# Eternal Inflation, Entropy Bounds and the Swampland

## TL;DR

This paper examines the compatibility of stochastic small field inflation models with string theory constraints, entropy bounds, and observational requirements, revealing fundamental conflicts and challenges for such models.

## Contribution

It demonstrates that stochastic inflation models either violate entropy bounds or swampland criteria, and face issues with achieving a graceful exit consistent with observations.

## Key findings

- Stochastic inflation models violate entropy bounds or swampland criteria.
- Models face a graceful exit problem with incompatible fluctuation spectra.
- Exiting patches of inflation cannot simultaneously explain CMB isotropy and spectral tilt.

## Abstract

It has been suggested that low energy effective field theories should satisfy given conditions in order to be successfully embedded into string theory. In the case of a single canonically normalized scalar field this translates into conditions on its potential and the derivatives thereof. In this Letter we revisit stochastic models of small field inflation and study the compatibility of the swampland constraints with entropy considerations. We show that stochastic inflation either violates entropy bounds or the swampland criterium on the slope of the scalar field potential. Furthermore, we illustrate that such models are faced with a graceful exit problem: any patch of space which exits the region of eternal inflation is either not large enough to explain the isotropy of the cosmic microwave background, or has a spectrum of fluctuations with an unacceptably large red tilt.