We study the dynamical effects of the non-coincidence gauge on the geometric dark energy within the framework of \(f\left( Q\right) \)-gravity. We specifically examine a spatially flat Friedmann–Lemaître–Robertson–Walker universe with a dust fluid source, employing \(f\left( Q\right) \)-theory of gravity. Our goal is to analyze the dynamical evolution of cosmological parameters to reconstruct the cosmological history. We reproduce previous findings; nevertheless due to the presence of the dust fluid, new asymptotic solutions exist. We emphasize the significance of selecting the appropriate connection, as it dramatically change the cosmological dynamics. Additionally, we show that for the simple power-law \(f\left( Q\right) \) model, there exist a non-coincidence connection where the field equation reconstruct the main epochs of the cosmological history, that is, inflation, radiation epoch, matter era and the late-time acceleration is attributed to the de Sitter solution, which is the unique attractor of the solution trajectories. Moreover this model support Big Rip singularities as unstable solutions in the past.

我们在\(f\left(Q\right)\)-引力框架内研究非重合规范对几何暗能量的动力学效应。我们采用引力理论专门研究了具有尘埃流体源的空间平坦的弗里德曼-勒梅特-罗伯逊-沃克宇宙。我们的目标是分析宇宙学参数的动态演化，以重建宇宙学历史。我们重现了之前的发现；然而，由于尘埃流体的存在，存在新的渐近解。我们强调选择适当连接的重要性，因为它极大地改变了宇宙动力学。此外，我们表明，对于简单的幂律\(f\left( Q\right) \)模型，存在非重合连接，其中场方程重建了宇宙学历史的主要历元，即暴胀，辐射纪元、物质纪元和后期加速归因于德西特解，它是解轨迹的唯一吸引子。此外，该模型支持 Big Rip 奇点作为过去的不稳定解。