Phantom singularities and their quantum fate: general relativity and beyond—a CANTATA COST action topic

  • Mariam Bouhmadi-López
  • Claus Kiefer
  • Prado Martín-MorunoEmail author
Invited Report: Introduction to Current Research


Cosmological observations allow the possibility that dark energy is caused by phantom fields. These fields typically lead to the occurrence of singularities in the late Universe. We review here the status of phantom singularities and their possible avoidance in a quantum theory of gravity. We first introduce phantom energy and discuss its behavior in cosmology. We then list the various types of singularities that can occur from its presence. We also discuss the possibility that phantom behavior is mimicked by an alternative theory of gravity. We finally address the quantum cosmology of these models and discuss in which sense the phantom singularities can be avoided.


Phantom energy Alternative theories of gravity Cosmological singularities Quantum cosmology 



The work of MBL is supported by the Basque Foundation of Science IKERBASQUE. She also wishes to acknowledge the partial support from the Basque government Grant No. IT956-16 (Spain) and and FONDOS FEDER under grant FIS2017- 85076-P (MINECO/AEI/FEDER, UE). PMM acknowledges financial support from the project FIS2016-78859-P (AEI/FEDER, UE). This article is based upon work from COST Action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology)


  1. 1.
    Starobinsky, A.A.: Spectrum of relict gravitational radiation and the early state of the universe. JETP Lett. 30, 682 (1979)ADSGoogle Scholar
  2. 2.
    Da̧browski, M.P., Kiefer, C., Sandhöfer, B.: Quantum phantom cosmology. Phys. Rev. D 74, 044022 (2006)ADSMathSciNetGoogle Scholar
  3. 3.
    Ade, P.A.R., et al.: [Planck Collaboration]: Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594, A13 (2016)Google Scholar
  4. 4.
    Aghanim, N., et al. [Planck Collaboration]: Planck 2018 results. VI. Cosmological parameters. arXiv:1807.06209 [astro-ph.CO]
  5. 5.
    Bouali, A., Albarran, I., BBouhmadi-L’opez, M., Ouali, T.: Cosmological constraints of phantom dark energy models. Phys. Dark Univ. 26, 100391 (2019)Google Scholar
  6. 6.
    Abbott, T.M.C., et al.: Dark energy survey year 1 results: cosmological constraints from galaxy clustering and weak lensing. Phys. Rev. D 98, 043526 (2018)ADSGoogle Scholar
  7. 7.
    Di Valentino, E., Linder, E.V., Melchiorri, A.: Vacuum phase transition solves the \(H_0\) tension. Phys. Rev. D 97, 043528 (2018)ADSGoogle Scholar
  8. 8.
    Amendola, L., et al.: [Euclid theory working group]: cosmology and fundamental physics with the Euclid satellite. Living Rev. Rel. 16, 6 (2013)Google Scholar
  9. 9.
    Starobinsky, A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99 (1980)ADSzbMATHGoogle Scholar
  10. 10.
    Kiefer, C.: Quantum Gravity, 3rd edn. Oxford University Press, Oxford (2012)zbMATHGoogle Scholar
  11. 11.
    Bouhmadi-López, M., Chen, C.Y.: Towards the quantization of eddington-inspired-Born-Infeld theory. J. Cosmol. Astropart. Phys. 11, 023 (2016)ADSMathSciNetGoogle Scholar
  12. 12.
    Albarran, I., Bouhmadi-López, M., Chen, C.Y., Chen, P.: Doomsdays in a modified theory of gravity: a classical and a quantum approach. Phys. Lett. B 772, 814 (2017)ADSzbMATHGoogle Scholar
  13. 13.
    Alonso-Serrano, A., Bouhmadi-López, M., Martín-Moruno, P.: \(f(R)\) quantum cosmology: avoiding the Big Rip. Phys. Rev. D 98, 104004 (2018)ADSMathSciNetGoogle Scholar
  14. 14.
    Bouhmadi-López, M., Chen, C.Y., Chen, P.: On the consistency of the Wheeler–DeWitt equation in the quantized eddington-inspired Born-Infeld gravity. J. Cosmol. Astropart. Phys. 12, 032 (2018)ADSMathSciNetGoogle Scholar
  15. 15.
