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The scaling density of axion strings

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Pith's one-line read The paper argues that the axion string density approaches a constant, $\zeta_0 = 1.19 \pm 0.20$, so recent reports of logarithmic growth are artifacts of initial conditions, and the implied axion dark matter mass rises by about 50%.

desk verdict New simulations and a promising diagnostic, but the claim that logarithmic growth is an artifact of initial conditions rests on a degeneracy the paper does not close. read the letter →

arxiv 1908.03522 v2 pith:7PY2LQY7 submitted 2019-08-09 astro-ph.CO hep-phhep-th

Mark Hindmarsh , Joanes Lizarraga , Asier Lopez-Eiguren , Jon Urrestilla This is my paper · ORCID
classification astro-ph.COhep-phhep-th
keywords axionstringsQCDdarkmattercosmicstringscalingPeccei-Quinnsymmetrybreakinglengthdensitypost-inflationarylatticefieldtheorysimulationsmass
topics Dark Matter
open problems Dark Matter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to settle whether the density of cosmic strings left over from the Peccei-Quinn symmetry breaking behind the QCD axion settles down to a constant, as predicted by the standard scaling picture, or grows logarithmically with time, as several recent simulation groups have reported. The answer matters because in the post-inflationary axion dark matter scenario the axion number density, and therefore the dark matter density, is proportional to the string length per unit volume at the QCD transition; an underestimated string density changes the axion mass needed for axions to make up all of the dark matter. Using new lattice simulations at three box sizes and three initial correlation lengths, the paper fits the mean string separation to linear growth in time and extracts $\zeta_0 = 1.19 \pm 0.20$. It concludes that apparent logarithmic and inverse-logarithmic corrections are artifacts of the initial conditions, absorbed by a single time offset, and that re-analysis of earlier axion number density estimates raises the required axion dark matter mass by about 50%.

What carries the argument

The central object is the string length density parameter $\zeta$ (equivalently the mean string separation $\xi = \sqrt{V/\ell_s} = t/\sqrt{\zeta}$), and the load-bearing identity is the standard scaling fit $\xi = 2\beta(t - t_0)$, with $\zeta_0 = 1/4\beta^2$. The time offset $t_0$ is the piece that carries the argument: it is the t-axis intercept of the linear fit and is interpreted as the entire memory of the initial conditions. The paper shows that for nonzero $t_0$ the raw estimator $\zeta = t^2/\xi^2$ approaches $\zeta_0$ slowly, behaving like $\zeta_0(1 + 2t_0/t)$, which a logarithmic fit misreads as growth with coefficient $\alpha_*(t_f) \simeq -2\zeta_0 (t_0/t_f)(1 - t_0/t_f)^{-1}$. In this way, the apparent logarithmic and inverse-logarithmic corrections are tied to the ratio $t_0/t_f$, making them properties of the initial setup rather than of the scaling network.

What would settle it

A longer simulation with $t_f/t_0 \gtrsim 20$ using the same shifted estimator would settle the claim: if the residual $\hat\zeta - \zeta_0$ stays flat within $|\alpha| < 0.94 \times 10^{-2}$, standard scaling stands; if it drifts upward at the rate $\alpha_* \simeq 0.2$ reported by the logarithmic-growth groups, the artifact explanation fails.

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Extended reading notes

Core claim

The paper's central claim is that the dimensionless string length density parameter $\zeta$, defined by writing the string length per unit volume as $\zeta/t^2$, reaches a constant $\zeta_0 = 1.19 \pm 0.20$ in the scaling regime, rather than growing logarithmically as claimed in Refs. [24–27]. The decisive evidence is that the mean string separation $\xi = t/\sqrt{\zeta}$ follows a straight line in time, $\xi = 2\beta(t - t_0)$, where the intercept $t_0$ encodes the initial string configuration. Fits of the logarithmic and inverse-logarithmic models give coefficients consistent with zero after the ratio $t_0/t_f$ is extrapolated to zero, and a direct fit to the residuals of the linear model gives $\alpha_r = (-0.04 \pm 0.30) \times 10^{-2}$. The paper concludes that no logarithmic correction larger than about 0.5 can occur between the Peccei-Quinn scale and the QCD scale, that constant-$\zeta$ extrapolation from simulations to cosmology is justified, and that the axion dark matter mass estimate should be revised upward by roughly 50%.

