Finding Horndeski theories with Einstein gravity limits
  The Horndeski action is the most general scalar-tensor theory with at most
second-order derivatives in the equations of motion, thus evading Ostrogradsky
instabilities and making it of interest when modifying gravity at large scales.
To pass local tests of gravity, these modifications predominantly rely on
nonlinear screening mechanisms that recover Einstein's Theory of General
Relativity in regions of high density. We derive a set of conditions on the
four free functions of the Horndeski action that examine whether a specific
model embedded in the action possesses an Einstein gravity limit or not. For
this purpose, we develop a new and surprisingly simple scaling method that
identifies dominant terms in the equations of motion by considering formal
limits of the couplings that enter through the new terms in the modified
action. This enables us to find regimes where nonlinear terms dominate and
Einstein's field equations are recovered to leading order. Together with an
efficient approximation of the scalar field profile, one can then further
evaluate whether these limits can be attributed to a genuine screening effect.
For illustration, we apply the analysis to both a cubic galileon and a
chameleon model as well as to Brans-Dicke theory. Finally, we emphasise that
the scaling method also provides a natural approach for performing
post-Newtonian expansions in screened regimes.
