Stability and attractor in a higher-dimensional cosmology. I

The stability condition of (the four-dimensional Friedmann universe)*(a compact internal space) (F⁴*K^D) is presented for a class of higher-dimensional theories, in which the effective potential depends only on a scale length of the internal space. The Candelas-Weinberg model (i.e. one-loop quantum correction+a cosmological constant Lambda ), eleven-dimensional supergravity+ Lambda , Einstein-Yang-Mills theory and six-dimensional Einstein-Maxwell theory are classified into this class. It is shown that the F⁴*K^D solution is stable against small perturbations in the above models. The stability against non-linear perturbation is also investigated. The author finds that the stable F⁴*K^D solution is an attractor for a finite range of initial conditions if the proper volume of the universe is increasing with time.