Bubble Nucleation and Eternal Inflation - cosmo 06

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Bubble Nucleation and Eternal Inflation - cosmo 06

Bubble Nucleation and

Eternal Inflation

Matthew C Johnson

In collaboration with Anthony Aguirre

at

University of California, Santa Cruz


nton is cut along the surface denoted by the black disc, which is

onto the hypersurface indicated in the conformal diagram. The

this hypersurface specify the initial conditions for the lorentzian

≪ 1, and

CDL Vacuum BubblesColeman 1977

x(s = 0) = x

pact) instanton, 0 (7)

like the one found in the absence

much ẋ(s faster = s max decay, ) = 0 so that

(8)

Callan and Coleman 1977

e bubble wall, indicated by the line with an arrow. Depending

Coleman and De Luccia 1980

l phase, this process describes the nucleation of a true or false

e. The zeros of ρ continue into the forward light cones indicated

E ≃ u

in the decay T,F

rate as V F → 0.

V (φ)

te that while

(9)

(10)

[1] are cor-

Instead 5we

x H

artitioned by

relines.

Minkowski

ss where the

– as argued

here R is the

x F

x T

the false vac- Figure 1: The potential V (φ), with

. The stabilly

the true vacuum x T , the false vac-

and the “Hawking-Moss”

metastable

False

uum xVacuum:

F

point x H labeled.

higher energy

S I = −

state

ago by Cole-

-wall limit and subsequently discussed by several

e instanton formalism, give approximate analytic

or of the instanton solutions in the limit where

umerical techniques.

S

After elucidating the actual

argue in Sec. 5 that BG = −

the Great Divide consists

in the V F → 0 limit, have static domain walls

∫ ∫

false stationary points of the potential 3 ; we also

σ = i

ubtraction becomes infinite, requiring an infinite instanton

probability.

t, have discovered this fact independently [2].

of Cvetic et. al. on singular domain walls and their relation



dtp ˙q = i

True Vacuum:

lower energy state

φ (11)

d 4 x √ gV (φ(x)) (12)

d 4 x √ gV (φ(x turnpt )) (13)

−σ 1 /¯h

dt [L E − L turnpt ] (14)


nton is cut along the surface denoted by the black disc, which is

CDL Vacuum Bubbles

pact) instanton, like the one found in the absence

≪ 1, and

onto x(s = the 0) = x 0 hypersurface indicated (7) in the conformal diagram. The

much ẋ(s faster = s max decay, ) = 0 so that

(8)

this hypersurface specify the initial conditions for the lorentzian

e bubble wall, indicated by the line with an arrow. Depending

l phase, this process describes the nucleation of a true or false

e. The zeros of ρ continue into the forward light cones indicated

E ≃ u

in the decay T,F

rate as V F → 0.

V (φ)

te that while

(9)

(10)

[1] are cor-

Instead 5we

x H

artitioned by

relines.

Minkowski

ss where the

– as argued

here R is the

x F

x T

the false vac- Figure 1: The potential V (φ), with

. The stabilly

metastable

the true vacuum x T , the false vacuum

x F and the “Hawking-Moss”

point x H labeled. ∫

ago by Cole-

-wall limit and subsequently discussed by several

S I = −

e instanton formalism, give approximate analytic

or of the instanton solutions in the limit where

umerical techniques.

S

After elucidating the actual

argue in Sec. 5 that BG = −

the Great Divide consists

in the V F → 0 limit, have static domain walls

∫ ∫

false stationary points of the potential 3 ; we also

σ = i

ubtraction becomes infinite, requiring an infinite instanton

probability.

t, have discovered this fact independently [2].

of Cvetic et. al. on singular domain walls and their relation


dtp ˙q = i

φ (11)

d 4 x √ gV (φ(x)) (12)

d 4 x √ gV (φ(x turnpt )) (13)

−σ 1 /¯h

dt [L E − L turnpt ] (14)


nton is cut along the surface denoted by the black disc, which is

CDL Vacuum Bubbles

pact) instanton, like the one found in the absence

≪ 1, and

onto x(s = the 0) = x 0 hypersurface indicated (7) in the conformal diagram. The

much ẋ(s faster = s max decay, ) = 0 so that

(8)

this hypersurface specify the initial conditions for the lorentzian

e bubble wall, indicated by the line with an arrow. Depending

l phase, this process describes the nucleation of a true or false

e. The zeros of ρ continue into the forward light cones indicated

E ≃ u

in the decay T,F

rate as V F → 0.

