×

Qualitative study of a model with Rastall gravity. (English) Zbl 1479.83284

Summary: We consider the Rastall theory for the flat Friedmann-Robertson-Walker Universe filled with a perfect fluid that satisfies a linear equation of state. The corresponding dynamical system is a two dimensional system of polynomial differential equations depending on four parameters. We show that this differential system is always Darboux integrable. In order to study the global dynamics of this family of differential systems we classify all their non-topological equivalent phase portraits in the Poincaré disc and we obtain 16 different dynamical situations for our spacetime.

MSC:

83F05 Relativistic cosmology
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
83E05 Geometrodynamics and the holographic principle
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Darabi F, Moradpour H, Licata I, Heydarzade Y and Corda C 2018 Einstein and rastall theories of gravitation in comparison Eur. Phys. J. C 78 25 · doi:10.1140/epjc/s10052-017-5502-5
[2] Fabris J C, Piattella O F, Batista C E M, Daouda M H and Rodrigues D C 2012 Rastall cosmology and the λ CDM model Phys. Rev. D 85 084008 · doi:10.1103/physrevd.85.084008
[3] Cruz M, Lepe S and Morales-Navarrete G 2019 A thermodynamics revision of Rastall gravity Class. Quantum Grav.36 225007 · Zbl 1478.83241 · doi:10.1088/1361-6382/ab45ab
[4] Birrell N D and Davies P C W 1982 Quantum Fields in Curved Space (Cambridge: Cambridge University Press) · Zbl 0972.81605 · doi:10.1017/CBO9780511622632
[5] Silva G F, Piattella O F, Fabris J C, Casarini L and Barbosa T O 2013 Bouncing solutions in Rastall’s theory with a barotropic fluid Gravit. Cosmol.19 156-62 · Zbl 1278.83046 · doi:10.1134/s0202289313030109
[6] Gibbons G W and Hawking S W 1977 Cosmological event horizons, thermodynamics, and particle creation Phys. Rev. D 15 2738 · doi:10.1103/physrevd.15.2738
[7] Ford L H 1987 Gravitational particle creation and inflation Phys. Rev. D 35 2955 · doi:10.1103/physrevd.35.2955
[8] Batista C E M, Fabris J C, Piattella O F and Velasquez-Toribio A M 2013 Observational constraints on Rastall’s cosmology Eur. Phys. J. C 73 2425 · doi:10.1140/epjc/s10052-013-2425-7
[9] Khyllep W and Dutta J 2019 Linear growth index of matter perturbations in Rastall gravity Phys. Lett. B 797 134796 · Zbl 1427.83070 · doi:10.1016/j.physletb.2019.134796
[10] Heydarzade Y, Moradpour H and Darabi F 2017 Black hole solutions in Rastall theory Can. J. Phys.95 1253-6 · doi:10.1139/cjp-2017-0254
[11] Moradpour H 2016 Thermodynamics of flat FLRW Universe in Rastall theory Phys. Lett. B 757 187 · Zbl 1360.83056 · doi:10.1016/j.physletb.2016.03.072
[12] Moradpour H, Heydarzadeand Y, Darabi F and Salako I G 2017 A generalization to the Rastall theory and cosmic eras Eur. Phys. J. C 77 259 · doi:10.1140/epjc/s10052-017-4811-z
[13] Smalley L L 1983 Rastall’s and related theories are conservative gravitational theories although physically inequivalent to general relativity J. Phys. A: Math. Gen.16 2179-85 · doi:10.1088/0305-4470/16/10/014
[14] Nojiri S I and Odintsov S D 2004 Gravity assisted dark energy dominance and cosmic acceleration Phys. Lett. B 599 137-42 · doi:10.1016/j.physletb.2004.08.045
[15] Koivisto T 2006 A note on covariant conservation of energy-momentum in modified gravities Class. Quantum Grav.23 4289 · Zbl 1096.83056 · doi:10.1088/0264-9381/23/12/n01
[16] Harko T and Lobo F 2014 Generalized curvature-matter couplings in modified gravity Galaxies2 410-65 · doi:10.3390/galaxies2030410
[17] Visser M 2018 Rastall gravity is equivalent to Einstein gravity Phys. Lett. B 782 83-6 · Zbl 1404.83009 · doi:10.1016/j.physletb.2018.05.028
[18] Rastall P 1972 Generalization of the Einstein theory Phys. Rev. D 6 3357-9 · Zbl 0959.83525 · doi:10.1103/physrevd.6.3357
[19] Hansraj S, Banerjee A and Channuie P 2019 Impact of the Rastall parameter on perfect fluid spheres Ann. Phys., NY400 320-45 · Zbl 1415.83043 · doi:10.1016/j.aop.2018.12.003
[20] Dumortier F, Llibre J and Artés J C 2006 Qualitative Theory of Planar Polynomial Systems (Berlin: Springer) · Zbl 1110.34002
[21] Lax P D 1968 Integrals of nonlinear equations of evolution and solitary waves Commun. Pure Appl. Math.21 467-90 · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[22] Olver P J 1986 Applications of Lie Groups to Differential Equations (Berlin: Springer) · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2
