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Non-Gaussian Expansion of Minkowski Tensors in Redshift Space

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Pith's one-line read This paper derives closed-form, leading-order non-Gaussian ensemble averages for the translation-invariant rank-2 Minkowski tensors of the redshift-space density field and shows they match dark-matter simulations to percent accuracy on…

desk verdict New non-Gaussian ensemble averages for two rank-2 Minkowski tensors in redshift space, with solid analytic anchoring but a validation that leans heavily on a single fitted Finger-of-God parameter. read the letter →

arxiv 2507.10091 v2 pith:KFTUHHUS submitted 2025-07-14 astro-ph.CO physics.data-an

Stephen Appleby , Christophe Pichon , Pravabati Chingangbam , Dmitri Pogosyan , Changbom Park This is my paper
classification astro-ph.COphysics.data-an
keywords Minkowskitensorsredshiftspacedistortionsnon-GaussiandensityfieldEdgeworthexpansioncosmologicalparameterestimationFinger-of-Godeffectlarge-scalestructureperturbationtheory
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 aims to turn two rank-2 Minkowski tensors—integrals over the boundary of a smoothed density field's excursion set, weighted by products of surface normals—into a perturbative cosmological probe of anisotropies introduced by redshift-space distortions. The authors derive closed-form ensemble averages for the translation-invariant tensors $W_1^{0,2}$ and $W_2^{0,2}$ for a density field whose joint probability distribution is Edgeworth-expanded through cubic cumulants, giving equations (14) and (21). They express the two- and three-point cumulants through the redshift-space matter power spectrum and bispectrum, including a phenomenological exponential damping for Finger-of-God velocity dispersion and a shot-noise prescription. Comparing with 100 dark-matter simulation snapshot boxes at $z=1$, they find the predicted tensor components match measurements at percent level for smoothing scales $R_G > 20\,h^{-1}\,{\rm Mpc}$, with the non-Gaussian part of the signal dominated by the expected odd and even Hermite polynomial terms. The payoff, if the predictions hold, is that measuring the tensors' skewness from galaxy surveys would yield joint constraints on the growth rate $f\sigma_8$ and the amplitude $b_1\sigma_8$.

What carries the argument

The machinery is the Edgeworth expansion of the joint probability density of the smoothed density field, its first derivatives, and its second derivatives, truncated at cubic cumulants. For $W_1^{0,2}$ the integrand is $\delta_D(\delta-\nu\sigma)\,\delta_i \delta_j/|\nabla\delta|$, and for $W_2^{0,2}$ the same quantity is weighted by mean curvature $G_2$; integrating these against the expanded PDF converts the ensemble averages into the closed forms (14) and (21), whose coefficients $A^{(1)}_{G\perp}, B^{(1)}_\perp, \ldots$ depend only on $\lambda$. The cumulants are then connected to cosmology through the redshift-space kernels $Z_1$ and $Z_2$, with Finger-of-God stochastic velocities inserted as the exponential factors $e^{-k^2\mu^2\sigma_v^2}$ and $e^{-(k_1^2\mu_1^2+k_2^2\mu_2^2)\sigma_B^2}$.

What would settle it

Compute the $W_1^{0,2}$ and $W_2^{0,2}$ $(3,3)$ components from the same dark-matter snapshots at $R_G=20\,h^{-1}\,{\rm Mpc}$ while fitting $\sigma_v$ and $\sigma_B$ independently, for example from the monopole power spectrum and from the three-point cumulant $\langle x x_3^2\rangle$, and compare with equations (14) and (21); a best-fit $\sigma_B$ differing from $4.9\,h^{-1}\,{\rm Mpc}$ by more than the reported uncertainty, or a residual that persists after the independent fit, would falsify the claim that the non-Gaussian tensor signal is described by tree-level perturbation theory plus this FoG prescription.

