Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 183

probability that the particle gets energyEis given by the usual Boltzmann factor
∝exp(−E/T).Taking into account that forE<V(qm)the tunneling amplitude is
given by (4.145), we obtain the following result for thetotalprobability of escape:

P∝

∑

E

exp

⎛

⎝−E

T

− 2 θ

b∫(E)

a(E)

√

2 M(V−E)dq

⎞

⎠, (4.149)

whereθ=1 forE<V(qm)andθ=0 otherwise. The sum in (4.149) can beesti-
matedusing the saddle point approximation. Taking the derivative of the expression
in the exponent, we find that this expression has its maximum value whenEsatisfies
the equation

1

T

= 2

b∫(E)

a(E)

√

M

2 (V−E)

dq= 2

b∫(E)

a(E)

dq

q ̇

. (4.150)

Hereq ̇=dq/dτis the “Euclidean velocity” along the trajectory with total Eu-
clidean energy−E.The term on the right hand side of (4.150) is equal to the period
of oscillation in the inverted potential(−V). Hence, for a given temperatureT,
the dominant contribution to the escape probability gives the periodic Euclidean
trajectory describing oscillations with period 1/Tin the potential(−V).
Let us consider two limiting cases. From (4.149) and ( 4.150), it follows that
forTV(qm)/SIthe main contribution to the escape probability comes from
the subbarrier trajectory withEV(qm)and as a result the rate of escape is
determined by the instanton. In the opposite case of very high temperatures,

T

V(qm)

SI

,

the “period of oscillation” tends to zero; hence the dominant trajectory comes
very close to the top of the potentialVand has energyE≈V(qm).The unstable
staticsolutionq=qm,which corresponds to the maximum ofV,is a prototype of
the field theory solutions called sphalerons (Greek name for “ready to fall”). For
the sphaleron, the second term in the exponent in (4.149) can be neglected and the
escape probability is given by

P∝exp

(

−

Esph

T

)

, (4.151)

whereEsph=V(qm)is the energy (or mass) of the sphaleron.We would like to
stress that at very high temperatures the main contribution to the escape probability