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Searching for Inflationary Physics with the CMB Trispectrum: 2. Code & Validation

T0 review · 2 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper presents PolySpec, a public code that estimates eleven primordial trispectrum amplitudes from the CMB's four-point correlations, and validates on simulations that the estimates are unbiased and near-minimum-variance.

desk verdict PolySpec is a genuinely useful public code with thorough Gaussian-level validation of new CMB trispectrum estimators, but the absolute calibration of the new exchange templates rests on internal consistency checks alone, so the abstract oversells the non-Gaussian validation. read the letter →

arxiv 2502.05258 v3 pith:ZXDSNGMN submitted 2025-02-07 astro-ph.CO gr-qchep-phhep-th

Oliver H. E. Philcox This is my paper · ORCID
classification astro-ph.COgr-qchep-phhep-th
keywords CMBtrispectrumprimordialnon-Gaussianityquarticestimatorscosmologicalcolliderdirection-dependenttrispectrapolarizationPolySpecvalidation
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's target is the CMB trispectrum—the four-point correlation function of temperature and polarization fluctuations on the sky—which carries non-Gaussian imprints of inflationary physics that the three-point bispectrum cannot fully probe. The author's claim is that this statistic is now observationally tractable: PolySpec, a public code, implements quasi-optimal quartic (fourth-order) estimators for eleven primordial templates, including local, constant, effective-field-theory, direction-dependent, and cosmological-collider shapes, plus lensing and point-source contaminants. An extensive simulation campaign shows the estimators return zero on Gaussian skies, recover injected amplitudes on non-Gaussian skies, and match the inverse Fisher matrix closely enough to be near-minimum-variance. The forecasts also show that many collider and direction-dependent templates are only weakly correlated with the local and effective-field-theory templates previously searched, so the companion Planck analysis can provide first meaningful constraints on phenomena such as spinning particle exchange during inflation.

What carries the argument

The load-bearing mechanism is the Hermite-expansion quartic estimator. Schematically, for each amplitude $A_\alpha$ it computes a numerator $\hat{N}_\alpha$ from filtered maps $h = S^{-1}d$: the four-field term $h^4$, a two-field term $-6h^2\langle h^2\rangle$, and a zero-field term $3\langle h^2\rangle^2$, then normalizes by the Fisher matrix $F_{\alpha\beta}$ estimated by stochastic trace estimation over Gaussian random fields. The templates are made computationally separable in one of two ways: contact templates factor into products of single-momentum functions with one radial integral over $r$; exchange templates factor into two quadratic pieces coupled by $F_L(r,r')$, involving two radial integrals. A greedy optimization step compresses the radial integrals to $N_{\rm opt} \approx 10$–$100$ points with weights, reproducing the idealized Fisher matrix within $f_{\rm thresh}$ (typically $10^{-3}$ to $10^{-4}$), which is what makes the code fast enough for Planck-resolution data. The collider templates additionally impose a collapsed-limit cut via $k > k_{\rm coll}$, $K < k_{\rm coll}$ with $k_{\rm coll} = 0.01\,{\rm Mpc}^{-1}$ and $L_{\rm max} \ge \ell_{\rm max}/4$.

What would settle it

Inject a collider signal with support at $K \sim k$ rather than $K \ll k$ into the non-Gaussian simulation suites and run PolySpec on the resulting maps; if the recovered $\tau_{\rm NL}$ amplitudes deviate from the input by more than the Monte Carlo error, the collapsed-limit truncation is losing real signal.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that PolySpec is ready for real data. For each of eleven primordial amplitudes, plus gravitational lensing and point sources, it constructs the optimal estimator of Paper 1—a Hermite-expansion numerator with four-, two-, and zero-field terms, normalized by a Fisher matrix obtained by Monte Carlo—and verifies three properties that together define readiness: no false signal on Gaussian maps, unbiased recovery of injected non-Gaussian amplitudes, and empirical covariances consistent with the inverse Fisher matrix across masks, scale cuts, and temperature-versus-polarization choices. The paper further finds that adding E-mode polarization tightens constraints by roughly 40% for local shapes and by 30–70% for direction-dependent and collider shapes, and that the collider and direction-dependent templates are only weakly correlated with the local and EFTI shapes, implying that prior Planck non-detections do not bound them. On this basis the code is presented as the foundation for the Planck trispectrum constraints of Paper 3.

Load-bearing premise

The load-bearing premise is that the primordial signals are exactly the factorized templates of Paper 1 and live entirely in the collapsed limit, so truncating at $k_{\rm coll} = 0.01\,{\rm Mpc}^{-1}$ and $L_{\rm max} \ge \ell_{\rm max}/4$ discards no real signal; if a genuine signal extends outside that regime, the recovered amplitudes would be biased or null.

