  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 $\Lambda$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.
