Time-reparametrization invariance in eternal inflation

TL;DR

This work analyzes how time reparametrization (time gauge) affects stochastic descriptions of eternal inflation in the FP formalism. Through a toy model, the author shows that equal-time distributions and volume-weighted observables are highly gauge-dependent, and that unbounded 3-volume growth can occur in some gauges while remaining finite in others, with a fractal dimension $\gamma=2-\alpha/H_0$ characterizing the inflating domain. It demonstrates that volume comparisons and equal-time cutoffs are not gauge-invariant and argues that there is no universally \'correct\' time parameter that yields unbiased results for all potentials. The findings clarify criticisms about gauge artifacts in eternal inflation, reaffirming that the standard picture is valid only when gauge choices and their limitations are properly accounted for.

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.

Figures (7)

• Figure 1: First steps in the construction of a random Sierpiński carpet with $N=5$ and $q=5/6$.
• Figure 2: Illustrative inflationary potential with a flat self-reproduction regime $\phi_{1}<\phi<\phi_{2}$ and deterministic regimes $\phi_{*}^{(1)}<\phi<\phi_{1}$ and $\phi_{2}<\phi<\phi_{*}^{(2)}$.
• Figure 3: The piecewise-null surface $S_{0}(x)$ having zero proper length.
• Figure 4: A null foliation of the de Sitter spacetime in conformal coordinates $\left(\eta,x\right)$. Thicker lines are the consecutively smaller copies of the saw-shaped surface $S_{0}(x)$, while the thinner lines in between show the continuous deformation procedure ("growing ridges"). The dashed line corresponds to an infinite future ($\eta\rightarrow0$).
• Figure 5: The "smooth corner" function $C(x)$ defined by Eq. (\ref{['eq:c0 def']}) with $A=0.01$, $B=0.5$, $C_{1}\approx0.6613$, $C_{2}\approx3.323$.