V (φ)

te that while

(9)

(10)

[1] are cor-

Instead 5we

x H

artitioned by

relines.

Minkowski

ss where the

– as argued

here R is the

x F

x T

the false vac- Figure 1: The potential V (φ), with

. The stabilly

metastable

the true vacuum x T , the false vacuum

x F and the “Hawking-Moss”

point x H labeled. ∫

ago by Cole-

-wall limit and subsequently discussed by several

S I = −

e instanton formalism, give approximate analytic

or of the instanton solutions in the limit where

umerical techniques.

S

After elucidating the actual

argue in Sec. 5 that BG = −

the Great Divide consists

in the V F → 0 limit, have static domain walls

∫ ∫

false stationary points of the potential 3 ; we also

σ = i

ubtraction becomes infinite, requiring an infinite instanton

probability.

t, have discovered this fact independently [2].

of Cvetic et. al. on singular domain walls and their relation


dtp ˙q = i

φ (11)

Tunneling = Bubble Nucleation

d 4 x √ gV (φ(x)) (12)

d 4 x √ gV (φ(x turnpt )) (13)

−σ 1 /¯h

dt [L E − L turnpt ] (14)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , the false vacuum

x F and the “Hawking-Moss”

etastable

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

stanton formalism, give approximate analytic

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

stanton formalism, give approximate analytic

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

FGG Mechanism

stanton formalism, give approximate analytic

Farhi, Guth, Guven 1990

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

FGG Mechanism

stanton formalism, give approximate analytic

Farhi, Guth, Guven 1990

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

Thermal Activation

Garriga and Megevand 2004

Gomberoff et. al. 2004

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

FGG Mechanism

stanton formalism, give approximate analytic

Farhi, Guth, Guven 1990

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

Thermal Activation

Garriga and Megevand 2004

Gomberoff et. al. 2004

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

Creation of a

universe from nothing

Vilenkin 1982

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

FGG Mechanism

stanton formalism, give approximate analytic

Farhi, Guth, Guven 1990

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 Hawking-Moss

limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

Hawking and Moss 1982

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


Instanton

dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

Thermal Activation

Garriga and Megevand 2004

Gomberoff et. al. 2004

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

Creation of a

universe from nothing

Vilenkin 1982

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

FGG Mechanism

stanton formalism, give approximate analytic

Farhi, Guth, Guven 1990

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 Hawking-Moss

limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

Hawking and Moss 1982

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


Instanton

dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

Thermal Activation

Garriga and Megevand 2004

Gomberoff et. al. 2004

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

Stochastic

Fluctuations

Starobinski

Linde

Creation of a

universe from nothing

Vilenkin 1982

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

FGG Mechanism

stanton formalism, give approximate analytic

Farhi, Guth, Guven 1990

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 Hawking-Moss

limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

Hawking and Moss 1982

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


Instanton

dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

Thermal Activation

Garriga and Megevand 2004

Gomberoff et. al. 2004

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

Stochastic

Fluctuations

Starobinski

Linde

Creation of a

universe from nothing

Vilenkin 1982

Others

(15)


to the hypersurface indicated in the conformal diagram. The

x(s = 0) = x 0 (7)

t) instanton, like the

False

one found in the absence

Vacuum Bubbles

(s h faster = s max decay, ) = 0 so that

(8)

his hypersurface specify the initial conditions for the lorentzian

E ≃ u

the decay T,F (9)

bubble

rate as

wall,

V F → 0.

indicated by the line with an arrow. Depending

V (φ) (10)

hat while

phase, this process describes the nucleation of a true or false

] are corstead

The 5we

zeros of xρ H

continue into the forward light cones indicated

itioned by

Lee and Weinberg 1987

nes.

