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Orbital Migration of Interacting Stellar Mass Black Holes in Disks around Supermassive Black Holes. III. Mass Distribution of Hierarchical Mergers

Through 360 N-body simulations of interacting stellar-mass black holes in active galactic nucleus disks, this study demonstrates that frequent hierarchical mergers within migration traps deplete low-mass black holes, populate the upper mass gap with a uniform distribution peaking near 70 solar masses, and generate a distinct intermediate-mass black hole population, suggesting AGN disks are a viable source for gravitational wave events difficult to explain via traditional stellar evolution.

Original authors: Katherine L. Gonglewski, Amy Secunda, Mordecai-Mark Mac Low, K. E. Saavik Ford, Barry McKernan, Fabian R. N. Schneider

Published 2026-08-17

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CC BY 4.0

Original authors: Katherine L. Gonglewski, Amy Secunda, Mordecai-Mark Mac Low, K. E. Saavik Ford, Barry McKernan, Fabian R. N. Schneider

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 the universe as a giant, chaotic dance floor where stars are born, live, and die. When massive stars die, they often collapse into black holes—invisible, gravity-sucking pits that are so dense not even light can escape. Usually, these black holes float alone or pair up with a partner, eventually spiraling into each other and crashing in a spectacular explosion that ripples through space-time. These ripples are called gravitational waves, and we have detectors on Earth (like LIGO and Virgo) that can "hear" them. But here's the mystery: some of the black holes we've heard crashing together are surprisingly heavy—too heavy to be made by the standard recipe of a single dying star. It's like finding a sumo wrestler in a room full of jockeys; the standard rules of how stars die say they shouldn't be that big.

This is where the story gets wild. Scientists have a theory that these heavyweights might be born in a very specific, chaotic neighborhood: the swirling, super-hot gas disks around supermassive black holes at the centers of galaxies. Think of these disks as cosmic highways. In these highways, there are special "traffic jams" called migration traps. Just like cars might get stuck in a specific spot on a highway due to road conditions, black holes drifting through the gas disk get stuck in these traps. Once they are stuck, they can't escape, and they start bumping into each other. If they bump hard enough, they merge. Then, the new, heavier black hole stays stuck in the trap and bumps into someone else. This process, called "hierarchical merging," is like a game of cosmic stacking, where black holes keep eating each other to get bigger and bigger, potentially explaining those mysterious heavyweights.

The Cosmic Stacking Game: How Black Holes Get Fat in AGN Disks

In this new study, a team of researchers decided to play a massive game of cosmic simulation to see if this "traffic jam" theory actually works. They didn't just watch one or two black holes; they ran 360 different simulations, each acting like a mini-universe where they could watch hundreds of black holes interact over millions of years. They used a super-computer code that tracks how these black holes move, not just because of gravity, but also because of the "wind" and "friction" from the gas disk they are swimming through.

The scientists wanted to answer a simple question: If we start with a bunch of normal-sized black holes in these gas disks, do they naturally grow into the heavy monsters we see in the gravitational wave detectors? To test this, they tried two different starting recipes for the black holes. One recipe was based on what we see in real star clusters (the "Salpeter" recipe), and the other was based on complex computer models of how stars actually die (the "Bimodal" recipe). They also tweaked the rules of the game to see how easily the black holes would merge, checking if the black holes needed to be practically touching to combine or if they could merge from a bit further away.

The Results: A Cosmic Buffet

The simulations showed that the "traffic jam" idea works surprisingly well. No matter which starting recipe they used, the black holes quickly found the migration traps and started merging. Here is what happened:

