Formation of multiple winding topological defects in the early universe

The formation probability of multiple winding topological defects is calculated by phase and flux distribution analysis based on the Kibble mechanism. The core size of defects is taken into account so that when it is much larger than the correlation length of the Higgs field, high winding configuration can be realized. For example, if the self coupling constant of the Higgs field is smaller than $4\times10^{-4}$, the topological inflation may occur at the grand unification energy scale by the multiple winding string produced during the GUT phase transition.