2.5. Global textures
Whenever a global non-Abelian symmetry is spontaneously and completely broken (e.g. at a grand unification scale), global defects called textures are generated. Theories where this global symmetry is only partially broken do not lead to global textures, but instead to global monopoles and non-topological textures. As we already mentioned global monopoles do not suffer the same constraints as their gauge counterparts: essentially, having no associated gauge fields, the long-range forces between pairs of monopoles lead to the annihilation of their eventual excess and as a result monopoles scale with the expansion. On the other hand, non-topological textures are a generalization that allows the broken subgroup H to contain non-Abelian factors. It is then possible to have 3 trivial as in, e.g., SO(5) SO(4) broken by a vector, for which case we have = S4, the four-sphere [Turok, 1989]. Having explained this, let us concentrate in global topological textures from now on.
Textures, unlike monopoles or cosmic strings, are not well localized in space. This is due to the fact that the field remains in the vacuum everywhere, in contrast to what happens for other defects, where the field leaves the vacuum manifold precisely where the defect core is. Since textures do not possess a core, all the energy of the field configuration is in the form of field gradients. This fact is what makes them interesting objects only when coming from global theories: the presence of gauge fields Aµ could (by a suitable reorientation) compensate the gradients of and yield Dµ = 0, hence canceling out (gauging away) the energy of the configuration (12).
One feature endowed by textures that really makes these defects peculiar is their being unstable to collapse. The initial field configuration is set at the phase transition, when develops a nonzero vacuum expectation value. lives in the vacuum manifold and winds around in a non-trivial way on scales greater than the correlation length, t. The evolution is determined by the nonlinear dynamics of . When the typical size of the defect becomes of the order of the horizon, it collapses on itself. The collapse continues until eventually the size of the defect becomes of the order of -1, and at that point the energy in gradients is large enough to raise the field from its vacuum state. This makes the defect unwind, leaving behind a trivial field configuration. As a result grows to about the horizon scale, and then keeps growing with it. As still larger scales come across the horizon, knots are constantly formed, since the field points in different directions on in different Hubble volumes. This is the scaling regime for textures, and when it holds simulations show that one should expect to find of order 0.04 unwinding collapses per horizon volume per Hubble time [Turok, 1989]. However, unwinding events are not the most frequent feature [Borrill et al., 1994], and when one considers random field configurations without an unwinding event the number raises to about 1 collapse per horizon volume per Hubble time.
12 This does not imply, however, that the classical dynamics of a gauge texture is trivial. The evolution of the - Aµ system will be determined by the competing tendencies of the global field to unwind and of the gauge field to compensate the gradients. The result depends on the characteristic size L of the texture: in the range m-1 << L << ma-1 ~ (e)-1 the behavior of the gauge texture resembles that of the global texture, as it should, since in the limit ma very small (e 0) the gauge texture turns into a global one [Turok & Zadrozny, 1990]. Back.