Pith. sign in

REVIEW 4 major objections 5 minor 2 cited by

After subtracting the string self-field, global-string axion radiation is exponentially suppressed, shifting the predicted axion dark-matter mass to roughly 125–160 μeV.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 07:37 UTC pith:NKBSQGAQ

load-bearing objection A careful, honest paper showing network spectra are likely contaminated by the string self-field; the specific exponential rate r≈2.5–2.9 is real but measured on a slimmer empirical base than the headline implies. the 4 major comments →

arxiv 2601.19463 v3 pith:NKBSQGAQ submitted 2026-01-27 hep-ph astro-ph.COhep-th

Spectrum of radiation from global strings and the relic axion density

Richard A. Battye , Lukasz P. Bunio , Steven J. Cotterill , Pranav B. Gangrekalve Manoj This is my paper
classification hep-ph astro-ph.COhep-th
keywords axion dark matterglobal cosmic stringsstring radiation spectrumself-field subtractionrelic axion densityexponential spectrumKalb-Ramond actionaxion mass prediction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that the axions radiated by a global cosmic string do not follow the hard power-law spectrum extracted from most network simulations; after removing the string's own self-field, the radiation from a single oscillating string falls exponentially with harmonic number, P_n ∝ e^{-rn} with r≈2.5–2.9. If that is right, the commonly used quantity ϕ∂_t α is dominated by the string's motion rather than by propagating axions, so spectral indices measured that way are suspect. A sympathetic reading of the relic-density calculation then pushes the axion dark-matter mass to about 125 μeV for direct emission from strings, or about 160 μeV if decay is via loops, rather than the ~4 μeV favored by a hard spectrum. The paper is explicit that uncertainties remain, including the extrapolation from one oscillating straight string to a network.

Core claim

On the paper's own terms, the central claim is that the spectrum of axion radiation from a perturbed straight global string is exponential, not hard: fitting harmonics n=2–5 gives P_n ∝ e^{-rn}, r≈2.5–2.9, with total power ∝ε^4 as expected. This is obtained only after decomposing the field phase into the static string ansatz θ plus perturbations Δα, and masking a cylinder of radius 33Δx; without that subtraction the spectrum of ϕ∂_t α peaks at the fundamental and falls only as k^{-2.25}, which the authors attribute to string motion. They therefore argue that large network simulations using ϕ∂_t α without self-field subtraction are likely measuring the self-field, and they use the exponential

What carries the argument

The key object is the phase decomposition α = α_str + Δα, separating the static string self-field (α_str≈θ, azimuthal around the string) from the propagating axion perturbation Δα. The authors compute the axion energy spectrum from the Fourier transform of ϕ∂_t(Δα) after subtracting the static field and excising a cylinder of radius r0=33Δx, with a mode-mixing kernel to correct for the mask. The analytic relic-density calculation is carried by the factor G2, which encodes the spectral shape's effect on axion number density, together with G1 for loop decay; this factorization isolates the spectrum as the dominant uncertainty.

Load-bearing premise

The load-bearing premise is that excising a cylinder of radius 33 grid cells around one oscillating string isolates the true propagating axion field in a way that carries over to dense networks, while the quantitative exponential fit itself rests on harmonics n=2–5 from a single box size analyzed for only about one oscillation after radiation first reaches the boundary.