[23] Cantrijn F and Sarlet W 1981 Generalizations of Noether’s theorem in classical mechanics SIAM Rev.23 467-94 · Zbl 0474.70014 · doi:10.1137/1023098
[24] Darboux G 1878 Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (mélanges) Bull. Sci. Math.2 60-96 · JFM 10.0214.01
[25] Bountis T C, Ramani A, Grammaticos B and Dorizzi B 1984 On the complete and partial integrability of non-Hamiltonian systems Physica A 128 268-88 · Zbl 0599.34002 · doi:10.1016/0378-4371(84)90091-8
[26] Morales-Ruiz J J 1999 Differential Galois theory and non-integrability of Hamiltonian systems Progress in Mathematics vol 179 (Basel: Birkhäuser) · Zbl 0934.12003
[27] Das D, Dutta S and Chakraborty S 2018 Cosmological consequences in the framework of generalized Rastall theory of gravity Eur. Phys. J. C 78 810 · doi:10.1140/epjc/s10052-018-6293-z
[28] Babichev E, Dokuchaev V and Eroshenko Y 2005 Dark energy cosmology with generalized linear equation of state Class. Quantum Grav.22 143-54 · Zbl 1060.83059 · doi:10.1088/0264-9381/22/1/010
[29] Rastall P 1976 A theory of gravity Can. J. Phys.54 66-75 · Zbl 1043.83558 · doi:10.1139/p76-008
[30] Wainwright J and Ellis G F R 1997 Dynamical Systems in Cosmology (Cambridge: Cambridge University Press) · Zbl 1072.83002 · doi:10.1017/CBO9780511524660
[31] Kofinas G, Leon G and Saridakis E N 2014 Class. Quantum Grav.31 175011 · Zbl 1300.83050 · doi:10.1088/0264-9381/31/17/175011
[32] Heisenberg L, Kimura R and Yamamoto K 2014 Cosmology of the proxy theory to massive gravity Phys. Rev.D89 103008 · doi:10.1103/physrevd.89.103008
[33] Goheer N, Goswami R and Dunsby P K S 2009 Dynamics of f(R)-cosmologies containing Einstein static models Class. Quantum Grav.26 105003 · Zbl 1166.83020 · doi:10.1088/0264-9381/26/10/105003
[34] Ganguly A, Gannouji R, Goswami R and Ray S 2015 Global structure of black holes via the dynamical system Class. Quantum Grav.32 105006 · Zbl 1328.83088 · doi:10.1088/0264-9381/32/10/105006
[35] Cruz M, Ganguly A, Gannouji R, Leon G and Saridakis E N 2017 Global structure of static spherically symmetric solutions surrounded by quintessence Class. Quantum Grav.34 125014 · Zbl 1367.83021 · doi:10.1088/1361-6382/aa70fc
[36] Moradpour H and Salako I G 2016 Thermodynamic analysis of the static spherically symmetric field equations in Rastall theory Adv. High Energy Phys.2016 5 · Zbl 1366.83083 · doi:10.1155/2016/3492796
[37] Yuan F-F and Huang P 2017 Emergent cosmic space in Rastall theory Class. Quantum Grav.34 077001 · Zbl 1368.83070 · doi:10.1088/1361-6382/aa61df
[38] Capone M, Cardone V F and Ruggiero M L 2010 The possibility of an accelerating cosmology in Rastall’s theory J. Phys.: Conf. Ser.222 012012 · doi:10.1088/1742-6596/222/1/012012
[39] Ziaie A H, Moradpour H and Ghaffari S 2019 Gravitational collapse in Rastall gravity Phys. Lett. B 793 276-80 · Zbl 1421.83092 · doi:10.1016/j.physletb.2019.04.055
[40] Christopher C, Llibre J and Pereira J V 2007 Multiplicity of invariant algebraic curves in polynomial vector fields Pacific J. Math.229 63-117 · Zbl 1160.34003 · doi:10.2140/pjm.2007.229.63
[41] Coppel W A 1966 A survey of quadratic systems J. Differ. Equ.2 293-304 · Zbl 0143.11903 · doi:10.1016/0022-0396(66)90070-2
[42] Oliveira A M, Velten H E S, Fabris J C and Casarini L 2015 Neutron stars in Rastall gravity Phys. Rev. D 92 044020 · doi:10.1103/physrevd.92.044020
[43] Hirsch M W, Pugh C C and Shub M 1977 Invariant Manifolds(Lecture Notes in Mathematics) (Berlin: Springer) · Zbl 0355.58009 · doi:10.1007/BFb0092042
[44] Markus L 1954 Global structure of ordinary differential equations in the plane Trans. Am. Math. Soc.76 127 · Zbl 0055.08102 · doi:10.1090/s0002-9947-1954-0060657-0
[45] Neumann D A 1975 Classification of continuous flows on 2-manifolds Proc. Am. Math. Soc.48 73 · Zbl 0307.34044 · doi:10.1090/s0002-9939-1975-0356138-6
[46] Li W, Llibre J, Nicolau M and Zhang X 2002 On the differentiability of first integrals of two dimensional flows Proc. Am. Math. Soc.130 2079-88 · Zbl 1010.34023 · doi:10.1090/s0002-9939-02-06310-4
[47] Peixoto M M 1973 Proccedings of a Symposium Held at the University of BahiaDynamical Systems (New York: Academic) 389-420
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.