  Short-period (P<1 hour) white dwarf binaries will be the most numerous
sources for the space-based gravitational wave detector LISA. Based on
thousands of resolved systems, we will be able to constrain binary evolution
and provide a new map of the Milky Way and its surroundings. Here we predict
the main properties of populations of different types of detached white dwarf
binaries detected by LISA. For the first time, we combine a high-resolution
cosmological simulation of a Milky Way-mass galaxy (from the FIRE project) with
a binary population synthesis model for low and intermediate mass stars. Our
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 disk, the bulge, the stellar halo, and
satellite galaxies and streams. With the simulation, we show how different
galactic components contribute differently to the gravitational wave signal,
due to their typical age and distance distributions. We find that the dominant
LISA sources will be He-He systems and He-CO systems with important
contributions from the thick disk and bulge but also a few systems in the
stellar halo. 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 localisation, becomes increasingly rich with long
observations, including an extended mission up to 8 years.
