Reassessing the spin of second-born black holes in coalescing binary black holes and its connection to the χ eff – q correlation

Context. The mass ratio q and effective inspiral spin χ eff of binary black hole mergers in the Gravitational-Wave Transient Catalog 4.0 (GWTC-4.0) have been reported to display a weaker anticorrelation compared to GWTC-3.0, a feature whose origin has been explored by several groups. For this work, within the isolated binary evolution framework, we adopted a recently proposed wind prescription for helium stars to systematically investigate the spin of the second-born black hole and its role in shaping this correlation. Aims. Our first goal was to investigate the main factors shaping the spin of the second-born black hole in a helium star–black hole binary, whether formed via a common-envelope or stable mass-transfer channel, and to further explore the potential correlation between the mass ratio q and the effective inspiral spin χ eff . Methods. Using the stellar and binary evolution code MESA, which includes a recently proposed helium star wind prescription alongside internal differential rotation and tidal interactions, we investigated how the initial conditions and fundamental physical processes shape the spin of the resulting black hole. We further employed rapid population synthesis calculations with COMPAS to predict the correlation between the mass ratio q and the effective inspiral spin χ eff . Results. We find that the recently proposed wind prescription for helium stars is substantially weaker than the standard Dutch wind scheme, particularly at subsolar metallicity. Using this scheme, we performed detailed binary modeling of a helium star with a black hole companion. Our results show that the spin magnitude of the resulting black hole is insensitive to the helium star’s evolutionary stage at the onset of tidal interactions or to the companion mass. Instead, wind mass loss plays the dominant role: more massive helium star progenitors produce lower-spinning black holes. The initial stellar rotation has only a minor effect, especially under strong tidal coupling, consistent with the common assumption of orbital synchronization. We then provide a fitting formula for the spin magnitude of the resulting second-born black hole. By contrast, the efficiency of angular momentum transport within helium stars can significantly alter the spin magnitude of the resulting black hole. Conclusions. Combining the fitting formula provided from the detailed binary evolution and rapid population synthesis with default model assumptions, we find that in the stable mass-transfer channel the majority (85.8%) of binary black holes undergo mass-ratio reversal, whereas in the common-envelope channel, only a small fraction (2.8%) exhibit mass-ratio reversal. Notably, we find no correlation between the mass ratio q and the effective spin parameter χ eff in either evolutionary channel. In future work, we plan to investigate how alternative physical prescriptions in population-synthesis models influence the relationship between q and χ eff , and to compare our predictions with coalescing binary black holes reported by the LIGO–Virgo–KAGRA Collaboration.