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arxiv: 2604.18378 · v1 · submitted 2026-04-20 · 🌌 astro-ph.CO

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Fourth-order galaxy-galaxy-lensing: Theoretical framework and direct estimation

Jonathan Oel , Lucas Porth , Peter Schneider , Elena Silvestre-Rosello
Authors on Pith no claims yet

Pith reviewed 2026-05-10 03:29 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords galaxy-galaxy lensingfour-point correlation functionaperture statisticshigher-order statisticsstage IV surveysnon-GaussianityFFT estimatortrispectrum
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The pith

Fourth-order galaxy-galaxy lensing aperture statistics can be measured with sub-percent accuracy and detected at signal-to-noise of nine in stage IV survey mocks.

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

This paper extends traditional galaxy-galaxy lensing, which uses two-point correlations between galaxy positions and gravitational shear to probe matter distributions, to fourth order so that non-Gaussian features become accessible. The authors define the four-point correlation function linking shear to triplets of foreground galaxies, relate it to the trispectrum, and convert the result into aperture statistics by deriving an analytical filter function. They implement this conversion through numerical integration and introduce a direct estimator based on fast Fourier transforms that works on pixelized maps to extract galaxy-mass aperture moments of any order. When tested on a realistic mock catalog covering 2000 square degrees, the connected part of the fourth-order aperture statistic reaches a signal-to-noise ratio of roughly nine on small scales while the numerical integration itself stays accurate to better than one percent. The framework therefore supplies both the theoretical bridge to higher-order information and a practical tool for extracting it from forthcoming large surveys.

Core claim

We define the four-point correlation function (4PCF) between the shear and the positions of triplets of foreground galaxies and derive its relation to the respective trispectrum. We convert the 4PCF to aperture statistics and derive the analytical form of the respective filter function, which we then implement in a numerical integration pipeline. Furthermore, we develop a direct estimator that allows us to measure galaxy-mass aperture moments of arbitrary order on pixelized data using a Fast-Fourier-Transform (FFT) algorithm. We show that the corresponding aperture measure ⟨N³ M_ap⟩ can be calculated with sub-percent accuracy on relevant aperture scales, θ, by means of numerical integration.

What carries the argument

The aperture statistic ⟨N³ M_ap⟩ obtained from the four-point correlation function of shear and galaxy triplets through an analytical filter function and computed on pixelized data by an FFT-based direct estimator.

Load-bearing premise

The mock catalog with a realistic stage IV survey setup accurately represents the non-Gaussian features and noise properties of real observations, and the analytical relations derived from the 4PCF to aperture statistics hold without significant biases from lensing approximations.

What would settle it

A comparison showing more than one percent difference between the numerical integration result for ⟨N³ M_ap⟩(θ) and the output of the FFT estimator on the same mock catalog at aperture scales of a few arcminutes would falsify the claimed sub-percent accuracy.

Figures

Figures reproduced from arXiv: 2604.18378 by Elena Silvestre-Rosello, Jonathan Oel, Lucas Porth, Peter Schneider.

Figure 1
Figure 1. Figure 1: Sketch of a galaxy-galaxy-galaxy-shear configuration which the 4PCF Ggggγ is sensitive to. The configuration is fully characterized by the radial separations ϑ1, ϑ2 and ϑ3 and two of the three opening angles ϕ12, ϕ13 and ϕ23 which are related by ϕ13 = ϕ12 + ϕ23. that also contains information about lower-order correlators. Us￾ing the fact that N/N¯ = 1 + κg and that the direction of pro￾jection of the rota… view at source ↗
Figure 2
Figure 2. Figure 2: Relative difference of the numerical integration results for ⟨N3Map⟩ according to Eq. (29) with respect to the expected results ob￾tained from the second-order aperture statistics. The integration was performed using a Gaussian 4PCF, and with different configurations of the integration range [ymin, ymax], all using ∆ln ≈ 0.11 and ∆ϕ/2π = 1/350. The dotted (dash-dotted) lines show that the results converge … view at source ↗
Figure 3
Figure 3. Figure 3: The real part of Eq. (29) as a function of the angular variables ϕ12 and ϕ23, with the integrations over the three radial variables already performed. All results are normalized by the respective maximum value. While the upper plots show the 2-D structure as a function of both ϕ12 and ϕ23, the lower plots show the results from the upper plots marginal￾ized over the ϕ23-axis. The left and right panel show d… view at source ↗
Figure 5
Figure 5. Figure 5: The aperture statistics ⟨M2 ap⟩(θ) (left) and ⟨N2 ⟩(θ) (right) com￾puted with the FFT-method, as well as with orpheus and the direct in￾tegration of the respective 2PCFs, applied to the same mock data. The lower plots show the relative difference of the results obtained with the FFT-method and orpheus w.r.t. to the integration of the 2PCF. In the lower plot of the right panel, the solid (dotted) blue line … view at source ↗
Figure 6
Figure 6. Figure 6: The aperture measure ⟨N3Map⟩ as a function of the aperture scale θ calculated via the FFT-estimator on simulated mock data. In the up￾per plot, the black curve shows the full ⟨N3Map⟩ statistic, the blue one the connected part, and the red one the Gaussian part. The lower plot shows the S/N of the connected part at the particular aperture scales. The computation was performed on 1632 different mock realizat… view at source ↗
read the original abstract

