  We show that the two basic assumptions of the model recently proposed by
Manrique and coworkers for the universal density profile of cold dark matter
(CDM) halos, namely that these objects grow inside out in periods of smooth
accretion and that their mass profile and its radial derivatives are all
continuous functions, are both well understood in terms of the very nature of
CDM. Those two assumptions allow one to derive the typical density profile of
halos of a given mass from the accretion rate characteristic of the particular
cosmology. This profile was shown by Manrique and coworkers to recover the
results of numerical simulations. In the present paper, we investigate its
behavior beyond the ranges covered by present-day N-body simulations. We find
that the central asymptotic logarithmic slope depends crucially on the shape of
the power spectrum of density perturbations: it is equal to a constant negative
value for power-law spectra and has central cores for the standard CDM power
spectrum. The predicted density profile in the CDM case is well fitted by the
3D S\'ersic profile over at least 10 decades in halo mass. The values of the
S\'ersic parameters depend on the mass of the structure considered. A practical
procedure is provided that allows one to infer the typical values of the best
NFW or S\'ersic fitting law parameters for halos of any mass and redshift in
any given standard CDM cosmology.
