⚛️ general relativity

Hierarchical Black Hole Mergers in Nuclear Star Clusters: A Combined Dynamical-Secular Channel for GW231123-like Events

This paper proposes that hierarchical black hole mergers like GW231123 naturally occur in nuclear star clusters through a combined channel of binary-single interactions and secular evolution driven by a central supermassive black hole, predicting a concurrent population of observable black hole-star binaries.

Original authors: Bin Liu, Dieran Wang, Dong Lai

Published 2026-07-17

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CC BY 4.0

Original authors: Bin Liu, Dieran Wang, Dong Lai

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, cosmic dance floor where the most extreme objects imaginable—black holes—are constantly bumping into each other. Sometimes, these invisible giants collide and merge, creating a new, heavier black hole and sending out ripples in space-time called gravitational waves. Scientists have been catching these ripples for years, but recently, they spotted a particularly strange dancer: a pair of black holes that were both incredibly heavy and spinning wildly fast. This discovery, named GW231123, is a puzzle because standard rules of stellar life say black holes shouldn't be this heavy or spin this fast unless they are the result of a previous merger. It's like finding a champion boxer who is also a heavyweight champion, but you can't figure out how they got so big without breaking the rules of the gym. To solve this mystery, scientists need to understand how these "second-generation" black holes find each other and crash together in the crowded, chaotic centers of galaxies.

This paper suggests a specific, chaotic dance routine that happens in the crowded cores of galaxies, known as nuclear star clusters, where a supermassive black hole sits like a giant, invisible bouncer at the center. The authors, Bin Liu, Dieran Wang, and Dong Lai, propose that the heavy, fast-spinning black holes we saw in GW231123 are likely "second-generation" (2G) objects—meaning they are the children of previous black hole mergers. The paper argues that these 2G black holes don't just randomly bump into each other; instead, they form a team through a series of specific interactions. First, a heavy black hole grabs a regular star to form a temporary duo. Then, another heavy black hole crashes into this duo, kicking the star out and leaving the two heavy black holes locked in a tight embrace. This new pair is then pushed by the gravity of the central supermassive black hole to spin faster and faster until they finally collide.

The authors used computer simulations and mathematical models to test three different ways these heavy pairs could form. They found that the most likely path is this "binary-single" interaction: a heavy black hole meets a star, they pair up, and then a second heavy black hole swaps places with the star. This process is much more efficient than the other two methods they considered: simply two heavy black holes crashing directly into each other (which is too rare) or a heavy black hole grabbing a star through a slow tidal pull (which usually ends with the star being eaten before a second heavy black hole can join the party). The paper suggests that once these heavy pairs are formed, they don't just sit there; the gravity of the central supermassive black hole acts like a cosmic slingshot, stretching their orbit and forcing them to merge quickly.

However, the paper also points out a major hurdle for this theory: if these heavy black holes merge again to form a third-generation object, the resulting explosion of energy (a "kick") is so strong that it likely throws the new black hole out of the galaxy entirely. This means that while we might see these second-generation mergers, it is very difficult for them to keep merging to become even heavier. The authors calculate that the rate of these mergers in galaxies like our own Milky Way matches the frequency of the GW231123 event, suggesting this "dance" is a plausible explanation. They also predict that this same process should create many pairs of heavy black holes and stars that haven't merged yet; these pairs might be spotted in the future as tiny, faint flashes of light or low-frequency gravitational waves. While the model relies on some simplified assumptions and is presented as a proof-of-concept rather than a final proof, it offers a compelling story for how the universe's heaviest black holes might be built, one chaotic collision at a time.

Technical Summary: Hierarchical Black Hole Mergers in Nuclear Star Clusters

Problem Statement
The detection of the binary black hole (BHB) merger GW231123 presents a significant challenge to standard stellar evolution models. With component masses ( $m_1 \approx 137 M_\odot$ , $m_2 \approx 103 M_\odot$ ) residing in the "high-mass gap" and high spin parameters ( $\chi_1=0.9, \chi_2=0.8$ ), the event strongly suggests a second-generation (2G) or higher origin, where the components are remnants of previous mergers. While Nuclear Star Clusters (NSCs) and Active Galactic Nuclei (AGN) disks are known environments for hierarchical growth, the specific physical mechanisms that efficiently produce merging 2G/2G BHBs with these properties remain unclear. Previous studies have often treated dynamical scattering and secular evolution separately, lacking a unified picture of how bound 2G/2G binaries form and merge.

