No-boundary prescriptions in Lorentzian quantum cosmology

TL;DR

This paper examines how different boundary conditions affect the Lorentzian gravitational path integral and the no-boundary wavefunction, revealing that current models require initial Euclidean momentum and challenge the sum over compact geometries.

Contribution

It introduces a simple implementation of the no-boundary proposal using the Einstein-Hilbert action without boundary terms and explores the effects of Robin boundary conditions on the wavefunction.

Findings

Adding no boundary term stabilizes fluctuations in the path integral.

Specifying the final Hubble rate alters the off-shell structure significantly.

Current models suggest abandoning the sum over compact geometries in favor of initial Euclidean momentum.

Abstract

We analyse the impact of various boundary conditions on the (minisuperspace) Lorentzian gravitational path integral. In particular we assess the implications for the Hartle-Hawking no-boundary wavefunction. It was shown recently that when this proposal is defined as a sum over compact metrics, problems arise with the stability of fluctuations. These difficulties can be overcome by an especially simple implementation of the no-boundary idea: namely to take the Einstein-Hilbert action at face value while adding no boundary term. This prescription simultaneously imposes an initial Neumann boundary condition for the scale factor of the universe and, for a Bianchi IX spacetime, Dirichlet conditions for the anisotropies. Another way to implement the no-boundary wavefunction is to use Robin boundary conditions. A sub-class of Robin conditions allows one to specify the Hubble rate on the boundary hypersurface, and we highlight the surprising aspect that specifying the final Hubble rate (rather than the final size of the universe) significantly alters the off-shell structure of the path integral. The conclusion of our investigations is that all current working examples of the no-boundary wavefunction force one to abandon the notion of a sum over compact and regular geometries, and point to the importance of an initial Euclidean momentum.