---
title: Third-Order Aperture-Mass Statistics
url: https://www.emergentmind.com/topics/third-order-aperture-mass-statistics
type: topic
---

# Third-Order Aperture-Mass Statistics

Third-order aperture-mass statistics, commonly denoted as ⟨M_ap^3⟩, are higher-order weak lensing observables that encapsulate the skewness, or non-Gaussianity, of the projected matter distribution via the cosmic shear field. By measuring the third moment of the aperture mass over a field of galaxies, these statistics access information in the bispectrum of matter fluctuations not captured by traditional second-order shear statistics, such as the power spectrum. As a result, third-order aperture-mass statistics break parameter degeneracies present at second order and significantly tighten cosmological constraints—particularly on key combinations like S₈ ≡ σ₈√(Ω_m/0.3)—in current and future weak lensing surveys.

## 1. Formalism and Core Definitions

The aperture mass M_ap(θ;θ_ap) at position θ and filter radius θ_ap is defined by applying a compensated, circular filter U_θ_ap(ϑ) to the convergence κ or, equivalently, to the tangential shear γₜ:
\[
M_{\rm ap}(\theta; \theta_{\rm ap}) = \int d^2\vartheta' \, U_{\theta_{\rm ap}}(|\vartheta'|)\, \kappa(\theta+\vartheta') = \int d^2\vartheta' \, Q_{\theta_{\rm ap}}(|\vartheta'|)\, \gamma_t(\theta+\vartheta')
\]
where the filter functions are related by:
\[
Q_{\theta_{\rm ap}}(\vartheta) = \frac{2}{\vartheta^2} \int_0^{\vartheta} d\vartheta' \, \vartheta' U_{\theta_{\rm ap}}(\vartheta') - U_{\theta_{\rm ap}}(\vartheta)
\]
A standard choice for the compensated filter is the Crittenden et al. (2002) “exponential” type:
\[
U_{\theta_{\rm ap}}(\vartheta) = \theta_{\rm ap}^{-2} \, u(\vartheta/\theta_{\rm ap}), \quad u(x) = \frac{1}{2\pi}\left(1-\frac{x^2}{2}\right) e^{-x^2/2}
\]
with Fourier transform:
\[
\hat{u}(\eta) = \frac{\eta^2}{2} e^{-\eta^2/2}
\]
The third-order aperture mass moment in tomographic bins i, j, k is:
\[
\langle M_{\rm ap}^3\rangle^{(ijk)}(\theta_1, \theta_2, \theta_3) = \left\langle M_{\rm ap}^{(i)}(\theta_1) M_{\rm ap}^{(j)}(\theta_2) M_{\rm ap}^{(k)}(\theta_3) \right\rangle
\]

## 2. Theoretical Modeling and Bispectrum Connection

The third-order moment is directly connected to the convergence bispectrum B_{κκκ}:
\[
\langle M_{\rm ap}^3\rangle^{(ijk)}(\theta_1,\theta_2,\theta_3) = \int \frac{d^2\ell_1}{(2\pi)^2} \int \frac{d^2\ell_2}{(2\pi)^2} B_{κκκ}^{(ijk)}(\ell_1, \ell_2, \ell_3) \hat{u}(\theta_1 |\ell_1|) \hat{u}(\theta_2 |\ell_2|) \hat{u}(\theta_3 |\ell_3|)
\]
with $\ell_3 = |\ell_1 + \ell_2|$.

Under the Limber approximation, the bispectrum projects as:
\[
B_{κκκ}^{(ijk)}(\ell_1, \ell_2, \ell_3) = \int_0^{\chi_{\rm max}} d\chi\, \frac{g^{(i)}(\chi) g^{(j)}(\chi) g^{(k)}(\chi)}{a^3(\chi) \chi} B_{\delta\delta\delta}\bigl(\ell_1/\chi, \ell_2/\chi, \ell_3/\chi; z(\chi)\bigr)
\]
where $g^{(i)}(\chi)$ is the lensing efficiency for bin $i$. The 3D matter bispectrum $B_{\delta\delta\delta}$ is modeled using BiHalofit [Takahashi et al. 2020], while the power spectrum for second-order statistics is usually given by the revised Halofit or HMcode2020 [2503.03964, 2309.08602]. This theoretical backbone supports both configuration-space and Fourier-space implementations, with all major analyses adopting these models over a broad dynamic range.

## 3. Tomographic Measurements and Data Compression

In surveys with multiple tomographic bins (n_tomo), all cross-statistics $\langle M_{\rm ap}^{(i)} M_{\rm ap}^{(j)} M_{\rm ap}^{(k)}\rangle$ are, in principle, measurable, yielding up to $n_\mathrm{tomo}^3$ bispectra. However, permutation symmetries and data compression are essential for tractable analyses:
- Equal-radius compression: Only statistics with θ₁ = θ₂ = θ₃ from a discrete set of scales (e.g., {4′, 8′, 14′, 32′} for KiDS-1000 [2309.08602], {7′, 14′, 25′, 40′} for DES Y3 [2508.14018]) are retained, reducing the data vector size by ≳60% with minimal loss (<8%) in Fisher information on key parameters.
- TreeCorr and multipole estimators: The spin-2 three-point correlation function (3PCF) is measured via multipole expansions and efficient algorithms, with conversion to aperture-mass skewness achieved through analytic convolution with the filter kernels [2309.08601, 2208.11686].
- Further data reduction is done using MOPED or principal component analysis when joint two- and three-point statistics are combined [2508.14019, 2508.14018].

