Initial condition of a gravitating thick loop cosmic string and linear perturbations

The initial data of the gravitational field produced by a thick loop string are considered. We show that a thick loop is not a geodesic on the initial hypersurface, while a loop conical singularity is. This suggests that there is a 'critical thickness' of a string, at which the linear perturbation theory with a flat space background fails to describe the gravity of a loop cosmic string. Using the above initial data, we also show that the linear perturbation around the flat space is plausible if the string thickness is larger than similar to5 x 10(-3)a, where a is the curvature radius of the loop.