Running of Running of the Spectral Index and WMAP Three-year data

Abstract

Three-year data of WMAP implies not only a negative running of the spectral index with large absolute value, but also a large positive running of running of the spectral index with order of the magnitude $10^{-2}$. We calculate the running of running in usual inflation model and noncommutative inflation model. A large tensor-scalar ratio $r\geq 1.23$ is needed in order to fit the WMAP data in the noncommutative inflation model, which roughly saturates the observational upper bound on it.