Possible quantum effects at the transition from cosmological deceleration to acceleration

The recent transition from decelerated to accelerated expansion can be seen as a reflection (or "bounce") in the connection variable, defined by the inverse comoving Hubble length ($b=\dot a$, on-shell). We study the quantum cosmology of this process. We use a formalism for obtaining relational time variables either through the demotion of the constants of Nature to integration constants, or by identifying fluid constants of motion. We extend its previous application to a toy model (radiation and $\Lambda$) to the realistic setting of a transition from dust matter to $\Lambda$ domination. In the dust and $\Lambda$ model two time variables may be defined, conjugate to $\Lambda$ and to the dust constant of motion, and we work out the monochromatic solutions to the Schr\"odinger equation representing the Hamiltonian constraint. As for their radiation and $\Lambda$ counterparts, these solutions exhibit "ringing", whereby the incident and reflected waves interfere, leading to oscillations in the amplitude. In the semi-classical approximation we find that, close to the bounce, the probability distribution becomes double-peaked, one peak following a trajectory close to the classical limit but with a Hubble parameter slightly shifted downwards, the other with a value of $b$ stuck at its minimum $b=b_\star$. Still closer to the transition, the distribution is better approximated by an exponential distribution, with a single peak at $b=b_\star$, and a (more representative) average $b$ biased towards a value higher than the classical trajectory. Thus, we obtain a distinctive prediction for the average Hubble parameter with redshift: slightly lower than its classical value when $z\approx 0$, but potentially much higher than the classical prediction around $z\sim 0.64$, where the bounce most likely occurred. The implications for the "Hubble tension" have not escaped us.