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RABBITS - III. Modelling relativistic accretion discs around spinning black holes in galaxy formation simulations

This paper presents a new geometrically thin relativistic accretion disc model within the RABBITS series that self-consistently evolves black hole mass and spin by analytically combining local relativistic solutions, thereby replacing stochastic Bondi prescriptions with first-principles physics to enable more accurate quasar energetic output predictions and direct spectral energy distribution generation in galaxy formation simulations.

Original authors: Dimitrios Irodotou, Shihong Liao, Theodoros Nakas, Geoffrey Compère, Roberto Oliveri, Jessica M. Hislop, Alexander Rawlings, Sonja Soininen, Aswin P. Vijayan

Published 2026-03-19

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CC BY 4.0

Original authors: Dimitrios Irodotou, Shihong Liao, Theodoros Nakas, Geoffrey Compère, Roberto Oliveri, Jessica M. Hislop, Alexander Rawlings, Sonja Soininen, Aswin P. Vijayan

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

Imagine a galaxy as a bustling, cosmic city. At the very center of this city sits a supermassive black hole, a gravitational "vacuum cleaner" so powerful that not even light can escape it. For decades, scientists trying to simulate how these galaxies grow and evolve have had to guess how this vacuum cleaner works. They used a "black box" approach: they'd say, "Okay, gas is near the black hole, so let's just make the black hole eat a little bit of it randomly."

This new paper, part of the RABBITS series (Resolving supermAssive Black hole Binaries In galacTic hydrodynamical Simulations), throws out the guesswork. Instead of a black box, the authors have built a detailed, 3D blueprint of the "dining room" right next to the black hole: the accretion disc.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Stochastic" Guess

In old simulations, the black hole's feeding was like a slot machine. You pulled the lever (ran the simulation), and sometimes it ate a lot, sometimes a little, based on random chance. This made it hard to predict exactly how much energy the black hole would release or how fast it would spin. It was like trying to predict the weather by flipping a coin.

2. The Solution: The "Relativistic Blueprint"

The authors created a new model that treats the accretion disc not as a blur, but as a structured, physical object. Think of the accretion disc as a giant, swirling pizza dough being spun around the black hole.

The Layers: Just like a pizza has a crust, sauce, and cheese, this "pizza" has different layers depending on how close you are to the center.
- Near the center: It's super hot and thin, dominated by radiation (light pressure).
- Further out: It's cooler and thicker, dominated by gas pressure.
The Physics: The authors didn't just guess the thickness or temperature. They used Einstein's theory of relativity to calculate exactly how the dough behaves. They combined different mathematical "recipes" (solutions) for different parts of the disc to build one giant, complete picture.

3. The "Spin" Factor

Black holes aren't just heavy; they spin. Imagine a figure skater pulling in their arms to spin faster.

Co-rotation: If the pizza dough spins the same way the skater spins, it's smooth sailing. The dough can get very close to the skater, releasing a lot of energy.
Counter-rotation: If the dough spins the opposite way, it's a chaotic mess. It crashes into the skater's path, creating friction and instability.
The Breakthrough: This model tracks the spin direction of the black hole and the disc in real-time. It calculates how the black hole's spin changes as it eats the dough. This is crucial because a spinning black hole acts differently than a stationary one, affecting how it heats up the surrounding gas and influences the galaxy's growth.

4. Why This Matters: From "Guessing" to "Predicting"

Because the model is so precise, it allows scientists to do two new things that were impossible before:

Measuring the Size: They can now predict exactly how big the "pizza" is (the size of the accretion disc) based on the black hole's mass and spin. They can compare this directly to real telescopes looking at distant quasars to see if their math matches reality.
Creating a "Soundtrack" (Spectral Energy Distribution): By knowing the temperature of every part of the disc, they can generate a "spectral energy distribution" (SED). Think of this as the black hole's soundtrack.
- Old models just said, "It's bright."
- This model says, "It's bright in blue light here, dim in red light there, and here's a specific hum in the ultraviolet."
- This allows astronomers to compare the simulation's "song" directly with the "song" recorded by telescopes like the James Webb Space Telescope.

