We present a reconstruction technique for models of $f (R)$ $f (R)$ gravity from the Chaplygin scalar field in flat de Sitter spacetimes. Exploiting the equivalence between $f (R)$ $f (R)$ gravity and scalar–tensor (ST) theories, and treating the Chaplygin gas (CG) as a scalar field model in a universe without conventional matter forms, the Lagrangian densities for the $f (R)$ $f (R)$ action are derived. Exact $f (R)$ $f (R)$ models and corresponding scalar field potentials are obtained for asymptotically de Sitter spacetimes in early and late cosmological expansion histories. It is shown that the reconstructed $f (R)$ $f (R)$ models all have General Relativity (GR) as a limiting solution.

Keywords:

AMSC: 83Fxx, 83Cxx

References

1. M. Kowalski et al., Improved cosmological constraints from new, old, and combined supernova data sets, Astrophys. J. 686(2) (2008) 749. Web of Science, Google Scholar
2. A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91(1) (1980) 99–102. Web of Science, Google Scholar
3. K. Sato, First-order phase transition of a vacuum and the expansion of the universel, Mon. Not. R. Astron. Soc. 195(3) (1981) 467–479. Web of Science, Google Scholar
4. D. Kazanas, Dynamics of the universe and spontaneous symmetry breaking, Astrophys. J. 241 (1980) L59–L63. Web of Science, Google Scholar
5. A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, 2000). Google Scholar
6. G. F. Smoot et al., Structure in the COBE differential microwave radiometer first-year maps, Astrophys. J. 396 (1992) L1–L5. Web of Science, Google Scholar
7. A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116(3) (1998) 1009. Web of Science, Google Scholar
8. V. Sahni and A. Starobinsky, The case for a positive cosmological $Λ$ $Λ$ -term, Int. J. Mod. Phys. D 9(04) (2000) 373–443. Link, Web of Science, Google Scholar
9. S. Jha et al., UBVRI light curves of 44 type Ia supernovae, Astron. J. 131(1) (2006) 527. Web of Science, Google Scholar
10. M. Tegmark et al., Cosmological parameters from SDSS and WMAP, Phys. Rev. D 69(10) (2004) 103501. Web of Science, Google Scholar
11. M. Tegmark et al., Cosmological constraints from the SDSS luminous red galaxies, Phys. Rev. D 74(12) (2006) 123507. Web of Science, Google Scholar
12. W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol, J. A. Peacock, A. C. Pope and A. S. Szalay, Measuring the baryon acoustic oscillation scale using the Sloan Digital Sky Survey and 2df galaxy redshift survey, Mon. Not. Roy. Astron. Soc. 381(3) (2007) 1053–1066. Web of Science, Google Scholar
13. E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. Limon, L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, S. S. Meyer, G. S. Tucker, J. L. Weiland, E. Wollack and E. L. Wright, Five-year Wilkinson microwave anisotropy probe observations: Cosmological interpretation, Astrophys. J. Suppl. Ser. 180(2) (2008) 330. Web of Science, Google Scholar
14. S. Chattopadhyay and U. Debnath, Interaction between phantom field and modified Chaplygin gas, Astrophys. Space Sci. 326(2) (2010) 155–158. Web of Science, Google Scholar
15. M. C. Bento, O. Bertolami and A. A. Sen, Generalized Chaplygin gas, accelerated expansion, and dark-energy-matter unification. Phys. Rev. D 66(4) (2002) 043507. Web of Science, Google Scholar
16. V. Gorini, A. Kamenshchik, U. Moschella and V Pasquier, The Chaplygin gas as a model for dark energy, The Tenth Merrel Grossman Meeting 20 (2003) 26. Google Scholar
17. A. Dev, J. S. Alcaniz and D. Jain, Cosmological consequences of a Chaplygin gas dark energy, Phys. Rev. D 67(2) (2003) 023515. Web of Science, Google Scholar
18. A. Kamenshchik, U. Moschella and V. Pasquier, An alternative to quintessence, Phys. Lett. B 511(2) (2001) 265–268. Web of Science, Google Scholar
