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Observational constraints on a phenomenological \(f\left( R,\partial R\right) \)-model

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Abstract

This paper analyses the cosmological consequences of a modified theory of gravity whose action integral is built from a linear combination of the Ricci scalar \(R\) and a quadratic term in the covariant derivative of \(R\). The resulting Friedmann equations are of the fifth-order in the Hubble function. These equations are solved numerically for a flat space section geometry and pressureless matter. The cosmological parameters of the higher-order model are fit using SN Ia data and X-ray gas mass fraction in galaxy clusters. The best-fit present-day \(t_{0}\) values for the deceleration parameter, jerk and snap are given. The coupling constant \(\beta \) of the model is not univocally determined by the data fit, but partially constrained by it. Density parameter \(\Omega _{m_0}\) is also determined and shows weak correlation with the other parameters. The model allows for two possible future scenarios: there may be either an eternal expansion or a Rebouncing event depending on the set of values in the space of parameters. The analysis towards the past performed with the best-fit parameters shows that the model is not able to accommodate a matter-dominated stage required to the formation of structure.

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Notes

  1. The authors of Ref. [58] use the same action (1) proposed here but their argument is based on different motivations.

  2. \(f_{\mathrm{gas}}=\frac{M_{\mathrm{gas}}}{M_{\mathrm{tot}}}\), where \(M_{\mathrm{gas}}\) and \(M_{\mathrm{tot}}\) are the gas mass and the total mass of the cluster respectively.

  3. \(\Omega _{b0}=\rho _{b}/\rho _{c}\) is the non-dimensional baryon density parameter.

  4. Notice that here \(s_{0}\) is not the snap, defined in Eq. (18). Even at risk of confusion, we decided to maintain the notation used in Ref. [15].

  5. We emphasize that \(\left\{ q_{0},j_{0},s_{0}\right\} \) are precisely the values of \(\left\{ -Q,J,S\right\} \) calculated at the present time \(t=t_{0}\).

  6. The result (51) is still valid even when radiation is added to the model.

References

  1. Spergel, D.N., et al.: First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parameters. Astrophys. J. Suppl. 148, 175 (2003)

    Article  ADS  Google Scholar 

  2. Komatsu, E., et al.: Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological interpretation. Astrophys. J. Suppl. 192, 18 (2011)

    Article  ADS  Google Scholar 

  3. Planck Collaboration: P.A.R. Ade et al.: Planck 2013 results. I. Overview of products and scientific results (2013). arXiv:1303.5062

  4. Planck Collaboration: P.A.R. Ade et al.: Planck 2013 results. XVI. Cosmological parameters (2013). arXiv:1303.5076

  5. Eisenstein, D.J., et al.: Detection of the Barion Acoustic Peak in the large-scale correlation function of SDSS luminous red galaxies. Astrophys. J. 633, 560 (2005)

    Article  ADS  Google Scholar 

  6. Percival, W.J., et al.: Baryon acoustic oscillations in the Sloan Digital Sky Survey data release 7 Galaxy Sample. Mon. Not. R. Astron. Soc. 401, 2148 (2010)

    Article  ADS  Google Scholar 

  7. Cole, S., et al.: The 2dF Galaxy Redshift Survey: power-spectrum analysis of the final dataset and cosmological implications. Mon. Not. R. Astron. Soc. 362, 505 (2005)

    Article  ADS  Google Scholar 

  8. Beutler, F.: The 6dF Galaxy Survey: baryon acoustic oscilations and the local Hubble constant. Mon. Not. R. Astron. Soc. 416, 3017 (2011)

    Article  ADS  Google Scholar 

  9. Astier, P., et al.: The Supernova legacy Survey: measurement of \(\Omega _{M}\), \(\Omega _{\Lambda }\) and \(w\) from the first year data set. Astron. Astrophys. 447, 31 (2006)

    Article  ADS  Google Scholar 

  10. 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 (2004)

    Article  ADS  Google Scholar 

  11. Riess, A.G., et al.: New Hubble Space Telescope discoveries of Type Ia Supernovae at \(z>1\): Narrowing constraints on the early behavior of dark energy. Astrophys. J. 659, 98 (2007)

    Article  ADS  Google Scholar 

  12. Wood-Vasey, W.M., et al.: Observational constraints on the nature of the dark energy: first cosmological results from the ESSENCE Supernova Survey. Astrophys. J. 666, 694 (2007)

    Article  ADS  Google Scholar 

  13. Amanullah, R., et al.: Spectra and Hubble Space Telescope light curves of six type Ia supervonae at and the Union2 Compilation. Astrophys. J. 716, 712 (2010)

