Skip to main content

Advertisement

SpringerLink
Log in

Kinematic model-independent reconstruction of Palatini f(R) cosmology

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

A kinematic treatment to trace out the form of f(R) cosmology, within the Palatini formalism, is discussed by only postulating the universe homogeneity and isotropy. To figure this out we build model-independent approximations of the luminosity distance through rational expansions. These approximants extend the Taylor convergence radii computed for usual cosmographic series. We thus consider both Padé and the rational Chebyshev polynomials. They can be used to accurately describe the universe late-time expansion history, providing further information on the thermal properties of all effective cosmic fluids entering the energy momentum tensor of Palatini’s gravity. To perform our numerical analysis, we relate the Palatini’s Ricci scalar with the Hubble parameter H and thus we write down a single differential equation in terms of the redshift z. Therefore, to bound f(R), we make use of the most recent outcomes over the cosmographic parameters obtained from combined data surveys. In particular our clue is to select two scenarios, i.e. (2, 2) Padé and (2, 1) Chebyshev approximations, since they well approximate the luminosity distance at the lowest possible order. We find that best analytical matches to the numerical solutions lead to \(f(R)=a+bR^n\) with free parameters given by the set \((a, b, n)=(-1.627, 0.866, 1.074)\) for (2, 2) Padé approximation, whereas \(f(R)=\alpha +\beta R^m\) with \((\alpha , \beta , m)=(-1.332, 0.749, 1.124)\) for (2, 1) rational Chebyshev approximation. Finally, our results are compared with the \(\Lambda \)CDM predictions and with previous studies in the literature. Slight departures from General Relativity are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. Although very successful in accounting for all the major cosmological observables, the \(\Lambda \)CDM model is plagued by fundamental issues, such as the coincidence problem [13] and the fine-tuning problem [14].

  2. We work in units of \(c=1\).

  3. This choice is perfectly consistent with the results of the Planck collaboration [6], within the \(1\sigma \) errors: \(\Omega _m=0.308 \pm 0.012\).

  4. One may, in principle, consider to relax this condition and allow for slight departures from G [68]. This would ensure that F(R) exactly recovers the \(\Lambda \)CDM behaviour at large curvatures [69, 70].

