Cosmological perturbations and the transition from contraction to expansion

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 $\hat{I}¨$ or $zeta$variablespassesregularlythroughthetransition.Theresultscanbeparameterizedthroughtheexponent$q$definingtherateofcontractionoftheuniverse.For$q \geq -1/2$wefindthattherearenostablecaseswherebothvariablesareregularduringthetransition.Inparticular,for$0<q\ll 1$,wefindthattheresultingspectralindexisclosetoscaleinvariantif$Ψ$isregular,whereasithasasteepbluebehaviorif$ζ$isregular.Wealsoshowthataslongas$q\leqslant 1$, 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.

30 pages, 16 figures. Version to appear in Phys. Rev. D. Slight modifications, but no change in the conclusion