For radiative efficiency $\epsilon$, luminosity and black-hole growth rate are related by
$$
L=\epsilon\dot M_{\rm in}c^2,
\qquad
\dot M_{\rm BH}=(1-\epsilon)\dot M_{\rm in}.
$$
If the absolute accretion rate is constant, the lifetime needed to reach $M_f=5\times10^8M_\odot$ is
$$
\boxed{t_Q=\frac{M_f-M_{\rm seed}}{\dot M_{\rm BH}}
=\frac{\epsilon c^2(M_f-M_{\rm seed})}{(1-\epsilon)L}.}
$$
Expressing the constant luminosity as $L=fL_{\rm Edd}(M_f)$ gives, when $M_{\rm seed}\ll M_f$,
$$
t_Q\simeq\frac{\epsilon}{1-\epsilon}\frac{t_{\rm Edd}}f
\simeq\frac{0.05\ {\rm Gyr}}f.
$$
Sub-Eddington feeding can therefore last a substantial fraction of the roughly $1\ {\rm Gyr}$ available by $z=6$, while super-Eddington feeding can assemble the hole much faster than $50\ {\rm Myr}$. Because an arbitrary constant rate can be traded against its starting time and <black-hole seed> mass, the final mass alone provides essentially no separate constraint on either seed mass or seeding redshift.

For continuously Eddington-limited growth the luminosity rises with the mass, so growth is exponential rather than linear:
$$
M(t)=M_{\rm seed}e^{t/t_{\rm Sal}},
\qquad
t_{\rm Sal}=\frac\epsilon{1-\epsilon}t_{\rm Edd}\simeq0.05\ {\rm Gyr}.
$$
Consequently
$$
\boxed{t_Q=t_{\rm Sal}\log\frac{M_f}{M_{\rm seed}}.}
$$
For seed masses from roughly $1M_\odot$ to $10^6M_\odot$, the supplied logarithms give about $20$ to $6.2$ e-foldings, or approximately $0.9$ to $0.3\ {\rm Gyr}$ if the common $t_{\rm Sal}\simeq0.045\ {\rm Gyr}$ convention neglects the small $(1-\epsilon)$ correction. Unlike constant absolute-rate growth, Eddington-limited growth therefore spends many e-folding times at low mass and strongly links viable seed mass to seeding epoch.

A sufficiently deep high-redshift quasar survey measures the numerous faint progenitors that precede the rare luminous objects. Its luminosity function and duty cycle can distinguish long exponential growth from short rapid episodes and constrain the distribution of seed masses, formation redshifts, Eddington ratios, and obscured growth.

Two useful host-mass arguments are the empirical <black-hole--bulge mass relation> and an independent dynamical or halo estimate. Applying the low-redshift ratio $M_{\rm BH}/M_{\rm bulge}\sim10^{-3}$--$10^{-2}$ suggests $M_*\sim5\times10^{10}$--$5\times10^{11}M_\odot$. Spatially resolved line widths and sizes give $M_{\rm dyn}\sim Rv^2/G$, while the abundance of such rare quasars and the cosmic baryon fraction constrain their halo's available stellar mass. The existence of these systems before the Universe is one gigayear old implies exceptionally rapid assembly of both stars and black holes; departures from the local mass relation are therefore plausible and scientifically informative.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 347 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-347/1/b/solution