arxiv: 2604.20063 · v2 · submitted 2026-04-21 · 🌌 astro-ph.CO · gr-qc· hep-ph

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Purely Quadratic Non-Gaussianity from Tachyonic Instability: Primordial Black Holes and Scalar-Induced Gravitational Waves

He-Xu Zhang , Mei Huang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:48 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-ph

keywords primordial black holesnon-Gaussianitytachyonic instabilityscalar-induced gravitational wavescurvature perturbationsinflationdark matter

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The pith

Narrow curvature spectra exponentially suppress primordial black hole formation while keeping gravitational waves strong.

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

The paper examines primordial black hole formation when curvature perturbations follow the purely quadratic non-Gaussian form arising from tachyonic instability in multi-field inflation. Using an extended Press-Schechter framework based on the compaction function, it derives the probability distribution of the linear compaction function and shows that the black hole abundance is exponentially sensitive to both the perturbation amplitude and the correlation coefficient between the smoothed field and its radial gradient. The spectral width of the curvature power spectrum controls this coefficient: broad spectra leave it mildly negative and allow significant black hole production, while sufficiently narrow spectra drive it toward minus one and produce strong exponential suppression. This separation permits a sizable scalar-induced gravitational wave signal to survive even as black hole overproduction is avoided, with thermal inflation serving as a benchmark case.

Core claim

In the tachyonic amplification scenario with curvature perturbations given by ζ = A(φ² − ⟨φ²⟩), the PBH abundance is exponentially sensitive to the amplitude of perturbations and the correlation coefficient ρ between the smoothed field and its radial gradient. Broad spectra yield mildly negative ρ and fail to suppress PBH formation, while sufficiently narrow spectra drive ρ toward −1, resulting in exponential suppression of the PBH abundance while maintaining a sizable gravitational-wave signal.

What carries the argument

The correlation coefficient ρ between the smoothed curvature field and its radial gradient, which sets the asymptotic exponential tail of the probability distribution for the linear compaction function.

If this is right

For sufficiently narrow spectra, PBH abundance can be exponentially suppressed while a sizable scalar-induced gravitational wave signal remains potentially detectable by future space-based interferometers.
Asteroid-mass PBHs could serve as dark matter in such narrow-spectrum cases without overproduction.
Broad spectra, as in typical thermal inflation, make it difficult to reconcile PTA observations with PBH constraints simultaneously.
The PBH formation probability depends exponentially on both the perturbation amplitude and the value of ρ fixed by spectral shape.

Where Pith is reading between the lines

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

Future gravitational wave observations at high frequencies could indirectly constrain the allowed width of the curvature power spectrum in tachyonic models.
The same exponential suppression might appear in other multi-field inflationary scenarios that generate quadratic non-Gaussianity, offering a route to satisfy both PBH and GW bounds without tuning amplitudes.
Direct probes of the curvature spectrum shape, if available, would provide a sharp test of whether observed GW signals align with the narrow-spectrum regime required for suppression.

Load-bearing premise

Curvature perturbations follow exactly the purely quadratic non-Gaussian form ζ = A(φ² − ⟨φ²⟩) from tachyonic instability, and the extended Press-Schechter framework based on the compaction function correctly calculates PBH abundance under this non-Gaussian statistics.

What would settle it

A measurement of the width of the primordial curvature power spectrum together with an observed PBH abundance that deviates from the predicted exponential dependence on ρ would test the mechanism.

Figures

Figures reproduced from arXiv: 2604.20063 by He-Xu Zhang, Mei Huang.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Posterior distributions for the parameters of the BPL model inferred from the NANOGrav 15-year dataset. The dark [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Comparison of the predicted SIGW energy density spectra with the first 14 frequency bins of the NANOGrav 15- [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Scalar-induced gravitational-wave (SIGW) spectra and corresponding PBH mass distribution. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

read the original abstract

We investigate primordial black hole (PBH) formation in a cosmological scenario where curvature perturbations follow purely quadratic non-Gaussianity, $\zeta = A(\phi^2-\langle\phi^2\rangle)$, arising from tachyonic instability in multi-component inflationary models. Within an extended Press-Schechter framework based on the compaction function, we derive the probability distribution of the linear compaction function and its asymptotic exponential tail, demonstrating that the PBH abundance is exponentially sensitive not only to the amplitude of perturbations but also to the correlation coefficient $\rho$ between the smoothed field and its radial gradient. We further find that, in this tachyonic amplification scenario, the spectral width of the curvature power spectrum plays a decisive role in avoiding PBH overproduction: broad spectra yield mildly negative $\rho$ and fail to suppress PBH formation, while sufficiently narrow spectra drive $\rho \to -1$, resulting in exponential suppression while maintaining a sizable gravitational-wave signal. Thermal inflation provides a useful benchmark scenario with asteroid-mass PBH dark matter and high-frequency scalar-induced gravitational waves potentially detectable by future space-based interferometers, but its typically broad spectra make it challenging to reconcile PTA observations with PBH constraints.