    Albarran, I., Bouhmadi-López, M., Chen, C.Y., Chen, P.: Quantum cosmology of Eddington–Born–Infeld gravity fed by a scalar field: the big rip case. Phys. Dark Univ. 23, 100255 (2019)Google Scholar
  16. 16.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1973)zbMATHGoogle Scholar
  17. 17.
    Hawking, S., Penrose, R.: The Nature of Space and Time. Princeton University Press, Princeton (1996)zbMATHGoogle Scholar
  18. 18.
    Senovilla, J.M.M.: Singularity theorems and their consequences. Gen. Relativ. Gravit. 30, 701 (1998)ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Ellis, G.F.R., Maartens, R., MacCallum, M.A.H.: Relativistic Cosmology. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  20. 20.
    Visser, M.: Lorentzian Wormholes: From Einstein to Hawking. AIP Press, New York (1996)Google Scholar
  21. 21.
    Martín-Moruno, P., Visser, M.: Classical and semi-classical energy conditions. Wormholes, warp drives and energy conditions. Fundam. Theor. Phys. 189, 193 (2017)ADSzbMATHGoogle Scholar
  22. 22.
    Reuter, M., Weyer, H.: Quantum gravity at astrophysical distances? J. Cosmol. Astropart. Phys. 12, 001 (2004)ADSGoogle Scholar
  23. 23.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)zbMATHGoogle Scholar
  24. 24.
    Abreu, G., Visser, M.: Some generalizations of the Raychaudhuri equation. Phys. Rev. D 83, 104016 (2011)ADSGoogle Scholar
  25. 25.
    Barceló, C., Visser, M.: Twilight for the energy conditions? Int. J. Mod. Phys. D 11, 1553 (2002)ADSMathSciNetzbMATHGoogle Scholar
  26. 26.
    Borde, A., Vilenkin, A.: Singularities in inflationary cosmology: a review. Int. J. Mod. Phys. D 5, 813 (1996)ADSMathSciNetGoogle Scholar
  27. 27.
    Molina-París, C., Visser, M.: Minimal conditions for the creation of a Friedman–Robertson–Walker universe from a ‘bounce’. Phys. Lett. B 455, 90 (1999)ADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Ellis, G.F.R., Maartens, R.: The emergent universe: inflationary cosmology with no singularity. Class. Quant. Grav. 21, 223 (2004)ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Cattoën, C., Visser, M.: Necessary and sufficient conditions for big bangs, bounces, crunches, rips, sudden singularities, and extremality events. Class. Quant. Grav. 22, 4913 (2005)ADSMathSciNetzbMATHGoogle Scholar
  30. 30.
    Carroll, S.M., Hoffman, M., Trodden, M.: Can the dark energy equation-of-state parameter w be less than -1? Phys. Rev. D 68, 023509 (2003)ADSGoogle Scholar
  31. 31.
    Fewster, C.J.: Quantum energy inequalities. Wormholes, warp drives and energy conditions. Fundam. Theor. Phys. 189, 215 (2017)ADSMathSciNetGoogle Scholar
  32. 32.
    Ford, L.H., Roman, T.A.: The quantum interest conjecture. Phys. Rev. D 60, 104018 (1999)ADSMathSciNetGoogle Scholar
  33. 33.
    Abreu, G., Visser, M.: Quantum interest in (3+1) dimensional Minkowski space. Phys. Rev. D 79, 065004 (2009)ADSMathSciNetGoogle Scholar
  34. 34.
    Martín-Moruno, P., Visser, M.: Semiclassical energy conditions for quantum vacuum states. J. High Energy Phys. 09, 050 (2013)ADSGoogle Scholar
  35. 35.
    Bouhmadi-López, M., Errahmani, A., Martín-Moruno, P., Ouali, T., Tavakoli, Y.: The little sibling of the big rip singularity. Int. J. Mod. Phys. D 24, 1550078 (2015)ADSMathSciNetGoogle Scholar
  36. 36.
    Bouhmadi-López, M., Lobo, F.S.N., Martín-Moruno, P.: Wormholes minimally violating the null energy condition. J. Cosmol. Astropart. Phys. 11, 007 (2014)ADSMathSciNetGoogle Scholar
  37. 37.
