On the quadratic action of the Hawking-Turok instanton

TL;DR

This paper analyzes the positivity of the quadratic action in the Hawking-Turok instanton, showing conditions under which negative modes appear or are absent, depending on the scalar potential and initial data.

Contribution

It provides an analytical proof of non-negativity of the quadratic action for certain potentials and numerically investigates the presence of negative modes for various scalar field configurations.

Findings

Quadratic action is non-negative for p>1 in well-behaved potentials.

Negative modes can occur for p=1 with specific initial data and potentials.

Monotonous potentials do not exhibit negative modes, while false vacuum potentials can have negative modes.

Abstract

Positive definiteness of the quadratic part of the action of the Hawking-Turok instanton is investigated. The Euclidean quadratic action for scalar perturbations is expressed in terms of a single gauge invariant quantity $q$. The mode functions satisfy a Schr\"odinger type equation with a potential $U$. It is shown that the potential $U$ tends to a positive constant at the regular end of the instanton. The detailed shape of $U$ depends on the initial data of the instanton, on parameters of the background scalar field potential $V$ and on a positive integer, $p$, labeling different spherical harmonics. For certain well behaved scalar field potentials it is proven analytically that for $p>1$ quadratic action is non-negative. For the lowest $p=1$ (homogeneous) harmonic numerical solution of the Schr\"odinger equation for different scalar field potentials $V$ and different initial data show that in some cases the potential $U$ is negative in the intermediate region. We investigated the monotonously growing potentials and a potential with a false vacuum. For the monotonous potentials no negative modes are found about the Hawking-Turok instanton. For a potential with the false vacuum the HT instanton is shown to have a negative mode for certain initial data.