Lyth Bound, eternal inflation and future cosmological missions
  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.
