Simplical minisuperspace. III. Integration contours in a five-simplex model

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Abstract
The no boundary proposal for the wave function of the universe is investigated in a minisuperspace model of pure gravity with cosmological constant. The model’s four geometries consist of five four-simplices joined together to make the surface of a five-simplex from which one four-simplex face has been removed. The model is further simplified by symmetrically choosing all the interior edges of equal length and all the edges of the four-simplex boundary of equal length. The wave function is thus a function of a single boundary squared edge length and is specified by an integral over the single interior edge length. The analytic properties of the action in the space of complex edge lengths are exhibited, its classical extrema are calculated, and the possible contours of integration defining the wave function of the universe are discussed. A descending contour of constant imaginary action is proposed along which the integral defining the wave function is convergent and which predicts classical space-time in the late universe. This contour is the analog for the model of the conformally rotated contour appropriate to Euclidean sums over asymptotically flat space-times. The wave function is evaluated numerically for this contour both directly and by semiclassical methods.