I was watching the film V for Vendetta the other day and realised that aside from its interesting dystopian commentary on modern political life, there is a scene which elegantly illustrates the concept of topological defects in general, and domain wall formation in particular.
The clip is shown below and the ‘domain wall’, played by the final standing domino, is formed at around second number 30.
The basic idea of domain wall formation is that one has a field that after a phase transition can fall into one of two vacuum states (typically the potential is Z2 symmetric) anywhere in space, in the same way that dominoes can fall left or right. Nearby dominoes can influence each other to fall in the same direction but when the dominoes are sufficiently well-separated one inevitably will find that some will fall left and some will fall right and hence, as in the clip, one will eventually be left with dominoes that cannot fall at all and are left standing. The domino left standing is the domain wall and in the context of field theory it is a region of space (a wall) which is forced to remain in its previous state, and hence lives at a much higher energy density than the true vacuum.
In cosmology the existence of domain walls, and higher order topological defects such as cosmic strings, monopoles and textures, after a phase transition is assured through the Kibble mechanism. Tom Kibble pointed out that if two regions of space are in causal contact then it is possible for the fields in those two regions to fall into the same vacuum (in the domino analogy they can fall in the same direction). However, if two regions are not causally in contact (they are separated by a distance larger than the distance light could have traveled in the appropriate time) then there is no way to make sure the field falls in the same direction in both places, since they could not have communicated their choices with each other.
Hence one should expect the field (the dominoes in this analogy) to fall randomly in each region and hence to form a defect at the boundary (standing dominoes in the clip), and on average one should expect roughly a density of one defect per horizon sized volume. Since domain walls quickly come to dominate the density of the universe they are disastrous for cosmology and one either has to avoid forming them in the first place (by avoiding Z2 symmetric phase transitions) or have a mechanism for getting rid of them once they have formed, of which inflation is a putative example.
Some further reading:
Topological defects, Department of Applied Mathematics and Theoretical Physics, Cambridge University, UK.