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arxiv: 2606.30087 · v1 · pith:W3KW74EXnew · submitted 2026-06-29 · 🌌 astro-ph.CO · gr-qc

Compaction function in stochastic inflation: a texttt{FOREST} of type I and II primordial black holes

Chiara Animali , Pierre Auclair , Baptiste Blachier , Eemeli Tomberg , Vincent Vennin This is my paper

Pith reviewed 2026-06-30 05:23 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords stochastic inflationprimordial black holescompaction functiontype I and II fluctuationsquantum diffusioneternal inflationbinary treescloud-in-cloud effects
0
0 comments X

The pith

The compaction function equals the sibling-to-child volume ratio on stochastic binary trees, making type-II primordial black holes outnumber type I when quantum diffusion dominates.

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

The paper establishes a method to compute the compaction function inside stochastic inflation by evolving the random field on binary trees, where the function is the volume ratio between sibling and child branches of each node. This ratio also decides whether a perturbation is type I or type II by checking if the areal radius grows monotonically. In a single-field toy model with constant potential slope, the classical regime produces a narrow PBH mass function with type II strongly suppressed, but the quantum and near-critical regimes produce a mass distribution spanning several orders of magnitude, raise the total abundance, and make type II more numerous than type I. A reader cares because these changes affect whether PBHs can explain dark matter or observed gravitational waves. The work notes that cloud-in-cloud effects become important in the quantum case.

Core claim

By solving the random field dynamics on stochastic binary trees, the compaction function is identified with the ratio of the volumes emerging from the sibling and child branches of a given node. This construction also determines whether or not the areal radius of a perturbation increases monotonically with the radial coordinate, thereby distinguishing between type-I and type-II fluctuations. In the classical regime the PBH mass function is narrowly distributed and type-II fluctuations are strongly suppressed, while in the quantum and near-critical regimes the PBH mass distribution spans several orders of magnitude, the overall PBH abundance is enhanced, and type-II fluctuations outnumber typ

What carries the argument

stochastic binary trees on which the random field evolves, with the compaction function given directly by the sibling-to-child volume ratio that also classifies the fluctuation type

If this is right

  • PBH mass functions become broad over many orders of magnitude once quantum diffusion is significant
  • Overall PBH abundance increases in the quantum and near-critical regimes
  • Type-II fluctuations become more numerous than type I near eternal inflation
  • Cloud-in-cloud effects must be modeled separately to obtain robust PBH predictions when stochastic effects matter

Where Pith is reading between the lines

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

  • The tree method could be applied to other potentials to test whether type-II dominance persists beyond the constant-slope case
  • If type-II collapse produces different gravitational-wave signatures, the enhanced type-II fraction would alter predicted backgrounds
  • The approach supplies a concrete way to quantify how stochastic effects change the threshold for PBH formation across regimes

Load-bearing premise

The direct mapping from sibling-to-child volume ratio on the tree to the compaction function reproduces the full spacetime dynamics of the perturbation without extra corrections from cloud-in-cloud evolution or general-relativistic effects.

What would settle it

A numerical relativity simulation of the collapse of type-II fluctuations seeded from a stochastic inflation tree that measures whether the predicted mass distribution and abundance match the tree-based compaction values.

