Cosmology in $f(Q)$ gravity: A unified dynamical systems analysis of the background and perturbations

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 the Lambda cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) 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 $\mathrm{\Lambda }\mathrm{CDM}$ concordance model.

Figure 1

The phase portrait of the system (16)–(18), for the power-law model I of (21) with $n=0.5$. This particular example exhibits the evolution $B_1\to A_1\to C_1$.

Figure 2

Upper: evolution of the density parameters of matter ${\mathrm{\Omega }}_m$ and of nonmetricity (dark energy) ${\mathrm{\Omega }}_Q$, as well as of the total, effective equation-of-state parameter $w_{\mathrm{eff}}$, as functions of the redshift, for the power-law model I of (21) with $n=0.5$. Lower: evolution of the perturbation variable $u=\frac{d(\ln \delta )}{d(\ln a)}$.

Figure 3

The evolution of $f{\sigma }_8$ as a function of the redshift, for the power-law model I of (21), for various values of $n$. The data are taken from [91, 92].

Figure 4

The phase portrait of the system (16)–(18), for the exponential model II of (22). This particular example exhibits the evolution $B_2\to A_2\to C_2$.

Figure 5

Upper: evolution of the density parameters of matter ${\mathrm{\Omega }}_m$ and of nonmetricity (dark energy) ${\mathrm{\Omega }}_Q$, as well as of the total, effective equation-of-state parameter $w_{\mathrm{eff}}$, as functions of the redshift, for the exponential model II of (22). Lower: evolution of the perturbation variable $u=\frac{d(\ln \delta )}{d(\ln a)}$.

Figure 6

The evolution of $f{\sigma }_8$ as a function of the redshift, for the exponential model II of (22). The data are taken from [91, 92].