Cosmological 3-Brane Solutions

TL;DR

Addresses recovering standard 4D Friedmann cosmology for a 3-brane in a 5D spacetime by introducing a nonzero bulk stress $\hat{T}^5_5$ and considering brane thickness. Develops thin-wall and thick-wall analyses showing how the extra dimension can be effectively compactified or stabilized, linking the 4D Planck scale to bulk/brane scales via $M_P^2 = M_5^3 (2 b |y_{min}|)$ or $M_P^2 = M_5^3 (\Delta b)$. Demonstrates that the usual $H^2 \propto \rho$ behavior can emerge without a second brane, provided the bulk stress cancels the problematic $\rho^2$ term. Discusses possible bulk sources proportional to the 4D trace and outlines implications for gravity localization and dimension stabilization.

Abstract

We analyze cosmological equations in the brane world scenario with one extra space-like dimension. We demonstrate that the cosmological equations can be reduced to the usual 4D Friedmann type if the bulk energy-momentum tensor is different from zero. We then generalize these equations to the case of a brane of finite thickness. We also demonstrate that when the bulk energy-momentum tensor is different from zero, the extra space-like dimension can be compactified with a single brane and show that the stability of the radius of compactification implies standard cosmology and vice versa. For a brane of finite thickness, we provide a solution such that the 4D Planck scale is related to the fundamental scale by the thickness of the brane. In this case, compactification of the extra dimension is unnecessary.