The Upper Mass Gap Gets Filled: In the universe, there is a "forbidden zone" for black hole sizes between about 40 and 120 times the mass of our Sun (solar masses). Standard star death can't make black holes this big. But in these simulations, the black holes in the migration traps ate each other so fast that they filled this gap. They created a steady supply of black holes ranging from 40 to 100 solar masses, with a little extra pile-up around 70 solar masses.
The "Resonant" Dancers: A key part of the story involves "resonant orbiters." Imagine a black hole getting stuck in a special orbit around the main trap, kind of like a satellite. As new black holes drift in from the outer disk, they bump into this satellite, making it heavier. The satellite keeps growing until it hits a specific weight limit—about 70 to 80 solar masses. At that point, it gets too heavy to stay in its special orbit and crashes into the main black hole in the trap. This process creates a distinct "peak" in the number of black holes around 70 solar masses, which matches a potential signal scientists are seeing in real data.
The Giant IMBHs: If the gas disk lasts long enough (about 2 million years or more), the main black hole in the trap keeps eating until it becomes a true giant. The simulations showed these giants growing to over 200 solar masses, and in some cases, reaching up to 1,000 solar masses. These are called Intermediate Mass Black Holes (IMBHs). The study suggests that if we look at the centers of galaxies with future space telescopes, we might find these giants hiding in the gas.
The Rate of Mergers: The team calculated that these migration-trap mergers happen at a rate of about 6 per cubic gigaparsec per year. This is a significant chunk of the total black hole mergers we detect. They found that about 40% of these mergers involve black holes of very different sizes (uneven mass ratios), and about 16% involve a primary black hole between 50 and 100 solar masses. This suggests that AGN disks could be the main factory for the heavy black holes that are hard to explain with normal star evolution.

What the Paper Says and Doesn't Say

The researchers are careful to point out that these are simulations, not direct observations. They found that the results are robust, meaning they hold true even when they changed the rules slightly (like making the black holes merge at different distances). However, they also noted that if the gas disk disappears too quickly (in less than half a million years), the black holes might not have enough time to grow into the heavyweights we see. But if the disks come and go in episodes, the heavy black holes from one episode could be the starting point for the next, allowing them to keep growing.

The study also highlights that while these simulations explain the heavy black holes well, they don't necessarily explain the lighter ones (under 40 solar masses) as perfectly as other theories do. The heavyweights seem to be the special product of the AGN disk "traffic jam," while the lighter ones might still be coming from the standard "dying star" recipe.

Why It Matters

This paper gives us a strong clue about where the universe's heaviest black holes come from. It suggests that the chaotic, gas-filled centers of galaxies are like cosmic construction sites where black holes can stack up and grow into giants. If future gravitational wave detectors confirm these patterns—especially the specific "peak" around 70 solar masses and the existence of the 1,000-solar-mass giants—it would prove that these migration traps are real and active. It also opens the door for future space missions, like LISA, to hunt for these intermediate-mass giants, which could help us understand how long these gas disks last and how they shape the galaxies we see today.

Technical Summary: Orbital Migration of Interacting Stellar Mass Black Holes in Disks around Supermassive Black Holes. III. Mass Distribution of Hierarchical Mergers

Problem Statement
The LIGO-Virgo-KAGRA (LVK) collaboration has detected over 150 binary black hole (BBH) mergers, with a significant fraction involving components in the "upper mass gap" ( $\sim40\text{--}120\,M_\odot$ ), a regime where traditional stellar evolution models struggle to form black holes (BHs). While hierarchical mergers in dense stellar environments (e.g., globular clusters, nuclear star clusters) are a proposed solution, Active Galactic Nucleus (AGN) disks offer a distinct channel where gas-driven migration can concentrate BHs. Previous population studies of AGN merger channels have been limited to Monte Carlo methods or 1-D N-body simulations, which evolve orbiters statistically rather than explicitly integrating their phase-space evolution under gravitational and gas forces. This paper addresses the need for a high-fidelity population study to determine if AGN disks can reproduce the observed LVK mass function (MF), specifically the upper mass gap and intermediate mass black hole (IMBH) populations.

Methodology
The authors performed 360 high-resolution 3-D N-body simulations using a Bulirsch-Stoer code augmented with analytic models for gas forces (migration torques, eccentricity/inclination damping, and turbulent density fluctuations). The simulations model a region from $200\text{--}1000\,R_g$ surrounding a migration trap at $\sim650\,R_g$ in an AGN disk orbiting a $10^8\,M_\odot$ supermassive black hole (SMBH).

Key methodological parameters include:

Duration: Each simulation runs for 10 Myr, representing the upper estimate of an AGN disk lifetime.
Initial Mass Functions (IMFs): Two distinct BH IMFs were tested:
1. A Salpeter (1955) IMF ( $dN/dm \propto M^{-2.35}$ ) with a range of $5\text{--}40\,M_\odot$ .
2. A bimodal IMF derived from Schneider et al. (2023) based on isolated binary stellar evolution, featuring peaks at $\sim9\,M_\odot$ and $\sim16\,M_\odot$ , also truncated at $40\,M_\odot$ .
Merger Criteria: The default criterion assumes a BBH forms and merges if the separation $\Delta R \le 1\,R_H$ (mutual Hill radius) and relative kinetic energy is below binding energy. To test sensitivity, additional runs utilized stricter ( $\Delta R \le 0.7\,R_H$ ) and looser ( $\Delta R \le 1.5\,R_H$ ) separation criteria.
Boundary Conditions: New BHs are injected at the outer boundary every $10^5$ years, with masses drawn from the respective IMF, to simulate migration from the outer disk and capture of inclined orbiters.

