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General Relativity and Gravitation

, Volume 43, Issue 6, pp 1733–1757 | Cite as

Filtering out the cosmological constant in the Palatini formalism of modified gravity

  • Florian BauerEmail author
Research Article

Abstract

According to theoretical physics the cosmological constant (CC) is expected to be much larger in magnitude than other energy densities in the universe, which is in stark contrast to the observed Big Bang evolution. We address this old CC problem not by introducing an extremely fine-tuned counterterm, but in the context of modified gravity in the Palatini formalism. In our model the large CC term is filtered out, and it does not prevent a standard cosmological evolution. We discuss the filter effect in the epochs of radiation and matter domination as well as in the asymptotic de Sitter future. The final expansion rate can be much lower than inferred from the large CC without using a fine-tuned counterterm. Finally, we show that the CC filter works also in the Kottler (Schwarzschild-de Sitter) metric describing a black hole environment with a CC compatible to the future de Sitter cosmos.

Keywords

Cosmological constant Modified gravity Palatini formalism Dark energy Vacuum energy 

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References

  1. 1.
    Bauer F., Sola J., Stefancic H.: Dynamically avoiding fine-tuning the cosmological constant: the ‘Relaxed Universe’. JCAP 1012, 029 (2010) arXiv:1006.3944 [hep-th]ADSGoogle Scholar
  2. 2.
    Weinberg S.: The cosmological constant problem. Rev. Mod. Phys. 61, 1–23 (1989)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertolami O.: The cosmological constant problem: a user’s guide. Int. J. Mod. Phys. D 18, 2303–2310 (2009) arXiv:0905.3110 [gr-qc]ADSzbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    WMAP Collaboration, Spergel D.N. et al.: Wilkinson Microwave Anisotropy Probe (WMAP) three year results: implications for cosmology. Astrophys. J. Suppl. 170, 377 (2007) arXiv:astro-ph/0603449ADSCrossRefGoogle Scholar
  5. 5.
    SupernovaCosmology Project Collaboration., Knop R.A. et al.: New constraints on ΩM, ΩΛ, and w from an independent set of eleven high-redshift supernovae observed with HST. Astrophys. J. 598, 102 (2003) arXiv:astro-ph/0309368CrossRefGoogle Scholar
  6. 6.
    SupernovaSearch Team Collaboration., Riess A.G. et al.: Type Ia supernova discoveries at z > 1 from the hubble space telescope: evidence for past deceleration and constraints on dark energy evolution. Astrophys. J. 607, 665–687 (2004) arXiv:astro-ph/0402512CrossRefGoogle Scholar
  7. 7.
    Carroll S.M.: The cosmological constant. Living Rev. Rel. 4, 1 (2001) arXiv:astro-ph/0004075Google Scholar
  8. 8.
    Peebles P.J.E., Ratra B.: The cosmological constant and dark energy. Rev. Mod. Phys. 75, 559–606 (2003) arXiv:astro-ph/0207347ADSzbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Padmanabhan T.: Cosmological constant: the weight of the vacuum. Phys. Rept. 380, 235–320 (2003) arXiv:hep-th/0212290ADSzbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nojiri S., Odintsov S.D.: Introduction to modified gravity and gravitational alternative for dark energy. ECONF C 0602061, 06 (2006) arXiv:hep-th/0601213Google Scholar
  11. 11.
