Unitarity, clock dependence and quantum recollapse in quantum cosmology

We continue our analysis of a quantum cosmology model describing a flat Friedmann–Lemaˆıtre–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 Pawlowski 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.