Reconstructing the inflaton potential for an almost flat COBE spectrum

TL;DR

This paper introduces a new method using the Hubble parameter formalism to reconstruct inflationary potentials based on observational data, applicable to various orders of the slow-roll approximation, and explores the relationship between spectral index and wavelength.

Contribution

Develops a novel Hubble parameter-based reconstruction method for inflationary potentials, accommodating higher-order slow-roll approximations and observational constraints.

Findings

Reconstructed inflationary potentials compatible with COBE data.

Identified constraints between spectral index and wavelength.

Showed no unique scalar spectral index value fits all potentials.

Abstract

Using the Hubble parameter as new `inverse time' coordinate ($H$-formalism), a new method of reconstructing the inflaton potential is developed also using older results which, in principle, is applicable to any order of the slow-roll approximation. In first and second order, we need three observational data as inputs: the scalar spectral index $n_s$ and the amplitudes of the scalar and the tensor spectrum. We find constraints between the values of $n_s$ and the corresponding values for the wavelength $\lambda $. By imposing a dependence $\lambda (n_s)$, we were able to reconstruct and visualize inflationary potentials which are compatible with recent COBE and other astrophysical observations. >From the reconstructed potentials, it becomes clear that one cannot find only one special value of the scalar spectral index $n_s$.