Towards Practical Field-Level Inference for Weak Lensing

Technical Summary: Towards Practical Field-Level Inference for Weak Lensing

Problem Statement
Weak gravitational lensing provides a statistical measure of the projected matter distribution in the Universe. However, as structure growth enters the nonlinear regime (roughly $z \sim 0.5$ for current surveys), the cosmic density field develops higher-order correlations and morphological features that cannot be fully captured by traditional two-point statistics (e.g., power spectra). While various alternative summary statistics (peak counts, Minkowski functionals, scattering transforms) have been proposed to extract non-Gaussian information, Field-Level Inference (FLI) offers a more general approach by inferring cosmological parameters directly from the observed weak lensing field, exploiting its full spatial information.

A critical gap exists in understanding the practical performance of FLI compared to standard two-point analyses when using physically motivated forward models. Previous FLI studies have relied on simplified statistical models (e.g., lognormal fields) or specific frameworks (e.g., BORG with Lagrangian Perturbation Theory) that may not fully capture the complexities of realistic simulations. Furthermore, there is a need to validate whether implicit (simulation-based) and explicit (sampling-based) inference methods yield consistent cosmological constraints when applied to matched data and simulations.

Methodology
The authors conduct a controlled comparison of three inference approaches using identical forward-modeling pipelines to generate weak lensing maps:

1. Power-Spectrum-Based Inference: A Simulation-Based Inference (SBI) approach using binned auto- and cross-power spectra as summary statistics.
2. Implicit FLI: An SBI approach where a neural density estimator learns the posterior directly from simulated data-field pairs, compressing the full field into a summary vector or learning the likelihood ratio.
3. Explicit FLI: A sampling-based approach (using Hamiltonian Monte Carlo/NUTS) that samples the joint posterior over cosmological parameters ($\vec{\theta}$) and the whitened latent initial conditions ($\vec{u}$).

Forward Models
The study utilizes two distinct forward models implemented in JAX to ensure differentiability for gradient-based sampling:

• Lagrangian Perturbation Theory (LPT): Specifically the Zel'dovich approximation (1LPT), which is computationally efficient but limited to the linear regime.
• Particle-Mesh (PM) N-body Evolution: Utilizing the BullFrog integrator (an LPT-informed leapfrog scheme) to evolve matter fields. This model captures nonlinear structure growth more accurately than LPT.

The simulations generate convergence maps for two narrow source redshift bins ($z=0.7$ and $z=0.8$) to ensure the lightcone remains within the simulation volume. The analysis employs conservative scale cuts ($\ell_{max} = 400$ and $\ell_{max} = 1000$) to mitigate finite-resolution artifacts inherent in the simulation grid.

Key Contributions

• Direct Comparison of FLI Paradigms: The paper establishes a controlled experiment comparing implicit and explicit FLI against power-spectrum-based inference using the same underlying physics (LPT and PM).
• Validation of Explicit FLI: The authors demonstrate that explicit FLI can successfully sample the high-dimensional joint posterior (cosmology + latent field) for both LPT and the more realistic PM models. They confirm that the implicit and explicit approaches yield closely consistent posteriors, validating the recovered field-level constraints.
• Quantification of Information Gain: The study quantifies the gain in cosmological information provided by map-level analysis over two-point statistics.
• Benchmarking: The authors provide computational benchmarks for the forward models on NVIDIA A100 GPUs, highlighting the trade-offs between resolution, computational cost, and the number of steps required for inference.

Results

• Consistency: The implicit and explicit FLI approaches produce consistent posterior distributions for cosmological parameters ($\Omega_m$ and $S_{8}^{lin}$). Coverage tests confirm the calibration of the implicit analyses.
• Information Gain: Relative to power-spectrum-based analyses, FLI shows significant gains in cosmological information. This gain is particularly pronounced when the PM-based forward model is used and small scales (up to $\ell_{max}=1000$) are included in the analysis.
• Model Dependence: The results indicate that the information gain is sensitive to the fidelity of the forward model. The PM model, which captures nonlinear structure, yields tighter constraints than the LPT model, especially at smaller scales where nonlinear effects dominate.
• Resolution Limits: The authors note that while the PM model agrees with nonlinear analytic predictions up to $\ell \simeq 500$, it becomes biased low on smaller scales due to finite simulation resolution. Consequently, the study imposes scale cuts to ensure robust inference.

Significance and Claims
The paper claims that Field-Level Inference is a viable and powerful method for extracting cosmological information from weak lensing surveys, offering significant improvements over traditional two-point statistics. The agreement between implicit and explicit methods provides confidence in the robustness of field-level constraints.

However, the authors remain modest regarding immediate application to observational data. They explicitly state that "remaining challenges must be addressed before PM-based explicit FLI can be applied to observational datasets." These challenges include the computational cost of high-resolution forward models and the need to handle systematics and model mismatches that arise when comparing simulations to real data. The work serves as a proof-of-concept demonstrating that with sufficiently accurate forward models, FLI can unlock the non-Gaussian information contained in weak lensing maps.