Physical Foundations of Cosmology

200 The very early universe

and

αw

(

q^2

)

≡

g^2

4 π



α^0 w

1 + 0. 265 α^0 wln

(

q^2 /q 02

), (4.199)

whereq 0 ∼100 GeV andαw^0 ≡αw(q 0 ) 1 / 29 .Comparing (4.31) with (4.199),
we find that the strong and weak coupling constants meet atq∼ 1017 GeV. This
suggests that above 10^17 GeV,strong and weak interactions may be unified in a large
gauge group and characterized by asinglecoupling constantgU.The difference in
their strength at low energies is then entirely explained by the different running of
the coupling constants within theSU( 3 )andSU( 2 )subgroups, after the symmetry
of this larger gauge group is broken at∼ 1017 GeV. The simplest single group
which incorporates all known fermions is theSU( 5 )group. It also contains aU( 1 )
subgroup and could therefore include all gauge interactions of the Standard Model.
However, to identify this subgroup with theU( 1 )of electroweak theory, one has to
verify that the properly normalizedU( 1 )fine structure constant,α 1 =( 5 / 3 )g′^2 / 4 π,
meets the other two constants at the correct energy scale. This is a highly nontrivial
task.

Problem 4.29Findα 1

(

q^2

)

.(HintDo not forget hypercharges entering the ver-
tices.)

With more accurate measurements ofαsandθwit has become clear that the
three constants do not quite meet at a point (Figure 4.19). This fact, along with

60

50

40

30

20

10

0

Standard Model

q (GeV)

105 1010 1015

αs−^1

1

αw−^1

α 1 −^1

Fig. 4.19.