read the original abstract

We show how to compute the compaction function within stochastic inflation, by solving the random field dynamics on stochastic binary trees. In this framework, the compaction function is directly related to the ratio of the volumes emerging from the sibling and child branches of a given node. This construction also determines whether or not the areal radius of a perturbation increases monotonically with the radial coordinate, thereby distinguishing between type-I and type-II fluctuations. As an application, we investigate primordial black hole (PBH) formation in a single-field toy model with a constant potential slope, using stochastic-tree realizations generated with the public code \texttt{FOREST}. In the classical regime, where quantum diffusion is subdominant, the PBH mass function is narrowly distributed and type-II fluctuations are strongly suppressed relative to type I. By contrast, in the quantum and near-critical (i.e. close to eternal inflation) regimes, the PBH mass distribution spans several orders of magnitude, the overall PBH abundance is enhanced, and type-II fluctuations outnumber type I. In that case cloud-in-cloud effects are also important, highlighting the need for a better understanding of the evolution and collapse of type-II fluctuations in order to obtain robust PBH predictions when stochastic effects are significant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a method to compute the compaction function in stochastic inflation by modeling the random field dynamics on stochastic binary trees generated with the FOREST code. The compaction function is identified with the ratio of volumes from sibling and child branches, which also determines if the areal radius increases monotonically, distinguishing type-I and type-II fluctuations. Applied to a single-field toy model with constant potential slope, the paper reports that in the classical regime the PBH mass function is narrow and type-II are suppressed, whereas in quantum and near-critical regimes the mass distribution is broad over several orders of magnitude, PBH abundance is enhanced, type-II outnumber type-I, and cloud-in-cloud effects are significant.

Significance. If the tree-based mapping to the compaction function holds, the work provides a new computational approach to PBH formation in the presence of stochastic effects, emphasizing differences across regimes and the potential importance of type-II fluctuations. The public availability of the FOREST code supports reproducibility and is a strength of the presentation.

major comments (1)
  1. [Abstract] Abstract: The central results on regime-dependent PBH abundances and type-II dominance rely on equating the compaction function directly to the sibling-to-child volume ratio on the stochastic tree. However, the abstract notes that 'cloud-in-cloud effects are also important, highlighting the need for a better understanding of the evolution and collapse of type-II fluctuations in order to obtain robust PBH predictions'. This indicates that the identification may omit GR corrections or cloud-in-cloud evolution, which could alter the reported trends in the quantum regime.
minor comments (1)
  1. The abstract states the method and resulting mass-function trends but supplies no error bars, convergence tests, or comparison against known analytic limits; these should be added in the results sections to support the claimed trends.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important caveat in our presentation. We address the major comment below.

read point-by-point responses

  1. Referee: The central results on regime-dependent PBH abundances and type-II dominance rely on equating the compaction function directly to the sibling-to-child volume ratio on the stochastic tree. However, the abstract notes that 'cloud-in-cloud effects are also important, highlighting the need for a better understanding of the evolution and collapse of type-II fluctuations in order to obtain robust PBH predictions'. This indicates that the identification may omit GR corrections or cloud-in-cloud evolution, which could alter the reported trends in the quantum regime.

    Authors: We agree that the mapping from the stochastic tree to the compaction function is an approximation that does not incorporate full general-relativistic evolution or detailed cloud-in-cloud dynamics. The identification follows from the definition of the compaction function in terms of the volume ratio on the binary tree and the associated condition for monotonicity of the areal radius; this is the central construction of the paper. The abstract already flags the importance of cloud-in-cloud effects and the need for further study of type-II collapse precisely because the present results are obtained within this framework. The reported differences between classical, quantum, and near-critical regimes are therefore indicative within the tree model rather than final predictions. We will revise the abstract to state more explicitly that the trends are derived under the tree-based identification and to underscore the associated caveats. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external code and new tree construction

full rationale

The paper defines a mapping from stochastic binary trees to the compaction function via sibling-to-child volume ratios and applies it to PBH counting in a toy model. This is presented as a computational framework rather than a fit or self-referential definition. FOREST is explicitly cited as public external code. No equations reduce the reported mass distributions or type-I/II counts to parameters defined by the same outputs, and no load-bearing self-citation chain is invoked. The results follow from the stated model assumptions and tree realizations without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; full list of modeling assumptions and any fitted parameters cannot be extracted.

axioms (1)
  • domain assumption Stochastic inflation dynamics on binary trees faithfully represent the underlying random field evolution
    Invoked to justify the volume-ratio definition of the compaction function.

