On the Euclidean Action of de Sitter Black Holes and Constrained Instantons

TL;DR

This paper calculates the Euclidean action of Schwarzschild-de Sitter black holes using constrained instantons, revealing finite actions and implications for black hole nucleation probabilities and non-perturbative effects in de Sitter space.

Contribution

It introduces a method to compute the on-shell Euclidean action of de Sitter black holes with singularities, linking it to horizon entropies and applying it to black hole nucleation probabilities.

Findings

On-shell action is finite and independent of Euclidean time periodicity.

The action equals minus the sum of black hole and cosmological horizon entropies.

Derived the probability of black hole nucleation in de Sitter space.

Abstract

We compute the on-shell Euclidean action of Schwarzschild-de Sitter black holes, and take their contributions in the gravitational path integral into account using the formalism of constrained instantons. Although Euclidean de Sitter black hole geometries have conical singularities for generic masses, their on-shell action is finite and is shown to be independent of the Euclidean time periodicity and equal to minus the sum of the black hole and cosmological horizon entropy. We apply this result to compute the probability for a nonrotating, neutral arbitrary mass black hole to nucleate spontaneously in empty de Sitter space, which separates into a constant and a "non-perturbative" contribution, the latter corresponding to the proper saddle-point instanton in the Nariai limit. We also speculate on some further applications of our results, most notably as potential non-perturbative corrections to correlators in the de Sitter vacuum.