Revealing the $χ_{\rm eff}$-$q$ correlation among Coalescing Binary Black Holes and { Tentative} Evidence for AGN-driven Hierarchical Mergers

Abstract: The origin of the correlation between the effective spins ($\chi_{\rm eff}$) and mass ratios ($q$) of LIGO-Virgo-KAGRA's binary black holes (BBHs) is still an open question. Motivated by the recent identification of two subpopulations of the BBHs, in this work we investigate the potential $\chi_{\rm eff}-q$ correlation for each subpopulation. Surprisingly, the $\chi_{\rm eff}$-$q$ correlation {either significantly weakens or disappears} for the low-mass subpopulation if we introduce a second $\chi_{\rm eff}$ distribution for the high-mass subpopulation, which likely originates from hierarchical mergers. {This suggests that the $\chi_{\rm eff}$-$q$ correlation in the overall population can be explained by the superposition of two distinct subpopulations.} {We find Bayesian evidence strongly favoring two separate $\chi_{\rm eff}$ distributions over a single mass-ratio-dependent distribution, with Bayes factors $\ln\mathcal{B}>4.2$.} The first subpopulation has a narrow $\chi_{\rm eff}$ distribution peaking at $\sim0.05$, whose primary-mass function {showing a rapid decline beyond} $\sim 40M_{\odot}$, in agreement with first-generation BBHs. The second $\chi_{\rm eff}$ distribution is broad and peaks at $\mu_{\chi,2} \sim 0.4$, aligning with predictions for hierarchical mergers in active galactic nucleus (AGN) disks. {However, we cannot exclude negative $\chi_{\rm eff}$values in the second subpopulation, suggesting hierarchical mergers might occur both in AGN disks and stellar clusters. Furthermore, the inferred second $\chi_{\rm eff}$ distribution might alternatively arise from other formation channels, such as stable mass transfer or chemically homogeneous evolution, if not interpreted as hierarchical mergers.}