Quantum analysis of the recent cosmological bounce in 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 to the fore quantum effects. 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. A detailed study of whether
these would be observable is left to further work. Nonetheless, even within our
approximations we are able to make an interesting prediction: for a period
surrounding the bounce, observations will be ruled by a double-peaked
distribution, one peak following a near-classical trajectory with a comoving
Hubble volume slightly shifted upwards, the other with a comoving Hubble volume
"stuck" at its maximum. This bias towards a larger Hubble volume could be
observable.
