Observational Constraints On the Growth Index Parameters in f ( Q ) $f(Q)$ Gravity

In this study, the authors analyze constraints on the growth index of matter perturbations, $\gamma$, within the framework of $f\left(Q\right)$ gravity, using recent cosmological observations, at the background and the perturbation levels, including ${\text{Pantheon}}^{+}$, Cosmic Chronometer (CC), and Redshift Space Distortion (RSD) datasets. The analysis focuses on quantifying the distortion parameter, which measures the deviation of the $f\left(Q\right)$ gravity model from the concordance $\Lambda$ CDM cosmology at the background level. Specifically, two cases of the growth index parameter are investigated: a constant $\gamma$ and a time-varying $\gamma \left(z\right)$. The various parametrizations of the growth index $\gamma$ are investigated, expressed as $\gamma ={\gamma }_0+{\gamma }_{1}y\left(z\right)$, where the function $y\left(z\right)$ assumes different forms, including constant (${\Gamma }_0$), Taylor expansion around $z=0$ (${\Gamma }_1$), Taylor expansion around the scale factor (${\Gamma }_2$), and an exponential form (${\Gamma }_3$). By employing the Akaike Information Criterion and Bayesian Information Criterion, authors find that the combined ${\text{Pantheon}}^{+}$+ CC+ RSD datasets impose stringent constraints on the value of the growth index. For the ${\Gamma }_0$ model, the results indicate that within the concordance $\Lambda$ CDM model, $\gamma$ is constrained to $0.545\pm 0.096$, showing strong agreement with the theoretical expectation of ${\gamma }_{\Lambda }=\frac{6}{11}$. However, within the framework of $f\left(Q\right)$ gravity, authors find $\gamma =0.{57}_{-0.110}^{+0.095}$, which is in a good agreement within $1\sigma$ errors of the $\Lambda$ CDM (with a difference of 0.18$\sigma$). Furthermore, when considering a time-varying growth index, the analysis reveals that the range of ${\gamma }_0$ spans from 0.596 to 0.62 across the ${\Gamma }_{1-3}$ models. Finally, authors compare the obtained results for the growth index parameters within $f\left(Q\right)$ with those from previous studies on modified gravity theories, such as $f\left(R\right)$ and $f\left(T\right)$.