The current accelerated universe could be produced by modified gravitational dynamics as it can be seen, in particular, in its Palatini formulation. We analyze here a specific nonlinear gravity-scalar system in the first-order Palatini formalism which leads to a Friedmann-Robertson-Walker cosmology different from the purely metric one. It is shown that the emerging Friedmann-Robertson-Walker cosmology may lead either to an effective quintessence phase (cosmic speed-up) or to an effective phantom phase. Moreover, the already known gravity assisted dark energy dominance occurs also in the first-order formalism. Finally, it is shown that a dynamical theory able to resolve the cosmological constant problem exists also in this formalism, in close parallel with the standard metric formulation.

Alternative gravitational theories described by Lagrangians depending on general functions of the Ricci scalar have been proven to give coherent theoretical models to describe the experimental evidence of the acceleration of the Universe at present time. In this paper we proceed further in this analysis of cosmological applications of alternative gravitational theories depending on (other) curvature invariants. We introduce Ricci squared Lagrangians in minimal interaction with matter (perfect fluid); we find modified Einstein equations and consequently modified Friedmann equations in the Palatini formalism. It is striking that both Ricci scalar and Ricci squared theories are described in the same mathematical framework and both the generalized Einstein equations and generalized Friedmann equations have the same structure. In the framework of the cosmological principle, without the introduction of exotic forms of dark energy, we thus obtain modified equations providing values of $w_{\mathrm{e}\mathrm{f}\mathrm{f}}<-1$ in accordance with the experimental data. The spacetime bi-metric structure plays a fundamental role in the physical interpretation of results and gives them a clear and very rich geometrical interpretation.

Two one-parameter families of twists providing $\kappa$-Minkowski $*$ product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfeld’s twisting tensor technique. The other one relies on an appropriate extension of “deformed realizations” of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us to calculate deformed dispersion relations for toy models ensuing from different twist parameters. In the Abelian case, one recovers $\kappa$-Poincaré dispersion relations having numerous applications in doubly special relativity. Jordanian twists provide a new type of dispersion relations which in the minimal case (related to Weyl-Poincaré algebra) takes an energy-dependent linear mass deformation form.

The evidence of the acceleration of universe at present time has led to investigate modified theories of gravity and alternative theories of gravity, which are able to explain acceleration from a theoretical viewpoint without the need of introducing dark energy. In this paper we study alternative gravitational theories defined by Lagrangians which depend on general functions of the Ricci scalar invariant in minimal interaction with matter, in view of their possible cosmological applications. Structural equations for the spacetimes described by such theories are solved and the corresponding field equations are investigated in the Palatini formalism, which prevents instability problems. Particular examples of these theories are also shown to provide, under suitable hypotheses, a coherent theoretical explanation of earlier results concerning the present acceleration of the universe and cosmological inflation. We suggest moreover a new possible Lagrangian, depending on the inverse of $\sinh \mathrm{}\left(R\right),$ which gives an explanation to the present acceleration of the universe.

A new systematic approach extending the notion of frames to the Palatini scalar–tensor theories of gravity in various dimensions [equation]is proposed. We impose frame transformation induced by the...

It has been shown recently that, in the first-order (Palatini) formalism, there is universality of the Einstein equations and the Komar energy-momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets the Einstein equations and Komar's expression for the energy-momentum complex. In this paper a similar analysis (also within the framework of the first-order formalism) is performed for all nonlinear Lagrangians depending on the (symmetrized) Ricci square invariant. The main result is that the universality of the Einstein equations and Komar's energy-momentum complex also extends to this case (modulo a conformal transformation of the metric).