Using $\Delta N_{\rm eff}$ to constrain preferred axion model dark matter

Andrew Cheek; Ui Min

REVIEW 1 major objections 5 minor 1 cited by

Using $\Delta N_{\rm eff}$ to constrain preferred axion model dark matter

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Planck's measurement of relativistic energy density already rules out parameter-space regions for 40% of canonical preferred axion models, with model E nearly excluded at $m_Q = f_a$.

desk verdict A clean, useful model-by-model calculation of ΔNeff from late heavy-quark decay in preferred axion models, whose headline exclusion of models D and E rests on one honest but unproven assumption that boosted axions stay decoupled. read the letter →

arxiv 2411.17320 v2 pith:ARGZEYDH submitted 2024-11-26 hep-ph

Andrew Cheek , Ui Min This is my paper · ORCID

classification hep-ph

keywords axiondarkmatterpreferredmodelsKSVZaxionsheavyquarkdecayradiationNeffCMBconstraintsPeccei-Quinnmechanism

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 tries to establish that the effective number of relativistic species, $N_{\rm eff}$, is a discriminating observable for the minimal hadronic axion models called preferred axion models. Tracking the heavy vectorlike quark $Q$ in the ten canonical models, the authors find that in several models $Q$ decays only after axions have thermally decoupled, so the axions produced in the decay are boosted and act as dark radiation. This raises $\Delta N_{\rm eff}$ above the Planck bound for a substantial share of parameter space: 40% of the canonical models have regions already excluded, with model E nearly excluded at $m_Q = f_a$. A sympathetic reader should care because this turns an existing CMB measurement into a model-selection tool for axion dark matter and gives future CMB surveys a concrete target signal to measure.

What carries the argument

The central object is the set of ten preferred axion models: minimal KSVZ-type hadronic QCD axion models with $N_{\rm DW}=1$, $f_a$ in $5\times10^{9}{-}3\times10^{11}$ GeV, and a heavy colored fermion $Q$ that must decay through operators of dimension $\le 5$. The mechanism is delayed decay: in models A, D and E, the dominant $Q$ decay channels have widths suppressed by $f_a/\Lambda$ or $m_Q^3/\Lambda^2$, so the decay temperature is two to four orders of magnitude below the axion decoupling temperature. The ratio $R(T)=3H/(n_i^{\rm eq}\langle\sigma v\rangle)=T_{\rm dax}/T$ (Eq. 4.10) encodes the decoupling argument: because $R\gg 1$ at decay time, the injected axions do not scatter back into the bath. The resulting dark-radiation energy is tracked with the Boltzmann equations (4.4), with the branching ratios of Table 1 determining how much of $Q$'s energy lands in axions.

What would settle it

Compute the scattering rate of a non-thermal axion with energy $\sim m_Q/2$ on quarks and gluons at the decay temperature $T_{\rm decay}$, using the full phase-space distribution of the injected axions. If that rate exceeds $3H$ at any time after injection, the boosted axions re-thermalize, the $\Delta N_{\rm eff}$ predictions of models D and E collapse to the thermal value $\approx 0.027$, and model E's exclusion by Planck disappears.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that most preferred axion models are not thermally quiet: their heavy quark $Q$ can decay after axions have decoupled from the Standard Model bath, injecting a population of boosted axions that behaves as dark radiation. For model E (and its KSVZ-II analogue), the dominant decay $Q \to a\, d$ puts $\Delta N_{\rm eff}$ above the Planck 2018 bound when $m_Q = f_a$, so the model is "all but excluded"; for model D the axion branching is about half, giving a smaller but still observable signal. Model A, by contrast, reheats only Standard Model states and slightly suppresses $\Delta N_{\rm eff}$ below the thermal-axion value. The authors conclude that Planck already excludes regions of parameter space for 40% of the canonical preferred axion models, and that $N_{\rm eff}$ measurements offer a way to distinguish among models that otherwise predict identical axion dark matter.

