{"id":"fddcf2a5-a404-42ad-8315-54148c87018e","arxiv_id":"2607.08738","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Shape dispersion around the average peak profile is a genuine statistical ingredient: rare deformed curvature profiles can dominate primordial black hole formation when the power spectrum is broad or non-Gaussianity is negative.","lead":"This paper develops a statistical method for including the full shape of primordial density ripples when predicting primordial black hole abundances. Spherical collapse examples show that rare, deformed profiles can dominate black-hole production and that ignoring them can bias abundance estimates by orders of magnitude.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Quantitative enhancements rest on one hand-picked radial split mode; no basis-convergence test establishes that it represents the infinite-dimensional residual shape space.","rationale":"The reader's weakest-assumption identification matches my own: the quantitative enhancement is computed for one hand-picked split-spectrum deformation and no basis-convergence test is performed. This is the single most load-bearing weakness because the central quantitative claims—orders-of-magnitude abundance enhancement and factor-~3 amplitude reduction—depend directly on the chosen direction in the infinite-dimensional residual shape space. The paper is transparent about this limitation (Sec. 5.1.2, Sec. 6) and even provides one internal consistency check (projected split mode in Sec. 5.2.1), which shows non-negligible sensitivity. The qualitative mechanism, however, is not undermined: the competition between Gaussian cost and threshold reduction is a general variational principle, and at least for some direction it demonstrably shifts the dominant profile away from the conditional mean. Thus the concern does not warrant rejection or a lowered verdict; it warrants retaining the CONDITIONAL verdict and requiring the basis-dependence test (and ideally a boundary scan for β_NG=-3) before the quantitative factors are used in phenomenological applications.","tokens_in":51973,"tokens_out":7695,"duration_ms":86563,"concrete_test":"For Case A (exponential spectrum, β_NG = -2 and -3), repeat the abundance calculation with at least two additional radial bases: (i) the q_x-orthogonal projected split mode of Eq. (5.35), and (ii) the next ℓ=0 action-normalized mode obtained by Gram-Schmidt orthogonalizing k^4 against q_ν and the split mode (and, if needed, against q_x). Recompute β_disp/β0, n*, and Q_A for each basis. If these quantities differ by more than an order of magnitude, or if the β_NG=-3 branch weight remains monotonic to the edge of the scan, then the reported enhancements are basis-dependent and should be presented only as a sensitivity bound. Also extend the β_NG=-3 scan beyond the current s-limit to check whether log Υ_br turns over.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The qualitative central claim—that PBH abundance is set by a competition between Gaussian action cost and threshold reduction—is well motivated and broadly supported. The load-bearing problem is quantitative: all reported orders-of-magnitude enhancements and amplitude-retuning factors (Tables 2–4) come from integrating a single action-normalized direction, q_0,split(k) = sign(k-kbar)/σ0, while every orthogonal residual mode is held at its mean (Sec. 5.1.2; Sec. 6). The collapse threshold μ_c(s) is a functional of the full profile, so a different orthonormal basis of radial modes can give a different threshold curve and therefore different β_disp/β0 and Q_A. The paper itself provides evidence of this sensitivity: in Sec. 5.2.1, projecting the split mode orthogonal to the BBKS curvature direction changes the R_F=50 abundance ratio from 3.0×10^3 to 5.2×10^3. Moreover, for β_NG=-3 the branch-weight log Υ_br has no internal maximum within the simulated s-range (Sec. 5.1.1), so the Table 2 values n* = -3.59 and Q_A = 0.339 are partly determined by the integration interval, not by a physical saddle point. Because the full residual space is infinite-dimensional, the reported numbers are illustrative of one direction rather than a converged prediction of the formalism.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a finite-action, Fourier–Bessel framework to describe coherent shape dispersion around reference curvature profiles in primordial black hole formation. The Gaussian power spectrum defines a metric on profile space; the BBKS peak variables are recovered as the lowest action-normalized directions, and orthogonal directions represent residual radial and angular deformations. The formalism is applied to two spherical numerical-collapse examples: a sharply peaked finite-width spectrum with logarithmic non-Gaussianity, and a finite-band scale-invariant spectrum with tunable bandwidth. In both cases a single split-spectrum radial mode is integrated against numerically determined collapse thresholds. The central qualitative claim is that the dominant PBH contribution is selected by a competition between the Gaussian cost of a coherent deformation and the threshold reduction it induces, rather than being the conditional-mean reference profile or the lowest-threshold profile. For negative non-Gaussianity and broad spectra, rare deformations can dominate, enhancing the integrated abundance by orders of magnitude and reducing the required power-spectrum amplitude by up to