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Extended reading notes

Core claim

The central claim is that equations (14) and (21) are the correct leading-order non-Gaussian ensemble averages of the translation-invariant rank-2 Minkowski tensors $W_1^{0,2}$ and $W_2^{0,2}$ in plane-parallel redshift space. Written as sums over Hermite polynomials with coefficients depending only on $\lambda = \sigma_{1\parallel}/\sigma_{1\perp}$, the averages are expressed through the field's two- and three-point cumulants, and those cumulants are in turn written in terms of the Kaiser-boosted linear power spectrum and the tree-level bispectrum with $b_1=1$, $b_2=0$ for dark matter. The paper validates this by measuring the tensors on $N=100$ redshift-space dark-matter snapshot boxes at $z=1$, smoothing with Gaussian kernels $R_G = 15,20,25,30\,h^{-1}\,{\rm Mpc}$, and finding that with an exponential Finger-of-God damping $\sigma_v=\sigma_B=4.9\,h^{-1}\,{\rm Mpc}$ the theory matches the perpendicular and line-of-sight components to percent level for $R_G>20\,h^{-1}\,{\rm Mpc}$. It also shows that setting the damping to zero leaves the perpendicular components well described but the line-of-sight $(3,3)$ components markedly off, and that the trace reproduces previous scalar Minkowski functionals in the isotropic limit and the $N_3$ result for the scalar surface-area statistic.

Load-bearing premise

The validation rests on the phenomenological Finger-of-God model: the stochastic velocity dispersion is put in as an exponential damping of the cumulant integrals, with $\sigma_B$ forced equal to the $\sigma_v$ value fitted to the monopole power spectrum, so if that damping does not carry over to the Minkowski-tensor cumulants—or $\sigma_B\neq\sigma_v$—the claimed percent-level match for the line-of-sight components collapses.

Editorial extensions

If this is right

  • For smoothing scales $R_G>20\,h^{-1}\,{\rm Mpc}$, the measured non-Gaussian part of $W_1^{0,2}$ and $W_2^{0,2}$ is captured by the cubic Edgeworth terms, so the tensors can be modeled without introducing free higher-order cumulants.
  • The skewness of the tensor curves scales with cumulants such as $\langle x^3\rangle/\sigma^3$, which are proportional to $b_1\sigma_8$; combined with the Gaussian amplitudes that carry $\Omega_m$, $n_s$, and $f/b$, a joint fit can constrain $f\sigma_8$ and $b_1\sigma_8$.
  • The line-of-sight $(3,3)$ components are strongly contaminated by Finger-of-God velocities, while the perpendicular components are not, so the perpendicular components are the safer target for parameter estimation from dark-matter-like fields.
  • Residual differences between theory and simulation at fixed smoothing scale trace mainly four-point cumulants; marginalizing over extra Hermite coefficients $h_{2,4,6}$ in a fit prevents biases when applying the formulas to data.
  • The trace of equation (21) gives the first derivation of $W_2$ to leading non-Gaussian order in redshift space, reducing in the isotropic limit to the previously known scalar Minkowski functional $W_2$.

Reading between the lines

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

  • If the exponential Finger-of-God damping is replaced by a velocity model with scale dependence, the $\sigma_B=\sigma_v$ degeneracy could be broken, and the line-of-sight tensor components might then constrain the velocity dispersion itself rather than only contaminate it.
  • The same Edgeworth machinery should extend to higher-rank Minkowski tensors and to spherical, radial redshift-space coordinates, where the tensor's off-diagonal elements become non-zero and could test statistical isotropy of the plane perpendicular to the line of sight.
  • Because galaxies have lower stochastic velocities than dark matter, the Finger-of-God contamination is weaker for galaxy surveys; the percent-level dark-matter agreement at $R_G>20\,h^{-1}\,{\rm Mpc}$ may be conservative, but galaxy bias and shot noise must then be added through the $b_1$, $b_2$, and $\bar{n}^{-1}$ terms already present in the cumulant integrals.
  • The proposed marginalization over $h_{2,4,6}$ coefficients is effectively a data-driven way to absorb trispectrum contributions; testing it on simulations would show how much cosmological information survives when those nuisance parameters are free.
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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 / 4 minor

Summary. This paper derives ensemble averages for the two translation-invariant, rank-2 Minkowski tensors W_1^{0,2} and W_2^{0,2} for a plane-parallel redshift-space matter density field. The derivation uses an Edgeworth expansion of the joint probability density function of the field and its derivatives, truncated at cubic-order cumulants, and results in equations (14) and (21), which express the tensor components in terms of Hermite polynomials, the anisotropy parameter lambda, and two- and three-point cumulants. The cumulants are evaluated with standard perturbation theory in redshift space, including an exponential Finger-of-God damping with parameters sigma_v and sigma_B, and shot noise is discussed but neglected for the dark-matter fields considered. The predictions are compared with measurements from 100 Quijote z=1 dark-matter snapshot boxes over smoothing scales R_G = 15, 20, 25, 30 h^-1 Mpc, finding percent-level agreement for R_G > 20 h^-1 Mpc when sigma_v = sigma_B = 4.9 h^-1 Mpc. The paper also discusses the breakdown of the Edgeworth expansion and proposes marginalizing over higher-order Hermite coefficients in future parameter estimation.

