Cosmic Strings and Defect Backgrounds

A cosmic-string background is an integral record of a network, not a spectrum fixed by the tension alone. Long strings approach scaling, intercommute, and produce loops; loops shrink while radiating through a distribution of harmonics; the expansion history redshifts every contribution. Consequently an inference of $G\mu$ is conditional on the network description, loop population, emission spectrum, and competing particle-loss channels.

Helpful background. The Einstein Equation supplies Einstein Equations for sourced tensor modes; Relativistic Matter and Compact Objects supplies stress-energy and defect matter; Gravitational Radiation and Waveforms supplies burst and stochastic conventions; Relativistic Cosmology supplies the expansion history; and Partial Differential Equations and Green Functions supplies retarded wave propagation.

Scaling networks and the loop transport equation

An ideal local string has tension and energy per unit length $\mu$. In the Nambu–Goto approximation its width is neglected, and the important gravitational parameter is the dimensionless tension $G\mu$. A scaling long-string network has a characteristic length proportional to cosmic time, so its density remains a fixed fraction of the background density rather than redshifting as a frustrated gas of strings.

Let $n(\ell,t)d\ell$ be the physical number density of non-self-intersecting loops with invariant lengths in $[\ell,\ell+d\ell]$. In a scaling era,

$$n(\ell,t)=t^{-4}\mathcal N(\ell/t).$$

The distribution follows a transport equation of the form

$$\frac{\partial n}{\partial t}+3Hn +\frac{\partial}{\partial\ell}(\dot\ell\,n) =f(\ell,t), \qquad \dot\ell=-\Gamma G\mu,$$

where $f$ is the loop-production function and $P_{\rm GW}=\Gamma G\mu^2$ is the assumed total gravitational luminosity. This equation makes three assumptions visible: gravitational radiation dominates loop decay, the average $\Gamma$ is meaningful for the population, and loop collisions or reconnection with long strings are negligible. The formalism and competing calibrated distributions are developed in Auclair et al. 2020, §§2–3, printed pp. 4–15 (Open PDF).

If a loop is born at $t_i$ with $\ell_i=\alpha t_i$, its approximate lifetime is

$$\tau_{\ell}=\frac{\alpha t_i}{\Gamma G\mu}.$$

Thus “large” loops with $\alpha\gg\Gamma G\mu$ survive across cosmological epochs, whereas loops born near the gravitational backreaction scale are short lived. This difference changes which era supplies the loops observed at a given frequency even when $G\mu$ and the total long-string energy loss are held fixed.

From loop harmonics to a stochastic background

Write the dimensionless emission weight in harmonic $m$ as

$$P_m=\frac{\Gamma}{\zeta(q)}m^{-q}, \qquad \sum_{m=1}^{\infty}P_m=\Gamma.$$

The physical luminosity in that harmonic is $G\mu^2P_m$. The Riemann zeta function $\zeta(q)=\sum_{m=1}^{\infty}m^{-q}$ normalizes the idealized infinite sum, with $q\gt1$.

Idealized cusp, kink, and kink–kink spectra use respectively $q=4/3$, $5/3$, and $2$. A smooth loop or a finite string width imposes a high-harmonic cutoff; extending a power law indefinitely can manufacture ultraviolet power. If a truncated model retains the same total gravitational luminosity, replace $\zeta(q)$ by $\sum_{m=1}^{m_{\max}}m^{-q}$. If the cutoff instead removes gravitational power or transfers it to another channel, change the total luminosity and loop-decay law consistently. A mode emitted at cosmic time $t$ by a loop of length $\ell$ is observed at

$$f_0=\frac{2m}{\ell}\frac{a(t)}{a_0}.$$

With $a_0=1$, one convenient population formula is

$$\Omega_{{\rm GW},0}(f) =\frac{8\pi(G\mu)^2}{3H_0^2} \sum_{m=1}^{\infty}P_m I_m(f),$$ $$I_m(f)=\frac{2m}{f}\int_{t_{\rm ini}}^{t_0}dt\, a^5(t)\, n\!\left(\frac{2m\,a(t)}{f},t\right).$$

