1 Introduction

To describe the inflationary phase in the early universe [1–4], many theories have been proposed which most of them are categorized into two classes: modified gravity models [5–9], and models with exotic fields dubbed inflaton [10–13]. These groups may related to each other through some conformal transformations [14–16].

In a well-known model, the responsible of the early accelerated expansion of the Universe is a canonical scalar field \(\varphi \), rolling down slowly a nearly flat potential. Inflation lasts as long as the slow roll conditions hold. In this paradigm we encounter a cold universe at the end of inflation. After the end of the slow roll, the scalar field begins a rapid coherent oscillation and decays to ultra-relativistic particles (radiation) reheating the Universe [17–19]. A natural candidate for this scalar field, as is proposed in [20], is the Higgs boson. In this context, adding a non-minimal coupling between the scalar field and scalar curvature is required for the renormalizability, and also consistency with the amplitude of density perturbations obtained via observations. Another model in which the inflaton is considered as the Higgs field is introduced in [21], where the scalar field has a non-minimal kinetic coupling term. This theory does not suffer from unitary violation and is safe of quantum corrections. In this framework, the inflation and the reheating of the Universe are discussed in the literature [22–25]. The same model, with a non-canonical scalar field dark energy, is also employed to describe the present acceleration of the Universe [26–32]. In the aforementioned model, inflation and reheating happen in two distinct eras, but one can unify them by assuming an appropriate dissipative coefficient which permits the decay of inflaton to radiation during inflation: Warm inflation was first introduced for minimal coupling model [33, 34]. Afterwards, numerous articles has been published in this subject [35–41]. A friction term for inflaton equation of motion is computed in [39–41]. Tachyonic warm inflationary universe models are considered in [42].

In this work we consider warm inflation in non-minimal derivative coupling model. In the second and third sections, based on our previous papers [22–25], we review the non-minimal derivative coupling model in the presence of an additional radiation sector and investigate slow roll conditions. In the fourth section, the perturbations in the model are studied, and the discussion is conducted in such a way that the parameter extractable from the observations such as the spectral index acquire more serviceable and more general compact form with respect to [43, 44], where the perturbations of this model were also discussed. By employing the Planck 2013 data, we use our results to obtain the temperature at the end of warm slow roll inflation.

We use units \(\hbar =c=8\pi G=1\) throughout the paper.

2 Preliminaries

The action of gravitational enhanced friction (GEF) theory is given by [21]

$$\begin{aligned} S=\int \bigg ({1\over 2 }R-{1\over 2}\Delta ^{\mu \nu }\partial _\mu \varphi \partial _{\nu } \varphi - V(\varphi )\bigg ) \sqrt{-g}\mathrm{d}^4x+S_\mathrm{int}+S_{r}, \end{aligned}$$
(1)

where \(\Delta ^{\mu \nu }=g^{\mu \nu }+{1\over M^2}G^{\mu \nu }\), \(G^{\mu \nu }=R^{\mu \nu }-{1\over 2}Rg^{\mu \nu }\) is Einstein tensor, M is a constant, \(S_{r}\) is the matter action and \(S_\mathrm{int}\) describes the interaction of the scalar field with all other ingredients. In the absence of terms containing more than two time derivatives, we have no additional degrees of freedom in this theory. We calculate the energy momentum tensor,

$$\begin{aligned} T_{\mu \nu }=T^{(\varphi )}_{\mu \nu }+T^{(r)}_{\mu \nu }, \end{aligned}$$
(2)

by variation of the action with respect to the metric [45]. \(T^{(r)}_{\mu \nu }\) is the radiation energy momentum tensor and \(T^{(\varphi )}_{\mu \nu }\) is the scalar field energy momentum tensor, consisting of parts coming from the minimal part: \(\mathcal {T}_{\mu \nu }\),

$$\begin{aligned} \mathcal {T}^{(\varphi )}_{\mu \nu }=\nabla _{\mu }\varphi \nabla _{\nu }\varphi -{1\over 2}g_{\mu \nu }{(\nabla \varphi )}^2-g_{\mu \nu }V(\varphi ), \end{aligned}$$
(3)

and parts coming from the non-minimal derivative coupling section, \(\Theta _{\mu \nu }\),

$$\begin{aligned}&\Theta _{\mu \nu }=-{1\over 2}G_{\mu \nu }{(\nabla \varphi )}^2-{1\over 2}R\nabla _{\mu }\varphi \nabla _{\nu }\varphi +R^{\alpha }_{\mu }\nabla _{\alpha }\varphi \nabla _{\nu }\varphi \nonumber \\&\quad +R^{\alpha }_{\nu }\nabla _{\alpha }\varphi \nabla _{\mu }\varphi +R_{\mu \alpha \nu \beta }\nabla ^{\alpha }\varphi \nabla ^{\beta }\varphi +\nabla _{\mu }\nabla ^{\alpha }\varphi \nabla _{\nu }\nabla _{\alpha }\varphi \nonumber \\&\quad -\nabla _{\mu }\nabla ^{\nu }\varphi \Box \varphi -{1\over 2}g_{\mu \nu }\nabla ^{\alpha }\nabla ^{\beta }\varphi \nabla _{\alpha }\nabla _{\beta }\varphi +{1\over 2}g_{\mu \nu }{(\Box \varphi )}^2 \nonumber \\&\quad -g_{\mu \nu }\nabla _{\alpha }\varphi \nabla _{\beta }\varphi R^{\alpha \beta }. \end{aligned}$$
(4)

By variation of the action (1) with respect to the scalar field \(\varphi \), the equation of motion for the homogeneous and isotopic scalar field in the presence of a dissipative term can be expressed as

$$\begin{aligned} \left( 1+{3H^2\over M^2}\right) \ddot{\varphi }+ & {} 3H\left( 1+{3H^2\over M^2}+{2\dot{H}\over M^2}\right) \dot{\varphi }\nonumber \\+ & {} V'(\varphi )+\Gamma \dot{\varphi }=0, \end{aligned}$$
(5)

where \(H={\dot{a}\over a}\) is the Hubble parameter, a dot is differentiation with respect to the cosmic time t, a prime is differentiation with respect to the scalar field \(\varphi \), and \(\Gamma \dot{\varphi }\) is the friction term adopted phenomenologically to describe the decay of the \(\varphi \) field and its energy transfer into the radiation bath. \(\Gamma \) in general is a function of \(\varphi \) and temperature [46, 47]. The Friedman equation for this model is given by

$$\begin{aligned} H^2={1\over 3}\left( \left( 1+{9H^2\over M^2}\right) {\dot{\varphi }^2\over 2}+V(\varphi )+\rho _{r}\right) , \end{aligned}$$
(6)

where \(\rho _{r}\) is the energy density of the radiation, which can be written as [46]

$$\begin{aligned} \rho _{r}={3\over 4}TS. \end{aligned}$$
(7)

