Conditions for Super-Eddington Accretion onto the First Black Holes

Britton D. Smith; Ricarda S. Beckmann; Sadegh Khochfar; Simone T. Gordon

REVIEW 3 major objections 6 minor 1 cited by

Conditions for Super-Eddington Accretion onto the First Black Holes

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Thermal feedback does not necessarily stop the first black holes from accreting at many times the Eddington rate, and sustained ~100 kyr super-Eddington bursts are possible in clumpy gas-rich mini-haloes.

desk verdict Careful, transparent simulation study showing super-Eddington accretion can survive weak thermal feedback in gas-rich clumpy conditions, though event selection from no-feedback runs limits how general the 'all environments' claim can be. read the letter →

arxiv 2412.06888 v2 pith:SGDIBENG submitted 2024-12-09 astro-ph.GA astro-ph.CO

Simone T. Gordon , Britton D. Smith , Sadegh Khochfar , Ricarda S. Beckmann This is my paper · ORCID

classification astro-ph.GAastro-ph.CO

keywords super-EddingtonaccretionfirstblackholesthermalfeedbackPopulationIIIseedsmini-haloesdiscfragmentationcosmologicalzoom-insimulationshigh-redshiftsupermassive

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the first black holes, born as roughly $$10^{3}$$-$$10^{5}$\,M_\odot$ seeds in dark-matter mini-haloes at $z\sim19$-26, can grow faster than the Eddington limit once they start heating the gas around them. The authors resimulate four accretion episodes identified in earlier no-feedback runs, injecting thermal feedback with radiative efficiencies $\epsilon_r=0.01$ or $0.1$, coupling efficiencies $\epsilon_f=0.0001$ to $0.05$, and feedback injection radii of 5 to 10 cell widths. They find that super-Eddington accretion can persist for about 100 kyr with very weak feedback in all environments, and with moderate feedback for two of the black holes, while trans-Eddington growth is possible for a $3\$times10^{3}$$-$6\$times10^{3}$\,M_\odot$ black hole at moderate efficiencies. The reason is that fragmented, clumpy discs are harder for thermal feedback to destroy, and a cold, dense inner disc can survive even when surrounding gas is heated to $$10^{8}$$ K. If many such episodes stack up, this growth channel could help explain the massive black holes observed at high redshift without invoking enormous seeds.

What carries the argument

The load-bearing mechanism is the clumpy, fragmented nuclear disc around the black hole. In these simulations a smooth disc is vulnerable to thermal feedback, while a disc broken into self-gravitating clumps keeps feeding the black hole because the clumps act as dense shields that absorb and re-radiate the injected heat. Feedback is implemented by depositing a fixed fraction of the accreted rest-mass energy, $dE_{\rm BH-feed}=\epsilon_f\epsilon_r\dot{M}_{\rm BH}$c^{2}$\,dt$, into a sphere of radius $r_{\rm fb}=5dx$, $7dx$, or $10dx$, with an energy-budgeting scheme that caps cell temperatures at $$10^{8}$$ K and releases surplus later. Resolution enters because the injection radius is measured in cell widths: lower-resolution runs spread the same energy over a larger physical volume and diffuse it, whereas higher-resolution runs resolve the dense clumps that withstand heating.

What would settle it

Resimulate the same four accretion events with thermal feedback active from the moment the seed is inserted, instead of only just before the event; if the clumpy disc fails to assemble and the Eddington fraction drops below one in all four cases, the paper's central claim would be refuted. The paper itself notes that even weak feedback quenched a $270\,M_\odot$ seed in its tests, making this the decisive check.

Watch

Extended reading notes

Core claim

The paper establishes that super-Eddington accretion onto the first black holes is compatible with thermal feedback under a specific set of conditions. With very weak feedback ($\epsilon_r=0.01$, $\epsilon_f=0.001$) and a feedback radius of $5dx$, accretion at more than ten times the Eddington rate persists in most simulated environments. For the $6\$times10^{4}$\,M_\odot$ black hole, moderate feedback ($\epsilon_r=0.01$, $\epsilon_f=0.05$) with a wider injection radius ($7dx$ or $10dx$) still allows super-Eddington growth because a compact inner sub-pc disc survives, and a $6\$times10^{3}$\,M_\odot$ black hole maintains an average $f_{\rm Edd}=1.58$ at $\epsilon_r=0.01$, $\epsilon_f=0.01$, $r_{\rm fb}=5dx$. The simulations also show that even black holes whose time-averaged accretion is only trans-Eddington can gain most of their mass during brief super-Eddington episodes, and that feedback destroys smaller clumps first, homogenizing the gas and lowering the clumping factor.

