# Lyth Bound, eternal inflation and future cosmological missions

## TL;DR

This paper generalizes the Lyth bound to relate inflaton field variation with observable parameters, analyzing implications for eternal inflation and future CMB experiments in a model-independent way.

## Contribution

It introduces a new, generalized expression for inflaton field variation during horizon crossing, extending the classical Lyth bound in a model-independent framework.

## Key findings

- Potential detection of tensor-to-scalar ratio $r$ constrains inflaton field variation.
- Upcoming experiments could improve estimates of $n_s$ and $eta_s$, impacting inflation models.
- The analysis links field variation to eternal inflation scenarios.

## Abstract

In this paper we provide a new expression for the variation of the inflaton field $\Delta\phi$ during the horizon crossing epoch in the context of single field slow roll inflationary models. Such an expression represents a generalization of the well-know Lyth bound. We also explore the consequences of a detection of permille order of the tensor-to-scalar ratio amplitude, $r$, as well as an improvement on the estimation of the scalar spectral index, $n_s$ and its running $\alpha_s$, by the upcoming CMB polarization experiments that will provide plausible constraints on the quantity $\Delta\phi$ during the horizon exit moment. In addition we discuss the relation between the local variation of the field and the possibilities of an eternal inflation. The results of the analysis are completely model independent.