Self-similar decomposition of the hierarchical merger tree of dark matter halos

In the $\Lambda$CDM universe, structure formation is generally not a self-similar process, while some self-similarity remains in certain statistics which can greatly simplify our description and understanding of the cosmic structures. In this work, we show that the merger tree of dark matter halos is approximately self-similar by investigating the universality of the subhalo peak mass function (PMF) describing the mass distribution of progenitor halos. Using a set of cosmological simulations and identifying subhalos of different merger levels with HBT+, we verify that the level-1 subhalo PMF is close to universal across halo mass, redshift, and cosmology. This approximate self-similarity allows us to analytically derive the subhalo PMF for subhalos accreted at any level (i.e., for sub-sub...halos) through self-convolutions of the level-1 PMF, and the resulting model shows good agreement with simulation measurements. We further derive a number of analytical properties on the hierarchical origin of subhalos. We show that higher-level subhalos dominate at progressively lower peak mass in the PMF and are more likely to originate from major mergers than lower-level ones. At a given merger mass ratio, the subhalo accretion rates at each level track the growth rate of the host halo. At a fixed final mass ratio, however, subhalos of higher-level, higher-mass-ratio and in more massive haloes tend to be accreted more recently. Matching subhalo peak mass to galaxy mass, these results have direct implications on the hierarchical origin of satellite galaxies.