Vacuum polarization of a scalar field in anisotropic multidimensional cosmology

An original method is proposed for calculating the semiclassical vacuum effective action for a scalar field in a multidimensional universe that is nonstationary in all dimensions, has metric that generalizes the Bianchi type I, and has toroidally compactified spacelike dimensions (some or all of them). It is noted that the number of terms that can be retained in the adiabatic expansion is restricted and proportional to the dimension of the open subspace. This is due to the presence of a “zero mode”, which also leads to the appearance of logarithmic terms. Examples with three-dimensional open subspace and one or two compact additional dimensions are considered as illustrations.