Lagrangian perturbation theory for modified gravity

We present a formalism to compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order. We emphasize the case of theories with scale dependent gravitational strengths, such as chameleons, but our formalism can be accommodated to other modified gravity theories. In the non-linear regime two qualitative features arise. One, as is well known, is that nonlinearities lead to a screening of the force mediated by the scalar field. The second is a consequence of the transformation of the Klein-Gordon equation from Eulerian to Lagrangian coordinates, producing frame-lagging terms that are important especially at large scales, and if not considered, the theory does not reduce to the $\Lambda$CDM model in that limit. We apply our formalism to compute the 1-loop power spectrum and the correlation function in $f(R)$ gravity by using different resummation schemes. We further discuss the IR divergences of these formalisms.