How Common Are Common Envelopes? Quantifying Their Role in Forming Gravitational-Wave Sources

A central goal of gravitational-wave astronomy is to use merging binary black hole (BBH), black hole-neutron star (BHNS), and binary neutron star (BNS) systems as fossils to reconstruct the formation and evolution of massive stars across cosmic time. In practice, this inference relies on population-synthesis models that map massive stellar binaries to merging compact objects. However, these models disagree on the dominant orbital-hardening mechanisms within isolated binary evolution, particularly on whether common-envelope (CE) evolution is required. To address this, we compile and systematically compare formation-channel predictions from more than 200 isolated-binary population-synthesis simulations, organized within a unified hierarchical taxonomy. We find that BBH and BHNS formation pathways span nearly the full allowed range from CE-dominated to without-CE-dominated evolution (0-100%), while often predicting similar merger rates, revealing a fundamental degeneracy: merger-rate measurements alone do not uniquely constrain the underlying evolutionary pathways. In contrast, BNS formation proceeds almost exclusively through channels involving at least one CE phase (>90-100%), suggesting CE evolution plays a qualitatively different role in BNS than in BBH and BHNS formation. The relative contributions of with-CE and without-CE pathways are governed primarily by assumptions controlling mass-transfer stability, angular-momentum loss, CE efficiency, and supernova physics, which often act non-linearly and in correlated fashion, such that trends from one-at-a-time parameter variations do not generalize across simulation frameworks. Robust interpretation of gravitational-wave populations will therefore require transparent formation-channel definitions, reproducible analysis pipelines, systematic cross-code comparisons, and observational constraints that extend beyond merger rates alone.