Brane Isotropisation in Extra-Dimensional Tolman-Bondi Universe

TL;DR

This paper studies how a 3-brane in a 5D Tolman-Bondi universe evolves from inhomogeneous, anisotropic conditions to an isotropic, homogeneous Friedmann-Lemaitre universe, highlighting a late-time attractor behavior.

Contribution

It demonstrates the isotropisation of the brane universe in a matter-dominated 5D setting, revealing convergence to standard cosmological models from inhomogeneous initial states.

Findings

Induced geometry becomes isotropic and homogeneous over time.

The 4D universe approaches a Friedmann-Lemaitre model in a radiation epoch.

Isotropisation acts as a late-time attractor in the model.

Abstract

We consider the dynamics of a 3-brane embedded in an extra-dimensional Tolman-Bondi Universe where the origin of space plays a special role. The embedding is chosen such that the induced matter distribution on the brane respects the spherical symmetry of matter in the extra dimensional space. The mirage cosmology on the probe brane is studied, resulting in an inhomogeneous and anisotropic four dimensional cosmology where the origin of space is also special. We then focus on the spatial geometry around the origin and show that the induced geometry, which is initially inhomogeneous and anisotropic, converges to an isotropic and homogeneous Friedmann-Lemaitre 4d space-time. For instance, when a 3-brane is embedded in a 5d matter dominated model, the 4d dynamics around the origin converge to a Friedmann-Lemaitre Universe in a radiation dominated epoch. We analyse this isotropisation process and show that it is a late time attractor.