References

  1. Perlmutter, S., et al.: Nature 391, 51 (1998)

    Article  ADS  Google Scholar 

  2. Riess, A.G., et al.: Astron. J. 116, 1009 (1998)

    Article  ADS  Google Scholar 

  3. Schmidt, B.P., et al.: Astrophys. J. 507, 46 (1998)

    Article  ADS  Google Scholar 

  4. Eisenstein, D.J., Hu, W., Tegmark, M.: Astrophys. J. 504, L57 (1998)

    Article  ADS  Google Scholar 

  5. Hinshaw, G., et al.: Astrophys. J. Suppl. Ser. 208, 19 (2013)

    Article  ADS  Google Scholar 

  6. Ade, P.A.R., et al.: [Planck Collaboration]. Astron. Astrophys. 594, A13 (2016)

  7. Haridasu, B.S., Lukovic, V.V., D’Agostino, R., Vittorio, N.: Astron. Astrophys. 600, L1 (2017)

    Article  ADS  Google Scholar 

  8. Bamba, K., Capozziello, S., Nojiri, S., Odintsov, S.D.: Astrophys. Space Sci. 342, 155 (2012)

    Article  ADS  Google Scholar 

  9. Joyce, A., Lombriser, L., Schmidt, F.: Annu. Rev. Nucl. Part. Sci. 66, 95 (2016)

    Article  ADS  Google Scholar 

  10. Kleidis, K., Spyrou, N.K.: Entropy 18, 3 (2016)

    Article  Google Scholar 

  11. Sahni, V., Starobinsky, A.: Int. J. Mod. Phys. D 9, 373 (2000)

    ADS  Google Scholar 

  12. Copeland, E.J., Sami, M., Tsujikawa, S.: Int. J. Mod. Phys. D 15, 1753 (2006)

    Article  ADS  Google Scholar 

  13. Zlatev, I., Wang, L.M., Steinhardt, P.J.: Phys. Rev. Lett. 82, 896 (1999)

    Article  ADS  Google Scholar 

  14. Weinberg, S.: Rev. Mod. Phys. 61, 1–23 (1989)

    Article  ADS  Google Scholar 

  15. Peebles, P.J.E., Ratra, B.: Phys. Rev. D 37, 3406 (1988)

    Article  ADS  Google Scholar 

  16. Padmanabhan, T.: Phys. Rev. D 66, 021301 (2002)

    Article  ADS  Google Scholar 

  17. Singh, P., Sami, M., Dadhich, N.: Phys. Rev. D 68, 023522 (2003)

    Article  ADS  Google Scholar 

  18. Chevallier, M., Polarski, D.: Int. J. Mod. Phys. D 10, 213 (2001)

    Article  ADS  Google Scholar 

  19. Linder, E.: Phys. Rev. Lett. 90, 091301 (2003)

    Article  ADS  Google Scholar 

  20. Jassal, H.K., Bagla, J.S., Padmanabhan, T.: Mon. Not. R. Astron. Soc. 356, L11 (2005)

    Article  ADS  Google Scholar 

  21. Bento, M.C., Bertolami, O., Sen, A.A.: Phys. Rev. D 66, 043507 (2002)

    Article  ADS  Google Scholar 

  22. Capozziello, S., D’Agostino, R., Luongo, O.: Phys. Dark Univ. 20, 1 (2018)

    Article  Google Scholar 

  23. Csaki, C., Graesser, M., Randall, L., Terning, J.: Phys. Rev. D 62, 045015 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  24. Maartens, R.: Liv. Rev. Rel. 7, 7 (2004)

    Article  Google Scholar 

  25. Sotiriou, T.P., Faraoni, V.: Rev. Mod. Phys. 82, 451 (2010)

    Article  ADS  Google Scholar 

  26. Capozziello, S., De Laurentis, M.: Phys. Rep. 509, 167321 (2011)

    Article  Google Scholar 

  27. Nojiri, S., Odintsov, S.D., Oikonomou, V.K.: Phys. Rep. 692, 1 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  28. Nojiri, S., Odintsov, S.D.: Phys. Rev. D 68, 123512 (2003)

    Article  ADS  Google Scholar 

  29. Carroll, S.M., Duvvuri, V., Trodden, M., Turner, M.S.: Phys. Rev. D 70, 043528 (2004)

    Article  ADS  Google Scholar 

  30. Olmo, G.J.: Phys. Rev. D 72, 083505 (2005)

    Article  ADS  Google Scholar 

  31. Amendola, L., Polarski, D., Tsujikawa, S.: Phys. Rev. Lett. 98, 131302 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  32. Capozziello, S., Cardone, V.F., Carloni, S., Troisi, A.: Int. J. Mod. Phys. D 12, 1969 (2003)

    Article  ADS  Google Scholar 

  33. Koivisto, T.: Phys. Rev. D 73, 083517 (2006)

    Article  ADS  Google Scholar 

  34. Santos, J., Alcaniz, J.S., Carvalho, F.C., Pires, N.: Phys. Lett. B 669, 14 (2008)

    Article  ADS  Google Scholar 

  35. Basilakos, S., Nesseris, S., Perivolaropoulos, L.: Phys. Rev. D 87, 123529 (2013)

    Article  ADS  Google Scholar 

  36. Nunes, R.C., Pan, S., Saridakis, E.N., Abreu, E.M.C.: J. Cosmol. Astrop. Phys. 1701, 005 (2017)

    Article  ADS  Google Scholar 

  37. Santos, B., Campista, M., Santos, J., Alcaniz, J.S.: Astron. Astrophys. 548, A31 (2012)

    Article  ADS  Google Scholar 

  38. Campista, M., Santos, B., Santos, J., Alcaniz, J.S.: Phys. Lett. B 699, 320 (2011)

    Article  ADS  Google Scholar 

  39. Capozziello, S., Cardone, Vincenzo F., Francaviglia, M.: Gen. Relativ. Gravit. 38, 711 (2006)

    Article  ADS  Google Scholar 

  40. Fay, S., Tavakol, R., Tsujikawa, S.: Phys. Rev. D 75, 063509 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  41. Tsujikawa, S., Uddin, K., Tavakol, R.: Phys. Rev. D 77, 043007 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  42. Pires, N., Santos, J., Alcaniz, J.S.: Phys. Rev. D 82, 067302 (2010)

