Scalar perturbations in regular two-component bouncing cosmologies

We consider a two-component regular cosmology bouncing from contraction to expansion, where, in order to include both scalar fields and perfect fluids as particular cases, the dominant component is allowed to have an intrinsic isocurvature mode. We show that the spectrum of the growing mode of the Bardeen potential in the pre-bounce is never transferred to the dominant mode of the post-bounce. The latter acquires at most a dominant isocurvature component, depending on the relative properties of the two fluids. Our results imply that several claims in the literature need substantial revision. PACS: 98.80.-k, 98.65.Dx, 98.80.Es, 11.25.Wx Bouncing cosmologies have been proposed as possible alternatives to standard inflation in string-inspired (e.g. Pre-Big Bang [1], ekpyrotic/cyclic [2]) scenarios. In order to compare the predictions of these scenarios with observations, it is crucial to follow the evolution of cosmological perturbations from the initial vacuum-normalized state, through the bounce, all the way until decoupling. Nevertheless, there is no general agreement on the true influence of a bounce on cosmological perturbations. In particular, it is not clear whether the Pre-Bounce growing mode of the Bardeen potential leaves any trace in the Post-Bounce, or whether it just completely matches a decaying mode [3]. Only in the former case can the claim that the ekpyrotic/cyclic scenario is in agreement with CMB data be defended. Studies of specific regular models apparently led to different results, suggesting that both possibilities can arise, depending on the specific model one assumes for the regular bounce. In spatially closed universes, a regular bounce from contraction to expansion can be triggered by the spatial curvature term in the Friedman equations [4]. In this case, the transfer matrix relating Post-Bounce modes to Pre-Bounce perturbations depends on the momentum scale k, suggesting that non-trivial matching conditions hold. These models have the advantage of being completely embedded into General Relativity, but they need fine-tuning in the initial conditions in order to have a sufficiently long collapse. Alternatively, one can start from a spatially flat universe, avoiding the complications of spatial curvature. However, Friedman equations imply that a bounce from contraction to expansion necessarily violates the null energy condition (NEC). This violation can be induced by high energy and high curvature corrections to General Relativity or can be directly introduced by a ghost field dominating during the bounce. In this class of models, contradictory results have emerged up to now. Peter & Pinto-Neto [5] and Finelli [6] find some mixing between the growing and decay modes of the Bardeen potential, while Gasperini, Giovannini & Veneziano [7], Cartier [8], and Allen & Wands [9] find that the growing mode of the Bardeen potential in the Pre-Bounce exactly matches the decaying mode in the Post-Bounce. In this letter we report the results of the study of a wide class of regular bouncing cosmological models, containing, as particular cases, all the above-mentioned two-source regular models. We confirm that the Pre-Bounce Bardeen growing mode matches the PostBounce decaying mode. The only possible mode mixing is between the two components that are present in the cosmological description, as a consequence of the normal interplay between adiabatic and isocurvature perturbations. We present hereafter the key arguments and give the main results, postponing most details to a later publication [10]. We consider a two-component cosmology, where the first component dominates at early and late times, while the secondary one has negative energy density and triggers the bounce from contraction to expansion in a fully regular fashion. It is well known that ghosts or negative energy density fluids induce instabilities at a quantum level, but this has nothing to do with the evolution of classical perturbations, which is completely under control, as