Extending the Velocity-dependent One-scale String Evolution Model

Cambridge); C.J.A.P. Martins; CMS; E.P.S. Shellard (DAMTP

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Extending the Velocity-dependent One-scale String Evolution Model

C.J.A.P. Martins, E.P.S. Shellard (DAMTP, CMS, Cambridge)

High Energy Physics - Phenomenology · Submitted 30 Mar 2000 · 27 references

Pith's verdict: A solid methods paper that does what it claims analytically, but its headline accuracy claim and one figure caption need fixing before it's citable.

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arXiv IDhep-ph/0003298 (version 1)

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CitationCambridge), C.J.A.P. Martins, CMS, E.P.S. Shellard (DAMTP. "Extending the Velocity-dependent One-scale String Evolution Model." arXiv:hep-ph/0003298 (2000). https://pith.science/paper/KUWKBGDT

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Editor's note

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Quick read on Martins & Shellard. It's a methods paper, not a results paper: it consolidates the VOS model and adds two things — a specified k(v) and a long-string radiation backreaction term. Both are real additions over the earlier VOS papers, where k was left as a second free parameter. Worth your time if you use VOS for anything quantitative; skip it if you don't work on strings at all. The analytic core is clean. The endpoints k(1/sqrt2)=0 and k_nr = 2sqrt2/pi are derived, not fitted, and the perturbed helicoidal equation (3.5) is written out so you can check it yourself. The scaling solutions (2.12) and (5.9) follow from the ODEs without hidden steps. The authors are also honest about what k is — they say plainly in §III that it's phenomenological and that the network measurement is deferred to ref. [16]. I'd rather see that than a fake derivation. Where I'd push back on the reader's framing: the worry isn't the interpolating function (3.9). At GUT velocities (v ~ 0.6) it and the relativistic form (3.10) differ by a few percent, so the hand-chosen (1 + 2sqrt2 v^3) factor isn't where the risk sits. The risk is the transfer — helicoidal strings are smooth, real network strings are kinked — and since gamma^2 is proportional to k(k + c-tilde), a 20% error in k moves gamma by ~10%, which is the size of the claimed agreement. That's acknowledged, not hidden, but it does mean the abstract's 'accurate description' [abstract:s3] rests entirely on companion papers listed as submitted or in preparation. Fine for a methods paper; a referee should still say so. The one thing that actually needs fixing: §IV says the Fig. 6 curves are Sigma = 0, 1.25, 5.5, and the caption says 0, 10, 50. That's an order of magnitude, and it decides whether the plotted curves illustrate the perturbative regime (4.3-4.4) or the asymptotic one (4.5-4.7). Also worth noting that Sigma_ax-sim ~ 3 is chosen to match Yamaguchi's densities, not predicted. Send it out; it deserves a referee, mostly to catch the caption and to press on the deferred comparison.

Referee report

Significance

The VOS model is the standard analytic tool for cosmic string network evolution, and reducing its phenomenological content to a single parameter c-tilde is genuinely useful for the many cosmological predictions that inherit it. The paper deserves credit for several things: the perturbation equation (3.5) is written out explicitly rather than quoted, the two endpoints of the k ansatz are derived rather than fitted (I verified that (3.9) reproduces both k(0)=2 sqrt2/pi and k(1/sqrt2)=0 exactly), and the backreaction extension yields a falsifiable comparison with published global-string simulation densities, Eq. (4.8)-(4.9). The authors are also candid that k is 'ab initio a phenomenological parameter'. The flat-space scaling solution (2.12) and the large-Sigma series (4.5)-(4.6) both check out algebraically, which is a good sign for the core of the construction.

Major objections (4)

