Journal of Physics: Conference Series FRW models in the conformal frame of fr gravity To cite this article: J Miritzis 2011 J. Phys.: Conf. Ser. 283 012024 View the article online for updates and enhancements. Related content - Energy exchange for homogeneous and isotropic universes with a scalar field coupled to matter Roberto Giambò and John Miritzis - Scalar-field cosmologies with an arbitrary potential John Miritzis - A conventional form of dark energy K Kleidis and N K Spyrou This content was downloaded from IP address 148.251.232.83 on 09/05/2018 at 03:43
Recent Developments in Gravity NEB XIV Journal of Physics: Conference Series 283 2011 012024 FRW models in the conformal frame of fr gravity 1 J Miritzis Department of Marine Sciences, University of the Aegean, University Hill, Mytilene 81100, Greece E-mail: imyr@aegean.gr Abstract. We study the late time evolution of Friedmann-Robertson-Walker FRW models with a perfect fluid matter source and a scalar field arising in the conformal frame of fr theories nonminimally coupled to matter. We prove using the approach of dynamical systems, that equilibria corresponding to non-negative local minima for V are asymptotically stable. We show that if γ, the parameter of the equation of state is larger than one, then there is a transfer of energy from the fluid to the scalar field and the latter eventually dominates. The results are valid for a large class of nonnegative potentials without any particular assumptions about the behavior of the potential at infinity. 1. Introduction In this paper we study the late time evolution of initially expanding flat FRW models with a scalar field having an arbitrary bounded from below potential function V φ. Since the nature of the scalar field supposed to cause accelerated expansion is unknown, it is important to investigate the general properties shared by all FRW models with a scalar field irrespectively of the particular choice of the potential. The scalar field is nonminimally coupled to ordinary matter described by a barotropic fluid with equation of state p = γ 1ρ, 0 < γ 2. The motivation comes from higher order gravity HOG theories [1] which are conformaly equivalent to the Einstein field equations with a scalar field φ as an additional matter source. 2. Flat and negatively curved FRW with an arbitrary non-negative potential For homogeneous and isotropic spacetimes the field equations, see for example [2, 3], reduce to the Friedmann equation, H 2 + k a 2 = 1 ρ + 1 3 2 φ 2 + V φ, 1 and the Raychaudhuri equation, 1 In collaboration with R. Giambò, University of Camerino, Italy. Ḣ = 1 2 φ 2 γ 2 ρ + k a2, 2 Published under licence by Ltd 1
Recent Developments in Gravity NEB XIV Journal of Physics: Conference Series 283 2011 012024 while the equation of motion of the scalar field becomes The Bianchi identities yield the conservation equation, φ + 3H φ + V φ = 4 3γ 6 ρ. 3 ρ + 3γρH = 4 3γ 6 ρ φ. 4 We adopt the metric and curvature conventions of [4]. at is the scale factor, an overdot denotes differentiation with respect to time t, H = ȧ/a and units have been chosen so that c = 1 = 8πG. Here V φ is the potential energy of the scalar field and V = dv/dφ. In the following we consider initially expanding solutions of 2-4, i.e. H0 > 0. For flat, k = 0, models the state vector of the system 2-4 is φ, φ,ρ,h, i.e. we have a four-dimensional dynamical system subject to the constraint 1. Defining y := φ and setting 4 3γ/ 6 =: α, we write the autonomous system as φ = y, ẏ = 3Hy V φ + αρ, ρ = 3γρH αρy, 5 Ḣ = 1 2 y2 γ 2 ρ, subject to the constraint 3H 2 = ρ + 1 2 y2 + V φ. 6 The function W defined by satisfies Wφ,y,ρ,H = H 2 1 3 1 2 y2 + V φ + ρ, 7 Ẇ = 2HW, 8 and therefore, the hypersurface {W = 0}, representing flat, k = 0, cosmologies is invariant under the flow of 5. Denoting by ǫ = 1 2 y2 + V φ, the energy density of the scalar field, we see that ǫ + ρ = 3Hy 2 + γρ, which implies that, for expanding models the total energy ǫ + ρ of the system decreases. The equilibria of 5 are given by φ = φ,y = 0,ρ = 0,H = ± V φ /3 where V φ = 0. Regarding their stability for expanding cosmologies, we note the following facts. Critical points of V with negative critical value are not equilibria and they rather allow for recollapse of the model. Moreover, nondegenerate maximum points with non negative critical value for V are unstable, as can be easily seen by linearizing system 5 at the corresponding equilibria and verifying the existence of at least one eigenvalue with positive real part. It is interesting to study what happens near local minima of the potential with non negative critical value, and a stability result can be given when k = 0, 1. 2