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.

Desk Editor's Note private letter to a colleague

Narrow spectra drive ρ near -1 to exponentially suppress PBHs in this quadratic non-Gaussian tachyonic setup while preserving GW signals, but the tail relies on unverified Press-Schechter approximations.

read the letter

The main thing to know is that the paper derives how purely quadratic non-Gaussianity from tachyonic instability produces a correlation ρ between the smoothed curvature perturbation and its radial gradient, and that narrow spectra can push ρ close to -1 for strong exponential suppression of PBH abundance without erasing the scalar-induced gravitational wave signal. Broad spectra leave ρ only mildly negative and fail to suppress enough. They use this to sketch a path for asteroid-mass PBH dark matter consistent with PTA constraints in models like thermal inflation, though they flag that typical thermal inflation spectra are too broad for the trick to work easily. This is the concrete new mapping they offer: spectral width controls ρ, which then controls the tail of the compaction function distribution inside an extended Press-Schechter calculation. The derivation of the probability distribution and its asymptotic exponential tail from the quadratic form ζ = A(φ² - ⟨φ²⟩) is the part that feels like a targeted step forward rather than a restatement. It gives a clear handle on why the width matters for avoiding overproduction while keeping GWs sizable. The soft spots are real but not fatal. The whole argument sits on the extended Press-Schechter framework with the compaction function, which already carries known model dependence even for Gaussian cases, and the paper does not show numerical simulations or independent checks to confirm the tail behavior holds when ρ approaches -1. The stress-test concern about possible sub-exponential corrections or rare configurations missed by a ρ-only parameterization looks worth examining, since the observables are quadratic and bilinear forms in the underlying Gaussian field. The assumption that the curvature perturbations stay exactly quadratic also feels model-dependent and could shift in full multi-field dynamics. No red flags on the citation pattern or internal contradictions, but the lack of validation keeps the soundness modest. This is for cosmologists who work on multi-field inflation, PBH formation, and scalar-induced GWs. A reader who wants a specific mechanism to tune the PBH-GW tension will find usable ideas here. It deserves peer review because the claim is sharp enough and the calculation is laid out clearly enough that referees can test the tail and the approximations directly.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that curvature perturbations with purely quadratic non-Gaussianity ζ = A(φ² − ⟨φ²⟩) arising from tachyonic instability in multi-component inflation lead to PBH formation whose abundance is exponentially sensitive to the correlation coefficient ρ between the smoothed curvature perturbation and its radial gradient. Using an extended Press-Schechter formalism based on the compaction function, it derives the probability distribution and asymptotic tail of the linear compaction function, showing that narrow spectra drive ρ → −1 and exponentially suppress PBH overproduction while preserving a sizable scalar-induced gravitational wave signal; thermal inflation is presented as a benchmark that struggles with broad spectra.

Significance. If the central derivations hold, the work identifies spectral width as a tunable parameter that controls non-Gaussian suppression of PBH formation, offering a concrete route to reconcile asteroid-mass PBH dark matter with PTA gravitational-wave data and future interferometer signals in tachyonic multi-field models. The explicit link between quadratic non-Gaussianity, ρ, and the exponential tail constitutes a useful addition to the literature on non-Gaussian PBH constraints.

major comments (2)

[Sections deriving the probability distribution and asymptotic tail (referenced in abstract)] The derivation of the ρ-dependent asymptotic exponential tail for the linear compaction function (central to the suppression claim) is presented without an explicit large-deviation calculation or the resulting functional form of the tail probability when the underlying field is quadratic in a multivariate Gaussian; this step is load-bearing for the statement that narrow spectra yield exponential suppression.
[Extended Press-Schechter framework section] The application of the extended Press-Schechter formalism with the compaction function to this quadratic non-Gaussian case is used to compute PBH abundance without any cross-check against numerical simulations or alternative methods; given that EPS is already an approximation whose accuracy is known to degrade for non-Gaussian tails, the quantitative reliability of the reported suppression factors remains unverified.

minor comments (3)

