  We show that a contracting universe which bounces due to quantum cosmological
effects and connects to the hot big-bang expansion phase, can produce an almost
scale invariant spectrum of perturbations provided the perturbations are
produced during an almost matter dominated era in the contraction phase. This
is achieved using Bohmian solutions of the canonical Wheeler-de Witt equation,
thus treating both the background and the perturbations in a fully quantum
manner. We find a very slightly blue spectrum ($n_{_\mathrm{S}}-1>0$). Taking
into account the spectral index constraint as well as the CMB normalization
measure yields an equation of state that should be less than $\omega\lesssim
8\times 10^{-4}$, implying $n_{_\mathrm{S}}-1 \sim \mathcal{O}(10^{-4})$, and
that the characteristic size of the Universe at the bounce is $L_0 \sim 10^3
\ell_\mathrm{Planck}$, a region where one expects that the Wheeler-DeWitt
equation should be valid without being spoiled by string or loop quantum
gravity effects.
