The impact of the non-coincidence gauge on the dark energy dynamics in f(Q)-gravity

We study the dynamical effects of the non-coincidence gauge on the geometric dark energy within the framework of fQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( Q\right) $$\end{document}-gravity. We specifically examine a spatially flat Friedmann–Lemaître–Robertson–Walker universe with a dust fluid source, employing fQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( Q\right) $$\end{document}-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 fQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( Q\right) $$\end{document} 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.