Is GW231123 a hierarchical merger?

1. Problem Statement

The gravitational wave event GW231123 represents a significant challenge to current astrophysical models of binary black hole (BBH) formation. It is the most massive event detected to date, with component masses $m_1 \approx 137 M_\odot$ and $m_2 \approx 103 M_\odot$. Furthermore, it exhibits extremal dimensionless spins ($\chi_1 \approx 0.90$, $\chi_2 \approx 0.80$).

• Stellar Physics Constraints: Standard isolated stellar evolution struggles to explain these parameters. Pair-instability supernovae typically create a "mass gap" ($50-130 M_\odot$) where black holes cannot form, and efficient angular momentum transport in massive stars usually results in low natal spins ($\chi < 0.01$).
• The Hierarchical Hypothesis: A leading alternative is hierarchical assembly, where black holes formed from previous mergers (second-generation, or 2G) merge again in dense stellar environments (e.g., globular clusters). These remnants inherit orbital angular momentum, naturally resulting in high spins ($\chi_f \approx 0.7$).
• The Core Question: Can the specific combination of high mass and extreme spins observed in GW231123 be explained by the merger of two second-generation (2G+2G) black holes in a globular cluster, or do the data rules out this scenario?

2. Methodology

The authors employ a statistical framework developed by Passenger et al. (2024) to test the consistency of GW231123's spin parameters against a theoretical population model of hierarchical mergers.

A. Population Model

• Source: The model utilizes results from Borchers et al. (2025), who used the Cluster Monte Carlo Code (CMC) to simulate collisional dynamics in globular clusters.
• Assumptions:
  • Natal black holes have a spin of $\chi_{birth} = 0.2$ (a conservative estimate allowing for inefficient angular momentum transport).
  • The model accounts for the distribution of remnant spins ($\chi_f$) after mergers, considering cluster escape velocities and mass ratios.
  • The focus is specifically on 2G+2G mergers (two second-generation black holes merging).

B. Statistical Formalism (Normalized Evidence)

Instead of a simple parameter comparison, the authors calculate a p-value based on the "normalized evidence" ($Z$):

1. Hypothesis: Compare the likelihood of the signal being drawn from the hierarchical population model ($\pi_{2G}$) versus a uniform prior ($\pi_U$).
2. Statistic ($Z$): Defined as the ratio of the Bayesian evidence for the population model against the uniform prior.
   $$Z \equiv \frac{\int d\theta \, \mathcal{L}(\tilde{d}|\theta) \pi_{2G}(\chi_1, \chi_2) \pi(\eta)}{\int d\theta \, \mathcal{L}(\tilde{d}|\theta) \pi_U(\chi_1, \chi_2) \pi(\eta)}$$
3. Simulation: The authors simulate $\sim100$ hierarchical merger events, inject them into Gaussian noise, and calculate the distribution of $Z$ values ($\pi_{pop}(Z)$).
4. Calculation: They calculate $Z_{event}$ for the actual GW231123 data and determine the p-value as the probability mass of the simulated distribution below $Z_{event}$.

C. Waveform Models

To address known systematics in high-spin regimes, the analysis is performed using two distinct waveform models:

1. IMRPhenomXO4a (XO4a): Favored by the original GW231123 detection (Abac et al. 2025) via Bayes factor, but less calibrated for extreme spins ($\chi > 0.8$).
2. NRSur7dq4 (NRSur): Calibrated against Numerical Relativity (NR) simulations, considered more physically robust for high spins, though less favored by the original detection's Bayes factor.

3. Key Results

The results are highly dependent on the choice of waveform model, highlighting the tension between model fit and physical calibration.

• Using IMRPhenomXO4a (XO4a):

  • The inferred spins ($\chi_1 \approx 0.90, \chi_2 \approx 0.80$) are extremely high.
  • The calculated p-value is $p \ll 1\%$.
  • Conclusion: The 2G+2G hierarchical merger scenario is strongly rejected under this waveform model. The spins are too extreme to be consistent with the simulated cluster population.

• Using NRSur7dq4 (NRSur):

  • The posterior distributions allow for slightly lower spins (specifically $\chi_1$ can be lower than in XO4a).
  • The calculated p-value is $p \approx 5\%$.
  • Conclusion: The 2G+2G hierarchical merger scenario is plausible. While not the most probable outcome, it is statistically consistent with the data (within a 5% tail probability).

• Waveform Systematics: The discrepancy arises because XO4a prefers $\chi_1 \gtrsim 0.8$, whereas NRSur allows $\chi_1$ to be smaller, bringing the event closer to the peak of the predicted hierarchical spin distribution ($\chi \approx 0.7$).

4. Limitations and Caveats

• Mass Uncertainty: The study focuses solely on spins. It does not model the mass distribution of 2G+2G mergers, which is highly uncertain. The extreme masses of GW231123 might further complicate the hierarchical interpretation.
• Noise Artifacts: GW231123 is a short signal susceptible to non-stationary noise and glitches (detected in H1 and L1). While BayesWave was used to mitigate glitches, residual non-Gaussian noise could bias spin measurements toward $\chi \approx 1$.
• Alternative Generations: The study focuses on 2G+2G. The authors note that 2G+1G or 3G+2G mergers might be more consistent with the data (especially the primary spin), but these scenarios have lower merger rates and require different environmental assumptions (e.g., nuclear star clusters).

5. Significance and Conclusion

• Validation of Hierarchical Channels: The paper demonstrates that, depending on the waveform model used, GW231123 could be a hierarchical merger. This supports the existence of dense stellar environments capable of retaining and merging multiple generations of black holes.
• Waveform Model Criticality: The results underscore a critical need for better-calibrated waveform models in the high-spin, high-mass regime. The conflicting conclusions ($p \ll 1\%$ vs. $p \approx 5\%$) based solely on the choice of waveform highlight that current systematics may obscure the true nature of extreme events.
• Future Directions: The authors suggest that if future events or improved noise modeling confirm the high-spin nature of GW231123, it will provide strong evidence for hierarchical assembly. Conversely, if the spins are lower due to noise bias, standard formation channels might need re-evaluation.

In summary, the paper concludes that GW231123 is a plausible candidate for a hierarchical merger of two second-generation black holes, provided the analysis relies on the numerically calibrated NRSur waveform model and accounts for current uncertainties in noise and waveform systematics.