No-boundary prescriptions in Lorentzian quantum cosmology

We analyze 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 wave function. 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, taking the Einstein-Hilbert action at face value while adding no boundary tenn. 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 wave function is to use Robin boundary conditions. A subclass 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 wave function force one to abandon the notion of a sum over compact and regular geometries, and point to the importance of an initial Euclidean momentum.