Self-interacting scalar field trapped in a Randall–Sundrum braneworld: The dynamical systems perspective

Abstract

We apply the dynamical systems tools to study the linear dynamics of a self-interacting scalar field trapped on a Randall-Sundrum brane. The simplest kinds of self-interaction potentials are investigated: a) constant potential, and b) exponential potential. It is shown that the dynamics of the Randall-Sundrum model significantly differs from the standard four-dimensional behavior at early times: in all cases of interest the (singular) empty universe is the past attractor for every trajectory in phase space, meanwhile the kinetic energy-dominated solution is always a saddle critical point. The late-time dynamics is not affected by the brane effects.

Phase space analysis of quintessence fields trapped in a Randall–Sundrum braneworld: a refined study

“…This class of potentials contains the previously studied potentials in [34] with Λ = 0. Thus our present results generalize those in [34].…”

“…They were investigated in [51] using an alternative compactification scheme. The asymptotic properties of a cosmological model with a scalar field with exponential potential have been investigated in the context of the General Relativity by the authors of [26,29], and in the context of the RS braneworlds by [34,52]. In both cases it was studied the pure exponential potential (Λ = 0).…”

Dynamical systems applied to cosmology: Dark energy and modified gravity

“…The first of these scenarios is known as Randall-Sundrum type II brane-world model (RS II) (Randall & Sundrum, 1999a,b (8.118) where λ is the brane tension, a constant associated with the properties of the brane in the higher dimensions, and ρ tot and p tot are respectively the total energy density and pressure of the matter fields living on the brane. The cosmological equations in the RS II model can be rewritten in the form of a dynamical system by a suitable choice of variables (Gonzalez et al, 2009;Leyva et al, 2009). Campos & Sopuerta (2001b) assumed perfect fluid matter living on a brane and studied attractor solutions with a bifurcation analysis, while Goheer & Dunsby (2002) repeated the same work for a scalar field on the brane with an exponential potential (for earlier related work see Copeland et al (2001)).…”

“…The case where both a (non inflaton) scalar field and matter live on a brane was considered with focus on scaling solutions by Savchenko & Toporensky (2003) and with different variables by Dutta & Zonunmawia (2015). Instead Gonzalez et al (2009), using a constant and exponential potential for the scalar field, showed that late time solutions are not influenced by the brane dynamics. The same system has been considered by Escobar et al (2012a) where the stability of de Sitter solutions employing the centre manifold theory was examined finding de Sitter stable solutions for a scalar field exponential potential.…”

Phase space analysis of quintessence fields trapped in a Randall–Sundrum braneworld: anisotropic Bianchi I brane with a positive dark radiation term

“…To the potential treatment we use the method developed in [17]. We extend previous results in [15,16,19,23,32] by considering scalar fields with arbitrary potentials in an anisotropic background and considering the effect of the projection of the five-dimensional Weyl tensor onto the three-brane in the form of a positive Dark Radiation term. In order to illustrate our analytical results we consider a toy model using the simple f -deviser given by f (s) = s + α.…”