Methodology
The authors propose a combined dynamical-secular channel operating within the nuclear star cluster of a Milky Way-like galaxy. The study employs semi-analytical models to evaluate three representative dynamical pathways for forming 2G/2G binaries, followed by merger analysis:

Model Setup: The NSC is modeled with four populations: single stars ( $m_s \approx 1 M_\odot$ ), stellar binaries, first-generation (1G) BHs ( $m_{1G} \lesssim 60 M_\odot$ ), and 2G BHs ( $m_{2G} \approx 100 M_\odot$ ). The stellar density follows a power-law cusp ( $n \propto R^{-\gamma}$ ) governed by the central Supermassive Black Hole (SMBH, $M_\bullet \approx 4 \times 10^6 M_\odot$ ).
Dynamical Channels:
- Channel 1 (GW Capture): Direct gravitational wave capture during 2G–2G encounters.
- Channel 2 (Tidal Capture): A 2G BH tidally captures a star to form a 2G/S binary, which subsequently undergoes an exchange interaction with another 2G BH to form a 2G/2G binary.
- Channel 3 (Binary-Single Interaction): A 2G BH interacts with a stellar binary, disrupting it to form a 2G/S binary, which then exchanges with a second 2G BH.
Merger Mechanisms: The evolution of formed binaries is tracked considering two merger drivers: direct Gravitational Wave (GW) emission and the von Zeipel–Lidov–Kozai (ZLK) effect induced by the central SMBH, which can drive eccentricity and accelerate coalescence.
Rate Calculation: A multi-stage evolutionary framework (governed by a set of differential equations) tracks the steady-state populations of single 2G BHs, intermediate 2G/S binaries, and final 2G/2G binaries to derive merger rates per galaxy.

Key Contributions and Results

Dominant Formation Channel: The analysis identifies the binary-single interaction (Channel 3) as the most efficient pathway for producing 2G/2G mergers.
- GW Capture is found to be inefficient due to long encounter timescales, yielding rates $10^2$ – $10^3$ times lower than the binary-single channel.
- Tidal Capture is largely ineffective for 2G/2G production because the intermediate 2G/S binaries tend to merge via GW emission before they can encounter a second 2G BH for an exchange.
- Binary-Single Interaction: This channel successfully produces 2G/2G binaries. While many formed binaries are "soft" and susceptible to disruption, a significant fraction possesses semi-major axes and eccentricities that allow them to merge rapidly.
Role of Secular Evolution: The ZLK effect driven by the central SMBH is crucial. For binaries formed with larger semi-major axes (the upper tail of the distribution), the ZLK mechanism significantly accelerates mergers, ensuring they coalesce before being disrupted by subsequent stellar scatterings.
Merger Rate Consistency: When scaled to the population of Milky Way-like galaxies, the predicted merger rate from the binary-single channel is consistent with the inferred rate of GW231123 within current model uncertainties.
Population Predictions:
- 2G/1G and 1G/1G Mergers: The model predicts that 2G/1G mergers are less efficient than 2G/2G due to longer GW merger timescales relative to disruption times. 1G/1G mergers are negligible in this specific dynamical-secular channel, implying that the 2G BHs required for this model must originate from other mechanisms (e.g., isolated binary evolution or AGN disk dynamics).
- 2G/Star Binaries: The model predicts an abundant population of 2G BH–star binaries. These may manifest observationally as micro tidal disruption events (micro-TDEs) or as low-frequency gravitational wave sources detectable by future space-based interferometers (LISA, Tianqin, Taiji).
Hierarchical Limits: The study highlights a constraint on hierarchical growth in NSCs. Mergers involving high-spin 2G BHs produce 3G remnants with large recoil velocities ( $>300$ km/s). Since the local escape velocity in the relevant NSC regions is often lower than this, most 3G BHs are ejected, suppressing further hierarchical growth (3G/3G mergers).

Significance
The paper provides a plausible dynamical-secular pathway for the formation of GW231123-like events in the nuclear star clusters of Milky Way-like galaxies. By combining sequential binary-single interactions with SMBH-induced secular evolution, the model resolves the difficulty of retaining and merging massive 2G BHs. The authors emphasize that their work serves as a "proof-of-concept" using simplified assumptions (fixed masses, idealized density profiles) to demonstrate the viability of this channel. The detection of the predicted 2G/star binaries (via micro-TDEs or low-frequency GWs) would provide crucial observational evidence to validate these dynamical assembly pathways. Furthermore, the framework offers a potential explanation for other recent GW events with high primary spins and mass ratios consistent with 2G/1G mergers.

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