Energy-momentum tensor of bouncing gravitons

TL;DR

This paper investigates the behavior of gravitons in a 5D brane-world model, deriving explicit formulas for their energy-momentum contributions from multiple bounces, and relates reheating temperature to the scale of extra dimensions.

Contribution

It introduces a method to calculate the energy-momentum tensor contributions from multiple graviton bounces in a brane-world scenario, with explicit formulas for up to three bounces.

Findings

Explicit expressions for graviton bounce contributions are derived.

Numerical estimates relate reheating temperature to extra dimension scale.

Reheating temperature of 10^6 GeV implies an extra dimension scale of about 10^{-9} GeV.

Abstract

In models of the Universe with extra dimensions gravity propagates in the whole space-time. Graviton production by matter on the brane is significant in the early hot Universe. In a model of 3-brane with matter embedded in 5D space-time conditions for gravitons emitted from the brane to the bulk to return back to the brane are found. For a given 5-momentum of graviton falling back to the brane the interval between the times of emission and return to the brane is calculated. A method to calculate contribution to the energy-momentum tensor from multiple graviton bouncings is developed. Explicit expressions for contributions to the energy-momentum tensor of gravitons which have made one, two and three bounces are obtained and their magnitudes are numerically calculated. These expressions are used to solve the evolution equation for dark radiation. A relation connecting reheating temperature and the scale of extra dimension is obtained. For the reheating temperature $T_R\sim 10^6 GeV$ we estimate the scale of extra dimension $\m$ to be of order $10^{-9} GeV\,\,\, (\m^{-1}\sim 10^{-5} cm )$.