The population of supermassive black holes (SMBHs) is composed by quiescent SMBHs, such as those seen in local galaxies including the Milky Ways, and active ones, resulting in quasars and active galactic nuclei (AGN). Outside our neighbourhood, all the information we have on SMBHs is derived from quasars and AGN, giving us a partial view. We study the evolution of the SMBH population, total and active, by the continuity equation, backwards in time from z=0 to z=4. Type-1 and type-2 AGN are differentiated in the model on the basis of the Eddington ratio distribution, chosen on the basis of observational estimates. The duty cycle is obtained by matching the luminosity function of quasars, and the average radiative efficiency is the only free parameter in the model. For higher radiative efficiencies (>~0.07) a large fraction of the SMBH population, most of them quiescent, must already be in place by z=4. For lower radiative efficiencies (~0.05), the duty cycle increases with the redshift and the SMBH population evolves dramatically since z=4. The mass function of active SMBHs does not depend on the choice of the radiative efficiency or of the local SMBH mass function, but it is mainly determined by the quasar luminosity function, once the Eddington ratio distribution is fixed. Only a direct measurement of the total BHMF at redshifts z>~2 could break these degeneracies giving important constraints on the average radiative efficiency. Focusing on type-1 AGN, for which observational estimates of the mass function and Eddington ratio distribution exist at various redshift, models with lower radiative efficiencies reproduce better the high-mass end of the mass function at high-z, but they tend to over-predict it at low-z, and vice-versa for models with higher radiative efficiencies.
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