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

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2607.08738 v2 pith:IAVIU3H6 submitted 2026-07-09 astro-ph.CO gr-qchep-phhep-th

Albert Escrivà This is my paper
classification astro-ph.COgr-qchep-phhep-th
keywords primordialblackholescurvature-profiledispersionpeaktheorycollapsethresholdfinite-actionformalismnon-Gaussianitypower-spectrumwidthcompactionfunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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

What would settle it

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-

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§5.1.1, Table 2 (β_NG = −3 row)] 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.
  2. [§5.1.2, Eq. (5.56), Tables 2–4] 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.
  3. [§5.1.2, Eq. (5.55)] 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.
minor comments (4)
  1. [§2.1 title] Typo: 'Multipolar F ourier–Bessel' should read 'Multipolar Fourier–Bessel'.
  2. [§3, Table 1] 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.
  3. [§5.1.2, Eq. (5.53)] 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.
  4. [§5.2, Fig. 20] 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.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: thresholds are independently simulated and no abundance is fitted; only a minor self-cited, openly-declared ansatz direction anchors the quantitative claims.

  1. other [Sec. 5.1, Eq. (5.13) and Sec. 5.1.2 (after Eq. 5.58); limitations in Sec. 6]
    "Following Ref. [54], we use a split-spectrum ansatz to construct a coherent finite-action deformation at fixed central amplitude. The split divides the spectrum into long- and short-wavelength parts with equal variance weight. This is not a cost-minimizing choice, (every action-normalized direction has the same quadratic cost n2), but rather a representative direction that maximizes the real-space profile dispersion ... Here we use it as a representative direction in shape space."

    This is not a hard circular reduction: the collapse thresholds mu_c(s) come from independent public numerical-relativity codes, the Gaussian cost n^2 is fixed by the assumed power spectrum, and the abundance integrals are genuine computations. The minor self-referential element is that the one-parameter family carrying every quantitative result (Tables 2-4: n*, beta_disp/beta0, Q_A) is fixed by a direction imported from the author's own prior work and chosen representatively rather than derived. The paper itself states the ratios 'read as the effect of the dominant radial direction, not as a full marginalization over the residual shape space' and that 'other residual shape directions ... should be explored'. The enhancement is therefore ansatz-conditional, and the self-citation is an origi

full rationale

Derivation chain: (i) assumed power spectrum defines the Gaussian-action metric and action-normalized modes (Eqs. 2.4-2.11); BBKS variables are recovered as the first such directions (Eq. 2.58 is shown to reproduce BBKS Eq. 7.8), a consistency check, not a renaming. (ii) The shape-dependent thresholds mu_c(s) are obtained from separate relativistic simulations with the public SPriBHoS codes (Refs. 16,57); they are inputs, not fits to the target abundance. (iii) The abundance integrates P_sh(n) times the BBKS peak count at the simulated threshold (Eqs. 2.90, 2.58); A_zeta is calibrated so the no-dispersion baseline is f=1, a normalization that does not force the enhancement. The qualitative 'competition between Gaussian cost and threshold gain' is, by construction, the structure of the integrand in Eq. 2.90, but the quantitative findings for negative beta_NG (n* = -3 to -6, enhancements up to 1e8, Q_A down to 0.339) are computed, not assumed. The amplitude-retuning Q_A is explicitly an equivalent re-expression of the abundance enhancement ('Equivalently...'), derived through the exponential sensitivity, not a separately fitted prediction. Self-citations (Refs. 53, 54, 70) are used for context, for an openly declared ansatz, and for a multiplicative Keff shown subdominant; no uniqueness theorem is imported and no result is smuggled via citation. Flagged limitations, weighed here as correctness risks rather than circularity: Sec. 5.1.1 admits 'the branch-weight diagnostic ... does not show a clear internal maximum within the simulated range' for beta_NG=-3, so the tabulated n* and Q_A are partly set by the integration interval; Sec. 5.1.2 stresses only the single split direction is integrated; Sec. 6 calls for convergence tests over other residual directions. These passages show the quantitative claims are honest but not basis-converged; they do not reduce the derivation to its own inputs.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new physical particles, forces, or conserved quantities are introduced. The 'split mode' and 'coherent deformations' are mathematical coordinates in the Gaussian profile space. The main free choices are the power-spectrum normalization (calibrated to a reference abundance), the effective critical-collapse prefactor, the hand-picked split direction, and the reference peak height used to define sigma units. The logarithmic non-Gaussian map and the spherical-symmetry truncation are the principal domain assumptions.

