Bouncing universes and their perturbations : a simple model revisited

TL;DR

This paper analyzes a simple bouncing universe model driven by a scalar field, deriving analytical solutions for perturbations and revealing how pre-bounce modes influence post-bounce spectra, with implications for early universe cosmology.

Contribution

It provides an analytical solution for matter perturbations in a bouncing universe model, clarifying the transfer of spectra through the bounce and introducing the concept of mode inversion.

Findings

Pre-bounce growing modes often dominate post-bounce spectra.

Analytical solutions are obtained at lowest order in slow-roll parameters.

Mode inversion occurs when pre-bounce modes are of similar magnitude.

Abstract

We reconsider the toy model studied in [1] of a spatially closed Friedmann-Lemaitre universe, driven by a massive scalar field, which deflates quasi-exponentially, bounces and then enters a period of standard inflation. We find that the equations for the matter density perturbations can be solved analytically, at least at lowest order in some "slow-roll" parameter. We can therefore give, in that limit, the explicit spectrum of the post-bounce perturbations in terms of their pre-bounce initial spectrum. Our result is twofold. If the pre-bounce growing and decaying modes are of the same order of magnitude at the bounce, then the spectrum of the pre-bounce growing modes is carried over to the post-bounce decaying modes ("mode inversion"). On the other hand, if, more likely, the pre-bounce growing modes dominate, then resolution at next order indicates that their spectrum is nicely carried over, with reduced amplitude, to the post-bounce growing modes.