{"id":"96db6484-5f05-4fdf-965c-01e7bfd5bc09","arxiv_id":"2607.09285","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"PBHs of ~1e3–1e7 solar masses can significantly grow by absorbing neutrinos before matter-radiation equality.","lead":"New calculations suggest that primordial black holes around 10^3–10^7 solar masses can grow by absorbing neutrinos in the early universe, challenging the long-standing idea that black holes do not gain mass during the radiation era. The result would shift predicted PBH mass peaks and dark-matter fractions, potentially affecting how PBHs are identified in observations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (11) mis-transforms the mass function under growth: multiplying by R(M) at fixed M instead of remapping M_init to M_final=R M_init makes the claimed peak shifts and new peak in Fig. 4 quantitatively unreliable.","rationale":"The paper's central claim has two parts: (1) PBHs of 10^3–10^7 M_sun can grow significantly via neutrino absorption, and (2) this growth shifts the e+e− peak and creates an intermediate-mass peak in the mass spectrum. My primary concern is with part (2) as computed: Eq. (11) applies the growth factor multiplicatively at fixed mass instead of mapping initial masses to final masses via the correct Jacobian. This is a concrete, checkable mathematical error, not just a modeling assumption. The correct transformation preserves the number of PBHs per comoving volume and shifts the support of the mass function; Eq. (11) does not. If the correct remapping were used, the positions and amplitudes of the peaks in Fig. 4 would change, potentially weakening or strengthening the claimed observable signatures. The integrated f_PBH in Tables I and II is correctly given by ∫ R f_i dln M_i, so the total dark-matter fraction may survive, but the spectral-shape claims are unreliable. The reader's flagged concern about the absorption transition (λ_mfp > r_S and the ad hoc weight function) is also valid and addresses the existence of growth itself; it remains an important open issue. However, the mass-function remapping is more immediately load-bearing because it directly invalidates the paper's presentation of its headline spectral results. A conditional accept is therefore appropriate: the mechanism is plausible and worth further work, but the spectral predictions must be recomputed with the correct number-conserving transformation, and the absorption-onset model needs a physical derivation or simulation.","tokens_in":12410,"tokens_out":17619,"duration_ms":153456,"concrete_test":"Recompute the updated mass function using the correct remapping: for the paper's R(M_init) (Fig. 2) and initial f_init, solve for M_f = R(M_i) M_i and evaluate f_f(M_f) = R(M_i) f_i(M_i) / (1 + d ln R/d ln M_i), then compare with Fig. 4 for γ=0.55, κ=1. Check whether the e+e− peak still shifts by the same amount and whether the intermediate-mass peak appears with similar amplitude. Also verify the integrated f_abs matches Table I. If the corrected spectra differ significantly, the headline results need revision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The updated spectrum in Eq. (11), d f_abs/d ln M = R(M) d f_init/d ln M, is not the correct number-conserving transformation for a mass-dependent growth factor R(M_init)=M_final/M_init. Number conservation demands f_f(M_f) d ln M_f = R(M_i) f_i(M_i) d ln M_i, with M_f=R(M_i)M_i, i.e. f_f(M_f)=R(M_i) f_i(M_i)/(1+d ln R/d ln M_i), where M_i solves M_f=R(M_i)M_i. The paper instead evaluates both R and f_i at the same mass M and multiplies, omitting the stretching of the mass axis (1+dlnR/dlnM_i) and evaluating the initial spectrum at the final mass rather than at the initial mass. For constant R the correct result is a pure shift, f_f(M)=R f_i(M/R), whereas Eq. (11) gives R f_i(M) with no shift. Since R rises sharply around 10^3–10^7 M_sun, Eq. (11) artificially tilts the spectrum at fixed masses and misplaces peaks. The claimed shifts of the e+e− peak and the additional intermediate-mass peak in Fig. 4 and the values in Table I/II are not trustworthy as presented; only the integrated f_PBH, which is ∫ R f_i dln M_i in both cases, is unchanged. No derivation or discussion of the Jacobian is given.