Uses of Complex Metrics in Cosmology

TL;DR

This paper examines the criteria for allowable complex metrics in cosmology, providing a method to quickly assess minisuperspace metrics and exploring implications for quantum transitions and boundary conditions in the gravitational path integral.

Contribution

It introduces a practical method to determine the allowability of minisuperspace metrics, facilitating the study of off-shell structures and boundary conditions in quantum cosmology.

Findings

Classical transitions are on the boundary of allowable metrics.

Quantum tunnelling transitions violate the allowability criterion.

No-boundary solutions are allowable and can be integrated explicitly.

Abstract

Complex metrics are a double-edged sword: they allow one to replace singular spacetimes, such as those containing a big bang, with regular metrics, yet they can also describe unphysical solutions in which quantum transitions may be more probable than ordinary classical evolution. In the cosmological context, we investigate a criterion proposed by Witten (based on works of Kontsevich & Segal and of Louko & Sorkin) to decide whether a complex metric is allowable or not. Because of the freedom to deform complex metrics using Cauchy's theorem, deciding whether a metric is allowable in general requires solving a complicated optimisation problem. We describe a method that allows one to quickly determine the allowability of minisuperspace metrics. This enables us to study the off-shell structure of minisuperspace path integrals, which we investigate for various boundary conditions. Classical transitions always reside on the boundary of the domain of allowable metrics, and care must be taken in defining appropriate integration contours for the corresponding gravitational path integral. Perhaps more surprisingly, we find that proposed quantum (`tunnelling') transitions from a contracting to an expanding universe violate the allowability criterion and may thus be unphysical. No-boundary solutions, by contrast, are found to be allowable, and moreover we demonstrate that with an initial momentum condition an integration contour over allowable metrics may be explicitly described in arbitrary spacetime dimensions.