Physical Foundations of Cosmology

4.4 “Symmetry restoration” and phase transitions 173

V(χ)

χ

T>T 1 T=T 1 T 1 >T >Tc T=Tc T<Tc

Fig. 4.10.

4.4.3 Symmetry restoration at high temperature

Potential (4.122) is shown in Figure 4.10 for a few different temperatures. For very
largeTit has only one minimum at ̄χ=0; the symmetry is restored and the gauge
bosons and fermions are massless. When the temperature drops below

T 1 =

T 0

√

1 −

(

9 e^4 / 16 π^2 λT 1

), (4.125)

the second minimum appears; first it is located at ̄χ 1 =

(

3 e^3 T 1 / 8 πλT

)

, and then it
moves to the right as the temperature drops. The values ofVeffat the two minima
become equal at the critical temperature

Tc=

T 0

√

1 −

(

e^4 / 2 π^2 λTc

). (4.126)

At this time the second minimum, located at

χ ̄c=

e^3 Tc

2 πλT

, (4.127)