[§IV, Eq. (4.4)] I am unable to reproduce Eq. (4.4) from Eqs. (2.3), (2.5) and (4.1). Writing x = v^2 and substituting gamma = k/(2 beta v) from the (unchanged) velocity equation into the length equation with the backreaction term gives 1 - beta = beta x (k + c-tilde)/k + 8 Sigma beta x^{7/2}/k. Linearising about the unperturbed x_0 = k(1-beta)/(beta(k+c-tilde)) gives delta x / x_0 = -8 Sigma v_0^5/(k+c-tilde), i.e. Delta = 8 Sigma v_0^5/(k+c-tilde) = 8 [k(1-beta)/(beta(k+c-tilde))]^{5/2} Sigma/(k+c-tilde), not 8 beta v_0^5 Sigma. The two agree only if beta(k+c-tilde) = 1, which fails for every realistic (k, c-tilde, beta). The same starting equation does reproduce (4.5)-(4.6) in the large-Sigma limit, which is why I believe the reading of (4.1) is correct and the slip is confined to (4.4). This matters quantitatively: for radiation-era parameters (beta = 1/2, k about 0.22, c-tilde = 0.23) the corrected coefficient is roughly 4.4 times larger, so the statement at the end of §IV that GUT-scale global strings would have their density reduced 'by less than 10%' becomes closer to 25%, and the value Sigma = 1.25 described as the upper limit of validity of (4.3)-(4.4) would move down. Please recheck and, if the correction stands, revise the numerical claims and the choice of curves in Fig. 6.
[§V, Eqs. (5.9)-(5.10)] Eqs. (5.9)-(5.10) do not appear to follow from (5.3)-(5.6). With L = gamma t, H = beta/t and 1/l_d = 2H, Eq. (5.6) gives v = w^2 k/(2 beta gamma), and Eq. (5.4) then gives 1 - beta = beta v^2 w^2 + c-tilde v/(2 gamma). Eliminating v yields gamma^2 = w^2 k (w^4 k + c-tilde)/(4 beta (1-beta)) and v^2 = w^2 k (1-beta)/(beta (w^4 k + c-tilde)); the printed (5.9) has k in place of w^4 k in both. The two forms coincide only at w = 1, that is Omega = 1, so the discrepancy is invisible in the flat case that is actually plotted. Separately, combining (5.3) with the printed (5.9) gives w = (sqrt(1+4A) - 1)/(2A) with A = (1-Omega) beta k (k+c-tilde)/(4(1-beta)), whereas (5.10) as printed equals that expression multiplied by (1-Omega)^2. Since §V is presented as one of the paper's extensions, the curved-universe scaling solution should be internally consistent before it is used; please recheck the algebra and, if necessary, state which of (5.3), (5.4), (5.6) or (5.9) is to be amended.
[§II.B and §V] Both sections assert that the scaling solutions (2.12) and (5.9) are stable ('it is easy to verify numerically that this solution is well-behaved and stable for all realistic parameter values') but no linearised analysis is given. The attractor property is precisely what licenses the use of these solutions to set scale-invariant initial conditions for structure formation, and the curved case (5.9)-(5.10) is new here, so the claim is load-bearing rather than decorative. A two-line linearisation of (2.3)-(2.5) about the fixed point, quoting the two eigenvalues and their signs for 0 < beta < 1, would settle the point at negligible cost and would also make the stability statement in §V checkable rather than asserted.
[§III and abstract] The paper is explicit that k is phenomenological and that the transfer of the ansatz to a realistic kinked network must be checked against ultra-high-resolution simulations, deferred to ref. [16]; I do not treat that acknowledged limitation as a defect in itself, and I note that at the velocities relevant for GUT strings (v about 0.6) the full ansatz (3.9) and the relativistic form (3.10) differ by only a few percent, so the hand-chosen (1 + 2 sqrt2 v^3) factor is not where the risk lies. What is missing is a sensitivity estimate. Since (2.12) gives gamma^2 proportional to k(k+c-tilde) and v^2 proportional to k/(k+c-tilde), a 20% uncertainty in k propagates to roughly 10% in gamma and in v^2, which is comparable to the accuracy claimed against simulations in the companion paper. A short paragraph quantifying this propagation, and stating which of the conclusions in §II and §IV survive it, would make the 'single free parameter' framing of abstract:s1 and the accuracy claim of abstract:s3 properly calibrated rather than conditional on an unstated error budget.

Minor objections (9)

[§IV and Fig. 6 caption] The text states that three curves are plotted for Sigma = 0, Sigma = 1.25 and Sigma = 5.5, while the Fig. 6 caption states Sigma = 0 (solid), Sigma = 10 (dashed) and Sigma = 50 (dotted). These cannot both be right; please reconcile, since the figure is the paper's main illustration of the backreaction timescale claim.
[§III, after Eq. (3.7)] The sentence 'in the relativistic limit A -> 1 this same calculation would give k = 4 sqrt2 (3/pi) about 0.6' contains a typographical error: (3.7) at A = 1 gives 4 sqrt2/(3 pi) about 0.600, not 4 sqrt2 times 3/pi.
[§II.D, Eq. (2.19)] Eq. (2.19) as printed, L v_infinity = (2 sqrt2/pi) H, is dimensionally inconsistent (the left side has dimensions of length, the right of inverse length). Solving (2.5) with L proportional to a and v proportional to 1/a in de Sitter gives L v_infinity = (2 sqrt2/pi)/H, so the inverse appears to have been dropped.
[§II.A] The definition 'rho_infinity = mu L^2' is dimensionally inconsistent with a density and with Eq. (2.11), which implies rho_infinity = mu/L^2. Please correct.
[§V, Eq. (5.1)] The Friedmann equation is written as H^2 = (a-dot/a) = ..., with the square missing on the right-hand side.
[§V, Eq. (5.4)] Typographical error: 'For simplicity wa have also defined a damping length' should read 'we'.
[§II.A, Eq. (2.1)] The energy integral is written with the scale factor evaluated at tau while the rest of the paper uses t; please make the time variable uniform. Similarly, Eq. (3.6) uses |x-dot^2| where (2.6) has (x-dot . u); a sentence explaining the curvature-only limit that connects the two would help.
[§IV] The remark that there is 'an interesting coincidence' between the v^6 power in the k ansatz (2.7) and in the backreaction term (4.1), and that the authors are 'unsure as yet whether this has any deeper significance', is speculative and could be cut or explicitly flagged as a numerical curiosity.
[References] Three of the citations supporting the paper's key claims, refs. [14], [15] and [16], are listed as in preparation, and the headline validation of abstract:s3 rests on ref. [5], listed as submitted. The manuscript would be clearer if it stated in the introduction that its own quantitative tests are confined to Fig. 4 and Fig. 6.

