The Atacama Cosmology Telescope: A Catalog of > 4000 Sunyaev-Zel'dovich Galaxy Clusters

We present a catalog of 4195 optically confirmed Sunyaev-Zel'dovich (SZ) selected galaxy clusters detected in 13,168 deg$^2$ of sky surveyed by the Atacama Cosmology Telescope (ACT). Cluster candidates were selected by applying a multi-frequency matched filter to 98 and 150 GHz maps constructed from all ACT observations obtained from 2008-2018, and subsequently confirmed using deep, wide-area optical surveys. The detected clusters span the redshift range 0.04 < z < 1.91, with median z = 0.52. The catalog contains 221 z > 1 clusters, and a total of 872 systems are new discoveries. Assuming an SZ-signal vs. mass scaling relation calibrated from X-ray observations, the sample has a 90% completeness mass limit of M500c > 3.8 x 10$^{14}$ MSun, evaluated at the median redshift of the cluster sample, for clusters detected at signal-to-noise ratio > 5 in maps filtered at an angular scale of 2.4'. The cluster sample is more than 22 times larger than the previous ACT cluster catalog, and is the largest homogeneous sample of SZ-selected clusters to date. The survey footprint has a large overlap with deep optical weak-lensing surveys that are being used to calibrate the SZ-signal mass-scaling relation, such as the Dark Energy Survey (4552 deg$^2$), the Hyper Suprime-Cam Subaru Strategic Program (468 deg$^2$), and the Kilo Degree Survey (823 deg$^2$). We briefly discuss some noteworthy objects in the sample, including potentially projected systems; clusters with potential strong lensing features; clusters with active central galaxies or star formation; and systems of two or more clusters that may be physically associated. The cluster catalog will be a useful resource for future cosmological analyses, and studying the evolution of the intracluster medium and the galaxy population in massive clusters over the past 10 Gyr....