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Rare profile shapes can dominate primordial black hole production

2026-07-10 02:01 UTC pith:IAVIU3H6

load-bearing objection Finite-action framework for PBH profile dispersion is well-built, but orders-of-magnitude abundance claims rest on a single shape direction the 3 major comments →

arxiv 2607.08738 v1 pith:IAVIU3H6 submitted 2026-07-09 astro-ph.CO gr-qchep-phhep-th

The statistics of curvature-profile dispersion in primordial black hole formation

Albert Escriv\`a This is my paper
classification astro-ph.CO gr-qchep-phhep-th PACS 98.80.Cq
keywords primordial black holescurvature perturbationpeak theoryprofile dispersioncollapse thresholdcompaction functionnon-GaussianityGaussian action
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.

The paper develops a statistical framework for treating curvature-profile dispersion in primordial black hole (PBH) formation. The central mechanism is a competition: coherent deformations of the curvature profile away from the mean peak configuration carry a Gaussian statistical cost (measured by their action), but they also change the collapse threshold. When the threshold reduction is large enough to compensate for the statistical rarity of the deformation, the dominant contribution to PBH abundance shifts away from the mean profile and toward rare shape branches. The author demonstrates this using a multipolar Fourier–Bessel decomposition that separates the standard BBKS peak variables (height, curvature, ellipsoidal deformation) from residual radial and angular shape modes, each normalized by its Gaussian action. Numerical collapse simulations in spherical symmetry show that for broad power spectra or negative logarithmic non-Gaussianity, the dominant branch can correspond to several-sigma shape fluctuations, enhancing the integrated abundance by orders of magnitude relative to the no-dispersion estimate. Equivalently, including shape dispersion can reduce the power-spectrum amplitude required to obtain a fixed PBH abundance.

Core claim

The dominant contribution to PBH abundance is selected by minimizing a combined cost: the Gaussian action of realizing a coherent shape deformation plus the squared collapse threshold along that branch. This competition can favor rare, several-sigma profile deformations over the mean profile, because the exponential sensitivity of PBH abundance to the collapse threshold can overwhelm the Gaussian suppression of the deformation. The effect is strongest for broad power spectra (where radial shape freedom is large) and for negative non-Gaussianity (where the threshold curve develops strongly shape-dependent branches). In the examples studied, the abundance enhancement reaches factors of 2 to 3,

What carries the argument

The finite-action deformation framework: the power spectrum defines a Gaussian-action metric on the space of curvature profiles, and coherent deformations are expanded in action-normalized modes orthogonal to the BBKS peak variables. Each mode carries a Gaussian cost n² (its action) and shifts the collapse threshold μ_c(n). The quantity being implicitly optimized is the critical action W_c(n) = n² + ν_c²(n), where ν_c = μ_c/σ₀. A deformed branch dominates when its lower threshold compensates its higher Gaussian cost.

Load-bearing premise

The dominance of rare shape deformations is demonstrated only within a one-parameter family of spherical radial deformations. The full shape space includes ellipticity, prolateness, and higher angular multipoles, all held at their mean values. If threshold variation along additional directions is smaller, or if the combined Gaussian cost of multi-mode deformations is higher, the enhancement from shape dispersion could be reduced.

What would settle it

If the collapse threshold were found to depend only weakly on residual radial shape modes once the BBKS height and curvature are fixed (i.e., if μ_c(n) were nearly flat across all action-normalized deformation directions), then the Gaussian cost n² would always dominate and the mean profile would always give the dominant contribution, falsifying the central claim.

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

If this is right

  • PBH abundance calculations that fix only the mean or representative profile may underestimate the true abundance by orders of magnitude, particularly for broad power spectra or negative non-Gaussianity.
  • The power-spectrum amplitude inferred from a given PBH abundance target can be significantly lower than no-dispersion estimates suggest (by factors of 0.3–0.6 in the examples studied).
  • For scenarios using negative non-Gaussianity to suppress PBH production while retaining scalar-induced gravitational waves, profile dispersion may strengthen the tension between PBH constraints and gravitational-wave signals.
  • The monochromatic-spectrum approximation eliminates radial shape dispersion entirely and may therefore be particularly misleading for abundance estimates.
  • Extending the framework to non-spherical modes (ellipticity, prolateness, higher multipoles) could further modify the dominant branch, especially at low peak heights where the high-peak near-spherical approximation is less restrictive.

