  The effective field theory (EFT) of dark energy relies on three functions of
time to describe the background dynamics. The viability of these functions is
investigated here by means of a thorough dynamical analysis. While the system
is underdetermined, and one can always find a set of functions reproducing any
expansion history, we are able to determine general compatibility conditions
for these functions by requiring a viable background cosmology. In particular,
we identify a set of variables that allows us to transform the non-autonomous
system of equations into an infinite-dimensional one characterized by a
significant recursive structure. We then analyze several autonomous
sub-systems, obtained truncating the original one at increasingly higher
dimension, that correspond to increasingly general models of dark energy and
modified gravity. Furthermore, we exploit the recursive nature of the system to
draw some general conclusions on the different cosmologies that can be
recovered within the EFT formalism and the corresponding compatibility
requirements for the EFT functions. The machinery that we set up serves
different purposes. It offers a general scheme for performing dynamical
analysis of dark energy and modified gravity models within the model
independent framework of EFT; the general results, obtained with this
technique, can be projected into specific models, as we show in one example. It
also can be used to determine appropriate ansatze for the three EFT background
functions when studying the dynamics of cosmological perturbations in the
context of large scale structure tests of gravity.
