The mass function and average mass-loss rate of dark matter subhaloes

We present a simple, semi-analytical model to compute the mass functions of dark matter subhaloes. The masses of subhaloes at their time of accretion are obtained from a standard merger tree. During the subsequent evolution, the subhaloes experience mass loss due to the combined effect of dynamical friction, tidal stripping, and tidal heating. Rather than integrating these effects along individual subhalo orbits, we consider the average mass-loss rate, where the average is taken over all possible orbital configurations. Under the Ansatz that the average distribution of orbits is independent of parent halo mass, this allows us to write the average mass-loss rate as a simple function that depends only on redshift and on the instantaneous mass ratio of subhalo and parent halo. After calibrating this model by matching the subhalo mass function (SHMF) of cluster-sized dark matter haloes obtained from high-resolution, numerical simulations, we investigate the predicted mass and redshift dependence of the SHMF. We find that, contrary to previous claims, the subhalo mass function is not universal. Instead, both the slope and the normalization depend on the ratio of the parent halo mass, M, and the characteristic non-linear mass M*. This simply reflects a halo formation time dependence; more massive parent haloes form later, thus allowing less time for mass loss to operate. We predict that galaxy-sized haloes, with a present-day mass of M ≃ 1012 h-1 M⊙, have an average mass fraction of dark matter subhaloes that is a factor of 3 lower than for massive clusters with M ≃ 1015 h-1 M⊙. We also analyse the halo-to-halo scatter in SHMFs, and show that the subhalo mass fraction of individual haloes depends most strongly on their accretion history in the last ∼1 Gyr. Finally we provide a simple fitting function for the average SHMF of a parent halo of any mass at any redshift and for any cosmology, and briefly discuss several implications of our findings.