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 / 7 minor

Summary. This paper develops a finite-action framework for describing curvature-profile dispersion in primordial black hole (PBH) formation. Using a multipolar Fourier–Bessel decomposition, the authors separate BBKS peak variables (height, curvature, ellipsoidal Hessian) from residual coherent deformations, each normalized by their Gaussian action cost. The framework is applied to two spherical numerical-collapse examples: (A) a sharply peaked finite-width spectrum with logarithmic local non-Gaussianity, and (B) a finite-band scale-invariant spectrum with tunable bandwidth. The central result is that the dominant contribution to the PBH abundance is selected by a competition between the Gaussian cost of realizing a coherent shape deformation and the exponential gain from a reduced collapse threshold—not necessarily by the mean profile or the profile with the lowest threshold. For broad spectra or negative non-Gaussianity, rare several-sigma shape deformations can enhance the integrated abundance by orders of magnitude. The theoretical framework is internally consistent: the Fourier–Bessel decomposition, action normalization, and recovery of the standard BBKS profile are clean. The numerical simulations use a public code (SPriBHoS) and threshold curves are compared against analytical estimates (EGS, HYK) with stated deviations (~4%).

Significance. The paper addresses a genuinely important gap in PBH abundance calculations: the statistical treatment of profile shape dispersion around reference peak profiles. The finite-action framework is a principled contribution that connects the Gaussian probability cost of coherent deformations to their effect on the collapse threshold, and the key insight—that the dominant profile is selected by a competition between statistical cost and threshold reduction—is well-motivated and potentially impactful. The use of a public, reproducible numerical code (SPriBHoS) and the comparison with analytical threshold prescriptions are commendable. The identification of specific regimes (broad spectra, negative non-Gaussianity) where shape dispersion can dominate the abundance is a falsifiable and quantitative prediction. However, the quantitative dominance claims rest on a restricted slice of the full shape space (a single radial deformation direction in spherical symmetry), which limits the scope of the conclusions at present.

major comments (3)
  1. §5.1.1, Fig. 13, and Table 2: The orders-of-magnitude abundance enhancements (e.g., β_disp/β0 ≈ 2.78×10^5 for β_NG = −2) depend entirely on the collapse threshold μ_c(s) computed along one specific deformation direction: the equal-variance split mode (Eq. 5.13). The paper states this mode is chosen to 'maximize the real-space profile dispersion,' not because it is a generic or typical direction in shape space. Since the threshold depends on the compaction function (involving radial derivatives of ζ), the direction that maximizes pointwise profile dispersion need not maximize threshold variation per unit Gaussian cost. The dominant branches occur at large deviations (n* ≈ −5.87 for β_NG = −2; n* ≈ −4.5 for R_F = 50), where the deformed profiles develop split-mode-specific features (oscillatory tails, branch switching). A different action-normalized shape direction could produce qualitativ
  2. §5.2.1, Eq. (5.104): The only consistency check provided is the projection of the split mode against the BBKS curvature direction q_x. This removes a component of the same split mode; it does not test an independent shape direction. The full integral (Eq. 2.91) is over all residual modes, but only one is sampled. The paper should more prominently acknowledge this limitation in the quantitative claims, or ideally test at least one additional independent radial deformation direction to assess whether the enhancement is a robust feature of the shape space or an artifact of the split-mode choice.
  3. §5.1.2, Eq. (5.46) and Table 2: For β_NG = −2 and β_NG = −3, the dominant branches occur at large negative s (n* ≈ −5.87 and −3.59 respectively), where the profiles develop features such as type-I/type-II transitions and central underdensities (Fig. 11). The logarithmic non-Gaussian map (Eq. 5.46) is used as a phenomenological template, and the paper notes (footnote 6) that deviations from the exact δN result can be ~8%. At these large deformations, the validity of the logarithmic template itself may be questionable. The paper should discuss whether the dominance result is robust to the non-Gaussian map prescription, or at minimum state this as a caveat on the quantitative enhancements.
minor comments (7)
  1. §2.4, Eq. (2.58): The standard BBKS profile is recovered, but the notation switches between γ and γ_BBKS in the surrounding equations. Using one symbol consistently would help the reader.
  2. §5.1, Eq. (5.34): The definition of the split mode G(x) in Eq. (5.31) involves the error function with complex argument. A brief note on how this is evaluated numerically would improve readability.
  3. Table 2: The column header 'f_shape_PBH / f^(n=0)_PBH' is somewhat ambiguous. Clarifying whether this is the mass-function-integrated ratio or the peak-abundance ratio would help.
  4. Fig. 10, upper-left panel: The threshold curves for different β_NG values are color-coded but the vertical dotted lines marking s_div are only shown for β_NG = −3 and −2. Adding these for all relevant β_NG values, or stating which curves have them, would improve clarity.
  5. §5.2, Eq. (5.60): The top-hat spectrum is introduced with Θ(k−k_min)Θ(k_max−k), but the relationship between R_F = k_max/k_min and the e-fold width L = ln R_F is only stated later (in the Conclusions). Stating this relationship at first introduction would help.
  6. §6, final paragraph: The statement that 'profile dispersion can become quantitatively important for enhanced power-spectrum features extending over several e-folds' could be made more concrete by specifying the approximate threshold width above which the effect becomes significant.
  7. The paper uses both 'logarithmic non-Gaussianity' and 'logarithmic local non-Gaussianity' in different places. Standardizing the terminology would improve consistency.