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The paper

The paper claims

Cosmic string networks may need only one adjustable parameter, not two, once a new momentum formula and radiation backreaction are included.

The reading

Cosmic strings are hypothetical one-dimensional defects left over from early-universe phase transitions, and their large-scale behaviour is often modelled by the velocity-dependent one-scale (VOS) model: two averaged equations for a correlation length $L$ and an rms velocity $v$, with a loop-chopping efficiency $�tilde c$ and a momentum parameter $k$. This paper asks whether the model can be made predictive with only one free parameter by replacing the previously free $k$ with a physically motivated function $k(v)$ and by adding energy loss from radiation backreaction. It then works out the resulting scaling solutions in radiation, matter, curvature, and cosmological-constant domination, and argues that the extended model captures the large-scale network properties when compared with simulations; the detailed comparison is reported in a companion paper. If that is right, analytic predictions for cosmic-string networks lose an adjustable knob and become more reliable across cosmic history, including the matter-radiation transition where the network is not scaling.

What carries the argument

The carrying object is the two-equation VOS model: $2�,dL/dt=2HL(1+v_�infty^2)+L v_�infty^2/�ell_f+�tilde c v_�infty$ in the flat case, and $dv_�infty/dt=(1-v_�infty^2)[k/L-(2H+1/�ell_f)v_�infty]$, with $k$ no longer a free function but the ansatz $k(v)=�frac{2�sqrt2}{�pi}(1-v^2)(1+2�sqrt2 v^3)�frac{1-8v^6}{1+8v^6}$. The ansatz is built to reproduce the relativistic limit $k_{�rm rel}=�frac{2�sqrt2}{�pi}�frac{1-8v^6}{1+8v^6}$ and the non-relativistic limit $k_{�rm nr}=�frac{2�sqrt2}{�pi}(1-v^2)$. Backreaction adds $8�Sigma v_�infty^6$ to the right-hand side of the $L$ equation, with $�Sigma=�tilde�Gamma G�mu$ for local strings and $�Sigma=�tilde�Gamma/[2�pi�ln(t/�delta)]$ for global strings; this shifts the scaling solution by $�Delta=8�beta v_0^5�Sigma$ and, for large $�Sigma$, gives $�gamma�propto�Sigma^{1/7}$ and $v�propto�Sigma^{-1/7}$.

Extended reading notes

Core claim

The central claim is that the VOS model, with $k$ fixed by Eq. (3.9) and with a radiation-backreaction term added to the $L$ equation, describes the large-scale features of cosmic-string networks using only the loop-chopping efficiency $�tilde c=0.23�pm0.04$. The paper derives the two endpoints of $k(v)$—$k(1/�sqrt2)=0$ and $k_{�rm nr}=2�sqrt2/�pi$—and supplies an interpolating formula; it then shows that scale-invariant attractors still exist with backreaction, though their scaling values shift, and it gives the scaling solutions for power-law FRW backgrounds and for curvature-dominated open universes. The authors also report that the low densities seen in global-string simulations can be accommodated by a backreaction parameter $�Sigma_{�rm ax-sim}�approx3$, while cautioning that cosmological global strings have much larger $�ln(t/�delta)$ and therefore much smaller backreaction.

Load-bearing premise

The load-bearing premise is that the interpolating formula for the momentum parameter in Eq. (3.9), which joins the two derived limits, is correct at intermediate velocities; the paper fixes that bridge by fitting and defers the decisive high-resolution simulation check to a companion paper.

