Dark matter haloes determine the masses of supermassive black holes

The energy and momentum deposited by the radiation from accretion flows on to 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 BH mass, m(BH), scales with halo mass as m(BH) proportional to m alpha(halo), with alpha approximate to 1.55 +/- 0.05, and that the scatter around the mean relation in part reflects the scatter in the halo concentration-mass relation.