Black hole growth and activity in a Λ cold dark matter universe

The observed properties of supermassive black holes suggest a fundamental link between their assembly and the formation of their host spheroids. We model the growth and activity of black holes in galaxies using Lambda cold dark matter cosmological hydrodynamic simulations by following the evolution of the baryonic mass component in galaxy potential wells. We find that the observed steep relation between black hole mass and spheroid velocity dispersion, M-BH proportional to sigma(4), is reproduced if the gas mass in bulges is linearly proportional to the black hole mass. To a good approximation, this is equivalent to assuming the conversion of a fixed fraction of gas mass into black hole mass. In this model, star formation and supernova feedback in the gas are sufficient for regulating and limiting the growth of the central black hole and of its gas supply. Black hole growth saturates because of the competition with star formation and, in particular, feedback, both of which determine the gas fraction available for accretion. Unless other processes also operate, we predict that the M-BH-sigma relation is not set in primordial structures but is fully established at low redshifts, z less than or similar to 2, and is shallower at earlier times. Once this relation is established, we find that central black hole masses are related to their dark matter halos simply via M-BH approximate toproportional to M-DM(4/3). We assume that galaxies undergo a quasar phase with a typical lifetime, t(Q) similar to 2 x 10(7) yr, the only free parameter of the model, and show that star formation - regulated depletion of gas in spheroids is sufficient to explain, for the most part, the decrease of the quasar population at redshift z < 3 in the optical blue band. However, with the simplest assumption of a redshift-independent quasar lifetime, the model slightly over-predicts optical quasar numbers at high redshifts, although it yields the observed evolution of number density of X-ray - faint quasars over the whole redshift range 1 < z < 6. Finally, we find that the majority of black hole mass is assembled in galaxies by z similar to 3 and that the black hole accretion rate density peaks in rough correspondence to the star formation rate density at z similar to 4-5.