Recent Developments in Gravity NEB XIV Journal of Physics: Conference Series 283 2011 012024 Proposition. Let φ a strict local minimum for V φ, possibly degenerate, with nonnegative V φ critical value. Then, p = φ,y = 0,ρ = 0,H = 3 is an asymptotically stable equilibrium point for expanding cosmologies in the open spatial topologies k = 0 and k = 1. Sketch of the proof. The proof consists in constructing a compact set Ω in R 4 and showing that it is positively invariant. Applying LaSalle s invariance theorem to the functions W and ρ + ǫ in Ω, it is shown that every trajectory in Ω is such that HW 0 and Hy 2 + γρ 0 as t +, which means y 0, ρ 0, and H 2 1 3V φ 0. Since H is monotone and admits a limit, V φ also admits a limit, V φ, thus the solution approaches the equilibrium point p. For details, see [5]. Similar results were proved in [6] for separately conserved scalar field and perfect fluid. 3. Energy exchange In the following we restrict ourselves to the flat, k = 0, case and study the energy transfer from the perfect fluid to the scalar field. We are interested in studying the late time behavior near V φ 3, which, by the previous Proposition the equilibrium point φ = φ,y = 0,ρ = 0,H = is asymptotically stable. We suppose that the initial data in the basin of attraction of this equilibrium are such that the fluid is the dominant matter component, i.e. ρ 0 > ǫ 0, and we investigate whether there is a time t 1 such that ǫt > ρt, t > t 1. 9 This question is relevant to the coincidence problem, that is, why dark energy and matter appear to have roughly the same energy density today see [7] and references therein. If V φ > 0, the transition 9 does always happen. In fact, it easily follows that if the critical value of the potential is strictly positive, it behaves as an effective cosmological constant and the energy of the scalar field tends to this value whereas the energy of the fluid tends to zero. We conclude that the relevant case is when V φ = 0 and hereafter we will focus on the case when φ is a nondegenerate minimum. Without loss of generality we suppose that φ = 0 and therefore, the general form of the potential studied can be written in a neighborhood of φ = 0 as V φ = 1 2 ω2 φ 2 + Oφ 3, ω > 0. 10 Integrating 4, we get ρt = ce αφt at 3γ. 11 From the Proposition in the previous Section we know a-priori that φ, φ,ρ,h 0,0,0,0 as t, hence we can write ρt cat 3γ as t. We now turn our attention to the equation of motion of the scalar field 3 with the potential 10, namely φ + 3H φ + ω 2 φ + Oφ 2 = αρ. 12 This equation can be solved by the Kryloff-Bogoliuboff KB approximation [8]. We present an outline of the method for the convenience of readers with no previous knowledge of the KB approximation. Consider the differential equation φ + ηf φ, φ + ω 2 φ = 0, 0 < η 1. 13 3
Recent Developments in Gravity NEB XIV Journal of Physics: Conference Series 283 2011 012024 If η = 0, the solution can be written as φt = 1 ω r cos ωt + χ and φt = r sin ωt + χ, where r and χ are arbitrary constants. We are looking for a solution of 13 which resembles to the form of the simple harmonic oscillator, that is, φt = 1 ω r t cos ωt + χt and φt = r tsin ωt + χt. 14 Setting θ t = ωt + χt and substituting 14 in 13 yields dr 1 dt = ηf ω dχ dt = η r f r cos θ,r sin θ 1 r cos θ,r sin θ ω sin θ, 15 cos θ. For η small, rt and χt are slowly varying functions of t. Consequently, in a time T = 2π/ω, rt and χt have not changed appreciably, while θ t = ωt+χt will increase approximately by 2π. Therefore, we replace the rhs of 15 by their average values over a range of 2π of θ, i.e. the amplitude rt is regarded as a constant in taking the average this is the essence of the KB approximation. This leads to the equations dr dt = η 