On Constraining the Growth History of Massive Black Holes via Their Distribution on the Spin–Mass Plane

The spin distribution of massive black holes (MBHs) contains rich information on the MBH growth history. In this paper, we investigate the spin evolution of MBHs by assuming that each MBH experiences two-phase accretion, with an initial phase of coherent accretion via either the standard thin disk or super-Eddington disk, followed by a chaotic-accretion phase composed of many episodes with different disk orientations. If the chaotic-accretion phase is significant to the growth of an MBH, the MBH spin quickly reaches the maximum value because of the initial coherent accretion, then changes to a quasi-equilibrium state and fluctuates around a value mainly determined by the mean ratio of the disk to the MBH mass (M-center dot) in the chaotic-accretion episodes, and further declines because of late chaotic accretion if M-center dot greater than or similar to (1 - 3) x 10(8) M-circle dot. The turning point to this decline is determined by the equality of the disk warp radius and disk size. By matching the currently available spin measurements with mock samples generated from the two-phase model(s) on the spin-mass plane, we find that MBHs must experience significant chaotic-accretion phase with many episodes and that the mass accreted in each episode is roughly 1%-2% of M-center dot or less. MBHs with M-center dot greater than or similar to 10(8) M-circle dot appear to have intermediate-to-high spins (similar to 0.5- 1), while lighter MBHs have higher spins (greater than or similar to 0.8). The best matches also infer that (1) the radiative efficiencies (eta) of those active MBHs appear to slightly decrease with M-center dot; however, the correlation between eta and M-center dot, if any, is weak; (2) the mean radiative efficiency of active MBHs is similar to 0.09 - 0.15, consistent with the global constraints.