Load-bearing premise

The load-bearing premise is that boosted axions emitted by $Q$ decay never re-enter thermal equilibrium with the Standard Model plasma, so they remain a separate dark-radiation component; proving this would require a phase-space scattering calculation beyond the paper's scope.

Editorial extensions

If this is right

For $m_Q = f_a$, model E's $\Delta N_{\rm eff}$ prediction sits above the Planck bound, so that model is nearly excluded as stated.
Model D's injection is roughly half of model E's, so it survives current data but would be a primary target for the next generation of CMB experiments.
Models B and C remain at the thermal-axion value $\Delta N_{\rm eff} \approx 0.027$, identical to each other and to standard axion cosmology.
Model A lowers $\Delta N_{\rm eff}$ slightly below 0.027, so a future positive detection of a deviation could separate it from models B and C.
An order-of-magnitude improvement in CMB sensitivity would make phenomenological discrimination among preferred axion models feasible.

Reading between the lines

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

Editorial extension: a phase-space calculation of boosted-axion scattering would close the main loophole; if re-thermalization occurs, the model E exclusion evaporates, so this is the first check a skeptic would run.
Editorial extension: the paper's expectation for dimension-6 decay operators is SM-only products and later decay, which implies $\Delta N_{\rm eff}$ well below 0.027; the sign of a future deviation could therefore point to different operator structures.
Editorial extension: the same Boltzmann treatment applies to any long-lived heavy particle whose decay injects a decoupled species, making $N_{\rm eff}$ a generic chronometer for the ordering of decoupling and decay temperatures.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: this paper identifies a genuinely overlooked channel. In the canonical preferred axion models, the heavy quark Q can decay after axions have decoupled, injecting boosted axions that count as dark radiation. The authors compute decay widths and run the Boltzmann equations model by model, and find that Planck data already cut into models D and E, with model E nearly excluded for mQ = fa. That is a real, within-subfield result, and it changes how people should think about the "preferred" list.

What is genuinely new and good: previous ΔNeff work on QCD axions stuck to thermal production. Here the late out-of-equilibrium decay channel is identified and quantified with a clear table of decay channels, standard freeze-out and decay evolution, and no fitting to the observable. The model A result, where Q decay heats the SM bath but not the decoupled axions and slightly suppresses ΔNeff relative to the thermal value, is a nice touch. The derivation is internally consistent and the comparison to Planck is transparent. The self-citation to Ref. [82] is appropriate; that paper supplied the late-decay picture, but the Boltzmann calculation here stands on its own.

The soft spot is exactly where the reader put the finger: Eq. (4.10) assumes that the scattering rate for axions with energy ~mQ/2 equals the thermal rate used to define decoupling. That is a dimensional-analysis guess, not a phase-space calculation. The authors say so themselves in the paragraph after Fig. 1. Since the entire exclusion claim depends on these boosted axions not rethermalizing, this is load-bearing. If they do rethermalize, the Q-decay energy is shared with the SM bath and ΔNeff returns to roughly the thermal value, 0.027, killing the model E exclusion and the 40% headline. I don't think this is a fatal flaw in the sense of an internal contradiction; it is an explicitly flagged approximation. But it is the single thing a referee should force to be checked, either by a real collision-term calculation or by a clearly stated parameter caveat on the headline.

Minor points: the benchmark choices Λ = mPl and O(1) couplings maximize the late decay and therefore the signal. The authors partially address this by showing mQ < fa for models D and E, but not with a full scan over the suppression scale. The Planck bound used is the standard 2σ one, fine for a first constraint.

Who this is for: axion phenomenologists and early-universe cosmologists who care about which axion models are still viable. It deserves a serious referee. I would send it to review, but with the explicit instruction that the rethermalization assumption be justified, softened, or clearly flagged as a conditional result. It is not a desk reject, and it is not ready to be taken as a settled exclusion until that piece is done.

