Mass transport in galaxy discs limits black hole growth to sub-Eddington rates
Pith reviewed 2026-05-25 01:46 UTC · model grok-4.3
The pith
Viscosity-driven gas inflow in galaxy discs caps seed black hole masses at 1.8e7 solar masses by redshift 6 in typical halos.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We model analytically the idealised growth of seed black holes, fed through gas inflow from growing proto-galaxy discs. The inflow depends on the disc gravitational stability and thus varies with black hole and disc mass. We find that for a typical host halo, the efficiency of angular momentum transport, as parametrised by the disc viscosity, is the limiting factor in determining the inflow rate and the black hole accretion rate. For our fiducial case we find an upper black hole mass estimate of M_• ∼ 1.8 × 10^7 M_⊙ at z=6. Only in the extreme case of ∼10^16 M_⊙ haloes at z=6 produces SMBH masses of ∼10^9 M_⊙. However, the number density of such haloes is many orders of magnitude below the 1
What carries the argument
Viscosity-parametrized angular momentum transport in gravitationally stable proto-galaxy discs that sets the rate of gas inflow to the central black hole.
If this is right
- Black hole accretion stays sub-Eddington because disc viscosity sets an upper bound on inflow.
- Typical halos at z=6 host black holes no larger than about 1.8 × 10^7 solar masses.
- Halos of 10^16 solar masses can reach 10^9 solar masses but occur at number densities far below the observed 1 Gpc^{-3}.
- Major mergers can deliver enough extra gas to overcome the viscosity limit and produce the observed masses, depending on orbital parameters.
Where Pith is reading between the lines
- High-redshift black hole growth probably depends on repeated mergers rather than continuous disc accretion alone.
- Galaxy formation simulations must track detailed merger histories to reproduce the black hole mass function at z greater than 6.
- Isolated discs at z=6 should contain systematically less massive black holes than those in recently merged systems.
Load-bearing premise
The inflow rate to the black hole is controlled only by disc gravitational stability and a fixed viscosity parameter without other transport processes dominating.
What would settle it
Discovery of a 10^9 solar mass black hole at z=6 residing in a halo well below 10^16 solar masses and lacking signs of recent major mergers would contradict the claimed growth limit.
read the original abstract
Super-massive black holes (SMBHs) observed to have masses of $M_\bullet \sim 10^9 \, \mathrm{M_\odot}$ at $z\gtrsim6$, $<1$ Gyr after the Big Bang, are thought to have been seeded by massive black holes which formed before growing concurrently with the formation of their host galaxies. We model analytically the idealised growth of seed black holes, fed through gas inflow from growing proto-galaxy discs. The inflow depends on the disc gravitational stability and thus varies with black hole and disc mass. We find that for a typical host halo, the efficiency of angular momentum transport, as parametrised by the disc viscosity, is the limiting factor in determining the inflow rate and the black hole accretion rate. For our fiducial case we find an upper black hole mass estimate of $M_\bullet \sim 1.8 \times 10^7 \, \mathrm{M_{\odot}}$ at $z=6$. Only in the extreme case of $\sim 10^{16}$ M$_{\odot}$ haloes at $z=6$ produces SMBH masses of $\sim 10^9$ M$_{\odot}$. However, the number density of such haloes is many orders of magnitude below the estimated 1 Gpc$^{-3}$ of SMBHs at $z=6$, indicating that viscosity driven accretion is too inefficient to feed the growth of seeds into $M_\bullet \sim 10^9 \, \mathrm{M_\odot}$ SMBHs by $z \sim 6$. We demonstrate that major mergers are capable of resolving the apparent discrepancy in black hole mass at $z=6$, with some dependence on the exact choice of orbital parameters of the merger.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents an analytical model for the growth of seed black holes fed by gas inflow from proto-galaxy discs. Inflow rate depends on disc gravitational stability (varying with black hole and disc mass) and is limited by angular-momentum transport parametrized by disc viscosity. For fiducial host halo mass and viscosity, the model yields an upper black hole mass of ~1.8×10^7 M_⊙ at z=6; reaching ~10^9 M_⊙ requires extreme ~10^16 M_⊙ haloes whose number density is far below the observed ~1 Gpc^{-3} for SMBHs at z=6. The paper concludes that viscosity-driven accretion is too inefficient and that major mergers can resolve the discrepancy.
