- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Eternal universes

Damien A. Easson*

Joseph E. Lesnefsky†

  • Department of Physics and Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe, Arizona 85287-1504, USA

  • *Contact author: easson@asu.edu
  • †Contact author: joseph.lesnefsky@pm.me
PDF

Phys. Rev. D 112, 063545 – Published 22 September, 2025

DOI: https://doi.org/10.1103/5mhz-m8bg

Abstract

We consider the possibility of a past and future eternal universe, constructing geodesically complete inflating, loitering, and bouncing spacetimes. We identify the constraints energy conditions that general relativity place on the building of eternal cosmological models. Inflationary and bouncing behavior are shown to be essential ingredients in all significant examples. Nontrivial complete spacetimes are shown to violate the null energy condition for at least some amount of time, as well as the average null energy condition, unless curvature is included. Ignoring the intractable subtleties introduced by quantum considerations, such as rare tunneling events and Boltzmann brains, we demonstrate that these universes need not have had a beginning or an end.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (123)

  1. A. H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347 (1981).
  2. A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. 108B, 389 (1982).
  3. A. Albrecht and P. J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48, 1220 (1982).
  4. Alexander Vilenkin, The birth of inflationary universes, Phys. Rev. D 27, 2848 (1983).
  5. Andrei D. Linde, Eternally existing selfreproducing chaotic inflationary Universe, Phys. Lett. B 175, 395 (1986).
  6. Alan H. Guth, Eternal inflation and its implications, J. Phys. A 40, 6811 (2007).
  7. Arvind Borde, Alan H. Guth, and Alexander Vilenkin, Inflationary space-times are incompletein past directions, Phys. Rev. Lett. 90, 151301 (2003).
  8. Damien A. Easson and Joseph E. Lesnefsky, Inflationary resolution of the initial singularity, arXiv:2402.13031.
  9. J. E. Lesnefsky, D. A. Easson, and P. C. W. Davies, Past-completeness of inflationary spacetimes, Phys. Rev. D 107, 044024 (2023).
  10. Carlos Barcelo and Matt Visser, Twilight for the energy conditions?, Int. J. Mod. Phys. D 11, 1553 (2002).
  11. Saugata Chatterjee, Damien A. Easson, and Maulik Parikh, Energy conditions in the Jordan frame, Classical Quantum Gravity 30, 235031 (2013).
  12. H. Epstein, V. Glaser, and A. Jaffe, Nonpositivity of energy density in quantized field theories, Nuovo Cimento 36, 1016 (1965).
  13. H. B. G. Casimir, On the attraction between two perfectly conducting plates, Indagat. Math 10, 261 (1948).
  14. Roger Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14, 57 (1965).
  15. Stephen Hawking, The occurrence of singularities in cosmology, Proc. R. Soc. A 294, 511 (1966).
  16. Stephen Hawking, The occurrence of singularities in cosmology. II, Proc. R. Soc. A 295, 490 (1966).
  17. Stephen Hawking, The occurrence of singularities in cosmology. III. Causality and singularities, Proc. R. Soc. A 300, 187 (1967).
  18. S. W. Hawking and R. Penrose, The singularities of gravitational collapse and cosmology, Proc. R. Soc. A 314, 529 (1970).
  19. Stephen W. Hawking and George F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2023).
  20. S. Dubovsky, T. Gregoire, A. Nicolis, and R. Rattazzi, Null energy condition and superluminal propagation, J. High Energy Phys. 03 (2006) 025.
