Comments on the Canonical Measure in Cosmology

TL;DR

This paper examines how the canonical measure in cosmology's mini-superspace is affected by phase space cutoffs, revealing their evolution under Hamiltonian flow and implications for inflation probability calculations.

Contribution

It analyzes the deformation of regularized phase space in cosmology and assesses the cutoff dependence of inflation probability in models with scalar fields and gauge fields.

Findings

Cutoff for the scale factor decreases backwards in time.

Inflation probability is cutoff dependent but not exponentially suppressed.

Deformation of phase space affects measure calculations in cosmology.

Abstract

In the mini-superspace approximation to cosmology, the canonical measure can be used to compute probabilities when a cutoff is introduced in the phase space to regularize the divergent measure. However, the region initially constrained by a simple cutoff evolves non-trivially under the Hamiltonian flow. We determine the deformation of the regularized phase space along the orbits when a cutoff is introduced for the scale factor of the universe or for the Hubble parameter. In the former case, we find that the cutoff for the scale factor varies in the phase space and effectively decreases as one evolves backwards in time. In the later case, we calculate the probability of slow-roll inflation in a chaotic model with a massive scalar, which turns out to be cutoff dependent but not exponentially suppressed. We also investigate the measure problem for non-abelian gauge fields giving rise to inflation.