1.4. A survey of topological defects

Different models for the Higgs field lead to the formation of a whole variety of topological defects, with very different characteristics and dimensions. Some of the proposed theories have symmetry breaking patterns leading to the formation of `domain walls' (mirror reflection discrete symmetry): incredibly thin planar surfaces trapping enormous concentrations of mass-energy which separate domains of conflicting field orientations, similar to two-dimensional sheet-like structures found in ferromagnets. Within other theories, cosmological fields get distributed in such a way that the old (symmetric) phase gets confined into a finite region of space surrounded completely by the new (non-symmetric) phase. This situation leads to the generation of defects with linear geometry called `cosmic strings'. Theoretical reasons suggest these strings (vortex lines) do not have any loose ends in order that the two phases not get mixed up. This leaves infinite strings and closed loops as the only possible alternatives for these defects to manifest themselves in the early universe ⁽⁴⁾.

Figure 2. In a simple model of symmetry breaking, the initial symmetric ground state of the Higgs field (yellow dot) can fall into the left- or right-hand valley of a double-well energy potential (light and dark dots). In a cosmic phase transition, regions of the new phase appear randomly and begin to grow and eventually merge as the transition proceeds toward completion (middle). Regions in which the symmetry has broken the same way can coalesce, but where regions that have made opposite choices encounter each other, a topological defect known as a domain wall forms (right). Across the wall, the Higgs field has to go from one of the valleys to the other (in the left panel), and must therefore traverse the energy peak. This creates a narrow planar region of very high energy, in which the symmetry is locally unbroken.

With a bit more abstraction scientists have even conceived other (semi) topological defects, called `textures'. These are conceptually simple objects, yet, it is not so easy to imagine them for they are just global field configurations living on a three-sphere vacuum manifold (the minima of the effective potential energy), whose non linear evolution perturbs spacetime. Turok [1989] was the first to realize that many unified theories predicted the existence of peculiar Higgs field configurations known as (texture) knots, and that these could be of potential interest for cosmology. Several features make these defects interesting. In contrast to domain walls and cosmic strings, textures have no core and thus the energy is more evenly distributed over space. Secondly, they are unstable to collapse and it is precisely this last feature which makes these objects cosmologically relevant, for this instability makes texture knots shrink to a microscopic size, unwind and radiate away all their energy. In so doing, they generate a gravitational field that perturbs the surrounding matter in a way which can seed structure formation.

⁴ `Monopole' is another possible topological defect; we defer its discussion to the next subsection. Cosmic strings bounded by monopoles is yet another possibility in GUT phase transitions of the kind, e.g., G K × U(1) K. The first transition yields monopoles carrying a magnetic charge of the U(1) gauge field, while in the second transition the magnetic field in squeezed into flux tubes connecting monopoles and antimonopoles [Langacker & Pi, 1980]. Back.