S is the entropy density and T is the temperature. The energy density and pressure of homogeneous and isotropic scalar field are given by

$$\begin{aligned} \rho _{\varphi }=\left( \left( 1+{9H^2\over M^2}\right) {\dot{\varphi }^2\over 2}+V(\varphi )\right) \end{aligned}$$
(8)

and

$$\begin{aligned} P_{\varphi }=\left( 1-{3H^2\over M^2}-{2\dot{H}\over M^2}\right) {\dot{\varphi }^2\over 2}-V(\varphi )-{2H\dot{\varphi }\ddot{\varphi }\over M^2}, \end{aligned}$$
(9)

respectively. By the continuity equation for the total system \(\dot{\rho }+3H(\rho +P)=0\), and also the equation of motion (5), we obtain

$$\begin{aligned} \dot{\rho _{r}}+4H\rho _{r}=\Gamma {\dot{\phi }}^2, \end{aligned}$$
(10)

which gives the rate of entropy production as

$$\begin{aligned} T(\dot{S}+3HS)=\Gamma \dot{\varphi }^2. \end{aligned}$$
(11)

3 Slow roll approximation

In the previous section we pointed out to the equations needed to describe the scalar field and radiation evolutions in an interacting non-minimal coupling model. Hereafter we consider the slow roll approximation:

$$\begin{aligned} \ddot{\varphi }\ll 3H\dot{\varphi } \quad \dot{H}\ll H^2 \quad \left( 1+{9H^2\over M^2}\right) {\dot{\varphi }^2\over 2}\ll V(\varphi ). \end{aligned}$$
(12)

The entropy density satisfies

$$\begin{aligned} TS\ll V(\varphi ) \quad \dot{S}\ll 3HS. \end{aligned}$$
(13)

For a positive potential, the slow roll conditions give rise to the inflation. Neglecting the second order derivative, we can write the equation of motion of the scalar field as

$$\begin{aligned} \dot{\varphi }\simeq -{V'(\varphi )\over {3HU(1+r)}}, \end{aligned}$$
(14)

where

$$\begin{aligned} U=1+{3 H^2\over M^2} \qquad r={\Gamma \over 3UH}. \end{aligned}$$
(15)

r is the ratio of thermal damping component to the expansion damping. During the slow roll warm inflation, the potential energy of the scalar field is dominant, and therefore the Friedman equation becomes

$$\begin{aligned} H^2\simeq {1\over 3} V(\varphi ). \end{aligned}$$
(16)

We have also

$$\begin{aligned} ST\simeq Ur\dot{\varphi }^2. \end{aligned}$$
(17)

By Eq. (16) we can write U as a function of the potential,

$$\begin{aligned} U=1+{V(\varphi )\over M^2 }. \end{aligned}$$
(18)

We employ the following set of parameters to characterize the slow roll:

$$\begin{aligned} \delta= & {} {1\over 2}{\left( {V'(\varphi )\over V(\varphi )}\right) }^2{1\over U(\varphi )},\end{aligned}$$
(19)
$$\begin{aligned} \eta= & {} {V''(\varphi )\over V(\varphi )}{1\over U(\varphi )},\end{aligned}$$
(20)
$$\begin{aligned} \beta= & {} {\Gamma '(\varphi )V'(\varphi )\over \Gamma (\varphi )V(\varphi )}{1\over U(\varphi )},\end{aligned}$$
(21)
$$\begin{aligned} \epsilon= & {} -{\dot{H}\over H^2}. \end{aligned}$$
(22)

To express slow roll conditions in terms of these parameters, we need to calculate \(\dot{U}\) and \(\dot{r}\). We have

$$\begin{aligned} \dot{U}={6\dot{H}H\over M^2}, \end{aligned}$$
(23)

therefore

$$\begin{aligned} {\dot{U}\over H}=-2\epsilon (U-1) \end{aligned}$$
(24)

and

$$\begin{aligned} {\dot{r}\over H}=-\beta {r\over r+1}+\epsilon r\left( 3-{2\over U}\right) . \end{aligned}$$
(25)

Using Eq. (16), one can obtain \(\epsilon \) as a function \(\delta \) and r,

$$\begin{aligned} \epsilon ={\delta \over 1+r}. \end{aligned}$$
(26)

From (14) we can derive

$$\begin{aligned} {\ddot{\varphi }\over H\dot{\varphi }}=-\eta {1\over r+1}+\delta \left( 3+{2\over U}\right) {1\over (1+r)^2}+\beta {r\over (1+r)^2}.\nonumber \\ \end{aligned}$$
(27)

The slow roll conditions can be expressed as

$$\begin{aligned} \epsilon \ll 1,\quad \delta \ll 1+r,\quad \eta \ll 1+r, \quad \beta \ll 1+r. \end{aligned}$$
(28)

Note that if \({H^2\over M^2}\rightarrow 0\) our model reduces to warm inflation in a minimal coupling model [33, 34], and if \(r\rightarrow 0\) and \({H^2\over M^2}\rightarrow 0\) we recover the standard slow roll inflation [48, 49]. By using Eqs. (19)–(21), we get

$$\begin{aligned} {1\over H}{\text {d}\ln (TS)\over \text {d}t}=\epsilon \left( 1+2{(3-{2\over U})\over 1+r}\right) +\beta {-1+r\over (1+r)^2}-2\eta {1\over (1+r)}. \end{aligned}$$
(29)

In our study, we take \(r\gg 1\) and consider the high friction limit,

$$\begin{aligned} {H^2\over M^2}\gg 1, \end{aligned}$$
(30)

therefore \(U\simeq {3H^2\over M^2}\gg 1\) and \({1\over H}{\text {d}\ln (TS)\over \text {d}t}={1\over H}\left( {\dot{T}\over T}+{\dot{S}\over S}\right) \ll 1\).

