Bouncing models with a cosmological constant
  Bouncing models have been proposed by many authors as a completion, or even
as an alternative to inflation for the description of the very early and dense
Universe. However, most bouncing models contain a contracting phase from a very
large and rarefied state, where dark energy might have had an important role as
it has today in accelerating our large Universe. In that case, its presence can
modify the initial conditions and evolution of cosmological perturbations,
changing the known results already obtained in the literature concerning their
amplitude and spectrum. In this paper, we assume the simplest and most
appealing candidate for dark energy, the cosmological constant, and evaluate
its influence on the evolution of cosmological perturbations during the
contracting phase of a bouncing model, which also contains a scalar field with
a potential allowing background solutions with pressure and energy density
satisfying p = w*rho, w being a constant. An initial adiabatic vacuum state can
be set at the end of domination by the cosmological constant, and an almost
scale invariant spectrum of perturbations is obtained for w~0, which is the
usual result for bouncing models. However, the presence of the cosmological
constant induces oscillations and a running towards a tiny red-tilted spectrum
for long wavelength perturbations.
