We show that an $n$ -dimensional generalized Robertson–Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress–energy tensor for any $f (R)$ gravity model. Furthermore, we prove that a conformally flat GRW space-time is still a perfect fluid in both $f (R)$ and quadratic gravity where other curvature invariants are considered.

Keywords:

AMSC: 83C40, 83C05, 83C10

References

1. Y. Choquet-Bruhat, Cauchy Problem, in Gravitation: An Introduction to Current Research, L. Witten edn. (Wiley, New York, 1962). Google Scholar
2. S. Capozziello and S. Vignolo, The Cauchy problem for $f (R)$ -gravity: An overview, Int. J. Geom. Methods Mod. Phys. 9 (2012) 1250006. Link, Web of Science, Google Scholar
3. S. Capozziello and S. Vignolo, On the well formulation of the initial value problem of metric-affine $f (R)$ -gravity, Int. J. Geom. Methods Mod. Phys. 6 (2009) 985. Link, Web of Science, Google Scholar
4. K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests, Astrophys. Space Sci. 342(1) (2012) 155–228. Crossref, Web of Science, Google Scholar
5. M. C. Chaki and R. K. Maity, On quasi Einstein manifolds, Publ. Math. Debrecen 57 (2000) 257–306. Crossref, Web of Science, Google Scholar
6. R. Deszcz, M. Glogowska, M. Hotloś and Z. Sentürk, On certain quasi-Einstein semisymmetric hypersurfaces, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 41 (1998) 151–164. Google Scholar
7. H. A. Buchdahl, Non-linear Lagrangians and cosmological theory, Mon. Not. Roy. Astron. Soc. 150(1) (1970) 1–8. Crossref, Web of Science, Google Scholar
8. A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91(1) (1980) 99–102. Crossref, Web of Science, Google Scholar
9. S. Capozziello, Curvature quintessence, Int. J. Mod. Phys. D 11(4) (2002) 483. Link, Web of Science, Google Scholar
10. S. Capozziello and M. Francaviglia, Extended theories of gravity and their cosmological and astrophysical applications, Gen. Relativ. Gravit. 40(2–3) (2008) 357–420. Crossref, Web of Science, Google Scholar
11. T. S. Sotiriou and V. Faraoni, $f (R)$ theories of gravity, Rev. Mod. Phys. 82 (2010) 451. Crossref, Web of Science, Google Scholar
12. S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Phys. Rep. 692 (2017) 1. Crossref, Web of Science, Google Scholar
13. S. Capozziello and M. De Laurentis, Extended theories of gravity, Phys. Rep. 509(4–5) (2011) 167–321. Crossref, Web of Science, Google Scholar
14. S. Deser and B. Tekin, Energy in generic higher curvature gravity theories, Phys. Rev. D 67 (2003) 084009. Crossref, Web of Science, Google Scholar
15. S. Capozziello, M. De Laurentis and G. Lambiase, Cosmic relic abundance and $f (R)$ gravity, Phys. Lett. B 715(1–3) (2012) 1–8. Crossref, Web of Science, Google Scholar
16. S. Capozziello, V. F. Cardone and A. Troisi, Dark energy and dark matter as curvature effects? J. Cosmol. Astropart. Phys. 0608 (2006) 001. Crossref, Google Scholar
17. L. Alías, A. Romero and M. Sánchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker space-times, Gen. Relativ. Gravit. 27(1) (1995) 71–84. Crossref, Web of Science, Google Scholar
18. M. Brozos-Vázquez, E. Garcia-Rio and R. Vázquez-Lorenzo, Some remarks on locally conformally flat static space-times, J. Math. Phys. 46 (2005) 022501. Crossref, Web of Science, Google Scholar
19. M. Gutiérrez and B. Olea, Global decomposition of a Lorentzian manifold as a generalized Robertson–Walker space, Differential Geom. Appl. 27 (2009) 146–156. Crossref, Web of Science, Google Scholar
