We have just demonstrated that feedback does not solve the overabundance of CDM satellites - clearly some form of stochastic biasing is required. A solution was proposed by Bullock etal (2001) in which only those dark matter halos that have virialised prior to re-ionisation can cool gas and form stars. Once the IGM has been reheated then the smallest CDM halos cannot capture or cool gas and they remain completely dark.
In Figure 5 we mark all the progenitor halos that satisfy the condition for cooling gas prior to z = 10, which we will take as the redshift of re-ionisation. We mark particles red if they lie within a region of overdensity larger than 1000. The locations of these particles are subsequently tracked to z = 0 and marked in the right panel of Figure 5. Roughly 100 satellites satisfy the density criteria at a redshift z = 10 and 80 of these physically merger together to form the very central region of the final galaxy halos. The remaining 20 survive intact and can be found orbiting within the virial radius of the two halos (see Figure 6). The mean radius of the surviving satellites is 80 kpc, which is a factor of 2.5 smaller than the half mass radius of the final halos.
Figure 5. The left panel shows the Local Group simulation at z = 10. Marked in red are all those particles that lie in regions with an overdensity larger than 1000. The right panel shows one of the high resolution halos at z = 0 and the locations of the red particles marked at z = 10.
Figure 6. The smoothed distribution of "starlight" in the Local Group at the present day. I plot only those stars that could form in dark matter halos prior to re-ionisation at z = 10. The distribution of these stars is highly biased. Roughly a dozen dark matter dominated satellites orbit within each of the parent halos and they have a spatial distribution that matches the real Local Group. Most of the population II/III stars lie at the very centers of the halos surrounding M31 and the Galaxy. Their half light radius is just a few kiloparsecs (c.f. White & Springel 1999) and their luminosity density falls as r-3 (c.f. Figure 7).
The final cumulative distribution of satellites within one of the simulated halos is shown in Figure 2 and provides a good match to the corrected observational data points. Several puzzles remain. Why don't we find any satellites in the Galactic halo with velocity dispersion less than ~ 7 km/s? Is cooling that inefficient below 10 km/s such that we do not find any dark matter dominated systems containing just a handful of stars?
The star formation histories of the Local Group satellites presents a further puzzle. Most of the satellites show evidence for several bursts of star formation, some continuing to the present day. Both re-ionisation and the "essential" feedback have been extremely inefficient at removing gas from these tiny halos that have masses 108 M.