Significance. If correct, the paper provides the first analytic non-Gaussian expressions for the translation-invariant rank-2 Minkowski tensors in redshift space, extending the scalar Minkowski functional results of Matsubara and Codis et al. to tensorial statistics. The consistency checks are genuine strengths: the isotropic limit of the trace reproduces Matsubara (2003), the W_1 trace matches N_3/6 of Codis et al. (2013), and the Gaussian limits agree with the authors' earlier work. The claimed connection between the non-Gaussian tensor signatures and parameters such as f sigma_8 and b_1 sigma_8 is interesting and timely for DESI/Euclid-type analyses. However, the validation is substantially conditional on a phenomenological Finger-of-God prescription, and the manuscript's own Figures 7 and 8 show that the line-of-sight agreement collapses when the FoG damping is removed. The central analytic derivation appears sound, but the empirical support for it needs strengthening before the paper can be accepted.

major comments (2)
  1. [Section 5.3, Appendix B, Figures 7 and 8] The percent-level validation of equations (14) and (21) is conditional on the exponential Finger-of-God damping in equations (29) and (30), with sigma_v = sigma_B = 4.9 h^-1 Mpc. Appendix B shows that sigma_v is fitted to the monopole power spectrum of the same Quijote z=1 suite, while Section 5.3 states that sigma_B is set equal to sigma_v without an independent determination. Figures 7 and 8 show that the (3,3) components of both tensors are poorly predicted when sigma_v = sigma_B = 0, so the agreement in Figures 2 and 4 largely rests on this phenomenological ansatz. This is load-bearing for the validation claim: the paper does not demonstrate that an exponential damping with sigma_B = sigma_v also describes the three-point cumulants, and it does not quantify how the comparison would change if sigma_B differs from sigma_v. I recommend that the authors either constrain sigma_B independently (for instance from the measured three-point cumulants shown in Figure 1), perform a robustness scan over sigma_B, or split the Quijote realizations into calibration and validation sets so that the test of the analytic predictions is not simultaneously calibrated and validated on the same data.
  2. [Section 4.2, equations (21) and (22)] Equation (21) is the central new technical result, but its component-level validation is weaker than the trace-level checks. The Gaussian limit in equation (23) and the isotropic trace comparison in Appendix A are useful, yet neither verifies the detailed dependence of the perpendicular and parallel components on the five cumulants entering (21). Given the complexity of the coefficients in (22), I ask the authors to include an independent component-level check, for example by Monte Carlo integration of the Edgeworth PDF for a synthetic field with prescribed cumulants, or by providing a machine-readable derivation notebook. This would also help rule out typographical errors in individual coefficients, which the trace test alone cannot detect.
minor comments (4)
  1. [Section 4.2, equation (22i)] The printed coefficient D_parallel^{(2)} does not appear to reduce to the isotropic limit 2/15 quoted in Appendix A when lambda^2 approaches 1/2; please verify the denominator and the argument of the inverse tangent in this expression.
  2. [Section 7, Figures 2 and 4] The statement that agreement is at the percent level would be more transparent if each panel included a quantitative goodness-of-fit measure, such as a reduced chi-square or a normalized residual RMS over the plotted threshold range, rather than only a visual comparison.
  3. [Appendix B] The quoted uncertainty sigma_v = 4.90 +/- 0.04 h^-1 Mpc is a statistical error from the power-spectrum fit and does not include uncertainty in the cosmological parameters or in the choice of the exponential FoG model; please state explicitly that this is an effective parameter for the specific model and data used.
  4. [Section 5.4] The shot-noise discussion is concise and clear, but the abstract's phrase 'addresses the effects of ... shot noise' overstates the treatment, since shot noise is only argued to be negligible for the dark-matter fields considered rather than modeled in the validation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Minkowski tensor predictions are derived from an Edgeworth expansion and perturbation-theory cumulants, while the Finger-of-God parameter is calibrated to the power spectrum, not to the tensor data.

