  Large dark matter overdensities can form around black holes of astrophysical
and primordial origin as they form and grow. This "dark dress" inevitably
affects the dynamical evolution of binary systems, and induces a dephasing in
the gravitational waveform that can be probed with future interferometers. In
this paper, we introduce a new analytical model to rapidly compute
gravitational waveforms in presence of an evolving dark matter distribution. We
then present a Bayesian analysis determining when dressed black hole binaries
can be distinguished from GR-in-vacuum ones and how well their parameters can
be measured, along with how close they must be to be detectable by the planned
Laser Interferometer Space Antenna (LISA). We show that LISA can definitively
distinguish dark dresses from standard binaries and characterize the dark
matter environments around astrophysical and primordial black holes for a wide
range of model parameters. Our approach can be generalized to assess the
prospects for detecting, classifying, and characterizing other environmental
effects in gravitational wave physics.
