Nihar Gupte, M. Coleman Miller, Rhiannon Udall, Sophie Bini, Alessandra Buonanno, Jonathan Gair, Aldo Gamboa, Lorenzo Pompili, Antoni Ramos-Buades, Maximilian Dax, Stephen R. Green, Annalena Kofler, Jakob Macke, Bernhard Schölkopf

Multiple formation pathways have been proposed for the origin of binary black holes (BBHs). These include isolated binary evolution and dynamical assembly in dense stellar environments such as nuclear or globular star clusters. Yet, the fraction of BBHs originating from each channel still remains uncertain. One way to constrain this fraction is by investigating the distribution of the orbital eccentricities of the BH coalescences detected by the LIGO-Virgo-KAGRA (LVK) Collaboration. In this work, we analyze 85 BBHs from the first part of the fourth LVK observing run (O4a) using a multipolar, eccentric, aligned-spin effective-one-body waveform model. We perform parameter inference with neural posterior estimation and nested sampling. After incorporating astrophysical prior odds and comparing to the quasicircular precessing-spin hypothesis, we find that no candidates reach a high enough statistical significance to claim a confident detection of eccentricity. We use these upper limits to explore a particular model, in which all O4a BBHs originate from single-single gravitational wave (GW) captures. We perform hierarchical inference on the velocity dispersion of the host environment and find $σ< 24.3\,\mathrm{km/s}$ (95% credible upper bound). This disfavors single-single capture in nuclear star clusters (~20-200 km/s) as the dominant source of all observed BBH mergers. We verify that this dispersion bound does not increase by repeating the inference on a synthetic catalog augmented with eccentric events motivated by analyses of the third observing run of the LVK (O3). Our results place improved constraints on the number of eccentric BBHs and highlight the importance of eccentricity measurements in disentangling compact-binary formation channels in current and future GW detectors.