Inflationary spectral indices and potential reconstruction

The Inflationary power spectrum plays an important role in modern cosmology. In this thesis, we studied both the experimental and theoretical aspects of the inflationary spectral index. By exploring the recent WMAP data, we found that the evidence for the running of the spectral index mainly comes from multipoles near l = 40. This fact allows a partial running spectrum to give as good a fit as the WMAP running spectrum. We gave some simple formulae for the inflationary spectral indices based on the Hamilton-Jacobi formulation of inflation. These simple formulae agree with the exact solutions in some special cases. The Hamilton-Jacobi formulation of inflation was also applied to reconstruct inflaton potentials from a given power spectrum. A simple and accurate reconstruction formulation was presented. All analytic potentials giving a constant spectral index are derived, which show that a nearly scale-invariant spectrum can give rise to slow-roll inflation during 60 e-foldings within sub-Planckian inflaton field values and a potential energy V1/4 ∼ 1015GeV. Potentials for large running of the spectral index and large tensor-to-scalar ratio were also constructed, which need super-Planckian field values and require that the slow-roll approximation breaks down before reaching 60 e-foldings. We have shown that for the cosmologically interesting scales, a renormalizable potential fits the reconstructed potential for a large running spectrum very well. Our reconstruction formulation also produces a self-consistent tensor spectrum once a scalar spectrum and the tensor-to-scalar ratio are given. Higher order corrections to the slow-roll approximation are also considered. We showed that they can be incorporated straightforwardly into our formulae for spectral indices and the reconstruction formalism.