Quantum analysis of the recent cosmological bounce in the comoving Hubble length

TL;DR

This paper explores quantum effects during a cosmological bounce in a simplified universe model, revealing wave interference phenomena and probability distribution features near the bounce, with implications for quantum cosmology.

Contribution

It introduces a quantum cosmology framework for a bouncing universe, analyzing ambiguities in quantum equations and solutions, and characterizing wave interference effects near the bounce.

Findings

Wave interference causes oscillations in probability distributions.

Near the bounce, the probability distribution becomes double-peaked.

Solutions show a classical-like trajectory with a shifted Hubble parameter.

Abstract

We formulate the transition from decelerated to accelerated expansion as a bounce in connection space and study its quantum cosmology, knowing that reflections are notorious for bringing quantum effects to the fore. We use a formalism for obtaining a time variable via the demotion of the constants of Nature to integration constants, and focus on a toy Universe containing only radiation and a cosmological constant $\Lambda$ for its simplicity. We find that, beside the usual factor ordering ambiguities, there is an ambiguity in the order of the quantum equation, leading to two distinct theories: one second, the other first order. In both cases two time variables may be defined, conjugate to $\Lambda$ and to the radiation constant of motion. We make little headway with the second-order theory, but are able to produce solutions to the first-order theory. They exhibit the well-known "ringing" whereby incident and reflected waves interfere, leading to oscillations in the probability distribution even for well-peaked wave packets. We also examine in detail the probability measure within the semiclassical approximation. Close to the bounce, the probability distribution becomes double-peaked, with one peak following a trajectory close to the classical limit but with a Hubble parameter slightly shifted downwards, and the other with a value of $b$ stuck at its minimum. An examination of the effects still closer to the bounce, and within a more realistic model involving matter and $\Lambda$, is left to future work.