Time-reparametrization invariance in eternal inflation

TL;DR

This paper investigates how different choices of time parametrization affect the description of eternal inflation, showing some physical quantities depend on the gauge while others remain invariant, and clarifies misconceptions about gauge dependence.

Contribution

The paper demonstrates that certain gauge-dependent quantities in eternal inflation can be made finite or infinite depending on the time foliation, and clarifies that no unique gauge provides an unbiased stationary distribution.

Findings

Relative abundance of regions is gauge-dependent.

Proper time gauge leads to unbounded 3-volume growth.

Finite-volume foliations exist with decreasing 3-volume.

Abstract

I address some recently raised issues regarding the time-parametrization dependence in stochastic descriptions of eternal inflation. To clarify the role of the choice of the time gauge, I show examples of gauge-dependent as well as gauge-independent statements about physical observables in eternally inflating spacetimes. In particular, the relative abundance of thermalized and inflating regions is highly gauge-dependent. The unbounded growth of the 3-volume of the inflating regions is found in certain time gauges, such as the proper time or the scale factor gauge. Yet in the same spacetimes there exist time foliations with a finite and monotonically decreasing 3-volume, which I demonstrate by an explicit construction. I also show that there exists no "correct" choice of the time gauge that would yield an unbiased stationary probability distribution for observables in thermalized regions.