60 v 1 2 2 Ju l 2 00 0 1 Numerical Analysis of the Wave Function of the Multidimensional Universe

In modern theories of unified physical interactions, e.g. superstring theory and M-theory, spacetime has more than four dimensions. There are many works on multidimensional cosmological models (e.g. Refs. 1)–6) and references therein). Some of them 7), 8), 9), 10), 11) are studied in the framework of the scenario of quantum creation of the universe. 12), 13) In the semi-classical approximation, the effect of classical Euclidean solutions is essential, and the wave function is estimated by the Euclidean action. However, a detailed analysis of the Euclidean Einstein equations in the multidimensional cosmological model remained to be done. In a previous paper, 14) we investigated quantum creation of the multidimensional universe with the cosmological constant Λ but without matter fields by the method of the instanton. In that paper, the metric was assumed to take the extended Friedmann-RobertsonWalker form. We found that there are three instanton solutions in the case that both the curvatures of the external and internal spaces are positive and Λ > 0. Furthermore, we found that the crossing point of three instanton solutions is a “quasiattractor” point from numerical analysis of the Euclidean solutions. Here, “quasiattractor” point is a point to which most trajectories in the a b plane circulate, where a and b are the scale factors of the external and internal spaces. We found that there are Lorentzian solutions which start near the quasi-attractor and three exact Lorentzian solutions corresponding to three instanton solutions in Ref. 14). Furthermore, from analyzing the classical solutions of the Einstein equations, we found that it is probable that three universes are nucleated, corresponding to these three solutions. The first solution represents the case in which the external space expands and the internal space contracts. The second solution represents the