Reheating in a modified teleparallel model of inflation

  • Parviz Goodarzi
  • H. Mohseni SadjadiEmail author
Open Access
Regular Article - Theoretical Physics


We study the cosmological inflation and reheating in a teleparallel model of gravity. Reheating is assumed to be due to the decay of a scalar field to radiation during its rapid oscillation. By using cosmological perturbations during inflation, and subsequent evolutions of the Universe, we calculate the reheating temperature as a function of the spectral index and the power spectrum.

1 Introduction

To solve problems arisen in the cosmological standard model such as the flatness, the absence of monopoles, the isotropy and homogeneity in large scale and so on, the inflation model was introduced [1, 2, 3, 4, 5, 6, 7]. Creation of small density inhomogeneity from quantum fluctuations in the early Universe is one of the most important predictions of the cosmic inflation [8]. In the standard inflation model, based on Einstein’s theory of general relativity, a canonical scalar field (inflaton) during its slow roll drives the cosmic acceleration. Afterward, the reheating era begins, during which the inflaton begins a coherent oscillation and generates radiation [9, 10, 11, 12, 13, 14]. At the end of reheating era, the Universe becomes radiation dominated. The temperature at this time is dubbed as the reheating temperature \(T_{rh}\). Constraints from the big bang nucleosynthesis (BBN), light elements abundance, and large scale structure and CMB put the lower bound \(4\, \mathrm{MeV}\) on reheating temperature[15]. In addition, as the reheating occurs after inflation, the reheating temperature must be less than the GUT energy scale which is around \(10^{16}\,\mathrm{GeV}\).

Recently, the theory of gravity in the teleparallel framework [16, 17, 18, 19] has attracted more attention[20, 21, 22, 23, 24, 25, 26, 27, 28]. This is due to the capacity of this model to describe the late time acceleration of the Universe [29, 30, 31, 32, 33, 34, 35, 36, 37, 38], as well as the inflation in the early Universe [39, 40]. In the teleparallel model, the curvatureless Weitzenbock connections are used instead of the torsionless Levi-Civita connections employed in the Einstein theory of gravity. Similar to the well-known extension of Einstein-Hilbert action to modified gravity ( f(R) model), the modified teleparallel gravity (f(T) model) is an extension of the teleparallel model [29, 30]. Scalar and tensor perturbations in teleparallel gravity were studied in [41, 42]. Power spectrum and spectral index for scalar and tensor modes in f(T) gravity have been calculated in [42].

In this paper, inspired by the above-mentioned models, we will consider inflation in the modified teleparallel model. In a pure teleparallel model, it is not clear how the Universe is warming up after the inflation, and how particles are created. So we consider also a scalar field which decays to ultra-relativistic particles after the inflation, in a period of its rapid oscillation [12, 13, 14, 43, 44]. By studying the evolution of the Universe, we compute the reheating temperature as a function of the observable parameters such as the spectral index and the power spectrum derived from Planck 2018 data [45].

The scheme of the paper is as follows: In the second section, first we introduce the model and after some preliminaries, we briefly review inflation and cosmological perturbations in the power law modified teleparallel cosmology. In the third section, which is the main part of the paper, by studying the evolution of the Universe, and by using the results of the second section, the reheating temperature is calculated. We use units \(\hbar =c=k_B=1\) through the paper.

2 Model introduction and preliminaries

To study the inflation and the subsequent reheating, we consider the modified teleparallel gravity with a canonical scalar field and radiation described by the following action [37]
$$\begin{aligned} S=\int {\bigg [{1\over 16\pi G}f(T)+ {1\over 2}\partial _\mu \varphi \partial ^{\mu } \varphi - V(\varphi )+\pounds _r\bigg ]}ed^4x , \end{aligned}$$
where \(e=det({e^A}_\mu )=\sqrt{-g}\), \(M_P=\sqrt{{1\slash (8\pi G)}}=2.4\times 10^{18}\,\mathrm{GeV}\) is the reduced Planck mass, \(\pounds _r\) is the radiation‘s lagrangian density, T is the torsion scalar which is constructed by contraction of the torsion tensor
$$\begin{aligned} T={1\over 4}T_\rho ^{\,\ \mu \nu }T^\rho _{\,\ \mu \nu }-{1\over 2}T_{\,\,\,\,\,\rho }^{\mu \nu }T^\rho _{\,\,\mu \nu }-T_{\,\,\rho \mu }^{\rho }T^{\nu \,\,\mu }_{\,\,\nu }. \end{aligned}$$
The torsion tensor is given by
$$\begin{aligned} T^\rho _{\,\ \mu \nu }=\Gamma ^\rho _{\mu \nu }-\Gamma ^\rho _{\nu \mu }=e_A^\rho \left( \partial _\mu e^A_\nu -\partial _\nu e^A_\mu \right) . \end{aligned}$$
Note that the radiation component becomes only relevant after inflation, i.e. in the reheating era.
The teleparallel is formulated with the veirbein fields \(e_A^\mu \), in terms of which, the metric is given by \(g_{\mu \nu }(x)=\eta ^{AB} e_{A}^{\mu } e_{B}^{\nu } \). For the spatially flat FLRW (Friedmann–Lemaître–Robertson–Walker) metric \(ds^2=dt^2-a^2(t)\delta _{ij}dx^idx^j\), the evolution of the scale factor, a(t), is given by the Friedmann equations
$$\begin{aligned} H^2= & {} {1\over 3M_P^2}(\rho _T+\rho _\varphi +\rho _{r}), \nonumber \\ \dot{H}= & {} -{1\over 2M_P^2}(\rho _T+\rho _\varphi +\rho _{r}+P_T+P_\varphi +P_{r}), \end{aligned}$$
where \( H=\dot{a}/a\) is the Hubble parameter, a “dot” is differentiation with respect to cosmic time t, and prime denotes differentiation with respect to the scalar field \(\varphi \). The torsion scalar is \(T=-6H^2\), and the energy density and the pressure of the scalar field are
$$\begin{aligned} \rho _\varphi= & {} {\dot{\varphi }^2\over 2}+V(\varphi ),\nonumber \\ P_\varphi= & {} {\dot{\varphi }^2\over 2}-V(\varphi ). \end{aligned}$$
\(\rho _T\) and \(P_T\) are determined by
$$\begin{aligned} \rho _T= & {} {M_P^2\over 2}\bigg (2Tf_{,T}-f-T \bigg ) ,\nonumber \\ P_T= & {} -{M_P^2\over 2}\bigg (-8\dot{H}Tf_{,TT}+(2T-4\dot{H})f_{,T}\nonumber \\&-f+4\dot{H}-T \bigg ). \end{aligned}$$
The equation of state (EoS) parameter is defined by \(w:=\gamma -1:=\frac{P}{\rho }\). The components involved in (4) satisfy the continuity equations
$$\begin{aligned}&\dot{\rho _T}+3H(\rho _T+P_T)=0 ,\nonumber \\&\dot{\rho _{\varphi }}+3H(\rho _{\varphi }+P_{\varphi })=-Q_0,\nonumber \\&\dot{\rho _r}+4H\rho _r=Q_0, \end{aligned}$$
where \(Q_0\) is the interaction term which becomes operative in the reheating era. In the inflationary epoch \(\varphi \) satisfies
$$\begin{aligned} \ddot{\varphi }+3H\dot{\varphi }+V'(\varphi )=0. \end{aligned}$$

