Bouncing Cosmology in Three Dimensions

We consider a dynamical two-brane in a four dimensional black hole background with scalar hair. At high temperature this black hole goes through a phase transition by radiating away the scalar. The end phase is a topological adS-Schwarzschild black hole. We argue here that for a sufficiently low temperature, the brane motion in this geometry is non-singular. This results in a universe which passes over from a contracting phase to an expanding one without reaching a singularity. Present address: The Institute of Mathematical Sciences, C. I. T Campus, Taramani, Chennai 600113, India. E-mail: [email protected] Electronic address: [email protected] In standard cosmology, the universe usually starts or/and ends in a cosmological singularity. Since near the singularity the gravitational interaction is strong, classical general relativity breaks down. Consequently, in order to study the fate of these singularities, one needs to look for a still illusive theory of quantum gravity. It is therefore of interest to search for a non-singular cosmological model by some how evading Penrose-Hawking singularity theorem [1]. If this is possible, one can at least perform some further computations around these cosmological backgrounds while trusting the perturbative theory of gravity. In the context of brane-world models, where the brane moves in a higher dimensional bulk, we often get exotic cosmologies on the brane (see [2, 3] for example). Since, in principle, in this scenario, bulk geometry can contribute a negative energy density on the brane [4], it may be possible to construct some non-singular cosmological models. Indeed, some models of this nature were constructed in [5, 6, 7, 8]. In particular, in [5], a non-singular cosmology was found by considering a three-brane in an electrically charged adS black hole background [9]. Here, the brane contracts to a finite size before expanding again. This happens because the background charge introduces a negative energy density on the brane world volume; hence preventing it to fall into the singularity. However, the bounce occurs inside the outer horizon of the black hole. As a result, regardless of the initial energy of the brane, the bulk space time collapses due to the back reaction at the bounce [10]. In our effort to search for other examples of non-singular cosmology, in this Letter, we study another model of bounce in three dimensions. As we will show below, in certain region of the parameter space of the bulk geometry, the brane makes a smooth transition from a contracting phase to an expanding phase without ever reaching to a singularity. The model of our interest consists of black holes in four dimensional gravity with self-interacting scalar field and a negative cosmological constant. The solutions are of topology R×Σ where Σ represents a two dimensional manifold with a constant negative curvature. These holes are parametrised by a single parameter associated with the mass of the black holes. Interestingly enough, the mass of the black hole here can take negative values without producing a naked singularity. We will take an advantage of this fact to construct a bouncing brane cosmology by considering a dynamical two-brane in this geometry. In the following, we describe the model in some detail and, subsequently, analyse the resulting cosmology. Let us start by considering the action