Topological defects are stable configurations of matter in the original, symmetric or old phase which persist after a phase transition to the asymmetric or new phase. The type of defect formed is determined by the symmetry properties of the matter and the nature of the phase transition.
In the laboratory, topological defects are commonly observed in condensed matter systems. Simple examples are the domains in a ferromagnet; regions in which the magnetic dipoles are aligned, separated by domain walls. Liquid crystals exhibit an array of topological defects, such as strings and monopoles.
Several types of topological defects are commonly produced in cosmological phase transitions:
These form when a discrete symmetry is broken. A network of domain walls partitions the universe into various cells. The gravitational field of a domain wall is repulsive rather than attractive.
These form when an axial or cylindrical symmetry is broken. Strings associated with grand unified particle physics models are very thin and may stretch across the visible universe. A 10 kilometre length of a GUT string would weigh as much as the earth itself.
Form when a spherical symmetry is broken. Monopoles are predicted to be supermassive and carry magnetic charge. The existence of monopoles is an inevitable prediction of grand unified theories (GUTs); this is one of the puzzles of the standard cosmology.
Form when large symmetry groups are completely broken. Textures are delocalized topological defects which are unstable to collapse. We shall return to there potential cosmological implications at the end of these pages.
If cosmic strings or other topological defects can form at a cosmological phase transition then they will form. This observed by Kibble and, in a cosmological context, the defect formation process is known as the Kibble mechanism. The simple fact is that causal effects in the early universe can only propagate (as at any time) as the speed of light c. This means that at a time t, regions of the universe separated by more than a distance d=ct can know nothing about each other. In a symmetry breaking phase transition, different regions of the universe will fall into different minima in the set of possible states (the vacuum manifold) and thus inevitably topological defects will form. For example, in a theory with two minima, plus + and minus -, then neighbouring regions separated by more than ct will tend to fall randomly into the different states (as shown below). Interpolating between these different minima will be a domain wall.
Cosmic strings will arise in slightly more complicated theories in which the minimum energy states possess `holes'. The strings will simply correspond to non-trivial `windings' around these holes (as illustrated below).