19. N. Bilić, G. B. Tupper and R. D. Viollier, Unification of dark matter and dark energy: The inhomogeneous Chaplygin gas, Phys. Lett. B 535(1) (2002) 17–21. Web of Science, Google Scholar
20. V. Gorini, A. Kamenshchik and U. Moschella, Can the Chaplygin gas be a plausible model for dark energy? Phys. Rev. D 67(6) (2003) 063509. Web of Science, Google Scholar
21. H. B. Benaoum, Modified Chaplygin gas cosmology, Adv. High Energy Phys. Article ID: 357802 2012 (2012) 12 pages. Web of Science, Google Scholar
22. S. M. Carroll, V. Duvvuri, M. S. Turner and M. Trodden, Is cosmic speed-up due to new gravitational physics? Phys. Rev. D 70(4) (2004) 043528. Web of Science, Google Scholar
23. S. Capozziello, V. F. Cardone and A. Troisi, Dark energy and dark matter as curvature effects? J. Cosmol. Astropart. Phys. 2006(08) (2006) 001. Google Scholar
24. S. Capozziello and M. De Laurentis, Extended theories of gravity, Phys. Rep. 509(4) (2011) 167–321. Web of Science, Google Scholar
25. S. Nojiri and S. D. Odintsov, Unified cosmic history in modified gravity: From $f (R)$ $f (R)$ theory to Lorentz non-invariant models, Phys. Rep. 505(2) (2011) 59–144. Web of Science, Google Scholar
26. S. Nojiri, S. D. Odintsov and D. Sáez-Gómez, Cyclic, ekpyrotic and little rip universe in modified gravity, in Towards New Paradigms: Proceeding of the Spanish Relativity Meeting 2011, Vol. 1458 (AIP Publishing, 2012), pp. 207–221. Google Scholar
27. S. Nojiri and S. D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, Int. J. Geom. Methods Mod. Phys. 4(01) (2007) 115–145. Link, Web of Science, Google Scholar
28. T. P. Sotiriou and V. Faraoni, $f (R)$ $f (R)$ theories of gravity, Rev. Mod. Phys. 82(1) (2010) 451. Web of Science, Google Scholar
29. T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology, Phys. Rep. 513(1) (2012) 1–189. Web of Science, Google Scholar
30. M. Elmardi, A. Abebe and A. Tekola, Chaplygin-gas solutions of $f (R)$ $f (R)$ Gravity, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1650120. Link, Web of Science, Google Scholar
31. S. Nojiri and S. D. Odintsov, Modified gravity with negative and positive powers of curvature: Unification of inflation and cosmic acceleration, Phys. Rev. D 68(12) (2003) 123512. Web of Science, Google Scholar
32. S. Nojiri and S. D. Odintsov, Unifying inflation with $Λ$ $Λ$ CDM epoch in modified $f (R)$ $f (R)$ gravity consistent with solar system tests, Phys. Lett. B 657(4) (2007) 238–245. Web of Science, Google Scholar
33. Y.-S. Song, W. Hu and I. Sawicki, Large scale structure of $f (R)$ $f (R)$ gravity, Phys. Rev. D 75(4) (2007) 044004. Web of Science, Google Scholar
34. A. Abebe, Á. de la Cruz-Dombriz and P. K. S. Dunsby, Large scale structure constraints for a class of $f (R)$ $f (R)$ theories of gravity, Phys. Rev. D 88(4) (2013) 044050. Web of Science, Google Scholar
35. A. de La Cruz-Dombriz, A. Dobado and A. L. Maroto, Evolution of density perturbations in $f (R)$ $f (R)$ theories of gravity, Phys. Rev. D 77(12) (2008) 123515. Web of Science, Google Scholar
36. S. Carloni, P. K. S. Dunsby and A. Troisi, Evolution of density perturbations in $f (R)$ $f (R)$ gravity, Phys. Rev. D 77(2) (2008) 024024. Web of Science, Google Scholar
37. K. N. Ananda, S. Carloni and P. K. S. Dunsby, Structure growth in $f (R)$ $f (R)$ theories of gravity with a dust equation of state, Class. Quantum Grav. 26(23) (2009) 235018. Web of Science, Google Scholar
38. A. Abebe and M. Elmardi, Irrotational-fluid cosmologies in fourth-order gravity, Int. J. Geom. Methods Mod. Phys. 12(10) (2015) 1550118. Link, Web of Science, Google Scholar
39. A. Abebe, P. K. S. Dunsby and D. Solomons, Integrability conditions of quasi-newtonian cosmologies in modified gravity, Int. J. Mod. Phys. D 26(6) (2016) 1750054. Link, Web of Science, Google Scholar
