Density contrast in the scalar-tensor extension of non-metricity gravity

We present a novel derivation of scalar cosmological perturbations in the scalar-tensor extension of non-metricity gravity, where the non-metricity scalar $Q$ is non-minimally coupled to a dynamical scalar field. While previous investigations of symmetric teleparallel gravity focused primarily on background evolution or specialised gauge choices, a complete treatment of scalar perturbations in this non-minimally coupled framework has remained unexplored. In this work, we derive the full set of perturbed field equations, impose the quasi-static approximation, and obtain the effective Poisson equation together with the corresponding modified gravitational constant $G_{\rm eff}$. These ingredients allow us to construct the density contrast evolution equation and analyse the matter growth rate and growth index. Through numerical analysis, we showed that the scalar non-metricity theory is comparable to the well-known $\Lambda CDM$ model to some extent. The results provide a foundation for testing scalar non-metricity theories against large-scale structure observations and open new avenues for constraining non-minimally coupled non-metricity cosmologies.