Extended Bose-Einstein condensate dark matter in $f(Q)$ gravity

In this article, we attempt to explore the dark sector of the universe i.e. dark matter and dark energy, where the dark energy components are related to the modified $f(Q)$ Lagrangian, particularly a power law function $f(Q)= \gamma \left(\frac{Q}{Q_0}\right)^n$, while the dark matter component is described by the Extended Bose-Einstein Condensate (EBEC) equation of state for dark matter, specifically, $p = \alpha \rho + \beta \rho^2$. We find the corresponding Friedmann-like equations and the continuity equation for both dark components along with an interacting term, specifically $\mathcal{Q} = 3b^2H \rho$, which signifies the energy exchange between the dark sector of the universe. Further, we derive the analytical expression of the Hubble function, and then we find the best-fit values of free parameters utilizing the Bayesian analysis to estimate the posterior probability and the Markov Chain Monte Carlo (MCMC) sampling technique corresponding to CC+Pantheon+SH0ES samples. In addition, to examine the robustness of our MCMC analysis, we perform a statistical assessment using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). Further from the evolutionary profile of the deceleration parameter and the energy density, we obtain a transition from the decelerated epoch to the accelerated expansion phase, with the present deceleration parameter value as $q(z=0)=q_0=-0.56^{+0.04}_{-0.03}$ ($68 \%$ confidence limit), that is quite consistent with cosmological observations. In addition, we find the expected positive behavior of the effective energy density. Finally, by examining the sound speed parameter, we find that the assumed theoretical $f(Q)$ model is thermodynamically stable.