Physical Foundations of Cosmology

182 The very early universe

InstantonsIn the special case ofE=0 (the particle is at rest at the local minimum
of the potential) we have

∫b(^0 )

0

√

2 MVdq=

∫τb

−∞

(

1

2

Mq ̇^2 +V

)

dτ≡Sb( 0 ), (4.146)

whereq(τ)satisfies the equation

Mq ̈+

∂(−V)

∂q

= 0. (4.147)

Integrating (4.147) once gives

Mq ̇^2 / 2 −V= 0.

Taking into account time-translational invariance, we can setτb=0 in (4.146)
without loss of generality.
Equation (4.147) describes the motion of the particle in the inverted potential
(−V)(Figure 4.12), and can be obtained from the original equation of motion if
we make the Wick rotationt→τ=it.Note that the formal substitutiont=−iτ
converts the Minkowski metric

ds^2 =dt^2 −dx^2

to the Euclidean metric

−ds^2 E=dτ^2 +dx^2.

Therefore,τ is called “Euclidean time.” The right hand side of (4.146) is the
Euclidean action calculated for the trajectory satisfying (4.147), with boundary
conditionsq(τ→−∞)=0 andq(τ= 0 )=b( 0 ). We can “close” this trajectory
by considering the “motion” back toq=0asτ→+∞.The corresponding so-
lution of (4.147), with boundary conditionsq(τ→±∞)= 0 ,is a baby version
of the Euclidean field theory solutions called instantons. It is clear from symmetry
that the instanton actionSIis just twice the actionSb( 0 ).Hence, for the ground
state(E= 0 ),the dominant contribution to the tunneling probability, which is the
square of the amplitude (4.145), is

PI∝exp(−SI). (4.148)

Thermal fluctuations and sphaleronsNow we consider a particle in equilibrium
with a thermal bath of temperatureT. The particle can acquire energy from the
thermal bath and if this energy exceeds the height of the barrier the particle escapes
from the potential well classically, without needing to “go under the barrier.” The