De Sitter space is sometimes not empty

A bstract 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 (S₃ₒ=A₇₎ₑ₈ₙ₎₍^dS/4G₍) S dS = A horizon dS / 4 G N is less than the entropy of the AdS black hole horizon (S₁₇=A₇₎ₑ₈ₙ₎₍^AdS/4G₍) S BH = A horizon AdS / 4 G N, i. e. , 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.