    Caldwell, R.R.: A Phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state. Phys. Lett. B 545, 23 (2002)ADSGoogle Scholar
  38. 38.
    Harada, T., Carr, B.J., Igata, T.: Complete conformal classification of the Friedmann–Lemaître–Robertson–Walker solutions with a linear equation of state. Class. Quantum Grav. 35, 105011 (2018)ADSzbMATHGoogle Scholar
  39. 39.
    Sushkov, S.V.: Wormholes supported by a phantom energy. Phys. Rev. D 71, 043520 (2005)ADSGoogle Scholar
  40. 40.
    Lobo, F.S.N.: Phantom energy traversable wormholes. Phys. Rev. D 71, 084011 (2005)ADSMathSciNetGoogle Scholar
  41. 41.
    Sbisà, F.: Classical and quantum ghosts. Eur. J. Phys. 36, 015009 (2015)Google Scholar
  42. 42.
    Cline, J.M., Jeon, S., Moore, G.D.: The phantom menaced: constraints on low-energy effective ghosts. Phys. Rev. D 70, 043543 (2004)ADSGoogle Scholar
  43. 43.
    Creminelli, P., D’Amico, G., Norena, J., Vernizzi, F.: The effective theory of quintessence: the \(w<-1\) side unveiled. J. Cosmol. Astropart. Phys. 02, 018 (2009)ADSGoogle Scholar
  44. 44.
    Rubakov, V.A.: The null energy condition and its violation. Phys. Usp. 57, 128 (2014)ADSGoogle Scholar
  45. 45.
    Rubakov, V.A.: The null energy condition and its violation. Usp. Fiz. Nauk 184, 137 (2014)Google Scholar
  46. 46.
    Capozziello, S., Faraoni, V.: Beyond Einstein gravity. Fundam. Theor. Phys. 170, 467 (2011)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Capozziello, S., Lobo, F.S.N., Mimoso, J.P.: Generalized energy conditions in extended theories of gravity. Phys. Rev. D 91, 124019 (2015)ADSMathSciNetGoogle Scholar
  48. 48.
    Baccetti, V., Martín-Moruno, P., Visser, M.: Null energy condition violations in bimetric gravity. J. High Energy Phys. 08, 148 (2012)ADSMathSciNetzbMATHGoogle Scholar
  49. 49.
    Albareti, F.D., Cembranos, J.A.R., de la Cruz-Dombriz, A., Dobado, A.: On the non-attractive character of gravity in \(f(R)\) theories. J. Cosmol. Astropart. Phys. 07, 009 (2013)ADSMathSciNetGoogle Scholar
  50. 50.
    Clarkson, C., Cortês, M., Bassett, B.A.: Dynamical dark energy or simply cosmic curvature? J. Cosmol. Astropart. Phys. 08, 011 (2007)ADSGoogle Scholar
  51. 51.
    Nojiri, S., Odintsov, S.D.: Unified cosmic history in modified gravity: from \(F(R)\) theory to Lorentz non-invariant models. Phys. Rep. 505, 59 (2011)ADSMathSciNetGoogle Scholar
  52. 52.
    Ludwick, K.J.: The viability of phantom dark energy: a review. Mod. Phys. Lett. A 32, 28 (2017)Google Scholar
  53. 53.
    Moffatt, H.K.: Singularities in fluid dynamics and their resolution. In: Berger, M.A., et al. (eds.) Lectures on Topological Fluid Mechanics, p. 157. Springer, Dordrecht (2009)Google Scholar
  54. 54.
    Nojiri, S., Odintsov, S.D., Tsujikawa, S.: Properties of singularities in (phantom) dark energy universe. Phys. Rev. D 71, 063004 (2005)ADSGoogle Scholar
  55. 55.
    Barrow, J.D., Galloway, G.J., Tipler, F.J.: The closed-universe recollapse conjecture. Mon. Not. R. Astr. Soc. 223, 835 (1986)ADSzbMATHGoogle Scholar
  56. 56.
    Starobinsky, A.A.: Future and origin of our universe: modern view. Grav. Cosmol. 6, 157 (2000)ADSzbMATHGoogle Scholar
  57. 57.
    Caldwell, R.R., Kamionkowski, M., Weinberg, N.N.: Phantom energy: dark energy with \(w<-1\) causes a cosmic doomsday. Phys. Rev. Lett. 91, 071301 (2003)ADSGoogle Scholar
  58. 58.
