Constraining supermassive black hole evolution through the continuity equation

The population of supermassive black holes (SMBHs) is split between those that are quiescent, such as those seen in local galaxies including the Milky Way, and those that are active, resulting in quasars and active galactic nuclei (AGN). Outside our neighborhood, 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 our model on the basis of their respective Eddington ratio distributions, 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 from z = 4 onwards. 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 direct measurement of the total black-hole mass function at redshifts z ≳ 2 could break these degeneracies, offering 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 redshifts, models with lower radiative efficiencies better reproduce the high-mass end of the mass function at high z, but tend to over-predict it at low z, and vice-versa for models with higher radiative efficiencies.