Dynamical instability criterion for circular (vorton) string loops

Abstract

Dynamic perturbation equations are derived for a generic stationary state of an elastic string model -- of the kind appropriate for representing a superconducting cosmic string -- in a flat background. In the case of a circular equilibrium (i.e. vorton) state of a closed string loop it is shown that the fundamental axisymmetric ($n=0$) and lowest order ($n=1$) nonaxisymmetric perturbation modes can never be unstable. However, stability for modes of higher order ($n\geq 2$) is found to be non-trivially dependent on the values of the characteristic propagation velocity, $c$ say, of longitudinal perturbations and of the corresponding extrinsic perturbation velocity, $v$ say. For each mode number the criterion for instability is the existence of nonreal roots for a certain cubic eigenvalue equation for the corresponding mode frequency. A very simple sufficient but not necessary condition for reality of the roots and therefore absence of instability is that the characteristic velocity ratio, $c/v$ be greater than or equal to unity. Closer examination of the low velocity (experimentally accessible) nonrelativistic regime shows that in that limit the criterion for instability is just that the dimensionless characteristic ratio $c/v$ be less than a critical value $χ_c$ whose numerical value is approximately $1\over 2$. In the relativistic regime that is relevant to superconducting cosmic strings the situation is rather delicate, calling for more detailed investigation that is postponed for future work.