Scalar perturbations in an $\alpha'$-regularised cosmological bounce
  We consider the evolution of scalar perturbations in a class of non-singular bouncing universes obtained with higher-order corrections to the low-energy bosonic string action. We show that previous studies have relied on a singular evolution equation for the perturbations. From a simple criterium we show that scalar perturbations cannot be described at all times by an homogeneous second-order perturbation equation in pre-big bang type universes if we are to regularise the background evolution with higher-order curvature and string coupling corrections, and we propose a new system of first-order coupled differential equations. Given a bouncing cosmological background with inflation driven by the kinetic energy of the dilaton field, we obtain numerically the final power spectra generated from the vacuum quantum fluctuations of the metric and the dilaton field during inflation. Our result shows that both Bardeen's potential, $\Phi(\eta,k)$, and the curvature perturbation in the uniform curvature gauge, ${\mathcal R}(\eta,k)$, lead to a blue spectral distribution long after the transition. 
Cyril Cartier