Level set method for the evolution of defect and brane networks

A theory for studying the dynamic scaling properties of branes and relativistic topological defect networks is presented. The theory, based on a relativistic version of the level set method, well known in other contexts, possesses self-similar “scaling” solutions, for which one can calculate many quantities of interest. Here, the length and area densities of cosmic strings and domain walls are calculated in Minkowski space, and radiation, matter, and curvature-dominated Friedmann-Robertson-Walker cosmologies with two and three space dimensions. The scaling exponents agree with the naive ones based on dimensional analysis, except for cosmic strings in three-dimensional Minkowski space, which are predicted to have a logarithmic correction to the naive scaling form. The scaling amplitudes of the length and area densities are a factor of approximately 2 lower than the results from numerical simulations of classical field theories. An expression for the length density of strings in the condensed matter literature is corrected.