IS THERE A MAXIMUM MASS FOR BLACK HOLES IN GALACTIC NUCLEI?

We here consider a model for a star-forming accretion disk around an SMBH with ${M}_{{\rm{BH}}}\sim {10}^{8}-{10}^{11}\,{\text{}}{M}_{\odot }$ based on Thompson et al. (2005, hereafter TQM05). In this model, the gas fueling rate from galactic scales (≳100 pc) to the nuclear region (≲1 pc) is estimated self-consistently, including gas depletion due to star formation. Because of star formation, the central BH is fed at a rate of $\lt {\dot{M}}_{{\rm{Edd}}}$, and thus the BH growth is limited. This is consistent with most observed AGNs/QSOs, whose Eddington ratios are inferred to be modest (e.g., $L/{L}_{{\rm{Edd}}}\sim 0.16$ for $0.35\lt z\lt 2.25$ and $L/{L}_{{\rm{Edd}}}\sim 0.43$ for z > 4; De Rosa et al. 2011; Shen et al. 2011). A few exceptionally bright QSOs at higher redshift are believed to accrete more rapidly, at or even somewhat above $\sim {\dot{M}}_{{\rm{Edd}}}$. In the high-rate case, fragmentation of a nuclear disk suppresses the BH feeding (see discussion in Section 4 and King 2016).

The TQM05 model assumes that radiation pressure from stars forming in the disk supports the gas against gravity in the vertical direction and keeps the disk marginally stable; the Toomre parameter is then

$$\begin{eqnarray}&&Q\simeq \displaystyle \frac{{c}_{{\rm{s}}}{\rm{\Omega }}}{\pi G{{\rm{\Sigma }}}_{{\rm{g}}}}\simeq 1,\end{eqnarray} \tag{ 1 }$$

where c_s is the sound speed, ${{\rm{\Sigma }}}_{{\rm{g}}}$ is the gas surface density, and Ω is the orbital frequency, given by

$$\begin{eqnarray}&&{\rm{\Omega }}={\left(\displaystyle \frac{{\mathrm{GM}}_{{\rm{BH}}}}{{r}^{3}}+\displaystyle \frac{2{\sigma }^{2}}{{r}^{2}}\right)}^{1/2}.\end{eqnarray} \tag{ 2 }$$

Here σ is the velocity dispersion characterizing the gravitational potential on galactic scales. From the continuity equation, the surface density is given by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{g}}}=\displaystyle \frac{\dot{M}}{2\pi {{rv}}_{{\rm{r}}}}=\displaystyle \frac{\dot{M}}{2\pi {{rmc}}_{{\rm{s}}}},\end{eqnarray} \tag{ 3 }$$

where $\dot{M}$ is the gas accretion rate through a radius of r, v_r is the radial velocity, and $m\,(={v}_{{\rm{r}}}/{c}_{{\rm{s}}})$ is the radial Mach number. Note that the viscosity in this model is specified by assuming a constant value of m (see below), instead of the α-prescription (Shakura & Sunyaev 1973).

The disk is supported vertically by both thermal gas pressure (${p}_{{\rm{gas}}}=\rho {k}_{{\rm{B}}}T/{m}_{{\rm{p}}}$) and radiation pressure due to stars in the disk, where $\rho ={{\rm{\Sigma }}}_{{\rm{g}}}/(2h)$ is the gas density, $h={c}_{{\rm{s}}}/{\rm{\Omega }}$ is the pressure scale height, and T is the gas temperature. The radiation pressure is given by

$$\begin{eqnarray}&&{p}_{{\rm{rad}}}=\epsilon {\dot{{\rm{\Sigma }}}}_{* }c\left(\displaystyle \frac{\tau }{2}+\xi \right),\end{eqnarray} \tag{ 4 }$$

