  Light bosons, proposed as a possible solution to various problems in
fundamental physics and cosmology, include a broad class of candidates for
beyond the Standard Model physics, such as dilatons and moduli, wave dark
matter and axion-like particles. If light bosons exist in nature, they will
spontaneously form "clouds" by extracting rotational energy from rotating
massive black holes through superradiance, a classical wave amplification
process that has been studied for decades. The superradiant growth of the cloud
sets the geometry of the final black hole, and the black hole geometry
determines the shape of the cloud. Hence, both the black hole geometry and the
cloud encode information about the light boson. For this reason, measurements
of the gravitational field of the black hole/cloud system (as encoded in
gravitational waves) are over-determined. We show that a single gravitational
wave measurement can be used to verify the existence of light bosons by model
selection, rule out alternative explanations for the signal, and measure the
boson mass. Such measurements can be done generically for bosons in the mass
range $[10^{-16.5},10^{-14}]$ eV using LISA observations of extreme mass-ratio
inspirals.

High Energy Astrophysical Phenomena;Instrumentation and Methods for Astrophysics;General Relativity and Quantum Cosmology