a factor ~3.","tokens_in":52377,"tokens_out":4330,"duration_ms":45272,"significance":"If the framework and the illustrative calculations are taken as a proof of principle, this is a valuable contribution. The algebraic derivation of the BBKS sector in action-normalized variables is clean and internally consistent, and the threshold curves are obtained from an established relativistic code with Hamiltonian-constraint monitoring, which gives confidence in the collapse dynamics. The paper is also commendably transparent about the limitations of its one-dimensional split-mode treatment, explicitly labeling the results as the effect of a single dominant direction rather than a full marginalization. The qualitative insight—that abundance is controlled by a cost-benefit competition and not simply by the reference profile—is well motivated and broadly supported by the examples. The quantitative enhancement factors and amplitude-retuning ratios, however, remain illustrative because they depend on a hand-picked deformation direction and, in one case, on an integration boundary.","major_comments":[{"comment":"The text states that for β_NG = −3 the branch-weight log Υ_br has no internal maximum within the simulated s-range; the weight keeps increasing towards the edge of the available threshold curve. Therefore the quoted n* = −3.59 and Q_A = 0.339 are partly determined by the integration interval [s_min, s_max] rather than by a physical saddle point. This makes the headline amplitude-retuning factor for the most extreme non-Gaussian case sensitive to an arbitrary numerical boundary. The authors should extend the threshold scan or demonstrate that the results converge as the integration range is enlarged.","section":"§5.1.1, Table 2 (β_NG = −3 row)"},{"comment":"All reported quantitative enhancements and amplitude-retuning factors arise from integrating a single hand-picked split mode q_0,split(k) = sign(k−k̄)/σ_0, while all orthogonal residual shape modes are held at their mean. The paper acknowledges this, but the abstract and conclusions present the factor ~3 amplitude reduction as a quantitative result. Since μ_c(s) is a functional of the full profile and the residual shape space is infinite-dimensional, a different orthonormal basis could yield different threshold curves and hence different β_disp/β_0 and Q_A. The projected-mode check in §5.2.1 is a useful consistency test, but it covers only the top-hat example and does not validate the non-Gaussian Case A. A basis-convergence test—e.g., including one or two additional radial modes—is needed before these numbers can be viewed as more than illustrative.","section":"§5.1.2, Eq. (5.56), Tables 2–4"},{"comment":"The mass-function calculation uses the height-only BBKS peak density N_BBKS^pk(ν), in which the curvature variable x_B has been integrated out, while the threshold curve μ_c(s) is computed for the unprojected split mode that changes x_B at fixed height. This mixes conditionings: the no-dispersion reference uses x_B = x_*(ν), but the deformed branch traces a one-parameter path that alters x_B, and the x_B marginalization is not performed coherently. The projected split mode in §5.2.1 partially addresses this for the top-hat spectrum, but no analogous consistency check is given for the non-Gaussian Case A. The size of the enhancements in Table 2 is therefore not fully controlled by the stated statistical prescription.","section":"§5.1.2, Eq. (5.55)"}],"minor_comments":[{"comment":"Typo: 'Multipolar F ourier–Bessel' should read 'Multipolar Fourier–Bessel'.","section":"§2.1 title"},{"comment":"In the text, 'T able 1' appears with a space; the same applies to other table references such as 'T able 2–4' in §5.1.2. Please correct the spacing.","section":"§3, Table 1"},{"comment":"The mass map M(μ,s) uses K_eff ≃6 from Ref. [70] with a power-law critical scaling. The paper notes that this is an effective choice, but it would be useful to state explicitly how the uncertainty in K_eff propagates into the abundance ratios, since K_eff enters multiplicatively and could affect f_PBH values even though the exponential threshold dominates.","section":"§5.1.2, Eq. (5.53)"},{"comment":"The comparison with the HYK threshold δ_HYK ≈ 0.41 and the EGS prediction is shown only for the unprojected top-hat family; a brief comment on how the projected family compares (e.g., in Fig. 23) would improve readability.","section":"§5.2, Fig. 20"}],"recommendation":"major_revision","confidential_remarks":"The paper's central qualitative claim is original and worth publishing, but the quantitative headline numbers are not yet robust against the choice of deformation basis. The author's transparency about the one-direction treatment is good, but the abstract and conclusions go somewhat beyond what the present evidence supports. The requested basis-convergence and integration-range checks are feasible within the paper's scope and would considerably strengthen it. The referee sees no issue with the self-referential use of the author's earlier framework [54]; it is properly credited and the present work clearly extends it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this carefully, and the reader's conditional verdict is about right. The central idea is well motivated: PBH abundance is set by the competition between the Gaussian cost of a coherent shape deformation and the threshold reduction it buys. The Fourier-Bessel action-normalized construction is a genuine extension of Ref. [54], and it cleanly recovers the BBKS peak variables as the first few action-normalized directions. The numerical work uses an established code with Hamiltonian-constraint monitoring, and the collapse thresholds are not fit to any target abundance, so the circularity burden is low.