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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 347 / 3 / c / Solution

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The Soltan argument compares the cosmological energy density radiated by active galactic nuclei with the present mass density in supermassive black holes. Their ratio constrains the mass-weighted mean radiative efficiency of black-hole growth.


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