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Notes on diffeomorphisms symmetry of $f(R)$ gravity in the cosmological context

We study the metric perturbations in the context of restricted $f(R)$ gravity, in which a parameter for deviation from the full diffeomorphisms of space-time is introduced. We demonstrate that one can choose the parameter to remove the induced anisotropic stress, which is present in the usual $f(R)$ gravity. Moreover, to prevent instability for the vector and tensor metric perturbations, some constraints on the model are obtained.