Bouncing cosmology inspired by regular black holes


In this article, we present a bouncing cosmology inspired by a family of regular black holes. This scale-dependent cosmology deviates from the cosmological principle by means of a scale factor which depends on the time and the radial coordinate as well. The model is isotropic but not perfectly homogeneous. That is, this cosmology describes a universe almost homogeneous only for large scales, such as our observable universe.

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Fig. 1


  1. 1.

    Besides the scalars, another criterion to determine the spacetime regularity depends on the nonexistence of incomplete geodesics. In this sense, a nonsingular or regular spacetime is defined as geodesically complete. As we can see in Wald’s book ([29], chapter 9), there are geometries with scalars which diverge but such spacetimes are geodesically complete. An example for this case is found in Ref. [30], where a wormhole geometry has problem with scalars but it is geodesically complete. However, the standard FLRW cosmology presents the two problems: the geodesics are incomplete and the scalars diverge.


  1. 1.

    Novello, M., Perez Bergliaffa, S.E.: Bouncing cosmologies. Phys. Rept. 463, 127 (2008)

    Article  Google Scholar 

  2. 2.

    Joshi, P.S.: Spacetime singularities. In: Ashtekar, A., Petkov, V. (eds.) Springer Handbook of Spacetime, p. 409. Springer, Berlin (2014)

    Google Scholar 

  3. 3.

    Riess, A.G., et al.: (Supernova search team collaboration): observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998)

    ADS  Article  Google Scholar 

  4. 4.

    Perlmutter, S., et al.: (Supernova cosmology project collaboration): measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565 (1999)

    ADS  Article  MATH  Google Scholar 

  5. 5.

    Bardeen, J. M.: Non-singular general-relativistic collapse. In: Conference proceedings of GR5, p. 174, Tbilisi, URSS (1968)

  6. 6.

    Ansoldi, S.: Spherical black holes with regular center: a review of existing models including a recent realization with Gaussian sources. In: Proceedings of BH2, dynamics and thermodynamics of black holes and naked singularities, Milano, Italy (2007)

  7. 7.

    Lemos, J.P.S., Zanchin, V.T.: Regular black holes: electrically charged solutions, Reissner–Nordström outside a de sitter core. Phys. Rev. D 83, 124005 (2011)

    ADS  Article  Google Scholar 

  8. 8.

    Sakharov, A.D.: Sov. Phys. JETP 22, 241 (1966)

    ADS  Google Scholar 

  9. 9.

    Gliner, E.B.: Sov. Phys. JETP 22, 378 (1966)

    ADS  Google Scholar 

  10. 10.

    Hayward, S.A.: Formation and evaporation of non-singular black holes. Phys. Rev. Lett. 96, 031103 (2006)

    ADS  Article  Google Scholar 

  11. 11.

    Neves, J.C.S.: Deforming regular black holes. Int. J. Mod. Phys. A 32, 1750112 (2017)

    ADS  Article  Google Scholar 

  12. 12.

    Smailagic, A., Spallucci, E.: “Kerrr” black hole: the lord of the string. Phys. Lett. B 688, 82 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Modesto, L., Nicolini, P.: Charged rotating noncommutative black holes. Phys. Rev. D 82, 104035 (2010)

    ADS  Article  Google Scholar 

  14. 14.

    Bambi, C., Modesto, L.: Rotating regular black holes. Phys. Lett. B 721, 329 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Toshmatov, B., Ahmedov, B., Abdujabbarov, A., Stuchlik, Z.: Rotating regular black hole solution. Phys. Rev. D 89, 104017 (2014)

    ADS  Article  MATH  Google Scholar 

  16. 16.

    Azreg-Ainou, M.: Generating rotating regular black hole solutions without complexification. Phys. Rev. D 90, 064041 (2014)

    ADS  Article  Google Scholar 

  17. 17.

    Neves, J.C.S.: Note on regular black holes in a brane world. Phys. Rev. D 92, 084015 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Neves, J.C.S., Saa, A.: Regular rotating black holes and the weak energy condition. Phys. Lett. B 734, 44 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Lehners, J.L.: Ekpyrotic and cyclic cosmology. Phys. Rept. 465, 223 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Brandenberger, R. H.: The matter bounce alternative to inflationary cosmology. arXiv:1206.4196

  21. 21.

    Ijjas, A., Steinhardt, P.J.: Inflationary paradigm in trouble after Planck 2013. Phys. Lett. B 723, 261 (2013)

    ADS  Article  Google Scholar 

  22. 22.

    Wu, K.K.S., Lahav, O., Rees, M.J.: The large-scale smoothness of the universe. Nature 397, 225 (1999)

    ADS  Article  Google Scholar 

  23. 23.

    Bolejko, K., Célérier, M.N., Krasinski, A.: Inhomogeneous cosmological models: exact solutions and their applications. Class. Quantum Grav. 28, 164002 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Ruiz, E., Senovilla, J.M.M.: General class of inhomogeneous perfect-fluid solutions. Phys. Rev. D 45, 1995 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Senovilla, J.M.M.: New class of inhomogeneous cosmological perfect-fluid solutions without big-bang singularity. Phys. Rev. Lett. 64, 2219 (1990)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Nemiroff, R.J., Joshi, R., Patla, B.R.: An exposition on friedmann cosmology with negative energy densities. JCAP 1506, 006 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    Hernandez Jr., W.C.: Material sources for the kerr metric. Phys. Rev. 159, 1070 (1967)

    ADS  Article  Google Scholar 

  28. 28.

    Kim, C.W., Song, J.: A proposed scale-dependent cosmology for the inhomogeneous cosmology. Int. J. Mod. Phys. D 5, 293 (1996)

    ADS  Article  Google Scholar 

  29. 29.

    Wald, R.: General Relativity. The University of Chicago Press, Chicago (1984)

    Google Scholar 

  30. 30.

    Olmo, G.J., Rubiera-Garcia, D., Sanchez-Puente, A.: Geodesic completeness in a wormhole spacetime with horizons. Phys. Rev. D 92, 044047 (2015)

    ADS  MathSciNet  Article  Google Scholar 

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This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil (Grant No. 2013/03798-3). I would like to thank Alberto Saa and an anonymous referee for comments and suggestions.

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Correspondence to J. C. S. Neves.

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Neves, J.C.S. Bouncing cosmology inspired by regular black holes. Gen Relativ Gravit 49, 124 (2017).

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  • Bouncing cosmology
  • Singularity
  • Regular black holes
  • Cosmological principle