  The tunneling wave function of the universe is investigated in a
minisuperspace framework of a de Sitter universe with a quantum scalar field,
treated as a perturbation. We consider three different approaches to defining
the tunneling wave function: (1) tunneling boundary conditions in superspace,
(2) Lorentzian path integral, and (3) quantum tunneling from initial universe
of a vanishing size. We show that the superspace approach requires Robin
boundary conditions for the scalar field modes, the path integral approach
requires adding an appropriate boundary term to the scalar field action, and
the initial universe approach requires the initial quantum state of the scalar
field to be Euclidean vacuum. We find that all three approaches yield identical
wave functions and that scalar field fluctuations are well behaved, contrary to
earlier claims in the literature.
