Orbital Eccentricity and Spin-Orbit Misalignment Are Evidence that Neutron Star-Black Hole Mergers Form through Triple Star Evolution

There is growing evidence that a substantial fraction of the neutron star-black holes (NSBHs) detected through gravitational waves merge with non-zero eccentricity or large BH spin-orbit misalignment. This is in tension with the leading formation scenarios to date. Residual eccentricity rules out formation through isolated binary star evolution, while NS natal kicks and the unequal masses of NSBHs inhibit efficient pairing in dense stellar environments. Here, we report that all observed properties-NSBH merger rate, eccentricity, and spin-orbit misalignment-are explained by the high prevalence of massive stellar triples in the field. Modelling their evolution from the ZAMS, we investigate NSBH mergers caused by gravitational perturbations from a tertiary companion. We show that the formation of the NS decisively impacts the triple stability, preferentially leaving behind surviving NSBHs in compact triple architectures. The rich three-body dynamics of compact, unequal-mass triples enables mergers across a wide range of orbital parameters without requiring fine-tuned highly inclined tertiary orbits and provides a natural explanation for an abundance of residual eccentricity and spin-orbit misalignment. We infer a total NSBH merger rate of $R\sim1-23,\rm Gpc^{-3},yr^{-1}$, with more than a few 10% exhibiting eccentricity $e_{20}>0.1$ or large spin-orbit misalignment $\cosθ_{\rm BH}<0$, consistent with current observations. Tertiary-driven NSBH mergers closely track the cosmic star formation rate due to their short delay times, include a substantial fraction of burst-like highly eccentric systems ($e_{20} > 0.9$), and almost universally retain eccentricities $e_{20}>10^{-3}$, potentially detectable by next-generation detectors. If evidence for eccentric and misaligned events solidifies, our results suggest that triple dynamics is the dominant formation channel of NSBH mergers.

💡 Research Summary

This paper addresses a growing tension between observations of neutron‑star–black‑hole (NSBH) mergers and the predictions of traditional formation channels. Several of the handful of NSBH events detected by LIGO‑Virgo‑KAGRA display either a measurable residual eccentricity (e ≳ 0.1 at 20 Hz) or a large misalignment between the black‑hole spin and the orbital angular momentum (cos θ_BH < 0). Such signatures are difficult to reconcile with isolated binary evolution, because gravitational‑wave emission efficiently circularises binaries long before merger, and they are also hard to produce in dense stellar environments where mass‑segregation and natal kicks tend to separate neutron stars from black holes.

The authors propose that the high prevalence of massive stellar triples in the field provides a natural solution. They construct a population synthesis model that follows 10⁷ hierarchical triples from the zero‑age main sequence (ZAMS) through the formation of a NSBH inner binary and the subsequent three‑body dynamics driven by a distant tertiary. The inner binary consists of a primary (≥ 10 M⊙) and a secondary (≥ 5 M⊙); the tertiary is less massive (≤ 5 M⊙). The evolution proceeds in two stages: (I) the primary collapses to a black hole while the secondary remains a massive star, forming a BH‑OB binary; (II) the secondary later undergoes core collapse, producing a neutron star and yielding a NSBH binary. The authors implement single‑star tracks (Klencki et al. 2020, 2025) at sub‑solar metallicity (Z ≈ 0.1 Z⊙), assume circular orbits at birth, and account for orbital widening due to stellar winds and mass loss. They apply stability checks (Mardling & Aarseth 2001) after each supernova, discarding systems that become dynamically unstable.

Mass transfer (MT) and common‑envelope (CE) phases are treated with detailed criteria based on donor envelope type (convective vs. radiative) and mass‑ratio thresholds. Stable MT is allowed for radiative donors only if the post‑MT orbit exceeds 20 R⊙ and the mass ratio exceeds a period‑dependent critical value; otherwise, CE evolution is invoked. These prescriptions reproduce the outcomes of more sophisticated binary‑evolution codes (e.g., COMBINE) while remaining computationally tractable for the large sample.

After the NSBH forms, the authors integrate the secular and direct three‑body dynamics, focusing on von Zeipel‑Kozai‑Lidov (ZKL) oscillations induced by the tertiary. Because the inner binaries are typically compact (a₁ ≲ 10 AU) and the outer-to-inner semi‑major‑axis ratio lies between ~10 and 10³, the systems occupy a regime where ZKL can drive the inner eccentricity to extreme values without requiring fine‑tuned mutual inclinations. The authors also include post‑Newtonian precession and gravitational‑wave radiation reaction, allowing them to capture the transition from high‑eccentricity ZKL cycles to rapid inspiral at pericentre.

The key results are:

1. Merger rate – The model predicts a NSBH merger rate of R ≈ 1–23 Gpc⁻³ yr⁻¹, consistent with the LIGO‑Virgo inferred range (≈ 9–84 Gpc⁻³ yr⁻¹). The rate is driven by the short delay times (10–100 Myr) of triple‑induced mergers, causing the merger history to trace the cosmic star‑formation rate.
2. Eccentricity distribution – More than a few % of mergers retain e₍₂₀₎ > 0.1 at 20 Hz, and a non‑negligible tail reaches e₍₂₀₎ > 0.9 (burst‑like events). Almost all mergers have e₍₂₀₎ > 10⁻³, a level potentially observable with next‑generation detectors (Einstein Telescope, Cosmic Explorer, LISA).
3. Spin‑orbit misalignment – Over 10 % of mergers exhibit cos θ_BH < 0, i.e., the black‑hole spin is tilted by more than 90° relative to the orbital angular momentum. This arises because the natal kick and mass loss during the supernova can reorient the inner orbital plane, and subsequent ZKL cycles further randomise the spin direction.
4. Parameter space – Surviving NSBH triples occupy a well‑defined region in (a₁, a₂/a₁) space bounded by constraints from supernova mass loss, natal kicks, and dynamical stability. The majority of successful mergers arise from systems that experienced stable mass transfer or a common‑envelope phase, which tightens the inner orbit enough to survive the second supernova.
5. Comparison with other channels – The triple channel yields rates and eccentricity/misalignment fractions that are orders of magnitude higher than those from dense‑cluster dynamics, ultra‑wide binaries, or active‑galactic‑nucleus environments, especially when realistic natal‑kick distributions are adopted.

The authors conclude that if future GW observations continue to reveal a significant fraction of NSBH mergers with measurable eccentricity or large spin‑orbit tilts, the triple‑star evolution channel will likely dominate the formation of these systems. They also highlight the need for improved constraints on massive‑star triple statistics, natal‑kick distributions, and the physics of mass transfer/CE phases to refine the quantitative predictions.