Shear-free anisotropic cosmological models in \(f(R)\) gravity. (English) Zbl 1338.83180
Summary: We study a class of shear-free, homogeneous but anisotropic cosmological models with imperfect matter sources in the context of \(f(R)\) gravity. We show that the anisotropic stresses are related to the electric part of the Weyl tensor in such a way that they balance each other. We also show that within the class of orthogonal \(f(R)\) models, small perturbations of shear are damped, and that the electric part of the Weyl tensor and the anisotropic stress tensor decay with the expansion as well as the heat flux of the curvature fluid. Specializing in locally rotationally symmetric spacetimes in orthonormal frames, we examine the late-time behaviour of the de Sitter universe in \(f(R)\) gravity. For the Starobinsky model of \(f(R)\), we study the evolutionary behavior of the Universe by numerically integrating the Friedmann equation, where the initial conditions for the expansion, acceleration and jerk parameters are taken from observational data.
MSC:
| 83F05 | Relativistic cosmology |
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
Keywords:
shear-free spacetimes; homogeneity; cosmic anisotropy; modified gravity; expansion; inflation; cosmological perturbationsReferences:
| [1] | Capozziello, S., De Laurentis, M.: Extended theories of gravity. Phys. Rep. 509, 167-321 (2011) · doi:10.1016/j.physrep.2011.09.003 |
| [2] | Modesto, L.: Super-renormalizable quantum gravity. Phys. Rev. D 86, 044005 (2012) · doi:10.1103/PhysRevD.86.044005 |
| [3] | Modesto, L., Rachwal, L.: Super-renormalizable and finite gravitational theories. Nucl. Phys. B 889, 228-248 (2014) · Zbl 1326.83061 · doi:10.1016/j.nuclphysb.2014.10.015 |
| [4] | Biswas, T., Gerwick, E., Koivisto, T., Mazumdar, A.: Towards singularity and ghost free theories of gravity. Phys. Rev. Lett. 108, 031101 (2012) · doi:10.1103/PhysRevLett.108.031101 |
| [5] | Clifton, T., Ferreira, P.G., Padilla, A., Skordis, C.: Modified gravity and cosmology. Phys. Rep. 513, 1-189 (2012) · doi:10.1016/j.physrep.2012.01.001 |
| [6] | De Felice, A., Tsujikawa, \[S.: f(R)\] f(R) theories. Living Rev. Rel. 13, 1002-4928 (2010) · Zbl 1215.83005 |
| [7] | Nojiri, S., Odintsov, S.D.: Introduction to modified gravity and gravitational alternative for dark energy. Int. J. Geom. Methods Mod. Phys. 4, 115-145 (2007) · Zbl 1112.83047 · doi:10.1142/S0219887807001928 |
| [8] | Nojiri, S., Odintsov, S.D.: Unified cosmic history in modified gravity: from f (r) theory to lorentz non-invariant models. Phys. Rep. 505, 59-144 (2011) · doi:10.1016/j.physrep.2011.04.001 |
| [9] | Buchdahl, H.A.: Non-linear Lagrangians and cosmological theory. Mon. Not. R. Astron. Soc. 150, 1 (1970) · doi:10.1093/mnras/150.1.1 |
| [10] | Starobinsky, A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99-102 (1980) · Zbl 1371.83222 · doi:10.1016/0370-2693(80)90670-X |
| [11] | Carroll, S., Duvvuri, V., Turner, M., Trodden, M.: Is cosmic speed-up due to new gravitational physics? Phys. Rev. D 70, 043528 (2004) · doi:10.1103/PhysRevD.70.043528 |
| [12] | Nojiri, S., Odintsov, S.D.: Modified gravity with negative and positive powers of curvature: unification of inflation and cosmic acceleration. Phys. Rev. D 68, 123512 (2003) · doi:10.1103/PhysRevD.68.123512 |
