In this work, we study the dynamics of an extreme mass-ratio inspiral (EMRI) embedded within a scalar cloud populated around the massive black hole. This cloud may be generated through the black hole superradiant process if the wavelength of the scalar particle is comparable to the size of the massive black hole. The EMRI motion perturbs the cloud, producing scalar radiation toward infinity and into the black hole horizon. In addition, the backreaction of the scalar radiation onto the orbit modifies the motion of the EMRI and induces an observable gravitational-wave phase shift for a range of system parameters. We quantify the scalar flux and the induced phase shift, as one of the examples of exactly solvable, environmental effects of EMRIs.

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• Dark matter
• Extensions of scalar sector
• General relativity
• Gravitational waves
• Classical black holes