Tachyon warm-intermediate inflation in the light of Planck data

We study the main properties of the warm inflationary model based on Barrow's solution for the scale factor of the universe. Within this framework we calculate analytically the basic slow roll parameters for different versions of warm inflation. We test the performance of this inflationary scenario against the latest observational data and we verify that the predicted spectral index and the tensor-to-scalar fluctuation ratio are in excellent agreement with those of {\it Planck 2015}. Finally, we find that the current predictions are consistent with those of viable inflationary models.


Introduction
Standard inflation driven by an inflaton field traces back to early efforts to alleviate the basic problems of the Big-Bang cosmology, namely horizon, flatness and monopoles [1,2]. The nominal inflationary paradigm contains the slowroll and the (P)reheating regimes. In the slow-roll phase the kinetic energy (which has the canonical form here) of the scalar field is negligible with respect to the potential energy V (φ ) which implies a deSitter expansion of the Universe. However, after the slow-roll epoch the kinetic energy becomes comparable to the potential energy and thus the inflaton field oscillates around the minimum and progressively the universe is filled by radiation [3,4].
Nevertheless, other theoretical patterns suggested a possible way to treat the physics of the early universe. For example, in the so-called warm inflationary scenario the radiation production occurs during the slow-roll epoch and the reheating period is avoided [5,6]. The nature of the warm inflationary scenario is different with respect to that of the a e-mail: vkamali@basu.ac.ir b e-mail:svasil@academyofathens.gr c e-mail: Mehrabi@basu.ac.ir standard cold inflation. Warm inflation satisfies the condition T > H, where T is the temperature and H is the Hubble parameter, which implies that the fluctuations of the inflaton field are thermal instead of quantum. An obvious consequence of the above inequality is that in the case of warm inflation density perturbations arise from thermal fluctuations rather than quantum fluctuations [7][8][9]. Specifically, thermal fluctuations are produced during the warm inflationary epoch and they play a central role toward describing the CMB anisotropies and thus providing the initial seeds for the formation of large scale structures. Of course, after this epoch the universe enters in the radiation dominated phase as it should [5,6]. In order to achieve warm inflation one may use a tachyon scalar field for which the kinetic term does not follow the canonical form (k-inflation [10]). It has been found that tachyon fields which are associated with unstable D-branes [11] can be responsible for the cosmic acceleration in early times [10,12,13].
Notice, that tachyon potentials have the following two properties: the maximum of the potential occurs when φ → 0 while the corresponding minimum takes place when φ → ∞. From the dynamical viewpoint one may obtain the equations of motion using a special Lagrangian [14] which is non-minimally coupled to gravity: Considering a spatially flat Friedmann-Robertson-Walker (hereafter FRW) space-time the stress-energy tensor is given by and Where φ is tachyon scalar field in unite of inverse Planck mass M −1 pl , and V (φ ) is potential associated with the tachyon field. In the past few years, there was an intense debate among cosmologists and particle physicists regarding those phenomenological models which can be produced in extra dimensions. For example, the reduction of higher-dimensional gravitational scale, down to TeV-scale, could be presented by an extra dimensional scenario [15][16][17]. In these scenarios, gravity field propagates in the bulk while standard models of particles are confined to the lower-dimensional brane.
In this framework, the extra dimension induces additional terms in the first Friedmann equation [18][19][20]. Especially, if we consider a quadratic term in the energy density then we can extract an accelerated expansion of the early universe [21][22][23][24][25]. In the current study we consider the tachyon warm inflation model in the framework of Randall-Sundrum II braneworld which contains a single, positive tension brane and a non-compact extra dimension.
Following the lines of Ref. [26], we attempt to study the main properties of the warm inflation in which the scale factor evolves as a(t) ∝ exp(At f ), where 0 < f < 1 ("intermediate inflation"). In this case cosmic expansion evolves faster than the power-law inflation (a ∝ t p , p > 1) and slower than the standard deSitter one, a(t) ∝ exp(H I t) [H(t) = H I =const.]. More details regarding the cosmic expansion in various inflationary solutions can be found in the paper of Barrow [27].
In the current work, we investigate the possibility of using the intermediate solution in the case of warm tachyon inflation. Specifically, the structure of the article is as follows: In section II we briefly discuss the main properties of the warm inflation, while in section III we provide the slow roll parameters. In section IV we study the performance of our predictions against the Planck 2015 data. Finally, the main conclusions are presented in section VI.

