Cosmological constraints on small-scale primordial non-Gaussianity

Di Wu; Jing-Zhi Zhou; Zhi-Chao Li

REVIEW 4 major objections 5 minor 1 cited by

Cosmological constraints on small-scale primordial non-Gaussianity

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Small-scale cosmic non-Gaussianity squeezed to -10 < f_NL < 1.2

desk verdict Solid Bayesian application of SIGW/PBH constraints to NANOGrav, but the quoted f_NL lower bound is a perturbativity cut at fixed A_zeta, not an observational limit. read the letter →

arxiv 2505.22614 v2 pith:KDQBKIDM submitted 2025-05-28 astro-ph.CO gr-qchep-phhep-th

Jing-Zhi Zhou , Zhi-Chao Li , Di Wu This is my paper

classification astro-ph.COgr-qchep-phhep-th

keywords primordialnon-Gaussianityscalar-inducedgravitationalwavesblackholespulsartimingarrayssmall-scalepowerspectrumlocal-typef_NLBayesfactorLISA

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 sets out to establish that the local-type primordial non-Gaussianity parameter $f_{\mathrm{NL}}$ can be meaningfully constrained on small scales, where the primordial power spectrum is otherwise largely unconstrained, rather than only on CMB scales. It combines pulsar timing array observations, the CMB+BAO upper bound on gravitational-wave energy density, a LISA signal-to-noise forecast, and the requirement that primordial black holes not overproduce dark matter. For a monochromatic primordial power spectrum the joint constraint is $-10.0 < f_{\mathrm{NL}} < 1.2$ at fixed amplitude $A_\zeta = 10^{-2}$. The same analysis reports that scalar-induced gravitational waves from that spectrum are preferred over supermassive-black-hole binaries as the source of the nHz background, with a Bayes factor of about 54. The contribution is thus a data-driven map of the small-scale $f_{\mathrm{NL}}$ region that explicitly depends on the assumed spectrum shape and amplitude.

What carries the argument

The central object is the local-type non-Gaussian curvature perturbation $\zeta = \zeta_g + (3/5) f_{\mathrm{NL}}\, \zeta_g^2$ in momentum space, whose higher correlators feed the energy density spectrum of scalar-induced gravitational waves. The computation keeps second-order and third-order gravitational waves, so the two-point correlator receives one-loop ($\sim A_\zeta^2$), two-loop ($\sim A_\zeta^3 f_{\mathrm{NL}}$ and $\sim f_{\mathrm{NL}}^2 A_\zeta^3$), and three-loop ($\sim f_{\mathrm{NL}}^4 A_\zeta^4$) contributions, with the odd-$f_{\mathrm{NL}}$ cross term supplying the sign asymmetry. Around this core sit the data handles: the kernel-density-estimator free-spectrum representation of the 15-year pulsar timing array dataset, the CMB+BAO bound $h^2 \rho_{\mathrm{GW}} < 2.9 \times 10^{-7}$, the primordial black hole abundance bound $f_{\mathrm{PBH}} < 1$, and a LISA signal-to-noise calculation, all joined in a Bayesian analysis over the spectrum amplitude $A_\zeta$, peak scale $f_*$, width indices, and $f_{\mathrm{NL}}$.

What would settle it

A measurement of the nHz gravitational-wave background spectral shape that matches the supermassive-black-hole binary prediction and excludes the scalar-induced gravitational wave peak predicted for $f_{\mathrm{NL}}$ inside the quoted interval would falsify the claim that monochromatic-spectrum scalar-induced gravitational waves dominate pulsar timing array observations. So would an independent primordial black hole abundance bound at the corresponding mass scale that violates $f_{\mathrm{PBH}} < 1$ for the model parameters the authors find.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that multi-scale current observations pin the local-type non-Gaussian parameter on small scales. With a monochromatic primordial power spectrum the allowed range is $-10.0 < f_{\mathrm{NL}} < 1.2$ when $A_\zeta = 10^{-2}$, and a Bayes-factor comparison favors these scalar-induced gravitational waves over supermassive-black-hole binaries for the pulsar timing array nHz background. The physical asymmetry comes from including third-order scalar-induced gravitational waves: the cross-correlation contribution proportional to $A_\zeta^3 f_{\mathrm{NL}}$ suppresses the total gravitational-wave energy spectrum for negative $f_{\mathrm{NL}}$, so the posterior is no longer symmetric under $f_{\mathrm{NL}} \to -f_{\mathrm{NL}}$ and the region near $-10 \lesssim f_{\mathrm{NL}} \lesssim -1$ is largely excluded. The result is explicitly shape-dependent, with log-normal, broken power-law, and monochromatic spectra giving different $f_{\mathrm{NL}}$ ranges in Table I.