\n\nWhat is new and valuable: for finite-width spectra and negative logarithmic non-Gaussianity, the dominant PBH contribution need not be the conditional-mean profile. The paper shows, using actual collapse thresholds, that a rare coherent deformation can beat the mean profile by orders of magnitude because it lowers the effective peak height. That is a real result with consequences for PTA-era abundance estimates. The paper also deserves credit for flagging its own limitation: Sec. 5.1.2 says explicitly that only one split direction is integrated and all orthogonal residual modes are held at their mean.\n\nThe soft spot is exactly the one the stress-test names. All the headline numbers—beta_disp/beta_0, Q_A, n_*—come from a single split-spectrum mode, and the threshold is a functional of the whole profile. Change the basis and you change the numbers. The paper itself shows this: projecting the split mode orthogonal to the BBKS curvature direction shifts the R_F=50 ratio from 3.0e3 to 5.2e3. That is a large sensitivity by the standards of the claimed orders of magnitude. And for beta_NG=-3, the branch weight has no internal maximum within the simulated s-range, so the quoted n*=-3.59 and Q_A=0.339 are partly set by where the integration stops. These are addressable weaknesses, not a broken argument. The qualitative conclusion—that profile dispersion can be statistically decisive—holds up.\n\nA revision should test a second orthonormal direction, do a low-dimensional multi-mode integration, or clearly reframe the quantitative claims as illustrative. A short section on threshold uncertainty from the code would also help.\n\nThis paper deserves a real referee. The formalism is likely to be reused, and the qualitative effect should be taken seriously even if the specific enhancements change. I would send it to peer review with a request for basis-dependence checks, and I would bring it to the reading group.","headline":"Strong finite-action framework with a real qualitative result; the headline enhancements rest on one hand-picked deformation direction and need basis-convergence testing before being taken as quantitative.","tokens_in":52772,"tokens_out":3763,"would_cite":true,"duration_ms":37852,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the profiles that dominate primordial black hole formation are selected by a competition between the Gaussian cost of coherent shape deformations and the exponential benefit of a lower collapse threshold—not by the av","keywords":["primordial black holes","curvature-profile dispersion","peak theory","collapse threshold","finite-action formalism","non-Gaussianity","power-spectrum width","compaction function"],"falsifier":"Run the same collapse-threshold calculation with a different orthonormal basis of residual radial modes—for example, include the second and third radial directions in the monopole sector, or construct a radial mode from a different spectral split—and check whether the integrated PBH abundance enhancement at the strongest negative-non-Gaussianity case remains near 10^5 or collapses back to order one. If the enhancement is not robust to basis choice, the dominant-branch mechanism is an artifact of the chosen deformation family. Alternatively, a non-spherical (3+1) simulation of the dominant low-","tokens_in":1440,"feed_emoji":"🕳️","tokens_out":1577,"duration_ms":63593,"temperature":0.7,"pith_summary":"This paper argues that in the standard curvature-perturbation scenario, the dominant contribution to primordial black hole (PBH) abundance is not the conditional-mean curvature profile of peak theory, nor simply the profile with the lowest collapse threshold. Instead, it is the profile that balances the Gaussian statistical cost of realizing a coherent shape deformation against the exponential gain from a lower collapse threshold. The paper develops a finite-action, power-spectrum-weighted decomposition of curvature profiles that turns every coherent deformation into a standard Gaussian variable, then combines numerically computed collapse thresholds with peak statistics. In the spherical examples, negative local non-Gaussianity and broad power spectra make rare deformations (several sigma away) dominate, enhancing integrated abundances by orders of magnitude and cutting the required power-spectrum amplitude by up to a factor of about three. The reader should care because abundance estimates that ignore profile dispersion systematically miss the dominant PBH-forming configurations when the spectrum is broad or non-Gaussianity is negative.","feed_headline":"Rare curved shapes, not average profiles, may form most PBHs","feed_subtitle":"A cost-benefit balance between Gaussian rarity and threshold drop can shift primordial-black-hole abundance estimates by orders of magnitude","key_machinery":"The central object is the Gaussian-action metric on the space of curvature profiles, induced by the primordial power spectrum, together with the multipolar decomposition into spherical Bessel radial envelopes and spherical harmonics. Coherent deformations are normalized by this metric so that each