2.1 Inflation

The inflationary phase is specified by \(\ddot{a}>0\). This is equivalent to \(\dot{H}+H^2>0\). In terms of the slow roll parameter \(\varepsilon _1\) defined by
$$\begin{aligned} \varepsilon _1\equiv -{\dot{H}\over H^2}, \end{aligned}$$
the inflation condition is \(\varepsilon <1\). We adopt a power law modified teleparallel model [42]
$$\begin{aligned} f(T)=CT^{1+\delta }, \end{aligned}$$
where \(\delta \) is a nonnegative integer number, and \(C=\frac{1}{M^{2\delta }}\), where M is a constant with mass dimension . Hence
$$\begin{aligned} \rho _T=-3M_P^2H^2\bigg ((1+2\delta )CT^\delta -1 \bigg ). \end{aligned}$$
From relation (4) we have
$$\begin{aligned} H^2={T^{-\delta }\over 3M_P^2C(1+2\delta )}\rho _\varphi . \end{aligned}$$
Inserting \(T=-6H^2\) in (12) gives
$$\begin{aligned} T^\delta =\bigg (-{2\over M_P^2C(1+2\delta )}\rho _\varphi \bigg )^{\delta \over \delta +1}. \end{aligned}$$
$$\begin{aligned} H^2={2^{{-\delta \over \delta +1}}\over 3}\bigg ({\rho _\varphi \over 3(-1)^\delta M_P^2C(1+2\delta )}\bigg )^{1\over 1+\delta }. \end{aligned}$$
For \(P_T\), we have
$$\begin{aligned} P_T= & {} -{M_P^2\over 2}\bigg (-4C(1+\delta )T^\delta \bigg (3H^2+\dot{H}(1+2\delta )\bigg )\nonumber \\&-CT^{1+\delta }+4\dot{H}+6H^2\bigg ). \end{aligned}$$
Using (15), one can rewrite the Eq. (4) as
$$\begin{aligned} \dot{H}=-{3\over 2}{H^2\over (1+\delta )}-{T^{-\delta }\over 2M_P^2C(1+\delta )(1+2\delta )}P_\varphi . \end{aligned}$$
Therefore the slow roll parameter becomes
$$\begin{aligned} \varepsilon _1=-{\dot{H}\over H^2}={3\over 2}{1\over 1+\delta }\bigg (1+{P_\varphi \over \rho _\varphi } \bigg ). \end{aligned}$$
For \(\delta =0\), this relation reduces to its well known form of the standard inflation model, i.e. \(\varepsilon _1=-{\dot{H}\over H^2}={3\over 2}\big (1+{P_\varphi \over \rho _\varphi } \big )\). \({P_\varphi \over \rho _\varphi }\simeq -1\), which is equivalent to the slow roll condition, guarantees \(\ddot{a}>0\). The slow roll condition implies:
$$\begin{aligned} \rho _\varphi\approx & {} V(\varphi )\nonumber \\ 3H\dot{\varphi }\approx & {} V'(\varphi ). \end{aligned}$$
Therefore (17) reduces to
$$\begin{aligned} \varepsilon _1\approx M_P^2{C(1+2\delta )T^\delta \over 1+\delta }\bigg ({V'(\varphi )\over V(\varphi )}\bigg )^2. \end{aligned}$$
In the standard inflation model (\(\delta =0\)), (19) becomes \(\varepsilon _1\approx {M_P^2\over 2}({V'(\varphi )\over V(\varphi )})^2 \). By substituting \(T^\delta \) from (13) in (19), we obtain
$$\begin{aligned} \varepsilon _1\approx -{\bigg (-M_P^2C(\delta +{1\over 2})\bigg )^{\big ({1\over 1+\delta }\big )}\over (1+\delta )}{V^{\prime }(\varphi )^2\over V(\varphi )^{\big ({\delta +2\over \delta +1}\big )}}. \end{aligned}$$
For our future computation we need also to the other slow roll parameter \(\varepsilon _2\) which is obtained as
$$\begin{aligned} \varepsilon _2\equiv {\dot{\varepsilon _1}\over H\varepsilon _1}\approx {\dot{T}\delta \over TH}-{2V(\varphi )\over 3H^2} \bigg ({V^{\prime \prime }(\varphi )\over V(\varphi )}- \bigg ({V^{\prime }(\varphi )\over V(\varphi )}\bigg )^2\bigg ). \end{aligned}$$
This can be rewritten
$$\begin{aligned} \varepsilon _2\approx -2M_P^2C(1+2\delta )T^\delta \bigg ({V^{\prime \prime }(\varphi )\over V(\varphi )}-{2+\delta \over 2+2\delta }\bigg ({V^{\prime }(\varphi )\over V(\varphi )}\bigg )^2\bigg ), \end{aligned}$$
which, as a function of \(\varepsilon _1\), is
$$\begin{aligned} \varepsilon _2\approx 2\varepsilon _1 \bigg ((2+\delta )-2(1+\delta )\bigg ({V^{\prime \prime }(\varphi )V(\varphi )\over V^{\prime }(\varphi )^2}\bigg )\bigg ). \end{aligned}$$
For the power law potential \(V(\varphi )=\lambda \varphi ^n\), \(\varepsilon _1\) and \(\varepsilon _2\) become:
$$\begin{aligned} \varepsilon _1&\approx -{n^2\left( -M_P^2C\left( \delta +{1\over 2}\right) \lambda ^\delta \right) ^{{1\over 1+\delta }}\over 1+\delta } \varphi ^{{n\delta \over \delta +1}+2}, \nonumber \\ \varepsilon _2&\approx 2\varepsilon _1{2+2\delta -n\delta \over n}. \end{aligned}$$
The number of e-folds from \(t_{*}\) in the inflation era, until the end of the inflation, \(t_{end}\), is given by
$$\begin{aligned} \mathcal {N}_I=\int _{t_{*}}^{t_{end}}Hdt\approx \int _{\varphi _{*}}^{\varphi _{end}}{H\over \dot{\varphi }}d\varphi \approx -\int _{\varphi _{*}}^{\varphi _{end}}{3H^2\over V^\prime (\varphi )}d\varphi . \end{aligned}$$
By using (14), (25) becomes
$$\begin{aligned} \mathcal {N}_I\approx {2^{-\delta \over 1+\delta }\over \left( M_P^2(-1)^\delta C(1+2\delta )\right) ^{1\over 1+\delta }}\int _{\varphi _{end}}^{\varphi _{*}}{V(\varphi )^{1\over 1+\delta }\over V^\prime (\varphi )}d\varphi . \end{aligned}$$

