{"id":"4340c360-d670-4b69-bac2-3a3b070cef41","arxiv_id":"2505.08859","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Simulated super-critical discs in large domains self-regulate to a net mass accretion rate near the Eddington limit, matching the critical disc model rather than the slim disc model.","lead":"Three of the largest and longest 3D simulations of matter falling onto a black hole at extreme rates all settle into the same pattern: the black hole swallows only about one Eddington mass rate, regardless of how much matter is fed in from far away. The simulations suggest outflows push back the surplus, supporting the 'critical disc' model and challenging earlier simulation results.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unresolved MRI (Q_theta ~ 1) may be setting the low alpha that produces the near-Eddington accretion; the paper's own high-resolution run shows mdot_BH trending upward, so the claimed limit may be a grid artifact.","rationale":"The paper is a serious, transparent numerical study with an interesting and potentially important result. The reader's weakest assumption correctly identifies the unresolved MRI as the most load-bearing issue: the claimed Eddington-limited accretion depends on the simulated balance between inflow and outflow, and the authors themselves state in Section 2 that they are not formally resolving the MRI and that the resulting low alpha values may be unreliable. The existing high-resolution extension already hints at a resolution-dependent increase in mdot_BH, which is a concrete red flag rather than a speculative one. I therefore agree with the reader that this is where the argument is least secure. I do not see grounds to reject the paper: the three simulations span a factor of ten in target mdot0, the measured profiles agree qualitatively with the critical disc model, and the authors honestly flag the resolution limitations and the need for additional work. However, the concern is load-bearing enough to keep the verdict conditional until a resolution-converged run demonstrates that mdot_BH ~ 1 is not an artifact of unresolved MRI. The lack of public code or data is a secondary reproducibility issue but not the core scientific weakness. The verdict should remain CONDITIONAL, so no change is made to the reader's assessment.","tokens_in":17215,"tokens_out":7459,"duration_ms":82294,"concrete_test":"Run simulation a9r20 at the next refinement level (L4) for at least 30,000 t_g after the L3 burn-in, verifying that Q_theta >= 10 and Q_phi >= 10 in the midplane, then compare the time-averaged mdot_BH and the mdot_in(r), mdot_out(r), and mdot_net(r) profiles to the L2 and L3 values over the same averaging window. If mdot_BH stays near 1 and the profiles are unchanged, the concern is resolved; if mdot_BH moves systematically above ~2 with resolution, the near-Eddington result is not converged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the inflow/outflow cancellation yielding mdot_BH ~ 1 is physical. This requires the simulated turbulent transport and wind driving to be accurate. Section 2 explicitly reports that the two-level meshes are 'not formally resolving the MRI' with Q_theta ~ 1 and Q_phi ~ 4, and that this 'may lead to our relatively low values for alpha... of 10^-3-10^-2.' Low alpha directly sets the radial drift speed and the balance between inflow and outflow; if unresolved MRI suppresses turbulent stress, the measured near-Eddington net accretion could reflect numerical transport rather than radiative self-limiting. The paper's one higher-resolution extension, a9r20L3, shows 'maybe a slight jump up in mdot_BH' (Section 3.1.1), exactly the sort of resolution dependence that would undermine the claim if it continues. No convergence test at commonly accepted MRI quality factors (Q >= 10) is presented, so the possibility that the Eddington-limited result is a resolution artifact remains open.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents three three-dimensional general-relativistic radiation-MHD simulations of super-critical accretion onto a rapidly spinning stellar-mass black hole, using the Cosmos++ code with an M1 closure scheme and starting from Novikov-Thorne thin-disc equilibria extended to large radii (r_max up to 1000 r_g) and evolved for long durations (up to ~166,000 t_g). The intended mass feeding rates span mdot0 = 1, 4, and 10 Eddington units. The central result is that, in the radial region that has reached inflow equilibrium (r_eq ~ 32-49 r_g), the measured net accretion rate onto the black hole is close to Eddington in all three runs, even though the inward mass flux at large radii can be orders of magnitude larger. The authors attribute this to a