    Chimento, L.P., Lazkoz, R.: Constructing phantom cosmologies from standard scalar field universes. Phys. Rev. Lett. 91, 211301 (2003)ADSGoogle Scholar
  59. 59.
    Da̧browski, M.P., Stachowiak, T., Szydłowski, M.: Phantom cosmologies. Phys. Rev. D 68, 103519 (2003)ADSGoogle Scholar
  60. 60.
    González-Díaz, P.F.: K-essential phantom energy: Doomsday around the corner? Phys. Lett. B 586, 1 (2004)ADSGoogle Scholar
  61. 61.
    González-Díaz, P.F.: Axion phantom energy. Phys. Rev. D 69, 063522 (2004)ADSGoogle Scholar
  62. 62.
    Albarran, I., Bouhmadi-López, M.: Quantisation of the holographic Ricci dark energy model. J. Cosmol. Astropart. Phys. 08, 051 (2015)ADSMathSciNetGoogle Scholar
  63. 63.
    Barrow, J.D.: Sudden future singularities. Class. Quant. Grav. 21, L79 (2004)ADSMathSciNetzbMATHGoogle Scholar
  64. 64.
    Gorini, V., Kamenshchik, A.Y., Moschella, U., Pasquier, V.: Tachyons, scalar fields and cosmology. Phys. Rev. D 69, 123512 (2004)ADSMathSciNetGoogle Scholar
  65. 65.
    Kamenshchik, A., Kiefer, C., Sandhöfer, B.: Quantum cosmology with big-brake singularity. Phys. Rev. D 76, 064032 (2007)ADSMathSciNetGoogle Scholar
  66. 66.
    Kamenshchik, A., Kiefer, C., Kwidzinski, N.: Classical and quantum cosmology of Born–Infeld type models. Phys. Rev. D 93, 083519 (2016)ADSMathSciNetGoogle Scholar
  67. 67.
    Bouhmadi-López, M., Kiefer, C., Sandhöfer, B., Vargas Moniz, P.: On the quantum fate of singularities in a dark-energy dominated universe. Phys. Rev. D 79, 124035 (2009)ADSGoogle Scholar
  68. 68.
    Barvinsky, A.O., Deffayet, C., Kamenshchik, A.Y.: Anomaly driven cosmology: big boost scenario and AdS/CFT correspondence. J. Cosmol. Astropart. Phys. 05, 020 (2008)ADSGoogle Scholar
  69. 69.
    Bouhmadi-López, M., González-Díaz, P.F., Martín-Moruno, P.: On the generalised Chaplygin gas: Worse than a big rip or quieter than a sudden singularity? Int. J. Mod. Phys. D 17, 2269 (2008)ADSzbMATHGoogle Scholar
  70. 70.
    Nojiri, S., Odintsov, S.D.: Final state and thermodynamics of a dark energy universe. Phys. Rev. D 70, 103522 (2004)ADSGoogle Scholar
  71. 71.
    Bouhmadi-López, M., González-Díaz, P.F., Martín-Moruno, P.: Worse than a big rip? Phys. Lett. B 659, 1 (2008)ADSzbMATHGoogle Scholar
  72. 72.
    Nojiri, S., Odintsov, S.D.: Inhomogeneous equation of state of the universe: phantom era, future singularity, and crossing the phantom barrier. Phys. Rev. D 72, 023003 (2005)ADSGoogle Scholar
  73. 73.
    Da̧browski, M.P., Denkiewicz, T.: Exotic-singularity-driven dark energy. AIP Conf. Proc. 1241, 561 (2010)ADSGoogle Scholar
  74. 74.
    Nojiri, S., Odintsov, S.D.: Future evolution and finite-time singularities in \(F(R)\)-gravity unifying the inflation and cosmic acceleration. Phys. Rev. D 78, 046006 (2008)ADSGoogle Scholar
  75. 75.
    Bamba, K., Nojiri, S., Odintsov, S.D.: The Universe future in modified gravity theories: approaching the finite-time future singularity. J. Cosmol. Astropart. Phys. 10, 045 (2008)ADSGoogle Scholar
  76. 76.
