The No-Boundary Measure of the Universe

TL;DR

This paper analyzes the no-boundary proposal for the universe, showing how volume weighting influences the probability of inflationary histories, and predicts a universe with large inflation starting near a de Sitter saddle point.

Contribution

It introduces a volume weighting factor exp(3N) to the no-boundary measure, linking it to eternal inflation and predicting large inflation in a landscape potential.

Findings

Bias towards small inflation without volume weighting

Volume weighting favors large inflation and de Sitter initial states

Predicts a universe with extensive inflation starting near a saddle point

Abstract

We consider the no-boundary proposal for homogeneous isotropic closed universes with a cosmological constant and a scalar field with a quadratic potential. In the semi-classical limit, it predicts classical behavior at late times if the initial scalar field is more than a certain minimum. If the classical late time histories are extended back, they may be singular or bounce at a finite radius. The no-boundary proposal provides a probability measure on the classical solutions which selects inflationary histories but is heavily biased towards small amounts of inflation. This would not be compatible with observations. However we argue that the probability for a homogeneous universe should be multiplied by exp(3N) where N is the number of e-foldings of slow roll inflation to obtain the probability for what we observe in our past light cone. This volume weighting is similar to that in eternal inflation. In a landscape potential, it would predict that the universe would have a large amount of inflation and that it would start in an approximately de Sitter state near a saddle-point of the potential. The universe would then have always been in the semi-classical regime.