    Copeland E.J., Sami M., Tsujikawa S.: Dynamics of dark energy. Int. J. Mod. Phys. D 15, 1753–1936 (2006) arXiv:hep-th/0603057ADSzbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Stefancic H.: The solution of the cosmological constant problem from the inhomogeneous equation of state—a hint from modified gravity?. Phys. Lett. B 670, 246–253 (2009) arXiv:0807.3692 [gr-qc]ADSCrossRefGoogle Scholar
  13. 13.
    Grande J., Sola J., Stefancic H.: LXCDM: a cosmon model solution to the cosmological coincidence problem?. JCAP 0608, 011 (2006) arXiv:gr-qc/0604057ADSGoogle Scholar
  14. 14.
    Bauer F., Sola J., Stefancic H.: Relaxing a large cosmological constant. Phys. Lett. B 678, 427–433 (2009) arXiv:0902.2215 [hep-th]ADSCrossRefGoogle Scholar
  15. 15.
    Bauer F.: Perturbations in the relaxation mechanism for a large cosmological constant. Class. Quant. Grav. 27, 055001 (2010) arXiv:0909.2237 [gr-qc]ADSCrossRefGoogle Scholar
  16. 16.
    Bonanno A., Reuter M.: Cosmology with self-adjusting vacuum energy density from a renormalization group fixed point. Phys. Lett. B 527, 9–17 (2002) arXiv:astro-ph/0106468ADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Nobbenhuis S.: Categorizing different approaches to the cosmological constant problem. Found. Phys. 36, 613–680 (2006) arXiv:gr-qc/0411093ADSzbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Barr S.M., Ng S.-P., Scherrer R.J.: Classical cancellation of the cosmological constant re-considered. Phys. Rev. D 73, 063530 (2006) arXiv:hep-ph/0601053ADSCrossRefGoogle Scholar
  19. 19.
    Diakonos F.K., Saridakis E.N.: A statistical solution to the cosmological constant problem in the brane world. JCAP 0902, 030 (2009) arXiv:0708.3143 [hep-th]ADSGoogle Scholar
  20. 20.
    Klinkhamer F.R., Volovik G.E.: Self-tuning vacuum variable and cosmological constant. Phys. Rev. D 77, 085015 (2008) arXiv:0711.3170 [gr-qc]ADSCrossRefGoogle Scholar
  21. 21.
    Batra P., Hinterbichler K., Hui L., Kabat D.N.: Pseudo-redundant vacuum energy. Phys. Rev. D 78, 043507 (2008) arXiv:0801.4526 [hep-th]ADSCrossRefGoogle Scholar
  22. 22.
    Dvali G., Hofmann S., Khoury J.: Degravitation of the cosmological constant and graviton width. Phys. Rev. D 76, 084006 (2007) arXiv:hep-th/0703027ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Patil S.P.: Degravitation, inflation and the cosmological constant as an afterglow. JCAP 0901, 017 (2009) arXiv:0801.2151 [hep-th]ADSGoogle Scholar
  24. 24.
    Demir D.A.: Vacuum energy as the origin of the gravitational constant. Found. Phys. 39, 1407–1425 (2009) arXiv:0910.2730 [hep-th]ADSzbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hassan S.F., Hofmann S., von Strauss M.: Brane induced gravity, its ghost and the cosmological constant problem. JCAP 1101, 020 (2011) arXiv:1007.1263 [hep-th]ADSGoogle Scholar
  26. 26.
    Unruh W.G.: A unimodular theory of canonical quantum gravity. Phys. Rev. D40, 1048 (1989)ADSMathSciNetGoogle Scholar
  27. 27.
    Henneaux M., Teitelboim C.: The cosmological constant and general covariance. Phys. Lett. B222, 195–199 (1989)ADSGoogle Scholar
  28. 28.
    Ng Y.J., van Dam H.: Possible solution to the cosmological constant problem. Phys. Rev. Lett. 65, 1972–1974 (1990)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ng Y.J., van Dam H.: Unimodular theory of gravity and the cosmological constant. J. Math. Phys. 32, 1337–1340 (1991)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Smolin L.: The quantization of unimodular gravity and the cosmological constant problem. Phys. Rev. D 80, 084003 (2009) arXiv:0904.4841 [hep-th]ADSCrossRefMathSciNetGoogle Scholar