Referee Report

1 major / 5 minor

Summary. The paper considers the ten 'preferred axion models' of Refs. [22,23] and computes the contribution of their heavy quark Q to the dark radiation density, as parameterized by ΔNeff. The authors derive the leading Q decay channels for each model, identify models in which Q decays after thermal axion decoupling, and solve a coupled set of Boltzmann equations for Q, axions, and SM radiation. They compare the resulting ΔNeff with the Planck 2018 constraint and conclude that existing limits exclude regions of parameter space for 40% of the canonical preferred axion models, with model E essentially excluded for mQ=fa.

Significance. If the central assumption holds, the paper provides a genuinely falsifiable connection between the model-selection criteria used to define preferred axion models and a cosmological observable. The decay-width derivations in Section 3 are explicit, the Boltzmann setup in Section 4 is transparent, and the comparison to Planck is straightforward; the predictions are derived without fitting to ΔNeff. The main weakness is concentrated in one load-bearing assumption about the non-thermalization of the boosted axions produced in Q decay, which the authors themselves flag as unproven.

major comments (1)

[Sec. 4.1, Eq. (4.10) and the paragraph after Fig. 1] The conclusion that boosted axions from Q decay remain decoupled is load-bearing for the paper's central claims, but it is supported only by a dimensional argument. Equation (4.10) defines R(T)=3H/(n_i^eq⟨σv⟩)=T_dax/T using a thermally averaged cross-section inferred from the equilibrium production rate γgg. That cross-section describes axions with thermal energies ~T, whereas the axions produced in Q→ad have energy ~mQ/2 and scatter at a bath temperature T_decay that is 2-4 orders of magnitude below T_dax. The actual collision term depends on the non-thermal phase-space distribution of the boosted axions and samples different Mandelstam variables; the rate could differ substantially from the thermal estimate. The authors explicitly state that formally proving non-thermalization 'would require a recalculation of γgg' with the non-thermal distribution and that this is 'beyond the scope of this work.' If the boosted axions re-thermalize with the SM plasma, the injected energy is shared with the SM bath and ΔNeff reverts to the thermal value ≈0.027, in which case the Planck exclusion of model E and the 40% headline no longer follow. A quantitative estimate of the boosted-axion collision rate, or a conservative upper bound on the resulting ΔNeff, is needed to establish the central claim.

minor comments (5)

[Section 1, after Eq. (2.5)] The condition 'TRH > T/PQ' should presumably read 'TRH > T_PQ'; the typesetting makes this confusing.
[Table 1] The table header and entries contain typographical artifacts ('T able 1' and a duplicated 'Md Md'); these should be cleaned up.
[Section 2, operator list] The text says 'for model B, y1,d = y2,d = 1', but the operators in Eqs. (2.6) and (2.7) use y1,q and y2,d; the notation should be harmonized.
[Eq. (4.14)] The dilution factor uses g⋆(T), whereas entropy conservation would suggest g⋆s(T); if the difference is intentional, a sentence of justification would help.
[Figure 1 caption] The phrase 'showing the comparison in temperatures' is awkward; the caption and legend would be easier to parse if the model labels were ordered consistently with the plotted lines.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Delta Neff predictions follow from the written Lagrangian, computed decay widths, freeze-out, and standard gamma_gg; the only self-citation [82] is motivational, not load-bearing.

full rationale

The central derivation is self-contained. Decay widths in Table 1 are computed from the operators in Eqs. (2.6) and (2.7) under explicit benchmark assumptions (all couplings unity, Lambda = m_Pl, m_Q = f_a); Q freeze-out is solved using Eqs. (4.11) and (4.12); axion decoupling uses the independently calculated thermal production rate gamma_gg of Refs. [29, 33, 119]; and the coupled Boltzmann system (4.4) is then integrated and compared with the Planck bound [127]. No parameter is fitted to Delta Neff, and the '40%' headline is a count over the ten tabulated models under stated benchmark assumptions, not a fit to the target observable. The only self-citation is Ref. [82], which shares an author with the present paper and supplies the qualitative late-decay picture and motivation; the Delta Neff numbers themselves are computed here, so the argument does not reduce to that citation. The weakest step is physical rather than circular: Eq. (4.10) assumes the thermal cross-section applies to boosted axions, and the paper explicitly states that a phase-space recalculation is beyond its scope. That is an unverified assumption and a correctness risk, but it is not an input-output identity or a fitted prediction disguised as a prediction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central prediction depends on the heavy-quark mass, the decay operator coefficients, and the suppression scale Λ=mPl; the first two are set to benchmark values and the third is a motivated maximal choice. The boosted-axion non-thermalization assumption is ad hoc to this paper's scenario.

