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

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 $10^5-10⁷ M_{\odot}$. This range is constrained by the COBE/FIRAS bound on the CMB $μ$-distortion, which limits the small-scale curvature variance to $σ_ζ² \lesssim 10<sup>{-4}$. 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 $δN$ formalism and relate asymptotic PDF tails to the global shape of the $δN$ 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.

• The paper shows that non-perturbative heavy-tailed non-Gaussianities enable sufficient primordial black hole formation to seed supermassive black holes while obeying CMB μ-distortion limits.
• It employs the δN formalism to derive and classify tail behaviors across various distributions, including stretched-exponential, power-law, and log-normal families.
• The analysis reveals that only genuinely non-Gaussian mechanisms with sub-exponential tails can bypass the Gaussian barrier, imposing novel constraints on inflationary models.

Evading the CMB $\mu$-Distortion Bound on Supermassive PBH Seeds with Non-Gaussian Tails

Introduction

The identification of supermassive black holes (SMBHs) at redshifts $z\gtrsim6$, with masses $\sim10^8$–$10^{10} M_\odot$, challenges canonical astrophysical formation scenarios due to the stringent requirements for early assembly within the first billion years. Standard Eddington-limited accretion onto stellar-mass seeds struggles to explain both the mass and number density of these SMBHs, motivating alternative paths such as primordial black hole (PBH) seeds in the range $10^{5}$–$10^{7} M_\odot$. However, the PBH abundance in this mass regime is tightly constrained by the COBE/FIRAS limits on CMB $\mu$-distortion, which translate into severe bounds on the amplitude of small-scale curvature perturbations. For Gaussian initial fluctuations, the so-called "Gaussian barrier" arises: the variance constraint simultaneously suppresses the tail of the probability distribution vital for PBH formation to negligible levels. This work systematically investigates whether non-Gaussian statistics—specifically, heavy tails generated through non-perturbative dynamics—can reopen the seed window while respecting the $\mu$-distortion bound.

The Gaussian Barrier and the Variance–Tail Connection

The spectral distortion constraint is formulated as an upper bound on the squared variance $\sigma_\zeta^2$ of curvature perturbations at seed masses via

$$\sigma_{\zeta,\max}^2(M) = \frac{\mu_{\mathrm{lim}}}{2.2\,W_\mu(M)}$$

where $z\gtrsim6$0 is a window function selecting the relevant comoving scales. Over $z\gtrsim6$1–$z\gtrsim6$2, the cap is $z\gtrsim6$3, corresponding to standardized thresholds $z\gtrsim6$4–$z\gtrsim6$5 for PBH collapse. For Gaussian statistics, the PBH formation probability $z\gtrsim6$6 becomes exponentially suppressed, rendering the PBH abundance unviable for seeding purposes.

Figure 1: (a) Maximum variance $z\gtrsim6$7 allowed by the FIRAS bound in the seed window; (b) Standardized PDFs of candidate non-Gaussian families highlight the relative weight in the formation-relevant tail.

This "Gaussian barrier" is a direct consequence of the fixed power–tail relationship in the Gaussian ensemble, not a limit on heavy-tail generations by genuinely non-Gaussian mechanisms.

Non-Perturbative Tail Dynamics and the $z\gtrsim6$8 Formalism

PBH formation is a tail-sensitive observable: the collapse probability depends on events tens of standard deviations away from the bulk. This precludes perturbative approaches based on lowest-order non-Gaussianity parameters (e.g., $z\gtrsim6$9), which cannot modulate the far tail independently of the variance. Instead, the non-perturbative $\sim10^8$0 formalism is employed: the PDF of $\sim10^8$1 is determined by the exact nonlinear $\sim10^8$2 map between Gaussian field fluctuations $\sim10^8$3 and curvature perturbations,

$\sim10^8$4

leading to tail features controlled by the structure of $\sim10^8$5. The asymptotic tail behavior is classified by the local log-slope $\sim10^8$6, distinguishing exponential ($\sim10^8$7) and "heavy" ($\sim10^8$8) sub-exponential or power-law tails.

Four Gaussian-cored families are systematically constructed based on the non-perturbative $\sim10^8$9 map:

• A. Generalized Normal: Symmetric, minimal deformation of Gaussian, demonstrates impact of tail index $10^{10} M_\odot$0.
• B. Gaussian/Stretched-Exponential: One-sided, as realized in single-field non-attractor transitions.
• C. Gaussian/Power-Law: Arises from $10^{10} M_\odot$1 maps with finite-field singularities, associated with fractional-potential models.
• D. Asymmetric Log-Normal: Phenomenological proxy for multiplicative mechanisms (e.g., multifield modulations).

These families are engineered to match the observed null CMB $10^{10} M_\odot$2-distortion while possessing flexible, non-Gaussian tails.

Tail Families: Structure and Abundance

Symmetric Generalized Normal Family

Empirically, the generalized normal (GN) distribution starkly illustrates how reducing the tail decay index $10^{10} M_\odot$3—even without changing core variance—can exponentially enhance the PBH formation probability at fixed variance. However, symmetric GN is unphysical as it also enhances voids, which are not associated with black hole formation.

