Third-Order Aperture-Mass Statistics

• Third-order aperture-mass statistics are higher-order weak lensing measures that quantify the skewness of the cosmic shear field and break degeneracies in key cosmological parameters.
• They utilize compensated circular filters and bispectrum integration, linking the third moment of the aperture mass to non-Gaussian matter fluctuations.
• Joint analyses with second-order statistics significantly tighten cosmological constraints on parameters like S₈ and Ωₘ in modern surveys.

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;θap) at position θ and filter radius θ_ap is defined by applying a compensated, circular filter Uθ_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 (Gomes et al., 5 Mar 2025, Burger et al., 2023). 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 (Burger et al., 2023), {7′, 14′, 25′, 40′} for DES Y3 (Gomes et al., 19 Aug 2025)) 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 (Porth et al., 2023, Heydenreich et al., 2022).
• Further data reduction is done using MOPED or principal component analysis when joint two- and three-point statistics are combined (Sugiyama et al., 19 Aug 2025, Gomes et al., 19 Aug 2025).

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 (Linke et al., 2022). 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 (Linke et al., 2022, Wielders et al., 24 Sep 2025). 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) (Shi et al., 2013). 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) (Secco et al., 2022, Semboloni et al., 2010).

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 (Burger et al., 2023, Gomes et al., 19 Aug 2025).
• 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 (Gomes et al., 19 Aug 2025, Sugiyama et al., 19 Aug 2025).

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 (Gomes et al., 19 Aug 2025).
• KiDS-1000: Adding $\langle M_{\rm ap}^3\rangle$ to COSEBIs tightens $S_8$ by ∼23%, yielding $S_8=0.772\pm0.022$, $\Omega_m=0.248^{+0.062}_{-0.055}$ (Burger et al., 2023).
• HSC Y3: Joint analysis improves $S_8$-$\Omega_m$ FoM by 80% and achieves $S_8=0.736\pm0.020$ (Sugiyama et al., 19 Aug 2025).
• 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% (Gomes et al., 5 Mar 2025).
• 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 (Secco et al., 2022).

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 (Peel et al., 2018).

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 (Porth et al., 2023, Heydenreich et al., 2022, Gomes et al., 5 Mar 2025, Sugiyama et al., 19 Aug 2025):

• 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/ (Heydenreich et al., 2022).

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 (Burger et al., 2023, Sugiyama et al., 19 Aug 2025, Gomes et al., 19 Aug 2025, Heydenreich et al., 2022).