  Many strongly coupled field theories admit a spectrum of gauge-invariant
bound states that includes scalar particles with the same quantum numbers as
the vacuum. The challenge naturally arises of how to characterise them. In
particular, how can a dilaton---the pseudo-Nambu-Goldstone boson associated
with approximate scale invariance---be distinguished from other generic light
scalars with the same quantum numbers? We address this problem within the
context of gauge-gravity dualities, by analysing the fluctuations of the
higher-dimensional gravitational theory. The diagnostic test that we propose
consists of comparing the results of the complete calculation, performed by
using gauge-invariant fluctuations in the bulk, with the results obtained in
the probe approximation. While the former captures the mixing between scalar
and metric degrees of freedom, the latter removes by hand the fluctuations that
source the dilatation operator of the boundary field-theory. Hence, the probe
approximation cannot capture a possible light dilaton, while it should fare
well for other scalar particles. We test this idea on a number of holographic
models, among which are some of the best known, complete gravity backgrounds
constructed within the top-down approach to gauge-gravity dualities. We compute
the spectra of scalar and tensor fluctuations, that are interpreted as bound
states (glueballs) of the dual field theory, and we highlight those cases in
which the probe approximation yields results close to the correct physical
ones, as well as those cases where significant discrepancies emerge. We
interpret the latter occurrence as an indication that identifying one of the
lightest scalar states with the dilaton is legitimate, at least as a
leading-order approximation.
