Energy Production in the Formation of a Finite Thickness Cosmic String

Abstract

The classical electromagnetic modes outside a long, straight, superconducting cosmic string are calculated, assuming the string to be surrounded by a superconducting cylindric surface of radius R. Thereafter, by use of a Bogoliubov-type argument, the electromagnetic energy W produced per unit length in the lowest two modes is calculated when the string is formed "suddenly". The essential new element in the present analysis as compared with prior work of Parker [Phys. Rev. Lett. {\bf 59}, 1369 (1987)] and Brevik and Toverud [Phys. Rev. D {\bf 51}, 691 (1995)], is that the radius {\it a} of the string is assumed finite, thus necessitating Neumann functions to be included in the fundamental modes. We find that the theory is changed significantly: W is now strongly concentrated in the lowest mode $(m,s)=(0,1)$, whereas the proportionality $W \propto (G\mu /t)^2$ that is characteristic for zero-width strings is found in the next mode (1,1). Here G is the gravitational constant, $\mu$ the string mass per unit length, and t the GUT time.