Dynamical evolution of domain walls in an expanding universe

Whenever the potential of a scalar field has two or more separated, degenerate minima, then domain walls form as the universe cools. We calculate the evolution of the resulting network of domain walls, for the case of two potential minima, by direct numerical integration of the partial differential equations of the scalar field, in two and three dimensions, thus including wall annihilation, crossing, and reconnection (intercommutation) effects. Three dimensional simulations were run on a 200 x 200 x 200 mesh, while two-dimensional simulations (three-dimensional equations but slab symmetry in one coordinate) were run on a mesh of 1024 x 1024. We find that the nature of the evolution is largely independent of the rate at which the universe expands, if time is measured in conformal units so that the wave operator in comoving space does not depend on the expansion. This means that our results should apply to walls both inside and outside the particle horizon. We find that, unlike cosmic strings, wall evolution does not leave behind many fragments or other topological detritus. Wall annihilation and reconnection occurs almost as fast as causality allows, so that the horizon volume is "swept clean" and contains, at any time, only about one, fairly smooth, wall. Quantitative statistics are given. The rms velocity of walls is ~ 0. 4c. Wall "bubbles" or "bags" are rare and collapse almost immediately, radiating away their energy as field excitations (schizons). Up to a logarithmic correction (discussed in some detail and related to the critical behavior of self-avoiding random walks in two-dimensions) the total (comoving) area of wall per (comoving) volume decreases as the first power of (conformal) time. In a matter dominated universe, this implies ρwall is proportional to alpha^^-3/2^^. The relative slowness of this decrease, along with the fact that the wall is smooth on the horizon scale, makes it impossible for walls (if they move freely and if they survive to the present) both to generate large-scale structure and to be consistent with quadrupole microwave background anisotropy limits.