Evidence for Low Universal Equilibrium Black Hole Spin in Luminous Magnetically Arrested Disks

TL;DR

This work investigates how black hole spin evolves in magnetically arrested disks (MADs) across a range of disk thicknesses and accretion regimes using high-resolution 3D GR(R)MHD and radiative simulations. The authors develop a semi-analytic thin-MAD spin-down model by decomposing horizon fluxes into hydrodynamic and electromagnetic components, finding that luminous thin MADs converge to a universal equilibrium spin of $a_{eq}^{\text{MAD,thin}}\approx0.3$, with a quadratic dependence on disk thickness: $a_{eq}^{\text{fit}}\simeq0.31-2.7(h/r)^2$ as $h/r\to0$. They show that the reduced spin-down in thin MADs arises from weaker jet-driven energy and angular momentum extraction, linked to jets becoming more monopolar and to a decoupling of magnetic forces from thermal disk structure at $h/r\lesssim0.1$. These results have broad implications for BH spin distributions in XRBs and LVK sources, AGN feedback, and the interpretation of spin measurements across radiative states, suggesting MADs drive BHs toward low spins regardless of high accretion rates given sufficient time. The study also provides a practical framework to predict spin evolution in thin MADs via a semi-analytic model anchored in horizon fluxes and jet physics, bridging simulations and observations.

Abstract

Relativistic collimated outflows, or jets, provide a crucial mode of active galactic nucleus feedback. Although jets extract their energy from the black hole (BH) rotation, their effect on the BH spin is poorly understood. Because the spin controls radiative and mechanical BH feedback, lack of first-principles models for spin evolution limits our ability to interpret observations, including the recent LIGO-Virgo-KAGRA spin constraints. Particularly important are luminous disks, which rapidly grow and strongly torque their BHs. Jetless and weakly magnetized standard luminous disks spin up their BHs to near-maximum spin, $a_{eq,NT}=0.998$. However, sufficient large-scale vertical magnetic flux can cause the inner disk to enter a magnetically arrested disk (MAD) state, whose jets can efficiently extract BH rotational energy and significantly spin down the BH. Lowell et al. (2024) found that nonradiative, thick MADs spin down their BHs to very low $a_{eq,MAD}^{thick}=0.07$. Their analytic model predicted that luminous, thin MADs also spin down their BHs to low $a_{eq,MAD}^{thin}\sim0.3\text{-}0.5$. To test this prediction, we perform 3D general relativistic (radiation) magnetohydrodynamic (GR(R)MHD) simulations of MADs across a wide range of BH spin ($-0.9\le{}a\le0.99$) and disk thickness ($0.03\le{}h/r\le0.3$, which corresponds to Eddington ratio, $0.35\le{}\dot{m}/\dot{m}_{Edd}\le\infty$). We find that luminous, thin MADs ($0.03\le{}h/r\le0.1$) efficiently spin down their BHs to a low universal equilibrium spin value, $a_{eq,MAD}^{thin}\approx0.3$: a maximally spinning BH ($a=1$) spins down to $a=0.5$ after accreting just $25\%$ of its initial mass. Our results follow quadratic convergence, $a_{eq,MAD}^{fit}\simeq0.3-2.7(h/r)^2\to0.3$ as $h/r\to0$, which we attribute to the aggressive cooling that renders disk thermodynamics irrelevant and magnetic forces insensitive to thermal $h/r$.

Figures (13)

