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 \( \left({S}_{\textrm{dS}}={A}_{\textrm{horizon}}^{\textrm{dS}}/4{G}_{\textrm{N}}\right) \) is less than the entropy of the AdS black hole horizon \( \left({S}_{\textrm{BH}}={A}_{\textrm{horizon}}^{\textrm{AdS}}/4{G}_{\textrm{N}}\right) \), i.e., SdS < SBH; (c) the entanglement entropy is finite when SdS > SBH. Thus, the de Sitter Hilbert space is effectively nontrivial only when SdS > SBH. 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.
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Acknowledgments
We thank Arjun Kar and Edgar Shaghoulian for discussions during the early stage of this research. VB was supported by the Department of Energy through DE-SC0013528 and QuantISED DE-SC0020360, as well as by the Simons Foundation through the It From Qubit Collaboration (Grant No. 38559). YN was supported in part by the Department of Energy, Office of Science, Office of High Energy Physics under QuantISED award DESC0019380 and contract DE-AC02-05CH11231 and in part by MEXT KAKENHI grant number JP20H05850, JP20H05860. TU was supported in part by JSPS Grant-in-Aid for Young Scientists 19K14716 and in part by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A “Extreme Universe” No.21H05184. VB thanks the Santa Fe Institute and the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452, for hospitality as this work was completed.
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Balasubramanian, V., Nomura, Y. & Ugajin, T. De Sitter space is sometimes not empty. J. High Energ. Phys. 2024, 135 (2024). https://doi.org/10.1007/JHEP02(2024)135
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DOI: https://doi.org/10.1007/JHEP02(2024)135