Statistics of dark matter substructure - I. Model and universal fitting functions

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. 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, uses a simple recipe to convert subhalo mass to maximum circular velocity, and considers subhalo disruption. The model is calibrated to accurately reproduce the average subhalo mass and velocity functions in numerical simulations. We demonstrate that, on average, the 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 a broad range in host halo mass, redshift and Lambda cold dark matter cosmology.