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