Editorial extensions

If this is right

A realistic GUT-scale string network is not scale-invariant from just before matter-radiation equality to the present, so predictions that assume pure scaling during that interval can be biased.
Electroweak-scale strings can remain friction-dominated until about three orders of magnitude after equality; for $�Omega_m=0.2,�Omega_�Lambda=0.8$ they are always non-relativistic today, which matters for superconducting-string and vorton studies.
A cosmological-constant-dominated future makes the network freeze out and be conformally stretched, with $L�propto a$ and $v_�infty�propto a^{-1}$, lowering density and velocity and affecting large-angle CMB anisotropies.
In an open, curvature-dominated universe the asymptotic solution is $L=At(�ln t)^{1/2}$ and $v_�infty=B(�ln t)^{-1/2}$, so the network is not scale-invariant and retains a record of when curvature domination began.
The global-string simulation densities $�zeta_{�rm rad}=0.9�pm0.1$ and $�zeta_{�rm mat}=0.5�pm0.1$ are consistent with the extended model for $�Sigma_{�rm ax-sim}�approx3$, but the same formula predicts much weaker backreaction for cosmological GUT-scale global strings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the one-parameter claim holds, a systematic drift in the fitted $�tilde c$ across radiation, matter, and transition simulations would be a clean diagnostic that the model is missing small-scale physics; conversely, a constant $�tilde c$ would support the idea that $k(v)$ absorbs the relevant small-scale effects.
The same $v^6$ power appears in the $k$ ansatz and in the backreaction term; a wiggly-string derivation could test whether both are symptoms of one underlying small-scale structure parameter, which the paper explicitly leaves open.
For CMB and structure-formation work, the non-scaling interval around equality suggests that bolting a scaling network onto a cosmological perturbation code is insufficient; one should integrate the VOS equations through the transition.
The large-$�Sigma$ regime seen in small global-string simulations could serve as a calibration target for simulation systematics, but extrapolating it to $�ln(t/�delta)�sim100$ should be treated as a separate cosmological calculation.

Original abstract

We provide a general overview of the velocity-dependent one-scale model for cosmic string evolution and discuss two further extensions to it. We introduce and justify a new ansatz for the momentum parameter $k$, and also incorporate the effect of radiation backreaction. We thus discuss the evolution of the basic large-scale features of cosmic string networks in all relevant cosmological scenarios, concentrating in particular on the `scaling' solutions relevant for each case. In a companion paper, we show, by comparing with numerical simulations, that this model provides an accurate description of the large-scale features of cosmic string networks.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,587 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.34 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We provide a general overview of the velocity-dependent one-scale model for cosmic string evolution and discuss two further extensions to it. We introduce and justify a new ansatz for the momentum parameter $k$, and also incorporate the effect of radiation backreaction. We thus discuss the evolution of the basic large-scale features of cosmic string networks in all relevant cosmological scenarios, concentrating in particular on the `scaling' solutions relevant for each case. In a companion paper, we show, by comparing with numerical simulations, that this model provides an accurate description of the large-scale features of cosmic string networks.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We provide a general overview of the velocity-dependent one-scale model for cosmic string evolution and discuss two further extensions to it.	contribution 0.77	0.80	0.08	asserted 0.92
s1	We introduce and justify a new ansatz for the momentum parameter $k$, and also incorporate the effect of radiation backreaction.	contribution 0.95	0.95	0.09	asserted 0.99
s2	We thus discuss the evolution of the basic large-scale features of cosmic string networks in all relevant cosmological scenarios, concentrating in particular on the `scaling' solutions relevant for each case.	contribution 0.91	0.80	0.16	asserted 0.75
s3	In a companion paper, we show, by comparing with numerical simulations, that this model provides an accurate description of the large-scale features of cosmic string networks.	future work 0.38	0.73	0.24	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from argument 0.53.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. The complete cosmological history of a GUT-scale cos

**Figure 2.** FIG. 2. Same as Fig. 1, but for an electroweak-scale cosmic st

**Figure 3.** FIG. 3. A close-upof Fig. 1, showing the recent evolution of t

**Figure 5.** FIG. 5. Comparing our full ansatz for the momentum parameter

**Figure 6.** FIG. 6. The effect of gravitational back-reaction on the appr

Figures reproduced from arXiv:hep-ph/0003298 by the authors.

References

Works cited (27)

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Gravitational Wave Bursts as Harbingers of Cosmic Strings Diluted by Inflation
hep-ph 2019-12 conditional novelty 6.0 of 10

A cosmic string network diluted by inflation can regrow to observable density by today, and its leading gravitational wave signal is resolvable bursts rather than a stochastic background.
Beyond the Standard Models with Cosmic Strings
hep-ph 2019-12 conditional novelty 6.0 of 10

Solving the velocity-dependent one-scale equations through cosmological transitions shifts the cosmic-string gravitational-wave turning point to lower frequency by a factor of about 22, letting future interferometers ...
Probing the gravitational wave background from cosmic strings with LISA
astro-ph.CO 2019-09 conditional novelty 6.0 of 10

LISA is projected to probe cosmic string tensions G mu down to about 1e-17 and to measure spectral features from early-universe physics.
The scaling density of axion strings
astro-ph.CO 2019-08 conditional novelty 6.0 of 10

Axion string networks show standard scaling with ζ0 = 1.19 ± 0.20, and the claimed logarithmic growth in string density is explained as an artifact of initial conditions.
Long-term dynamics of cosmological axion strings
hep-ph 2018-06 conditional novelty 6.0 of 10

Extended field-theoretic simulations show the axion-string scaling parameter grows logarithmically in time and varies non-monotonically with the Peccei-Quinn breaking scale, contradicting the standard scaling solution.
Scaling from gauge and scalar radiation in Abelian Higgs string networks
astro-ph.CO 2017-03 conditional novelty 6.0 of 10