What would settle it

Extract the spectrum from a network simulation using masks large enough to exclude the string self-field, and compare the power-law versus exponential shape; a hard spectrum with p≈1 surviving the correction would refute the claim, as would an exponential fit that fails to stabilize at r≈2.5–2.9 when a single-string run is extended beyond the boundary-limited window.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The relic axion density from global strings is highly sensitive to the radiation spectrum, captured by a single factor G2, so predictions for m_a shift by orders of magnitude between hard and exponential spectra.
  • Spectra of ϕ∂_t α without self-field subtraction—the method used by large network simulations—are dominated by string motion, so previously inferred hard spectral indices do not reflect the propagating axion population.
  • Using the measured exponential spectrum, the axion mass required to match the observed dark-matter density is roughly 125 μeV for direct string emission and 160 μeV for loop-dominated soft emission, with detection frequencies around 30–38 GHz.
  • The total radiated power follows ε^4 and the harmonic powers fall as P_n ∝ e^{-rn} with r≈2.5–2.9, consistent with Nambu-like Kalb-Ramond expectations.
  • A substantial mask radius, about 33 grid spacings, is required to isolate axion radiation—far larger than masks previously applied in network simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If self-field contamination is as strong in networks as in this single-string case, past axion mass predictions from hard spectra may be systematically low; reanalyzing existing network simulations with larger masks would test this directly.
  • The exponential spectrum for a single string suggests the Kalb-Ramond/Nambu description of global strings may be more accurate than recent network simulations imply; if confirmed, the debate over loop production and scaling density could shift back toward Nambu-like behavior.
  • A natural next step is to apply the same subtraction-and-mask procedure to oscillating loops; if loops also emit exponentially, the loop-dominated scenario converges on m_a≈160 μeV, provided the runtime limitation from boundary reflections is overcome.
  • The authors' analytic factorization of the relic density into cosmology-dependent and spectrum-dependent factors could be used to convert any future measured spectrum, from network or loop simulations, directly into a mass prediction without rerunning the cosmology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies the spectrum of axion radiation from global strings. It first develops a parametric model for the relic axion density showing that the result is very sensitive to the radiation spectrum. It then presents numerical simulations of a perturbed straight global string, arguing that the common practice of using ϕ∂_t α is dominated by the string self-field. After subtracting the self-field and applying a cylindrical mask, the authors claim the spectrum is exponential, P_n ∝ e^{-r n} with r ≈ 2.5–2.9, and that the total power scales as ε^4. Using this spectrum in their relic-density framework gives m_a ≈ 125 μeV for direct long-string emission and ≈160 μeV for loop-dominated decay, much higher than the ~4 μeV obtained from a hard power-law spectrum.

Significance. If the central claim is correct, the paper makes an important contribution: it identifies a concrete and plausible reason why large network simulations may measure a hard spectrum (self-field contamination), and it provides a transparent analytic framework separating the spectral function G_2 from the cosmological normalization. The demonstration that the self-field subtraction qualitatively changes the spectrum (Figs. 7–8) is convincing and should be of interest to the axion-string community. The exact evaluation of G_1 and the explicit dependence on the spectrum parameters in Eqs. (14)–(26) are useful improvements over earlier treatments. However, the quantitative claim of an exponential spectrum with r≈2.5–2.9 currently rests on a narrow numerical analysis that is partially inconsistent with the paper's own stated boundary limitations, and the extrapolation to a network is explicitly not demonstrated. The paper is therefore valuable as a methodological warning and a parametric framework, but its headline mass predictions need stronger numerical support.

major comments (4)
  1. [§III, Appendix A, Figs. 9–11] The central quantitative claim, r≈2.5–2.9, is extracted from the n_x=801 runs at times up to t≈1000 Δt, while Appendix A states that t_b = n_x dx/(2dt) ≈ 934 Δt and that "we cannot perform any analysis for times later than t_b" because the boundary reflects a small fraction of radiation that can interfere with the spectrum. Fig. 9 explicitly includes "the first such time step after the radiation hits the boundary," and Fig. 11 shows the r(t) curves for L=40,50 reaching the quoted band only at late times, with L=80,100 not stabilizing. This means the quoted r may be contaminated by boundary reflections. Please provide fits restricted to t<t_b, quantify the reflection amplitude, or run larger boxes so that the spectral peaks form before t_b.
  2. [§III, Fig. 11] The exponential fit P_n ∝ e^{-r n} is based on only harmonics n=2–5 from a single box size (n_x=801), with no error bars. Appendix A shows that the n_x=201 and n_x=401 simulations do not exhibit clean harmonic peaks, so there is no cross-box convergence check. The preference for an exponential over a power law is only illustrated in an inset for one time and one L; no quantitative goodness-of-fit or stability under different fitting ranges (e.g., n=2–4 vs n=2–5, different time windows) is given. Please report error bars on P_n, fits over multiple harmonic ranges and time windows, and a statistical comparison (e.g., Δχ²) between the exponential and power-law forms.
  3. [§V and Abstract] The paper explicitly states in §V that a simple oscillating string is not a network and "we are not suggesting that the spectra will necessarily be similar," yet the abstract presents m_a≈125 μeV "using our spectra" as a concrete scenario. The mass prediction therefore goes beyond what the simulations actually demonstrate. Please either provide a network-level test or clearly label the m_a≈125 μeV and ≈160 μeV values as illustrative single-string results that indicate sensitivity, not as network predictions.
  4. [Appendix B] The mode-mixing kernel is derived for ensemble-averaged spectra, but the analysis uses it for a single realization without ensemble averaging. The authors note this approximation may fail at low k, but the harmonics n=2–5 at k=4π/L–10π/L are not obviously in the regime where the approximation is guaranteed to hold. No validation of the reconstruction is shown on a synthetic field with a known spectrum. Please validate the reconstruction procedure on a controlled test case and quantify the systematic uncertainty in P_n for the harmonic range used.
minor comments (5)
  1. [Fig. 9 caption] The caption says the spectra are shown "until and including the first such time step after the radiation hits the boundary," which conflicts with Appendix A's statement that no analysis can be performed after t_b. Please remove or justify this inconsistency.
  2. [Abstract] The abstract says "we find that this leads to a range of possible axion masses... albeit that they are typically higher" and gives m_a≈125 μeV and ≈160 μeV; please mark explicitly that these are based on a single oscillating string, not a network.
  3. [Eq. (35)] The notation P=βε^4 is used, but later β/μ≈0.16 is quoted. Please clarify the units and define μ consistently (tension per unit length vs f_a^2).
  4. [Fig. 3 caption] The phrase "the nx case" appears to be missing a subscript; it should be "the n_x=801 case."
  5. [General] Several instances of "ran" should be "run" (e.g., "we ran until t≈1000 Δt"). Also, the abbreviation "nx" is used inconsistently with "n_x".

