  Recently, various families of black holes (BHs) with synchronised hair have
been constructed. These are rotating BHs surrounded, as fully non-linear
solutions of the appropriate Einstein-matter model, by a non-trivial bosonic
field in synchronised rotation with the BH horizon. Some families bifurcate
globally from a bald BH ($e.g.$ the Kerr BH), whereas others bifurcate only
locally from a bald BH ($e.g.$ the $D=5$ Myers-Perry BH). It would be desirable
to understand how generically synchronisation allows hairy BHs to bifurcate
from bald ones. However, the construction and scanning of the domain of
existence of the former families of BHs can be a difficult and time consuming
(numerical) task. Here, we first provide a simple perturbative argument to
understand the generality of the synchronisation condition. Then, we observe
that the study of Q-clouds is a generic tool to establish the existence of BHs
with synchronised hair bifurcating (globally or locally) from a given bald BH
without having to solve the fully non-linear coupled system of Einstein-matter
equations. As examples, we apply this tool to establish the existence of
synchronised hair around $D=6$ Myers-Perry BHs, $D=5$ black rings and $D=4$
Kerr-AdS BHs, where $D$ is the spacetime dimension. The black rings case
provides an example of BHs with synchronised hair beyond spherical horizon
topology, further establishing the generality of the mechanism.

General Relativity and Quantum Cosmology;High Energy Physics - Theory