---
title: "Recompute the radiation-era plateau of a cosmic-string background"
description: "Rerun a fixed Nambu–Goto loop model, compare it with the published plateau formula and see which checks fail when one step is wrong."
canonical_url: "https://sandboxphysics.com/stories/cosmic-strings/recompute-the-radiation-plateau/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/cosmic-strings/recompute-the-radiation-plateau/"
last_modified: "2026-11-12"
---

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

# Recompute the radiation-era plateau of a cosmic-string background

Sandbox Physics · 2026-11-12

This guide recomputes one published number and one published formula with a fixed Nambu–Goto loop model. The number is the coefficient 8.04 in eq. 5.2 of Auclair and colleagues, the height of the radiation-era plateau in units of $\Omega_{\mathrm r}\sqrt{G\mu/\Gamma}$. The formula is their eq. 6.1 for the plateau as a function of loop size. Both come out within a fraction of a percent. Along the way it records two printed values that disagree with the same model, and four ways to get a wrong answer.

Read [From a loop population to the background spectrum](https://sandboxphysics.com/stories/cosmic-strings/from-loop-population-to-background/) first; it derives every equation used here. The paper is [Auclair et al., 2020, arXiv version 1](https://arxiv.org/abs/1909.00819v1), published in JCAP 04 (2020) 034.

## Get the materials and run them

Download the [scripts, inputs and saved outputs](https://sandboxphysics.com/stories/cosmic-strings/materials/cosmic-string-background.zip) and extract the archive. It contains six Python files, the three saved JSON files and a short README. The scripts use only the Python standard library. The saved run used Python 3.9.6 and took 37.4 seconds.

```
cd cosmic-string-background/scripts
python3 reproduce_benchmarks.py ../my-run
```

The run writes inputs.json, outputs.json and failure-cases.json into the folder you name. Compare them with the saved copies in the archive's benchmarks folder; the same files are also served as [inputs](https://sandboxphysics.com/stories/cosmic-strings/materials/inputs.json), [outputs](https://sandboxphysics.com/stories/cosmic-strings/materials/outputs.json) and [failure cases](https://sandboxphysics.com/stories/cosmic-strings/materials/failure-cases.json). Run the command from the scripts folder: the main script imports its five siblings from there.

## Fix the inputs and units

Every time and length is in units of $1/H_0$ with $c=1$, frequencies are in hertz, and densities are reported as $h^2\Omega_{\mathrm{gw}}$. The fixed inputs are:

**Table 5.1** Fixed inputs of the saved run.

| Quantity | Value | Source |
| --- | --- | --- |
| Hubble parameter $h$ | 0.678 | Auclair eq. 2.7 |
| $\Omega_{\mathrm m}$ and $\Omega_{\mathrm r}$ | 0.308 and $9.1476\times10^{-5}$ | Auclair eq. 2.7 |
| Change in particle species | none, factor held at 1 | teaching choice |
| Loop power $\Gamma$ | 50 | Auclair Section 4.1 |
| $A_{\mathrm r}$ and $A_{\mathrm m}$ | 0.54 and 0.039 | Auclair eqs. 3.16 and 3.17 |
| $\xi_{\mathrm r}$ and $\xi_{\mathrm m}$ | 0.271 and 0.625 | Auclair eq. 3.11 |
| Earliest redshift | $10^{26}$ | teaching choice |
| Highest harmonic | $10^{5}$ | as in NANOGrav Section 5.4 |

The model blends the radiation and matter loop amplitudes and birth sizes linearly in $\rho_{\mathrm r}/(\rho_{\mathrm r}+\rho_{\mathrm m})$. That blend is a teaching choice. The paper solves the one-scale equations through the transition instead, so any result that depends on the transition is a property of this model only. [NANOGrav Collaboration, 2023, Section 5.4](https://arxiv.org/abs/2306.16219v1) describes the same Model I family with $\alpha_{\mathrm L}\simeq0.37$ and $F=0.1$.

## Two independent routes to the same spectrum

The first route is the redshift integral of eqs. 4.4 and 4.5. It evaluates $C_1$ by Simpson's rule in $u=\ln(1+z)$ with 800 intervals, from today back to the birth boundary of the loops that can emit at the requested frequency. It uses the loop density $\mathsf n(\ell,t)$ and its Jacobian.

The second route never builds a loop density. It follows cohorts of loops from birth, lets each loop shrink at $\Gamma G\mu$, deposits the energy it radiates in its fundamental mode into logarithmic frequency bins after redshifting, and divides by today's critical density. An error in the Jacobian or in the $(1+z)$ powers of the first route would not appear in the second.

```
from loop_population import LoopPopulation
from background_by_redshift_integral import fundamental_only_density
import string_cosmology as cosmology

table = cosmology.CosmicTimeTable()
loops = LoopPopulation(table, 1e-10, 0.1)
print(fundamental_only_density(loops, 1e4))
```

This prints about $4.537\times10^{-10}$, the fundamental-mode plateau for $G\mu=10^{-10}$ and $\alpha=0.1$.

