---
title: "How a cosmic-string loop radiates and shrinks"
description: "Follow one Nambu–Goto loop: its period, the cusps that beam bursts, the harmonics it emits and the steady shrinking that ends its life."
canonical_url: "https://sandboxphysics.com/stories/cosmic-strings/how-a-loop-radiates-and-shrinks/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/cosmic-strings/how-a-loop-radiates-and-shrinks/"
last_modified: "2026-11-12"
---

[Cosmic strings and the gravitational-wave background](https://sandboxphysics.com/stories/cosmic-strings/)

# How a cosmic-string loop radiates and shrinks

Sandbox Physics · 2026-11-12

A loop of length $\ell$ repeats its motion every $\ell/2$, so it radiates gravitational waves at the harmonics $f_n=2n/\ell$. Cusps, points that briefly reach the speed of light, and kinks, sharp bends, keep the high harmonics strong: the power in harmonic $n$ falls only as $n^{-4/3}$ or $n^{-5/3}$. The total power $\Gamma G\mu^2$ does not depend on the loop's size, so the loop shortens at the constant rate $\Gamma G\mu$ until all of its rest energy has become gravitational waves.

This chapter follows the previous one, [What the tension of a cosmic string measures](https://sandboxphysics.com/stories/cosmic-strings/what-string-tension-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,

$$
\boldsymbol X(\sigma,t)=\frac12\left[\boldsymbol X_-(\sigma_-)+\boldsymbol X_+(\sigma_+)\right],\qquad \sigma_\pm=t\pm\sigma,\qquad |\boldsymbol X_\pm'|=1 .
$$

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](https://arxiv.org/abs/1909.00819v1)

Differentiating gives the velocity and the tangent of the string:

$$
\partial_t\boldsymbol X=\frac12\left(\boldsymbol X_-'+\boldsymbol X_+'\right),\qquad \partial_\sigma\boldsymbol X=\frac12\left(\boldsymbol X_+'-\boldsymbol X_-'\right).
$$

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

$$
f_n=\frac{n}{T}=\frac{2n}{\ell},\qquad n=1,2,3,\ldots
$$

## 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.

Worked example: the loop drawn in the Lab

The Lab draws a Kibble–Turok loop with $\boldsymbol X_-'(u)=(\cos u,\sin u,0)$ and $\boldsymbol X_+'(v)=(\cos v,\cos\psi\sin v,\sin\psi\sin v)$, using $\psi=1.05$. Both curves are great circles. The first lies in the horizontal plane, the second in a plane tilted by $\psi$ about the first axis, so they cross at $\pm(1,0,0)$. Each crossing happens once per period, which gives two cusps per oscillation moving in opposite directions along that axis. The loop has $\ell=2\pi$ and $T=\pi$ in these units. A direct numerical evaluation of $|\partial_t\boldsymbol X|$ at the crossing gives 1.0000.

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](https://arxiv.org/abs/1909.00819v1)

## 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,

$$
P_n=\frac{\Gamma}{\zeta(q)}\,n^{-q},\qquad \sum_{n=1}^{\infty}P_n=\Gamma,
$$

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](https://arxiv.org/abs/1909.00819v1)

Worked example: how much power sits in the low harmonics

$\zeta(4/3)=3.601$ and $\zeta(5/3)=2.124$. The fundamental carries $1/\zeta(q)$ of the power: $27.8\,\%$ for cusps and $47.1\,\%$ for kinks. Summing the first ten terms of the series gives $62.0\,\%$ for cusps and $85.3\,\%$ for kinks. A kinky loop puts more of its energy into its lowest notes.

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](https://arxiv.org/abs/2306.16219v1)

## 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

$$
\frac{\mathrm d\ell}{\mathrm dt}=-\Gamma G\mu,\qquad \ell(t)=\ell_{\mathrm i}-\Gamma G\mu\,(t-t_{\mathrm i}),\qquad \tau=\frac{\ell_{\mathrm i}}{\Gamma G\mu}.
$$

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.

Worked example: a loop born one second after the Big Bang

Take $\alpha=0.1$, so the loop is born with $\ell=0.1$ light-seconds, or $3.0\times10^{7}\,\mathrm m$. Its fundamental frequency at emission is $2c/\ell=20\,\mathrm{Hz}$. In the flat model with a constant number of particle species used in this course, the universe was one second old at $1+z=4.88\times10^{9}$. The wave arrives today at $20\,\mathrm{Hz}/4.88\times10^{9}=4.1\times10^{-9}\,\mathrm{Hz}$, inside the pulsar-timing band. With $G\mu=10^{-10}$ the loop lives $2\times10^{7}\,\mathrm s$, about eight months, so it radiates almost at the redshift where it was born.

## 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?

Answers

1. $f_1=2c/\ell=2/(1\,\mathrm{yr})=6.3\times10^{-8}\,\mathrm{Hz}$.
2. $\theta_{\mathrm b}\approx[2/(1.732\times10^{5})]^{1/3}=0.0226\,\mathrm{rad}$, about $1.3^\circ$. At $n=10^3$ the cone is about $6^\circ$ wide, so higher harmonics are beamed more tightly.
3. $\tau/t_{\mathrm i}=\ell_{\mathrm i}/(\Gamma G\mu\,t_{\mathrm i})=\alpha/(\Gamma G\mu)$. For $\alpha=0.1$ and $\Gamma=50$ this is $0.002/G\mu$.
4. The total power is fixed at $\Gamma G\mu^2$ and the lifetime at $\mu\ell_{\mathrm i}/(\Gamma G\mu^2)$, so the radiated energy is always $\mu\ell_{\mathrm i}$. Changing $q$ only moves that energy between harmonics.

## [Try it in the cosmic-string Lab](https://sandboxphysics.com/labs/cosmic-string-loops/)

Watch the cusps flash, then switch to kinks and compare the rings.

## Sources

1. [Auclair et al., 2020, Appendix A.1](https://arxiv.org/abs/1909.00819v1)
2. [NANOGrav Collaboration, 2023, Section 5.4](https://arxiv.org/abs/2306.16219v1)

## Keep reading

Cosmic strings and the gravitational-wave background · Reading path

### From one shrinking loop to a background spectrum

1. [Article **Why could a thin cosmic string shape a whole gravitational-wave spectrum?**](https://sandboxphysics.com/stories/cosmic-strings/thin-strings-whole-spectrum/)
2. [Article **What the tension of a cosmic string measures**](https://sandboxphysics.com/stories/cosmic-strings/what-string-tension-measures/)
3. [You are here **How a cosmic-string loop radiates and shrinks**](https://sandboxphysics.com/stories/cosmic-strings/how-a-loop-radiates-and-shrinks/)
4. [Article **From a loop population to the background spectrum**](https://sandboxphysics.com/stories/cosmic-strings/from-loop-population-to-background/)
5. [Article **Recompute the radiation-era plateau of a cosmic-string background**](https://sandboxphysics.com/stories/cosmic-strings/recompute-the-radiation-plateau/)
6. [Article **Reading pulsar-timing and LISA results on cosmic strings**](https://sandboxphysics.com/stories/cosmic-strings/reading-pta-and-lisa-string-limits/)