Skip to main content

Islands in cosmology

A preprint version of the article is available at arXiv.

Abstract

A quantum extremal island suggests that a region of spacetime is encoded in the quantum state of another system, like the encoding of the black hole interior in Hawking radiation. We study conditions for islands to appear in general spacetimes, with or without black holes. They must violate Bekenstein’s area bound in a precise sense, and the boundary of an island must satisfy several other information-theoretic inequalities. These conditions combine to impose very strong restrictions, which we apply to cosmological models. We find several examples of islands in crunching universes. In particular, in the four-dimensional FRW cosmology with radiation and a negative cosmological constant, there is an island near the turning point when the geometry begins to recollapse. In a two-dimensional model of JT gravity in de Sitter spacetime, there are islands inside crunches that are encoded at future infinity or inside bubbles of Minkowski spacetime. Finally, we discuss simple tensor network toy models for islands in cosmology and black holes.

References

  1. [1]

    J.D. Bekenstein, A Universal Upper Bound on the Entropy to Energy Ratio for Bounded Systems, Phys. Rev. D 23 (1981) 287 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  2. [2]

    H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  3. [3]

    W. Fischler and L. Susskind, Holography and cosmology, hep-th/9806039 [INSPIRE].

  4. [4]

    R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  5. [5]

    R. Bousso, The Holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  6. [6]

    E.E. Flanagan, D. Marolf and R.M. Wald, Proof of classical versions of the Bousso entropy bound and of the generalized second law, Phys. Rev. D 62 (2000) 084035 [hep-th/9908070] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  7. [7]

    A. Strominger and D.M. Thompson, A Quantum Bousso bound, Phys. Rev. D 70 (2004) 044007 [hep-th/0303067] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  8. [8]

    R. Bousso, E.E. Flanagan and D. Marolf, Simple sufficient conditions for the generalized covariant entropy bound, Phys. Rev. D 68 (2003) 064001 [hep-th/0305149] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  9. [9]

    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Proof of a Quantum Bousso Bound, Phys. Rev. D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].

    ADS  Google Scholar 

  10. [10]

    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Entropy on a null surface for interacting quantum field theories and the Bousso bound, Phys. Rev. D 91 (2015) 084030 [arXiv:1406.4545] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  11. [11]

    A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  12. [12]

    G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  13. [13]

    A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  14. [14]

    G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [INSPIRE].

  15. [15]

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  16. [16]

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, arXiv:2006.06872 [INSPIRE].

  17. [17]

    N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].

    ADS  Google Scholar 

  18. [18]

    A.C. Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  19. [19]

    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  20. [20]

    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].

    ADS  Google Scholar 

  21. [21]

    X. Dong, X.-L. Qi, Z. Shangnan and Z. Yang, Effective entropy of quantum fields coupled with gravity, JHEP 10 (2020) 052 [arXiv:2007.02987] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  22. [22]

    Y. Chen, V. Gorbenko and J. Maldacena, Bra-ket wormholes in gravitationally prepared states, arXiv:2007.16091 [INSPIRE].

  23. [23]

    C. Krishnan, Critical Islands, arXiv:2007.06551 [INSPIRE].

  24. [24]

    K. Narayan, On extremal surfaces and de Sitter entropy, Phys. Lett. B 779 (2018) 214 [arXiv:1711.01107] [INSPIRE].

    ADS  MATH  Google Scholar 

  25. [25]

    C. Arias, F. Diaz and P. Sundell, de Sitter Space and Entanglement, Class. Quant. Grav. 37 (2020) 015009 [arXiv:1901.04554] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  26. [26]

    K. Narayan, de Sitter future-past extremal surfaces and the entanglement wedge, Phys. Rev. D 101 (2020) 086014 [arXiv:2002.11950] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  27. [27]

    T. Hertog and G.T. Horowitz, Towards a big crunch dual, JHEP 07 (2004) 073 [hep-th/0406134] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  28. [28]

    T. Hertog and G.T. Horowitz, Holographic description of AdS cosmologies, JHEP 04 (2005) 005 [hep-th/0503071] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  29. [29]

    B. Freivogel, Y. Sekino, L. Susskind and C.-P. Yeh, A Holographic framework for eternal inflation, Phys. Rev. D 74 (2006) 086003 [hep-th/0606204] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  30. [30]

    J. Maldacena, Vacuum decay into Anti de Sitter space, arXiv:1012.0274 [INSPIRE].

  31. [31]

    D. Harlow and L. Susskind, Crunches, Hats, and a Conjecture, arXiv:1012.5302 [INSPIRE].

  32. [32]

    A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE].

  33. [33]

    A. Almheiri, R. Mahajan and J.E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020) 001 [arXiv:1911.09666] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  34. [34]

    M. Mezei and J. Virrueta, Exploring the Membrane Theory of Entanglement Dynamics, JHEP 02 (2020) 013 [arXiv:1912.11024] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  35. [35]

    T. Anous, J. Kruthoff and R. Mahajan, Density matrices in quantum gravity, SciPost Phys. 9 (2020) 045 [arXiv:2006.17000] [INSPIRE].