Figure 2: $10^{10} M_\odot$4-distortion versus PBH mass fraction $10^{10} M_\odot$5 for generalized-normal tail; decreasing $10^{10} M_\odot$6 boosts $10^{10} M_\odot$7 significantly within the FIRAS limit.

Stretched-Exponential (Single-Sided) Family and Physical Models

The physically motivated stretched-exponential family captures the output of single-field inflation with transient non-attractor dynamics. For all canonical single-field dynamics, the tail never exceeds the exponential form ($10^{10} M_\odot$8). Even with maximal flattening ($10^{10} M_\odot$9), the spectral distortion cap ensures $10^{5}$0 remains orders of magnitude below the seed window. Only strictly sub-exponential cases ($10^{5}$1) enable seed-relevant abundances, but these require nonstandard, non-perturbative inflationary mechanisms.

Figure 3: Family B: PDFs and $10^{5}$2 for the matched stretched-exponential model. The exponential tail ($10^{5}$3) is insufficient; sub-exponential tails are necessary.

Power-Law (Algebraic) Tail Family

The heaviest physical tails derive from non-attractor classical inflationary dynamics on potentials with fractional power corrections. These yield PBH abundances that are almost insensitive to changes in the seed mass, since algebraic decay lacks exponential suppression. Viable power-law models correspond to potential corrections of the form $10^{5}$4 with $10^{5}$5.

Figure 4: Family C: PDFs and $10^{5}$6 for Gaussian core with power-law tail; even moderate indices ($10^{5}$7) yield substantial seed abundances for $10^{5}$8.

Asymmetric Log-Normal Family

The log-normal family represents dynamics where the curvature perturbation is built multiplicatively, possibly realized in multifield or weighted measures of expansion. This family yields the heaviest tails: for moderate shape parameters ($10^{5}$9), the abundance easily exceeds the seed threshold. However, concrete single-field models producing such tails are not provided, so this case is treated as phenomenologically motivated.

Figure 5: Family D: PDFs and corresponding $10^{7} M_\odot$0 for standardized log-normal tails. Multiplicative amplification of tail events ensure strong survival of seed-forming probability.

Parameter Space and Abundance Mapping

A comprehensive scan shows the abundance is controlled almost exclusively by asymptotic tail properties rather than core features, as variance is already capped by FIRAS. In the power-law and log-normal parameterizations, the dependence on the mass scale nearly disappears for physical indices over $10^{7} M_\odot$1–$10^{7} M_\odot$2, with heavy-tail families achieving $10^{7} M_\odot$3 required for seeding.

Figure 6: Distortion-capped PBH abundance in tail index parameter space, mapping contours of present-day $10^{7} M_\odot$4 at fixed seed mass.

Theoretical and Astrophysical Implications

Astrophysical Requirements: The minimum seeding fraction merely requires one PBH per high-$10^{7} M_\odot$5 quasar $10^{7} M_\odot$6, whereas a more ambitious scenario replaces the entire SMBH population $10^{7} M_\odot$7. The distortion-evading heavy-tail mechanisms described are easily compatible with these requirements and avoid robust exclusions from current dynamical, accretion, and lensing constraints.

Inflationary Model Constraints: The results delineate sharp boundaries for viable early-universe mechanisms:

• Single-field ultra-slow-roll or sharp potential features: Exponential tails too light; cannot evade the Gaussian barrier even with maximal enhancement.
• Fractional-potential and non-canonical kinetic sectors: Capable of power-law tails that can evade all current and projected distortion limits.
• Multiplicative/multifield scenarios: The heaviest viable tails, but require explicit model construction.

The analysis demonstrates that only genuinely non-Gaussian (heavy-tail) mechanisms with $10^{7} M_\odot$8 can produce sufficient seeds under the spectral-distortion cap. This insight highlights the need for precise, non-perturbative modeling of curvature perturbation statistics in inflationary cosmology, especially when connecting PBH constraints to inflationary model-building.

Future Directions

Several avenues for extension are identified:

• Model Realizations: Explicit construction of inflationary setups (e.g., Dirac–Born–Infeld, sound-speed resonance, multifield, or curvaton models) that dynamically realize power-law or log-normal tails with sufficient amplitude.
• Non-Gaussian Dissipation Corrections: Calculation of corrections to the energy injection ($10^{7} M_\odot$9) from higher-order correlators for strongly non-Gaussian statistics.
• Improved Collapse Criteria: Incorporation of realistic compaction functions and critical collapse thresholds, especially with nonlinearity between $\mu$0 and the density contrast.
• Next-Generation Distortion Probes: Anticipation of future improvements from experiments such as PIXIE, which can lower the $\mu$1 limit, tightening the allowed region for heavy-tail models.

Conclusion

When the tail of the curvature perturbation PDF is decoupled from its variance, PBH formation for supermassive seeds can evade the CMB $\mu$2-distortion bound. However, the underlying inflationary mechanism must go beyond both perturbative non-Gaussianity and the standard single-field non-attractor constructions. The possibility of seeding the first SMBHs with PBHs is not excluded by present distortion data—but only if the inflationary small-scale fluctuations are dominated by genuinely non-Gaussian, heavy-tailed statistics. This imposes strong theoretical constraints on viable early-universe models and foregrounds the necessity of full, non-perturbative one-point statistics in connecting CMB observables to PBH-based inflationary constraints.