Can we tunnel up

inkowski

where the x F

x T

L(ee)W(einberg) Bubbles

as argued

R is the

false vache

stabil-

Figure 1: The potential V (φ), φ with

(11)

the true vacuum x T , false vacuum

x F and the “Hawking-Moss”

etastable

But, not the only way to make False Vacuum...

point x H labeled. ∫

by Colel

limit and subsequently discussed by several

S I = −

FGG Mechanism

stanton formalism, give approximate analytic

Farhi, Guth, Guven 1990

f the instanton solutions in the limit where

rical techniques.

S

After elucidating the actual

ue in Sec. 5 that BG = −

the Great Divide consists

the V F → 0 limit, have static domain walls

∫ ∫

e stationary points of the potential 3 ; we also

σ = i

action becomes infinite, requiring an infinite instanton

ability.

ve discovered this fact independently [2].

vetic et. al. on singular domain walls and their relation


dtp ˙q = i

Ψ 1

= e−σ 1/¯h

d 4 x √ gV (φ(x)) (12)

Thermal Activation

Garriga and Megevand 2004

Gomberoff et. al. 2004

d 4 x √ gV (φ(x turnpt )) (13)

dt [L E − L turnpt ] (14)

= e −S E/¯h

Creation of a

universe from nothing

Vilenkin 1982

Four can be unified in the Thin-wall limit!

There are implications for Eternal Inflation

(15)


Vacuum Bubbles

3 Ingredients for Classical Dynamics

True or False

Vacuum exterior

Bubble Wall

Tension

k

True or False

Vacuum interior


Vacuum Bubbles

3 Ingredients for Classical Dynamics

True or False

Vacuum exterior

Bubble Wall

Tension

k

True or False

Vacuum interior

Condition Formalism

space times across an infinitesimally thin spherical shell of mass.

ic continuity across boundary.

n’s equations.

-1990) - Is it possible to create a universe in the lab

CDL/LW: wall and volume energy cancels, so M=0.

n make universes via fluctuations (DKS and AS).

time will be de Sitter with a cosmological constant .

e time will be Schwarzschild de Sitter (spherically symmetric

itter space) with a cosmological constant .

!

dS

+

SdS

Can also consider massive bubbles:

Birkhoff’s Theorem: exterior Schwarzschild-de Sitter.

orals – p.4/41


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% &

!'

!!(

!'(

!! %)*

!!

%

Continues

forever

% &


Schwarzschild-de Sitter

Continues

forever

% &

!"

Black Hole

event horizon

!!((

% &

% #$

!!

#$

Cosmological

event horizon

!!!(( !!!

!

!!!(

!'((

%)*

% &

!"

% &

% &

% #$

% #$

% &

% &

!'

!!(

!'(

!! %)*

!!

%

Continues

forever

% &


Require metric continuity across boundary.

Israel Junction Conditions

Solve Einstein’s equations.

Guth et. al. (1987-1990) - Is it possible to create a universe in the lab

de Sitter space can make universes via fluctuations (DKS and AS).

Interior space time will be de Sitter with a cosmological constant .

Exterior space time will be Schwarzschild de Sitter (spherically symmetric

mass in de Sitter space) with a cosmological constant .

Israel 1966, 1967

Blau et. al 1987

Aurilia et. al. 1989

Berezin et. al 1983, 1987

Sato 1986

!

dS

+

SdS

Wall Dynamics from

1-D potential:

M

orals – p.4/41

Monotonic

Bound

Unbound

Complete catalogue:

Aguirre and Johnson gr-qc/0508093 and gr-qc/0512034

R


Allowed Spacetimes

II

II

II''

r BH

II'

r=0

III

IV

I

III''

III

Bound Solution

IV''

r

BH

IV

I

IV'

III'

II

II''

II

II'

r=0

III

I

III''

III

r

r BH

BH

I

III'

IV

IV''

IV

IV'

r=0

de Sitter

II

interior

r

C

III

I

III''

II''

III

II

I

r C

II'

III'

IV

IV''

IV

r

BH

IV'

r C

II

II''

II

r

C

II'

r=0

III

I

III''

III

I

III'

r C

IV

IV''

IV

r C

IV'


Allowed Spacetimes

II

II

II''

r BH

II'

r=0

III

IV

I

III''