  21. Alberto Nicolis, Riccardo Rattazzi, and Enrico Trincherini, Energy’s and amplitudes’ positivity, J. High Energy Phys. 05 (2010) 095; 11 (2011) 128(E).
  22. Tsutomu Kobayashi, Masahide Yamaguchi, and Jun’ichi Yokoyama, G-inflation: Inflation driven by the Galileon field, Phys. Rev. Lett. 105, 231302 (2010).
  23. Cedric Deffayet, Oriol Pujolas, Ignacy Sawicki, and Alexander Vikman, Imperfect dark energy from kinetic gravity braiding, J. Cosmol. Astropart. Phys. 10 (2010) 026.
  24. Ignacy Sawicki and Alexander Vikman, Hidden negative energies in strongly accelerated universes, Phys. Rev. D 87, 067301 (2013).
  25. V. A. Rubakov, The null energy condition and its violation, Phys. Usp. 57, 128 (2014).
  26. Yong Cai, Youping Wan, Hai-Guang Li, Taotao Qiu, and Yun-Song Piao, The effective field theory of nonsingular cosmology, J. High Energy Phys. 01 (2017) 090.
  27. Paolo Creminelli, David Pirtskhalava, Luca Santoni, and Enrico Trincherini, Stability of geodesically complete cosmologies, J. Cosmol. Astropart. Phys. 11 (2016) 047.
  28. Yong Cai, Hai-Guang Li, Taotao Qiu, and Yun-Song Piao, The effective field theory of nonsingular cosmology: II, Eur. Phys. J. C 77, 369 (2017).
  29. Damien A. Easson and Tucker Manton, Stable cosmic time crystals, Phys. Rev. D 99, 043507 (2019).
  30. Jean Alexandre and Janos Polonyi, Tunnelling and dynamical violation of the null energy condition, Phys. Rev. D 103, 105020 (2021).
  31. Jean Alexandre and Drew Backhouse, Null energy condition violation: Tunneling versus the Casimir effect, Phys. Rev. D 107, 085022 (2023).
  32. Damien A. Easson, Ignacy Sawicki, and Alexander Vikman, G-bounce, J. Cosmol. Astropart. Phys. 11 (2011) 021.
  33. Yi-Fu Cai, Damien A. Easson, and Robert Brandenberger, Towards a nonsingular bouncing cosmology, J. Cosmol. Astropart. Phys. 08 (2012) 020.
  34. Damien A. Easson and Alexander Vikman, The phantom of the new oscillatory cosmological phase, arXiv:1607.00996.
  35. Anna Ijjas and Paul J. Steinhardt, Classically stable nonsingular cosmological bounces, Phys. Rev. Lett. 117, 121304 (2016).
  36. Anna Ijjas and Paul J. Steinhardt, Fully stable cosmological solutions with a non-singular classical bounce, Phys. Lett. B 764, 289 (2017).
  37. Yong Cai and Yun-Song Piao, A covariant Lagrangian for stable nonsingular bounce, J. High Energy Phys. 09 (2017) 027.
  38. Jean Alexandre and Silvia Pla, Cosmic bounce and phantom-like equation of state from tunnelling, J. High Energy Phys. 05 (2023) 145.
  39. Robert M. Wald and U. Yurtsever, General proof of the averaged null energy condition for a massless scalar field in two-dimensional curved space-time, Phys. Rev. D 44, 403 (1991).
  40. Noah Graham and Ken D. Olum, Achronal averaged null energy condition, Phys. Rev. D 76, 064001 (2007).
  41. Thomas Hartman, Sandipan Kundu, and Amirhossein Tajdini, Averaged null energy condition from causality, J. High Energy Phys. 07 (2017) 066.
  42. M. S. Morris, K. S. Thorne, and U. Yurtsever, Wormholes, time machines, and the weak energy condition, Phys. Rev. Lett. 61, 1446 (1988).
  43. John L. Friedman, Kristin Schleich, and Donald M. Witt, Topological censorship, Phys. Rev. Lett. 71, 1486 (1993); 75, 1872(E) (1995).
  44. Aron C. Wall, Proving the achronal averaged null energy condition from the generalized second law, Phys. Rev. D 81, 024038 (2010).
  45. Noah Graham and Ken D. Olum, Plate with a hole obeys the averaged null energy condition, Phys. Rev. D 72, 025013 (2005).
  46. Christopher J. Fewster, Ken D. Olum, and Michael J. Pfenning, Averaged null energy condition in spacetimes with boundaries, Phys. Rev. D 75, 025007 (2007).
