Measuring eccentricity and gas-induced perturbation from gravitational waves of LISA massive black hole binaries

We assess the possibility of detecting both eccentricity and gas effects (migration and accretion) in the gravitational wave (GW) signal from LISA massive black hole binaries at redshift $z=1$. Gas induces a phase correction to the GW signal with an effective amplitude (Cg) and a semimajor axis dependence (assumed to follow a power-law with slope ng). We use a complete model of the LISA response and employ a gas-corrected post-Newtonian inspiral-only waveform model TaylorF2Ecc. By using the Fisher formalism and Bayesian inference, we constrain Cg together with the initial eccentricity e₀, the total redshifted mass Mz, the primary-to-secondary mass ratio q, the dimensionless spins χ 1,2 of both component BHs, and the time of coalescence tc. We find that simultaneously constraining Cg and e₀ leads to worse constraints on both parameters with respect to when considered individually. For a standard thin viscous accretion disc around Mz=10⁵~ M, $q=8$, χ 1,2=0.9, and tc=4 years MBHB, we can confidently measure (with a relative error of < 50 per cent) an Eddington ratio fEdd~ 0.1 for a circular binary and fEdd~ 1 for an eccentric system assuming O(10) stronger gas torque near-merger than at the currently explored much-wider binary separations. The minimum measurable eccentricity is e₀ 10-2.75 in vacuum and e₀ 10⁻² in gas. A weak environmental perturbation (fEdd 1) to a circular binary can be mimicked by an orbital eccentricity during inspiral, implying that an electromagnetic counterpart would be required to confirm the presence of an accretion disc.