Black Holes and Galactic Density Cusps III 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.