  Future gravitational wave interferometers such as LISA, Taiji, DECIGO, and
TianQin, will enable precision studies of the environment surrounding black
holes. In this paper, we study intermediate and extreme mass ratio binary black
hole inspirals, and consider three possible environments surrounding the
primary black hole: accretion disks, dark matter spikes, and clouds of
ultra-light scalar fields, also known as gravitational atoms. We present a
Bayesian analysis of the detectability and measurability of these three
environments. Focusing for concreteness on the case of a detection with LISA,
we show that the characteristic imprint they leave on the gravitational
waveform would allow us to identify the environment that generated the signal,
and to accurately reconstruct its model parameters.