Load-bearing premise

The conclusion rests on the assumption that one fitted time offset in the linear string separation fully captures the effect of the initial conditions, so that any residual logarithmic behaviour is just the disguised approach to scaling; if a genuine logarithmic drift could hide inside that offset over the fitted time window, the claim that $\zeta$ is constant would collapse.

Editorial extensions

If this is right

  • The string network can be extrapolated at constant $\zeta$ from the Peccei-Quinn transition to the QCD transition, so no large enhancement of string density near the QCD transition is expected.
  • The $3\sigma$ bound $|\alpha| < 0.94 \times 10^{-2}$ on the residual logarithmic coefficient limits any drift in $\zeta$ to less than about 0.5 over the full cosmological range.
  • The axion dark matter mass in the post-inflationary singlet scenario should be increased by about 50% relative to estimates based on the larger string densities.
  • For network models, the energy loss rate per unit length of string must rise at the same rate as the effective string tension, otherwise constant $\zeta$ would be unsustainable.
  • Frustrated-string simulations with $\kappa \simeq 50$ produce $\zeta \simeq 20$, far above the $O(1)$ value allowed here, so their large-separation extrapolations of axion number density should not be used.

Reading between the lines

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

  • Editorial extension: the $t_0/t_f$ diagnostic provides a cheap retest of previously published data; if those runs are re-fitted with a shifted estimator and their reported $\alpha_* \simeq 0.2$ survives the extrapolation to $t_0/t_f \to 0$, the artifact explanation would be wrong.
  • Editorial extension: the same residual analysis could be applied to local (gauge) cosmic string networks, whose tension does not grow logarithmically, to test whether initial-condition offsets are the generic source of apparent scaling violations.
  • Editorial extension: replacing the paper's figure-based reading of $n_{\rm ax}/n_{\rm mis} \simeq 0.5$ at $\kappa = 1$ with a direct simulation measurement would turn the 50% axion mass revision into a sharper prediction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. This paper presents new lattice simulations of global axion strings and compares three phenomenological models for the string-density parameter ζ: standard scaling with a constant ζ0, a logarithmic correction ζ=ζ0*+α*log(ηt), and an inverse-logarithmic correction. Fitting the mean string separation ξ to ξ=2β(t−t0), the authors find ζ0=1.19±0.20 and report a residual logarithmic coefficient αr=(−0.04±0.30)×10^-2, concluding that the apparent logarithmic growth seen in earlier work is an artifact of initial conditions and that a constant-ζ extrapolation from the Peccei-Quinn scale to the QCD scale is justified. The paper also re-analyzes previous axion-number-density estimates and suggests that the axion dark-matter mass should be revised upward by about 50%.

Significance. If the central conclusion holds, the paper resolves an active controversy: recent simulations reporting logarithmic growth of ζ would imply substantially larger axion string densities and a different axion dark-matter mass, whereas a constant ζ0≈1.2 supports the traditional scaling extrapolation. The paper's strengths include a careful decomposition of statistical and systematic uncertainties, multiple box sizes and initial correlation lengths, an infinite-volume extrapolation, a physical-width versus constant-comoving-width cross-check, and explicit comparison of the three competing models. The result is also falsifiable: the quoted 3σ bound |α|<0.94×10^-2 gives a concrete target for independent simulations.

