Fourth-Order Aperture Mass Statistics

• Fourth-order aperture mass statistics are advanced measures that capture non-Gaussian features in weak-lensing data through compensated filtering of convergence and shear fields.
• They employ varied filters, such as starlet and Gaussian, to achieve explicit E/B-mode separation and robust multi-scale analysis of the convergence trispectrum.
• Estimation methods include both map-based and catalogue-based approaches, delivering computational gains while highlighting challenges in non-Gaussian covariance and binning effects.

Searching arXiv for recent and foundational papers on fourth-order aperture mass statistics and related weak-lensing/aperture-mass formalism. Fourth-order aperture mass statistics are fourth-order moments or cumulants of the aperture-filtered weak-lensing field, defined with compensated filters applied to convergence or shear and used to access trispectrum-level non-Gaussian information with explicit $E/B$ separation. In the literature, the term encompasses several related but non-identical objects: the non-standardised fourth central moment of aperture-mass maps, called “kurtosis” in some map-based analyses; the connected fourth-order aperture-mass cumulant $M_{\rm c}^4$, which is a filtered projection of the convergence trispectrum; and equal-scale or multiscale fourth-order aperture measures obtained by integrating the shear four-point correlation function against explicit kernels (Peel et al., 2018, Porth et al., 2021, Silvestre-Rosello et al., 9 Sep 2025).

1. Definitions and formal status of the fourth-order statistic

The aperture mass is the compensated filtering of the convergence field,

$$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$

with $U$ radially symmetric and compensated,

$$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$

The same quantity can be written in terms of tangential shear with a corresponding filter $Q$,

$$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$

where

$$Q(\vartheta)\equiv \frac{2}{\vartheta^2}\int_0^\vartheta \vartheta' U(\vartheta')\,{\rm d}\vartheta' - U(\vartheta).$$

This compensated construction underlies the standard claim that aperture mass is $E/B$-decomposed by construction (Leonard et al., 2012, Porth et al., 2021).

At fourth order, a central conceptual distinction is required. In the map-based modified-gravity analysis of Giocoli et al., the quantity labelled “kurtosis” is

$$\langle M^4_{\rm ap}\rangle(\vartheta)=\frac{1}{N}\sum_k\left[M_{\rm ap}(\boldsymbol{\theta}_k;\vartheta)-\overline{M_{\rm ap}(\vartheta)}\right]^4,$$

which is a central moment, non-standardised, not connected, and not the excess kurtosis $M_{\rm c}^4$0. By contrast, the direct-estimator and 4PCF-based formalisms emphasize the connected fourth-order cumulant, for which, when $M_{\rm c}^4$1,

$M_{\rm c}^4$2

A persistent misconception is therefore that all “fourth-order aperture mass” measurements are interchangeable. They are not: a raw fourth moment, a fourth central moment, a normalized kurtosis, and a connected cumulant encode different combinations of Gaussian and non-Gaussian contributions (Peel et al., 2018, Porth et al., 2021).

2. Filters, scale selection, and wavelet representations

Filter choice is not ancillary. The aperture-mass formalism requires compensation, but the localization properties of the chosen filter strongly affect fourth-order observables because tails, rare peaks, and mode mixing are more consequential at fourth order than for the variance. Leonard et al. showed that aperture mass is formally identical to a wavelet transform at a specific scale, and that many commonly used aperture-mass filters are non-local in both real and Fourier space. The same work emphasized that the starlet wavelet is localized in both spaces and can be computed with very fast algorithms, with reported speed-up factors of $M_{\rm c}^4$3 to $M_{\rm c}^4$4 for aperture radii in the range $M_{\rm c}^4$5 to $M_{\rm c}^4$6 pixels on a $M_{\rm c}^4$7 image (Leonard et al., 2012).

This wavelet equivalence was used directly in the DUSTGRAIN-pathfinder analysis, where starlet-filtered convergence maps were treated as aperture-mass maps,

$M_{\rm c}^4$8

For $M_{\rm c}^4$9 maps with pixel scale $$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$0 arcsec, the dyadic scales were

$$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$1

The starlet filter was highlighted as compensated, compactly supported, non-oscillatory in both real and Fourier space, and admitting exact reconstruction; those properties reduce frequency leakage when one measures higher moments on filtered maps (Peel et al., 2018).

Catalog-based and 4PCF-based formalisms have adopted different filters. The direct-estimator work of 2021 used the Schneider et al. polynomial filter,

$$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$2

which has compact support (Porth et al., 2021). The 2025 shear-4PCF program and the 2026 fourth-order galaxy-galaxy-lensing formalism instead adopted the Gaussian compensated filter of Crittenden et al.,

$$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$3

whose Fourier transform is used as the aperture filter in trispectrum expressions (Silvestre-Rosello et al., 9 Sep 2025, Oel et al., 20 Apr 2026). A plausible implication is that “fourth-order aperture mass statistics” should be understood as a family of filter-dependent summaries rather than a unique observable.