Circularity Check

0 steps flagged

No significant circularity; framework is derived from first principles and thresholds come from independent numerical simulations

full rationale

The paper's central framework is constructed from standard, independently verifiable ingredients: the Gaussian action (Eq. 2.5), the Fourier-Bessel decomposition (Eq. 2.2), and BBKS peak theory (Eqs. 2.80-2.84). The collapse thresholds μ_c(s) are obtained from independent relativistic numerical simulations using the SPriBHoS code, not from the statistical framework itself. The key quantitative results (Tables 2-3) combine these independently determined thresholds with the Gaussian action cost n², which is a direct consequence of the orthonormality condition (Eq. 2.8) — a standard construction, not a fitted or self-defined relation. The split-mode deformation (Eq. 5.13) is an explicit, action-normalized spectral direction satisfying the orthonormality constraint; it is not defined in terms of the abundance result it is used to compute. Self-citations (Refs. [37], [52], [58], [62]) provide the numerical code and threshold fitting formulas, but these are independently reproducible tools, not circular dependencies. The projected mode in Sec. 5.2.1 provides a partial consistency check using a different (though related) shape direction, yielding qualitatively similar results. The paper's limitation — that only one shape direction is sampled — is a scope restriction honestly acknowledged in Sec. 6, not a circularity. The derivation chain does not reduce to its inputs by construction. The skeptic's concern that the split mode may be atypical is a validity/scope issue, not a circularity issue: the threshold curves are computed from simulations, not assumed or fitted to reproduce the abundance enhancement. Score 2 reflects minor self-citation of the author's own numerical code and threshold formulas, which are not load-bearing for the logical structure of the argument but provide tools used in the computation.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

No new physical entities are introduced. The framework uses standard Gaussian field theory, Fourier-Bessel decomposition, and BBKS peak theory. The split mode and action-normalized deformation directions are mathematical constructions within the existing statistical framework, not new physical postulates.

free parameters (4)
  • K_eff = 6
    Effective critical-collapse constant used in the mass-function calculation (Eq. 5.49). Stated as an approximation motivated by Ref. [65], not fitted to the current data, but introduces an unquantified systematic.
  • k_peak = 1.1e13 Mpc^-1
    Choice of peak wavenumber placing the power spectrum in the asteroid-mass range (Sec. 5.1.2). Sets the mass scale but does not affect the abundance enhancement ratios.
  • nu_ref = 8.45
    Reference peak height used to calibrate the no-dispersion amplitude to unit PBH fraction (Sec. 5.1.2). This is a normalization choice for the comparison, not a physical parameter.
  • beta_NG = various (-3 to 3)
    Logarithmic non-Gaussianity parameter, treated as a phenomenological input scanned over a range. The paper states this is a template, not derived from a specific inflationary model.
axioms (4)
  • domain assumption The primordial curvature field is Gaussian to leading order, with non-Gaussianity introduced via a local logarithmic map (Eq. 5.46).
    Sec. 5.1. The logarithmic template is phenomenological; the paper notes it is not an exact relation from a specific inflationary model.
  • domain assumption PBH formation threshold is determined by the maximum of the compaction function and can be found via spherical numerical collapse for the radial shape modes studied.
    Sec. 5. The restriction to spherical symmetry is a modeling choice; non-spherical collapse thresholds may differ.
  • ad hoc to paper The split-mode direction is a representative direction in shape space that captures the dominant radial dispersion effect.
    Sec. 5.1, 5.2. The split mode is chosen as a representative direction, not derived as the optimal or dominant deformation. The paper acknowledges this is not cost-minimizing but maximizes real-space dispersion.
  • standard math Critical collapse scaling M ~ (mu - mu_c)^gamma_cr with gamma_cr ~ 0.356 applies to the shape-dispersed profiles.
    Eq. 2.93. Standard critical collapse result from the gravitational collapse literature, applied here to shape-dispersed initial data.

