Bubble Nucleation and Eternal Inflation - cosmo 06

Tunneling
There are many possible transitions
Exponents
to consider, corresponding
to the many qualitatively different spacetimes
shown Figs. 5 and 6. In each case, the tunneling probability
in the WKB approximation is given by
The integral outside the bubbl
F O
Z 1
r w
dr i L RR
1
LogP k 2
More
Probable
P R 1 ! R 2
R 2
R 1
2
’ e
2i 0 R 2 R 1 ; (33)
which evaluated between the tw
two turning points (for an examp
Z R2
F O R 2 R 1 dR
0where [R 2 R 1 ] represents evaluation between the two
R 1
turning points of the classical motion, and 0 is obtained
At the turning point, R
by integrating Eq. (32). The plan of attack is to split the
inside
2
by solving Eqs. (25) and (26) fo
integral into three parts: one over the interior of the bubble,
one over the exterior, and one in the neighborhood of the R 0 r w La 1=2
ds
wall. We thus write:
4
Therefore, the inverse
= R
cosine in
6
i 0 F I R 2 R 1 F O R 2 R 1 F w R 2 R 1 : and (38) are either 0 when R 0 is
= L
negative. To perform these integ
(34)
wall along the tunneling hypersu
8
is positive if the coordinate r
II
II’
II’’ direction normal to the wall and
0 0.005 0.01 0.015 0.02 0.025 Therefore, 0.03 the sign of R 0 is equ
I
III
M k III’
moves along the tunneling hype
Eq. (36) and (38) will be zero in
IV
IV’
IV’’
in regions of negative .
False Vacuum Bubbles 10 −50 < k < 10 −5
Shown in Table I are the valu
weak the possible transitions ~Planck where t
FIG. 10 (color online). Tunneling from a bound solution to an the left, on the conformal diagr
unbound solution which exists outside the cosmological horizon. (for example, the process show
r
BH
r C