Modern trends in multidimensional gravity and cosmology

The role of multidimensional gravitational models with different matter sources in solving the basic problems of cosmology is stressed. The choice, nature, classification and precision of determination of fundamental physical constants are described. The problem of their temporal variations is also discussed, temporal and range variations of G in particular. A need for their more precise measurements is pointed out. The novel multipurpose space project SEE, aimed for measuring G and its stability in space and time 3-4 orders better than at present, may answer many important questions posed by gravitation, cosmology and unified theories. A short review of multidimensional integrable gravitational and cosmological models is given. As an example, D-dimensional cosmological model describing the evolution of a perfect fluid with negative pressure (x-fluid) and a. fluid possessing both shear and bulk viscosity in n Ricci-flat spaces is investigated. The second equations of state are chosen in some special form of metric dependence of the shear and bulk viscosity coefficients. The equations of motion are integrated and the, dynamical properties of the exact solutions are studied. The possibility is shown to resolve the cosmic coincidence problem when the x-fluid plays role of quintessence and the viscous fluid is used as a cold dark matter.