    Bouhmadi-López, M., Kiefer, C., Krämer, M.: Resolution of type IV singularities in quantum cosmology. Phys. Rev. D 89, 064016 (2014)ADSGoogle Scholar
  77. 77.
    Da̧browski, M.P., Marosek, K., Balcerzak, A.: Standard and exotic singularities regularized by varying constants. Mem. Soc. Ast. It. 85, 44 (2014)ADSGoogle Scholar
  78. 78.
    Da̧browski, M.P., Denkiewicz, T.: Barotropic index \(w\)-singularities in cosmology. Phys. Rev. D 79, 063521 (2009)ADSMathSciNetGoogle Scholar
  79. 79.
    Albarran, I., Bouhmadi-López, M., Morais, J.: Cosmological perturbations in an effective and genuinely phantom dark energy Universe. Phys. Dark Univ. 16, 94 (2017)Google Scholar
  80. 80.
    Beltrán Jiménez, J., Rubiera-Garcia, D., Sáez-Gómez, D., Salzano, V.: Cosmological future singularities in interacting dark energy models. Phys. Rev. D 94, 123520 (2016)ADSMathSciNetGoogle Scholar
  81. 81.
    Chimento, L.P., Richarte, M.G.: Interacting realization of cosmological singularities with variable vacuum energy. Phys. Rev. D 92, 043511 (2015)ADSGoogle Scholar
  82. 82.
    Frampton, P.H., Ludwick, K.J., Scherrer, R.J.: Pseudo-rip: cosmological models intermediate between the cosmological constant and the little rip. Phys. Rev. D 85, 083001 (2012)ADSGoogle Scholar
  83. 83.
    Ruzmaikina, T., Ruzmaikin, A.A.: Quadratic corrections to the Lagrangian density of the gravitational field and the singularity. Sov. Phys. JETP 30, 372 (1970)ADSGoogle Scholar
  84. 84.
    Barrow, J.D.: Graduated inflationary universes. Phys. Lett. B 235, 40 (1990)ADSMathSciNetGoogle Scholar
  85. 85.
    Štefančić, H.: Expansion around the vacuum equation of state: sudden future singularities and asymptotic behavior. Phys. Rev. D 71, 084024 (2005)ADSGoogle Scholar
  86. 86.
    Bouhmadi-López, M.: Phantom-like behaviour in dilatonic brane-world scenario with induced gravity. Nucl. Phys. B 797, 78 (2008)ADSGoogle Scholar
  87. 87.
    Frampton, P.H., Ludwick, K.J., Scherrer, R.J.: The little rip. Phys. Rev. D 84, 063003 (2011)ADSGoogle Scholar
  88. 88.
    Brevik, I., Elizalde, E., Nojiri, S., Odintsov, S.D.: Viscous little rip cosmology. Phys. Rev. D 84, 103508 (2011)ADSGoogle Scholar
  89. 89.
    Bouhmadi-López, M., Chen, P., Liu, Y.-W.: Tradeoff between smoother and sooner ‘little rip’. Eur. Phys. J. C 73, 2546 (2013)ADSGoogle Scholar
  90. 90.
    Albarran, I., Bouhmadi-López, M., Kiefer, C., Marto, J., Vargas Moniz, P.: Classical and quantum cosmology of the little rip abrupt event. Phys. Rev. D 94, 063536 (2016)ADSMathSciNetGoogle Scholar
  91. 91.
    Barrow, J.D.: The deflationary Universe: an instability of the de Sitter universe. Phys. Lett. B 180, 335 (1986)ADSMathSciNetGoogle Scholar
  92. 92.
    Barrow, J.D.: String-driven inflationary and deflationary cosmological models. Nucl. Phys. B 310, 743 (1988)ADSMathSciNetGoogle Scholar
  93. 93.
    Albarran, I., Bouhmadi-López, M., Cabral, F., Martín-Moruno, P.: The quantum realm of the ‘little sibling’ of the big rip singularity. J. Cosmol. Astropart. Phys. 11, 044 (2015)ADSMathSciNetGoogle Scholar
  94. 94.
    Fernández-Jambrina, L., Lazkoz, R.: Geodesic behaviour of sudden future singularities. Phys. Rev. D 70, 121503 (2004)ADSMathSciNetGoogle Scholar
  95. 95.
