Lyth Bound, eternal inflation and future cosmological missions

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.