2π 2π dχ dt = η 2πr 0 2π 0 1 f f r cos θ,r sinθ ω 1 r cos θ,r sin θ ω If we could apply the KB approximation to 12 with ηf then we should obtain for the amplitude of φ The integral sin θdθ, cos θdθ. φ, φ = 3H φ + Oφ 2 αρ, 16 dr dt = 1 2π 3Hωr cos θ + Or 2 cos 2 θ αce α/ωr cos θ a 3γ sin θdθ. 17 2π 0 2π 0 e α/ωr cos θ sin θdθ, vanishes and since H is a slow varying function compared to the rapid oscillations of the scalar field, it can be considered as constant during one period, T = 2π/ω. Therefore, 17 becomes dr dt = 3 2 Hr + Or3, 18 since the integration of the quadratic terms vanish. Ignoring third-order terms, eq. 18 can be integrated to give rt = r 0 ta 3/2, where r 0 t is bounded and bounded away from zero uniformly as t +. 4
Recent Developments in Gravity NEB XIV Journal of Physics: Conference Series 283 2011 012024 Hence we find that for large t the amplitude of the scalar field varies as a 3/2. Since the amplitude of φ has by 14 the same time dependence as the amplitude of φ and in our approximation it follows that ǫ = 1 2 φ 2 + 1 2 ω2 φ 2, ǫ = a 3. Comparing this result with the time dependence of the fluid density ρ = a 3γ, we arrive at the following picture for the evolution of the universe. If γ < 1 the energy density ρ eventually dominates over the energy density of the scalar field ǫ and this universe follows the classical Friedmannian evolution. For γ > 1, since ρ decreases faster than the energy density ǫ of the scalar field, ǫ eventually dominates over ρ. A rigorous but long proof of these results is given in [5]. The main obstruction to apply the KB approximation is due to 16, indeed, the above argument applies if ρ/r goes to zero. Otherwise, the rhs in 15 is not infinitesimal and so χt could in principle be comparable to ωt. 4. Summary We analysed the late time evolution of flat and negatively curved expanding FRW models having a scalar field coupled to matter. Equilibria corresponding to non negative local minima of V are asymptotically stable. In case the minimum is positive, say V φ > 0, the energy density, ǫ, of the scalar field eventually rules over the energy density of the fluid, ρ, and the asymptotic state has an effective cosmological constant V φ. In case the minimum is zero and nondegenerate, then ρ eventually dominates over ǫ if γ < 1 and ǫ dominates over ρ if γ > 1. These results could be interesting in investigations of cosmological scenarios in which the energy density of the scalar field mimics the background energy density. For viable dark energy models, it is necessary that the energy density of the scalar field remains insignificant during most of the history of the universe and emerges only at late times to account for the current acceleration of the universe. The above results can be rigorously proved only assuming that critical points are finite, and that V φ is non-negative as φ ±. It must be remarked that the latter assumption does not enter in the study of the late time behavior around a critical point φ, because for that situation only the behavior of the potential near φ is important and no growth at infinity assumptions on V are actually needed. The cases studied of course do not cover all possible situations. In particular, one can take into account degenerate minima for the potential. For closed cosmologies further investigation is needed. We believe that a closed model cannot avoid recollapse, unless the minimum of the potential is strictly positive. In that case, the asymptotic state must be de Sitter space. References [1] Capozziello S and Francaviglia M 2008 Gen. Rel. Grav. 40 357 [2] Bean R, Bernat Pogosian D L, Silvestri A and Trodden M 2007 Phys. Rev. D 75 064020 [3] Magnano G, Soko lowski L M 1994 Phys. Rev. D 50 5039 [4] Wainwright J and Ellis G F R 1997 Dynamical Systems in Cosmology Cambridge University Press 5
Recent Developments in Gravity NEB XIV Journal of Physics: Conference Series 283 2011 012024 [5] Giambò R and Miritzis J 2010 Class. Quantum Grav. 27 095003 [6] Miritzis J 2003 Class. Quantum Grav. 20 2981 [7] Amendola L, Quartin M, Tsujikawa S and Waga I 2006 Phys. Rev. D 74 023525 [8] Kryloff N and Bogoliuboff N 1943 Introduction to Nonlinear Mechanics Princeton University Press 6