Inverse-Scattering Reconstruction of Inflation from Scalar and Tensor Primordial Spectra

Abstract

We develop an inverse-scattering framework to reconstruct the effective inflationary potentials governing scalar and tensor perturbations. By recasting the Mukhanov--Sasaki equation as a Schrödinger-like problem on the half-line, we identify the Bunch--Davies initial condition with the asymptotic Jost solution and show that the freeze-out amplitude of the growing mode is encoded in the corresponding Jost function. This allows the scalar and tensor primordial power spectra to be written in terms of $F^{(s)}_{ν_s-\frac12}(k)$ and $F^{(t)}_{ν_t-\frac12}(k)$, respectively, and leads to an inverse-scattering expression for the tensor-to-scalar ratio as a ratio of Jost amplitudes. We then test the reconstruction in the large-$k$ regime using the Born approximation, where the Marchenko equation becomes linear. As benchmarks, we consider a smooth quadratic potential and a step potential that transiently violates slow roll and generates localized features in the primordial spectra. The reconstructed effective potentials reproduce the dominant behavior of $z^{\prime\prime}/z$ and $a^{\prime\prime}/a$ for smooth slow-roll evolution, while localized discrepancies arise in the scalar sector when sharp features induce stronger scattering. Our results show that inverse scattering provides a physically transparent method for connecting features in the primordial spectra to the underlying inflationary dynamics, and that the Jost function acts as a sensitive diagnostic of departures from canonical slow-roll evolution.