  In gravity theories that exhibit spontaneous scalarization, astrophysical
objects are identical to their general relativistic counterpart until they
reach a certain threshold in compactness or curvature. Beyond this threshold,
they acquire a non-trivial scalar configuration, which also affects their
structure. The onset of scalarization is controlled only by terms that
contribute to linear perturbation around solutions of general relativity. The
complete set of these terms has been identified for generalized scalar-tensor
theories. Stepping on this result, we study the onset on scalarization in
generalized scalar-tensor theories and determine the relevant thresholds in
terms of the contributing coupling constants and the properties of the compact
object.
