  In cosmological scenarios such as the pre-big bang scenario or the ekpyrotic
scenario, a matching condition between the metric perturbations in the pre-big
bang phase and those in the post big-bang phase is often assumed. Various
matching conditions have been considered in the literature. Nevertheless
obtaining a scale invariant CMB spectrum via a concrete mechanism remains
impossible. In this paper, we examine this problem from the point of view of
local causality. We begin with introducing the notion of local causality and
explain how it constrains the form of the matching condition. We then 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. Moreover we demonstrate
that the magnitude of the spectrum is compatible with observations if the
bounce is assumed to occur at an energy scale which is a few orders of
magnitude below the Planckian energy scale.
