Non-Gaussianity and Alternative Formation Channels

Enhanced, nearly Gaussian curvature perturbations collapsing during radiation domination are only one route to primordial black holes. Non-Gaussian tails can alter how often rare peaks occur, while an early matter era, a phase transition, a defect, or scalar fragmentation can alter both the collapse dynamics and characteristic mass. The mechanisms can be compared meaningfully only after fixing the expansion history and asking what correlated relics accompany the black holes.

Helpful background. Einstein Equations supplies gravitational collapse criteria; Relativistic Matter and Compact Objects supplies equations of state and stresses; Black-Hole Dynamics and Collapse supplies nonlinear endpoints; Relativistic Cosmology supplies primordial fluctuations and thermal history; and Probability and Statistical Inference supplies rare-tail and population reasoning.

Non-Gaussianity changes the rare tail

A familiar local model writes the primordial superhorizon curvature $\mathcal R\simeq\zeta$ as a nonlinear function of a Gaussian field,

$$\mathcal R=\mathcal R_G+\frac35 f_{\rm NL} \left(\mathcal R_G^2-\langle\mathcal R_G^2\rangle\right)+\cdots.$$

For ordinary fluctuations the correction may be perturbatively small. PBH formation instead samples curvature amplitudes of order unity, many standard deviations into the distribution. The relevant probability is an integral over that far tail, so a truncated cumulant expansion can fail precisely where it is used. Ultra-slow-roll or stochastic-inflation models can generate exponential or heavier tails, but “stochastic inflation” alone does not fix one distribution: the quantum state, coarse-graining window and scale, phase-space prescription, and first-passage or end boundary must be declared. The resulting tail is an imported model output, not a universal consequence of stochastic dynamics. Escrivà, Kühnel, and Tada 2025, § II.J, printed pp. 64–79 show why the nonlinear map and nonperturbative tail must be propagated into the peak calculation.

Non-Gaussianity also changes clustering. A long mode that modulates small-scale collapse can correlate rare PBH sites even when the underlying narrow Gaussian spectrum would give little clustering beyond discreteness. Therefore abundance, spatial statistics, and induced gravitational waves must be predicted from the same stochastic model; tuning only the one-point tail is not a complete scenario.

The background expansion changes collapse

During radiation domination, pressure gradients oppose collapse and produce a nonzero profile-dependent threshold. During an early pressureless era, density perturbations grow and pressure no longer supplies the same barrier. A concrete calculation using the Zel’dovich approximation, a hoop criterion, and the Doroshkevich distribution finds, for the horizon-entry rms linear density contrast $\sigma_H\lesssim0.01$,

$$\beta_0\simeq0.056\,\sigma_H^5.$$

Here $\beta_0$ is a production probability, not automatically the later energy fraction. The power law replaces the radiation-era Gaussian exponential only inside the calculation’s pressureless, sufficiently long matter-era, Gaussian-deformation, and nonspherical-collapse assumptions. Finite duration, angular momentum, central caustics, reheating dilution, and the requirement that collapse complete before reheating can suppress it further. Harada et al. 2016, §§ III.2–III.3, Eqs. (37)–(40) derive the result; Green and Kavanagh 2021, § II.B, printed p. 6 summarize its cosmological boundary.

The duration condition can be estimated before attempting a collapse calculation. In a pressureless growing mode, $\delta\propto a$ and $a\propto t^{2/3}$. An initial horizon-entry contrast $\delta_H\ll1$ reaches order unity at

$$\frac{a_{\rm nl}}{a_H}\sim\delta_H^{-1}, \qquad \frac{t_{\rm nl}}{t_H}\sim\delta_H^{-3/2}.$$

Thus reheating after $t_H$ is not enough: the matter era must last at least until this nonlinear time, and usually longer to complete collapse. This is a necessary growth estimate, not a sufficient black-hole criterion; a nonspherical region can form a caustic or bound structure instead.

A temporary softening of the equation of state near a thermal transition is a related but distinct effect. It lowers pressure support for modes entering during a limited interval and can imprint features in the mass function. The result depends on the time-dependent $w(T)$ and sound speed, not merely on naming the QCD or another transition.

