Self-Adjoint Wheeler-DeWitt Operators, the Problem of Time and the Wave Function of the Universe

We discuss minisuperspace aspects a non empty Robertson-Walker universe containing 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 non-trivial cosmological implications since it is related with 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 non trivial 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 {\em single} ray. On the other hand, if time is parametrized by the scale factor both types of wave functions give rise to non trivial domains for the self adjoint WDW operators, and no new parameters appear in them.