Time as a test-field: the no-boundary universe in motion and a smooth radiation bounce

Abstract

The proper time of an observer can be introduced as a degree of freedom in quantum cosmology, additional to the existing fields. We review two arguments for using the Schr\"odinger equation to evolve the corresponding wavefunction. We restrict to solutions in which time acts as a component with negligible backreaction on the metric -- that is, it plays the role of a test field. We apply this idea to various minisuperspace models. In the semiclassical regime we recover expected results: the wavefunction peaks on the classical solution and, in models with a scalar field, the variance of $\zeta$ (a mini-superspace analogue of the comoving curvature perturbation) is conserved. Applied to the no-boundary wavefunction, our model recovers the bouncing behavior of classical global de Sitter space, with small corrections associated to the evolving variance of the wavefunction. Other bouncing solutions do not have any classical analogue. This is the case of a radiation dominated universe, which classically leads to a big-bang singularity but corresponds quantum mechanically to an $s$-wave scattering off a central potential of the form $-r^{-2/3}$. As much as the hydrogen atom, this potential is famously made stable by the Heisenberg uncertainty principle. We study the unitary evolution of the wavepacket numerically. During the bounce, the uncertainty and the expectation value of the scale factor become comparable. By selecting a large initial variance, the bounce can be made arbitrarily smooth, the mean value of the Hubble parameter correspondingly soft.