On hilltop and brane inflation after Planck

Understand

Hilltop inflation models are often described by potentials $V = V_{0}(1-{\phi^{n}\over m^{n}}+...)$.

• The omitted terms indicated by ellipsis do not affect inflation for $m \lesssim 1$, but the most popular models with $n =2$ and $4$ for $m \lesssim 1$ are ruled out observationally.
• Meanwhile in the large $m$ limit the results of the calculations of the tensor to scalar ratio $r$ in the models with $V = V_{0}(1-{\phi^{n}\over m^{n}})$, for all $n$, converge to $r= 4/N \lesssim 0.07$, as in chaotic inflation with $V \sim \phi$, suggesting a reasonably good fit to the Planck data.
• We show, however, that this is an artifact related to the inconsistency of the model $V = V_{0}(1-{\phi^{n}\over m^{n}})$ at $\phi > m$.