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that primordial black holes (PBHs) in the mass range roughly 10^3–10^7 M_sun can grow substantially during the radiation era by absorbing neutrinos, contrary to the usual Carr–Hawking conclusion. The author models the onset of absorption by comparing the neutrino mean free path to the Schwarzschild radius, introduces an exponential weight function W for the transition, solves a mass-growth ODE, and applies the resulting growth factor R(M) to an extended PBH mass spectrum generated from the thermal history (QCD and e+e− features). The main reported consequences are a shift of the e+e− peak, a possible new intermediate-mass peak, and an increase in f_PBH from 0.1 to about 0.11–0.13 for γ=0.55.","tokens_in":12825,"tokens_out":8648,"duration_ms":79157,"significance":"If the mechanism is correct, it would overturn a long-standing assumption that PBHs do not grow during the radiation era and would affect interpretations of intermediate-mass BHs, JWST 'little red dots', and constraints based on the initial fluctuation scale. The paper is transparent about its toy-model nature, uses the CosEoS code for the thermal history, and provides an explicit ODE and analytic solution in Appendix B. However, the quantitative results as presented are not yet trustworthy because of an incorrect spectrum remapping and an inconsistency between the main-text ODE and the appendix derivation; the physical transition function is also ad hoc. The paper is therefore promising as a proposal, but it needs substantial revision before the central claims can be accepted.","major_comments":[{"comment":"Eq. (11), d f_abs/dlnM = R(M) d f_init/dlnM, is not a valid remapping under a mass-dependent growth factor R(M_i)=M_f/M_i. Properly, each initial bin at M_i moves to M_f=R(M_i)M_i, so f_f(M_f)dlnM_f = R(M_i) f_i(M_i)dlnM_i, i.e. f_f(M_f)=R(M_i) f_i(M_i)/(1+dlnR/dlnM_i), with M_i determined implicitly. The paper evaluates both R and f_i at the same mass M and omits the Jacobian. For constant R this would give a pure shift f_f(M)=R f_i(M/R), whereas Eq. (11) gives R f_i(M) with no shift. Since R rises sharply across 10^3–10^7 M_sun, the claimed peak shifts and additional peak in Fig. 4 are quantitatively unreliable. The integrated f_PBH is unchanged by this error, but the spectral-shape claims in Section IV are not supported.","section":"Section IV, Eq. (11)"},{"comment":"The main text writes dR/dx = 12πγ_eff δ_hf/α x^3 R^3 W, but Appendix B derives dR/dx = g(x) R^2 and solves R=1/(1−G). The R^2 form follows from dM/dt=σ_hf ρ_R ∝ M^2 T^4 and dt/dx ∝ x. As printed, Eq. (8) is inconsistent with the equation actually solved and with the stated solution. The author must state which ODE was integrated; if R^3 was used, all growth factors, Tables I–II, and the γ bounds change; if R^2 was used, Eq. (8) must be corrected.","section":"Section III Eq. (8) vs Appendix B Eq. (B4)"},{"comment":"The absorption criterion is λ_mfp ≥ r_S, with an ad hoc weight W = exp(−κ r_S/λ_mfp), κ in {0.5, 1, 2}. No derivation from kinetic theory or plasma physics is provided. The geometric-optics cross section Eq. (B1) assumes particles reach the horizon, whereas the mean-free-path condition concerns the ambient plasma; a particle with λ_mfp ≫ r_S may also pass through the hole's vicinity without being captured. Because the magnitude of R, the position of the extra peak, and the γ_div bound all depend on this transition (see Table I), the mechanism needs either a transport-equation calculation for neutrinos in a Schwarzschild background or a quantitative comparison with the Bondi/Carr–Hawking regime. As it stands, the central predictions are conditional on an unvalidated parametrization.","section":"Section III, Eqs. (6)–(7) and Fig. 1"},{"comment":"The large-growth branch is obtained with γ=0.55, which is close to the divergence value from Eq. (10). Since γ is treated as a free parameter and is not independently constrained, the statement that PBHs can grow significantly is a demonstration of the vicinity of the pole rather than a robust prediction. A physical prior on γ from critical-collapse/numerical-relativity fits would be needed to turn this into a falsifiable claim; otherwise the conclusions for IMBHs and LRDs rest on a tuned input.","section":"Section IV, Fig. 2 and Table II"}],"minor_comments":[{"comment":"Typos and formatting: 'predictede+e−peak' and the truncated 's' in the conclusion (which should read 'little red dots') should be fixed throughout.","section":"Abstract, Section V, and conclusion"},{"comment":"The caption refers to a red step function but the colors of the curves are not described; please add a legend or explicit color labels.","section":"Fig. 2 caption"},{"comment":"The amplitude A is introduced as a free normalization, but its relation to the primordial power spectrum amplitude is not stated; please clarify dimensions and typical values.","section":"Appendix A, Eq. (A2)"},{"comment":"The notation ρ_R and g_ρ in δ_hf is confusing because only neutrinos are absorbed. Please write g_ρ ≡ g_ν^ρ explicitly in the derivation to avoid the factor ambiguity.","section":"Appendix B, Eq. (B2)–(B3)"},{"comment":"Several bibliographic entries have inconsistent or missing year fields (e.g. Refs. [18], [22], [23], [38], [40]); please check against the journal style.","section":"References"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know first: this paper proposes something genuinely new. The author argues that in the radiation era, PBHs in the ~10^3–10^7 M_sun range can absorb neutrinos because the neutrino mean free path can exceed the Schwarzschild radius, overturning the standard Carr–Hawking no-growth picture. That is a real idea, and it deserves to be taken seriously. The paper is also refreshingly transparent: it uses CosEoS, explains the absorption regime, and explicitly lists many simplifications.