**2.6. Evolution of global textures**

We mentioned earlier that the breakdown of any non-Abelian global
symmetry led to the formation of textures. The simplest possible
example involves the breakdown of a global SU(2) by a complex
doublet ^{a},
where the latter may be expressed as a
four-component scalar field, *i.e.*, *a* = 1 ... 4. We may
write the
Lagrangian of the theory much in the same way as it was done in
Eq. (1.7), but now we drop the gauge fields (thus the
covariant derivatives become partial derivatives). Let us take the
symmetry breaking potential as follows,
*V*() =
/4
(||^{2} -
^{2})^{2}. The situation in which a
global SU(2) in broken by a complex doublet with this potential *V*
is equivalent to the theory where SO(4) is broken by a
four-component vector to SO(3), by making
^{a} take on a vacuum
expectation value. We then have the vacuum manifold
given
by SO(4) / SO(3) = *S*^{3}, namely, a three-sphere with
^{a}
_{a} =
^{2}.
As _{3}
(*S*^{3})
**1** (in fact,
_{3}
(*S*^{3}) =
) we see we will have
non-trivial solutions of the field
^{a} and global
textures will arise.

As usual, variation of the action with respect to the
field ^{a} yields
the equation of motion

(15) |

where primes denote
derivatives with respect to conformal time and
is computed in
comoving coordinates. When the symmetry in broken three of the
initially four degrees of freedom go into massless Goldstone bosons
associated with the three directions tangential to the vacuum
three-sphere. The `radial' massive mode that remains (*m*_{} ~
^{1/2}
) will not be
excited, provided we concentrate on length scales much larger than
*m*_{}^{-1}.

To solve for the dynamics of the field
^{b}, two
different approaches have been implemented in the literature. The first one
faces directly the full equation (15), trying to solve it
numerically. The alternative to this exploits the fact that, at
temperatures smaller than *T*_{c}, the field is constrained
to live in the true vacuum. By implementing this fact via a Lagrange
multiplier ^{(13)} we get

(16) |

with ^{µ}
the covariant derivative operator.
Eq. (16) represents a non-linear sigma model
for the interaction of the three massless modes
[Rajaraman, 1982].
This last approach is only valid when probing length scales
larger than the inverse of the mass *m*_{}^{-1}.
As we mentioned before, when this condition is not met the
gradients of the field are strong enough to make it leave
the vacuum manifold and unwind.

The approach (cf. Eqs. (16)) is suitable for analytic
inspection. In fact, an exact flat space solution was found assuming a
spherically symmetric ansatz. This solution represents the collapse
and subsequent conversion of a texture knot into massless Goldstone
bosons, and is known as the spherically symmetric self-similar (SSSS)
exact unwinding solution. We will say no more here with regard to the
this solution, but just refer the interested reader to the original
articles [see, *e.g.*,
Turok & Spergel,
1990;
Notzold, 1991].
Simulations taking full account of the energy stored in gradients of
the field, and not just in the unwinding events, like in
Eq. (15), were performed, for example, in
[Durrer & Zhou, 1995].
^{(14)}

^{(13)} In fact, in the action the coupling
constant
of the `Mexican hat'
potential is interpreted as the Lagrange multiplier.
Back.

^{(14)} Simulations of the collapse of
`exotic' textures can be found at
http://camelot.mssm.edu/~ats/texture.html
Back.