5. The "RABBITS" Toolkit

The authors didn't just write a paper; they built a toolkit for the whole scientific community:

C Version: A module that can be plugged directly into massive galaxy simulations to run in real-time.
Python Version: A tool for scientists to take existing simulation data and "post-process" it to build detailed 3D models of the discs afterward.
Updated Relagn: An upgraded tool to generate the "soundtracks" (SEDs) for any set of black hole parameters.

The Bottom Line

Before this paper, simulating a galaxy was like trying to direct a movie where the main character's actions were decided by a dice roll. Now, thanks to this new model, we have a script. We know exactly how the black hole eats, how it spins, how big its "dining room" is, and what kind of light it emits. This brings us one giant step closer to understanding the true relationship between these cosmic monsters and the galaxies they call home.

1. Problem Statement

Galaxy formation simulations have historically relied on sub-grid models to describe black hole (BH) accretion and feedback, primarily using the Bondi-Hoyle-Lyttleton (BHL) prescription. These classical approaches suffer from several critical limitations:

Stochasticity: Accretion rates are often determined stochastically based on local gas properties, leading to numerical noise rather than physical variability.
Lack of Relativistic Physics: Most models use the Newtonian Shakura & Sunyaev (1973) prescription, which neglects general relativistic effects crucial near spinning (Kerr) black holes, such as the Innermost Stable Circular Orbit (ISCO) and frame-dragging.
Inaccurate Spin Evolution: Without a physically motivated accretion disc structure, the evolution of the BH spin vector (magnitude and direction) is poorly constrained, affecting predictions for gravitational wave signatures and radiative efficiency.
Missing Observables: Standard models cannot directly predict accretion disc sizes or Spectral Energy Distributions (SEDs) from first principles, hindering direct comparison with observational data (e.g., microlensing and reverberation mapping).

2. Methodology

The authors develop and implement a geometrically thin, relativistic accretion disc (AD) model that self-consistently evolves the mass and spin vector of supermassive black holes (SMBHs) within Smoothed Particle Hydrodynamics (SPH) galaxy formation simulations.

Core Components

Relativistic Local Solutions: The model employs a suite of analytic solutions for steady-state accretion discs based on the Novikov & Thorne (1973) framework, extended by Compère & Oliveri (2017).
- Regimes: The disc structure is constructed by piece-wise combining three local regimes based on which pressure and opacity mechanisms dominate:
  - Gas-ES: Gas pressure dominated, electron scattering opacity.
  - Rad-ES: Radiation pressure dominated, electron scattering opacity.
  - Gas-FF: Gas pressure dominated, free-free absorption opacity.
- Intra-ISCO Region: The model incorporates recent analytic solutions by Mummery & Balbus (2023) to smoothly connect the region inside the ISCO (plunging region) to the standard disc, relaxing the unphysical "zero-torque" boundary condition of the original Novikov & Thorne model.
Global Disc Construction:
- The global surface density profile is built by identifying the radial boundaries where the inequalities defining the validity of each regime (Gas-ES, Rad-ES, Gas-FF) are satisfied.
- The outer radius ( $R_{outer}$ ) is determined by the relativistic Toomre parameter ( $Q$ ), ensuring the disc remains stable against self-gravity.
- The inner radius is the photon radius ( $R_{ph}$ ), and the transition to the plunging region occurs at the ISCO ( $R_{isco}$ ).
Spin and Angular Momentum Evolution:
- Alignment: The model explicitly calculates the total angular momentum of the disc ( $J_{\odot}$ ) by integrating the surface density profile. It uses the King et al. (2005) criterion to determine if the disc is co-rotating or counter-rotating relative to the BH spin.
- Bardeen-Petterson Effect: The model calculates the torque exerted by the spinning BH on the disc (and vice versa) to evolve the BH spin direction, accounting for Lense-Thirring precession.
- Mass Accretion: Mass flows from the disc to the BH at the ISCO rate, updating the BH mass and spin magnitude.
Feedback Implementation:
- Radiative efficiency ( $\varepsilon_r$ ) is calculated self-consistently, including spin-dependent terms and torque contributions at the ISCO.
- Thermal feedback energy is deposited into the surrounding gas based on this efficiency.
Software Implementation:
- A C version is provided for integration as a module in galaxy formation simulations (using the Gadget-3/Sphgal code).
- A Python version is available for post-processing simulations to generate mock accretion discs and SEDs.

