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Anisotropic Einstein-aether cosmological models

B. Alhulaimi, A. Coley, P. Sandin

2013
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Journal of Mathematical Physics
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We investigate a class of spatially anisotropic cosmological models in Einsteinaether theory with a scalar field in which the self-interaction potential depends on the timelike aether vector field through the expansion and shear scalars. We derive the evolution equations in terms of expansion-normalized variables, which reduce to a dynamical system. We study the local stability of the equilibrium points of the dynamical system corresponding to physically realistic solutions, and find that there
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... are always ranges of values of the parameters of the models for which there exists an inflationary attractor. C 2013 AIP Publishing LLC. [http://dx.to 194.94.224.254. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 042503-2 Alhulaimi, Coley, and Sandin J. Math. Phys. 54, 042503 (2013) a scalar inflation which would dominate in any inflationary epoch), with a self-interaction potential V , V can now be a function of φ and the expansion rate and even the shear. 1 A. Exponential potentials Exponential potentials V 0 e −λφ arise naturally in various higher dimensional frameworks, such as in Kaluza-Klein theories and supergravity. 11 Scalar fields with exponential potentials in GR do not yield exponential inflation as with (for example) the harmonic potential. 5 However, they do lead to a power-law inflation provided the potential is not too steep. The limitation of steep potentials can be alleviated using multiple fields, which can cooperate via Hubble damping to yield assisted inflation. 12 In addition, for sufficiently flat potentials, exponential potentials yield a scaling solution which is a late time attractor. 13, 14 The dynamical properties of the positive exponential potentials leading to inflation in the Friedmann-Robertson-Walker (FRW) model have been widely studied 15-17 (also see Appendix A). The classical solution for the scale factor can be written as a power law, a ∝ t n , with n = 2/λ 2 ; in order to have an inflationary phase with the exponential potential, one requires the steepness parameter λ to satisfy λ 2 < 2. If the potential is steep, then it does not support inflation. The effective dynamics with an exponential potential have been more widely studied. 13, 14 The equilibrium points corresponding to sources consist of a subset of the (non-oscillatory) Jacobs anisotropic Bianchi I non-vacuum (massless scalar field) solutions. 14 Negative exponential potentials also lead to a rich physics, such as in the Ekpyrotic scenario. 18 In this paper, we are interested in the qualitative features of cosmological models in Einstein aether theory (and in particular in the presence of curvature and shear). A. The potential If the universe contains a single self-interacting scalar field φ (e.g., a scalar inflation which would dominate in any inflationary epoch), the self-interaction potential V is a function of φ but Downloaded 15 May 2013 to 194.94.224.254. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 042503-3 Alhulaimi, Coley, and Sandin J. Math. Phys. 54, 042503 (2013) Alhulaimi, Coley, and Sandin J. Math. Phys. 54, 042503 (2013) Alhulaimi, Coley, and Sandin J. Math. Phys. 54, 042503 (2013) FIG. 16.λ = 1 2 , b = 0; q for p 3 . J = +582784b 10 + 471552b 8 + 345984b 12 − 2b 28 + 127584b 14 , F 1 =J − 894b 22 − 124b 24 + 42b 26 + 28128b 16 + 312b 18 , J 3 = 2b 2 (−164864b 2 − 205824b 4 − 61440 − 82752b 8 − 156160b 6 + J ), J 4 = −294912 − 983040b 4 − 909312b 2 − 23142b 6 + F 1 , J 5 = 18b 6 + b 10 + 48b 4 + 48b 2 + 3b 8 , J 6 = −3b 10 − 30b 8 + b 12 − 80b 6 + 192b 2 + 128. There are also two cases for p 4 . First, when a = 0 andλ = 1 2 , the value of is

doi:10.1063/1.4802246
fatcat:wulyvszs75hgxcopwngjq4dnuq