Can dark energy explain a high growth index?

A promising way to test the physics of the accelerated expansion of the Universe is by studying the growth rate of matter fluctuations, which can be parametrized by the matter energy density parameter to the power $\gamma$, the so-called growth index. It is well-known that the $\Lambda$CDM cosmology predicts $\gamma=0.55$. However, using observational data, Nguyen et al. (2023) measured a much higher $\gamma=0.633^{+0.025}_{-0.024}$, excluding the $\Lambda$CDM value within $3.7\sigma$. In this work, we analyze whether Dark Energy (DE) with the Equation of State (EoS) parameter described by the CPL parametrization can significantly modify $\gamma$ with respect to the predicted $\Lambda$CDM one. Besides the usual Smooth DE (SDE) scenario, where DE perturbations are neglected on small scales, we also consider the case of Clustering Dark Energy (CDE), which has more potential to impact the growth of matter perturbations. In order to minimally constrain the background evolution and assess the largest meaningful $\gamma$ distribution, we use data from $32$ Cosmic Chronometers, $H(z$), data points. In this context, we found that both SDE and CDE models described by the CPL parametrization have almost negligible probability of providing $\gamma>0.6$. However, given that the measured $\gamma$ value assumes the $\Lambda$CDM background, a direct statistical measure of the incompatibility between theory and the measured value can not be done for other backgrounds. Thus, we devise a method in order to make a quick estimation of the $\gamma$ constraints for CPL background. This method indicates that, when using DESI DR2 BAO data to constrain background parameters, no significant changes in the $\gamma$ central value and uncertainty is observed. (Abridged)