Significance. If the central analytical limit holds, the work supplies a quantitative demonstration that standard disc accretion cannot produce the observed z~6 SMBH population without rare haloes or additional channels such as mergers. The concrete mass and halo-mass thresholds constitute falsifiable predictions that can be tested against simulations or high-z observations.
major comments (2)
- [Abstract / model description] The reported upper mass limit of 1.8×10^7 M_⊙ (abstract) is produced once the fiducial disc viscosity and halo mass are inserted; these are free input parameters rather than derived quantities. The manuscript must show how the mass limit and the conclusion that viscosity is the limiting factor change when viscosity and halo mass are varied over observationally plausible ranges.
- [Analytical model] The functional dependence of inflow rate on black hole and disc mass (via gravitational stability) is load-bearing for the entire mass-limit result, yet the explicit derivation or governing equations are not reproduced in the supplied abstract. The full manuscript must provide the step-by-step relation between stability criterion, viscosity, and accretion rate so that the numerical upper bound can be reproduced.
minor comments (1)
- [Abstract] The abstract states that major mergers 'resolve the apparent discrepancy' with some dependence on orbital parameters; the manuscript should quantify the range of orbital parameters that succeed and the resulting merger rates at z~6.
Simulated Author's Rebuttal
We thank the referee for their constructive comments, which have helped improve the clarity of the manuscript. We address each major comment below and have made revisions where the points identify areas for enhancement.
read point-by-point responses
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Referee: [Abstract / model description] The reported upper mass limit of 1.8×10^7 M_⊙ (abstract) is produced once the fiducial disc viscosity and halo mass are inserted; these are free input parameters rather than derived quantities. The manuscript must show how the mass limit and the conclusion that viscosity is the limiting factor change when viscosity and halo mass are varied over observationally plausible ranges.
Authors: We agree that the reported limit is tied to the fiducial choices. The manuscript already demonstrates that only extreme ~10^16 M_⊙ haloes reach 10^9 M_⊙ and notes the low number density of such objects. To strengthen this, we have added an appendix with explicit variations: viscosity parameter α from 0.001–0.1 and halo masses 10^11–10^13 M_⊙ at z=6. These confirm the upper limit stays ≲ few ×10^7 M_⊙ for observationally motivated ranges, while the conclusion that mergers are required remains unchanged. The number-density comparison is retained as the key constraint. revision: yes
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Referee: [Analytical model] The functional dependence of inflow rate on black hole and disc mass (via gravitational stability) is load-bearing for the entire mass-limit result, yet the explicit derivation or governing equations are not reproduced in the supplied abstract. The full manuscript must provide the step-by-step relation between stability criterion, viscosity, and accretion rate so that the numerical upper bound can be reproduced.
Authors: The governing equations and stability criterion appear in Section 2 of the full manuscript, but we accept that a more explicit, self-contained derivation improves reproducibility. We have revised Section 2 to include a numbered step-by-step derivation: (i) Toomre Q = c_s κ/(π G Σ) < 1 for instability, (ii) resulting critical Σ_crit(M_•, M_disc), (iii) viscous inflow velocity v_r derived from ν = α c_s H with angular-momentum transport, and (iv) the integrated Ṁ_inflow(M_•) that yields the 1.8×10^7 M_⊙ cap for fiducial inputs. A summary table of the key equations has also been added. revision: yes
Circularity Check
No significant circularity identified
full rationale
The paper presents an analytical model deriving upper limits on black hole mass from parametrised disc viscosity and gravitational stability in growing proto-galaxies. The reported fiducial upper mass of 1.8e7 Msun at z=6 is a direct numerical output of the model once input parameters (viscosity, halo mass) are inserted; this is standard model evaluation rather than a prediction that reduces to its own inputs by construction. No equations, self-citations, or uniqueness theorems are quoted that would create self-definitional loops, fitted-input predictions, or load-bearing self-citation chains. The derivation is self-contained against external benchmarks as a parametric calculation whose assumptions (viscosity range, stability criterion) are stated independently of the target SMBH mass result.
Axiom & Free-Parameter Ledger
free parameters (2)
- disc viscosity
- fiducial host halo mass
axioms (1)
- domain assumption Inflow rate depends on disc gravitational stability which varies with black hole and disc mass
discussion (0)
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