Dark Gravitational Field on Riemannian and Sasaki Spacetime

The aim of this paper is to provide the geometrical structure of a gravitational field that includes the addition of dark matter in the framework of a Riemannian and a Riemann–Sasaki spacetime. By means of the classical Riemannian geometric methods we arrive at modified geodesic equations, tidal forces, and Einstein and Raychaudhuri equations to account for extra dark gravity. We further examine an application of this approach in cosmology. Moreover, a possible extension of this model on the tangent bundle is studied in order to examine the behavior of dark matter in a unified geometric model of gravity with more degrees of freedom. Particular emphasis shall be laid on the problem of the geodesic motion under the influence of dark matter.

Noncommutative Friedmann equations in effective LQC

In this work, we construct a noncommutative version of the Friedmann equations in the framework of effective loop quantum cosmology, extending and applying the ideas presented in a previous proposal by some of the authors. The model under consideration is a flat FRW spacetime with a free scalar field. First, noncommutativity in the momentum sector is introduced. We establish the noncommutative equations of motion and obtain the corresponding exact solutions. Such solutions indicate that the bounce is preserved, in particular, the energy density is the same as in the standard LQC. We also construct an extension of the modified Friedmann equations arising in effective LQC which incorporates corrections due to noncommutativity, and argue that an effective potential is induced. This, in turn, leads us to investigate the possibility of an inflationary era. Finally, we obtain the Friedmann and the Raychaudhuri equations when implementing noncommutativity in the configuration sector. In this case, no effective potential is induced.

Congruence kinematics in conformal gravity

In this paper we calculate the kinematical quantities possessed by Raychaudhuri equations, tocharacterize a congruence of time-like integral curves, according to the vacuum radial solution of Weyl theory of gravity. Also the corresponding flows are plotted for denfinite values of constants.

Raychaudhuri equation in the Finsler–Randers space-time and generalized scalar-tensor theories

In this work, we obtain the Raychaudhuri equations for various types of Finsler spaces as the Finsler–Randers (FR) space-time and in a generalized geometrical structure of the space-time manifold which contains two fibers that represent two scalar fields [Formula: see text]. We also derive the Klein–Gordon equation for this model. In addition, the energy conditions are studied in a FR cosmology and are correlated with FRW model. Finally, we apply the Raychaudhuri equation for the model [Formula: see text], where M is a FRW-space-time.

Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.