Self-adjoint Wheeler-DeWitt operators, the problem of time, and the wave function of the Universe

We discuss minisuperspace aspects of a nonempty Robertson-Walker universe containing a scalar matter field. The requirement that the Wheeler-DeWitt (WDW) operator be self-adjoint is a key ingredient in constructing the physical Hilbert space and has nontrivial cosmological implications since it is related to the problem of time in quantum cosmology. Namely, if time is parametrized by matter fields we find two types of domains for the self-adjoint WDW operator: a nontrivial domain is comprised of zero current (Hartle-Hawking-type) wave functions and is parametrized by two new parameters, whereas the domain of a self-adjoint WDW operator acting on tunneling (Vilenkin-type) wave functions is a single ray. On the other hand, if time is parametrized by the scale factor both wave function types give rise to nontrivial domains for the self-adjoint WDW operators, and no new parameters appear in them.