The Origin and Bifurcation of the Space-Time Defects in the Early Universe

In the early universe, a new topological invariant is interpreted as the space-time dislocation flux and is quantized in the topological level. By extending to a topological current of dislocations, the dynamic form of the defects is obtained under the condition that the Jacobian determinant D(φ/u) ≠ 0. When D(φ/u) = 0, it is shown that there exists the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the origin and bifurcation of the space-time dislocations are detailed in the neighborhoods of the limit points and bifurcation points of φ-mapping, respectively. It is pointed out that, since the dislocation current is identically conserved, the total topological quantum numbers of the branched dislocation fluxes will remain constant during their origin and bifurcation processes, which are important in the early universe because of spontaneous symmetry breaking.