Cosmological perturbations and the transition from contraction to expansion

TL;DR

This paper analyzes how scalar perturbations evolve during a smooth transition from contraction to expansion in cosmological models, revealing how the spectral index depends on the regularity of variables and the contraction rate.

Contribution

It provides a combined analytical and numerical study of perturbation evolution during the contraction-expansion transition, highlighting the dependence on the variable regularity and contraction rate.

Findings

Spectral index depends on which variable, Ψ or ζ, remains regular during transition.

For 0<q<<1, near scale-invariance occurs if Ψ is regular.

Perturbations from the Bardeen potential stay small for q ≤ 1.

Abstract

We investigate both analytically and numerically the evolution of scalar perturbations generated in models which exhibit a smooth transition from a contracting to an expanding Friedmann universe. We find that the resulting spectral index in the late radiation dominated universe depends on which of the $\Psi$ or \$zeta$ variables passes regularly through the transition. The results can be parameterized through the exponent $q$ defining the rate of contraction of the universe. For $q \geq -1/2$ we find that there are no stable cases where both variables are regular during the transition. In particular, for $0<q\ll 1$, we find that the resulting spectral index is close to scale invariant if $\Psi$ is regular, whereas it has a steep blue behavior if $\zeta$ is regular. We also show that as long as $q\leqslant 1$, perturbations arising from the Bardeen potential remain small during contraction in the sense that there exists a gauge in which all the metric and matter perturbation variables are small.