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Heavy non-Gaussian tails can produce supermassive primordial black-hole seeds without violating the CMB μ-distortion bound that kills Gaussian seeds.

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2026-07-14 16:34 UTC pith:GIRMITUL

load-bearing objection Clean variance-capped comparison: standard single-field exponential tails still fail for SMBH seeds; only algebraic and heavy log-normal reopen the window, with the usual μ caveats for strong NG. the 3 major comments →

arxiv 2607.03138 v2 pith:GIRMITUL submitted 2026-07-03 astro-ph.CO gr-qchep-phhep-th

Evading the CMB μ-distortion bound on Supermassive Primordial Black Hole seeds with Non-Gaussian tails

Sanket Dave , Sheng-Feng Yan , Amara Ilyas , Yi-Fu Cai This is my paper
classification astro-ph.CO gr-qchep-phhep-th
keywords primordial black holesCMB μ-distortionnon-Gaussian tailsδN formalismsupermassive black hole seedsfractional potentiallog-normal tailsGaussian barrier
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

High-redshift quasars need black-hole seeds of roughly 10^5–10^7 solar masses, a range that under Gaussian statistics is ruled out by the COBE/FIRAS limit on CMB μ-distortion. That limit caps the small-scale curvature variance at σ_ζ^{2} ≲ 10^{-4}; for a Gaussian the same variance fixes the far tail, so the collapse probability is negligible—the Gaussian barrier. The paper shows the barrier is not fundamental: if the one-point PDF has a sufficiently heavy non-Gaussian tail, the variance that sets the distortion can be decoupled from the tail probability that controls collapse. Working non-perturbatively with the δN map, the authors standardize four Gaussian-cored families to unit variance, impose the FIRAS cap, and map the resulting seed abundance across tail-shape parameters. Ordinary exponential tails from standard single-field non-attractor dynamics remain too light; algebraic tails from fractional-potential dynamics and sufficiently heavy log-normal tails can still supply seed-relevant abundances. A reader who cares about the origin of early supermassive black holes therefore has a concrete, observationally bounded target: only a narrow class of early-universe dynamics survives.

Core claim

After each Gaussian-cored family is standardized to unit variance and the FIRAS μ-cap is imposed, the ordinary exponential tail generic to single-field non-attractor dynamics is still too light to reopen the 10^5–10^7 M_⊙ seed window, whereas algebraic tails (D_∞ = 0) from fractional-potential dynamics and sufficiently heavy log-normal tails can yield seed-relevant PBH abundances while respecting the distortion bound.

What carries the argument

The standardized collapse integral β_max(M; θ) = 2 S(u_c(M); θ), with u = ζ/σ_ζ. By locking every family to unit variance the FIRAS bound saturates σ_ζ while the asymptotic shape of the δN map alone controls the far-tail probability that sets the PBH abundance, thereby decoupling the second-moment distortion observable from the deep-tail collapse criterion.

Load-bearing premise

The ordinary acoustic-damping formula that turns variance into a μ-distortion still supplies a trustworthy absolute cap even for the strongly non-Gaussian heavy tails that reopen the seed window.

