Reconstructing homospectral inflationary potentials

TL;DR

This paper develops an algorithm to reconstruct scalar field potentials from expansion histories and demonstrates that multiple potentials can produce identical curvature spectra, highlighting challenges in uniquely determining inflationary models.

Contribution

It introduces a general method for potential reconstruction from expansion histories and explores the non-uniqueness of potentials in homospectral inflationary cosmologies.

Findings

Homospectral expansion histories can correspond to different potentials.

The class of homospectral potentials depends on two free parameters.

Observations of primordial gravitational waves could help fix the initial conditions.

Abstract

Purely geometrical arguments show that there exist classes of homospectral inflationary cosmologies, i.e. different expansion histories producing the same spectrum of comoving curvature perturbations. We develop a general algorithm to reconstruct the potential of minimally-coupled single scalar fields from an arbitrary expansion history. We apply it to homospectral expansion histories to obtain the corresponding potentials, providing numerical and analytical examples. The infinite class of homospectral potentials depends on two free parameters, the initial energy scale and the initial value of the field, showing that in general it is impossible to reconstruct a unique potential from the curvature spectrum unless the initial energy scale and the field value are fixed, for instance through observation of primordial gravitational waves.