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The problem of boundary conditions for the cosmological wave function is analyzed using a simple model as an example. The model has two degrees of freedom: the scale factor and a homogeneous scalar field. A tunneling wave function is found for this model which describes an ensemble of universes tunneling from ``nothing'' to a de Sitter space, then evolving along the lines of an inflationary scenario and eventually collapsing to a singularity. A boundary condition in superspace is proposed which selects the tunneling solution for. This boundary condition can be thought of as a quantum version of the Penrose condition in classical general relativity. The probability current, the concept of time, and the possibility of an experimental test of quantum cosmology are also discussed.
Alexander Vilenkin (Sun,) studied this question.