Primordial perturbations in a non singular bouncing universe model.

We construct a simple non singular cosmological model in which the currently observed expansion phase was preceded by a contraction. This is achieved, in the framework of pure general relativity, by means of a radiation fluid and a free scalar field having negative energy. We calculate the power spectrum of the scalar perturbations that are produced in such a bouncing model and find that, under the assumption of initial vacuum state for the quantum field associated with the hydrodynamical perturbation, this leads to a spectral index n=-1. The matching conditions applying to this bouncing model are derived and shown to be different from those in the case of a sharp transition. We find that if our bounce transition can be smoothly connected to a slowly contracting phase, then the resulting power spectrum will be scale invariant.