Growth Index in the $\gamma\delta$CDM model

To better distinguish the nature of $H_0$ and $S_8$ tensions, it is necessary to separate the effects of expansion and the growth of structure. The growth index $\gamma$ was identified as the most important parameter that characterizes the growth of density fluctuations independently of the effects of cosmic expansion. In the $\Lambda$CDM model, analyses performed with various cosmological datasets indicate that the growth index has to be larger than its theoretically predicted value. Cosmological models based on $f(R)$ gravity theories have scale-dependent growth indices, whose values are even more at odds with the growth rate data. In this work, we evaluate the growth index in the $\gamma\delta$CDM model both theoretically and numerically. Although based on $f(R)$ gravity theory, we show through several analyses with different combinations of datasets that the growth index in the $\gamma\delta$CDM model is very close in value to the $\Lambda$CDM and the $\omega$CDM models. The growth of structure is suppressed in the $\gamma\delta$CDM model, which is formulated with the extended gravitational growth framework. Upon analyzing cosmological data, we ascertain that the $\gamma\delta$CDM model is equally competitive as the $\Lambda$CDM and the $\omega$CDM models.