---
title: "From a loop population to the background spectrum"
description: "Build the loop number density, integrate its emission over redshift and derive why the radiation-era plateau grows as the square root of tension."
canonical_url: "https://sandboxphysics.com/stories/cosmic-strings/from-loop-population-to-background/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/cosmic-strings/from-loop-population-to-background/"
last_modified: "2026-11-12"
---

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

# From a loop population to the background spectrum

Sandbox Physics · 2026-11-12

The background is a sum over every loop that ever radiated. Count the loops of each length alive at each time, let each one emit its harmonics, redshift the waves to today and add them up; the result is a single integral over redshift. Deep in the radiation era the loop population has a closed form, and the integral gives a flat plateau whose height grows as $\sqrt{G\mu\,\alpha/\Gamma}$ when loops are born large.

This chapter uses the two before it, [What the tension of a cosmic string measures](https://sandboxphysics.com/stories/cosmic-strings/what-string-tension-measures/) and [How a cosmic-string loop radiates and shrinks](https://sandboxphysics.com/stories/cosmic-strings/how-a-loop-radiates-and-shrinks/). You need the Dirac delta function, a change of variables in an integral and the radiation-era relations $a\propto t^{1/2}$ and $H=1/(2t)$. Units have $c=1$; times are measured in units of $1/H_0$ when numbers appear.

## Counting loops of each length

Write the number of loops born per unit volume and time as $R(t_{\mathrm i})$. In Model I every loop is born with length $\alpha t_{\mathrm i}$, and the rate is

$$
R(t_{\mathrm i})=\frac{A}{\alpha\,t_{\mathrm i}^{4}},\qquad A_{\mathrm r}=\frac{F}{f_{\mathrm r}}\,\frac{\tilde c\,\bar v_{\mathrm r}}{\xi_{\mathrm r}^{3}}=0.54 .
$$

Here $\tilde c=0.23$ is the chopping efficiency, $\bar v_{\mathrm r}=0.662$ and $\xi_{\mathrm r}=0.271$ describe the long strings, $f_{\mathrm r}=\sqrt2$ removes the energy that goes into the loops' bulk motion, and $F=0.1$ accounts for the spread of real birth sizes. [Auclair et al., 2020, Section 3.1, eqs. 3.12 to 3.16](https://arxiv.org/abs/1909.00819v1)

A loop of length $\ell$ seen at time $t$ was born at the time that satisfies $\alpha t_{\mathrm i}-\Gamma G\mu\,(t-t_{\mathrm i})=\ell$:

$$
t_{\mathrm i}=\frac{\ell+\Gamma G\mu\,t}{\alpha+\Gamma G\mu},\qquad \frac{\mathrm dt_{\mathrm i}}{\mathrm d\ell}=\frac{1}{\alpha+\Gamma G\mu}.
$$

Loops born in an interval $\mathrm dt_{\mathrm i}$ occupy a length interval $\mathrm d\ell=(\alpha+\Gamma G\mu)\,\mathrm dt_{\mathrm i}$ today, and their density has been diluted by $(a_{\mathrm i}/a)^3$. So

$$
\mathsf n(\ell,t)=\frac{R(t_{\mathrm i})}{\alpha+\Gamma G\mu}\left(\frac{a(t_{\mathrm i})}{a(t)}\right)^{3}.
$$

In the radiation era $(a_{\mathrm i}/a)^3=(t_{\mathrm i}/t)^{3/2}$. Substituting $t_{\mathrm i}$ and writing $x=\ell/t$ gives

$$
\mathsf n(\ell,t)=t^{-4}\,\frac{A_{\mathrm r}\,(\alpha+\Gamma G\mu)^{3/2}}{\alpha\,(x+\Gamma G\mu)^{5/2}},\qquad x\le\alpha ,
$$

which is eq. 3.16 of Auclair and colleagues. NANOGrav writes the same density for any era with the extra term $\dot\alpha_\ast t_\ast$ in the Jacobian, needed when the birth size changes with time. [NANOGrav Collaboration, 2023, Section 5.4, eqs. 43 and 44](https://arxiv.org/abs/2306.16219v1)

