  The presence of dark matter overdensities surrounding a black hole can
influence the evolution of a binary system. The gravitational wave signals
emitted by a black hole binary offer a promising means to probe the dark matter
environments near a black hole. The dense region of dark matter can lead to the
dephasing of gravitational waveforms, which can be detected by upcoming
experiments such as the Laser Interferometer Space Antenna (LISA). The dark
matter density profile around the black hole can vary for different dark matter
models. Our study specifically investigates the impact of the ultralight
self-interacting scalar dark matter (SIDM) on the gravitational wave signals
emitted by black hole binaries. A distinctive characteristic of SIDM
surrounding a black hole, as opposed to collisionless dark matter, is the
formation of a soliton core. We perform a Fisher matrix analysis to estimate
the size of the soliton and the corresponding SIDM parameter space that future
LISA-like gravitational wave experiments can explore.