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discussion (0)

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Reference graph

Works this paper leans on

97 extracted references · 2 canonical work pages

  1. [1]

    M. M. Ivanov, M. Simonovi´ c and M. Zaldarriaga,Cosmological Parameters from the BOSS Galaxy Power Spectrum,JCAP05(2020) 042, [1909.05277]

    arXiv 2020

  2. [2]

    Hawking,Gravitationally collapsed objects of very low mass,Mon

    S. Hawking,Gravitationally collapsed objects of very low mass,Mon. Not. Roy. Astron. Soc. 152(1971) 75. – 29 –

    1971

  3. [3]

    B. J. Carr and S. W. Hawking,Black holes in the early Universe,Mon. Not. Roy. Astron. Soc. 168(1974) 399–415

    1974

  4. [4]

    B. J. Carr,The Primordial black hole mass spectrum,Astrophys. J.201(1975) 1–19

    1975

  5. [5]

    Pattison, V

    C. Pattison, V. Vennin, H. Assadullahi and D. Wands,Quantum diffusion during inflation and primordial black holes,JCAP10(2017) 046, [1707.00537]

    arXiv 2017

  6. [6]

    J. M. Ezquiaga, J. Garc´ ıa-Bellido and V. Vennin,The exponential tail of inflationary fluctuations: consequences for primordial black holes,JCAP03(2020) 029, [1912.05399]

    arXiv 2020

  7. [7]

    A. A. Starobinsky,Stochastic De Sitter (inflationary) stage in the early universe,Lect. Notes Phys.246(1986) 107–126

    1986

  8. [8]

    D. S. Salopek and J. R. Bond,Nonlinear evolution of long wavelength metric fluctuations in inflationary models,Phys. Rev. D42(1990) 3936–3962

    1990

  9. [9]

    Sasaki and E

    M. Sasaki and E. D. Stewart,A General analytic formula for the spectral index of the density perturbations produced during inflation,Prog. Theor. Phys.95(1996) 71–78, [astro-ph/9507001]

    Pith/arXiv arXiv 1996

  10. [10]

    Wands, K

    D. Wands, K. A. Malik, D. H. Lyth and A. R. Liddle,A New approach to the evolution of cosmological perturbations on large scales,Phys. Rev. D62(2000) 043527, [astro-ph/0003278]

    Pith/arXiv arXiv 2000

  11. [11]

    D. H. Lyth and D. Wands,Conserved cosmological perturbations,Phys. Rev. D68(2003) 103515, [astro-ph/0306498]

    Pith/arXiv arXiv 2003

  12. [12]

    G. I. Rigopoulos and E. P. S. Shellard,The separate universe approach and the evolution of nonlinear superhorizon cosmological perturbations,Phys. Rev. D68(2003) 123518, [astro-ph/0306620]

    Pith/arXiv arXiv 2003

  13. [13]

    D. H. Lyth and Y. Rodriguez,The Inflationary prediction for primordial non-Gaussianity, Phys. Rev. Lett.95(2005) 121302, [astro-ph/0504045]

    Pith/arXiv arXiv 2005

  14. [14]

    Artigas, J

    D. Artigas, J. Grain and V. Vennin,Hamiltonian formalism for cosmological perturbations: the separate-universe approach,JCAP02(2022) 001, [2110.11720]

    arXiv 2022

  15. [15]

    J. H. P. Jackson, H. Assadullahi, A. D. Gow, K. Koyama, V. Vennin and D. Wands,The separate-universe approach and sudden transitions during inflation,JCAP05(2024) 053, [2311.03281]

    arXiv 2024

  16. [16]

    Vennin and A

    V. Vennin and A. A. Starobinsky,Correlation Functions in Stochastic Inflation,Eur. Phys. J. C75(2015) 413, [1506.04732]

    Pith/arXiv arXiv 2015

  17. [17]

    Lesgourgues, D

    J. Lesgourgues, D. Polarski and A. A. Starobinsky,Quantum to classical transition of cosmological perturbations for nonvacuum initial states,Nucl. Phys. B497(1997) 479–510, [gr-qc/9611019]

    Pith/arXiv arXiv 1997

  18. [18]

