We quantify the importance of mass accretion during AGN phases in the growth of supermassive black holes ( \mathrm { BH } ) by comparing the mass function of black holes in the local universe with that expected from AGN relics , which are black holes grown entirely with mass accretion during AGN phases .
The local \mathrm { BH } mass function ( \mathrm { BHMF } ) is estimated by applying the well-known correlations between \mathrm { BH } mass , bulge luminosity and stellar velocity dispersion to galaxy luminosity and velocity functions .
We find that different correlations provide the same \mathrm { BHMF } only if they have the same intrinsic dispersion .
The density of supermassive black holes in the local universe which we estimate is \rho _ { \mathrm { BH } } = 4.6 _ { -1.4 } ^ { +1.9 } h _ { 0.7 } ^ { 2 } \times 10 ^ { 5 } ~ { } \mathrm { M } _ % { \odot } \mathrm { Mpc } ^ { -3 } .
The relic \mathrm { BHMF } is derived from the continuity equation with the only assumption that AGN activity is due to accretion onto massive \mathrm { BH } ’ s and that merging is not important .
We find that the relic \mathrm { BHMF } at z = 0 is generated mainly at z < 3 where the major part of \mathrm { BH } ’ s growth takes place .
Moreover , the \mathrm { BH } growth is anti-hierarchical in the sense that smaller \mathrm { BH } ’ s ( M _ { BH } < 10 ^ { 7 } M _ { \odot } ) grow at lower redshifts ( z < 1 ) with respect to more massive one ’ s ( z \sim 1 - 3 ) .
Unlike previous work , we find that the \mathrm { BHMF } of AGN relics is perfectly consistent with the local \mathrm { BHMF } indicating the local black holes were mainly grown during AGN activity .
This agreement is obtained while satisfying , at the same time , the constraints imposed from the X-ray background .
The comparison between the local and relic \mathrm { BHMF } ’ s also suggests that the merging process is not important in shaping the relic \mathrm { BHMF } , at least at low redshifts ( z < 3 ) , and allows us to estimate the average radiative efficiency ( \varepsilon ) , the ratio between emitted and Eddington luminosity ( \lambda ) and the average lifetime of active \mathrm { BH } ’ s .
Our analysis thus suggests the following scenario : local black holes grew during AGN phases in which accreting matter was converted into radiation with efficiencies \varepsilon = 0.04 - 0.16 and emitted at a fraction \lambda = 0.1 - 1.7 of the Eddington luminosity .
The average total lifetime of these active phases ranges from \simeq 4.5 \times 10 ^ { 8 } yr for M _ { BH } < 10 ^ { 8 } M _ { \odot } to \simeq 1.5 \times 10 ^ { 8 } yr for M _ { BH } > 10 ^ { 9 } M _ { \odot } but can become as large as \sim 10 ^ { 9 } \mathrm { yr } for the lowest acceptable \epsilon and \lambda values .