SPT Clusters with DES and HST Weak Lensing. I. Cluster Lensing and Bayesian Population Modeling of Multi-Wavelength Cluster Datasets

We present a Bayesian population modeling method to analyze the abundance of galaxy clusters identified by the South Pole Telescope (SPT) with a simultaneous mass calibration using weak gravitational lensing data from the Dark Energy Survey (DES) and the Hubble Space Telescope (HST). We discuss and validate the modeling choices with a particular focus on a robust, weak-lensing-based mass calibration using DES data. For the DES Year 3 data, we report a systematic uncertainty in weak-lensing mass calibration that increases from 1\% at $z=0.25$ to 10\% at $z=0.95$, to which we add 2\% in quadrature to account for uncertainties in the impact of baryonic effects. We implement an analysis pipeline that joins the cluster abundance likelihood with a multi-observable likelihood for the SZ, optical richness, and weak-lensing measurements for each individual cluster. We validate that our analysis pipeline can recover unbiased cosmological constraints by analyzing mocks that closely resemble the cluster sample extracted from the SPT-SZ, SPTpol~ECS, and SPTpol~500d surveys and the DES Year~3 and HST-39 weak-lensing datasets. This work represents a crucial prerequisite for the subsequent cosmological analysis of the real dataset.