where $\tau =\kappa {{\rm{\Sigma }}}_{{\rm{g}}}/2$ is the optical depth, κ is the dust opacity (Semenov et al. 2003), ${\dot{{\rm{\Sigma }}}}_{* }$ is the star formation rate per unit disk surface area, and is the matter–radiation conversion efficiency, which depends on the mass function of stars. The first term on the right-hand side of Equation (4) is the radiation pressure on dust grains in the optically thick limit ($\tau \gg 1$), and the second term represents stellar UV radiation pressure and turbulent support by supernovae in the optically thin limit ($\tau \ll 1$), which is characterized by the nondimensional value of ξ. Energy balance between cooling and heating is given by

$$\begin{eqnarray}&&{\sigma }_{{\rm{SB}}}{T}_{{\rm{eff}}}^{4}=\displaystyle \frac{1}{2}\epsilon {\dot{{\rm{\Sigma }}}}_{* }{c}^{2}+\displaystyle \frac{3}{8\pi }\dot{M}{{\rm{\Omega }}}^{2},\end{eqnarray} \tag{ 5 }$$

where the effective temperature ${T}_{{\rm{eff}}}$ is given by

$$\begin{eqnarray}&&{T}^{4}=\displaystyle \frac{3}{4}{T}_{{\rm{eff}}}^{4}\left(\tau +\displaystyle \frac{2}{3\tau }+\displaystyle \frac{4}{3}\right).\end{eqnarray} \tag{ 6 }$$

In this disk, a fraction of gas forms stars at a rate of ${\dot{{\rm{\Sigma }}}}_{* }(r)$ and the gas accretion rate decreases inward, given by

$$\begin{eqnarray}&&\dot{M}(r)={\dot{M}}_{{\rm{out}}}-{\int }_{{R}_{{\rm{out}}}}^{r}2\pi r{\dot{{\rm{\Sigma }}}}_{* }{dr}.\end{eqnarray} \tag{ 7 }$$

Equations (1)–(7) determine the radial profiles of all physical quantities, once the five parameters ${M}_{{\rm{BH}}}$, σ, m, , and ξ and the outer boundary conditions of the accretion rate ${\dot{M}}_{{\rm{out}}}$ at the radius ${R}_{{\rm{out}}}$ are chosen. As our fiducial model, we set $\epsilon ={10}^{-3}$, appropriate for a Salpeter initial mass function (IMF) with $1\mbox{--}100\,{\text{}}{M}_{\odot }$, and $\xi =1$, appropriate when turbulent support by supernovae is negligible, as in a high-$\dot{M}$ (or ρ) disk. The velocity dispersion is set to $\sigma =400\,{\text{km s}}^{-1}$, motivated by the empirical correlation between BH mass and σ of its host galaxy for ${M}_{{\rm{BH}}}\sim {10}^{10}\,{\text{}}{M}_{\odot }$ (e.g., Tremaine et al. 2002; Gültekin et al. 2009). Note that the dependence of our results on the choice of σ is very weak because the stellar gravitational potential is subdominant at r ≲ GM_BH/(2σ²) ≃ 140 pc (M_BH/10¹⁰ M_⊙)(σ/400 km s⁻¹)⁻², where the BH feeding rate is determined.

We consider a very high accretion rate of ${\dot{M}}_{{\rm{out}}}\,={10}^{3}\,{M}_{\odot }\,{{\rm{yr}}}^{-1}$ at the boundary, in order to give a conservative estimate on the maximum BH mass. From cosmological simulations (e.g., Genel et al. 2009; Fakhouri et al. 2010), the maximum gas accretion rate onto a dark matter halo is estimated as $\lesssim {10}^{3}\,{M}_{\odot }\,{{\rm{yr}}}^{-1}\,{({M}_{{\rm{halo}}}/{10}^{12}{\text{}}{M}_{\odot })}^{1.1}{[(1+z)/7]}^{5/2}$, where ${M}_{{\rm{h}}}$ is the halo mass. This could be exceeded only for brief periods during major merger events in the early universe (Mayer et al. 2015), and for ${M}_{{\rm{h}}}\gtrsim {10}^{12}\,{\text{}}{M}_{\odot }$, gas heating by a virial shock prevents cold gas supply because of inefficient radiative cooling (Birnboim & Dekel 2003; Dekel & Birnboim 2006).