One can derive the scale-factor and harmonic weights directly. At emission, the spectral power per physical volume is $G\mu^2\sum_mP_m\int d\ell\,n(\ell,t) \delta(f_s-2m/\ell)$, where $f_s=f/a$. The length integral supplies $2m/f_s^2$. Redshifting an emitted energy density supplies $a^4$, while $df_s=df/a$ supplies $a^{-1}$. Multiplying by $f$ to form energy per $d\ln f$ therefore gives $a^3f(2m/f_s^2)=(2m/f)a^5$, as in $I_m$. The density $n$ is already physical; its evolution already accounts for number dilution. The expression also shows why a nonstandard expansion era changes a string spectrum: it alters both the time–redshift map and the loop population inherited by later epochs. Auclair et al. 2020, §§2 and 4–5, printed pp. 4–7 and 15–27 (Open PDF) derives equivalent background and burst representations and their frequency structure.

Cusps and kinks can also produce individually resolvable bursts. Calling every event stochastic double-counts rare loud signals. A population analysis must choose a rate criterion that separates an approximately Gaussian overlap from infrequent bursts, then use the same $n(\ell,t)$ and beaming law in both pieces. The split is analysis-dependent near the popcorn regime; a stochastic integral alone cannot predict the burst catalog.

Which network model is being propagated?

Network or emission picture	What it controls	Failure boundary
Nambu–Goto, large-loop population	Long-lived loop abundance and broad radiation-era plateau	Neglects finite width and direct field emission
Nambu–Goto, small-loop population	High-frequency abundance near the backreaction scale	Sensitive to numerical resolution and energy-balance extrapolation
Abelian–Higgs field simulation	Core fields, massive radiation, and long-string scaling	Present dynamic ranges do not directly reach cosmological tension and time ratios
Cusp/kink harmonic model	Burst beaming and high-harmonic power	Backreaction smooths features and sets a cutoff
Metastable strings	Network disappearance after monopole-pair nucleation or another break process	Requires an additional decay scale; stable-string plateaus no longer apply
Superstrings with reconnection probability $p\lt1$	Network density and loop production	Junctions and multi-tension species invalidate a one-species rescaling

Nambu–Goto simulations and field-theory simulations agree that long networks can scale, but differ sharply over where the lost energy goes. In field theory, massive radiation can dominate at accessible resolution; in thin-string models, long-lived loops can dominate gravitational emission. Extrapolating either result over many unresolved scale decades is a physical hypothesis, not a numerical convergence theorem. Auclair et al. compare the analytic, BOS simulation-inferred, LRS simulation-inferred, and Abelian–Higgs pictures in Auclair et al. 2020, §3, printed pp. 7–15 (Open PDF).

A matched expansion-history comparison

The decisive formulation axis is the source population: the loop density cannot be absorbed into a redefinition of tension.

To isolate loop uncertainty, hold $a(t)$, $G\mu$, $\Gamma$, formation time, and harmonic spectrum fixed, and evaluate $I_m$ twice: once with a calibrated large-loop distribution $n_L$ and once with a calibrated small-loop distribution $n_S$. The background ratio is then

$$\frac{\Omega_L(f)}{\Omega_S(f)} =\frac{\sum_m mP_m\int dt\,a^5 n_L(2ma/f,t)} {\sum_m mP_m\int dt\,a^5 n_S(2ma/f,t)}.$$

The common $2/f$ cancels, but the harmonic-dependent factor $m$ must stay inside each sum. No factor of $(G\mu)^2$ survives this ratio. Any difference is therefore a population effect. The same pair must next be passed through the burst-rate integral, rather than renormalized separately to agree with the background. Finally, impose a common short-loop particle-emission prescription. A model that gains high-frequency power from numerous tiny loops but discards their massive-particle losses is not a controlled alternative to the large-loop case. This matched calculation is the clean counterexample to quoting a network-independent tension bound.

