On hilltop and brane inflation after Planck

Hilltop inflation models are often described by potentials V = V₀ (1−$\varphi$ⁿ/mⁿ+... ). 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 = V₀ (1−$\varphi$ⁿ/mⁿ), for all n, converge to r = 4/N $\lesssim$ 0.07, as in chaotic inflation with V ∼ $\varphi$, 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 = V₀ (1−$\varphi$ⁿ/mⁿ) at $\varphi$ > 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 = V₀(1−mⁿ/$\varphi$ⁿ), but consistent generalizations of D-brane inflation models may successfully complement α-attractors in describing most of the area in the (n_s, r) space favored by Planck 2018.