On Brane World Cosmological Perturbations

Abstract

We discuss the scalar cosmological perturbations in a 3-brane world with a 5D bulk. We first show explicitly how the effective perturbed Einstein's equations on the brane (involving the Weyl fluid) are encoded into Mukohyama's master equation. We give the relation between Mukohyama's master variable and the perturbations of the Weyl fluid, we also discuss the relation between the former and the perturbations of matter and induced metric on the brane. We show that one can obtain a boundary condition on the brane for the master equation solely expressible in term of the master variable, in the case of a perfect fluid with adiabatic perturbations on a Randall-Sundrum (RS) or Dvali-Gabadadze-Porrati (DGP) brane. This provides an easy way to solve numerically for the evolution of the perturbations as well as should shed light on the various approximations done in the literature to deal with the Weyl degrees of freedom.

Figures (1)

• Figure 1: Schematic representation of the bulk space-time with the brane trajectory in the characteristic $X,T$ coordinates (of equation (\ref{['MINKOCOS']})) for the DGP model. The left figure is the case with a Big Bang (with e.g. a radiation dominated universe), while the right one corresponds to a case with inflation. The $X=0$ line is a coordinate singularity. The gray (cyan) part is cutoff and the complete bulk space-time is made out of the remaining part, glued to a copy of itself along the brane. If a given initial instant is chosen along the brane from which one wishes to evolve cosmological perturbations, one needs to specify a boundary condition along the brane (AB), and initial data in the bulk (e.g along (AD)) that may be in part characteristic data (as along (AEF)). The future of the brane events with cosmic time larger than initial time lies above the dotted line (AC).