    Grain and V

    J. Grain and V. Vennin,Stochastic inflation in phase space: Is slow roll a stochastic attractor?, JCAP05(2017) 045, [1703.00447]

    arXiv 2017

  19. [19]

    A. A. Starobinsky,Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations,Phys. Lett. B117(1982) 175–178

    1982

  20. [20]

    A. A. Starobinsky,Multicomponent de Sitter (Inflationary) Stages and the Generation of Perturbations,JETP Lett.42(1985) 152–155

    1985

  21. [21]

    Sasaki and T

    M. Sasaki and T. Tanaka,Superhorizon scale dynamics of multiscalar inflation,Prog. Theor. Phys.99(1998) 763–782, [gr-qc/9801017]

    Pith/arXiv arXiv 1998

  22. [22]

    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]. – 30 –

    Pith/arXiv arXiv 2005

  23. [23]

    Finelli, G

    F. Finelli, G. Marozzi, A. Starobinsky, G. Vacca and G. Venturi,Generation of fluctuations during inflation: Comparison of stochastic and field-theoretic approaches,Phys. Rev. D79 (2009) 044007, [0808.1786]

    Pith/arXiv arXiv 2009

  24. [24]

    Finelli, G

    F. Finelli, G. Marozzi, A. A. Starobinsky, G. P. Vacca and G. Venturi,Stochastic growth of quantum fluctuations during slow-roll inflation,Phys. Rev. D82(2010) 064020, [1003.1327]

    Pith/arXiv arXiv 2010

  25. [25]

    Enqvist, S

    K. Enqvist, S. Nurmi, D. Podolsky and G. Rigopoulos,On the divergences of inflationary superhorizon perturbations,JCAP04(2008) 025, [0802.0395]

    Pith/arXiv arXiv 2008

  26. [26]

    Fujita, M

    T. Fujita, M. Kawasaki, Y. Tada and T. Takesako,A new algorithm for calculating the curvature perturbations in stochastic inflation,JCAP12(2013) 036, [1308.4754]

    Pith/arXiv arXiv 2013

  27. [27]

    Panagopoulos and E

    G. Panagopoulos and E. Silverstein,Primordial Black Holes from non-Gaussian tails, 1906.02827

    arXiv 1906

  28. [28]

    D. G. Figueroa, S. Raatikainen, S. Rasanen and E. Tomberg,Non-Gaussian Tail of the Curvature Perturbation in Stochastic Ultraslow-Roll Inflation: Implications for Primordial Black Hole Production,Phys. Rev. Lett.127(2021) 101302, [2012.06551]

    arXiv 2021

  29. [29]

    D. G. Figueroa, S. Raatikainen, S. Rasanen and E. Tomberg,Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation,JCAP05(2022) 027, [2111.07437]

    arXiv 2022

  30. [30]

    Pattison, V

    C. Pattison, V. Vennin, D. Wands and H. Assadullahi,Ultra-slow-roll inflation with quantum diffusion,JCAP04(2021) 080, [2101.05741]

    arXiv 2021

  31. [31]

    Tomberg,A numerical approach to stochastic inflation and primordial black holes,J

    E. Tomberg,A numerical approach to stochastic inflation and primordial black holes,J. Phys. Conf. Ser.2156(2021) 012010, [2110.10684]

    arXiv 2021

  32. [32]

    Rigopoulos and A

    G. Rigopoulos and A. Wilkins,Inflation is always semi-classical: diffusion domination overproduces Primordial Black Holes,JCAP12(2021) 027, [2107.05317]

    arXiv 2021

  33. [33]

    Achucarro, S

    A. Achucarro, S. Cespedes, A.-C. Davis and G. A. Palma,The hand-made tail: non-perturbative tails from multifield inflation,JHEP05(2022) 052, [2112.14712]

    arXiv 2022

  34. [34]

    J. M. Ezquiaga, J. Garc´ ıa-Bellido and V. Vennin,Massive Galaxy Clusters Like El Gordo Hint at Primordial Quantum Diffusion,Phys. Rev. Lett.130(2023) 121003, [2207.06317]

    arXiv 2023

  35. [35]