Abelian Higgs cosmic string networks lose energy to scalar and gauge radiation at a scale-independent rate, so the network scales and thin-string models overestimate loop densities and gravitational wave signals.
CMB constraints on cosmic strings and superstrings
astro-ph.CO 2016-03 conditional novelty 6.0 of 10

Joint MCMC fits to Planck 2015 data give Gμ < 1.1e-7 for cosmic strings and Gμ_F < 2.8e-8 for superstrings, marginalizing over network and string-theory parameters.
Planck 2015 results. XIII. Cosmological parameters
astro-ph.CO 2015-02 accept novelty 6.0 of 10

Full-mission Planck temperature and polarization data are consistent with six-parameter flat Lambda-CDM, yielding H0 = 67.8 +/- 0.9, Omega_m = 0.308 +/- 0.012, ns = 0.968 +/- 0.006, and tau = 0.066 +/- 0.016.
Frustrated Expectations: Defect Networks and Dark Energy
astro-ph 2006-02 conditional novelty 6.0 of 10

Domain wall networks can only explain dark energy if their curvature parameter is below 10^-4, implying severe fine-tuning; the authors conjecture such frustrated lattices do not emerge naturally from phase transitions.
Phenomenological implications of cosmic (super)strings with time-varying tension
astro-ph.CO 2026-10 conditional novelty 5.0 of 10

Time-varying cosmic string tension models fit the NANOGrav background as well as the constant-tension superstring model, but their spectral features are not distinctive enough to be identified with current data.
Cosmological Consequences of a Highly Decoupled Right-handed Neutrino
hep-ph 2026-10 conditional novelty 5.0 of 10

A long-lived, weakly coupled right-handed neutrino can drive an early matter-dominated phase whose decay dilutes the baryon asymmetry and imprints a frequency break on the cosmic-string gravitational-wave spectrum.
The friction-era VOS amplitude of a $\mathbb{Z}_2$ string network from nematic disclination data
cond-mat.soft 2026-08 conditional novelty 5.0 of 10

Digitized 1991 disclination data yield friction-era VOS amplitudes A=10.0 (234 µm cell) and A=3.0 (158 µm cell), giving a sink-coefficient curve c̃(λ) that exceeds oriented U(1) values unless λ is small.
Cosmic string gravitational wave backgrounds at LISA: II. Reconstruction of conventional signals over astrophysical foregrounds
astro-ph.CO 2026-07 conditional novelty 5.0 of 10

When realistic astrophysical foregrounds are included, LISA can reconstruct the cosmic-string tension to 10% precision only for Gμ ≳ 10^{-11}, 10^5 times larger than foreground-free forecasts.
Gravitational waves from cosmic strings with friction: analytical approximations and parameter space
astro-ph.CO 2026-05 unverdicted novelty 5.0 of 10

Analytical approximations for the ultra-high-frequency GW peak sourced by friction-era cosmic string loops, with expanded observable parameter space.
High-Quality Axion Dark Matter at Gravitational Wave Interferometers
hep-ph 2025-09 unverdicted novelty 5.0 of 10

In gauged U(1) completions enabling high-quality axion dark matter, cosmic string loops generate a stochastic gravitational wave background with an infrared break frequency that exceeds foregrounds above 10^14 GeV bre...
Cosmic string gravitational wave backgrounds at LISA: I. Signal survey, template reconstruction, and model comparison
astro-ph.CO 2025-08 unverdicted novelty 5.0 of 10

As provided, the manuscript body (random lasing) does not correspond to the abstract (cosmic string gravitational wave backgrounds at LISA), leaving the abstract's quantitative claims unsupported by any accessible text.
Gravitational waves from metastable cosmic strings in the delayed scaling scenario
astro-ph.CO 2025-01 conditional novelty 5.0 of 10

Metastable cosmic strings with delayed scaling can explain the PTA gravitational wave background with tension up to Gmu ~ 3e-5 while evading LVK bounds, producing a plateau and an f^-1/3 tail detectable by future inte...
What is the source of the PTA GW signal?
astro-ph.CO 2023-08 conditional novelty 5.0 of 10

A common-likelihood comparison of seven proposed sources of the nanohertz gravitational wave background finds that phase transitions, domain walls, scalar-induced and first-order gravitational waves fit the NANOGrav 1...
Gravitational Waves from Global Cosmic Strings and Cosmic Archaeology
hep-ph 2021-06 conditional novelty 5.0 of 10

A calibrated analytic model of global cosmic string networks predicts a gravitational wave spectrum that falls as a log-cubed power law, and whose shape maps observed frequencies onto cosmic temperatures as high as 10^8 GeV.
Exploring the Early Universe with Gaia and THEIA
hep-ph 2021-04 conditional novelty 5.0 of 10

Gaia and its proposed successor THEIA can detect gravitational waves in the frequency gap between pulsar timing arrays and LISA, a band where dark-sector confinement transitions and the QCD transition are expected to radiate.
Baryogenesis from ultralight primordial black holes and strong gravitational waves from cosmic strings
hep-ph 2020-12 conditional novelty 5.0 of 10