The nature of the angular momentum transfer, allowing gas to flow from large scales to the inner subparsec regions, remains uncertain. Following TQM05, we assume that the transfer is provided by global density waves and also that the radial Mach number m is constant (independent of radius). Since our results depend on the choice of m, we here consider three cases of m = 0.05, 0.1, and 0.2. These values are motivated by analytical arguments yielding the limit $m\lesssim 0.2$ (Goodman 2003). Note that in terms of the standard α-prescription as a model for the disk viscosity (Shakura & Sunyaev 1973), the viscous parameter is related to the Mach number as $\alpha =m(2r/3h)$.

Figure 1 shows radial profiles of the gas accretion rate (solid) and star formation rate (dashed) for three different BH masses and for m = 0.1. For the lowest BH mass (${M}_{{\rm{BH}}}={10}^{9}\,{\text{}}{M}_{\odot };$ in red), star formation is inefficient at $r\gtrsim 10$ pc, and the accretion rate remains close to its value at the outer boundary. At $r\lesssim 10$ pc, vigorous star formation depletes most of the gas, and the accretion rate rapidly decreases inward. In this domain, the disk temperature reaches the dust sublimation temperature (${T}_{{\rm{dust,sub}}}\simeq {10}^{3}$ K), above which the dust opacity drops rapidly. Since ${\dot{{\rm{\Sigma }}}}_{* }\propto {{\rm{\Sigma }}}_{{\rm{g}}}/\kappa$ in the optically thick limit (see Equation (15) in the Appendix), a higher star formation rate is required to maintain the marginally stable disk structure with $Q\simeq 1$ when the opacity decreases. Note that since dust is composed of multiple species and each has a different sublimation temperature (Semenov et al. 2003), the jagged radial profile for the star formation rate at $r\sim 10\mbox{--}100$ pc is caused by small drops of dust opacity at the corresponding sublimation temperature. Within $r\lesssim 0.5$ pc, the disk becomes stable, star formation ceases, and the gas accretion rate approaches a constant value. This accretion rate in the nuclear region does not depend on the value of ${\dot{M}}_{{\rm{out}}}$, as long as ${\dot{M}}_{{\rm{out}}}\gt {\dot{M}}_{{\rm{crit}}}\simeq 280\,{M}_{\odot }\,{{\rm{yr}}}^{-1}$ (see Equation (25) in the Appendix), and thus hardly on the model parameters of the star-forming disk except the radial Mach number m (see Section 2.2). For higher SMBH masses, the accretion rate in the nuclear region gradually increases. However, for ${M}_{{\rm{BH}}}\gtrsim 6\times {10}^{10}\,{\text{}}{M}_{\odot }$, the accretion rate at $\lt 1$ pc decreases again, because gas is depleted more efficiently due to star formation at larger radii ($r\sim 100$ pc), where evaporation of volatile organics decreases the dust opacity moderately ($T\gtrsim 400$ K) and the star formation rate increases in the optically thick starburst disk (${\dot{{\rm{\Sigma }}}}_{* }\propto {{\rm{\Sigma }}}_{{\rm{g}}}/\kappa$).