Load-bearing premise

The load-bearing premise is that the headline interval is evaluated at one fixed amplitude, $A_\zeta = 10^{-2}$, and for a single assumed shape of the small-scale primordial power spectrum; allow the amplitude to vary or choose another shape and the quoted $f_{\mathrm{NL}}$ range changes.

Editorial extensions

If this is right

Under the monochromatic scalar-induced gravitational wave interpretation, the nHz background has a sharp predicted peak whose high-frequency side falls in the LISA band with a signal-to-noise ratio that depends on $f_*$ and the spectrum width.
When third-order gravitational waves are included, negative $f_{\mathrm{NL}}$ suppresses the total spectrum, so the mirror-symmetric region of negative values is excluded and future pulsar timing array fits can be sensitive to the sign of $f_{\mathrm{NL}}$.
If scalar-induced gravitational waves do not dominate the pulsar timing array band, the observed spectrum still acts as an upper bound on their energy density, so the amplitude $A_\zeta$ at each $f_{\mathrm{NL}}$ must lie below the derived PTA regions.
For inflationary models that produce log-normal, broken power-law, or monochromatic small-scale spectra, the computed constraints directly restrict the model parameter space.

Reading between the lines

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

Because the headline interval is fixed at $A_\zeta = 10^{-2}$, reading it as a universal small-scale bound would overstep: the paper's own Table I shows the interval changes to $-9.5 < f_{\mathrm{NL}} < 2.9$ for a log-normal spectrum and $-5.0 < f_{\mathrm{NL}} < -0.1$ for a broken power law, and a fully marginalized amplitude would presumably widen these ranges.
The Bayes factor compares scalar-induced gravitational wave models against a single supermassive-black-hole binary alternative; adding other nHz sources such as cosmic strings or a first-order phase transition would likely reshuffle the odds.
The odd-$f_{\mathrm{NL}}$ cross term is a direct probe of the sign of small-scale non-Gaussianity, whereas second-order-only analyses are insensitive to it; future multi-frequency pulsar timing and LISA data could target that term specifically.
A direct primordial black hole search at the masses tied to the gravitational-wave peak scale would test the joint model independently of the gravitational-wave spectrum, since the black hole abundance estimates used here are explicitly model-dependent.

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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

Good news first: this is a solid, workmanlike application of the SIGW/PBH machinery to the NANOGrav 15-yr data, comparing three PPS shapes and computing Bayes factors. The genuinely new bit is the third-order SIGW cross-correlation term, which is odd in f_NL and breaks the f_NL <-> -f_NL symmetry of the second-order spectrum; that sign-breaking effect drives the asymmetry in the posterior and is worth having. The Bayes factor favoring the monochromatic spectrum over SMBHB (about 54 in Fig. 8) is a concrete, citable result, and the paper is honest about the model-dependence of the f_NL constraints in the text.

The soft spot is the headline. The abstract says 'current observational data' constrain -10.0 < f_NL < 1.2, but the lower bound is exactly the perturbativity cut (f_NL)^2 A_zeta < 1 evaluated at the fixed A_zeta = 10^-2. That is not a data limit; it is a theoretical consistency condition the authors impose. Table I does disclose the A_zeta assumption, and the text states the constraints depend on amplitude and shape, but the abstract and the phrase 'Constraints from current cosmological observations' in Table I are misleading. A referee should ask for the 2D posterior in (A_zeta, f_NL) with the perturbativity cut marked separately, and ideally for a version that marginalizes over A_zeta or shows how the interval shifts if the cut is relaxed.

Other soft spots are minor: the third-order kernels are cited rather than shown (they are in the authors' earlier papers), no code is shipped, and the perturbativity condition is a heuristic scaling argument rather than a rigorous bound. These are not fatal—the machinery is published and the analysis follows standard practice.