near-cancellation between inward and outward mass fluxes and interpret it as evidence that these discs obey the Eddington limit and are described by the critical-disc model rather than the slim-disc model. They also compare radiative and kinetic luminosities with analytic expectations and discuss why their results differ from most earlier global simulations, which typically find mdot_BH ~ 10 or more.","tokens_in":17489,"tokens_out":4392,"duration_ms":44076,"significance":"If the central claim holds, this is a significant result. It would indicate that large, steady super-critical discs fed by thin Keplerian discs at large radius self-regulate their net accretion to near-Eddington through a finely balanced outflow, in agreement with the critical-disc scenario and in tension with the conclusions of most previous global simulations. The paper also speaks directly to the growth of high-redshift black holes and to the observed properties of ultraluminous X-ray sources. The strengths of the study include the unusually large radial domains and long evolution times, the choice to initialize from a Shakura-Sunyaev solution rather than a small finite torus, and the fact that the near-Eddington net accretion rate is a measured outcome rather than an imposed parameter. The comparisons with the analytic predictions of Fukue (2004) and Poutanen et al. (2007) provide independent diagnostics that go beyond a simple visual match. The authors are also explicit about several limitations, including the modest MRI resolution and the fact that the critical radius is not captured on the grid.","major_comments":[{"comment":"The central claim in the abstract and Section 5 that 'these simulated discs obey the Eddington limit' and are 'locally Eddington limited at all radii' extrapolates beyond the region actually shown to be in equilibrium. Table 1 reports r_eq = 32-49 r_g, and Section 6 explicitly states that the critical radius r_cr is not captured and that no plateau in mdot_in(r) is seen within the equilibrated region. Because the outer disc is not in steady state and the cancellation between mdot_in and mdot_out there is not demonstrated, the Eddington-limited conclusion should be restricted to r < r_eq, with a clear discussion of how the non-equilibrated outer region could or could not alter the inner balance.","section":"Section 3.1, Table 1, Section 6"},{"comment":"The MRI is formally unresolved in these simulations: the paper reports Q_theta ~ 1 and Q_phi ~ 4, and attributes the low measured alpha ~ 10^-3 - 10^-2 to this under-resolution. Since the near-Eddington net accretion emerges from a delicate cancellation between large inward and outward fluxes, the value of alpha directly controls the radial drift speed and hence that balance. The one higher-resolution extension, a9r20L3, shows a slight upward jump in mdot_BH, which is the direction one would expect if resolved turbulence increases transport. No convergence test at commonly accepted quality factors (e.g., Q >= 10) is presented. This leaves open the possibility that the Eddington-limited result is, at least in part, a resolution artifact rather than a robust physical self-regulation.","section":"Section 2, Section 3.1.1"},{"comment":"The claimed 'reasonable quantitative agreement' with the critical-disc model relies on the analytic curve mdot_in(r) = [mdot_in(r_cr) - mdot_BH] r / r_cr, but the value of r_cr used to draw this curve is not stated. Since r_cr is not captured in the simulations and Section 6 states that it must lie beyond r_eq, this curve is not an independent prediction but depends on a choice of r_cr that may be adjusted to match the data. The authors should state explicitly how r_cr was chosen for each panel and show the sensitivity of the comparison to that choice; otherwise the agreement is partly by construction.","section":"Section 4.2, Figure 3"},{"comment":"The fine-tuned cancellation between mdot_in and mdot_out is characterized by large statistical fluctuations, with mdot_net often changing sign, and much of the outflow is not unbound by the Bernoulli criterion (mdot_out significantly exceeds mdot_un). The paper notes that the total mass in the domain drops by less than 8%, but this does not by itself establish that the observed cancellation is a steady-state attractor rather than a transient sloshing of matter within a domain that has not reached equilibrium at large radii. The ultimate fate of the bound outflow is explicitly uncertain. The conclusions should therefore be tempered: the simulations demonstrate near-Eddington net accretion in the equilibrated inner region, but whether this constitutes a global Eddington limit requires either longer evolutions, larger domains, or a