On the Hawking Turok solution to the Open Universe wave function

Hawking and Turok have recently published a solution to the WKB “wavefunction for the universe” which they claim leads in a natural way to an open universe as the end point of the evolution for a universe dominated by a scalar field. They furthermore argue that their solution a preferred solution under the rules of the game. This paper will, I hope, clarify their solution and the limits of validity of their argument. Hawking and Turok [1] (hereafter HT) recently published a fascinating paper in which they claimed to have a solution of the (complex) Einstein equations which satisfy the rules for a WKB solution for the “wave-function of the universe” as laid down by Hartle and Hawking [4], but which has as the future 3-geometry a spatially open hyperbolic homogeneous universe. Thus this “instanton” solutions allows them to calculate the probability that the universe will form as a homogeneous open universe. Using the anthropic arguments, they also claimed to find a preferred value for the present day Ω, the ratio of density to critical density, of 0.01, rather than the value of 1 as usually expected of inflationary models. This of course is “too small” in comparison with experimental evidence, just as the usual prediction of unity may be “too large” [2]. Linde [3] has also criticized their derivation. However, their derivation was somewhat elliptic and clarification of what they calculated seems desirable. This paper will I hope provide such clarification and will also raise some questions about the interpretation of their solution. Let us first review the rules [4]. The wave function for the universe Ψ( G) is a wave function defined on Euclidean non-singular three universes G. It is defined (at least formally) by means of a path integral, where the path integral is to be taken over all four geometries, including complex four geometries, which are 4 dimensional manifolds whose only boundary is the three geometry of interest. This path integral prescription is poorly defined, and thus its principle application has been via a semi-classical approximation. One chooses as one’s allowed 4 geometries only those which extremize the Einstein-matter action, and have the given three geometry as their only boundary. One hopes that there are only a few, or one, of these solutions. One then takes as the approximation to the wave-function just Σje iSj( G) where Sj is the action for the j th 4-geometry solution. The real part of Sj is then the phase of the wave function, while the imaginary part gives the probability for