Load-bearing premise

The dense, clumpy disc that sustains super-Eddington accretion is assembled with feedback switched off, and thermal feedback is only turned on immediately before each accretion event, so a real black hole that had been heating its surroundings all along might never build that reservoir.

Editorial extensions

If this is right

Thermal feedback alone does not cap the growth of first-generation black holes at the Eddington rate, so gas-rich mini-haloes can host repeated ~100 kyr super-Eddington bursts.
At very low feedback efficiency ($\epsilon_r=0.01$, $\epsilon_f=0.001$), super-Eddington growth occurs in nearly all environments, making the gas supply more important than feedback for these seeds.
For intermediate-mass seeds near $6\times10^4\,M_\odot$, moderate feedback with a larger injection radius still permits super-Eddington accretion, connecting this channel to massive-seed formation pathways.
Brief super-Eddington episodes can supply most of a black hole's mass even when the average rate is only trans-Eddington, so short bursts matter for building high-redshift supermassive black holes.
If sustained, the simulated growth tracks can reach the $10^8$-$10^9\,M_\odot$ black holes observed at $z\sim6$-10 from $10^3$-$10^5\,M_\odot$ seeds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is to activate thermal feedback from the moment the seed is inserted, rather than immediately before each accretion event; the authors' own caveat that even weak feedback quenched a $270\,M_\odot$ seed suggests this may be the decisive test of whether the clumpy disc is a feedback-free artifact.
If fragmented discs are the real accretion state, super-Eddington growth should be episodic and variable on $\sim$kyr timescales, which could appear as strong variability in high-redshift black hole candidates rather than as steady ultra-luminous emission.
The paper's correlation between feedback intensity per unit volume and the mass remaining in the disc implies that lower-resolution cosmological simulations may need to lower effective feedback efficiencies to avoid numerical overcooling, a calibration that could be tested by re-running one accretion event at several resolutions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a look. This is a solid, well-documented extension of the authors' no-feedback study, adding thermal feedback in sub-pc cosmological zoom-ins and systematically scanning radiative efficiency, coupling efficiency, and injection radius. The new quantitative results are real: super-Eddington survival at very weak feedback (eps_r=0.01, eps_f=0.001) in most environments, and at moderate feedback for a 6e4 Msun BH when the energy is spread over r_fb=7-10 cells. The resolution comparison in Event 1 cleanly shows that clumpy discs resist thermal feedback while smooth discs do not; the injection-radius runs as a probe of numerical overcooling are a nice touch. Analysis scripts are public, which helps.

The main soft spot is a selection effect. The four events are chosen from no-feedback runs, and feedback is turned on only just before each event. The dense clumpy discs that make super-Eddington growth possible are assembled with feedback off. Section 5 admits that activating even weak feedback at the 270 Msun seed stage quickly quenched accretion. For a real seed the feedback is on from the start, so the favorable structures may never form. That weakens the abstract's 'in all environments'. The authors' own hedge, 'in a gas-rich environment', is what the simulations actually support. I don't see this as fatal — the protocol is transparent and the paper is explicitly a first step — but it is an external-validity caveat, not an internal flaw.

Other gaps are acknowledged by the authors: no radiative or kinetic feedback, no HD cooling, no spin evolution. Fair enough. Minor issues: the BHL boost factor alpha in eq. 2 is never given a value, and there are small inconsistencies between the abstract and Section 3.4 ('all' vs 'almost all environments') and in some Table 2 labels. These are cosmetic.

Bottom line: this paper is for people working on first BH seeds, JWST high-z SMBH detections, and sub-grid feedback recipes. It deserves a serious referee; I'd send it out. My own verdict is conditional — reword the abstract, make the seed-stage limitation more prominent — but the work is honest and the results are a genuine contribution.