    Article  ADS  Google Scholar 

  14. Suzuki, N., et al.: The Hubble Space Telescope Cluster Supernova Survey: V. Improving the dark energy constraints above and building an early-type-hosted supernova sample. Astrophys. J. 746, 85 (2012)

    Article  ADS  Google Scholar 

  15. Allen, S.W., et al.: Improved constraints on dark energy from Chandra X-ray observations of the largest relaxed galaxy clusters. Mon. Not. R. Astron. Soc. 383, 879 (2008)

    Article  ADS  Google Scholar 

  16. Schindler, S.: \(\Omega _{M}\)-different ways to determine the matter density of the universe. Space Sci. Rev. 100, 299 (2002)

    Article  ADS  Google Scholar 

  17. Weinberg, S.: The cosmological constant problem. Rev. Mod. Phys. 61, 1 (1989)

    Article  ADS  MATH  Google Scholar 

  18. Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998)

    Article  ADS  Google Scholar 

  19. Perlmutter, S., et al.: Measurements of \(\Omega \) and \(\Lambda \) from 42 high-redshift supernovae. Astrophys. J. 517, 565 (1999)

    Article  ADS  Google Scholar 

  20. Amendola, L., Tsujikawa, S.: Dark Energy: Theory and Observations. Cambridge University Press, New York (2010)

    Book  Google Scholar 

  21. Ratra, B., Peebles, P.J.E.: Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev. D 37, 3406 (1988)

    Article  ADS  Google Scholar 

  22. Caldwell, R.R., Dave, R., Steinhardt, P.J.: Cosmological imprint of an energy component with general equation-of-state. Phys. Rev. Lett. 80, 1582 (1998)

    Article  ADS  Google Scholar 

  23. Carrol, S.M.: Quintessence and the rest of the world. Phys. Rev. Lett. 81, 3067 (1998)

    Article  ADS  Google Scholar 

  24. Hebecker, A., Wetterich, C.: Natural quintessence? Phys. Lett. B 497, 281 (2001)

    Article  ADS  MATH  Google Scholar 

  25. Amendola, L., Campos, G.C., Rosenfeld, R.: Consequences of dark matter-dark energy interaction on cosmological parameters derived from SN Ia data. Phys. Rev. D 75, 083506 (2007)

    Article  ADS  Google Scholar 

  26. Chiba, T., Okabe, T., Yamaguchi, M.: Kinetically driven quintessence. Phys. Rev. D 62, 023511 (2000)

    Article  ADS  Google Scholar 

  27. Armendariz-Picon, C., Mukhanov, V.F., Steinhardt, P.J.: Essentials of k-essence. Phys. Rev. D 63, 103510 (2001)

    Article  ADS  Google Scholar 

  28. Kamenshchik, A.Y., Moschella, U., Pasquier, V.: An alternative to quintessence. Phys. Lett. B 511, 265 (2001)

    Article  ADS  MATH  Google Scholar 

  29. Bento, M.C., Bertolami, O., Sen, A.A.: Generalized Chaplygin gas, accelerated expansion and dark energy-matter unification. Phys. Rev. D 66, 043507 (2002)

    Article  ADS  Google Scholar 

  30. Capozziello, S., De Laurentis, M.: Extended theories of gravity. Phys. Rep. 509, 167 (2011)

    Article  ADS  Google Scholar 

  31. Nojiri, S., Odintsov, S.D.: Unified cosmic history in modified gravity: from \(f\left( R\right) \) theory to Lorentz non-invariant models. Phys. Rep. 505, 59 (2011)

    Article  ADS  Google Scholar 

  32. Dvali, G.R., Gabadadze, G., Porrati, M.: 4D gravity on a brane in 5D minkowski space. Phys. Lett. B 485, 208 (2000)

    Article  ADS  MATH  Google Scholar 

  33. Sahni, V.: Shtanov, : Braneworld models of dark energy. JCAP 0311, 014 (2003)

    Article  ADS  Google Scholar 

  34. Bartolo, N., Pietroni, M.: Scalar tensor gravity and quintessence. Phys. Rev. D 61, 023518 (2000)

    Article  ADS  Google Scholar 

  35. Perrota, F., Baccigalupi, C., Matarrese, S.: Extended quintessence. Phys. Rev. D 61, 023507 (2000)

    Article  ADS  Google Scholar 

  36. Chang, Z., Li, X.: Modified Friedmann model in Randers-Finsler space of approximate Berwald type as a possible alternative to dark energy hypothesis. Phys. Lett. B 676, 173 (2009)

