**2.1. Local and global monopoles and domain walls**

Monopoles are yet another example of stable topological defects.
Their formation stems from the fact that the vacuum expectation value
of the symmetry breaking Higgs field has random orientations
(<^{a}>
pointing in different directions in group space) on
scales greater than the horizon. One expects therefore to have a
probability of order unity that a monopole configuration will result
after the phase transition (cf. the Kibble mechanism). Thus, about
one monopole per Hubble volume should arise and we have for the number
density *n*_{monop} ~ 1 / *H*^{-3} ~
*T*_{c}^{6} /
*m*_{P}^{3}, where *T*_{c} is
the critical temperature and *m*_{P} is Planck mass,
when the transition occurs. We also know
the entropy density at this temperature, *s* ~
*T*_{c}^{3}, and so the
monopole to entropy ratio is *n*_{monop} / *s*
100
(*T*_{c} / *m*_{P})^{3}.
In the absence of non-adiabatic processes after monopole creation
this constant ratio determines their present abundance. For the
typical value *T*_{c} ~ 10^{14} GeV we have
*n*_{monop} / *s* ~ 10^{-13}. This
estimate leads to a present
_{monop}
*h*^{2}
10^{11}, for the superheavy monopoles
*m*_{monop}
10^{16} GeV
that are created ^{(6)}.
This value
contradicts standard cosmology and the presently most attractive way
out seems to be to allow for an early period of inflation: the massive
entropy production will hence lead to an exponential decrease of the
initial *n*_{monop} / *s* ratio, yielding
_{monop}
consistent with observations.
^{(7)} In summary, the
broad-brush picture one has in mind is that of a mechanism that could
solve the monopole problem by `weeping' these unwanted relics out of
our sight, to scales much bigger than the one that will eventually
become our present horizon today.

Note that these arguments do not apply for global monopoles as these (in the absence of gauge fields) possess long-range forces that lead to a decrease of their number in comoving coordinates. The large attractive force between global monopoles and antimonopoles leads to a high annihilation probability and hence monopole over-production does not take place. Simulations performed by Bennett & Rhie [1990] showed that global monopole evolution rapidly settles into a scale invariant regime with only a few monopoles per horizon volume at all times.

Given that global monopoles do not represent a danger for cosmology
one may proceed in studying their observable consequences. The
gravitational fields of global monopoles may lead to matter clustering
and CMB anisotropies. Given an average number of monopoles per horizon
of ~ 4,
Bennett & Rhie [1990]
estimate a scale invariant spectrum of fluctuations
( /
)_{H} ~ 30 *G*
^{2} at
horizon crossing ^{(8)}.
In a subsequent paper they
simulate the large-scale CMB anisotropies and, upon normalization
with *COBE*-DMR, they get roughly *G*
^{2} ~
6 × 10^{-7} in agreement with a GUT energy scale
[Bennett & Rhie,
1993].
However, as we will see in the CMB sections below,
current estimates for the angular power spectrum of global defects do
not match the most recent observations, their main problem being the
lack of power on the degree angular scale once the spectrum is
normalized to *COBE* on large scales.

Let us concentrate now on domain walls, and briefly try to show why
they are not welcome in any cosmological context (at least in the
simple version we here consider - there is always room for more
complicated (and contrived) models). If the symmetry breaking pattern
is appropriate at least one domain wall per horizon volume will be
formed. The mass per unit surface of these two-dimensional objects is
given by ~ ^{1/2}
^{3},
where as usual
is the coupling constant in the symmetry breaking potential for the
Higgs field. Domain walls are generally horizon-sized and therefore
their mass is given by ~
^{1/2}
^{3}
*H*^{-2}. This implies a mass energy density roughly given by
_{DW} ~
^{3}
*t*^{-1} and we may readily see now how the problem
arises: the critical density goes as
_{crit}
~ *t*^{-2} which
implies _{DW}(*t*) ~
( /
*m*_{P})^{2}
*t*.
Taking a typical GUT value for
we get
_{DW}(*t*
~ 10^{-35}sec) ~ 1 *already* at the time of the phase
transition. It is not hard to imagine that today this will be at
variance with observations; in fact we get
_{DW}(*t*
~ 10^{18}sec) ~ 10^{52}. This indicates that models where
domain walls are produced are tightly constrained, and the general
feeling is that it is best to avoid them altogether [see
Kolb & Turner, 1990
for further details; see also
Dvali et al., 1998,
Pogosian &
Vachaspati,2000
^{(9)} and
Alexander et al., 1999
for an alternative solution].

^{6} These are the actual figures for a gauge
SU(5) GUT second-order phase transition.
Preskill [1979]
has shown that in this case monopole antimonopole annihilation is not
effective to reduce their abundance.
Guth & Weinberg [1983]
did the case for a
first-order phase transition and drew qualitatively similar
conclusions regarding the excess of monopoles.
Back.

^{7} The inflationary expansion reaches an end
in the so-called reheating process, when the enormous vacuum energy
driving inflation is transferred to coherent oscillations of the
inflaton field. These oscillations will in turn be damped by the
creation of light particles (*e.g.*, via preheating)
whose final fate is to thermalise and
reheat the universe.
Back.

^{8} The spectrum of density
fluctuations on smaller scales has also been computed. They normalize
the spectrum at 8 *h*^{-1} Mpc and agreement with
observations lead
them to assume that galaxies are clustered more strongly than the
overall mass density, this implying a `biasing' of a few [see
Bennett, Rhie &
Weinberg, 1993
for details].
Back.

^{9} Animations of monopoles colliding with
domain walls can be found in `LEP' page at
http://theory.ic.ac.uk/~LEP/figures.html
Back.