Referee Report

3 major / 6 minor

Summary. The paper investigates whether super-Eddington accretion onto the first black holes can be sustained when thermal feedback is included. Using sub-pc resolution cosmological zoom-in simulations of two mini-haloes, the authors identify four accretion events from no-feedback runs and resimulate each with thermal feedback activated just before the event, varying the radiative efficiency (eps_r = 0.01-0.1), the feedback coupling efficiency (eps_f = 0.0001-0.05), and the feedback injection radius (r_fb = 5-10 dx). They report that super-Eddington accretion can be sustained for ~100 kyr under very weak feedback (eps_r = 0.01, eps_f = 0.001) in almost all environments, and under moderate feedback for a 6x10^4 Msun BH with larger injection radii and for a 6x10^3 Msun BH at eps_r = 0.01, eps_f = 0.01. The paper also finds that clumpy, fragmented discs are more resistant to thermal feedback than smooth discs, and that brief super-Eddington episodes dominate the mass growth in most simulations. The authors provide an extensive list of caveats, including the absence of radiative and kinetic feedback, the neglect of spin evolution, and the fact that the simulated accretion events were selected from no-feedback runs.

Significance. The paper's positive result, that thermal feedback at the low-efficiency end of the explored range does not necessarily quench super-Eddington accretion in dense, clumpy environments, is an important proof-of-concept for early BH growth scenarios. The simulations are technically demanding, achieving sub-pc resolution in a cosmological context and resolving the Bondi/Hoyle-Lyttleton radius, which is rare in this field. The authors are transparent about their protocol's limitations, and the availability of the ENZO fork and analysis scripts supports reproducibility. However, the external validity of the result is limited by the selection protocol: the dense discs are assembled in no-feedback runs, and the abstract overstates the generality of the findings relative to the body of the paper.

major comments (3)

[Section 2 (protocol) and Section 5 (caveats)] The simulation protocol selects accretion events from no-feedback runs and activates thermal feedback only immediately before each event (Section 2, Table 1), so the dense, clumpy discs that enable super-Eddington accretion are assembled with feedback absent; the authors' own test in Section 5 shows that even weak feedback on a 270 Msun seed quickly quenched accretion. Consequently, the manuscript demonstrates that super-Eddington accretion can persist for ~100 kyr given a pre-formed dense disc, but it does not demonstrate that such discs form when feedback is active throughout the assembly. The abstract's claim that super-Eddington growth is possible 'in all environments' therefore overstates the external validity of the results, and I ask the authors to rephrase the abstract and conclusions to make the conditional nature explicit (e.g., 'given a pre-existing dense, clumpy disc') and to qualify or remove 'in all environments'.
[Abstract and Section 3.4] The abstract claims super-Eddington growth is possible with very weak thermal feedback 'in all environments', but Section 3.4 states this occurs 'in almost all environments', with the exception of 1L14- ϵr 0.01- ϵf 0.001-5dx accreting at a trans-Eddington rate on average. Moreover, Table 3 classifies all ϵr = 0.01, ϵf = 0.001 simulations as having ineffective thermal feedback, meaning the very weak feedback cases are cases in which feedback does not significantly alter the gas; the abstract should distinguish between sustained super-Eddington growth with ineffective feedback and growth in the presence of effective thermal feedback, which is the paper's more interesting claim.
[Table 1 and Section 5] The accretion events start at BH masses of 1.1x10^3 to 6x10^4 Msun, far above the 270 Msun Pop III seed mass, and the paper does not demonstrate that a seed can grow to these masses while thermal feedback is active; Section 5 reports that even weak feedback at the seed stage quenched accretion in the authors' tests. The abstract's suggestion that the results apply to 'black holes formed from the first stars' is therefore not directly supported, unless the authors provide a plausible pathway (e.g., quiescent growth between gas-rich episodes) or explicitly limit the claims to BHs that have already assembled a dense disc.

minor comments (6)

