Galactic and extragalactic probe of dark matter with LISA's binary black holes

The upcoming LISA mission will be able to detect gravitational waves from galactic and extragalactic compact binaries. Here, we report on LISA's capability to probe dark matter around these binaries if the latter constitute black holes. By analyzing the variation in the chirp mass of the binary, we show that depending on the black hole masses, LISA should be able to probe their surrounding dark matter to a luminosity distance of $\approx 1$ Gpc if such binaries are observed within the inner $\approx 10$ pc of their galactic center for particle-like dark matter or near the galactic solitonic core for wave-like dark matter. In the case a null result is recorded during the course of observation of \emph{well-localized} binaries, one can still rule out certain parameter spaces of dark matter as being the dominant contributor to the matter budget of the Universe.