Consequences of energy conservation violation: Late time solutions of Λ(T) CDM subclass of f(R,T) gravity using dynamical system approach

Very recently, the authors of [PRL {\bf 118} (2017) 021102] have shown that violation of energy-momentum tensor (EMT) could result in an accelerated expansion state via appearing an effective cosmological constant, in the context of unimodular gravity. Inspired by this outcome, in this paper we investigate cosmological consequences of violation of the EMT conservation in a particular class of $f(R,T)$ gravity when only the pressure-less fluid is present. In this respect, we focus on the late time solutions of models of the type $f(R,T)=R+\beta \Lambda(-T)$. As the first task, we study the solutions when the conservation of EMT is respected and then we proceed with those in which violation occurs. We have found, provided that the EMT conservation is violated, there generally exist two accelerated expansion solutions which their stability properties depend on the underlying model. More exactly, we obtain a dark energy solution for which the effective equation of state (EoS) depend on model parameters and a de Sitter solution. We present a method to parametrize $\Lambda(-T)$ function which is useful in dynamical system approach and has been employed in the herein model. Also, we discuss the cosmological solutions for models with $\Lambda(-T)=8\pi G(-T)^{\alpha}$ in the presence of the ultra relativistic matter.