In this note we explore a modified theory of gravitation that is not based on the least action principle, but on a natural generalization of the original Einstein's field equations. This approach leads to the non-covariant conservation of the stress-energy tensor, a feature shared with other Lagrangian theories of gravity such as the $f(R,T)$ case. We consider the cosmological implications of a pair of particular models within this theory, and we show that they have some interesting properties. In particular, for some of the studied models we find that the density is bounded from above, and cannot exceed a maximum value that depends on certain physical constants. In the last part of the work we compare the theory to the $f(R,T)$ case and show that they lead to different predictions for the motion of test particles.
Publisher URL: http://arxiv.org/abs/1801.00284