pith-pipeline@v1.1.0-glm · 53427 in / 2517 out tokens · 366172 ms · 2026-07-10T02:01:00.354699+00:00 · methodology

0 comments
read the original abstract

In the standard curvature-perturbation scenario, PBHs form from the collapse of superhorizon curvature fluctuations after horizon re-entry. The predicted abundance is exponentially sensitive to the collapse threshold and hence to the shape of the primordial curvature profile. In this work we develop a finite-action framework to describe curvature-profile dispersion around representative peak profiles. Using a multipolar Fourier--Bessel decomposition, we separate the local peak variables of the Gaussian field from residual radial and angular deformations, normalized by their Gaussian action. We apply the formalism to spherical numerical-collapse examples in order to isolate the effect of radial shape dispersion. For finite-width spectra, and in the presence of logarithmic local non-Gaussianity, we compute the collapse threshold as a function of a coherent shape variable and combine the result with peak statistics. We find that the dominant contribution to the PBH abundance is not necessarily the mean profile, nor simply the profile with the lowest threshold. Instead, it is selected by a competition between the Gaussian cost of realizing a coherent deformation and the exponential gain associated with lowering the collapse threshold. Broad spectra and negative non-Gaussianity can make rare shape deformations dominate the abundance. In the examples studied here, the dominant branches can correspond to several-sigma coherent shape fluctuations while enhancing the integrated abundance by orders of magnitude. Equivalently, including shape dispersion can reduce the power-spectrum amplitude required to obtain a fixed PBH abundance. Our results show that residual profile dispersion is a genuine statistical ingredient in PBH formation and can be quantitatively important for accurate abundance estimates.

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Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages · 59 internal anchors

  1. [1]

    Hawking,Gravitationally Collapsed Objects of Very Low Mass,Monthly Notices of the Royal Astronomical Society152(1971) 75

    S. Hawking,Gravitationally Collapsed Objects of Very Low Mass,Monthly Notices of the Royal Astronomical Society152(1971) 75

    work page 1971

  2. [2]

    Carr and S.W

    B.J. Carr and S.W. Hawking,Black Holes in the Early Universe,Monthly Notices of the Royal Astronomical Society168(1974) 399

    work page 1974

  3. [3]

    Primordial Black Holes - Perspectives in Gravitational Wave Astronomy -

    M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama,Primordial Black Holes: Perspectives in Gravitational Wave Astronomy,Classical and Quantum Gravity35(2018) 063001 [1801.05235]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  4. [4]

    B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama,Constraints on primordial black holes,Rept. Prog. Phys.84(2021) 116902 [2002.12778]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  5. [5]

    Primordial Black Holes as a dark matter candidate

    A.M. Green and B.J. Kavanagh,Primordial Black Holes as a Dark Matter Candidate,Journal of Physics G48(2021) 043001 [2007.10722]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  6. [6]

    Primordial Black Holes

    A. Escriv` a, F. Kuhnel and Y. Tada,Primordial Black Holes,2211.05767

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity

    M. Shibata and M. Sasaki,Black Hole Formation in the Friedmann Universe: Formulation and Computation in Numerical Relativity,Physical Review D60(1999) 084002 [gr-qc/9905064]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  8. [8]

    Dynamics of Primordial Black Hole Formation

    J.C. Niemeyer and K. Jedamzik,Dynamics of primordial black hole formation,Phys. Rev. D 59(1999) 124013 [astro-ph/9901292]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  9. [9]

    Computations of primordial black hole formation

    I. Musco, J.C. Miller and L. Rezzolla,Computations of Primordial Black-Hole Formation, Classical and Quantum Gravity22(2005) 1405 [gr-qc/0412063]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  10. [10]

    Primordial black hole formation in the radiative era: investigation of the critical nature of the collapse

    I. Musco, J.C. Miller and A.G. Polnarev,Primordial black hole formation in the radiative era: Investigation of the critical nature of the collapse,Class. Quant. Grav.26(2009) 235001 [0811.1452]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  11. [11]

    Formalism for Primordial Black Hole Formation in Spherical Symmetry

    J. Bloomfield, D. Bulhosa and S. Face,Formalism for Primordial Black Hole Formation in Spherical Symmetry,1504.02071

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    H. Deng, J. Garriga and A. Vilenkin,Primordial black hole and wormhole formation by domain walls,JCAP04(2017) 050 [1612.03753]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  13. [13]

