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

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 $γ$ radiated in gravitational waves. We evaluate $γ$ 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 $π-2πM/N$; (3) ${\bf b}$ an isosceles triangle with semi-angle $θ$. We prove that case (1) with $M=1$ is the absolute minimum of $γ$ within our special class of loops, and identify all the stationary points of $γ$ in this class.