40. A. Abebe, Anti-newtonian cosmologies in $f (R)$ $f (R)$ gravity, Class. Quantum Grav. 31(11) (2014) 115011. Web of Science, Google Scholar
41. A. Abebe, D. Momeni and R. Myrzakulov, Shear-free anisotropic cosmological models in $f (R)$ $f (R)$ gravity, Gen. Relativ. Gravit. 48(4) (2016) 1–17. Web of Science, Google Scholar
42. S. Capozziello and M. Francaviglia, Extended theories of gravity and their cosmological and astrophysical applications, Gen. Relativ. Gravit. 40(2) (2008) 357–420. Web of Science, Google Scholar
43. A. de la Cruz-Dombriz, Some cosmological and astrophysical aspects of modified gravity theories, preprint (2010), arXiv:1004.5052. Google Scholar
44. T. Chiba, T. L. Smith and A. L. Erickcek, Solar system constraints to general $f (R)$ $f (R)$ gravity, Phys. Rev. D 75(12) (2007) 124014. Web of Science, Google Scholar
45. M. A. Resco, Á. de la Cruz-Dombriz, F. J. Llanes Estrada and V. Z. Castrillo, On neutron stars in $f (R)$ $f (R)$ theories: Small radii, large masses and large energy emitted in a merger, Phys. Dark Universe 13 (2016) 147–161. Web of Science, Google Scholar
46. T. P. Sotiriou, $f (R)$ $f (R)$ gravity and scalar–tensor theory, Class. Quantum Grav. 23(17) (2006) 5117. Web of Science, Google Scholar
47. V. Faraoni, de sitter space and the equivalence between $f (R)$ $f (R)$ and scalar-tensor gravity, Phys. Rev. D 75(6) (2007) 067302. Web of Science, Google Scholar
48. T. Faulkner, M. Tegmark, E. F. Bunn and Y. Mao, Constraining $f (R)$ $f (R)$ gravity as a scalar-tensor theory, Phys. Rev. D 76(6) (2007) 063505. Web of Science, Google Scholar
49. J. Ntahompagaze, A. Abebe and M. Mbonye, On $f (R)$ $f (R)$ gravity in scalar–tensor theories, Int. J. Geom. Methods Mod. Phys. 14(7) (2017) 1750107. Link, Web of Science, Google Scholar
50. T. P. Sotiriou and V. Faraoni, $f (R)$ $f (R)$ theories of gravity, Rev. Mod. Phys. 82(1) (2010) 451. Web of Science, Google Scholar
51. A. Abebe, Beyond Concordance Cosmology (Scholars Press, 2015). Google Scholar
52. A. V. Frolov, Singularity problem with $f (R)$ $f (R)$ models for dark energy, Phys. Rev. Lett. 101(6) (2008) 061103. Web of Science, Google Scholar
53. T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology, Phys. Rep. 513(1) (2012) 1–189. Web of Science, Google Scholar
54. G. J. Olmo, Post-newtonian constraints on $f (R)$ $f (R)$ cosmologies in metric and palatini formalism, Phys. Rev. D 72(8) (2005) 083505. Web of Science, Google Scholar
55. B. Li and J. D. Barrow, Cosmology of $f (R)$ $f (R)$ gravity in the metric variational approach, Phys. Rev. D 75(8) (2007) 084010. Web of Science, Google Scholar
56. S. Chakraborty and S. SenGupta, Solving higher curvature gravity theories, Eur. Phys. J. C 76(10) (2016) 552. Web of Science, Google Scholar
57. D. Sáez-Gómez, Modified $f (R)$ $f (R)$ gravity from scalar–tensor theory and inhomogeneous eos dark energy, Gen. Relativ. Gravit. 41(7) (2009) 1527–1538. Web of Science, Google Scholar
58. S. Nojiri, S. D. Odintsov and D. Sáez-Gómez, Cosmological reconstruction of realistic modified $F (R)$ $F (R)$ gravities, Phys. Lett. B 681(1) (2009) 74–80. Web of Science, Google Scholar
59. W. Zimdahl and J. C. Fabris, Chaplygin gas with non-adiabatic pressure perturbations, Class. Quantum Grav. 22 (2005) 4311–4324. Web of Science, Google Scholar
60. J. Lu, Cosmology with a variable generalized chaplygin gas, Phys. Lett. B 680(5) (2009) 404–410. Web of Science, Google Scholar
61. G. F. R. Ellis and M. S. Madsen, Exact scalar field cosmologies, Class. Quantum Grav. 8(4) (1991) 667. Web of Science, Google Scholar
62. V. Méndez, Exact scalar field cosmologies with a fluid, Class. Quantum Grav. 13(12) (1996) 3229. Web of Science, Google Scholar