III

Bound Solution

IV''

r

BH

IV

I

IV'

III'

II

II''

II

II'

r=0

III

I

III''

III

r

r BH

BH

I

III'

IV

IV''

IV

IV'

r=0

r

C

III

II

I

III''

Schwarzschild-de II''

II

Sitter II'

exterior

III

I

r C

III'

IV

IV''

IV

r

BH

IV'

r C

II

II''

II

r

C

II'

r=0

III

I

III''

III

I

III'

r C

IV

IV''

IV

r C

IV'


Allowed Spacetimes

II

II

II''

r BH

II'

r=0

III

IV

I

III''

III

Bound Solution

IV''

r

BH

IV

I

IV'

III'

II

II''

II

II'

r=0

III

I

III''

III

r

r BH

BH

I

III'

IV

IV''

IV

IV'

r=0

r

C

III

II

IV

II''

Bubble wall:

I III''

III

identify these points

IV''

II

IV

r

BH

I

r C

II'

IV'

III'

r C

II

II''

II

r

C

II'

r=0

III

I

III''

III

I

III'

r C

IV

IV''

IV

r C

IV'


Allowed Spacetimes

II

II

II''

r BH

II'

r=0

III

IV

I

III''

III

Bound Solution

IV''

r

BH

IV

I

IV'

III'

II

II''

II

II'

r=0

III

I

III''

III

r

r BH

BH

I

III'

IV

IV''

IV

IV'

r=0

r

C

III

II''

II

II

I III''

III

I

Only shaded regions physical!

IV

IV''

IV

r

BH

r C

II'

IV'

III'

r C

II

II''

II

r

C

II'

r=0

III

I

III''

III

I

III'

r C

IV

IV''

IV

r C

IV'


Allowed Spacetimes

II

II

II''

r BH

II'

r=0

III

IV

I

III''

III

Bound Solution

IV''

r

BH

IV

I

III'

classically buildable

IV'

Farhi and Guth 1987

II

II''

II

II'

7

r

r=0

III

I

III''

III

BH

I

III'

r

C

II

IV

II’’

IV''

r C

II

IV

r BH

II’

IV'

Cell 4

r=0

r

III

r

C

C

IV II

I

III’’

III

II''

II

Unbound IV’’ Solution IV

r BH

I

III’

Not II'

classically IV’ buildable!

r C

r=0

III

I

III''

III

I

III'

Cell 5

r=0

r

C

r

III

C

r C

II

IV

IV II

I

III’’

II’’

IV''

II''

IV’’

r C

r C

III

II

IV

II

IV

r

BH

I

r

C

II’

IV'

II'

IV’

III’

r=0

III

I

III''

III

I

III'

r C

r C

II

IV

II’’

IV''

r C

II

IV

r C

II’

IV'


orals – p.14/41

Got Tunneling

Got Tunneling

Can This Happen

Can This Happen

Farhi, Guth, Guven 1990

Fischler et. al. 1990

The FGG Mechanism

r=0

r=0

r=1/H

III

r=1/H

III

r=1/H

r=1/H

II

IV

J+

II

IV

J−

J+

I

I

III’’

III’’

II’’

II’’

IV’’

IV’’

r C

r C

III

III

II

II

IV

IV

r BH

r BH

I

I

II’

IV’

II’

IV’

III’

III’

J−

orals – p.14/41


ell 4

r=0

r

C

r

Solution 12

An Observation

II

II

III I III’’

III

I

III’

For IVevery unbound solution behind III I

III

IV’ a worm hole,

C

II

II’’

IV’’

r C

r BH

II

IV

r=0

r

C

II’

r=0

r r

C

Solution 13

I

ell 5

r=0

r

C

r

III

C

II

IV

I

III’’

II’’

IV’’

r C

r C

III

II

IV

I

IV

IV

II’

III’

Solution 14 Solution 15

IV’

ell 6

r=0

r

C

r

C

IV

IV’’

r C

r C

FIG. 6: II’ Conformal diagrams for the

II

II’’

II

there is an identical solution outside the cosmological

III I III’’

III

I

III’

horizon (SdS non-compact).