2.2 Cosmological perturbations

Cosmological perturbations of this model have been considered in [41, 42], where the spectral index and power spectrum have been calculated. Here we briefly review and list the main results. The spectral index is [42]
$$\begin{aligned} n_s-1\approx -2\varepsilon _1-\varepsilon _2. \end{aligned}$$
This relation gives the spectral index as a function of the slow roll parameters. The power spectrum is given by
$$\begin{aligned} \mathcal {P}_s(k_0)\approx {H^2\over 8\pi ^2M_P^2c_s^3\varepsilon _1}\bigg |_{c_sk=aH}. \end{aligned}$$
This relation must be evaluated at the horizon crossing for which \(c_sk=aH\). The sound speed, \(c_s\), is
$$\begin{aligned} c_s^2={f_{,T}\over f_{,T}-12H^2f_{,TT}}. \end{aligned}$$
For \(f(T)=CT^{1+\delta }\), the sound speed becomes
$$\begin{aligned} c_s^2={1\over 1+2\delta }, \end{aligned}$$
which is a constant. For \(\delta >0\) this speed is upper luminal. By using Eqs. (19) and (29) we obtain
$$\begin{aligned} \mathcal {P}_s(k_0)\approx -{T^{-2\delta }(1+\delta )\over 12\pi ^2C^2M_P^6\sqrt{1+2\delta }}{V(\varphi )^3\over V^{\prime }(\varphi )^2}\bigg |_{c_sk=aH}. \end{aligned}$$
By using (13) and (20), we obtain \(\mathcal {P}_s(k_0)\), at the horizon crossing, as
$$\begin{aligned} \mathcal {P}_s(k_0)\approx -{\left( -M_P^2C\left( \delta +{1\over 2}\right) \right) ^{2\delta \over 1+\delta }(1+\delta )\over 12\pi ^2C^2M_P^6\sqrt{1+2\delta }}{V(\varphi )^{3+\delta \over 1+\delta }\over V^\prime (\varphi )^2}\bigg |_{c_sk=aH}. \end{aligned}$$
The spectral index as a function of the scalar field potential is obtained as
$$\begin{aligned} 1-n_s\approx 2\varepsilon _1 \bigg ((3+\delta )-2(1+\delta )\bigg ({V^{\prime \prime }(\varphi )V(\varphi )\over V^{\prime }(\varphi )^2}\bigg )\bigg ). \end{aligned}$$

3 Reheating temperature

After giving a glimpse of the model and a summary of the required equations, hereafter we begin our main discussion, and try to obtain the reheating temperature. This reheating is assumed to be due to the inflaton decay to ultra-relativistic particles during its coherent rapid oscillation. We follow the method used in [43, 46], and consider the following distinct epochs: 1 – The inflation era, from \(t_\star \) (exit time of a pivot scale from the Hubble radius) until the end of inflation \(t_{end}\). In this time the main density is \(\rho _\varphi \). 2 – Rapid oscillation period, from \(t_{end}\) until \(t_{reh}\), where the (thermal) radiation becomes dominant. 3 – From \(t_{reh}\) until the recombination \(t_{rec}\) and finally from \(t_{rec}\) until the present time \(t_0\). The e-folds number from horizon crossing until now is then given by
$$\begin{aligned} \mathcal {N}= & {} \ln {\left( {a_0\over a_\star }\right) }=\ln {\left( {a_0\over a_{rec}}\right) }+\ln {\left( {a_{rec}\over a_{reh}}\right) }\nonumber \\&+\ln {\left( {a_{reh}\over a_{end}}\right) }+\ln {\left( {a_{end}\over a_{\star }}\right) }\nonumber \\:= & {} \mathcal {N}_{4}+\mathcal {N}_{3}+\mathcal {N}_{2}+\mathcal {N}_{1} \end{aligned}$$
In the following subsections we will derive \(\mathcal {N}\) for each period.

It is important to note that during rapid oscillation if one adopts the perturbative approach, then the radiation (ultra-relativistic particles)becomes dominant at \(a_{reh}\), i.e. thermalization occurs when \(a=a_{RD}\), that is when the radiation dominates. But if we consider preheating, the perturbative approach fails and \(a_{RD}\ne a_{reh}\). This is due to the fact that we may have a large number of non-thermal produced particles shortly after the beginning of rapid oscillation. This issue will be discussed in the second subsection.

