Adiabatic perturbations in pre-big bang models: Matching conditions and scale invariance

At low energy, the four-dimensional effective action of the ekpyrotic model of the universe is equivalent to a slightly modified version of the pre-big bang model. We discuss cosmological perturbations in these models. In particular we address the issue of matching the perturbations from a collapsing to an expanding phase. We show that, under certain physically motivated and quite generic assumptions on the high energy corrections, one obtains $n=0\mathrm{}$ for the spectrum of scalar perturbations in the original pre-big bang model (with a vanishing potential). With the same assumptions, when an exponential potential for the dilaton is included, a scale invariant spectrum $\mathrm{}(n=1\mathrm{})\mathrm{}$ of adiabatic scalar perturbations is produced under very generic matching conditions, both in a modified pre-big bang and ekpyrotic scenario. We also derive the resulting spectrum for arbitrary power law scale factors matched to a radiation-dominated era.