• Figure 1: Smaller $h/r$ MADs (yellow) are thinner and result in wider, less collimated polar jets (dark purple), as seen in the vertical slices through the instantaneous fluid-frame density for our simulations with different disk scale heights, $h/r = 0.3$ (model H3a0.94hr), $0.1$ (model H1a0.94), and $0.03$ (model Ra0.94), as labeled on the panels. Black lines show the axisymmetric poloidal magnetic flux contours and demonstrate that the vertical magnetic flux floods both the BH and inner disk, as expected in MADs.
• Figure 2: Thin MADs spin down their BHs to a low equilibrium spin, $a_\mathrm{eq,MAD}^\mathrm{thin}\approx 0.3$: although it is higher than for thick MADs, which have $a_\mathrm{eq,MAD}^\mathrm{thick}\approx 0.07$, it is still much much lower than unity. We reveal this through the plots of the spin-up parameter, $s$, vs BH spin, $a$, for several disk aspect ratios ($h/r= 0.03, 0.05, 0.1, 0.3$). The positive values of the Novikov-Thorne (NT) disk's dot-dashed blue spin-up curve indicate sustained spin up to the equilibrium spin, $a_\mathrm{eq,NT}=1$. We show the nonradiative $h/r=0.3$ MADs with $\Gamma=4/3$ with gray crosses (simulation results) and dashed gray curve (semi-analytic model of lowell_rapid_2023). We show the $h/r=0.1$ MADs with filled purple circles, the $h/r=0.05$ MAD with a green diamond, and the $h/r=0.03$ MADs with orange squares for the radiative 2T and a black x symbol for cooled 1T simulations. We also show the higher-resolution radiatively inefficient MADs with $\Gamma=13/9$ with brown triangles. We indicate the equilibrium spin values, where the spin-up curves inferred for each of the simulation families vanish, with vertical semi-transparent lines of the corresponding colors. The values of $a_\mathrm{eq}$ for all thin MADs, $h/r \lesssim 0.1$, cluster around $a_\mathrm{eq,MAD}^\mathrm{thin} \simeq 0.3$, which is much smaller than $a_\mathrm{eq,NT} \approx 1$ and much larger than $a_\mathrm{eq,MAD}^\mathrm{thick} \approx 0.07$.
• Figure 3: MAD equilibrium spin tends to a universal value, $a_\mathrm{eq,MAD}^\mathrm{thin} \approx 0.31$, in the limit of luminous, thin MADs, as seen in the plot of the equilibrium BH spin, $a_\mathrm{eq,MAD}$, vs disk scale height, $h/r$. Data point colors correspondingly match those in Figure \ref{['fig:spinup']}. As the disk thickness decreases from $h/r=0.3$ to $h/r=0.1$, the equilibrium spin increases. However, it levels off for even thinner disks, $h/r=0.05$ and $0.03$, and approximately follows a quadratic fit, $a_\mathrm{eq,MAD}^\mathrm{fit} \approx 0.31 - 2.7(h/r)^2$, which we show with the dashed black line.
• Figure 4: Our new analytic model (purple lines) well describes the HD and EM components of the specific angular momentum flux supplied by thin MADs ($0.03 \le h/r \le 0.1$), as seen in the plots of the specific angular momentum flux vs spin for different disk thermal scale heights: $h/r=0.03$ (2T red squares and 1T black hexagons), $h/r=0.05$ (green diamonds), $h/r=0.1$ (purple circles), $h/r=0.3$ with $\Gamma = 4/3$ (gray plus symbols), and $h/r=0.3$ with $\Gamma = 13/9$ (brown triangles). [panel (a)]: Hydrodynamic angular momentum flux component, $l_\mathrm{HD}$, in MADs is much lower than in the NT disk (blue dashed line). Whereas $l_\mathrm{HD}$ in thick MADs (gray) remains roughly independent of spin, in thinner MADs $l_\mathrm{HD}$ decreases with increasing $a$. For $h/r=0.1$ (purple), the shapes of $l_\mathrm{HD}(a)$ and $l_\mathrm{NT}(a)$ curves are similar, so we fit $l_\mathrm{HD}(a)$ with a (purple dashed) $40\% \, l_\mathrm{NT}$ curve, which captures the data well. Interestingly, our thinnest disks ($h/r=0.03$) show very similar $l_\mathrm{HD}$ values (open red squares and filled black hexagons) to our $h/r=0.1$ results. Filled red squares show $l_\mathrm{HD} + l_\mathrm{rad}$. (We note that the $l_\mathrm{rad}$ measurement for Ra0.4, its sum with $l_\mathrm{HD}$ shown with the light filled red square, does not include a $\sigma$-cutoff to account for the numerical floors; the effects of $\sigma$-cutoffs for Ra0.3 and Ra0.94 imply that $l_\mathrm{rad}$ for Ra0.4 would be lower, shown with the dark red filled square.) [panel (b)]: Electromagnetic (EM) angular momentum flux component, $l_\mathrm{EM}$, is largest in thick MADs (gray) and is lower -- and consistent -- across all thin MADs ($h/r\leq0.1$). We also show the semi-analytic EM models for $h/r=0.3$ (gray) and $h/r=0.1$ (purple dashed curves). The reduced values of $s$ for $h/r=0.3$ with $\Gamma=13/9$ (brown) relative to $\Gamma=4/3$ (gray) in Figure \ref{['fig:spinup']} reflect the lower contributions from both $l_\mathrm{HD}$ and $l_\mathrm{EM}$ shown here.
• Figure 5: Our new analytic model (purple lines) well describes the HD and EM components of the specific energy flux supplied by all our thin MADs ($0.03 \le h/r \le 0.1$), as seen through the plots of the specific energy flux on the event horizon vs BH spin for different thermal scale heights: $h/r=0.3$ with $\Gamma=4/3$ (gray plus signs), $h/r=0.3$ with $\Gamma=13/9$ (brown triangles), $h/r=0.1$ (purple circles), $h/r=0.05$ (green diamonds), and $h/r=0.03$ (2T red squares and 1T black hexagons). [panel (a)]: Spin dependence of HD specific energy flux, $e_\mathrm{HD}$, shows that thin MADs with $h/r=0.1$ contribute less HD energy to the BH than thick MADs with $h/r = 0.3$. The values of $e_\mathrm{HD}$ for $h/r=0.1$ appear to follow a similar shape vs $a$ as for the NT disk. We plot $85 \%$ the $e_\mathrm{NT}$ curve with the purple dashed line. The $e_\mathrm{HD}$ points for $h/r=0.03$ (red and black) follow the same trend as $e_\mathrm{HD}$ of $h/r=0.1$. Filled red squares show $e_\mathrm{HD} + e_\mathrm{rad}$ and open red squares show $e_\mathrm{HD}$. (The $e_\mathrm{rad}$ measurement for Ra0.4, its sum with $e_\mathrm{HD}$ shown with the light filled red square, does not include a $\sigma$-cutoff to account for numerical floors; the effects of $\sigma$-cutoffs for Ra0.3 and Ra0.94 indicate that $e_\mathrm{rad}$ for Ra0.4 would be lower, shown with the dark filled red square). We find little difference in the $e_\mathrm{HD}$ of thin $h/r=0.1$ and thinnest $h/r\leq0.03$ MAD. [panel (b)]: The dashed purple line, which well represents the EM specific energy flux of thin MADs, is simply half of the thick MAD model (gray dashed line, Eq. \ref{['etafit']}). MADs extract EM energy from spinning BHs when $e_\mathrm{EM} < 0$. For $h/r=0.3$, both $e_\mathrm{HD}$ and $e_\mathrm{EM}$ show little difference between the $\Gamma=4/3$ (gray) and $\Gamma=13/9$ (brown) models, indicating that the spin-up difference in Figure \ref{['fig:spinup']} arises from lower HD and EM angular momentum flux values (Figure \ref{['fig:lHD_lEM']}). Thin MADs -- all our disks with $h/r \leq 0.1$ -- extract a similar, universal, amount of EM energy, which is approximately half of thick MADs.