Results and limitations through 2026-08-19

Formulation or evidence	Result	Scope or limitation
Auclair et al. 2020 analytic and simulation synthesis	Propagates several loop distributions and emission spectra through a common cosmology	Forecast differences remain large because loop production and small-scale structure are not settled
Nambu–Goto network simulations	Support scaling and long-lived loop populations in the thin-string description	Finite dynamic range and gravitational-backreaction modeling control the smallest loops
Abelian–Higgs simulations	Resolve string cores and direct radiation into field modes	Extrapolation to the enormous cosmological width-to-horizon hierarchy remains unresolved
NANOGrav 15-year new-physics analysis	A low-frequency background can be fit by several stable or metastable string templates	The collaboration explicitly does not treat the model comparison as evidence for new physics; SMBHB and string priors are consequential
LVK O1–O4a cosmological-background search	Finds no evidence for a string component in its tested likelihoods	Resulting tension exclusions vary strongly among loop models and burst-feature assumptions
Particle and CMB constraints	Limit energy injection or defect stress through observables other than gravitational waves	Translation depends on branching fractions, network type, and expansion history

The NANOGrav analysis and its model definitions are documented in Afzal et al. 2023, §5.4, printed pp. 23–30, and §7, printed pp. 40–43 (Open PDF). The O1–O4a null search, likelihood, and explicit loop-model dependence appear in LIGO–Virgo–KAGRA 2025, §I, printed pp. 1–3, and §III, printed pp. 5–8 (Open PDF). These observations appraise specified templates; neither selects a universal network model.

What could change the conclusion. The interpretation could change after a new network simulation resolves the loop-production and backreaction scales, an observational likelihood jointly compares burst and stochastic channels, or a cross-band analysis materially shifts tension inferences under matched loop and expansion models.

Scope and observational interpretation

This page explains how a physical string model maps to cosmological wave production. Instrument calibration, overlap-reduction functions, timing-noise models, observing-run products, and mutable numerical upper limits belong to Gravitational-Wave Astronomy. A possible background must also be separated from astrophysical populations on the Cosmological and Astrophysical Background Separation page.

Exercises

Take $\Gamma=50$ and $G\mu=10^{-11}$. Compare a loop born with $\alpha_L=0.1$ with one born at $\alpha_S=\Gamma G\mu$. Find their lifetimes in units of $t_i$. At a fixed emission time and harmonic, how does doubling the loop length change the observed frequency?

Solution

The gravitational shrinkage scale is $\Gamma G\mu=5\times10^{-10}$. Therefore

$$\frac{\tau_L}{t_i} =\frac{0.1}{5\times10^{-10}}=2\times10^8, \qquad \frac{\tau_S}{t_i} =\frac{5\times10^{-10}}{5\times10^{-10}}=1.$$

The first loop retains memory across many expansion times, while the second decays on the order of its formation time. Since $f_0=2ma(t)/(\ell a_0)$, replacing $\ell$ by $2\ell$ sends $f_0\to f_0/2$. Thus identical tension does not imply identical frequency weighting: the birth-size distribution and survival time enter before any instrument comparison.

As a separate normalization check, retain harmonics $m=1,2$ with equal $P_m$. Suppose the two time integrals $\int dt\,a^5n_L$ are $(1,3)$ in common units, while the corresponding $n_S$ integrals are $(1,1)$. Find $\Omega_L/\Omega_S$.

Solution

The harmonic weights give $(1+2\cdot3)/(1+2\cdot1)=7/3$. Dropping $m$ would incorrectly give $(1+3)/(1+1)=2$. Common tension and emission coefficients cancel only outside the sum; a frequency-dependent population integral retains its harmonic Jacobian.

References

• Afzal, Adeela, et al. 2023. “The NANOGrav 15-Year Data Set: Search for Signals from New Physics.” Astrophysical Journal Letters 951, L11. DOI.
• Auclair, Pierre, et al. 2020. “Probing the Gravitational Wave Background from Cosmic Strings with LISA.” Journal of Cosmology and Astroparticle Physics 04, 034. DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. 2025. “Cosmological and High Energy Physics Implications from Gravitational-Wave Background Searches in LIGO–Virgo–KAGRA’s O1–O4a Runs.” arXiv:2510.26848v2. Open preprint.