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ABSTRACT  We present a comprehensive synthesis model for the active galactic nuclei (AGN) evolution and the growth of supermassive black hole (SMBH) in the Universe. We assume that black holes accrete in just three distinct physical states, or ‘modes’: at low Eddington ratio, only a radiatively inefficient, kinetically dominated mode is allowed [low kinetic (LK)]; at high Eddington ratio, instead, AGN may display both a purely radiative [radio quiet, high radiative (HR)] and a kinetic [radio loud, high kinetic (HK)] mode. We solve the continuity equation for the black hole mass function using the locally determined one as a boundary condition, and the hard X‐ray luminosity function as tracer of the AGN growth rate distribution, supplemented with a luminosity‐dependent bolometric correction and an absorbing column distribution. Differently from most previous semi‐analytic and numerical models for black hole growth, we do not assume any specific distribution of Eddington ratios, rather we determine it empirically by coupling the mass and luminosity functions and a set of fundamental relations between observables in the three accretion modes. SMBH always show a very broad accretion rate distribution, and we discuss the profound consequences of this fact for our understanding of observed AGN fractions in galaxies, as well as for the empirical determination of SMBH mass functions with large surveys. We confirm previous results and clearly demonstrate that, at least for   z ≲ 1.5 , SMBH mass function evolves antihierarchically, i.e. the most massive holes grew earlier and faster than less massive ones. For the first time, we find hints of a reversal of such a downsizing behaviour at redshifts above the peak of the black hole accretion rate density  ( z ≈ 2) . We also derive tight constraints on the (mass‐weighted) average radiative efficiency of AGN: under the simplifying assumption that the mass density of both high redshift  ( z ∼ 5)  and ‘wandering’ black holes ejected from galactic nuclei after merger events are negligible compared to the local mass density, we find that  0.065 &lt; ξ 0 〈ε rad 〉 &lt; 0.07 , where ξ 0  is the local SMBH mass density in units of  4.3 × 10 5  M ⊙  Mpc −3  . We trace the cosmological evolution of the kinetic luminosity function of AGN, and find that the overall efficiency of SMBH in converting accreted rest mass energy into kinetic power,  ε kin  , ranges between  ε kin ≃ (3–5) × 10 −3  , depending on the choice of the radio core luminosity function. Such a ‘kinetic efficiency’ varies however strongly with SMBH mass and redshift, being maximal for very massive holes at late times, as required for the AGN feedback by many galaxy formation models in cosmological contexts.

A synthesis model for AGN evolution: supermassive black holes growth and feedback modes