Reconstructing homospectral inflationary potentials

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.

Automated summary

Purely geometrical arguments demonstrate the existence of homospectral inflationary cosmologies, which are different expansion histories resulting in the same spectrum of comoving curvature perturbations. A general algorithm is developed to reconstruct the potential of minimally coupled single scalar fields from arbitrary expansion histories. The infinite class of homospectral potentials depends on two free parameters, the initial energy scale and the initial value of the field. However, it is generally impossible to reconstruct a unique potential from the curvature spectrum unless the initial energy scale and field value are fixed.