The Decay of Nearly Flat Space

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The Decay of Nearly Flat Space

• Matthew Lippert
• with Raphael Bousso
• and Ben Freivogel
• hep-th/0603105

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Motivation
Landscape Many Vacua Probability of each
vacuum ? hard
Eternal Inflation Semi-classical Large L dS
dominates?
Ergotic Evolution (Banks Johnson
hep-th/0512141) Lmin gt 0 true ground, all
others are fluctuations Probability Lifetime
Entropy G ? 0 for L ? 0 to stabilize Lmin dS
but, G ? 0 (discontinuous) at L 0
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What we did

• Investigate CdL equations
• Consider singular solutions
• General properties
• Map solution space
• ? continuous as ? ? 0
• If ? ? 0, ? 0 limit is stable

(See also Banks, Johnson, Aguirre
hep-th/0603107)
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CdL Tunneling Review
Scalar coupled to gravity
V(?)
?T
VF
Euclidean instanton ? exp(-SI SBG) SO(4)
symmetry metric
?F
VT
S3
Lorentzian dynamics expanding bubble of true
vacuum
VT gt 0 ? dS VT 0 ? open FRW VT lt 0 ? big crunch
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Equations of Motion
Particle in potential -V(?) with friction
Coupled to FRW
Boundary Conditions at ? 0 poles
? Continuous
? Smooth
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Solutions - Noncompact
-V
V
Vf 0
?
?
?0
R4 topology, one pole at t 0
May not reach ?f False vac. stable
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Solutions - Compact
-V
V
Vf gt 0
?
?
t 0
S4 EdS, two poles at t 0, tmax
t tmax
Always tunneling solution
Multiple passes - P 0

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Properties of Solutions
Solution - solve with (VF, ?0 )
singular or regular

• Generically compact with singularity at tmax
• ? ? 8 for singular solutions
• Across reg. compact soln DP 1
• gt 0 ? ? ? -8
• lt 0 ? ? ? 8, extra pass
• Across non-compact soln DP ?
• Between ?01 and ?02 with DP ? 0, reg. soln

VF
?0
DE
Singular solution
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Solution space No L0 tunneling
Reg. Solns
Passes
P 1 Instanton
Vmax 0 (HM flat)
?F
?T
HM
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Solution spaceL0 tunneling
P 1 Noncompact Instanton
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VF ? 0 Limit
Stable VF 0 False Vacuum No noncompact
solution (by assumption)
Reg. Compact VF gt 0
Reg. Compact VF 0
T
F
F
T
Flat
Big dS
SI finite
SI finite (SF 8 ) ? G 0
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VF ? 0 Limit
Unstable VF 0 False Vacuum Noncompact solution
exists (by assumption) Limit discontinuous -
hard to perturb
Noncompact VF 0
Large singular compact
Large Reg. Compact VF ?
??0
VF ? ?
interpolate
as ? ? 0 SI ? 8 ? gt 0
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Summary

• Smooth VF ? 0 limit
• ? ? 0 ? stable flat space
• Ergotic landscape doubtful
• Solution space - rich structure