Analytic Results for the Gravitational Radiation from a Class of Cosmic String Loops

TL;DR

This paper derives exact analytic expressions for gravitational wave power emitted by a specific class of cosmic string loops, revealing minimal and stationary configurations within that class.

Contribution

It provides the first closed-form solutions for gravitational radiation from a particular class of cosmic string loops and identifies minimal and stationary configurations.

Findings

Exact closed-form expressions for gravitational wave power b3 for specific loop configurations

Identification of the loop configuration with minimal b3 within the class

Determination of all stationary points of b3 in the class

Abstract

Cosmic string loops are defined by a pair of periodic functions ${\bf a}$ and ${\bf b}$, which trace out unit-length closed curves in three-dimensional space. We consider a particular class of loops, for which ${\bf a}$ lies along a line and ${\bf b}$ lies in the plane orthogonal to that line. For this class of cosmic string loops one may give a simple analytic expression for the power $\gamma$ radiated in gravitational waves. We evaluate $\gamma$ exactly in closed form for several special cases: (1) ${\bf b}$ a circle traversed $M$ times; (2) ${\bf b}$ a regular polygon with $N$ sides and interior vertex angle $\pi-2\pi M/N$; (3) ${\bf b}$ an isosceles triangle with semi-angle $\theta$. We prove that case (1) with $M=1$ is the absolute minimum of $\gamma$ within our special class of loops, and identify all the stationary points of $\gamma$ in this class.