Late time dynamics of $f(R, T, R_{\mu\nu}T^{\mu\nu})$ gravity

Dynamical behavior and future singularities of $f(R, T,R_{\mu\nu}T^{\mu\nu})$ gravitational theory are investigated. This gravitational model is a more complete form of the $f(R,T)$ gravity which can offer new dynamics for the universe. We investigate this gravitational theory for the case $f = R + \alpha R_{\mu\nu}T^{\mu\nu}$ using the method of autonomous dynamical systems and by assuming an interaction between matter and dark energy. The fixed points are identified and the results are consistent with standard cosmology and show that for small $\alpha$, the radiation dominated era is an unstable fixed point of the theory and the universe will continue its procedure toward matter era which is a saddle point of the theory and allows the evolution to dark energy dominated universe. Finally the dark energy dominated epoch is a stable fixed point and will be the late time attractor for the universe. We also consider future singularities for the two $f = R + \alpha R_{\mu\nu}T^{\mu\nu}$ and $f = R +\alpha RR_{\mu\nu}T^{\mu\nu}$ cases and for $w = 0,\dfrac{1}{3},1$ and $-1$. Our results show that for the case of $f = R + \alpha R_{\mu\nu}T^{\mu\nu}$, the future singularities of the universe will happen in the same condition as do for the Einstein-Hilbert FRW universe. However, a new type of singularity is obtained for $f = R +\alpha RR_{\mu\nu}T^{\mu\nu}$ that is captured by $t\rightarrow t_s; a \rightarrow a_s; \rho\rightarrow \infty;$ and $ |p| \rightarrow 0$.