  The energy and momentum deposited by the radiation from accretion onto the
supermassive black holes (BHs) that reside at the centres of virtually all
galaxies can halt or even reverse gas inflow, providing a natural mechanism for
supermassive BHs to regulate their growth and to couple their properties to
those of their host galaxies. However, it remains unclear whether this
self-regulation occurs on the scale at which the BH is gravitationally
dominant, on that of the stellar bulge, the galaxy, or that of the entire dark
matter halo. To answer this question, we use self-consistent simulations of the
co-evolution of the BH and galaxy populations that reproduce the observed
correlations between the masses of the BHs and the properties of their host
galaxies. We first confirm unambiguously that the BHs regulate their growth:
the amount of energy that the BHs inject into their surroundings remains
unchanged when the fraction of the accreted rest mass energy that is injected,
is varied by four orders of magnitude. The BHs simply adjust their masses so as
to inject the same amount of energy. We then use simulations with artificially
reduced star formation rates to demonstrate explicitly that BH mass is not set
by the stellar mass. Instead, we find that it is determined by the mass of the
dark matter halo with a secondary dependence on the halo concentration, of the
form that would be expected if the halo binding energy were the fundamental
property that controls the mass of the BH. We predict that the logarithmic
slope of the relation between dark matter halo mass and black hole mass is
1.55+/-0.05 and that the scatter around the mean relation in part reflects the
scatter in the halo concentration-mass relation.
