Formation of topological defects in phase transitions

We argue, and find through numerical work, that the results of nondynamical Monte Carlo computer simulations cannot be applied to describe the formation of topological defects when the correlation length at the Ginzburg temperature is significantly smaller than the horizon size, the case which was originally considered. To test the current hypothesis that ‘‘infinite’’ strings at formation are essentially described by Brownian walks of size the correlation length at the Ginzburg temperature, we examine equilibrated fields at the Ginzburg temperature. We find that no ‘‘infinite’’ structure exists in equilibrium for reasonable definitions of the Ginzburg temperature, and that horizons must be included in a proper treatment. A phase transition, from small-scale to large-scale string or domain-wall structure, is found to occur very close to the Ginzburg temperature, in agreement with recent work. We also investigate the formation process of domain walls and global strings through the breaking of initially ordered states. To mimic conditions in the early Universe, cooling times are chosen so that horizons exist in our sample volume when topological structure formation occurs. The classical fields are evolved in real time by the numerical solution of Langevin equations of motion on a three-dimensional spatial lattice. Our results indicate that it is possible for most of the string energy to be in small loops, rather than in long strings, at formation.