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

Cosmic string loops are defined by a pair of periodic functions $\mathbf{a}$ and $\mathbf{b}$, which trace out unit-length closed curves in three-dimensional space. We consider a particular class of loops, for which $\mathbf{a}$ lies along a line and $\mathbf{b}$ 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) $\mathbf{b}$ a circle traversed $M$ times; (2) $\mathbf{b}$ a regular polygon with $N$ sides and interior vertex angle $\pi -2\pi M\mathrm{/}N$; (3) $\mathbf{b}$ 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.