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. The quadratic chaotic model, Starobinsky model, and Higgs inflation or α-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 inflaton potential V(φ). We find that for ns - 1 = -2/N, V(φ) is either tanh2(γ φ/2) ("T-model") or φ2 (chaotic inflation) to the leading order in the slow-roll approximation. γ is the ratio of 1/V at N → ∞ to the slope of 1/V at a finite N and is related to "α" in the α-attractors by γ2 = 2/3α. The tensor-to-scalar ratio r is r = 8/N(γ2 N + 1). The implications for the reheating temperature are also discussed.We also derive formulas for ns - 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 ns(N).