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 ¼ _ 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 Λ) to the realistic setting of a transition from dust matter to Λ domination. In the dust and Λ model two time variables may be defined, conjugate to Λ and to the dust constant of motion, and we work out the monochromatic solutions to the Schrödinger equation representing the Hamiltonian constraint. As for their radiation and Λ counterparts, these solutions exhibit "ringing," whereby the incident and reflected waves interfere, leading to oscillations in the amplitude. In the semiclassical 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 ¼ b ⋆ . Still closer to the transition, the distribution is better approximated by an exponential distribution, with a single peak at b ¼ b ⋆ , 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 ≈ 0, but potentially much higher than the classical prediction around z ∼ 0.64, where the bounce most likely occurred. The implications for the "Hubble tension" have not escaped us.

We revisit a recent proposal for a definition of time in quantum cosmology, to investigate the effects of having more than one possible type of clock "at the same time." We use as a test tube an extension of Einstein gravity with a massless scalar field in which the gravitational coupling G N is only a constant on-shell, mimicking the procedure for Λ in unimodular gravity. Hence we have two "simultaneous" clocks in the theory: a scalar field clock, and the conjugate of G N . We find that attempts to use two coherent clocks concurrently are disastrous for recovering the classical limit. The Heisenberg relations, instead of being saturated, are always realized abundantly above their bound, with strong quantum effects expected at least in parts of the trajectory. Semiclassical states always result from situations where we effectively impose a single clock, either by making the other clock a failed clock (i.e., by choosing a state where its conjugate constant is infinitely sharp) or by choosing a basis of constants where all clocks but one are redundant; i.e., motion or change in phase space does not occur with the passing of their "times." We show how this conclusion generalizes to fluids with any equation of state. It also applies to systems where "subclocks" of the same type could be used, for example, in mixtures of different fluids with the same equation of state.

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