Tachyon warm inflation
Let us assume a flat, homogeneous and isotropic Friedmann-Robertson-Walker (FRW) universe, in which the radiation era is endowed with the scalar field described by the Lagrangian (1) in the context of the Randall-Sundrum II (RSII) brane [28] . Following the notations of [5,6,[18][19][20]26] one may check that the basic cosmological equations are where Γ is the dissipation coefficient, in unit of M 5 pl . The latter two equations (6),(7) imply the continuity equation, namelyρ + 3H(ρ + p) = 0. Notice, that Eqs.(6) and (7) have been proposed by various authors such as [29][30][31][32][33][34]. In these studies the quantity Γφ 2 is the dissipation term which is introduced phenomenologically in order to describe the nearly-thermal radiation bath that is the outcome of the warm inflationary scenario. It is well known that Tachyon inflation in its standard picture (cold inflation) suffers from a serious problem. In particular, reheating and matter creation are both problematic because the tachyon fields in such theories do not oscillate around the minimum of the potential [35]. This problem can be alleviated in the context of warm inflation. In this scenario radiation production occurs during the slow-roll era which implies that reheating is avoided and thus the universe heats up and finally it enters in the radiation era [5] (See Eqs.(6) and (7)).
In the above set of equations, an over-dot denotes derivative with respect to time, ρ = ρ φ + ρ γ and p = p φ + p γ (p γ = ρ γ /3) are the total density and pressure, ρ φ and ρ γ are the scalar field and radiation densities, H =ȧ/a is the Hubble parameter. Notice, that λ is the brane tension which obeys the following restriction λ ≥ (10TeV ) 4 [22,36,37]. Obviously, substituting equations (3),(4) in Eq. (6) it is easy to derive the modified Klein-Gordon equation which describes the time evolution of the tachyon field. This is where V (φ ) = dV /dφ . The above cosmological equations imply that the model is strongly affected by the quantity Γ . This is due to the fact that radiation is exchanging energy with the tachyon field and this is reflected in the corresponding behavior of dissipation coefficient Γ , which is negligible in the classical inflationary paradigm by definition. Although, the precise functional form of Γ is still an open issue, a number of different parametrizations have been proposed in the literature treating the functional form of Γ (see [7,[38][39][40]). In the current work we use the well known parametrization of where T is the temperature and Γ c is constant.
During the warm inflationary epoch, the energy density of the scalar field dominates the total fluid (stable regime [41]) and thus Eq.(5) becomes or Another important quantity in this kind of studies is the dimensionless dissipation parameter which characterizes the type of inflation and it is defined as (for more details see appendix) The above definition is presented for warm tachyon inflation in several papers [30,32,33,[42][43][44]. Notice, that for the canonical scalar field model of warm inflation, the corresponding dimensionless ratio is defined as Γ 3H . In the weak dissipation regime, the ratio R tends to zero (Γ /3Hρ φ 1), however, in the strong dissipation regime, the coefficient Γ guides the damped evolution of the scalar field. Now using Eq.(8) and Eq.(11) in the high-dissipation regime (Γ 3Hρ φ ) to prove thaṫ Owing to the fact that during inflation the parameters H, Γ and φ are slowly varying functions the production of radiation become quasi-stable whenρ 4Hρ γ , andρ γ Γφ 2 [5][6][7]. Under these conditions, using Eqs. (8) and (13) we write the radiation density as follows: The latter formula can be identified with the equation relating ρ γ with the radiation temperature T . Indeed, under of adiabatic condition we may write where C γ = π 2 g * 30 and g * is the degrees of freedom of the created massless modes [45]. Combining Eq. (14) and (15) we obtain the temperature Lastly, with the aid of Eqs. (11) and (14) we obtain the potential of the scalar field

