Vacuum polarization of scalar field in anisotropic multidimensional cosmology

An original method is suggested for calculating the semiclassical vacuum effective action for scalar field in multidimensional nonstationary in all directions Universe with the metric of the generalized Bianchi I-type and toroidally compactified (some or all) space-like dimensions. It is pointed out that the number of the terms which can be preserved in the adiabatic expansion is restricted and proportional to the open subspace dimension because of the zero mode presence which leads also to the appearance of logarithmic terms. As an illustration, examples with three-dimensional open subspace and one or two compact extra dimensions are considered