Statistics of Dark Matter Substructure: I. Model and Universal Fitting Functions

Abstract: We present a new, semi-analytical model describing the evolution of dark matter subhaloes. The model uses merger trees constructed using the method of Parkinson et al. (2008) to describe the masses and redshifts of subhaloes at accretion, which are subsequently evolved using a simple model for the orbit-averaged mass loss rates. The model is extremely fast, treats subhaloes of all orders, accounts for scatter in orbital properties and halo concentrations, and uses a simple recipe to convert subhalo mass to maximum circular velocity. The model accurately reproduces the average subhalo mass and velocity functions in numerical simulations. The inferred subhalo mass loss rates imply that an average dark matter subhalo loses in excess of 80 percent of its infall mass during its first radial orbit within the host halo. We demonstrate that the total mass fraction in subhaloes is tightly correlated with the `dynamical age' of the host halo, defined as the number of halo dynamical times that have elapsed since its formation. Using this relation, we present universal fitting functions for the evolved and unevolved subhalo mass and velocity functions that are valid for any host halo mass, at any redshift, and for any {\Lambda}CDM cosmology.