Quantum cosmology and the initial state of the Universe

Cosmological wave functions are found in a minisuperspace model (1) with ``tunneling'' and (2) with Hartle-Hawking boundary conditions. The probability distributions for the initial states of the Universe corresponding to the two wave functions are calculated and compared. It is shown that the tunneling wave function predicts initial states that lead to inflation, while the Hartle-Hawking wave function does not. Small perturbations about the minisuperspace model are considered and it is argued that both wave functions predict that the Universe nucleates with quantum fields in de Sitter-invariant vacuum states.