Closed Form Expression for the Momentum Radiated from Cosmic String Loops

TL;DR

This paper derives an exact analytic formula for the momentum radiation rate from cosmic string loops, extending previous energy loss formulas, and provides tools for more accurate future calculations in cosmic string research.

Contribution

It introduces a new analytic formula for momentum radiation from cosmic string loops, improving upon previous energy loss models and aiding future computational studies.

Findings

Derived an exact formula for momentum radiation rate

Provided explicit calculations for specific loop configurations

Corrected previous numerical results in the literature

Abstract

We modify the recent analytic formula given by Allen and Casper for the rate at which piecewise linear cosmic string loops lose energy to gravitational radiation to yield the analogous analytic formula for the rate at which loops radiate momentum. The resulting formula (which is exact when the effects of gravitational back-reaction are neglected) is a sum of O(N^4) polynomial and log terms where, N is the total number of segments on the piecewise linear string loop. As illustration, we write the formula explicitly for a simple one-parameter family of loops with N=5. For most loops the large number of terms makes evaluation ``by hand" impractical, but, a computer or symbolic manipulator may by used to yield accurate results. The formula has been used to correct numerical results given in the existing literature. To assist future work in this area, a small catalog of results for a number of simple string loops is provided.