    Primordial black hole formation by vacuum bubbles

    H. Deng and A. Vilenkin,Primordial black hole formation by vacuum bubbles,JCAP12(2017) 044 [1710.02865]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    Simulation of primordial black hole formation using pseudo-spectral methods

    A. Escriv` a,Simulation of primordial black hole formation using pseudo-spectral methods,Phys. Dark Univ.27(2020) 100466 [1907.13065]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  15. [15]

    C.-M. Yoo, T. Harada and H. Okawa,Threshold of Primordial Black Hole Formation in Nonspherical Collapse,Phys. Rev. D102(2020) 043526 [2004.01042]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  16. [16]

    C.-M. Yoo, T. Harada, S. Hirano, H. Okawa and M. Sasaki,Primordial black hole formation from massless scalar isocurvature,Phys. Rev. D105(2022) 103538 [2112.12335]. – 64 –

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  17. [17]

    Simulation of Primordial Black Holes with large negative non-Gaussianity

    A. Escriv` a, Y. Tada, S. Yokoyama and C.-M. Yoo,Simulation of primordial black holes with large negative non-Gaussianity,JCAP05(2022) 012 [2202.01028]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  18. [18]

    Primordial Black Hole Formation during a Strongly Coupled Crossover

    A. Escriv` a and J.G. Subils,Primordial black hole formation during a strongly coupled crossover,Phys. Rev. D107(2023) L041301 [2211.15674]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  19. [19]

    Simulations of PBH formation at the QCD epoch and comparison with the GWTC-3 catalog

    A. Escriv` a, E. Bagui and S. Clesse,Simulations of PBH formation at the QCD epoch and comparison with the GWTC-3 catalog,JCAP05(2023) 004 [2209.06196]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  20. [20]

    Primordial black hole formation during the QCD phase transition: threshold, mass distribution and abundance

    I. Musco, K. Jedamzik and S. Young,Primordial black hole formation during the QCD phase transition: Threshold, mass distribution, and abundance,Phys. Rev. D109(2024) 083506 [2303.07980]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  21. [21]

    Spinning primordial black holes formed during a matter-dominated era

    E. de Jong, J.C. Aurrekoetxea, E.A. Lim and T. Fran¸ ca,Spinning primordial black holes formed during a matter-dominated era,JCAP10(2023) 067 [2306.11810]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  22. [22]

    Primordial Black Holes and Induced Gravitational Waves from a Smooth Crossover beyond Standard Model

    A. Escriva, Y. Tada and C.-M. Yoo,Primordial black holes and induced gravitational waves from a smooth crossover beyond standard model theories,Phys. Rev. D110(2024) 063521 [2311.17760]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  23. [23]

    Uehara, A

    K. Uehara, A. Escriv` a, T. Harada, D. Saito and C.-M. Yoo,Numerical simulation of type II primordial black hole formation,JCAP01(2025) 003 [2401.06329]

    work page arXiv 2025

  24. [24]

    Primordial black hole formation from a nonspherical density profile with a misaligned deformation tensor

    C.-M. Yoo,Primordial black hole formation from a nonspherical density profile with a misaligned deformation tensor,Phys. Rev. D110(2024) 043526 [2403.11147]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  25. [25]

    Escriv` a and C.-M

    A. Escriv` a and C.-M. Yoo,Simulations of ellipsoidal primordial black hole formation,Phys. Rev. D112(2025) 083518 [2410.03452]

    work page arXiv 2025

  26. [26]

    Primordial Black Hole Formation in a Scalar Field Dominated Universe: Investigation of the Critical nature of the Collapse

    L.E. Padilla, E. Milligan, D.J. Mulryne and J.C. Hidalgo,Primordial Black Hole formation in a scalar field dominated Universe: Investigation of the critical nature of the collapse,JCAP04 (2026) 049 [2509.10431]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  27. [27]

    Ning, X.-X

    Z. Ning, X.-X. Zeng, Z.-Y. Yuwen, S.-J. Wang, H. Deng and R.-G. Cai,Sound waves from primordial black hole formations,Phys. Rev. D113(2026) 024020 [2504.12243]

    work page arXiv 2026

  28. [28]

    Primordial Black Hole Formation in a Scalar Field Dominated Universe

    E. Milligan, L.E. Padilla, D.J. Mulryne and J.C. Hidalgo,Primordial black hole formation in a scalar field dominated universe,JCAP10(2025) 025 [2504.02600]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  29. [29]