[Section 2.4, Eq. (6)] The mapping between ϵr and η is stated incorrectly: the text reads 'We set ϵr ∈ [0.01, 0.1], which correspond to η = 0.11 and η = 0.01, respectively', but since ϵr = η/(1-η), ϵr = 0.1 corresponds to η ≈ 0.091 and ϵr = 0.01 to η ≈ 0.0099; please correct this sentence.
[Table 2] Several rows have entries for ϵr and ϵf that do not match the simulation names; for example, '1L16- ϵr 0.01- ϵf 0.05-5dx' is listed as ϵr = 0.1, ϵf = 0.05, '2E14- ϵr 0.01- ϵf 0.05-7dx' is listed as ϵr = 0.1, ϵf = 0.05, and '3E16- ϵr 0.01- ϵf 0.05-5dx' is listed as ϵr = 0.1, ϵf = 0.05; please verify that the numeric entries correspond to the names, or rename the simulations accordingly.
[Section 2.3, Eq. (2)] The BHL boost factor α is introduced in Eq. (2) but its value or range is never stated in this manuscript; since the accretion rate is proportional to α, the quantitative results depend on this choice, so please state the value used and cite the relevant section of G24.
[Section 2.4] The stated range of overall accretion efficiency, 1x10^-5 ≤ ϵf ϵr ≤ 5x10^-3, appears to have the lower bound off by an order of magnitude if the ϵf = 0.0001 run (2E14- ϵr 0.01- ϵf 0.0001-5dx) is included, as its product is 1x10^-6; please check the intended range.
[Table 1] The column header '⟨ΔṀ_t/Ṁ_{t-x/t+x}⟩' is difficult to parse; the definition in the caption ('average increase in accretion rate during the event relative to the accretion rate t = 200 kyr before and after the event') should be written as a clear formula, for example ⟨(Ṁ(t) - Ṁ_bg)/Ṁ_bg⟩ over the event duration.
[Section 3.5, Fig. 8] The trendline in Fig. 8 is based on a small number of points (about a dozen) with considerable scatter; please report the number of points and the correlation coefficient (with significance) or present the trend as indicative only.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the super-Eddington-with-feedback result is an emergent outcome of transparently parameterized simulations, not a by-construction consequence of the no-feedback event selection.

full rationale

I checked the paper's argued chain from event selection (Section 2, Table 1) through the feedback injection model (equations 2-7) to the conclusions in Sections 3.4-3.5. The accretion events are defined as short periods of elevated accretion 'identified in the simulations without feedback', and those events are then resimulated with thermal feedback switched on. The finding that super-Eddington accretion is sustained with very weak feedback (epsilon_r = 0.01, epsilon_f = 0.001) and with moderate feedback only for Events 2 and 4 at larger injection radii is not guaranteed by construction: the same protocol quenches accretion in most moderate- and high-efficiency runs (e.g., 1L14-epsilon_r0.1-epsilon_f0.05-5dx 'falls rapidly to f_Edd < 1e-3'), so the parameter dependence is the result rather than the input. No fitted parameter is renamed as a prediction: alpha, epsilon_r, epsilon_f, and r_fb are adopted or systemically varied, and equation (5) is the standard thermal feedback injection formula, not a fit to f_Edd. Self-citations to G24 transfer the halo selection, SmartStars seeding module, BHL sub-grid model, and Bondi/HL resolution checks; these are legitimate methodological transfers from the authors' prior work, not an invoked uniqueness theorem or a smuggled ansatz that forces the super-Eddington conclusion. The paper also discloses its principal external-validity limitation in Section 5: 'Immediately introducing even weak feedback to the 270 M_sun direct-collapse seed quickly quelled accretion and we did not have the computational resources to continue the simulations at high resolution until a significant injection of gas occurred.' That statement confirms the dense clumpy disc is assembled without feedback, so the 'all environments' claim inherits a selection bias from the no-feedback event definition; this is an honestly flagged representativeness/correctness risk, not an internal by-construction reduction of the claimed result to its inputs. Verdict: no significant circularity; score 2 reflects the one minor, non-load-bearing self-citation to the authors' G24 setup.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim depends mainly on modeled feedback efficiencies and the choice to activate feedback only at selected dense episodes; these are hand-chosen or post-hoc rather than derived. The simulation relies on standard cosmology and hydrodynamics from ENZO and GRACKLE, with no new physical entities postulated.

free parameters (6)

BHL boost factor alpha = Not stated in paper
Multiplies the Bondi-Hoyle accretion rate in Eq. 2; the absolute accretion rates and hence Eddington fractions depend on it, but its value is not given here, only in G24.
Feedback coupling efficiency eps_f = 0.05, 0.001, 0.0001 (grid)
Hand-selected range from galaxy formation literature plus weaker values; low values are the ones that produce the paper's main super-Eddington results.
Radiative efficiency factor eps_r = 0.01 and 0.1 (grid)
Chosen to represent thin-disc and slim-disc efficiencies; central to the feedback energy injection rate in Eq. 5.
Feedback injection radius r_fb = 5dx, 7dx, 10dx
Adopted multiples of cell width to probe numerical overcooling; larger radii reduce feedback intensity and allow Event 2 to stay super-Eddington.
Maximum temperature cap T_max = 1e8 K
Numerical energy-budget cap in Eq. 7 that delays feedback energy injection; affects how feedback couples to gas.
Initial BH seed mass = 270 Msun, growing to 1e3 to 6e4 Msun by the event
Seeded via modified SmartStars; the seed mass is too high for standard Pop III core collapse, as acknowledged in Section 5.

