Saidov T. The compactification problems of additional dimensions in multidimensional cosmological theories.

The thesis is devoted to study of freezing stabilization and compactification of additional dimensions in nonlinear multidimensional gravitational models and to investigation of the inflation in such models. Several nonlinear multidimensional gravitational models are studied in this thesis, such as pure gravitational models with nonlinearities of type and form-field containing models with linear, quadratic, quartic and of type curvatures. Adding forms to the model of gives possibility to reach positive minimum of the effective potential. Then the freezing stabilization of the internal spaces is provided, which lets avoiding of the problem for the fundamental constant variation in multidimensional models. At the same time, the stage of the cosmic acceleration is achieved too. However, fine tuning is required to get parameters of the present-day accelerating expansion. Additionally the existence of domain walls was found for this model, which separates regions with different vacua in the Universe without providing of inflation, because the effective potential is not flat enough around the saddle point. The pure gravitational model with curvature-quadratic plus quartic correction terms allows for the stable compactification of the internal space for certain region of stability. The inflation has been investigated in the models being linear and with nonlinearities of type and with a monopole form field, D-dimensional bare cosmological constant. As result, the equivalent linear model has two scalar fields: one corresponds to the scale factor of the internal space and another nonlinearity of the original models. The dynamical behavior of the scalaron field and the Universe was studied in quadratic plus quartic nonlinear models. The scalaron potential can be a multi-valued function consisting of number of branches, being connected in the branching points or in the monotonic points. It appears that the monotonic points are penetrable for scalaron field while in the vicinity of the branching points scalaron has the bouncing behavior and cannot cross these points. Also branching points where found, where scalaron bounces an infinite number of times with decreasing amplitude, approaching asymptotically de-Sitter stage. This king of accelerating behavior is called bouncing inflation. This represents a new type of inflation, which takes place both in the Einstein and Brans-Dicke frames and play an important role during the early stages of the Universe evolution.