Reconstructing the inflaton potential from the spectral index

Abstract

Recent cosmological observations are in good agreement with the scalar spectral index ns with ns 1 2=N, where N is the number of e-foldings. Quadratic chaotic model, Starobinsky model and Higgs ination or -attractors connecting them are typical examples predicting such a relation. We consider the problem in the opposite: given ns as a function of N, what is the inaton potential V ( ). We nd that for ns 1 = 2=N, V ( ) is either tanh 2 (= 2) (T-model) or 2 (chaotic ination) to the leading order in the slow-roll approximation. is the ratio of 1=V at N!1 to the slope of 1=V at a nite N and is related to in the -attractors by 2 = 2=3 . The tensor-to-scalar ratio r is r = 8=N( 2 N + 1). The implications for the reheating temperature are also discussed. We also derive formulas for ns 1 = p=N. We nd that if the potential is