assumptions (6)

domain assumption Bondi-Hoyle-Lyttleton accretion with a constant boost factor alpha (Eq. 2)
Sub-grid prescription for the unresolved accretion flow; the conclusion that feedback can be survived depends on this accretion model.
domain assumption Thermal feedback injects a fixed fraction eps_f times eps_r of accreted rest-mass energy isotropically within r_fb, independent of environment and accretion rate (Eq. 5)
Model choice; radiative and kinetic feedback are not included, so the paper only tests thermal coupling.
standard math Eddington rate definition with eta and the relation eps_r = eta/(1-eta) (Eq. 1 and Section 2.4)
Standard definition; the paper's application is routine, though the eta/eps_r correspondence sentence contains a typo.
domain assumption GRACKLE primordial chemistry without HD cooling and with optically thin radiative losses
The authors note HD cooling is omitted and could become important after shock heating (Section 5), which could change disc fragmentation and feedback effectiveness.
standard math Cosmological initial conditions from WMAP7 and MUSIC (Section 2)
Standard input parameters from prior literature; not load-bearing for the qualitative result.
ad hoc to paper Feedback is disabled during pre-event evolution and enabled only at t_init
This is the central selection assumption; it creates the gas-rich disc that the central claim depends on. The paper acknowledges this in Section 5.

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Pith. "Pith review of Conditions for Super-Eddington Accretion onto the First Black Holes." pith.science (2026). https://pith.science/paper/SGDIBENG

@misc{pith2026241206888,
  author       = {Pith},
  title        = {Pith review of: Conditions for Super-Eddington Accretion onto the First Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGDIBENG}},
  note         = {Machine review of arXiv:2412.06888}
}

abstract

Observations of supermassive black holes at high redshift challenge our understanding of the evolution of the first generation of black holes (BHs) in proto-galactic environments. One possibility is that they grow much more rapidly than current estimates of feedback and accretion efficiency permit. Following our previous analysis of super-Eddington accretion onto stellar-mass black holes in mini-haloes under no-feedback conditions, we now investigate whether this can be sustained when thermal feedback is included. We use four sets of cosmological simulations at sub-pc resolution with initial black hole masses varying from $1 \times 10^3 - 6 \times 10^4 M_\odot$, exploring a range of feedback efficiencies. We also vary the feedback injection radius to probe the threshold of numerical overcooling. We find that super-Eddington growth sustained on the order of $\sim$$100 \, \rm kyr$ is possible with very weak thermal feedback efficiency in all environments and moderate efficiency for two of the BHs. Trans-Eddington growth is possible for a $3 \times 10^3 - 6 \times 10^3 M_\odot$ BH at moderate feedback efficiencies. We discuss the effectiveness of thermal feedback in heating the gas, suppressing accretion, and driving outflows at these parameter configurations. Our results suggest that super-Eddington growth may be possible in the presence of thermal feedback for black holes formed from the first stars.

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1 extracted references · 1 linked inside Pith · cited by 1 Pith paper

[1]

, Bryan G

Abel T. , Bryan G. L., Norman M. L., 2002, Science , 295, 93 Adams N. J. et al., 2022, MNRAS , 518, 4755 Agarwal B. , Smith B., Glo v er S., Natarajan P., Khochfar S., 2016, MNRAS , 459, 4209 Alvarez M. A. , Bromm V. , Shapiro P. R., 2006, ApJ , 639, 621 Alvarez M. A. , Wise J. H., Abel T., 2009, ApJ , 701, L133 Atek H. et al., 2023, MNRAS , 519, 1201 Bec...

arXiv 2025

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Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

, Bryan G

Abel T. , Bryan G. L., Norman M. L., 2002, Science , 295, 93 Adams N. J. et al., 2022, MNRAS , 518, 4755 Agarwal B. , Smith B., Glo v er S., Natarajan P., Khochfar S., 2016, MNRAS , 459, 4209 Alvarez M. A. , Bromm V. , Shapiro P. R., 2006, ApJ , 639, 621 Alvarez M. A. , Wise J. H., Abel T., 2009, ApJ , 701, L133 Atek H. et al., 2023, MNRAS , 519, 1201 Bec...

arXiv 2025