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.

在本文中，我们尝试探索宇宙的暗区，即暗物质和暗能量，其中暗能量成分与修正的 f(Q) 拉格朗日量有关，特别是幂律函数 \(f(Q)= \ gamma \left( \frac{Q}{Q_0}\right) ^n\)，而暗物质成分由暗物质的扩展玻色-爱因斯坦凝聚 (EBEC) 状态方程描述，具体来说， \(p = \alpha \rho + \beta \rho ^2\)。我们找到了相应的弗里德曼方程和两个暗分量的连续性方程以及一个相互作用项，特别是 \(\mathcal {Q} = 3b^2H \rho \)，它表示暗部分之间的能量交换宇宙。进一步推导了哈勃函数的解析表达式，然后利用贝叶斯分析估计后验概率以及CC+Pantheon+对应的马尔可夫链蒙特卡罗（MCMC）采样技术找到自由参数的最佳拟合值SH0ES 样品。此外，为了检查 MCMC 分析的稳健性，我们使用赤池信息准则 (AIC) 和贝叶斯信息准则 (BIC) 进行统计评估。进一步从减速参数和能量密度的演化曲线，我们得到从减速期到加速膨胀期的转变，当前减速参数值为 \(q(z=0)=q_0=-0.56^{ +0.04}_{-0.03}\) (\(68 \%\) 置信极限)，这与宇宙学观测结果相当一致。此外，我们发现了有效能量密度的预期积极行为。最后，通过检查声速参数，我们发现假设的理论 f(Q) 模型是热力学稳定的。