= Solution
The <Soltan argument> compares the time-integrated luminosity density of the cosmological <active galactic nucleus>[AGN] population with the present comoving mass density in dormant <supermassive black hole>[supermassive black holes]. If $U_{\rm AGN}$ is the emitted energy density corrected for obscuration and bolometric output, accretion with population-averaged efficiency $\bar\eta$ predicts
$$
\rho_{\rm BH}c^2
=\frac{1-\bar\eta}{\bar\eta}U_{\rm AGN},
\qquad
\bar\eta=\frac{U_{\rm AGN}}{U_{\rm AGN}+\rho_{\rm BH}c^2}.
$$
In practice $U_{\rm AGN}$ comes from integrating AGN <luminosity function>[luminosity functions] over luminosity and <cosmological redshift>, with corrections for obscured sources and missed wavebands, while $\rho_{\rm BH}$ 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 accretion flow>[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.
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