  The tidal deformability of compact objects by a companion has a detectable
imprint in the gravitational waves emitted by a binary system. This effect is
governed by the so-called tidal Love numbers. For a particular theory of
gravity, these depend solely on the object internal structure and they vanish
for black holes in general relativity. A measurement compatible with nonzero
tidal Love numbers could thus provide evidence of new physics in the
strong-field regime. However, in realistic astrophysical scenarios, compact
objects are surrounded by a nonvacuum environment. In this work, we study the
tidal deformability of configurations of black holes immersed in matter,
focusing on two analytical models representing an anisotropic fluid and a
thin-shell of matter around a black hole. We then apply our results to the
astrophysically relevant case of a black hole surrounded by an accretion disk,
in the parameter region of interest of the upcoming LISA mission. Our results
indicate that there are challenges to overcome concerning tests of strong-field
gravity using tidal Love numbers.