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We next consider the stable nuclear (subparsec) region of the accretion disk, which is embedded by the galactic star-forming disk. The properties of this disk are determined primarily by the BH mass and the gas accretion rate from larger scales ($\gtrsim 1$ pc). Figure 2 shows the accretion rate into the nuclear region for three different Mach numbers m = 0.05 (blue), 0.1 (red), and 0.2 (black). Up to a turnover at a critical ${M}_{{\rm{BH}}}$, the accretion rates in panels (a) and (b) are fit by the single power laws

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{BH}}}\simeq 4.2\,{m}_{0.1}{M}_{{\rm{BH,10}}}^{0.3}\,{M}_{\odot }\,{{\rm{yr}}}^{-1},\end{eqnarray} \tag{ 8 }$$

or

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{M}}_{{\rm{BH}}}}{{\dot{M}}_{{\rm{Edd}}}}\simeq 1.8\times {10}^{-2}\,{m}_{0.1}{M}_{{\rm{BH,10}}}^{-0.7},\end{eqnarray} \tag{ 9 }$$

where ${m}_{0.1}\equiv m/0.1$ and ${M}_{{\rm{BH,10}}}\equiv {M}_{{\rm{BH}}}/({10}^{10}\,{\text{}}{M}_{\odot })$. These scaling relations can be explained by an analytical argument in the Appendix. Assuming a constant radiative efficiency η, we can integrate Equation (8) over the age of the universe and obtain ${M}_{{\rm{BH,10}}}\simeq 7.4\,{m}_{0.1}^{10/7}{(1-\eta )}^{10/7}$. This suggests that SMBHs would not grow above $\sim {10}^{11}\,{\text{}}{M}_{\odot }$ within the finite age of the universe.

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The above argument yields a maximum BH mass, which comes close to the largest observed masses. Here we discuss further constraints on the maximum value, considering properties of accretion flows in the vicinity of the BH. In panel (b), the normalized rate for ${M}_{{\rm{BH}}}\simeq {10}^{9}\,{\text{}}{M}_{\odot }$ is ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}\sim 0.1$. For this value, a standard geometrically thin, optically thick nuclear disk can form (Shakura & Sunyaev 1973). Through the disk, the BH grows via accretion at a rate given by Equation (8). On the other hand, the normalized rate decreases with BH mass and reaches the critical value of ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}\lesssim {10}^{-2}$, at which the nuclear disk cannot remain thin, because of inefficient radiative cooling (Narayan & McClintock 2008). The inner disk would then likely be replaced by a radiatively inefficient ADAF (advection-dominated accretion flow; Narayan & Yi 1994, 1995).³ Adopting the critical rate to be ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}={10}^{-2}$, we find that the transition occurs at ${M}_{{\rm{BH}}}\gtrsim {M}_{{\rm{tr}}}=2.3\times {10}^{10}{m}_{0.1}^{10/7}\,{\text{}}{M}_{\odot }$ for $0.05\lesssim m\lesssim 0.2$. We note that the transition BH mass ${M}_{{\rm{tr}}}$ does not depend on the model parameters of the star-forming disk, except on the Mach number m, while the behavior of the accretion rate at ${M}_{{\rm{BH}}}\gt {M}_{{\rm{tr}}}$ depends on the choices of the other model parameters.

