Semiclassical wave function of the Universe at small three-geometries

Of particular interest is the behavior of the wave function of minimum excitation for compact cosmologies as the three-geometry becomes small. Information on its qualitative behavior is contained in the scaling behavior of the semiclassical approximation of the wave function. For small three-geometries, the scaling of the semiclassical wave function is determined by the scaling of the prefactor. We expect that the scaling of the prefactor for small arbitrary three-geometries is qualitatively the same as that of the prefactor for the simple geometry of a three-sphere. We compute the scaling of the prefactor for the geometry of Euclidean flat space with a three-sphere boundary; this geometry corresponds to that for a three-sphere boundary with a positive cosmological constant in the small-volume limit. For this geometry, the scaling of the prefactor entirely determines its form