3.1 Slow roll inflation

During the slow roll inflation, the Hubble parameter varies slowly \(\frac{\dot{H}}{H^2}\ll 1\), and the energy density of the scalar field is dominant. by relation (26) the number of e-folds for the potential \(V(\varphi )=\lambda \varphi ^n \) is
$$\begin{aligned} \mathcal {N}_1\approx {(\delta +1)\left( -M_P^2C\left( \delta +{1\over 2}\right) \lambda ^\delta \right) ^{\big (-{1\over 1+\delta }\big )}\over n(2+2\delta -n\delta )}\varphi _\star ^{\big ({2+2\delta -n\delta \over 1+\delta }\big )}, \end{aligned}$$
where \(\varphi (t_\star )=\varphi _\star \). For the special case \(V(\varphi )={1\over 2}m^2\varphi ^2\) we have
$$\begin{aligned} \mathcal {N}_1\approx -{(\delta +1)\over 4(-C(1+2\delta ))^{1\over 1+\delta }(m^2)^{\delta \over 1+\delta }}\bigg ({\varphi _\star \over M_P}\bigg )^{2\over 1+\delta }. \end{aligned}$$
Using (33), we write the spectral index for power law potential as
$$\begin{aligned} 1-n_s\approx -{2n(2+2\delta +n-n\delta )\left( -M_P^2C\left( \delta +{1\over 2}\right) \lambda ^\delta \right) ^{\big ({1\over 1+\delta }\big )}\over (1+\delta )} \varphi _\star ^{\big ({n\delta \over \delta +1}-2\big )}. \end{aligned}$$
From (36) and (37) we obtain
$$\begin{aligned} \mathcal {N}_1\approx \bigg ({2+2\delta +n-n\delta \over 2+2\delta -n\delta }\bigg ){1\over 1-n_s}, \end{aligned}$$
which for \(V(\varphi )={1\over 2}m^2\varphi ^2\) reduces to \(\mathcal {N}_I\approx {1/(1-n_s)}\). The relation (32), for the power law potential, becomes
$$\begin{aligned} \mathcal {P}_s(k_0)\approx -{\left( -M_P^2C\left( \delta +{1\over 2}\right) \right) ^{2\delta \over 1+\delta }(1+\delta )\lambda ^{\big ({1-\delta \over 1+\delta }\big )} \over 12\pi ^2n^2C^2M_P^6\sqrt{1+2\delta }}\varphi _\star ^{\big [n\big ({1-\delta \over 1+\delta }\big )+2\big ]}, \end{aligned}$$
where \(\varphi _\star \) is the scalar field at the horizon crossing. The Hubble parameter at the horizon crossing is
$$\begin{aligned} H^2\approx & {} 8\pi ^2M_P^2c_s^3\varepsilon _1\mathcal {P}_s(k_0)\bigg |_{c_sk=aH}\nonumber \\\approx & {} 4\pi ^2M_P^2{n(1-n_s)(1+2\delta )^{-3\over 2}\over (2+2\delta +n-n\delta )}\mathcal {P}_s(k_0)\bigg |_{c_sk=aH}. \end{aligned}$$
At the end of inflation we have \(\varepsilon _1\approx 1\), therefore
$$\begin{aligned} \varphi _{end}\approx {\bigg ({-M_P^2C\lambda ^\delta \left( \delta +{1\over 2}\right) }\bigg )}^{\big ({ 1\over 2+2\delta -n\delta }\big )} {\bigg (-{n^2\over 1+\delta }\bigg )}^{\big ({1+\delta \over 2+2\delta -n\delta }\big )}, \end{aligned}$$
$$\begin{aligned} \rho _{end}\approx & {} \lambda \varphi _{end}^n \approx {\bigg ({-M_P^2C\left( \delta +{1\over 2}\right) }\bigg )}^{\big ({n\over 2+2\delta -n\delta }\big )}\nonumber \\&\times {\bigg (-{n^2\over 1+\delta }\bigg )}^{\big ({n(1+\delta )\over 2+2\delta -n\delta }\big )}\lambda ^{\big ({2(1+\delta )\over 2+2\delta -n\delta }\big )}. \end{aligned}$$

3.2 Rapid oscillation

Based on the CMB anisotropies measurements and the relative abundances of light elements, we know that at the beginning of the big-bang nucleosynthesis (BBN) the Universe was in thermal equilibrium in a radiation dominated era with a temperature satisfying \(T_{reh}>T_{BBN}\). This thermalization occured in a period after inflation which we call reheating epoch which ended at \(a=a_{reh}\). We begin this part in the context of the original pertubative approach [2, 11, 46], then point out briefly the required modifications in the presence of preheating.

