Observational constraints on the growth index parameters in $f(Q)$ gravity

In this study, we analyse constraints on the growth index of matter perturbations, $\gamma$, within the framework of $f(Q)$ gravity, using recent cosmological observations, at the background and the perturbation levels, including Pantheon${}^{+}$, Cosmic Chronometer (CC), and Redshift Space Distortion (RSD) datasets. Our analysis focuses on quantifying the distortion parameter, which measures the deviation of the $f(Q)$ gravity model from the concordance $\mathrm{\Lambda }$CDM cosmology at the background level. Specifically, we investigate two cases of the growth index parameter: a constant $\gamma$ and a time-varying $\gamma (z)$. We investigate various parametrizations of the growth index $\gamma$, expressed as $\gamma ={\gamma }_0+{\gamma }_{1}y(z)$, where the function $y(z)$ assumes different forms, including constant (${\mathrm{\Gamma }}_0$), Taylor expansion around $z=0$ (${\mathrm{\Gamma }}_1$), Taylor expansion around the scale factor (${\mathrm{\Gamma }}_2$), and an exponential form (${\mathrm{\Gamma }}_3$). By employing the Akaike Information Criterion and Bayesian Information Criterion, we find that the combined Pantheon${}^{+}$+ CC+ RSD datasets impose stringent constraints on the value of the growth index. For the ${\mathrm{\Gamma }}_0$ model, our results indicate that within the concordance $\mathrm{\Lambda }$CDM model, $\gamma$ is constrained to $0.545\pm 0.096$, showing strong agreement with the theoretical expectation of ${\gamma }_{\mathrm{\Lambda }}=\frac{6}{11}$. However, within the framework of $f(Q)$ gravity, we observe $\gamma =0.571_{-0.110}^{+0.095}$, slightly exceeding the $\mathrm{\Lambda }$CDM value by 4.66 $\mathrm{\%}$. Furthermore, when considering a time-varying growth index, our analysis reveals that the range of ${\gamma }_0$ spans from $0.596$ to $0.62$ across the ${\mathrm{\Gamma }}_{1-3}$ models.