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  • Open Access

Horizon Edge Partition Functions in Λ>0 Quantum Gravity

Y. T. Albert Law*

Varun Lochab†

  • *Contact author: ytalaw@stanford.edu
  • †Contact author: vv2338@columbia.edu

Phys. Rev. Lett. 136, 151601 – Published 13 April, 2026

DOI: https://doi.org/10.1103/c683-tqw8

Abstract

We obtain the spectra of codimension-2 horizon “edge” degrees of freedom for gravity and higher-spin gauge fields in de Sitter space and in the static Nariai spacetime, advancing previous Lorentzian and Euclidean analyses of one-loop thermodynamics. The edge spectra exhibit universal shift symmetries, revealing a novel symmetry-breaking structure in one-loop partition functions with a positive cosmological constant. For the graviton, these modes admit a geometric interpretation as fluctuations of the cosmic horizon, which also persists in the Nariai case.

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Physics Subject Headings (PhySH)

See Also

de Sitter horizon edge partition functions

Y. T. Albert Law
Phys. Rev. D 113, 086005 (2026)

Gravitons on Nariai edges

Y. T. Albert Law and Varun Lochab
Phys. Rev. D 113, 086006 (2026)

Article Text

References (159)

  1. A. A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30, 682 (1979), http://jetpletters.ru/ps/1370/article_20738.pdf.
  2. A. H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347 (1981).
  3. A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108, 389 (1982).
  4. A. Albrecht and P. J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48, 1220 (1982).
  5. S. Perlmutter et al. (Supernova Cosmology Project Collaboration), Measurements of the cosmological parameters Ω and Λ from the first 7 supernovae at z≥0.35, Astrophys. J. 483, 565 (1997).
  6. A. G. Riess et al. (Supernova Search Team Collaboration), Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009 (1998).
  7. M. Kowalski et al. (Supernova Cosmology Project Collaboration), Improved cosmological constraints from new, old and combined supernova data sets, Astrophys. J. 686, 749 (2008).
  8. A. Loeb, The long-term future of extragalactic astronomy, Phys. Rev. D 65, 047301 (2002).
  9. L. M. Krauss and R. J. Scherrer, The return of a static universe and the end of cosmology, Gen. Relativ. Gravit. 39, 1545 (2007).
  10. G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).
  11. T. Banks, Cosmological breaking of supersymmetry?, Int. J. Mod. Phys. A 16, 910 (2001).
  12. T. Banks, Some Thoughts on the Quantum Theory of De Sitter Space, arXiv:astro-ph/0305037.
  13. E. Witten, Quantum gravity in de Sitter space, arXiv:hep-th/0106109.
  14. D. Anninos, T. Bautista, and B. Mühlmann, The two-sphere partition function in two-dimensional quantum gravity, J. High Energy Phys. 09 (2021) 116.
  15. E. Coleman, E. A. Mazenc, V. Shyam, E. Silverstein, R. M. Soni, G. Torroba, and S. Yang, De Sitter microstates from TT¯+Λ2 and the Hawking-Page transition, J. High Energy Phys. 07 (2022) 140.
  16. D. Anninos and B. Mühlmann, The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe), J. High Energy Phys. 12 (2021) 206.
  17. D. Anninos, D. A. Galante, and B. Mühlmann, Finite features of quantum de Sitter space, Classical Quantum Gravity 40, 025009 (2023).
  18. N. Bobev, T. Hertog, J. Hong, J. Karlsson, and V. Reys, Microscopics of de Sitter entropy from precision holography, Phys. Rev. X 13, 041056 (2023).
  19. S. Collier, L. Eberhardt, and B. Mühlmann, A microscopic realization of dS3, SciPost Phys. 18, 131 (2025).
  20. R. Figari, R. Hoegh-Krohn, and C. R. Nappi, Interacting relativistic boson fields in the de Sitter universe with two space-time dimensions, Commun. Math. Phys. 44, 265 (1975).
  21. G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15, 2738 (1977).
  22. 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.
  23. Our convention for Zedge differs from [22] by inversion. We adopt the present notation because recent Lorentzian analyses naturally interpret Zedge itself as a partition function, at least for p-form gauge theories [24, 25].