    Kitajima,Primordial black hole formation from collapsing domain walls with full general relativity,2510.22759

    N. Kitajima,Primordial black hole formation from collapsing domain walls with full general relativity,2510.22759

    work page arXiv

  30. [30]

    Germani and L

    C. Germani and L. Montell` a,Trichotomy of primordial black holes initial conditions,Phys. Rev. D113(2026) 064054 [2510.02006]

    work page arXiv 2026

  31. [31]

    Baumgarte, K

    T.W. Baumgarte, K. Clough and J.T. Giblin, Jr.,Restrictions on initial conditions in cosmological scenarios and implications for simulations of primordial black holes and inflation, Phys. Rev. D112(2025) 123528 [2509.26470]

    work page arXiv 2025

  32. [32]

    Primordial Black Holes in a Radiation-Dominated Universe

    T.W. Baumgarte, K. Clough, M. Gerhardinger, J.T. Giblin and A. Miller,Primordial Black Holes in a Radiation-Dominated Universe,2606.30641

    work page internal anchor Pith review Pith/arXiv arXiv

  33. [33]

    Numerical simulations of primordial black hole formation via delayed first-order phase transitions

    Z. Ning, X.-X. Zeng, R.-G. Cai and S.-J. Wang,Numerical simulations of primordial black hole formation via delayed first-order phase transitions,2601.21878

    work page internal anchor Pith review Pith/arXiv arXiv

  34. [34]

    Primordial black hole formation in bulk-viscous cosmology

    Z.-Y. Yuwen, C. Joana, S.-J. Wang and R.-G. Cai,Primordial black hole formation in bulk-viscous cosmology,2606.26532

    work page internal anchor Pith review Pith/arXiv arXiv

  35. [35]

    PBH formation from spherically symmetric hydrodynamical perturbations: a review

    A. Escriv` a,PBH Formation from Spherically Symmetric Hydrodynamical Perturbations: A Review,Universe8(2022) 66 [2111.12693]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  36. [36]

    Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations

    I. Musco,Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations,Phys. Rev. D100(2019) 123524 [1809.02127]. – 65 –

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  37. [37]

    A universal threshold for primordial black hole formation

    A. Escriv` a, C. Germani and R.K. Sheth,A Universal Threshold for Primordial Black Hole Formation,Physical Review D101(2020) 044022 [1907.13311]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  38. [38]

    Bardeen, J.R

    J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay,The Statistics of Peaks of Gaussian Random Fields,The Astrophysical Journal304(1986) 15

    work page 1986

  39. [39]

    A new calculation of the mass fraction of primordial black holes

    A.M. Green, A.R. Liddle, K.A. Malik and M. Sasaki,A New calculation of the mass fraction of primordial black holes,Phys. Rev. D70(2004) 041502 [astro-ph/0403181]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  40. [40]

    The abundance of primordial black holes depends on the shape of the inflationary power spectrum

    C. Germani and I. Musco,Abundance of Primordial Black Holes Depends on the Shape of the Inflationary Power Spectrum,Phys. Rev. Lett.122(2019) 141302 [1805.04087]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  41. [41]

    C.-M. Yoo, T. Harada, J. Garriga and K. Kohri,Primordial black hole abundance from random Gaussian curvature perturbations and a local density threshold,PTEP2018(2018) 123E01 [1805.03946]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  42. [42]

    Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation

    S. Young, I. Musco and C.T. Byrnes,Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation,JCAP 11(2019) 012 [1904.00984]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  43. [43]

    Non-linear statistics of primordial black holes from gaussian curvature perturbations

    C. Germani and R.K. Sheth,Nonlinear statistics of primordial black holes from Gaussian curvature perturbations,Phys. Rev. D101(2020) 063520 [1912.07072]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  44. [44]

    Abundance of primordial black holes with local non-Gaussianity in peak theory

    C.-M. Yoo, J.-O. Gong and S. Yokoyama,Abundance of primordial black holes with local non-Gaussianity in peak theory,JCAP09(2019) 033 [1906.06790]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  45. [45]

    Application of peaks theory to the abundance of primordial black holes

    S. Young and M. Musso,Application of peaks theory to the abundance of primordial black holes, JCAP11(2020) 022 [2001.06469]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  46. [46]

    The statistics of primordial black holes in a radiation dominated Universe -- recent and new results

