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

TL;DR

The paper presents a novel sub-grid model for BH mass and spin evolution in the super-Eddington regime, embedded within the GIZMO code. It combines an inner photon-trapping disc with an outer three-region thin α-disc to regulate mass growth and angular momentum transfer, with a spin evolution that switches between Bardeen-Petterson alignment and inner thick-disc precession depending on the Eddington ratio f_Edd,16. The results demonstrate that misaligned gas inflows can trigger sharp super-Eddington episodes, enabling substantial BH growth and notable spin changes, potentially explaining the rapid emergence of massive BHs at high redshift. This framework provides a physically motivated path to seeding and growing BHs to the masses required to power bright high-redshift quasars, offering insights into SMBH assembly in dynamically active galactic nuclei.

Abstract

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.

Figures (18)

• Figure 1: Surface density profile (first and third panels) and specific angular momentum profile (second and fourth panels) for accretion discs with $f_{\rm Edd, 16} = 10$ and 0.1, assuming $M_{\rm BH,6}=1$, $\alpha_{0.1} = 1$, and $a_{\rm BH} = 0$. The blue solid lines represent the profile constructed using our model, renormalised to ensure continuity (RN in the figure). The green dashed lines correspond to the fitting formulae from Kitaki_et_al_2018, i.e. Equations \ref{['eq:sigma_trap']} and \ref{['eq:L_trap']}. The red dotted lines show the surface density profile of regions (a) and (b) from Kato_et_al_2008, i.e. Equations \ref{['eq:sigma_a']} and \ref{['eq:sigma_b']}. We also plot the surface density profile from Frank_et_al_2002, which includes only region (c) of the thin $\alpha$-disc and has been commonly used in previous sub-grid models. The black vertical lines in the first and third panels indicate $R_{\rm ISCO}$ (solid), $R_{\rm trap}$ (dashed), $R_{\rm ab}$ (dashed-dotted), and $R_{\rm bc}$ (dotted). The black vertical lines in the second and fourth panels indicate $R_{\rm ISCO}$ and $R_{\rm trap}$. For $f_{\rm Edd, 16} = 0.1$, $R_{\rm trap} < R_{\rm ISCO}$, meaning that the photon-trapping region does not exist. Texts "PT", "(a)", "(b)", and "(c)" indicate the photon-trapping region and regions (a), (b), and (c) of the disc, respectively. The value of $R_{\rm tran}/R_{\rm g}$, which is $10^{-5}$ for $f_{\rm Edd, 16} = 10$ and $10^{-1}$ for $f_{\rm Edd, 16} = 0.1$, is not shown in this figure due to its small value. $R_{\rm sg}/R_{\rm g}$ (for $Q_{\rm min} = 1$) is equal to $4 \times 10^4$ [in region (b)] and $2.9 \times 10^5$ [in region (c)] for $f_{\rm Edd, 16} = 10$ and 0.1, respectively.
• Figure 2: Characteristic radii as a function of $f_{\rm Edd, 16}$ for different values of $M_{\rm BH}$: $R_{\rm trap}$ (dark blue line), $R_{\rm ab}$ (medium blue line), $R_{\rm bc}$ (light blue line), $R_{\rm sg}$ (green line), $R_{\rm ISCO}$ (grey line), and $R_{\rm tran}$ (solid black line). The warp radius ($R_{\rm warp}$, red dashed line) is defined in Section \ref{['sec:angular_momentum']}. The top, middle, and bottom panels correspond to $M_{\rm BH,6} = 0.1, 1$, and 10, respectively, with $\alpha_{0.1} = 1$, $Q_{\rm min} = 1$, and $a_{\rm BH} = 0.8$ (prograde disc). The shaded regions indicate the parameter space corresponding to different disc regions. The vertical black dotted lines represent $\hat{f}_{\rm Edd, 16}$, the critical value of $f_{\rm Edd, 16}$ distinguishing the two torque models (see Section \ref{['sec:angular_momentum']} for more details). The vertical black dashed-dotted lines represent $f_{\rm Edd, 16, max}$ (defined in Section \ref{['sec:mass_accretion']}), assuming $M_{\rm disc} = M_{\rm sg}$.
• Figure 3: The warp radius, $R_{\rm warp}$, as a function of $f_{\rm Edd,16}$ for $a_{\rm BH}=0.8$ (top row) and 0.2 (bottom row) and $M_{\rm BH, 6}$ = 0.1 (left-hand column), 1 (central column), and 10 (right-hand column), with $\alpha_{0.1} = 1$ and $\xi=0.7$. For each region, $R_{\rm warp}$ is calculated by assuming a power-law surface density profile, using Equations \ref{['eq:r_warp_a']}, \ref{['eq:r_warp_b']}, and \ref{['eq:r_warp_c']}, properly readjusted after the renormalisation of the surface density and specific angular momentum profiles. Dark red, red, and pink lines correspond to regions (a), (b), and (c), respectively. Black lines indicate $R_{\rm bc}$ (dotted), $R_{\rm ab}$ (dashed), and $\hat{f}_{\rm Edd, 16}$ (dash-dotted). When $f_{\rm Edd, 16} < \hat{f}_{\rm Edd, 16}$, $R_{\rm warp} = R_{\rm warp, b}$ if $R_{\rm warp, b} < R_{\rm bc}$; otherwise, $R_{\rm warp} = R_{\rm warp, c}$. When $f_{\rm Edd, 16} > \hat{f}_{\rm Edd, 16}$, $R_{\rm warp, a} > R_{\rm ab}$ and $R_{\rm warp, b} < R_{\rm ab}$. In this case, the Bardeen-Petterson configuration cannot be reached. For $f_{\rm Edd, 16} > \hat{f}_{\rm Edd, 16}$, we extrapolate $R_{\rm warp, b}$ and $R_{\rm warp, a}$ beyond their regions of validity to better illustrate this in the figure.
• Figure 4: Critical Eddington ratio, $\hat{f}_{\rm Edd, 16}$ (Equation \ref{['eq:f_edd_crit']}), as a function of the BH spin parameter, $a_{\rm BH}$, with $\alpha_{0.1} = 1$ and $\xi = 0.7$. The different curves correspond to $M_{\rm BH,6} = 0.1$ (cyan), 1 (blue), and 10 (dark blue).
• Figure 5: Time-scales $t_{\rm align}$ (red line), $t_{\rm gm}$ (dashed dark red line), and $t_{\rm prec}$ for prograde ($t_{\rm prec, pro}$; dark blue line) and retrograde ($t_{\rm prec, ret}$; light blue line) discs as a function of $f_{\rm Edd, 16}$, displayed for $a_{\rm BH} = 0.8$ (top row) and $0.2$ (bottom row), and for $M_{\rm BH, 6} = 0.1$ (left-hand column), 1 (central column), and 10 (right-hand column), with $\alpha_{0.1} = 1$ and $\xi = 0.7$. The vertical black dash-dotted line marks $\hat{f}_{\rm Edd, 16}$. We find that $t_{\rm prec} \sim t_{\rm gm}$, whereas $t_{\rm align} \gg t_{\rm gm}$ at $f_{\rm Edd, 16} = \hat{f}_{\rm Edd, 16}$.