Dynamical analysis of logarithmic energy-momentum squared gravity

We perform the dynamical system analysis of a cosmological model in the energy-momentum squared gravity (EMSG) of the form $f(T_{\mu\nu} T^{\mu\nu})=\alpha\ln({\lambda}T_{\mu\nu} T^{\mu\nu})$, which is known as energy-momentum log gravity (EMLG). In particular, we show that the analytical cosmological solution of EMLG presented by Akarsu {\it et al.} (Eur. Phys. J. C 79:846, 2019) is a future attractor. It includes new terms in the right-hand side of the Einstein field equations, which yield constant inertial mass density and provide a dynamical dark energy with a density passing below zero at large redshifts, accommodating a mechanism for screening $\Lambda$ in the past for $\alpha<0$, suggested for alleviating some cosmological tensions. We show that the second law of thermodynamics requires $\alpha\leq0$ that allows the screening mechanism to take place. We also show that the model gives rise to an entire class of new stable late-time solutions with $H\rightarrow\sqrt{(\Lambda+2\alpha)/3}$ as $a\rightarrow\infty$, where the new term is due to the constant effective inertial mass density that arises from EMLG contribution of dust, whereas $H\rightarrow\sqrt{\Lambda/3}$ as $a\rightarrow\infty$ in the $\Lambda$CDM model. We also show the existence of new interesting features and trajectories that are absent in $\Lambda$CDM with or without spatial curvature.