Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory

Abstract

In this paper we investigate the evolution of a Jordan-Brans-Dicke scalar field, Φ, with a power-law potential in the presence of a second scalar field, φ, with an exponential potential, in both the Jordan and the Einstein frames. We present the relation of our model with the induced gravity model with power-law potential and the integrability of this kind of models is discussed when the quintessence field φ is massless, and has a small velocity. The fact that for some fine-tuned values of the parameters we may get some integrable cosmological models, makes our choice of potentials very interesting. We prove that in Jordan-Brans-Dicke theory, the de Sitter solution is not a natural attractor. Instead, we show that the attractor in the Jordan frame corresponds to an "intermediate accelerated" solution of the form a(t) ≃ e α 1 t p 1 , as t → ∞ where α 1 > 0 and 0 < p 1 < 1, for a wide range of parameters. Furthermore, when we work in the Einstein frame we get that the attractor is also an "intermediate accelerated" solution of the form a(t) ≃ e α 2 t p 2 as t → ∞ where α 2 > 0 and 0 < p 2 < 1, for the same conditions on the parameter space as in the Jordan frame. In the special case of a quadratic potential in the Jordan frame, or for a constant potential in the Einstein's frame, the above intermediate solutions are of saddle type. These results were proved using the center manifold theorem, which is not based on linear approximation. Finally, we present a specific elaboration of our extension of the induced gravity model in the Jordan frame, which corresponds to a particular choice of a linear potential of Φ. The dynamical system is then reduced to a two dimensional one, and the late-time attractor is linked with the exact solution found for the induced gravity model. In this example the "intermediate accelerated" solution does not exist, and the attractor solution has an asymptotic de Sitter-like evolution law for the scale factor. Apart from some fine-tuned examples such as the linear, and quadratic potential U (Φ) in the Jordan frame, it is true that "intermediate accelerated" solutions are generic late-time attractors in a modified Jordan-Brans-Dicke theory.

Tachyon Warm Intermediate and Logamediate Inflation in the Brane World Model in the Light of Planck Data

“…In one sector of the present work, we would like to consider warm tachyon model on the brane in the context of "intermediate inflation". This scenario is one of the exact solutions of inflationary field equation in the Einstein theory with scale factor a(t) = a 0 exp(At f ) (A > 0, 0 < f < 1), this solution of the scale factor in the context of a modified tensor-scalar theory has been found in [33]. The study of this model is motivated by string/M-theory [34].…”

Dynamical systems applied to cosmology: Dark energy and modified gravity

“…The presence of a non-trivial matter field can change the features of the late time asymptotics. For example, Cid et al (2016) showed that in the case of the presence of a second, minimally coupled scalar field one can prove that a de Sitter solution is not the asymptotic state for the cosmological model.…”

Interacting quintessence in light of generalized uncertainty principle: cosmological perturbations and dynamics

“…The latter system has been the case of study of various analysis in the literature [49][50][51][52][53][54][55][56][57][58][59][60] where Q = Q (ρ m , ρ d ) has been assumed to be either a linear function [46][47][48][49][61][62][63][64][65][66] and many others. Additionally, with the above choices for the coupling function, various functional forms of the scalar field potential function V (φ) can also be chosen [33,35,[50][51][52]60,[67][68][69][70][71][72].…”