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arxiv: 2509.07973 · v1 · submitted 2025-09-09 · 🌌 astro-ph.CO

Towards an application of fourth-order shear statistics I. The information content of langle M_ap⁴ rangle

Elena Silvestre-Rosello , Lucas Porth , Peter Schneider , Laila Linke , Jonas Krueger , Sebastian Grandis , Jonathan Oel This is my paper

Pith reviewed 2026-05-18 17:27 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords aperture mass statisticsshear four-point correlationweak gravitational lensinghigher-order statisticsFisher forecastnon-Gaussian informationcosmological constraints
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The pith

Fourth-order aperture statistics add only minimal extra constraining power on cosmology beyond second- and third-order statistics in non-tomographic analyses.

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

The paper develops the theoretical framework linking fourth-order aperture mass statistics to the shear four-point correlation function and convergence polyspectra. It derives the corresponding filter functions, implements a numerical integration pipeline, and validates it to two-percent precision on Gaussian random fields. A Fisher-matrix forecast for a DES-Y3-like survey then shows that adding the fourth-order statistic to a joint second-plus-third-order analysis produces only minimal improvement in parameter constraints when using equal-scale, non-tomographic apertures.

Core claim

The central claim is that fourth-order aperture statistics contain some non-Gaussian information from the projected matter field but deliver only a minimal gain in cosmological constraining power when combined with lower-order aperture statistics for current-generation survey setups.

What carries the argument

The fourth-order aperture statistics ⟨M_ap^4⟩, obtained by integrating the shear four-point correlation function against derived filter functions that connect directly to the convergence polyspectra.

If this is right

  • Including fourth-order statistics does not substantially tighten cosmological parameter constraints beyond those from second- and third-order aperture statistics in equal-scale non-tomographic setups.
  • The integration pipeline reaches the precision level required for Stage IV surveys because two-percent accuracy lies well below the expected noise budget.
  • Non-Gaussian information accessible through higher-order shear statistics remains present yet contributes limited additional leverage under the analysis choices examined.

Where Pith is reading between the lines

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

  • Tomographic binning or unequal-scale combinations of apertures could extract more information from the fourth-order statistic than the equal-scale non-tomographic case tested here.
  • Surveys with lower shape noise might see larger relative gains from fourth-order statistics even if the absolute improvement stays modest.
  • The same numerical framework could be applied to cross-correlations with other tracers to test whether the minimal-gain result is specific to auto-correlations of galaxy shear.

Load-bearing premise

The novel method for estimating derivatives from a simulation suite with arbitrarily distributed cosmological sets produces accurate Fisher-matrix elements.

What would settle it

Recomputing the Fisher matrix with a conventional finite-difference derivative estimator on a regularly spaced cosmological grid and checking whether the resulting error ellipses differ significantly from those obtained with the new estimator.

Figures

Figures reproduced from arXiv: 2509.07973 by Elena Silvestre-Rosello, Jonas Krueger, Jonathan Oel, Laila Linke, Lucas Porth, Peter Schneider, Sebastian Grandis.

