We study linear perturbations about non rotating black hole solutions in
scalar-tensor theories, more specifically Horndeski theories. We consider two
particular theories that admit known hairy black hole solutions. The first one,
Einstein-scalar-Gauss-Bonnet theory, contains a Gauss-Bonnet term coupled to a
scalar field, and its black hole solution is given as a perturbative expansion
in a small parameter that measures the deviation from general relativity. The
second one, known as 4-dimensional-Einstein-Gauss-Bonnet theory, can be seen as
a compactification of higher-dimensional Lovelock theories and admits an exact
black hole solution. We study both axial and polar perturbations about these
solutions and write their equations of motion as a first-order (radial) system
of differential equations, which enables us to study the asymptotic behaviours
of the perturbations at infinity and at the horizon following an algorithm we
developed recently. For the axial perturbations, we also obtain effective
Schr\”odinger-like equations with explicit expressions for the potentials and
the propagation speeds. We see that while the Einstein-scalar-Gauss-Bonnet
solution has well-behaved perturbations, the solution of the
4-dimensional-Einstein-Gauss-Bonnet theory exhibits unusual asymptotic
behaviour of its perturbations near its horizon and at infinity, which makes
the definition of ingoing and outgoing modes impossible. This indicates that
the dynamics of these perturbations strongly differs from the general
relativity case and seems pathological.
