Stars or gas? Constraining the hardening processes of massive black-hole binaries with LISA

Massive black-hole binaries will be the loudest sources detectable by LISA. These systems are predicted to form during the hierarchical assembly of cosmic structures and coalesce by interacting with the surrounding environment. The hardening phase of their orbit is driven by either stars or gas and encodes distinctive features into the binary black holes that can potentially be reconstructed with gravitational-wave observations. We present a Bayesian framework to assess the likelihood of massive mergers being hardened by either gaseous or stellar interactions. We use state-of-the-art astrophysical models tracking the cosmological evolution of massive black-hole binaries and construct a large number of simulated catalogs of sources detectable by LISA. From these, we select a representative catalog and run both parameter estimation assuming a realistic LISA response as well model comparison capturing selection effects. Our results suggest that, at least within the context of the adopted models, future LISA observations can confidently constrain whether stars or gas are responsible for the binary hardening. We stress that accurate astrophysical modeling of the black-hole spins and the inclusion of subdominant emission modes in the adopted signal might be crucial to avoid systematic biases.