In the framework of the beyond Horndeski action, we consider three sub-theories, that support scalarised black-hole solutions, and look for modified characteristics compared to GR. We first study the propagation of massless scalar and vector test fields in the fixed background of an analytical spherically-symmetric black hole derived in the context of a parity-symmetric beyond Horndeski theory, and show that the profiles of the effective gravitational potentials, greybody factors, absorption cross-sections and quasinormal frequencies exhibit distinct modifications as we move away from the GR limit. We then turn our attention to the perturbations of the gravitational field itself and adopt a full-theory analysis that takes into account the backreaction of the scalar field on the metric. Employing as background solutions scalarised black holes arising in the shift-symmetric Horndeski theory and in the Quadratic-Quartic-scalar-Gauss-Bonnet theory, we compute the greybody factors and QNMs of the axial sector. In both theories, in direct correspondence to the form of the gravitational potential which features multiple extremal points, we find modified (suppressed or non-monotonic) greybody curves and altered quasinormal frequencies (smaller oscillating frequencies and larger damping times), especially as the hair-sourcing parameter increases.
Antoniou et al. (Wed,) studied this question.