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Bridging Scales in Black Hole Accretion and Feedback: Subgrid Prescription from First Principles

This paper presents long-duration, multizone GRMHD simulations spanning from the event horizon to the Bondi scale to derive first-principles, spin-dependent subgrid prescriptions for accretion and feedback in low-Eddington flows, revealing that black hole spins remain effectively frozen during quiescent phases and cautioning against extrapolating results from small-scale simulations to galactic models.

Original authors: Hyerin Cho, Ben S. Prather, Ramesh Narayan, Kung-Yi Su, Angelo Ricarte, Priyamvada Natarajan, Antonio J. Porras-Valverde

Published 2026-02-18

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CC BY 4.0

Original authors: Hyerin Cho, Ben S. Prather, Ramesh Narayan, Kung-Yi Su, Angelo Ricarte, Priyamvada Natarajan, Antonio J. Porras-Valverde

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Black Hole "Thermostat"

Imagine a supermassive black hole at the center of a galaxy as a giant, cosmic vacuum cleaner. Its job is to suck in gas and dust (the "food") from the surrounding galaxy. But this isn't just a one-way street. As the black hole eats, it also burps out massive jets of energy and particles (the "feedback") that push back against the galaxy.

This creates a delicate dance:

If the black hole eats too much, it burps too hard, blowing the gas away and stopping itself from eating more.
If it eats too little, the gas piles up, and it starts eating again.

For decades, scientists have struggled to simulate this dance because the scales are impossible to match. The black hole's "mouth" (the event horizon) is tiny, while the gas cloud it eats from (the Bondi radius) is huge—like trying to simulate a single grain of sand falling into a hurricane while also tracking the hurricane's path across a continent.

The Problem: The "Zoom-In" Trap

Previous computer simulations were like taking a magnifying glass to the black hole. They looked incredibly closely at the immediate neighborhood of the black hole (the "event horizon") but had to guess what was happening far away.

The authors of this paper argue that this approach is flawed. It's like trying to understand how a person digests a meal by only looking at their stomach, ignoring the fact that they are sitting at a dinner table with a waiter bringing food and a host trying to stop them from eating too much. The "stomach" (the black hole) behaves differently depending on what's happening at the "table" (the galaxy).

The Solution: The "Multizone" Method

The team developed a new way to run simulations called the "Multizone Method."

Think of it like a smart thermostat system in a house with many rooms:

The Kitchen (The Black Hole): This room changes temperature and pressure incredibly fast. The thermostat here checks every millisecond.
The Living Room (The Galaxy): This room changes slowly. The thermostat here checks once an hour.
The Connection: Instead of forcing the whole house to update every millisecond (which would take forever to compute), the system updates the Kitchen fast and the Living Room slow, but they constantly talk to each other.

By using this method, the team ran simulations that lasted for a very long time (in computer time), allowing the system to settle into a natural, stable rhythm. They didn't just look at the black hole; they watched the whole "dinner party" evolve.

The Key Discoveries

Here is what they found when they finally let the simulation run long enough to reach "dynamical equilibrium" (the natural state of the system):

1. The "Spin" Doesn't Matter as Much as You Think

Black holes can spin like tops. Scientists thought a fast-spinning top would eat differently than a slow one.

The Old View: Fast spin = huge energy jets.
The New View: While fast spins do create stronger jets, the amount of gas the black hole eats is actually suppressed by magnetic fields in a way that is surprisingly consistent, regardless of how fast it spins or how the gas was initially moving. The magnetic fields act like a brake, slowing down the gas before it even reaches the black hole.

2. The "Flickering" Light Bulb

In the past, scientists assumed the black hole ate at a steady, average rate.

The Reality: The black hole is like a flickering light bulb. It doesn't eat steadily; it gorges itself, then stops, then gorges again.
The team found that the amount of food eaten and the energy released follows a specific statistical pattern (called a "lognormal distribution"). This means the black hole is highly unpredictable in the short term, even if its long-term average is stable. This is crucial for cosmologists who need to know if a black hole will suddenly blow up a galaxy or stay quiet.

3. The "Frozen" Spin

One of the most surprising findings is about the black hole's spin speed over billions of years.

The Analogy: Imagine a spinning top. Usually, we think friction slows it down quickly. But in the "hot, quiet" phase of a galaxy (where the black hole isn't eating much), the black hole's spin is effectively frozen.
The math shows that it would take longer than the current age of the universe for the spin to change significantly during these quiet periods. The spin only changes dramatically when the black hole is in a "feast" mode (eating rapidly) or when two black holes crash into each other.

