  Current- and next-generation gravitational-wave observatories may reveal new,
ultralight bosons. Through the superradiance process, these theoretical
particle candidates can form clouds around astrophysical black holes and result
in detectable gravitational-wave radiation. In the absence of detections,
constraints$-$contingent on astrophysical assumptions$-$have been derived using
LIGO-Virgo-KAGRA data on boson masses. However, the searches for ultralight
scalars to date have not adequately considered self-interactions between
particles. Self-interactions that significantly alter superradiance dynamics
are generically present for many scalar models, including axion-like dark
matter candidates and string axions. We implement the most complete treatment
of particle self-interactions available to determine the gravitational-wave
signatures expected from superradiant scalar clouds and revisit the constraints
obtained in a past gravitational-wave search targeting the black hole in Cygnus
X-1. We also project the reach of next-generation gravitational-wave
observatories to scalar particle parameter space in the mass-coupling plane. We
find that while proposed observatories have insufficient reach to
self-interactions that can halt black hole spin-down, next-generation
observatories are essential for expanding the search beyond gravitational
parameter space and can reach a mass and interaction scale of $\sim
10^{-13}-10^{-12}$ eV/c$^2$ and $\gtrsim 10^{17}$ GeV, respectively.
