Kinetic dominance and psi series in the Hamilton-Jacobi formulation of inflaton models

TL;DR

This paper develops a method to compute psi series solutions for inflaton models in the Hamilton-Jacobi framework, aiding in understanding the onset and duration of inflation and perturbation analysis.

Contribution

It introduces a systematic approach to derive psi series for key cosmological quantities in inflaton models, including the Hubble parameter and scale factor, near the singularity.

Findings

Derived explicit psi series for various inflaton potentials.

Estimated the start point and duration of inflation using psi series.

Applied method to generalized Starobinsky models.

Abstract

Single-field inflaton models in the kinetic dominance period admit formal solutions given by generalized asymptotic expansions called psi series. We present a method for computing psi series for the Hubble parameter as a function of the inflaton field in the Hamilton-Jacobi formulation of inflaton models. Similar psi series for the scale factor, the conformal time and the Hubble radius are also derived. They are applied to determine the value of the inflaton field when the inflation period starts and to estimate the contribution of the kinetic dominance period to calculate the duration of inflation. These psi series are also used to obtain explicit two-term truncated psi series near the singularity for the potentials of the Mukhanov-Sasaki equation for curvature and tensor perturbations. The method is illustrated with wide families of inflaton models determined by potential functions combining polynomial and exponential functions as well as with generalized Starobinsky models.