Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution

Distinguish the rest-mass supply ${\dot{M}}_{\mathrm{in}}$ from the actual black-hole mass growth. With the usual radiative-efficiency definition,At the Eddington limit the growth is exponential:The available time is the difference of the supplied ages, $\Delta t=650\mathrm{Myr}$; no new cosmological age-redshift relation is needed.

Using the printed ${\sigma }_e=6.6\times 10^{-29}{\mathrm{m}}^2$, together with $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $m_p\simeq 1.67262\times 10^{-27}\mathrm{kg}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ and a Julian year of $31557600\mathrm{s}$, givesThereforeThis is about $13.09$ e-folds. The constants just listed are evaluation constants, not additional observational inputs asserted to be on the paper.

Sometimes an elementary growth convention identifies ${\dot{M}}_{\mathrm{BH}}$ directly with $L/(ϵc^2)$ and drops the radiated fraction of the incoming rest mass. That approximation instead gives $t_{\mathrm{Sal}}\simeq 44.69\mathrm{Myr}$ and $M_0\simeq 9.7\times 10^2{M}_{\odot }$. The first result retains the physically distinct supplied and retained masses; the convention must be stated because exponential growth magnifies this difference.

The main implication is already severe in the optimistic calculation: a low-mass stellar-remnant seed has insufficient time under the specified efficiency and strictly Eddington-limited accretion. A heavier seed, earlier formation, a different efficiency, mergers or episodes beyond that idealized luminosity limit can relax the inference. Any duty cycle below unity or sub-Eddington episodes require a larger initial mass within the adopted model. The reported high-redshift quasar motivating this calculation was identified spectroscopically at $z=7.085$; its published age estimate need not replace the ages supplied here.^[ref]