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

We explore the eccentricity measurement threshold of LISA for gravitational waves radiated by massive black hole binaries (MBHBs) with redshifted BH masses $M_z$ in the range $10^{4.5}\mbox{-}10^{7.5}~{\rm M}_\odot$ 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 ($e_{\rm min}$, defined at 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 $e_{\rm min}$ has a strong dependence on $M_z$ and a weak dependence on mass ratio and extrinsic parameters. Match-based signal-to-noise ratio criterion suggest that LISA 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 MBHBs with a $90\%$ confidence. Bayesian inference with Fisher initialization and a zero noise realization pushes this limit to $e_{\rm min}\sim10^{-2.75}$ for lower-mass binaries assuming a $<50\%$ relative error. Bayesian inference can recover injected eccentricities of $0.1$ and $10^{-2.75}$ for a $10^5~{\rm M}_\odot$ system with a $\sim10^{-2}\%$ and a $\sim10\%$ relative errors, respectively. 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 to prepare for unlocking the mysteries of the MBHB evolution.