What would settle it

An explicit δN construction that realises a power-law or heavy log-normal tail at the FIRAS-capped variance, evaluated with the full spectral integral for μ including non-Gaussian corrections to the dissipated energy, and shown either to reach or to fall short of β ~ 10^{-15} across 10^5–10^7 M_⊙.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Standard single-field ultra-slow-roll or upward-step models (exponential tails) cannot seed the observed high-z SMBHs under the FIRAS bound.
  • Fractional inflaton potentials with 2 < m < 2.5 reopen a viable algebraic-tail window for seed-mass PBHs.
  • Multiplicative or multifield dynamics that generate sufficiently heavy log-normal tails can also supply seed-relevant abundances.
  • A PIXIE-level μ measurement would push the required tail weight deeper into the heavy regime and further restrict viable dynamics.
  • The same scales source a second-order stochastic gravitational-wave background that can independently test these non-Gaussian tails.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Radiatively corrected inflection points and non-canonical (DBI-type) kinetic sectors become the natural single-field targets if algebraic tails are required.
  • Refining the collapse criterion from a fixed ζ threshold to a compaction-function treatment will change absolute abundances but is unlikely to reverse the ordering of exponential versus power-law families.
  • A non-detection of μ by next-generation spectrometers together with a seed-mass SGWB detection would strongly favour D_∞ = 0 tails over any exponential construction.
  • The concurrent self-interacting-curvaton mechanism noted in the paper is one concrete multifield realisation of the multiplicative class the taxonomy flags as viable.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper argues that the COBE/FIRAS μ-distortion bound, which caps the small-scale curvature variance at σ_ζ^{2} ≲ 10^{-4} over the 10^5–10^7 M_⊙ PBH seed window, creates a 'Gaussian barrier': under Gaussian statistics the same variance fixes the far tail of the one-point PDF and drives the formation fraction β far below any seed-relevant level. The authors implement a variance-capped, standardized collapse integral β_max = 2 S(u_c; θ) with u_c = ζ_c/σ_ζ,max in the non-perturbative δN formalism, organize admissible asymptotic tails via a δN map dictionary, and scan four Gaussian-cored families (generalized-normal, stretched-exponential, power-law, log-normal). Their main result is that the ordinary exponential tail (p = 1) generic to single-field non-attractor dynamics remains too light to reopen the seed window, whereas algebraic tails (D_∞ = 0) from fractional-potential dynamics and sufficiently heavy log-normal tails (treated as a phenomenological proxy for multiplicative dynamics) can yield seed-relevant β_max while saturating the FIRAS cap.

Significance. If the ordering of tail shapes under a fixed variance cap holds, the work cleanly separates a robust negative result (standard single-field exponential tails do not seed SMBHs under FIRAS) from a more provisional positive window (fractional-potential algebraic tails and heavy multiplicative tails). The standardized collapse integral and the δN tail dictionary provide a reusable, model-agnostic language that connects the tail taxonomy of Hooshangi et al. and Pi–Sasaki to spectral-distortion analyses, and the parameter-space maps in Fig. 6 make the seed-relevant region falsifiable once a concrete map is specified. The negative p = 1 result is particularly useful for the community, as it rules out a large class of existing non-attractor constructions for this mass window without requiring a full model-by-model scan.

major comments (3)
  1. Sec. II, Eqs. (4)–(6) and the claim that non-Gaussian corrections to dissipated energy 'can rescale σ_ζ,max by a common factor but cannot change the ordering of tail shapes': this assertion is least secure precisely for the D_∞ = 0 families that carry the positive result. Refs. [79–82] are cited for mild NG; the paper itself notes that corrections are 'model dependent only in the strongly non-Gaussian regime' yet retains the leading-order μ formula for all families, including power-law and log-normal at u_c ∼ 50–100. If the effective variance cap tightens more for heavy tails than for exponential ones, the reopening of the seed window in Figs. 4–6 and Sec. IV A can shrink while the negative p = 1 result remains. A quantitative bound (or an explicit shape-dependent correction estimate) for the heavy-tail regime, or a clear scoping of the positive claim as conditional on leading-order μ, i
  2. Sec. III B 4 and Sec. IV A: Family D (asymmetric log-normal) is openly phenomenological and lacks a derived single-field δN map, yet the abstract and Sec. IV A present 'sufficiently heavy log-normal tails' on equal footing with the fractional-potential algebraic tails of Family C (which do have a concrete potential, Eq. (26)). The seed-relevant floor for s ≳ 0.5 in Fig. 5 is therefore an existence statement about a PDF shape, not a dynamical prediction. The manuscript should either (i) demote Family D more clearly to a diagnostic upper envelope in the abstract and results, or (ii) supply at least one explicit multifield/multiplicative construction that realizes the map (29) at the required amplitude, so that the positive claim is not carried by an unanchored proxy.
  3. Sec. II and Sec. IV (narrow-feature baseline): the entire β_max(M) analysis saturates μ ≃ 2.2 σ_ζ^{2} W_μ(M) for a narrow or sharply peaked spectrum. For the fractional-potential and multiplicative dynamics that are supposed to generate the viable tails, the spectrum need not be narrow; a broad enhancement would replace Eq. (4) by the full integral (3) and shift σ_ζ,max by an O(1) mass-dependent factor. Because u_c already sits at 50–100, even an O(1) change in the cap moves β_max by many orders of magnitude for stretched and power-law families. The paper defers this to Sec. V; for the seed-window claims it should either recompute β_max with a representative broad feature or show that the viable (q, x_×) and (p, x_×) regions of Fig. 6 remain non-empty under a conservative O(1) degradation of the cap.
minor comments (5)
  1. Sec. III B 1 / Family A: the symmetric generalized-normal is used only as a diagnostic, which is appropriate, but Fig. 2 is the only place it appears in the results. A short sentence in Sec. IV A stating that Family A is not used for the seed-window conclusion would avoid any impression that symmetric heavy tails are being advocated as physical.
  2. Eq. (8) and Table II: the conversion from β to f_PBH uses γ ≃ 0.2 and g_* = 10.75; these are standard but should be stated once in the table caption so that the one-seed-per-quasar numbers are reproducible without hunting through the text.
  3. Fig. 1(b) and Figs. 3–5: the standardized PDFs are plotted to u = 100, which is helpful, but the vertical u_c lines for 10^5, 10^6, 10^7 M_⊙ are only labeled in some panels. Adding them consistently (or a shared legend) would make the collapse sampling of the far tail immediately visible.
  4. Sec. V, note added: the concurrent curvaton paper [146] is acknowledged; a one-sentence comparison of which tail shape that mechanism realizes (exponential vs algebraic vs log-normal) would help the reader place the two works relative to the dictionary in Table I.
  5. Typographical: abstract and title use both 'μ-distortion' and 'CMBµ-distortion' (mixed macro); Sec. I 'Salpeter time is independent of the black hole mass' is fine but the displayed t_s formula could note the ϵ/(1-ϵ) convention explicitly for non-specialists.

