Scalar perturbation and density contrast evolution in $f(Q,C)$ gravity

The symmetric teleparallel theory offers an alternative gravitational formulation which can elucidate events in the early and late universe without requiring the physical existence of dark matter or dark energy. In this formalism, $f(Q,C)$ gravity has been recently introduced by incorporating the boundary term $C$ with the non-metricity scalar $Q$. In this paper, we develop the theory of cosmological scalar perturbation for $f(Q,C)$ gravity, and retrieve that of $f(\mathring{R})$ and $f(Q)$ gravity from our result. The analysis assumes a model-independent approach within these theories that adheres to the conventional continuity equation at the background level. We derive the density contrast equation by employing some standard cosmological approximations, where the $f(Q,C)$ theory is encoded in the effective Newtonian constant $G_{eff}$. Finally, we derive the evolution equation of density growth $f_g$.