Cosmological dynamics and observational constraints on a viable $f(Q)$ non-metric gravity model

Inspired by an exponential $f(R)$ gravity model studied in the literature, in this work we introduce a new and viable $f(Q)$ gravity model, which can be represented as a perturbation of $\Lambda$CDM. Typically, within the realm of $f(Q)$ gravity, the customary approach to investigate cosmological evolution involves employing a parametrization of the Hubble expansion rate in terms of the redshift, $H(z)$, among other strategies. In this work we have implemented a different strategy, deriving an analytical approximation for $H(z)$, from which we deduce approximated analytical expressions for the parameters $w_{\rm{DE}}$, $w_{\rm{eff}}$, and $\Omega_{\rm{DE}}$, as well as the deceleration parameter $q$. In order to verify the viability of this approximate analytical solution, we examined the behavior of the these parameters in the late-time regime\textbf, in terms of the free parameter of the model, $b$. We find that for $b>0$, $w_{\rm{DE}}$ shows a quintessence-like behavior, while for $b<0$, it shows a phantom-like behavior. However, regardless of the sign of $b$, $w_{\rm{eff}}$ exhibits a quintessence-like behavior. Furthermore, it has been deduced that as the magnitude of the parameter $b$ increases, the present model deviates progressively from $\Lambda$CDM. We have also performed a Markov Chain Monte Carlo statistical analysis to test the model predictions with the Hubble parameter, the Pantheon supernova (SN) observational data, and the combination of those samples, obtaining constraints on the parameters of the model and the current values of the Hubble parameter and the matter density. Our findings indicate that this $f(Q)$ gravity model is indeed a viable candidate for describing the late-time evolution of the Universe at the background level.