Observational constraints on growth of massive black holes

We study the observational constraints on the growth of massive black holes (BHs) in galactic nuclei. We use the velocity dispersions of early-type galaxies obtained by the Sloan Digital Sky Survey and the relation between BH mass and velocity dispersion to estimate the local BH mass density to be rho.(z = 0) similar or equal to (2.5 +/- 0.4) x 10(5) h(0.65)(2) M. Mpc(-3). We also use the quasi-stellar object (QSO) luminosity function from the 2dF Redshift Survey to estimate the BH mass density accreted during optically bright QSO phases. The local BH mass density is consistent with the density accreted during optically bright QSO phases if QSOs have a mass-to-energy conversion efficiency epsilon similar or equal to 0.1. By studying the continuity equation for the BH mass distribution, including the effect of BH mergers, we find relations between the local BH mass function and the QSO luminosity function. If the BH mass is assumed to be conserved during BH mergers, comparison of the predicted relations with the observations suggests that luminous QSOs (L-bol greater than or similar to 10(46) erg s(-1)) have a high efficiency (e.g. epsilon similar to 0.2, which is possible for thin-disc accretion on to a Kerr BH) and the growth of high-mass BHs (greater than or similar to 10(8) M.) comes mainly from accretion during optically bright QSO phases, or that luminous QSOs have a super-Eddington luminosity. If luminous QSOs are not accreting with super-Eddington luminosities and the growth of low-mass BHs also occurs mainly during optically bright QSO phases, less luminous QSOs must accrete with a low efficiency, <0.1; alternatively, they may accrete with high efficiency, but a significant fraction should be obscured. We estimate that the mean lifetime of luminous QSOs (L-bol greater than or similar to 10(46) erg s(-1)) is (3-13) x 10(7) yr, which is comparable to the Salpeter time. We also investigate the case in which total BH mass decreases during BH mergers due to gravitational radiation; in the extreme case in which total BH entropy is conserved, the observations again suggest that BHs in most luminous QSOs are Kerr BHs accreting with an efficiency greater than or similar to 0.1.