Parametrizing Theories of Gravity on Large and Small Scales in Cosmology

We present a link between parameterizations of alternative theories of gravity on large and small scales in cosmology. This relationship is established using theoretical consistency conditions only. We find that in both limits the "slip" and "effective Newton's constant" can be written in terms of a set of four functions of time, two of which are direct generalizations of the $\alpha$ and $\gamma$ parameters from post-Newtonian physics. To the best of our knowledge, this is the first time that a link between parameterizations of gravity on these very different scales has been established. We expect our result to facilitate the imposition of observational constraints, by drastically reducing the number of functional degress of freedom required to consistently test gravity on multiple scales in cosmology.