  47. Matt Visser, Gravitational vacuum polarization. 2: Energy conditions in the Boulware vacuum, Phys. Rev. D 54, 5116 (1996).
  48. B. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Am. Math. Soc. 145, 1 (1969).
  49. Barrett O’Neill, Semi-Riemannian Geometry with Applications to Relativity, edited by Samuel Eilenberg and Hyman Bass (Academic Press, New York, NY, 1983), p. 468.
  50. Miguel Sanchez, On the geometry of generalized Robertson Walker spacetimes: Geodesics, Gen. Relativ. Gravit. 30, 915 (1998).
  51. William H. Kinney, Suvashis Maity, and L. Sriramkumar, The Borde-Guth-Vilenkin theorem in extended de Sitter spaces, Phys. Rev. D 109, 043519 (2024).
  52. Ghazal Geshnizjani, Eric Ling, and Jerome Quintin, On the initial singularity and extendibility of flat quasi-de Sitter spacetimes, J. High Energy Phys. 10 (2023) 182.
  53. Robert R. Caldwell, Marc Kamionkowski, and Nevin N. Weinberg, Phantom energy and cosmic doomsday, Phys. Rev. Lett. 91, 071301 (2003).
  54. George F. R. Ellis and Roy Maartens, The emergent universe: Inflationary cosmology with no singularity, Classical Quantum Gravity 21, 223 (2004).
  55. Tirthabir Biswas, Anupam Mazumdar, and Warren Siegel, Bouncing universes in string-inspired gravity, J. Cosmol. Astropart. Phys. 03 (2006) 009.
  56. Tirthabir Biswas, Alexey S. Koshelev, Anupam Mazumdar, and Sergey Yu. Vernov, Stable bounce and inflation in non-local higher derivative cosmology, J. Cosmol. Astropart. Phys. 08 (2012) 024.
  57. A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. 91B, 99 (1980).
  58. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd ed. (Wiley and Sons, New York, 2004), p. 932.
  59. Massimo Giovannini, Averaged energy conditions and bouncing universes, Phys. Rev. D 96, 101302 (2017).
  60. Robert H. Brandenberger, Costas Kounnas, Hervé Partouche, Subodh P. Patil, and N. Toumbas, Cosmological perturbations across an S-brane, J. Cosmol. Astropart. Phys. 03 (2014) 015.
  61. M. A. Markov, Possible state of matter just before the collapse stage, Pisma Zh. Eksp. Teor. Fiz 36, 214 (1982).
  62. V. F. Mukhanov and R. H. Brandenberger, A nonsingular universe, Phys. Rev. Lett. 68, 1969 (1992).
  63. R. H. Brandenberger, V. F. Mukhanov, and A. Sornborger, A cosmological theory without singularities, Phys. Rev. D 48, 1629 (1993).
  64. R. Moessner and M. Trodden, Singularity—free two-dimensional cosmologies, Phys. Rev. D 51, 2801 (1995).
  65. R. H. Brandenberger, Nonsingular cosmology and Planck scale physics, arXiv:gr-qc/9503001.
  66. R. H. Brandenberger, R. Easther, and J. Maia, Nonsingular dilaton cosmology, J. High Energy Phys. 08 (1998) 007.
  67. D. A. Easson and R. H. Brandenberger, Nonsingular dilaton cosmology in the string frame, J. High Energy Phys. 09 (1999) 003.
  68. D. A. Easson, Towards a stringy resolution of the cosmological singularity, Phys. Rev. D 68, 043514 (2003).
  69. Damien A. Easson, The accelerating universe and a limiting curvature proposal, J. Cosmol. Astropart. Phys. 02 (2007) 004.
  70. Ali H. Chamseddine and Viatcheslav Mukhanov, Resolving cosmological singularities, J. Cosmol. Astropart. Phys. 03 (2017) 009.
  71. V. P. Frolov, M. A. Markov, and V. F. Mukhanov, Through a black hole into a new universe?, Phys. Lett. B 216, 272 (1989).
  72. V. P. Frolov, M. A. Markov, and V. F. Mukhanov, Black holes as possible sources of closed and semiclosed worlds, Phys. Rev. D 41, 383 (1990).
  73. D. Morgan, Black holes in cutoff gravity, Phys. Rev. D 43, 3144 (1991).
