Reconstructing the Inflaton Potential---Perturbative Reconstruction to Second Order

One method to reconstruct the scalar field potential of inflation is a perturbative approach, where the values of the potential and its derivatives are calculated as an expansion in departures from the slow-roll approximation. They can then be expressed in terms of observable quantities, such as the square of the ratio of the gravitational wave amplitude to the density perturbation amplitude, the deviation of the spectral index from the Harrison--Zel'dovich value, etc. Here, we calculate complete expressions for the second-order contributions to the coefficients of the expansion by including for the first time corrections to the standard expressions for the perturbation spectra. As well as offering an improved result, these corrections indicate the expected accuracy of the reconstruction. Typically the corrections are only a few percent.