Revealing the ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$-$q$ correlation among Coalescing Binary Black Holes and { Tentative} Evidence for AGN-driven Hierarchical Mergers

The origin of the correlation between the effective spins (${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$) 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 ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}-q$ correlation for each subpopulation. Surprisingly, the ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$-$q$ correlation {either significantly weakens or disappears} for the low-mass subpopulation if we introduce a second ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ distribution for the high-mass subpopulation, which likely originates from hierarchical mergers. {This suggests that the ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$-$q$ correlation in the overall population can be explained by the superposition of two distinct subpopulations.} {We find Bayesian evidence strongly favoring two separate ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ distributions over a single mass-ratio-dependent distribution, with Bayes factors $\ln \mathcal{B}>4.2$.} The first subpopulation has a narrow ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ distribution peaking at $\sim 0.05$, whose primary-mass function {showing a rapid decline beyond} $\sim 40M_{\odot }$, in agreement with first-generation BBHs. The second ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ 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 ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$values in the second subpopulation, suggesting hierarchical mergers might occur both in AGN disks and stellar clusters. Furthermore, the inferred second ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ distribution might alternatively arise from other formation channels, such as stable mass transfer or chemically homogeneous evolution, if not interpreted as hierarchical mergers.}