Time-reparametrization invariance in eternal inflation

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 choice of the time gauge that would yield an unbiased stationary probability distribution for observables in thermalized regions.

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 ${\varphi }_1<\varphi <{\varphi }_2$ and deterministic regimes ${\varphi }_{*}^{(1)}<\varphi <{\varphi }_1$ and ${\varphi }_2<\varphi <{\varphi }_{*}^{(2)}$.

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 ($\eta$, $x$). 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 ($\eta \to 0$).

Figure 5

The ”smooth corner” function $C(x)$ defined by Eq. (b8) with $A=0.01$, $B=0.5$, $C_1\approx 0.6613$, $C_2\approx 3.323$.

Figure 6

The rescaled “smooth corner” functions $C_{\delta }(x)$ (corner down) and ${\tilde{C}}_{\delta }(x)$ (corner up) defined by Eq. (b12).

Figure 7

A typical piecewise-null hypersurface from the foliation in Fig. 4.