  The discovery of cosmic acceleration motivated extensive studies of dynamical
dark energy and modified gravity models. Of particular interest are the
scalar-tensor theories, with a scalar field dark energy non-minimally coupled
to matter. Cosmological constraints on these models often employ the
quasi-static approximation (QSA), in which the dynamics of the scalar field
perturbations is proportional to the perturbation in the matter density. Using
the QSA simplifies the physical interpretation of the phenomenology of
scalar-tensor theories, and results in substantial savings of computing time
when deriving parameter constraints. Focusing on the symmetron model, which is
a well-motivated scalar-tensor theory with a screening mechanism, we compare
the exact solution of the linearly perturbed field equations to those obtained
under the QSA and identify the range of the model parameters for which the QSA
is valid. We find that the evolution of background scalar field is most
important, namely, whether it is dominated by the Hubble friction or the scalar
field potential. This helps us derive a criterion for the symmetron model, but
same argument can be applied to other scalar-tensor theories of generalized
Brans-Dicke type. We consider two scenarios, one where the scalar field is only
coupled to dark matter and where it couples to all of the matter.
