Joint constraints on the field-cluster mixing fraction, common envelope efficiency, and globular cluster radii from a population of binary hole mergers via deep learning

The recent release of the second Gravitational-Wave Transient Catalog (GWTC-2) has increased significantly the number of known GW events, enabling unprecedented constraints on formation models of compact binaries. One pressing question is to understand the fraction of binaries originating from different formation channels, such as isolated field formation versus dynamical formation in dense stellar clusters. In this paper, we combine the $\texttt{COSMIC}$ binary population synthesis suite and the $\texttt{CMC}$ code for globular cluster evolution to create a mixture model for black hole binary formation under both formation scenarios. For the first time, these code bodies are combined self-consistently, with $\texttt{CMC}$ itself employing $\texttt{COSMIC}$ to track stellar evolution. We then use a deep-learning enhanced hierarchical Bayesian analysis to constrain the mixture fraction $f$ between formation models, while simultaneously constraining the common envelope efficiency $\alpha$ assumed in $\texttt{COSMIC}$ and the initial cluster virial radius $r_v$ assumed in $\texttt{CMC}$. Under specific assumptions about other uncertain aspects of isolated binary and globular cluster evolution, we report the median and $90\%$ confidence interval of three physical parameters $(f,\alpha,r_v)=(0.20^{+0.32}_{-0.18},2.26^{+2.65}_{-1.84},2.71^{+0.83}_{-1.17})$. This simultaneous constraint agrees with observed properties of globular clusters in the Milky Way and is an important first step in the pathway toward learning astrophysics of compact binary formation from GW observations.