An Asymptotic Method for Selection of Inflationary Modes

TL;DR

This paper develops an asymptotic method using Hankel functions to analyze inflationary modes, providing generalized solutions that include non-linear corrections and align with standard results in de Sitter limits.

Contribution

It introduces a novel asymptotic expansion approach for inflationary modes considering non-linear Hankel function solutions with arbitrary index $ u$.

Findings

Derived general inflationary mode solutions with non-linear Hankel function corrections.

Provided scale factor and equation of state expansions consistent with standard models.

Included non-linear corrections to the power spectrum for early universe cosmology.

Abstract

We present some features of early universe cosmology in terms of Hankel functions index ($\nu$). Actually, the recent data from observational cosmology indicate that our universe was nearly de Sitter space-time in the early times which results in an approximate scale-invariant spectrum. This imposes some constrains on index $\nu$ [1]. These constrains stimulate us to use general solution of inflaton field equation for $\nu\neq{\frac{3}{2}}$. To obtain the general solution for the inflationary background, we use asymptotic expansion of Hankel functions up to non-linear order of $\frac{1}{k\eta}$. We consider the non-linear modes as the fundamental modes for early universe during the inflation. In this paper, we obtain the general form of the inflationary modes, scale factor expansion, equation of state and some non-linear corrections of power spectrum in terms of index \nu. These results are general and in the quasi-de Sitter and de Sitter limit confirm the conventional results.