    Article  ADS  Google Scholar 

  43. Capozziello, S., Cardone, V.F., Troisi, A.: Phys. Rev. D 71, 043503 (2005)

    Article  ADS  Google Scholar 

  44. Capozziello, S., D’Agostino, R., Luongo, O.: J. Cosmol. Astrop. Phys. 1805, 008 (2018)

    Article  ADS  Google Scholar 

  45. Gruber, C., Luongo, O.: Phys. Rev. D 89, 103506 (2014)

    Article  ADS  Google Scholar 

  46. Aviles, A., Bravetti, A., Capozziello, S., Luongo, O.: Phys. Rev. D 90, 043531 (2014)

    Article  ADS  Google Scholar 

  47. Capozziello, S., D’Agostino, R., Luongo, O.: Mon. Not. R. Astron. Soc. 476, 3924 (2018)

    Article  ADS  Google Scholar 

  48. Capozziello, S., Ruchika Sen, A.A.: arXiv:1806.03943 [astro-ph.CO] (2018)

  49. Capozziello, S.: Int. J. Mod. Phys. D 11, 483–492 (2002)

    Article  ADS  Google Scholar 

  50. Allemandi, G., Borowiec, A., Francaviglia, M.: Phys. Rev. D 70, 103503 (2004)

    Article  ADS  Google Scholar 

  51. Carloni, S., Dunsby, P.K.S., Capozziello, S., Troisi, A.: Class. Quant. Grav. 22, 4839 (2005)

    Article  ADS  Google Scholar 

  52. Vollick, D.N.: Phys. Rev. D 68, 063510 (2003)

    Article  ADS  Google Scholar 

  53. Will, C.M.: Theory and Experiment in Gravitational Physics. Cambridge University Press, New York (1981)

    Google Scholar 

  54. Saini, T.D., Raychaudhury, S., Sahni, V., Starobinsky, A.A.: Phys. Rev. Lett. 85, 1162 (2000)

    Article  ADS  Google Scholar 

  55. Visser, M.: Class. Quantum Grav. 21, 2603 (2004)

    Article  ADS  Google Scholar 

  56. Cattoen, C., Visser, M.: Class. Quantum Grav. 24, 5985 (2007)

    Article  ADS  Google Scholar 

  57. Carvalho, J.C., Alcaniz, J.S.: Mon. Not. R. Astron. Soc. 418, 1873 (2011)

    Article  ADS  Google Scholar 

  58. Luongo, O.: Mod. Phys. Lett. A 26, 1459 (2011)

    Article  ADS  Google Scholar 

  59. Busti, V.C., et al.: Phys. Rev. D 92, 123512 (2015)

    Article  ADS  Google Scholar 

  60. Aviles, A., Gruber, C., Luongo, O., Quevedo, H.: Phys. Rev. D 86, 123516 (2012)

    Article  ADS  Google Scholar 

  61. Baker Jr., G.A., Graves-Morris, P.: Padé Approximants. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  62. Litvinov, G.: Appl. Russ. J. Math. Phys. 1, 313 (1993)

    Google Scholar 

  63. Gerald, C.F., Wheatley, P.O.: Applied Numerical Analysis. Prentice Hall College Div, New Jersey (2003)

    MATH  Google Scholar 

  64. Obsieger, B.: Numerical Methods III—Approximations of Functions. University of Rijeka, Rijeka (2013)

    Google Scholar 

  65. Baghram, S., Rahvar, S.: Phys. Rev. D 80, 124049 (2009)

    Article  ADS  Google Scholar 

  66. Dick, R.: Gen. Relativ. Gravit. 36, 217 (2004)

    Article  ADS  Google Scholar 

  67. Dominguez, A.E., Barraco, D.E.: Phys. Rev. D 70, 043505 (2004)

    Article  ADS  Google Scholar 

  68. Martins, C.J.A.P.: Rep. Prog. Phys. 80, 12 (2017)

    Article  Google Scholar 

  69. Hu, W., Sawicki, I.: Phys. Rev. D 76, 064004 (2007)

    Article  ADS  Google Scholar 

  70. Appleby, S.A., Battye, R.A.: Phys. Lett. B 654, 7 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  71. de la Cruz-Dombriz, A., Dunsby, P.K.S., Luongo, O., Reverberi, L.: J. Cosmol. Astrop. Phys. 1612, 042 (2016)

    Article  Google Scholar 

  72. Dunsby, P.K.S., Luongo, O.: Int. J. Geom. Method Mod. Phys. 13, 1630002 (2016)

    Article  Google Scholar 

  73. Basilakos, S., Nesseris, S., Perivolaropoulos, L.: Phys. Rev. D 87, 123529 (2013)

    Article  ADS  Google Scholar 

  74. Pan, S., Chakraborty, S.: Int. J. Mod. Phys. D 23, 1450092 (2014)

    Article  ADS  Google Scholar 

  75. Borowiec, A., Godlowski, W., Szydlowski, M.: Phys. Rev. D 74, 043502 (2006)

    Article  ADS  Google Scholar 

  76. Amarzguioui, M., Elgaroy, O., Mota, D.F., Multamaki, T.: Astron. Astrophys. 454, 707 (2006)

    Article  ADS  Google Scholar 

  77. Li, B., Chan, K.C., Chu, M.C.: Phys. Rev. D 76, 024002 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  78. Carvalho, F.C., Santos, E.M., Alcaniz, J.S., Santos, J.: J. Cosmol. Astropart. Phys. 0809, 008 (2008)

    Article  ADS  Google Scholar 

  79. Ruggiero, M. L.: [\(n\) Collaboration], Gen. Relativ. Gravit. 41, 1497 (2009)

  80. Yang, X.J., Chen, D.M.: Mon. Not. R. Astron. Soc. 394, 1449 (2009)

    Article  ADS  Google Scholar 

  81. Roshan, M., Shojai, F.: Phys. Lett. B 668, 238 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  82. Capozziello, S., De Felice, A.: J. Cosmol. Astropart. Phys. 0808, 016 (2008)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This paper is based upon work from COST action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology).

