**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
=
*S*^{4}, 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 *A*_{µ} could
(by a suitable
reorientation) compensate the gradients of
and yield
*D*_{µ} =
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
- *A*_{µ}
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* <<
*m*_{a}^{-1} ~
(*e*)^{-1} the behavior of the gauge texture resembles
that
of the global texture, as it should, since in the limit
*m*_{a} very
small (*e*
0) the gauge texture turns into a global one
[Turok & Zadrozny,
1990].
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