Circularity Check

0 steps flagged

No significant circularity: abundances follow from externally capped, standardized PDF tails scanned over free shape parameters.

full rationale

The derivation chain is: (i) FIRAS supplies an external variance cap σ²_ζ,max(M) via the leading acoustic-damping formula; (ii) each family is standardized to unit variance so that shape parameters alone control the far-tail integral S(u_c); (iii) β_max = 2 S(u_c) is evaluated and compared to an external seed bookkeeping requirement. Shape parameters (p, q, s, x×) are scanned, not fitted to force a target β; the result that D_∞=0 tails can clear the seed window while p=1 cannot is a numerical consequence of those integrals, not a definitional identity. The δN tail dictionary cites independent literature (Pi–Sasaki logarithmic duality; Hooshangi et al. for algebraic tails). Family D is explicitly labeled phenomenological rather than derived, which is a scope limitation, not circularity. Related-author inflation/PBH citations appear only as background mechanisms and do not define or force the abundance ordering. The skeptic concern about non-Gaussian corrections to μ for heavy tails is a correctness/assumption risk, not a reduction of the claim to its inputs by construction. No equation equates a claimed prediction to a fitted input or to a self-citation uniqueness theorem.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 2 invented entities

The central claim rests on standard cosmological tools (δN, Press–Schechter, FIRAS μ formula, PBH mass–scale relation) plus phenomenological PDF families standardized by hand. Free parameters are the tail-shape and transition parameters scanned to map viable regions, plus conventional thresholds (ζ_c, γ, μ_lim). No new particles or forces; the ‘invented’ objects are the four Gaussian-cored standardized families used as representatives of the δN dictionary, with Family D explicitly provisional.

