The population of dark matter subhaloes: mass functions and average mass-loss rates

Using a cosmological N-body simulation and a sample of resimulated cluster-like haloes, we study the mass-loss rates of dark matter subhaloes, and interpret the mass function of subhaloes at redshift zero in terms of the evolution of the mass function of systems accreted by the main halo progenitor (hereafter called the `unevolved subhalo mass function'). When expressed in terms of the ratio between the mass of the subhalo at the time of accretion, mv, and the present day host mass, M0, the unevolved subhalo mass function is found to be universal, in that it is independent of the mass of the host halo. However, the subhalo mass function at redshift zero (hereafter called the `evolved subhalo mass function') clearly depends on M0, in that more massive host haloes host more subhaloes. In order to relate the unevolved and evolved subhalo mass functions, we measure the subhalo mass-loss rate as a function of host mass and redshift. We find that the average, specific mass-loss rate of dark matter subhaloes depends mainly on redshift, with only a very weak dependence on the instantaneous ratio between the mass of the subhalo, msb, and that of the host halo at that time Mv. In fact, to good approximation, subhalo masses `decay' exponentially, with a decay time that is proportional to the instantaneous dynamical time of the host halo. Combined with the fact that more massive haloes assemble later, these results suggest a pleasingly simple picture for the evolution and mass dependence of the evolved subhalo mass function. Less massive host haloes accrete their subhaloes earlier, which are thus subjected to mass-loss for a longer time. In addition, their subhaloes are typically accreted by denser hosts, which causes an additional boost of the mass-loss rate. To test the self-consistency of this picture, we use semi-analytical merger trees constructed using the extended Press-Schechter formalism, and evolve the subhalo populations using the average mass-loss rates obtained from our simulations. The resulting subhalo mass functions are found to be in good agreement with the simulations. Our model can be applied to semi-analytical methods of galaxy formation, to accurately follow the time evolution of subhalo masses.