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

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

the possible transitions where t

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

integral into Must three parts: fix onesign over theof interior exponent

by solving Eqs. (25) **and** (26) fo

of the bubble,

one over the exterior, **and** one in the neighborhood of the R

by h**and**!

r w La 1=2

ds

4

wall. We thus write:

Therefore, the inverse

= R

cosine in

6

i

e 2iΣ 0 F I R 2 R 1 F O R0[R 2 R 21 −RF w 1 R]

2 R 1 : **and** (38) are either 0 when R

= L

is

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 **Bubble**s Shown in Table I are the valu

FIG. 10 (color online). Tunneling from a bound solution to an the left, on the conformal diagr

unbound solution which exists outside the **cosmo**logical horizon. (for example, the process show

r

BH

r C