Among many alternative gravitational theories to General Relativity (GR), $f(R,T)$ gravity (where $R$ is the Ricci scalar and $T$ the trace of the energy-momentum tensor) has been widely studied recently. By adding a matter contribution to the gravitational Lagrangian, $f(R,T)$ theories have become an interesting extension to GR displaying a broad phenomenology in astrophysics and cosmology. In this paper, we discuss, however, the difficulties appearing in explaining a viable and realistic cosmology within the $f(R,T)$ class of theories. Our results challenge the viability of $f(R,T)$ as an alternative modification of gravity.

Apart from the familiar structure firmly-rooted in the general relativistic field equations where the energy–momentum tensor has a null divergence i.e., it conserves, there exists a considerable number of extended theories of gravity allowing departures from the usual conservative framework. Many of these theories became popular in the last few years, aiming to describe the phenomenology behind dark matter and dark energy. However, within these scenarios, it is common to see attempts to preserve the conservative property of the energy–momentum tensor. Most of the time, it is done by means of some additional constraint that ensures the validity of the standard conservation law, as long as this option is available in the theory. However, if no such extra constraint is available, the theory will inevitably carry a non-trivial conservation law as part of its structure. In this work, we review some of such proposals discussing the theoretical construction leading to the non-conservation of the energy–momentum tensor.

We formulate a theory combining the principles of scalar–tensor gravity and Rastall’s proposal of a violation of the usual conservation laws. We obtain a scalar–tensor theory with two parameters \(\omega \) and \(\lambda \) , the latter quantifying the violation of the usual conservation laws ( \(\lambda = 1\) corresponding to the General Relativity limit). The only exact spherically symmetric solution is that of Robinson–Bertotti besides the Schwarzschild solution. A PPN analysis reveals that General Relativity results are reproduced when \(\lambda = 0\) . The cosmological case displays a possibility of deceleration/acceleration or acceleration/deceleration transitions during the matter dominated phase depending on the values of the free parameters.

The possibility of dark matter being a dissipative component represents an option for the standard view where cold dark matter (CDM) particles behave on large scales as an ideal fluid. By including a physical mechanism in the dark matter description like viscosity, we construct a more realistic model for the Universe. Also, the known small scale pathologies of the standard CDM model either disappear or become less severe. We study clustering properties of a $\mathrm{\Lambda }\mathrm{CDM}$-like model in which dark matter is described as a bulk viscous fluid. The linear power spectrum, the nonlinear spherical "top hat" collapse, and the mass functions are presented. We use the analysis with such structure formation tools in order to place an upper bound on the magnitude of the dark matter’s viscosity.

The viability of achieving gravitational consistent braneworld models in the framework of a $f(R)$ theory of gravity is investigated. After a careful generalization of the usual junction conditions encompassing the embedding of the 3-brane into a $f(R)$ bulk, we provide a prescription giving the necessary constraints in order to implement the projected second-order effective field equations on the brane.