- Open Access
Remarks on the de Sitter double cone
Phys. Rev. D 113, 126012 – Published 8 June, 2026
DOI: https://doi.org/10.1103/jync-pmnz
Abstract
We study the double cone geometry proposed by Saad-Shenker-Stanford in de Sitter space. We show that with the inclusion of static patch observers, the double cone leads to a linear ramp consistent with random matrix behavior, where the ramp comes from the relative timeshift between two clocks sitting at opposite static patches.
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References (79)
- L. Dyson, M. Kleban, and L. Susskind, Disturbing implications of a cosmological constant, J. High Energy Phys. 10 (2002) 011.
- N. Goheer, M. Kleban, and L. Susskind, The trouble with de Sitter space, J. High Energy Phys. 07 (2003) 056.
- P. Saad, S. H. Shenker, and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840.
- Y. Chen, V. Ivo, and J. Maldacena, Comments on the double cone wormhole, J. High Energy Phys. 04 (2024) 124.
- E. Witten, Gravity and the crossed product, J. High Energy Phys. 10 (2022) 008.
- V. Chandrasekaran, G. Penington, and E. Witten, Large N algebras and generalized entropy, J. High Energy Phys. 04 (2023) 009.
We use the convention that the subscribe means insertion of window function in the partition function.
- V. Chandrasekaran, R. Longo, G. Penington, and E. Witten, An algebra of observables for de Sitter space, J. High Energy Phys. 02 (2023) 082.
- X. Dong, E. Silverstein, and G. Torroba, De Sitter holography and entanglement entropy, J. High Energy Phys. 07 (2018) 050.
- E. Silverstein, Black hole to cosmic horizon microstates in string/M theory: Timelike boundaries and internal averaging, J. High Energy Phys. 05 (2023) 160.
- M. Kaplan, D. Marolf, X. Yu, and Y. Zhao, De Sitter quantum gravity and the emergence of local algebras, J. High Energy Phys. 04 (2025) 171.
- D. K. Kolchmeyer and H. Liu, Chaos and the emergence of the cosmological horizon, arXiv:2411.08090.
- D. Tietto and H. Verlinde, A microscopic model of de Sitter spacetime with an observer, arXiv:2502.03869.
- J. Maldacena, Real observers solving imaginary problems, arXiv:2412.14014.
- V. Ivo, J. Maldacena, and Z. Sun, Physical instabilities and the phase of the Euclidean path integral, J. High Energy Phys. 04 (2026) 118.
- X. Shi and G. J. Turiaci, The phase of the gravitational path integral, J. High Energy Phys. 07 (2025) 047.
Throughout the paper, means asymptotes at large or large .
It is denoted as in [14].
The factor arises because the time shift U(1) symmetry in has not been quotiented out.
- A. M. Polyakov, Gauge fields and strings, Contemp. Concepts Phys. 3, 1 (1987).
- R. Jackiw, Lower dimensional gravity, Nucl. Phys. B252, 343 (1985).
- C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. 126B, 41 (1983).
- K. Jensen, Chaos in holography, Phys. Rev. Lett. 117, 111601 (2016).
- J. Maldacena, D. Stanford, and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-Sitter space, Prog. Theor. Exp. Phys. 2016, 12C104 (2016).
- J. Engelsöy, T. G. Mertens, and H. Verlinde, An investigation of backreaction and holography, J. High Energy Phys. 07 (2016) 139.
- A. Svesko, E. Verheijden, E. P. Verlinde, and M. R. Visser, Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity, J. High Energy Phys. 08 (2022) 075.
- A. A. Rahman, dS JT gravity and double-scaled SYK, arXiv:2209.09997.
The de Sitter vacuum may also decay, but for our current purposes we assume an eternal de Sitter space.
In same spirit as the situation of typical state in black holes [31, 32, 33, 34, 35, 36, 37, 38, 39, 40].
- D. Stanford and Z. Yang, Firewalls from wormholes, arXiv:2208.01625.
- A. Almheiri, D. Marolf, J. Polchinski, D. Stanford, and J. Sully, An apologia for firewalls, J. High Energy Phys. 09 (2013) 018.
- D. Marolf and J. Polchinski, Gauge/gravity duality and the black hole interior, Phys. Rev. Lett. 111, 171301 (2013).
- L. Susskind, Singularities, firewalls, and complementarity, arXiv:1208.3445.
- L. Susskind, The typical-state paradox: Diagnosing horizons with complexity, Fortschr. Phys. 64, 84 (2016).
- J. de Boer, R. Van Breukelen, S. F. Lokhande, K. Papadodimas, and E. Verlinde, On the interior geometry of a typical black hole microstate, J. High Energy Phys. 05 (2019) 010.