Circularity Check

0 steps flagged

No significant circularity; r is an empirical fit and the relic-density mapping uses it as an input. Score 1 only for minor reliance on previous self-work that is not load-bearing; the main caveats are data-quality issues, not circularity.

full rationale

The paper's derivation chain is linear: (i) simulate perturbed straight global strings; (ii) separate the propagating axion phase Δα from the string self-field by subtracting the boosted static-field ansatz and applying a cylindrical mask; (iii) measure the harmonic powers P_n directly from the simulation and fit P_n ∝ e^{-rn} (or n^{-p}); (iv) insert the fitted spectrum g(z)=exp[-rz/4π] into the analytic Ω_a integral and invert Ω_a h²≈0.12 for m_a. No equation defines the output in terms of the input: r is not set by the relic-density model, the exponential is tested against the simulation data and compared with a power law, and the mass calculation is an independent mapping from assumed network parameters and the fitted spectrum to m_a. The authors do rely on their earlier work (refs [24]-[26]) for KR-action expectations (P∝ε^4, n=2 dominance, loop formulas), but these are consistency checks/inputs and are re-validated against the new simulations, so the central result does not reduce to a self-citation. I therefore identify no circular step. I do explicitly flag, as robustness rather than circularity, the boundary-time contradiction: Appendix A says "we cannot perform any analysis for times later than t_b" and that "the boundary reflects a small fraction of the radiation back into the volume... interfering with the formation of the spectrum", yet the main n_x=801 runs "were ran until t≈1000 Δt, that is within one period of the string's oscillation after t_b"; Fig. 11 also shows L=80,100 not reaching a stable r. These are correctness risks for r≈2.5-2.9 and for the derived masses, but they are not input-output circularity.

Axiom & Free-Parameter Ledger

8 free parameters · 6 axioms · 0 invented entities

The central spectral claim is a direct measurement; the analytic density model is a parametric wrapper. The free parameters are all inputs from previous simulations, not fitted to the new data. The axioms include standard cosmology and the paper-specific ansatz subtraction and masking assumptions.

