Towards an application of fourth-order shear statistics I. The information content of $\langle M_\mathrm{ap}^4 \rangle$

1. Problem Statement

Cosmic shear (weak gravitational lensing) has traditionally relied on second-order statistics (power spectrum, two-point correlation functions) to constrain cosmological parameters. While these capture Gaussian information, they miss the non-Gaussian features of the matter distribution that arise from non-linear structure formation.

• The Gap: Third-order statistics (bispectrum, three-point correlation functions) have been successfully applied to capture some non-Gaussianity. However, fourth-order statistics (trispectrum, four-point correlation functions) remain largely unexplored in observational cosmology.
• The Challenge: Fourth-order aperture statistics ($\langle M_{ap}^4 \rangle$) offer a way to decompose the shear field into $E$-modes (lensing signal) and $B$-modes (systematics/noise). However, computing them requires integrating complex four-point correlation functions (4PCF) over a four-dimensional configuration space, which introduces significant numerical challenges regarding filter design, integration accuracy, and computational cost.
• The Goal: This paper aims to establish the theoretical framework for fourth-order aperture statistics, develop a robust numerical integration pipeline, and forecast their information content for a survey similar to the Dark Energy Survey Year 3 (DES-Y3).

2. Methodology

Theoretical Framework

The authors extend the formalism of $n$-th order aperture statistics to the fourth order.

• Relation to 4PCF: They derive the functional relationship between the fourth-order aperture mass moments ($\langle M_{ap}^4 \rangle$, $\langle M_{ap}^3 M_\times \rangle$, etc.) and the shear 4PCF.
• Filter Derivation: They derive explicit filter functions ($K_\mu$) for the 4PCF. These filters depend on the relative positions of four points forming a quadrilateral. The authors define a specific projection (the "$\times$-projection") to handle the shear components, reducing the 8-dimensional integral to a 5-dimensional one by integrating over the center of mass and orientation.
• $E/B$-Mode Separation: They provide the linear combinations of the eight complex aperture measures required to separate pure $E$-modes (lensing) from $B$-modes (systematics) and parity-violating modes.

Numerical Integration Pipeline

To test the theoretical filters, the authors developed a numerical integration pipeline:

• Test Case: They used a Gaussian Random Field (GRF) as a testbed. In a GRF, all higher-order statistics are determined entirely by the second-order statistics (via Wick's theorem). This allows them to verify the fourth-order calculation by comparing the integrated result against the theoretical expectation derived from second-order aperture statistics ($\langle M_{ap}^4 \rangle = 3\langle M_{ap}^2 \rangle^2$).
• Integration Scheme: They implemented a Riemann sum integration over radial and angular coordinates.
  • Bin Geometry: They identified that "mixed pairs" (configurations where points are close together) dominate the signal. To handle sharp peaks near zero angles, they employed hypercubical bins (where $\Delta \psi \approx \Delta \ln \vartheta$) and logarithmic radial binning.
  • Mitigation of Finite-Field Effects: They analyzed how finite integration ranges and binning accuracy cause $E/B$-mode mixing. They found that integrating the 4PCF over coarse radial bins (simulating realistic bin-averaging) significantly reduces numerical dispersion.

Fisher Forecast

• Data Vector: They constructed a data vector combining second-, third-, and fourth-order equal-scale aperture statistics for a DES-Y3-like setup.
• Derivative Estimation: A novel method was developed to estimate derivatives of observables with respect to cosmological parameters ($\Omega_m, S_8, h, w_0$) from the cosmo-SLICS simulation suite. Instead of unstable finite differences, they used a loss-function minimization to fit a tangent plane to the simulation data, accommodating irregular sampling (Latin Hypercube).
• Covariance: The covariance matrix was derived from full-sky ray-tracing simulations (Takahashi et al. 2017).

3. Key Contributions

1. Analytical Derivation: Provided the first explicit derivation of filter functions for fourth-order aperture statistics in terms of the 4PCF and the convergence trispectrum.
2. Numerical Pipeline: Developed and validated a numerical integration pipeline capable of computing fourth-order aperture statistics from 4PCFs with 2% precision for a Gaussian field. This precision is well below the noise budget of Stage IV surveys.
3. Error Analysis: Systematically quantified numerical biases arising from finite integration ranges and binning schemes, specifically identifying how these lead to artificial $E/B$-mode mixing.
4. Novel Derivative Method: Introduced a robust, simulation-based method for estimating cosmological derivatives that handles irregular parameter sampling and noise better than standard finite differences.

4. Results

Numerical Performance

• The pipeline achieves a 2% level precision on fourth-order aperture statistics for a GRF when using a radial integration range of $\vartheta \in [e^{-2'}, e^{5'}]$ and a logarithmic binning accuracy of $\Delta(\ln \vartheta) = 0.07$.
• Using a "coarse binning" approach (integrating the 4PCF within bins before applying the filter) further improves stability, allowing for a binning accuracy of $\Delta(\ln \vartheta) = 0.10$ while maintaining high precision. This is crucial for computational efficiency.

Cosmological Forecast (DES-Y3-like)

• Constraining Power: When performing a Fisher forecast for a joint analysis of $\langle M_{ap}^2 \rangle$, $\langle M_{ap}^3 \rangle$, and $\langle M_{ap}^4 \rangle_c$ (connected part) on a DES-Y3-like setup, the inclusion of fourth-order statistics provided only a minimal improvement in constraining power for parameters like $\Omega_m$ and $S_8$.
• Reasons for Limited Gain:
  • Gaussian Assumption: The Fisher forecast assumes Gaussian likelihoods, which is a poor approximation for higher-order statistics (which are non-Gaussian).
  • Scale Limitations: The simulations used for derivatives lacked small scales ($\theta < 4'$) where fourth-order statistics carry the most information.
  • Parameter Degeneracies: The analysis marginalized over $h$ and $w_0$ without tomography, reducing the relative impact of higher-order statistics.
  • Correlation: Strong correlations between neighboring scales and orders in the covariance matrix mean that equal-scale statistics capture only a fraction of the total information.

5. Significance and Conclusion

• Validation for Stage IV: While the gain for current Stage III surveys (like DES) is minimal, the paper establishes that fourth-order statistics are computationally feasible and can be measured with high precision.
• Systematics Control: Even if fourth-order statistics are degenerate with lower orders in terms of cosmological parameters, they offer a unique consistency check for systematics (e.g., intrinsic alignments, baryonic physics) because they have different degeneracy directions for nuisance parameters.
• Future Outlook: The authors conclude that the true potential of fourth-order statistics lies in Stage IV surveys (e.g., Euclid, Rubin Observatory). The unprecedented signal-to-noise ratio of these surveys will make higher-order statistics significant, particularly when combined with tomographic bins, potentially breaking degeneracies between cosmological parameters that lower-order statistics cannot resolve.

In summary, this paper provides the essential theoretical and numerical toolkit required to move fourth-order shear statistics from theory to observation, paving the way for their application in next-generation cosmological surveys.