An Accurate Comprehensive Approach to Substructure: III. Masses and
  Formation Times of the Host Haloes
  With this Paper we complete a comprehensive study of substructure in dark
matter haloes. In Paper I we derived the radial distribution and mass function
(MF) of accreted subhaloes (scaled to the radius and mass of the host halo) and
showed they are essentially universal. This is not the case, however, for those
of stripped subhaloes, which depend on halo mass and assembly history. In Paper
II we derived these latter properties in the simplest case of purely accreting
haloes. Here we extend the study to ordinary haloes having suffered major
mergers. After showing that all the properties of substructure are encoded in
the mean truncated-to-original subhalo mass ratio profile, we demonstrate that
the dependence of the subhalo MF on halo mass arises from their mass-dependent
concentration, while the shape of the subhalo radial distribution depends on
the time of the last major merger of the host halo. In this sense, the latter
property is a better probe of halo formation time than the former.
Unfortunately, this is not the case for the radial distribution of satellites
as this profile is essentially disconnected from subhalo stripping and the
properties of accreted subhaloes are independent of the halo formation time.
