Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 187

Forφx(τ)=φ(r ̃)(4.155) simplifies to the ordinary differential equation

d^2 φ

dr ̃^2

+

3

r ̃

dφ

dr ̃

−

∂V

∂φ

= 0. (4.156)

The solution of this equation, with boundary conditionsφ→0asr ̃→∞and
dφ/dr ̃=0atr ̃= 0 ,gives the desired instanton. The second boundary condition
is needed to avoid a singularity at the center of the bubble.
The vacuum decay rate per unit time and unit volume is

Aexp(−SI), (4.157)

whereAis a pre-exponential factor that is very difficult to calculate. We can obtain
a rough estimate forAusing dimensional arguments. In units wherec==1 and
G= 1 ,the decay rate has the dimension cm−^4 and hence

A∼O

(

R−I^4 ,V,φφ^2 ,···

)

, (4.158)

whereRIis the size of the instanton and one uses typical instanton values for the
derivatives ofV. Usually all these quantities have the same order of magnitude.

Problem 4.23Verify that the Euclidean action for the instanton can be reduced to

SI= 2 π^2

∫(

1

2

(

dφ

dr ̃

) 2

+V

)

r ̃^3 dr ̃. (4.159)

(HintThe Euclidean actionSEis related to the Lorentzian actionSL,given in
(4.152), bySL→iSEast→τ=it.)

Decay via sphaleronsThe above consideration is applicable only at low or zero
temperature. If the temperature is very high (we will specify later what this means),
the dominant contribution to the vacuum decay is given by over-barrier classical
transitions. To estimate their rate we have to find the extremum of the potentialV
with the minimal possible energy. This extremum is reached for thestaticscalar
field configuration (sphaleron), satisfying the equation

δV/δφ=−φ+V,φ= 0. (4.160)

For a spherical bubble this becomes

d^2 φ

dr^2

+

2

r

dφ

dr

−

∂V

∂φ

= 0 , (4.161)

whereφ=φ(r)andr=|x|is the radial coordinate inthree-dimensional space. The
boundary conditions we need to impose are similar to those for (4.156), namely,
φ→0asr→∞anddφ/dr=0atr= 0 .Note that the sphaleron, in contrast