The Hartle-Hawking Wavefunction and de Sitter Black Holes

The Schwarzschild-de Sitter solution can be used to compute the Hartle-Hawking wavefunction of the universe on a spatial circle times a sphere. Its norm can be evaluated in the semiclassical limit and compared to the on-shell action of the Euclidean Nariai geometry. I will describe new results clarifying the role that quantum corrections to the wavefunction play in this derivation. Next, I will present a calculation of the phase of the path integral on Nariai to test whether, at the quantum level, it can be interpreted as a norm. In the process, I will derive a simple formula for the phase of the path integral on any solution of pure gravity. Finally, I will describe ongoing work based on an alternative interpretation of the Nariai geometry as computing the nucleation rate of black holes in de Sitter.