On the 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, D-dimensional bare cosmological constant and tensions of branes located in fixed points. In the spirit of the Universal Extra Dimensions models, the Standard Model fields are not localized on branes but can move in the bulk. We define conditions which ensure the stable compactification of the internal space in zero minimum of the effective potentials. Such effective potentials may have rather complicated form with a number of local minima, maxima and saddle points. Then, we investigate inflation in these models. It is shown that $R^2$ and $R^4$ models can have up to 10 and 22 e-foldings, respectively. These values are not sufficient to solve the homogeneity and isotropy problem but big enough to explain the recent CMB data. Additionally, $R^4$ model can provide conditions for eternal topological inflation. However, the main drawback of the given inflationary models consists in a value of spectral index $n_s$ which is less than observable now $n_s\approx 1$. For example, in the case of $R^4$ model we find $n_s \approx 0.61$. 
Tamerlan Saidov and Alexander Zhuk