2013ApJ...779..127C

A growing number of galaxy clusters at z = 1-2 is being discovered as part of deep optical, IR, X-ray, and Sunyaev-Zel'dovich effect surveys. For a complete picture of cluster formation, however, it is important that we also start probing the much earlier epoch, between redshifts of about 2 and 7, during which these clusters and their galaxies first began to form. Because the study of these so-called proto-clusters is currently quite limited by small number statistics, widely varying selection techniques, and many assumptions, we have performed a large systematic study of cluster formation utilizing cosmological simulations. We use the Millennium Simulations to track the evolution of dark matter and galaxies in about 3000 clusters from the earliest times to z = 0. We define an effective radius R_e for proto-clusters and characterize their growth in size and mass with cosmic time. We show that the progenitor regions of galaxy clusters (ranging in mass from ∼10¹⁴ to a few times 10¹⁵ M_☉) can already be identified in galaxy surveys at very early times (at least up to z ∼ 5), provided that the galaxy overdensities are measured on a sufficiently large scale (R_e∼ 5-10 Mpc comoving) and with sufficient statistics. We present the overdensities in matter, dark matter halos, and galaxies as functions of present-day cluster mass, redshift, bias, and window size that can be used to interpret the wide range of structures found in real surveys. We also derive the probability that a structure having a galaxy overdensity δ_gal, defined by a set of observational selection criteria, is indeed a proto-cluster, and we show how their z = 0 masses can already be estimated long before virialization. We present overdensity profiles as a function of radius, and we further show how the projected surface overdensities of proto-clusters decrease as the uncertainties in redshift measurements increase. We provide a table of proto-cluster candidates selected from the literature and discuss their properties in light of our simulation predictions. This paper provides the general framework that will allow us to extend the study of cluster formation out to much higher redshifts using the large number of proto-clusters that are expected to be discovered in, e.g., the upcoming HETDEX and Hyper Suprime-Cam surveys.