tokens: CGs spanning all redshifts, the only available path forward is to utilize cosmic simulations of galaxy evolution. The Millennium Simulation (Springel et al. 2005 ; Lemson Virgo Consortium 2006 ) provides a tool to study the history of CGs in the universe. It used 2160 3 particles with mass M = 8.6 × 10 8 h −1 M ⊙ to trace the evolution of a comoving cube with side length 500 h −1 Mpc, where h is the Hubble constant parameterization such that H 0 = 100 h km s −1 Mpc −1 . The Millennium Simulation used a ΛCDM model with cosmological parameters Ω M = 0.25, Ω b =0.045, Ω Λ = 0.75, and h = 0.73, consistent with observational results from the Carnegie Hubble Program (Freedman et al. 2012 ) and the WMAP (Hinshaw et al. 2013 ) mission. 10 10 In this paper all distances and masses are scaled to h = 1. The Millennium Simulation used only dark matter particles to which associated properties were later assigned. Dark matter halos were identified using a friends-of-friends algorithm as described in Springel et al. ( 2005 ). Galaxies were added to the simulation post facto using semianalytic techniques (e.g., De Lucia Blaizot 2007 ; Guo et al. 2010 , 2011 , 2013 ). Galaxies are initially associated with individual dark matter halos, which may become subhalos of larger structures over time. The techniques of De Lucia Blaizot ( 2007 ), for example, expanded on the methods of Croton et al. ( 2006 ) and incorporated the effects of rapid star formation and gas loss in galaxy mergers, changes in galaxy properties with varying stellar mass functions and stellar populations, and attenuation due to dust. The simulation results were saved in a collection of 64 redshift “snapshots.” The identified dark matter halos were then assigned galaxy properties using the semianalytic techniques discussed in the references above. The snapshots, combined with the high resolution of the simulation and the sophisticated semianalytic galaxy formation and evolution models, provide an excellent tool for studying the history of CGs over cosmic time. Throughout this paper, we adopt the De Lucia Blaizot ( 2007 ) catalog of galaxies created through this semianalytic approach. 1.2. Previous Work A number of previous studies have utilized the Millennium Simulation galaxy catalogs to investigate CGs. These studies have been limited to relatively low redshifts to facilitate comparison to observations, such as the well-known Hickson catalog (Hickson 1982 ; Hickson et al. 1992 ). By using a light cone in the Millennium Simulation galaxy catalogs (Henriques et al. 2012 ), it has also been possible to create “mock” catalogs of CGs, which can be compared to observations and are subject to analogous limitations (e.g., interloping foreground or background galaxies; McConnachie et al. 2008 ; Díaz-Giménez Mamon 2010 ; Díaz-Giménez Zandivarez 2015 ; Farhang et al. 2017 ). By comparing mock CGs in projection in the Millennium Simulation to observed Hickson compact groups (HCGs), McConnachie et al. (
ner_tags: O O O O O O O O O O O O O O O O O O B-Model I-Model B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Model I-Model O O B-Model I-Model O O O B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O B-Survey I-Survey I-Survey B-Citation I-Citation I-Citation I-Citation O O O B-Telescope B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Model I-Model O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Model I-Model O O O O O O B-Citation I-Citation I-Citation I-Citation