Quadratic action of the Hawking-Turok instanton

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 Schrödinger 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 values 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 Schrödinger equation for different scalar field potentials $V$ and different initial values show that in some cases the potential $U$ is negative in the intermediate region. We investigated the monotonous 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 Coleman-De Luccia bounce is shown to have a negative mode and the HT instanton for certain initial values is shown to have a negative mode as well