Euclid preparation. LXXXV. Toward a DR1 application of higher-order weak lensing statistics

1. Problem Statement

The primary goal of the Euclid mission is to constrain cosmological parameters, specifically the dark energy equation of state ($w_0$) and matter fluctuations ($\sigma_8$), by mapping the large-scale structure (LSS) of the Universe via weak gravitational lensing (WL).

• Limitation of Standard Methods: Traditional WL analyses rely on two-point correlation functions (2PCFs), which capture only the Gaussian properties of the matter density field. However, the non-linear evolution of matter introduces significant non-Gaussian features (skewness, kurtosis, connectivity) on small scales, which 2PCFs fail to capture.
• The Challenge: While Higher-Order Statistics (HOS) can extract this non-Gaussian information, their application to real data (specifically Euclid's Data Release 1, DR1) faces challenges:
  • Lack of theoretical models for many HOS, requiring reliance on numerical simulations.
  • The need for realistic photometric redshifts ($z_{ph}$) and tomographic binning.
  • The impact of observational masks (e.g., bright stars) and mass reconstruction algorithms.
  • Ensuring the statistical validity of the Fisher formalism, which assumes a Gaussian likelihood (often violated by HOS).

2. Methodology

The authors developed a comprehensive pipeline to forecast the performance of HOS for a Euclid-like DR1 setup, moving beyond the non-tomographic analysis of their previous work (Paper I).

A. Simulations and Mocks

• N-body Simulations: Utilized the SLICS (924 realizations) for covariance estimation and DUSTGRAIN-pathfinder (256 realizations with varied cosmological parameters) for derivative estimation.
• Mock Construction: Created realistic mock catalogs by sampling from the Euclid Flagship (FS) simulation to assign true redshifts and photometric redshifts ($z_{ph}$).
• Configurations: Defined DR1-opt (optimistic) and DR1-pes (pessimistic) galaxy density scenarios based on ground-based photometry cuts. The analysis focuses on DR1-opt.
• Tomography: Implemented five equi-populated redshift bins ($0.2 \le z_{ph} \le 2.5$) and generated combined bins (pairs, triplets, quadruplets) to form 31 tomographic configurations.

B. Mass Mapping and Masking

• Algorithms: Compared three mass reconstruction techniques used in the Euclid pipeline:
  1. Kaiser–Squires (KS): Standard Fourier-space inversion.
  2. Aperture Mass (APM): Real-space convolution with a Starlet wavelet filter.
  3. KS+: KS with inpainting (sparsity-based) to handle masked/empty pixels.
• Masks: Applied realistic masks based on Gaia DR2 star catalogs, covering $\approx 5\%$ of the area, to simulate bright star exclusion zones.

C. Statistics and Data Vector (DV)

• Selected HOS: Five statistics satisfying the Gaussian likelihood assumption for the Fisher formalism:
  1. One-point: Probability Density Function (PDF) and $\ell_1$-norm.
  2. Topological: Peak counts, Minkowski Functionals (MFs), and Betti Numbers (BNs).
• Excluded: Higher-order moments, persistent homology, and scattering transforms were discarded as they failed the Gaussianity test.
• Filtering: Applied consistent smoothing scales ($2.34', 4.68', 9.38'$) across all statistics to ensure fair comparison.
• Gaussianity Check: Used the SMAPE (Symmetrized Mean Absolute Percentage Error) test to iteratively trim the tails of the data vector, ensuring the likelihood is sufficiently Gaussian for the Fisher matrix inversion.

D. Analysis Techniques

• Fisher Formalism: Used to forecast constraints on $(\sigma_8, w_0)$. Addressed ill-conditioning (numerical instability in matrix inversion) by focusing on the 2D parameter plane and optimizing the number of realizations.
• Tomographic Optimization: Employed a greedy algorithm to select the optimal subset of redshift bins that maximizes the Figure of Merit (FoM) while minimizing dimensionality.
• BNT Transformation: Tested the Bernardeau–Nishimichi–Taruya (BNT) nulling scheme to disentangle redshift scales and mitigate baryonic feedback, comparing standard BNT against a new "BNT smoothing" approach.

