A Closed-Form Expression for the Gravitational Radiation Rate from Cosmic Strings

TL;DR

This paper introduces an exact, closed-form formula for calculating the gravitational radiation rate from cosmic strings, applicable to all piecewise-linear loops, improving accuracy and computational efficiency over previous methods.

Contribution

The authors derive a comprehensive closed-form expression for gravitational radiation from cosmic string loops, correcting past errors and simplifying calculations for all piecewise-linear configurations.

Findings

The formula is exact when neglecting gravitational back-reaction.

It is more accurate and convenient than previous numerical methods.

Discrepancies in earlier work were identified and corrected.

Abstract

We present a new formula for the rate at which cosmic strings lose energy into gravitational radiation, valid for all piecewise-linear cosmic string loops. At any time, such a loop is composed of $N$ straight segments, each of which has constant velocity. Any cosmic string loop can be arbitrarily-well approximated by a piecewise-linear loop with $N$ sufficiently large. The formula is a sum of $O(N^4)$ polynomial and log terms, and is exact when the effects of gravitational back-reaction are neglected. For a given loop, the large number of terms makes evaluation ``by hand" impractical, but a computer or symbolic manipulator yields accurate results. The formula is more accurate and convenient than previous methods for finding the gravitational radiation rate, which require numerical evaluation of a four-dimensional integral for each term in an infinite sum. It also avoids the need to estimate the contribution from the tail of the infinite sum. The formula has been tested against all previously published radiation rates for different loop configurations. In the cases where discrepancies were found, they were due to errors in the published work. We have isolated and corrected both the analytic and numerical errors in these cases. To assist future work in this area, a small catalog of results for some simple loop shapes is provided.