 Solution

The Soltan argument compares the time-integrated luminosity density of the cosmological AGN population with the present comoving mass density in dormant supermassive black holes. If $U_{\mathrm{AGN}}$ is the emitted energy density corrected for obscuration and bolometric output, accretion with population-averaged efficiency $\bar{\eta }$ predictsIn practice $U_{\mathrm{AGN}}$ comes from integrating AGN luminosity functions over luminosity and cosmological redshift, with corrections for obscured sources and missed wavebands, while ${\rho }_{\mathrm{BH}}$ is inferred from local galaxy--black-hole scaling relations.

The inferred efficiency is of order the canonical thin-disk value, about ten per cent, so most cosmic black-hole mass was accumulated in radiatively efficient, optically thick accretion episodes. Radiatively inefficient flows can dominate low-luminosity activity or brief extreme phases, and mergers redistribute existing mass, but neither naturally accounts for the observed integrated AGN radiation while supplying most of the final mass.