full rationale

The central derivation is self-contained. Equations (14) and (21) are obtained by Edgeworth-expanding the joint PDF of the field and its derivatives, with the ensemble averages expressed in terms of cumulants; the cumulants are then computed from the linear matter power spectrum and standard perturbation-theory kernels in equations (24)-(27). The analytic predictions are tested against Quijote simulations, which are an external benchmark rather than a product of the present derivation. The Finger-of-God velocity dispersion sigma_v is fitted to the monopole power spectrum of the same simulation suite in Appendix B, and sigma_B is set equal to sigma_v; this is a phenomenological calibration that affects the validation, but it is not a fit to the Minkowski tensor measurements and does not make the tensor prediction equivalent to its input by construction. No parameter is adjusted to the tensor data, and the paper explicitly acknowledges that the four-point cumulants and the FoG ansatz limit the accuracy of the comparison. Self-citations to Appleby et al. (2018, 2019, 2022a) are methodological or concern already-established Gaussian limits and do not carry the non-Gaussian derivation. No equation or definition reduces the claimed prediction to its own input, so there is no significant circularity.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The paper's analytic contribution is the conversion of two- and three-point cumulants into Minkowski tensor ensemble averages. It imports the Edgeworth machinery and cumulant integrals from Codis et al. (2013), the linear matter power spectrum from CAMB, and the Gaussian-limit tensor framework from the authors' own prior work. The only fitted quantity is the Finger-of-God velocity dispersion, fit to the same Quijote simulation suite. No new physical entities are introduced.

free parameters (1)
  • sigma_v = sigma_B (Finger-of-God velocity dispersion) = sigma_v = 4.90 +/- 0.04 h^-1 Mpc; sigma_B set equal to 4.9 h^-1 Mpc
    Fit to the monopole redshift-space power spectrum of the Quijote z=1 fiducial boxes over 31 k-bins (Appendix B, equations B5-B6). Enters the two-point cumulants via exp(-k^2 mu^2 sigma_v^2) (equation 29) and the three-point integrand via sigma_B (equation 30).
assumptions (7)
  • domain assumption Ergodicity: volume averages measured in simulation boxes equal ensemble averages exactly.
    Stated in Section 4: 'we simply assume ergodicity holds exactly'. Finite-volume uncertainty and potential biases are not quantified.
  • domain assumption Plane-parallel approximation: every tracer obeys a single line of sight e_3.
    Section 2. The actual redshift-space distortion is radial relative to the observer; the paper cites its own earlier work (Appleby et al. 2022a) for the spherical treatment and defers adaptation.
  • domain assumption Linear Kaiser mapping: delta_tilde(k) = (1 + f mu^2) delta(k).
    Section 2, equation (2). Assumes a linear real-to-redshift-space mapping of the density field, ignoring non-linear velocity terms beyond the phenomenological FoG patch.
  • standard math Edgeworth expansion of the joint PDF of the field and its first and second derivatives, truncated at cubic order.
    Section 4, equation (11), following Codis et al. (2013). The truncation at three-point cumulants is a modeling choice that the paper shows breaks down at negative thresholds and small smoothing scales.
  • domain assumption Statistical isotropy in the plane perpendicular to the line of sight.
    Section 4. Used to set off-diagonal tensor components to zero and to reduce the gradient degrees of freedom to q^2_perp and x_3.
  • domain assumption Tree-level perturbation theory kernels Z_1 and Z_2 and the cumulant integrals of Codis et al. (2013).
    Section 5, equations (26)-(27) and Table 1. The two- and three-point cumulants are imported from prior literature, with the linear matter power spectrum from CAMB as input.
  • ad hoc to paper Exponential FoG damping form with sigma_v = sigma_B.
    Section 5.3, equations (29)-(30). Phenomenological and acknowledged to be too steep on small scales; the equality of sigma_v and sigma_B is assumed without independent constraint.

