Uncertainties of predictions in models of eternal inflation
Serge Winitzki, Alexander Vilenkin
TL;DR
The paper analyzes predictions in eternal inflation models by focusing on two sources of uncertainty: the choice of time parametrization and the factor-ordering in the diffusion equation that governs the evolution of the inflaton field. By reformulating the diffusion problem into a self-adjoint, Schrödinger-like system, it derives ground-state eigenvalues gamma and tilde_gamma and expresses the volume ratio r between thermalized vacua in terms of these eigenvalues and ground-state wavefunctions. Analytic estimates based on a quadratic expansion near the potential maximum are validated against numerical results for symmetric and asymmetric inflaton potentials, showing that the dependencies on alpha and beta are of order H0^2 and that r remains largely invariant to these choices within the model’s accuracy. The findings suggest a form of time-parametrization independence at the level of the diffusion model and highlight the intrinsic limitations imposed by the diffusion approximation, with implications for more fundamental quantum-cosmological treatments.
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.