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

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