Cosmological Perturbations Through a General Relativistic Bounce

The ekpyrotic and cyclic universe scenarios have revived the idea that the density perturbations apparent in today's universe could have been generated in a `pre-singularity' epoch before the big bang. These scenarios provide explicit mechanisms whereby a scale invariant spectrum of adiabatic perturbations may be generated without the need for cosmic inflation, albeit in a phase preceding the hot big bang singularity. A key question they face is whether there exists a unique prescription for following perturbations through the bounce, an issue which is not yet definitively settled. This goal of this paper is more modest, namely to study a bouncing Universe model in which neither General Relativity nor the Weak Energy Condition is violated. We show that a perturbation which is pure growing mode before the bounce does not match to a pure decaying mode perturbation after the bounce. Analytical estimates of when the comoving curvature perturbation varies around the bounce are given. It is found that in general it is necessary to evaluate the evolution of the perturbation through the bounce in detail rather than using matching conditions.