A new covariant generalization of Einstein’s general relativity is developed which allows the existence of a term proportional to $T_{\alpha \beta }{T}^{\alpha \beta }$ in the action functional of the theory ($T_{\alpha \beta }$ is the energy-momentum 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 spacetime, we find that there is a maximum energy density ${\rho }_{\max }$, and correspondingly a minimum length $a_{\min }$, at the 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. We also argue that, although in the context of the standard cosmological model the cosmological constant $\mathrm{\Lambda }$ 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 in resolving the singularity.

The standard relativistic de-Broglie--Bohm theory has the problems of tacyonic solutions and the incorrect non-relativistic limit. In this paper we obtain a relativistic theory, not decomposing the relativistic wave equations but looking for a generalization of non-relativistic Bohmian theory in such a way that the correct non-relativistic limit emerges. In this way we are able to construct a relativistic de-Broglie--Bohm theory both for a single particle and for a many-particle system. At the end, the theory is extended to the curved space-time and the connection with quantum gravity is discussed.Comment: 13 pages, RevTeX. To appear in Physica Scripta, 200

We study Palatini $f(R)$ cosmology using Noether symmetry approach for the matter-dominated universe. In order to construct a point-like Lagrangian in the flat FRW space–time, we use the dynamical equivalence between $f(R)$ gravity and scalar–tensor theories. The existence of Noether symmetry of the cosmological $f(R)$ Lagrangian helps us to find out the form of $f(R)$ and the exact solutions for cosmic scale factor. We show that this symmetry always exist for $f(R)\sim R^n$ and the Noether constant is a function of the Newton's gravitational constant and the current matter content of the universe.