On perturbative constraints for vacuum f(R) gravity

Perturbative techniques are important for modified theories of gravity since they allow to calculate deviations from General Relativity without recurring to exact solutions, which can be difficult to find. When applied to models such as $f(R)$ gravity, these techniques introduce corrections in the field equations that involve higher order derivatives. Such corrections must be handled carefully to have a well defined perturbative scheme, and this can be achieved through the method of perturbative constraints, where the coefficient of the additional term in the action is used as expansion parameter for the quantities of interest. In this work, we implement a perturbative framework that compares solutions in modified theories of gravity with solutions of the Einstein field equations, by following the guidelines of perturbation theory constructed in General Relativity together with the perturbative constraints rationale. By using this formalism, we demonstrate that a consistent $f(R)$ perturbation theory in vacuum, for an important class of $f(R)$ functions, produces no additional effects with respect to what is expected from the perturbation theory of General Relativity. From this result, we argue that there are fundamental limitations that explain why the solutions of some $f(R)$ models can be disconnected from their general relativistic counterparts, in the sense that the limit that leads from the $f(R)$ action to General Relativity does not transform the solutions accordingly.