Why This Matters for the Future

This paper provides a new "recipe" (subgrid prescription) for scientists building models of the entire universe.

Before: They had to guess how black holes behaved, often using data from tiny, short simulations that didn't capture the full picture.
Now: They have a set of rules derived from "first principles" (physics from the ground up) that accounts for the black hole's spin and the size of the gas cloud.

The Takeaway:
If you want to simulate the history of the universe, you can no longer treat the black hole as a simple, steady vacuum cleaner. You have to treat it as a flickering, magnetic, spin-regulated engine that interacts with its galaxy over vast distances. The authors have handed the scientific community the manual on how to program this engine correctly.

In a Nutshell

The universe is a complex place where the very small (black holes) and the very large (galaxies) are tightly linked. This paper says, "Stop looking at the black hole in isolation." By using a new method that connects the tiny scales to the huge scales, they found that black holes are more chaotic, more magnetic, and more "frozen" in their spin than we previously thought. This new understanding will help us build better movies of how galaxies grow and die.

1. Problem Statement

The coupling between supermassive black holes (BHs) and their host galaxies spans an enormous dynamic range of spatial and temporal scales (from the event horizon to the Bondi radius and beyond). This poses a significant computational challenge:

Small-scale simulations: General Relativistic Magnetohydrodynamic (GRMHD) simulations can resolve horizon-scale physics and jet launching but typically only reach steady states out to a few hundred gravitational radii ( $r_g$ ). They often fail to capture the long-term dynamical equilibrium required for galactic-scale feedback.
Large-scale simulations: Cosmological and galaxy formation simulations require "subgrid prescriptions" to model unresolved BH accretion and feedback. Current prescriptions often rely on highly idealized analytic models (e.g., Bondi accretion, Blandford-Znajek mechanism) or are calibrated on short-duration, small-domain GRMHD simulations.
The Gap: It remains untested whether small-scale GRMHD results reliably extrapolate to galactic scales. Specifically, the dependence of accretion rates and feedback efficiency on BH spin ( $a_*$ ) and the Bondi radius ( $R_B$ ) in a self-consistent, long-term equilibrium is poorly constrained.

2. Methodology

The authors employ the multizone method, a framework developed by their group to bridge scales across $\sim 8$ orders of magnitude.

Simulation Framework:
- Code: KHARMA (Kokkos-based High-Accuracy Relativistic Magnetohydrodynamics).
- Domain: Extends from the event horizon ( $r \sim r_g$ ) to beyond the Bondi radius ( $R_B$ ).
- Technique: The method iterates between small and large timesteps across nested zones (a "V-cycle"). This allows the inner horizon-scale physics to evolve on its characteristic short timescale while the outer galactic-scale gas evolves on a longer timescale, ensuring bidirectional communication and global dynamical equilibrium.
Parameter Space:
- BH Spin ( $a_*$ ): A grid of values: $0, 0.1, 0.3, 0.5, 0.7, 0.9$ (and one run at $0.97$).
- Bondi Radius ( $R_B$ ): $4\times 10^2$ to $2\times 10^6$ $r_g$ .
- Physics: Hot, radiatively inefficient, advection-dominated accretion flows (ADAFs) with low Eddington ratios ( $f_{\text{Edd}} \lesssim 10^{-3}$ ).
- Initial Conditions (ICs): Primarily spherically symmetric Bondi flow with no initial gas angular momentum. Additional runs tested prograde/retrograde rotation and varying magnetic field strengths ( $\beta$ ) to verify robustness.
Duration: Simulations run for $700 t_B$ (Bondi timescales), significantly longer than typical GRMHD runs, allowing the system to reach a true statistical equilibrium.

3. Key Contributions

The paper provides first-principles, spin-dependent subgrid prescriptions derived directly from long-duration GRMHD simulations. These are designed for direct adoption in cosmological simulations and Semi-Analytic Models (SAMs).

The contributions are twofold:

Time-Averaged Prescriptions: Compact analytic fits for mean accretion rates and feedback power.
Stochastic Prescriptions: Characterization of the full probability distributions of accretion and feedback, acknowledging their intrinsic variability.
Spin Evolution: A derived law for BH spin evolution in the hot accretion mode.

