Nucleation of true vacuum
59
2.9 Nucleation of true vacuum
In sections 2.4 and 2.6, we have found that first-order phase transitions may occur
as the universe cools. If the phase transition is first order, it will be necessary for
the universe to tunnel out of the metastable minimum [19-21] (false vacuum) to
reach the absolute minimum (true vacuum). If the tunnelling rate is small, this
may occur at temperatures very much lower than the temperature Tc1 of (2.76)
at which the energy of the zero- (and low-) temperature vacuum drops below
that of the high-temperature vacuum. In what follows, we shall approximate the
tunnelling rate by its T = 0 value and study, for simplicity, the case of a single
scalar field cp.
In the semi-classical limit (small h) the probability per unit time per unit
volume for formation of a bubble of true vacuum r is given by [19]
r = Ae- B / Ii
(2.167)
where
B = SE
(2.168)
with SE the Euclidean action for a solution of the Euclidean Euler-Lagrange
equations which satisfies the boundary conditions that cp approaches the false
vacuum (metastable minimum) as the Euclidean time tE ~ ±oo, and with zero
Euclidean time derivative at tE = 0, where
tE == it.
(2.169)
This is referred to as the 'bounce' solution (because it turns around and bounces
back to the false vacuum.) The tunnelling is dominated by the solution for cp that
gives the smallest value of SE. The derivation of this result is by studying the
imaginary part of the effective potential in the false vacuum. The coefficient A
is, in general, more difficult to calculate [20]. However, since it does not appear
in an exponent, an estimate on dimensional grounds is sufficient. At T = 0, we
may expect A to be of order M4 where M is an appropriate mass scale, such as
the height of the potential barrier to be tunnelled through or (see [21]) the value
of (d 2 V /dcp2) 1/2 at the metastable minimum, which will usually be of the same
order of magnitude.
For a single real scalar field. the Euclidean action takes the form
SE = f d 4 x (!aIlCPallcp + V(cp»
(2.170)
with the metric the positive-definite metric of four-dimensional Euclidean space,
and the Euclidean Euler-Lagrange equation is
a 2 cp
0/lollq, = -
+ V2q, = V'(q,).
(2.171)
at 2
E
59
2.9 Nucleation of true vacuum
In sections 2.4 and 2.6, we have found that first-order phase transitions may occur
as the universe cools. If the phase transition is first order, it will be necessary for
the universe to tunnel out of the metastable minimum [19-21] (false vacuum) to
reach the absolute minimum (true vacuum). If the tunnelling rate is small, this
may occur at temperatures very much lower than the temperature Tc1 of (2.76)
at which the energy of the zero- (and low-) temperature vacuum drops below
that of the high-temperature vacuum. In what follows, we shall approximate the
tunnelling rate by its T = 0 value and study, for simplicity, the case of a single
scalar field cp.
In the semi-classical limit (small h) the probability per unit time per unit
volume for formation of a bubble of true vacuum r is given by [19]
r = Ae- B / Ii
(2.167)
where
B = SE
(2.168)
with SE the Euclidean action for a solution of the Euclidean Euler-Lagrange
equations which satisfies the boundary conditions that cp approaches the false
vacuum (metastable minimum) as the Euclidean time tE ~ ±oo, and with zero
Euclidean time derivative at tE = 0, where
tE == it.
(2.169)
This is referred to as the 'bounce' solution (because it turns around and bounces
back to the false vacuum.) The tunnelling is dominated by the solution for cp that
gives the smallest value of SE. The derivation of this result is by studying the
imaginary part of the effective potential in the false vacuum. The coefficient A
is, in general, more difficult to calculate [20]. However, since it does not appear
in an exponent, an estimate on dimensional grounds is sufficient. At T = 0, we
may expect A to be of order M4 where M is an appropriate mass scale, such as
the height of the potential barrier to be tunnelled through or (see [21]) the value
of (d 2 V /dcp2) 1/2 at the metastable minimum, which will usually be of the same
order of magnitude.
For a single real scalar field. the Euclidean action takes the form
SE = f d 4 x (!aIlCPallcp + V(cp»
(2.170)
with the metric the positive-definite metric of four-dimensional Euclidean space,
and the Euclidean Euler-Lagrange equation is
a 2 cp
0/lollq, = -
+ V2q, = V'(q,).
(2.171)
at 2
E