free parameters (5)
  • stretched-exponential index p and transition x× (Family B)
    Scanned shape parameters controlling asymptotic tail weight and onset; p=1 is theoretically preferred for single-field non-attractor, but viable seed abundances require p≃0.6 and early onset.
  • power-law index q and transition x× (Family C)
    q>2 imposed for finite variance; q and x× set the nearly mass-independent β floor. Linked to potential exponent m=2+1/q but not derived from a full model spectrum.
  • log-normal shape s (Family D)
    Single free shape after standardization; seed-relevant abundances require s≳0.5. No derived δN map fixes s.
  • collapse threshold ζ_c ≃ 0.67
    Fiducial fixed threshold in Press–Schechter integral; absolute β is sensitive to this choice though family ordering is not.
  • collapse mass fraction γ ≃ 0.2 and FIRAS μ_lim = 9×10^{-5}
    Standard literature inputs that set M(k) and the absolute variance cap σ_ζ,max(M).
axioms (6)
  • domain assumption Leading-order μ-distortion is controlled by the curvature power spectrum / variance via the narrow-feature formula μ≃2.2 σ_ζ² W_μ(M).
    Sec. II Eqs. (3)–(6); non-Gaussian corrections acknowledged but not included in the cap.
  • domain assumption PBH abundance is given by the Press–Schechter tail integral β≃2∫_{ζ_c}^∞ P_ζ dζ with fixed ζ_c.
    Sec. II Eq. (7); used as proxy for compaction-function treatment throughout.
  • domain assumption On super-horizon scales ζ=δN(δφ,δπ) with Gaussian field fluctuations, so PDF tails are fixed by the global δN map.
    Sec. III A–B; organizes Table I dictionary.
  • ad hoc to paper Finite-variance algebraic tails require q>2 (fractional potential 2<m<2.5) and can be matched C¹ to a Gaussian core.
    Family C construction Eqs. (25)–(28); classical δN only, diffusion excluded.
  • ad hoc to paper A log-normal tail is an admissible phenomenological proxy for multiplicative/multifield dynamics even without a derived single-field δN map.
    Sec. III B 4 and Sec. V; Family D treated as indicative of the heaviest regime.
  • standard math Standardization to zero mean and unit variance before applying the FIRAS cap correctly decouples amplitude from tail shape.
    Sec. II–III C Eqs. (10)–(14), (32)–(33).
invented entities (2)
  • Four standardized Gaussian-cored tail families (GN, stretched-exponential, power-law, asymmetric log-normal) as variance-capped representatives no independent evidence
    purpose: Provide a finite taxonomy spanning the δN tail dictionary so distortion-limited β_max can be mapped in shape-parameter space.
    Families A–C are motivated by existing maps; Family D is explicitly phenomenological. They are modeling constructs, not new physical fields.
  • Gaussian barrier (named obstruction) independent evidence
    purpose: Label the statement that under Gaussian statistics the FIRAS variance cap forces negligible seed-mass β.
    Naming device for a known Gaussian-tail consequence; not a new physical entity.

pith-pipeline@v1.1.0-grok45 · 27240 in / 4033 out tokens · 43941 ms · 2026-07-14T16:34:50.064834+00:00 · methodology

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read the original abstract

Supermassive black holes (SMBHs) powering quasars at $z \gtrsim 6$ are difficult to grow from stellar mass remnants, motivating seeds from primordial black holes (PBHs) with masses $10^5-10^7 M_{\odot}$. This range is constrained by the COBE/FIRAS bound on the CMB $\mu$-distortion, which limits the small-scale curvature variance to $\sigma_\zeta^2 \lesssim 10^{-4}$. For Gaussian perturbations, the variance fixes the far tail of the one-point probability distribution function (PDF), making the PBH abundance negligible. We call this the Gaussian barrier. The barrier can be evaded only if the variance probed by the distortion is decoupled from the tail probability controlling collapse. We implement this idea in the non-perturbative $\delta N$ formalism and relate asymptotic PDF tails to the global shape of the $\delta N$ map. Four Gaussian-cored families are analyzed: generalized-normal, stretched-exponential, power-law, and log-normal tails. After standardizing each family to unit variance, we impose the FIRAS cap and compute the distortion-limited PBH abundance in the tail-shape parameter space. The ordinary exponential tail produced by standard single-field non-attractor dynamics is still too light to reopen the seed window. Algebraic tails from fractional-potential dynamics, and sufficiently heavy log-normal tails treated as a phenomenological proxy for multiplicative dynamics, can supply seed-relevant abundances while respecting the distortion bound.

Figures

Figures reproduced from arXiv: 2607.03138 by Amara Ilyas, Sanket Dave, Sheng-Feng Yan, Yi-Fu Cai.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Family B (Gaussian core with stretched-exponential tail). [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Family C (Gaussian core with power-law tail). [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Family D (asymmetric log-normal). [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Distortion-capped abundance for the stretched-exponential (left) Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

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