In the reheating era, the Universe was composed of a rapid oscillating scalar field (\(\varphi \)), and particles to which \(\varphi \) decayed. The radiation (ultra-relativistic particles)stress tensor is
$$\begin{aligned} T^{\mu \nu }_r=(\rho _r+P_r)u^{\mu }u^{\nu }+P_r g^{\mu \nu }. \end{aligned}$$
where \(u^{\mu }\) is the four velocity of the radiation. We take the energy transfer as [11]
$$\begin{aligned} Q_{\mu }=-\Gamma u^{\nu }\partial _{\mu }\varphi \partial _{\nu }\varphi , \end{aligned}$$
$$\begin{aligned} \nabla _{\mu }T^{\mu \nu }_r=Q^{\nu }. \end{aligned}$$
Similarly the continuity equation for \(\varphi \) gives
$$\begin{aligned} \nabla _{\mu }T^{\mu \nu }_{\varphi }=-Q^{\nu }. \end{aligned}$$
In a comoving frame, (44) becomes \(Q_{0}=\Gamma \dot{\varphi }^2\) and the Eq. (45) reduces to
$$\begin{aligned} \dot{\rho }_r +3H(\rho _r+P_r)=\Gamma \dot{\varphi }^2. \end{aligned}$$
In the same way, (46) reduces to
$$\begin{aligned} \dot{\rho }_{\varphi } +3H(\rho _{\varphi }+P_{\varphi })=-\Gamma \dot{\varphi }^2. \end{aligned}$$
The inflaton’s equation of motion is then
$$\begin{aligned} \ddot{\varphi }+(3H+\Gamma )\dot{\varphi }+V'(\varphi )=0. \end{aligned}$$
During its rapid coherent oscillation, the inflaton decayed to ultra-relativistic bosonic and fermionic particles. Following [11, 47, 48], we have specified this decay by inserting the phenomenological friction term \(\Gamma \dot{\varphi }^2\) into the main equation of motion. A fundamental derivation of this term requires a fuller understanding of the nature of inflaton and its interactions. If we consider a three-legged interaction of the form \(S_{int.}=\int \sqrt{-g}d^4x(-\sigma \varphi \chi ^2-h\varphi \bar{\psi }\psi )\), where \(\chi \) and \(\psi \) are bosonic and fermionic fields, the decay rate in tree level is derived as \(\Gamma =\frac{\sigma ^2}{8\pi m}+\frac{h^2m}{8\pi }\) [2]. This multiplies the scalar field solution by an exponential decay factor \(e^{- \gamma \Gamma t/2}\) [2], which is the same as the effect of the friction term in (49).
During the inflation the scalar field decreases very slowly, and then after the slow roll, starts a rapid oscillation, through which generates relativistic particles. This quasiperiodic oscillation was discussed in [12, 13, 14] and is described by
$$\begin{aligned} \varphi =\Phi (t)\sin \bigg (\int \omega (t) dt\bigg ). \end{aligned}$$
The scalar field EoS parameter, \(w_\varphi \), is derived as [12, 43, 44]
$$\begin{aligned} \gamma= & {} w_\varphi +1={\langle \rho _\varphi +P_\varphi \rangle \over \langle \rho _\varphi \rangle }={\langle \dot{\varphi }^2 \rangle \over \langle \rho _\varphi \rangle }\nonumber \\= & {} {2 \langle \rho _\varphi -V(\varphi ) \rangle \over V(\Phi )}\nonumber \\= & {} 2{\int _{-\Phi }^{\phi }\sqrt{\rho _\varphi -V(\varphi )}d\varphi \over {\int _{-\Phi }^{\phi }{1\over \sqrt{\rho _\varphi -V(\varphi )}}d\varphi }} \nonumber \\= & {} {2n\over n+2}. \end{aligned}$$
The average is taken over an oscillation (For more details see [12, 43, 44]).
For the power law potential, one can show that \(\langle \dot{\varphi }^2 \rangle \simeq \gamma \rho _{\varphi }\) [43]. Hence (48) may be rewritten as
$$\begin{aligned} {\dot{\rho }}_{\varphi }+3H\gamma \rho _{\varphi }+\gamma \Gamma \rho _{\varphi }=0 \end{aligned}$$
In the beginning of oscillations the scalar field is the dominant component of the Universe and in addition we take \(\Gamma \ll 3H\) [43, 48], but later, as H decreases, this approximation fails and the third term in (52) gains the same order of magnitude as the second term. For \(\Gamma \ll 3H\), by ignoring the interaction term we approximate
$$\begin{aligned} \rho _\varphi = \rho _{\varphi ,0}a^{-3\gamma }(t). \end{aligned}$$
From (14) and (53) we deduce
$$\begin{aligned} a(t)\propto t^{{2(\delta +1)\over 3\gamma }}. \end{aligned}$$
In this era the Hubble parameter is approximated by
$$\begin{aligned} H\approx {2(\delta +1)\over 3\gamma t}. \end{aligned}$$
By putting this back into (52), the scalar field energy density, in the next approximation, is derived as
$$\begin{aligned} \rho _\varphi= & {} \rho _{end}\left( \frac{t_{end}}{t}\right) ^{2(\delta +1)}e^{-\Gamma \gamma (t-t_{end})}\nonumber \\ \rho _\varphi= & {} \rho _{end}\left( \frac{a_{end}}{a}\right) ^{3\gamma }e^{-\gamma \Gamma (t-t_{end})}, \end{aligned}$$
where \(t_{end}\) is the end of the slow roll, i.e. when the oscillation begins. The term \(\left( \frac{t_{end}}{t}\right) ^{2(\delta +1)}\) shows the density reduction due to the redshift, and the exponential term corresponds to \(\varphi \)’s decay to ultra-relativistic particles (radiation). The radiation energy density is obtained as
$$\begin{aligned} \rho _r=\frac{4\Gamma M_P^2(\delta +1)^2}{(5+8\delta )t}\left( 1-\left( \frac{t}{t_{osc}}\right) ^{{-\frac{5}{3}-\frac{8}{3}\delta }}\right) . \end{aligned}$$
Note that for \(t\sim \frac{1}{\Gamma }\), we have \(\Gamma \sim 3H\) and \(\rho _r\sim \rho _{\varphi }\) (for details see ([48]). In the above approach we have assumed that during rapid oscillation, the main ingredient of the Universe is the scalar field. But by gradual decay of the inflaton to the radiation, after some time (when \(\Gamma \sim 3\gamma H\)), the produced relativistic particles become dominant and compose a fluid in thermal equilibrium. We specify \(t_{reh}\) by the time at which \(\rho _r(t_{reh})\simeq \rho _{\varphi }(t_{reh})\). The value of the temperature at \(t_{reh}\) is denoted by the reheating temperature \(T_{reh}\), similar notation will be employed for other parameters at \(t_{reh}\). At \(t_{reh}\), we have \(H^2=\simeq \frac{1}{3M_P^2}\rho _{reh}\). The radiation density and the temperature are related by \(\rho _r={g\over 30}\pi ^2T^4\), where g is the number of relativistic degrees of freedom, therefore [46, 48]
$$\begin{aligned} \Gamma ^2\simeq \frac{\pi ^2}{10 M_P^2}g_{reh} T_{reh}^4. \end{aligned}$$
During the rapid oscillation until the radiation dominance, i.e. from \(t_{end}\) until \(t_{reh}\), the main contribution in the energy density is coming from the scalar field, and \(\Gamma \precsim 3H\), hence from (53) we derive
$$\begin{aligned} \mathcal {N}_{2}=\ln \bigg ({a_{reh}\over a_{end}}\bigg )=-{1\over 3\gamma }\ln \bigg ({\rho _{reh}\over \rho _{end}}\bigg ), \end{aligned}$$
which can be rewritten as
$$\begin{aligned} \mathcal {N}_{2}= & {} -{1\over 3\gamma }\ln \nonumber \\&\times \left( {{g_{reh}\over 30}\pi ^2T_{reh}^4\over {\left( {-M_P^2C(\delta +{1\over 2})}\right) }^{\left( {n\over 2+2\delta -n\delta }\right) } {\left( -{n^2\over 1+\delta }\right) }^{\left( {n(1+\delta )\over 2+2\delta -n\delta }\right) }\lambda ^{\left( {2(1+\delta )\over 2+2\delta -n\delta }\right) }}\right) ,\nonumber \\ \end{aligned}$$
where \(g_{reh}\) is the number of relativistic degrees of freedom at \(t_{reh}\), and we have used (42). Using (37), we can obtain the scalar field at the horizon crossing
$$\begin{aligned} \varphi _\star\approx & {} \bigg ({2n(2+2\delta +n-n\delta )\over (1+\delta )(1-n_s)}\bigg )^{\big ({1+\delta \over 2+2\delta -n\delta }\big )}\nonumber \\&\times \left( -M_P^2C\left( \delta +{1\over 2}\right) \lambda ^\delta \right) ^{\big ({1\over 2+2\delta -n\delta }\big )}. \end{aligned}$$
From (39) \(\lambda \) may be derived as
$$\begin{aligned} \lambda\approx & {} (-1)^{n\over 2}{(12\pi ^2\mathcal {P}_s\sqrt{1+2\delta })^{\big ({2\delta +2-n\delta \over 2}\big )} (1+\delta )^{n\over 2}\over \big ({2(2+2\delta +n-n\delta ) \over n(1-n_s)}\big )^{\big ({n-n\delta +2\delta +2\over 2}\big )}\left( -\left( \delta +{1\over 2}\right) \right) ^{\big ({n+(1+2\delta )(2+2\delta -n\delta )\over -2(1+\delta )}\big )}}\nonumber \\&\times C^{\left( {2-n\over 2}\right) }M_P^{(2\delta +4-n\delta -n)}. \end{aligned}$$
By inserting this relation in Eq. (42) \(\rho _{end} \) becomes
$$\begin{aligned} \rho _{end}\approx & {} {(12\pi ^2\mathcal {P}_s\sqrt{1+2\delta })^{(\delta +1)}(-(\delta +{1\over 2}))^{-\big (n+1+2\delta +{n\over n\delta -2\delta -2}\big )}\over n^{\big ({(\delta +1)(3n-n\delta +2\delta +2)\over n\delta -2\delta -2}\big )}\big ({2(2+2\delta +n-n\delta )\over (1-n_s)}\big )^{\big ({(n-n\delta +2\delta +2)(\delta +1)\over 2+2\delta -n\delta }\big )}} \nonumber \\&\times CM_P^{2(2+\delta )}, \end{aligned}$$
and finally
$$\begin{aligned}&\mathcal {N}_{2} = -{1\over 3\gamma }\nonumber \\&\quad \times \ln {{{g_{reh}\over 30}\pi ^2T_{reh}^4 n^{\big ({(\delta +1)(3n-n\delta +2\delta +2)\over n\delta -2\delta -2}\big )}\left( {2(2+2\delta +n-n\delta )\over {1-n_s}}\right) ^{{(n-n\delta +2\delta +2)(\delta +1)\over 2+2\delta -n\delta }} }\over {(12\pi ^2\mathcal {P}_s \sqrt{1+2\delta })^{(\delta +1)}}\left( -(\delta +{1\over 2})\right) ^{\big (n+1+2\delta +{n\over n\delta -2\delta -2}\big )}CM_P^{2(2+\delta )}}.\nonumber \\ \end{aligned}$$
Our above analysis about reheating was based on the simple original studies of perturbative reheating after inflation. There are some problems with this simple model, limiting the range of its applicability, such as collective effects like Bose condensation which alters the decay rate [47]. The decay rate \(\Gamma _\chi \equiv \Gamma _{\varphi \rightarrow \chi \chi }=\frac{\sigma ^2}{8\pi m}\), derived from the aforementioned three legged interaction, changes when the phase space of bosonic \(\chi \) particles is occupied by previously produced bosons. In this situation, we have \(\Gamma _{eff.}\simeq \Gamma _\chi (1+2n_k)\), where \(n_k\) is the occupation numbers of \(\chi \) particles with momentum \(\vec {k}\) and \(-\vec {k}\): \(n_{\vec {k}}=n_{-\vec {k}}=n_k\). For large occupation number, this enhances significantly the decay rate. To get an estimation, by neglecting for a moment the Universe expansion, one finds the simple expression \(n_{\chi }\propto e^{\frac{\pi \sigma \Phi t}{2m}}\). This can be derived more precisely from the Mathieu equation corresponding to the equation of motion of the modes of the bosonic field \(\chi \) [47, 49]. Note that the Universe expansion, the back-reaction, and re-scattering of created particles reduce the decay rate enhancement, so the effect of Bose condensation is actually less than the naive aforementioned estimation [50].

