Slay the Shear: A Unified Statistical Framework for Weak Gravitational Lensing Shear Estimation

Abstract

Weak gravitational lensing shear measurements are fundamentally limited by shape noise arising from the intrinsic diversity of galaxy morphologies. Upcoming surveys such as Rubin/LSST, Euclid, and Roman demand more flexible, statistically optimal approaches that can fully exploit high-dimensional image information. In this work, we develop a unified theoretical framework for shear estimation that connects classical response-based methods, shape noise, and modern machine-learning estimators through the concept of the score function -- the gradient of the image likelihood with respect to shear. We show that, for a general spin-2 ellipticity definition, the ensemble shear response corresponds to an inner product between the estimator and the score function, and that the score provides the minimum-variance unbiased shear estimator. By incorporating response into the classical inverse-variance weight, we prove that the response-weighted inverse-variance weight is a general shape-noise-minimizing weight, independent of the intrinsic shape distribution. Furthermore, we propose Response-weighted Denoising Score Matching (RDSM) that exploits the remaining structure to reduce shape noise by $\sim 17.5\%$ relative to moment-based methods at LSST 10-year depth while maintaining a multiplicative shear estimation bias below $2\times 10^{-3}$. Our result clarifies the optimality of existing calibration techniques while revealing a principled pathway for constructing improved estimators via nonlinear shape transformations and learned representations.

24 pages, 4 figures, and 3 tables. Submitting to APJ