## What the recomputation gives

The pure-radiation integral of the simulation-based loop density, eqs. 3.18 and 3.19, gives a plateau coefficient of 8.0425. The paper prints 8.04 in eq. 5.2. An analytic evaluation of the same integral gives 8.0425 as well.

Eq. 6.1 matches the numerical fundamental-mode plateau at $10^4\,\mathrm{Hz}$ within $0.06\,\%$ for 24 of 25 cases spanning $G\mu=10^{-8}$ to $10^{-17}$ and loop sizes from $0.1$ down to $10^{-3}\,\Gamma G\mu$. The one exception, $G\mu=10^{-17}$ with $\alpha=5\times10^{-19}$, is $4.6\,\%$ low. There the loops heard at $10^4\,\mathrm{Hz}$ can only have been emitting at $1+z$ below about $6\times10^4$, close to matter–radiation equality at $z\approx3366$, so that frequency is not yet deep in the radiation plateau.

The cohort histogram and the redshift integral agree within $0.9\,\%$ at 27 frequencies from $1.1\times10^{-12}$ to $4\times10^{3}\,\mathrm{Hz}$, for $G\mu=10^{-10}$ and $10^{-17}$. Summing all the energy radiated by every cohort, without any spectrum, agrees with the frequency integral of the spectrum to $0.3\,\%$ for both tensions.

As the tension falls toward zero the plateau divided by $\sqrt{G\mu}$ stays at $4.537\times10^{-5}$ from $G\mu=10^{-12}$ to $10^{-16}$. At $10^{-18}$ and $10^{-20}$ the ratio rises to $4.549\times10^{-5}$ and $5.16\times10^{-5}$, because the spectrum reaches its plateau at higher frequency when the tension is lower and $10^4\,\mathrm{Hz}$ is no longer on it. At one cycle per year the background falls as $(G\mu)^2$ at small tension, reaching $8.0\times10^{-27}$ at $G\mu=10^{-20}$. The background goes to zero with the tension, as it must.

## Two printed values the model does not reproduce

Eq. 6.1 and eq. 6.2 print the prefactor $(128\pi/9)A_{\mathrm r}\Omega_{\mathrm r}h^2$ as about $1.02\times10^{-2}$. With the paper's own inputs it is $1.0146\times10^{-3}$, a factor of $10.05$ smaller. Eq. 6.5 shows the same factor of ten. The symbolic form agrees with eq. 5.2 and with both numerical routes, so the scripts use it. They do not edit the paper; the discrepancy is recorded as unresolved in outputs.json under printedPlateauCoefficient.

Section 6.1, in the text after figure 8, quotes a plateau of $h^2\Omega\approx6.04\times10^{-14}$ for $G\mu=10^{-17}$. Eq. 5.2 with constant particle species gives $1.51\times10^{-13}$, so the quoted value is 0.40 of it. The paper's figure 4 shows that the Standard Model's changing species count alters the high-frequency part of the spectrum, which may account for some of the difference. The statement is recorded as unresolved, and left uncorrected, until that effect is computed.

## Four ways to get it wrong

The saved failure-cases.json records four mistakes and their size.

1. Dropping the Jacobian, using $1/\alpha$ for $\mathrm dt_{\mathrm i}/\mathrm d\ell$ when $\alpha=\Gamma G\mu$, doubles the plateau: the ratio is 2.000.
2. Using the plateau formula at one cycle per year for $G\mu=10^{-10}$ underestimates the fundamental-mode background by a factor of 2.43.
3. Assuming $\sum P_n=\Gamma$ while summing only $10^5$ cusp harmonics misses $1.79\,\%$ of the power.
4. Starting the network at $z=10^{10}$ instead of $10^{26}$ removes the high-frequency plateau: at $1\,\mathrm{Hz}$ the result falls from $4.54\times10^{-10}$ to $2.6\times10^{-14}$.

## What this does and does not establish

The recomputation shows that one model agrees with itself: its loop density, its spectrum and its energy budget agree with each other and with two published closed forms. It does not test whether cosmic strings exist, whether Model I describes them, or whether constant particle species is a good approximation. It fits no data. Comparing it with pulsar-timing results needs the full likelihood analysis described in the [guide to reading pulsar-timing and LISA results](https://sandboxphysics.com/stories/cosmic-strings/reading-pta-and-lisa-string-limits/).

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

Compare the five presets with the saved reference values.

## Sources

1. [Auclair et al., 2020, arXiv version 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. [Article **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. [You are here **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/)