A first-order transition can additionally nucleate bubbles. In one explicit supercooled collision model, a bubble of radius $R$ with collision efficiency $\kappa_{\rm coll}$ and overlap solid angle $\Delta\Omega$ must satisfy

$$1\lesssim\kappa_{\rm coll}HR \sqrt{\frac{\Delta\Omega}{4\pi}}, \qquad M_{\rm PBH}\approx \frac{8\pi}{3\kappa_{\rm coll}}H^{-3}\Delta V,$$

with $H^2=8\pi G\Delta V/3$. Thus the relevant bubbles are at least Hubble-sized, must carry a macroscopically thick fluid shell, and are born much earlier than typical nucleation; the predicted narrow mass scale and abundance are exponentially sensitive to the early-time nucleation history. These are model conditions, not a generic result for every first-order transition Jung and Okui 2024, §§ II.B–II.E, Eqs. (12)–(13), and § III, Eqs. (14), (31)–(33).

Defects, walls, bubbles, and condensates

Alternative channels should be compared through outputs rather than labels. Fix one background history $H(a)$, the reference Hubble mass $M_H=(2GH)^{-1}$, and the start, end, and reheating conditions; each mechanism must then supply its own source scale, collapse probability, mass distribution, and correlated relics.

Mechanism	Background and characteristic mass	Abundance control, relic, or failure mode
Early matter domination	Hubble masses for modes entering early enough to collapse before reheating	$\beta_0\simeq0.056\sigma_H^5$ only for the stated small-$\sigma_H$ pressureless model; spin, nonsphericity, duration, and dilution alter it
First-order transition and bubble collisions	Transition scale and rare $R\gtrsim H^{-1}$ bubbles; one controlled model predicts a narrow threshold mass	Thick-shell dynamics, early nucleation, percolation, and gravitational waves must be computed together
Cosmic-string loops	$M\sim\mu L$ for a loop of invariant length $L$	Collapse requires exceptional loop shapes; the loop distribution and stochastic string-wave background constrain the same network
Cosmic-string cusps	A generic cusp is not an independent Hubble-mass channel	A rest-frame hoop analysis finds no significant collapse for realistic $G\mu$ without extreme fine-tuning; do not combine cusps with rare whole-loop collapse
Closed domain walls	Wall tension and enclosed radius can yield broad and clustered masses	Wall domination, topology, and collapse efficiency depend on the underlying field model
Spectator or inflaton condensate fragmentation	Oscillon, Q-ball, or lump clusters can be well below the Hubble radius during a temporary matter era	The potential must permit attractive instability and long-lived lumps; fragmentation statistics, decay, mergers, and collapse require model-specific nonlinear control

These channels do not inherit the density-peak constraint map automatically. For example, a limit on a primordial curvature spectrum from induced gravitational waves applies directly only when that spectrum produced the PBHs; it does not exclude a defect-created population with the same mass. Conversely, defects and phase transitions bring their own correlated radiation, relic, and consistency constraints.

String cusps provide a useful negative control. The invariant mass of a candidate cusp segment, rather than its boosted laboratory energy, must enter a hoop test. For realistic $G\mu$, the resulting condition can be met only by extreme alignment; the expected fraction is suppressed at least parametrically as $(G\mu)^2$ in that analysis Blanco-Pillado, Olum, and Vilenkin 2021, printed pp. 2–3, Eqs. (9)–(11). Scalar fragmentation supplies a different primary construction: attractive self-interactions fragment an inflaton or spectator condensate into long-lived lumps that redshift as matter, and PBHs form only if the lumps live long enough to dominate and cluster Cotner et al. 2019, §§ 2.1–2.3, printed pp. 4–8, and § 3, printed pp. 11–24. Broader channel assumptions are reviewed in Green and Kavanagh 2021, § II.D, printed pp. 10–11 and Escrivà, Kühnel, and Tada 2025, §§ II.K–II.N, printed pp. 79–93.

Results and limitations through 2026-08-19

As of 2026-08-19, none of these mechanisms supplies a model-independent PBH abundance or characteristic mass. The status remains unresolved at the level needed to infer a channel from a black-hole population alone.

