Reconstructing the Inflaton Potential --- an Overview

TL;DR

This overview analyzes how inflationary physics links the scalar and tensor perturbation spectra to the inflaton potential and how one can reconstruct V(φ) from observations using a perturbative, slow-roll framework. It derives lowest-order and next-order expressions for the spectra, establishes a hierarchy of consistency relations, and presents a practical perturbative reconstruction approach that relates observables (A_S, A_T, n, dn/dlnk) to the potential and its derivatives. The work highlights the central role of tensor measurements to break degeneracies, demonstrates reconstruction on simulated data, and discusses additional constraints from the end of inflation, gravitational-wave detection prospects, primordial black holes, and spectral distortions. The findings indicate that, with sufficiently accurate data, inflationary potentials can be constrained at the level of the first few derivatives around the field value corresponding to observable scales, offering a path to probe physics at energies far beyond terrestrial experiments.

Abstract

We review the relation between the inflationary potential and the spectra of density (scalar) perturbations and gravitational waves (tensor perturbations) produced, with particular emphasis on the possibility of reconstructing the inflaton potential from observations. The spectra provide a potentially powerful test of the inflationary hypothesis; they are not independent but instead are linked by consistency relations reflecting their origin from a single inflationary potential. To lowest-order in a perturbation expansion there is a single, now familiar, relation between the tensor spectral index and the relative amplitude of the spectra. We demonstrate that there is an infinite hierarchy of such consistency equations, though observational difficulties suggest only the first is ever likely to be useful. We also note that since observations are expected to yield much better information on the scalars than on the tensors, it is likely to be the next-order version of this consistency equation which will be appropriate, not the lowest-order one. If inflation passes the consistency test, one can then confidently use the remaining observational information to constrain the inflationary potential, and we survey the general perturbative scheme for carrying out this procedure. Explicit expressions valid to next-lowest order in the expansion are presented. We then briefly assess the prospects for future observations reaching the quality required, and consider a simulated data set that is motivated by this outlook.