Reconstructing the inflaton potential from the spectral index

Recent cosmological observations are in good agreement with the scalar spectral index n(s) with n(s) - 1 similar to -2/N, where N is the number of e-foldings. The quadratic chaotic model, Starobinsky model, and Higgs inflation or alpha-attractors connecting them are typical examples predicting such a relation. We consider the problem in the opposite: given n(s) as a function of N, what is the inflaton potential V(phi). We find that for n(s) -1 = -2/N, V(phi) is either tanh(2) (gamma phi/2) ("T-model") or phi(2) (chaotic inflation) to the leading order in the slow-roll approximation. gamma is the ratio of 1/V at N -> infinity to the slope of 1/V at a finite N and is related to "alpha" in the alpha-attractors by gamma(2) = 2/3 alpha. The tensor-to-scalar ratio r is r = 8/N(gamma N-2 + 1). The implications for the reheating temperature are also discussed. We also derive formulas for n(s) - 1 = -p/N. We find that if the potential is bounded from above, only p > 1 is allowed. Although r depends on a parameter, the running of the spectral index is independent of it, which can be used as a consistency check of the assumed relation of n(s)(N).