  We have previously presented a reconstruction of Horndeski scalar-tensor
theories from linear cosmological observables. It includes free nonlinear terms
which can be added onto the reconstructed covariant theory without affecting
the background and linear dynamics. After discussing the uniqueness of these
correction terms, we apply this nonlinear freedom to a range of different
applications. First we demonstrate how the correction terms can be configured
to endow the reconstructed models with screening mechanisms such as the
chameleon, k-mouflage and Vainshtein effects. A further implication is the
existence of classes of Horndeski models that are degenerate with standard
cosmology to an arbitrary level in the cosmological perturbations. Particularly
interesting examples are kinetically self-accelerating models that mimic the
dynamics of the cosmological constant to an arbitrary degree in perturbations.
Finally, we develop the reconstruction method further to the level of
higher-order effective field theory, which under the restriction to a luminal
propagation speed of gravitational waves introduces two new free functions per
order. These functions determine the corresponding correction terms in the
linearly reconstructed action at the same order. Our results enable the
connection of linear cosmological constraints on generalised modifications of
gravity and dark energy with the nonlinear regime and astrophysical probes for
a more global interpretation of the wealth of forthcoming cosmological survey
data.