IV

IV’

r

C

II

r C

II’’

II

II’

ell 7

r=0

III

I

III’’

III

I

III’

III’

r

C

IV

r C

IV’’

IV

IV’

J+

FIG. 7: Solutions can be to the right of region I instead of


=0

C

r

III’’

III I

C

III’’

IV

II’’

IV’’

r C

III''

II’’

r C

Two

III

Tunnels

I

III'

II

IV''

IV

IV'

r C

BH

r BH

rBH

BH

[1] A. Aguirre and M. C. Johnson, Phys. Rev. D72, 103525 (2005), gr-

[2] A. Aguirre and M. C. Johnson (2005), gr-qc/0512034.

FIG. 10: Conformal diagrams for the thermalon spacetime

[3] E. Farhi, A. H. Guth, and J. Guven, Nucl. Phys. I B339, 417 (1990)

[4] S. R. ColemanIII

and F. De Luccia, Phys. Rev. D21, 3305 (1980).

[5] K.-M. Lee and E. J. Weinberg, Phys. Rev. D36, 1088 (1987).

II

II’

[6]

IV’’

D. J. Gross, M. J. Perry, andIV

L. G. Yaffe, Phys. Rev. D25,

IV’

330 (19

[7] P. H. Ginsparg and M. J. Perry, Nucl. Phys. B222, 245 (1983).

I

III’

[8] R. III Bousso and A. Chamblin, Phys. Rev. D59, 084004 (1999), gr-qc

[9] S. W. Hawking and I. G. Moss, Phys. Lett. B110, 35 (1982).

FIG. IV 11: Tunneling through IV’ the worm hole.

[10] T. Vachaspati and M. Trodden, Phys. Rev. D61, 023502 (2000), gr

[11] S. R. Coleman, Phys. Rev. D15, 2929 (1977).

[12] U. Gen and M. Sasaki, Phys. Rev. D61, 103508 (2000), gr-qc/9912

[13] D. Lindley, Nucl. Phys. B236, 522 (1984).

FIG. 11: Tunneling through the worm hole.

R Tunneling Geometry: goes through dS horizon.

[14] R. II Basu, A. II’ H. Guth, and II’’ A. Vilenkin, Phys. Rev. D44, 340 (1991)

[15] I

III A. Aurilia, M. Palmer, III’ and E. Spallucci, Phys. Rev. D40, 2511 (198

III

r

BH

r C

L Tunneling Geometry: goes through wormhole.

II

IV

r

r C

BH

[18] T. FIG. R. 12: Choudhury Second option for tunneling. and T. Padmanabhan (2004), gr-qc/0404091.

[20] B. Freivogel 9 et al. (2005), hep-th/0510046.

I

II’

[16] S. K. Blau, E. I. Guendelman, and A. H. Guth, Phys. Rev. D35, 17

IV

IV’

IV’’

[17] T. Tanaka and M. Sasaki, Prog. Theor. Phys. 88, 503 (1992).

IV’

[19] F.-L. Lin and C. Soo, Class. Quant. Grav. 16, 551 (1999), gr-qc/97

r BH

r C

III’

FIG. 12: Second option for tunneling.

II’’

IV’’

II’


=0

r

Zero Mass Limit:

III

IV

L Geometry

r C

III

C

II

IV

I

III''

II''

IV''

r C

III

r BH BH BH

r C

FIG. 10: Conformal diagrams for the thermalon spacetime

rBH

BH BH BH BH BH

II

IV

I

II'

IV'

III'

II’’

II

II’

r C

III’’

III

I

III’

r BH BH BH

IV’’

IV

IV’

FIG. 11: Tunneling through the worm hole.

II

II’

II’’

III

r

BH BH BH

I

r C

III’

IV

IV’

IV’’


=0

Zero Mass Limit:

III

IV

L Geometry

r

r C

III

C

II

IV

I

III''

II''

IV''

r C

III

r BH BH

r C

FIG. 10: Conformal diagrams for the thermalon spacetime

rBH

BH BH BH

II

IV

I

II'

IV'

III'

II’’

II

II’

r C

III’’

III

I

III’

r BH BH

IV’’

IV

IV’

FIG. 11: Tunneling through the worm hole.

