On perturbative constraints for vacuum f(R) gravity

In this study, we utilize a perturbative framework that compares solutions in modified gravity theories with solutions of the Einstein field equations. We adhere to the perturbation theory constructed in General Relativity alongside the perturbative constraints rationale. By applying this formalism, we demonstrate that, for a significant class of f(R) functions, a consistent perturbation theory in vacuum produces no additional effects beyond what is expected from the perturbation theory of General Relativity. Based on this finding, we assert that fundamental limitations exist, which elucidate why solutions of certain f(R) models can be disconnected from their general relativistic counterparts. This implies that the transformation that leads from the f(R) action to General Relativity does not transform the solutions accordingly. Some conjectures and results are also presented in other extended theories of gravity.