    Article  ADS  Google Scholar 

  37. Adhav, K.S.: LRS Bianchi type-I universe with anisotropic dark energy in Lyra geometry. Int. J. Astr. Astrophys. 1(4), 204 (2011). doi:10.4236/ijaa.2011.14026

    Article  Google Scholar 

  38. Casana, R., de Melo, C.A.M., Pimentel, B.M.: Massless DKP field in a Lyra manifold. Class. Quantum Grav. 24, 723 (2007)

    Article  ADS  MATH  Google Scholar 

  39. Capozziello, S.: Curvature quintessence. Int. J. Mod. Phys. D 11, 483 (2002)

    Article  ADS  MATH  Google Scholar 

  40. De Felice, A., Tsujikawa, S.: \(f\left( R\right) \) theories. Liv. Rev. Rel. 13, 3 (2010)

    Google Scholar 

  41. Sotiriou, T.P., Faraoni, V.: \(f\left( R\right) \) theories of gravity. Rev. Mod. Phys. 82, 451 (2010)

    Article  ADS  MATH  Google Scholar 

  42. Santos, J., et al.: Latest supernovae constraints on \(f\left( R\right) \) cosmologies. Phys. Lett. B 669, 14 (2008)

    Article  ADS  Google Scholar 

  43. Pires, N., Santos, J., Alcaniz, J.S.: Cosmographic constraints on a class of Palatini \(f\left( R\right) \) gravity. Phys. Rev. D 82, 067302 (2010)

    Article  ADS  Google Scholar 

  44. Nojiri, S., Odintsov, S.D.: Introduction to modified gravity and gravitational alternative for dark energy. Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)

    Article  MATH  Google Scholar 

  45. Bengochea, G., Ferraro, R.: Dark torsion as the cosmic speed-up. Phys. Rev. D 79, 124019 (2009)

    Article  ADS  Google Scholar 

  46. Linder, E.V.: Einstein’s other gravity and the acceleration of the universe. Phys. Rev. D 81, 127301 (2010)

    Article  ADS  Google Scholar 

  47. Bamba, K., et al.: Equation of state for dark energy in \(f\left( T\right) \) gravity. JCAP 1101, 021 (2011)

    Article  ADS  Google Scholar 

  48. Aldrovandi, R., Pereira, J.G.: Teleparallel Gravity: An Introduction. Springer, Dordrecht (2013)

    Book  Google Scholar 

  49. SDSS-III Collaboration: The eighth data release of the Sloan Digital Sky Survey: first data from SDSS-III. APJS 193, 29 (2011). arXiv:1101.1559

  50. Tolman, R.C.: Effect of inhomogeneity on cosmological models. Proc. Natl. Acad. Sci. 20, 169 (1934)

    Article  ADS  Google Scholar 

  51. Bondi, H.: Spherically symmetrical models in general relativity. MNRAS 107, 410 (1947)

    Article  ADS  MATH  Google Scholar 

  52. Buchert, T.: On average properties of inhomogeneous fluids in general relativity: dust cosmologies. Gen. Relativ. Gravit. 32, 105 (2000)

    Article  ADS  MATH  Google Scholar 

  53. Buchert, T., Räsänen, S.: Backreaction in late-time cosmology. Annu. Rev. Nucl. Part. Sci. 62, 57 (2012)

    Article  ADS  Google Scholar 

  54. Räsänen, S.: Backreaction: directions of progress. Class. Quant. Grav. 28, 16 (2011)

    Article  Google Scholar 

  55. Wiltshire, D.L.: Exact solution to the averaging problem in cosmology. Phys. Rev. Lett. 99, 25 (2007)

    Article  Google Scholar 

  56. Wiltshire, D.L.: Cosmic structure, averaging and dark energy. In: Perez Bergliaffa S.E., Novello M. (eds.) Proceedings of the 15th Brazilian School on Cosmology and Gravitation (2013). arXiv:1311.3787

  57. Saulder, C., Mieske, S., Zeilinger, W.W.: Observational aspects of inhomogeneous cosmology. In: Proceedings of the VIII International Workshop on the Dark Side of the Universe (2012). arXiv:1211.1926

  58. Gottlöber, S., Schmidt, H.-J., Starobinsky, A.A.: Sixth-order gravity and conformal transformations. Class. Quantum Gravity 7, 893 (1990)

    Article  ADS  MATH  Google Scholar 

  59. Biswas, T., Mazumdar, A., Siegel, W.: Bouncing universes in string-inspired gravity. JCAP 0603, 009 (2006)

    Google Scholar 

  60. Biswas, T., Koivisto, T., Mazumdar, A.: Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity. JCAP 1011, 008 (2010)