Editorial extensions

If this is right

  • Paper 3 will place first Planck constraints on spinning massive-particle exchange and direction-dependent trispectra, because those templates are not already bounded by previous local and EFTI searches.
  • Future trispectrum analyses should include E-mode polarization, since the validation shows up to 40% tighter constraints on local shapes and 30–70% tighter constraints on direction-dependent and collider shapes.
  • The odd-parity direction-dependent templates with odd $n_1,n_3$ are excluded from the Planck analysis because they fail $k$-resolution convergence, so the public template list is not the full set of inflationary models.
  • At Planck-like resolution ($\ell_{\rm max} = 2048$), estimator numerators take seconds and Fisher-matrix realizations minutes per realization, so the analysis is computationally feasible on current clusters.
  • Using $k_{\rm coll} = 0.01\,{\rm Mpc}^{-1}$ and $L_{\rm max} \ge \ell_{\rm max}/4$ retains the majority of the Fisher information for every collider template, which validates the collapsed-limit strategy.

Reading between the lines

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

  • Beyond the paper, the weak correlations among templates suggest that a joint multi-template search could substantially outperform separate single-template analyses; this is testable in the Paper 3 dataset by comparing joint and marginalized constraints.
  • If a real inflationary signal has support in non-collapsed tetrahedra, the collapsed-limit truncation would make PolySpec miss it even though the estimator is validated on collapsed-limit injections; a natural extension is to build non-collapsed templates or relax $k_{\rm coll}$ and check whether constraints degrade.
  • The radial-point optimization used here to compress Fisher information could be transplanted to other optimal estimators, such as bispectrum or 21cm polyspectrum analyses, yielding similar speed-ups.
  • The failure of odd $n_1,n_3$ templates to converge when the $k$-grid is doubled suggests that convergence testing is a cheap diagnostic for whether a new template can be meaningfully constrained.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper presents PolySpec, a public CMB trispectrum estimation code implementing quasi-optimal quartic estimators for a wide set of inflationary trispectrum amplitudes: local (g^loc_NL, tau^loc_NL), constant (g^con_NL), EFTI (g^{dot-sigma^4}_NL, g^{dot-sigma^2(d sigma)^2}_NL, g^{(d sigma)^4}_NL), direction-dependent (tau^{n1 n3 n}_NL, tau^{n,even}_NL, tau^{n,odd}_NL), cosmological collider (light and heavy spin-s exchange), plus lensing and point-source amplitudes. The paper describes the code architecture, the radial-integral optimization algorithm, computational scaling, and an extensive validation suite: Gaussian null tests, empirical variances versus Fisher-matrix predictions, comparisons with analytic Fisher computations and published scaling forecasts, hyperparameter stability checks, and non-Gaussian injection tests for gloc_NL, tau_loc_NL, and lensing. It also provides forecasts emphasizing the information content of polarization and the weak correlations among many of the new templates versus previously studied shapes.

Significance. If the validation is accepted, PolySpec is a substantial and carefully engineered public code that should enable first direct Planck constraints on several inflationary trispectrum families, including cosmological collider signals. The paper's strengths are its breadth, the extensive Gaussian validation, the comparisons with analytic calculations and published forecasts, the explicit reporting of failed convergence for some direction-dependent templates, and the public availability of the code. The principal weakness is that unbiasedness in the presence of a signal is demonstrated end-to-end for only three estimators (gloc_NL, tau_loc_NL, Alens); for the EFTI, direction-dependent, and collider families the validation is Gaussian-only, which leaves a specific class of template-normalization errors undetected. This gap is load-bearing for the abstract's central claim and needs to be addressed or explicitly re-scoped.