    C. Germani and R.K. Sheth,The Statistics of Primordial Black Holes in a Radiation-Dominated Universe: Recent and New Results,Universe9(2023) 421 [2308.02971]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  47. [47]

    S. Pi, M. Sasaki, V. Takhistov and J. Wang,Primordial Black Hole formation from power spectrum with finite-width,JCAP09(2025) 045 [2501.00295]

    work page arXiv 2025

  48. [48]

    V. Atal, J. Cid, A. Escriv` a and J. Garriga,PBH in single field inflation: the effect of shape dispersion and non-Gaussianities,JCAP05(2020) 022 [1908.11357]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  49. [49]

    Escriv` a, J

    A. Escriv` a, J. Garriga and S. Pi,Inflationary relics from an ultra-slow-roll plateau,JCAP03 (2026) 018 [2512.04986]

    work page arXiv 2026

  50. [50]

    Observation of critical phenomena and self-similarity in the gravitational collapse of radiation fluid

    C.R. Evans and J.S. Coleman,Observation of critical phenomena and selfsimilarity in the gravitational collapse of radiation fluid,Phys. Rev. Lett.72(1994) 1782 [gr-qc/9402041]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  51. [51]

    Escriv` a and C.-M

    A. Escriv` a and C.-M. Yoo,Non-spherical effects on the mass function of primordial black holes, Phys. Rev. D112(2025) L081304 [2410.03451]

    work page arXiv 2025

  52. [52]

    Escriv` a,A new approach for simulating PBH formation from generic curvature fluctuations with the Misner-Sharp formalism,Phys

    A. Escriv` a,A new approach for simulating PBH formation from generic curvature fluctuations with the Misner-Sharp formalism,Phys. Dark Univ.50(2025) 102177 [2504.05813]

    work page arXiv 2025

  53. [53]

    Albert escriva github repository

    A. Escriva, “Albert escriva github repository.”https://github.com/albert-escriva, 2026. Accessed: 2026-04-20

    work page 2026

  54. [54]

    How well do we know the primordial black hole abundance? The crucial role of nonlinearities when approaching the horizon

    V. De Luca, A. Kehagias and A. Riotto,How well do we know the primordial black hole abundance: The crucial role of nonlinearities when approaching the horizon,Phys. Rev. D108 (2023) 063531 [2307.13633]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  55. [55]

    Non-Gaussianities and Primordial Black Holes

    S. Pi,Non-Gaussianities and Primordial Black Holes, (2025), DOI [2404.06151]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  56. [56]

    V. Atal, J. Garriga and A. Marcos-Caballero,Primordial black hole formation with non-Gaussian curvature perturbations,JCAP09(2019) 073 [1905.13202]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  57. [57]

    Logarithmic Duality of the Curvature Perturbation

    S. Pi and M. Sasaki,Logarithmic Duality of the Curvature Perturbation,Phys. Rev. Lett.131 (2023) 011002 [2211.13932]. – 66 –

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  58. [58]

    Escriv` a, V

    A. Escriv` a, V. Atal and J. Garriga,Formation of trapped vacuum bubbles during inflation, and consequences for PBH scenarios,JCAP10(2023) 035 [2306.09990]

    work page arXiv 2023

  59. [59]

    Primordial Black Hole Formation from Type II Fluctuations with Primordial Non-Gaussianity

    M. Shimada, A. Escriv´ a, D. Saito, K. Uehara and C.-M. Yoo,Primordial black hole formation from type II fluctuations with primordial non-Gaussianity,JCAP02(2025) 018 [2411.07648]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  60. [60]

    R. Inui, C. Joana, H. Motohashi, S. Pi, Y. Tada and S. Yokoyama,Primordial black holes and induced gravitational waves from logarithmic non-Gaussianity,JCAP03(2025) 021 [2411.07647]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  61. [61]

    M. Kopp, S. Hofmann and J. Weller,Separate Universes Do Not Constrain Primordial Black Hole Formation,Phys. Rev. D83(2011) 124025 [1012.4369]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  62. [62]

    Escriv` a,Threshold for PBH formation in the type-II region and its analytical estimation, Phys

    A. Escriv` a,Threshold for PBH formation in the type-II region and its analytical estimation, Phys. Rev. D112(2025) 103527 [2504.05814]

    work page arXiv 2025

  63. [63]

    Threshold of primordial black hole formation

    T. Harada, C.-M. Yoo and K. Kohri,Threshold of primordial black hole formation,Phys. Rev. D88(2013) 084051 [1309.4201]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  64. [64]