The number of e-folds during slow roll warm inflation is

$$\begin{aligned} \mathcal {N}=\int ^{t_\mathrm{end}}_{t_{\star }} H\text {d}t=\int ^{\varphi _\mathrm{end}}_{\varphi _{\star }}{H\over \dot{\varphi }}\text {d}\varphi = -\int ^{\varphi _\mathrm{end}}_{\varphi _{\star }}{3H^2U(1+r)\over V'(\varphi )}\text {d}\varphi , \end{aligned}$$
(31)

where \(\varphi _{\star }=\varphi (t_{\star })\) and \(\varphi _\mathrm{end}=\varphi (t_\mathrm{end})\) are the values of the scalar field at the horizon crossing (\(t_{\star }\)), and at the end of inflation, (\(t_\mathrm{end}\)). By horizon crossing (or horizon exit) we mean the time at which a pivot scale exited the Hubble radius during inflation. Using the Friedman equation the above relation becomes

$$\begin{aligned} \mathcal {N}=\int ^{\varphi _\mathrm{end}}_{\varphi _{\star }}{V(\varphi )\over V'(\varphi )}U(1+r) \text {d}\varphi . \end{aligned}$$
(32)

At the end of this section, by choosing the form of \(\Gamma \) and the potential, we derive more specific results. We adopt the (general) damping term proposed in [46]

$$\begin{aligned} \Gamma =\Gamma _0{\left( {\varphi \over \varphi _0}\right) }^p, \end{aligned}$$
(33)

where p is an arbitrary real number and \(\varphi _0,\Gamma _0\) are constant, and we consider the power law potential

$$\begin{aligned} V(\varphi )=\lambda \varphi ^n, \end{aligned}$$
(34)

where n and \(\lambda \) are two constants. By using Eq. (18), and in the high friction limit for \(r\gg 1\), after some computations we obtain

$$\begin{aligned} \rho _r={\Gamma \dot{\varphi }^2\over 4H}. \end{aligned}$$
(35)

By inserting \(\dot{\varphi }\) from (14) into the above equation we obtain

$$\begin{aligned} \rho _r={V'(\varphi )^2\over 4H\Gamma }={\sqrt{3} V'(\varphi )^2\over 4\Gamma \sqrt{V(\varphi )}}. \end{aligned}$$
(36)

Using (33) and (34), \(\rho _r\) is obtained as

$$\begin{aligned} \rho _r={\sqrt{3} n^2 \lambda ^{3\over 2}\varphi _0^p \over 4\Gamma _0}\varphi ^{({3n\over 2}-2-p)}. \end{aligned}$$
(37)

We can write radiation energy density as a function of temperature,

$$\begin{aligned} \rho _r={g\pi ^2\over 30}T^4, \end{aligned}$$
(38)

where g is the number of degree of freedom for ultra-relativistic particles. By Eqs. (37) and (38) the temperature of the universe may derived as a function of \(\varphi \),

$$\begin{aligned} T=A \varphi ^{{3n-4-2p\over 8}}, \end{aligned}$$
(39)

where in this relation A is given by

$$\begin{aligned} A={\left( {15\sqrt{3} n^2 \lambda ^{3\over 2}\varphi _0^p\over 2\Gamma _0 g\pi ^2}\right) }^{1\over 4}. \end{aligned}$$
(40)

The slow roll parameters may now be expressed as

$$\begin{aligned} \delta= & {} {M^2 n^2\over 2\lambda }{1\over \varphi ^{n+2}},\end{aligned}$$
(41)
$$\begin{aligned} \eta= & {} {M^2 n(n-1)\over \lambda }{1\over \varphi ^{n+2}}, \end{aligned}$$
(42)

and the number of e-folds is given by

$$\begin{aligned} \mathcal {N}={1\over 3}\int _{\varphi _\mathrm{end}}^{\varphi _\star } {V(\varphi )\Gamma \over V'(\varphi )H}\text {d}\varphi , \end{aligned}$$
(43)

where \(\varphi _{\star }=\varphi (t_{\star })\) and \(t_{\star }\) is the time at the horizon crossing. By using (39) and assuming \(\varphi _{\star }\ll \varphi _\mathrm{end}\), the number of e-folds becomes

$$\begin{aligned} \mathcal {N}={4\Gamma _0\over \sqrt{3}{n\sqrt{\lambda }\varphi _0^p}}\times {\varphi _\star ^{4p-2n+8\over 4}\over 4p-2n+8}. \end{aligned}$$
(44)

The equation \({\ddot{a}\over a}=H^2(1-\epsilon )\) implies that the inflation ends when \(\epsilon \sim 1\) (\(\sim \) denotes the order of magnitude). Putting \(\epsilon \sim 1\) back into (26) gives \(\delta \sim 1+r \) and if \(r\gg 1 \), at the end of warm inflation we have \(\delta \sim r\).

4 Cosmological perturbations

In this section we consider the evolution equation for the first order cosmological perturbations of a system containing inflaton and radiation. In the Newtonian gauge, scalar perturbations of the metric can be written as [50]

$$\begin{aligned} \text {d}s^2=-(1+2\Phi )\text {d}t^2+a^2(1-2\Psi )\delta _{ij}\text {d}x^i\text {d}x^j. \end{aligned}$$
(45)

The energy momentum tensor splits into radiation \(T^{\mu \nu }_r\) and an inflaton part \(T^{\mu \nu }_{\varphi }\),

$$\begin{aligned} T^{\mu \nu }=T^{\mu \nu }_r+T^{\mu \nu }_{\varphi }. \end{aligned}$$
(46)

