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 _ { { } _ { S } } = -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 .