Non-thermal leptogenesis from evaporating ultralight primordial black holes, combined with cosmic strings from the same U(1) breaking, predicts a strong stochastic gravitational-wave background with a spectral break near MHz.
From NANOGrav to LIGO with metastable cosmic strings
astro-ph.CO 2020-09 conditional novelty 5.0 of 10

Metastable cosmic strings from a B-L breaking GUT model can explain the NANOGrav signal and predict a gravitational-wave background detectable by ground-based interferometers.
Cosmic String Interpretation of NANOGrav Pulsar Timing Data
astro-ph.CO 2020-09 conditional novelty 5.0 of 10

A cosmic string network with tension Gμ between 4×10^-11 and 10^-10 reproduces the NANOGrav 12.5-year common-spectrum signal at 68% confidence, and fits it better than a supermassive-black-hole slope.
Gravitational waves and proton decay: complementary windows into GUTs
hep-ph 2020-05 conditional novelty 5.0 of 10

Proton-decay and cosmic-string gravitational-wave bounds, taken together, favour SO(10) breaking through SU(5)xU(1) or Pati-Salam and disfavour flipped SU(5)xU(1) or plain SU(5), conditional on the NANOGrav signal bei...
BSM with Cosmic Strings: Heavy, up to EeV mass, Unstable Particles
hep-ph 2019-12 conditional novelty 5.0 of 10

Detecting a turning point in the cosmic-string gravitational-wave spectrum would extend Big Bang nucleosynthesis limits on heavy relic lifetimes by up to 15 orders of magnitude and constrain moduli, gravitationally pr...
Stochastic Gravitational Wave Background from Global Cosmic Strings
hep-ph 2019-10 conditional novelty 5.0 of 10

Gravitational waves from global cosmic strings can be observable by planned detectors for symmetry-breaking scales above 10^14 GeV, and their spectral shape probes the pre-BBN expansion history.
Probing the Early Universe with Axion Physics and Gravitational Waves
astro-ph.CO 2019-04 conditional novelty 5.0 of 10

In non-standard early cosmologies the dark-matter axion mass shifts by orders of magnitude, and in an early matter-dominated phase the gravitational-wave background from axion strings sits at the edge of LISA and BBO ...
Probing the pre-BBN universe with gravitational waves from cosmic strings
hep-ph 2018-08 conditional novelty 5.0 of 10

A cosmic string network's gravitational wave spectrum maps observed frequency onto the cosmic temperature at emission, so a measured spectrum could test radiation domination up to temperatures far above Big Bang nucle...
Cosmic Archaeology with Gravitational Waves from Cosmic Strings
hep-ph 2017-11 conditional novelty 5.0 of 10

The gravitational wave spectrum from a scaling cosmic string network carries a break at a frequency set by the temperature at which any non-standard cosmological era ended, so future detectors could measure the univer...
New CMB constraints for Abelian Higgs cosmic strings
astro-ph.CO 2016-09 conditional novelty 5.0 of 10

Using 64x larger Abelian Higgs string simulations and improved radiation-matter transition modelling, the cosmic string tension limit from Planck 2015 tightens to G mu < 2.0e-7 at 95% confidence, about 17% stronger th...
Extending the velocity-dependent one-scale model for domain walls
hep-ph 2016-02 conditional novelty 5.0 of 10

Domain wall networks lose energy primarily through scalar radiation rather than chopping, and a velocity-dependent momentum parameter lets the one-scale model fit simulations across twelve expansion rates.
Axion dark matter from topological defects
hep-ph 2014-12 conditional novelty 5.0 of 10

Updated field-theoretic simulations of axion strings and domain walls place axion dark matter at m_a = (0.8-1.3)x10^-4 eV for N_DW = 1, and at 10^-4 to 10^-2 eV for N_DW > 1 with a tuned bias term.
Stochastic Gravitational Wave Background generated by Cosmic String Networks: Velocity-Dependent One-Scale model versus Scale-Invariant Evolution
astro-ph.CO 2013-04 conditional novelty 5.0 of 10

Using the velocity-dependent one-scale model instead of assuming scale-invariant network evolution raises the predicted peak amplitude and broadness of the cosmic string gravitational wave background, which would tigh...
Scaling Properties of Domain Wall Networks
hep-ph 2011-10 conditional novelty 5.0 of 10

Large domain-wall simulations confirm scale-invariant network evolution and calibrate the two free parameters of the velocity-dependent one-scale model to c_w = 0.5 and k_w = 1.1.
Cosmic string loop distribution on all length scales and at any redshift
astro-ph.CO 2010-06 conditional novelty 5.0 of 10

Cosmic string loops settle into a universal scaling distribution on all length scales, with a scale-independent plateau below the gravitational backreaction cutoff, and today's loop density is bounded in terms of the ...
Impact of cosmic evolution on the gravitational wave spectrum from cosmic strings
hep-ph 2026-10 conditional novelty 4.0 of 10

