Physical Foundations of Cosmology

176 The very early universe

χ

V(χ)

T 3 T 2 <T 3 T 1 <T 2

Fig. 4.11.

4.4.5 Electroweak phase transition

The above considerations can easily be generalized to study the electroweak phase
transition in the early universe. In electroweak theory, the equation for the scalar
field is obtained by variation ofχ-dependent terms in the electroweak Lagrangian
given in (4.59), (4.79) and (4.84). If we assume that the Higgs mass is small and
neglect the scalar particles, then the equation for the homogeneous field ̄χis

χ ̄;α;α+V′(χ ̄)−

g^2 +g′^2

4

χ ̄

〈

ZμZμ

〉

−

g^2

2

χ ̄

〈

Wμ+W−μ

〉

+ft〈t ̄t〉= 0. (4.131)

We have retained here only the top quarks, which dominate over the contributions
from the other fermions because of their large Yukawa coupling constant ft. The
contributions ofZandWbosons toVeffcan immediately be written down using
the formulae derived in the previous section. We simply note that the chargedW
bosons have twice as many degrees of freedom as the neutral vector field and hence
give twice the contribution toVeff.