Author information

Authors and Affiliations

  1. Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II”, Via Cinthia, 80126, Naples, Italy

    Salvatore Capozziello

  2. Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli, Via Cinthia, Naples, Italy

    Salvatore Capozziello

  3. Gran Sasso Science Institute, Via F. Crispi 7, 67100, L’Aquila, Italy

    Salvatore Capozziello

  4. Dipartimento di Fisica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Rocco D’Agostino

  5. Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Rocco D’Agostino

  6. Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044, Frascati, Italy

    Orlando Luongo

  7. School of Science and Technology, University of Camerino, 62032, Camerino, Italy

    Orlando Luongo

  8. Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, AP 70543, 04510, Mexico, DF, Mexico

    Orlando Luongo

Authors
  1. Salvatore Capozziello
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rocco D’Agostino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Orlando Luongo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Orlando Luongo.

Appendix: Rational approximations of the luminosity distance

Appendix: Rational approximations of the luminosity distance

We here report the (2, 2) Padé and the (2, 1) rational Chebyshev approximations of \(d_L(z)\), respectively:

$$\begin{aligned} P_{2,2}(z)&=\dfrac{1}{H_0}(6 z (10 + 9 z - 6 q_0^3 z + s_0 z - 2 q_0^2 (3 + 7 z) - q_0 (16 + 19 z) \nonumber \\&\quad + j_0 (4 + (9 + 6 q_0) z))\Big /(60 + 24 z + 6 s_0 z - 2 z^2 \nonumber \\&\quad + 4 j_0^2 z^2 - 9 q_0^4 z^2 - 3 s_0 z^2 + 6 q_0^3 z (-9 + 4 z) + q_0^2 (-36 - 114 z + 19 z^2), \end{aligned}$$
(A.1)
$$\begin{aligned} R_{2,1}(z)&=\dfrac{1}{H_0}(-((3 (16 (-1 - j_0 + q_0 + 3 q_0^2) (7 - j_0 + q_0 + 3 q_0^2) - (18 + 5 j_0 (1 + 2 q_0) \nonumber \\&\quad - 3 q_0 (6 + 5 q_0 (1 + q_0)) + s_0) (14 \nonumber \\&\quad + 5 j_0 (1 + 2 q_0) - q_0 (14 + 15 q_0 (1 + q_0)) + s_0)))/(14 + 5 j_0 (1 + 2 q_0)\nonumber \\&\quad - q_0 (14 + 15 q_0 (1 + q_0)) + s_0))+4 (47 - j_0 \nonumber \\&\quad + q_0 + 3 q_0^2 - (12 (-1 + q_0) (1 + j_0 - q_0 (1 + 3 q_0)))/(14 + 5 j_0 (1 + 2 q_0) \nonumber \\&\quad - q_0 (14 + 15 q_0 (1 + q_0)) + s_0)) z \nonumber \\&\quad -(4 (12 (-1- j_0 + q_0 + 3 q_0^2) (7 - j_0 + q_0 + 3 q_0^2) \nonumber \\&\quad + 4 (1 + j_0 - q_0 (1 + 3 q_0))^2 - (14 + 5 j_0 (1 + 2 q_0) - q_0 (14 \nonumber \\&\quad + 15 q_0 (1 + q_0))+ s_0)^2) (-1 + 2 z^2))/(14 + 5 j_0 (1 + 2 q_0)\nonumber \\&\quad - q_0 (14 + 15 q_0 (1 + q_0)) + s_0))\Big /(192 (1 + (4 (1 + j_0 \nonumber \\&\quad - q_0 (1 + 3 q_0)) z)/( 14+5 j_0 (1 + 2 q_0) - q_0 (14 + 15 q_0 (1 + q_0)) + s_0))). \end{aligned}$$
(A.2)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Capozziello, S., D’Agostino, R. & Luongo, O. Kinematic model-independent reconstruction of Palatini f(R) cosmology. Gen Relativ Gravit 51, 2 (2019). https://doi.org/10.1007/s10714-018-2483-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-018-2483-0

Keywords

Search

Navigation