Lyth bound, eternal inflation, and future cosmological missions

In this paper we provide a new expression for the variation of the inflaton field Δϕ during the horizon crossing epoch in the context of single field slow roll inflationary models. Such an expression represents a generalization of the well-known 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, ns and its running αs, by the upcoming cosmic microwave background (CMB) polarization experiments that will provide plausible constraints on the quantity Δϕ 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.