# Simulating the symmetron

*Extra videos to accompany the paper: "simulating the symmetron"*, arXiv:1409.6570

*Contents*

### Introduction

The idea is to understand the dynamics of domain walls in the presence of symmetry-restoring defects. Domain walls are topological defects which form from phase transitions when the potential has multiple minima: here are much more rigorous definitions and explanations available, but imagine a set of small balls being thrown into an egg-box. Each of the balls is attached to a location in space, and so nearby locations in space may be attached to balls in different depressions in the egg-box -- these are "minima" of the energy. There is an "energy cost" associated with balls being in different minima, and these manifest in space as lines, or sheets, of energy. These are *domain walls*.

The point of the paper is to begin to understand and catalogue what happens (or, what could happen) if domain walls form in a system which isn't perfect; the imperfections are modelled by "impurities". These impurities can have different physical origins, depending on where you are imagining the evolution to take place. We have two applications in mind: (1) field theory: understanding the dynamics in their own right, (2) cosmology: the imperfections corresponds to structures in the Universe.

** Note on the numerics** The movies here are evolved on grids with P = 512

^{2}lattice sites in either direction (this is a smaller grid than I've used in the paper, and its use is limited to making movies). Damped dynamics is used until t = 20 to smooth out the initial conditions. After that, it is the full relativistic equation of motion which takes hold. The simulations terminate at "light-crossing-time". For full details, see the paper.

### Single impurity

We have put a single impurity at the origin (it has internal density ρ_{0} = 15), and let the domain walls evolve from random initial conditions. The movie shows the evolution of the energy density.

### Rings of impurities

This is a movie of random walls evolving in the presence of N = 10 impurities of internal density ρ_{0} = 15. The impurities are evenly distributed on a circle. On the left you see the scalar field evolving, and on the right the energy density.

There are walls which pin to the impurities, and they move along the surface of the impurities -- this slows down the usual rate of collapse of the network. In a high-res version you can see that as the walls collapse, waves of radiation get released. Indeed, the energy density in the interior of the impurities is in turmoil.