    Article  ADS  Google Scholar 

  61. Biswas, T., et al.: Stable bounce and inflation in non-local higher derivative cosmology. JCAP 1208, 024 (2012)

    Article  ADS  Google Scholar 

  62. Arkani-Hamed, N., et al.: Non-local modification of gravity and the cosmological constant problem (2002). arXiv:hep-th/0209227

  63. Barvinsky, A.O.: Nonlocal action for long-distance modifications of gravity theory. Phys. Lett. B 572, 109 (2003)

  64. Cuzinatto, R.R., de Melo, C.A.M., Pompeia, P.J.: Second order gauge theory. Ann. Phys. 322, 1211 (2007)

    Article  ADS  MATH  Google Scholar 

  65. Cuzinatto, R.R., de Melo, C.A.M., Medeiros, L.G., Pompeia, P.J.: Gauge formulation for higher order gravity. Eur. Phys. J. C 53, 99 (2008)

    Article  ADS  MATH  Google Scholar 

  66. Cuzinatto, R.R., de Melo, C.A.M., Medeiros, L.G., Pompeia, P.J.: Cosmic acceleration from second order gauge gravity. Astrophys. Space Sci. 332, 201 (2011)

    Article  ADS  MATH  Google Scholar 

  67. Visser, M.: Jerk, snap, and the cosmological equation of state. Class. Quantum Gravity 21, 2603 (2004)

    Article  ADS  MATH  Google Scholar 

  68. Medeiros, L.G.: Realistic cyclic magnetic universe. Int. J. Mod. Phys. D 21, 1250073 (2012)

    Article  ADS  Google Scholar 

  69. Holz, D.E., Linder, E.V.: Safety in numbers: gravitational lensing degradation of the Luminosity Distance-Redshift relation. Astrophys. J. 631, 678 (2005)

    Article  ADS  Google Scholar 

  70. Sasaki, S.: A new method to estimate cosmological parameters using the Baryon fraction of clusters of galaxies. Publ. Astron. Soc. Jpn. 48, L119 (1996)

    ADS  Google Scholar 

  71. Kirkman, D., et al.: The cosmological baryon density from the deuterium to hydrogen ratio towards QSO absorption systems: D/H towards Q1243+3047. Astrophys. J. Suppl. 149, 1 (2003)

    Article  ADS  Google Scholar 

  72. Riess, A.G., et al.: A 3% solution: determination of the Hubble constant with the Hubble Space Telescope and Wide Field Camera 3. Astrophys. J. 730, 119 (2011)

    Article  ADS  Google Scholar 

  73. Beringer, J., et al.: Review of particle physics. Phys. Rev. D 86, 010001 (2012)

    Article  ADS  Google Scholar 

  74. Mukhanov, V.: Physical Foundations of Cosmology. Cambridge University Press, New York (2005)

    Book  MATH  Google Scholar 

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Acknowledgments

This paper is dedicated to Prof. Mario Novello on the ocasion of his 70th birthday. RRC thanks FAPEMIG-Brazil (Grant CEX–APQ–04440-10) for financial support. CAMM is grateful to FAPEMIG-Brazil for partial support. LGM acknowledges FAPERN-Brazil for financial support. The authors would like to thank two anonymous referees for the valuable comments that helped to improve the paper.

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Authors and Affiliations

  1. Instituto de Ciência e Tecnologia, Universidade Federal de Alfenas, Rod. José Aurélio Vilela (BR 267), km 533, no 11999, Poços de Caldas, MG, CEP 37701-970, Brazil

    R. R. Cuzinatto & C. A. M. de Melo

  2. Instituto de Física Teórica, Universidade Estadual Paulista, Rua Bento Teobaldo Ferraz 271 Bloco II, P.O. Box 70532-2, São Paulo, SP, CEP 01156-970, Brazil

    C. A. M. de Melo

  3. Escola de Ciência e Tecnologia, Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Campus Universitário, s/n - Lagoa Nova, Natal, RN, CEP 59078-970, Brazil

    L. G. Medeiros

  4. Instituto de Fomento e Coordenação Industrial, Praça Mal. Eduardo Gomes 50, São José dos Campos, SP, CEP 12228-901, Brazil

    P. J. Pompeia

  5. Instituto Tecnológico de Aeronaútica, Praça Mal. Eduardo Gomes 50, São José dos Campos, SP, CEP 12228-900, Brazil

    P. J. Pompeia

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  1. R. R. Cuzinatto
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Cuzinatto, R.R., de Melo, C.A.M., Medeiros, L.G. et al. Observational constraints on a phenomenological \(f\left( R,\partial R\right) \)-model. Gen Relativ Gravit 47, 29 (2015). https://doi.org/10.1007/s10714-015-1862-z

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