Searching for new physics in the neighborhood of a black hole: fundamental interactions, plasmas and tests of gravity

A plethora of theoretical arguments and experimental observations suggest evidence for new physics beyond the standard model and general relativity. A promising way to detect such deviations is to analyze the possible signatures of these theories around compact astrophysical objects. In this spirit, this thesis is devoted to the study of different phenomenological imprints of new physics around black holes. In the first part of the thesis, I analyze in detail superradiant instabilities, focusing in particular on the role of interactions of the superradiant field with astrophysical plasma. I show that such interactions are pivotal to obtain a detailed description of the systems, and may either lead to promising observables or hamper the instability itself. In the second part of the thesis, I focus on tests of general relativity using gravitational waves. In particular, I discuss in detail the possible detectability of fundamental dipoles in extreme mass-ratio inspirals, and introduce a formalism to probe theories beyond general relativity using pulsar-timing arrays. In both cases, I adopt a model-agnostic approach, encapsulating different theories in the analysis.