224 The very early universe
thismonopole problem. Inflationary cosmology provides us with such a solution.
If the monopoles were produced in the very early universe, then a subsequent
inflationary stage would drastically dilute their number density, leaving less than
one monopole per present horizon scale. Of course, this solution works only if the
reheating temperature after inflation does not exceed the Grand Unified Theory
scale; otherwise monopoles are produced in unacceptable amounts after the end of
inflation. We will see in the following sections that this assumption about the energy
scale of inflation is in agreement with contemporary ideas. Moreover, according
to inflationary scenarios, it is likely that the temperature in the universe was never
larger than the Grand Unified Theory scale and hence that monopoles were never
produced according to the mechanism described above. This, however, does not
mean that primordial Grand Unified Theory monopoles do not exist. In principle
they could be produced during a preheating phase after inflation (see Section 5.5)
in amounts allowed by present cosmological bounds. The search for primordial
monopoles remains important.
TexturesThe other possible defect−a texture−arises when the symmetry is
broken with four real scalar fieldsφi,i= 1 ,..., 4 .In this case the vacuum manifold
is a 3-sphereS^3 and the textures are classified by the homotopy groupπ 3 (M).
Because four equations
φi(x^1 ,x^2 ,x^3 )= 0for three variablesx^1 ,x^2 ,x^3 generically have no solutions, regions of false vacuum
are not formed during the phase transition. However, the fieldsφi are uncorre-
lated on superhorizon scales and therefore(∂φ)^2 is generally different from zero
even ifφ^2 =σ^2 andV(φ)=0. The resulting stable structure has positive energy
and is called aglobaltexture. Some time ago global textures were considered a
compelling mechanism for explaining the structure of the universe. However, the
texture scenario is in contradiction with measurements of CMB fluctuations and
hence textures cannot play any significant role in structure formation.
Static textures do not “survive” in local gauge theories. In this case the gauge
fieldsexactlycompensate the spatial gradients of the scalar fields; as distinct from
strings and monopoles, textures correspond to the true vacuum everywhere. As a
result(Dφ)^2 vanishes and the total energy of a local texture is equal to zero.
In the Standard Model, electroweak symmetry is broken with a doublet of com-
plex scalar fields, or equivalently, with four real scalar fields. Hence, the only
topological defects which could occur are textures. However, because this theory
possesses local gauge invariance, the corresponding static textures have zero energy
and they are not very interesting.