We perform a detailed investigation of the simplest possible cosmological
model in which a bounce can occur, namely that where the dynamics is led by a
simple massive scalar field in a general self-interacting potential and a
background spacetime with positively curved spatial sections. By means of a
phase space analysis, we give the conditions under which an initially
contracting phase can be followed by a bounce and an inflationary phase lasting
long enough (i.e., at least 60-70 e-folds) to suppress spatial curvature in
today's observable universe. We find that, quite generically, this realization
requires some amount of fine-tuning of the initial conditions. We study the
effect of this background evolution on scalar perturbations by propagating an
initial power-law power spectrum through the contracting phase, the bounce and
the inflationary phase. We find that it is drastically modified, both
spectrally (k-mode mixing) and in amplitude. It also acquires, at leading
order, an oscillatory component, which, once evolved through the radiation and
matter dominated eras, happens to be compatible with the WMAP data.Comment: Updated references, improved figure resolutio