Figure 1
Figure 1. Figure 1: Parametrization of a quadrilateral using the ×-projection. Equivalent to Fig. (1) from P25 by defining 𝜙12 = 𝜓12 and 𝜙13 = 𝜓12 + 𝜓23. The apertures measures related to Γ0, Γ1 and Γ5 read (and similarly for Γ2−4 and Γ6−7) ⟨𝑀4 ⟩ (𝜃) = Ö 3 𝑗=1 ∫ d𝜗𝑗 𝜃 𝜗𝑗 𝜃  ∫ d𝜓12 2𝜋 ∫ d𝜓23 2𝜋 × 𝐾0  𝒒0 𝜃 , 𝒒1 𝜃 , 𝒒2 𝜃 , 𝒒3 𝜃  𝑃 cart,× 0 Γ × 0 (𝜗1, 𝜗2, 𝜗3, 𝜓12, 𝜓23) , ⟨𝑀∗𝑀3 ⟩ (𝜃) = Ö 3 𝑗=1 ∫ d𝜗𝑗 𝜃 𝜗𝑗 𝜃  ∫ d𝜓12 2𝜋 ∫ d𝜓23 … view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of the two sets of coordinates for the projected trispectrum. However, we do not recommend this approach on observa￾tional data because the conversion from observable (real) space to Fourier space involves an integral over the whole real space, which is not practical due to finite survey sizes. Moreover, the Article number, page 5 of 19 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Integrand for the eight complex aperture measures once the radial integration is performed (first and third rows) and once all but one angular integral are performed (second and fourth rows), computed on 126 radial bins and 113 angular bins. At small aperture radii, 𝜃 = 2 ′ (top two rows), we find a higher signal, while at large aperture radii, 𝜃 = 20 ′ (bottom two rows), the signal is more strongly peaked… view at source ↗
Figure 4
Figure 4. Figure 4: Effect of a finite radial integration range on the fourth-order aperture statistics. (Top) Aperture statistics for varying radial in￾tegration ranges, computed on a grid with a logarithmic bin width of Δ(ln 𝜗) = 0.07 and Δ𝜓 = 0.07 radm alongside with the con￾sidered ground truth (see text). Dash-dotted lines show the effect of 𝐸- and 𝐵-mode mixing in the second-order aperture statistics for a radial integr… view at source ↗
Figure 5
Figure 5. Figure 5: Effect of the binning accuracy on the fourth-order aperture statistics. (Top-middle) Equivalent to [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Increase on the precision on the fourth-order aperture statis￾tics by integrating the 4PCF within a bin with a Riemann sum. Equivalent to [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Joint forecast of second-, third-, and fourth-order aperture statistics, with 1𝜎 contours from the Fisher co￾variance matrix F −1 . The cloud of points in the two￾dimensional distribution corresponds to the closest sets of cosmological parameters to the fiducial cSfid value, marked with a cross. like DES-Y3 (Abbott et al. 2022), would lead only to a minimal improvement on the cosmological constraints. We d… view at source ↗
read the original abstract

Higher-order shear statistics contain part of the non-Gaussian information of the projected matter field and therefore can provide additional constraints on the cosmological parameters when combined with second-order statistics. We aim to provide the theoretical framework for studying shear four-point correlation functions (4PCF) using fourth-order aperture statistics and develop a numerical integration pipeline to compute them. Finally, we forecast the information content of fourth-order aperture statistics. We begin by giving the relation of the $n$-th order aperture statistics, $\langle M_\mathrm{ap}^n\rangle$, to the shear $n$PCF and to the convergence polyspectra. We then focus on the fourth-order case, where we derive the functional form of their filters and test the behavior of these filters by numerically integrating over the 4PCF of a Gaussian random shear field (GRF). Finally, we perform a Fisher forecast on the constraining power of $\langle M_\mathrm{ap}^4\rangle_\rm{c}$, where we develop a novel method to estimate derivatives from a simulation suite with arbitrarily distributed cosmological sets. By analyzing and mitigating numerical effects within the integration pipeline, we achieve a two-percent-level precision on the fourth-order aperture statistics for a GRF, which remains well below the noise budget of Stage IV surveys. We report a minimal improvement in the constraining power of the aperture statistics when including fourth-order statistics to a $\langle M_\mathrm{ap}^2\rangle + \langle M_\mathrm{ap}^3\rangle$ joint analysis for a DES-Y3-like setup, using non-tomographic equal-scale aperture statistics.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops the theoretical framework relating n-th order aperture statistics ⟨M_ap^n⟩ to shear n-point correlation functions and convergence polyspectra, with a focus on the fourth-order case. It derives the corresponding filter functions, implements and validates a numerical integration pipeline achieving ~2% precision on Gaussian random field (GRF) integrals, introduces a novel derivative estimator for simulation suites with arbitrarily distributed cosmological parameters, and performs a Fisher-matrix forecast for a DES-Y3-like non-tomographic equal-scale setup. The central result is that adding the connected fourth-order aperture statistic ⟨M_ap^4⟩_c yields only minimal improvement in cosmological constraining power relative to a joint ⟨M_ap^2⟩ + ⟨M_ap^3⟩ analysis.

Significance. If the forecast result holds, the work supplies a practical pipeline and theoretical relations that could enable extraction of additional non-Gaussian information from weak-lensing surveys. The 2% GRF precision demonstrates control over the integration step, and the derivative estimator addresses a practical need for irregular simulation grids; however, the reported minimal improvement implies that, for the specific non-tomographic equal-scale configuration, fourth-order statistics add limited new information beyond lower-order aperture statistics.

