Observing the inspiral of coalescing massive black hole binaries with LISA in the era of Multi-Messenger Astrophysics

Massive black hole binaries (MBHBs) of $10^5 \, \rm M_\odot - 3 \times 10^7 \, \rm M_\odot $ merging in low redshift galaxies ($z\le4$) are sufficiently loud to be detected weeks before coalescence with the Laser Interferometer Space Antenna (LISA). This allows us to perform the parameter estimation $on$ $the$ $fly$, i.e. as a function of the time to coalescence during the inspiral phase, relevant for early warning of the planned LISA protected periods and for searches of electromagnetic signals. In this work, we study the evolution of the sky position, luminosity distance, chirp mass and mass ratio uncertainties as function of time left before merger. Overall, light systems with total intrinsic mass $\rm M_{\rm tot} = 3 \times 10^5 \, \rm M_\odot$ are characterized by smaller uncertainties than heavy ones ($\rm M_{\rm tot} = 10^7 \, \rm M_\odot$) during the inspiral. Luminosity distance, chirp mass and mass ratio are well constrained at the end of the inspiral. Concerning sky position, at $z=1$, MBHBs with $\rm M_{\rm tot} = 3 \times 10^5 \, \rm M_\odot$ can be localized with a median precision of $\simeq 10^2 \, \rm deg^2 (\simeq 1 \, \rm deg^2)$ at 1 month (1 hour) from merger, while the sky position of heavy MBHBs can be determined to $10 \, \rm deg^2$ only 1 hour before merger. However the uncertainty around the median values broadens with time, ranging in between 0.04 -- 20 $\rm deg^2$ (0.3 -- 3 $\times 10^3 \, \rm deg^2$) for light (heavy) systems at 1 hour before merger. At merger the sky localization improves down to $\simeq 10^{-1} \, \rm deg^2$ for all masses. For the benefit of the observer community, we provide the full set of data from our simulations and simple and ready-to-use analytical fits to describe the time evolution of uncertainties in the aforementioned parameters, valid for systems with total mass between $10^5$--$10^7 \, \rm M_\odot$ and redshift $0.3$--$3$.