major comments (1)
  1. [§Measurements and results, Eqs. (5), (9)–(10), Table III] The conclusion that the residual logarithmic coefficient is tightly bounded rests on the assumption that the free intercept t0 in Eq. (5) absorbs only initial-condition effects. This assumption is not tested, and the concern lands: if the true density parameter were ζ(t)=ζ0*+α log(ηt), then the estimator in Eq. (9) gives ζhat(t)=(1−t0/t)^2[ζ0*+α log(ηt)] ≈ ζ0*+α log(ηt)−2ζ0*t0/t for small |t0/t|. Over a finite fitting window, a fitted value t0≈−α t_m/(2ζ0*), where t_m is the midpoint of the window, cancels the logarithmic trend to second order. Thus the small αr in Table III is exactly what a genuine log correction with α≈0.2 would produce under this pipeline. Equation (8) demonstrates only the direction from t0 to an apparent logarithmic coefficient; the reverse direction is not excluded. Since the central claims of standard scaling (Eq. (11)) and of a constant-ζ extrapolation to the QCD scale both depend on this non-degeneracy, I ask for an explicit test: fit a nested model that allows both t0 and a logarithmic correction, or perform an injection-recovery test in which a known α log(ηt) term is added to the scaling model and the residual procedure is required to recover it.
minor comments (3)
  1. [Table III and accompanying text] The statement that the direct log-model fits are 'apparently inconsistent' with α*≃0.2 is stronger than the numbers in Table III support: for the 4k box α*=0.11±0.11, so 0.2 lies within 1σ. The tight exclusion of α*≈0.2 comes mainly from the residual bound in Eq. (12), which is the degenerate test flagged in the major comment. Please present the direct-fit and residual evidence separately when characterizing the constraint.
  2. [Eq. (5) and footnote 1] The notation is confusing because Eq. (5) is written in terms of physical time t and ξ, while footnote 1 defines β as the slope of comoving separation ξ/a against conformal time τ=2t/a. Please state explicitly which time variable enters the plotted and fitted quantities, and define all symbols consistently.
  3. [Conclusions, paragraph beginning 'An implication...'] The suggested upward revision of the axion dark-matter mass by about 50% is based on an estimate read from a figure in Ref. [30], with no propagated uncertainty. Please label this as indicative or provide an error estimate, since it appears in the abstract as a quantitative conclusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the scaling claim is an empirical model comparison with independently fitted parameters, and the cited self-work is not load-bearing.

full rationale

The central claim (standard scaling with ζ0 = 1.19 ± 0.20) rests on direct linear fits to the measured mean string separation ξ(t) with two free parameters (β, t0), Eq. (5). The alternative logarithmic and inverse-logarithmic models in Eq. (6) are fitted separately with their own parameters, and their coefficients are found to be consistent with zero. The apparent-α conversion in Eq. (8) is an algebraic identity explaining how a non-zero t0 can masquerade as a logarithmic term; the paper tests this against simulations with different initial conditions, showing that α* tracks t0/tf and vanishes as t0/tf → 0. The residual test in Eq. (10) is a standard goodness-of-fit check: it subtracts the fitted ζ0 and tests for a remaining trend, but αr is not forced to zero by construction because a genuine logarithmic trend would survive in the residuals unless absorbed by t0, and the finite-window degeneracy is addressed empirically by varying initial conditions and fit ranges. Self-citations [21] and [36] are consistency/method references, not load-bearing assumptions. No equation reduces to its input by definition; the conclusion is an empirical measurement and model comparison, not a derivation from the premise. The skeptical degeneracy concern is a statistical identifiability caveat, not circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles or forces. The central numbers are empirical fits from lattice simulations, so the free parameters are the model fit parameters. The most fragile domain assumption is that t0 captures all initial-condition effects, because the paper's dismissal of log growth rests on it.