Overall: the central analysis appears sound, the model comparison is standard, and the main flaw is framing, not the math. This paper deserves peer review; I would recommend major revision with the headline fixed and the A_zeta-dependence made explicit rather than buried in a table caption. For a reading group, it is a good case study in how prior cuts can masquerade as observational constraints.

Referee Report

4 major / 5 minor

Summary. The paper studies observational constraints on the local-type primordial non-Gaussianity parameter f_NL on small scales, using scalar-induced gravitational waves (SIGWs), primordial black hole (PBH) abundance, PTA data (NANOGrav 15-year), CMB+BAO bounds, and projected LISA SNR. It considers log-normal, broken power-law, and monochromatic primordial power spectra, including second-order and (for the log-normal case) third-order SIGW contributions. A Bayesian analysis with bilby/dynesty yields posterior distributions, Bayes factors between SIGW and SMBHB interpretations of the PTA background, and a headline constraint -10.0 < f_NL < 1.2 for the monochromatic spectrum, quoted for a fixed amplitude A_zeta = 10^-2.

Significance. If the headline constraint were a robust observational bound, it would be a valuable addition to the small-scale primordial perturbation literature, complementing large-scale f_NL constraints. The paper correctly identifies that small-scale f_NL is poorly constrained and that SIGW+PBH observations can probe it. Strengths include the use of publicly available NANOGrav KDE data, a standard nested-sampling pipeline (bilby/dynesty), and explicit Bayes factors for model comparison. The inclusion of third-order SIGW corrections and their sign-dependent effect on the spectrum is an interesting extension. However, as detailed in the major comments, the quoted f_NL interval is substantially conditioned on a fixed amplitude and a theoretical perturbativity cut, so the claim as stated in the abstract is not fully supported.

major comments (4)

[Abstract and Table I] The headline interval -10.0 < f_NL < 1.2 is presented as 'constraints from current cosmological observations,' but the lower bound -10.0 is exactly the theoretical perturbativity cut (f_NL)^2 A_zeta < 1 imposed in Sec. III, evaluated at the fixed amplitude A_zeta = 10^-2 stated in Table I. With A_zeta = 10^-2, this condition gives |f_NL| < 10, so the lower endpoint is not determined by PTA, CMB, BAO, or PBH data. The abstract and the table should either explicitly state that the quoted interval includes an a priori convergence assumption, or the analysis should be repeated with A_zeta marginalized so that the reported interval is data-driven. As written, the wording 'rigorously constrain the parameter space' and 'constraints from current cosmological observations' is misleading.
[Sec. III, Eq. (10) and perturbativity condition] The condition (f_NL)^2 A_zeta < 1 is introduced heuristically from the scaling of loop contributions in the SIGW spectrum, but the paper does not provide the explicit third-order SIGW kernels or the full expression for the cross-correlation term Omega^(3,2)_GW, instead referring to Refs. [99,100]. Because this condition directly sets the lower bound of the main result, the manuscript should justify the convergence criterion more rigorously (e.g., by showing that including the next-order terms does not shift the boundary significantly) or at least quantify how the quoted interval would change if the cut were relaxed or replaced by a data-driven prior. Without this, the lower bound -10.0 is an assumption rather than a measurement.
[Table I vs. Fig. 12 and Fig. 7b] For the monochromatic spectrum, the posterior distribution in Fig. 12 shows median log10(A_zeta) around -1.3, yet Table I fixes A_zeta = 10^-2 and quotes f_NL = (-10.0, 1.2). The paper does not explain how this interval is derived from the joint posterior or why the amplitude is fixed at a value that is not the posterior peak. Since the SIGW spectrum depends on A_zeta through A_zeta^2, f_NL^2 A_zeta^3, and f_NL^4 A_zeta^4, the allowed f_NL range can shift substantially with A_zeta. The authors should present the f_NL constraint as a function of A_zeta (or as a joint posterior) and state explicitly that the Table I values are conditional on a chosen amplitude, not marginalized constraints.
[Sec. II.A, paragraph after Fig. 1b] The text states that after including third-order SIGWs, 'the parameter interval f_NL in [-10, -1] is significantly excluded,' but Table I reports intervals that all include f_NL values in this range (e.g., -10.0 < f_NL < 1.2 for the delta peak, -9.5 < f_NL < 2.9 for LN). This apparent inconsistency should be resolved. If the exclusion applies only to the Omega_tot_GW model for the LN spectrum, that should be stated clearly; if it applies broadly, then Table I's intervals contradict it. The reader cannot currently tell which f_NL regions are actually excluded by the third-order analysis.

