  We study a cosmological scenario in which inflation is preceded by a bounce.
In this scenario, the primordial singularity, one of the major shortcomings of
inflation, is replaced by a non-singular bounce, prior to which the universe
undergoes a phase of contraction. Our starting point is the bouncing cosmology
investigated in Falciano et al. (2008), which we complete by a detailed study
of the transfer of cosmological perturbations through the bounce and a
discussion of possible observational effects of bouncing cosmologies. We focus
on a symmetric bounce and compute the evolution of cosmological perturbations
during the contracting, bouncing and inflationary phases. We derive an
expression for the Mukhanov-Sasaki perturbation variable at the onset of the
inflationary phase that follows the bounce. Rather than being in the
Bunch-Davies vacuum, it is found to be in an excited state that depends on the
time scale of the bounce. We then show that this induces oscillations
superimposed on the nearly scale-invariant primordial spectra for scalar and
tensor perturbations. We discuss the effects of these oscillations in the
cosmic microwave background and in the matter power spectrum. We propose a new
way to indirectly measure the spatial curvature energy density parameter in the
context of this model.
