Bouncing universes and their perturbations: A simple model reexamined

We reconsider the toy model studied by Gordon and Turok of a spatially closed Friedmann-Lemaitre universe, driven by a massive scalar field, which deflates quasiexponentially, bounces, and then enters a period of standard inflation. We find that the equations for the matter density perturbations can be solved analytically, at least at lowest order in some "slow-roll" parameter. We can therefore give, in that limit, the explicit spectrum of the postbounce perturbations in terms of their prebounce initial spectrum. Our result is twofold. If the prebounce growing and decaying modes are of the same order of magnitude at the bounce, then the spectrum of the prebounce growing mode is carried over to the postbounce decaying mode. On the other hand, if, more likely, the prebounce growing mode dominates, then resolution at next order indicates that its spectrum is nicely carried over, with reduced amplitude, to the postbounce growing mode.