clearer identification of the physical mechanism enforcing the cancellation.","section":"Section 3.1.2, Section 6"}],"minor_comments":[{"comment":"In Section 4.1 the text says 'we see significant mass outflow ... in Figure 2', and in Section 4.2 it says 'which is exactly what we see in Figure 2'; in both places the radial mass-flux profiles are shown in Figure 3, not Figure 2, which shows the time history of mdot_BH.","section":"Section 4.1, Section 4.2"},{"comment":"The displayed expression for the radiation four-force density G^mu appears to have unbalanced parentheses and missing terms; please re-derive and reformat this expression so that the coupling terms are unambiguous.","section":"Equations (2)-(4)"},{"comment":"The phrase 'these simulated discs obey the Eddington limit' should be qualified to 'in the radial region that has reached inflow equilibrium', to be consistent with the admitted absence of equilibrium at large radii and the lack of capture of r_cr.","section":"Abstract, Section 5"},{"comment":"The sentence 'we stand by our finding' is a statement of authorial conviction rather than an argument; consider replacing it with a summary of the specific simulation features that distinguish this work from prior studies.","section":"Section 5, bulleted list"},{"comment":"Figure 2 shows shaded 1-sigma standard deviations obtained after moving-average smoothing; please clarify in the caption whether the shaded regions are the standard deviation of the smoothed or unsmoothed time series, since this affects the interpretation of the secular trends.","section":"Figure 2 and Table 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within the scope of MNRAS and addresses an important problem. The main risk is overclaiming: the Eddington-limited conclusion is currently stretched beyond the equilibrated radial range and rests on under-resolved MRI and on an r_cr-dependent comparison that is not fully specified. These issues are fixable in revision by narrowing the claims, adding a convergence-oriented discussion, and reporting how r_cr was obtained for the analytic curves. I saw no indication of citation or attribution problems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is the only large-domain, long-duration 3D GRRMHD study of super-critical discs that starts from a Shakura-Sunyaev-like disc rather than a finite torus, and all three runs—fed at 1, 4, and 10 times Eddington—settle to net accretion rates within a factor of two of Eddington onto the black hole. That directly contradicts essentially every prior global simulation, which typically find mdot_BH >= 10. Second, the paper is unusually candid about its own limits: it states plainly that the MRI is not formally resolved (Q_theta ~ 1, Q_phi ~ 4), that equilibrium is only established out to 30–50 r_g, and that the critical radius r_cr is not captured on the grid. The authors flag these issues themselves, which says good things about their scientific judgment.\n\nThe strongest part is the model comparison. The simulated radial profiles of m_dot_in, m_dot_out, L_out, and V_phi all track the Fukue/Poutanen critical-disc scalings, and the slim-disc model is rejected using distinct diagnostics—Keplerian rotation, small trapping radius, large outflow. These use independent analytic predictions, not fitted parameters, so the comparison is meaningful.\n\nThe soft spots are the same ones the authors acknowledge, but I weigh them a bit more heavily. The unresolved MRI is load-bearing: the low alpha values (10^-3 to 10^-2) may reflect the grid rather than physics, and alpha sets the radial drift speed and hence the inflow/outflow balance. The one higher-resolution run shows a slight upward creep in mdot_BH, which is exactly the resolution dependence you would worry about, and there is no convergence test at commonly accepted quality factors. So the claim that these discs \"obey the Eddington limit\" is not yet secure beyond the limited region of inflow equilibrium. The phrase \"at all radii\" overstates what the data actually show; a more defensible formulation would be \"in the region where the disc has reached inflow equilibrium, the net rate is close to Eddington.\" Also, the net rate is a small difference of large, fluctuating inward and outward fluxes, which weakens the measurement precision even where a steady state has been claimed; the authors discuss this but it remains a concern. Minor: the data are only \"available upon reasonable request,\" which slows independent checking.