\n\nNow the soft spots. Two are load-bearing. First, the growth equation is internally inconsistent: Eq. (8) has dR/dx ∝ R^3, while Appendix B derives dR/dx = g(x) R^2 and solution R = 1/(1-G). Those are different equations with different runaway conditions. The paper never says which one was solved. Second, Eq. (11) mis-transforms the mass function under mass-dependent growth. Number conservation requires mapping M_i to M_f = R(M_i)M_i and including the Jacobian 1 + d ln R/d ln M_i. The paper instead multiplies the initial spectrum by R(M) at the same mass. That is wrong, and it undermines the claimed peak shifts and the new intermediate-mass peak in Fig. 4. The integrated f_PBH is unchanged by the Jacobian, which is probably why the error slipped through, but the differential spectrum is not.\n\nBeyond that, the large growth only appears for gamma close to the divergence (0.55), and the transition factor W = exp(-kappa r_S / lambda_mfp) is an ad hoc interpolation. The author acknowledges these limitations. Still, as presented, the main quantitative claims are not robust.\n\nWho is this for? People working on PBH mass evolution, PBH dark matter constraints, and early-universe plasma physics. They would get a useful prompt to think about neutrino absorption, but they should not take the spectra or tables at face value.\n\nI would send this to peer review rather than desk-reject. The mechanism is novel, the topic is important, and the flaws are fixable: correct the ODE, fix the mass-function transformation, and scan gamma away from the runaway. But I would not cite the quantitative results until that happens.","headline":"A novel neutrino-absorption growth mechanism for intermediate-mass PBHs, but the quantitative claims are undermined by an incorrect mass-function transformation and an internal ODE inconsistency.","tokens_in":13315,"tokens_out":3329,"would_cite":false,"duration_ms":32222,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.-k","95.35.+d","97.60.Lf"],"model":"deepseek-v4-flash","headline":"Primordial black holes in the roughly 10^3 to 10^7 solar-mass range can grow substantially during the radiation era by absorbing neutrinos, overturning the usual assumption that PBH masses are frozen until matter domination.","keywords":["primordial black holes","neutrino absorption","radiation era","mass growth","thermal bath","QCD transition","dark matter","mass spectrum"],"falsifier":"Compute, for a Schwarzschild black hole immersed in a thermal neutrino bath, the exact absorption probability for neutrinos with momenta near the thermal peak and compare it with the geometric-optics value used in the mass-growth equation; if the probability is substantially below the geometric value when r_S is comparable to the mean free path, the predicted growth factors are correspondingly overestimated.","tokens_in":1554,"feed_emoji":"🕳️","tokens_out":1863,"duration_ms":72211,"temperature":0.7,"pith_summary":"This paper challenges a long-standing assumption in primordial black hole physics: that black holes cannot gain mass during the radiation-dominated era because radiation is too tightly coupled to flow into them. It argues that neutrinos are different, because once the neutrino mean free path exceeds the black hole's horizon radius, the hole acts as a blackbody absorber of neutrinos and can grow substantially. For black holes formed near the QCD transition, with masses around 10^3 to 10^7 solar masses and a collapse fraction gamma = 0.55, the growth is large enough to shift the predicted electron-positron annihilation peak in the mass spectrum, create an additional intermediate-mass peak, and raise the dark matter fraction carried by PBHs. If the mechanism is real, heavy PBHs no longer require very large primordial density fluctuations, which relaxes constraints from spectral distortions and scalar-induced gravitational waves while changing how observational bounds must be applied.","feed_headline":"Neutrino absorption lets early black holes grow","feed_subtitle":"Mass growth around 10^3–10^7 solar masses shifts the PBH mass spectrum and raises the dark matter fraction.","key_machinery":"The load-bearing mechanism is the geometric-optics absorption cross section of a Schwarzschild black hole, σ = (27/64π) M^2/M_P^4, applied to neutrinos whose mean free path λ_ν = 1/(n_ν σ_weak) exceeds the horizon radius r_S. A weight function