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ABSTRACT 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 ($$C_{\rm g}$$) and a semimajor 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 inspiral-only waveform model TaylorF2Ecc. By using the Fisher formalism and Bayesian inference, we 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. For a standard thin viscous accretion disc around $$M_z=10^5~{\rm M}_{\odot }$$, $q=8$, $$\chi _{1,2}=0.9$$, and $$t_c=4$$ years MBHB, we can confidently measure (with a relative error of $$\lt 50$$ per cent) an Eddington ratio $${\rm f}_{\rm Edd}\sim 0.1$$ for a circular binary and $${\rm f}_{\rm Edd}\sim 1$$ for an eccentric system assuming $$\mathcal {O}(10)$$ stronger gas torque near-merger than at the currently explored much-wider binary separations. The minimum measurable eccentricity is $$e_0\gtrsim 10^{-2.75}$$ in vacuum and $$e_0\gtrsim 10^{-2}$$ in gas. A weak environmental perturbation ($${\rm f}_{\rm Edd}\lesssim 1$$) 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. 

ABSTRACT We explore the eccentricity measurement threshold of Laser Interferometer Space Antenna (LISA) for gravitational waves radiated by massive black hole binaries (MBHBs) with redshifted BH masses Mz in the range 104.5–107.5 M⊙ at redshift z = 1. The eccentricity can be an important tracer of the environment where MBHBs evolve to reach the merger phase. To consider LISA’s motion and apply the time delay interferometry, we employ the lisabeta software and produce year-long eccentric waveforms using the inspiral-only post-Newtonian model taylorf2ecc. We study the minimum measurable eccentricity (emin, defined one year before the merger) analytically by computing matches and Fisher matrices, and numerically via Bayesian inference by varying both intrinsic and extrinsic parameters. We find that emin strongly depends on Mz and weakly on mass ratio and extrinsic parameters. Match-based signal-to-noise ratio criterion suggest that LISA will be able to detect emin ∼ 10−2.5 for lighter systems (Mz ≲ 105.5 M⊙) and ∼10−1.5 for heavier MBHBs with a 90 per cent confidence. Bayesian inference with Fisher initialization and a zero noise realization pushes this limit to emin ∼ 10−2.75 for lower-mass binaries, assuming a <50 per cent relative error. Bayesian inference can recover injected eccentricities of 0.1 and 10−2.75 for a 105 M⊙ system with an ∼10−2 per cent and an ∼10 per cent relative errors, respectively. Stringent Bayesian odds criterion ($$\ln {\mathcal {B}}\gt 8$$) provides nearly the same inference. Both analytical and numerical methodologies provide almost consistent results for our systems of interest. LISA will launch in a decade, making this study valuable and timely for unlocking the mysteries of the MBHB evolution.