Model computations show a kination era boosts the cosmic-string gravitational-wave spectrum, while matter, vacuum, and extra-radiation epochs suppress it, and dark energy has negligible effect.
Bayesian Forecasts on Cosmic Superstring Searches with LISA
hep-ph 2026-08 conditional novelty 4.0 of 10

With simulated LISA data, cosmic superstring parameters Gµ and P remain strongly degenerate except in loud, small-P regions where a cusp-kink shape mixture can be distinguished.
Constraints on Cosmic Strings from the Curl-Mode CMB Lensing Power Spectrum measured by ACT DR6
astro-ph.CO 2026-06 unverdicted novelty 4.0 of 10

ACT DR6 curl lensing yields 2σ bounds Gμ P^{-1} ≤ 3.5×10^{-5} (small-P) and Gμ ≤ 5.0×10^{-5} (P=1), tightened further with Planck 2013 data.
Magnetic monopoles and high frequency gravitational waves from quasi-stable strings
hep-ph 2026-03 conditional novelty 4.0 of 10

SO(10) breaking through flipped SU(5) or Pati-Salam subgroups can produce GUT monopoles from merging monopole-antimonopole pairs, while the intervening quasi-stable strings emit gravitational waves from Hz to kHz.
Cosmic Strings as Dynamical Dark Energy: Novel Constraints
astro-ph.CO 2025-05 conditional novelty 4.0 of 10

Cosmic string networks are constrained to less than ~1% of the energy density using CMB+BAO+SN data, with some models preferring mildly negative densities but no Bayesian evidence favoring them over LambdaCDM.
C-parity, magnetic monopoles and higher frequency gravitational waves
hep-ph 2025-02 conditional novelty 4.0 of 10

The C-string-wall network from SO(10) breaking with unbroken C-parity produces a gravitational wave background peaking between 100 Hz and 100 kHz for intermediate breaking scales around 10^10 to 10^12 GeV.
The nanohertz stochastic gravitational wave background from cosmic string loops and the abundant high redshift massive galaxies
astro-ph.HE 2023-06 conditional novelty 4.0 of 10

A two-parameter fit of cosmic string loop models to the NANOGrav 15-year spectrum succeeds, but the same parameters underpredict the JWST high-redshift stellar mass density by orders of magnitude, so no single loop po...
Cosmic Superstrings Revisited in Light of NANOGrav 15-Year Data
astro-ph.CO 2023-06 conditional novelty 4.0 of 10

Cosmic superstrings with tension 10^-12 to 10^-11 and intercommutation probability 10^-3 to 10^-1 fit the NANOGrav 15-year data, and the resulting high-frequency signal stays visible to planned detectors even under mo...
Has NANOGrav found first evidence for cosmic strings?
astro-ph.CO 2020-09 conditional novelty 4.0 of 10

Cosmic-string networks with tension between roughly 10^-11 and 10^-7 and loop sizes from 10^-8 to 0.1 can reproduce the NANOGrav 12.5-year common-spectrum signal.
Indirect dark matter searches with neutrino telescopes via energetic cosmic showers
hep-ph 2025-09 reject novelty 3.0 of 10

Bounds on Z'-mediated dark matter are derived from neutrino telescopes, freeze-in, and cosmic strings, but the telescope bounds rest on comparing a Galactic-center flux to faraway sources.
Metastable cosmic strings
hep-ph 2023-07 accept novelty 3.0 of 10

A network of metastable cosmic strings with GUT-scale tension and lifetime parameter sqrt(kappa) around 8 can fit the pulsar-timing gravitational-wave signal at 3 nHz and predict an observable signal at LISA and groun...
Signals of Inflationary Models with Cosmic Strings
astro-ph.CO 2011-06 accept novelty 3.0 of 10

A review of the observational signatures of cosmic strings produced at the end of inflation, concluding that CMB data already bound the string tension G-mu below roughly 10^-6, while the loop population and hence the ...
Waveform Modelling for the Laser Interferometer Space Antenna
gr-qc 2023-11 unverdicted novelty 2.0 of 10

A review of existing waveform models for LISA sources and the challenges that must still be overcome.
The landscape of QCD axion models
hep-ph 2020-03 unverdicted novelty 2.0 of 10

Review classifies QCD axion models extending the standard mass-coupling window and updates bounds from cosmology, astrophysics, and experiments.

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146 later papers cite it

Evidence: some. 124 of the citing papers share no author with it, and 68 of those carry a confident label from Pith's citation classifier.

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1 paper by the same authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 47 papers with this label · 1 by the same authors

By the same authors

Used the momentum parameter given by this paper in the system of equations it numerically integrated.

Cosmic strings and other topological defects in nonscaling regimes (2017) (same authors) · 0.85 · paraphrased: the licence does not allow quoting

How the literature uses it

137 arXiv papers cite this in 297 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 174 sentences without a confident label

Contests it 1 paper

Our larger simulations do not appear to follow a spiral approach close to the fixed point as predicted by the two-parameter VOS model of [ref] , and our new value for the fixed point velocity is about $5%$ lower than the model predicts, using 4k simulations [ref] .

