86
Topological defects
When H is SU(3) x SU(2) x U(l). which will be the case for a phase
transition in which the grand unified theory breaks spontaneously to the standard
model. then 1rt(H) is just :ll'1(U(1» = Z. because :11' 1 (SU(3» and :II'}(SU(2»
are both trivial. Thus. for such a spontaneous symmetry breaking. the resulting
second homotopy group is Z/:II'I (G). In particular. if :11' 1 (G) is trivial. as is the
case for the SUeS) grand unified group discussed in section 2.6. then :ll'2(G/ H) =
Z and we have magnetic monopole solutions.
We now ask what masses are possessed by the magnetic monopoles in grand
unified theories. By analogy with (3.92). the magnetic monopole mass will be of
order 4:11''1/ ga. where" is the expectation value of the Higgs scalar responsible for
breaking the grand unified symmetry and ga is the value of the gauge coupling
constant for the grand unified group at the unification scale. In the case of the
SUeS) grand unified theory of section 2.6. ". which is identified with "' c. is of
order 10 15 GeV and ga is of order 1. Thus. we expect the magnetic monopole
mass mM to be of order 10 16 GeV. In the case of the supersymmetric SUeS) grand
unified theory of section 2.7. with a unification scale of 2 x 10 16 GeV. which is
1.5 orders of magnitude greater than in the non-supersymmetric case. a magnetic
monopole mass of order 10 11 _10 18 GeV is to be expected.
3.10 Abundance of magnetic monopoles
Magnetic monopoles form as the phase transition from the SUeS) symmetric
phase to the standard model SU(3) x SU(2) x U(l) phase occurs. This is
the result of the expectation values of the Higgs field only being correlated over
some finite distance. The expectation values of the Higgs field at different points
in space will not be aligned to produce a uniform Higgs field over distances
greater than this. Thus, we can expect topologically non-trivial configurations
to be produced. in particular. magnetic monopoles. The number of magnetic
monopoles formed [16. 17] should be determined as to order of magnitude by
the distance over which the Higgs expectation values are correlated [1,9].
There are two effects which can limit the range over which this correlation
occurs. The first is the statistical-mechanical thermal average over the product
of the two Higgs fields. For a second-order phase transition. this correlation
length is of order T c - I but can be larger for a first-order phase transition, which
proceeds through the formation of bubbles of the low-temperature phase which
then coalesce. The second effect is the general-relativistic particle horizon dH.
Correlations cannot occur over distances greater than the distance dH that light
has been able to travel since the big bang. For a Friedman-Robertson-Walker
(FRW) universe. as discussed in section 1.2, the proper distance at time t from
any point to the particle horizon is
I
dt'
dH(t) = R(t) 10 R(t')'
(3.115)
Topological defects
When H is SU(3) x SU(2) x U(l). which will be the case for a phase
transition in which the grand unified theory breaks spontaneously to the standard
model. then 1rt(H) is just :ll'1(U(1» = Z. because :11' 1 (SU(3» and :II'}(SU(2»
are both trivial. Thus. for such a spontaneous symmetry breaking. the resulting
second homotopy group is Z/:II'I (G). In particular. if :11' 1 (G) is trivial. as is the
case for the SUeS) grand unified group discussed in section 2.6. then :ll'2(G/ H) =
Z and we have magnetic monopole solutions.
We now ask what masses are possessed by the magnetic monopoles in grand
unified theories. By analogy with (3.92). the magnetic monopole mass will be of
order 4:11''1/ ga. where" is the expectation value of the Higgs scalar responsible for
breaking the grand unified symmetry and ga is the value of the gauge coupling
constant for the grand unified group at the unification scale. In the case of the
SUeS) grand unified theory of section 2.6. ". which is identified with "' c. is of
order 10 15 GeV and ga is of order 1. Thus. we expect the magnetic monopole
mass mM to be of order 10 16 GeV. In the case of the supersymmetric SUeS) grand
unified theory of section 2.7. with a unification scale of 2 x 10 16 GeV. which is
1.5 orders of magnitude greater than in the non-supersymmetric case. a magnetic
monopole mass of order 10 11 _10 18 GeV is to be expected.
3.10 Abundance of magnetic monopoles
Magnetic monopoles form as the phase transition from the SUeS) symmetric
phase to the standard model SU(3) x SU(2) x U(l) phase occurs. This is
the result of the expectation values of the Higgs field only being correlated over
some finite distance. The expectation values of the Higgs field at different points
in space will not be aligned to produce a uniform Higgs field over distances
greater than this. Thus, we can expect topologically non-trivial configurations
to be produced. in particular. magnetic monopoles. The number of magnetic
monopoles formed [16. 17] should be determined as to order of magnitude by
the distance over which the Higgs expectation values are correlated [1,9].
There are two effects which can limit the range over which this correlation
occurs. The first is the statistical-mechanical thermal average over the product
of the two Higgs fields. For a second-order phase transition. this correlation
length is of order T c - I but can be larger for a first-order phase transition, which
proceeds through the formation of bubbles of the low-temperature phase which
then coalesce. The second effect is the general-relativistic particle horizon dH.
Correlations cannot occur over distances greater than the distance dH that light
has been able to travel since the big bang. For a Friedman-Robertson-Walker
(FRW) universe. as discussed in section 1.2, the proper distance at time t from
any point to the particle horizon is
I
dt'
dH(t) = R(t) 10 R(t')'
(3.115)