- J. De Boer, R. Van Breukelen, S. F. Lokhande, K. Papadodimas, and E. Verlinde, Probing typical black hole microstates, J. High Energy Phys. 01 (2020) 062.
- L. Susskind, Black holes at exp-time, arXiv:2006.01280.
- A. Blommaert, C.-H. Chen, and Y. Nomura, Firewalls at exponentially late times, J. High Energy Phys. 10 (2024) 131.
- L. V. Iliesiu, A. Levine, H. W. Lin, H. Maxfield, and M. Mezei, On the non-perturbative bulk Hilbert space of JT gravity, J. High Energy Phys. 10 (2024) 220.
After diving that represents gauging the zero mode of the clock.
We use the convention that is positive.
- Z. Yang, Comments on the Saad wormhole, https://groups.oist.jp/exu-oist/recorded-talks (2024).
Note added in V2: After published the first version, we learned similar ideas have been thought independently by [45].
- E. Shaghoulian, The central dogma and horizons in quantum cosmology, https://cds.cern.ch/record/2871349.
That said, [1, 2, 47] has proposed that a statistical description could still be viable in such scenarios, providing a potential pathway to address these conceptual challenges.
- L. Susskind, De Sitter holography: Fluctuations, anomalous symmetry, and wormholes, Universe 7, 464 (2021).
We thank Douglas Stanford for emphasizing this interpretation.
- A. Kitaev and S. J. Suh, Statistical mechanics of a two-dimensional black hole, J. High Energy Phys. 05 (2019) 198.
- Z. Yang, The quantum gravity dynamics of near extremal black holes, J. High Energy Phys. 05 (2019) 205.
- G. Penington and E. Witten, Algebras and states in JT gravity, arXiv:2301.07257.
- D. L. Jafferis and D. K. Kolchmeyer, Entanglement entropy in Jackiw-Teitelboim gravity, arXiv:1911.10663.
.
Otherwise one could always find a frame such that the two observers are in the same static patch, and they will not satisfy the corresponding boost constraint.
More precisely, the coinvariant Hilbert space.
- S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions (American Mathematical Society, Providence, NJ, 2022), Vol. 83.
- L. Susskind, A paradox and its resolution illustrate principles of de Sitter holography, J. Hologr. App. Phys. 5, 1 (2025).
- D. Harlow and T. Numasawa, Gauging spacetime inversions in quantum gravity, J. High Energy Phys. 01 (2026) 098.
The easy way to see is to notice that their difference is a linear combination of the constraints (2.43)–(2.47).
- J. M. Maldacena, Eternal black holes in anti-de Sitter, J. High Energy Phys. 04 (2003) 021.
- P. Saad, Late time correlation functions, baby universes, and ETH in JT gravity, arXiv:1910.10311.
- P. Petersen, Riemannian Geometry (Springer, New York, 2006), Vol. 171.
- Y. Chen, V. Gorbenko, and J. Maldacena, Bra-ket wormholes in gravitationally prepared states, J. High Energy Phys. 02 (2021) 009.
- A. Fumagalli, V. Gorbenko, and J. Kames-King, De Sitter bra-ket wormholes, J. High Energy Phys. 05 (2025) 074.
- D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions, J. High Energy Phys. 01 (2022) 088.
- P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115.
Here we are talking about the expansion of the classical trajectory of the observer. Do not confuse this with the quantum Lyapunov exponent associated with the de Sitter horizon.
- F. Haake, Quantum Signatures of Chaos (Springer, New York, 1991).
We thank Douglas Stanford for suggesting us this analogy.
- D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, J. High Energy Phys. 10 (2017) 008.
- D. Kapec, R. Mahajan, and D. Stanford, Matrix ensembles with global symmetries and ’t Hooft anomalies from 2D gauge theory, J. High Energy Phys. 04 (2020) 186.
We thank Yiming Chen for asking this question.
We thank Yiming Chen and Douglas Stanford for raising this question.
- D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, Adv. Theor. Math. Phys. 24, 1475 (2020).
- N. Engelhardt, G. Penington, and A. Shahbazi-Moghaddam, Twice upon a time: Timelike-separated quantum extremal surfaces, J. High Energy Phys. 01 (2024) 033.
- P. Saad, D. Stanford, Z. Yang, and S. Yao, A convergent genus expansion for the plateau, J. High Energy Phys. 09 (2024) 033.
- L. Aalsma and G. Shiu, Chaos and complementarity in de Sitter space, J. High Energy Phys. 05 (2020) 152.
- Y. Chen, D. Stanford, H. Tang, and Z. Yang, On the phase of the de Sitter density of states, arXiv:2511.01400.