2.3. Cosmic strings
Cosmic strings are without any doubt the topological defect most thoroughly studied, both in cosmology and solid-state physics (vortices). The canonical example, also describing flux tubes in superconductors, is given by the Lagrangian
(7) |
with Fµ = [µ A], where A is the gauge field and the covariant derivative is Dµ = µ + i e Aµ, with e the gauge coupling constant. This Lagrangian is invariant under the action of the Abelian group G = U(1), and the spontaneous breakdown of the symmetry leads to a vacuum manifold that is a circle, S1, i.e., the potential is minimized for = exp(i), with arbitrary 0 2. Each possible value of corresponds to a particular `direction' in the field space.
Now, as we have seen earlier, due to the overall cooling down of the universe, there will be regions where the scalar field rolls down to different vacuum states. The choice of the vacuum is totally independent for regions separated apart by one correlation length or more, thus leading to the formation of domains of size ~ -1. When these domains coalesce they give rise to edges in the interface. If we now draw a imaginary circle around one of these edges and the angle varies by 2 then by contracting this loop we reach a point where we cannot go any further without leaving the manifold . This is a small region where the variable is not defined and, by continuity, the field should be = 0. In order to minimize the spatial gradient energy these small regions line up and form a line-like defect called cosmic string.
The width of the string is roughly m-1 ~ (1/2 )-1, m being the Higgs mass. The string mass per unit length, or tension, is µ ~ 2. This means that for GUT cosmic strings, where ~ 1016 GeV, we have Gµ ~ 10-6. We will see below that the dimensionless combination Gµ, present in all signatures due to strings, is of the right order of magnitude for rendering these defects cosmologically interesting.
There is an important difference between global and gauge (or local) cosmic strings: local strings have their energy confined mainly in a thin core, due to the presence of gauge fields Aµ that cancel the gradients of the field outside of it. Also these gauge fields make it possible for the string to have a quantized magnetic flux along the core. On the other hand, if the string was generated from the breakdown of a global symmetry there are no gauge fields, just Goldstone bosons, which, being massless, give rise to long-range forces. No gauge fields can compensate the gradients of this time and therefore there is an infinite string mass per unit length.
Just to get a rough idea of the kind of models studied in the literature, consider the case G = SO(10) that is broken to H = SU(5) × 2. For this pattern we have 1() = 2, which is clearly non trivial and therefore cosmic strings are formed [Kibble et al., 1982]. (10)
10 In the analysis one uses the fundamental theorem stating that, for a simply-connected Lie group G breaking down to H, we have 1(G / H) 0(H); see [Hilton, 1953]. Back.