  24. A. Ball, Y. T. A. Law, and G. Wong, Dynamical edge modes and entanglement in Maxwell theory, J. High Energy Phys. 09 (2024) 032.
  25. A. Ball and Y. T. A. Law, Dynamical edge modes in p-form gauge theories, J. High Energy Phys. 02 (2025) 182.
  26. Z. Sun, Higher spin de Sitter quasinormal modes, J. High Energy Phys. 11 (2021) 025.
  27. M. Grewal and Y. T. A. Law, Real-time observables in de Sitter thermodynamics, J. High Energy Phys. 10 (2025) 052.
  28. G. W. Gibbons and S. W. Hawking, Classification of gravitational instanton symmetries, Commun. Math. Phys. 66, 291 (1979).
  29. A. Rios Fukelman, M. Sempé, and G. A. Silva, Notes on gauge fields and discrete series representations in de Sitter spacetimes, J. High Energy Phys. 01 (2024) 011.
  30. Y. T. A. Law, companion paper, de Sitter horizon edge partition functions, Phys. Rev. D 113, 086005 (2026).
  31. J. Mukherjee, Entanglement entropy and the boundary action of edge modes, J. High Energy Phys. 06 (2024) 113.
  32. D. N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B453, 281 (1995).
  33. J. S. Dowker, Entanglement entropy for even spheres, arXiv:1009.3854.
  34. C. Eling, Y. Oz, and S. Theisen, Entanglement and thermal entropy of gauge fields, J. High Energy Phys. 11 (2013) 019.
  35. S. N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665, 305 (2008).
  36. H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of holographic entanglement entropy, J. High Energy Phys. 05 (2011) 036.
  37. W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85, 085004 (2012).
  38. W. Donnelly and A. C. Wall, Do gauge fields really contribute negatively to black hole entropy?, Phys. Rev. D 86, 064042 (2012).
  39. D. Radicevic, Notes on entanglement in Abelian gauge theories, arXiv:1404.1391.
  40. W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Classical Quantum Gravity 31, 214003 (2014).
  41. W. Donnelly and A. C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114, 111603 (2015).
  42. K.-W. Huang, Central charge and entangled gauge fields, Phys. Rev. D 92, 025010 (2015).
  43. S. Ghosh, R. M. Soni, and S. P. Trivedi, On the entanglement entropy for gauge theories, J. High Energy Phys. 09 (2015) 069.
  44. L.-Y. Hung and Y. Wan, Revisiting entanglement entropy of lattice gauge theories, J. High Energy Phys. 04 (2015) 122.
  45. S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba, and H. Tasaki, On the definition of entanglement entropy in lattice gauge theories, J. High Energy Phys. 06 (2015) 187.
  46. W. Donnelly and A. C. Wall, Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D 94, 104053 (2016).
  47. D. Radičević, Entanglement in weakly coupled lattice gauge theories, J. High Energy Phys. 04 (2016) 163.
  48. M. Pretko and T. Senthil, Entanglement entropy of U(1) quantum spin liquids, Phys. Rev. B 94, 125112 (2016).
  49. R. M. Soni and S. P. Trivedi, Aspects of entanglement entropy for gauge theories, J. High Energy Phys. 01 (2016) 136.
  50. F. Zuo, A note on electromagnetic edge modes, arXiv:1601.06910.
  51. R. M. Soni and S. P. Trivedi, Entanglement entropy in (3+1)−d free U(1) gauge theory, J. High Energy Phys. 02 (2017) 101.
  52. C. Delcamp, B. Dittrich, and A. Riello, On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity, J. High Energy Phys. 11 (2016) 102.
  53. A. Agarwal, D. Karabali, and V. P. Nair, Gauge-invariant variables and entanglement entropy, Phys. Rev. D 96, 125008 (2017).
