  During hierarchical clustering, smaller masses generally collapse earlier
than larger masses and so are denser on the average. The core of a small mass
halo could be dense enough to resist disruption and survive undigested, when it
is incorporated into a bigger object. We explore the possibility that a nested
sequence of undigested cores in the center of the halo, which have survived the
hierarchical, inhomogeneous collapse to form larger and larger objects,
determines the halo structure in the inner regions. For a flat universe with
$P(k) \propto k^n$, scaling arguments then suggest that the core density
profile is, $\rho \propto r^{-\alpha}$ with $\alpha = (9+3n)/(5+n)$. But
whether such behaviour obtains depends on detailed dynamics. We first examine
the dynamics using a fluid approach to the self-similar collapse solutions for
the dark matter phase space density, including the effect of velocity
dispersions. We highlight the importance of tangential velocity dispersions to
obtain density profiles shallower than $1/r^2$ in the core regions. If
tangential velocity dispersions in the core are constrained to be less than the
radial dispersion, a cuspy core density profile shallower than 1/r cannot
obtain, in self-similar collapse. We then briefly look at the profiles of the
outer halos in low density cosmological models where the total halo mass is
convergent. Finally, we analyze a suite of dark halo density and velocity
dispersion profiles obtained in cosmological N-body simulations of models with
n= 0, -1 and -2. We find that the core-density profiles of dark halos, show
considerable scatter in their properties, but nevertheless do appear to reflect
a memory of the initial power spectrum, with steeper initial spectra producing
flatter core profiles. (Abridged)
