de Sitter space is sometimes not empty

TL;DR

This paper investigates the entanglement entropy between de Sitter and anti-de Sitter spaces, revealing conditions under which the de Sitter Hilbert space appears nontrivial due to non-perturbative quantum gravity effects involving wormholes.

Contribution

It demonstrates the existence of wormholes connecting de Sitter and AdS spaces and establishes when the de Sitter Hilbert space becomes effectively nontrivial based on entropy comparisons.

Findings

Wormholes connecting de Sitter and AdS spaces exist.

Entanglement entropy vanishes if S_dS < S_BH.

Entanglement entropy is finite when S_dS > S_BH.

Abstract

Multiple lines of evidence suggest that the Hilbert space of an isolated de Sitter universe is one dimensional but can appear larger when probed by a gravitating observer. To test this idea, we compute the von Neumann entropy of a field theory in a two-dimensional de Sitter universe which is entangled in a thermal-like state with the same field theory on a disjoint, asymptotically anti-de Sitter (AdS) black hole. Previously, it was shown that the replica trick for computing the entropy of such entangled gravitating systems requires the inclusion of a non-perturbative effect in quantum gravity -- novel wormholes connecting the two spaces. Here we show that: (a) the expected wormholes connecting de Sitter and AdS universes exist, avoiding a no-go theorem via the presence of sources on the AdS boundary; (b) the entanglement entropy vanishes if the nominal entropy of the de Sitter cosmological horizon (S_dS = A_horizon^dS / 4G_N) is less than the entropy of the AdS black hole horizon (S_BH = A_horizon^AdS / 4G_N), i.e., S_dS < S_BH; (c) the entanglement entropy is finite when S_dS > S_BH. Thus, the de Sitter Hilbert space is effectively nontrivial only when S_dS > S_BH. The AdS black hole we introduce can be regarded as an ``observer'' for de Sitter space. In this sense, our result is a non-perturbative generalization of the recent perturbative argument that the algebra of observables on the de Sitter static patch becomes nontrivial in the presence of an observer.