A unified model for the spatial and mass distribution of subhaloes

N-body simulations suggest that the substructures that survive inside dark matter haloes follow universal distributions in mass and radial number density. We demonstrate that a simple analytical model can explain these subhalo distributions as resulting from tidal stripping which increasingly reduces the mass of subhaloes with decreasing halo-centric distance. As a starting point, the spatial distribution of subhaloes of any given infall mass is shown to be largely indistinguishable from the overall mass distribution of the host halo. Using a physically motivated statistical description of the amount of mass stripped from individual subhaloes, the model fully describes the joint distribution of subhaloes in final mass, infall mass and radius. As a result, it can be used to predict several derived distributions involving combinations of these quantities including, but not limited to, the universal subhalo mass function, the subhalo spatial distribution, the gravitational lensing profile, the dark matter annihilation radiation profile and boost factor. This model clarifies a common confusion when comparing the spatial distributions of galaxies and subhaloes, the so called "anti-bias", as a simple selection effect. We provide a Python code SubGen for populating haloes with subhaloes at http://icc.dur.ac.uk/data/