Galaxy-Point Spread Function correlations as a probe of weak-lensing systematics with UNIONS data

Weak gravitational lensing requires precise measurements of galaxy shapes and therefore an accurate knowledge of the PSF model. The latter can be a source of systematics that affect the shear two-point correlation function. A key stake of weak lensing analysis is to forecast the systematics due to the PSF. Correlation functions of galaxies and the PSF, the so-called $\rho$- and $\tau$-statistics, are used to evaluate the level of systematics coming from the PSF model and PSF corrections, and contributing to the two-point correlation function used to perform cosmological inference. Our goal is to introduce a fast and simple method to estimate this level of systematics and assess its agreement with state-of-the-art approaches. We introduce a new way to estimate the covariance matrix of the $\tau$-statistics using analytical expressions. The covariance allows us to estimate parameters directly related to the level of systematics associated with the PSF and provides us with a tool to validate the PSF model used in a weak-lensing analysis. We apply those methods to data from the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS). We show that the semi-analytical covariance yields comparable results than using covariances obtained from simulations or jackknife resampling. It requires less computation time and is therefore well suited for rapid comparison of the systematic level obtained from different catalogs. We also show how one can break degeneracies between parameters with a redefinition of the $\tau$-statistics. The methods developed in this work will be useful tools in the analysis of current weak-lensing data but also of Stage IV surveys such as Euclid, LSST or Roman. They provide fast and accurate diagnostics on PSF systematics that are crucial to understand in the context of cosmic shear studies.