    Fernández-Jambrina, L., Lazkoz, R.: Classification of cosmological milestones. Phys. Rev. D 74, 064030 (2006)ADSMathSciNetGoogle Scholar
  96. 96.
    Fernández-Jambrina, L.: Hidden past of dark energy cosmological models. Phys. Lett. B 656, 9 (2007)ADSMathSciNetzbMATHGoogle Scholar
  97. 97.
    Fernández-Jambrina, L., Lazkoz, R.: Singular fate of the universe in modified theories of gravity. Phys. Lett. B 670, 254 (2009)ADSMathSciNetzbMATHGoogle Scholar
  98. 98.
    Fernández-Jambrina, L.: \(w\)-cosmological singularities. Phys. Rev. D 82, 124004 (2010)ADSGoogle Scholar
  99. 99.
    Ellis, G.F.R., Schmidt, B.G.: Singular space-times. Gen. Relativ. Gravit. 8, 915 (1977)ADSMathSciNetzbMATHGoogle Scholar
  100. 100.
    Tipler, F.J.: Singularities in conformally flat spacetimes. Phys. Lett. A 64, 8 (1977)ADSMathSciNetGoogle Scholar
  101. 101.
    Clarke, C.J.S., Królak, A.: Curvature conditions for the occurrence of a class of spacetime singularities. J. Geom. Phys. 2, 17 (1985)MathSciNetGoogle Scholar
  102. 102.
    Królak, A.: Towards the proof of the cosmic censorship hypothesis. Class. Quant. Grav. 3, 267 (1986)ADSMathSciNetzbMATHGoogle Scholar
  103. 103.
    Bouhmadi-López, M., Chen, C.Y., Chen, P.: Eddington–Born–Infeld cosmology: a cosmographic approach, a tale of doomsdays and the fate of bound structures. Eur. Phys. J. C 75, 90 (2015)ADSGoogle Scholar
  104. 104.
    Puetzfeld, D., Obukhov, Y.N.: Generalized deviation equation and determination of the curvature in general relativity. Phys. Rev. D 93, 044073 (2016)ADSMathSciNetGoogle Scholar
  105. 105.
    Morais, J., Bouhmadi-López, M., Capozziello, S.: Can \(f(R)\) gravity contribute to (dark) radiation? J. Cosmol. Astropart. Phys. 09, 041 (2015)ADSMathSciNetGoogle Scholar
  106. 106.
    Chimento, L.P., Lazkoz, R., Maartens, R., Quiros, I.: Crossing the phantom divide without phantom matter. J. Cosmol. Astropart. Phys. 09, 004 (2006)ADSGoogle Scholar
  107. 107.
    Bouhmadi-López, M., Vargas Moniz, P.: Phantom-like behaviour in a brane-world model with curvature effects. Phys. Rev. D 78, 084019 (2008)ADSGoogle Scholar
  108. 108.
    Bouhmadi-López, M., Jiménez Madrid, J.A.: Escaping the big rip? J. Cosmol. Astropart. Phys. 05, 005 (2005)ADSGoogle Scholar
  109. 109.
    Brown, R.A., Maartens, R., Papantonopoulos, E., Zamarias, V.: A late-accelerating universe with no dark energy- and a finite-temperature big bang. J. Cosmol. Astropart. Phys. 11, 008 (2005)ADSGoogle Scholar
  110. 110.
    Bouhmadi-López, M., Tavakoli, Y., Vargas Moniz, P.: Appeasing the phantom menace? J. Cosmol. Astropart. Phys. 04, 016 (2010)ADSGoogle Scholar
  111. 111.
    Bañados, M., Ferreira, P.G.: Eddington’s theory of gravity and its progeny. Phys. Rev. Lett. 105, 011101 (2010). Erratum: [Phys. Rev. Lett. 113, 119901 (2014)]ADSMathSciNetGoogle Scholar
  112. 112.
    DeWitt, B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160, 1113 (1967)ADSzbMATHGoogle Scholar
  113. 113.
    Kamenshchik, A.Y.: Quantum cosmology and late-time singularities. Class. Quantum Grav. 30, 173001 (2013)ADSMathSciNetzbMATHGoogle Scholar
  114. 114.
