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Aims:Calibrating the point spread function (PSF) is a fundamental part of weak gravitational lensing analyses. Even with corrected galaxy images, imperfect calibrations can introduce biases. We propose an analytical framework for quantifying PSF-induced systematics as diagnostics for cross-correlation measurements of weak lensing with density tracers, e.g., galaxy-galaxy lensing. We show how those systematics propagate to physical parameters of the density tracers. Those diagnostics only require a shape catalogue of PSF stars and foreground galaxy positions. Methods:We consider the PSF-induced multiplicative bias, and introduce three second-order statistics as additive biases. We compute both biases for the weak-lensing derived halo mass of spectroscopic foreground galaxy samples, in particular, their effect on the tangential shear and fitted halo mass as a function of stellar mass. In addition, we assess their impact on the recently published black-hole - halo-mass relation for type I Active Galactic Nuclei (AGNs). Results:Using weak-lensing catalogues from the Ultraviolet Near Infrared Optical Northern Survey (UNIONS) and Dark Energy Survey (DES), we find the multiplicative biases in the tangential shear to be less than $0.5\%$. No correlations between additive bias and galaxy properties of the foreground sample are detected. The combined PSF systematics affect low-mass galaxies and small angular scales; halo mass estimates can be biased by up to 18$\%$ for a sample of central galaxies in the stellar mass range 9.0 $\leq$ log $M_*/\rm M_{\odot}$ < 9.5. Conclusions:The PSF-induced multiplicative bias is a subdominant contribution to current studies of weak-lensing - density cross-correlations, but might become significant for upcoming Stage-VI surveys. For samples with a low tangential shear, additive PSF systematics can induce a significant bias on derived properties such as halo mass., Comment: 11 pages, 5 figures

This letter presents, for the first time, direct constraints on the black-hole-to-halo-mass relation using weak gravitational lensing measurements. We construct type I and type II Active Galactic Nuclei (AGNs) samples from the Sloan Digital Sky Survey (SDSS), with a mean redshift of 0.4 0.1 for type I (type II) AGNs. This sample is cross-correlated with weak lensing shear from the Ultraviolet Near Infrared Northern Survey (UNIONS). We compute the excess surface mass density of the halos associated with $36,181$ AGNs from $94,308,561$ lensed galaxies and fit the halo mass in bins of black-hole mass. We find that more massive AGNs reside in more massive halos. We see no evidence of dependence on AGN type or redshift in the black-hole-to-halo-mass relationship when systematic errors in the measured black-hole masses are included. Our results are consistent with previous measurements for non-AGN galaxies. At a fixed black-hole mass, our weak-lensing halo masses are consistent with galaxy rotation curves, but significantly lower than galaxy clustering measurements. Finally, our results are broadly consistent with state-of-the-art hydro-dynamical cosmological simulations, providing a new constraint for black-hole masses in simulations., Comment: 14 pages, 5 figures. Submitted to ApJL