A new covariant generalization of Einstein's general relativity is developed which allows the existence of a term proportional to T αβ T αβ in the action functional of the theory (T αβ is the energymomentum tensor). Consequently the relevant field equations are different from general relativity only in the presence of matter sources. In the case of a charged black hole, we find exact solutions for the field equations. Applying this theory to a homogeneous and isotropic space-time, we find that there is a maximum energy density ρmax, and correspondingly a minimum length amin, at early universe. This means that there is a bounce at early times and this theory avoids the existence of an early time singularity. Moreover we show that this theory possesses a true sequence of cosmological eras. Also, we argue that although in the context of the standard cosmological model the cosmological constant Λ does not play any important role in the early times and becomes important only after the matter dominated era, in this theory the "repulsive" nature of the cosmological constant plays a crucial role at early times for resolving the singularity.

We present a new approach to find exact solutions for cosmological models. By requiring the existence of a symmetry transformation vector for the equations of motion of the given cosmological model (without using either Lagrangian or Hamiltonian), one can find corresponding Hojman conserved quantities. With the help of these conserved quantities, the analysis of the cosmological model can be simplified. In the case of quintessence scalar-tensor models, we show that the Hojman conserved quantities exist for a wide range of V (φ)-potentials and allow to find exact solutions for the cosmic scale factor and the scalar field. Finally, we investigate the general cosmological behavior of solutions by adopting a phase-space view.

When bars form within galaxy formation simulations in the standard cosmological context, dynamical friction with dark matter (DM) causes them to rotate rather slowly. However, almost all observed galactic bars are fast in terms of the ratio between corotation radius and bar length. Here, we explicitly display an 8σ tension between the observed distribution of this ratio and that in the EAGLE simulation at redshift 0. We also compare the evolution of Newtonian galactic discs embedded in DM haloes to their evolution in three extended gravity theories: Milgromian Dynamics (MOND), a model of non-local gravity, and a scalar-tensor-vector gravity theory (MOG). Although our models start with the same initial baryonic distribution and rotation curve, the long-term evolution is different. The bar instability happens more violently in MOND compared to the other models. There are some common features between the extended gravity models, in particular the negligible role played by dynamical friction − which plays a key role in the DM model. Partly for this reason, all extended gravity models predict weaker bars and faster bar pattern speeds compared to the DM case. Although the absence of strong bars in our idealized, isolated extended gravity simulations is in tension with observations, they reproduce the strong observational preference for ‘fast’ bar pattern speeds, which we could not do with DM. We confirm previous findings that apparently ‘ultrafast’ bars can be due to bar-spiral arm alignment leading to an overestimated bar length, especially in extended gravity scenarios where the bar is already fast.

MOdified Gravity (MOG) is a covariant modification of Einstein's general relativity. This theory is one of the current alternatives to dark matter models. We describe dynamics of collisionless self-gravitating systems in the context of MOG. By studying the weak field approximation of this theory, we derive the equations governing the dynamics of the self-gravitating systems. More specifically, we consider the Jeans instability for self-gravitating fluid and stellar systems, and derive new Jeans mass limit $\tilde{M}_J$ and wave-number $\tilde{k}_J$. Furthermore, considering the gravitational instability in star forming regions, we show that MOG has not a significant difference with general relativity on this astrophysical scale. However, at larger scales such as intergalactic space MOG may lead to different galaxy and structure formation processes