Physical Foundations of Cosmology

184 The very early universe

comes from the states which surmount the barrier classically. ForT>Esphthere is
no exponential suppression and the particle very quickly leaves the potential well.
So far we have considered a particle with only one degree of freedom. For a
system withNdegrees of freedom the potential can depend on all coordinates
q≡(q 1 ,q 2 ,...,qN).The generalization to this case, however, is rather straightfor-
ward. To calculate the tunneling probability at low temperatures we have to find an
instanton with the minimal Euclidean actionSI. If the energy is normalized so that
V=0 at the bottom of the potential well,the dominant contribution to the tun-
neling probability is proportional to exp(−SI).At very high temperatures we need
to find the local extrema of the potential through which the particle can escape.
The extremum with the minimal value of the potential determines the dominant
contribution to the escape probability. This probability is given by (4.151), where
Esphis the value of the potential at the corresponding extremum.

4.5.2 Decay of the metastable vacuum

We consider a real scalar field in this section using the standard notationφinstead of
χand for simplicity neglect the expansion of the universe. Let us assume that at the
time relevant for the transition the potentialV(φ)has the shape shown in Figure 4.13.
For convenience we normalize the energy such thatV( 0 )=0 andV(φ 0 )=−< 0.
Obviously the stateφ=0 is metastable and decays. If the transition takes place
efficiently when the temperature is small, we can neglect thermal fluctuations and

φ 0

V

0

−

φ

Fig. 4.13.