\(T^{\mu \nu }_{\varphi }\) is the energy momentum tensor of the inflaton, introduced in the second section. We have modeled the radiation field as a perfect barotropic fluid. We have

$$\begin{aligned} T^{\mu \nu }_r=(\rho _{r}+P_r)u_{\mu }u_{\nu }+P_rg_{\mu \nu }, \end{aligned}$$
(47)

where \(u_r\) is the four-velocity of the radiation fluid and \(\overline{u}_i=0\) and \(\overline{u}_0=-1\). A bar denotes unperturbed quantities. By considering the normalization condition \(g^{\mu \nu }u_{\mu }u_{\nu }=-1\), we obtain

$$\begin{aligned} \delta u^0=\delta u_0={h_{00}\over 2}. \end{aligned}$$
(48)

\(\delta u^i\) is an independent dynamical variable. We can define \(\delta u_i=\partial _i\delta u\) [50]. Energy transfer between the two components is described by a flux term [51],

$$\begin{aligned} Q_{\mu }=-\Gamma u^{\nu }\partial _{\mu }\varphi \partial _{\nu }\varphi , \end{aligned}$$
(49)

associated to the field equations

$$\begin{aligned} \nabla _{\mu }T^{\mu \nu }_r=Q^{\nu } \end{aligned}$$
(50)

and

$$\begin{aligned} \nabla _{\mu }T^{\mu \nu }_{\varphi }=-Q^{\nu }. \end{aligned}$$
(51)

From Eq. (49) we deduce \(Q_{0}=\Gamma \dot{\varphi }^2\), so the unperturbed equation (50) becomes \(Q_{0}=\dot{\rho }_r +3H(\rho _r+P_r)\), which is the continuity equation for radiation field in the presence of interaction. Similarly, Eq. (51) becomes \(-Q_{0}=\dot{\rho }_{\varphi } +3H(\rho _{\varphi }+P_{\varphi })\). Perturbations to the energy momentum transfer are described by the energy transfer

$$\begin{aligned} \delta Q_{0}=-\delta \Gamma \dot{\varphi }^2+\Phi \Gamma \dot{\varphi }^2-2\Gamma \dot{\varphi }\dot{\delta \varphi } \end{aligned}$$
(52)

and the momentum flux

$$\begin{aligned} \delta Q_{i}=-\Gamma \dot{\varphi }\partial _i{\delta \varphi }. \end{aligned}$$
(53)

By variation of Eq. (50) as \(\delta (\nabla _{\mu }T^{\mu \nu }_r)=\delta Q^{\nu }\), for the zeroth (0–0) component we obtain

$$\begin{aligned} \dot{\delta \rho _r}+4H\delta \rho _r+{4\over 3}\rho _r\nabla ^2\delta u-4\dot{\Psi }\rho _r= & {} -\Phi \Gamma {\dot{\varphi }}^2+\delta \Gamma {\dot{\varphi }}^2\nonumber \\&+2\Gamma \dot{\delta \varphi }\dot{\varphi }, \end{aligned}$$
(54)

and for the ith component we derive

$$\begin{aligned}&4\rho _r\dot{\delta u^i}+4\dot{\rho _r}\delta u^i+20 H\rho _r\delta u^i\nonumber \\&\quad =-a^2[3\Gamma \dot{\varphi }\partial _i\delta \varphi +\partial _i\delta \rho _r+4\rho _r\partial _i\Phi ]. \end{aligned}$$
(55)

The equation of motion for perturbation of the scalar field can be calculated by variation of (51) as \(\delta (\nabla _{\mu }T^{\mu \nu }_{\varphi })=-\delta Q^{\nu }\) giving

$$\begin{aligned}&\left( 1+{3H^2\over M^2}\right) \ddot{\delta \varphi }+\left[ \left( 1+{3H^2\over M^2}+{2\dot{H}\over M^2}\right) 3H+\Gamma \right] \dot{\delta \varphi } \nonumber \\&\quad +\,\delta V'(\varphi )+\dot{\varphi }\delta \Gamma \end{aligned}$$
(56)
$$\begin{aligned}&-\left( 1+{3H^2\over M^2}+{2\dot{H}\over M^2}\right) {\nabla ^2\delta \varphi \over a^2} \nonumber \\&\quad = - \left[ 2V'(\varphi )+3\Gamma \dot{\varphi }-{6H\dot{\varphi }\over M^2}(3H^2+2\dot{H}) -{6H^2\ddot{\varphi }\over M^2}\right] \Phi \nonumber \\&\qquad + \left( 1+{9H^2\over M^2}\right) \dot{\varphi }\dot{\Phi }+{2H\dot{\varphi }\over M^2}{\nabla ^2\Phi \over a^2}\nonumber \\&\qquad +3 \left( 1+{9H^2\over M^2}+{2\dot{H}\over M^2}+{2H\ddot{\varphi }\over M^2}\right) \dot{\Psi }\nonumber \\&\qquad +{6H\dot{\varphi }\over M^2}\ddot{\Psi }-{2(\ddot{\varphi }+H\dot{\varphi })\over M^2}{\nabla ^2\Psi \over a^2}, \end{aligned}$$
(57)

for the zeroth component. By using a perturbation to the Einstein field equation \(G_{\mu \nu }=-T_{\mu \nu }\) (note that we have taken \(8\pi G=1\)), one can obtain the evolution equation for the perturbation parameters, which for the 0–0 component is

$$\begin{aligned}&-3H\dot{\Psi }-3H^2\Phi +{\nabla ^2\Psi \over a^2} \nonumber \\&\quad = {1\over 2}\bigg [-\bigg (1+{18H^2\over M^2}\bigg ){\dot{\varphi }}^2\Phi -{9H{\dot{\varphi }}^2\over M^2}\dot{\Psi } \nonumber \\&\qquad +{{\dot{\varphi }}^2\over M^2}{\nabla ^2\Psi \over a^2}+\acute{V(\varphi )}\delta \varphi +\left( 1+{9H^2\over M^2}\right) \dot{\varphi }\dot{\delta \varphi } \nonumber \\&\qquad - {2H\dot{\varphi }\over M^2}{\nabla ^2{(\delta \varphi )}\over a^2}+\delta \rho _r\bigg ], \end{aligned}$$
(58)

