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

T0 review · 1 major / 1 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read 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.

desk verdict The paper maps compaction to volume ratios on stochastic binary trees to classify type-I/II PBHs and shows broader mass functions plus higher abundance in the quantum regime, but the central mapping is flagged by the authors themselves as approximate. read the letter →

arxiv 2606.30087 v1 pith:W3KW74EX submitted 2026-06-29 astro-ph.CO gr-qc

Chiara Animali , Pierre Auclair , Baptiste Blachier , Eemeli Tomberg , Vincent Vennin This is my paper · ORCID
classification astro-ph.COgr-qc
keywords stochasticinflationprimordialblackholescompactionfunctiontypeIandIIfluctuationsquantumdiffusioneternalbinarytreescloud-in-cloudeffects
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

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.

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

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.

Watch

Extended reading notes

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

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.

Editorial extensions

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

Reading between the lines

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
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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

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

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

assumptions (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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0 comments
Cite this review

Pith. "Pith review of Compaction function in stochastic inflation: a \texttt{FOREST} of type I and II primordial black holes." pith.science (2026). https://pith.science/paper/W3KW74EX

@misc{pith2026260630087,
  author       = {Pith},
  title        = {Pith review of: Compaction function in stochastic inflation: a \textttFOREST of type I and II primordial black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W3KW74EX}},
  note         = {Machine review of arXiv:2606.30087}
}
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.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Primordial black holes forming during kination: the trapped, the overdense, and the void

    astro-ph.CO 2026-07 conditional novelty 6.0 of 10

    In one non-attractor inflation model, initially similar field fluctuations produce three distinct black-hole formation channels—trapped, overdense, void—with collapse thresholds determined by the full density profile,...

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Reviewed June 30, 2026 · model on record in the stance chip above.