Worked example: where the 0.54 comes from

$F/f_{\mathrm r}=0.1/1.414=0.0707$ and $\tilde c\,\bar v_{\mathrm r}/\xi_{\mathrm r}^3=0.23\times0.662/0.0199=7.65$. Their product is $0.541$, the printed $A_{\mathrm r}=0.54$. Dropping the Jacobian, that is using $1/\alpha$ instead of $1/(\alpha+\Gamma G\mu)$, gives exactly twice the right density when $\alpha=\Gamma G\mu$; the saved failure cases of the method chapter record this.

## Adding up the emission

A loop of length $\ell$ at time $t$ emits power $G\mu^2P_n$ at frequency $2n/\ell$, which arrives today at $f=2n/[(1+z)\ell]$. Energy in gravitational waves dilutes as $(1+z)^{-4}$, and a frequency interval is stretched by $(1+z)$. The energy density per unit frequency today is therefore

$$
\frac{\mathrm d\rho_{\mathrm{gw}}}{\mathrm df}=G\mu^2\sum_n P_n\int\mathrm dt\,(1+z)^{-3}\int\mathrm d\ell\;\mathsf n(\ell,t)\,\delta\!\left(f(1+z)-\frac{2n}{\ell}\right).
$$

The delta function picks $\ell_n=2n/[(1+z)f]$ with weight $\ell_n^2/(2n)$. With $\mathrm dt=\mathrm dz/[(1+z)H]$ and $\Omega_{\mathrm{gw}}=(8\pi G/3H_0^2)\,f\,\mathrm d\rho_{\mathrm{gw}}/\mathrm df$,

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

eqs. 4.4 and 4.5 of Auclair and colleagues. [Auclair et al., 2020, Section 4.1](https://arxiv.org/abs/1909.00819v1) Because $C_n(f)=C_1(f/n)/n$, one table of the fundamental $C_1$ gives every harmonic.

## The radiation-era plateau

At high frequency the integral is dominated by loops from the radiation era, where $H=H_0\sqrt{\Omega_{\mathrm r}}\,(1+z)^2$ and $t=1/(2H)$. Change variables from $z$ to $x=\ell_n/t$:

$$
x=\frac{4nH_0\sqrt{\Omega_{\mathrm r}}\,(1+z)}{f},\qquad t^{-4}=16H_0^4\Omega_{\mathrm r}^2(1+z)^8 .
$$

Every power of $(1+z)$ cancels, and $C_n(f)=(8H_0^2\Omega_{\mathrm r}/f)\int\mathsf n_{\mathrm r}(x)\,\mathrm dx$. The factor $f$ cancels too: the spectrum is flat. Summing $\sum_nP_n=\Gamma$ and integrating $\mathsf n_{\mathrm r}$ from $0$ to $\alpha$,

$$
\int_0^\alpha\mathsf n_{\mathrm r}\,\mathrm dx=\frac{2A_{\mathrm r}}{3\alpha}\left[(1+\epsilon)^{3/2}-1\right],\qquad \epsilon=\frac{\alpha}{\Gamma G\mu},
$$

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

This is eq. 6.1. For large loops, $\epsilon\gg1$, it becomes $(128\pi/9)A_{\mathrm r}\Omega_{\mathrm r}h^2\sqrt{G\mu\,\alpha/\Gamma}$. [Auclair et al., 2020, Section 6.2.1](https://arxiv.org/abs/1909.00819v1)

Worked example: the plateau for the Lab default

With $\Omega_{\mathrm r}=9.1476\times10^{-5}$ and $h=0.678$, the prefactor is $(128\pi/9)\times0.54\times4.205\times10^{-5}=1.015\times10^{-3}$. For $G\mu=10^{-10}$ and $\alpha=0.1$, $\sqrt{G\mu\,\alpha/\Gamma}=4.47\times10^{-7}$, so $h^2\Omega_{\mathrm{plateau}}=4.54\times10^{-10}$. The Lab's harmonic sum at $25\,\mathrm{Hz}$ reads $4.46\times10^{-10}$, lower by the $1.8\,\%$ of cusp power beyond the $10^5$th harmonic. The paper prints the prefactor as about $1.02\times10^{-2}$, ten times the value its own symbols give; the method chapter records this as an unresolved discrepancy and uses the symbolic form.

