Cyclostationary signals in LISA: a practical application to Milky Way satellites

One of the primary sources of gravitational waves (GWs) anticipated to be detected by the Laser Interferometer Space Antenna (LISA) are Galactic double white dwarf binaries (DWDs). However, most of these binaries will be unresolved, and their GWs will overlap incoherently, creating a stochastic noise known as the Galactic foreground. Similarly, the population of unresolved systems in the Milky Way's (MW) satellites is expected to contribute to a stochastic gravitational wave background (SGWB). Due to their anisotropy and the annual motion of the LISA constellation, both the Galactic foreground and the satellite SGWB fall into the category of cyclostationary processes. Leveraging this property, we develop a purely frequency-based method to study LISA's capability to detect the MW foreground and SGWBs from the most promising MW satellites. We analyze both mock data generated by an astrophysically motivated SGWB spectrum, and realistic ones from a DWD population generated via binary population synthesis. We are able to recover or put constrains on the candidate foregrounds, reconstructing -- in the presence of noise uncertainties -- their sky distribution and spectrum. Our findings highlight the significance of the interplay between the astrophysical spectrum and LISA's sensitivity to detect the satellites' SGWB. Considering an astrophysically motivated prior on the satellite positions improves their detectability, which becomes otherwise challenging in the presence of the Galactic foreground. Furthermore, we explore the potential to observe a hypothetical satellite located behind the Galactic disk. Our results suggest that a Large Magellanic Cloud-like satellite could indeed be observable by LISA.