free parameters (5)
  • power-spectrum normalization A_zeta (per branch) = Calibrated so f_PBH,tot^(0)=1; e.g. A_zeta=(mu_c(0)/8.45)^2
    Sets the no-dispersion reference abundance to unity; the reported enhancement ratios and dominant n* values depend on this normalization.
  • critical-collapse mass prefactor K_eff = 6
    Adopted from the O(1-10) range in Ref. [70]; affects the mass-function normalization but not the threshold-dominated abundance ratios.
  • split-mode equal-variance scale y = y ~ 1.087652 (k ~ 1.538 kappa-tilde)
    Determines the single hand-picked radial deformation direction; not cost-minimizing, chosen to maximize real-space profile dispersion. Quantitative results are conditional on this direction.
  • reference peak height nu_ref = ~8.45
    Used together with A_zeta to set the no-dispersion reference f=1; effectively sets the amplitude scale used when reporting n* in units of sigma.
  • spectral peak wavenumber k_peak = 1.1e13 Mpc^-1
    Places PBH masses in the asteroid-mass range; sets the mass axis but does not affect the abundance-enhancement ratios.
assumptions (6)
  • domain assumption The primordial curvature fluctuation zeta_G is a homogeneous, isotropic Gaussian field with power spectrum P_zetaG(k), and the Gaussian probability functional is exp(-W/2) with W = sum |B|^2/P.
    Foundation of the action metric; standard in peak theory but a physical assumption about the statistics of primordial fluctuations.
  • domain assumption BBKS peak density formulas (Eqs. 2.79-2.86) correctly count peaks and their curvature and ellipsoid distributions.
    The abundance integrals rely on BBKS peak theory as prior literature; no independent derivation is given in this paper.
  • domain assumption Gradient-expansion initial conditions and Misner-Sharp spherical evolution with p = w rho describe PBH collapse in the radiation era.
    The numerical machinery in Appendix A.2 is standard but is a modeling assumption for the threshold determination.
  • ad hoc to paper The logarithmic local non-Gaussian map zeta = -(1/beta_NG) log(1 - beta_NG zeta_G) is a valid phenomenological template.
    The paper explicitly flags this as a template rather than a derived relation; results are conditional on it.
  • domain assumption Critical-collapse scaling M = K M_H (mu - mu_c)^gamma with gamma ~ 0.356, and the volume-averaged compaction criterion Cc ~ 2/5, apply across all shape branches.
    These are taken from the PBH critical-collapse literature and used to convert thresholds into mass functions and abundance comparisons.
  • ad hoc to paper The chosen split-spectrum mode is a representative direction in the infinite-dimensional residual shape space; all other orthogonal directions can be neglected or behave similarly.
    The weakest assumption: the quantitative enhancements are computed for one hand-picked split direction and no basis-dependence test is performed.