free parameters (8)
  • ζ (long-string density parameter) = 13 (Nambu) or 0.5–1.5 (field theory), fiducial 1
    Sets the string energy density per horizon volume in Eq. (1); taken from prior simulations, varied as an input.
  • ⟨v²⟩ (rms string velocity) = ≈0.35
    Input from Nambu/network simulations; enters chopping efficiency (Eq. 3).
  • Γ̂a = β Γ_a (loop radiation coefficient) = ≈46
    Adopted from refs. 25–26 for Nambu-like loops.
  • Γ̃a = 4π γ log(Δ/δ) (long-string radiation strength) = 200–1000
    Inferred from γ≈ζ^{-1/2} and log(Δ/δ)≈70; varied over the range.
  • α/κ (loop-size to decay parameter) = 0.1–10
    Varies in Figs. 12–14; no first-principles value.
  • Fℓ (fraction of energy into loops) = 0–1
    Chosen across scenarios; key uncertainty.
  • q, n* / p, m* (spectral index and UV cutoff) or exponential r = q≈0.9–2, n*≈10^3–10^9; p≈1, m*≈e^70; or r≈2.5–2.9 (measured)
    Parameters of g(z); the paper's own r is a measured fit to simulation.
  • β1 (QCD instanton amplitude in m_a(T)) = ≈0.026 (DIGA), varying by ~3 orders between lattice groups
    Determines T_AD via Eq. (45); dominates QCD uncertainty (m_a ∝ β1^{-1/7}).
axioms (6)
  • domain assumption One-scale model for string network evolution (Eqs. 2–3)
    The long-string density, chopping efficiency and loop production are described by a single length scale L=ζ^{-1/2}t; standard but an idealization.
  • domain assumption The emission spectrum g(z) is time-independent and the same functional form applies to loops and long strings
    Stated in §IIB before Eq. (23); in reality the spectrum may evolve and loop/long-string emission may differ.
  • standard math Axion number-to-entropy ratio conserved from T_AD to today
    Assumes axions are decoupled and entropy conserved, standard cosmology.
  • domain assumption Static-string ansatz subtraction with Lorentz-boost approximation θ′≈θ, r′≈r is accurate
    Used in Eqs. (43)–(44) and §III to isolate Δα; retardation effects are neglected.
  • ad hoc to paper Mode-mixing kernel derived for ensemble averages can be applied to a single realization
    Acknowledged in Appendix B: 'we use the above relation without the ensemble averages', which fails for low-k bins but is argued not to affect the extraction.
  • ad hoc to paper Masking with r_0=33 Δx and reconstruction (Appendix B) yields an unbiased estimate of the true spectrum
    The mask radius is chosen to make ϕ∂_t α and ϕ∂_t Δα agree over 4π/L ≲ k ≲ 10π/L, a calibration that could introduce selection.

pith-pipeline@v1.3.0-alltime-deepseek · 25964 in / 14346 out tokens · 152232 ms · 2026-08-03T07:37:18.754261+00:00 · methodology

0 comments
read the original abstract

We discuss key aspects of the nature of radiation from global strings and its impact on the relic axion density. Using a simple model we demonstrate the dependence on the spectrum of radiation emitted by strings. We then study the radiation emitted by perturbed straight strings paying particular attention to the difference between the overall phase of the field and the small perturbations about the string solution which are the axions. We find that a significant correction is required to be sure that one is analyzing the axions and not the self-field of the string. Typically this requires one to excise a sizeable region around the string - something which is not usually done in the case of numerical field theory simulations of string networks. We have measured the spectrum of radiation from these strings and find that it is compatible with an exponential, as predicted by the Nambu-like Kalb-Ramond action, and in particular is not a ``hard'' spectrum often found in string network simulations. We conclude by attempting to assess the uncertainties on relic density and find that this leads to a range of possible axion masses when compared to the measured density from the Cosmic Microwave Background, albeit that they are typically higher than what is predicted by the Initial Misalignment Mechanism. If the decay is via a ``soft spectrum'' from loops produced close to the backreaction scale we find that $m_{\rm a}\approx 160\,\mu{\rm eV}$ and a detection frequency $f\approx 38\,{\rm GHz}$. If axions are emitted directly by the string network, and we use emission spectra reported in field theory simulations, then $m_{\rm a}\approx 4\,\mu{\rm eV}$ and $f\approx 1\,{\rm GHz}$, however this increases to $m_a \approx 125\,\mu{\rm eV}$ and $f\approx 30\,{\rm GHz}$ using our spectra for the case of an oscillating string. In all scenarios there are significant remaining uncertainties that we delineate.

Figures

Figures reproduced from arXiv: 2601.19463 by Lukasz P. Bunio, Pranav B. Gangrekalve Manoj, Richard A. Battye, Steven J. Cotterill.

Figure 1
Figure 1. Figure 1: FIG. 1. The function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustrations of the perturbed string configurations [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Relative amplitude, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The axion radiation field, ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. In the left-hand panel we present the field [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. In the right-hand panel we present the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. In the top panel of the column on the right we present the axion field [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The evolution of the axion energy spectrum (42) for a simulation with [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. In the left-hand panel, we present the power emitted in each of the peaks shown in Fig. 9 for the simulation with [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. In the left-hand panel we present the evolution of the parameter [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. In the left-hand figure we present predictions for the axion mass, [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. In the left-hand panel we present the predictions for the axion mass, [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Predictions for the axion mass, [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. A comparison of predicted mass ratios [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. In the top row we present the evolution of the spectrum for the simulations with [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Whispers of Supergravity in Gravitational Wave Backgrounds: Determining the Gravitino Mass from Cosmic Thermal History

    astro-ph.CO 2026-05 unverdicted novelty 5.0

    Gravitino masses in the 100 TeV to 10^10 TeV range can be inferred from two frequency features in the stochastic gravitational wave spectrum produced by an early matter-dominated phase.