major comments (2)
  1. [Fisher forecast section] § on Fisher forecast / derivative estimator: the headline conclusion of minimal improvement rests on Fisher-matrix elements computed with a new derivative estimator for arbitrarily distributed cosmological parameter sets. This estimator is introduced after the GRF validation and is not cross-checked against analytic derivatives or standard finite-difference methods on a regular grid; any systematic bias or excess variance in the estimated derivatives would propagate directly into the reported information gain and the 'minimal improvement' statement.
  2. [Abstract and forecast results] Abstract and results section: the minimal-improvement claim is obtained exclusively for non-tomographic, equal-scale aperture statistics. The manuscript does not quantify how the conclusion changes under tomographic binning or multi-scale combinations, which are the configurations most relevant for Stage-IV analyses and could alter the relative information content of ⟨M_ap^4⟩_c.
minor comments (2)
  1. [Notation and figures] Notation for the connected fourth-order statistic should be used consistently (⟨M_ap^4⟩_c versus ⟨M_ap^4⟩) in all equations, tables, and figure captions.
  2. [Methods] A brief comparison of the new derivative estimator's computational cost or stability relative to existing interpolation-based methods would help readers assess its practical utility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the theoretical framework, numerical pipeline, and the practical relevance of the derivative estimator. Below we respond to each major comment. We have revised the manuscript to address the concerns where possible.

read point-by-point responses

  1. Referee: [Fisher forecast section] § on Fisher forecast / derivative estimator: the headline conclusion of minimal improvement rests on Fisher-matrix elements computed with a new derivative estimator for arbitrarily distributed cosmological parameter sets. This estimator is introduced after the GRF validation and is not cross-checked against analytic derivatives or standard finite-difference methods on a regular grid; any systematic bias or excess variance in the estimated derivatives would propagate directly into the reported information gain and the 'minimal improvement' statement.

    Authors: We agree that additional validation of the derivative estimator would strengthen the robustness of the Fisher forecast results. The estimator was developed specifically to handle simulation suites with irregularly distributed cosmological parameters, which is a common practical constraint. While the integration pipeline itself was validated to 2% precision on Gaussian random fields, we acknowledge that a direct comparison of the derivative estimator to standard finite-difference methods on a regular grid was not performed. In the revised manuscript we will add a new subsection that performs such a cross-check on a regular-grid subset of the available simulations, quantifying any differences in the resulting Fisher-matrix elements and confirming that the reported minimal improvement is not affected by estimator bias. revision: yes

  2. Referee: [Abstract and forecast results] Abstract and results section: the minimal-improvement claim is obtained exclusively for non-tomographic, equal-scale aperture statistics. The manuscript does not quantify how the conclusion changes under tomographic binning or multi-scale combinations, which are the configurations most relevant for Stage-IV analyses and could alter the relative information content of ⟨M_ap^4⟩_c.

    Authors: We acknowledge that the forecast demonstrating minimal additional constraining power is restricted to the non-tomographic, equal-scale configuration, as stated in the abstract and Section on Fisher forecast. This choice isolates the information content in a controlled DES-Y3-like setting while focusing on the new theoretical relations and numerical methods. Tomographic binning and multi-scale combinations are indeed more relevant for Stage-IV surveys and could change the relative contribution of the connected fourth-order term. Performing the full tomographic forecast lies beyond the scope of the present work, which is intended as a methodological first step. In the revised version we will expand the discussion and conclusions to explicitly note this limitation, discuss how the framework can be extended to tomographic analyses, and outline the expected computational requirements for such an extension. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives the relation of nth-order aperture statistics to shear nPCF and convergence polyspectra analytically, then specializes to fourth-order filters whose behavior is tested by direct numerical integration over a GRF 4PCF, reaching 2% precision after mitigating pipeline effects. A separate novel derivative estimator is introduced for simulation suites with arbitrary cosmological sampling and is used to populate the Fisher matrix for the forecast. The headline result (minimal improvement when adding fourth-order statistics) is the numerical output of that forecast rather than a quantity defined to equal its inputs by construction. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the chain; the analytic content and numerical pipeline remain independent of the final forecast numbers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the accuracy of the new derivative estimator, the validity of the Gaussian random field test for the non-Gaussian pipeline, and the assumption that equal-scale non-tomographic statistics capture the relevant information content.

axioms (1)
  • domain assumption The numerical integration over the 4PCF of a Gaussian random shear field validates the pipeline for the subsequent non-Gaussian forecast
    Invoked when the authors state that 2% precision on GRF remains well below the noise budget of Stage IV surveys

pith-pipeline@v0.9.0 · 5848 in / 1365 out tokens · 37228 ms · 2026-05-18T17:27:15.694549+00:00 · methodology

discussion (0)

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Reference graph

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