Introduction to Cosmology

The Primordial Hot Plasma 113

space-time which may be passing through regions of different vacua. In the simple
model of symmetry breaking by a real scalar field휑the푇-dependence may be put in
explicitly, as well as other dependencies (denoted by dots),

푉(휑, 푇 ,etc.)=−^1

2

휇^2 휑^2 +^1

4

휆휑^4 +^1

8

휆푇^2 휑^2 +···. (6.5)

As time decreases푇increases, the vacuum expectation value휙 0 decreases, so that
finally, in the early Universe, the true minimum of the potential is the trivial one at
휑=0. This occurs above acritical temperatureof

푇c= 2 휇∕

√

휆. (6.6)

An example of this behavior is illustrated by the potentials in Figure 6.1. The dif-
ferent curves correspond to different temperatures. At the highest temperature, the
only minimum is at휑=0 but, as the temperature decreases, a new minimum devel-
ops spontaneously. If there is more than one minimum, only one of these is stable. A
classical example of an unstable minimum is a steam bubble in boiling water.

0

a

V(φ) – V(0)

bc

φ 0

φ

Figure 6.1 Effective scalar potentials. The different curves correspond to different temper-
atures. At the highest temperature (a) the only minimum is at휑=0 but, as the temperature
decreases (b), a new minimum develops spontaneously. Finally, in (c), a stable minimum is
reached at휑 0.