3. Estimation strategies from maps and shape catalogues

One implementation route computes fourth order directly on filtered maps. In the modified-gravity study, aperture-mass statistics were measured on starlet-filtered convergence maps produced by ray-tracing simulations, not from galaxy ellipticity catalogues. The averaging was over all valid pixels after excluding those whose distance to the map border was smaller than the filter diameter, and the statistic was then summarized over $$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$4 realizations for each cosmology, scale, and source redshift. Most of that analysis used noise-free maps; in the Appendix, adding Gaussian shape noise with a Euclid-like prescription changed the kurtosis distributions substantially and made them strongly non-Gaussian and asymmetric (Peel et al., 2018).

A second route works directly on shape catalogues. The direct-estimator formalism of Halder et al. rewrites the unequal-index $$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$5-tuple sums algebraically so that general aperture-mass statistics can be obtained with a linear-order algorithm once apertures have been populated with galaxies. For fourth order, the single-aperture estimator becomes

$$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$6

where $$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$7 and $$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$8 are one-pass power sums over weighted galaxy ellipticities and filter values. The same paper generalized to arbitrary order with complete Bell polynomials and showed that multiscale fourth order can still be computed from a finite set of one-pass multivariate sums; for $$M_{\rm ap}(\boldsymbol{\theta}_0;\vartheta)=\int {\rm d}^2\theta_1\,\kappa(\boldsymbol{\theta}_1)\,U(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$9, the multiscale basis contains $U$0 elements (Porth et al., 2021).

The computational gain is decisive. Brute-force aperture filtering from shear scales as $U$1 on an $U$2 image, whereas the starlet transform scales as $U$3. In catalogue space, the direct-estimator method avoids explicit galaxy quadruplet enumeration and empirically shows linear dependence on the total number of galaxies for fixed order and binning. This is the technical precondition for any practical fourth-order analysis on contemporary survey volumes (Leonard et al., 2012, Porth et al., 2021).

4. Trispectrum and shear-4PCF formulations

The connected fourth-order aperture statistic has a clean Fourier-space interpretation. In the cumulant formalism,

$U$4

so the fourth-order connected aperture mass is explicitly a filtered convergence trispectrum (Porth et al., 2021). The 2025 formal treatment generalized this to arbitrary $U$5 and wrote the equal-scale fourth-order moment as an integral over the convergence polyspectrum $U$6, while emphasizing that the observationally relevant route is the real-space transform from the shear 4PCF rather than direct trispectrum estimation on finite masked surveys (Silvestre-Rosello et al., 9 Sep 2025).

In real space, a shear 4PCF consists of eight independent natural components $U$7, $U$8, once homogeneity and isotropy are imposed. For equal-scale fourth order, the aperture measures are reduced from eight-dimensional integrals to five-dimensional ones over three side lengths and two angles, with explicit kernels $U$9. The simplest kernel is

$$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$0

and the remaining kernels are more elaborate oscillatory combinations with Gaussian envelopes. The eight complex fourth-order aperture measures can then be combined into mode-pure real statistics; for a pure $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$1-mode lensing field, only $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$2 should survive (Silvestre-Rosello et al., 9 Sep 2025).

The numerical transform from 4PCF to fourth-order aperture measures is nontrivial. The 2025 pipeline identified strong sensitivity to angular sampling near nearly degenerate quadrilateral configurations, finite radial integration range, and bin-averaging effects. With logarithmic radial binning, nearly symmetric angular sampling, $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$3 angular bins, $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$4 radial bins, and $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$5, the fourth-order aperture statistics were recovered to about the two-percent level over $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$6 in Gaussian random fields, a precision stated to remain well below the noise budget of Stage IV surveys (Silvestre-Rosello et al., 9 Sep 2025). This suggests that the current technical bottleneck is not the aperture transform itself but the accurate measurement and covariance modeling of the underlying 4PCF.

5. Cosmological role, detectability, and information content

The principal cosmological motivation is that weak-lensing non-Gaussianity generated by nonlinear structure formation is not exhausted by second-order statistics. This became explicit in the modified-gravity application to $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$7 models with massive neutrinos, where some models could mimic $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$8CDM in the matter power spectrum and in convergence two-point statistics, motivating the use of higher-order aperture-mass observables. In that setting, the fourth-order statistic showed mean deviations from $$\int_0^\theta \vartheta\,U(\vartheta)\,{\rm d}\vartheta = 0.$$9CDM of up to about $Q$0, roughly $Q$1 more than the variance, and a characteristic scale dependence with a local minimum near $Q$2 for neutrino cases. Yet the same analysis found that kurtosis was a poor discriminator, “not exceeding 55% at any filter scale or redshift,” and that peak counts computed in aperture-mass maps outperformed third- and fourth-order moments (Peel et al., 2018).