  2. Echoes of Global Cosmic Strings

    hep-ph 2026-04 unverdicted novelty 4.0

    Global cosmic strings from symmetry breaking produce Nambu-Goldstone bosons whose cosmological signatures can be constrained by current and upcoming CMB and large-scale structure observations.

Reference graph

Works this paper leans on

72 extracted references · 25 linked inside Pith · cited by 2 Pith papers

  1. [1]

    If we had used (11) we would have deduced that G1(x)∝(1 +x) 3/2−1 which has the same dependence on xasx→0, but it is very different for largex

    We note that, within the context of the calculation, this limit is not the same as all the energy being emitted from the long strings due to influence of the spectrum - ifα→ ∞all the axions would be emitted with very high frequencies, reducingn a for the same amount of energy emitted. If we had used (11) we would have deduced that G1(x)∝(1 +x) 3/2−1 which...

  2. [2]

    R. D. Peccei and H. R. Quinn, Constraints imposed by cp conservation in the presence of pseudoparticles, Physical Review D16, 1791 (1977)

    1977

  3. [3]

    Weinberg, A new light boson?, Physical Review Letters 40, 223 (1978)

    S. Weinberg, A new light boson?, Physical Review Letters 40, 223 (1978)

    1978

  4. [4]

    D. J. Marsh, Axion cosmology, Physics Reports643, 1 (2016)

    2016

  5. [5]

    M. Dine, W. Fischler, and M. Srednicki, A simple solution to the strong CP problem with a harmless axion, Physics Letters B104, 199 (1981)

    1981

  6. [6]

    A. R. Zhitnitsky, On Possible Suppression of the Axion Hadron Interactions. (In Russian), Sov. J. Nucl. Phys. 31, 260 (1980), [Yad. Fiz.31,497(1980)]

    1980

  7. [7]

    Wilczek, Problem of strong P and T invariance in the presence of instantons, Physical Review Letters40, 279 (1978)

    F. Wilczek, Problem of strong P and T invariance in the presence of instantons, Physical Review Letters40, 279 (1978)

    1978

  8. [8]

    Topological defects in Cosmology

    In our simulations, this led to a markedly different spectrum, one which we have shown to agree well with the expectations from the Nambu-Goto action — the total power followsP∝ε 4 and the power emitted in each mode goes likeP n ∝e −rn withr≈2.5−2.9 — albeit with a noticeable relaxation time. If we were to take this seriously and consider axion emission t...

  9. [9]

    M. S. Turner, Thermal Production of Not SO Invisible Axions in the Early Universe, Phys. Rev. Lett.59, 2489 (1987), [Erratum: Phys.Rev.Lett. 60, 1101 (1988)]

    1987

  10. [10]

    J. E. Kim, Weak-interaction singlet and strong CP invariance, Physical Review Letters43, 103 (1979)

    1979

  11. [11]

    M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can confinement ensure natural CP invariance of strong interactions?, Nuclear Physics B166, 493 (1980)

    1980

  12. [12]

    Preskill, M

    J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the invisible axion, Physics Letters B120, 127 (1983). 20

    1983

  13. [13]

    Dine and W

    M. Dine and W. Fischler, The not-so-harmless axion, Physics Letters B120, 137 (1983)

    1983

  14. [14]

    Abbott and P

    L. Abbott and P. Sikivie, A cosmological bound on the invisible axion, Physics Letters B120, 133 (1983)

    1983

  15. [15]

    Wantz and E

    O. Wantz and E. P. S. Shellard, Axion cosmology revisited, Phys. Rev. D82, 123508 (2010), arXiv:0910.1066 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  16. [16]

    Sikivie, Of Axions, Domain Walls and the Early Universe, Phys

    P. Sikivie, Of Axions, Domain Walls and the Early Universe, Phys. Rev. Lett.48, 1156 (1982)

    1982

  17. [17]

    R. L. Davis, Cosmic Axions from Cosmic Strings, Phys. Lett. B180, 225 (1986)

    1986

  18. [18]

    Borsanyiet al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature539, 69 (2016), arXiv:1606.07494 [hep-lat]

    S. Borsanyiet al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature539, 69 (2016), arXiv:1606.07494 [hep-lat]

    Pith/arXiv arXiv 2016

  19. [19]

