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We explore the eccentricity measurement threshold of LISA for gravitationalwaves radiated by massive black hole binaries (MBHBs) with redshifted BH masses$M_z$ in the range $10^{4.5}$-$10^{7.5}~{\rm M}_\odot$ at redshift $z=1$. Theeccentricity can be an important tracer of the environment where MBHBs evolveto reach the merger phase. To consider LISA's motion and apply the time delayinterferometry, we employ the lisabeta software and produce year-long eccentricwaveforms using the inspiral-only post-Newtonian model TaylorF2Ecc. We studythe minimum measurable eccentricity ($e_{\rm min}$, defined one year before themerger) analytically by computing matches and Fisher matrices, and numericallyvia Bayesian inference by varying both intrinsic and extrinsic parameters. Wefind that $e_{\rm min}$ strongly depends on $M_z$ and weakly on mass ratio andextrinsic parameters. Match-based signal-to-noise ratio criterion suggest thatLISA will be able to detect $e_{\rm min}\sim10^{-2.5}$ for lighter systems($M_z\lesssim10^{5.5}~{\rm M}_\odot$) and $\sim10^{-1.5}$ for heavier MBHBswith a $90$ per cent confidence. Bayesian inference with Fisher initializationand a zero noise realization pushes this limit to $e_{\rm min}\sim10^{-2.75}$for lower-mass binaries, assuming a $<50$ per cent relative error. Bayesianinference can recover injected eccentricities of $0.1$ and $10^{-2.75}$ for a$10^5~{\rm M}_\odot$ system with a $\sim10^{-2}$ per cent and a $\sim10$ percent relative errors, respectively. Stringent Bayesian odds criterion($\ln{B}>8$) provides nearly the same inference. Both analytical and numericalmethodologies provide almost consistent results for our systems of interest.LISA will launch in a decade, making this study valuable and timely forunlocking the mysteries of the MBHB evolution.

The minimum measurable eccentricity from gravitational waves of LISA massive black hole binaries