de Sitter space is sometimes not empty

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.