Unitarity, clock dependence and quantum recollapse in quantum cosmology
  We continue our analysis of a quantum cosmology model describing a flat
Friedmann--Lema\^itre--Robertson--Walker universe filled with a (free) massless
scalar field and an arbitrary perfect fluid. For positive energy density in the
scalar and fluid, each classical solution has a singularity and expands to
infinite volume. When quantising we view the cosmological dynamics in
relational terms, using one degree of freedom as a clock for the others. Three
natural candidates for this clock are the volume, a time variable conjugate to
the perfect fluid, and the scalar field. We have previously shown that
requiring unitary evolution in the "fluid" time leads to a boundary condition
at the singularity and generic singularity resolution, while in the volume time
semiclassical states follow the classical singular trajectories. Here we
analyse the third option of using the scalar field as a clock, finding further
dramatic differences to the previous cases: the boundary condition arising from
unitarity is now at infinity. Rather than singularity resolution, this theory
features a quantum recollapse of the universe at large volume, as was shown in
a similar context by Paw{\l}owski and Ashtekar. We illustrate the properties of
the theory analytically and numerically, showing that the ways in which the
different quantum theories do or do not depart from classical behaviour
directly arise from demanding unitarity with respect to different clocks. We
argue that using a Dirac quantisation would not resolve the issue. Our results
further illustrate the problem of time in quantum gravity.