  54. A. Blommaert, T. G. Mertens, H. Verschelde, and V. I. Zakharov, Edge state quantization: Vector fields in Rindler, J. High Energy Phys. 08 (2018) 196.
  55. A. Blommaert, T. G. Mertens, and H. Verschelde, Edge dynamics from the path integral—Maxwell and Yang-Mills, J. High Energy Phys. 11 (2018) 080.
  56. L. Freidel and D. Pranzetti, Electromagnetic duality and central charge, Phys. Rev. D 98, 116008 (2018).
  57. W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, J. High Energy Phys. 09 (2016) 102.
  58. M. Geiller, Edge modes and corner ambiguities in 3d Chern–Simons theory and gravity, Nucl. Phys. B924, 312 (2017).
  59. A. J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, J. High Energy Phys. 02 (2018) 021.
  60. M. Geiller, Lorentz-diffeomorphism edge modes in 3d gravity, J. High Energy Phys. 02 (2018) 029.
  61. L. Freidel, E. R. Livine, and D. Pranzetti, Gravitational edge modes: from Kac–Moody charges to Poincaré networks, Classical Quantum Gravity 36, 195014 (2019).
  62. V. Benedetti and H. Casini, Entanglement entropy of linearized gravitons in a sphere, Phys. Rev. D 101, 045004 (2020).
  63. T. Takayanagi and K. Tamaoka, Gravity edges modes and hayward term, J. High Energy Phys. 02 (2020) 167.
  64. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, J. High Energy Phys. 11 (2020) 026.
  65. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part II. Corner metric and Lorentz charges, J. High Energy Phys. 11 (2020) 027.
  66. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part III. Corner simplicity constraints, J. High Energy Phys. 01 (2021) 100.
  67. W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, Gravitational edge modes, coadjoint orbits, and hydrodynamics, J. High Energy Phys. 09 (2021) 008.
  68. L. Ciambelli and R. G. Leigh, Isolated surfaces and symmetries of gravity, Phys. Rev. D 104, 046005 (2021).
  69. S. Carrozza and P. A. Hoehn, Edge modes as reference frames and boundary actions from post-selection, J. High Energy Phys. 02 (2022) 172.
  70. L. Ciambelli, R. G. Leigh, and P.-C. Pai, Embeddings and integrable charges for extended corner symmetry, Phys. Rev. Lett. 128, 171302 (2022).
  71. J. R. David and J. Mukherjee, Entanglement entropy of gravitational edge modes, J. High Energy Phys. 08 (2022) 065.
  72. S. Carrozza, S. Eccles, and P. A. Hoehn, Edge modes as dynamical frames: Charges from post-selection in generally covariant theories, SciPost Phys. 17, 048 (2024).
  73. L. Ciambelli and R. G. Leigh, Universal corner symmetry and the orbit method for gravity, Nucl. Phys. B986, 116053 (2023).
  74. T. G. Mertens, J. Simón, and G. Wong, A proposal for 3d quantum gravity and its bulk factorization, J. High Energy Phys. 06 (2023) 134.
  75. G. Wong, A note on the bulk interpretation of the quantum extremal surface formula, J. High Energy Phys. 04, (2024) 024.
  76. W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, Matrix quantization of gravitational edge modes, J. High Energy Phys. 05 (2027) 163.
  77. L. Ciambelli, From asymptotic symmetries to the corner proposal, Proc. Sci. Modave2022 (2023) 002 [arXiv:2212.13644].
  78. V. Balasubramanian and C. Cummings, The entropy of finite gravitating regions, arXiv:2312.08434.
  79. K.-S. Lee, A. Sivakumar, and J. Yoon, Gravitational edge mode in N=1 Jackiw-Teitelboim supergravity, J. High Energy Phys. 08 (2024) 011.
  80. A. Blommaert and S. Colin-Ellerin, Gravitons on the edge, J. High Energy Phys. 03 (2025) 116.
