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An analytic calculation of the growth index for $f(R)$ dark energy model

We derive the analytic formula of the growth index for $f(R)$ dark energy model where the effect on the growth of matter density perturbation $δ_m$ from modified gravity (MG) is encoded in the effective Newton coupling constant $G_{\rm eff}$ in MG (or equivalently $g\equiv {G_{\rm eff}/ G}$). Based on the analytic formula, we propose that the parameter $g$ can be directly figured out by comparing the observed growth rate $f_g\equiv d\lnδ_m/d\ln a$ to the prediction of $f_g$ in general relativity.