Reconstructing Horndeski theories from phenomenological modified gravity and dark energy models on cosmological scales

Recently we have derived a set of mapping relations that enables the reconstruction of the family of Horndeski scalar-tensor theories which reproduce the background dynamics and linear perturbations of a given set of effective field theory of dark energy coefficients. In this paper we present a number of applications of this reconstruction. We examine the form of the underlying theories behind different phenomenological parametrizations of modified gravity and dark energy used in the literature, as well as examine theories that exhibit weak gravity, linear shielding, and minimal self-acceleration. Finally, we propose a new inherently stable parametrization basis for modified gravity and dark energy models.