major comments (2)
  1. [Abstract; Secs. IV.D-IV.F] The abstract's claim that the estimators are 'unbiased and minimum-variance, both in Gaussian and non-Gaussian regimes' is stronger than the evidence provided. Non-Gaussian injection tests are performed only for gloc_NL and tau_loc_NL (Fig. 7) and for lensing (Fig. 23); Sec. IV.D states explicitly that 'non-Gaussian simulations have not been generated' for the EFTI shapes, and Secs. IV.E and IV.F likewise validate the direction-dependent and collider estimators only on Gaussian simulations. This is not a purely cosmetic gap: a constant normalization error in the implemented template derivative Q_alpha is invisible to Gaussian-only tests. In Eq. (3), if Q -> c Q, then the numerator N scales as c, the Fisher matrix F scales as c^2, the estimated amplitude bA = F^{-1} N is biased by a factor 1/c, while the predicted covariance F^{-1} scales as 1/c^2 and the empirical-variance check against F^{-1} still passes. The internal consistency check d†Q = tau compares the numerator and Fisher code paths against each other, not against an independent definition of the template. The analytic Fisher comparisons in Secs. IV.C and IV.D are reassuring only if the analytic calculation is genuinely independent of the code's Q_alpha construction. I request at least one non-Gaussian injection test (or an equivalent independent normalization check) for a representative template in each of the EFTI, direction-dependent, and collider families, or, failing that, a clear restriction of the unbiasedness claim to the tested templates in both the abstract and the conclusions.
  2. [Sec. IV.E; Sec. VI] The direction-dependent estimators with odd n1,n3 (and the corresponding parity-even/odd combinations with mismatched parity) do not pass the k-space convergence tests and are explicitly said to 'cannot be meaningfully constrained from the CMB'; they are excluded from Paper 3. This is an honest and useful statement, but it conflicts with the blanket wording in Sec. VI that 'All templates have passed these checks except for certain direction-dependent tauNL shapes.' For these shapes the optimization algorithm is not guaranteed to converge, so the fthresh-based error bound does not apply. The paper should state explicitly that the validation, forecast, and unbiasedness claims are restricted to the well-behaved subset of direction-dependent templates, and should phrase the abstract's 'wide variety of templates' so that it does not imply validated estimators for all listed shapes.
minor comments (6)
  1. [Sec. IV.C.4, Fig. 6] The masked tau_loc_NL case at lmax=64 shows a clear mismatch between the empirical variance and the Fisher prediction; the text attributes this to the approximate S^{-1} weighting. This is not fatal for Paper 3 if the analysis runs at lmax=2048, but it should be listed explicitly as a known limitation of the minimum-variance claim for that configuration.
  2. [Sec. V.D] The sentence 'the more complex estimators can be applied to high-resolution Planck data prohibitive computational costs' appears to be missing the word 'without' before 'prohibitive'.
  3. [Fig. 20 caption] The word 'agremeent' in the caption should be 'agreement'.
  4. [Sec. IV.E] The text says the estimators are built for '(small) positive integer ni', but templates with ni=0 are included (e.g., tau^{000}_NL); 'non-negative integer' would be more accurate.
  5. [Sec. IV.B, Eq. (13)] The filter in Eq. (13) includes a beam B_ell, while Sec. IV.B says 'do not otherwise include a beam'; please clarify whether the fiducial analysis sets B_ell = 1 or whether a beam is included in the filter.
  6. [General] Consider adding a reproducibility note giving the exact version/commit of PolySpec used for the validation numbers, and a list of the random seeds or a statement on how they were drawn, to make the Monte Carlo results fully reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the estimator pipeline is validated against external analytic forecasts and non-Gaussian simulations where available, and the remaining Gaussian-only validation of EFTI, direction-dependent, and collider estimators is a disclosed limitation rather than a circular reduction.

full rationale

The paper's derivation chain is bA = F^{-1} bN from Eq. (3), implemented from template derivatives defined in Paper 1 and from the independent optimal-estimation formalism of Smith, Senatore, and Zaldarriaga [21]. The templates are not defined in terms of the quantities being predicted, and the claimed predictions are not fitted to the data used for validation. Contact-template Fisher matrices are checked against independent analytic calculations (e.g., Fig. 4 and Fig. 10 circles), empirical variances are compared with F^{-1} predictions from large sets of Gaussian simulations, and the local and lensing estimators recover injected non-Gaussian amplitudes in Figs. 7 and 23. Reliance on Paper 1 (same author) for template definitions and optimization algorithms is legitimate self-citation because the estimators are additionally tested against external scaling forecasts (e.g., Kalaja et al. 2021; Kogo and Komatsu 2006) and through explicit grid-doubling convergence tests. The only substantive gap is that EFTI, direction-dependent, and collider estimators are not tested with non-Gaussian injections; the paper explicitly states this, e.g., in Section IV D: 'non-Gaussian simulations have not been generated, thus we must validate our estimators in the Gaussian limit,' with analogous statements in Sections IV E and IV F. Gaussian-only tests cannot detect a constant template-normalization error because both numerator and Fisher matrix inherit the same proportionality to the template derivative Q, so bA = F^{-1}bN is changed by 1/c while the predicted covariance F^{-1} changes by 1/c^2 and can still match the empirical variance. That is a genuine validation limitation for the unbiasedness claim extended to all eleven primordial templates, but it is not circularity: the estimator is not equivalent to its inputs by construction, the limitation is acknowledged in the manuscript, and no load-bearing step reduces to a self-citation or a fitted parameter renamed as a prediction.