    K. J. Bae, J.-H. Huh, and J. E. Kim, Updating the axion cold dark matter energy density, Journal of Cosmology and Astro-Particle Physics2008, 005 (2008), arXiv:0806.0497 [hep-ph]

    Pith/arXiv arXiv 2008

  20. [20]

    Both works calculate the topological susceptibility, which we have converted to mass ratios using the value forχ(0) provided in [18]

    we use the fitting formula that they provide (note that it is extended to higher temperatures as the data that the fit was based upon only reaches up to 512 MeV) while the Borsanyi et al.plot uses the direct numerical results that are provided in Table S9 of the Supplementary Information [18]. Both works calculate the topological susceptibility, which we ...

  21. [21]

    Borsanyi, M

    S. Borsanyi, M. Dierigl, Z. Fodor, S. D. Katz, S. W. Mages, D. Nogradi, J. Redondo, A. Ringwald, and K. K. Szabo, Axion cosmology, lattice QCD and the dilute instanton gas, Phys. Lett. B752, 175 (2016), arXiv:1508.06917 [hep-lat]

    Pith/arXiv arXiv 2016

  22. [22]

    Bazavov, T

    A. Bazavov, T. Bhattacharya, M. Cheng, C. DeTar, H.-T. Ding, S. Gottlieb, R. Gupta, P. Hegde, U. M. Heller, F. Karsch, E. Laermann, L. Levkova, S. Mukherjee, P. Petreczky, C. Schmidt, R. A. Soltz, W. Soeldner, R. Sugar, D. Toussaint, W. Unger, and P. Vranas (HotQCD Collaboration), Chiral and deconfinement aspects of the qcd transition, Phys. Rev. D85, 054...

    2012

  23. [23]

    Chen, T.-W

    Y.-C. Chen, T.-W. Chiu, and T.-H. Hsieh (TWQCD Collaboration), Topological susceptibility in finite temperature qcd with physical (u/d, s, c) domain-wall quarks, Phys. Rev. D106, 074501 (2022)

    2022

  24. [24]

    Bonati, M

    C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro, F. Sanfilippo, and G. Villadoro, Axion phenomenology andθ-dependence fromN f = 2+1 lattice QCD, JHEP03, 155, arXiv:1512.06746 [hep-lat]

    Pith/arXiv arXiv

  25. [25]

    Petreczky, H.-P

    P. Petreczky, H.-P. Schadler, and S. Sharma, The topological susceptibility in finite temperature qcd and axion cosmology, Physics Letters B762, 498 (2016)

    2016

  26. [26]

    R. L. Davis and E. P. S. Shellard, DO AXIONS NEED INFLATION?, Nucl. Phys. B324, 167 (1989)

    1989

  27. [27]

    Battye and E

    R. Battye and E. Shellard, Global string radiation, Nuclear Physics B423, 260 (1994)

    1994

  28. [28]

    R. A. Battye and E. P. S. Shellard, Axion string constraints, Phys. Rev. Lett.73, 2954 (1994), [Erratum: Phys.Rev.Lett. 76, 2203–2204 (1996)], arXiv:astro-ph/9403018

    Pith/arXiv arXiv 1994

  29. [29]

    Battye and E

    R. Battye and E. Shellard, Recent perspectives on axion cosmology, Arxiv preprint astro-ph/9706014 (1997)

    Pith/arXiv arXiv 1997

  30. [30]

    Vilenkin and T

    A. Vilenkin and T. Vachaspati, Radiation of Goldstone Bosons From Cosmic Strings, Phys. Rev. D35, 1138 (1987)

    1987

  31. [31]

    Kalb and P

    M. Kalb and P. Ramond, Classical direct interstring action, Physical Review D9, 2273 (1974)

    1974

  32. [32]

    Nambu, Strings, monopoles, and gauge fields, Physical Review D10, 4262 (1974)

    Y. Nambu, Strings, monopoles, and gauge fields, Physical Review D10, 4262 (1974)

    1974

  33. [33]

    R. L. Davis and E. P. S. Shellard, Antisymmetric Tensors and Spontaneous Symmetry Breaking, Phys. Lett. B214, 219 (1988)

    1988

  34. [34]

    Harari and P

    D. Harari and P. Sikivie, On the evolution of global strings in the early universe, Physics Letters B195, 361 (1987)

    1987

  35. [35]

    Hagmann, P

    C. Hagmann, P. Sikivie, N. S. Sullivan, and D. B. Tanner, Results from a search for cosmic axions, Phys. Rev. D42, 1297 (1990)

