Freeze-free cosmological evolution with a non-monotonic internal clock

Given the lack of an absolute time parameter in general relativistic systems, quantum cosmology often describes the expansion of the universe in terms of relational changes between two degrees of freedom, such as matter and geometry. However, if clock degrees of freedom (self-)interact non-trivially, they in general have turning points where their momenta vanish. At and beyond a turning point, the evolution of other degrees of freedom is no longer described directly by changes of the clock parameter because it stops and then turns back, while time is moving forward. Previous attempts to describe quantum evolution relative to a clock with turning points have failed and led to frozen evolution in which degrees of freedom remain constant while the clock parameter, interpreted directly as a substitute for monotonic time, is being pushed beyond its turning point. Here, a new method previously used in oscillator systems is applied to a tractable cosmological model, given by an isotropic universe with spatial curvature and scalar matter. The re-collapsing scale factor presents an example of a clock with a single turning point. The method succeeds in defining unitary and freeze-free evolution by unwinding the turning point of the clock, introducing an effective monotonic time parameter that is related to but not identical with the non-monotonic clock degree of freedom. Characteristic new quantum features are found around the turning point, based on analytical and numerical calculations.