Traditional galaxy-galaxy lensing is a well-established method of probing the statistical properties of the Universe's matter and galaxy distribution. However, this measure does not carry all the statistical information, provided the matter and galaxy distribution contain non-Gaussian features. In order to study these non-Gaussianities, it is necessary to consider higher-order statistical measures. The aim of this work is to extend the analytical basis describing the statistical correlations between galaxies and shear to the fourth order, with special emphasis on the associated aperture statistics. In order to include fourth-order statistics in future analysis of the relation between mass and galaxies, we further investigate whether we can expect to detect these statistics from observations of stage IV surveys. We define the four-point correlation function (4PCF) between the shear and the positions of triplets of foreground galaxies and derive its relation to the respective trispectrum. We convert the 4PCF to aperture statistics and derive the analytical form of the respective filter function, which we then implement in a numerical integration pipeline. Furthermore, we develop a direct estimator that allows us to measure galaxy-mass aperture moments of arbitrary order on pixelized data using a Fast-Fourier-Transform (FFT) algorithm. We show that the corresponding aperture measure $\langle\mathcal{N}^3 M_\mathrm{ap}\rangle$ can be calculated with sub-percent accuracy on relevant aperture scales, $\theta$, by means of numerical integration. Furthermore, we apply the FFT-based direct estimator to a mock catalog with a realistic stage IV survey setup on a sky area of $2000~\mathrm{deg}^2$, and detect the connected part of the aperture statistics $\langle\mathcal{N}^3 M_\mathrm{ap}\rangle(\theta)$ with a signal-to-noise ratio of roughly nine on small aperture scales.

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 / 3 minor

Summary. The paper extends galaxy-galaxy lensing to fourth order by defining the 4PCF between shear and triplets of foreground galaxies, relating it to the trispectrum via standard Fourier transforms, deriving the corresponding aperture statistics ⟨N³ M_ap⟩ with explicit filter functions, implementing a numerical integration pipeline that achieves sub-percent accuracy, and developing an FFT-based direct estimator for pixelized data. Application to a 2000 deg² stage-IV mock catalog yields a detection of the connected part of ⟨N³ M_ap⟩(θ) at SNR ≈ 9 on small aperture scales.

Significance. If the central claims hold, the work supplies a practical theoretical and computational framework for extracting non-Gaussian information from galaxy-mass correlations beyond the two-point level. The sub-percent numerical accuracy of the aperture filters and the reported SNR ~9 detection on realistic mocks indicate that fourth-order aperture statistics could become a viable observable for stage-IV surveys, provided the mock-based validation is tightened. The derivations follow standard relations without circularity or free parameters.