II

II’

II’’

III

r

BH BH

I

r C

III’

IV

IV’

IV’’


=0

Zero Mass Limit:

III

IV

L Geometry

r

r C

III

C

II

IV

I

III''

II''

IV''

r C

III

r BH BH

r C

FIG. 10: Conformal diagrams for the thermalon spacetime

rBH

BH BH BH

II

IV

I

II'

IV'

III'

II’’

II

II’

r C

III’’

III

I

III’

r BH BH

IV’’

IV

IV’

Universe

“from nothing”

containing a LW

II

bubble

III

FIG. 11: Tunneling through the worm hole.

r

BH BH

I

II’

r C

III’

Undisturbed

universe

II’’

IV

IV’

IV’’


Zero Mass Limit:

FIG. 11: Tunneling through the worm hole.

R Geometry

II

II’

II’’

III

r

BH BH BH

I

r C

III’

IV

IV’

IV’’

FIG. 12: Second option for tunneling.


Zero Mass Limit:

FIG. 11: Tunneling through the worm hole.

R Geometry

II

II’

II’’

III

r

BH

I

r C

III’

IV

IV’

IV’’

FIG. 12: Second option for tunneling.


Zero Mass Limit:

FIG. 11: Tunneling through the worm hole.

R Geometry

II

II’

II’’

III

r

BH

I

r C

III’

IV

IV’

IV’’

LW bubble

FIG. 12: Second option for tunneling.


dS to Bubble Spacetime

IV

IV’’

FIG. 11: Tunneling through the worm hole.

IV’

Three step process:

III

II

r

BH

I

II’

r C

III’

II’’

IV

IV’

IV’’

FIG. 12: Second option for tunneling.

9


dS to Bubble Spacetime

IV

IV’’

FIG. 11: Tunneling through the worm hole.

IV’

Three step process:

III

II

r

BH

I

II’

r C

III’

II’’

IV

IV’

IV’’

FIG. 12: Second option for tunneling.

• A bound solution is formed in its 9 expanding

phase.


dS to Bubble Spacetime

IV

IV’’

FIG. 11: Tunneling through the worm hole.

IV’

Three step process:

III

II

r

BH

I

II’

r C

III’

II’’

IV

IV’

IV’’

FIG. 12: Second option for tunneling.

• A bound solution is formed in its 9 expanding

phase.

• The bound solution evolves to the classical

turning point.

Bound solutions are unstable.

(Aguirre & Johnson, Phys. Rev. D72, 103525)


dS to Bubble Spacetime

IV

IV’’

FIG. 11: Tunneling through the worm hole.

IV’

Three step process:

III

II

r

BH

I

II’

r C

III’

II’’

IV

IV’

IV’’

FIG. 12: Second option for tunneling.

• A bound solution is formed in its 9 expanding

phase.

• The bound solution evolves to the classical

turning point.

Bound solutions are unstable.

(Aguirre & Johnson, Phys. Rev. D72, 103525)

• The bound solution tunnels through the

effective potential to an unbound solution.


dS

Ψ ()

to Bubble Spacetime

(18)

ˆπ


N tΨ = ˆπ N rΨ = 0,

(19b)

Tunneling rate calculated using Dirac quantization

Ĥ t Ψ =

in ĤrΨ the WKB

= 0, horizon.

approximation, assuming

(19a)

a spherically

symmetric mini-superspace. Fischler et. al. 1990

ˆπ N tΨ = ˆπ N rΨ = 0, R 1 (19b) (20)

Can This Happen

Fig. 7. This process, which can occur (19a) only in the p

of a a positive exterior cosmological constant, is d

in Fig. 10. For every transition which goes thro

wormhole, as in the FGG mechanism, there is

transition which instead goes out the cosm

Ĥ t Ψ = ĤrΨ = 0,

There are many possible transitions to conside

sponding to the many qualitatively different spa

shown in Figs. 5 and 6. In each case, the tunneling

bility in the WKB approximation is given by

R 2

P R 1 ! R 2

2 ’ e

2i 0 R 2 R 1 ;

R 1

R 1 R 2 (20) (21)

r=0

r=1/H

III

r=1/H

II

IV

J−

J+

II’’

II

II’

∞∑ I III’’ R 2

I

III’

III integral into three(21)

parts: one over the interior of the

a n (q)

IV’’

IV one over IV’ the exterior, and one in the neighborhoo

n=0

r C

( ) n dσ

= H(q, dσ

dq

dq ) = E (22)

r BH

where [R 2 R 1 ] represents evaluation between

turning points of theCalculate.....

classical motion, and 0 is o

by integrating Eq. (32). The plan of attack is to

wall. We thus write:

orals – p.14/41


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

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


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


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

LW exponent!

