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Figure 3.9. Evolution of radii rM enclosing a fixed mass versus cosmic time or scale factor a. The enclosed mass grows in constant amounts of 0.3 x 1012M0 from bottom to top. Initially all spheres are growing in the physical (non-comoving) units used here. Inner shells turn around, collapse, and stabilize, whereas the outermost shells are still expanding today. Solid circles: points of maximum expansion at the turnaround time imax. Open squares: time after turnaround when rM first contracts within 20% of the final value. These mark the approximate epoch of "stabilization."

Figure 3.9. Evolution of radii rM enclosing a fixed mass versus cosmic time or scale factor a. The enclosed mass grows in constant amounts of 0.3 x 1012M0 from bottom to top. Initially all spheres are growing in the physical (non-comoving) units used here. Inner shells turn around, collapse, and stabilize, whereas the outermost shells are still expanding today. Solid circles: points of maximum expansion at the turnaround time imax. Open squares: time after turnaround when rM first contracts within 20% of the final value. These mark the approximate epoch of "stabilization."

equivalently, r200), which depends on the cosmic background density at the time. As the latter decreases with the Hubble expansion, formal virial radii and masses grow with cosmic time even for stationary halos. Studying the transformation of halo properties within rv;r (or some fraction of it) mixes real physical change with apparent evolutionary effects caused by the growing radial window, and makes it hard to disentangle the two. Figure 3.9 shows the formation of Via Lactea where radial shells enclosing a fixed mass, rM, have been used instead. Unlike rvir, rM stops growing as soon as the mass distribution of the host halo becomes stationary on the corresponding scale. Note that mass and substructure are constantly exchanged between these shells, since rM is not a Lagrangian radius enclosing the same material at all times, just the same amount of it. The fraction of material belonging to a given shell in the past that still remains within the same shell today is shown in Figure 3.10. The mixing is larger before stabilization, presumably because of shell crossing during collapse, and smaller near the center, where most of the mass is in a dynamically cold, concentrated old component (Diemand et al. 2005b). Outer shells numbers 9 and 10, for example, retain today less than 25% of the particles that originally belonged to them at a < 0.4.

Note that the collapse times also appear to differ from the expectations of spherical top-hat behavior. Shell number 5, for example, encloses a mean density of about 100,0° today, has a virial mass of 1.5 x 1012M0, and should have virialized just now according to the spherical top-hat model. It did so instead much earlier, at a = 0.6. Even the

Figure 3.10. The fraction of material belonging to shell i at epoch a that remains in the same shell today. Shells are the same as in Figure 3.9, numbered from 1 (inner) to 10 (outer). Solid circles: time of maximum expansion. Open squares: stabilization epoch. Mass mixing generally decreases with time and toward the halo center.

Figure 3.10. The fraction of material belonging to shell i at epoch a that remains in the same shell today. Shells are the same as in Figure 3.9, numbered from 1 (inner) to 10 (outer). Solid circles: time of maximum expansion. Open squares: stabilization epoch. Mass mixing generally decreases with time and toward the halo center.

next-larger shell with 1.8 x 1012Mq stabilized before a = 0.8. It appears that the spherical top-hat model provides only a crude approximation to the virialized regions of simulated galaxy halos.

To understand the mass-accretion history of the Via Lactea halo it is useful to analyze the evolution of mass within fixed physical radii. Figure 3.11 shows that the mass within all radii from the resolution limit of kpc up to 100 kpc grows during a series of major mergers before a = 0.4. After this phase of active merging and growth by accretion the halo mass distribution remains almost perfectly stationary at all radii. Only the outer regions (—400 kpc) experience a small amount of net mass accretion after the last major merger. The mass within 400 kpc increases only mildly, by a factor of 1.2 from z = 1 to the present. During the same time the mass within radii of 100 kpc and smaller, the peak circular velocity, and the radius at which this is reached all remain constant to within 10%. The fact that mass definitions inspired by the spherical top-hat model fail to accurately describe the real assembly of galaxy halos is clearly seen in Figure 3.11, where M2oo is shown to increase at late times even when the halo's physical mass remains the same. This is just an artificial effect caused by the growing radial windows rvir and r200 as the background density decreases. For Via Lactea M200 increases by a factor of 1.8 from z = 1 to the present, while the real physical mass within a 400-kpc sphere grows by only a factor of 1.2 during the same time interval, and by an even smaller factor at smaller radii.

3.3.6 Smallest SUSY-CDM microhalos

As already mentioned above, the key idea of the standard cosmological paradigm for the formation of structure in the Universe, namely that primordial density fluctuations grow by gravitational instability driven by cold, collisionless dark matter, is constantly being elaborated upon and explored in detail through supercomputer simulations, and tested

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Figure 3.11. The mass-accretion history of Via Lactea. Masses within spheres of fixed physical radii centered on the main progenitor are plotted against the cosmological expansion factor a. The thick solid lines correspond to spheres with radii given by the labels on the left. The thin solid lines correspond to nine spheres of intermediate radii that are 1.3, 1.6, 2.0, 2.5, 3.2, 4.0, 5.0, 6.3, and 7.9 times larger than the next-smaller labeled radius. Dashed line: M2oo. The halo is assembled during a phase of active merging before a ~ 0.37 (z ~ 1.7) and its net mass content remains practically stationary at later times.