    1990

  36. [36]

    Aghanimet al.(Planck), Planck 2018 results

    N. Aghanimet al.(Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  37. [37]

    Yamaguchi, J

    M. Yamaguchi, J. Yokoyama, and M. Kawasaki, Evolution of a global string network in a matter dominated universe, Phys. Rev. D61, 061301 (2000), arXiv:hep-ph/9910352

    Pith/arXiv arXiv 2000

  38. [38]

    Kawasaki, T

    M. Kawasaki, T. Sekiguchi, M. Yamaguchi, and J. Yokoyama, Long-term dynamics of cosmological axion strings, PTEP2018, 091E01 (2018), arXiv:1806.05566 [hep-ph]

    Pith/arXiv arXiv 2018

  39. [39]

    Saikawa, J

    K. Saikawa, J. Redondo, A. Vaquero, and M. Kaltschmidt, Spectrum of global string networks and the axion dark matter mass, arXiv preprint arXiv:2401.17253 (2024)

    Pith/arXiv arXiv 2024

  40. [40]

    Kaltschmidt, J

    M. Kaltschmidt, J. Redondo, K. Saikawa, and A. Vaquero, The Spectrum of Global Axion Strings (2025) arXiv:2502.02398 [hep-ph]

    Pith/arXiv arXiv 2025

  41. [41]

    Buschmann, J

    M. Buschmann, J. W. Foster, and B. R. Safdi, Early-universe simulations of the cosmological axion, Physical review letters124, 161103 (2020)

    2020

  42. [42]

    Buschmann, J

    M. Buschmann, J. W. Foster, A. Hook, A. Peterson, D. E. Willcox, W. Zhang, and B. R. Safdi, Dark matter from axion strings with adaptive mesh refinement, Nature communications13, 1049 (2022)

    2022

  43. [43]

    J. N. Benabou, M. Buschmann, J. W. Foster, and B. R. Safdi, Axion mass prediction from adaptive mesh refinement cosmological lattice simulations, Phys. Rev. Lett.134, 241003 (2025)

    2025

  44. [44]

    Hindmarsh, J

    M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and J. Urrestilla, Scaling density of axion strings, Physical Review Letters124, 021301 (2020)

    2020

  45. [45]

    V. B. . Klaer and G. D. Moore, The dark-matter axion mass, JCAP11, 049, arXiv:1708.07521 [hep-ph]

    Pith/arXiv arXiv

  46. [46]

    V. B. Klaer and G. D. Moore, Global cosmic string networks as a function of tension, JCAP06, 021, arXiv:1912.08058 [hep-ph]

    Pith/arXiv arXiv 1912

  47. [47]

    Hindmarsh, J

    M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and J. Urrestilla, Approach to scaling in axion string networks, Phys. Rev. D103, 103534 (2021), arXiv:2102.07723 [astro-ph.CO]

    Pith/arXiv arXiv 2021

  48. [48]

    H. Kim, J. Park, and M. Son, Axion dark matter from cosmic string network, JHEP07, 150, arXiv:2402.00741 [hep-ph]

    Pith/arXiv arXiv

  49. [49]

    Hiramatsu, M

    T. Hiramatsu, M. Kawasaki, T. Sekiguchi, M. Yamaguchi, and J. Yokoyama, Improved estimation of radiated axions from cosmological axionic strings, Phys. Rev. D83, 123531 (2011)

    2011

  50. [50]

    Correia, M

    J. Correia, M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, K. Rummukainen, and J. Urrestilla, Scaling density of axion strings in terasite simulations, Phys. Rev. D111, 063532 (2025)

    2025

  51. [51]

    Gorghetto, E

    M. Gorghetto, E. Hardy, and G. Villadoro, More axions from strings, SciPost Phys.10, 050 (2021), arXiv:2007.04990 [hep-ph]

    Pith/arXiv arXiv 2021

  52. [52]

    Kim and M

    H. Kim and M. Son, More scalings from cosmic strings (2024), arXiv:2411.08455 [hep-ph]

    arXiv 2024

  53. [53]

    Gorghetto, E

    M. Gorghetto, E. Hardy, and G. Villadoro, Axions from Strings: the Attractive Solution, JHEP07, 151, arXiv:1806.04677 [hep-ph]

    Pith/arXiv arXiv

  54. [54]

    Correia, M

    J. Correia, M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, K. Rummukainen, and J. Urrestilla, The spectrum of axions in a scaling string network 21 (2025), arXiv:2512.13653 [hep-ph]

    arXiv 2025

  55. [55]

