Super-Eddington Growth Ceiling: Analytic Constraints on the Rapid Growth of Light-Seed Black Holes in Massive Clumps

TL;DR

The study interrogates whether light-seed black holes can reach the scales needed for early supermassive black hole formation via recurrent clump encounters. By merging analytic constraints with toy-model simulations that incorporate Bondi inflow, radiative feedback, gas dynamical friction, and forward acceleration from ionised shells, the work identifies three concurrent requirements—mass doubling, clump lifetime, and BH trapping—that carve a narrow sweet spot in $n_{ m H}$–$T$ space where growth is possible. Even within this region, growth is limited (e.g., a $10^{3}\,M_\odot$ seed can reach only $\sim4\times10^{3}\,M_\odot$; the maximum growth factor decreases with $M_{ m BH}$ and becomes negligible above $\sim10^{4}\,M_\odot$), and the forward-acceleration effect generally prevents trapping unless photon trapping holds. Consequently, BH-clump capture alone is unlikely to bridge the gap to $>10^{5}$–$10^{6}\,M_\odot$ seeds, favoring alternative pathways such as heavy-seed formation for the rapid emergence of high-redshift SMBHs.

Abstract

The existence of $\sim10^{7-8}{\rm M}_\odot$ supermassive black holes at $z\gtrsim 8$ challenges conventional growth channels. One attractive possibility is that light seeds ($M_{\rm BH}\lesssim10^{3}{\rm M}_\odot$) undergo short, super-Eddington episodes when they cross, and are captured by, dense massive gas clumps. We revisit this ``BH-clump-capture'' model using analytic arguments supported by toy-model simulations that follow Bondi-scale inflow, radiative feedback, gas dynamical friction and the recently discovered forward acceleration effect caused by the ionised bubble. For substantial growth the black hole must remain trapped for many dynamical times, which imposes three simultaneous constraints. The clump must be heavier than the black hole (mass doubling condition); its cooling time must exceed the super-Eddington growth time (lifetime condition); and dynamical friction must dominate shell acceleration (BH-trapping condition). These requirements confine viable clumps to a narrow density-temperature region, $n_{\rm H}\simeq10^{7-8}{\rm cm}^{-3}$ and $T\simeq(2-6)\times10^{3}{\rm K}$, for a $10^{3}{\rm M}_\odot$ seed. Even inside this sweet spot a $10^{3}{\rm M}_\odot$ seed grow up at most $4\times10^{3}{\rm M}_\odot$; the maximum growth ratio $M_{\rm BH, fin}/M_{\rm BH}$ falls approximately as $M_{\rm BH}^{-0.4}$ and is negligible once $M_{\rm BH}\gtrsim10^{4}{\rm M}_\odot$. The forward-acceleration effect is essential, expelling the black hole whenever photon trapping fails. We conclude that BH-clump-capture model, and potentially broad super-Eddington models, cannot produce the $>10^{4}{\rm M}_\odot$ seeds required for subsequent Eddington-limited growth, suggesting that alternative pathways, such as heavy seed formation, remain necessary.

Figures (7)

• Figure 1: A schematic illustration of the set-up and conceptual framework of our toy-model simulation, designed to examine the BH-clump interaction in the context of early galactic nuclei. In the nuclear disc or cluster at the center of the first galaxies, numerous dense gas clumps are expected to form due to some effects, such as gravitational instabilities. We model the interaction between a single gas clump and a BH as a two-body dynamical problem embedded within a static background gravitational potential. This simplified approach allows us to determine whether the BH is captured by the clump and undergoes rapid, super-Eddington accretion, or escapes with negligible growth. The toy model is specified by five input parameters: the initial BH mass $M_{\rm BH}$, the gas number density $n_{\rm H}$, the gas temperature $T$, the initial relative velocity $v_{\rm in}$, and the impact parameter $b$. By scanning these parameters, we delineate the conditions under which BH trapping and efficient growth occur.
• Figure 2: Time evolution of key physical quantities in a representative run in which the BH is successfully trapped within the clump, selected from the fiducial setup ($M_{\rm BH} = 10^3\ \mathrm{M}_{\odot},\ v_{\rm in} \to 0,\ b \to 0$). The clump parameters are $n_{\rm H} = 1.8 \times 10^7\ \mathrm{cm}^{-3}$ and $T = 5.6 \times 10^3\ {\rm K}$. The horizontal axis shows time normalised by the free-fall timescale $t_{\rm ff}$, with $t=0$ defined as the moment when the BH first enters the clump ($r=R_{\rm cl}$). The evolution is shown up to the end of the super-Eddington accretion phase. Top: Radial distance of the BH from the clump centre, normalised by $R_{\rm cl}$ (red solid line); the black dashed line marks $r = R_{\rm cl}$. Middle: Mechanical energy of the BH, normalised by $E_{\rm norm} \equiv \sqrt{G(M_{\rm cl}+M_{\rm BH})/R_{\rm cl}}$. Negative values indicate the BH is gravitationally bound to the clump. Bottom: BH mass. The red solid line shows the actural growth in the simulation, while the grey dashed line represents Eddington-limited growth for comparison.
• Figure 3: Same as Figure \ref{['fig:tevo_o']}, but for a representative run in which the BH is not trapped by the clump. The clump parameters are $n_{\rm H} = 1.5 \times 10^7\ \mathrm{cm}^{-3}$ and $T = 5.6 \times 10^3\ {\rm K}$. Red solid lines show the fiducial run including ionised-bubble acceleration, while blue dashed lines correspond to a comparison run with the same initial parameters but with the ionised-bubble acceleration artificially turned off.
• Figure 4: Summary of the fiducial runs in the clump density-temperature plane. Each dot represents an individual simulation. Red points: the BH is trapped and undergoes super-Eddington accretion. Light-blue points: the BH is not trapped. Grey points: the run terminates when the clump reaches the end of its lifetime while the BH is still traversing it. The black star marks the run with the greatest BH growth, yielding $M_{\rm BH,fin} = 3.9 \times 10^{3}\ \mathrm{M}_{\odot}$. Solid and dashed curves indicate analytically derived boundaries for the BH-clump-capture model: - Black solid: mass doubling condition (Equation (\ref{['eq:doubling_condition']})); - Red solid: clump lifetime condition (Equations (\ref{['eq:lifetime_condition_easy']}) and (\ref{['eq:lifetime_condition']})); - Blue solid: BH-trapping condition (Equation (\ref{['eq:hyper-Eddington_condition_BHL']})); - Blue dashed: hyper-Eddington accretion condition evaluated at zero relative velocity (Equation (\ref{['eq:hyper-Eddington_condition']})).
• Figure 5: Same as Figure \ref{['fig:rho-T_m3_v0']}, but for simulations with different initial BH masses: $M_{\rm BH} = 10^{2}\ \mathrm{M}_{\odot}$ (left) and $M_{\rm BH} = 10^{4}\ \mathrm{M}_{\odot}$ (right). The orbital parameters are fixed at $v_{\rm in} \to 0$ and $b \to 0$ as in the fiducial setup. The allowed region for super-Eddington accretion shifts to higher densities and expands in the left panel ($M_{\rm BH} = 10^{2}\ \mathrm{M}_{\odot}$), while it moves to lower densities and narrows significantly in the right panel ($M_{\rm BH} = 10^{4}\ \mathrm{M}_{\odot}$).