General matching conditions in bouncing cosmology and CMB spectrum

To clarify the issue of obtaining the scale invariant CMB spectrum in bouncing cosmology, we examine the matching condition between the metric perturbations before and after the bounce. We prove a no-go theorem: independent of the details of the matching condition, a scale invariant spectrum is impossible as long as the local causality condition is satisfied. In our framework, it is easy to show that a violation of local causality around the bounce is needed in order to give a scale invariant spectrum. We study a specific scenario of this possibility by considering a nonlocal effective theory inspired by noncommutative geometry around the bounce and show that a scale invariant spectrum is possible.