    ADS  Google Scholar 

  36. [36]

    A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  37. [37]

    D. Harlow, The Ryu–Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  38. [38]

    X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  39. [39]

    Y. Chen, Pulling Out the Island with Modular Flow, JHEP 03 (2020) 033 [arXiv:1912.02210] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  40. [40]

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  41. [41]

    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  42. [42]

    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  43. [43]

    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].

    ADS  MATH  Google Scholar 

  44. [44]

    T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  45. [45]

    M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell and D. Wakeham, Information radiation in BCFT models of black holes, JHEP 05 (2020) 004 [arXiv:1910.12836] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  46. [46]

    X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations, and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  47. [47]

    R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  48. [48]

    H. Casini, M. Huerta, R.C. Myers and A. Yale, Mutual information and the F-theorem, JHEP 10 (2015) 003 [arXiv:1506.06195] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  49. [49]

    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  50. [50]

    R.C. Myers and A. Singh, Comments on Holographic Entanglement Entropy and RG Flows, JHEP 04 (2012) 122 [arXiv:1202.2068] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  51. [51]

    P. Calabrese and J.L. Cardy, Time-dependence of correlation functions following a quantum quench, Phys. Rev. Lett. 96 (2006) 136801 [cond-mat/0601225] [INSPIRE].

    ADS  Google Scholar 

  52. [52]

    P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench: a conformal field theory approach, J. Stat. Mech. 0710 (2007) P10004 [arXiv:0708.3750] [INSPIRE].

    MATH  Google Scholar 

  53. [53]

    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  54. [54]

    H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].

    ADS  Google Scholar 

  55. [55]

    H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems, Phys. Rev. D 89 (2014) 066012 [arXiv:1311.1200] [INSPIRE].

    ADS  Google Scholar 

  56. [56]

    R. Easther, R. Flauger and J.B. Gilmore, Delayed Reheating and the Breakdown of Coherent Oscillations, JCAP 04 (2011) 027 [arXiv:1003.3011] [INSPIRE].

    ADS  Google Scholar 

  57. [57]

    R. Bousso, Holography in general space-times, JHEP 06 (1999) 028 [hep-th/9906022] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  58. [58]

    C. Akers, N. Engelhardt, G. Penington and M. Usatyuk, Quantum Maximin Surfaces, JHEP 08 (2020) 140 [arXiv:1912.02799] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  59. [59]

    M. Mezei and J. Virrueta, The Quantum Null Energy Condition and Entanglement Entropy in Quenches, arXiv:1909.00919 [INSPIRE].

  60. [60]

    F.F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, Page Curve for an Evaporating Black Hole, JHEP 05 (2020) 091 [arXiv:2004.00598] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  61. [61]

    T. Anegawa and N. Iizuka, Notes on islands in asymptotically flat 2d dilaton black holes, JHEP 07 (2020) 036 [arXiv:2004.01601] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  62. [62]

    T. Hartman, E. Shaghoulian and A. Strominger, Islands in Asymptotically Flat 2D Gravity, JHEP 07 (2020) 022 [arXiv:2004.13857] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  63. [63]

    D.A. Lowe, Comments on a covariant entropy conjecture, JHEP 10 (1999) 026 [hep-th/9907062] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  64. [64]

    S. Cooper, M. Rozali, B. Swingle, M. Van Raamsdonk, C. Waddell and D. Wakeham, Black Hole Microstate Cosmology, JHEP 07 (2019) 065 [arXiv:1810.10601] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  65. [65]

    H.Z. Chen, R.C. Myers, D. Neuenfeld, I.A. Reyes and J. Sandor, Quantum Extremal Islands Made Easy. Part I. Entanglement on the Brane, JHEP 10 (2020) 166 [arXiv:2006.04851] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  66. [66]

    V. Balasubramanian, A. Kar, O. Parrikar, G. Sárosi and T. Ugajin, Geometric secret sharing in a model of Hawking radiation, arXiv:2003.05448 [INSPIRE].

  67. [67]

    H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, Evaporating Black Holes Coupled to a Thermal Bath, arXiv:2007.11658 [INSPIRE].

  68. [68]

    M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, JHEP 05 (2017) 065 [arXiv:1608.05101] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  69. [69]

    M. Mezei, On entanglement spreading from holography, JHEP 05 (2017) 064 [arXiv:1612.00082] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  70. [70]

    M. Mezei, Membrane theory of entanglement dynamics from holography, Phys. Rev. D 98 (2018) 106025 [arXiv:1803.10244] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  71. [71]

    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  72. [72]

    H. Casini, H. Liu and M. Mezei, Spread of entanglement and causality, JHEP 07 (2016) 077 [arXiv:1509.05044] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  73. [73]

    T. Hartman and N. Afkhami-Jeddi, Speed Limits for Entanglement, arXiv:1512.02695 [INSPIRE].

  74. [74]

    J. Maldacena and G.L. Pimentel, Entanglement entropy in de Sitter space, JHEP 02 (2013) 038 [arXiv:1210.7244] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  75. [75]

    S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  76. [76]

    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  77. [77]

    C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  78. [78]

    R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].

