Stability and attractor in a higher-dimensional cosmology. I

The stability condition of (the four-dimensional Friedmann universe) x (a compact internal space) (F exp 4 x K exp 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 plus a cosmological constant Lamda), eleven-dimensional supergravity plus Lambda, Einstein-Yang-Mills theory and six-dimensional Einstein-Maxwell theory are classified into this class. It is shown that the F exp x K exp D solution is stable against small perturbations in the above models. The stability against non-linear perturbation is also investigated. It is found that the stable F exp 4 x K exp D solution is an attractor for a finite range of initial conditions if the proper volume of the universe is increasing with time.