and the i–i components are

$$\begin{aligned}&(3H^2+2\dot{H})\Phi +H(3\dot{\Psi }+\dot{\Phi })+{\nabla ^2(\Phi -\Psi )\over 3a^2}+\ddot{\Psi } \nonumber \\&\quad = {1\over 2}\bigg [\bigg ({(3H^2+2\dot{H}){2\dot{\varphi }^2\over M^2}-{\dot{\varphi }}^2+{8H\dot{\varphi }\ddot{\varphi }\over M^2}}\bigg )\Phi + {3H{\dot{\varphi }}^2\over M^2}\dot{\Phi } \nonumber \\&\qquad +{{\dot{\varphi }}^2\over M^2}{\nabla ^2\Phi \over 3a^2}+\bigg ({3H{\dot{\varphi }}^2\over M^2}+{2\dot{\varphi }\ddot{\varphi }\over M^2}\bigg )\dot{\Psi }+{{\dot{\varphi }}^2\over M^2}\ddot{\Psi }+{{\dot{\varphi }}^2\over M^2}{\nabla ^2\Psi \over 3a^2} \nonumber \\&\qquad -\acute{V(\varphi )}\delta \varphi -\bigg [\bigg (-1+{3H^2\over M^2}+{2\dot{H}\over M^2}\bigg )\dot{\varphi }+{2H\ddot{\varphi }\over M^2}\bigg ]\dot{\delta \varphi } \nonumber \\&\qquad -{2H\dot{\varphi }\over M^2}\ddot{\delta \varphi } +{2(\ddot{\varphi }+H\dot{\varphi })\over M^2}{\nabla ^2{(\delta \varphi )}\over 3a^2}+\delta P_r\bigg ]. \end{aligned}$$
(59)

By the relation \(-H\partial _i\Phi -\partial _i\dot{\Psi }={1\over 2}(\rho +P)\partial _i\delta u \), from the 0–i component of the field equation we have

$$\begin{aligned} H\Phi +\dot{\Psi }= & {} {1\over 2}\bigg [{3H\dot{\varphi }^2\over M^2}\Phi +{\dot{\varphi }^2\over M^2}\dot{\Psi }+\bigg (1+{3H^2\over M^2}\bigg )\dot{\varphi }\delta \varphi \nonumber \\&-{2H\dot{\varphi }\over M^2}\dot{\delta \varphi }+(\rho _r+P_r)\delta u\bigg ]. \end{aligned}$$
(60)

The six equations (54)–(60) generally describe the evolution of perturbations.

We consider the quantities in momentum space via Fourier transform; therefore the spatial parts of these quantities are \(e^{ikx}\) where k is the wave number of the corresponding mode. So by replacing \(\partial _j\rightarrow ik_j \) and \(\nabla ^2\rightarrow -k^2 \), and defining

$$\begin{aligned} \delta u={-a\over k}v e^{ikx}, \end{aligned}$$
(61)

we can write Eq. (55) as

$$\begin{aligned} \rho _r\dot{v}+\dot{\rho _r}v+4H\rho _r v={k\over a}\bigg [\rho _r\Phi +{\delta \rho _r\over 4}+{3\over 4}\Gamma \dot{\varphi }\delta \varphi \bigg ]. \end{aligned}$$
(62)

During warm inflation the background and perturbation satisfy the slow roll approximation. In other words the background and perturbations vary slowly in time (e.g. \(\dot{\Phi }\ll H\Phi \)). We consider modes with wavenumbers satisfying \({k\over a}\ll H\). By applying these conditions to Eq. (54) and considering the high friction regime, we obtain

$$\begin{aligned} {\delta \rho _r\over \rho _r} =-\Phi +{\delta \Gamma \over \Gamma }. \end{aligned}$$
(63)

Similarly, (62) reduces to

$$\begin{aligned} v={k\over 4aH}\bigg [\Phi +{\delta \rho _r\over 4\rho _r}+{3\Gamma \dot{\varphi }\delta \varphi \over 4\rho _r}\bigg ], \end{aligned}$$
(64)

and Eq. (56) takes the form

$$\begin{aligned}&\bigg [\bigg (1+{3H^2\over M^2}\bigg )3H+\Gamma \bigg ]\dot{\delta \varphi } +\delta V'(\varphi )+\dot{\varphi }\delta \Gamma \nonumber \\&\quad =-\bigg [2V'(\varphi )+3\Gamma \dot{\varphi }-{3H^2\over M^2}(6H\dot{\varphi })\bigg ]\Phi . \end{aligned}$$
(65)

We derive also

$$\begin{aligned} H\Phi ={1\over 2}\bigg [{3H\dot{\varphi }^2\over M^2}\Phi +\bigg (1+{3H^2\over M^2}\bigg )\dot{\varphi }\delta \varphi -{4a\over 3k}\rho _rv\bigg ]. \end{aligned}$$
(66)

From Eqs. (63–66) we can calculate \(\delta \varphi \) as a function of H, \(\Gamma \), and \(V(\varphi )\),

$$\begin{aligned} \delta \varphi \approx C V' \exp {(\mathfrak {I}(\varphi ))}, \end{aligned}$$
(67)

where \(\mathfrak {I}(\varphi )\) is defined as

$$\begin{aligned}&\mathfrak {I}(\varphi )\equiv -\int \bigg ({\Gamma '\over \Gamma }{r\over 1+r}+{V'\over V}{2+5r\over 2(r+1)^2}\nonumber \\&\quad \times {\bigg [1+{3r\over 4}-{\beta r\over 16(1+r)}\bigg ]}\bigg )\mathrm{d}\varphi . \end{aligned}$$
(68)

The density perturbation is then [35, 42, 46]

$$\begin{aligned} \delta _H={16\pi \over 5}{\exp {(-\mathfrak {I}(\varphi ))}\over V'}\delta \varphi . \end{aligned}$$
(69)

In this relation \(\delta \varphi \) is the fluctuation of the scalar field during the warm inflation [33–35]

$$\begin{aligned} \delta \varphi ^2={k_FT\over 2\pi ^2}, \end{aligned}$$
(70)

where \(k_F\) is the freeze out scale. To calculate \(k_F\), we must determine the time at which the damping rate of relation (56) falls below the expansion rate H. At the freeze out time, \(t_F\), the freeze out wavenumber, \(k_F={k\over a(t_F)}\), is given by

$$\begin{aligned} k_F=\sqrt{\Gamma H+3H^2\left( 1+{3H^2\over M^2}\right) }=\sqrt{3H^2U(1+r)}, \end{aligned}$$
(71)

therefore the density perturbation becomes

$$\begin{aligned} \delta _H^2= \left( {128\over 25}\right) \left( {\exp (-2\mathfrak {I}(\varphi ))\over {V'(\varphi )}^2}\right) \sqrt{3H^2U(1+r)}T. \end{aligned}$$
(72)

