Abundance of magnetic monopoles
87
Barring cosmological inflation (to be discussed in later chapters). the growth of
R(t) with time is according to the power law
R(I) ,..,. t n
(3.116)
(with n = ! for a radiation-dominated universe). Then
t
dH(t) = - -
(3.117)
I-n
provided n :f:. O. Thus. dH(t) is of order t. We require the particle horizon at
time te that the phase transition is completed. A slightly different discussion is
required for second-order (or weakly first-order) and first-order phase transitions.
For a second-order phase transition. the phase transition is completed at the
critical temperature Tc. For the radiation-dominated era of the FRW universe.
there is the connection (see section 1.3) between the time I since the big bang and
temperature T:
t ~ 0.3N;I/2 m P
(3.118)
T2
where mp is the Planck mass (_10 19 GeV) and N. is the effective number of
degrees of freedom at temperature T:
N. = NB + jNF.
(3.119)
NB and N F are, respectively, the numbers of bosonic and fermionic degrees of
freedom for particles with mass small compared to T. in the sense described after
(2.19). For approximately one monopole per horizon volume, the number density
nM of monopoles should be
nM(T,.) - (dH(te»-3 - 1;3"" N;/2Te6mi,3(0.6)-3
(3.120)
where we have taken
dH(IC> = 2te
(3.121 )
for the radiation-dominated era. If we compare this with the entropy density of
(2.21),
2
_ 27r N.T 3
(3.122)
s - 45
then
nM(T e ) "'- 10 6N 1/ 2T3 -3
• •
emp.
(3.123)
seTt")
At temperatures below Tt" but above the electroweak phase transition. the
appropriate value of N. is that for the SU(3) x SU(2) x U(I) standard model:
N. = 106.75.
(3.124)