    Animali and V

    C. Animali and V. Vennin,Primordial black holes from stochastic tunnelling,JCAP02(2023) 043, [2210.03812]

    arXiv 2023

  36. [36]

    Cai, X.-H

    Y.-F. Cai, X.-H. Ma, M. Sasaki, D.-G. Wang and Z. Zhou,Highly non-Gaussian tails and primordial black holes from single-field inflation,JCAP12(2022) 034, [2207.11910]

    arXiv 2022

  37. [37]

    A. D. Gow, H. Assadullahi, J. H. P. Jackson, K. Koyama, V. Vennin and D. Wands, Non-perturbative non-Gaussianity and primordial black holes,EPL142(2023) 49001, [2211.08348]

    arXiv 2023

  38. [38]

    Tomberg,Stochastic constant-roll inflation and primordial black holes,Phys

    E. Tomberg,Stochastic constant-roll inflation and primordial black holes,Phys. Rev. D108 (2023) 043502, [2304.10903]

    arXiv 2023

  39. [39]

    Briaud and V

    V. Briaud and V. Vennin,Uphill inflation,JCAP06(2023) 029, [2301.09336]

    arXiv 2023

  40. [40]

    Vennin and D

    V. Vennin and D. Wands,Quantum Diffusion and Large Primordial Perturbations from Inflation. 2025.2402.12672. 10.1007/978-981-97-8887-3 8

    work page doi:10.1007/978-981-97-8887-3 2025

  41. [41]

    R. Inui, H. Motohashi, S. Pi, Y. Tada and S. Yokoyama,Constant roll and non-Gaussian tail in light of logarithmic duality,JCAP02(2025) 042, [2409.13500]

    arXiv 2025

  42. [42]

    Sharma,Stochastic inflation and non-perturbative power spectrum beyond slow roll,JCAP 03(2025) 017, [2411.08854]

    D. Sharma,Stochastic inflation and non-perturbative power spectrum beyond slow roll,JCAP 03(2025) 017, [2411.08854]

    arXiv 2025

  43. [43]

    Ando and V

    K. Ando and V. Vennin,Power spectrum in stochastic inflation,JCAP04(2021) 057, [2012.02031]. – 31 –

    arXiv 2021

  44. [44]

    Tada and V

    Y. Tada and V. Vennin,Statistics of coarse-grained cosmological fields in stochastic inflation, JCAP02(2022) 021, [2111.15280]

    arXiv 2022

  45. [45]

    Animali and V

    C. Animali and V. Vennin,Clustering of primordial black holes from quantum diffusion during inflation,JCAP08(2024) 026, [2402.08642]

    arXiv 2024

  46. [46]

    Animali, P

    C. Animali, P. Auclair, B. Blachier and V. Vennin,Harvesting primordial black holes from stochastic trees with FOREST,JCAP05(2025) 019, [2501.05371]

    arXiv 2025

  47. [47]

    A. D. Linde, D. A. Linde and A. Mezhlumian,From the Big Bang theory to the theory of a stationary universe,Phys. Rev. D49(1994) 1783–1826, [gr-qc/9306035]

    Pith/arXiv arXiv 1994

  48. [48]

    Jain and M

    M. Jain and M. P. Hertzberg,Statistics of Inflating Regions in Eternal Inflation,Phys. Rev. D 100(2019) 023513, [1904.04262]

    arXiv 2019

  49. [49]

    Auclair,Forest: Fortran recursive exploration of stochastic trees, Apr., 2025

    P. Auclair,Forest: Fortran recursive exploration of stochastic trees, Apr., 2025. 10.5281/zenodo.15235932

    work page doi:10.5281/zenodo.15235932 2025

  50. [50]

    Jedamzik,The Cloud in cloud problem in the Press-Schechter formalism of hierarchical structure formation,Astrophys