3. Key Contributions

First Principles Accretion: The model derives every episode of BH mass accretion from the physical structure of the disc, eliminating the stochasticity inherent in Bondi-type prescriptions.
Relativistic Consistency: It is the first model in this context to fully incorporate Kerr geometry, spin-dependent ISCO radii, and the Bardeen-Petterson effect within a cosmological simulation framework.
Predictive Power for Observables:
- Disc Sizes: The model can predict the half-light radius of accretion discs directly from simulation parameters.
- SED Generation: By extracting radial profiles of temperature and opacity, the model can generate Spectral Energy Distributions (SEDs) that can be compared with quasar observations.
Open Source Tools: The authors released the model as a public C and Python module, along with an updated version of the Relagn model for SED generation.

4. Results

The authors tested the model through parameter space exploration and a simulation of an isolated disc galaxy (NGC 4038 analog).

Parameter Sensitivity:
- Viscosity ( $\alpha$ ): Higher viscosity leads to lower surface densities but larger disc extents (more stable against self-gravity).
- Black Hole Mass ( $M_\bullet$ ): Higher mass BHs have larger ISCO and photon radii, shifting the entire disc structure outward.
- Accretion Rate ( $\dot{M}$ ): High accretion rates trigger the Rad-ES regime (radiation pressure dominated) near the BH, potentially driving outflows.
- Spin ( $\alpha_\bullet$ ): Spin significantly affects the ISCO radius and radiative efficiency but has a weaker effect on the global disc structure compared to mass and accretion rate.
- Orientation: Counter-rotating discs have larger ISCO radii and different stability profiles compared to co-rotating discs.
Simulation Outcomes (Isolated Galaxy):
- Stochasticity Reduction: The model produces smooth, physically motivated accretion rates, unlike the noisy fluctuations seen in standard BHL models.
- Spin Evolution: The model reveals that "chaotic" accretion (random disc orientation) does not necessarily lead to spin-down. Due to the explicit calculation of the total disc angular momentum, large discs often satisfy the alignment criterion for co-rotation, leading to spin-up.
- Feedback Efficiency: Higher spin BHs (resulting from specific alignment histories) exhibit higher radiative efficiencies, leading to stronger feedback that suppresses further gas accretion, creating a self-regulating loop.
- Disc Sizes: The simulated half-light radii are systematically larger than standard Shakura & Sunyaev predictions but align better with some observational datasets (though still showing discrepancies with microlensing data, a known issue in the field).
- SEDs: The generated SEDs show distinct differences from standard blackbody models, particularly in the UV/optical "big blue bump" and the infrared tail, due to the piece-wise temperature profile.

5. Significance

This work represents a paradigm shift in how black hole accretion is treated in galaxy formation simulations.

Bridging Scales: It successfully bridges the gap between unresolved sub-grid physics and resolved galactic scales, allowing simulations to capture relativistic effects without requiring prohibitively expensive global MHD simulations.
Multi-Messenger Astronomy: By accurately modeling spin evolution and feedback, the model improves predictions for gravitational wave event rates and recoil velocities from BH mergers.
Observational Constraints: The ability to generate SEDs and predict disc sizes directly from simulations allows for rigorous testing of accretion physics against observational data (e.g., from the Event Horizon Telescope, microlensing surveys, and reverberation mapping).
Future Applications: The framework provides a flexible foundation for future extensions, including radiatively inefficient flows (ADAFs), jet-mode feedback, and the evolution of supermassive black hole binaries.

In summary, the RABBITS-III model provides a robust, physically motivated, and computationally efficient tool to study the co-evolution of black holes and galaxies, offering a path toward resolving long-standing discrepancies between theory and observation in AGN physics.