Cosmology on a brane in Minkowski bulk

Abstract

We discuss the cosmology of a 3-brane embedded in a 5D bulk space–time with a cosmological constant when an intrinsic curvature Ricci scalar is included in the brane action. After deriving the ‘brane-Friedmann’ equations for a Z₂ symmetrical metric, we focus on the case of a Minkowski bulk. We show that there exist two classes of solutions, close to the usual Friedmann–Lemaı̂tre–Robertson–Walker cosmology for small enough Hubble radii. When the Hubble radius gets larger one either has a transition to a fully 5D regime or to a self-inflationary solution which produces a late accelerated expansion. We also compare our results with a perturbative approach and eventually discuss the embedding of the brane into the Minkowski space–time. This latter part of our discussion also applies when no intrinsic curvature term is included.

Introduction

A lot of interest has recently been raised for field theories where the standard model of high-energy physics is assumed to live on a surface (called generically a brane) embedded in a larger space–time. The (super)gravitational fields are in contrast usually considered to live in the whole space–time. Models coming from string-M theory like the Hořava–Witten walls [1], or D-branes, as well as from a more phenomenological approach [2], [3] have been extensively studied in particular in cosmology.

A question, which naturally arises, is how to recover standard gravity in its well tested perturbative regime. A first approach is to assume that the dimensions transverse to the brane are compact, in which case the usual Kaluza–Klein results allow to recover 4D gravity on scales which are larger than the size of the extra dimensions. The main difference with the old picture being there that, because the standard model fields are assumed to be brane-localized, one can have very large extra-dimensions in comparison to the length scales probed by high-energy physics [2]. On scales smaller than the size of the extra dimensions, on the other hand, gravity enters a higher-dimensional regime. Another way of recovering usual 4D gravity on the brane for large distances is to embed a positive tension 3-brane into an AdS₅ bulk [3] in which case the crossover scale between 4D and 5D gravity is set by the AdS radius. In this latter case, the extra dimension has also a finite size.

We will be mainly here interested in a more recent approach advocated by Dvali et al. [4], [5]. In this approach, the 3-brane is embedded in a space–time with infinite-size extra dimensions, with the hope that this picture could shed new light on the standing problem of the cosmological constant as well as on supersymmetry breaking [4], [6]. The recovery of the usual gravitational laws is obtained by adding to the action of the brane an Einstein–Hilbert term computed with the brane intrinsic curvature. The presence of such a term in the action is generically induced by quantum corrections coming from the bulk gravity and its coupling with matter living on the brane and should be included for a large class of theories for self-consistency (see, e.g., [5], [7]). In the particular case of a 3-brane embedded in a 5D Minkowski space–time, Dvali, Gabadadze and Porrati have shown that one recovers a standard 4D newtonian potential for small distances, whereas gravity is in a 5D regime for large distances. The tensorial structure of the graviton propagator in this theory has in contrast been shown to be higher-dimensional which is likely to rule out the theory from a phenomenological point of view. For a brane embedded in a bulk with 2 or more extra dimensions, one can show that the theory is always 4-dimensional [5] for a zero thickness brane.¹

Our purpose is here to study the cosmology of these models in the case of a 5D bulk. Although, as mentioned above, such a theory has serious phenomenological problems, its cosmology can help to have a better understanding of this kind of model and of the idea of gravity localization through an intrinsic curvature term on the brane. In particular, exact cosmological solutions, as given in this work, provide a unique way to test the theory in its fully nonlinear regime. The Friedman-like equations governing the cosmological evolution of a brane possessing an intrinsic curvature term in its action have already been derived and discussed for an AdS–Schwarzschild bulk space–time [9], [10], [11]. In the first part of this paper we derive similar equations (valid whenever the bulk matter is a pure cosmological constant) in a slightly different way, following the work of Binétruy et al. [12], [13]. For this purpose we will adopt a brane-based coordinate system and specialize to a Z₂ symmetrical metric. We then discuss these equations for a vanishing bulk cosmological constant as well as for a vanishing Schwarzschild mass parameter. We show that there exist two possible types of cosmology which are both similar to the usual 4D Friedman–Lemaı̂tre–Robertson–Walker (FLRW) cosmology when the Hubble radius is small. We point out, however, a discrepancy between the Newton's constant inferred by cosmology in that regime, and the one defined by a Cavendish-like experiment. For larger Hubble radius, the cosmology either evolves to the brane-typical linear relationship between the Hubble parameter and energy density, or to a brane self-inflationary solution previously noticed by Shtanov [10]. In this latter case, one has a late-time accelerated expansion sourced by the intrinsic curvature term of the brane itself (and not by its tension). In the last part of this paper we interpret the found two-folded cosmology by looking at the brane embedding into the bulk space–time. We give in particular the change of coordinates between our brane-based metric and a canonical minkowskian bulk metric.