Caustic rings and cold dark matter

2.2. Hierarchical growth

CDM halos form via a series of many mergers and accretions of dark matter clumps along highly filamentary mass distributions. The assumptions of symmetry and locally cold flows is an incorrect over-simplification of the true hierarchical growth.

At a given point in a CDM halo, the "smooth" dark matter background arises from material that has been tidally stripped from less massive halos. The infalling background of CDM is a hierarchy of tiny halos that are unresolved in current numerical simulations. Globally, the fraction of CDM in bound objects is expected to be 100%, whereas current simulations that can resolve just one level of the hierarchy find a value of ~ 50%.

If the power spectrum was cutoff on small scales, e.g. warm or hot dark matter, then locally cold material would fall onto halos giving rise to caustic structures. These features would be phase wrapped only a few times and it is possible that they have observable consequences. The power spectrum of fluctuations in the CDM model allows small dense dark matter halos to collapse at very early epochs - the cutoff scale from the free streaming of neutralinos is approximately 10^-12 M. The first halos to collapse will have prominent caustics although on a tiny scale compared to the solar system. Infalling material will be scattered by various processes such as tidal fields and halo-halo encounters which smear out the caustic sheets of CDM mini-halos (e.g. Tremaine 1999, Helmi and White 1999).

We have used numerical simulations to investigate the structure of dark matter halos across a range of mass scales from 10⁷ M to 10¹⁵ M. Rather than run many simulations at low resolution we have simulated several at very high resolution. One of the key results that we find is that substructure within hierarchical models is essentially scale free. The distribution and orbital properties of "halos-within-halos" is independent of halo mass - CDM galaxy halos contain literally thousands of halos more massive than those that surround the dwarf satellite galaxies in our own halo, Draco and Ursa-Minor.

Figure 2 shows the formation of a single CDM halo resolved with 10⁶ particles and force resolution < 0.001R₂₀₀. The substructure visible within the CDM halo in Figure 2 is nearly self-similar. If we could zoom into a single subhalo it would appear like a scaled version of the entire system.

Figure 2. The hierarchical collapse of a CDM halo. The grey scale shows the local density of dark matter at the indicated redshifts. The size of each panel is 4 comoving Mpc for H = 50 km/s/Mpc.

Figure 3. From top left to bottom right we plot the spherically averaged rotation curve and and 1d velocity dispersion profile of a CDM halo, the short to long axis ratios of two CDM halos, the angular momentum vector of these two halos and the anisotropy parameter within a single halo simulated at different resolutions. From these plots it is clear that the global properties of halos are complex and axial symmetry is not preserved.

Finally, we show the structure in velocity space as a function of location within a CDM halo. We have taken 3 cubes at different positions in the halo that are smooth in density space. Each cube contains 2000 particles and has no bound subhalos within them as can been seen in the central panels. The left panel shows the histogram of one component of the distribution of particle speeds in each cube - the curve shows a Maxwellian distribution with set equal to the spherically averaged value at 0.1R₂₀₀. The right panels show the x and y components of velocity plotted against each other. In the outer halo streams of tidally stripped particles have made just one or two complete orbits and they remain very distinct and cold features in velocity space. As one moves from the outer to the inner halo, the streams are significantly more wound up in phase space and the velocity distribution becomes close to a Maxwellian. However this is purely a resolution effect since higher resolution would resolve smaller halos and a network of cold streams at the solar radius.

Figure 4. The left panels show velocity histograms of particles extracted from small patches of the halo where the particles are smoothly distributed in density space (center panel) but show complex velocity distributions (right panels).