Phase space solutions in scalar-tensor cosmological models

Abstract

An analysis of the solutions for the field equations of generalized scalar-tensor theories of gravitation is performed through the study of the geometry of the phase space and the stability of the solutions, with special interest in the Brans-Dicke model. Particularly, we believe to be possible to find suitable forms of the BransDicke parameter ω and potential V of the scalar field, using the dynamical systems approach, in such a way that they can be fitted in the present observed scenario of the Universe. I. SCALAR-TENSOR THEORIES OF GRAVITATIONIn a homogeneous and isotropic space, described by the Friedmann-Lematre-Robertson-Walker metricwhere a is the scale factor and K is the spatial curvature index, gravitation can be described by an action of the kindwhere S m is the action of usual matter, g is the determinant of the metric tensor, ω is a coupling function (which we will eventually assume to be a constant, known as Brans-Dicke parameter) and V (φ) is the potential of the scalar field φ [2]. From (2), we obtain for the field equations:where H ≡ȧ/a is the Hubble parameter and ρ m and P m are the energy density and the pressure of the material fluid.As usual, we parameterize the equation of state for the fluid as P m = (γ − 1)ρ m with γ a constant chosen to indicate a variety of fluids that are predominantly responsible for the energy density of the Universe. We can see that through the energy conservation equationρ m + 3H(ρ m + P m ) = 0 we obtain ρ m = ρ 0 /a 3γ , with ρ 0 a constant. (7) [2]. The "hole" in the surface indicates the region forbidden for the orbits of the solutions. II. THE CASE FORIn the referred paper [2], the author proceeds to show the phase space allowed for the orbits of solutions for these field equations in several cases with different potentials and parameters ω. For example, in the case of vacuum, flat space (K = 0) and potential V = which has the solutionṡThe assumption of flat space is required in order to reduce the dimensionality of the phase space. We want to analyze qualitatively the geometry of the phase space (H, φ,φ), expecting to infer the form of the functions ω(φ) and V (φ) to fit better the available data on the structure of the Universe.The phase space for this situation is composed of a 2-d surface with two sheets, related to the lower and upper signs in eq. (7). Figures 1-3 show the phase space for the choice ω = 10.The fixed points for this dynamical system, obtained makingḢ =φ = 0, are de Sitter solutions, given by H 0 = ± φ 0 /12, with constant H 0 and φ 0 .

Dynamical systems applied to cosmology: Dark energy and modified gravity

“…de Souza & Saa (2005) considered the linear coupling F = φ plus a linear or a quadratic self interacting potential, while Jarv et al (2010) assumed the same linear coupling but delivered a general work on potential dominated cosmological solutions. Finally Skugoreva et al (2014) looked at the model with power-law functions for both F and V , while Agarwal & Bean (2008) performed a general analysis of scalar-tensor cosmological dynamics in both the Jordan and Einstein frame, finding transient and stable accelerated solutions.…”

Asymptotically Static Universe Dominated by Phantom Energy

“…The theory that we describe here generalises the chameleonic Brans-Dicke gravity [11,12] characterised by a non-minimal coupling between the scalar field and matter and differs as well from the author's work [13] characterised by the presence of an extra dimension. It is, in fact, a generalisation of the work done by de Souza and Braz and Jamil et al [14,15] by the facts that a scalar field coupled to the matter is present in the theory, the EoS parameter is time dependent and, in addition, we introduce a particular form of a scalar field-dependent Hubble parameter. We show that these constraints lead to an interesting cosmological scenario and a consistent scale factor with the present accelerating universe.…”

“…Accordingly, (13) and a possible solution is given by φ(t) = k + be -t where k and b are constants. After replacing this solution into (14) we obtain effortlessly 2 2…”

Dark Matter and Dark Energy from a Kaluza-Klein inspired Brans-Dicke Gravity with Barotropic Fluid

“…As a result, the gravitational constant is promoted to a scalar field which coupled to both mass and geometry. A huge number of studies for this particular model can be found here [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”