# Reheating in a modified teleparallel model of inflation

## Abstract

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

*T*is the torsion scalar which is constructed by contraction of the torsion tensor

*a*(

*t*), is given by the Friedmann equations

*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

### 2.1 Inflation

*M*is a constant with mass dimension . Hence

### 2.2 Cosmological perturbations

## 3 Reheating temperature

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

### 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.

*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

*g*is the number of relativistic degrees of freedom, therefore [46, 48]

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.

### 3.3 Recombination era

### 3.4 Reheating temperature

*M*is a mass scale, and \(\alpha _n\)’s are defined by

*A*is

*n*, \(\bar{\gamma }\), as well as the scale

*M*. If we take a quadratic potential, i.e. \(n=2\), we obtain

## 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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