    Hindmarsh, J

    M. Hindmarsh, J. Lizarraga, J. Urrestilla, D. Daverio, and M. Kunz, Scaling from gauge and scalar radiation in Abelian Higgs string networks, Phys. Rev. D96, 023525 (2017), arXiv:1703.06696 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  56. [56]

    J. N. Moore, E. P. S. Shellard, and C. J. A. P. Martins, Evolution of abelian-higgs string networks, Phys. Rev. D 65, 023503 (2001)

    2001

  57. [57]

    Hindmarsh, S

    M. Hindmarsh, S. Stuckey, and N. Bevis, Abelian Higgs Cosmic Strings: Small Scale Structure and Loops, Phys. Rev. D79, 123504 (2009), arXiv:0812.1929 [hep-th]

    Pith/arXiv arXiv 2009

  58. [58]

    Hindmarsh, J

    M. Hindmarsh, J. Lizarraga, A. Urio, and J. Urrestilla, Loop decay in Abelian-Higgs string networks, Phys. Rev. D104, 043519 (2021), arXiv:2103.16248 [astro-ph.CO]

    Pith/arXiv arXiv 2021

  59. [59]

    Allen and E

    B. Allen and E. P. S. Shellard, Cosmic string evolution: a numerical simulation, Phys. Rev. Lett.64, 119 (1990)

    1990

  60. [60]

    E. P. S. Shellard and B. Allen, On the evolution of cosmic strings, inNuffield Workshop: Symposium on the Formation and Evolution of Cosmic Strings(1989)

    1989

  61. [61]

    D. P. Bennett and F. R. Bouchet, Evidence for a Scaling Solution in Cosmic String Evolution, Phys. Rev. Lett. 60, 257 (1988)

    1988

  62. [62]

    D. P. Bennett and F. R. Bouchet, Cosmic string evolution, Phys. Rev. Lett.63, 2776 (1989)

    1989

  63. [63]

    D. P. Bennett and F. R. Bouchet, High resolution simulations of cosmic string evolution. 1. Network evolution, Phys. Rev. D41, 2408 (1990)

    1990

  64. [64]

    Drew and E

    A. Drew and E. Shellard, Radiation from global topological strings using adaptive mesh refinement: Methodology and massless modes, Physical Review D 105, 063517 (2022)

    2022

  65. [65]

    Saurabh, T

    A. Saurabh, T. Vachaspati, and L. Pogosian, Decay of Cosmic Global String Loops, Phys. Rev. D101, 083522 (2020), arXiv:2001.01030 [hep-ph]

    Pith/arXiv arXiv 2020

  66. [66]

    Vilenkin and E

    A. Vilenkin and E. P. S. Shellard,Cosmic strings and other topological defects(Cambridge University Press, 1994)

    1994

  67. [67]

    M. S. Turner, Cosmic and Local Mass Density of Invisible Axions, Phys. Rev. D33, 889 (1986)

    1986

  68. [68]

    Hadjidimos, Successive overrelaxation (sor) and related methods, Journal of Computational and Applied Mathematics123, 177 (2000)

    A. Hadjidimos, Successive overrelaxation (sor) and related methods, Journal of Computational and Applied Mathematics123, 177 (2000)

    2000

  69. [69]

    A. Drew, T. Kinowski, and E. Shellard, Axion string source modelling, arXiv preprint arXiv:2312.07701 (2023)

    Pith/arXiv arXiv 2023

  70. [70]

    Di Luzio, M

    L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, The landscape of QCD axion models, Phys. Rept.870, 1 (2020), arXiv:2003.01100 [hep-ph]

    Pith/arXiv arXiv 2020

  71. [71]

    Hivon, K

    E. Hivon, K. M. Gorski, C. B. Netterfield, B. P. Crill, S. Prunet, and F. Hansen, Master of the cosmic microwave background anisotropy power spectrum: a fast method for statistical analysis of large and complex cosmic microwave background data sets, Astrophys. J. 567, 2 (2002), arXiv:astro-ph/0105302

    Pith/arXiv arXiv 2002

  72. [72]

    Program Generation, Optimization, and Platform Adaptation

    M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE 93, 216 (2005), special issue on “Program Generation, Optimization, and Platform Adaptation”. Appendix A: T ests of the calculations of the power forn x = 200and400 To investigate the influence of the simulations’ finite volume on the behavior, we observe, we compa...

    2005