Accelerating Universe and Dynamical Compactification of Extra Dimensions

We study a $(4+D)$-dimensional Kaluza-Klein cosmology with a Robertson-Walker type metric having two scale factors $a$ and $R$, corresponding to $D$-dimensional internal space and 4-dimensional universe, respectively. By introducing an exotic matter as the space-time part of the higher dimensional energy-momentum tensor, a 4-dimensional decaying cosmological term is appeared as $\mathrm{\Lambda }\sim R^{-2}$, playing the role of an evolving dark energy in the universe. The resulting field equations yield the exponential solutions for the scale factors. These exponential behaviors may account for the dynamical compactification of extra dimensions and the accelerating expansion of the 4-dimensional universe in terms of the Hubble parameter. The acceleration of universe may be explained by the negative pressure of the exotic matter. It is shown that the rate of compactification of higher dimensions depends on the dimension, $D$. We then obtain the Wheeler-DeWitt equation and find the general exact solutions in $D$-dimensions. A good correspondence between the classical solutions and the Wheeler-DeWitt solutions, in any dimension $D$, is obtained.