Stability analysis of f(Q) gravity models using dynamical systems

In recent years, the modified theory of gravity known as $f(Q)$ gravity has drawn interest as a potential alternative to general relativity. According to this theory, the gravitational force is determined by a function of the so-called ``non-metricity" tensor $Q$, which expresses how far a particle space-time is from the metric geometry. In contrast to general relativity, which describes the gravitational field using the curvature tensor, $f(Q)$ gravity builds a theory of gravity using the non-metricity tensor. For this class of theories, dynamical system analysis of the background and perturbation equations has been carried out in this work to determine how various models behave cosmologically. Here, the critical points are determined for two $f(Q)$ models from the literature: the power law, $f(Q)=Q+mQ^{n}$, and the logarithmic, $f(Q)=\alpha+\beta log Q$ models. The stability behaviour and corresponding cosmology are displayed for each critical point. For the power law model, we achieve a matter-dominated saddle point with the right matter perturbation growth rate. For the logarithmic model, we get a saddle point dominated by the geometric component of the $f(Q)$ model with perturbations in the decomposition of matter. For both models, we later achieved a stable and accelerating Universe with constant matter perturbations.