\n\nWho is this for? Anyone working on super-critical accretion, ULXs, or early supermassive black hole growth. It is a careful, well-organized paper that deserves expert referee time. Send it to review, with the expectation that the authors either soften the \"all radii\" language or add a genuinely higher-resolution run that tests whether the near-Eddington outcome persists. The paper is honest, novel, and probably pointing in the right direction, but the central conclusion needs another level of numerical evidence before it becomes a solid result.","headline":"A genuinely new numerical result—large, long-duration GRRMHD super-critical discs settling to net near-Eddington accretion—presented honestly, but the unresolved MRI and limited equilibrium region mean the headline claim is provisional; still worth serious refereeing.","tokens_in":18019,"tokens_out":2775,"would_cite":true,"duration_ms":31436,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Super-critical discs cap their own feeding near the Eddington rate, even when the outer supply is many times Eddington.","keywords":["accretion discs","super-critical accretion","Eddington limit","radiation magnetohydrodynamics","black hole accretion","ultra-luminous X-ray sources","critical disc model","slim disc model"],"falsifier":"Run the same initial conditions with a resolution that resolves the MRI (quality factors $Q_\\theta, Q_\\phi \\gtrsim 10$) and measure $\\dot{m}_\\mathrm{BH}$; if the net accretion rises well above $\\dot{M}_\\mathrm{Edd}$ once turbulence is resolved, the near-Eddington result is a grid artifact rather than a physical limit.","tokens_in":17028,"feed_emoji":"🕳️","tokens_out":5620,"duration_ms":52897,"temperature":0.7,"pith_summary":"Three long-duration, large-domain 3D radiation-MHD simulations of a rapidly spinning stellar-mass black hole fed above its Eddington rate find that the black hole's net accretion settles to approximately the Eddington accretion rate, regardless of whether the outer feeding rate is 1, 4, or 10 times Eddington. The mechanism is a near-perfect cancellation: at every radius where the discs have reached equilibrium, the outward mass flux rises to match the inward flux, leaving a net inward rate $\\dot{m}_\\mathrm{net} = \\dot{m}_\\mathrm{in} - \\dot{m}_\\mathrm{out} \\approx 1$. The authors read this as evidence that $\\dot{M}_\\mathrm{Edd}$ is a true limit for steady, large super-critical discs fed by thin Keplerian discs at large radius, in agreement with the critical disc model and in tension with most earlier global simulations. If correct, the result would explain why ultra-luminous X-ray sources can appear so bright while their black holes grow at essentially Eddington-limited rates, and it would sharpen the puzzle of how the first supermassive black holes grew so quickly.","feed_headline":"Super-critical discs cap their own feeding near Eddington","feed_subtitle":"Outflow cancels inflow at every radius, so black holes stay near the Eddington rate even when fed ten times over.","key_machinery":"The load-bearing object is the radial profile of mass flux through spherical shells: the inward flux $\\dot{M}_\\mathrm{in}$ and outward flux $\\dot{M}_\\mathrm{out}$, each integrated over the full $4\\pi$ solid angle, and their difference $\\dot{M}_\\mathrm{net}$. In a steady disc $\\dot{M}_\\mathrm{net}$ must be independent of radius, and the simulations achieve this with $\\dot{m}_\\mathrm{net} \\approx 1$ out to $r_\\mathrm{eq} \\sim 30$-$50\\,r_g$; the delicate balance between two large opposing fluxes is what enforces the Eddington limit. The simulations also track the unbound outflow fraction through the Bernoulli parameter and the radiative luminosity through spherical shells, and compare those against the analytic scalings of the critical disc model.","core_discovery":"The central claim is that large, steady super-critical accretion discs obey the Eddington limit locally: all the simulations converge to $\\dot{m}_\\mathrm{BH} \\approx 1$ over the radii where they have reached equilibrium, even though the inward mass flux at large radius $\\dot{m}_\\mathrm{in}$ can exceed $1000 \\dot{M}_\\mathrm{Edd}$. This happens because the outflowing flux $\\dot{m}_\\mathrm{out}$ adjusts to almost exactly cancel $\\dot{m}_\\mathrm{in}$, so the net accretion $\\dot{m}_\\mathrm{net}$ stays near unity at all radii. The authors show that the simulated profiles of mass flux, radiative luminosity versus radius and polar angle, and angular velocity all match the critical disc model, whereas the slim disc model predicts no significant outflow, a trapping radius about twenty times larger than observed, and sub-Keplerian rotation. They argue that most earlier torus-based simulations started with their critical radius outside the domain, which forced an advective rather than an outflow-dominated outcome.","pith_inferences":["If confirmed at higher resolution, the result