Generalized Friedmann branes

TL;DR

This paper establishes a comprehensive formalism linking bulk Einstein equations with brane gravity in generalized Randall-Sundrum models, including non-symmetric bulk and arbitrary matter, and applies it to cosmological scenarios with various bulk geometries.

Contribution

It provides a complete set of equations connecting bulk Einstein equations with brane dynamics for non-$Z_{2}$-symmetric models with arbitrary matter content.

Findings

Derived generalized Friedmann and Raychaudhuri equations independent of embedding and bulk matter.

Identified two new functions governing brane cosmology depending on bulk matter and embedding.

Extended analysis to Reissner-Nordström-Anti de Sitter and Vaidya-Anti de Sitter bulk spacetimes.

Abstract

We prove that for a large class of generalized Randall-Sundrum II type models the characterization of brane-gravity sector by the effective Einstein equation, Codazzi equation and the twice-contracted Gauss equation is equivalent with the bulk Einstein equation. We give the complete set of equations in the generic case of non-$Z_{2}$-symmetric bulk and arbitrary energy-momentum tensors both on the brane and in the bulk. Among these, the effective Einstein equation contains a varying cosmological constant and two new source terms. The first of these represents the deviation from $Z_{2}$ symmetry, while the second arises from the bulk energy-momentum tensor. We apply the formalism for the case of perfect fluid on a Friedmann brane embedded in a generic bulk. The generalized Friedmann and Raychaudhuri equations are given in a form independent of both the embedding and the bulk matter. They contain two new functions obeying a first order differential system, both depending on the bulk matter and the embedding. Then we focus on Friedmann branes separating two non-identical (inner or outer) regions of Reissner-Nordstr\"om-Anti de Sitter bulk space-times, generalizing previous non-$Z_{2}$-symmetric treatments. Finally the analysis is repeated for the Vaidya-Anti de Sitter bulk space-time, allowing for both ingoing and outgoing radiation in each region.