Eccentric Signatures of Stellar-Mass Binary Black Holes with Circumbinary Disks in LISA

Binary black holes may have circumbinary disks if formed through common-envelope evolution or within gaseous environments. Disks can drive binaries into wider and more eccentric orbits, while gravitational waves harden and circularise them. We combine cutting-edge evolution prescriptions for disk-driven binaries with well-known equations for gravitational-wave-driven evolution, and study the evolution of stellar-mass binary black holes. We find that binaries are driven by their disk to an equilibrium eccentricity, 0.2 eₑq 0.5, that dominates their evolution. Once they transition to the GW-dominated regime their eccentricity decreases rapidly; we find that binaries with long-lived disks will likely be observed in LISA with detectable eccentricities ~ 10⁻² at $0.01$ Hz, with the precise value closely correlating with the binary's initial mass ratio. This may lead binary black holes with CBDs observed in LISA to be confused with dynamically-formed binary black holes.