Once the transition to an ADAF occurs, the accretion flow through the disk becomes hot because of inefficient cooling. The hot gas near the BH would launch strong outflows and jets, suppressing the feeding of the BH (e.g., Blandford & Begelman 1999). The location of the transition radius ${R}_{{\rm{tr}}}$, inside which a thin disk turns into a hot ADAF, has been discussed by several authors (e.g., Yuan & Narayan 2014, and references therein). Although this location is uncertain, the theoretical models suggest ${R}_{{\rm{tr}}}/{R}_{{\rm{Sch}}}\gtrsim 300\mbox{--}{10}^{3}\,{[{\dot{M}}_{{\rm{BH}}}/({10}^{-2}{\dot{M}}_{{\rm{Edd}}})]}^{-q}$ ($q\gt 0$) for ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}\lesssim {10}^{-2}$. This value is consistent with results obtained from fitting the spectra of observed BH accretion systems (${R}_{{\rm{tr}}}\sim 100\,{R}_{{\rm{Sch}}}$ for ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}\sim {10}^{-2};$ Yuan & Narayan 2004). Moreover, numerical simulations of ADAFs suggest that the gas accretion rate decreases inward within the transition radius as ${\dot{M}}_{{\rm{BH}}}\propto {(r/{R}_{{\rm{tr}}})}^{s}$ (Igumenshchev & Abramowicz 1999; Stone et al. 1999; Hawley & Balbus 2002; McKinney & Gammie 2004; see also Blandford & Begelman 1999). The power-law index is estimated as $0.4\lesssim s\lesssim 0.6$ at $2\lesssim r/{R}_{{\rm{Sch}}}\lesssim {10}^{4}$, independent of the strength of viscosity and magnetic field (Yuan et al. 2012). For a conservative estimate, we set s = 0.3 and ${R}_{{\rm{tr}}}=100\,{R}_{{\rm{Sch}}}$ for ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}\,\leqslant {10}^{-2}$. This reduces the BH feeding rate by a factor of ${({R}_{{\rm{Sch}}}/{R}_{{\rm{tr}}})}^{0.3}\simeq 0.25$ from the original accretion rate at ${R}_{{\rm{tr}}}$ (Equation (8)). As a result, the BH growth time is roughly given by $\sim 16\,{[{\dot{M}}_{{\rm{BH}}}/({10}^{-2}{\dot{M}}_{{\rm{Edd}}})]}^{-1}$ Gyr, and we conclude that once an SMBH reaches the critical mass of ${M}_{{\rm{BH,max}}}\simeq {M}_{{\rm{tr}}}\,\simeq (0.9\mbox{--}6.2)\times {10}^{10}\,{\text{}}{M}_{\odot }$, it cannot gain significant mass within the Hubble time.

Among observed SMBHs, the brightest QSOs with $\gtrsim 2\times {10}^{47}$ erg s⁻¹, which are inferred to have Eddington ratios near unity ($L\sim {L}_{{\rm{Edd}}}$), would grow at a rate of $\sim {\dot{M}}_{{\rm{Edd}}}$. In the TQM05 model we adopted, this would require a high radial Mach number ($m\gtrsim 1$). However, such a large m is unlikely to be realized by global spiral waves in a marginally stable disk $Q\simeq 1$. Instead, this rapid inflow could be triggered by a major galaxy merger and sustained for a few dynamical timescales of a few $\times {10}^{7}$ yr (Hopkins & Quataert 2010, 2011). After a brief burst phase, the BH feeding rate would decrease to the value given by Equation (8). As long as these major-merger-triggered inflows are sufficiently rare and brief, the SMBH masses will remain limited by the physics of the star-forming disks, as discussed in Section 2.2.

We next argue that BH growth at ${M}_{{\rm{BH}}}\gtrsim {10}^{10}\,{\text{}}{M}_{\odot }$ would also be suppressed by fragmentation of the nuclear disk, even at the higher accretion rates of $\sim {\dot{M}}_{{\rm{Edd}}}$. In this case, the disk becomes cold and thin instead of a hot ADAF. Such a thin disk is better described by the standard α-viscosity prescription (Shakura & Sunyaev 1973). The α-disk becomes self-gravitating and unstable at large radii, where $Q\lesssim 1$,

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{{\rm{sg}}}}{{R}_{{\rm{Sch}}}}=\left\{\begin{array}{l}8.1\,{\alpha }_{0.1}^{14/27}{\dot{m}}_{{\rm{BH}}}^{-8/27}{M}_{{\rm{BH,10}}}^{-26/27}\quad ({p}_{{\rm{gas}}}\gt {p}_{{\rm{rad}}}),\\ \\ 85.3\,{\alpha }_{0.1}^{1/3}{\dot{m}}_{{\rm{BH}}}^{1/6}{M}_{{\rm{BH,10}}}^{-1/2}\quad ({p}_{{\rm{rad}}}\gt {p}_{{\rm{gas}}}),\end{array}\right.\end{eqnarray} \tag{ 10 }$$

where ${\alpha }_{0.1}\equiv \alpha /0.1$ and ${\dot{m}}_{{\rm{BH}}}\equiv {\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}$ (Goodman & Tan 2004). The top (bottom) expression is valid when gas (radiation) pressure dominates. Figure 4 shows the fragmentation radius ${R}_{{\rm{sg}}}$ as a function of the BH mass (solid curve). The filled circle marks the location where ${p}_{{\rm{gas}}}={p}_{{\rm{rad}}}$, inside of which radiation pressure dominates (${M}_{{\rm{BH}}}\gtrsim 4\times {10}^{7}\,{\text{}}{M}_{\odot }$).

