The shortest cut in brane cosmology

TL;DR

This paper investigates the shortest null paths for photons on the brane and gravitons in the bulk within brane cosmology, deriving differential equations and comparing travel times and redshifts for different cosmological models.

Contribution

It derives the differential equation for the shortest path in the bulk for a 1+4 cosmological metric and analyzes the existence of shortest paths in various brane cosmology solutions.

Findings

No shortest path exists in the bulk for the Randall-Sundrum vacuum brane.

No shortest path in the bulk for the linear cosmological solution with specific parameters.

Some expanding brane universes also lack shortest bulk paths.

Abstract

We consider brane cosmology studying the shortest null path on the brane for photons, and in the bulk for gravitons. We derive the differential equation for the shortest path in the bulk for a 1+4 cosmological metric. The time cost and the redshifts for photons and gravitons after traveling their respective path are compared. We consider some numerical solutions of the shortest path equation, and show that there is no shortest path in the bulk for the Randall-Sundrum vacuum brane solution, the linear cosmological solution of Bin\'etruy, et al for $\omega = -1, -{2/3}$, and for some expanding brane universes.