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

The central claim (unbiased, near-optimal estimators for many templates) rests on the estimator formalism, the factorizability of the templates, the validity of collapsed-limit truncation for collider shapes, and several numerical approximation parameters. No new physical entities are introduced; all templates represent previously proposed inflationary physics.

free parameters (5)
  • kcoll = 0.01 Mpc^-1
    Truncation scale for cosmological collider templates, chosen to capture the majority of Fisher information (Section IV F 1, Fig. 16). Affects bias and signal-to-noise.
  • fthresh = 10^-4 (primary), 10^-3 (some exchange templates)
    Optimization tolerance for radial integration; controls the approximation error in the Fisher matrix, roughly sqrt(fthresh).
  • Nopt = Varies by template, 30 to roughly 100
    Number of optimized radial integration points selected by the greedy optimization algorithm to reach fthresh. Larger values improve accuracy at higher computational cost.
  • Nfish = 50 to 100 (varies by test)
    Number of Gaussian random fields used to estimate the Fisher matrix. Larger values reduce Monte Carlo noise in the inverse Fisher matrix.
  • Ndisc = 50 to 100 (varies by test)
    Number of simulations used to subtract the disconnected trispectrum. Chosen by hand; larger values reduce bias from covariance mismatch.
assumptions (5)
  • domain assumption The optimal estimator (Eq. 3) is unbiased and minimum-variance when the data covariance is known and the inverse-variance filter equals P^dagger C^-1.
    Stated in Section II A; this is a standard result from Paper 1 and Smith et al. 2015.
  • domain assumption All templates are exactly separable into contact (Eq. 6) or exchange (Eq. 10) forms.
    Derived in Paper 1; if not exactly separable, the O(Npix log Npix) scaling breaks and the estimators would be intractable.
  • domain assumption The theoretical trispectra for collider particles are valid only in the collapsed limit, and the kcoll truncation removes contamination from non-collapsed regimes.
    Section IV F relies on cosmological collider literature (Arkani-Hamed and Maldacena, Lee et al.); this assumption is load-bearing for the collider estimators.
  • domain assumption Monte Carlo averages over Ndisc simulations with covariance matching the data remove the disconnected trispectrum exactly.
    Eq. (4); if the simulation covariance does not match the data covariance, the estimator gains a bias.
  • ad hoc to paper The optimized radial basis (Nopt points and weights) reproduces the full Fisher matrix to accuracy sqrt(fthresh).
    Section II B 2; this numerical approximation is validated for well-behaved templates but fails for odd-parity direction-dependent shapes, as reported in Section IV E 1.

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Cite this review

Pith. "Pith review of Searching for Inflationary Physics with the CMB Trispectrum: 2. Code & Validation." pith.science (2026). https://pith.science/paper/ZXDSNGMN

@misc{pith2026250205258,
  author       = {Pith},
  title        = {Pith review of: Searching for Inflationary Physics with the CMB Trispectrum: 2. Code & Validation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZXDSNGMN}},
  note         = {Machine review of arXiv:2502.05258}
}
read the original abstract

To unlock the vast potential of the CMB trispectrum, we require both robust estimators and efficient computational tools. In this work, we introduce the public code PolySpec: a suite of quartic estimators designed to measure the amplitudes of a wide variety of inflationary templates, including local non-Gaussianity, effective field theory models, direction-dependent trispectra, spinning massive particle exchange, and weak gravitational lensing. PolySpec includes a python/cython implementation of each estimator derived in Paper 1 and has been carefully optimized to ensure efficient use of computational resources. We perform a broad range of validation tests, which demonstrate that the estimator is unbiased and minimum-variance, both in Gaussian and non-Gaussian regimes. In addition, we forecast constraints on various types of trispectra; this highlights the utility of CMB polarization and demonstrates that many models of primordial physics are poorly correlated with the simple templates considered in previous studies. This work lays the foundation for the Planck trispectrum analyses performed in Paper 3.

Figures

Figures reproduced from arXiv: 2502.05258 by the authors.