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Gas clumps formed in the unstable region ($r\gtrsim {R}_{{\rm{sg}}}$) subsequently grow via gas accretion from the ambient disk, and the gas near the clump within $\sim {f}_{{\rm{H}}}{R}_{{\rm{H}}}$ is depleted, where ${R}_{{\rm{H}}}$ is the clump's Hill radius and ${f}_{{\rm{H}}}\sim O(1)$. Assuming that the clump grows until a density gap is created, the mass reaches a substantial fraction of the isolation mass (Goodman & Tan 2004)

$$\begin{eqnarray}&&{M}_{{\rm{c,iso}}}\simeq 1.3\times {10}^{9}\,{\alpha }_{0.1}^{-1/2}{\dot{m}}_{{\rm{BH}}}^{5/4}{M}_{{\rm{BH,10}}}^{7/4}\,{f}_{{\rm{H}}}^{3/2}{\text{}}{M}_{\odot },\end{eqnarray} \tag{ 11 }$$

where the clump location is set to $r={R}_{{\rm{sg}}}$. The width of the gap is estimated as ${R}_{{\rm{gap}}}\approx {f}_{{\rm{H}}}{R}_{{\rm{H}}}\approx {f}_{{\rm{H}}}{R}_{{\rm{sg}}}{({M}_{{\rm{c,iso}}}/3{M}_{{\rm{BH}}})}^{1/3}$, and thus

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{{\rm{gap}}}}{{R}_{{\rm{sg}}}}\approx 0.36\,{f}_{{\rm{H}}}^{3/2}{\alpha }_{0.1}^{-1/6}{{\dot{m}}_{{\rm{BH}}}}^{5/12}{M}_{{\rm{BH,10}}}^{1/4}.\end{eqnarray} \tag{ 12 }$$

Figure 4 shows the value of (${R}_{{\rm{sg}}}-{R}_{{\rm{gap}}}$) for ${f}_{{\rm{H}}}=1.5$ (dashed blue curve). A stable disk can exist only below this line, down to the innermost stable circular orbit (ISCO), ${R}_{{\rm{ISCO}}}\simeq 3{R}_{{\rm{Sch}}}$. The size of the stable region shrinks with increasing BH mass and disappears entirely at ${M}_{{\rm{BH}}}=4.5\times {10}^{10}\,{\text{}}{M}_{\odot }$ (i.e., ${R}_{{\rm{sg}}}-{R}_{{\rm{gap}}}\approx {R}_{{\rm{ISCO}}}$). Subsequently, the BH could not be fed via a stable disk. Instead, the BH could be fed stars from a nuclear star cluster, forming by the gravitational collapse of a massive clump at ${R}_{{\rm{sg}}}$ with ${M}_{{\rm{c,iso}}}$. The stellar feeding occurs on the timescale of $\simeq {t}_{{\rm{relax}}}\mathrm{ln}(2/{\theta }_{{\rm{lc}}})$ (e.g., Frank & Rees 1976; Syer & Ulmer 1999), where ${t}_{{\rm{relax}}}$ is the (two-body) relaxation timescale, estimated as

$$\begin{eqnarray}&&{t}_{{\rm{relax}}}\simeq \displaystyle \frac{0.34\,{\sigma }_{* }^{3}}{{G}^{2}{M}_{* 2}{\rho }_{* }\mathrm{ln}({M}_{{\rm{BH}}}/{M}_{* })}\simeq 6\,{\rm{Gyr}}\end{eqnarray} \tag{ 13 }$$

