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Combining large-deviation theory with excursion-set walks, this paper derives analytic first-passage-time distributions for halos and voids from non-Gaussian exponential tails, predicting strongly enhanced counts of rare massive halos and l

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2026-08-01 21:11 UTC pith:VCTZ7PBA

load-bearing objection Clever proof of concept for LDP plus excursion sets, but the central t^{q/2} scaling in Eq. (4.5) relies on an unstated equal-variance assumption that breaks the stated contract. the 3 major comments →

arxiv 2607.16152 v1 pith:VCTZ7PBA submitted 2026-07-17 astro-ph.CO hep-ph

Large deviations for halos and voids: beyond perturbative non-gaussianities

Martin Teuscher , Ruth Durrer , Julien Grain , Killian Martineau , Aur\'elien Barrau This is my paper
classification astro-ph.CO hep-ph
keywords primordial non-Gaussianitylarge deviation principleexcursion set formalismhalo mass functionvoid size functionfirst-passage timeexponential tailsnon-perturbative
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the large deviation principle, combined with excursion-set theory, gives fully analytical predictions for the abundance of rare dark-matter halos and cosmic voids when primordial fluctuations have strongly non-Gaussian, exponentially decaying tails. For densities with per-mode distributions p(δ_k) ∝ exp(−|δ_k|^q/σ_k^q), q≥1, it derives closed-form first-passage-time distributions for random walks—Eq. (4.6) for the one-barrier (halo) problem and Eq. (5.6) for the two-barrier (void-in-cloud) problem. These reduce to the standard Gaussian results at q=2, and for q<2 they predict dramatically more very massive halos and large voids, for example up to 10^5 times more 10^13 M⊙ halos at z=8. The authors present this as a proof of concept that large-deviation techniques can connect inflationary tails to late-time structure counts non-perturbatively.

Core claim

Starting from the observation that a symmetric, Markovian, scale-invariant random walk obeys the first-passage identity f_FPT(t)=b t^{−3/2} p_{Δ(1)}(b/√t) without any Gaussian assumption, the paper computes p_{Δ(1)} for the exponential-tail family by applying the large-deviation contraction principle to independent Fourier modes. The optimization reduces to a Hölder-inequality lemma that yields, for q≥1, p_Δ(δ) ∝ exp(−(|δ|γ/√t)^q), with γ fixed by the variance. Inserting this into the first-passage identity gives a new halo mass function, and a Laplace-transform inversion of the two-barrier problem gives a new void size function. The paper's quantitative claims are that q<2 enhances the abun

What carries the argument

The argument rests on two pieces. First, the large-deviation rate function for each Fourier mode, I_k(|δ_k|)=|δ_k|^q, whose sum over independent half-space modes is minimized under the linear constraint that defines the smoothed density; a Hölder-inequality lemma solves this infinite-dimensional optimization exactly for q≥1, producing the exponential-tail distribution (4.5). Second, the non-Gaussian first-passage identity f_FPT(t) = (b/t^{3/2}) p_{Δ(1)}(b/√t), valid for any symmetric Markovian walk with the diffusion property, which transposes the field distribution into crossing-time statistics; for voids, the same distribution is fed through a Laplace-transform relation for the two-barrier

Load-bearing premise

The calculation assumes that the non-Gaussian Fourier modes at different scales stay mutually independent once grouped in a half-space, which makes the random walk Markovian; realistic inflationary models usually generate Fourier-space correlations, and if those correlations are not negligible the predicted rate function and all downstream abundances shift.

What would settle it

Simulate a random field with per-mode exponential tails (e.g. q=1) and a fixed power spectrum, smooth it with a Fourier top-hat, and measure the first-passage distribution of the smoothed contrast across scales. If the measured f_FPT(t) deviates from Eq. (4.6) in the regime t ≪ δ_c^2, the assumption of independent Fourier modes (Markovianity) is violated, or sub-exponential prefactors fixed by the large-deviation principle are substantial.

