$f(Q,T)$ gravity, its covariant formulation, energy conservation and phase-space analysis

In the present article we analyze the matter-geometry coupled $f(Q,T)$ theory of gravity. We offer the fully covariant formulation of the theory, with which we construct the correct energy balance equation and employ it to conduct a dynamical system analysis in a spatially flat Friedmann-Lemaître-Robertson-Walker spacetime. We consider three different functional forms of the $f(Q,T)$ function, specifically, $f(Q,T)=αQ+ βT$, $f(Q,T)=αQ+ βT^2$, and $f(Q,T)=Q+ αQ^2+ βT$ . We attempt to investigate the physical capabilities of these models to describe various cosmological epochs. We calculate Friedmann-like equations in each case and introduce some phase space variables to simplify the equations in more concise forms. We observe that the linear model $f(Q,T)=αQ+ βT$ with $β=0$ is completely equivalent to the GR case without cosmological constant $Λ$. Further, we find that the model $f(Q,T)=αQ+ βT^2$ with $β\neq 0$ successfully depicts the observed transition from decelerated phase to an accelerated phase of the universe. Lastly, we find that the model $f(Q,T)= Q+ αQ^2+ βT$ with $α\neq 0$ represents an accelerated de-Sitter epoch for the constraints $β< -1$ or $ β\geq 0$.