W = exp(-κ r_S/λ_ν), with κ = 0.5, 1, or 2, smoothly interpolates between absorbing and non-absorbing regimes, and the mass evolution is carried by the equation dR/dx = 12π γ_eff (δ_hf/α) x^3 R^3 W. The physics that decides how much growth occurs is the ratio δ_hf/α, the neutrino energy density relative to the total radiation density, which peaks after the QCD transition and creates the 'sweet spot' where growth is maximal.","core_discovery":"The central claim is that the standard criterion for whether a PBH can absorb surrounding radiation, namely that the radiation mean free path exceeds the Schwarzschild radius, is satisfied for neutrinos at the relevant epochs even though it fails for photons. Starting from the weak-interaction neutrino mean free path, the paper defines a start temperature at which absorption becomes possible and solves a semi-classical mass-growth equation, dR/dx = 12π γ_eff (δ_hf/α) x^3 R^3 W, using the geometric-optics cross section σ = (27/64π) M^2/M_P^4 and a smooth weight function W = exp(-κ r_S/λ_ν) to model the transition from non-absorbing to absorbing regimes. With a collapse fraction γ = 0.55, PBHs","pith_inferences":["The same absorption logic should apply to any weakly interacting relic that decouples early; if a feebly interacting species carries a non-negligible energy density at PBH formation, it would contribute to growth and could shift the optimal mass window.","The γ ≲ 0.55 bound is derived using the geometric cross section and an ad hoc transition weight; a first-principles calculation of neutrino absorption probabilities in the Schwarzschild metric near r_S ~ λ_ν would determine whether the growth factors are over- or under-estimated.","Because growth is fastest shortly after formation, PBHs formed with super-critical overdensities (δ > δ_c) would grow more than the δ = δ_c approximation used here, moving the additional peak to higher masses.","A clean observational discriminator is the intermediate-mass black hole mass spectrum: if the 10^3 to 10^7 solar-mass window remains empty in gravitational-wave and microlensing searches despite a large initial f_PBH, then either γ is lower or the absorption efficiency is suppressed."],"forward_implications":["PBHs in the 10^3 to 10^7 solar-mass range can grow substantially in the radiation era, with the largest growth for PBHs formed near the QCD transition when neutrinos carry a peak share of the radiation density.","The e+e- annihilation peak in the PBH mass spectrum shifts to larger masses, and an additional peak can appear in the intermediate-mass range depending on the collapse fraction γ.","The dark matter fraction in PBHs increases relative to its initial value; for γ = 0.55, the step-function transition raises f_PBH from 0.1 to 0.126.","To avoid runaway absorption, the collapse fraction must satisfy γ ≲ 0.55, otherwise the growth equation diverges within the model.","Observational constraints on PBHs must be remapped from formation mass to final mass; CMB and accretion bounds are not relieved and may be tightened for this mass window, while constraints tied to primordial fluctuations are partially evaded."],"fun_headline_variants":["Neutrinos power black hole growth in early cosmos","Black holes bulk up on neutrinos in radiation era","Neutrino diet grows primordial black holes","Early black holes feast on neutrinos, altering dark matter"],"cache_read_input_tokens":14464,"weakest_assumption_plain":"The whole result rests on one premise: a black hole absorbs neutrinos as a perfect blackbody once the neutrino mean free path exceeds the hole's horizon radius, with a smoothly guessed transition in between; if any significant fraction of neutrinos scatter before reaching the horizon, the predicted growth largely disappears.","fun_headline_variants_meta":{"raw":{"variants":["Neutrinos power black hole growth in early cosmos","Black holes bulk up on neutrinos in radiation era","Neutrino diet grows primordial black holes","Early black holes feast on neutrinos, altering dark matter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1104,"prompt_tokens":697,"completion_tokens":407,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":345}},"tokens_in":441,"tokens_out":407,"duration_ms":4330,"temperature":1.0,"reasoning_tokens":345,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:47:32.485188+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a Schwarzschild black hole immersed in a thermal neutrino bath, the exact absorption probability for neutrinos with momenta near the thermal peak and compare it with the geometric-optics value used in the mass-growth equation; if the probability is substantially below the geometric value when r_S is comparable to the mean free path, the predicted growth factors are correspondingly overestimated.","supporting_citations":[],"review_version":2}