QUADRATIC ACTION OF THE HAWKING-TUROK INSTANTON - ART. NO. 063505

The positive definiteness of the quadratic part of the action of the H awking-Turok instanton is investigated. The Euclidean quadratic action for scalar perturbations is expressed in terms of a single gauge inva riant quantity q. The mode functions satisfy a Schrodinger-type equati on with a potential U. It is shown that the potential U tends to a pos itive constant at the regular end of the instanton. The detailed shape of U depends on the initial data of the instanton, on parameters of t he background scaler field potential V, and on a positive integer, p, labeling different spherical harmonics. For certain well behaved scala r field potentials it is proven analytically that for p>1 quadratic ac tion is non-negative. For the lowest p=1 (homogeneous) harmonic, the n umerical solution of the Schrodinger equation for different scalar fie ld potentials V and different initial data shows that in some cases th e potential U is negative in the intermediate region. We investigate t he monotonically growing, potentials and a potential with a false vacu um. For the monotonic potentials no negative modes are found about the Hawking-Turok instanton. For a potential with a false vacuum the HT i nstanton is shown to have a negative mode for certain initial data. [S 0556-2821(98)05518-0].