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 (MBHBs) at redshift $z=1$. Gas induces a phase correction to the GW signal with an effective amplitude ($C_{\rm g}$) and a semi-major axis dependence (assumed to follow a power-law with slope $n_{\rm g}$). We use a complete model of the LISA response, and employ a gas-corrected post-Newtonian in-spiral-only waveform model \textsc{TaylorF2Ecc}. By using the Fisher formalism and Bayesian inference, we explore LISA's ability to constrain $C_{\rm g}$ together with the initial eccentricity $e_0$, the total redshifted mass $M_z$, the primary-to-secondary mass ratio $q$, the dimensionless spins $\chi_{1,2}$ of both component BHs, and the time of coalescence $t_c$. We find that simultaneously constraining $C_{\rm g}$ and $e_0$ leads to worse constraints on both parameters with respect to when considered individually. Assuming a standard thin viscous accretion disc, for $M_z=10^6~{\rm M}_\odot$, $q=8$, $\chi_{1,2}=0.9$, and $t_c=4$ years, we can confidently measure (with a relative error of $<50 $ per cent) an Eddington ratio as small as ${\rm f}_{\rm Edd}\sim0.1$ for a circular binary while for an eccentric system only ${\rm f}_{\rm Edd}\gtrsim1$ can be inferred. The minimum measurable eccentricity is $e_0\gtrsim10^{-2.75}$ in vacuum and $e_0\gtrsim10^{-2}$ in the presence of a circumbinary disc. A weak environmental perturbation (${\rm f}_{\rm Edd}\lesssim1$) to a circular binary can be mimicked by an orbital eccentricity during in-spiral, implying that an electromagnetic counterpart would be required to confirm the presence of an accretion disc.