Figure 1
Figure 1. FIG. 1. Sample [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Contributions to the ideal [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Breakdown of contributions to the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: additionally shows the dependence of σ(g loc NL) on scale-cuts and field content. Adding E-modes improves constraints by up to 40% (particularly at larger ℓmax); this suggests that the excision of polarization in the official Planck g loc NL analyses led to significant…
Figure 6
Figure 6. Figure 6: FIG. 6. Optimality test for the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Parameter recovery test for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Optimality test for the [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Optimality test for the three EFTI estimators, as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Joint analysis of the three EFTI templates, as applied to the mean of 50 Gaussian simulations. The blue (red) contours [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Correlations between the parity-even (top left) and [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Optimality of the [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Dependence of the light (top) and heavy (bottom) spin-zero cosmological collider constraints on the truncation [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Optimality of the spin-zero cosmological collider estimators, as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Dependence of the spin-zero collider non-Gaussianity constraints on the maximum external mode, [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Correlations between the spin-zero cosmological col [PITH_FULL_IMAGE:figures/full_fig_p024_20.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Fisher matrix for the EFTI templates (top left), [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Validation of the [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Runtime of the [PITH_FULL_IMAGE:figures/full_fig_p027_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Scaling of the [PITH_FULL_IMAGE:figures/full_fig_p028_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Dependence of [PITH_FULL_IMAGE:figures/full_fig_p030_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Breakdown of computation time for analyzing the numerator (left) and Fisher matrix (right) of three trispectrum [PITH_FULL_IMAGE:figures/full_fig_p031_27.png]

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Reference graph

Works this paper leans on

100 extracted references · 18 canonical work pages · cited by 5 Pith papers

  1. [1]

    Optimization First, we demonstrate the efficacy of the optimization scheme discussed in §II. Despite the additional τ integral, the optimization proceeds similarly to that of the gloc NL template, starting from a finely-spaced array of Ns = 4161 (ri, τi) points, as described in §III. In Fig. 9, we show the diagonal of the ideal g ˙σ4 NL Fisher derivative ...

  2. [2]

    By default, we use the hyperparameter set {Ndisc = 50, Nfish = 50, ℓmin = 2, fthresh = 10−4, Nside = 256}, analyzing 50 simulations

    Performance on Gaussian Simulations The three EFTI estimators can be validated using a large suite of Gaussian simulations, generated at the Planck -like signal-plus-noise power spectrum described in §IV B. By default, we use the hyperparameter set {Ndisc = 50, Nfish = 50, ℓmin = 2, fthresh = 10−4, Nside = 256}, analyzing 50 simulations. We use a lower Nf...

  3. [3]

    Consistency Tests Finally, we test the dependence of our results on hyperparameter choices, focusing on the masked temperature-plus- polarization analysis at ℓmax = 512. Halving Nfish induces an error below 0 .15% in 1/ √ F , corresponding to < 0.25% in the errorbars and < 0.001σ in the mean, implying that our Monte Carlo Fisher matrix is highly converged...

  4. [4]

    General Direction-Dependence We first validate the τ n1n3n NL estimator by comparing its empirical variance to the Fisher matrix prediction, using 50 Gaussian simulations. Noting that computational costs scale quadratically with nmax, we restrict to ni ∈ {0, 1, 2}, but consider each non-trivial combination of n1, n3, nobeying the triangle conditions, spli...

  5. [5]

    For completeness, we perform additional tests in this section, which allow us to verify the scalings with ℓmax discussed in the literature

    Even & Odd Templates As discussed in Paper 1, the optimal estimators for τ n,even NL and τ n,odd NL can be formed as a linear combination of the τ n1n3n NL , and thus are implicitly validated by the above tests. For completeness, we perform additional tests in this section, which allow us to verify the scalings with ℓmax discussed in the literature. In th...

  6. [6]

    not-too-collapsed

    Dependence on kcoll Choosing kcoll requires balancing two constraints: (1) low kcoll leads to less contamination from equilateral regimes (since the external legs are dominated by large ℓ ∼ kχ∗); (2) high kcoll leads to increased signal-to-noise. To assess this, we perform a suite of spin-zero Fisher forecasts ( i.e. we compute F ) for various choices of ...

  7. [7]

    Our key results are shown in Fig

    Spin-Zero Results We now validate the spin-zero estimators using Gaussian simulations. Our key results are shown in Fig. 17, comparing the empirical and theoretical errorbars on τ light,heavy NL across a wide range of mass parameters ν0 and µ0. The two sets of errorbars agree within 1 .8σ (accounting for the expected scatter), validating our estimators. T...

  8. [8]

    Due to the Higuchi bound, m2/H 2 ≥ s(s − 1) [83], which implies that the light template is restricted to νs ∈ [0, 1/2], avoiding the local-type divergence at ν0 → 3/2

    Higher-Spin Results The most complex templates included in PolySpec are the light and heavy collider templates with spin s >0. Due to the Higuchi bound, m2/H 2 ≥ s(s − 1) [83], which implies that the light template is restricted to νs ∈ [0, 1/2], avoiding the local-type divergence at ν0 → 3/2. As such, we expect that higher-spin trispectra will be harder ...

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.