minor comments (5)

[Sec. II, Eq. (3)] The seven loop contributions Omega^G, Omega^H, Omega^C, Omega^Z, Omega^R, Omega^P, Omega^N are named but not individually defined; a brief description of which diagram each corresponds to, or a reference to the relevant figure in Ref. [93], would improve readability.
[Throughout] There are numerous typographical issues, e.g., 'PT A observations' and 'T A data' instead of 'PTA'. These should be corrected in a final proofreading pass.
[Table I] The column header 'f_NL Bayesian factors' mixes two different quantities (the f_NL interval and the Bayes factor relative to SMBHB). Please split these into separate columns with clear captions, and note explicitly that the f_NL intervals are for A_zeta = 10^-2.
[Fig. 5 caption and Sec. III] The caption of Fig. 5 repeats 'Omega^(2)_GW' twice in the legend description; the text also uses 'Omega tot_GW' with inconsistent subscripts. Please standardize notation and verify that the shaded regions correspond to the correct models.
[Sec. IV, paragraph after Fig. 8] The sentence 'when the parameter f_NL using the energy density spectrum in Eq. (10), the current PTA observations cannot be dominated by SIGWs' appears to have a grammatical error and is confusing in context, given that Fig. 8 reports Bayes factors for Omega_tot_GW,LN models. Please rephrase and clarify the intended meaning.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the headline f_NL interval is a stated conditional result, with the lower bound partly reflecting the paper's own perturbativity cut rather than a hidden reuse of fitted data.

full rationale

The paper's derivation chain is self-contained against external data. The local-type f_NL expansion in Eq. (1), the second- and third-order SIGW spectra in Eqs. (3) and (10), the PTA likelihood built from NANOGrav KDE free spectra, the CMB+BAO bound in Eq. (18), and the PBH abundance calculation in Eq. (22) are all external inputs or standard calculations. The headline interval in Table I is explicitly conditioned on A_zeta = 10^-2, and the paper states that the constraints depend on both the spectrum shape and the amplitude. The lower bound -10.0 coincides with the perturbativity condition (f_NL)^2 A_zeta < 1 at that amplitude, so that part of the quoted interval is a theoretical consistency cut rather than a purely data-driven limit; however, the paper identifies this condition and the chosen amplitude explicitly, so the result is a well-posed conditional statement rather than a circular prediction. The Bayes-factor comparison with the SMBHB model is a standard model-evidence calculation on the same PTA data and does not reuse a fitted parameter as a prediction. Self-citations for third-order SIGW kernels and PBH threshold parameters provide parameter-free, externally calculable results and are not load-bearing in a circular way. Overall, no step reduces the claimed constraint to its own input by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper's constraints are obtained by applying published SIGW and PBH formulas to NANOGrav free-spectrum data. The main free parameters are the PPS amplitude, shape parameters, and f_NL. The headline f_NL interval is conditional on a fixed amplitude and one PPS shape, so it is an inference under model assumptions rather than a parameter-free derivation.

free parameters (6)

A_zeta (amplitude of primordial power spectrum) = log10(A_zeta) posterior median about -1.0 to -1.4 depending on model; set to 10^-2 for Table I
Controls the amplitude of SIGWs and PBH abundance; the headline f_NL constraint is conditional on A_zeta = 10^-2.
f_NL (local non-Gaussian parameter) = posterior medians near -0.5 to 10.7 across models; headline interval -10.0 < f_NL < 1.2 for the delta peak
Parameter of interest; fitted to PTA data rather than derived from first principles.
f_star (peak frequency of PPS) = posterior median log10(f_star/Hz) near -6 to -7 depending on model
Peak location fitted to the PTA band; prior ranges vary by PPS shape.
sigma (log-normal width) = posterior median around 1.3 to 1.4; fixed to 1 for some fits
Width of the log-normal spectrum; affects the shape of the SIGW spectrum.
alpha, beta (broken power-law indices) = posterior medians near 1.9 to 2.0
Slopes of the broken power-law spectrum; fitted with priors [1,3].
A_BHB, gamma_BHB (SMBHB background parameters) = posterior medians log10(A_BHB) about -15.7 and gamma_BHB about 4.6
Astrophysical foreground model fitted jointly; prior from NANOGrav analysis [68].

