The evolution of networks of domain walls and cosmic strings

When a complex scalar field has a potential with two or more degenerate minima, then a network of domain walls joined together by cosmic strings can form and evolve as the universe expands. We numerically compute the evolution of such wall-string networks, for potentials with 1-6 minima. We give results for two-dimensional simulations on a 1024 x 1024 grid, and for fully three-dimensional simulations on a 100 x 100 x 100 grid. We find that the rms velocity of the walls is about 0.4c, independent of the number of minima in the potential. The number of walls within the horizon is constant, to within a logarithmic factor. The small number of walls within the horizon implies that a wall network produced in a late (postrecombination) phase transition cannot generate large-scale structure without violating anisotropy constraints on the cosmic microwave background. Particle physics models with multiple discrete minima, such as the DFSZ axion model, are cosmologically disastrous, since they generate wall networks which would have dominated the early universe at high redshift. Models with only one discrete minimum produce walls terminating on strings; these walls quickly disappear and are cosmologically innocuous.