A volume-weighted measure for eternal inflation

TL;DR

This work addresses the measure problem in eternal inflation by proposing the reheating-volume (RV) cutoff, which conditions on finite reheating volumes to define a well-defined, gauge-invariant, volume-weighted probability measure. The approach relies on finitely produced reheated-volume (FPRV) distributions and generating-function techniques within a slow-roll diffusion framework, enabling computation of the large-volume limit $p(Q) = \lim_{\mathcal{V}\to\infty} p(Q|\mathcal{E}_{\mathcal{V}})$ for observables $Q$. In slow-roll inflation, the key tool is nonlinear diffusion equations for generating functions $g(z;\phi)$ and $\tilde{g}(z,q;\phi,Q)$, whose boundary conditions encode reheating and Planck-scale limits; the asymptotics of these functions determine the RV distributions and ensure independence from initial conditions, thereby avoiding the youngness paradox. The RV framework is then extended to landscape models via an inflation-in-a-box formulation, yielding nonlinear algebraic equations for bubble-count generating functions and providing a concrete, albeit model-dependent, prediction for terminal-vacua abundances (e.g., a corrected ratio $p(1)/p(2)$) while highlighting the sensitivity to slow-roll expansion and the potential to apply RV to general observables in the multiverse.

Abstract

I propose a new volume-weighted probability measure for cosmological "multiverse" scenarios involving eternal inflation. The "reheating-volume (RV) cutoff" calculates the distribution of observable quantities on a portion of the reheating hypersurface that is conditioned to be finite. The RV measure is gauge-invariant, does not suffer from the "youngness paradox," and is independent of initial conditions at the beginning of inflation. In slow-roll inflationary models with a scalar inflaton, the RV-regulated probability distributions can be obtained by solving nonlinear diffusion equations. I discuss possible applications of the new measure to "landscape" scenarios with bubble nucleation. As an illustration, I compute the predictions of the RV measure in a simple toy landscape.

Figures (2)

• Figure 1: A 1+1-dimensional slice of the spacetime in an eternally inflating universe (numerical simulation in Ref. Vanchurin:1999iv). Shades of different color represent different regions where reheating took place. The reheating surface is the line separating the white (inflating) domain and the shaded domains.
• Figure 2: A schematic representation of the ensemble $E_{\mathcal{V}}$ in comoving coordinates $\left(t,x\right)$. Lightly shaded vertical strips represent the comoving future of various initial $H$-regions from $E_{\mathcal{V}}$; dark shades represent reheated domains; the boundary of the dark-shaded domains is the reheating surface. Vertical dashed lines are the eternally inflating comoving worldlines that never cross the reheating surface. The 3-volumes of the reheating surfaces in the comoving future of the pictured $H$-regions are large but finite because these $H$-regions are located near eternal worldlines but do not contain any such worldlines.