[Notation and definitions] Clarify the precise definition and numerical evaluation of the correlation coefficient ρ for different spectral shapes, including any assumptions on the window function used for smoothing.
[Thermal inflation benchmark] The discussion of thermal inflation as a benchmark would benefit from explicit values of the spectral width parameter and the resulting ρ to make the tension with PTA/PBH constraints quantitative.
[Introduction and references] Add references to prior literature on quadratic non-Gaussianity in PBH formation and on the validity of compaction-based EPS for non-Gaussian fields.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive assessment of its significance, and the constructive major comments. We address each point below and have revised the manuscript to incorporate clarifications and additional details where feasible.

read point-by-point responses

Referee: [Sections deriving the probability distribution and asymptotic tail (referenced in abstract)] The derivation of the ρ-dependent asymptotic exponential tail for the linear compaction function (central to the suppression claim) is presented without an explicit large-deviation calculation or the resulting functional form of the tail probability when the underlying field is quadratic in a multivariate Gaussian; this step is load-bearing for the statement that narrow spectra yield exponential suppression.

Authors: We thank the referee for this observation. The probability distribution and asymptotic tail in the manuscript are obtained by starting from the joint Gaussian distribution of the smoothed curvature perturbation and its radial gradient (which are correlated with coefficient ρ), noting that the linear compaction function is a quadratic form in these variables due to the purely quadratic non-Gaussianity. The tail is then extracted by evaluating the relevant integral over the super-threshold region. To make the derivation fully explicit, we have added a new appendix that performs the large-deviation calculation for this quadratic form in the multivariate Gaussian case and states the resulting functional form of the tail probability. revision: yes
Referee: [Extended Press-Schechter framework section] The application of the extended Press-Schechter formalism with the compaction function to this quadratic non-Gaussian case is used to compute PBH abundance without any cross-check against numerical simulations or alternative methods; given that EPS is already an approximation whose accuracy is known to degrade for non-Gaussian tails, the quantitative reliability of the reported suppression factors remains unverified.

Authors: We agree that the extended Press-Schechter (EPS) formalism with the compaction function is an approximation whose accuracy can degrade for strongly non-Gaussian tails, and that direct numerical simulations would provide valuable verification. In the revised manuscript we have added an expanded discussion of the known limitations of EPS for non-Gaussian PBH calculations, together with references to existing literature that compares EPS results against simulations for comparable non-Gaussian models. We also stress that the reported exponential suppression is a qualitative trend driven by the ρ dependence rather than a precise numerical prediction. Performing dedicated N-body or lattice simulations for this specific tachyonic quadratic non-Gaussianity lies beyond the scope of the present theoretical study. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation of compaction tail from quadratic mapping is self-contained

full rationale

The paper derives the probability distribution and asymptotic exponential tail of the linear compaction function directly from the quadratic non-Gaussian mapping ζ = A(φ² − ⟨φ²⟩) by considering the joint statistics of the smoothed curvature perturbation and its radial gradient. The correlation coefficient ρ emerges as a calculable property of the power spectrum shape rather than a fitted parameter renamed as a prediction. The extended Press-Schechter application is presented as an approximation whose accuracy is not claimed to be proven within the paper itself, but no step reduces the central result to its inputs by construction, self-citation, or ansatz smuggling. The link between spectral width and ρ is obtained from explicit integrals over the spectrum and does not constitute a tautology.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption of purely quadratic non-Gaussianity arising from tachyonic instability and the applicability of the compaction-function-based Press-Schechter formalism; no new entities are postulated and the amplitude A and spectral parameters function as free parameters.

free parameters (3)

amplitude A
Scales the quadratic non-Gaussianity and is chosen to set perturbation strength.
correlation coefficient ρ
Determines the tail of the compaction distribution and is controlled by spectral width.
spectral width parameter
Tuned to drive ρ toward -1 for suppression while preserving GW amplitude.

axioms (2)

domain assumption Curvature perturbations obey the purely quadratic form ζ = A(φ² - ⟨φ²⟩) generated by tachyonic instability in multi-component inflation.
Invoked in the opening sentence as the starting point for the entire analysis.
domain assumption The extended Press-Schechter formalism based on the compaction function accurately captures PBH formation probability under this non-Gaussianity.
Used to derive the probability distribution and asymptotic exponential tail.

pith-pipeline@v0.9.0 · 5520 in / 1705 out tokens · 36442 ms · 2026-05-10T00:48:57.414271+00:00 · methodology

discussion (0)

Reference graph

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