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


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 hand!

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 Bubbles 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 cosmological horizon. (for example, the process show

r

BH

r C


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

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 Maximum Mass

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

= 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 Bubbles 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 cosmological horizon. (for example, the process show

r

BH

r C


Maximum Mass

< M max <

Planck Scale

FV

Weak Scale

FV


would be a

ple, if the

fact forbidwall

false-

tunnel to one of the unbound solutions. In the absence

probability to go from empty dS to the spacetime

of a detailed theory of the nature of these fluctuations,

ing an expanding we assume that vacuum the probability bubble of fluctuating is givena by solution the

Nucleation Probability

of a given mass is given by the exponential of the entropy

change due to the change in

P ’ CP seed e 2i the area of the

0

Ce S exterior de

E

Sitter horizon in the presence of a mass [2] :

[ ( )] 3

P seed = exp −π − RC

2 , (44)

Λ +

11

where R C is the radius of curvature of the cosmological

horizon in SdS.

Once the bound solution has been fluctuated, it must

survive until it reaches the turning point of the classical

motion. The authors have shown [1] that any solution

with a turning point is unstable against non-spherical

perturbations. Even quantum fluctations present on the

bubble wall at the time of nucleation will go nonlinear

over some range of initial size and mass. To avoid this

instability, the seed bubbles must form very near the

turning point and be almost spherically symmetric. It is

unclear how asphericities will affect the tunneling mechanism

discussed in the previous section, but this may be

a significant correction to these processes.

Assuming that the seed bubble is still reasonable spherically

symmetric when it reaches the turning point, the

probability to go from empty de Sitter to the spacetime

containing an expanding vacuum bubble is given by the


0

2

4

would be a

ple, if the

fact forbidwall

false-

tunnel to one of the unbound solutions. In the absence

probability to go from empty dS to the spacetime

of a detailed theory of the nature of these fluctuations,

ing an expanding we assume that vacuum the probability bubble of fluctuating is givena by solution the

Nucleation Probability

11

of a given mass is given by the exponential of the entropy

change due to the change in

P ’ CP seed e 2i the area of the

0

Ce S exterior de

E

Sitter horizon in the presence of a mass [2] :

[ ( )] 3

P seed = exp −π − RC

2 , (44)

Λ +

where R C is the radius of curvature of the cosmological

horizon in SdS.

Once the bound 0 solution has been fluctuated, it must

survive until it reaches the turning point of the classical

0.1

motion. The authors have shown [1] that any solution

with a turning point is unstable against non-spherical

0.2

perturbations. Even quantum fluctations present on the

bubble wall at

0.3the time of nucleation will go nonlinear

over some range of True initial size Vacuum and mass. Bubbles To avoid this

instability, the 0.4seed bubbles must form very near the

turning point and be almost spherically symmetric. It is

unclear how asphericities will affect the tunneling mechanism

discussed in the previous section, M k but this may be

a significant correction = R to these processes.

Assuming that the seed bubble is still reasonable spherically

symmetric = when L it reaches the turning point, the

probability to go from empty de Sitter to the spacetime

containing an expanding vacuum bubble is given by the

LogP k 2 6 False Vacuum Bubbles

8

0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.0005 0.001 0.0015 0.002

M k


Nucleation Probability

Zero mass limit always most probable!