    Analytical thresholds for black hole formation in general cosmological backgrounds

    A. Escriv` a, C. Germani and R.K. Sheth,Analytical Thresholds for Black Hole Formation in General Cosmological Backgrounds,Journal of Cosmology and Astroparticle Physics01(2021) 030 [2007.05564]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  65. [65]

    Effects of the shape of curvature peaks on the size of primordial black holes

    A. Escriv` a and A.E. Romano,Effects of the shape of curvature peaks on the size of primordial black holes,JCAP05(2021) 066 [2103.03867]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  66. [66]

    Primordial black hole tower: Dark matter, earth-mass, and LIGO black holes

    Y. Tada and S. Yokoyama,Primordial black hole tower: Dark matter, earth-mass, and LIGO black holes,Phys. Rev. D100(2019) 023537 [1904.10298]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  67. [67]

    Primordial Black Hole formation from overlapping cosmological fluctuations

    A. Escriv` a and C.-M. Yoo,Primordial Black hole formation from overlapping cosmological fluctuations,JCAP04(2024) 048 [2310.16482]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  68. [68]

    Primordial black hole compaction function from stochastic fluctuations in ultra-slow-roll inflation

    S. Raatikainen, S. R¨ as¨ anen and E. Tomberg,Primordial Black Hole Compaction Function from Stochastic Fluctuations in Ultraslow-Roll Inflation,Phys. Rev. Lett.133(2024) 121403 [2312.12911]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  69. [69]

    Raatikainen, S

    S. Raatikainen, S. Rasanen and E. Tomberg,Effect of stochastic kicks on primordial black hole abundance and mass via the compaction function,JCAP03(2026) 063 [2510.09303]. [70]NANOGravcollaboration,The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,Astrophys. J. Lett.951(2023) L8 [2306.16213]. [71]EPTA, InPTA:collaboration,The secon...

    work page arXiv 2026

  70. [70]

    The recent gravitational wave observation by pulsar timing arrays and primordial black holes: the importance of non-gaussianities

    G. Franciolini, A.J. Iovino, V. Vaskonen and H. Veerm¨ ae,The recent gravitational wave observation by pulsar timing arrays and primordial black holes: the importance of non-Gaussianities,2306.17149

    work page internal anchor Pith review Pith/arXiv arXiv

  71. [71]

    Implications of Pulsar Timing Array Data for Scalar-Induced Gravitational Waves and Primordial Black Holes: Primordial Non-Gaussianity $f_{\mathrm{NL}}$ Considered

    S. Wang, Z.-C. Zhao, J.-P. Li and Q.-H. Zhu,Implications of Pulsar Timing Array Data for Scalar-Induced Gravitational Waves and Primordial Black Holes: Primordial Non-Gaussianity fNL Considered,2307.00572

    work page internal anchor Pith review Pith/arXiv arXiv

  72. [72]

    Primordial non-Gaussianity as a saviour for PBH overproduction in SIGWs generated by Pulsar Timing Arrays for Galileon inflation

    S. Choudhury, K. Dey, A. Karde, S. Panda and M. Sami,Primordial non-Gaussianity as a saviour for PBH overproduction in SIGWs generated by Pulsar Timing Arrays for Galileon inflation,2310.11034. – 67 –

    work page internal anchor Pith review Pith/arXiv arXiv

  73. [73]

    The Detected Stochastic Gravitational Waves and Subsolar-Mass Primordial Black Holes

    K. Inomata, K. Kohri and T. Terada,The Detected Stochastic Gravitational Waves and Subsolar-Mass Primordial Black Holes,2306.17834

    work page internal anchor Pith review Pith/arXiv arXiv

  74. [74]

    A general proof of the conservation of the curvature perturbation

    D.H. Lyth, K.A. Malik and M. Sasaki,A General proof of the conservation of the curvature perturbation,JCAP05(2005) 004 [astro-ph/0411220]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  75. [75]

    Gradient expansion approach to nonlinear superhorizon perturbations II -- a single scalar field --

    Y. Tanaka and M. Sasaki,Gradient expansion approach to nonlinear superhorizon perturbations. II. A Single scalar field,Prog. Theor. Phys.118(2007) 455 [0706.0678]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  76. [76]

    Revisiting compaction functions for primordial black hole formation

    T. Harada, C.-M. Yoo and Y. Koga,Revisiting compaction functions for primordial black hole formation,Phys. Rev. D108(2023) 043515 [2304.13284]. – 68 –

    work page internal anchor Pith review Pith/arXiv arXiv 2023