Slow-roll parameters
Let us present here the main quantities of the tachyonic inflation. In particular, the basic slow-roll parameters are given by equation In this context, the number of e-folds is written as Also, the power spectrum of the scalar fluctuations is given by [5,6] equation An important feature of the warm inflationary model is related with the fact that the origin of δ φ is thermal and not quantum as we consider in the nominal inflationary paradigm.
In the case of warm inflation it has been found [5][6][7] that scalar perturbations are written as where the wave number ≥ H corresponds to the freeze-out scale at the special point when, the dissipation damps out to thermally excited fluctuations of inflaton ( V V < Γ H V ) [46]. Notice, that Eq. (22) is valid in the high-dissipation regime R 1. As we have already mentioned in the previous section we study our model via Eq. (13) in the context of high-dissipation regime, which means that for scalar perturbations we can utilize Eq. (22).
Inserting the freeze-out wave-number and Eq.(22) into Eq.(21) we find after some simple calculations that the power spectrum of the tachyonic scalar field is given by We would like to point out that in the case of canonical scalar fields within the framework of warm inflation one can find other forms of the power spectrum P s in Refs. [40,47,48]. The corresponding spectral index n s is defined in terms of the P s slow-roll parameters, as usual [49], by On the other hand, it has been found [50] that the power spectrum of the tensor perturbations which are defined on the brane takes the form arises from normalization of zero-mod of a graviton [50]. Therefore, using the so called tensor-to-scalar ratio we arrive at In order to proceed with the analysis it would help to know the functional form of the scale factor a(t). Barrow [27] showed that under of specific conditions we can have an intermediate inflation in which the scale factor satisfies the following exponential form: which provides where f satisfies the restriction 0 < f < 1. The above expansion evolves faster than the power-law inflation (a ∝ t p , p > 1) and slower than the standard deSitter one, a(t) This parametrization has been used for non-SUSY models [52,53]. Q5: (III)-the case where Γ = Γ 0 is a positive constant (hereafter Γ 0 -parametrization: see [30,32,33,[42][43][44][54][55][56]) which implies that the pair (b, c) in Eq. (9) is strictly equal to (0, 0) and finally (IV) we utilize the so called high temperature regime (hereafter Γ 1 -parametrization) in which we select (b, c) = (0, 1) and thus Γ ∝ T (see also [57]).
In this paper, we are going to focus on parameterizations (III) and (IV) in order to calculate the slow roll parameters. Lastly, we remind the reader that in the framework of warm inflationary model thermal fluctuations dominate over the quantum fluctuations.
Combining the latter argument with the fact that thermal fluctuations are proportional to temperature T while quantum fluctuations are proportional to H, one can easily derive the condition T > H. Obviously, if we consider our model in the high temperature regime (Γ ∝ T ) then the aforesaid restriction (T > H) is satisfied. For more details we refer the reader the work of [57].

Γ 0 -parametrization
In this inflationary scenario (Γ = Γ 0 =const.) with the aid of Eq.(28) we integrate Eq.(13) and we obtain the evolution of the scalar field in terms of the hyper-geometric function [58,59] where and Γ (n) is the normal Gamma-function. Notice, that without losing the generality we have set φ 0 = 0. Now we can derive the Hubble parameter H and the associated potential V (φ ) in the limit ofφ 2 V (φ ) Clearly, if we substitute Eq.(31) in the slow-roll parameters then we have Notice, that in order to extract the latter equalities in Eqs(32), Plugging φ in into Eq.(32) and using the constraint ε(φ in ) 1 we find In order to proceed with the analysis we need to know the values of N and φ in . Firstly, it is natural to consider that the number of e-folds is 50 or 60. Secondly, using the condition ε(φ in ) = 1 and Eqs. (34), (4) we can estimate the slow-roll parameters. Now we focus on the power spectrum formulas. Specifically, inserting the appropriate expressions into Eq.(23) we define the scalar power spectrum bining the definition of the spectral index n s (24) and the above equation we obtain where in the derivation of the above equality we have used Lastly, based on Eq.(26) we compute the tensor-to-scalar ratio parameter r = r 1 I(N) where r 1 = ( 1 In the literature sometimes we replace φ by φ which denotes the value at the horizon crossing.

