We explore the dynamical restrictions on the structure of dark matter halos through a study of cosmological self-similar gravitational collapse solutions. A fluid approach to the collisionless dynamics of dark matter is developed, and the resulting closed set of moment equations are solved numerically, including the effect of halo velocity dispersions (both radial and tangential), for a range of spherically averaged initial density profiles. Our results highlight the importance of tangential velocity dispersions for obtaining density profiles shallower than 1/r~2 in the core regions, and for retaining a memory of the initial density profile, in self-similar collapse. For an isotropic core velocity dispersion, only a partial memory of the initial density profile is retained. 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 hold in self-similar collapse.
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