A cosmological model with two global internal times shows that time reparametrization invariance, and therefore covariance, is not guaranteed by deparametrization. In particular, it is impossible to derive unambiguous proper-time effective equations from a single deparametrized model if quantum corrections from fluctuations and higher moments are included. The framework of effective constraints shows how different choices of internal time can be compared, and it suggests a new definition of coordinate time in quantum cosmology which is amenable to a notion of time reparametrization invariance. In addition to corrections from moments such as quantum fluctuations, factor ordering corrections may appear also. The latter generically lead to inequivalent internal-time formulations owing to quantization ambiguities.

• Received 24 July 2018

DOI:https://doi.org/10.1103/PhysRevD.98.066001