assumptions (5)

domain assumption The primordial curvature perturbation has the local-type non-Gaussian form zeta = zeta_g + (3/5) f_NL zeta_g^2 (Eq. 1).
Defines the non-Gaussian model; standard in the literature but not derived here.
domain assumption The second-order and third-order SIGW energy density spectra from Refs. [93,99,100] are correct and complete.
The paper adopts these formulas without rederiving them; the third-order kernels come from the authors' previous papers.
domain assumption PBH abundance is computed with the compaction-function and scaling-law approach with K=4.4, gamma=0.38 and correlators from Refs. [117-122].
The PBH bound f_PBH < 1 is used to cut parameter space; this calculation is model-dependent as the paper itself notes.
domain assumption The perturbative expansion in (f_NL)^2 A_zeta must satisfy (f_NL)^2 A_zeta < 1 for convergence.
Used to restrict parameter space (red curve in Fig. 5); reasonable but not rigorously justified.
domain assumption For the main PTA constraints, SIGWs are assumed to dominate the nHz gravitational wave background, or are combined with an SMBHB component whose priors come from Ref. [68].
The central inference is conditional on this source model; Bayes factors are used to compare models.

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Cite this review

Pith. "Pith review of Cosmological constraints on small-scale primordial non-Gaussianity." pith.science (2026). https://pith.science/paper/KDQBKIDM

@misc{pith2026250522614,
  author       = {Pith},
  title        = {Pith review of: Cosmological constraints on small-scale primordial non-Gaussianity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDQBKIDM}},
  note         = {Machine review of arXiv:2505.22614}
}

abstract

In contrast to the large-scale primordial power spectrum $\mathcal{P}_{\zeta}(k)$ and primordial non-Gaussianity $f_{\mathrm{NL}}$, which are strictly constrained, the small-scale $\mathcal{P}_{\zeta}(k)$ and $f_{\mathrm{NL}}$ remain less restricted. Considering local-type primordial non-Gaussianity, we study the PBH and SIGW caused by large-amplitude small-scale primordial power spectrum. By analyzing current observational data from PTA, CMB, BAO, and abundance of PBH, and combining them with the SNR analysis of LISA, we rigorously constrain the parameter space of $\mathcal{P}_{\zeta}(k)$ and $f_{\mathrm{NL}}$. Furthermore, we examine the effects of different shapes of the primordial power spectrum on these constraints and comprehensively calculate the Bayes factors for various models. Our results indicate that SIGW generated by a monochromatic primordial power spectrum are more likely to dominate current PTA observations, with the corresponding constraint on the primordial non-Gaussian parameter being $-10.0<f_{\mathrm{NL}}<1.2$.

Figures

Figures reproduced from arXiv: 2505.22614 by the authors.

**Figure 1.** FIG. 1: The corner plot of the posterior [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 4.** FIG. 4: The upper limits of the amplitude of LN [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (9 more)

**Figure 5.** Figure 5: FIG. 5: The constraints on [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: , when SIGWs lie within the observational frequency range of PTA, the SGWB generated by the corresponding binary PBH mergers falls within LISA’s detection frequency band, which may affect the SNR of LISA. 10−9 10−7 10−5 10−3 10−1 101 103 f/Hz 10−16 10−14 10−12 10−10 10…

**Figure 7.** Figure 7: b, respectively. The corresponding posterior distributions are given in the Appendix. A. For the monochromatic power spectrum, the prior distributions of log(f∗/Hz), log(Aζ ), and fNL are uniformly distributed over the ranges [−10, −5], [−3, 0], and [−30, 30] respecti…

**Figure 8.** Figure 8: FIG. 8: The Bayes factors between different models. The vertical axis represents the Bayes factor of [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: FIG. 9: Posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: FIG. 10: Posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: FIG. 11: Posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 12.** Figure 12: FIG. 12: Posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

**Figure 14.** Figure 14: FIG. 14: Posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

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