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Large, high-resolution simulations permit detailed predictions of the distribution and properties of galaxies and clusters. From 2005-2010, the benchmark simulations were Millennium-I Springel et al. (2005) and Millennium-II Boylan-Kolchin et al. (2009), which have been the basis for more than 400 papers. However, these simulations used first-year Wilkinson Microwave Anisotropy Probe (WMAP) cosmological parameters, including σ8 = 0.90, that are now in serious disagreement with the latest observations. Improved cosmological parameters, simulation codes, and computer power have permitted more accurate simulations Kuhlen et al. (2012), Skillman et al. (2014) including Bolshoi Klypin et al. (2011) and BigBolshoi/MultiDark Prada et al. (2012), Riebe et al. (2013), which have recently been rerun using the Planck cosmological parameters Klypin et al. (2014b). 2

Dark matter halos can be characterized in a number of ways. A common one is by mass, but the mass attributed to a halo depends on a number of factors including how the outer edge of the halo is defined; popular choices include the spherical radius within which the average density is either 200 times critical density or the virial density (which depend on redshift). Properties of all the halos in many stored time steps of both the Bolshoi and BigBolshoi/MultiDark simulations are available on the web in the MultiDark database.2 For many purposes it is more useful to characterize halos by their maximum circular velocity Vmax, which is defined as the maximum value of [G M(< r) / r]1/2, where G is Newton's constant and M(< r) is the mass enclosed within radius r. The reason this is useful is that Vmax is reached at a relatively low radius rmax, closer to the central region of a halo where stars or gas can be used to trace the velocity of the halo, while most of the halo mass is at larger radii. Moreover, the measured internal velocity of a galaxy (line of sight velocity dispersion for early-type galaxies and rotation velocity for late-type galaxies) is closely related to its luminosity according to the Faber-Jackson and Tully-Fisher relations. In addition, after a subhalo has been accreted by a larger halo, tidal stripping of its outer parts can drastically reduce the halo mass but typically decreases Vmax much less. (Since the stellar content of a subhalo is thought to be determined before it was accreted, some authors define Vmax to be the peak value at any redshift for the main progenitor of a halo.) Because of the observational connection between larger halo internal velocity and brighter galaxy luminosity, a common simple method of assigning galaxies to dark matter halos and subhalos is to rank order the galaxies by luminosity and the halos by Vmax, and then match them such that the number densities are comparable Kravtsov et al. (2004), Tasitsiomi et al. (2004), Conroy et al. (2006). This is called "halo abundance matching" or (more modestly) "sub-halo abundance matching" (SHAM) Reddick et al. (2014). Halo abundance matching using the Bolshoi simulation predicts galaxy-galaxy correlations (which are essentially counts of the numbers of pairs of galaxies at different separation distances) that are in good agreement with the Sloan Digital Sky Survey (SDSS) observations Trujillo-Gomez et al. (2011), Reddick et al. (2013).

Abundance matching with the Bolshoi simulation also predicts galaxy velocity-mass scaling relations consistent with observations Trujillo-Gomez et al. (2011), and a galaxy velocity function in good agreement with observations for maximum circular velocities Vmax ≳ 100 km/s, but higher than the HI Parkes All Sky Survey (HIPASS) and the Arecibo Legacy Fast ALFA (ALFALFA) Survey radio observations Zwaan et al. (2010), Papastergis et al. (2011) by about a factor of 2 at 80 km/s and a factor of 10 at 50 km/s. This either means that these radio surveys are increasingly incomplete at lower velocities, or else ΛCDM is in trouble because it predicts far more small-Vmax halos than there are observed low-V galaxies. A deeper optical survey out to 10 Mpc found no disagreement between Vmax predictions and observations for Vmax ≥ 60 km/s, and only a factor of 2 excess of halos compared to galaxies at 40 km/s Klypin et al. (2014a). This may indicate that there is no serious inconsistency with theory, since for V ≲ 30 km/s reionization and feedback can plausibly explain why there are fewer observed galaxies than dark matter halos Bullock et al. (2000), Somerville (2002), Benson et al. (2003), Kravtsov (2010), Wadepuhl and Springel (2011), Sawala et al. (2012), and also the observed scaling of metallicity with galaxy mass Dekel and Woo (2003), Woo et al. (2008), Kirby et al. (2011).

The radial dark matter density distribution in halos can be approximately fit by the simple formula ρNFW = 4ρs x−1(1 + x)−2, where xr / rs Navarro et al. (1996), and the "concentration" of a dark matter halo is defined as C = Rvir / Rs where Rvir is the virial radius of the halo. When we first understood that dark matter halos form with relatively low concentration C ∼ 4 and evolve to higher concentration, we suggested that "red" galaxies that shine mostly by the light of red giant stars because they have stopped forming stars should be found in high-concentration halos while "blue" galaxies that are still forming stars should be found in younger low-concentration halos Bullock et al. (2001). This idea was recently rediscovered by Hearin and Watson, who used the Bolshoi simulation to show that this leads to remarkably accurate predictions for the correlation functions of red and blue galaxies Hearin and Watson (2013), Hearin et al. (2014).

The Milky Way has two rather bright satellite galaxies, the Large and Small Magellanic Clouds. It is possible using sub-halo abundance matching with the Bolshoi simulation to determine the number of Milky-Way-mass dark matter halos that have subhalos with high enough circular velocity to host such satellites. It turns out that about 55% have no such subhalos, about 28% have one, about 11% have two, and so on Busha et al. (2011a). Remarkably, these predictions are in excellent agreement with an analysis of observations by the Sloan Digital Sky Survey (SDSS) Liu et al. (2011). The distribution of the relative velocities of central and bright satellite galaxies from SDSS spectroscopic observations is also in very good agreement with the predictions of the Millennium-II simulation Tollerud et al. (2011), and the Milky Way's lower-luminosity satellite population is not unusual Strigari and Wechsler (2012). Considered in a cosmological context, the Magellanic clouds are likely to have been accreted within about the last Gyr Besla et al. (2012), and the Milky Way halo mass is 1.2−0.4+0.7(stat.) ± 0.3(sys.) × 1012 M Busha et al. (2011b).

2 The web address for the MultiDark simulation data center is Back.

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