The Decay of Nearly Flat Space

1. The Decay of Nearly Flat Space Matthew Lippert with Raphael Bousso and Ben Freivogel hep-th/0603105

2. 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 > 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

3. 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)

4. 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 > 0  dS VT = 0  open FRW VT < 0  big crunch

5. Equations of Motion Particle in potential -V() with friction ~ Coupled to FRW Boundary Conditions at  = 0 poles  Continuous  Smooth

6.   f as t  ∞ EAdS (VF < 0) or Flat (VF = 0) F T SBG ∞ Need SI ∞ for G > 0 Solutions - Noncompact -V V Vf ≤ 0 :   0 R4 topology, one pole at t = 0 May not reach f False vac. stable

7. F T T T crosses 0 at equator E (anti-friction) F P=2 Solutions - Compact -V V Vf > 0 :   t = 0 S4 ~EdS, two poles at t = 0, tmax t = tmax Always tunneling solution Multiple passes - P ≥ 0

8. -V  T F Properties of “Solutions” “Solution” - solve with (VF, 0 ) singular or regular • Generically compact with singularity at tmax •   ±∞ for singular “solutions” • Across reg. compact “sol’n” DP = 1 • > 0    -∞ • < 0    ∞, extra pass • Across non-compact soln DP = ? • Between 01 and 02 with DP ≠ 0, reg. sol’n VF 0 DE Singular “solution”

9. Solution space No L=0 tunneling Reg. Sol’ns # = Passes P =1 Instanton Vmax = 0 (HM flat) F T HM

10. Solution spaceL=0 tunneling P =1 Noncompact Instanton

11. VF 0 Limit Stable VF = 0 False Vacuum No noncompact solution (by assumption) Reg. Compact VF > 0 Reg. Compact VF = 0 F T F T Flat Big dS SI finite SI finite (SF = ∞ ) G = 0

12. T F F T T F 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 ∞  > 0

13. Summary • Smooth VF 0 limit •  0  stable flat space • Ergotic landscape doubtful • “Solution” space - rich structure