  Constraining the Dark Energy (DE) equation of state, w, is one of the primary
science goals of ongoing and future cosmological surveys. In practice, with
imperfect data and incomplete redshift coverage, this requires making
assumptions about the evolution of w with redshift z. These assumptions can be
manifested in a choice of a specific parametric form, which can potentially
bias the outcome, or else one can reconstruct w(z) non-parametrically, by
specifying a prior covariance matrix that correlates values of w at different
redshifts. In this work, we derive the theoretical prior covariance for the
effective DE equation of state predicted by general scalar-tensor theories with
second order equations of motion (Horndeski theories). This is achieved by
generating a large ensemble of possible scalar-tensor theories using a Monte
Carlo methodology, including the application of physical viability conditions.
We also separately consider the special sub-case of the minimally coupled
scalar field, or quintessence. The prior shows a preference for tracking
behaviors in the most general case. Given the covariance matrix, theoretical
priors on parameters of any specific parametrization of w(z) can also be
readily derived by projection.
