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. 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.