Stellar black hole binaries from two common envelope evolution phases in triple stellar systems

Technical Summary: Stellar Black Hole Binaries from Two Common Envelope Evolution Phases in Triple Stellar Systems

Problem Statement
Gravitational wave detectors (LIGO-Virgo-KAGRA) have observed approximately 300 binary black hole (BBH) merger events. A key observational feature is the distribution of the effective inspiral spin parameter, $\chi_{\text{eff}}$. The data reveals an asymmetric distribution: while the mean is slightly positive ($\langle \chi_{\text{eff}} \rangle \approx 0.054$), there is a significant tail of systems with negative $\chi_{\text{eff}}$.
Existing formation channels struggle to reproduce this asymmetry in isolation:

• Isolated binary evolution typically predicts $\chi_{\text{eff}}$ narrowly peaked near zero.
• Pure dynamical assembly (e.g., in globular clusters or AGN disks) predicts a distribution symmetric about zero, failing to account for the observed positive mean.
• Hybrid scenarios exist, but the specific origin of the negative tail and the positive mean simultaneously remains an open question.

Methodology
The authors propose and model a specific evolutionary channel within the framework of the Jittering-Jets Explosion Mechanism (JJEM) for core-collapse supernovae (CCSNe). The scenario involves a hierarchical triple-star system undergoing two distinct Common Envelope Evolution (CEE) phases.

1. Evolutionary Scenario:

   • Initial State: A hierarchical triple system consisting of a massive primary ($M_1$), a massive secondary ($M_2$), and a low-mass tertiary ($M_3 \lesssim 5 M_\odot$).
   • First CEE: As the primary evolves off the main sequence, it engulfs the tertiary. The tertiary spirals into the primary's core, tidally disrupting and spinning up the core. This rapid rotation is required by the JJEM to form the first black hole (BH1) via a successful CCSN. The spin of BH1 ($\vec{\chi}_1$) aligns with the inner orbital angular momentum ($\vec{L}_{\text{orb,in}}$), which is inclined relative to the outer orbit.
   • Second CEE: The secondary star evolves into a Red Supergiant (RSG) and engulfs BH1. BH1 spirals in, spinning up the secondary's core. The secondary then collapses to form the second black hole (BH2). The spin of BH2 ($\vec{\chi}_2$) aligns with the binary orbital angular momentum ($\vec{L}_{\text{orb}}$).
   • Result: The final BBH system possesses a spin distribution where $\vec{\chi}_2$ is aligned with the orbit, but $\vec{\chi}_1$ may be highly misaligned (or even anti-aligned) due to the initial inclination between the inner and outer orbits. Accretion during the second CEE may partially align $\vec{\chi}_1$, but significant misalignment remains possible, creating the negative $\chi_{\text{eff}}$ tail.

2. Numerical Approach:

   • The study employs a semi-analytical approach rather than full population synthesis to estimate the viability of the channel.
   • Draw Probability ($P_{\text{draw}}$): Calculated using empirically calibrated initial distributions (Moe & Di Stefano 2017) for massive stars, including mass, period, mass ratio, and eccentricity. Constraints are applied to ensure the system meets the evolutionary requirements (e.g., $M_1 \geq 25 M_\odot$, specific orbital period ranges).
   • Success Fraction ($\beta$): Defined as the conditional probability that a system satisfying the initial ZAMS conditions successfully evolves through the full sequence (two CEEs, two CCSNe, and a final tight orbit) to merge within a Hubble time.
   • Rate Normalization: The calculated merger rate is normalized against the observed local CCSN rate ($R_{\text{CCSN}} \approx 6.9 \times 10^4 \text{ Gpc}^{-3} \text{ yr}^{-1}$) and the observed BBH merger rate ($R_{\text{BBH}} \approx 17.4 - 31.3 \text{ Gpc}^{-3} \text{ yr}^{-1}$).
   • Stability Check: Dynamical stability is enforced using the Aarseth & Mardling (2001) criterion. Additionally, the authors use the Multiple Stellar Evolution (MSE) code to verify that the proposed evolutionary sequence is dynamically realizable and to refine the allowed inner-orbit period ranges.

Key Results

• Parameter Constraints: The viable parameter space requires a primary mass $M_1 \in [25, 150] M_\odot$, a secondary mass $M_2 \in [18, M_1] M_\odot$, a tertiary mass $M_3 \in [1.25, 4.5] M_\odot$, an inner period $\log(P_{\text{in}}/\text{day}) \in [0.7, 1.5]$, and an outer period $\log(P_{\text{out}}/\text{day}) \in [2.3, 3.5]$.
• Draw Probability: For the fiducial triple-star parameters, the probability of drawing a system with these initial properties is $P_{\text{draw}}^{\text{triple}} \approx 1.5 \times 10^{-3}$.
• Success Fraction ($\beta$):
  • Under fiducial constraints ($M_1 \geq 25 M_\odot$), the required success fraction to match the observed BBH rate is $\beta_{\text{triple}} \lesssim 0.67$.
  • When relaxing the primary mass constraint to the theoretical BH formation floor ($M_1 \geq 20 M_\odot$), the required success fraction drops to $\beta_{\text{triple}} \approx 0.47$.
• Comparison to Binary Channel: The success fraction for the triple channel is significantly higher than that of a comparable isolated binary channel ($\beta_{\text{binary}} \approx 0.023$), though the triple channel is rarer due to the "multiplicity penalty" (the low probability of forming a stable triple with the specific required parameters).

Significance and Claims
The paper claims that this triple-star evolutionary channel is a viable and non-negligible contributor to the population of merging BBHs.

• Qualitative Fit: The mechanism naturally explains the observed asymmetric $\chi_{\text{eff}}$ distribution. The misalignment of the first-born BH's spin (due to the triple system's geometry) generates the negative tail, while the alignment of the second-born BH's spin contributes to the positive mean.
• Rate Viability: The calculated success fraction ($\beta < 1$) indicates that the channel is not excluded by rate constraints. The authors explicitly state that while the channel could theoretically produce the entire observed BBH population (if $\beta$ is close to unity), it is more likely a significant contributor alongside other channels.
• Limitations: The authors do not claim this channel dominates the BBH population. They note that the current work does not derive the quantitative distribution of $\chi_{\text{eff}}$ nor determine the exact fraction of the population this channel supplies. The high required success fraction ($\sim 0.5-0.7$) suggests that while the channel is plausible, it may not be the sole formation mechanism.

In summary, the study proposes a physically motivated triple-star pathway that resolves the tension between the positive mean and negative tail of observed BBH spins, demonstrating that such systems can form at rates consistent with current gravitational wave observations.