  31. 31.
    Bauer F., Sola J., Stefancic H.: The Relaxed Universe: towards solving the cosmological constant problem dynamically from an effective action functional of gravity. Phys. Lett. B 688, 269–272 (2010) arXiv:0912.0677 [hep-th]ADSCrossRefGoogle Scholar
  32. 32.
    Vollick D.N.: On the viability of the Palatini form of 1/R gravity. Class. Quant. Grav. 21, 3813–3816 (2004) arXiv:gr-qc/0312041ADSzbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Allemandi G., Borowiec A., Francaviglia M.: Accelerated cosmological models in Ricci squared gravity. Phys. Rev. D 70, 103503 (2004) arXiv:hep-th/0407090ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    Sotiriou T.P.: The nearly Newtonian regime in non-linear theories of gravity. Gen. Relativ. Gravit. 38, 1407–1417 (2006) arXiv:gr-qc/0507027ADSzbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Barausse E., Sotiriou T.P., Miller J.C.: Curvature singularities, tidal forces and the viability of Palatini f(R) gravity. Class. Quant. Grav. 25, 105008 (2008) arXiv:0712.1141 [gr-qc]ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    Li B., Barrow J.D., Mota D.F.: The cosmology of Ricci-tensor-squared gravity in the Palatini variational approach. Phys. Rev. D 76, 104047 (2007) arXiv:0707.2664 [gr-qc]ADSCrossRefMathSciNetGoogle Scholar
  37. 37.
    Sotiriou T.P., Faraoni V.: f(R) theories of gravity. Rev. Mod. Phys. 82, 451–497 (2010) arXiv:0805.1726 [gr-qc]ADSzbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Exirifard Q., Sheikh-Jabbari M.M.: Lovelock gravity at the crossroads of Palatini and metric formulations. Phys. Lett. B 661, 158–161 (2008) arXiv:0705.1879 [hep-th]ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    Tsujikawa S., Uddin K., Tavakol R.: Density perturbations in f(R) gravity theories in metric and Palatini formalisms. Phys. Rev. D 77, 043007 (2008) arXiv:0712.0082 [astro-ph]ADSCrossRefMathSciNetGoogle Scholar
  40. 40.
    Bauer F., Demir D.A.: Inflation with non-minimal coupling: metric vs. Palatini formulations. Phys. Lett. B 665, 222–226 (2008) arXiv:0803.2664 [hep-ph]ADSCrossRefGoogle Scholar
  41. 41.
    Borunda M., Janssen B., Bastero-Gil M.: Palatini versus metric formulation in higher curvature gravity. JCAP 0811, 008 (2008) arXiv:0804.4440 [hep-th]ADSGoogle Scholar
  42. 42.
    Capozziello, S., De Laurentis, M., Faraoni, V.: A bird’s eye view of f(R)-gravity. Invited Review for The Special Issue in Cosmology, The Open Astronomy Journal, Eds. S.D. Odintsov et al. (2009). arXiv:0909.4672 [gr-qc]Google Scholar
  43. 43.
    De Felice A., Tsujikawa S.: f(R) theories. Living Rev. Rel. 13, 3 (2010) arXiv:1002.4928 [gr-qc]Google Scholar
  44. 44.
    Goenner H.F.M.: Alternative to the Palatini method: a new variational principle. Phys. Rev. D 81, 124019 (2010) arXiv:1003.5532 [gr-qc]ADSCrossRefMathSciNetGoogle Scholar
  45. 45.
    Meng X.-H., Wang P.: Gravitational potential in Palatini formulation of modified gravity. Gen. Relativ. Gravit. 36, 1947–1954 (2004) arXiv:gr-qc/0311019ADSzbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Sotiriou T.P.: Constraining f(R) gravity in the Palatini formalism. Class. Quant. Grav. 23, 1253–1267 (2006) arXiv:gr-qc/0512017ADSzbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Li B., Mota D.F., Shaw D.J.: Indistinguishable macroscopic behaviour of Palatini gravities and general relativity. Class. Quant. Grav. 26, 055018 (2009) arXiv:0801.0603 [gr-qc]ADSCrossRefMathSciNetGoogle Scholar
