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-DeWitt 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 cosmic microwave background 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 curvature scale of the Universe at the bounce is $L_0\sim {10}^3{\ell }_{\mathrm{Pl}}$, a region where one expects that the Wheeler-DeWitt equation should be valid without being spoiled by string or loop quantum gravity effects. We have also obtained a consistency relation between the tensor-to-scalar ratio $T/S$ and the scalar spectral index as $T/S\sim 4.6\times {10}^{-2}\sqrt{n_{\mathrm{S}}-1}$, leading to potentially measurable differences with inflationary predictions.

• Received 18 October 2006

DOI:https://doi.org/10.1103/PhysRevD.75.023516