**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 *F*_{µ}
= _{[µ}
*A*_{]}, where
*A*_{} is the
gauge field and the covariant derivative is *D*_{µ} =
_{µ} +
*i* *e* *A*_{µ}, 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, *S*^{1}, *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
~
10^{16} GeV, we have
*G**µ* ~ 10^{-6}. We will see below that the
dimensionless
combination *G**µ*, 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
*A*_{µ} 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].
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