  74. M. Trodden, V. F. Mukhanov, and R. H. Brandenberger, A nonsingular two-dimensional black hole, Phys. Lett. B 316, 483 (1993).
  75. D. A. Easson, Hawking radiation of nonsingular black holes in two-dimensions, J. High Energy Phys. 02 (2003) 037.
  76. Damien A. Easson, Nonsingular Schwarzschild–de Sitter black hole, Classical Quantum Gravity 35, 235005 (2018).
  77. A. H. Chamseddine and V. Mukhanov, Nonsingular black hole, Eur. Phys. J. C 77, 183 (2017).
  78. Robert Brandenberger, Lavinia Heisenberg, and Jakob Robnik, Through a black hole into a new universe, Int. J. Mod. Phys. D 30, 2142001 (2021).
  79. Valeri P. Frolov and Andrei Zelnikov, Spherically symmetric black holes in the limiting curvature theory of gravity, Phys. Rev. D 105, 024041 (2022).
  80. Varun Sahni, Hume Feldman, and Albert Stebbins, Loitering universe, Astrophys. J. 385, 1 (1992).
  81. S. Alexander, Robert H. Brandenberger, and D. A. Easson, Brane gases in the early universe, Phys. Rev. D 62, 103509 (2000).
  82. Robert Brandenberger, Damien A. Easson, and Dagny Kimberly, Loitering phase in brane gas cosmology, Nucl. Phys. B623, 421 (2002).
  83. Brandon Melcher, Arnab Pradhan, and Scott Watson, Waiting for inflation: A new initial state for the universe, Phys. Rev. D 110, 063517 (2024).
  84. Roger Penrose, On the gravitization of quantum mechanics 2: Conformal cyclic cosmology, Found. Phys. 44, 873 (2014).
  85. Paul J. Steinhardt and Neil Turok, Cosmic evolution in a cyclic universe, Phys. Rev. D 65, 126003 (2002).
  86. Paul J. Steinhardt, Neil Turok, and N. Turok, A cyclic model of the universe, Science 296, 1436 (2002).
  87. Justin Khoury, Paul J. Steinhardt, and Neil Turok, Designing cyclic universe models, Phys. Rev. Lett. 92, 031302 (2004).
  88. Anna Ijjas and Paul J. Steinhardt, A new kind of cyclic universe, Phys. Lett. B 795, 666 (2019).
  89. Jean-Luc Lehners and Paul J. Steinhardt, Planck 2013 results support the cyclic universe, Phys. Rev. D 87, 123533 (2013).
  90. D. Easson and J. Lesnefsky (to be published).
  91. Paolo Creminelli, Alberto Nicolis, and Enrico Trincherini, Galilean genesis: An alternative to inflation, J. Cosmol. Astropart. Phys. 11 (2010) 021.
  92. A. A. Starobinskii, On a nonsingular isotropic cosmological model, Sov. Astron. Lett. 4, 82 (1978).
  93. Peter W. Graham, Bart Horn, Shamit Kachru, Surjeet Rajendran, and Gonzalo Torroba, A simple harmonic universe, J. High Energy Phys. 02 (2014) 029.
  94. Nathan Burwig and Damien A. Easson (to be published).
  95. John C. Mather et al., A preliminary measurement of the cosmic microwave background spectrum by the cosmic background explorer (COBE) satellite, Astrophys. J. Lett. 354, L37 (1990).
  96. R. A. Daly, Spectral distortions of the microwave background radiation resulting from the damping of pressure waves, Astrophys. J. 371, 14 (1991).
  97. John C. Mather et al., Measurement of the cosmic microwave background spectrum by the COBE FIRAS instrument, Astrophys. J. 420, 439 (1994).
  98. Wayne Hu, Douglas Scott, and Joseph Silk, Power spectrum constraints from spectral distortions in the cosmic microwave background, Astrophys. J. Lett. 430, L5 (1994).
  99. Andrei D. Linde and Viatcheslav F. Mukhanov, Non-Gaussian isocurvature perturbations from inflation, Phys. Rev. D 56, R535 (1997).
  100. Takeo Moroi and Tomo Takahashi, Effects of cosmological moduli fields on cosmic microwave background, Phys. Lett. B 522, 215 (2001); 539, 303(E) (2002).
  101. David H. Lyth and David Wands, Generating the curvature perturbation without an inflaton, Phys. Lett. B 524, 5 (2002).