## Where the plateau formula stops

The plateau assumes radiation domination for the whole life of the loops that contribute. At low frequency the signal comes from large loops that radiate after matter takes over, the spectrum rises into a bump, and the formula fails. For $G\mu=10^{-10}$ the numerical fundamental-mode spectrum at one cycle per year is $2.43$ times the plateau formula. The Lab blends the radiation and matter loop densities by the radiation share of the energy density, a teaching choice; Auclair and colleagues instead solve the one-scale model through the transition.

## Lab task

In the cosmic-string Lab, drag the loop size from its default down to the smallest value while watching the dashed plateau line. The line follows the curve at high frequency for every size. Then choose Loops born tiny and compare the plateau with the small-loop limit in exercise 3.

## Exercises

1. Derive the matter-era density $\mathsf n_{\mathrm m}$ from the same steps, using $a\propto t^{2/3}$.
2. By what factor does the large-loop plateau change between $G\mu=10^{-10}$ and $G\mu=10^{-16}$ at fixed $\alpha$?
3. Show that for $\epsilon\ll1$ the plateau no longer depends on $\alpha$, and evaluate it for $G\mu=10^{-10}$.
4. The Lab's tiny-loop preset has $\alpha=10^{-8}$ and $G\mu=10^{-10}$. Evaluate eq. 6.1 and compare with the Lab's $25\,\mathrm{Hz}$ readout of $2.09\times10^{-13}$.

Answers

1. $(a_{\mathrm i}/a)^3=(t_{\mathrm i}/t)^2$, so $\mathsf n=A_{\mathrm m}t_{\mathrm i}^{-2}t^{-2}/[\alpha_{\mathrm m}(\alpha_{\mathrm m}+\Gamma G\mu)]$. Substituting $t_{\mathrm i}$ gives $\mathsf n_{\mathrm m}=t^{-4}A_{\mathrm m}(\alpha_{\mathrm m}+\Gamma G\mu)/[\alpha_{\mathrm m}(x+\Gamma G\mu)^2]$, eq. 3.17 of the paper, with $A_{\mathrm m}=0.039$.
2. The plateau scales as $\sqrt{G\mu}$, so it falls by $\sqrt{10^6}=1000$: from $4.54\times10^{-10}$ to $4.54\times10^{-13}$. In the Lab the dashed plateau line drops by three decades between the two tensions.
3. $(1+\epsilon)^{3/2}-1\approx3\epsilon/2$, so $h^2\Omega_{\mathrm{plateau}}\approx(64\pi/3)A_{\mathrm r}\Omega_{\mathrm r}h^2\,G\mu$. Loops that die almost at birth radiate all their energy at once, whatever their size. For $G\mu=10^{-10}$ this is $1.52\times10^{-13}$. The paper's eq. 6.5 prints the coefficient as $1.52\times10^{-2}$, again ten times the symbolic value.
4. $\epsilon=10^{-8}/(5\times10^{-9})=2$, and $1.015\times10^{-3}\times(10^{-10}/2)\times(3^{3/2}-1)=2.13\times10^{-13}$. Removing the $1.8\,\%$ truncated cusp power gives $2.09\times10^{-13}$, the Lab value. Neither limit of exercise 3 applies at $\epsilon=2$.

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

Drag the loop size and check the dashed plateau formula against the curve.

## Sources

1. [Auclair et al., 2020, Section 3.1, eqs. 3.12 to 3.16](https://arxiv.org/abs/1909.00819v1)
2. [NANOGrav Collaboration, 2023, Section 5.4, eqs. 43 and 44](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. [Article **How a cosmic-string loop radiates and shrinks**](https://sandboxphysics.com/stories/cosmic-strings/how-a-loop-radiates-and-shrinks/)
4. [You are here **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/)