An Accurate Comprehensive Approach to Substructure: I. Accreted
  Subhaloes
  This is the first of a series of three Papers devoted to the study of halo
substructure in hierarchical cosmologies by means of the CUSP formalism. In the
present Paper we derive the properties of subhaloes and diffuse dark matter
(dDM) accreted onto haloes and their progenitors. Specifically, we relate the
dDM present at any time in the inter-halo medium of the real Universe or a
cosmological simulation with the corresponding free-streaming mass or the halo
resolution mass, respectively, and establish the link between subhaloes and
their seeds in the initial density field. By monitoring the collapse and
virialisation of haloes, we derive from first principles and with no single
free parameter the abundance and radial distribution of dDM and subhaloes
accreted onto them. Our predictions are in excellent agreement with the results
of simulations, but for the predicted fraction of accreted dDM, which is larger
than reported in previous works as they only count the dDM accreted onto the
final halo, not onto its progenitors. The derivation pursued here clarifies the
origin of some key features of substructure. Overall, our results demonstrate
that CUSP is a powerful tool for understanding halo substructure and extending
the results of simulations to haloes with arbitrary masses, redshifts and
formation times in any hierarchical cosmology endowed with random Gaussian
density perturbations.
