Cosmological perturbations and the transition from contraction to expansion

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. If the perturbation equations are formulated as second order equations for either the Bardeen potential $\Psi$ or the curvature perturbation on uniform comoving hypersurfaces $\zeta ,$ at best one of them can stay regular during the transition. We find that the resulting spectral index in the late radiation dominated universe depends on which of these two variables passes regularly through the transition. The results can be parametrized by the exponent q defining the rate of contraction of the universe, or equivalently through the equation of state $w=\left(2\mathrm{}-q\right)/3q$ of the background fluid. For $q>~-\frac{1}{2}$ we find that there are no stable cases where both $\Psi$ and $\zeta$ 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 $q<~1$ and we are in the regime where corrections can be ignored, perturbations remain small during contraction in the sense that there exists a gauge in which all the metric and matter perturbation variables are small. This work has important implications for the current debate concerning the nature of perturbations evolving through a collapsing regime into an expanding one: it shows that if in the ekpyrotic model, where $0<q\ll 1,$ the Bardeen potential passes regularly through the transition, this leads to a nearly scale invariant spectrum with $n=1\mathrm{}-2q,$ whereas in the case of dilaton-driven string cosmology we have the opposite situation. There it is assumed that $\zeta$ passes regularly through the transition, leading to a very blue spectrum of highly suppressed perturbations.