free parameters (6)
  • t0 (string evolution intercept) = Table I: about -8.9 to -34.5 for 4k runs depending on fit range
    In Eq. (5), ξ = 2β(t - t0); the paper interprets nonzero t0 as an initial-condition artifact and uses it to explain apparent log growth via Eq. (8). If t0 is not a complete description of initial conditions, the central conclusion weakens.
  • β (slope of ξ(t) linear fit) = Table II: 0.499±0.042 (1k), 0.486±0.036 (2k), 0.467±0.037 (4k)
    Defines the scaling density ζ0 = 1/(4β²) in the standard scaling model; the quoted ζ0 = 1.19±0.20 comes from a linear extrapolation in 1/L of β.
  • α* (log-growth coefficient in alternative model) = 4k: 0.11±0.11
    Fitted parameter in Eq. (6), ζ(t) = ζ*0 + α* log(ηt); consistency with zero is used to reject logarithmic growth claims.
  • α' (inverse-log coefficient in alternative model) = 4k: -3.6±3.8
    Fitted parameter in Eq. (6), ζ(t) = ζ'0 + α'/log(ηt); consistency with zero supports constant-ζ scaling.
  • α_r (residual log coefficient) = (-0.04±0.30)×10^-2
    Coefficient in Eq. (10) for residuals after subtracting the standard scaling fit; the tight upper bound |α|<0.94×10^-2 is the main evidence against log growth.
  • ζ*0 and ζ'0 (intercepts of alternative models) = 4k: 0.42±0.59 and 1.66±0.68
    Constant terms in Eq. (6) are fitted and found consistent with standard scaling; used to argue the alternative models reduce to scaling.
assumptions (6)
  • domain assumption The axion field obeys the U(1) symmetric action Eq. (1) and equations of motion Eq. (2) in a radiation-dominated FLRW background with a ∝ τ.
    All simulations implement this model; deviations from the standard axion/string picture, such as gauge interactions or finite axion mass before QCD, are neglected.
  • ad hoc to paper Gaussian random initial fields with power spectrum PΦ(k)=A[1+(k lφ)^2]^-1 and a diffusion preparation phase drive the network into the scaling regime within the simulation time.
    The claim that apparent log growth is an artifact of initial conditions depends on these initial conditions being representative and on the network having relaxed by the fitting window.
  • domain assumption The physical string length is correctly measured by counting phase-winding plaquettes with a 2/3 Manhattan correction.
    ζ and ξ are derived from this measurement; a bias here would shift ζ0.
  • domain assumption The single time offset t0 in Eq. (5) fully accounts for initial-condition effects in the standard scaling fit.
    This is the structural foundation of the conclusion that logarithmic corrections are artifacts; if a physical log correction merely mimics a t0 term over the fitted window, the conclusion fails.
  • domain assumption The QCD axion dark matter scenario with post-inflationary Peccei-Quinn breaking, strings, QCD domain walls, and axion number density proportional to ζ applies to the real universe.
    The cosmological significance of ζ0 and the axion mass revision are only meaningful in this scenario.
  • domain assumption Linear extrapolation in 1/L to infinite volume is valid.
    Final ζ0 and αr are obtained by a linear fit in inverse box size; nonlinear finite-volume effects would change the results.

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Cite this review

Pith. "Pith review of The scaling density of axion strings." pith.science (2026). https://pith.science/paper/7PY2LQY7

@misc{pith2026190803522,
  author       = {Pith},
  title        = {Pith review of: The scaling density of axion strings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7PY2LQY7}},
  note         = {Machine review of arXiv:1908.03522}
}
abstract

In the QCD axion dark matter scenario with post-inflationary Peccei-Quinn symmetry breaking, the number density of axions, and hence the dark matter density, depends on the length of string per unit volume at cosmic time $t$, by convention written $\zeta/t^2$. The expectation has been that the dimensionless parameter $\zeta$ tends to a constant $\zeta_0$, a feature of a string network known as scaling. It has recently been claimed that in larger numerical simulations $\zeta$ shows a logarithmic increase with time, while theoretical modelling suggests an inverse logarithmic correction. Either case would result in a large enhancement of the string density at the QCD transition, and a substantial revision to the axion mass required for the axion to constitute all of the dark matter. With a set of new simulations of global strings we compare the standard scaling (constant-$\zeta$) model to the logarithmic growth and inverse-logarithmic correction models. In the standard scaling model, by fitting to linear growth in the mean string separation $\xi = t/\sqrt{\zeta}$, we find $\zeta_0 = 1.19 \pm 0.20$. We conclude that the apparent corrections to $\zeta$ are artefacts of the initial conditions, rather than a property of the scaling network. The residuals from the constant-$\zeta$ (linear $\xi$) fit also show no evidence for logarithmic growth, restoring confidence that numerical simulations can be simply extrapolated from the Peccei-Quinn symmetry-breaking scale to the QCD scale. Re-analysis of previous work on the axion number density suggests that recent estimates of the axion dark matter mass in the post-inflationary symmetry-breaking scenario we study should be increased by about 50%.

Figures

Figures reproduced from arXiv: 1908.03522 by the authors.

Figure 1
Figure 1. FIG. 1. Mean string separation [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Top: string length density parameter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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    The coefficients of the logarithm and inverse loga- rithm are consistent with zero at t0/tf = 0

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    The constant terms in logarithm and inverse loga- rithm models are consistent with standard scaling at t0/tf = 0. 8 0 50 100 150 t 0 50 100 150ξ 0 50 100 150 200 250 300 t 0 100 200 300ξ 0 100 200 300 400 500 t 0 100 200 300 400 500ξ 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 log(ηt) 0.0...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.