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

Eemeli Tomberg, Konstantinos Dimopoulos

TL;DR

This paper investigates eternal inflation in single-field inflection-point models designed to produce primordial black holes. By recasting the perturbation statistics in terms of a Fokker–Planck equation with absorbing boundaries, it derives a sufficient condition for eternal inflation: $\lambda_1 \le 3$, where $\lambda_1$ is the leading FP eigenvalue. The authors obtain analytic expressions for $\lambda_1$ in simplified sub-potentials: a constant-drift (linear diffusion) and a linear-drift (quadratic) toy model, including slow-roll and constant-roll generalizations. Applying these results to three concrete PBH potentials, they show eternal inflation is generically unavoidable near the inflection-point feature, with baby universes forming inside black-hole horizons and dominating the reheated volume. The work highlights potential tensions with large-scale structure and CMB observations under volume-weighted measures, and provides a practical method to test non-eternal inflection-point PBH models, emphasizing the fractal nature of eternally inflating PBH spacetimes.

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 $λ_1 \leq 3$, where $λ_1$ is the index of the `exponential tail,' the lowest eigenvalue of the Fokker--Planck equation over a bounded region. We write $λ_1$ in terms of the model parameters for linear and quadratic regions. Wide quadratic regions inflate eternally if the second slow-roll parameter $η_V \geq -6$. 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.

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

TL;DR

This paper investigates eternal inflation in single-field inflection-point models designed to produce primordial black holes. By recasting the perturbation statistics in terms of a Fokker–Planck equation with absorbing boundaries, it derives a sufficient condition for eternal inflation: , where is the leading FP eigenvalue. The authors obtain analytic expressions for in simplified sub-potentials: a constant-drift (linear diffusion) and a linear-drift (quadratic) toy model, including slow-roll and constant-roll generalizations. Applying these results to three concrete PBH potentials, they show eternal inflation is generically unavoidable near the inflection-point feature, with baby universes forming inside black-hole horizons and dominating the reheated volume. The work highlights potential tensions with large-scale structure and CMB observations under volume-weighted measures, and provides a practical method to test non-eternal inflection-point PBH models, emphasizing the fractal nature of eternally inflating PBH spacetimes.

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 , where is the index of the `exponential tail,' the lowest eigenvalue of the Fokker--Planck equation over a bounded region. We write in terms of the model parameters for linear and quadratic regions. Wide quadratic regions inflate eternally if the second slow-roll parameter . 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.

Paper Structure

This paper contains 35 sections, 101 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fokker--Planck equation and eternal inflation
  3. Eternal inflation
  4. Late-time behaviour
  5. Studying a subsection of the potential
  6. Analytical solutions for sub-potentials
  7. Constant drift
  8. Slow roll.
  9. Linear drift
  10. Negative $\boldsymbol\beta$.
  11. Positive $\boldsymbol\beta$.
  12. Example distribution.
  13. Eternal inflation.
  14. Small $\boldsymbol\beta$: slow roll.
  15. Large $\boldsymbol\beta$: constant roll.
  16. ...and 20 more sections

Figures (5)

  • Figure 1: The function $g$ defined in \ref{['eq:g_defined']}, with the asymptotic limits \ref{['eq:g_asymptotics']}.
  • Figure 2: Parameter region for eternal inflation, $\lambda_1 \leq 3$, in the case of linear drift.
  • Figure 3: The potentials \ref{['eq:model_I_potential']}, \ref{['eq:model_II_potential']}, and \ref{['eq:model_III_potential']}. The right-hand panels are zoomed into the feature and also depict quadratic approximations around the maximum and minimum (dashed lines).
  • Figure 4: A sketch of the eternal inflation fractal. The grey regions (E) are inflating, the blue regions (U1, U2, U3) have reheated. The circles inside the blue regions are black holes; eternal inflation takes place inside some of them. The arrows indicate the 'zoom-in direction' inside the fractal. See main text for further explanation.
  • Figure 5: Asymptotic behaviour of the $g$ function and its apporximations.