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Attempts at constraining theories of late time accelerated expansion oftenassume broad priors for the parameters in their phenomenological description.Focusing on shift-symmetric scalar-tensor theories with standard gravitationalwave speed, we show how a more careful analysis of their dynamical evolutionleads to much narrower priors. In doing so, we propose a simple and accurateparametrisation of these theories, capturing the redshift dependence of theequation of state, $w(z)$, and the kinetic braiding parameter, $\alpha_{\rmB}(z)$, with only two parameters each, and derive their statisticaldistribution (a.k.a. theoretical priors) that fit the cosmology of theunderlying model. We have considered two versions of the shift-symmetric model,one where the energy density of dark energy is given solely by the scalarfield, and another where it also has a contribution from the cosmologicalconstant. By including current data, we show how theoretical priors can be usedto improve constraints by up to an order of magnitude. Moreover, we show thatshift-symmetric theories without a cosmological constant are observationallyviable. We work up to quartic order in first derivatives of the scalar in theaction and our results suggest this truncation is a good approximation to moregeneral shift-symmetric theories. This work establishes an actionable linkbetween phenomenological parameterisations and Lagrangian-based theories, thetwo main approaches to test cosmological gravity and cosmic acceleration.

Theoretical priors in scalar-tensor cosmologies: Shift-symmetric Horndeski models