would predict that the observed luminosity and variability of ultra-luminous X-ray sources track $\\dot{M}_\\mathrm{Edd}$ rather than the external supply rate, a testable distinction from slim-disc models.","The authors' own speculation implies that the outcome depends on the ratio $r_\\mathrm{cr}/r_\\mathrm{cir}$; one could test this directly by re-running a torus-based simulation with its pressure maximum placed beyond the critical radius and checking whether $\\dot{m}_\\mathrm{BH}$ drops toward unity.","The near-Eddington attractor may apply only to discs whose outer boundary is a thin Keplerian disc; in transients such as tidal disruption events, where the circularization radius can lie inside the critical radius, the same outflow-inflow cancellation need not operate."],"forward_implications":["Super-Eddington accretion need not imply super-Eddington black hole growth: a large, aligned, steady disc will instead launch most of the supplied mass outward, keeping the hole near $\\dot{M}_\\mathrm{Edd}$.","The trapping radius where advection dominates sits close to the hole, around $5$-$8\\,r_g$, so advection alone cannot explain super-Eddington discs beyond roughly $20\\,r_g$.","The discs yield high radiative efficiencies, $\\eta \\sim 0.3$-$0.7$, which the authors present as upper limits because some outward radiation remains trapped in the wind.","The luminosity is strongly concentrated toward the poles, matching the appearance of many ultra-luminous X-ray sources and explaining why the same source can look sub-Eddington from some viewing angles.","Growth of the first supermassive black holes cannot rely on steady, long-term accretion from a large, aligned, Keplerian disc if the Eddington limit holds as found here."],"supporting_citations":[{"why":"Provides the critical disc model whose predicted mass-flux, luminosity-radius, and luminosity-angle scalings the simulations are compared against.","marker":"Fukue 2004"},{"why":"Supplies the analytic $\\dot{m}_\\mathrm{in}(r)$ profile that the measured inward flux follows inside the critical radius.","marker":"Poutanen et al. 2007"},{"why":"Defines the slim disc model whose predictions of negligible outflow, a large trapping radius, and sub-Keplerian rotation the simulations rule out.","marker":"Abramowicz et al. 1988"},{"why":"Provides the thin-disc solution used as initial conditions and the original outflow-based resolution of the super-Eddington problem.","marker":"Shakura & Sunyaev 1973"},{"why":"Offers the Bernoulli-parameter definition of unbound outflow and a prior torus-based simulation whose higher $\\dot{m}_\\mathrm{BH}$ contrasts with the present result.","marker":"Sądowski & Narayan 2016"},{"why":"One of the earlier high-resolution super-Eddington simulations listed in the comparison table, which found $\\dot{m}_\\mathrm{BH} \\sim 22$.","marker":"Jiang et al. 2014"}],"fun_headline_variants":["Overfed black holes still obey the Eddington limit","Outflow cancels inflow, keeping accretion at Eddington","Super-critical discs self-regulate: accretion stays at Eddington","Inflow up to 1000x Eddington, net accretion stays at 1","Simulations show discs cap accretion at Eddington via outflows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results rest on the assumption that the under-resolved magnetorotational instability still produces physically realistic angular momentum transport and wind launching; the paper states it does not formally resolve the MRI, and higher-resolution turbulence could alter the outflow-inflow balance that produces near-Eddington net accretion.","fun_headline_variants_meta":{"raw":{"variants":["Overfed black holes still obey the Eddington limit","Outflow cancels inflow, keeping accretion at Eddington","Super-critical discs self-regulate: accretion stays at Eddington","Inflow up to 1000x Eddington, net accretion stays at 1","Simulations show discs cap accretion at Eddington via outflows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1712,"prompt_tokens":1097,"completion_tokens":615,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":713,"completion_tokens_details":{"reasoning_tokens":522}},"tokens_in":713,"tokens_out":615,"duration_ms":5609,"temperature":1.0,"reasoning_tokens":522,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:45:52.959879+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same initial conditions with a resolution that resolves the MRI (quality factors $Q_\\theta, Q_\\phi \\gtrsim 10$) and measure $\\dot{m}_\\mathrm{BH}$; if the net accretion rises well above $\\dot{M}_\\mathrm{Edd}$ once turbulence is resolved, the near-Eddington result is a grid artifact rather than a physical limit.","supporting_citations":[],"review_version":1}