Predictions in eternal inflation

TL;DR

This work analyzes how to make predictions in eternally inflating spacetimes by formulating a stochastic inflation framework based on Langevin dynamics and Fokker-Planck equations for $P(\phi,t)$ and $P_V(\phi,t)$. It highlights gauge dependence issues, the distinction between diffusion- and tunneling-driven self-reproduction, and the role of regularization in deriving observable distributions; it also contrasts volume-based and worldline-based measures, including their behavior in the string theory landscape. The paper provides both general analytical structures (largest eigenvalues $\gamma,\tilde{\gamma}$, stationary distributions) and specific prescriptions (spherical vs equal-time cutoffs; comoving vs worldline weighting) for extracting predictions like the distribution of fields $\chi_a$ at reheating. Overall, it clarifies how eternal inflation can generate statistically diverse observers and outlines concrete methodologies to translate this diversity into testable predictions, even in highly inhomogeneous, multi-vacua settings.

Abstract

In generic models of cosmological inflation, quantum fluctuations strongly influence the spacetime metric and produce infinitely many regions where the end of inflation (reheating) is delayed until arbitrarily late times. The geometry of the resulting spacetime is highly inhomogeneous on scales of many Hubble sizes. The recently developed string-theoretic picture of the "landscape" presents a similar structure, where an infinite number of de Sitter, flat, and anti-de Sitter universes are nucleated via quantum tunneling. Since observers on the Earth have no information about their location within the eternally inflating universe, the main question in this context is to obtain statistical predictions for quantities observed at a random location. I describe the problems arising within this statistical framework, such as the need for a volume cutoff and the dependence of cutoff schemes on time slicing and on the initial conditions. After reviewing different approaches and mathematical techniques developed in the past two decades for studying these issues, I discuss the existing proposals for extracting predictions and give examples of their applications.

Figures (9)

• Figure 1: A qualitative diagram of self-reproduction during inflation. Shaded spacelike domains represent Hubble-size regions with different values of the inflaton field $\phi$. The time step is of order $H^{-1}$. Dark-colored shades are regions undergoing reheating ($\phi=\phi_{*}$); lighter-colored shades are regions where inflation continues. On average, the number of inflating regions grows with time.
• Figure 2: A 1+1-dimensional slice of the spacetime structure in an eternally inflating universe (numerical simulation in Ref. Vanchurin:1999iv). Shades of different color represent different, causally disconnected regions where reheating took place. The reheating surface is the line separating the white (inflating) domain and the shaded domains.
• Figure 3: A spacetime diagram of a bubble interior. The infinite, spacelike reheating surface is shown in darker shade. Galaxy formation is possible within the spacetime region indicated.
• Figure 4: A schematic representation of the "landscape of string theory," consisting of a large number of local minima of an effective potential. The variable $X$ collectively denotes various fields and $\Lambda$ is the effective cosmological constant. Arrows show possible tunneling transitions between vacua.
• Figure 5: A conformal diagram of the spacetime where self-reproduction occurs via bubble nucleation. Regions labeled "5" are asymptotically flat ($\Lambda=0$).