An upper limit to the central density of dark matter haloes from
  consistency with the presence of massive central black holes
  We study the growth rates of massive black holes in the centres of galaxies
from accretion of dark matter from their surrounding haloes. By considering
only the accretion due to dark matter particles on orbits unbound to the
central black hole, we obtain a firm lower limit to the resulting accretion
rate. We find that a runaway accretion regime occurs on a timescale which
depends on the three characteristic parameters of the problem: the initial mass
of the black hole, and the volume density and velocity dispersion of the dark
matter particles in its vicinity. An analytical treatment of the accretion rate
yields results implying that for the largest black hole masses inferred from
QSO studies ($>10^{9} M_{\odot}$), the runaway regime would be reached on time
scales which are shorter than the lifetimes of the haloes in question for
central dark matter densities in excess of $250 M_{\odot}$pc$^{-3}$. Since
reaching runaway accretion would strongly distort the host dark matter halo,
the inferences of QSO black holes in this mass range lead to an upper limit on
the central dark matter densities of their host haloes of $\rho_{0} < 250
M_{\odot} $pc$^{-3}$. This limit scales inversely with the assumed central
black hole mass. However, thinking of dark matter profiles as universal across
galactic populations, as cosmological studies imply, we obtain a firm upper
limit for the central density of dark matter in such structures.