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

If this is right

  • For exponential-like tails (1 ≤ q < 2), the halo mass function is enhanced at high masses by orders of magnitude, with e.g. a 10^5-fold increase for ~10^13 M⊙ halos at z=8 relative to Gaussian.
  • Large voids become far more abundant: at z=8 the abundance of ~2 Mpc voids is enhanced by 10^4, and at z=0 by 10^5 for ~40 Mpc voids.
  • For q > 2, rare massive halos and large voids are suppressed relative to the Gaussian case.
  • The log-normal asymmetric case predicts 10^3–10^5 times more massive halos while depleting large underdensities, offering a non-perturbative route to explaining overabundant high-redshift massive galaxies.
  • All formulas reduce to the known Gaussian excursion-set results at q=2 and α=0, and the validity of the large-deviation tail is restricted to t < δ_c^2, i.e., large objects.

Where Pith is reading between the lines

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

  • The single-parameter family q is a ready-made observational discriminant: fitting q to high-mass cluster counts or the void size function could constrain the tail shape independently of bispectrum-style analyses.
  • The paper's Fourier-space independence assumption is in tension with the real-space δN form of inflationary non-Gaussianity; the authors' own estimate shows O(u) corrections to the rate function from scale correlations, so the next natural step is to fold an Edgeworth-style correlation term into the contraction, which the framework allows.
  • Since the gambler's ruin normalization fails for q≠2 outside the large-deviation regime, the void-size-function prediction likely needs a matching or interpolation to the Gaussian behavior at t ~ δ_c^2; detecting where the series stops converging in simulations would map the true domain of validity.
  • The prediction that q<2 boosts both halos and voids at high redshift is a concrete, testable target for forthcoming wide-area surveys and could be checked by measuring the void size function at z ≳ 1, where the large-deviation validity extends to smaller scales.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper applies large-deviation theory (LDP) to excursion-set theory, aiming to derive first-passage-time distributions for cosmological density fluctuations whose Fourier modes have strongly non-Gaussian tails p(|δ_k|) ∼ exp(−|δ_k|^q/σ^q). Under the assumptions of independent Fourier modes and Markovian random walks, it obtains a smoothed-density distribution (Eq. 4.5), a halo mass function via the first-passage-time formula (Eq. 4.6), a log-normal extension (Section 4.2), and a void size function from the two-barrier problem (Eq. 5.6). The results reduce to known Gaussian results at q = 2 and predict strong enhancement of rare massive halos and large voids for q < 2.

Significance. If the derivation from the stated assumptions were fully valid, this would be a useful non-perturbative contribution: it provides closed-form, analytically checkable formulas for a class of exponential-tail models, it recovers the Gaussian limit, and it makes specific falsifiable predictions for high-mass and large-void abundances. The mathematical core — the optimization lemma in Appendix B.2 and the Laplace-transform treatment of the two-barrier problem — is sound and is a genuine asset. The paper also honestly labels itself a proof of concept and identifies directions for future work. However, as detailed below, the main formulas rely on an unstated equal-variance assumption and on promoting an LDP tail to a full normalized density with a free exponent α; these points are load-bearing for the claimed physical predictions.

