Black Holes and Galactic Density Cusps : From Black Hole to Bulge
  Aims. In this paper we continue our study of density cusps that may contain
central black holes. Methods. We recall our attempts to use distribution
functions with a memory of self-similar relaxation, but mostly they apply only
in restricted regions of the global system. We are forced to consider related
distribution functions that are steady but not self-similar. Results. One
remarkably simple distribution function that has a filled loss cone describes a
bulge that transits from a near black hole domain to an outer 'zero flux'
regime where$\rho\propto r^{-7/4}$. The transition passes from an initial
inverse square profile through a region having a 1/r density profile. The
structure is likely to be developed at an early stage in the growth of a
galaxy. A central black hole is shown to grow exponentially in this background
with an e-folding time of a few million years. Conclusions. We derive our
results from first principles, using only the angular momentum integral in
spherical symmetry. The initial relaxation probably requires bar instabilities
and clump-clump interactions.
