Volume Weighting the Measure of the Universe from Classical Slow-Roll Expansion

One of the most frustrating issues in early universe cosmology centers on how to reconcile the vast choice of universes in string theory and in its most plausible high energy sibling, eternal inflation, that jointly generate the string landscape with the fine-tuned and hence relatively small number of universes that have undergone a large expansion and can accommodate observers and, in particular, galaxies. We show that such observations are highly favored for any system whereby physical parameters are distributed at a high energy scale, due to the conservation of the Liouville measure and the gauge nature of volume, asymptotically approaching a period of large isotropic expansion characterised by $w=-1$. Our interpretation predicts that all observational probes for deviations from $w=-1$ in the foreseeable future are doomed to failure. The purpose of this paper is not to introduce a new measure for the multiverse, but rather to show how what is perhaps the most natural and well known measure, volume weighting, arises as a consequence of the conservation of the Liouville measure on phase space during the classical slow-roll expansion.