Detectability is a distinct question from discriminatory power. The direct-estimator study of 2021 validated equal-scale and multiscale aperture-mass statistics up to tenth order on Gaussian mocks and found that a KiDS-1000-like survey should have sufficient fidelity to detect aperture-mass statistics up to fourth order. In the SLICS-based analysis, the signal-to-noise peaked near $Q$3, and higher orders, including fourth order, required an aperture oversampling factor $Q$4 to capture nearly all available signal (Porth et al., 2021).

Forecasted information gain from fourth order has been more cautious. In the 2025 shear-4PCF study, a DES-Y3-like, non-tomographic, equal-scale Fisher analysis used the joint data vector

$Q$5

with $Q$6 and reported only a minimal improvement when adding $Q$7 to a $Q$8 analysis. The paper explicitly restricted that conclusion to non-tomographic equal-scale aperture statistics in a DES-Y3-like setup (Silvestre-Rosello et al., 9 Sep 2025). The combined literature therefore supports a sharper statement: fourth-order aperture mass is detectable, formally well-defined, and physically distinct, but its incremental cosmological leverage is highly implementation-dependent.

6. Survey measurements, sampling distributions, and extensions

A direct observational milestone was reached with the DES Year 3 application of efficient shear-4PCF estimation. That analysis used the DES Y3 metacalibration catalogue, with $Q$9 million galaxies over $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$0, mean redshift $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$1, and weighted number density $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$2, in a non-tomographic setup chosen both because the 4PCF estimator scales quartically with the number of tomographic bins and because mock signal-to-noise studies indicated that the non-tomographic setup gives the largest S/N for this statistic. The resulting fourth-order aperture statistics showed a significant $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$3-mode signal, no convincing evidence for significant $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$4-modes or parity-violating modes, and a very strong rejection of the null hypothesis of no connected cosmological $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$5-mode signal, with

$$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$6

The same paper found the sampling distribution of the fourth-order aperture mass to be significantly skewed and showed that internal covariance estimates underestimate the variance, a point attributed mainly to the non-Gaussian, skewed sampling distribution (Porth et al., 9 Sep 2025).

That skewness is not merely a statistical technicality. It bears directly on likelihood construction, covariance estimation, and comparisons with simulations. The DES Y3 $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$7-mode amplitude was reported to be significantly lower than in the T17 mocks, but no full cosmological interpretation was attempted because the sampling distribution is highly non-Gaussian and the amplitude discrepancy is not straightforward to interpret. The same work therefore frames fourth-order aperture mass as already measurable on Stage-III data, but not yet reducible to a routine Gaussian-likelihood summary statistic (Porth et al., 9 Sep 2025).

The formalism has also been generalized beyond pure cosmic shear. In fourth-order galaxy-galaxy lensing, the mixed aperture statistic

$$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$8

correlates the aperture-smoothed foreground galaxy field cubed with the aperture mass and probes the projected galaxy-galaxy-galaxy-mass trispectrum. The 2026 framework derived the explicit five-dimensional kernel for converting the relevant 4PCF into $$M_{\rm ap/\times}(\boldsymbol{\theta}_0;\vartheta)\equiv \int {\rm d}^2\theta_1\,\gamma_{\rm t/\times}(\boldsymbol{\theta}_1;\boldsymbol{\theta}_0)\,Q(|\boldsymbol{\theta}_1-\boldsymbol{\theta}_0|;\vartheta),$$9, developed an FFT-based direct estimator for mixed aperture moments of arbitrary order on pixelized data, achieved sub-percent accuracy for equal-scale $$Q(\vartheta)\equiv \frac{2}{\vartheta^2}\int_0^\vartheta \vartheta' U(\vartheta')\,{\rm d}\vartheta' - U(\vartheta).$$0 over $$Q(\vartheta)\equiv \frac{2}{\vartheta^2}\int_0^\vartheta \vartheta' U(\vartheta')\,{\rm d}\vartheta' - U(\vartheta).$$1, and reported that in a realistic stage-IV mock setup on $$Q(\vartheta)\equiv \frac{2}{\vartheta^2}\int_0^\vartheta \vartheta' U(\vartheta')\,{\rm d}\vartheta' - U(\vartheta).$$2 the connected part of $$Q(\vartheta)\equiv \frac{2}{\vartheta^2}\int_0^\vartheta \vartheta' U(\vartheta')\,{\rm d}\vartheta' - U(\vartheta).$$3 is detected with signal-to-noise ratio of roughly nine on small aperture scales (Oel et al., 20 Apr 2026).

Taken together, these results establish a mature but non-uniform picture. Fourth-order aperture mass statistics are now available as map moments, direct catalogue estimators, 4PCF transforms, and mixed galaxy-shear aperture observables. They are physically tied to four-point clustering, technically enabled by compensated filtering and multiscale compression, and empirically measurable in Stage-III and Stage-IV survey regimes. At the same time, the literature repeatedly identifies the same limiting issues: the distinction between full and connected fourth order, strong sensitivity to masks and binning, non-Gaussian sampling distributions, and the possibility that detectability outpaces cosmological information gain (Porth et al., 2021, Silvestre-Rosello et al., 9 Sep 2025, Porth et al., 9 Sep 2025).