Optical galaxy cluster mock catalogs with realistic projection effects: Validations with the SDSS clusters

Galaxy clusters identified in optical imaging surveys suffer from projection effects: Physically unassociated galaxies along a cluster's line of sight can be counted as its members and boost the observed richness (the number of cluster members). To model the impact of projection on cluster cosmology analyses, we apply a halo occupation distribution model to N-body simulations to simulate the red galaxies contributing to cluster members, and we use the number of galaxies in a cylinder along the line of sight (counts in cylinders) to model the impact of projection on cluster richness. We compare three projection models: uniform, quadratic, and Gaussian, and we convert between them by matching their effective cylinder volumes. We validate our mock catalogs using SDSS redMaPPer clusters' data vectors, including counts vs. richness, stacked lensing signal, spectroscopic redshift distribution of member galaxies, and richness remeasured on a redshift grid. We find the former two are insensitive to the projection model, while the latter two favor a quadratic projection model with a width of approximately 180 Mpc/h (equivalent to the volume of a uniform model with a width of 100 Mpc/h and a Gaussian model with a width of 110 Mpc/h, or a Gaussian redshift error of 0.04). Our framework provides an efficient and flexible way to model optical cluster data vectors, paving the way for a simulation-based joint analysis for clusters, galaxies, and shear.