Abstract. The kinetic motion of the stars of a galaxy is considered within the framework of a relativistic scalar theory of gravitation. This model, even though unphysical, may represent a good laboratory where to study in a rigorous, mathematical way those problems, like the influence of the gravitational radiation on the dynamics, which are still beyond our present understanding of the physical model represented by the Einstein-Vlasov system. The present paper is devoted to derive the equations of the model and to prove the existence of spherically symmetric equilibria with finite radius.

The dynamics of cosmological models with isotropic matter sources (perfect fluids) is extensively studied in the literature; in comparison, the dynamics of cosmological models with anisotropic matter sources is not. In this paper we consider spatially homogeneous locally rotationally symmetric solutions of the Einstein equations with a large class of anisotropic matter models including collisionless matter (Vlasov), elastic matter, and magnetic fields. The dynamics of models of Bianchi types I, II, and IX are completely described; the two most striking results are the following: (i) There exist matter models, compatible with the standard energy conditions, such that solutions of Bianchi type IX (closed cosmologies) need not necessarily recollapse; there is an open set of forever expanding solutions. (ii) Generic type IX solutions associated with a matter model like Vlasov matter exhibit oscillatory behavior toward the initial singularity. This behavior differs significantly from that of vacuum/perfect fluid cosmologies; hence "matter matters". Finally, we indicate that our methods can probably be extended to treat a number of open problems, in particular, the dynamics of Bianchi type VIII and Kantowski-Sachs solutions.

A new model to describe the dynamics of particles undergoing diffusion in general relativity is proposed. The evolution of the particle system is described by a Fokker-Planck equation without friction on the tangent bundle of spacetime. It is shown that the energy-momentum tensor for this matter model is not divergence-free, which makes it inconsistent to couple the Fokker-Planck equation to the Einstein equations. This problem can be solved by postulating the existence of additional matter fields in spacetime or by modifying the Einstein equations. The case of a cosmological scalar field term added to the left hand side of the Einstein equations is studied in some details. For the simplest cosmological model, namely the flat Robertson-Walker spacetime, it is shown that, depending on the initial value of the cosmological scalar field, which can be identified with the present observed value of the cosmological constant, either unlimited expansion or the formation of a singularity in finite time will occur in the future. Future collapse into a singularity also takes place for a suitable small but positive present value of the cosmological constant, in contrast to the standard diffusion-free scenario. * calogero@ugr.es

The Nordström-Vlasov system describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordström scalar theory of gravitation. We prove existence and uniqueness of classical solutions of the Cauchy problem in three dimensions and establish a condition which guarantees that the solution is global in time. Moreover, we show that if one changes the sign of the source term in the field equation, which heuristically corresponds to the case of a repulsive gravitational force, then solutions blow up in finite time for a large class of initial data. Finally, we prove global existence of classical solutions for the one dimensional version of the system with the correct sign in the field equation.

We construct a viable cosmological model based on velocity diffusion of matter particles. In order to ensure the conservation of the total energy-momentum tensor in the presence of diffusion, we include a cosmological scalar field φ which we identify with the dark energy component of the universe. The model is characterized by only one new degree of freedom, the diffusion parameter σ. The standard ΛCDM model can be recovered by setting σ = 0. If diffusion takes place (σ > 0) the dynamics of the matter and of the dark energy fields are coupled. We argue that the existence of a diffusion mechanism in the universe may serve as a theoretical motivation for interacting models. We constrain the background dynamics of the diffusion model with Supernovae, H(z) and BAO data. We also perform a perturbative analysis of this model in order to understand structure formation in the universe. We calculate the impact of diffusion both on the CMB spectrum, with particular attention to the integrated Sachs-Wolfe signal, and on the matter power spectrum P (k). The latter analysis places strong constraints on the magnitude of the diffusion mechanism but does not rule out the model.