The spin is an important but poorly constrained parameter for describing supermassive black holes ( SMBHs ) .
Using the continuity equation of SMBH number density , we explicitly obtain the mass-dependent cosmological evolution of the radiative efficiency for accretion , which serves as a proxy for SMBH spin .
Our calculations make use of the SMBH mass function of active and inactive galaxies ( derived in the first paper of this series ) , the bolometric luminosity function of active galactic nuclei ( AGNs ) , corrected for the contribution from Compton-thick sources , and the observed Eddington ratio distribution .
We find that the radiative efficiency generally increases with increasing black hole mass at high redshifts ( z \gtrsim 1 ) , roughly as \eta \propto M _ { \bullet } ^ { 0.5 } , while the trend reverses at lower redshifts , such that the highest efficiencies are attained by the lowest mass black holes .
Black holes with M _ { \bullet } \gtrsim 10 ^ { 8.5 } M _ { \odot } maintain radiative efficiencies as high as \eta \approx 0.3 - 0.4 at high redshifts , near the maximum for rapidly spinning systems , but their efficiencies drop dramatically ( by an order of magnitude ) by z \approx 0 .
The pattern for lower mass holes is somewhat more complicated but qualitatively similar .
Assuming that the standard accretion disk model applies , we suggest that the accretion history of SMBHs and their accompanying spins evolve in two distinct regimes : an early phase of prolonged accretion , plausibly driven by major mergers , during which the black hole spins up , then switching to a period of random , episodic accretion , governed by minor mergers and internal secular processes , during which the hole spins down .
The transition epoch depends on mass , mirroring other evidence for “ cosmic downsizing ” in the AGN population ; it occurs at z \approx 2 for high-mass black holes , and somewhat later , at z \approx 1 , for lower-mass systems .