On Einstein equations with cosmological constant in braneworld models

TL;DR

This paper analyzes Einstein equations with a cosmological constant in braneworld models, deriving warp functions and solutions that exhibit exponential or oscillating behaviors, relevant for warp inflation and brane waves.

Contribution

It provides exact solutions for warp functions in DGP models with cosmological constants, highlighting differences from RS models and their physical implications.

Findings

In RS models, bulk Einstein equations reduce to vacuum equations on the brane.

In DGP models, bulk Einstein equations are affected by the cosmological constant and solved exactly.

Solutions include exponential and oscillating warp functions indicating warp inflation and brane waves.

Abstract

In this paper, we investigate the Einstein equations with cosmological constant for Randall-Sundrum (RS) and Dvali-Gabadadze-Porrati (DGP) models to determine the warp functions in the context of warp product spacetimes. In RS model, it is shown that Einstein's equation in the bulk is reduced into the brane as a vacuum equation, having vacuum solution, which is not affected by the cosmological constant in the bulk. In DGP model, it is shown that the Einstein's equation in the bulk is reduced into the brane and along the extra dimension, where both equations are affected by the cosmological constant in the bulk. We have solved these equations in DGP model, subject to vanishing cosmological constants on the brane and along extra dimension, and obtained exact solutions for the warp functions. The solutions depend on the typical values of cosmological constant in the bulk as well as the dimension of the brane. So, corresponding to the typical values, some solutions have exponential behaviours which may be set to represent warp inflation on the brane, and some other solutions have oscillating behaviours which may be set to represent warp waves or branes waves along the extra dimension.