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 \lambda \sim 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 \eta _ { rad } .
Motivated by Volonteri , Sikora & Lasota ’ s study on the spin evolution of massive black holes , we assume the radiative efficiency to be dependent on black hole mass , i.e. , \eta _ { rad } is low for M _ { bh } < 10 ^ { 8 } { M } _ { \odot } and it increases with black hole mass for M _ { bh } \geq 10 ^ { 8 } { M } _ { \odot } .
We find that the BHMF of AGN relics can roughly reproduce the local BHMF of galaxies if \eta _ { rad } \simeq 0.08 for M _ { bh } < 10 ^ { 8 } { M } _ { \odot } and it increases to \ga 0.18 for M _ { bh } \ga 10 ^ { 9 } { M } _ { \odot } , which implies that most massive black holes ( \ga 10 ^ { 9 } { M } _ { \odot } ) are spinning very rapidly .