## 4. Covariance Estimation and Cross-Order Covariance

The covariance of $\langle M_{\rm ap}^3\rangle$ is non-Gaussian and nontrivial. Analytic models show the full covariance comprises Gaussian (disconnected), bispectrum-squared, trispectrum (power×trispectrum), and pentaspectrum (genuine six-point) contributions [2212.04485]. Accurate joint analyses require cross-covariances between second- and third-order statistics, which separate into terms governed by the power spectrum, bispectrum, and tetraspectrum. Finite-field ("supersample") effects are significant, and the tetraspectrum (five-point function in configuration or multipole space) typically dominates the cross-covariance for small filter scales.

Computation is validated against large suites of mock catalogs (e.g. SLICS, CosmoGrid, T17 full-sky) and analyzed with Student-t likelihoods to propagate sampling uncertainty [2212.04485, 2509.20443]. Analytical covariances agree at the tens of percent level with empirical mocks, enabling their use for cosmological inference in Stage III-IV surveys.

## 5. Systematics, E/B-mode Separation, and Validation

Aperture-mass statistics offer robust E/B-mode separation at third order. Explicit construction of mixed and pure B-mode statistics via filter convolution yields negligible parity-violating or leakage terms above cutoffs θ > 10 θ_min (where θ_min is the smallest measurable scale set by the survey resolution) [1311.7035]. For realistic (ground/space-based) cutoffs, this corresponds to θ ≳ 0.5′–2′—comfortably within the regime of current analyses. Comprehensive null tests in all major surveys (DES, KiDS, HSC) show B-modes, parity-violation, and PSF/modeling systematics to be subdominant (<1% of the E-mode signal) [2201.05227, 1005.4941].

Systematic error models include:
- Intrinsic alignment (IA): Treated via the non-linear alignment (NLA) model with redshift-dependent amplitude $f_{\rm IA}(z)$, modifying all relevant shear correlations.
- Baryonic effects: Marginalized via hydrodynamical response functions or avoided by scale cuts; validated by comparing hydrodynamic and DMO mock ratios [2309.08602, 2508.14018].
- Photometric redshift: Treated by shifting $n(z)$ in each bin and, importantly, self-calibrated by exploiting the complementary redshift dependence in second- and third-order signals [2508.14018, 2508.14019].

## 6. Cosmological Impact and Results

Inclusion of third-order aperture-mass statistics in joint cosmological inference considerably enhances constraining power. Results from recent large surveys include:
- DES Y3: Joint $\xi_\pm+\langle M_{\rm ap}^3\rangle$ yields $S_8=0.780\pm0.015$, $\Omega_m=0.266^{+0.039}_{-0.040}$, a 111% Figure-of-Merit (FoM) gain over second-order alone. In $w$CDM, $w_0=-1.39\pm0.31$ and a 22% joint $S_8$-$w_0$ improvement [2508.14018].
- KiDS-1000: Adding $\langle M_{\rm ap}^3\rangle$ to COSEBIs tightens $S_8$ by ∼23%, yielding $S_8=0.772\pm0.022$, $\Omega_m=0.248^{+0.062}_{-0.055}$ [2309.08602].
- HSC Y3: Joint analysis improves $S_8$-$\Omega_m$ FoM by 80% and achieves $S_8=0.736\pm0.020$ [2508.14019].
- Simulated DES Y3: Addition of $\langle M_{\rm ap}^3\rangle$ brings an 83% improvement in the $\Omega_m$–$S_8$ FoM, with marginal $S_8$ errors shrinking by $\sim$36% [2503.03964].
- Reduced skewness $S(\theta;z) = \langle M_{\rm ap}^3\rangle / \langle M_{\rm ap}^2\rangle^2$ shows expected redshift and scale evolution, carrying independent cosmological information [2201.05227].

These constraints are robust to baryonic, IA, and photo-z systematics at the current measurement precision. The third-order statistic also enables internal calibration of high-z photo-z errors and can help distinguish subtle non-Gaussian features of the large-scale structure, including those arising in modified gravity or massive neutrino cosmologies [1805.05146].

## 7. Algorithms, Codebases, and Practical Recommendations

State-of-the-art implementations use multipole expansions and FFT-based convolution for rapid measurement and modeling of the shear 3PCF, with neural network emulators (e.g., CosmoPower, TensorFlow PCA+ResNet) accelerating theory evaluation to practical speeds for MCMC inference [2309.08601, 2208.11686, 2503.03964, 2508.14019]:
- The “fastnc” code enables O(N log N) computation of Map^3 via multipole decomposition.
- Binning strategies restrict to equal-radius apertures and employ as few as 4–8 scales, preserving information with tractable data vectors.
- Data compression (MOPED, PCA) is recommended for high-dimensional vector stability, with typical final vectors of length ∼100–300.
- All major pipelines and validation suites are public or reproducible, e.g., https://github.com/sheydenreich/threepoint/releases/ [2208.11686].

Aperture-mass measurement is robust to E/B leakage so long as θ≫ θ_min, and baryonic feedback can be cleanly marginalized or avoided via conservative scale cuts. Future Stage IV surveys (LSST, Euclid, Roman) are expected to benefit strongly from joint second- and third-order analyses, with third-order statistics now established as a practical, validated, and information-rich tool for cosmological weak lensing [2309.08602, 2508.14019, 2508.14018, 2208.11686].

Source: https://www.emergentmind.com/topics/third-order-aperture-mass-statistics