amplitude is a standard Gaussian variable and the statistical cost is the sum of squared amplitudes; the usual peak-theory variables (height, gradient, Hessian) emerge as the first action-normalized directions. The split-spectrum ansatz—an equal-variance division of the spectrum into long- and short-wavelength halves—supplies a concrete one-dimensional residual radial mode used in the numerical col","core_discovery":"The central claim is that residual profile dispersion—the infinitely many curvature configurations sharing the same local peak height, gradient, and Hessian—is a genuine statistical ingredient in PBH formation. The paper shows that the collapse threshold is a functional of the full profile, and that accounting for it turns PBH abundance into a competition: rare coherent deformations cost Gaussian action n^2 but can lower the threshold enough to win. In the spherical cases studied, with a logarithmic local non-Gaussian map, negative non-Gaussianity shifts the dominant branch to several-sigma deformations and enhances abundance by factors up to about 10^8; in finite top-hat spectra, broadening","pith_inferences":["Editorial inference: The single split-mode direction was chosen for maximal real-space dispersion; a full infinite-dimensional marginalization could either strengthen the effect (if many modes lower thresholds) or weaken it (if the chosen mode is unusually efficient). Basis dependence is therefore a decisive test of generality.","Editorial inference: The same cost-benefit logic should apply to angular multipoles beyond the quadrupole: there may exist rare non-spherical deformations that dominate over both the spherical reference and ellipsoidal profiles, but their thresholds require full 3+1 simulations.","Editorial inference: The near-threefold reduction of the required amplitude in the most negative non-Gaussianity example suggests that constraints on primordial non-Gaussianity derived from PBH overproduction may need to be revisited in models with finite-width spectra.","Editorial inference: A direct cross-check would be to recompute the sharply-peaked-spectrum enhancement with a deformation projected to keep both height and curvature fixed; if the enhancement persists, curvature leakage is not the driver."],"forward_implications":["Abundance estimates that evaluate collapse only on the conditional-mean profile underestimate PBH production when the spectrum is broad or local non-Gaussianity is negative.","For a fixed target PBH abundance, the inferred primordial power-spectrum amplitude can be reduced substantially (down to about a third in the strongest negative-non-Gaussianity example), weakening the usual tension with pulsar-timing-array gravitational-wave constraints.","The monochromatic-spectrum approximation, which leaves no independent radial shape freedom, is the least favorable regime for dispersion; finite-width spectra open statistically available radial modes that can dominate.","The dominant PBH-forming configurations can have effective peak heights as low as a few, where the high-peak near-spherical approximation breaks down and angular (non-spherical) modes should be included.","Mass-function predictions in realistic finite-width enhanced-spectrum models require the full shape-dispersed integral, not a single reference-profile evaluation."],"fun_headline_variants":["Rare shape deformations dominate black hole formation","Deformed profiles, not the average, set PBH abundance","Cost-benefit balance picks rare shapes for PBH dominance","Shape dispersion shifts PBH abundance by orders of magnitude","Why rare curvature shapes may dominate PBH yields"],"cache_read_input_tokens":54144,"weakest_assumption_plain":"The quantitative enhancements rest on treating one hand-picked split-spectrum radial deformation as representative of the entire infinite-dimensional residual shape space, with all other orthogonal modes held at their mean; if other radial or angular modes respond to the collapse threshold differently, the reported orders-of-magnitude enhancements could change.","fun_headline_variants_meta":{"raw":{"variants":["Rare shape deformations dominate black hole formation","Deformed profiles, not the average, set PBH abundance","Cost-benefit balance picks rare shapes for PBH dominance","Shape dispersion shifts PBH abundance by orders of magnitude","Why rare curvature shapes may dominate PBH yields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000369,"raw_usage":{"total_tokens":1846,"prompt_tokens":805,"completion_tokens":1041,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":964}},"tokens_in":549,"tokens_out":1041,"duration_ms":8468,"temperature":1.0,"reasoning_tokens":964,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:46:15.584899+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same collapse-threshold calculation with a different orthonormal basis of residual radial modes—for example, include the second and third radial directions in the monopole sector, or construct a radial mode from a different spectral split—and check whether the integrated PBH abundance enhancement at the strongest negative-non-Gaussianity case remains near 10^5 or collapses back to order one. If the enhancement is not robust to basis choice, the dominant-branch mechanism is an artifact of the chosen deformation family. Alternatively, a non-spherical (3+1) simulation of the dominant low-","supporting_citations":[],"review_version":2}