  102. Tristan L. Smith, Marc Kamionkowski, and Asantha Cooray, Direct detection of the inflationary gravitational wave background, Phys. Rev. D 73, 023504 (2006).
  103. Tristan L. Smith, Hiranya V. Peiris, and Asantha Cooray, Deciphering inflation with gravitational waves: Cosmic microwave background polarization vs. direct detection with laser interferometers, Phys. Rev. D 73, 123503 (2006).
  104. G. Hinshaw et al. (WMAP Collaboration), Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological parameter results, Astrophys. J. Suppl. Ser. 208, 19 (2013).
  105. James B. Dent, Damien A. Easson, and Hiroyuki Tashiro, Cosmological constraints from CMB distortion, Phys. Rev. D 86, 023514 (2012).
  106. Jessica L. Cook, Emanuela Dimastrogiovanni, Damien A. Easson, and Lawrence M. Krauss, Reheating predictions in single field inflation, J. Cosmol. Astropart. Phys. 04 (2015) 047.
  107. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  108. Damien A. Easson and Brian A. Powell, Identifying the inflaton with primordial gravitational waves, Phys. Rev. Lett. 106, 191302 (2011).
  109. Damien A. Easson and Brian A. Powell, Optimizing future experimental probes of inflation, Phys. Rev. D 83, 043502 (2011).
  110. Damien A. Easson and Brian A. Powell, The degeneracy problem in non-canonical inflation, J. Cosmol. Astropart. Phys. 03 (2013) 028.
  111. Vincent Vennin, Kazuya Koyama, and David Wands, Encyclopædia curvatonis, J. Cosmol. Astropart. Phys. 11 (2015) 008.
  112. Swagat S. Mishra, Varun Sahni, and Alexei A. Starobinsky, Curing inflationary degeneracies using reheating predictions and relic gravitational waves, J. Cosmol. Astropart. Phys. 05 (2021) 075.
  113. Ido Ben-Dayan and Udaykrishna Thattarampilly, Requiem to ”Proof of Inflation” or sourced fluctuations in a non-singular bounce, J. Cosmol. Astropart. Phys. 06 (2024) 004.
  114. Kurt Godel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems (Dover Publications, New York, NY USA, 1992), p. 72.
  115. John K. Beem, Paul Ehrlich, and Kevin Easley, Global Lorenztian Geometry, 2nd ed. (CRC Press, Boca Raton, 1996), p. 656.
  116. Alfonso Romero and Miguel Sanchez, On completeness of certain families of semi-Riemannian manifolds, Geometriae Dedicata 53, 103 (1994).
  117. Peter Peterson, Riemannian Geometry, Graduate Texts in Mathematics, Vol. 171 (Springer, New York, 2006).
  118. Andrei D. Polyanin and Zaitsev Valentin F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed. (Chapman and Hall/CRC, Boca Raton, Flordia, USA, 2003), p. 816.
  119. Milton Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover Books on Mathematics (Dover Publications, Mineola, NY, 1965).
  120. Joseph Edward Lesnefsky, Geodesic completeness of inflationary spacetimes, Ph.D. thesis, Arizona State University, 2024.
  121. John M. Lee, Introduction to Smooth Manifolds, 2nd ed., edited by S. Axler and K. Ribet, Graduate Texts in Mathematics Vol. 218 (Springer, New York, NY, 2012), p. 708.
  122. Barak Shoshany, Ogre: An object-oriented general relativity package for mathematica, J. Open Source Softwaare 6, 3416 (2021).
  123. Jeff Tupper, Reliable two-dimensional graphing methods for mathematical formulae with two free variables, Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 2001 (2001), pp. 77–86.

Outline

Information

Phys. Rev. D 112, 063545– Published 22 September, 2025
  • Received 4 April 2024
  • Accepted 20 August 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation