Rapidly spinning massive black holes in active galactic nuclei: evidence   from the black hole mass function
  The comparison of the black hole mass function (BHMF) of active galactic nuclei (AGN) relics with the measured mass function of the massive black holes in galaxies provides strong evidence for the growth of massive black holes being dominated by mass accretion. We derive the Eddington ratio distributions as functions of black hole mass and redshift from a large AGN sample with measured Eddington ratios given by Kollmeier et al. We find that, even at the low mass end, most black holes are accreting at Eddington ratio ~0.2, which implies that the objects accreting at extremely high rates should be rare or such phases are very short. Using the derived Eddington ratios, we explore the cosmological evolution of massive black holes with an AGN bolometric luminosity function (LF). It is found that the resulted BHMF of AGN relics is unable to match the measured local BHMF of galaxies for any value of (constant) radiative efficiency. Motivated by Volonteri, Sikora & Lasota's study on the spin evolution of massive black holes, we assume the radiative efficiency to be dependent of black hole mass, i.e., it is low for M<10^8 solar masses and it increases with black hole mass for black holes with >10^8 solar masses. We find that the BHMF of AGN relics can roughly reproduce the local BHMF of galaxies if the radiative efficiency ~0.08 for the black holes with <10^8 solar masses and it increases to ~0.18 for black holes with >10^9 solar masses, which implies that most massive black holes (>10^9 solar masses) are spinning very rapidly. 
Xinwu Cao, Fan Li