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

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 Λ for 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 order, and one first order. In both cases two time variables may be defined, conjugate to Λ and 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 Λ, is left to future work.