The Universal Specific Merger Rate of Dark Matter Halos

We employ a set of high resolution N-body simulations to study the merger rate of dark matter halos. We define a specific merger rate by normalizing the average number of mergers per halo with the logarithmic mass growth change of the hosts at the time of accretion. Based on the simulation results, we find that this specific merger rate, $\mathrm{d}N_{\mathrm{merge}}(\xi|M,z)/\mathrm{d}\xi/\mathrm{d}\log M(z)$, has a universal form, which is only a function of the mass ratio of merging halo pairs, $\xi$, and does not depend on the host halo mass, $M$, or redshift, $z$, over a wide range of masses ($10^{12}\lesssim M \lesssim10^{14}\,M_\odot/h$) and merger ratios ($\xi\ge 1e-2$). We further test with simulations of different $\Omega_m$ and $\sigma_8$, and get the same specific merger rate. The universality of the specific merger rate shows that halos in the universe are built up self-similarly, with a universal composition in the mass contributions and an absolute merger rate that grows in proportion to the halo mass growth. As a result, the absolute merger rate relates with redshift and cosmology only through the halo mass variable, whose evolution can be readily obtained from the universal mass accretion history (MAH) model of \cite{2009ApJ...707..354Z}. Lastly, we show that this universal specific merger rate immediately predicts an universal un-evolved subhalo mass function that is independent on the redshift, MAH or the final halo mass, and vice versa.