  48. 48.
    Li B., Mota D.F., Shaw D.J.: Microscopic and macroscopic behaviors of palatini modified gravity theories. Phys. Rev. D 78, 064018 (2008) arXiv:0805.3428 [gr-qc]ADSCrossRefMathSciNetGoogle Scholar
  49. 49.
    Olmo G.J., Sanchis-Alepuz H., Tripathi S.: Dynamical aspects of generalized Palatini theories of gravity. Phys. Rev. D 80, 024013 (2009) arXiv:0907.2787 [gr-qc]ADSCrossRefMathSciNetGoogle Scholar
  50. 50.
    Barragan C., Olmo G.J.: Isotropic and anisotropic bouncing cosmologies in Palatini gravity. Phys. Rev. D 82, 084015 (2010) arXiv:1005.4136 [gr-qc]ADSCrossRefGoogle Scholar
  51. 51.
    Olmo G.J., Singh P.: Effective action for loop quantum cosmology a la Palatini. JCAP 0901, 030 (2009) arXiv:0806.2783 [gr-qc]ADSGoogle Scholar
  52. 52.
    Vitagliano V., Sotiriou T.P., Liberati S.: The dynamics of generalized Palatini theories of gravity. Phys. Rev. D 82, 084007 (2010) arXiv:1007.3937 [gr-qc]ADSCrossRefGoogle Scholar
  53. 53.
    Woodard R.P.: Avoiding dark energy with 1/R modifications of gravity. Lect. Notes Phys. 720, 403–433 (2007) arXiv:astro-ph/0601672ADSCrossRefGoogle Scholar
  54. 54.
    Bekenstein J.D.: The relation between physical and gravitational geometry. Phys. Rev. D 48, 3641–3647 (1993) arXiv:gr-qc/9211017ADSCrossRefMathSciNetGoogle Scholar
  55. 55.
    Bekenstein J.D.: Relativistic gravitation theory for the MOND paradigm. Phys. Rev. D 70, 083509 (2004) arXiv:astro-ph/0403694ADSCrossRefGoogle Scholar
  56. 56.
    Zumalacarregui M., Koivisto T.S., Mota D.F., Ruiz-Lapuente P.: Disformal scalar fields and the dark sector of the universe. JCAP 1005, 038 (2010) arXiv:1004.2684 [astro-ph.CO]ADSGoogle Scholar
  57. 57.
    Steigman, G.: Primordial nucleosynthesis: the predicted and observed abundances and their consequences. Revised version to appear in the Proceedings of the 11th Symposium on Nuclei in the Cosmos (NIC XI), to be published by Proceedings of Science (PoS, SISSA) (2010). arXiv:1008.4765 [astro-ph.CO]Google Scholar
  58. 58.
    Izotov Y.I., Thuan T. X.: The primordial abundance of 4He: evidence for non-standard big bang nucleosynthesis. Astrophys. J. 710, L67–L71 (2010) arXiv:1001.4440 [astro-ph.CO]ADSCrossRefGoogle Scholar
  59. 59.
    Gonzalez-Garcia M.C., Maltoni M., Salvado J.: Robust cosmological bounds on neutrinos and their combination with oscillation results. JHEP 08, 117 (2010) arXiv:1006.3795 [hep-ph]ADSCrossRefGoogle Scholar
  60. 60.
    Hamann J., Hannestad S., Raffelt G.G., Tamborra I., Wong Y.Y.Y.: Cosmology seeking friendship with sterile neutrinos. Phys. Rev. Lett. 105, 181301 (2010) arXiv:1006.5276 [hep-ph]ADSCrossRefGoogle Scholar
  61. 61.
    Einstein A., Straus E.G.: The influence of the expansion of space on the gravitation fields surrounding the individual stars. Rev. Mod. Phys. 17, 120–124 (1945)ADSzbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Balbinot R., Bergamini R., Comastri A.: Solution of the Einstein-strauss problem with a lambda term. Phys. Rev. D38, 2415–2418 (1988)ADSMathSciNetGoogle Scholar
  63. 63.
    Barausse E., Sotiriou T.P., Miller J.C.: A no-go theorem for polytropic spheres in Palatini f(R) gravity. Class. Quant. Grav. 25, 062001 (2008) arXiv:gr-qc/0703132ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.High Energy Physics Group, Department of ECM, and Institut de Ciències del CosmosUniversitat de BarcelonaBarcelona, CataloniaSpain

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