Scaling density of axion strings in terasite simulations · Discussion and conclusions · contest 0.74

Builds on it 6 papers

We derived basic equations for the evolution of string segments in cosmological backgrounds by extending the velocity-dependent one-scale model with the intercommuting probability $P≠1$ [ref] .

Skewness in CMB temperature fluctuations from curved cosmic (super-)strings · Introduction · 0.92
To this aim, we extended the Velocity One Scale (VOS) model of [ref] to describe a string network in the time-dependent case.

Cosmic (super)strings with a time-varying tension · Outlook · 1.00
This work directly builds on previous extensions of the VOS model [ref] for describing superstring networks, which can be found in [ref] .

Dynamics of junctions and the multi-tension velocity-dependent one-scale model · Introduction · 0.90

Uses its method or tool 47 papers

We calculated these templates numerically by solving the string-network evolution via the velocity-dependent one-scale model [ref] , shutting off the loop production after $H = m_a$, and calculating the SGWB following Ref. [ref] .

Constraining Postinflationary Axions with Pulsar Timing Arrays · Case i) $N_DW = 1$ · 1.00
All GWB spectra in this work are calculated numerically by following [ref] (see also [ref] ) where $n_loop$ relies on the velocity-dependent one-scale (VOS) model [ref] .

Ultrahigh frequency primordial gravitational waves beyond the kHz: The case of cosmic strings · General properties of GWB from cosmic strings · 1.00
In order to obtain the macroscopic variables for the VOS network evolution model, one needs to integrate over the spacelike variable $σ$ along all the strings in the network [ref] .

Generalised velocity-dependent one-scale model for current-carrying strings · Key Modelling Assumptions · 0.85

Uses its data or values 23 papers

The last term accounts for energy loss into loops with $c̃$ the efficiency of chopping loops from the network, and found numerically to be $c̃ ≃0.23$ [ref] .

Gravitational waves from cosmic superstrings and gauge strings · Dynamics of long strings and loops · 1.00
Nambu-Goto simulations are well described by $=0.23$ and a momentum parameter of the form [ref] [equation] see Ref. [ref] for the latest calibration of these parameters from the simulation of Abelian-Higgs cosmic strings.

CMB anisotropies generated by cosmic string loops · Cosmic string network evolution and loop production · 0.91
We fix $c̅ = 0.23$ based on numerical simulations [ref] , and we use the ansatz of Ref. [ref] for the function $k()$: [equation] In terms of $ξ$ and $$, the energy density $ρ_L$ of the long string network is ρ_L = μ/(ξ t)^2 , while the rate of energy loss needed to maintain scaling is dρ_L/dt = c̅ ρ_L/(ξ t) .

Probing the pre-BBN universe with gravitational waves from cosmic strings · Cosmic string scaling and loop production · 0.89

Compares with it 3 papers

We wish to compare the standard VOS model [ref] and the model with distinguished $v_(2)$, $v_(4)$ and $v_(6)$.

Generalised velocity-dependent one-scale model for current-carrying strings · $⟨Ẋ^4⟩$ and $⟨Ẋ^2⟩^2$ differences for standard VOS model · 0.99
The bounds derived for the root-mean squared string velocity can be compared to theoretical expectations from recent high-resolution simulations of Nambu–Goto and Abelian–Higgs string networks, which predict velocities in the range $v_s ∼0.3$–$0.6$ during both the radiation- and matter-dominated eras [ref] .

Cosmic Strings as Dynamical Dark Energy: Novel Constraints · Constraints on Model 2: Positive energy, velocity-dependent string network. · 0.98
If one assumes a curvature parameter $k_m$ of order unity as in the case of Goto-Nambu strings [ref] , our estimated velocities are comparable to (though possibly somewhat lower than) the ones typically encountered in other field theory defect simulations [ref] .

Evolution of Semilocal String Networks: I. Large-scale Properties · Simulation results and comparison to analytic models · 0.90

Cites it as background 26 papers

The Velocity-dependent One-Scale (VOS) model [ref] comprises two coupled differential equations, describing how two network averaged quantities—the mean correlation length or characteristic length scale and the root mean squared velocity—evolve over cosmic time.

High resolution calibration of the cosmic strings velocity dependent one-scale model · Extended Semi-analytical modelling · 0.98
The canonical one is the Velocity-dependent One-Scale (VOS) model, first introduced for cosmic strings [ref] and then subsequently developed for several other types of defects—for a recent overview, see [ref] .

Extending and Calibrating the Velocity dependent One-Scale model for Cosmic Strings with One Thousand Field Theory Simulations · Introduction · 0.87
The standard analytic description of late-time coarsening is the velocity-dependent one-scale (VOS) model of Martins and Shellard [ref] .

The friction-era VOS amplitude of a $\mathbb{Z}_2$ string network from nematic disclination data · The VOS model and its calibration · 0.81

Discussion

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