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 } \left ( 10 ^ { -4 } \right ) , and that the characteristic size 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 .