  81. J. R. Fliss, A. Frenkel, S. A. Hartnoll, and R. M. Soni, Minimal areas from entangled matrices, SciPost Phys. 18, 171 (2025).
  82. C. Tsukamoto, Spectra of Laplace-Beltrami operators on SO(n+2)/SO(2)×SO(n) and Sp(n+1)/Sp(1)×Sp(n), Osaka J. Mathemat. 18, 407 (1981).
  83. T. E. Clark, S. T. Love, M. Nitta, and T. ter Veldhuis, AdSd+1→AdSd, J. Math. Phys. (N.Y.) 46, 102304 (2005).
  84. T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Oscillating p-branes, Phys. Rev. D 76, 105014 (2007).
  85. G. Goon, K. Hinterbichler, and M. Trodden, Symmetries for Galileons and DBI scalars on curved space, J. Cosmol. Astropart. Phys. 07 (2011) 017.
  86. G. Goon, K. Hinterbichler, and M. Trodden, A new class of effective field theories from embedded branes, Phys. Rev. Lett. 106, 231102 (2011).
  87. C. Burrage, C. de Rham, and L. Heisenberg, de Sitter Galileon, J. Cosmol. Astropart. Phys. 05 (2011) 025.
  88. D. Anninos, T. Hartman, and A. Strominger, Higher spin realization of the dS/CFT correspondence, Classical Quantum Gravity 34, 015009 (2017).
  89. D. Anninos, F. Denef, and D. Harlow, Wave function of Vasiliev’s universe: A few slices thereof, Phys. Rev. D 88, 084049 (2013).
  90. D. Anninos, F. Denef, G. Konstantinidis, and E. Shaghoulian, Higher spin de Sitter holography from functional determinants, J. High Energy Phys. 02 (2014) 007.
  91. D. Anninos, F. Denef, R. Monten, and Z. Sun, Higher spin de Sitter Hilbert space, J. High Energy Phys. 10 (2019) 071.
  92. C. Fronsdal, Massless fields with integer spin, Phys. Rev. D 18, 3624 (1978).
  93. Y. T. A. Law, A compendium of sphere path integrals, J. High Energy Phys. 12 (2021) 213.
  94. S. Deser and R. I. Nepomechie, Anomalous propagation of gauge fields in conformally flat spaces, Phys. Lett. 132B, 321 (1983).
  95. S. Deser and R. I. Nepomechie, Gauge invariance versus masslessness in de Sitter spaces, Ann. Phys. (N.Y.) 154, 396 (1984).
  96. A. Higuchi, Forbidden mass range for spin-2 field theory in de Sitter space-time, Nucl. Phys. B282, 397 (1987).
  97. L. Brink, R. R. Metsaev, and M. A. Vasiliev, How massless are massless fields in AdS(d), Nucl. Phys. B586, 183 (2000).
  98. S. Deser and A. Waldron, Gauge invariances and phases of massive higher spins in (A)dS, Phys. Rev. Lett. 87, 031601 (2001).
  99. S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B607, 577 (2001).
  100. S. Deser and A. Waldron, Stability of massive cosmological gravitons, Phys. Lett. B 508, 347 (2001).
  101. S. Deser and A. Waldron, Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations, Phys. Lett. B 513, 137 (2001).
  102. Y. M. Zinoviev, On massive high spin particles in AdS, arXiv:hep-th/0108192.
  103. K. Hinterbichler and A. Joyce, Manifest duality for partially massless higher spins, J. High Energy Phys. 09 (2016) 141.
  104. H. Nariai, On some static solutions of Einstein’s gravitational field equations in a spherically symmetric case, Sci. Rep. Tohoku Univ. Eighth Ser. 34, 160 (1950).
  105. Y. T. A. Law and V. Lochab, companion paper, Gravitons on Nariai edges, Phys. Rev. D 113, 086006 (2026).
  106. J. Mukherjee, Quasinormal bulk-edge characters of gravitons in Nariai geometry, arXiv:2506.07556.
  107. J. Bonifacio, K. Hinterbichler, A. Joyce, and R. A. Rosen, Shift symmetries in (Anti) de Sitter space, J. High Energy Phys. 02 (2019) 178.
  108. J. Bonifacio, K. Hinterbichler, L. A. Johnson, and A. Joyce, Shift-symmetric spin-1 theories, J. High Energy Phys. 09 (2019) 029.