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Pith. "Pith review of The statistics of curvature-profile dispersion in primordial black hole formation." pith.science (2026). https://pith.science/paper/IAVIU3H6

@misc{pith2026260708738,
  author       = {Pith},
  title        = {Pith review of: The statistics of curvature-profile dispersion in primordial black hole formation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IAVIU3H6}},
  note         = {Machine review of arXiv:2607.08738}
}
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 conditional-mean reference 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

84 extracted references · 66 linked inside Pith · cited by 1 Pith paper

  1. [54]

    Escriv` a, J

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

    arXiv 2026

  2. [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. – 67 –

    1971

  3. [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

    1974

  4. [3]

    Sasaki, T

    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]

    arXiv 2018

  5. [4]

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

    arXiv 2021

  6. [5]

    Green and B.J

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

    arXiv 2021

  7. [6]

    Escriv` a, F

    A. Escriv` a, F. K¨ uhnel and Y. Tada,Primordial black holes, inBlack Holes in the Era of Gravitational-Wave Astronomy, pp. 261–377, Elsevier (2024), DOI [2211.05767]

    arXiv 2024

  8. [7]

    Shibata and M

    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]

    arXiv 1999

Show all 84 references
  1. [8]

    Niemeyer and K

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

    1999 arXiv

  2. [9]

    Hawke and J.M

    I. Hawke and J.M. Stewart,The dynamics of primordial black hole formation,Class. Quant. Grav.19(2002) 3687

    2002

  3. [10]

    Musco, J.C

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

    2005 arXiv

  4. [11]

    Musco, J.C

    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]

    2009 arXiv

  5. [12]

    Nakama, T

    T. Nakama, T. Harada, A.G. Polnarev and J. Yokoyama,Identifying the most crucial parameters of the initial curvature profile for primordial black hole formation,JCAP01(2014) 037 [1310.3007]

    2014 arXiv

  6. [13]

    Bloomfield, D

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

    arXiv

  7. [14]

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

    2017 arXiv

  8. [15]

    Deng and A

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

    2017 arXiv

  9. [16]

    Escriv` a,Simulation of primordial black hole formation using pseudo-spectral methods,Phys

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

    2020 arXiv

  10. [17]

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

    2020 arXiv

  11. [18]

    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]

    2022 arXiv

  12. [19]

    Escriv` a, Y

    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]

    2022 arXiv

  13. [20]

    Escriv` a and J.G

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

    2023 arXiv

  14. [21]

    Escriv` a, E

    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]. – 68 –

    2023 arXiv

  15. [22]

    Musco, K

    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]

    2024 arXiv

  16. [23]

    de Jong, J.C

    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]

    2023 arXiv

  17. [24]

    Escriva, Y

    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]

    2024 arXiv

  18. [25]

    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]

    2025

  19. [26]

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

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

    2024 arXiv

  20. [27]

    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]

    2025

  21. [28]

    Padilla, E

    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]

    2026 arXiv

  22. [29]

    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]

    2026

  23. [30]

    Milligan, L.E

    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]

    2025 arXiv

  24. [31]

    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

  25. [32]

    Germani and L

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

    2026

  26. [33]

    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]

    2025

  27. [34]

    Cheng, P

    C. Cheng, P. Giannadakis, L. Heurtier and E.A. Lim,Non-linear dynamics and primordial black hole formation during kination,JCAP07(2026) 048 [2507.19166]

    2026 arXiv

  28. [35]

    Baumgarte, K

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

    arXiv

  29. [36]

    Ning, X.-X

    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

    arXiv

  30. [37]

    Yuwen, C

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

    arXiv

  31. [38]

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

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

    2022 arXiv

  32. [39]

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

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

    2019 arXiv

  33. [40]

    Escriv` a, C

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

    2020 arXiv

  34. [41]

    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. – 69 –

    1986

  35. [42]

    Green, A.R

    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]

    2004 arXiv

  36. [43]

    Germani and I

    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]

    2019 arXiv

  37. [44]

    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]

    2018 arXiv

  38. [45]

    Young, I

    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]

    2019 arXiv

  39. [46]

    Germani and R.K

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

    2020 arXiv

  40. [47]

    Yoo, J.-O

    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]

    2019 arXiv

  41. [48]

    Young and M

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

    2020 arXiv

  42. [49]