(Binney & Tremaine 2008; Kocsis & Tremaine 2011), where ${\sigma }_{* }={[G({M}_{{\rm{BH}}}+{M}_{{\rm{c,iso}}})/{R}_{{\rm{sg}}}]}^{1/2}$ is the stellar velocity dispersion, ${M}_{* 2}$ is the ratio of the mean-square stellar mass to the mean stellar mass of the stars, and ${\rho }_{* }=3{M}_{{\rm{c,iso}}}/(4\pi {R}_{{\rm{sg}}}^{3})$ is the stellar density of the cluster. Assuming the Salpeter IMF with ${M}_{{\rm{min(max)}}}=1\,(100)\,{\text{}}{M}_{\odot }$, we obtain ${M}_{* 2}\simeq 11\,{\text{}}{M}_{\odot }$. Since the angular size of the loss cone is estimated as ${\theta }_{{\rm{lc}}}=\sqrt{2{R}_{{\rm{Sch}}}{{GM}}_{{\rm{BH}}}}/({\sigma }_{* }{R}_{{\rm{sg}}})\sim 0.19$ and $\mathrm{ln}(2/{\theta }_{{\rm{lc}}})\sim 2.4$, the stellar feeding time for ${M}_{{\rm{BH}}}\gt 4.5\times {10}^{10}\,{\text{}}{M}_{\odot }$ exceeds the age of the universe (at z = 0). Therefore, we expect disk fragmentation to suppress BH growth above this mass (placing the corresponding upper limit of $L\simeq {L}_{{\rm{Edd}}}\simeq 6\times {10}^{48}$ erg s⁻¹ on the luminosity).

We note that King (2016) recently proposed the existence of an upper limit on the masses of SMBHs, due to fragmentation of the nuclear disk. King (2016) suggests that the maximum mass is the one for which the fragmentation radius is located at the ISCO. This is very similar to our discussion of the case of the ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}(\simeq 1)$ α-disk above. The main difference is that King (2016) adopts a gas-pressure-dominated disk, though the radiation pressure in fact dominates at ${R}_{{\rm{sg}}}$ for ${M}_{{\rm{BH}}}\gtrsim 4\times {10}^{7}\,{\text{}}{M}_{\odot }$ (below the dot-dashed line in Figure 4). King (2016) argues that a large radiation-pressure-dominated disk extending all the way out to ${R}_{{\rm{sg}}}$ would be thermally unstable and cannot form at all. Then, the BH mass limit is estimated as $\simeq 3\times {10}^{10}\,{\text{}}{M}_{\odot }$ from ${R}_{{\rm{sg}}}\simeq {R}_{{\rm{ISCO}}}$ assuming ${p}_{{\rm{gas}}}\gt {p}_{{\rm{rad}}}$. As the implications of this instability are not yet understood, we here conservatively assumed that a radiation-pressure-dominated disk could still feed the central BH, as long as it is gravitationally stable. This, in principle, would greatly increase the fragmentation radius (see red solid curve in Figure 4). However, we argued that the large physical size of the clumps in this case prevents a stable disk from forming all the way down to smaller radii, comparable to the ${R}_{{\rm{sg}}}$ in the fiducial gas-dominated case (see dashed blue curve in Figure 4). As a result, our main conclusion agrees with that of King (2016).

In the star-forming disk model, a high accretion rate is required to feed the central BH (Sections 2 and 3). This fact means that a large number of stars would form around the SMBHs. We briefly discuss the stellar mass of massive galaxies hosting the most massive BHs with $\sim {10}^{10}\,{\text{}}{M}_{\odot }$.

