Analytical Weak-lensing Shear Responses of Galaxy Properties and Galaxy Detection

Shear estimation bias from galaxy detection and blending identification is now recognized as an issue for ongoing and future weak lensing surveys. Currently, the empirical approach to correcting for this bias involves numerically shearing every observed galaxy and rerunning the detection and selection process. In this work, we provide an analytical correction for this bias that is accurate to subpercent level and far simpler to use. With the interpretation that smoothed image pixel values and galaxy properties are projections of the image signal onto a set of basis functions, we analytically derive the linear shear responses of both the pixel values and the galaxy properties (i.e., magnitude, size and shape) using the shear responses of the basis functions. With these derived shear responses, we correct for biases from shear-dependent galaxy detection and galaxy sample selection. With the analytical covariance matrix of measurement errors caused by image noise on pixel values and galaxy properties, we correct for the noise biases in galaxy shape measurement and the detection/selection process to the second-order in noise. The code used for this paper can carry out the detection, selection, and shear measurement for ~1000 galaxies per CPU second. The algorithm is tested with realistic image simulations, and we find, after the analytical correction (without relying on external image calibration) for the detection/selection bias of about $-4\%$, the multiplicative shear bias is $-0.12 \pm 0.10\%$ for isolated galaxies; and $-0.3 \pm 0.1\%$ for blended galaxies with Hyper Suprime-Cam observational condition.