Harvesting primordial black holes from stochastic trees with $\texttt{FOREST}$

Baptiste Blachier; Chiara Animali; Pierre Auclair; Vincent Vennin

REVIEW 3 major objections 4 minor 3 cited by

Harvesting primordial black holes from stochastic trees with $\texttt{FOREST}$

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that inflationary spacetime can be simulated as a branching stochastic tree, making primordial black hole statistics—including nested collapse—computable without a fixed lattice.

desk verdict FOREST delivers a genuinely new numerical tool for stochastic inflation and the tree machinery is well validated; the PBH mass-function numbers rest on a heuristic proxy that needs a sensitivity check before being used. read the letter →

arxiv 2501.05371 v2 pith:VBDQZOSL submitted 2025-01-09 astro-ph.CO gr-qc

Chiara Animali , Pierre Auclair , Baptiste Blachier , Vincent Vennin This is my paper · ORCID

classification astro-ph.COgr-qc

keywords stochasticinflationdelta-Nformalismprimordialblackholestreescurvatureperturbationcloud-in-cloudeffectquantumdiffusionbranchingprocesses

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 the stochastic-δN program can be implemented on stochastic trees, where each Hubble patch branches into two independent patches at volume doubling, and that this makes real-space maps of the curvature perturbation numerically cheap and straightforward to build. The payoff is a criterion for primordial black hole formation based on unbalanced nodes, with nested collapse (the cloud-in-cloud effect) handled automatically by the tree recursion. Running the accompanying FOREST code on the quantum-well toy model, the authors find broad PBH mass distributions with mild power laws terminated by exponential tails. They also show that the total PBH abundance roughly agrees with earlier approximations, while the detailed mass distribution does not, because PBHs form in the far tail of the volume distribution where central-limit reasoning fails.

What carries the argument

The central object is the stochastic tree: a binary branching structure whose root is a single Hubble patch and whose nodes are Hubble patches at volume-doubling times, each child carrying an independent realization of the Langevin equation, with leaves at the end-of-inflation hypersurface. The load-bearing identity is the unbalance index, built from the coarse-shelled curvature perturbation $\zeta_{\ell i}/\zeta_{\ell i,c} = \frac{2}{\ln(V_i/V_\ell)}\frac{V_m}{V_i}(W_\ell - W_m)$, which after window-function matching and the threshold $C_c=0.5$ becomes the criterion for PBH formation. This index is what lets the code flag PBHs at unbalanced nodes and, through upward recursion, automatically apply the cloud-in-cloud exclusion so that only the outermost collapsing regions survive.

What would settle it

Run FOREST on the quantum-well model while replacing the window-function proxy with an exact coarse-grained density contrast or compaction function computed from the reconstructed comoving maps; if the resulting PBH mass fraction and mass function differ substantially from the paper's figures, the unbalance-index criterion is not reliable. A sharper version is to vary the fixed threshold $C_c$ across the known range $2/5$ to $2/3$ and check whether the reported broad power laws and exponential tails survive.

Watch

Extended reading notes

Core claim

The central claim is that the causal structure of inflating spacetime is faithfully encoded in a binary tree: every Hubble-sized patch evolves stochastically via the Langevin equation and, after $\Delta N = \ln(2)/3$ e-folds, doubles and splits into two independent patches, with leaves marking where inflation ends. Using the $\delta N$ formalism, the curvature perturbation coarse-grained over any node is $N_{1\to i} + W_i - W_1$, computed from leaf volumes and volume-averaged expansions propagated upward through the tree. PBH formation is then decided at each node by the coarse-shelled curvature perturbation $\zeta_{\ell i} = \zeta_\ell - \zeta_i$, which stands in for the compaction function after an approximate window-function matching and with a fixed threshold $C_c = 0.5$; nodes whose two child subtrees expand very unequally are said to be unbalanced and form PBHs, and when PBHs are nested the outer one discards the inner ones. In the quantum-well model, FOREST yields a PBH mass fraction $f_{\mathrm{PBH,end}}$ that increases with the quantum-diffusion parameter $\mu$ and approaches nearly the whole universe near criticality, with mass distributions showing mild power laws followed by exponential tails.

Load-bearing premise

The entire PBH statistics rest on replacing the sibling-branch expansion difference with the compaction function through an approximate window-function matching and a fixed threshold $C_c=0.5$; if that proxy is inaccurate, the predicted mass fractions and mass distributions do not describe real black hole formation.

Editorial extensions

If this is right

PBH mass fractions and mass functions can be computed without a fixed spatial lattice and without requiring inflation to continue everywhere until the grid is post-processed.
The cloud-in-cloud problem is resolved automatically by the tree recursion, since outer PBHs discard inner ones as information propagates from leaves to root.
Volume-weighted first-passage-time distributions reconstructed from independent Langevin realizations match the tree distributions, validating the volume-weighting procedure used in analytical approximations.
The large-volume (central-limit) approximation reproduces the total PBH abundance to within about an order of magnitude but fails to capture the mass distribution, because PBHs form from anomalously large subtrees far beyond the mean volume.
The tree method can probe one-point statistics down to probabilities around $10^{-10}$ and scale to populations above $10^{12}$ trees, opening the way to non-perturbative maps and multi-point statistics.

