Application of the Abel Equation of the 1st kind to an inflation analysis of non-exactly solvable cosmological models

TL;DR

This paper explores how the Abel equation of the first kind can be applied to analyze inflation in flat Friedmann universes with scalar fields, providing conditions for inflation and slow-roll regimes through numerical methods.

Contribution

It demonstrates a novel application of the Abel equation to inflationary cosmology, offering a new approach to analyze non-exactly solvable models with polynomial potentials.

Findings

Derived necessary and sufficient conditions for inflation.

Established the relationship between slow-roll and inflation.

Numerical analysis of inflation conditions based on initial scalar field values.

Abstract

In this paper we revisit the relationship between the Einstein--Friedman and the Abel equations to demonstrate how it might be applied to the inflationary analysis in a flat Friedman universe filled with a real-valued scalar field. The analysis is performed for three distinct cases of polynomial potentials. As a result of a numeric integration of Abel equation, the necessary and sufficient conditions for both slow-rolling and inflation proper are estimated with respect to the initial value of the field. In addition, the relationship between the slow-rolling condition and the inflation is ascertained.