  109. K. Jensen, J. Sorce, and A. J. Speranza, Generalized entropy for general subregions in quantum gravity, J. High Energy Phys. 12 (2023) 020.
  110. J. De Vuyst, S. Eccles, P. A. Hoehn, and J. Kirklin, Gravitational entropy is observer-dependent, J. High Energy Phys. 07 (2025) 146.
  111. S. Ali Ahmad, W. Chemissany, M. S. Klinger, and R. G. Leigh, Quantum reference frames from top-down crossed products, Phys. Rev. D 110, 065003 (2024).
  112. S. Ali Ahmad, W. Chemissany, M. S. Klinger, and R. G. Leigh, Relational quantum geometry, Nucl. Phys. B1015, 116911 (2025).
  113. J. Kirklin, Generalised second law beyond the semiclassical regime, J. High Energy Phys. 07 (2025) 192.
  114. J. De Vuyst, S. Eccles, P. A. Hoehn, and J. Kirklin, Crossed products and quantum reference frames: On the observer-dependence of gravitational entropy, J. High Energy Phys. 07 (2025) 063.
  115. C. J. Fewster, D. W. Janssen, L. D. Loveridge, K. Rejzner, and J. Waldron, Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory, Commun. Math. Phys. 406, 19 (2025).
  116. C. J. Fewster, D. W. Janssen, and K. Rejzner, Semi-local observables, edge modes and quantum reference frames in quantum electromagnetism: An algebraic approach, arXiv:2508.20939.
  117. D. Anninos, S. A. Hartnoll, and D. M. Hofman, Static patch solipsism: Conformal symmetry of the de Sitter worldline, Classical Quantum Gravity 29, 075002 (2012).
  118. C. Goeller, P. A. Hoehn, and J. Kirklin, Diffeomorphism-invariant observables and dynamical frames in gravity: Reconciling bulk locality with general covariance, arXiv:2206.01193.
  119. V. Chandrasekaran, R. Longo, G. Penington, and E. Witten, An algebra of observables for de Sitter space, J. High Energy Phys. 02 (2023) 082.
  120. C.-H. Chen and G. Penington, A clock is just a way to tell the time: Gravitational algebras in cosmological spacetimes, arXiv:2406.02116.
  121. J. Maldacena, Real observers solving imaginary problems, arXiv:2412.14014.
  122. G. W. Gibbons, S. W. Hawking, and M. J. Perry, Path integrals and the indefiniteness of the gravitational action, Nucl. Phys. B138, 141 (1978).
  123. J. Polchinski, The Phase of the Sum Over Spheres, Phys. Lett. B 219, 251 (1989).
  124. X. Shi and G. J. Turiaci, The phase of the gravitational path integral, J. High Energy Phys. 07 (2025) 047.
  125. V. Ivo, J. Maldacena, and Z. Sun, Physical instabilities and the phase of the Euclidean path integral, arXiv:2504.00920.
  126. B. Banihashemi and T. Jacobson, The enigmatic gravitational partition function, Classical Quantum Gravity 57, 43 (2025).
  127. G. T. Horowitz, D. Marolf, and J. E. Santos, Constraints are not enough, J. High Energy Phys. 10 (2025) 031.
  128. D. Marolf, Gravitational thermodynamics without the conformal factor problem: Partition functions and Euclidean saddles from Lorentzian path integrals, J. High Energy Phys. 07 (2022) 108.
  129. B. Banihashemi and T. Jacobson, Thermodynamic ensembles with cosmological horizons, J. High Energy Phys. 07 (2022) 042.
  130. D. Anninos, D. A. Galante, and C. Maneerat, Cosmological observatories, Classical Quantum Gravity 41, 165009 (2024).
  131. E. Silverstein and G. Torroba, Timelike-bounded dS4 holography from a solvable sector of the T2 deformation, J. High Energy Phys. 03 (2025) 156.
  132. D. Anninos, C. Baracco, S. Brian, and F. Denef, Features of the partition function of a Λ>0 universe, J. High Energy Phys. 01 (2026) 141.