0

0

2

0.1

4

0.2

6

LogP k 2 0 0.0005 0.001 0.0015 0.002

False Vacuum Bubbles

0.3

True Vacuum Bubbles

8

0.4

0 0.005 0.01 0.015 0.02 0.025 0.03

M k

= R

= L

M k


Nucleation Probability

0

Creation from nothing

dominates

0

2

0.1

4

0.2

6

LogP k 2 0 0.0005 0.001 0.0015 0.002

False Vacuum Bubbles

0.3

True Vacuum Bubbles

8

0.4

0 0.005 0.01 0.015 0.02 0.025 0.03

M k

= R

= L

M k


Nucleation Probability

CDL Bubbles dominate

0

0

2

0.1

4

0.2

6

LogP k 2 0 0.0005 0.001 0.0015 0.002

False Vacuum Bubbles

0.3

True Vacuum Bubbles

8

0.4

0 0.005 0.01 0.015 0.02 0.025 0.03

M k

= R

= L

M k


Eternal Inflation

Two types of eternal inflation:

1) False Vacuum eternal inflation: bubble nucleation, no percolation.

2) Slow-roll eternal inflation: driven by stochastic fluctuations.

Deciding which transition mechanisms are

physical is relevant for both!

L and R L or R

L nor R


Measures for Eternal Inflation

We need a probability measure for eternal inflation!

Many options:

• Physical volume weighting (gauge dependent).

• Count transitions along worldlines.


• Comoving volume weighting.

• More.......

Other gauge invariant cutoff methods (CHC).


Measures for Eternal Inflation

We need a probability measure for eternal inflation!

Many options:

• Comoving volume weighting.


Comoving Volume Weighting

• Weight depends on initial conditions.

(if terminal vacua)


Comoving Volume Weighting

• Weight depends on initial conditions.

(if terminal vacua)

Initially, the entire comoving volume is inflating.


Comoving Volume Weighting

• Weight depends on initial conditions.

• Evolve using R geometry alone.

(if terminal vacua)

• Use FP formalism to find the distribution as t → ∞

Initially, the entire comoving volume is inflating.


Comoving Volume Weighting

t

• Weight depends on initial conditions.

• Evolve using R geometry alone.

• Use FP formalism to find the distribution as t → ∞

Distribution as t → ∞

(if terminal vacua)

Initially, the entire comoving volume is inflating.


L+R Geometries


L+R Geometries

Original comoving volume.


L+R Geometries

Behind the

wormhole.

Original comoving volume.


L+R Geometries

Behind the

wormhole.

Initial conditions fixed by L geometry.

Original comoving volume.


L+R Geometries

Behind the

wormhole.

Original comoving volume.


L+R Geometries

Behind the

wormhole.

Original comoving volume.


Adjusted Measure

• Initial comoving volume is diluted.


Distribution at future boundary of all pockets

determined by initial conditions of L geometry.

• Average over all regions, and:


Comoving volume distribution becomes

independent of initial conditions!


Slow-Roll Eternal Inflation

Driven by large V or small V’:

Condition for eternal inflation:

V

m 2 pV ′2/3 > 1

Jump in a Hubble volume over a Hubble time:

(Coarse-graining)

δφ = H 2π


Slow-Roll Eternal Inflation

Concentrate on large V:

V f

V i


Slow-Roll Eternal Inflation

Concentrate on large V:

V f

V i

δφ = H 2π

But, what

∆V

does this correspond to


Slow-Roll Eternal Inflation

Concentrate on large V:

V f

V i

δφ = H 2π

But, what

∆V

does this correspond to

V = 1 2 m2 φ 2

−→ ∆V

V i

∼ m2 p

m 2 ≫ 1

Typically, fluctuations in energy density are large!


Slow-Roll Eternal Inflation

Model overdense regions as thin-wall bubbles.

V

R

If

∆V

V i

> 2 − 3


Vf

m 2 p

V f

V i

Then surviving fluctuations are:

horizon-size (current formalism OK) and/or

behind a wormhole (current formalism not OK).


Conclusions

• There are a variety of transition mechanisms!

• This has implications for eternal inflation.

• What is allowed

• We may need more than semi-classical methods to

find the answer.

• Detailed balance and quantum gravity.

• ADS/CFT and holography.

• Initial conditions for inflation.

• High mass limit......

• Interpolating geometry.....

Frievogel et. al. 2005

Bousso 2005

Banks 2002

Banks and Johnson 2005

Albrecht and Sorbo 2004

Dyson et. al. 2002

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