Reading between the lines

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

A natural extension, not made in the paper, is to stress-test the window-function proxy by comparing the unbalance index against exact coarse-grained density contrasts or compaction functions computed from the reconstructed maps, since all reported PBH statistics inherit the proxy's accuracy.
The method's efficiency suggests it could be combined with importance sampling to reach even deeper tails, potentially clarifying behavior as the eternal-inflation boundary is approached.
Because the tree construction carries the full causal genealogy of leaves, tree-generated curvature maps could be fed into PBH-clustering or N-body pipelines to test whether the unbalance criterion predicts the same spatial clustering as exact compaction-function peaks.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Marie — read this one. It is the first real implementation of stochastic inflation on branching trees, and the tree part holds up well. The code FOREST is public, scales to 10^11 trees, and the paper validates it against the analytical volume-weighted first-passage-time distribution (fig. 7) and against de Sitter QFT correlators at large scales (sec. 5.3). That is real work, and it makes the framework credible. Cloud-in-cloud handling is elegant and built into the recursive structure, not bolted on.

The novelty is honest: branching-process representations go back to Linde et al. and Jain & Hertzberg, but the authors develop the full stochastic-delta-N program on trees — real-space maps, coarse-grained curvature, PBH harvesting — which I have not seen done before. The large-volume approximation comparison (sec. 5.2) is also a nice sanity check: the Press-Schechter estimate gets the abundance within an order of magnitude, while the mass distribution cannot be captured.

The soft spot is exactly where the stress-test note points. The PBH results in sec. 4.3 rest on eq. (3.7): the coarse-shelled curvature perturbation is identified with the compaction function through a window-function replacement, with alpha and beta matched in a heuristic way, plus a fixed threshold Cc=0.5. The footnote claims results 'do not depend much' on Cc, but no scan is shown, and there is no direct comparison of the unbalance index against the actual compaction function on the reconstructed maps, nor against a lattice/STOLAS calculation in the same model. The mass distributions in fig. 8 are the paper's main physical output; as they stand they are suggestive, not yet established. This is the single load-bearing gap.

A minor point: the maps in figs. 3-4 use a splitting prescription that breaks symmetries at small scales. The authors know this and show the artefact decays at large scales; it does not threaten the one-point statistics they focus on.

If I were the editor I would send it to referees. The tree framework deserves publication even if the PBH numbers are later revised. The authors should be asked to add a Cc scan, error bars, and ideally a compaction-function check or a STOLAS comparison in a companion paper. That is a condition, not a reason to reject.

Bottom line: for anyone working on stochastic inflation and PBHs, this is worth citing and worth a serious referee. I would bring it to the group — it will generate a good discussion.

Referee Report

3 major / 4 minor

Summary. The paper introduces a numerical framework, FOREST, that implements the stochastic-δN formalism on binary stochastic trees. Each Hubble patch splits after ΔN = ln(2)/3 e-folds, and the two daughter patches evolve as independent Langevin realizations; the tree structure is then used to compute volume-weighted expansion factors, the curvature perturbation ζ at leaves and at coarse-grained nodes, and real-space maps via a recursive plane-splitting prescription. For primordial black holes, the authors define a 'coarse-shelled' curvature perturbation ζ_ℓi (Section 3.1) and, through an approximate window-function replacement (Eq. 3.7), identify it with the compaction function. Nodes whose unbalance index exceeds a threshold derived from Cc = 0.5 are counted as PBHs, with cloud-in-cloud effects handled by discarding inner PBHs when a larger one is found. The framework is applied to the flat quantum-well model, yielding PBH mass fractions and mass distributions with mild power laws and exponential tails (Section 4.3). The rest of the paper checks volume weighting against an analytic first-passage-time formula (Section 5.1), compares with the large-volume approximation (Section 5.2), and analyzes discretization artefacts against QFT correlators (Section 5.3).

Significance. If the PBH mapping is validated, the paper would provide a substantially more efficient alternative to lattice stochastic inflation for generating non-Gaussian curvature maps and PBH statistics, with cloud-in-cloud accounting built into the tree recursion. The paper has concrete strengths: FOREST is publicly released under GPLv3, strong- and weak-scaling tests are reported (Appendix A), the volume-weighted first-passage-time distribution matches the analytic formula in Fig. 7, and the large-scale two-point function agrees with QFT in the light-test-field limit (Section 5.3). The counting-function derivation in Appendix C is also a useful closed-form check. However, the core PBH results — f_PBH and the mass distributions in Fig. 8 — rest on a window-matching heuristic and a fixed threshold whose robustness is asserted but not demonstrated. The tree machinery itself is well supported; the gap is specifically in the conversion from tree-node contrasts to PBH formation.

major comments (3)