Formulation or evidence	Result	Scope and limitation
Non-Gaussian inflationary tails	Can enhance or suppress PBH abundance by many orders of magnitude at fixed two-point power	The state, coarse graining, far-tail distribution, profile statistics, and perturbative truncation must be controlled
Harada et al. pressureless calculation	Gives $\beta_0\simeq0.056\sigma_H^5$ for $\sigma_H\lesssim0.01$	Zel’dovich, hoop, Gaussian deformation, long-duration, spin-free assumptions delimit the result
Jung–Okui bubble-collision model	Requires rare Hubble-sized bubbles with thick fluid shells and predicts a narrow threshold mass	Curved-spacetime collision dynamics and exponentially sensitive early nucleation remain uncalibrated
Blanco-Pillado–Olum–Vilenkin cusp null	Finds no significant black-hole production from realistic string cusps	It is a negative result for cusp segments, not a proof that specially shaped whole loops never collapse
Cotner et al. fragmentation framework	Connects spectator or inflaton fragmentation, temporary matter domination, and subhorizon lump clusters	Potential, fragmentation, lump lifetime, occupation statistics, and nonlinear collapse are model-specific
Correlated relic searches	Gravitational waves, spectral distortions, or spatial clustering can discriminate mechanisms	Null results constrain only the joint source model and transfer history used to predict that relic

The competing positions concern whether enhanced non-Gaussian perturbations, matter-era growth, phase transitions, cosmic strings or domain walls, bubble collisions, spectator fields, and fragmentation can yield a substantial population without violating their correlated constraints. A common expansion history, probability measure, mass definition, and selection model are required before their predictions can be ranked.

The interpretation could change after a controlled relativistic formation simulation calibrates one channel’s mass and spin distribution together with its relics, or after an observed correlated signature separates that channel from ordinary density-peak collapse under a jointly specified likelihood.

Diagnose a tail approximation

Two toy models have the same characteristic width $0.1$ and superhorizon-curvature threshold $\mathcal R_c=1$. One estimates the tail by $e^{-\mathcal R_c^2/(2\sigma^2)}$; the other by $e^{-\mathcal R_c/\lambda}$ with $\sigma=\lambda=0.1$. How different are their leading suppressions?

Solution

The Gaussian leading factor is $e^{-50}$, whereas the exponential-tail factor is $e^{-10}$. Their ratio is

$$\frac{e^{-10}}{e^{-50}}=e^{40}\simeq2.35\times10^{17}.$$

Matching a nominal width does not match a rare tail. Normalization and profile effects would change the prefactor, but not the lesson: one cannot infer PBH abundance from the two-point variance without specifying and validating the tail model.

Scope boundary

This comparison does not build a UV-complete particle model or declare any channel realized in nature. It provides the axes needed to test proposals. Detailed inflation, defect evolution, phase-transition microphysics, and gravitational-wave transfer remain with their dedicated treatments.

References

• Blanco-Pillado, Jose J., Ken D. Olum, and Alexander Vilenkin. 2021. “No Black Holes from Cosmic String Cusps.” arXiv:2101.05040v1. Open PDF.
• Cotner, Eric, Alexander Kusenko, Misao Sasaki, and Volodymyr Takhistov. 2019. “Analytic Description of Primordial Black Hole Formation from Scalar Field Fragmentation.” Journal of Cosmology and Astroparticle Physics 10: 077. DOI.
• Escrivà, Albert, Florian Kühnel, and Yuichiro Tada. 2025. “Primordial Black Holes.” arXiv:2211.05767v5. Open PDF.
• Green, Anne M., and Bradley J. Kavanagh. 2021. “Primordial Black Holes as a Dark Matter Candidate.” Journal of Physics G 48: 043001. DOI.
• Harada, Tomohiro, Chul-Moon Yoo, Kazunori Kohri, Ken-ichi Nakao, and Sanjay Jhingan. 2016. “Primordial Black Hole Formation in the Matter-Dominated Phase of the Universe.” The Astrophysical Journal 833: 61. DOI.
• Jung, Tae Hyun, and Takemichi Okui. 2024. “Primordial Black Holes from Bubble Collisions during a First-Order Phase Transition.” Physical Review D 110: 115014. DOI.