Predicting the LISA white dwarf binary population in the Milky Way with cosmological simulations

White dwarf binaries with orbital periods below 1 h will be the most numerous sources for the space-based gravitational wave detector Laser Interferometer Space Antenna (LISA). Based on thousands of individually resolved systems, we will be able to constrain binary evolution and provide a new map of the Milky Way and its close surroundings. In this paper we predict the main properties of populations of different types of detached white dwarf binaries detected by LISA over time. For the first time, we combine a high-resolution cosmological simulation of a MilkyWay-mass galaxy (taken from the FIRE project) with a binary population synthesis model for low- and intermediate-mass stars. Our Galaxy model therefore provides a cosmologically realistic star formation and metallicity history for the Galaxy and naturally produces its different components such as the thin and thick disc, the bulge, the stellar halo, and satellite galaxies and streams. Thanks to the simulation, we show how different Galactic components contribute differently to the gravitational wave signal, mostly due to their typical age and distance distributions. We find that the dominant LISA sources will be He-He double white dwarfs (DWDs) and He-CO DWDs with important contributions from the thick disc and bulge. The resulting sky map of the sources is different from previous models, with important consequences for the searches for electromagnetic counterparts and data analysis. We also emphasize that much of the science-enabling information regarding white dwarf binaries, such as the chirp mass and the sky localization, becomes increasingly rich with long observations, including an extended mission up to 8 yr.