Uncertainties of predictions in models of eternal inflation

Abstract

In a previous paper \cite{MakingPredictions}, a method of comparing the volumes of thermalized regions in eternally inflating universe was introduced. In this paper, we investigate the dependence of the results obtained through that method on the choice of the time variable and factor ordering in the diffusion equation that describes the evolution of eternally inflating universes. It is shown, both analytically and numerically, that the variation of the results due to factor ordering ambiguity inherent in the model is of the same order as their variation due to the choice of the time variable. Therefore, the results are, within their accuracy, free of the spurious dependence on the time parametrization.Comment: 30 pages, RevTeX, figure included, added some references and Comments on recent proposal (gr-qc/9511058) of alternative regularization schemes, to appear in Phys. Rev.

What does inflation really predict?

“…This is equivalent to assigning statistical weights to the different pockets once the ordering and hence the p-distribution has been determined within each pocket. As detailed below, many attempts to solve the ordering problem have implicitly involved grouping the reference objects by the value of some physical parameter at their spacetime location [52][53][54][55][56][57][58][59][60][61], say by proper time t (Figure 4, top panel), scale factor a ( Figure 4, middle panel) or density ρ (roughly the black curves in top two panels).…”

“…As detailed below, this first class of orderings is intimately linked to the infamous problem of chosing a time variable in the inflationary Fokker-Planck equation, and suffers from serious problems [48][49][50][52][53][54][55][56][57][58][59][60][61]. For all cases where the total number of reference objects grows exponentially over time, any asymptotic t → ∞ distribution for reference objects at fixed time will be identical to the asymptotic distribution for reference objects formed at time ≤ t, because older objects which formed at time ≪ t are exponentially underrepresented.…”

“…[57,58,61]. As an interesting example giving qualitatively different predictions from t-ordering, let us consider ordering by the cosmic scale factor a.…”

“…(If this largest eigenvalue is negative, inflation is not eternal.) This eigenvalue problem can be transformed into a manifestly Hermitean form, and has been shown to reduce to a Schrödinger equation [57].…”

Fisher's Arrow of "Time" in Cosmological Coherent Phase Space

“…Winitzki and Vilenkin [27]. These models are based on FP equations for an inflaton probability distribution P (φ, t)dφ which is interpreted as the (either comoving or physical) volume of regions with a particular value of the inflaton scalar field in the infinitesimal interval dφ.…”