major comments (3)
  1. [§3.2, Eq. (3.24); §4.1, Eqs. (4.2)–(4.5)] The joint rate function is not Σ_k I_k when the mode speeds differ. Section 3.1.1 defines ε = sup_k ε_k, and Eq. (4.3) sets ε_k = σ_k^q with σ_k varying for any realistic power spectrum. For independent modes, p_joint = ∏ exp(−I_k/ε_k) = exp(−(1/ε) Σ_k (ε/ε_k) I_k), so Eq. (3.24) requires ε_k = ε for all k. The optimization lemma B.2 minimizes the unweighted norm ||f||_q^q; with mode-dependent ε_k the contraction problem is weighted by σ_k^q and gives a different exponent and scale dependence. Thus Eqs. (4.4) and (4.5), and hence Eqs. (4.6) and (5.6), are valid only under an additional equal-variance assumption that is not stated or derived.
  2. [§4.1, Eqs. (4.4)–(4.5); §5 and App. C.2] The LDP fixes only the asymptotic tail, not the full normalized density. Eq. (4.5) promotes the tail to a complete probability distribution with variance t and introduces α > −1 as a free parameter. The paper acknowledges that α is not determined by the LDP and later chooses α = (q−2)/2 in Section 5 and Appendix C.2 to make the Laplace transform tractable. Since Eq. (4.5) is used as the exact input to the void two-barrier calculation, the final formulas are not unique LDP predictions; they depend on an extra modeling choice. A sensitivity analysis or explicit discussion of how the results vary with α is needed before Eqs. (4.6) and (5.6) can be presented as predictions.
  3. [§3.1.2, Eqs. (3.15)–(3.21); §3.2, Eq. (3.22)] The paper's own estimate in Eq. (3.21) states that mode correlations can give O(u) corrections to the rate function, so they are not negligible in the ε → 0 limit. Realistic inflationary non-Gaussianity is usually defined in real space, leading to Fourier correlations; the independent-mode model (3.22) is an ansatz, not a consequence of the physics. The text acknowledges this as a proof of concept, but the abstract and conclusions describe the results as a bridge to inflationary scenarios. The manuscript should state explicitly that Eqs. (4.6) and (5.6) are predictions of the independent-mode model, and should quantify the regime (if any) in which the O(u) corrections are small enough not to alter the conclusions.
minor comments (4)
  1. [Throughout] There are several typos: 'presnt' (§1), 'radily' (§2.2), 'adressed' (§3.2), 'aforemetioned' (§1), 'orrectly' (footnote, §2.1), 'intuitevely' (App. B.1), and 'this enough is not enough' (App. A). The manuscript would benefit from a careful proofread.
  2. [§4.1, Eq. (4.4)] The displayed formula is ambiguous: it is unclear whether the factor 2πV/(3R^3) multiplies the exponential or sits in the denominator. Please add parentheses and define all symbols (V appears both as a Fourier-space volume and as a real-space volume).
  3. [§4.2 and Fig. 2] The decimal separator changes between '1.686' and '1,686' (also '1,686' near Eq. (4.12)). Use a consistent convention. In Fig. 2, the quantity νf_FPT(ν) is stated but the transformation from t to ν is not shown on the figure; a one-line reminder would help.
  4. [§5, Eq. (5.6)] The notation f_FPT(t|b=c_n, ...) in Eq. (5.8) is introduced without defining the conditional-style argument. It would be clearer to write f_FPT(t; c_n, ...) or to define the parameters explicitly.

Circularity Check

1 steps flagged

Central contraction assumes equal Fourier-mode variances, making the headline FPT prediction reduce to an unstated input; no self-citation circularity.

specific steps
  1. other [Section 3.2, Eq. (3.24) and Section 4.1, Eqs. (4.2)-(4.3)]
    "Their joint rate function (3.2) then breaks down into Ijoint(|z|) = Σ_{k∈K+(R)} I_k(|z_k|) ... Ik(|δk|)≡−lim_{εk→0} εk lnp∆k(|δk|)=|δk|^q, where we have identified the small parameter of the LDP as εk=σ^q_k."

    With independent modes, ∏_k p_Δk ≍ exp(−Σ|z_k|^q/σ_k^q). The paper's own joint LDP (3.2) fixes the global speed as ε=sup_k ε_k, so the correct joint rate is Σ(ε/ε_k)|z_k|^q, not Σ|z_k|^q. Eq. (3.24) is therefore valid only when σ_k is k-independent. The contraction (3.25) then invokes lemma (B.14) on the unweighted norm ||f||_q; consequently the exponent in Eq. (4.4) and the t^{q/2} scaling of Eqs. (4.5)-(4.6) are consequences of an unstated equal-variance model rather than of the stated σ_k-dependent densities (4.1). The prediction is in this part built by construction from a hidden input; the paper's own §3.1.2 estimate that correlations contribute O(u) corrections makes it additionally conditional.

full rationale

There is no load-bearing self-citation chain: the reflection principle, the two-barrier Laplace relation (C.11) from Ref. [48], and the LDP theorems from Dembo-Zeitouni/Touchette are external, independently checkable results. The main derivation is mathematical and self-contained given the stated assumptions. However, the central contraction is not executed from the stated mode-dependent LDP: Eq. (3.24) silently drops the ε_k = σ_k^q speeds, so Eqs. (4.4)-(4.6) and the void result (5.6) effectively rely on an equal-variance assumption that is neither stated nor derived. The paper also acknowledges that Fourier-space correlations can produce O(u) corrections to the rate function (§3.1.2) and that real-space inflationary distributions are not yet connected (§6), so the halo/void predictions are conditional on the independence assumption. Finally, α in Eq. (4.5) is unconstrained by the LDP and is chosen in Appendix C.2 as α=(q−2)/2 to simplify the Laplace inversion; this shapes the void-size-function prediction, though it is an acknowledged free parameter rather than a fit. These are assumption and construction gaps rather than self-citation circularity, so the score is moderate: one central prediction partially reduces by construction to a hidden input.