    C.-M. Yoo, T. Harada, S. Hirano and K. Kohri,Abundance of Primordial Black Holes in Peak Theory for an Arbitrary Power Spectrum,PTEP2021(2021) 013E02 [2008.02425]

    2021 arXiv

  43. [50]

    Germani and R.K

    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]

    2023 arXiv

  44. [51]

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

    2025

  45. [52]

    Fumagalli, J

    J. Fumagalli, J. Garriga, C. Germani and R.K. Sheth,Unexpected shape of the primordial black hole mass function,Phys. Rev. D111(2025) 123518 [2412.07709]

    2025 arXiv

  46. [53]

    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]

    2020 arXiv

  47. [55]

    Evans and J.S

    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]

    1994 arXiv

  48. [56]

    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]

    2025

  49. [57]

    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]

    2025

  50. [58]

    Albert escriva github repository

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

    2026

  51. [59]

    De Luca, A

    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]

    2023 arXiv

  52. [60]

    Pi,Non-gaussianities and primordial black holes, inPrimordial Black Holes, pp

    S. Pi,Non-gaussianities and primordial black holes, inPrimordial Black Holes, pp. 155–200, Springer Singapore (2025), DOI [2404.06151]

    2025 arXiv

  53. [61]

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

    2019 arXiv

  54. [62]

    Pi and M

    S. Pi and M. Sasaki,Logarithmic Duality of the Curvature Perturbation,Phys. Rev. Lett.131 (2023) 011002 [2211.13932]. – 70 –

    2023 arXiv

  55. [63]

    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]

    2023

  56. [64]

    Shimada, A

    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]

    2025 arXiv

  57. [65]

    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]

    2025 arXiv

  58. [66]

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

    2011 arXiv

  59. [67]

    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]

    2025

  60. [68]

    Harada, C.-M

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

    2013 arXiv

  61. [69]

    Escriv` a, C

    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]

    2021 arXiv

  62. [70]

    Escriv` a and A.E

    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]

    2021 arXiv

  63. [71]

    Tada and S

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

    2019 arXiv

  64. [72]

    Escriv` a and C.-M

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

    2024 arXiv

  65. [73]

    Raatikainen, S

    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]

    2024 arXiv

  66. [74]

    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]. [75]NANOGravcollaboration,The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,Astro...

    2026

  67. [77]

    Reardon et al.,Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array,Astrophys

    D.J. Reardon et al.,Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array,Astrophys. J. Lett.951(2023) L6 [2306.16215]

    2023 arXiv

  68. [78]

    Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,Res

    H. Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,Res. Astron. Astrophys.23(2023) 075024 [2306.16216]

    2023 arXiv

  69. [79]

    Franciolini, A

    G. Franciolini, A. Iovino, Junior., V. Vaskonen and H. Veermae,Recent Gravitational Wave Observation by Pulsar Timing Arrays and Primordial Black Holes: The Importance of Non-Gaussianities,Phys. Rev. Lett.131(2023) 201401 [2306.17149]

    2023 arXiv

  70. [80]

    Wang, Z.-C

    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,Phys. Rev. Res.6(2024) L012060 [2307.00572]

    2024 arXiv

  71. [81]

    Choudhury, K

    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. – 71 –

    arXiv

  72. [82]

    Inomata, K

    K. Inomata, K. Kohri and T. Terada,Detected stochastic gravitational waves and subsolar-mass primordial black holes,Phys. Rev. D109(2024) 063506 [2306.17834]

    2024 arXiv

  73. [83]

    Lyth, K.A

    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]

    2005 arXiv

  74. [84]

    Tanaka and M

    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]

    2007 arXiv

  75. [85]

    Harada, C.-M

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

    2023 arXiv

  76. [86]

    Harada, C.-M

    T. Harada, C.-M. Yoo, T. Nakama and Y. Koga,Cosmological long-wavelength solutions and primordial black hole formation,Phys. Rev. D91(2015) 084057 [1503.03934]. – 72 –

    2015 arXiv

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