If the coupling constants or the inflaton amplitude become large, the perturbative method fails, and higher order Feynman diagrams become relevant. In this situation, the main role in the production of particles is due to the parametric resonance in the preheating era, leading to explosive particles production [49, 50, 51]. This effect must be studied non-perturbatively. After the inflation, the produced matter field evolves from an initial vacuum state in the background of the oscillating inflaton field. A result of this oscillating background, is a time dependent frequency for the bosonic field (\(\chi \)) which satisfies the Hill’s equation [50]. Following Floquet analysis, this may result in a broad parametric resonance and a quick growth of matter in the background of the oscillating inflaton [49, 50, 51]. By defining q as \(q=\frac{4\sigma \Phi }{m^2}\), one can show that the broad resonance occurs for \(q\gtrsim 1\) and we have \(n_k\sim e^{2\mu _kmt}\), where \(n_k\) is the occupation number for the bosonic mode k, and \(\mu _k\sim \mathcal {O}(1)\) is the parameter of instability. For \(q\ll 1\) we obtain the narrow resonance \(n_k\sim e^{2\mu _kmt}\), where \(\mu _k\ll 1\) [50]. The main part of the initial energy is transferred to matter field via the parametric resonance and at the end of broad resonance only a small amount of the initial energy still stored in the inflaton field. Particles created in the preheating era were initially far from thermal equilibrium state, but they reached local thermal equilibrium before BBN. The precise details of reheating era is largely uncertain, also the realistic picture of preheating faces more complications than the simple aforementioned models.