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Cite this review

Pith. "Pith review of Non-Gaussian Expansion of Minkowski Tensors in Redshift Space." pith.science (2026). https://pith.science/paper/KFTUHHUS

@misc{pith2026250710091,
  author       = {Pith},
  title        = {Pith review of: Non-Gaussian Expansion of Minkowski Tensors in Redshift Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFTUHHUS}},
  note         = {Machine review of arXiv:2507.10091}
}
abstract

This paper focuses on extending the use of Minkowski Tensors to analyze anisotropic signals in cosmological data, focusing on those introduced by redshift space distortion. We derive the ensemble average of the two translation-invariant, rank-2 Minkowski Tensors ($W_1^{0,2}$ and $W_2^{0,2}$) for a matter density field that is perturbatively non-Gaussian in redshift space. This is achieved through the Edgeworth expansion of the joint probability density function of the field and its derivatives, expressing the ensemble averages in terms of cumulants up to cubic order. Our goal is to connect these theoretical predictions to the underlying cosmological parameters, allowing for parameter estimation by measuring them from galaxy surveys. The work builds on previous analyses of Minkowski Functionals in both real and redshift space and addresses the effects of Finger-of-God velocity dispersion and shot noise. We validate our predictions by matching them to measurements of the Minkowski Tensors from dark matter simulation data, finding that perturbation theory is a qualified success. Non-perturbative Finger-of-God effects remain significant at relatively large scales $R_G \lesssim 20 \, h^{-1} \, {\rm Mpc}$ and are particularly pronounced in the components parallel to the line of sight.

Figures

Figures reproduced from arXiv: 2507.10091 by the authors.

Figure 1
Figure 1. [Top panels] Numerically measured two-point (left panel) and three-point (right panel) cumulants from a set of z = 1 dark matter snapshot boxes in redshift space (points/error bars). The dashed lines are the corresponding ensemble expectation values constructed in Section 5, taking σv = σB = 4.9 h −1 Mpc. Error bars are the error on the mean. The scaling of most cumulants is well captured by perturbation theory, not… view at source ↗
Figure 2
Figure 2. The Minkowski tensor components W0,2 1 |1 1 and W0,2 1 |3 3 (green/red points and error bars) extracted from N = 100 realisations of quijote z = 1 dark matter snapshot boxes in redshift space. The black dashed lines are the theoretical predictions for the ensemble averages ⟨w1 1 ⟩ and ⟨w3 3 ⟩ inferred from Section 4.1, with Finger of God velocity dispersion σv = σB = 4.9 h −1 Mpc. The grey dotted lines are the Gauss… view at source ↗
Figure 3
Figure 3. Residuals W0,2 1 |i i − ⟨wi i ⟩G (red/green dashed lines) and ⟨wi i ⟩ − ⟨wi i ⟩G (red/green solid lines), where W0,2 1 |i i is the mean value of the Minkowski tensor components from the snapshot boxes, ⟨wi i ⟩ is the non-Gaussian prediction obtained in this work and ⟨wi i ⟩G the Gaussian limit of the prediction. The residuals are dominated by the odd Hermite polynomials H1(ν) and H3(ν). As expected the larger the sm… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The Minkowski tensor components W0,2 2 |1 1 and W0,2 2 |3 3 (green/red points and 1σ error bars) extracted from N = 100 realisations of quijote z = 1 dark matter snapshot boxes in redshift space. The black dashed lines are the theoretical predictions for the ensemble a…
Figure 5
Figure 5. Figure 5: Residuals W0,2 2 |i i − ⟨vi i ⟩G (red/green dashed lines) and ⟨vi i ⟩ − ⟨vi i ⟩G (red/green solid lines), where W0,2 2 |i i is the mean value of the Minkowski tensor components from the snapshot boxes, ⟨vi i ⟩ is the non-Gaussian prediction obtained in this work and ⟨v…
Figure 6
Figure 6. Figure 6: [Left panel] A comparison of the trace of wi j calculated in this work (black solid lines) and the total area of iso-field surfaces N3 in S. Codis et al. (2013) (yellow dashed lines). The curves in descending amplitude are for smoothing scales RG = 15, 20, 25, 30 h −1 …
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: [Top Panels] The difference between measured values of the Minkowski tensors (wi j (left) and vi j (right)) and their Gaussian expectation values in redshift space, for a field smoothed using RG = 15 h −1Mpc. The green/red data are reproduced from the top left panels o…

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