Cosmology in $f(Q)$ gravity: A unified dynamical system analysis at background and perturbation levels

Motivated by the fact that cosmological models based on $f(Q)$ gravity are very efficient in fitting observational datasets at both background and perturbation levels, we perform a combined dynamical system analysis of both background and perturbation equations in order to examine the validity of this result through an independent method. We examine two studied $f(Q)$ models of the literature, namely the power-law and the exponential ones. For both cases, we obtain a matter-dominated saddle point characterized by the correct growth rate of matter perturbations, followed by the transition to a stable dark-energy dominated accelerated universe in which matter perturbations remain constant. Furthermore, analyzing the behavior of $f \sigma_8$, we find that the models fit observational data successfully, obtaining a behavior similar to that of $\Lambda$CDM scenario, although the exponential model does not possess the latter as a particular limit. Hence, through the independent approach of dynamical systems, we do verify the results of observational confrontation, namely that $f(Q)$ gravity can be considered as a very promising alternative to the $\Lambda$CDM concordance model.