  We examine the effect of primordial dark matter velocity dispersion and/or
particle self-interactions on the structure and stability of galaxy halos,
especially with respect to the formation of substructure and central density
cusps. Primordial velocity dispersion is characterised by a ``phase density''
$Q\equiv \rho/<v^2>^{3/2}$, which for relativistically-decoupled relics is
determined by particle mass and spin and is insensitive to cosmological
parameters. Finite $Q$ leads to small-scale filtering of the primordial power
spectrum, which reduces substructure, and limits the maximum central density of
halos, which eliminates central cusps. The relationship between $Q$ and halo
observables is estimated. The primordial $Q$ may be preserved in the cores of
halos and if so leads to a predicted relation, closely analogous to that in
degenerate dwarf stars, between the central density and velocity dispersion.
Classical polytrope solutions are used to model the structure of halos of
collisional dark matter, and to show that self-interactions in halos today are
probably not significant because they destabilize halo cores via heat
conduction. Constraints on masses and self-interactions of dark matter
particles are estimated from halo stability and other considerations.
