  The most general covariant action describing gravity coupled to a scalar
field with only second order equations of motion, Horndeski's theory (also
known as "Generalized Galileons"), provides an all-encompassing model in which
single scalar dark energy models may be constrained. However, the generality of
the model makes it cumbersome to manipulate. In this paper, we demonstrate that
when considering linear perturbations about a Friedmann-Robertson-Walker
background, the theory is completely specified by only six functions of time,
two of which are constrained by the background evolution. We utilise the ideas
of the Effective Field Theory of Inflation/Dark Energy to explicitly construct
these six functions of time in terms of the free functions appearing in
Horndeski's theory. These results are used to investigate the behavior of the
theory in the quasistatic approximation. We find that only four functions of
time are required to completely specify the linear behavior of the theory in
this limit, which can further be reduced if the background evolution is fixed.
This presents a significantly reduced parameter space from the original
presentation of Horndeski's theory, giving hope to the possibility of
constraining the parameter space. This work provides a cross-check for previous
work on linear perturbations in this theory, and also generalizes it to include
spatial curvature.
