Unstable growth of curvature perturbation in non-singular bouncing cosmologies

We consider non-singular bouncing cosmologies, such as the new ekpyrotic model, in which the universe undergoes a slow contraction phase with equation of state $w \gg 1$, followed by a bounce that occurs at a finite scale factor when quantum gravity corrections are still negligible. Such a non-singular bounce requires a violation of the null energy condition in which $w$ falls below -1 at some time before the bounce. In this paper, we show that a component of the adiabatic curvature perturbations, though decaying and negligible during the ekpyrotic phase, is exponentially amplified just before $w$ approaches -1, enough to spoil the scale-invariant perturbation spectrum. We discuss how the problem may be avoided, for example, in singular bounces.