Two Wave Functions and dS/CFT on S^1 x S^2

TL;DR

This paper evaluates the tunneling and Hartle-Hawking wave functions on S^1 x S^2 in Einstein gravity, analyzing their classical predictions, divergences, and holographic formulations, especially in relation to Schwarzschild-de Sitter black holes.

Contribution

It provides a detailed analysis of wave functions on S^1 x S^2, including their classical limits, divergences, and a holographic reformulation using Euclidean AdS saddle points.

Findings

Classical predictions include Schwarzschild-de Sitter black holes.

The Hartle-Hawking measure converges in the small S^1 limit.

Divergences in the tunneling state are linked to unphysical saddle points.

Abstract

We evaluate the tunneling and Hartle-Hawking wave functions on S^1 x S^2 boundaries in Einstein gravity with a positive cosmological constant. In the large overall volume limit the classical predictions of both wave functions include an ensemble of Schwarzschild-de Sitter black holes. We show that the Hartle-Hawking tree level measure on the classical ensemble converges in the small S^1 limit. A divergence in this regime can be identified in the tunneling state. However we trace this to the contribution of an unphysical branch of saddle points associated with negative mass black holes. Using a representation in which all saddle points have an interior Euclidean anti-de Sitter region we also generalise the holographic formulation of the semiclassical Hartle-Hawking wave function to S^1 x S^2 boundaries.