  133. D. N. Page, A compact rotating gravitational instanton, Phys. Lett. 79B, 235 (1978).
  134. G. W. Gibbons, H. Lu, D. N. Page, and C. N. Pope, The general Kerr-de Sitter metrics in all dimensions, J. Geom. Phys. 53, 49 (2005).
  135. A. Castro, N. Lashkari, and A. Maloney, A de Sitter Farey Tail, Phys. Rev. D 83, 124027 (2011).
  136. Y. T. A. Law, Sphere Partition Functions and Quantum De Sitter Thermodynamics, Ph.D. thesis, Columbia University, 2021.
  137. F. Denef, S. A. Hartnoll, and S. Sachdev, Black hole determinants and quasinormal modes, Classical Quantum Gravity 27, 125001 (2010).
  138. Y. T. A. Law and K. Parmentier, Black hole scattering and partition functions, J. High Energy Phys. 10 (2022) 039.
  139. A. Castro, C. Keeler, and P. Szepietowski, Tweaking one-loop determinants in AdS3, J. High Energy Phys. 10 (2017) 070.
  140. C. Keeler, V. L. Martin, and A. Svesko, Connecting quasinormal modes and heat kernels in 1-loop determinants, SciPost Phys. 8, 017 (2020).
  141. C. Keeler, V. L. Martin, and A. Svesko, BTZ one-loop determinants via the Selberg zeta function for general spin, J. High Energy Phys. 10 (2020) 138.
  142. M. Grewal, Y. T. A. Law, and K. Parmentier, Black hole horizon edge partition functions, J. High Energy Phys. 06 (2023) 025.
  143. D. Kapec, Y. T. A. Law, and C. Toldo, Quasinormal corrections to near-extremal black hole thermodynamics, J. High Energy Phys. 06 (2025) 069.
  144. G. W. Gibbons and M. J. Perry, Quantizing gravitational instantons, Nucl. Phys. B146, 90 (1978).
  145. S. M. Christensen and M. J. Duff, Quantizing gravity with a cosmological constant, Nucl. Phys. B170, 480 (1980).
  146. E. S. Fradkin and A. A. Tseytlin, One loop effective potential in gauged O(4) supergravity, Nucl. Phys. B234, 472 (1984).
  147. B. Allen, Phase transitions in de Sitter space, Nucl. Phys. B226, 228 (1983).
  148. T. R. Taylor and G. Veneziano, Quantum gravity at large distances and the cosmological constant, Nucl. Phys. B345, 210 (1990).
  149. P. A. Griffin and D. A. Kosower, Curved spacetime one-loop gravity in a physical gauge, Phys. Lett. B 233, 295 (1989).
  150. P. O. Mazur and E. Mottola, Absence of phase in the sum over spheres, Technical Report LA-UR-89-2118, 1989.
  151. D. V. Vassilevich, One loop quantum gravity on de Sitter space, Int. J. Mod. Phys. A 08, 1637 (1993).
  152. M. S. Volkov and A. Wipf, Black hole pair creation in de Sitter space: A Complete one loop analysis, Nucl. Phys. B582, 313 (2000).
  153. A. Lopez-Ortega, Quasinormal modes of D-dimensional de Sitter spacetime, Gen. Relativ. Gravit. 38, 1565 (2006).
  154. M. A. Rubin and C. R. Ordónez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, J. Math. Phys. (N.Y.) 25, 2888 (1984).
  155. A. Higuchi, Symmetric tensor spherical harmonics on the N sphere and their application to the De Sitter Group SO(N,1), J. Math. Phys. (N.Y.) 28, 1553 (1987); 43, 6385(E) (2002).
  156. L. Vanzo and S. Zerbini, Asymptotics of quasinormal modes for multihorizon black holes, Phys. Rev. D 70, 044030 (2004).
  157. A. Lopez-Ortega, Electromagnetic quasinormal modes of D-dimensional black holes. II, Gen. Relativ. Gravit. 40, 1379 (2008).
  158. A. Lopez-Ortega, The Dirac equation in D-dimensional spherically symmetric spacetimes, arXiv:0906.2754.
  159. J. Venâncio and C. Batista, Spin-2 quasinormal modes in generalized Nariai spacetimes, Phys. Rev. D 101, 084037 (2020).

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