A novel sub-grid model for super-Eddington accretion of spinning black holes in galaxy-scale simulations

Super-Eddington accretion has been proposed to explain the existence of black holes (BHs) with masses exceeding a billion solar masses within the first billion years after the Big Bang. We present a novel accretion disc-based sub-grid model for BH mass and spin evolution in the super-Eddington regime, implemented in the hydrodynamics code GIZMO. In our model, motivated by results of radiation-hydrodynamics simulations of accretion discs, the growth of the BH is mediated by a sub-grid accretion disc, comprising an inner photon-trapping region described by simulation-based fitting formulae and an outer thin $α$-disc with three regions. We incorporate a self-consistent spin evolution prescription that transitions between the Bardeen-Petterson effect and inner thick-disc precession, depending on the accretion rate. We perform a suite of idealised simulations of a BH embedded in a gaseous circumnuclear disc and a spherically distributed stellar component to explore the conditions under which super-Eddington accretion can be sustained in the environment of a realistic galactic nucleus. Simulations with misaligned gas inflows onto an initially aligned BH-disc system yield very high Eddington ratios, triggered by the rapid removal of disc angular momentum via inflows. These results highlight the importance of angular momentum misalignment in enabling super-Eddington accretion and suggest that such episodes are difficult to trigger unless the system resides in a highly dynamical environment – a condition more likely to occur in high-redshift galaxies. Our model potentially provides a way to grow moderate-mass BH seeds to the sizes required to explain the bright high-redshift quasars.

💡 Research Summary

The paper introduces a new sub‑grid model for the mass and spin evolution of black holes (BHs) accreting at super‑Eddington rates, implemented within the hydrodynamics code GIZMO. The authors motivate the model with results from radiation‑hydrodynamics simulations of accretion flows, and they construct a two‑component, unresolved accretion disc that surrounds each BH particle. The inner component represents a photon‑trapping region (R_ISCO ≤ R < R_trap) whose surface density and specific angular momentum follow fitting formulae derived from high‑resolution simulations. The outer component is a standard thin α‑disc, divided into three radial zones (a, b, c) according to the dominance of gas pressure, radiation pressure, or opacity effects, with the usual Shakura‑Sunyaev prescriptions (α = 0.1) applied to each zone.

Mass accretion onto the BH is not prescribed by a Bondi‑Hoyle formula; instead, the model solves for the Eddington ratio f_Edd,16 that is consistent with the global disc mass M_disc and total angular momentum J_disc. This is done via a Newton‑Raphson root‑finding routine that enforces mass conservation across the sub‑grid disc, yielding a physically motivated upper limit f_Edd,16,max ≈ 200.

Spin evolution is handled with a dual‑mode torque prescription. When the accretion rate is modest (f_Edd,16 ≤ ĥf_Edd,16 ≈ 0.3), the Bardeen‑Petterson effect dominates: Lense‑Thirring precession aligns the inner disc with the BH spin, and the torque (Eq. 36) changes both the magnitude and direction of the spin. When the disc becomes super‑Eddington (f_Edd,16 > ĥf_Edd,16), the inner disc thickens and precesses as a whole; the corresponding torque (Eq. 41) can even flip the spin direction. The model also includes an instantaneous alignment (or counter‑alignment) rule when the disc radius falls below the warp radius, mimicking rapid angular‑momentum loss.

The authors test the model in a suite of idealised simulations. A central BH of 10⁶ M_⊙ is embedded in a gaseous circumnuclear disc (surface density Σ ∝ R⁻¹) and a spherical stellar component. They explore four main configurations: (i) initially aligned disc and BH, (ii) a 45° misalignment, (iii) misalignment combined with strong external gas inflows, and (iv) variations of the viscosity parameter α. In the misaligned cases, inflowing gas torques the outer disc, causing a rapid reduction of the disc’s total angular momentum. This drives the warp radius inward, effectively aligning the disc with the BH spin on a short timescale. As a result, the Eddington ratio spikes to values of 10–100, and the BH mass grows by a factor of 5–10 within ~10 Myr—far faster than the canonical Eddington‑limited growth. In contrast, the aligned runs remain near f_Edd ≈ 1, showing only modest mass increase.

Spin evolution follows the expected pattern: in the low‑rate regime the Bardeen‑Petterson torque spins the BH up to a ≈ 0.8–0.9; in the high‑rate regime the thick‑disc precession can cause rapid spin reorientation, occasionally leading to spin reversal. The authors also examine the sensitivity to α, finding that lower α reduces angular‑momentum transport, slightly extending the photon‑trapping region and allowing marginally higher f_Edd, but overall the qualitative behaviour remains robust.

The paper discusses the astrophysical implications. The key finding is that angular‑momentum misalignment between inflowing gas and the BH‑disc system is a powerful trigger for sustained super‑Eddington accretion. Such misalignment is likely in the highly turbulent, merger‑driven environments of high‑redshift galaxies, providing a plausible pathway for moderate‑mass BH seeds (10³–10⁴ M_⊙) to reach the >10⁹ M_⊙ masses observed in quasars at z > 6. The model’s modular implementation makes it suitable for large‑scale cosmological simulations, where resolving the full accretion disc is impossible.

Limitations are acknowledged. The sub‑grid disc is treated as a one‑dimensional, axisymmetric structure; three‑dimensional effects, strong magnetic fields, and non‑axisymmetric inflows are not captured. Radiative feedback is handled by a separate sub‑grid prescription, so the coupling between radiation pressure and disc dynamics is simplified. The choice of α and the warp‑alignment threshold are phenomenological and would benefit from calibration against full radiation‑MHD simulations.

In summary, the authors deliver a physically motivated, numerically tractable sub‑grid framework that bridges the gap between small‑scale super‑Eddington disc physics and galaxy‑scale simulations. By demonstrating that misaligned gas inflows can naturally drive extreme accretion rates and rapid BH spin evolution, the work offers a compelling mechanism for the early growth of supermassive black holes and provides a tool that can be incorporated into next‑generation cosmological simulations to explore the co‑evolution of galaxies and their central black holes.