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 ${\delta }_m$ from modified gravity (MG) is encoded in the effective Newton coupling constant $G_{\mathrm{e}\mathrm{f}\mathrm{f}}$ in MG (or equivalently $g\equiv G_{\mathrm{e}\mathrm{f}\mathrm{f}}\mathrm{/}G$). Based on the analytic formula, we propose that the parameter $g$ can be directly figured out by comparing the observed growth rate $f_g\equiv d\ln {\delta }_m\mathrm{/}d\ln a$ to the prediction of $f_g$ in general relativity.