  In two previous papers (Salvador-Sol\'e 2012a,b), it was shown that: i) the
typical structural and kinematic properties of haloes in (bottom-up)
hierarchical cosmologies endowed with random Gaussian density perturbations of
dissipationless collisionless dark matter emerge naturally from the typical
properties of peaks in the primordial density field and ii) halo statistics are
well described by the peak formalism. In the present paper, we use these
results to model halo substructure. Specifically, making use of the peak
formalism and the fact that accreting haloes evolve from the inside-out, we
derive the subhalo mass abundance and number density profile per infinitesimal
mass for subhaloes of different masses, as a function of the subhalo maximum
circular velocity or mass, before and after the tidal truncation of subhaloes
by the host potential well. The subhalo properties obtained by assuming that
subhaloes are mainly made of diffuse particles are in very good agreement with
those found in current high-resolution N-body simulations. We also predict the
subhalo properties in the opposite extreme case, likely better suited for the
real universe in CDM cosmologies, that haloes are made of subhaloes within
subhaloes at all scales.