major comments (2)
  1. [results section] Application to mock catalog (results section): the SNR ≈ 9 claim for the connected ⟨N³ M_ap⟩(θ) on small scales rests on the mock faithfully reproducing the connected trispectrum contribution, survey mask, shape noise, and the 3D-to-observed mapping. No end-to-end closure test is reported that compares the numerical integration prediction (from the derived filter applied to the trispectrum) directly against the FFT estimator output on the identical mock; without this, percent-level biases from Born or lens-lens approximations (known to grow at fourth order on arcminute scales) cannot be ruled out.
  2. [results section] Error analysis (results section): the quoted signal-to-noise ratio lacks a full covariance propagation that includes the estimator variance, mask effects, and cosmic variance from the finite mock volume; this directly affects the reliability of the detection significance.
minor comments (3)
  1. [introduction] The notation for the galaxy number density N and the aperture mass M_ap is introduced without an explicit reminder of their relation to the convergence and shear fields in the opening paragraphs.
  2. [numerical integration] Figure captions for the numerical integration tests should state the exact aperture scales θ and the integration method (e.g., quadrature order) used to achieve the sub-percent accuracy.
  3. [discussion] A brief comparison to existing third-order galaxy-galaxy lensing results (e.g., aperture statistics from earlier works) would help place the fourth-order extension in context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review, positive assessment of the work, and recommendation for minor revision. The comments identify valuable opportunities to strengthen the validation and error analysis. We address each major comment below and will incorporate the necessary additions in the revised manuscript.

read point-by-point responses

  1. Referee: [results section] Application to mock catalog (results section): the SNR ≈ 9 claim for the connected ⟨N³ M_ap⟩(θ) on small scales rests on the mock faithfully reproducing the connected trispectrum contribution, survey mask, shape noise, and the 3D-to-observed mapping. No end-to-end closure test is reported that compares the numerical integration prediction (from the derived filter applied to the trispectrum) directly against the FFT estimator output on the identical mock; without this, percent-level biases from Born or lens-lens approximations (known to grow at fourth order on arcminute scales) cannot be ruled out.

    Authors: We appreciate the referee's emphasis on rigorous end-to-end validation. The numerical integration pipeline was validated independently to sub-percent accuracy on the derived filter functions, and the FFT estimator was tested on controlled inputs. The stage-IV mock incorporates full non-linear clustering from the underlying simulation, thereby including connected trispectrum contributions along with survey mask, shape noise, and projection effects. Nevertheless, we acknowledge that a direct comparison—applying the numerical trispectrum-based prediction to the same mock and contrasting it with the FFT estimator output—has not been performed. This leaves room for small residual biases from higher-order effects such as lens-lens coupling or Born approximation violations. We will add this closure test to the revised manuscript by estimating the trispectrum from the mock, computing the theoretical ⟨N³ M_ap⟩, and comparing it directly to the estimator measurements on identical data. revision: yes

  2. Referee: [results section] Error analysis (results section): the quoted signal-to-noise ratio lacks a full covariance propagation that includes the estimator variance, mask effects, and cosmic variance from the finite mock volume; this directly affects the reliability of the detection significance.

    Authors: We agree that a more complete covariance treatment is required for robust significance claims. The reported SNR ≈ 9 was obtained from the scatter across the available mock realizations, which already incorporates cosmic variance and shape noise for the 2000 deg² area. However, we did not construct an explicit covariance matrix that fully propagates mask-induced mode coupling and the intrinsic variance of the FFT estimator. In the revised version we will augment the error analysis with jackknife or bootstrap resampling of the mock catalog to include these contributions, recompute the covariance, and update the detection significance accordingly. This will provide a more conservative and transparent assessment of the SNR. revision: yes

Circularity Check

0 steps flagged

Derivation from 4PCF definition through trispectrum and aperture filters uses standard Fourier and filter integrals with no reduction to fitted inputs or self-referential equations.

full rationale

The paper defines the 4PCF for shear and galaxy triplets, relates it to the trispectrum via standard Fourier transforms, converts to aperture statistics by deriving the filter function through explicit integration over the aperture filters, and implements numerical evaluation of that integral. The FFT direct estimator is a computational implementation of the same aperture moments on pixelized data. Neither step invokes fitted parameters, renames a prior result as a prediction, nor relies on a load-bearing self-citation whose content is unverified outside the paper. The SNR estimate on the mock catalog is an application test, not part of the derivation chain itself. The chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard weak-lensing assumptions such as the Born approximation and the validity of the trispectrum relation; no new free parameters, axioms, or invented entities are introduced beyond those in conventional lensing theory.

axioms (1)
  • domain assumption Standard assumptions of weak gravitational lensing theory hold, including the Born approximation and negligible higher-order lensing effects.
    Implicit in the derivation of the 4PCF from shear and galaxy positions and its conversion to aperture statistics.

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