Probing the astrophysical environment with millihertz Gravitational Waves

With the expected launch of LISA in 2035, gravitational waves in the millihertz band will be detected. One of the most promising avenue which will be opened is the study of the astrophysical environment surrounding massive black holes and stellar-mass compact objects through the wave signal alone. Given the complexity and the many possible physical cases resulting in the interplay between matter and black hole binaries, it is of the utmost importance to develop pipelines and formalisms that are able to correctly capture the presence of an environment in an agnostic way, and to minimize any possible bias coming from the neglect of these additional physical properties. This thesis works in this direction, mainly focusing on two classes of sources in this band: extreme mass ratio inspirals, which are the most promising sources for the detection of beyond-GR physics, and massive black hole binaries. In the first case, we did a systematic study through the injection of realistic signals, in which the environmental contribution is generated through numerical simulations including stochastic features, against agnostic templates. We found that little to no systematics is present even with the simple agnostic templates. In a similar direction we also developed a consistency test for long lived signals able to detect missing physical properties. In the massive black hole binary case, instead, we performed a population study on different scenarios, and we assessed the ability of LISA to measure deviations coming from the astrophysical environment. We also assessed LISA's ability to distinguish between a modified GR and an environment injection for a full population catalogue, finding that the most preferred population description is always the vacuum one.