Key Contributions and Results

Mass Distribution Evolution:
- Depletion of Low Masses: Frequent hierarchical mergers rapidly deplete BHs with masses $\lesssim 10\,M_\odot$ within $<0.5$ Myr, as the merger rate exceeds the replenishment rate from migration and disk capture.
- Filling the Upper Mass Gap: Regardless of the initial IMF or merger criteria, the simulations fill the upper mass gap ( $40\text{--}100\,M_\odot$ ) with a roughly uniform distribution. A distinct peak emerges around $\sim70\,M_\odot$ , attributed to resonant orbiters that merge with the primary BH in the migration trap.
- IMBH Formation: After $\sim2$ Myr, a separate population of IMBHs ( $>200\,M_\odot$ ) emerges. By 10 Myr, the migration trap hosts a massive IMBH population peaking at $\sim1000\,M_\odot$ ( $1000 \pm 100\,M_\odot$ ), driven by oligarchic growth.
Resonant Orbiters and the "IMBH Mass Gap":
- The simulations identify a specific population of 1–3 resonant orbiters trapped in first-order resonances ( $k+1:k$ ) with the migration trap BH.
- These orbiters grow from $\sim15\,M_\odot$ to a threshold of $\sim80\,M_\odot$ through mergers with incoming BHs. Once they reach this threshold, resonant overlap occurs, leading to a merger with the migration trap BH.
- This mechanism creates a "mass gap" in the primary mass distribution between $\sim80\,M_\odot$ and $\sim800\,M_\odot$ , as resonant orbiters rarely exceed $\sim80\,M_\odot$ before merging with the trap.
Merger Rates and Mass Ratios:
- The calculated average merger rate for migration-trap-aided BBHs is $\sim6.2\,\text{Gpc}^{-3}\,\text{yr}^{-1}$ , accounting for approximately 25–50% of the total LVK observed rate.
- Approximately 40% of mergers are uneven mass ratio events ( $q < 0.5$ ), and 16% involve a primary mass in the range $[50, 100]\,M_\odot$ .
- The bimodal IMF simulations produce a primary MF with peaks at $10\,M_\odot$ and $15\text{--}19\,M_\odot$ , which aligns more closely with the low-mass features of the GWTC-4 catalog than the Salpeter runs, though neither IMF reproduces the $\sim35\,M_\odot$ peak observed in GWTC-4.
Robustness to Merger Criteria:
- Varying the merger separation criterion ( $\Delta R$ ) alters the speed of IMBH growth and the distinctness of the mass gaps but does not change the qualitative outcomes. Stricter criteria ( $\Delta R = 0.7\,R_H$ ) slow growth, while looser criteria ( $\Delta R = 1.5\,R_H$ ) accelerate it, yet both result in upper mass gap filling and IMBH formation.

Significance and Claims
The paper claims that AGN disks are a viable and efficient source for the high-mass BBH mergers observed by LVK, particularly those in the upper mass gap and the $\sim70\,M_\odot$ range, which are difficult to produce via isolated stellar evolution. The study suggests that:

The observed LVK mass spectrum in the upper mass gap could be largely explained by migration-trap-assisted hierarchical mergers in AGN disks.
The specific peak at $\sim70\,M_\odot$ is a robust prediction of the migration trap mechanism, arising from the physics of resonant orbiters rather than the initial IMF.
Future gravitational wave detectors (LISA, Cosmic Explorer, Einstein Telescope) can constrain AGN disk models by detecting the predicted population of IMBHs ( $>200\,M_\odot$ ) and intermediate mass ratio inspirals (IMRIs). Specifically, the ratio of light IMRIs (stellar mass BH + IMBH) to heavy IMRIs (IMBH + SMBH) could serve as a probe for the average lifetime of AGN disks.

The authors maintain that while their 3-D N-body approach is computationally expensive and limited in parameter space compared to Monte Carlo methods, it provides a necessary explicit integration of phase-space evolution that confirms the robustness of hierarchical merger outcomes in AGN environments.

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