88
Topological defects
Thus,
nM(T(") '" 102 (~)3
(3.125)
s(T(")
mp
Assuming that the expansion of the universe for T < Te is adiabatic, then
S cc R- 3 and the ratio nM(T)/s(T) does not change. As a consequence, the
monopole contribution OMh 2 to Omh2 today is predicted to be many orders
of magnitude greater than the observational bound of about 0.15. For a nonsupersymmetric GUT theory with Te of order 10 15 Ge V and a magnetic monopole
mass mM of order 10 16 GeV, OMh 2 is 14 orders of magnitude greater than this
upper bound. For a supersyrrunetric GUT theory with Te of order 10 16 GeV and
a magnetic monopole mass mM of order 10 17 _10 18 GeV, the situation is even
worse with OMh 2 some 18-19 orders of magnitude greater than the upper bound
(exercise 10).
In the case of a first-order phase transition, the transition does not proceed
until some temperature below Te at which the bubble nucleation rate for bubbles
of the low-temperature phase is of the same order as the expansion rate H for
the universe. We expect the Higgs expectation values to be correlated within
a bubble but uncorrelated between any two bubbles. Thus, the number density
of monopoles (or antimonopoles) produced should be of the order of (~lrTl)-I,
where rb is the average radius of a bubble at a time when the bubbles have
expanded to just fill the whole of space. The universe supercools at the firstorder phase transition but reheats when the bubbles coalesce, so that the entropy
density after reheating is 21r 2 N.T; /45, as in the second-order case. Thus, for a
first-order phase transition,
nM
45 _I -3 (4
(3.126)
-;- '" 211'2 N. Te
31rTb
3)-1
The value of rb has been estimated [18] leading to a value of OMh 2 even larger
than in the second-order phase transition case.
In either case, if magnetic monopoles form at a grand unified phase
transition, some mechanism is required to dilute the monopole density by many
orders of magnitude. The most obvious mechanism would be annihilation of
monopoles and antimonopoles. However, this has been estimated [16] and there
is no significant effect for nM/s ;5 10- 10 and, for larger values of nM/s,
the annihilation process cannot reduce nM/s much below 10- 10 • For the nonsupersymmetric case, this mechanism is ineffective and, for the supersymmetric
case, it can do no more than reduce the monopole abundance closer to that for
the non-supersymmetric case. A possible mechanism that can do the trick is
cosmological inflation, which will be discussed in later chapters.
After some mechanism has reduced the monopole abundance to a value
compatible with the bound on Omh2, any residual monopole density can have
important astrophysical consequences [19]. For instance, because the expectation
value of the grand unified Higgs field approaches zero as the centre of the
Topological defects
Thus,
nM(T(") '" 102 (~)3
(3.125)
s(T(")
mp
Assuming that the expansion of the universe for T < Te is adiabatic, then
S cc R- 3 and the ratio nM(T)/s(T) does not change. As a consequence, the
monopole contribution OMh 2 to Omh2 today is predicted to be many orders
of magnitude greater than the observational bound of about 0.15. For a nonsupersymmetric GUT theory with Te of order 10 15 Ge V and a magnetic monopole
mass mM of order 10 16 GeV, OMh 2 is 14 orders of magnitude greater than this
upper bound. For a supersyrrunetric GUT theory with Te of order 10 16 GeV and
a magnetic monopole mass mM of order 10 17 _10 18 GeV, the situation is even
worse with OMh 2 some 18-19 orders of magnitude greater than the upper bound
(exercise 10).
In the case of a first-order phase transition, the transition does not proceed
until some temperature below Te at which the bubble nucleation rate for bubbles
of the low-temperature phase is of the same order as the expansion rate H for
the universe. We expect the Higgs expectation values to be correlated within
a bubble but uncorrelated between any two bubbles. Thus, the number density
of monopoles (or antimonopoles) produced should be of the order of (~lrTl)-I,
where rb is the average radius of a bubble at a time when the bubbles have
expanded to just fill the whole of space. The universe supercools at the firstorder phase transition but reheats when the bubbles coalesce, so that the entropy
density after reheating is 21r 2 N.T; /45, as in the second-order case. Thus, for a
first-order phase transition,
nM
45 _I -3 (4
(3.126)
-;- '" 211'2 N. Te
31rTb
3)-1
The value of rb has been estimated [18] leading to a value of OMh 2 even larger
than in the second-order phase transition case.
In either case, if magnetic monopoles form at a grand unified phase
transition, some mechanism is required to dilute the monopole density by many
orders of magnitude. The most obvious mechanism would be annihilation of
monopoles and antimonopoles. However, this has been estimated [16] and there
is no significant effect for nM/s ;5 10- 10 and, for larger values of nM/s,
the annihilation process cannot reduce nM/s much below 10- 10 • For the nonsupersymmetric case, this mechanism is ineffective and, for the supersymmetric
case, it can do no more than reduce the monopole abundance closer to that for
the non-supersymmetric case. A possible mechanism that can do the trick is
cosmological inflation, which will be discussed in later chapters.
After some mechanism has reduced the monopole abundance to a value
compatible with the bound on Omh2, any residual monopole density can have
important astrophysical consequences [19]. For instance, because the expectation
value of the grand unified Higgs field approaches zero as the centre of the