Towards Reliably Measuring the Mass-Size Relation of Dark Matter Halos: Impact of projection, miscentering and Eddington bias

In the conventional spherical overdensity definition of a dark matter halo, masses and radii, e.g. $M_{200{\rm m}}$ and $r_{200{\rm m}}$, are equivalent. However, halos do exhibit a physical boundary at the splashback radius $r_{\rm sp}$, which manifests as a sharp drop in density. Splashback radii are now routinely measured in stacked weak lensing profiles, but the expected trend of decreasing $r_{\rm sp}/r_{200{\rm m}}$ with mass has not been verified in observed galaxy groups or clusters. In this work, we show that several observational effects conceal the predicted mass-size relation. In particular, we quantify the impact of systematics on the mean density and lensing profiles of mock-observed halo samples, which are created by applying a realistic cluster-finding procedure to dark matter-only simulations. We reproduce the known impact of projection: preferentially observing systems in overdense regions. But, differing from previous works, we show that the same projection effect, combined with scatter in observed luminosities, causes a significant Eddington bias in the derived mass-observable relation and a complex suppression in mean density profiles over all radial scales. The magnitude of this effect is mass-dependent and coupled to the cluster-finding procedure and survey properties, making it challenging to model simply and motivating full forward-modeling of observed group and cluster samples. However, we demonstrate that a simple mitigation approach, effectively a weak lensing mass calibration, does well in approximately recovering the true mass-size relation. Finally, we show that only severe miscentering can bias the measurement of the halo size, but any miscentering can increase the measurement uncertainty.