Unitary and non-unitary transitions around a cosmological bounce

In this work, we investigate the notion of time and unitarity in the vicinity of a bounce in quantum cosmology, that is, a turning point for the scale factor. Because WKB methods drastically fail near a turning point, the scale factor cannot play the role of time in such scenarios. We overcome this difficulty by studying the dynamics of matter transitions when using its conjugate momentum as a time. We find precise conditions so as to recover unitarity, and hence, a consistent notion of probability. We then compute transitions in a concrete example, extract the specific feature of a bounce and argue about the necessity of the conjugate momentum representation to go beyond the background field approximation. Our analysis is also equally relevant for a closed Universe undergoing a recollapsing phase.