20. C. A. Mantica, L. G. Molinari and U. C. De, A condition for a perfect fluid space-time to be a generalized Robertson–Walker space-time, J. Math. Phys. 57(2) (2016) 022508; Erratum 57 (2016) 049901. Google Scholar
21. C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker space-times, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1650079. Link, Web of Science, Google Scholar
22. C. A. Mantica and L. G. Molinari, On the Weyl and the Ricci tensors of generalized Robertson–Walker space-times, J. Math. Phys. 57(10) (2016) 102502. Crossref, Web of Science, Google Scholar
23. A. Romero, R. N. Rubio and J. J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker space-times, Class. Quantum Gravit. 30(11) (2013) 115007. Crossref, Web of Science, Google Scholar
24. M. Sánchez, On the geometry of generalized Robertson–Walker space-times: Curvature and killing fields, Gen. Relativ. Gravit. 31 (1999) 1–15. Crossref, Web of Science, Google Scholar
25. C. A. Mantica and S. Shenawy, Einstein-like warped product manifolds, Int. J. Geom. Methods Mod. Phys. 14(11) (2017) 1750166. Link, Web of Science, Google Scholar
26. M. Sánchez, On the geometry of generalized Robertson–Walker space-times: Geodesics, Gen. Relativ. Gravit. 30 (1998) 915–932. Crossref, Web of Science, Google Scholar
27. C. A. Mantica and L. G. Molinari, Generalized Robertson–Walker space-times, a survey, Int. J. Geom. Methods Mod. Phys. 14(3) (2017) 1730001. Link, Web of Science, Google Scholar
28. B.-Y. Chen, A simple characterization of generalized Robertson–Walker space-times, Gen. Relativ. Gravit. 46 (2014) 1833. Crossref, Web of Science, Google Scholar
29. B.-Y. Chen, Differential Geometry of Warped Product Manifolds and Submanifolds (World Scientific, Singapore, 2017). Link, Google Scholar
30. C. A. Mantica and L. G. Molinari, Riemann compatible tensors, Colloq. Math. 128 (2012) 197–210. Crossref, Web of Science, Google Scholar
31. C. A. Mantica and L. G. Molinari, Weyl compatible tensors, Int. J. Geom. Methods Mod. Phys. 11(8) (2014) 1450070. Link, Web of Science, Google Scholar
32. C. A. Mantica and L. G. Molinari, Twisted Lorentzian manifolds: A characterization with torse-forming time-like unit vectors, Gen. Relativ. Gravit. 49 (2017) 51. Crossref, Web of Science, Google Scholar
33. S. Capozziello, M. De Laurentis and O. Luongo, Connecting early and late universe by $f (R)$ gravity, Int. J. Mod. Phys. D 24(4) (2014) 1541002. Link, Web of Science, Google Scholar
34. R. M. Wald, General Relativity (The University of Chicago Press, Chicago, 1984). Crossref, Google Scholar
35. J. D. Barrow and S. Cotsakis, Inflation and the conformal structure of higher-order gravity theories, Phys. Lett. B 214(4) (1988) 515–518. Crossref, Web of Science, Google Scholar
36. K. Yano, On the torseforming directions in Riemannian Spaces, Proc. Imp. Acad. Tokyo 20 (1944) 340–345. Crossref, Google Scholar
37. M. De Laurentis and A. J. Lopez-Revelles, Newtonian, Post Newtonian and parameterized post Newtonian limits of $f (R, G)$ gravity, Int. J. Geom. Methods Mod. Phys. 11 (2014) 1450082. Link, Web of Science, Google Scholar
38. S. Capozziello, M. De Laurentis and S. D. Odintsov, Noether symmetry approach in Gauss–Bonnet cosmology, Mod. Phys. Lett. A 29(30) (2014) 1450164. Link, Web of Science, Google Scholar
39. P. K. S. Dunsby and O. Luongo, On the theory and applications of modern cosmography, Int. J. Geom. Methods Mod. Phys. 13(3) (2016) 1630002. Link, Web of Science, Google Scholar