Classical and Quantum Cosmology of an Accelerating Model Universe with 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 in the form of perfect fluid with an special equation of state, as the space-time part of the higher dimensional energy-momentum tensor, a four dimensional effective decaying cosmological term appears as $\lambda \sim R^{-m}$ with $0 \leq m\leq 2$, playing the role of an evolving dark energy in the universe. By taking $m=2$, which has some interesting implications in reconciling observations with inflationary models and is consistent with quantum tunneling, the resulting Einstein's field equations yield the exponential solutions for the scale factors $a$ and $R$. These exponential behaviors may account for the dynamical compactification of extra dimensions and the accelerating expansion of the 4-dimensional universe in terms of Hubble parameter, $H$. The acceleration of the universe may be explained by the negative pressure of the exotic matter. It is shown that the rate of compactification of higher dimensions as well as expansion of 4-dimensional universe depends on the dimension, $D$. We then obtain the corresponding Wheeler-DeWitt equation and find the general exact solutions in $D$-dimensions. A good correspondence between the solutions of classical Einstein's equations and the solutions of quantum Wheeler-DeWitt equation in any dimension, $D$, is obtained based on Hartle's point of view concerning the classical limits of quantum cosmology.