Observational Constraints on Growth Index with Cosmography

In the literature, it was proposed that the growth index $\gamma$ is useful to distinguish the scenarios of dark energy and modified gravity. In the present work, we consider the constraints on the growth index $\gamma$ by using the latest observational data. To be model-independent, we use cosmography to describe the cosmic expansion history, and also expand the general $\gamma(z)$ as a Taylor series with respect to redshift $z$ or $y$-shift, $y=z/(1+z)$. We find that the present value $\gamma_0=\gamma(z=0)\simeq 0.42$ (for most of viable $f(R)$ theories) is inconsistent with the latest observational data at high confidence level (C.L.). On the other hand, $\gamma_0\simeq 0.55$ (for dark energy models in GR) can be consistent with the latest observational data at $1\sigma$ C.L. in 5 of the 9 cases under consideration, but is inconsistent beyond $2\sigma$ C.L. in the other 4 cases (while it is still consistent within the $3\sigma$ region). Thus, we can say nothing firmly about $\gamma_0\simeq 0.55$. We also find that a varying $\gamma(z)$ is favored.