Generalised Asymptotic Solutions for the Inflaton in the Oscillatory Phase of Reheating

Gabriel Álvarez,Elena Medina,Luis Martínez Alonso

doi:10.3390/universe7100390

Abstract

We determine generalised asymptotic solutions for the inflaton field, the Hubble parameter, and the equation-of-state parameter valid during the oscillatory phase of reheating for potentials that close to their global minima behave as even monomial potentials. For the quadratic potential, we derive a generalised asymptotic expansion for the inflaton with respect to the scale set by inverse powers of the cosmic time. For the quartic potential, we derive an explicit, two-term generalised asymptotic solution in terms of Jacobi elliptic functions, with a scale set by inverse powers of the square root of the cosmic time. In the general case, we find similar two-term solutions where the leading order term is defined implicitly in terms of the Gauss hypergeometric function. The relation between the leading terms of the instantaneous equation-of-state parameter and different averaged values is discussed in the general case. Finally, we discuss the physical significance of the generalised asymptotic solutions in the oscillatory regime and their matching to the appropriate solutions in the thermalization regime.

Highlights

The reheating period in inflationary cosmology is a transition period connecting the end of inflation to the radiation-dominated era [1,2,3,4,5,6,7,8]
We discuss separately the quartic potential. This is a relevant potential because the corresponding oscillatory phase of reheating is radiationdominated, and the universe would be radiation-dominated since the end of inflation with an ensuing reduction of the uncertainty on the period were the observable perturbations of the universe were produced [15]
As we mentioned in the introduction, we look for a generalised asymptotic expansion of φ(τ ) with respect to the asymptotic sequence τ −n, (n = 1, 2, 3, . . .)

Summary

Introduction

The reheating period in inflationary cosmology is a transition period connecting the end of inflation to the radiation-dominated era [1,2,3,4,5,6,7,8]. The first phase of reheating can be described as an oscillatory phase around the true minimum of the inflaton potential, wherein the inflaton is not yet effectively coupled to the radiation fields, and the friction term ΓΦin Equation (1) can be neglected During this period, Equations (1) and (6) reduce to. This is a relevant potential because the corresponding oscillatory phase of reheating is radiationdominated, and the universe would be radiation-dominated since the end of inflation with an ensuing reduction of the uncertainty on the period were the observable perturbations of the universe were produced [15]. We obtain explicit asymptotic formulas for the inflaton field and derived quantities in terms of Jacobi elliptic functions, which have appeared recently in perturbative approaches to inflationary cosmology due to its relation to Hill’s Equation (16). The paper ends with some brief remarks on possible developments and applications of these generalised solutions

The Harmonic Potential

The Quartic Potential

The General Case

Connection between the Oscillatory Period and the Thermalization Period

Conclusions