Axiom & Free-Parameter Ledger

2 free parameters · 8 axioms · 0 invented entities

The central result rests on two modelling choices not fixed by data or by inflation: the tail exponent q and the factorized independent-mode ansatz. The mathematical machinery (LDP, Hölder, Laplace, reflection principle) is standard. Alpha is an extra degree of freedom used for tractability. No new physical entities are introduced.

free parameters (2)
  • q
    Tail exponent of the Fourier-mode distribution in Eq. (4.1); a free parameter of the model, varied between 1 and 2.5 in the figures and not derived from any specific inflation model.
  • alpha = alpha = (q-2)/2 for voids; alpha = 0 for the halo plots
    Polynomial prefactor exponent in Eq. (4.5), undetermined by the LDP; chosen in Appendix C.2 to simplify the Laplace transform of the first-passage distribution.
axioms (8)
  • domain assumption Statistical homogeneity implies uniform Fourier phases and independence of phase and modulus (Appendix A).
    Used throughout to write the joint distribution of Fourier modes in terms of moduli only, and to justify the factorization over the half-space in the Gaussian case.
  • ad hoc to paper Non-Gaussian Fourier modes in K_+(R) are mutually independent, e.g. via Delta_k = F[Delta^G_k] (Eq. 3.22).
    This postulate is introduced in Section 3.2 to restore Markovianity and factorize the joint rate function; the authors acknowledge it is one extreme of possible non-Gaussian structures.
  • domain assumption Fourier-mode distributions have Weibull tails p(|delta_k|) ~ exp(-|delta_k|^q / sigma_k^q), q >= 1 (Eq. 4.1).
    The input model family for all subsequent predictions; it is motivated by inflationary exponential tails but not derived from a specific model.
  • standard math Contraction principle and Gärtner-Ellis lemma from large-deviation theory (Appendix B).
    Standard results used to map the joint rate function of Fourier modes to the rate function of the smoothed density contrast.
  • standard math Reflection principle and first-passage-time relation Eq. (2.8) hold for symmetric Markov self-similar random walks without Gaussianity.
    Valid for Lévy-type processes; used to convert the distribution of Delta(t) into the halo first-passage-time distribution.
  • ad hoc to paper The asymptotic LDP tail can be promoted to the full normalized density Eq. (4.5) with variance t.
    The LDP fixes only the exponential tail; the paper reabsorbs subexponential factors and uses the generalized Gaussian density for all t in the FPT and mass-function plots.
  • domain assumption Two-barrier Laplace-transform relation of Sheth and van de Weygaert, Eq. (5.3), applies.
    External result used for voids; relies on Markovianity and symmetry but not Gaussianity, as stated in the paper.
  • standard math Laplace's method and large-s inversion capture the relevant first-passage-time behavior in the LDP regime.
    Used in Appendix C.2 to derive the void size function series; the paper notes the result is valid for small t / large fluctuations.

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read the original abstract

The excursion-set formalism provides a key connection between primordial density fluctuations and the abundance of cosmic structures such as dark matter halos and voids, traditionally assuming Gaussian random walks. In this work, we extend this framework to fluctuations whose distribution presents strongly non-Gaussian tails. Such tails are beyond the reach of perturbative approaches to primordial non-Gaussianity based on moment expansion. We address the problem with rigorous, analytical derivations relying on the large deviation principle, suited for the study of rare fluctuations. We derive new first-passage time distributions for random walks with non-Gaussian statistics and obtain updated predictions for the halo mass function. We also study the two-barrier problem relevant to cosmic void formation, leading to a new analytical prediction for the void size function, with improved accuracy on large scales. Our results demonstrate the potential of large deviation techniques as a bridge between inflationary scenarios, often leading to strongly non-Gaussian tails, and late-Universe observables.

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