On hilltop and brane inflation after Planck

Abstract: 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 $\phi > m$. Consistent generalizations of this model in the large $m$ limit typically lead to a much greater value $r= 8/N$, which negatively affects the observational status of hilltop inflation. Similar results are valid for D-brane inflation with $V = V_{0}(1-{m^{n}\over \phi^{n}})$, but consistent generalizations of D-brane inflation models may successfully complement $\alpha$-attractors in describing most of the area in the ($n_{s}$, $r$) space favored by Planck 2018.