  Superradiant instabilities of spinning black holes can be used to impose
strong constraints on ultralight bosons, thus turning black holes into
effective particle detectors. However, very little is known about the
development of the instability and whether its nonlinear time evolution accords
to the linear intuition. For the first time, we attack this problem by studying
the impact of gravitational-wave emission and gas accretion on the evolution of
the instability. Our quasi-adiabatic, fully-relativistic analysis shows that:
(i) gravitational-wave emission does not have a significant effect on the
evolution of the black hole, (ii) accretion plays an important role and (iii)
although the mass of the scalar cloud developed through superradiance can be a
sizeable fraction of the black-hole mass, its energy-density is very low and
backreaction is negligible. Thus, massive black holes are well described by the
Kerr geometry even if they develop bosonic clouds through superradiance. Using
Monte Carlo methods and very conservative assumptions, we provide strong
support to the validity of the linearized analysis and to the bounds of
previous studies.
