Eternal inflation near inflection points: a challenge to primordial black hole models

TL;DR

This paper investigates the conditions under which eternal inflation occurs near inflection points in inflationary models for primordial black holes, showing it is a common and potentially problematic feature that challenges their observational viability.

Contribution

It derives a criterion for eternal inflation in inflection point models and demonstrates its widespread applicability, questioning the viability of such models for explaining primordial black holes.

Findings

Eternal inflation occurs if the eigenvalue λ₁ ≤ 3.

Quadratic regions with η_V ≥ -6 tend to inflate eternally.

Eternal inflation leads to inhomogeneous baby universes dominating the multiverse.

Abstract

Inflation with an inflection point potential is a popular model for producing primordial black holes. The potential near the inflection point is approximately flat, with a local maximum next to a local minimum, prone to eternal inflation. We show that a sufficient condition for eternal inflation is $\lambda_1 \leq 3$, where $\lambda_1$ is the index of the `exponential tail,' the lowest eigenvalue of the Fokker--Planck equation over a bounded region. We write $\lambda_1$ in terms of the model parameters for linear and quadratic regions. Wide quadratic regions inflate eternally if the second slow-roll parameter $\eta_V \geq -6$. We test example models from the literature and show this condition is satisfied; we argue eternal inflation is difficult to avoid in inflection point PBH models. Eternally inflating regions correspond to type II perturbations and form baby universes, hidden behind black hole horizons. These baby universes are inhomogeneous on large scales and dominate the multiverse's total volume. We argue that, if volume weighting is used, eternal inflation makes inflection point primordial black hole models incompatible with large-scale structure observations.