The spectral index for the scalar perturbation is given by

$$\begin{aligned} n_s-1={\text {d}\ln {\delta _H^2}\over \text {d}\ln {k}}, \end{aligned}$$
(73)

where this derivative is computed at the horizon crossing \(k\approx aH\). Finally we obtain

$$\begin{aligned} n_s-1= & {} {2\eta \over (1+r)}-{\delta \over 2(1+r)}-{\beta (1+5r)\over 2{(1+r)}^2} \nonumber \\&-{\delta (2+5r)(4+3r)\over 2(1+r)^2} +{\delta \beta r(2+5r)\over 8(1+r)^4}. \end{aligned}$$
(74)

Using (33) and (34), one can see that the slow roll parameters are

$$\begin{aligned} \delta \sim {n^2\over 2}\alpha ,\quad \eta \sim n(n-1)\alpha ,\quad \beta \sim pn\alpha , \end{aligned}$$
(75)

where

$$\begin{aligned} \alpha ={M^2\over \lambda }\varphi ^{-(n+2)}. \end{aligned}$$
(76)

For \(r\gg 1\), we have

$$\begin{aligned} \delta _H^2=\left( {128\over 25\times 3^{1\over 4}}\right) \left( {V^4\Gamma ^{5\over 2}\over {V'}^2}\right) T. \end{aligned}$$
(77)

With our power law choices for the potential and dissipation coefficient, (77) reduces to

$$\begin{aligned} \delta _H^2= \left( {128\lambda ^2\over 25\times 3^{1\over 4}n^2}\right) {\left( {\Gamma _0\over \varphi _0^p}\right) }^{5\over 2}\varphi ^{(2n+2+{5p\over 2})}T. \end{aligned}$$
(78)

The spectral index is

$$\begin{aligned} n_s-1={2\eta \over r}-{8\delta \over r}-{5\beta \over 2r}. \end{aligned}$$
(79)

We can rewrite this relation as

$$\begin{aligned} n_s-1=-{n\alpha \over r}\left( 2n+2+{5p\over 2}\right) , \end{aligned}$$
(80)

where r is given by

$$\begin{aligned} r={\Gamma _0 M^2\over \sqrt{3}\varphi _0^p\lambda ^{3\over 2}}\varphi ^{(p-{3n\over 2})}. \end{aligned}$$
(81)

By inserting the value of \(\varphi \) at the horizon crossing in (80) we get

$$\begin{aligned} n_s-1=-{n\sqrt{3\lambda }\varphi _0^p\over \Gamma _0}\left( 2n+2+{5p\over 2}\right) \varphi ^{-(p+2-{n\over 2})}. \end{aligned}$$
(82)

5 Evolution of the universe and temperature of the warm inflation

In this section, using our previous results, we intend to calculate the temperature of warm inflation as a function of observational parameters via the method introduced in [52]. By the temperature of warm inflation, we mean the temperature of the universe at the end of warm inflation. For this purpose, we divide the evolution of the universe into three parts as follows:

  1. (I)

    from \(t_\star \) (horizon exit) until the end of slow roll warm inflation, denoted by \(t_e\). In this era, the potential of the scalar field is the dominant term in the energy density.

  2. (II)

    from \(t_e\) until recombination era, denoted by \(t_\mathrm{rec}\).

  3. (III)

    from \(t_\mathrm{rec}\) until the present time \(t_0\).

Therefore the number of e-folds from horizon crossing until now becomes

$$\begin{aligned} \mathcal {N}= & {} \ln {\bigg ({a_0\over a_\star }\bigg )}=\ln {\bigg ({a_0\over a_\mathrm{rec}}\bigg )}+\ln {\bigg ({a_\mathrm{rec}\over a_{e}}\bigg )}+\ln {\bigg ({a_e\over a_{\star }}\bigg )} \nonumber \\= & {} \mathcal {N}_\mathrm{I}+\mathcal {N}_\mathrm{II}+\mathcal {N}_\mathrm{III}. \end{aligned}$$
(83)

5.1 Slow roll

During the slow roll warm inflation, the scalar field rolls down to the minimum of the potential and ultra-relativistic particles are generated. In this period the positive potential energy of the scalar field is dominant and therefore expansion of the universe is accelerated. By Eqs. (44) and (82), for high damping term \(r\gg 1\), the number of e-folds during warm inflation becomes

$$\begin{aligned} \mathcal {N}_\mathrm{I}={2n+2+{5p\over 2}\over (p+2-{n\over 2})(1-n_s)}. \end{aligned}$$
(84)

We need to calculate the scalar field and the temperature at the end of slow roll. Inflation ends at the time when \(r(\varphi _\mathrm{end})\sim \delta (\varphi _\mathrm{end})\). From Eqs. (75) and (81), we can calculate the scalar field at the end of inflation as

$$\begin{aligned} \varphi _\mathrm{end}^{-(p+2-{n\over 2})}\simeq {2\Gamma _0\over n^2\sqrt{3\lambda }\varphi _0^p}. \end{aligned}$$
(85)

At the end of inflation the radiation energy density becomes of the same order as the energy density of the scalar field,

$$\begin{aligned} \rho _\mathrm{end}\simeq V(\varphi _\mathrm{end})=\lambda {\Big ({n^2\sqrt{3\lambda }\varphi _0^p\over 2\Gamma _0}\Big )}^{n\over p+2-{n\over 2}}. \end{aligned}$$
(86)

From Eq. (38) we deduce that the temperature of the universe at the end of inflation is

$$\begin{aligned} T_\mathrm{end}\simeq \left( {30\lambda \over g \pi ^2}\right) ^{1\over 4}\left( {2\Gamma _0\over n^2\sqrt{3\lambda }\varphi _0^p}\right) ^{-n\over 4(p+2-{n\over 2})}. \end{aligned}$$
(87)

