Observing the inspiral of coalescing massive black hole binaries with LISA in the era of multimessenger astrophysics

Massive black hole binaries (MBHBs) of 10⁵ M_-3×10⁷ M_ merging in low redshift galaxies (z≤4) 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 Mₜₒₜ=3×10⁵ M_ are characterized by smaller uncertainties than heavy ones (Mₜₒₜ=10⁷ M_) 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 Mₜₒₜ=3×10⁵ M_ can be localized with a median precision of 10² deg²(1 deg²) at 1 month (1 hour) from merger, while the sky position of heavy MBHBs can be determined to 10 deg² only 1 hour before merger. However the uncertainty around the median values broadens with time, ranging in between 0.04--20 deg² (0.3--3×10³ deg²) for light (heavy) systems at 1 hour before merger. At merger the sky localization improves down to 10^-1 deg² 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 10⁵--10⁷ M_ and redshift 0.3--3.