The bound r ≤ 16ǫ on the primordial tensor perturbation

I recall the well-known sufficient conditions for the bound r < 16ǫ on the spectrum of the primordial tensor perturbation. Two recent papers claim a violation of this bound, without stating explicitely any violation of the sufficient conditions. The following conditions are sufficient for the bound r < 16ǫ on the primordial tensor perturbation. • Inflation is of the slow-roll variety. • The curvature perturbation is generated from the vacuum fluctuation of one or more light scalar fields. • Einstein gravity holds during and after inflation. Light fields are here defined as those whose vacuum fluctuations generate classical perturbations a few Hubble times after horizon exit, which are uncorrelated and have a spectrum well-approximated by (H/2π). According to the second condition, these perturbations provide the initial condition for the evolution of the curvature perturbation. Taking into account also the first condition, the spectrum of the curvature perturbation is of the form