Tachyonic Instability and Darboux Transformation

TL;DR

This paper explores how Darboux transformation can generate scalar field potentials with flat fluctuation spectra and blue metric perturbations, and addresses the inverse problem of reconstructing potentials from known one-loop functions.

Contribution

It introduces a method to construct potentials with specific spectral properties and solves the inverse problem of potential reconstruction from one-loop data.

Findings

Generated potentials with flat scalar fluctuation spectra

Produced metrics with blue spectrum of perturbations

Provided a framework for potential reconstruction from one-loop functions

Abstract

Using Darboux transformation one can construct infinite family of potentials which lead to the flat spectrum of scalar field fluctuations with arbitrary multiple precision, and, at the same time, with "essentially blue" spectrum of perturbations of metric. Besides, we describe reconstruction problem: find classical potential V(phi) starting from the known "one-loop potential" u(t) = d^2V(phi(t))/d phi(t)^2.