    Kleinschmidt, A., Koehn, M., Nicolai, H.: Supersymmetric quantum cosmological billiards. Phys. Rev. D 80, 061701 (2009)ADSMathSciNetGoogle Scholar
  115. 115.
    Hájiček, P., Kiefer, C.: Singularity avoidance by collapsing shells in quantum gravity. Int. J. Mod. Phys. D 10, 775 (2001)ADSMathSciNetzbMATHGoogle Scholar
  116. 116.
    Kiefer, C.: Quantum black hole without singularity. In: Bianchi, M., Jantzen, R.T., Ruffini, R. (eds.) The Fourteenth Marcel Grossmann Meeting, p. 1685. World Scientific, Singapore (2017)Google Scholar
  117. 117.
    Calcagni, G.: Classical and Quantum Cosmology. Springer, Cham (2017)zbMATHGoogle Scholar
  118. 118.
    Giulini, D., Kiefer, C.: Wheeler–DeWitt metric and the attractivity of gravity. Phys. Lett. A 193, 21 (1994)ADSGoogle Scholar
  119. 119.
    Kiefer, C.: On the meaning of path integrals in quantum cosmology. Ann. Phys. (N.Y.) 207, 53 (1991)ADSMathSciNetzbMATHGoogle Scholar
  120. 120.
    Kiefer, C., Zeh, H.D.: Arrow of time in a recollapsing quantum universe. Phys. Rev. D 51, 4145 (1995)ADSMathSciNetGoogle Scholar
  121. 121.
    Kiefer, C., Kwidzinski, N., Piontek, D.: Singularity avoidance in Bianchi I quantum cosmology. Eur. Phys. J. C 79, 686 (2019)ADSGoogle Scholar
  122. 122.
    Bouhmadi-López, M., Vargas Moniz, P.: FRW quantum cosmology with a generalized Chaplygin gas. Phys. Rev. D 71, 063521 (2005)ADSGoogle Scholar
  123. 123.
    Bojowald, M.: Quantum Cosmology. Springer, New York (2011)zbMATHGoogle Scholar
  124. 124.
    Sami, M., Singh, P., Tsujikawa, S.: Avoidance of future singularities in loop quantum cosmology. Phys. Rev. D 74, 043514 (2006)ADSMathSciNetGoogle Scholar
  125. 125.
    Singh, P., Vidotto, F.: Exotic singularities and spatially curved loop quantum cosmology. Phys. Rev. D 83, 064027 (2011)ADSGoogle Scholar
  126. 126.
    Wilson-Ewing, E.: The loop quantum cosmology bounce as a Kasner transition. Class. Quantum Grav. 35, 065005 (2018)ADSMathSciNetzbMATHGoogle Scholar
  127. 127.
    Vilenkin, A.: Classical and quantum cosmology of the Starobinsky inflationary model. Phys. Rev. D 32, 2511 (1985)ADSMathSciNetGoogle Scholar
  128. 128.
    Hawking, S.W., Luttrell, J.C.: Higher derivatives in quantum cosmology (I). The isotropic case. Phys. Lett. B 247, 250 (1984)MathSciNetGoogle Scholar
  129. 129.
    Horowitz, G.T.: Quantum cosmology with a positive-definite action. Phys. Rev. D 31, 1169 (1985)ADSMathSciNetGoogle Scholar
  130. 130.
    Brizuela, D., Kiefer, C., Krämer, M.: Quantum-gravitational effects on gauge-invariant scalar and tensor perturbations during inflation: the slow-roll approximation. Phys. Rev. D 94, 123527 (2016)ADSMathSciNetGoogle Scholar
  131. 131.
    Bouhmadi-López, M., Kraemer, M., Morais, J., Robles-Pérez, S.: The interacting multiverse and its effect on the cosmic microwave background. J. Cosmol. Astropart. Phys. 02, 057 (2019)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Theoretical PhysicsUniversity of the Basque Country UPV/EHUBilbaoSpain
  2. 2.IKERBASQUE, Basque Foundation for ScienceBilbaoSpain
  3. 3.Institute for Theoretical PhysicsUniversity of CologneCologneGermany
  4. 4.Departamento de Física Teórica and Instituto de Física de Partículas y del Cosmos (IPARCOS)Universidad Complutense de MadridMadridSpain

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