The Phase Transition to Eternal Inflation

TL;DR

This work defines slow-roll eternal inflation via a sharp transition in the reheating-volume statistics, identifying the key control parameter Ω ≡ (2π^2/3) dotφ^2/(H^4) and establishing the transition at Ω=1 (dotφ^2/H^4 = 3/(2π^2)). By modeling the inflaton as a Gaussian, free field in de Sitter space and treating its long-wavelength fluctuations as a classical diffusion with a moving absorbing barrier, the authors compute the average and higher moments of the reheating-volume distribution, showing the mean diverges at Ω>1 while the variance diverges earlier at Ω=9/8, and that a finite probability of infinite reheating volume emerges at Ω=1. To robustly confirm the transition, they introduce a discrete bacteria-branching model that reproduces the continuum results and demonstrates that the onset of eternal inflation corresponds to a nonzero probability of infinite reheating volume, consistent with a phase-transition-like behavior. The analysis is carried at leading order in slow-roll and $H^2/M_{Pl}^2$, and the authors discuss generalizations to more realistic models, smoothing schemes, and potential limitations in non-minimal inflation where EFT may break down in the eternal regime.

Abstract

For slow-roll inflation we study the phase transition to the eternal regime. Starting from a finite inflationary volume, we consider the volume of the universe at reheating as order parameter. We show that there exists a critical value for the classical inflaton speed, \dotφ^2/H^4 = 3/(2 π^2), where the probability distribution for the reheating volume undergoes a sharp transition. In particular, for sub-critical inflaton speeds all distribution moments become infinite. We show that at the same transition point the system develops a non-vanishing probability of having a strictly infinite reheating volume, while retaining a finite probability for finite values. Our analysis represents the exact quantum treatment of the system at lowest order in the slow-roll parameters and H^2/M_Pl^2.

Figures (8)

• Figure 1: Left: For standard, non-eternal inflation the inflationary phase (shaded region) is nearly de Sitter, with slow-roll corrections and small inhomogenities; the reheating surface (red curve) is slightly bent; the post inflationary phase is nearly FRW, with small perturbations inherited from the reheating surface. Right: In the eternal inflation regime, the inflationary phase is very well approximated by de Sitter geometry, with corrections of order $H^2/M_{\rm Pl}^2$, but the reheating surface has wild fluctuations; as a consequence the post-inflationary phase is highly inhomogeneous, and intractable.
• Figure 2: At any given comoving point, the local inflaton value undergoes a random walk. Left: for $\phi$ the reheating barrier is held fixed at $\phi = \phi_r$, but there is classical drift towards it. Right: for the fluctuation $\psi$ there is no net drift, but the barrier is approaching at a speed $\dot \phi$.
• Figure 3: Our two-step approximation. The local inflaton fluctuations in two nearby comoving points are assumed to be totally correlated until the physical distance between the two points exits the UV cutoff ($t<t_*$), and totally uncorrelated afterwards ($t>t_*$).
• Figure 4: The branching process.
• Figure 5: The gaussian model described in the text.