Six-dimensional Einstein-Gauss-Bonnet gravity (with a linear Gauss-Bonnet term) is investigated. This theory is inspired by basic features of results coming from string and M-theory. Dynamical compactification is carried out and it is seen that a four-dimensional accelerating FRW universe is recovered, when the two-dimensional internal space radius shrinks. A nonperturbative structure of the corresponding theory is identified which has either three or one stable fixed points, depending on the Gauss-Bonnet coupling being positive or negative. A much richer structure than in the case of the perturbative regime of the dynamical compactification recently studied by Andrew, Bolen, and Middleton is exhibited. 1. Introduction. To realize that the expansion of the universe is accelerating was one of the most important scientific discoveries of the last century. There are several alternative explanations of this remarkable fact (what already means that it is not well understood yet). A quite appealing possibility for the gravitational origin of the dark energy responsible for this accelerated expansion is the modification of General Relativity (GR) or the corresponding Einsteinian gravity (see [1] for a review of these approaches). Such a well-established and successful theory cannot be modified without a very good reason, and much less in an arbitrary way. But observe that there is no compelling reason why standard GR should be trusted at cosmological scales. For a rather minimal modification of the same, one assumes that the gravitational action might contain some additional terms which would start to grow slowly with decreasing curvature. A particularly interesting formulation is obtained by using well-grounded geometrical arguments, specifically when the modified gravity action is endowed with a function of the Gauss-Bonnet (GB) topological invariant, G, as it was suggested in [2]. It must be noted that different types of dark energy may actually show up in different ways, at large distances. Cold dark matter is known to be localized near galaxy clusters but, quite on the contrary, dark energy distributes uniformly in the universe. The reason for that behavior could be explained by a difference in the equation of state parameter w = p/ρ. Moreover, the effect of gravity on the cosmological fluid turns out to depend on w and it so happens that, even when −1 < w < 0, gravity can act sometimes as a repulsive force. The effect of gravity on matter with −1 < w < 0 can be shown to be opposite to that on usual matter, which becomes dense near a star, while matter with −1 < w < 0 becomes less dense when approaching a star [2]. Dark energy contributes uniformly throughout the universe, which would be indeed consistent, since the equation of state parameter of dark energy is almost −1. If dark energy is of phantom nature (w < −1), its density becomes large near the cluster but if dark energy is of quintessence type (−1 < w < −1/3), then its density becomes smaller. PACSAnother very important argument in favor of the GB ...

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All classes of spatially homogeneous space-time models are found that allow the integration of the equations of motion of test particles and the eikonal equation by the method of complete separation of variables according to type (2.1). Four classes of model data are obtained. The resulting models can be applied in any modified metric theories of gravity. Two of the above models allow solutions of the Einstein equations with a cosmological constant and radiation. For the models of a spatially homogeneous Universe with a cosmological constant and radiation obtained in Einstein's theory of gravity, the Hamilton-Jacobi equations of motion of the test particles and the eikonal equation for radiation are integrated by the method of separation of variables.

The characteristics of dust matter in space-time models, admitting the existence of privilege coordinate systems are given, where the single-particle Hamilton-Jacobi equation can be integrated by the method of complete separation of variables. The resulting functional form of the 4-velocity field and energy density of matter for all types of spaces under consideration is presented.

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