| Abstract: | For the minimally coupled scalar field in Einstein's theory of gravitation we look for the space of solutions within the class of closed Friedmann universe models. We prove D ≥ 1, where D ≥ is the dimension of the set of solutions which can be integrated up to t → ∞ (D > 0 was conjectured by PAGE (1984)). We discuss concepts like “the probability of the appearance of a sufficiently long inflationary phase” and argue that it is primarily a probability measure μ in the space V of solutions (and not in the space of initial conditions) which has to be applied. μ is naturally defined for Bianchi-type I cosmological models because V is a compact cube. The problems with the closed Friedmann model (which led to controversial claims in the literature) will be shown to originate from the fact that V has a complicated non-compact non-Hausdorff Geroch topology: no natural definition of μ can be given. We conclude: the present state of our universe can be explained by models of the type discussed, but thereby the anthropic principle cannot be fully circumvented. |