[Section 3.1, Eq. (3.7)]
[Section 3.2, footnote 3]
[Section 5.3.2]

minor comments (4)

[Section 3.1]
[Figures 6–8]
[Section 3.3, Eq. (3.12)]
[Abstract and Section 6]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FOREST's tree statistics are benchmarked against external analytic results, and the PBH criterion is an openly stated heuristic proxy rather than a fitted quantity.

full rationale

The main derivation chain is self-contained: starting from the Langevin equation (2.1), the tree is generated by volume doubling and independent branch evolution; the curvature perturbation (2.5)-(2.6) and the coarse-shelled quantity (3.1) are definitions following the δN formalism, not outputs fitted to the target quantities. The PBH criterion uses the window-function replacement (3.7) with matching parameters and a fixed threshold Cc=0.5 (Section 3.2); this is an explicit approximation (the paper calls ζℓi a 'proxy for the compaction function'), so any error in it is a robustness/correctness risk, not circularity. The main results are numerical outputs of FOREST (figs. 6-8), and the paper validates the tree machinery against independent benchmarks: the volume-weighted first-passage-time distribution matches the analytic formula (5.1) in fig. 7, and the discretized two-point function reproduces QFT results in de Sitter (Section 5.3). Citations to the authors' prior stochastic-δN work ([51], [60], [68], [69], etc.) supply analytic comparison curves and standard relations, but the new PBH mass distributions and f_PBH values are not derived from those citations; the figures are simulation products. The fixed Cc and the untested real-space accuracy of the coarse-shelled proxy are legitimate concerns, but they are not cases where a 'prediction' is equivalent to its inputs by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities; the tree is a numerical representation. The main externally chosen inputs are the PBH threshold Cc and the tree-splitting idealization.

free parameters (2)

Cc (compaction threshold) = 0.5
Assumed fixed threshold for PBH formation in eq. (3.9), chosen within the literature range 2/5 to 2/3. The paper asserts results do not depend much on it, but shows no scan. It directly sets the PBH formation criterion zeta_li,c = ln(V_i/V_l)/2.
kappa (adaptive step-size control) = 5
Controls the adaptive integration step near the absorbing boundary in eq. (B.1). It is chosen by hand for numerical convergence and affects the first-passage time only in a controlled 5-sigma window, so it is not a physical fit.

assumptions (6)

domain assumption Separate-universe picture: Hubble patches become causally disconnected and evolve independently after they split.
Foundational to the tree model; invoked in section 2.1 and section 1 (refs [36-43]).
domain assumption The Langevin equation (2.1) with white noise describes the coarse-grained inflaton dynamics.
Standard stochastic-inflation formalism, imported from refs [47-51].
ad hoc to paper Each Hubble patch splits into exactly two equal-volume patches after Delta N = ln(2)/3 e-folds.
A modeling idealization that defines the binary tree; section 2.1. The authors show in section 5.3.1 that the precise branching time is degenerate with the coarse-graining scale, but the binary split itself is assumed.
ad hoc to paper The coarse-shelled curvature perturbation zeta_li is a valid proxy for the compaction function with matching condition (3.7).
Used to set the PBH formation criterion in eqs. (3.8)-(3.9); the window-matching is approximate and is the weakest link for PBH statistics.
domain assumption Cloud-in-cloud effects are handled by discarding PBHs nested inside a larger collapsing region.
Standard Press-Schechter-style counting, implemented in section 3.4; assumes the larger region's collapse swallows smaller ones.
domain assumption Instantaneous reheating and radiation domination (w = 1/3) at PBH formation.
Used to set the equation of state in eqs. (3.8)-(3.9) and the mass estimate (3.12).

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Cite this review

Pith. "Pith review of Harvesting primordial black holes from stochastic trees with $\texttt{FOREST}$." pith.science (2026). https://pith.science/paper/VBDQZOSL

@misc{pith2026250105371,
  author       = {Pith},
  title        = {Pith review of: Harvesting primordial black holes from stochastic trees with $\textttFOREST$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBDQZOSL}},
  note         = {Machine review of arXiv:2501.05371}
}

abstract

We introduce a novel framework to implement stochastic inflation on stochastic trees, modelling the inflationary expansion as a branching process. Combined with the $\delta N$ formalism, this allows us to generate real-space maps of the curvature perturbation that fully capture quantum diffusion and its non-perturbative backreaction during inflation. Unlike lattice methods, trees do not proceed on a fixed background since new spacetime units emerge dynamically as trees unfold, naturally incorporating metric fluctuations. The recursive structure of stochastic trees also offers remarkable numerical efficiency, and we develop the FOrtran Recursive Exploration of Stochastic Trees ($\texttt{FOREST}$) tool and demonstrate its performance. We show how primordial black holes blossom at unbalanced nodes of the trees, and how their mass distribution can be obtained while automatically accounting for the "cloud-in-cloud" effect. In the "quantum-well" toy model, we find broad mass distributions, with mild power laws terminated by exponential tails. We finally compare our results with existing approximations in the literature and discuss several prospects.

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Forward citations

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