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 comoving curvature perturbations. We develop a general algorithm to reconstruct potential minimally coupled single scalar fields from an arbitrary history. apply it obtain corresponding potentials, providing numerical and analytical examples. The infinite class potentials depends on two free parameters, initial energy scale value field, showing that, in general, is impossible unique unless field are fixed, for instance, through observation primordial gravitational waves.