Figure 5 shows radial profiles of the gas accretion rate and star formation rate for the two different values of ${\dot{M}}_{{\rm{out}}}=400$ and ${10}^{3}\,{M}_{\odot }\,{{\rm{yr}}}^{-1}$ (for ${M}_{{\rm{BH}}}={10}^{10}\,{\text{}}{M}_{\odot }$ and m = 0.1). We note that ${\dot{M}}_{{\rm{out}}}=400\,{M}_{\odot }\,{{\rm{yr}}}^{-1}\,(\gt {\dot{M}}_{{\rm{crit}}})$ is sufficient to maintain the universal feeding rate in Equation (8). In the case with ${\dot{M}}_{{\rm{out}}}=400\,{M}_{\odot }\,{{\rm{yr}}}^{-1}$, a population of stars with total mass $\sim {10}^{12}\,{\text{}}{M}_{\odot }$ forms within ∼200 pc in the mass doubling time of the ${10}^{10}\,{\text{}}{M}_{\odot }$ BH (∼2.5 Gyr). Note that such a compact star-forming region is consistent with observed ultraluminous infrared galaxies, where stars form in a few times 10² pc nuclear disk at a rate up to several $100\,{M}_{\odot }\,{{\rm{yr}}}^{-1}$ (Medling et al. 2014).

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The most massive elliptical galaxies are as old as $\gtrsim 8$ Gyr (e.g., Bernardi et al. 2003; Thomas et al. 2005). Thus, we can observe stars with masses of $\lt 1.1\,{\text{}}{M}_{\odot }$, whose lifetimes are longer than the age of the galaxies (at least 8 Gyr). Although the IMF of stars around the most massive BHs is highly uncertain, many authors have discussed the possibility that stars forming in SMBH disks, including those observed in the Galactic center, have a top-heavy IMF (e.g., Paumard et al. 2006; Levin 2007; Nayakshin et al. 2007; see also Goodman & Tan 2004). Assuming the Salpeter IMF with ${M}_{{\rm{min(max)}}}=1\,(100)\,{\text{}}{M}_{\odot }$, 4% of the stars in mass live in longer lifetimes of $\gt 8\,{\rm{Gyr}}$ and can be observed in the most massive elliptical galaxies. Therefore, we can estimate the stellar mass surface density as $\sim 3\times {10}^{11}\,{\text{}}{M}_{\odot }\,{{\rm{kpc}}}^{-2}$, which is consistent with a maximum value of dense stellar systems within a factor of three (Hopkins et al. 2010; see also Lauer et al. 2007).

We briefly mention rapid growth of intermediate-mass BHs. According to Equation (9), the BH feeding rate in the Eddington units ${\dot{M}}_{{\rm{BH}}}/{\dot{M}}_{{\rm{Edd}}}(\propto {M}_{{\rm{BH}}}^{-0.7})$ exceeds unity at ${M}_{{\rm{BH}}}\lesssim 3.2\times {10}^{7}\,{m}_{0.1}^{10/7}\,{\text{}}{M}_{\odot }$. In the regime of ${\dot{M}}_{{\rm{BH}}}\gtrsim {\dot{M}}_{{\rm{Edd}}}$, the nuclear accretion disk transits to an optically thick ADAF solution, the so-called slim disk, where super-Eddington accretion would be possible (e.g., Abramowicz et al. 1988; Jiang et al. 2014; Sa̧dowski et al. 2015). However, radiation heating suppresses gas supply from larger scales, which results in a lower accretion rate $\sim {\dot{M}}_{{\rm{Edd}}}$ (e.g., Ciotti & Ostriker 2001; Novak et al. 2011; Park & Ricotti 2012). For intermediate-mass BHs with ${M}_{{\rm{BH}}}\lesssim {10}^{4}\,{\text{}}{M}_{\odot }$, the BH feeding rate becomes ${\dot{M}}_{{\rm{BH}}}\gtrsim 3000\,{m}_{0.1}{L}_{{\rm{Edd}}}/{c}^{2}$, where hyper-Eddington accretion could be realized unimpeded by radiation feedback, and the massive BHs would grow rapidly (Inayoshi et al. 2016; Ryu et al. 2016).