4. Key Results

A. Time-Avered Accretion and Feedback (Prescription 1)

The simulations reveal that the final state is largely insensitive to initial gas rotation or magnetic field strength, but strongly dependent on $R_B$ and $a_*$ .

Accretion Rate ( $\dot{M}$ ): The accretion rate is suppressed relative to the classic Bondi rate ( $\dot{M}_B$ $\dot{M}_{B}$ ) due to magnetic fields.
- Scaling: $\dot{M}/\dot{M}_B \propto (R_B/r_g)^{-0.5}$ .
- Spin Dependence: Higher spin leads to lower accretion (stronger jet feedback).
- Formula: $\frac{\dot{M}}{\dot{M}_B} \approx (-3|a_*| + 4.7) \left(\frac{R_B}{r_g}\right)^{-0.5}$ .
Feedback Power ( $\dot{E}_{fb}$ ):
- Scaling: $\dot{E}_{fb} \propto (R_B/r_g)^{-0.6}$ .
- Spin Dependence: Feedback power scales linearly with spin ( $|a_*|$ ), contrary to the quadratic dependence ( $a_*^2$ ) often assumed in force-free jet models.
- Formula: $\frac{\dot{E}_{fb}}{\dot{M}_B c^2} \approx (0.98|a_*| + 0.13) \left(\frac{R_B}{r_g}\right)^{-0.6}$ .
Efficiency ( $\eta$ ): For rapidly spinning BHs ( $a_* \gtrsim 0.7$ ), the feedback efficiency is roughly constant ( $\eta \approx 0.3$ ) regardless of $R_B$ . For slow spins, efficiency drops slightly with increasing $R_B$ .

B. Statistical Distributions (Prescription 2)

Accretion and feedback are highly time-variable, especially for large $R_B$ and high spins.

Distribution Type: Both the accretion rate ( $\dot{M}$ ) and feedback efficiency ( $\eta$ ) follow lognormal distributions.
Variability: The width (standard deviation) of these distributions increases with larger Bondi radii.
Implication: Using only time-averaged values may miss the "bursty" nature of feedback. The paper provides the means and variances of $\ln(\dot{M})$ and $\ln(\eta)$ for stochastic modeling.

C. Spin Evolution

Spindown: In the hot accretion mode, BHs undergo spindown (loss of angular momentum) regardless of initial gas rotation direction. The system evolves to a state where the jet power intermittently flips, erasing memory of the initial spin orientation.
Spinup Parameter ( $s$ ): Defined as $s = \frac{da_*}{dt} \frac{M}{\dot{M}}$ $s = \frac{d a _{*}}{d t} \frac{M}{M ˙}$ . The simulations find a linear relationship:
- $s(a_*) \approx -3.7 a_*$ .
Timescale: The spindown timescale is extremely long:
- $t_s \approx 12 \left(\frac{10^{-3}}{f_{\text{Edd}}}\right) \text{ Gyr}$ .
- For typical low-accretion systems ( $f_{\text{Edd}} \lesssim 10^{-3}$ ), the spin is effectively frozen over cosmological times. Spin evolution occurs primarily during high-accretion phases or mergers.

D. Comparison with Small-Scale GRMHD

The authors demonstrate that short-duration, small-domain GRMHD simulations (e.g., those using torus ICs and running for $< 10^5 t_g$ ) fail to capture the true long-term equilibrium.

Small-scale runs often show distinct prograde/retrograde behaviors and higher feedback efficiencies for prograde spins.
The multizone method reveals that over long timescales, the system enters a highly variable state where these distinctions vanish, and feedback efficiency becomes independent of initial rotation. Direct extrapolation from small-scale simulations to galactic scales is cautioned against.

5. Significance

Robust Subgrid Models: This work provides the first set of subgrid prescriptions for BH accretion and feedback derived from first-principles GRMHD simulations that self-consistently span horizon-to-galaxy scales.
Spin-Dependence: It offers a physically motivated, spin-dependent framework for cosmological simulations, moving beyond simple analytic approximations.
Stochasticity: By quantifying the lognormal distributions, it enables more realistic modeling of "bursty" AGN feedback, which is crucial for regulating star formation in massive galaxies.
Spin Evolution: The finding that BH spins are "frozen" during quiescent phases challenges models that assume continuous spin evolution in low-accretion regimes, suggesting spin changes are rare events driven by major accretion episodes or mergers.
Methodological Validation: It validates the multizone approach as essential for capturing the global dynamical equilibrium of accretion flows, highlighting the limitations of conventional small-domain simulations.