  Pulsar timing and gravitational-wave (GW) detectors are superb laboratories
to study gravity theories in the strong-field regime. Here we combine those
tools to test the mono-scalar-tensor theory of Damour and Esposito-Far{\`e}se
(DEF), which predicts nonperturbative scalarization phenomena for neutron stars
(NSs). First, applying Markov-chain Monte Carlo techniques, we use the absence
of dipolar radiation in the pulsar-timing observations of five binary systems
composed of a NS and a white dwarf, and eleven equations of state (EOSs) for
NSs, to derive the most stringent constraints on the two free parameters of the
DEF scalar-tensor theory. Since the binary-pulsar bounds depend on the NS mass
and the EOS, we find that current pulsar-timing observations leave
scalarization windows, i.e., regions of parameter space where scalarization can
still be prominent. Then, we investigate if these scalarization windows could
be closed and if pulsar-timing constraints could be improved by
laser-interferometer GW detectors, when spontaneous (or dynamical)
scalarization sets in during the early (or late) stages of a binary NS (BNS)
evolution. For the early inspiral of a BNS carrying constant scalar charge, we
employ a Fisher matrix analysis to show that Advanced LIGO can improve
pulsar-timing constraints for some EOSs, and next-generation detectors, such as
the Cosmic Explorer and Einstein Telescope, will be able to improve those
bounds for all eleven EOSs. Using the late inspiral of a BNS, we estimate that
for some of the EOSs under consideration the onset of dynamical scalarization
can happen early enough to improve the constraints on the DEF parameters
obtained by combining the five binary pulsars. Thus, in the near future the
complementarity of pulsar timing and direct observations of GWs on the ground
will be extremely valuable in probing gravity theories in the strong-field
regime.
