Accretion history of super-massive black holes

We show that the luminosity function of the actively star-forming Lyman break galaxies and the B-band quasar luminosity function at $z = 3$ can be fit reasonably well with the mass function of collapsed galaxy scale dark matter haloes predicted by viable variants of hierarchical cold dark matter dominated cosmological models for lifetimes $t_Q$ of the optically bright phase of QSOs in the range $10^{6}$ to $10^{8}$ yr. There is a strong correlation between $t_Q$ and the required degree of non-linearity in the relation between black hole and host halo mass. Such a non-linear relation is motivated by suggesting that the mass of supermassive black holes may be limited by the back-reaction of the emitted energy on the accretion flow in a self-gravitating disc. This would imply a relation of black hole to halo mass of the form $M_{\rm bh} \propto v_{\rm halo}^5 \propto M_{\rm halo}^{5/3}$ and a typical duration of the optically bright QSO phase of the order of the Salpeter time, $\sim 10^{7}$ yr. The high integrated local mass density of black holes inferred from recent kinematic determinations of black hole masses in nearby galaxies seem to indicate that the overall efficiency of supermassive black holes for producing blue light is lower than was previously assumed. We discuss three possible accretion modes with low optical emission efficiency: (i) accretion well above the Eddington rate, (ii) accretion obscured by dust, and (iii) accretion below the critical rate leading to an advection dominated accretion flow lasting for a Hubble time. We further argue that accretion with low optical efficiency might be closely related to the origin of the hard X-ray background.