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Figure 3.11. The mass-accretion history of Via Lactea. Masses within spheres of fixed physical radii centered on the main progenitor are plotted against the cosmological expansion factor a. The thick solid lines correspond to spheres with radii given by the labels on the left. The thin solid lines correspond to nine spheres of intermediate radii that are 1.3, 1.6, 2.0, 2.5, 3.2, 4.0, 5.0, 6.3, and 7.9 times larger than the next-smaller labeled radius. Dashed line: M2oo. The halo is assembled during a phase of active merging before a ~ 0.37 (z ~ 1.7) and its net mass content remains practically stationary at later times.

against a variety of astrophysical observations. The leading candidate for dark matter is the neutralino, a weakly interacting massive particle predicted by the supersymmetry (SUSY) theory of particle physics. While in a SUSY-CDM scenario the mass of bound dark-matter halos may span about twenty orders of magnitude, from the most-massive galaxy clusters down to Earth-mass clumps (Green et al. 2004), it is the smallest microhalos that collapse first, and only these smallest scales are affected by the nature of the relic dark-matter candidate.

Recent numerical simulations of the collapse of the earliest and smallest gravitationally bound CDM clumps (Diemand et al. 2005a; Gao et al. 2005) have shown that tiny virialized microhalos form at redshifts above 50 with internal density profiles that are quite similar to those of present-day galaxy clusters. At these epochs a significant fraction of neutralinos has already been assembled into non-linear Earth-mass over-densities. If this first generation of dark objects were to survive gravitational disruption during the early hierarchical merger and accretion process - as well as late tidal disruption from stellar encounters (Zhao et al. 2007) - then over 1015 such clumps may populate the halo of the Milky Way. The nearest microhalos may be among the brightest sources of y-rays from neutralino annihilation. Since the annihilation rate increases quadratically with the matter density, small-scale clumpiness may enhance the total y-ray flux from nearby extragalactic systems (like M31), making them detectable by the forthcoming GLAST satellite or the next-generation of air Cerenkov telescopes.

The possibility of observing the fingerprints of the smallest-scale structure of CDM in direct and indirect dark-matter searches hinges on the ability of microhalos to survive the hierarchical clustering process as substructure within the larger halos that form at

Figure 3.12. Local dark-matter-density maps. The left panels illustrate the almost simultaneous structure formation in the SUSY run at different epochs, within a sphere of physical radius r including a mass of 0.014M0. A galaxy-cluster halo (right panels) forms in the standard hierarchical fashion: the dark-matter distribution is shown within a sphere of radius r including a mass of 5.9 x 1014M0. The SUSY and cluster halos have concentration parameters for an NFW profile of c = 3.7 and c = 3.5, respectively. In each image the logarithmic color scale ranges from 10 to 106 times pc(z).

later times. In recent years high-resolution Y-body simulations have enabled the study of gravitationally bound subhalos with Msub/M > 10—6 on galaxy (and galaxy-cluster) scales (e.g. Moore et al. 1999; Klypin et al. 1999; Stoehr et al. 2003). The main differences between these subhalos - the surviving cores of objects that fell together during the hierarchical assembly of galaxy-sized systems - and the tiny sub-microhalos discussed here is that on the smallest CDM scale the effective index of the linear power spectrum of mass-density fluctuations is close to —3. In this regime typical halo-formation times

Figure 3.13. A phase-space density (p/a3, where a is the one-dimensional velocity dispersion) map for a z = 75 SUSY halo (left) and a z = 0 galaxy cluster (right). Note the different color scales: relative to the average phase-space density, the logarithmic color scale ranges from 10 to 105 in the SUSY halo and from 10 to 107 in the cluster halo.

depend only weakly on halo mass, the capture of small clumps by larger ones is very rapid, and sub-microhalos may be more easily disrupted.

In Diemand et al. (2006) we presented a large Y-body simulation of early substructure in a SUSY-CDM scenario characterized by an exponential cutoff in the power spectrum at 10-6M0. The simulation resolves a 0.014M0 parent "SUSY" halo at z = 75 with 14 million particles. Compared with a z = 0 galaxy cluster, substructure within the SUSY host is less evident both in phase space and in physical space (see Figures 3.12 and 3.13), and it is less resistant against tidal disruption. As the Universe expands by a factor of 1.3, between 20 and 40 percent of well-resolved SUSY substructure is destroyed, compared with only percent in the low-redshift cluster. Nevertheless SUSY substructure is just as abundant as in z = 0 galaxy clusters, i.e. the normalized mass and circular velocity functions are very similar.

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