Problem of inflation in nonlinear multidimensional cosmological models

We consider a multidimensional cosmological model with nonlinear quadratic R-2 and quartic R-4 actions. As a matter source, we include a monopole form field, a D-dimensional bare cosmological constant and the tensions of branes located at fixed points. In the spirit of the universal extra dimension model, the standard model fields are not localized on branes, but rather they can move in the bulk. We define conditions that ensure stable compactification of the internal space in zero minima of the effective potentials. Such effective potentials may have a rather complicated form with a number of local minima, maxima, and saddle points. We investigate inflation in such models. It is shown that the R-2- and R-4 models can produce up to 10 and 22 e-foldings, respectively. These values are not sufficient to solve the homogeneity and isotropy problem, but they are large enough to explain recent cosmic microwave background data. Additionally, the R-4 model can provide conditions for eternal topological inflation. The main drawback of the obtained inflationary models consists in a spectral index n(s) that is less than the presently observed n(s)approximate to 1. For the R-4 model we find, e.g., n(s)approximate to 0.61.