free parameters (3)

yQ (Q Yukawa coupling) = 1 (figures effectively use mQ = fa, i.e. yQ = sqrt(2))
Sets the heavy quark mass mQ = yQ fa/sqrt(2). The paper assumes yQ=1 throughout and explores variations of mQ/fa in fig. 3.
decay operator coefficients (y1,q, y2,d, y3,d, λd, λ'd, λ1,d, λ2,d, λ2,q, λ3,d) = set to 1 for all
Decay widths scale as the square of these couplings; the paper assumes family-universal O(1) values. Smaller couplings would delay decays further, larger couplings would make decays earlier and weaken the ΔNeff signal.
Λ (suppression scale of dimension-5 operators) = mPl = 1.22e19 GeV
Chosen equal to the Planck scale, motivated by quantum-gravity breaking of global symmetries and the axion quality problem. This maximizes the Q lifetime and hence the late-time decay and ΔNeff enhancement; lower Λ would weaken the constraints.

assumptions (5)

domain assumption Axion-gluon effective coupling L = (αs/8π)(a/fa) G Gtilde (Eq. 2.1)
Standard QCD axion EFT; drives both strong-CP solution and axion production rates.
domain assumption Preferred axion model criteria from Refs [22,23]: NDW=1, 5e9 GeV < fa < 3e11 GeV, Q decay via operators of dimension ≤5, no Landau pole below Planck
Defines the model class under study; adopted from prior literature.
domain assumption The heavy quark Q is in thermal equilibrium at high temperatures via strong interactions
Basis for the freeze-out calculation in Eq. (4.11)-(4.12).
standard math Thermal axion production rate γgg from Refs [29,33] with F3(T) taken as constant above the electroweak scale
Determines the axion decoupling temperature; the paper notes Ref [119] finds a factor-2 difference in F3 but claims insensitivity.
ad hoc to paper Boosted axions from Q decay have the same scattering cross-section as thermal axions, so the decoupling ratio R(T) = T_dax/T applies (Eq. 4.10)
This is the weakest premise; the paper provides a dimensional-analysis argument and explicitly defers a phase-space calculation to future work. If violated, the dark-radiation predictions fail.

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Cite this review

Pith. "Pith review of Using $\Delta N_{\rm eff}$ to constrain preferred axion model dark matter." pith.science (2026). https://pith.science/paper/ARGZEYDH

@misc{pith2026241117320,
  author       = {Pith},
  title        = {Pith review of: Using $\Delta N_\rm eff$ to constrain preferred axion model dark matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARGZEYDH}},
  note         = {Machine review of arXiv:2411.17320}
}

read the original abstract

Preferred axion models are minimal realizations of the Peccei-Quinn solution to the strong CP problem while providing a dark matter candidate. These models invoke new heavy quarks that interact strongly with the Standard Model bringing them into thermal equilibrium in the early Universe. We show that for a number of these models, the heavy quarks will decay after axions have decoupled from the Standard Model thermal bath. As a consequence, any axion products in the decay form a component of dark radiation. This provides the potential to differentiate between preferred axion models through measurements of the number of relativistic degrees of freedom. The most sensitive of which comes from the Planck collaboration's measurements of the Cosmic Microwave Background. We find that existing constraints allow us to rule out regions of parameter space for 40% of the canonical preferred axion models.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

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