The massive scalar field in a closed Friedmann universe model - new rigorous results

TL;DR

This paper rigorously analyzes the solution space of a minimally coupled scalar field in closed Friedmann universe models, revealing a fractal dimension structure and discussing implications for cosmic inflation probabilities.

Contribution

It proves that the fractal dimension of the solution set is either 1 or greater than 1, clarifies the measure problem in cosmology, and addresses misconceptions in the literature.

Findings

The solution set has fractal dimension D=1 or D>1.

The probability measure should be defined in the space of solutions, not initial conditions.

The topology of the solution space complicates defining a natural probability measure.

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 that D = 1 or D > 1, where D is the (fractal) dimension of the set of solutions which can be integrated up to t to infinity. (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 q in the space V of solutions (and not in the space of initial conditions) which has to be applied. q 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 q 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.