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On the oscillations and future asymptotics of locally rotationally symmetric Bianchi type III cosmologies with a massive scalar field. (English) Zbl 1478.83248

Summary: We analyse spatially homogenous cosmological models of locally rotationally symmetric Bianchi type III with a massive scalar field as matter model. Our main result concerns the future asymptotics of these spacetimes and gives the dominant time behaviour of the metric and the scalar field for all solutions for late times. This metric is forever expanding in all directions, however, in one spatial direction only at a logarithmic rate, while at a power-law rate in the other two. Although the energy density goes to zero, it is matter dominated in the sense that the metric components differ qualitatively from the corresponding vacuum future asymptotics. Our results rely on a conjecture for which we give strong analytical and numerical support. For this we apply methods from the theory of averaging in nonlinear dynamical systems. This allows us to control the oscillations entering the system through the scalar field by the Klein-Gordon equation in a perturbative approach.

MSC:

83F05 Relativistic cosmology
53C30 Differential geometry of homogeneous manifolds
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
83E05 Geometrodynamics and the holographic principle
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
70K65 Averaging of perturbations for nonlinear problems in mechanics
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[1] Alho A, Bessa V and Mena F C 2020 Global dynamics of the Einstein-Euler-Yang-Mills system in flat Robertson-Walker cosmologies J. Math. Phys.61 032502 · Zbl 1441.83005 · doi:10.1063/1.5139879
[2] Alho A, Hell J and Uggla C 2015 Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids Class. Quantum Grav.32 145005 · Zbl 1327.83011 · doi:10.1088/0264-9381/32/14/145005
[3] Alho A and Uggla C 2015 Global dynamics and inflationary center manifold and slow-roll approximants J. Math. Phys.56 012502 · Zbl 1309.83029 · doi:10.1063/1.4906081
[4] Andréasson H 2011 The Einstein-Vlasov system/kinetic theory Living Rev. Relativ.14 · Zbl 1316.83021 · doi:10.12942/lrr-2011-4
[5] Barzegar H, Fajman D and Heissel G 2020 Isotropization of slowly expanding spacetimes Phys. Rev. D 101 044046 · doi:10.1103/physrevd.101.044046
[6] Calogero S and Heinzle J M 2011 Bianchi cosmologies with anisotropic matter: Locally rotationally symmetric models Phys. D 240 636-69 · Zbl 1219.37058 · doi:10.1016/j.physd.2010.11.015
[7] Carr J 1981 Applications of Cetnre Manifold Theory(Applied Mathematical Sciences vol 35) (New York, NY: Springer) · Zbl 0464.58001 · doi:10.1007/978-1-4612-5929-9
[8] Coley A A 2003 Dynamical Systems and cosmology(Astrophysics and Space Science Library vol 291) (Dordrecht: Kluwer Academic Publishers) · Zbl 1055.83001 · doi:10.1007/978-94-017-0327-7
[9] Fajman D and Heissel G 2019 Kantowski-Sachs cosmology with Vlasov matter Class. Quantum Grav.36 135002 · Zbl 1477.83094 · doi:10.1088/1361-6382/ab2425
[10] Fajman D and Wyatt Z 2019 Attractors of the Einstein-Klein-Gordon system (arXiv:1901.10378)
[11] Horwood J T, Hancock M J, The D and Wainwright J 2003 Late-time asymptotic dynamics of Bianchi VIII cosmologies Class. Quantum Grav.20 1757-77 · Zbl 1034.83029 · doi:10.1088/0264-9381/20/9/312
[12] Ionescu A D and Pausader B 2019 The Einstein-Klein-Gordon coupled system: global stability of the Minkowski solution (arXiv:1911.10652)
[13] Lee H, Nungesser E and Tod P 2019 On the future of solutions to the massless Einstein-Vlasov system in a Bianchi I cosmology (arXiv:1911.04937)
[14] Lefloch P G and Ma Y 2016 The global nonlinear stability of minkowski space for self-gravitating massive fields Commun. Math. Phys.346 603-65 · Zbl 1359.83003 · doi:10.1007/s00220-015-2549-8
[15] Nilsson U S, Hancock M J and Wainwright J 2000 Non-tilted Bianchi VII0 models—the radiation fluid Class. Quantum Grav.17 3119-34 · Zbl 1073.83503 · doi:10.1088/0264-9381/17/16/303
[16] Perko L 2001 Differential Equations and Dynamical Systems(Texts in Applied Mathematics vol 7) 3 edn (New York, NY: Springer) · Zbl 0973.34001 · doi:10.1007/978-1-4613-0003-8
[17] Rendall A D 2002 Cosmological models and centre manifold theory Gen. Relativ. Gravit.34 1277-94 · Zbl 1016.83044 · doi:10.1023/a:1019734703162
[18] Rendall A D 2004 The Einstein-Vlasov System The Einstein Equations and the Large Scale Behavior of Gravitational Fields ed Chruściel P T and Friedrich H (Basel: Birkhäuser) pp 231-50 · Zbl 1064.83007 · doi:10.1007/978-3-0348-7953-8_6
[19] Rendall A D 2008 Partial differential equations in General Relativity(Oxford Graduate Texts in Mathematics vol 16) (Oxford: Oxford University Press) · Zbl 1148.35002
[20] Rendall A D and Uggla C 2000 Dynamics of spatially homogeneous locally rotationally symmetric solutions of the Einstein-Vlasov equations Class. Quantum Grav.17 4697 · Zbl 0988.83034 · doi:10.1088/0264-9381/17/22/310
[21] Ringström H 2013 On the Topology and Future Stability of the Universe(Oxford Mathematical Monographs) (Oxford: Oxford University Press) · Zbl 1270.83005 · doi:10.1093/acprof:oso/9780199680290.001.0001
[22] Ryan M P and Shepley L C 1975 Homogenous Relativistic Cosmologies (Princeton, NJ: Princeton University Press)
[23] Sanders J A, Verhulst F and Murdock J 2007 Averaging Methods in Nonlinear Dynamical Systems(Applied Mathematical Sciences vol 59) 2 edn (Berlin: Springer) · Zbl 1128.34001
[24] Wainwright J 2000 Asymptotic self-similarity breaking in cosmology Gen. Relativ. Gravit.32 1041-54 · Zbl 1073.83534 · doi:10.1023/a:1001917610163
[25] Wainwright J and Ellis G F R (ed) 1997 Dynamical Systems in cosmology (Cambridge: Cambridge University Press) · Zbl 1072.83002 · doi:10.1017/CBO9780511524660
[26] Wainwright J, Hancock M J and Uggla C 1999 Asymptotic self-similarity breaking at late times in cosmology Class. Quantum Grav.16 2577-98 · Zbl 0946.83067 · doi:10.1088/0264-9381/16/8/302
[27] WANG J 2019 Future stability of the 1 + 3 Milne model for the Einstein-Klein-Gordon system Class. Quantum Grav.36 225010 · Zbl 1478.83215 · doi:10.1088/1361-6382/ab4dd3
[28] Wang Q 2016 An intrinsic hyperboloid approach for Einstein Klein-Gordon equations (arXiv:1607.01466)
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