Cuspidal Singularities in Collapsing Domain Walls

Abstract

Domain wall networks have attracted renewed interest, particularly in relation to the dynamics of network collapse. Accurately describing this process is challenging and typically requires large scale numerical simulations. Here we adopt a complementary approach by studying the collapse of individual closed domain walls, extending previous thin wall analyses and comparing them with adaptive mesh refinement field theory simulations. Firstly, we show that collapsing domain walls generically develop worldvolume singularities of two types: cuspidal edge singularities, consisting of one dimensional singular edges that propagate along the wall surface at the speed of light for a finite time, and cuspidal vertex singularities, which are spike like and instantaneous events where the wall moves momentarily at the speed of light. Both types of features arise generically from smooth initial conditions, and their formation and evolution follow the universal patterns of singularity theory. We show that these structures are captured both by the Nambu-Goto equations and by an eikonal like approximation valid in the relativistic regime. Furthermore, we demonstrate that the same singular structures are reproduced qualitatively in full field theory simulations, establishing that they are not artifacts of the thin wall approximation but robust features of realistic domain wall dynamics. Naturally, such focusing effects in the field theory simulations result in localized regions of high energy density. We briefly discuss possible phenomenological implications.