    K. Jedamzik,The Cloud in cloud problem in the Press-Schechter formalism of hierarchical structure formation,Astrophys. J.448(1995) 1–17, [astro-ph/9408080]

    Pith/arXiv arXiv 1995

  51. [51]

    Shibata and M

    M. Shibata and M. Sasaki,Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity,Phys. Rev. D60(1999) 084002, [gr-qc/9905064]

    Pith/arXiv arXiv 1999

  52. [52]

    J. C. Niemeyer and K. Jedamzik,Near-critical gravitational collapse and the initial mass function of primordial black holes,Phys. Rev. Lett.80(1998) 5481–5484, [astro-ph/9709072]

    Pith/arXiv arXiv 1998

  53. [53]

    Musco, J

    I. Musco, J. C. Miller and L. Rezzolla,Computations of primordial black hole formation,Class. Quant. Grav.22(2005) 1405–1424, [gr-qc/0412063]

    Pith/arXiv arXiv 2005

  54. [54]

    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]

    Pith/arXiv arXiv 2014

  55. [55]

    Harada, C.-M

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

    Pith/arXiv arXiv 2013

  56. [56]

    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]

    Pith/arXiv arXiv 2015

  57. [57]

    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]

    arXiv 2019

  58. [58]

    Escriv` a, C

    A. Escriv` a, C. Germani and R. K. Sheth,Universal threshold for primordial black hole formation,Phys. Rev. D101(2020) 044022, [1907.13311]

    arXiv 2020

  59. [59]

    Musco, V

    I. Musco, V. De Luca, G. Franciolini and A. Riotto,Threshold for primordial black holes. II. A simple analytic prescription,Phys. Rev. D103(2021) 063538, [2011.03014]

    arXiv 2021

  60. [60]

    Escriv` a, C

    A. Escriv` a, C. Germani and R. K. Sheth,Analytical thresholds for black hole formation in general cosmological backgrounds,JCAP01(2021) 030, [2007.05564]

    arXiv 2021

  61. [61]

    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]

    arXiv 2024

  62. [62]

    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]

    arXiv 2026

  63. [63]

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

    Pith/arXiv arXiv 2011

  64. [64]

    B. J. Carr and T. Harada,Separate universe problem: 40 years on,Phys. Rev. D91(2015) 084048, [1405.3624]. – 32 –

    Pith/arXiv arXiv 2015

  65. [65]

    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]

    arXiv 2023

  66. [66]

    Harada,Primordial Black Holes: Formation, Spin and Type II,Universe10(2024) 444, [2409.01934]

    T. Harada,Primordial Black Holes: Formation, Spin and Type II,Universe10(2024) 444, [2409.01934]

    arXiv 2024

  67. [67]

    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]

    arXiv 2025

  68. [68]

    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]

    arXiv 2025

  69. [69]

    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]

    arXiv 2025

  70. [70]

    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]

    arXiv 2025

  71. [71]

    Uehara, A

    K. Uehara, A. Escriv` a, T. Harada, D. Saito and C.-M. Yoo,Primordial black hole formation from a type II perturbation in the absence and presence of pressure,JCAP08(2025) 042, [2505.00366]

    arXiv 2025

  72. [72]

    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

  73. [73]

    J. M. Bardeen, J. R. Bond, N. Kaiser and A. S. Szalay,The Statistics of Peaks of Gaussian Random Fields,ApJ304(1986) 15

    1986

  74. [74]

    Escriv` a and C.-M

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

    arXiv 2025

  75. [75]

    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]

    arXiv 2025

  76. [76]

    Germani and R

    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]

    arXiv 2023

  77. [77]

    Germani and R

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

    arXiv 2020

  78. [78]

    Harada, H

    T. Harada, H. Iizuka, Y. Koga and C.-M. Yoo,Geometrical origin for the compaction function for primordial black hole formation,Phys. Rev. D111(2025) 023537, [2409.05544]

    arXiv 2025

  79. [79]

    Germani and L

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

    arXiv 2026

  80. [80]

    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]

    arXiv 2025

Showing first 80 references.