5.2 Radiation dominated and recombination eras

At the end of the warm inflation, the universe enters a radiation dominated epoch. During this era the universe is filled of ultra-relativistic particles which are in thermal equilibrium, and experiences an adiabatic expansion during which the entropy per comoving volume is conserved: \(\mathrm{d}S=0\) [53]. In this era the entropy density, \(s=Sa^{-3}\), is derived as [53]

$$\begin{aligned} s={2\pi ^2\over 45}g T^3, \end{aligned}$$
(88)

So we have

$$\begin{aligned} {a_\mathrm{rec}\over a_\mathrm{end}}={T_\mathrm{end}\over T_\mathrm{rec}}\left( {g_\mathrm{end}\over g_\mathrm{rec}}\right) ^{1\over 3}. \end{aligned}$$
(89)

In the recombination era, \(g_\mathrm{rec}\) is related to photons degrees of freedom and as a consequence \(g_\mathrm{rec}=2\). Hence

$$\begin{aligned} \mathcal {N}_\mathrm{II}= \ln \left( {T_\mathrm{end}\over T_\mathrm{rec}}\left( {g_\mathrm{end}\over 2}\right) ^{1\over 3}\right) . \end{aligned}$$
(90)

By the expansion of the universe, the temperature diminishes: \(T(z)=T(z=0)(1+z)\), where z is the redshift parameter. So we can describe \(T_\mathrm{rec}\) in terms of \(T_\mathrm{CMB}\) as

$$\begin{aligned} T_\mathrm{rec}=(1+z_\mathrm{rec})T_\mathrm{CMB}. \end{aligned}$$
(91)

We have also

$$\begin{aligned} {a_0\over a_\mathrm{rec}}=(1+z_\mathrm{rec}), \end{aligned}$$
(92)

hence

$$\begin{aligned} \mathcal {N}_\mathrm{II}+\mathcal {N}_\mathrm{III}=\ln \left( {T_\mathrm{end}\over T_\mathrm{CMB}}\left( {g_\mathrm{end}\over 2}\right) ^{1\over 3}\right) . \end{aligned}$$
(93)

5.3 Temperature in the warm inflation

We have determined the number of e-folds appearing in the right hand side of (83). To determine the warm inflation temperature we are required to determine \(\mathcal {N}\) in (83). By assuming \(a_0=1\), the number of e-folds from the horizon crossing until the present time is obtained as \(\mathcal {N}=\ln (\Delta )\), where

$$\begin{aligned} \Delta ={1\over a_*}={H_*\over k_0}={V(\varphi _\star )^{1\over 2}\over \sqrt{3}k_0}. \end{aligned}$$
(94)

By Eqs. (83), (84), and (93) we can obtain \(T_\mathrm{end}\),

$$\begin{aligned} T_\mathrm{end}=T_\mathrm{CMB}{\left( {2\over g_\mathrm{end}}\right) }^{1\over 3}\exp {\left( \mathcal {N}-{(2n+2+{5p\over 2})\over (p+2-{n\over 2})(1-n_s)}\right) }, \end{aligned}$$
(95)

which by using Eq. (94) can be expressed as

$$\begin{aligned} T_\mathrm{end}=T_\mathrm{CMB}{\left( {2\over g_\mathrm{end}}\right) }^{1\over 3}{\lambda ^{1\over 2}\varphi _\star ^{n\over 2}\over \sqrt{3}k_0}\exp {\left( -{(2n+2+{5p\over 2})\over (p+2-{n\over 2})(1-n_s)}\right) }. \end{aligned}$$
(96)

With the help of the relations \(\mathcal {P}_s(k_0)={25\over 4}\delta ^2_H(k_0)\) (\(k_0\) is the pivot scale) and (78) we express the power spectrum as

$$\begin{aligned} \mathcal {P}_s(k_0)\approx {\left( {32\over 3^{{1\over 4}}n^2}\right) } {\left( {\Gamma _0\over \varphi _0^p}\right) }^{5\over 2}\lambda ^2\varphi _\star ^{(2n+2+{5p\over 2})}T_\star . \end{aligned}$$
(97)

In the above equation \(T_\star \) is the temperature of the universe at the horizon crossing where Eq. (39) holds, thus

$$\begin{aligned} \mathcal {P}_s(k_0)\approx {32\lambda ^{19\over 8}\over n^{3\over 2}}{\left( {\left( {\Gamma _0\over \varphi _0^p}\right) }^9\times {15\over 2\sqrt{3} g_\mathrm{end}\pi ^2}\right) }^{1\over 4}\varphi _\star ^{({19n+12+18p\over 8})}. \end{aligned}$$
(98)

From (82) and (98) we have

$$\begin{aligned} \varphi _*=\left( {\mathcal {P}_s(k_0)\over \Omega (1-n_s)^{19\over 4}\left( {\Gamma _0\over \varphi _0^p}\right) ^7}\right) ^{1\over 7p+1}, \end{aligned}$$
(99)

where

$$\begin{aligned} \Omega ={2^{9.5}\times 3^{-2.25}\times 5^{0.25}\over n^{{25\over 4}}(4n+5p+4)^{{19\over 4}}(\pi ^2g_\mathrm{end})^{1\over 4}}. \end{aligned}$$
(100)

In addition from (82) and (98) and (99)

$$\begin{aligned} \sqrt{\lambda }={\mathcal {P}_s(k_0)^{p-{n\over 2}+2\over 7p+1}\left( {\Gamma _0\over \varphi _0^p}\right) ^{7n-26\over 2(1+7p)}(1-n_s)^{18p+19n-68\over 8(7p+1)}\over \sqrt{3}n\Omega ^{p+2-{n\over 2}\over 7p+1}(2n+{5p\over 2}+2)}. \end{aligned}$$
(101)

By inserting (99) and (101) in (96), we derive

$$\begin{aligned} T_\mathrm{end}= & {} B{T_\mathrm{CMB}\over k_0}(1-n_s)^{9p-34\over 28p+4}\mathcal {P}_s(k_0)^{p+2\over 7p+1}\left( {\Gamma _0\over \varphi _0^p}\right) ^{-13\over 7p+1}\nonumber \\&\times \exp \left( -{2n+{5p\over 2}+2\over (p-{n\over 2})(1-n_s)}\right) , \end{aligned}$$
(102)

where B is defined by

$$\begin{aligned} B={2^{1\over 3}\Omega ^{-{p+2\over 7p+1}}\over 3ng_\mathrm{end}^{1\over 3}(2n+{5p\over 2}+2)}. \end{aligned}$$
(103)

