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Figure 3.5. An example of a merger tree obtained from an N-body simulation of a 9 x 1012h-1M0 halo at redshift z = 0. Each circle represents a dark-matter halo identified in the simulation, the area of the circle being proportional to halo mass. The vertical position of each halo on the plot is determined by the redshift z at which the halo is identified; the horizontal positioning is arbitrary. The solid lines connect halos to their progenitors. The solid line in the panel on the left-hand side shows the fraction of the final mass contained in resolved progenitors as a function of redshift. The dotted line shows the fraction of the final mass contained in the largest progenitor as a function of redshift. From Helly et al. (2003.)

Figure 3.5. An example of a merger tree obtained from an N-body simulation of a 9 x 1012h-1M0 halo at redshift z = 0. Each circle represents a dark-matter halo identified in the simulation, the area of the circle being proportional to halo mass. The vertical position of each halo on the plot is determined by the redshift z at which the halo is identified; the horizontal positioning is arbitrary. The solid lines connect halos to their progenitors. The solid line in the panel on the left-hand side shows the fraction of the final mass contained in resolved progenitors as a function of redshift. The dotted line shows the fraction of the final mass contained in the largest progenitor as a function of redshift. From Helly et al. (2003.)

3.3.4 Dark-halo mergers The assumption that virialized objects form from smooth spherical collapse, while providing a useful framework for thinking about the formation histories of gravitationally bound dark-matter halos, does not capture the real nature of structure formation in CDM theories. In these models galaxies are assembled hierarchically through the merging of many smaller subunits that formed in a similar manner at higher redshift (see Figure 3.5). Galaxy halos experience multiple mergers during their lifetime, with those between comparable-mass systems ("major mergers") expected to result in the formation of elliptical galaxies (e.g. Barnes 1988; Hernquist 1992). Figure 3.6 shows the number of major mergers (defined as mergers for which the mass ratio of the progenitors is >0.3) per unit redshift bin experienced by halos of various masses. For galaxy-sized halos this quantity happens to peak in the redshift range 2-4, corresponding to the epoch when the observed space density of optically selected quasars also reaches a maximum.

The merger between a large parent halo and a smaller satellite system will evolve under two dynamical processes: dynamical friction, that causes the orbit of the satellite to decay toward the central regions; and tidal stripping, that removes material from the satellite and adds it to the diffuse mass of the parent. Since clustering is hierarchical, the satellite will typically form at earlier times and have a higher characteristic density

Figure 3.6. Mean numbers of major mergers experienced per unit redshift by halos with masses >1O1oM0. Solid line: progenitors of an M0 = 2 x 1Û13M0 halo at z = 0. Dashed line: the same for M0 = 3 x 1O12M0. Dotted line: the same for M0 = 1O11M0. From Volonteri et al. (2003.)

and a smaller characteristic radius. The study of the assembly of dark-matter halos by repeated mergers is particularly well suited to the A-body methods that have been developed in the past two decades. Numerical simulations of structure formation by dissipationless hierarchical clustering from Gaussian initial conditions indicate a roughly universal spherically averaged density profile for the resulting halos (Navarro et al. 1997, hereafter NFW):

where x = r/rvir and the characteristic density ps is related to the concentration parameter c by

This function fits the numerical data presented by NFW over a radius range of about two orders of magnitude. Equally good fits are obtained for high-mass (rich galaxy cluster) and low-mass (dwarf) halos. Power-law fits to this profile over a restricted radial range have slopes that steepen from —1 near the halo center to —3 at large cr/rvil. Bullock et al. (2001) found that the concentration parameter follows a log-normal distribution such that the median depends on the halo mass and redshift, where M* is the mass of a typical halo collapsing today. The halos in the simulations by Bullock et al. were best described by a = —0.13 and c* = 9.0, with a scatter around the median of aiog c = 0.14 dex.

Figure 3.7. Top: the logarithmic slope of the density profile of the "Via Lactea" halo, as a function of radius. Densities were computed in 50 radial logarithmic bins, and the local slope was determined by a finite-difference approximation using one neighboring bin on each side. The thin line shows the slope of the best-fit NFW profile with concentration c = 12. The vertical dotted line indicates the estimated convergence radius: local densities (but not necessarily the logarithmic slopes) should be correct to within 10% outside of this radius. Bottom: the residuals in percent between the density profile and the best-fit NFW profile, as a function of radius. Here r200 is the radius within which the enclosed average density is 200 times the background value, r200 = 1.35rvir in the adopted cosmology. From Diemand et al. (2007a).

Figure 3.7. Top: the logarithmic slope of the density profile of the "Via Lactea" halo, as a function of radius. Densities were computed in 50 radial logarithmic bins, and the local slope was determined by a finite-difference approximation using one neighboring bin on each side. The thin line shows the slope of the best-fit NFW profile with concentration c = 12. The vertical dotted line indicates the estimated convergence radius: local densities (but not necessarily the logarithmic slopes) should be correct to within 10% outside of this radius. Bottom: the residuals in percent between the density profile and the best-fit NFW profile, as a function of radius. Here r200 is the radius within which the enclosed average density is 200 times the background value, r200 = 1.35rvir in the adopted cosmology. From Diemand et al. (2007a).

"Via Lactea," the highest-resolution A-body simulation to date of the formation of a Milky Way-sized halo (Diemand et al. 2007a, b), shows that the fitting formula proposed by NFW with concentration c =12 provides a reasonable approximation to the density profile down a convergence radius of rconv = 1.3 kpc (Figure 3.7). Within the region of convergence, deviations from the best-fit NFW matter density are typically less than 10%. From 10 kpc down to rconv Via Lactea is actually denser than predicted by the NFW formula. Near rconv the density approaches the NFW value again while the logarithmic slope is shallower (-1.0 at rconv) than predicted by the NFW fit.

3.3.5 Assembly history of a Milky Way halo The simple spherical top-hat collapse ignores shell crossing and mixing, accretion of self-bound clumps, triaxiality, angular momentum, random velocities, and large-scale tidal forces. It is interesting at this stage to use "Via Lactea" and study in more details, starting from realistic initial conditions, the formation history of a Milky Way-sized halo in a ACDM cosmology. The Via Lactea simulation was performed with the PKDGRAV tree-code (Stadel 2001) using the best-fit cosmological parameters from the WMAP 3-year data release. The galaxy-forming region was sampled with 234 million particles of mass 2.1 X 104M0, evolved from redshift 49 to the present with a force resolution of 90 pc and adaptive time-steps as short as 68 500 yr, and centered on an isolated halo that had

Figure 3.8. Projected dark-matter density-squared map of our simulated Milky Way-halo region ("Via Lactea") at various redshifts, from z = 10.6 to the present. The image covers an area of 400 x 300 physical kiloparsecs.

no major merger after z = 1.7, making it a suitable host for a Milky Way-like disk galaxy (see Figure 3.8). The number of particles is an order of magnitude larger than had been used in previous simulations. The run was completed in 320 000 CPU hours on NASA's Project Columbia supercomputer, currently one of the fastest machines available. (More details about the Via Lactea run are given in Diemand et al. (2007a, b). Movies, images, and data are available at http://www.ucolick.org/~diemand/vl.) The host halo mass at z = 0 is M200 = 1.8 x 1012M0 within a radius of r200 = 389 kpc.

Following the spherical top-hat model, the common procedure used to describe the assembly of a dark-matter halo is to define at each epoch a virial radius rvir (or, a = 1/(1 + z)

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