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We present a formalism to compute Lagrangian displacement fields for a widerange 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 modifiedgravity theories. In the non-linear regime two qualitative features arise. One,as is well known, is that nonlinearities lead to a screening of the forcemediated by the scalar field. The second is a consequence of the transformationof the Klein-Gordon equation from Eulerian to Lagrangian coordinates, producingframe-lagging terms that are important especially at large scales, and if notconsidered, 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 correlationfunction in $f(R)$ gravity by using different resummation schemes. We furtherdiscuss the IR divergences of these formalisms.

Lagrangian perturbation theory for modified gravity