{"id":"68d4c9a3-eb6f-4bc7-b35b-faa8aae35d35","arxiv_id":"2411.17320","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In preferred axion models with slow heavy-quark decays, axions produced after decoupling form dark radiation that can exceed Planck's ΔNeff bound, excluding much of the parameter space of models D and E.","lead":"Some axion dark matter models contain new heavy quarks that can decay long after axions have stopped interacting with ordinary matter, injecting an extra population of axions. This extra radiation changes the cosmic expansion and lets existing Planck measurements rule out parts of 40% of the preferred axion models.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Boosted-axion rethermalization (Eq. 4.10) is load-bearing: if Q-decay axions scatter back into the SM bath, model E/D exclusions collapse to thermal ΔNeff≈0.027. A phase-space collision-term calculation is required.","rationale":"Good-faith reading: this is a phenomenological study of ΔNeff from late Q decays in the ten 'preferred axion' models. The decay widths in Table 1 follow from the operators in Eqs. (2.6)–(2.7); the freeze-out and entropy evolution are standard; the Planck comparison is transparent. The central claim hinges on the assumption that boosted axions produced at T_decay remain collisionless. The evidence for this is Eq. (4.10), which extrapolates from the equilibrium production rate γgg to a temperature-only ratio R(T)=T_dax/T. This uses a thermal axion distribution, whereas the actual decay products have E~mQ/2 and interact through matrix elements that depend on s≈mQ T. The paper explicitly delegates the required phase-space calculation to future work (paragraph after Fig. 1). If the boosted axions thermalize, the injected energy is shared with the SM bath and the axion sector freezes out from equilibrium again, giving the standard thermal value and removing the model E exclusion. This is exactly the reader's weakest assumption. I also considered benchmark dependence: Λ=mPl and O(1) couplings maximize the effect, but that alone would not invalidate the claimed 'regions of parameter space'; rethermalization would. Hence I agree with the CONDITIONAL verdict and propose a concrete phase-space test that would settle the issue. If the test shows rethermalization, the exclusion claims should be removed; if it confirms free-streaming, the central claim stands under the stated benchmarks.","tokens_in":122,"tokens_out":14436,"duration_ms":262408,"concrete_test":"Compute the momentum-dependent collision term C[f_a](p) for the dominant processes a(p)+g(k)→g+g and a+g→q+qbar using the KSVZ interaction in Eq. (4.8), with f_a(p)=(N_a/n_a)δ(p−mQ/2) normalized to the Q-decay injection rate at T_decay. Evaluate the scattering rate Γ_scat(E_a=mQ/2, T_decay)=(1/n_a)∫dΠ C[f_a] and compare it with H(T_decay). If Γ_scat>H, rethermalization occurs and the model E/D exclusions must be replaced by the thermal ΔNeff≈0.027; if Γ_scat<H, the paper's decoupled treatment is validated. A simpler limiting check: repeat the R(T) estimate using the high-energy limit of the axion-gluon cross-section (√s≈√(mQ T)) instead of the thermal ⟨σv⟩; if R(T_decay)<1 under that cross-section, the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central exclusion claim — 'model E is all but excluded by Planck for mQ=fa' and the 40% headline — requires that boosted axions produced in Q→a d decays (energy ~mQ/2) stream freely after production. The paper's argument for this is dimensional: Eq. (4.10) defines R(T)=3H/(n_i^eq⟨σv⟩_gg)=T_dax/T, using the thermally averaged cross-section inferred from the equilibrium production rate γgg. That cross-section is derived for axions with thermal energies ~T, not for axions with E~mQ/2 scattering on gluons/quarks at T_decay~10^4–10^5 GeV. The collision term depends on f_a(p), and a monoenergetic boosted distribution samples very different Mandelstam variables; the rate could be much larger or smaller than the thermal estimate. The paper explicitly flags this gap in the paragraph after Fig. 1: 'To formally prove that this is the case would require a recalculation of γgg but now taking into account the non-thermal boosted axion distribution. This is beyond the scope of this work.' If the boosted axions do re-thermalize, the Q-decay energy is shared with the SM bath and the axion population freezes out again from equilibrium, producing the standard thermal value ΔNeff≈0.027 for all models; the Planck exclusion and the 40% claim disappear. This is therefore the single load-bearing assumption. Benchmark choices (Λ=mPl, O(1) couplings) affect the size of the excluded regions but do not threaten the existence of the signal the way rethermalization does.