ &{} \quad \mathrm{for~~any}~~ q. \end{array} \right. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{aligned}$$

(5.587)

For the inverse cosh potential we get the following consistency relations:

$$\begin{aligned} n_{\zeta , \star }-1 +\frac{r_{\star }}{4}\approx & {} \left\{ \begin{array}{lll} \displaystyle \displaystyle \displaystyle \left( 2\bar{\eta }_{V}-8\bar{\epsilon }_{V}\right) +\cdots , &{} \quad \mathrm{for}~~ q=1/2, \\ \displaystyle \displaystyle \displaystyle \sqrt{\frac{2}{q}}\bar{\eta }_{V}-\left( \frac{1}{q}+3\sqrt{\frac{2}{q}}\right) \bar{\epsilon }_{V}+\cdots , &{} \quad \mathrm{for~~any}~~ q. \end{array}\right. \end{aligned}$$

(5.588)

6 Conclusion

In this paper, we have explored various cosmological consequences from the GTachyonic field. We start with the basic introduction of tachyons in the context of non-BPS string theory, where we also introduce the GTachyon field, in the presence of which the tachyon action is modified, and one can quantify the amount of the modification via a superscript q instead of 1 / 2. This modification exactly mimics the role of effective field theory operators and, studying the various cosmological features from this theory, one of the final objectives is to constrain the index q and a specific combination ($\propto \alpha ^{\prime }M^4_s/g_s$) of the Regge slope parameter $\alpha ^{\prime }$, the string coupling constant $g_{s}$ and the mass scale of tachyon $M_s$, from the recent Planck 2015 and Planck+BICEP2/Keck Array joint data. To serve this purpose, we introduce various types of tachyonic potentials: the inverse cosh, logarithmic, exponential and inverse polynomials, using which we constrain the index q. To explore this issue in detail, we start with the detailed characteristic features of each of the potentials. Next we discuss the dynamics of GTachyon as well as usual tachyon for single, assisted and multi-field scenario. We also derive the dynamical solutions for various phases of the universe, including two situations, $T<<T_0$ and $T>>T_0$, where $T_0$ is interpreted as the minimum or the mass scale of the tachyon. Next we have explicitly studied the inflationary paradigm from single field, assisted field and multi-field tachyon set-ups. Specifically for the single field and assisted field cases we have derived the results in the quasi-de Sitter background in which we have utilized the details of: (1) cosmological perturbations and quantum fluctuations for scalar and tensor modes, (2) slow-roll prescription up to all orders. In this context we have derived the expressions for all inflationary observables using any arbitrary vacuum and also for the Bunch–Davies vacuum by exactly solving the Mukhanov–Sasaki equation as obtained from the fluctuation of scalar and tensor modes. For the single field and assisted field cases in the presence of a GTachyon we have derived the inflationary Hubble flow and potential dependent flow equations, new sets of consistency relations, which are valid in the slow-roll regime and also derived the expression for the field excursion formula for the tachyon in terms of inflationary observables from both of the solutions obtained from arbitrary and the Bunch–Davies initial conditions for inflation. Particularly the derived formula for the field excursion for the GTachyon can be treated as one of the probes through which one can: (1) test the validity and the applicability of the effective field theory prescription within the present set-up, (2) distinguish between various classes of models from an effective field theory point of view. Also we have shown that in the case of assisted tachyon inflation the validity of the effective field theory prescription is much better compared to the single field case. This is because of the fact that the field excursion for assisted inflation is expressed in terms of the field excursion for the single field, provided the multiplicative scaling factor is $1/\sqrt{M}$, where M is the number of identical tachyons participating in assisted inflation. This derived formula also suggests that if M is a very large number then it is very easy to validate effective theory techniques, as in that case $|\Delta T|_\mathrm{Assisted}<<M_p$. Next using the explicit form of the tachyonic potentials we have studied the inflationary constraints and quantified the allowed range of the generalized index q for each of the potentials. Hence using each of the specific forms of the tachyonic potentials in the context of the single field scenario, we have studied the features of CMB angular power spectrum from TT, TE and EE correlations from scalar fluctuations within the allowed range of q for each of the potentials. We also put in the constraints from the Planck temperature anisotropy and polarization data, which shows that our analysis matches well with the data. We have additionally studied the features of the tensor contribution in the CMB angular power spectrum from TT, BB, TE and EE correlations, which will give more interesting information in the near future once one can detect the signature of primordial B-modes. Further, using the $\delta N$ formalism we have derived the expressions for the inflationary observables in the context of multi-field tachyons. We have also demonstrated the results for the two-field tachyonic case to understand the cosmological implications of the results in a better way.

The future prospects of our work are appended below:

• Throughout this analysis one finds each an every detail, but one can further study the features of primordial non-Gaussianity from the single field, assisted field and multi-field cases [96, 97, 140–144]. Due to the presence of a generalized index q one would expect that the consistency relations, which connect non-Gaussian parameters with the inflationary observables, are getting modified in the slow-roll regime of inflation and consequently one can generate large amount of primordial non-Gaussianity from the GTachyonic set-up. Also our aim is generalize the results as well as the consistency relations in all order of cosmological perturbations by incorporating the effect of sound speed $c_{S}$ and the generalized parameter q. Using this set-up one can also compute the cosmological Ward identities for inflation [143–145], through which one can write down the recursion relation between correlation functions. This will help to derive the modified consistency relations in the present context. Additionally, one can also explore the possibility of devising cosmological observables which violate Bell’s inequalities recently pointed out in Ref. [146]. Such observables could be used to argue that cosmic scale features were produced by quantum mechanical effects in the very early universe, specifically in the context of inflation. We are planning to report on these issues explicitly very soon.

• One would also like to ask what happened when we add additional higher derivatives in the gravity sector within tachyonic set-up, which are important around the string scale $M_s$. How the higher derivative gravity sector will change the cosmological dynamics and details of the primordial non-Gaussianity in the presence of a non-canonical GTachyonic sector is completely unknown. We have also plans to extend this project in that direction.

• The generation of the seed (primordial) magnetic field from the inflationary sector [129, 147] is a long standing issue in primordial cosmology. To address this well-known issue one can also study the cosmological consequences from the effective field theory of inflationary magnetogenesis from the GTachyon within the framework of Type-IIA/IIB string theory.

• Also, our present analysis has been performed for five different potentials motivated from tachyonic string theory, specifically in the context of single field and assisted field inflation. For the multi-field we have quoted the results only for a specific kind of separable potential. It would be really very interesting to check whether we can study the behavior of non-separable potentials in the present context [148, 149]. Also it is interesting to reconstruct any general form of a tachyonic inflationary potential, using which one can study the applicability of the effective field theory framework within this present set-up.

• One can also carry forward our analysis in the context of non-minimal set-up where the usual Einstein–Hilbert term in the gravity sector is coupled with the GTachyonic field. By applying the conformal transformation in the metric one can construct an equivalent representation of the gravity sector¹¹ in which the gravity sector is decoupled from the tachyonic sector and the usual tachyonic matter sector modified in the presence of an extra conformal factor. This clearly implies that the tachyonic potentials that we have studied in this paper all get modified via the additional conformal factor. Consequently it is expected that the cosmological dynamics will be modified in this context. Also one can check the applicability of the slow-roll prescription, which helps further to understand the exact behavior of cosmological perturbations in the presence of conformally rescaled modified tachyonic potentials.

• The detailed study of the generation of the dark matter using effective field theory framework,¹² other crucial particle physics issues i.e., the reheating phenomenon, leptogenesis and baryogenesis from the GTachyonic set-up is also an unexplored issue till date.