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 $\ensuremath{\gamma}$, the so-called growth index. It is well known that the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ cosmology predicts $\ensuremath{\gamma}=0.55$. However, using observational data, Nguyen et al. [Evidence for suppression of structure growth in the concordance cosmological model, Phys. Rev. Lett. 131, 111001 (2023)] measured a much higher $\ensuremath{\gamma}=0.63{3}_{\ensuremath{-}0.024}^{+0.025}$, excluding the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ value within $3.7\ensuremath{\sigma}$. In this work, we analyze whether dark energy (DE) with the equation-of-state parameter described by the Chevallier-Polarski-Linder (CPL) parametrization can significantly modify $\ensuremath{\gamma}$ with respect to the predicted $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ one. In addition to 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 $\ensuremath{\gamma}$ 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 $\ensuremath{\gamma}>0.6$. However, given that the measured $\ensuremath{\gamma}$ value assumes the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ background, a direct statistical measure of the incompatibility between theory and the measured value cannot be done for other backgrounds. Thus, we devise a method in order to make a quick estimation of the $\ensuremath{\gamma}$ constraints for CPL background. This method indicates that, when using Dark Energy Spectroscopic Instrument data release 2 baryon acoustic oscillation data to constrain background parameters, no significant changes in the $\ensuremath{\gamma}$ central value and uncertainty is observed. Consequently, both SDE and CDE described by CPL parametrization shall have a comparable level of tension with the expected $\ensuremath{\gamma}$ measurements. Moreover, we present new fitting functions for $\ensuremath{\gamma}$, which are more accurate and general than the one proposed by Linder [Cosmic growth history and expansion history, Phys. Rev. D 72, 043529 (2005)] for SDE and, for the first time, fitting functions for CDE models.