60
Phase transitions in the early universe
It can be shown that the bounce which minimizes SE is 0(4) symmetric, i.e. tP is
a function of the four-dimensional radial variable p alone, where
p2;: t~ +x 2 •
(2.172)
Then
SE ;:z,,' f dpp' [H:Y +V(~)] (2.173)
and the equation of motion is (exercise 4)
d 2 tP + ~ dtP = V'(tP).
(2.174)
dp2
p dp
In terms of this variable, the boundary conditions for a bounce solution are
tP -+ tP+ as p -+ 00, and dtP dp = 0 when p = 0, where tP+ is the value of
tP at the metastable minimum.
An explicit bounce solution is most easily obtained in the so-called 'thinwall' approximation [19] which treats the energy difference E between the two
vacua as small compared with the height of the potential barrier between them.
Then, we write
V(tP) = Vo(tP) + O(E)
(2.175)
where Vo(tP) is the effective potential in the limit that we neglect the energy
difference between the two vacua, and
E = V(tP+> - V(tP-)
(2.176)
where tP+ and tP- are, respectively, the values of tP at the metastable and absolute
minima. The Euclidean equation of motion is approximated first by replacing
V'(tP) by V~(tP) in (2.174). We shall see later that, in this approximation, it is also
correct to neglect ~~. in which case the equation of motion to be solved for the
bounce solution becomes
d 2 tP
,
- 2 = Vo(tP).
(2.177)
dp
It is not difficult to show that the solution is
dtP
f
(2.178)
p = J2Vo(tP)'
A simple example is obtained by taking
). , (
2)2
Vo(tP) = 8 tP 2 - ~
(2.179)