How a cosmic-string loop radiates and shrinks

This chapter follows the previous one, What the tension of a cosmic string measures. You need vectors, derivatives and a geometric series. Units have $c=1$, and the loop sits in flat space, which is accurate while it is much smaller than the distance light has travelled since the Big Bang.

A loop is two curves on a sphere

In a suitable gauge, the general solution of the Nambu–Goto equations in flat space is a sum of a left-moving and a right-moving wave,

For a loop of invariant length $\ell$ both functions are periodic with period $\ell$. Their derivatives are unit vectors, so as $\sigma_\pm$ runs over one period each derivative traces a closed curve on the unit sphere. Because the loop's centre of mass is at rest, each curve averages to zero. Auclair et al., 2020, Appendix A.1

Differentiating gives the velocity and the tangent of the string:

At a fixed point $\sigma$, after a time $\ell/2$ the arguments $\sigma_-$ and $\sigma_+$ have each moved by $\ell/2$, and together they return the configuration to itself. The loop's period is $T=\ell/2$, and it radiates at the harmonics

Cusps: where the two curves cross

Two closed curves on a sphere that both average to zero generically cross. At a crossing, $\boldsymbol X_-'=\boldsymbol X_+'$. There the tangent $\partial_\sigma\boldsymbol X$ vanishes and the speed $|\partial_t\boldsymbol X|$ equals one: the string doubles back on itself and moves at the speed of light. That event is a cusp. A kink is a jump in one of the two derivatives, left behind when strings reconnect.

At a cusp the emission is beamed. At harmonic $n$ the burst is concentrated in a cone of opening angle $\theta_{\mathrm b}\approx[2/(\sqrt3\,n)]^{1/3}$ around the cusp's direction of motion. Auclair et al., 2020, Appendix A.2

How the power is shared among harmonics

Averaged over the loops in a network, the share of power in harmonic $n$ is modelled as a power law,

with $q=4/3$ for cusps, $5/3$ for kinks and $2$ for collisions between kinks. The Riemann zeta function $\zeta(q)$ makes the shares add up to $\Gamma$. Simulations of realistic loops find total powers sharply peaked near $\Gamma\approx50$. Auclair et al., 2020, Section 4.1

A computer has to stop the sum somewhere. Summing $10^5$ cusp harmonics keeps $98.2\,\%$ of the power and misses $1.8\,\%$; for kinks the first $10^5$ terms keep $99.967\,\%$. The NANOGrav analysis also stops at $10^5$ harmonics. NANOGrav Collaboration, 2023, Section 5.4

Steady shrinking and a finite life

The power $\Gamma G\mu^2$ contains no length. Energy conservation, $\mathrm d(\mu\ell)/\mathrm dt=-\Gamma G\mu^2$, then gives

Integrating the power over the lifetime returns $\Gamma G\mu^2\tau=\mu\ell_{\mathrm i}$: all of the rest energy leaves as gravitational waves. As the loop shrinks, its harmonics rise in frequency. A small loop therefore sings ever higher until it vanishes.

Lab task

In the cosmic-string Lab, keep the default and watch two cusps flash in opposite directions each period. Then tick kinks instead of cusps. The flat high-frequency readouts hardly change, because the plateau depends on the total power $\Gamma$ and not on how it is shared. The value at one cycle per year changes, because there the low harmonics of large loops matter.

Exercises

1. Find the fundamental frequency at emission of a loop one light-year long.
2. What is the cusp beaming angle at harmonic $n=10^{5}$?
3. Show that a loop born at time $t_{\mathrm i}$ with $\ell_{\mathrm i}=\alpha t_{\mathrm i}$ lives $\alpha/(\Gamma G\mu)$ times its birth age.
4. Why does doubling the share of power in the fundamental leave the total radiated energy of a loop unchanged?