  Black holes in certain modified gravity theories that contain a scalar field
coupled to curvature invariants are known to possess (monopole) scalar hair
while non-black-hole spacetimes (like neutron stars) do not. Therefore, as a
neutron star collapses to a black hole, scalar hair must grow until it settles
to the stationary black hole solution with (monopole) hair. In this paper, we
study this process in detail and show that the growth of scalar hair is tied to
the appearance and growth of the event horizon (before an apparent horizon
forms), which forces scalar modes that would otherwise (in the future) become
divergent to be radiated away. We prove this result rigorously in general first
for a large class of modified theories, and then we exemplify the results by
studying the temporal evolution of the scalar field in scalar Gauss-Bonnet
gravity in two backgrounds: (i) a collapsing Oppenheimer-Snyder background, and
(ii) a collapsing neutron star background. In case (i), we find an exact scalar
field solution analytically, while in case (ii) we solve for the temporal
evolution of the scalar field numerically, with both cases supporting the
conclusion presented above. Our results suggest that the emission of a burst of
scalar field radiation is a necessary condition for black hole formation in a
large class of modified theories of gravity.
