Measures for a Transdimensional Multiverse

TL;DR

This paper explores probability measures for a multiverse with varying dimensions, finding that a volume factor cutoff measure aligns better with observations than a naive scale factor cutoff extension.

Contribution

It introduces a generalized volume factor cutoff measure for transdimensional multiverses, improving prediction consistency over the scale factor cutoff.

Findings

Scale factor cutoff disfavors slow-roll inflation in transdimensional settings.

Volume factor cutoff retains desirable properties and aligns predictions with observations.

Naive extension of scale factor cutoff leads to conflicts with cosmological data.

Abstract

The multiverse/landscape paradigm that has emerged from eternal inflation and string theory, describes a large-scale multiverse populated by "pocket universes" which come in a huge variety of different types, including different dimensionalities. In order to make predictions in the multiverse, we need a probability measure. In $(3+1)d$ landscapes, the scale factor cutoff measure has been previously shown to have a number of attractive properties. Here we consider possible generalizations of this measure to a transdimensional multiverse. We find that a straightforward extension of scale factor cutoff to the transdimensional case gives a measure that strongly disfavors large amounts of slow-roll inflation and predicts low values for the density parameter $\Omega$, in conflict with observations. A suitable generalization, which retains all the good properties of the original measure, is the "volume factor" cutoff, which regularizes the infinite spacetime volume using cutoff surfaces of constant volume expansion factor.