| [13] | Sotiriou, T.P., Liberati, S.: Metric-affine \[f(R)\] f(R) theories of gravity. Ann. Phys. 322, 935-966 (2007) · Zbl 1112.83049 · doi:10.1016/j.aop.2006.06.002 |
| [14] | Nojiri, S., Odintsov, S.D.: Modified \[f(R)\] f(R) gravity consistent with realistic cosmology: from a matter dominated epoch to a dark energy universe. Phys. Rev. D 74, 086005 (2006) · doi:10.1103/PhysRevD.74.086005 |
| [15] | Starobinsky, A.A.: Disappearing cosmological constant in \[f(R)\] f(R) gravity. JETP Lett. 86, 157-163 (2007) · doi:10.1134/S0021364007150027 |
| [16] | Mimoso, J.P., Crawford, P.: Shear-free anisotropic cosmological models. Class. Quantum Gravity 10, 315 (1993) · doi:10.1088/0264-9381/10/2/013 |
| [17] | Ellis, G.: Dynamics of pressure-free matter in general relativity. J. Math. Phys. 8, 1171 (1967) · doi:10.1063/1.1705331 |
| [18] | G ö del, K.: Rotating universes in general relativity theory . In: Graves, L.M. et al. Proceedings of the International Congress of Mathematicians, vol. 1, p. 175. Cambridge, Mass (1952) · Zbl 1087.83053 |
| [19] | Goldberg, J., Sachs, R.: A theorem on petrov type (field equations for proving theorem identifying geometrical properties of null congruence with existence of algebraically special riemann tensor). 1966. 13-23 (1962) · Zbl 1096.83053 |
| [20] | Robinson, I., Schild, A.: Generalization of a theorem by goldberg and sachs. J. Math. Phys. 4, 484 (1963) · Zbl 0114.43903 · doi:10.1063/1.1703980 |
| [21] | Ellis, G., MacCallum, M.A.: A class of homogeneous cosmological models. Commun. Math. Phys. 12, 108-141 (1969) · Zbl 0177.57601 · doi:10.1007/BF01645908 |
| [22] | MacCallum, M.: A class of homogeneous cosmological models iii: asymptotic behaviour. Commun. Math. Phys. 20, 57-84 (1971) · doi:10.1007/BF01646733 |
| [23] | Collins, C.: Shear-free fluids in general relativity. Can. J. Phys. 64, 191-199 (1986) · doi:10.1139/p86-034 |
| [24] | Barrow, J.D., Matzner, R.A.: The homogeneity and isotropy of the universe. Mon. Not. R. Astron. Soc. 181, 719-727 (1977) · doi:10.1093/mnras/181.4.719 |
| [25] | MacCallum, M., Stewart, J., Schmidt, B.: Anisotropic stresses in homogeneous cosmologies. Commun. Math. Phys. 17, 343-347 (1970) · doi:10.1007/BF01646029 |
| [26] | Koivisto, T.S., Mota, D.F., Quartin, M., Zlosnik, T.G.: Possibility of anisotropic curvature in cosmology. Phys. Rev. D 83, 023509 (2011) · doi:10.1103/PhysRevD.83.023509 |
| [27] | Barrow, J.D., Ottewill, A.C.: The stability of general relativistic cosmological theory. J. Phys. A Math. Gen. 16, 2757 (1983) · Zbl 0518.53029 · doi:10.1088/0305-4470/16/12/022 |
| [28] | Barrow, J.D., Clifton, T.: Exact cosmological solutions of scale-invariant gravity theories. Class. Quant. Gravity 23, L1 (2006) · Zbl 1087.83053 · doi:10.1088/0264-9381/23/1/L01 |
| [29] | Clifton, T., Barrow, J.D.: Further exact cosmological solutions to higher-order gravity theories. Class. Quant. Gravity 23, 2951 (2006) · Zbl 1096.83053 · doi:10.1088/0264-9381/23/9/011 |
| [30] | Middleton, J.: On the existence of anisotropic cosmological models in higher order theories of gravity. Class. Quant. Gravity 27, 225013 (2010) · Zbl 1204.83073 · doi:10.1088/0264-9381/27/22/225013 |
| [31] | Abebe, A., Goswami, R., Dunsby, P.K.S.: Shear-free perturbations of \[f(R)\] f(R) gravity. Phys. Rev. D 84, 124027 (2011) · doi:10.1103/PhysRevD.84.124027 |
| [32] | Abebe, A.: Beyond concordance cosmology. Ph.D. thesis, UCT (University of Cape Town) (2013) · Zbl 1065.83048 |
