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The upcoming LISA mission offers the unique opportunity to study the MilkyWay through gravitational wave radiation from Galactic binaries. Among thevariety of Galactic gravitational wave sources, LISA is expected toindividually resolve signals from $\sim 10^5$ ultra-compact double white dwarf(DWD) binaries. DWDs detected by LISA will be distributed across the Galaxy,including regions that are hardly accessible to electromagnetic observationssuch as the inner part of the Galactic disc, the bulge and beyond. Wequantitatively show that the large number of DWD detections will allow us touse these systems as tracers of the Milky Way potential. We demonstrate thatdensity profiles of DWDs detected by LISA may provide constraints on the scalelength parameters of the baryonic components that are both accurate andprecise, with statistical errors of a few percent to $10$ percent level.Furthermore, the LISA sample is found to be sufficient to disentangle betweendifferent (commonly used) disc profiles, by well covering the disc out tosufficiently large radii. Finally, up to $\sim 80$ DWDs can be detected throughboth electromagnetic and gravitational wave radiation. This enablesmulti-messenger astronomy with DWD binaries and allows one to extract theirphysical properties using both probes. We show that fitting the Galacticrotation curve constructed using distances inferred from gravitational waves{\it and} proper motions from optical observations yield a unique andcompetitive estimate of the bulge mass. Instead robust results for the stellardisc mass are contingent upon knowledge of the Dark Matter content.

A multi-messenger study of the Milky Way's stellar disc and bulge with LISA, Gaia and LSST