In our initial perturbative approach, the ultra-relativistic particles are gradually produced (see (57)), and until \(\rho _r\simeq \rho _\varphi \) (\(a=a_{RD}=a_{reh}\)) where thermal radiation begins its domination, the Universe is nearly governed by the oscillating inflaton field \(\varphi \), whose EoS parameter is given by (51). By considering preheating, this assumption fails [51]. In this situation, as we have explained briefly, the Universe, besides the oscillating inflaton, is composed of largely produced particles via parametric resonance. Hence we must modify the equation of state (EoS) parameter, and consequently the number of e-folds obtained in (64). To do so, we employ the method used in [50], and instead of assuming \(w\simeq w_\varphi =\frac{n-2}{n+2}\), we consider an effective equation of state parameter
$$\begin{aligned} \bar{w}_{eff.}=\frac{\int _{t_{end}}^{t_{reh}}w_{eff.}dt}{t_{reh}-t_{end}}. \end{aligned}$$
which yields
$$\begin{aligned} \mathcal {N}_{2}= & {} -{1\over 3\bar{\gamma }}\nonumber \\&\times \ln {{{g_{reh}\over 30}\pi ^2T_{reh}^4 n^{\big ({(\delta +1)(3n-n\delta +2\delta +2)\over n\delta -2\delta -2}\big )}\left( {2(2+2\delta +n-n\delta )\over {1-n_s}}\right) ^{{(n-n\delta +2\delta +2)(\delta +1)\over 2+2\delta -n\delta }} }\over {(12\pi ^2\mathcal {P}_s \sqrt{1+2\delta })^{(\delta +1)}}\left( -(\delta +{1\over 2})\right) ^{\big (n+1+2\delta +{n\over n\delta -2\delta -2}\big )}CM_P^{2(2+\delta )}},\nonumber \\ \end{aligned}$$
where \(\bar{\gamma }= \bar{w}_{eff.}+1\) (instead of \(\gamma = \frac{2n}{n+2}\)). We do not know the exact form of \(\bar{\gamma }\). But using the fact that after inflation until the thermalization, the Hubble parameter should satisfy \(\dot{H}+H^2<0\), we obtain \(\bar{w}_{eff.}>-\frac{1}{3}+\frac{2\delta }{3}\) leading to \(\bar{\gamma }>\frac{2}{3}(1+\delta )\). In the original context where the inflaton gradually decays to particles, we have \(\bar{\gamma }\simeq \gamma _\varphi =\frac{2n}{n+2}\) which for the quadratic potential gives \(\bar{\gamma }=1\). By considering preheating, we expect that \(\bar{\gamma }_{eff.}\) begins from \(\bar{\gamma }_{eff.}=1\) and ends to \(\bar{\gamma }\simeq \frac{4}{3}\), when the Universe thermalizes. The evolution of EoS between these values depends on the coupling, and also the effective masses of produced particles. This issues has been studied numerically in [51, 52]. By considering preheating and an instantaneous thermalization of ultra-relativistic particles, we obtain \(\bar{\gamma }\simeq \frac{4}{3}\) immediately after rapid oscillation [52].

3.3 Recombination era

After the coherent oscillation, the Universe contains ultra-relativistic particles in thermal equilibrium and experiences an adiabatic expansion for which [46, 48]
$$\begin{aligned} {a_{rec}\over a_{reh}}={T_{reh}\over T_{rec}}\left( {g_{reh}\over g_{rec}}\right) ^{1\over 3}. \end{aligned}$$
In the recombination era, only photons have relativistic degrees of freedom, therefore \(g_{rec}=2\), and
$$\begin{aligned} \mathcal {N}_{3}= \ln \left( {T_{reh}\over T_{rec}}\left( {g_{reh}\over 2}\right) ^{1\over 3}\right) . \end{aligned}$$
The temperature redshifts as, \(T(z)=T(z=0)(1+z)\), so we can write \(T_{rec}\) in terms of \(T_{CMB}\) as
$$\begin{aligned} T_{rec}=(1+z_{rec})T_{CMB}. \end{aligned}$$
$$\begin{aligned} \mathcal {N}_{3}+\mathcal {N}_{4}=\ln \left( {T_{reh}\over T_{CMB}}\left( {g_{reh}\over 2}\right) ^{1\over 3}\right) , \end{aligned}$$
where \(\mathcal {N}_{4}\) is the number of e-folds after the recombination era.