    ADS  Google Scholar 

  79. [79]

    D. Anninos and D.M. Hofman, Infrared Realization of dS2 in AdS2, Class. Quant. Grav. 35 (2018) 085003 [arXiv:1703.04622] [INSPIRE].

    ADS  MATH  Google Scholar 

  80. [80]

    D. Anninos, D.A. Galante and D.M. Hofman, de Sitter Horizons \& Holographic Liquids, JHEP 07 (2019) 038 [arXiv:1811.08153] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  81. [81]

    J. Maldacena, G.J. Turiaci and Z. Yang, Two dimensional Nearly de Sitter gravity, arXiv:1904.01911 [INSPIRE].

  82. [82]

    J. Cotler, K. Jensen and A. Maloney, Low-dimensional de Sitter quantum gravity, JHEP 06 (2020) 048 [arXiv:1905.03780] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  83. [83]

    L. Susskind, The Census taker’s hat, arXiv:0710.1129 [INSPIRE].

  84. [84]

    S.R. Coleman and F. De Luccia, Gravitational Effects on and of Vacuum Decay, Phys. Rev. D 21 (1980) 3305 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  85. [85]

    C.G. Callan Jr., S.B. Giddings, J.A. Harvey and A. Strominger, Evanescent black holes, Phys. Rev. D 45 (1992) 1005 [hep-th/9111056] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  86. [86]

    C. Akers, N. Engelhardt and D. Harlow, Simple holographic models of black hole evaporation, JHEP 08 (2020) 032 [arXiv:1910.00972] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  87. [87]

    Y. Zhao, A quantum circuit interpretation of evaporating black hole geometry, JHEP 07 (2020) 139 [arXiv:1912.00909] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  88. [88]

    P. Hayden and G. Penington, Learning the Alpha-bits of Black Holes, JHEP 12 (2019) 007 [arXiv:1807.06041] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  89. [89]

    A.R. Brown, H. Gharibyan, G. Penington and L. Susskind, The Python’s Lunch: geometric obstructions to decoding Hawking radiation, JHEP 08 (2020) 121 [arXiv:1912.00228] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  90. [90]

    H.F. Jia and M. Rangamani, Petz reconstruction in random tensor networks, JHEP 10 (2020) 050 [arXiv:2006.12601] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  91. [91]

    Y. Zhao, Petz map and Python’s lunch, JHEP 20 (2020) 038 [arXiv:2003.03406] [INSPIRE].

    Google Scholar 

  92. [92]

    H. Liu and S. Vardhan, A dynamical mechanism for the Page curve from quantum chaos, arXiv:2002.05734 [INSPIRE].

  93. [93]

    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].

    ADS  Google Scholar 

  94. [94]

    X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, JHEP 10 (2019) 240 [arXiv:1811.05382] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  95. [95]

    C. Akers and P. Rath, Holographic Renyi Entropy from Quantum Error Correction, JHEP 05 (2019) 052 [arXiv:1811.05171] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  96. [96]

    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  97. [97]

    T. Faulkner, S. Hollands, B. Swingle and Y. Wang, Approximate recovery and relative entropy I. general von Neumann subalgebras, arXiv:2006.08002 [INSPIRE].

  98. [98]

    S. Lloyd, The Computational universe: Quantum gravity from quantum computation, quant-ph/0501135 [INSPIRE].

  99. [99]

    N. Bao, C. Cao, S.M. Carroll and L. McAllister, Quantum Circuit Cosmology: The Expansion of the Universe Since the First Qubit, arXiv:1702.06959 [INSPIRE].

  100. [100]

    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].

    MATH  Google Scholar 

  101. [101]

    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  102. [102]

    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  103. [103]

    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  104. [104]

    T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].

  105. [105]

    C. Akers, J. Koeller, S. Leichenauer and A. Levine, Geometric Constraints from Subregion Duality Beyond the Classical Regime, arXiv:1610.08968 [INSPIRE].

  106. [106]

    T.M. Fiola, J. Preskill, A. Strominger and S.P. Trivedi, Black hole thermodynamics and information loss in two-dimensions, Phys. Rev. D 50 (1994) 3987 [hep-th/9403137] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yikun Jiang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2008.01022

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hartman, T., Jiang, Y. & Shaghoulian, E. Islands in cosmology. J. High Energ. Phys. 2020, 111 (2020). https://doi.org/10.1007/JHEP11(2020)111

Download citation

Keywords

  • AdS-CFT Correspondence
  • Black Holes