Measures for a transdimensional multiverse

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 Ω, 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.