Stability analysis of power-law cosmological models

TL;DR

This paper investigates the stability of power-law cosmological models by incorporating second-order corrections and different temporal variables, revealing that contraction solutions are unstable in cosmic time but stable in e-folds, challenging previous assumptions.

Contribution

The study introduces a second-order analysis and considers different temporal variables, providing new insights into the stability of power-law cosmological models beyond standard methods.

Findings

Power-law contraction is not an attractor in cosmic time.

Second-order corrections cause significant deviations from exact solutions.

In e-folds, the system remains an attractor, unlike in cosmic time.

Abstract

In this paper, we revisit the stability of power-law models, focusing on an alternative approach that differs significantly from the standard approaches used in studying power-law models. In the standard approach, stability is studied by reducing the system of background FRW equations to a one-dimensional system for a new background variable $X$ in terms of the number of e-foldings. However, we rewrote the equations, incorporating $H$ into the system and went on to do the calculations up to the second order. We demonstrate by computing the deviations from the power-law exact solution to second-order in time and show that power-law contraction is never an attractor in time, regardless of parameter values. Our analysis shows that while first-order corrections align with existing interpretations, second-order corrections introduce significant deviations that cannot be explained by a simple time shift that explains the first-order diverging terms. With importance, we note that in the number of e-folds, the system remains an attractor, while in cosmic time, it is unstable. We also support our claim with numerical results. This new insight has broader implications for the study of attractor behaviour of differential equation solutions and raises questions about the stability of scenarios like the ekpyrotic bounce driven by an exponential potential. Our work also hints that the different temporal variables we use might not be equivalent.