Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 185

the metastable vacuum decays via quantum tunneling. If thermal fluctuations are
not negligible on the other hand, they can push the field to the top of the potential
and the transition occurs classically without tunneling. In both cases critical bubbles
are formed, filled with the new phaseφ=0. If the bubble nucleation rate exceeds
the expansion rate of the universe, the bubbles collide and finally fill all space with
the new phase. Let us now calculate the decay rate of the metastable (false) vacuum.
The scalar fieldφ(x,t)is a system with an infinite number of degrees of freedom.
We can treat the spatial coordinatesxas a continuous index enumerating the degrees
of freedom. In this caseφ(x,t)≡φx(t) plays the same role asqn(t)in the preceding
discussion (the correspondence is obvious:φ⇐⇒qandx⇐⇒n).The action for
the scalar field can be written as

S=

∫

(K−V)dt, (4.152)

where

K≡

∑

x

1

2

(

∂φx

∂t

) 2

=

1

2

∫(

∂φx

∂t

) 2

d^3 x (4.153)

is the kinetic energy and

V(φx)≡

∫(

1

2

(∂iφx)^2 +V(φx)

)

d^3 x (4.154)

is the potential energy orpotentialof the scalar field configurationφx.As usual,∂i
denotes the partial derivative with respect to the spatial coordinatexiwhich, in the
language adopted in this section, is the derivative with respect to the continuous
index. We must stress that in the study of tunneling in field theory the potentialV,
andnotthe scalar field potentialV(φ), plays the role ofV(q)from the previous
discussion. To avoid confusion, the reader must distinguish between them carefully.
The potentialV(φx)is afunctional.It depends on the infinite number of variables
φxand takes a definite numerical value only when the field configurationφ(x)is
completely specified.

Decay via instantonsFor the scalar field potentialV(φ),shown in Figure 4.13, the
stateφ(x)=0 corresponds to a local minimum of the potentialVwithV( 0 )= 0.
The other static configurationφ(x)=φ 0 has negative energyV(φ 0 )=−×vol-
ume. Therefore the stateφ=0 is metastable and should decay. This decay can
be described, in complete analogy with the preceding discussion, as an “escape”
via tunneling of the infinitely many degrees of freedom from the local poten-
tial well inV. The dominant contribution to the semiclassical tunneling probabil-
ity, proportional to exp(−SI),comes from the instanton with the actionSI.This