Cosmic String Loops and the Gravitational-Wave Background

How can a thin line of trapped energy shape a whole gravitational-wave spectrum? A string network keeps chopping off loops in every era. Each loop vibrates, sends out gravitational waves at a pitch set by its size and slowly shrinks. Loops from earlier eras were smaller and their waves were stretched by expansion, so many generations of loops together cover frequencies from pulsar timing up to ground-based detectors.

Drag a loop’s birth size and change the string tension. Watch the loop shrink as it radiates, and read the background spectrum built from the same loops.

Loop lifetime / birth time $2\times10^{7}\,t_{\mathrm{i}}$

h^2\Omega_{\mathrm{gw}}

1\,\mathrm{yr}^{-1}

$1.07\times10^{-9}$

h^2\Omega_{\mathrm{gw}}

3\,\mathrm{mHz}

$4.57\times10^{-10}$

h^2\Omega_{\mathrm{gw}}

25\,\mathrm{Hz}

$4.46\times10^{-10}$

Model assumptions

The loop population follows the one-scale Nambu–Goto model used in the LISA cosmic-string study by Auclair and colleagues (2020): every loop is born at one fixed fraction of cosmic time, the production rate includes the paper’s efficiency factor of one tenth for the spread of real birth sizes, and every loop radiates fifty times its tension squared as power and shrinks at a constant rate. The radiation-era and matter-era loop densities are the published ones; between the two eras the model blends them by the radiation share of the energy density, which is a teaching choice. The background sums harmonics up to the hundred-thousandth mode with power falling as the four-thirds power of mode number for cusps and the five-thirds power for kinks. The expansion history is the flat Lambda-CDM model with the parameters listed by Auclair and colleagues, and the number of particle species is held constant; their figure 4 shows that the Standard Model’s changing species count alters the high-frequency part of the spectrum. The network is assumed to scale from redshift ten to the twenty-sixth onwards. Lowering the reconnection probability divides the whole spectrum by that probability, the simple rescaling used by NANOGrav for superstrings. Detector bands are frequency ranges only, not sensitivity curves. The stage is schematic: loop sizes are not to scale and the oscillation is slowed.

Background spectrum today

Loops alive deep in the radiation era

When the waves we hear were emitted

What each curve is

Violet curve: model output, integrated in your browser from the same loop population that drives the stage. It is not measured data.
Dashed line: an ideal reference, the published radiation-era plateau formula. It is computed separately and never fitted to the curve.
Shaded bands: frequency ranges of pulsar timing arrays, LISA and ground-based detectors. They are not sensitivity curves and contain no observed signal.
Loop-size chart: loops per volume multiplied by the fourth power of cosmic time. The gold line is the birth size; the dashed line is the length a loop loses in one cosmic age.
Epoch chart: each curve is scaled to a peak of one, so the two heights cannot be compared.

Many generations of shrinking loops make one broad spectrum

BackgroundA cosmic string is a line of trapped field energy left behind if the early universe passed through a symmetry-breaking transition. Where long strings cross they can reconnect and cut off closed loops. A loop oscillates, radiates gravitational waves and shrinks until it disappears.

Why it mattersOne loop sings at a pitch set by its length. The network keeps making loops in every era, and expansion stretches each era’s waves by a different amount, so the sum spreads over many decades of frequency.

Focus question: Why does a smaller tension lower the background but push its rise to higher frequency?
One-sentence intuition: Tension sets both how bright each loop is and how long it lives; long-lived loops pile up and emit late, at low frequency.

Core mathematical model

A loop shrinks at a constant rate

\ell(t)=\alpha t_{\mathrm i}-\Gamma G\mu\,(t-t_{\mathrm i}),\qquad \frac{\tau}{t_{\mathrm i}}=\frac{\alpha}{\Gamma G\mu}

A loop born at time $t_{\mathrm i}$ with length $\alpha t_{\mathrm i}$ loses energy at the fixed power $\Gamma G\mu^2 c$ , with $\Gamma=50$ . The lifetime readout is this ratio; the radiated energy equals the loop’s rest energy.

Each loop sings harmonics of its length

f_{\mathrm{emit}}=\frac{2n}{\ell},\qquad f=\frac{f_{\mathrm{emit}}}{1+z},\qquad P_n\propto n^{-q}

Mode $n$ carries a share of the power falling as $n^{-4/3}$ for cusps and $n^{-5/3}$ . Expansion divides the emitted frequency by one plus the redshift at emission.

The background integrates the same loop population

\Omega_{\mathrm{gw}}(f)=\frac{8\pi(G\mu)^2}{3H_0^2}\,f\sum_{n}\Gamma P_n\,C_n(f),\qquad C_n(f)=\frac{2n}{f^2}\int_0^{z_\ast}\frac{\mathrm dz}{H(z)(1+z)^6}\,\mathsf n\!\left(\frac{2n}{(1+z)f},\,t(z)\right)

$\mathsf n(\ell,t)$ is the number of loops per volume per unit length, the curve in the loop-size chart. The Lab evaluates this redshift integral for every frequency; an independent cohort-by-cohort energy count agrees within one percent.

The radiation-era plateau

h^2\Omega_{\mathrm{plateau}}=\frac{128\pi}{9}A_{\mathrm r}\,\Omega_{\mathrm r}h^2\,\frac{G\mu}{\epsilon}\left[(1+\epsilon)^{3/2}-1\right],\qquad \epsilon=\frac{\alpha}{\Gamma G\mu}

This closed form, eq. 6.1 of Auclair et al. (2020) with $A_{\mathrm r}=0.54$ , is the dashed reference line. For large loops it grows as $\sqrt{G\mu}$ . It describes only the flat high-frequency part.

Common difficulties

A free spectral fit is not a string detection

Typical misconceptionIf a string curve can pass through the pulsar-timing signal, the signal comes from strings.

Better mental modelSupermassive black-hole binaries also make a background in that band. In the NANOGrav 15-year analysis, stable-string models scored Bayes factors of about 0.28 to 0.38 against the binary model, so the data preferred the binaries. The Lab draws model curves and frequency bands, not measured data.

Smaller tension does not shift the plateau down uniformly

Typical misconceptionHalving the tension halves the spectrum at every frequency.

Better mental modelThe plateau falls only as $\sqrt{G\mu}$ , because lighter loops radiate more weakly but live longer and pile up. Their final, low-frequency emission moves out of the pulsar-timing band, so the low-frequency part falls much faster.

Run the experiment

01
Watch the default loop
Leave the default running. Follow the bright flashes on the loop and compare the length readout on the stage with the lifetime readout.
What to observe: Two cusps flash every oscillation and send out rings. The loop shrinks steadily, and the spectrum below rises through the pulsar-timing band to a flat plateau.
02
Drag the loop size
Drag inside the horizon circle toward the centre, then choose Loops born tiny.
What to observe: Tiny loops last only about twice their birth time. The plateau drops about two thousand times, and the dashed formula still sits on the curve.
03
Lower the tension
Choose Faint, only at high pitch and compare the epoch chart with the default.
What to observe: The pulsar-timing readout collapses while the 3 mHz readout falls far less. The 3 mHz signal comes from loops that radiated much earlier.

Sources and model scope

The loop population, harmonic power and plateau formula follow Auclair et al. (2020), sections 3 to 6. The superstring rescaling and the stable-string comparison with pulsar-timing data follow the NANOGrav 15-year new-physics paper, section 5.4. The Lab’s spectrum is recomputed by an independent Python script that saves its inputs, outputs and failure cases.

Study the cosmic-string course: from a thin defect to a whole spectrum