3. Key Contributions

1. First Tomographic HOS Forecast for DR1: The first study to apply HOS to a realistic DR1 setup with photometric redshifts, moving from $(\Omega_m, \sigma_8)$ to the critical $(\sigma_8, w_0)$ plane.
2. Unified Comparison: Provided a fair, apples-to-apples comparison of five distinct HOS using identical mass maps, smoothing scales, and filtering, decoupling the estimator performance from the mass reconstruction method.
3. Mass Mapping Benchmark: Systematically evaluated KS, APM, and KS+ under realistic masking conditions, quantifying the trade-off between inpainting accuracy (KS+) and computational cost/noise correlation.
4. BNT Smoothing Validation: Demonstrated that standard BNT degrades HOS performance due to shape noise propagation, but the novel BNT smoothing technique successfully preserves non-Gaussian information while allowing for physical scale cuts.
5. Optimal DV Selection: Developed an iterative scheme to reduce the dimensionality of the data vector without significant loss of cosmological information.

4. Key Results

• Performance Gain: Every individual HOS outperforms the standard shear two-point correlation functions ($\gamma$-2PCFs) by a factor of 2.5 in constraining $w_0$.
  • Improvements on $\sigma_8$: 33–50%.
  • Improvements on $w_0$: 50–64%.
• Redundancy with 2PCFs: Combining HOS with $\gamma$-2PCFs yields only marginal additional gains. This suggests that the multi-scale HOS analysis already captures the full Gaussian information content of the field, making the addition of 2PCFs largely redundant in this specific setup.
• HOS Comparison:
  • One-point statistics (PDF, $\ell_1$-norm) generally provide tighter constraints on $w_0$ (better by 5–22% over topological estimators).
  • Topological statistics (MFs, BNs) perform slightly better on $\sigma_8$ (MFs are best).
  • Minkowski Functionals (MFs) and Betti Numbers (BNs) show partially uncorrelated information, suggesting potential for combination in future MCMC analyses.
• Mass Mapping:
  • KS and APM show comparable performance ($\approx 10\%$ difference, within uncertainty limits).
  • KS+ (inpainting) better recovers the true field at borders but introduces correlated noise that slightly degrades constraints in simulation-based forecasts. It is computationally expensive ($\sim 10^4$ times slower than KS).
  • Masking reduces constraining power by $\approx 5\%$, proportional to the masked area, without introducing significant bias in the parameter ellipse orientation.
• BNT Results: Standard BNT application to HOS inflates error contours due to shape noise. The BNT smoothing method (smoothing before inverse transformation) successfully mitigates this, preserving the non-Gaussian signal.
• Multi-scale Analysis: Combining multiple smoothing scales improves constraints by 10–20% and helps break degeneracies, though adding a third scale (beyond the two smallest) offers negligible gains.

5. Significance and Future Outlook

• Cosmological Impact: The results strongly validate the inclusion of HOS in the Euclid DR1 analysis strategy. The substantial improvement in $w_0$ constraints (a primary science goal) justifies the computational cost and complexity of implementing these non-Gaussian probes.
• Methodological Guidance: The paper provides a "recipe" for DR1 analysis:
  • Use merged bins (contiguous redshift slices) for one-point statistics and two-point-like setups for topological statistics.
  • Adopt KS for simulation-based forecasts due to speed and stability, but consider KS+ for analytical likelihoods where leakage must be minimized.
  • Apply BNT smoothing if physical scale cuts (to remove baryonic effects) are required.
• Limitations & Next Steps: The study relies on the Fisher formalism and assumes a Gaussian likelihood (after tail cuts). The authors note that some statistics still exhibit non-Gaussian behavior. Future work must transition to Markov Chain Monte Carlo (MCMC) analyses to fully exploit the non-Gaussian tails and include a broader set of nuisance parameters and systematics (e.g., intrinsic alignments, baryonic feedback) for a robust DR1 analysis.

In summary, this paper establishes that higher-order weak lensing statistics are not only viable but essential for maximizing the scientific return of the Euclid mission, offering a 2.5-fold improvement in dark energy constraints over traditional methods.