Theoretical priors in scalar-tensor cosmologies: Shift-symmetric
  Horndeski models
  Attempts at constraining theories of late time accelerated expansion often
assume broad priors for the parameters in their phenomenological description.
Focusing on shift-symmetric scalar-tensor theories with standard gravitational
wave speed, we show how a more careful analysis of their dynamical evolution
leads to much narrower priors. In doing so, we propose a simple and accurate
parametrisation of these theories, capturing the redshift dependence of the
equation of state, $w(z)$, and the kinetic braiding parameter, $\alpha_{\rm
B}(z)$, with only two parameters each, and derive their statistical
distribution (a.k.a. theoretical priors) that fit the cosmology of the
underlying model. We have considered two versions of the shift-symmetric model,
one where the energy density of dark energy is given solely by the scalar
field, and another where it also has a contribution from the cosmological
constant. By including current data, we show how theoretical priors can be used
to improve constraints by up to an order of magnitude. Moreover, we show that
shift-symmetric theories without a cosmological constant are observationally
viable. We work up to quartic order in first derivatives of the scalar in the
action and our results suggest this truncation is a good approximation to more
general shift-symmetric theories. This work establishes an actionable link
between phenomenological parameterisations and Lagrangian-based theories, the
two main approaches to test cosmological gravity and cosmic acceleration.
