order to allow enough e-foldings of expansion and to obtain a nearly scale invariant spectrum of perturbations. This motivated the so-called *natural inflation* model [@Freese:1990rb; @Adams:1992bn], where the inflaton enjoys a (broken) axionic shift symmetry, $\varphi\rightarrow\varphi+const.$, that protects its potential from large quantum corrections. The simplest implementation of this idea, namely a single field $\varphi$ evolving in a potential $V \propto \cos \frac{\varphi}{f}$, agrees with observations only if the axion decay constant $f$ is of the order of or greater than the Planck mass [@Savage:2006tr]. It has been debated in the literature whether a trans-Planckian breaking may be compatible with gravity, in the case in which the symmetry is global [@f-Mp-global]. The shift symmetry may emerge from a gauge symmetry, as typically in string theory. However, also in this case a trans-Planckian $f$ is regarded as problematic, since all known controlled string theory constructions are characterized by $f<M_{p}$ [@Banks:2003sx; @Svrcek:2006yi].

A scale $f < M_p$ can be compatible with inflation through a number of mechanisms, several of which also lead to an interesting phenomenology (see [@review5] for a recent review). One can, for instance, consider more than one axion [@sol1; @sol2], require nontrivial compactifications in string theory [@sol3], couple the 