Synergies between Gaia and LISA missions for Galactic multi-messenger studies

Ultra-short period Galactic binaries are unique multi-messenger tracers of the Milky Way. They can be detected in large numbers through electromagnetic radiation by Gaia and through gravitational waves by the upcoming LISA mission. First, we revise the current census of known multi-messenger Galactic binaries by computing their GW signals using updated distances from the Gaia Data Release 2. Our work confirms thirteen guaranteed multi-messenger sources: nine AM CVns, three detached double white dwarfs (DWD) and one hot subdwarf. Next, we forecast the detection prospects for DWDs with both Gaia and LISA using a binary population synthesis technique. We predict respectively hundreds and tens of thousands detections by Gaia and LISA, with an overlap of several tens.  We show that synergies between Gaia and LISA observations of DWDs allow the study of the Milky Way baryonic structure. The success of this synergy is due to LISA's ability to localise binaries through virtually the whole Galactic plane, thus mapping its shape. While observations of LISA's electromagnetic counterparts observed by Gaia yield the information on their motion; tracing the underlying total enclosed mass. We envisage that multi-messenger observations will ensure the best science return of the LISA mission for Galactic studies.