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Modeling $f(R)$ gravity in terms of mass dilation rate

We review the conformal equivalence in describing the background expansion of the universe by $f(R)$ gravity both in the Jordan frame and the Einstein frame. In the Jordan frame, we present the general analytic expression for $f(R)$ models that have the same expansion history as the $Λ$CDM model. This analytic form can provide further insights on how cosmology can be used to test the $f(R)$ gravity at the largest scales. Moreover we present a systematic and self-consistent way to construct the viable $f(R)$ model in Jordan frame using the mass dilation rate function from the Einstein frame through the conformal transformation. In addition, we extend our study to the linear perturbation theories and we further exhibit the equivalence of the $f(R)$ gravity presented in the Jordan frame and Einstein frame in the perturbed space-time. We argue that this equivalence has solid physics root.