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19371,"tokens_out":10449,"duration_ms":100860,"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":[{"comment":"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.","section":"Sec. 4.1, Eq. (4.10) and the paragraph after Fig. 1"}],"minor_comments":[{"comment":"The condition 'TRH > T/PQ' should presumably read 'TRH > T_PQ'; the typesetting makes this confusing.","section":"Section 1, after Eq. (2.5)"},{"comment":"The table header and entries contain typographical artifacts ('T able 1' and a duplicated 'Md Md'); these should be cleaned up.","section":"Table 1"},{"comment":"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.","section":"Section 2, operator list"},{"comment":"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.","section":"Eq. (4.14)"},{"comment":"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.","section":"Figure 1 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about the main gap, but the gap is not cosmetic: the headline exclusion and the 40% claim depend on the assumption that the boosted axions do not re-thermalize. I would recommend requiring a quantitative estimate of the boosted-axion collision rate, or a conservative bound on its effect, before publication. The self-citation to Ref. [82] is used appropriately; the present Boltzmann calculation is independent of that earlier qualitative argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nMinor 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.\n\nWho 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.","headline":"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.","tokens_in":19962,"tokens_out":1541,"would_cite":true,"duration_ms":18496,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $m_Q = f_a$.","keywords":["axion dark matter","preferred axion models","KSVZ axions","heavy quark decay","dark radiation","Neff","CMB constraints","Peccei-Quinn mechanism"],"falsifier":"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.","tokens_in":18745,"feed_emoji":"🌌","tokens_out":9859,"duration_ms":82405,"temperature":0.7,"pith_summary":"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.","feed_headline":"40% of preferred axion models now hit Planck limits","feed_subtitle":"Late heavy-quark decays inject fast axions, inflating ΔNeff and separating otherwise identical models.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"defines the preferred axion model criteria and the axion dark matter window the paper works within.","marker":"[22]"},{"why":"enumerates the ten canonical models and their charge assignments and decay operators.","marker":"[23]"},{"why":"determines the KSVZ-I and KSVZ-II charge assignments, the upper $f_a$ bound, and the decay operator list used in Table 1.","marker":"[38]"},{"why":"shows dimension-5 heavy-quark decays can be long-lived and alter axion relic abundances, motivating the late-decay scenario.","marker":"[82]"},{"why":"supplies the thermal axion production rate and thermal function used to locate axion decoupling.","marker":"[29]"},{"why":"provides the thermal QCD axion production rate $\\gamma_{gg}$ across thresholds used to compute the axion decoupling temperature.","marker":"[33]"},{"why":"reports the Planck 2018 $N_{\\rm eff}$ measurement whose upper bound sets the $\\Delta N_{\\rm eff} \\le 0.276$ constraint.","marker":"[127]"},{"why":"establishes the thermal axion contribution $\\Delta N_{\\rm eff}\\approx 0.027$ that models B and C reproduce.","marker":"[126]"}],"fun_headline_variants":["Planck’s ΔNeff rule out 40% of preferred axion models","Late decay injects axion radiation, pinching favored dark matter models","Preferred axion models: 40% already excluded by CMB","Axion dark matter models face Planck’s ΔNeff squeeze","Heavy quark decays boost axions, tripping Planck limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Planck’s ΔNeff rule out 40% of preferred axion models","Late decay injects axion radiation, pinching favored dark matter models","Preferred axion models: 40% already excluded by CMB","Axion dark matter models face Planck’s ΔNeff squeeze","Heavy quark decays boost axions, tripping Planck limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1446,"prompt_tokens":883,"completion_tokens":563,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":468}},"tokens_in":499,"tokens_out":563,"duration_ms":4737,"temperature":1.0,"reasoning_tokens":468,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:15:10.877947+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}