Γ 1 -parametrization
Using the same methodology as in the previous section we provide the basic slow-roll parameters in the case of Γ 1parametrization, namely Γ = Γ 1 T , where Γ 1 is constant. In particular, from Eqs. (28,13) the tachyon field is written as whereF(t) andK are given by the following expressions .
Also here we have set φ 0 = 0. Now the Hubble parameter, the potential and the corresponding slow-roll parameters are given by whereF −1 (Kφ ) is inverse function ofF(t). In the current case the number of e-folds becomes and following standard lines the end of inflation takes place when The scalar power-spectrum can be easily identified by comparing the current cosmological expressions with Eq.(23), and we find P s = p 2 I(N) 8 . If we take the aforementioned P s formula we find the following spectral index where Finally, the tensor-to-scalar ratio [see Eq.26)] takes the form r = r 2 I(N) .

Comparison with observation
The analysis of Planck [62] and BICEP2/Keck Array [63] data sets has provided a new constraint on inflationary scenarios [64]. In particular, the comprehensive analysis of Planck data [62] indicates that single scalar-field models of slowroll inflation have a very low tensor-to-scalar fluctuation ratio r = P t /P s 1, a scalar spectral index n s = 0.968 ± 0.006 and no appreciable running. The upper bound set by the Planck team and the joint analysis of BICEP2/Keck Array/Planck [63] on tensor-to-scalar fluctuation ratio is r < 0.11. In this section we attempt to test the performance of the warm inflationary model against the above observational results. Let us now concentrate on our results. Notice, that in the case of warm inflation the number of degrees of freedom becomes g * 200 (C γ 70, [65]) Also, for the rest of the paper we have set λ = 10 −14 . Concerning the number of e-folds, it is natural to consider that N lies in the interval [50,60]. Here, we have set it either to 50 or 60. In figures 1 (Γ 0 parametrization) and 2 (Γ 1 parametrization) we present the A − f allowed region in which our (n s , r) results satisfy the above restrictions of Planck within 1σ uncertainties. In the case of Γ 0 model, we observe that for various values of the dissipation coefficient there is a narrow region in the A − f plane which is consistent with the observed values of n s and r. The absence of A − f pair solutions and thus of (n s , r), appear for Γ 0 ≤ 10 −10 .
For the Γ 1 parametrization the situation is slightly different. Figure (2) shows broader A − f regions with respect to those of Γ 0 parametrization. Also in this case we verify that for Γ 1 ≤ 10 −10 , there is no A − f pairs which satisfy the observational criteria. The theoretical curves of cold intermediate inflation model in Einstein General Relativity in n s − r plane are well outside of the 95% C.L region. Our aim here is to test the viability of the warm inflation, involving the latest Planck2015 data. In figure (3) we present the confidence contours in the (n s , r) plane. On top of figure (3) we provide the solid stars for the individual sets of (n s , r) which are based on the Γ 0 parametrization, whereas in the same figure we display the corresponding solid points in the case of Γ 1 parametrization. From the comparison it becomes clear that our (n s , r) results are in excellent agreement with those of Planck 2015.
Indeed, we find: Bellow we compare the current predictions with those of viable literature potentials. This can help us to understand the variants of the warm inflationary model from the observationally viable inflationary scenarios.
