We consider the scenario emerging from the dynamics of a generalized Born-Infeld theory. The equation of state describing this system is given in terms of the energy density $\rho$ and pressure p by the relationship $p=-A/{\rho }^{\alpha },$ where A is a positive constant and $0<\mathrm{}\alpha <~1.$ We discuss the conditions under which homogeneity arises and show that this equation of state describes the evolution of a universe evolving from a phase dominated by nonrelativistic matter to a phase dominated by a cosmological constant via an intermediate period where the effective equation of state is given by $p=\alpha \rho .$

The equation of motion for massive particles in $f(R)$ modified theories of gravity is derived. By considering an explicit coupling between an arbitrary function of the scalar curvature, $R$, and the Lagrangian density of matter, it is shown that an extra force arises. This extra force is orthogonal to the four-velocity and the corresponding acceleration law is obtained in the weak-field limit. Connections with MOND and with the Pioneer anomaly are further discussed.

In this work, we consider different forms of relativistic perfect fluid Lagrangian densities that yield the same gravitational field equations in general relativity (GR). A particularly intriguing example is the case with couplings of the form $[1+f_2(R)]{\mathcal{L}}_m$, where $R$ is the scalar curvature, which induces an extra force that depends on the form of the Lagrangian density. It has been found that, considering the Lagrangian density ${\mathcal{L}}_m=p$, where $p$ is the pressure, the extra-force vanishes. We argue that this is not the unique choice for the matter Lagrangian density, and that more natural forms for ${\mathcal{L}}_m$ do not imply the vanishing of the extra force. Particular attention is paid to the impact on the classical equivalence between different Lagrangian descriptions of a perfect fluid.

We show that the Abell cluster A586 exhibits evidence of the interaction between dark matter and dark energy and argue that this interaction implies a violation of the equivalence principle. This violation is found in the context of two different models of dark energy–dark matter interaction. We also argue, based on the spherical symmetry of the Abell cluster A586 that skewness is not the most general quantity to test the equivalence principle

Recent studies of type Ia supernovas with redshifts up to about $z\lesssim 1$ reveal evidence for a cosmic acceleration in the expansion of the Universe. The most straightforward explanation to account for this acceleration is a cosmological constant dominating the recent history of our Universe; however, a more interesting suggestion is to consider an evolving vacuum energy. Several proposals have been put forward along these lines, most of them in the context of general relativity. In this work we analyze the conditions under which the dynamics of a self-interacting Brans-Dicke field can account for this accelerated expansion of the Universe. We show that accelerated expanding solutions can be achieved with a quadratic self-coupling of the Brans-Dicke field and a negative coupling constant ω.