Impact of Property Covariance on Cluster Weak lensing Scaling Relations

We present an investigation into a hitherto unexplored systematic that affects the accuracy of galaxy cluster mass estimates with weak gravitational lensing. Specifically, we study the covariance between the weak lensing signal, $\Delta\Sigma$, and the "true" cluster galaxy number count, $N_{\rm gal}$, as measured within a spherical volume that is void of projection effects. By quantifying the impact of this covariance on mass calibration, this work reveals a significant source of systematic uncertainty. Using the MDPL2 simulation with galaxies traced by the SAGE semi-analytic model, we measure the intrinsic property covariance between these observables within the 3D vicinity of the cluster, spanning a range of dynamical mass and redshift values relevant for optical cluster surveys. Our results reveal a negative covariance at small radial scales ($R \lesssim R_{\rm 200c}$) and a null covariance at large scales ($R \gtrsim R_{\rm 200c}$) across most mass and redshift bins. We also find that this covariance results in a $2-3\%$ bias in the halo mass estimates in most bins. Furthermore, by modeling $N_{\rm gal}$ and $\Delta\Sigma$ as multi-(log)-linear equations of secondary halo properties, we provide a quantitative explanation for the physical origin of the negative covariance at small scales. Specifically, we demonstrate that the $N_{\rm gal}$-$\Delta\Sigma$ covariance can be explained by the secondary properties of halos that probe their formation history. We attribute the difference between our results and the positive bias seen in other works with (mock)-cluster finders to projection effects. These findings highlight the importance of accounting for the covariance between observables in cluster mass estimation, which is crucial for obtaining accurate constraints on cosmological parameters.