Stability and attractor in a higher-dimensional cosmology. II

For pt.I see ibid., vol.3, p.233 (1986). In most higher-dimensional theories, the effective matter Lagrangian after compactification is described not only by the radius of the internal space but also by a set of scalar fields (e.g. a dilaton). The author presents stability conditions for (the four-dimensional Friedmann universe (F⁴))*(a constant internal space (K)) in the above class. The stability against non-linear (large) perturbations and the attractor property of the F⁴*K solution are also investigated in order to explain why our universe is the present one. If the local zero minimum of the effective four-dimensional potential U is isolated, the stable F⁴*K solution is always one of the attractors when the 3-space is expanding. In the case that U has a degenerate zero minimum, which appears in theories involving some scale invariance, sufficient conditions for the F⁴*K solution to be the attractor are presented. Both the S² monopole compactification in the six-dimensional, N=2 supergravity theory and the Calabi-Yau compactification in the ten-dimensional, N=1 supergravity theory with or without the gluino condensation SUSY breaking potential are the cases in question. As an example of an isolated zero minimum, the model by Chapline and Gibbons (1984) (a coset compactification in the ten-dimensional, N=1 supergravity theory plus the SUSY breaking mass term) with a cosmological constant is also discussed.