We use (87) and (101) to obtain a second equation for the temperature in terms of dissipation factor as

$$\begin{aligned} T_\mathrm{end}= & {} C(1-n_s)^{-{(p+2)(18p+19n-68)\over 8(7p+1)(-2p+n-4)}}\nonumber \\&\times P_s(k_0)^{2+p\over 2(7p+1)}\left( {\Gamma _0\over \varphi _0^p}\right) ^{-{13\over 2(7p+1)}}, \end{aligned}$$
(104)

where C is defined by

$$\begin{aligned} C= & {} \left( {30\over \pi ^2g_\mathrm{end}}\right) ^{1\over 4}\left( {2\over \sqrt{3}n^2}\right) ^{-{n\over 4\left( p-{n\over 2}+2\right) }}\nonumber \\&\times \Omega ^{-{{p+2}\over 2(7p+1)}}\left( \sqrt{3}n\left( 2n+{5p\over 2}+2\right) \right) ^{-{{p+2\over 2p-n+4}}}. \end{aligned}$$
(105)

By combining (104) and (102), we can determine the temperature in terms p, n, and the spectral index

$$\begin{aligned} T_\mathrm{end}= & {} K{k_0\over T_\mathrm{CMB}}(1-n_s)^{n\over 2p+4-n}\nonumber \\&\times \exp \left( {2n+{5p\over 2}+2\over (p-{n\over 2}+2)(1-n_s)}\right) M_P, \end{aligned}$$
(106)

where

$$\begin{aligned} K={\sqrt{90}(4n+5p+4)\over 2^{-{2(p+n+2)\over 3(n-2p-4)}}\pi g_\mathrm{end}^{1\over 6}n^{n\over -2p+n-4}\left( 2n+{5p\over 2}+2\right) ^{2(p+2)\over 2p-n+4}}. \end{aligned}$$
(107)

\(M_P=2.4\times 10^{18} \ \mathrm{GeV}=8\pi G\) is the reduced Planck mass. Hereafter we reset the natural units. Equation (106) is completely different from the result obtained for temperature in reheating era in the ordinary (cold) minimal inflation obtained in [52] for a quadratic potential,

$$\begin{aligned} T_\mathrm{end}=0.085\sqrt{{(1-n_s)\over \mathcal {P}_s}}\left( k_0\over T_\mathrm{CMB}\right) ^3\exp \left( {6\over 1-n_s}\right) M_P. \end{aligned}$$
(108)

Up to a first order Taylor expansion, the relative uncertainty in our result is

$$\begin{aligned} {\sigma (T_\mathrm{end})\over T_\mathrm{end}}=\sqrt{{\sigma ^2(n_s)\over T_\mathrm{end}^2}\left( {\partial T\over \partial n_s}\right) ^2}. \end{aligned}$$
(109)

The two conditions that we have used for calculation of the temperature, i.e. \(r\gg 1\) and \({H^2\over M^2}\gg 1\), lead to

$$\begin{aligned}&{(2n+{5p\over 2}+2)^3\over 3n\Omega ^{-2p-6\over 7p+1}}\mathcal {P}_s(k_0)^{-{2p-6\over 7p+1}}(1-n_s)^{51-23p\over 2(7p+1)}\nonumber \\&\quad \times \left( {\Gamma _0\over \varphi _0^p}\right) ^{40\over 7p+1}M^2\gg M_P^{42-26p\over 7p+1} \end{aligned}$$
(110)

and

$$\begin{aligned} {g_\mathrm{end}\pi ^2\over 90}T_\mathrm{end}^4\gg M^2M_P^2, \end{aligned}$$
(111)

respectively. Equation (110) was derived from (81) and (111) was obtained using \({1\over 3M_P^2}\rho _r\gg M^2\).

To calculate \(T_\mathrm{end}\), we set \(g_\mathrm{end}=106.75\), which is the ultra-relativistic degree of freedom at the electroweak energy scale. From Planck 2013 for the pivot scale \(k_0=0.05~\mathrm{Mpc}^{-1}\) in one sigma level, we set \(\mathcal {P}_s(k_0)=(2.20\pm 0.056)\times 10^{-9}\) and \(n_s=0.9608\pm 0.0054\) [54–57]. Note that \({k_0\over T_\mathrm{CMB}}={0.05 Mpc^{-1}\over 2.725 K}=0.05\times 10^{-26}\). After fixing these parameters, the temperature depends entirely on p and n. As an example if one takes \(p=-1\) [(this choice gives a positive power for \(\Gamma _0\) in (102)] and \(n=0.8\) (for non-integer values of n see [58, 59]), one obtains

$$\begin{aligned} 5.01\times 10^7 \ \mathrm{GeV} <T_\mathrm{end} < 2.11\times 10^{13} \ \mathrm{GeV}. \end{aligned}$$
(112)

For \(n_s=0.9608\), the temperature is \(T_\mathrm{end}=1.32\times 10^{10} \ \mathrm{GeV}\), whose relative uncertainty is \({\sigma (T_\mathrm{end})\over T_\mathrm{end}}=6.35\).

The range of the temperature must lie below the upper bound scale assumed in the literature, which is about the GUT scale \(T_\mathrm{max}\simeq 10^{16} \ \mathrm{GeV}\). By considering the big bang nucleosynthesis (BBN), and on the base of the data derived from large scale structure and also cosmic microwave background (CMB), a lower bound, \(T_\mathrm{min}\simeq 4 \ \mathrm{MeV}\), is obtained in [60], which is consistent with our example.

6 Summary

We considered warm inflation in the framework of non-minimal derivative coupling model in high friction regime. After an introduction to the model, we studied the slow roll conditions and e-folds number and then specified them in terms of the parameters of the model for a power law potential and a general power law dissipation factor. By studying the cosmological perturbations, we obtained the power spectrum and the spectral index. We used these quantities to determine the temperature of the universe in terms of \(T_\mathrm{CMB}\) and the spectral index.