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 + 3 n ) / ( 5 + n ) .
For any n < 1 , the signature of undigested cores is a core density profile shallower than \rho \propto 1 / r ^ { 2 } and dependent on the power spectrum .
For typical objects formed from a CDM like power spectrum the effective value of n is close to -2 and thus \alpha could typically be near 1 , the NFW ( see text ) value .
Also velocity dispersions should deviate from a constant value to decrease with decreasing radius in the core .
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 can not 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 .
We find a limiting r ^ { -4 } outer profile for the open case and a limiting outer profile for the \Lambda dominated case , which at late times has the form [ 1 - ( r / \bar { r } _ { \lambda } ) ^ { -3 \epsilon } ] ^ { 1 / 2 } , where 3 \epsilon is the logarithmic slope of the initial density profile .
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 .
These results apply as well for low density cosmological models ( \Omega _ { matter } = 0.2 - 0.3 ) , since high density cores were formed early where \Omega _ { matter } \approx 1 .