3.4 Reheating temperature

To determine the reheating temperature we make use of (34). The e-folds from the horizon crossing until the present time, which we take \(a_0=1\), is \(\ln \frac{1}{a_*}\). By using (27), (28), and (30) we obtain
$$\begin{aligned} \ln {1\over a_*}= & {} \ln {H_*\over k_0}\nonumber \\\approx & {} \ln \left( {2\pi M_P\over k_0}\sqrt{ {n(1-n_s)(1+2\delta )^{-3\over 2}\over (2+2\delta +n-n\delta )}\mathcal {P}_s(k_0)}\right) .\nonumber \\ \end{aligned}$$
By putting this in the left hand side of (34), and summing (38), (66) and (70) in the right hand side, we derive our main result i.e. \(T_{reh}\) as
$$\begin{aligned} T_{reh}&\approx A \pi ^{\alpha _1}\left( \frac{M}{M_P}\right) ^{\alpha _2} g_{reh}^{\alpha _3}\nonumber \\&\quad \times \left( {T_{CMB}\over k_0}\right) ^{\alpha _4}(1-n_s)^{\alpha _5} \mathcal {P}_s^{\alpha _6} e^{\left( \frac{\alpha _7}{1-n_s}\right) }M_P, \end{aligned}$$
where we have used \(C=M^{-2\delta }\), where M is a mass scale, and \(\alpha _n\)’s are defined by
$$\begin{aligned} \alpha _1= & {} \frac{3\bar{\gamma }-2\delta }{3\bar{\gamma }-4} \nonumber \\ \alpha _2= & {} \frac{2\delta }{3\bar{\gamma }-4}\nonumber \\ \alpha _3= & {} \frac{-\bar{\gamma }+1}{3\bar{\gamma }-4}\nonumber \\ \alpha _4= & {} \frac{3\bar{\gamma }}{3\bar{\gamma }-4}\nonumber \\ \alpha _5= & {} -{\frac{ \left( n\delta -2\delta -n-2 \right) \left( \delta +1 \right) }{\left( n\delta -2\delta -2 \right) \left( 3\bar{\gamma }-4 \right) }}+3{\frac{\bar{\gamma }}{6\bar{\gamma }-8}} \nonumber \\ \alpha _6= & {} \frac{3\bar{\gamma }-2\delta -2}{6\bar{\gamma }-8} \nonumber \\ \alpha _7= & {} 3{\frac{\left( -\delta n+2\delta +n+2 \right) \bar{\gamma }}{ \left( - \delta n+2\,\delta +2 \right) \left( 3\bar{\gamma }-4 \right) }} \end{aligned}$$
and the coefficient A is
$$\begin{aligned} A= & {} \left( -1 \right) ^{-{\frac{2{\delta }^{2}n+\delta {n}^{2}-4{ \delta }^{2}-n\delta -6\delta -n-2}{3\delta \bar{\gamma } n-6\delta \bar{\gamma }-4n\delta +8\delta -6\bar{\gamma }+8}}}{2}^{{\frac{{\delta }^{2}n+4 \delta \bar{\gamma } n+\delta {n}^{2}-2{\delta }^{2}-8\delta \bar{\gamma }- 4n\delta -8\bar{\gamma }-2n+2}{ \left( n\delta -2\delta -2 \right) \left( 3\bar{\gamma }-4 \right) }}}\nonumber \\&\times {3}^{{\frac{2+\delta }{4-3\bar{\gamma }}}}{ 5}^{\frac{1}{4-3\bar{\gamma }}}{n}^{{\frac{-{\delta }^{2}n+3 \delta \bar{\gamma } n+2{\delta }^{2}-6\delta \bar{\gamma }-2n\delta +12 \delta -6\bar{\gamma }+3n+10}{ \left( n\delta -2\delta -2\right) \left( 3\bar{\gamma }-4 \right) }}}\nonumber \\&\times \left( 1+2\delta \right) ^{-{\frac{10 {\delta }^{2}n+9\,\delta \bar{\gamma } n+4\delta {n}^{2}-20{\delta }^{ 2}-18\delta \bar{\gamma }-2n\delta -32\delta -18\bar{\gamma }-4n-12}{4 \left( n\delta -2\delta -2 \right) \left( 3\bar{\gamma }-4 \right) }}}\nonumber \\&\times \left( \left( -n+2 \right) \delta +n+2 \right) ^{{\frac{ \left( 2n -4\right) {\delta }^{2}+ \left( -8+ \left( -3n+6 \right) \bar{\gamma } \right) \delta -2n+6\,\bar{\gamma }-4}{2\, \left( n-2 \right) \left( 3 \bar{\gamma }-4 \right) \delta -12\bar{\gamma }+16}}}. \end{aligned}$$
The temperature (72), depends on the parameters of the model: \(\delta \), n, \(\bar{\gamma }\), as well as the scale M. If we take a quadratic potential, i.e. \(n=2\), we obtain
$$\begin{aligned} \frac{T_{reh,\delta \ne 0}}{T_{reh,\delta =0}}=\beta (1-n_s)^{\frac{8\delta }{16-12\bar{\gamma }}}\mathcal {P}_s^{\frac{\delta }{4-3\bar{\gamma }}}\left( \frac{M_P}{M}\right) ^{\frac{2\delta }{4-3\bar{\gamma }}}, \end{aligned}$$
where \(\beta =\left( -\pi \right) ^{{\frac{2\delta }{4-3\bar{\gamma }}}}{4}^{{\frac{ \delta }{3\bar{\gamma }-4}}}{3}^{{\frac{\delta }{4-3\bar{\gamma }}}} \left( 1+2 \delta \right) ^{{\frac{-10-10\delta -9\,\bar{\gamma }}{12\,\bar{\gamma }-16}}}\). We may obtain a simple expression, if we take \(\bar{\gamma }=1\)
$$\begin{aligned} \frac{T_{reh,\delta \ne 0}}{T_{reh,\delta =0}}=\left( \frac{3\pi ^2}{4}\right) ^\delta (1+2\delta )^{\frac{19+10\delta }{4}} (1-n_s)^{2\delta }\mathcal {P}_s^{\delta }\left( \frac{M_P}{M}\right) ^{2\delta }. \end{aligned}$$
For \(n=2\), \(\delta =0\) and \(\bar{\gamma }=1\), the reheating temperature (72) reduces to
$$\begin{aligned} T_{reh}\approx \frac{45}{2\pi ^3} \sqrt{{(1-n_s)\over \mathcal {P}_s}}\exp {\Big ({6\over 1-n_s}\Big )}\bigg ({k_0\over T_{CMB}}\bigg )^3M_P, \end{aligned}$$
which is the temperature obtained in [46] in the context of the standard general relativity. In this situation, and for \(\delta =1\), we have
$$\begin{aligned} {T_{reh,\delta =1}\over T_{reh,\delta =0}}\approx 21306 \mathcal {P}_s(1-n_s)^2{M_P^2\over M^2}. \end{aligned}$$
By taking \(g_{reh}=106.75\), \(k_0=0.05Mpc^{-1}\), and by using \((68\% CL, TT,TE,EE+lowE+lensing)\) [45]
$$\begin{aligned} \ln \Big (10^{10}\mathcal {P}_s(k_0)\Big )= & {} 3.044\pm 0.014\nonumber \\ n_s= & {} 0.9645\pm 0.0042, \end{aligned}$$
(78) reduces to
$$\begin{aligned} {T_{reh,\delta =1}\over T_{reh,\delta =0}}\approx 5.635\times 10^{-8}{M_P^2\over M^2}. \end{aligned}$$
So the temperature is much less than \(T_{reh,\delta =0}\) unless \(M_P\gg M\). For example, by taking \(M=10^{-4}M_P\) we obtain \(T_{reh}\approx 5.635T_{reh,\delta =0}\sim 6\times 10^{14}\,\mathrm{GeV}\).

4 Conclusion

We considered inflation in a modified teleparallel model of gravity (see (1)), in which a scalar field is responsible to reheat the Universe after the inflationary era. To determine the reheating temperature, we used the cosmological perturbations to find the number of e-folds from the horizon exit of a pilot scale, until now . In addition, we divided the evolution of the Universe into different segments and obtained the corresponding efolds in each segment and summed over them. By equating efolds numbers derived from these two methods, we achieved to obtain an expression for the reheating temperature in terms of the CMB temperature, the spectral index, the power spectrum and the parameters of the model.


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Authors and Affiliations

  1. 1.Department of ScienceA. O. Boroujerdi UniversityBoroujerdIran
  2. 2.Department of PhysicsCollege of Science, University of TehranTehranIran

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