Dynamical stability analysis of modified gravity with various interactions

Abstract

This study investigates the dynamical stability of cosmological models within the special case of $f\left(Q\right)$ gravity, a modified theory of gravity where non-metricity $Q$ governs gravitational interactions. We analyze the phase space dynamics of a flat Friedmann–Robertson–Walker (FRW) universe, incorporating three distinct non-linear interaction models between dark energy and dark matter: $Q_1=3Hbq\left({\rho }_D+{\rho }_m+\frac{{\rho }_D{\rho }_m}{{\rho }_D+{\rho }_m}\right)$, $Q_2=3Hbq\left({\rho }_D+{\rho }_m+\frac{{\rho }_D^2}{{\rho }_D+{\rho }_m}\right)$, and $Q_3=3H\xi {\rho }_{D}exp\left(\frac{{\rho }_D}{{\rho }_m}-1\right)$. By deriving autonomous dynamical equations and evaluating critical points, we assess their stability through eigenvalue analysis. The first two models exhibit stable attractor behavior in the quintessence phase for both dust (${\omega }_m=0$) and radiation (${\omega }_m=1/3$) scenarios, with all eigenvalues being negative. The third model demonstrates stability across phantom, $\Lambda$CDM, and quintessence phases, with critical points consistently converging to equilibrium states. Our results highlight the role of non-linear interactions in shaping cosmic evolution, offering viable alternatives to explain late-time acceleration without relying solely on dark energy. The findings underscore the robustness of $f\left(Q\right)$ gravity in describing cosmological dynamics while providing new insights into the universe’s large-scale structure and evolution.