Asymptotic analysis of perturbed dust cosmologies to second order

TL;DR

This paper analyzes the late-time behavior of second-order nonlinear perturbations in dust cosmologies with a positive cosmological constant or spatial curvature, revealing stability conditions and divergence in Einstein-de Sitter models.

Contribution

It provides a detailed asymptotic analysis of second-order perturbations in dust cosmologies, highlighting the stabilizing effects of spatial curvature and cosmological constant.

Findings

Density perturbations approach finite limits at late times in models with positive cosmological constant or curvature.

Decaying modes in scalar perturbations persist and influence asymptotic behavior when the cosmological constant is positive.

Einstein-de Sitter universe exhibits divergence and instability in perturbations, unlike models with curvature or cosmological constant.

Abstract

Nonlinear perturbations of Friedmann-Lemaitre cosmologies with dust and a positive cosmological constant have recently attracted considerable attention. In this paper our first goal is to compare the evolution of the first and second order perturbations by determining their asymptotic behaviour at late times in ever-expanding models. We show that in the presence of spatial curvature K or a positive cosmological constant, the density perturbation approaches a finite limit both to first and second order, but the rate of approach depends on the model, being power law in the scale factor if the cosmological constant is positive but logarithmic if it is zero and and K<0. Scalar perturbations in general contain a growing and a decaying mode. We find, somewhat surprisingly, that if the cosmological constant is positive the decaying mode does not die away, i.e. it contributes on an equal footing as the growing mode to the asymptotic expression for the density perturbation. On the other hand, the future asymptotic regime of the Einstein-de Sitter universe (for which the cosmological constant and the spatial curvature are both zero) is completely different, as exemplified by the density perturbation which diverges; moreover, the second order perturbation diverges faster than the first order perturbation, which suggests that the Einstein-de Sitter universe is unstable to perturbations, and that the perturbation series do not converge towards the future. We conclude that the presence of spatial curvature or a cosmological constant stabilizes the perturbations. Our second goal is to derive an explicit expression for the second order density perturbation that can be used to study the effects of including a cosmological constant and spatial curvature.