-The chaotic inflation [66]: In this inflationary model the potential is V (φ ) ∝ φ k . Therefore, the basic slow-roll parameters are written as ε = k/4N, η = (k −1)/2N which implies n s = 1 − (k + 2)/2N and r = 4k/N. It has been found that monomial potentials with k ≥ 2 can not accommodate the Planck priors [62]. For example, using k = 2 and N = 50 we obtain n s 0.96 and r 0.16. For N = 60 we have n s 0.967 and r 0.133. It is interesting to mention that the chaotic inflation also corresponds to the slow-roll regime of intermediate inflation [60,61,67,68] with Hubble rate during inflation given by H ∝ t k/(4−k) with n s = 1 − (k + 2)r/8k and k = −2 gives n s = 1 exactly to first order. -The R 2 inflation [69]: In Starobinsky inflation the asymptotic behavior of the effective potential becomes which provides the following slow-roll predictions [70,71]: n s ≈ 1 − 2/N and r ≈ 8/B 2 N 2 , where B 2 = 2/3. Therefore, if we select N = 50 then we obtain (n s , r) ≈ (0.96, 0.0048). For N = 60 we find (n s , r) ≈ (0.967, 0.0033). It has been found that the Planck data [62] favors the Starobinsky inflation. Obviously, our results (see figure 3) are consistent with those of R 2 inflation. -Hyperbolic inflation [72]: In hyperbolic inflation the potential is given by V (φ ) ∝ sinh b (φ / f 1 ). Initially, Rubano and Barrow [73] proposed this potential in the context of dark energy. Recently, Basilakos & Barrow [72] investigated the properties of this scalar field potential back in the inflationary epoch. Specifically, the slow-roll param- where φ end f 2 ln θ +1 θ −1 . Comparing this model with the data Basilakos & Barrow [72] found n s 0.968, r 0.075, 1 < b ≤ 1.5 and f 1 ≥ 11.7M pl .
-Other inflationary models: The origin of brane [74,75] and exponential [76,77] inflationary models are motivated by the physics of extra dimentions and supergravity respectively. It has been found that these models are in agreement with the Planck data although the R 2 inflation is the winner from the comparison [62].
At this point we would like to mention that in the highdissipation regime R 1, there is always a region in A − f plane which is consistent with the warm inflation condition T > H. To clarify this issue we plot in Fig.(4) the diagram of log 10 T H in the A − f plane. The solid line corresponds to the boundary limit T = H. Clearly, based on the condition T > H we can reduce the parameter space and thus producing one of the strongest existing constraints (to our knowledge) on A and f . Note that in order to produce the above   diagram we have fixed the initial values of T and H to those at the beginning of inflation. After the triggering of inflation the inflaton/photon interaction takes place which leads to radiation production and thus it guarantees that the above condition holds during the inflationary era. Finally, we investigate the possibility to treat λ as a free parameter. In fact there are three main conditions which we need to use in order to provide a viable limit on the λ . These are: (a) the high dissipation regime R 1, (b) the warm in- flation condition T > H and (c) to recover the Planck2015 (n s , r) observational constraints. Our investigation shows that λ is correlated with the (A, f ) pair. For example for (A, f ) = (0.5, 0.25) we find λ > 6 × 10 −16 which is consistent with above conditions while for (A, f ) = (0.4, 0.2) we obtain λ > 1.5×10 −20 . In general we verify that it is not possible to find a lower value of λ for all pairs of (A, f ).

Appendix
In this paper we have studied our model in natural unit ( h 2π = c = 1) therefore we have ( Using Eq.(5) we have

Conclusions
In this article we investigate the warm inflation for the Friedmann-Robertson-Walker spatially flat cosmological model in which the scale factor of the universe satisfies the form of Barrow [27], namely a(t) = a I exp(At f ) (0 < f < 1). Within this context, we estimate analytically the slow-roll parameters and we compare our predictions with those of other inflationary models as well as we test the performance of warm inflation against the observational data. We find that currently warm inflationary model is consistent with the results given by Planck 2015 within 1σ uncertainties.