| [33] | Abebe, A.: Anti-newtonian cosmologies in \[f(R)\] f(R) gravity. Class. Quant. Gravity 31, 115011 (2014) · Zbl 1293.83011 · doi:10.1088/0264-9381/31/11/115011 |
| [34] | Abebe, A., Elmardi, M.: Irrotational-fluid cosmologies in fourth-order gravity. arXiv preprint arXiv:1411.6394 (2014) · Zbl 1331.83109 |
| [35] | Ellis, GFR; Elst, H.; Lachiéze-Rey, M. (ed.), Cosmological models, 1-116 (1999), The Netherlands · doi:10.1007/978-94-011-4455-1_1 |
| [36] | Maartens, R., Triginer, J.: Density perturbations with relativistic thermodynamics. Phys. Rev. D 56, 4640 (1997) · doi:10.1103/PhysRevD.56.4640 |
| [37] | Betschart, G.: General relativistic electrodynamics with applicantions in cosmology and astrophysics. Ph.D. thesis, University of Cape Town (2005) · Zbl 1326.83061 |
| [38] | Carloni, S., Dunsby, P., Troisi, A.: Evolution of density perturbations in \[f(R)\] f(R) gravity. Phys. Rev. D 77, 024024 (2008) · doi:10.1103/PhysRevD.77.024024 |
| [39] | Ellis, G., Bruni, M.: Covariant and gauge-invariant approach to cosmological density fluctuations. Phys. Rev. D 40, 1804 (1989) · doi:10.1103/PhysRevD.40.1804 |
| [40] | Ellis, G., Maartens, R., MacCallum, M.A.: Relativistic Cosmology. Cambridge University Press, Cambridge (2012) · Zbl 1242.83001 · doi:10.1017/CBO9781139014403 |
| [41] | Maartens, R.: Covariant velocity and density perturbations in quasi-newtonian cosmologies. Phys. Rev. D 58, 124006 (1998) · doi:10.1103/PhysRevD.58.124006 |
| [42] | Israel, W.: Nonstationary irreversible thermodynamics: a causal relativistic theory. Ann. Phys. 100 (1976). http://gen.lib.rus.ec/scimag/index.php?s=10.1016/0003-4916(76)90064-6 |
| [43] | Novella, M.: Cosmology and Gravitation Two, vol. 2, Atlantica S é guier Fronti è res, (1996) |
| [44] | Maartens, R. Causal thermodynamics in relativity. arXiv preprint astro-ph/9609119 (1996) · Zbl 1054.83046 |
| [45] | Rezzolla, L., Zanotti, O.: Relativistic Hydrodynamics. Oxford University Press, Oxford (2013) · Zbl 1297.76002 · doi:10.1093/acprof:oso/9780198528906.001.0001 |
| [46] | Coley, A.A., Van den Hoogen, R.: Qualitative analysis of viscous fluid cosmological models satisfying the israel-stewart theory of irreversible thermodynamics. Class. Quant. Gravity 12, 1977 (1995) · Zbl 0828.58048 · doi:10.1088/0264-9381/12/8/015 |
| [47] | Visser, M.: Jerk, snap and the cosmological equation of state. Class. Quant. Gravity 21, 2603 (2004) · Zbl 1054.83046 · doi:10.1088/0264-9381/21/11/006 |
| [48] | Nojiri, S., Odintsov, S, & Tsujikawa, S.: Properties of singularities in (phantom) dark energy universe. Phys. Rev. d71 063004 (2005). arXiv preprint hep-th/0501025 |
| [49] | Brandenberger, R. H.: The matter bounce alternative to inflationary cosmology. arXiv preprint arXiv:1206.4196 (2012) |
| [50] | Cai, Y.-F.: Exploring bouncing cosmologies with cosmological surveys. Sci. China Phys. Mech. Astron. 57, 1414-1430 (2014) · doi:10.1007/s11433-014-5512-3 |
| [51] | Bamba, K., Nojiri, S., Odintsov, S.D.: The future of the universe in modified gravitational theories: approaching a finite-time future singularity. J. Cosmol. Astropart. Phys. 2008, 045 (2008) · doi:10.1088/1475-7516/2008/10/045 |
| [52] | Barrow, J.D.: More general sudden singularities. Class. Quant. Gravity 21, 5619 (2004) · Zbl 1065.83048 · doi:10.1088/0264-9381/21/23/020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
