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

Motivation Landscape Many Vacua Probability of each vacuum  hard Eternal Inflation Semi-classical Large  dS dominates? Ergotic Evolution (Banks & Johnson hep-th/ )  min > 0 “true” ground, all others are fluctuations Probability ~ Lifetime ~ Entropy   0 for   0 to stabilize  min dS but,  ≠ 0 (discontinuous) at  = 0

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/ )

CdL Tunneling Review Scalar coupled to gravity Euclidean instanton  ~ exp(-S I + S BG ) SO(4) symmetry metric: TT FF VFVF Lorentzian dynamics expanding bubble of true vacuum V T > 0  dS V T = 0  open FRW V T < 0  big crunch V(  ) VTVT S3S3

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

Solutions - Noncompact R 4 topology, one pole at t = 0    f as t  ∞ EAdS (V F < 0) or Flat (V F = 0) May not reach  f False vac. stable S BG  ∞ Need S I  ∞ for  > 0 V f ≤ 0 : V  -V 00 F T

Solutions - Compact S 4 ~EdS, two poles at t = 0, t max crosses 0 at equator E (anti-friction) Always tunneling solution V f > 0 : V  -V Multiple passes - P ≥ 0 F T P=2 F T T t = 0 t = t max

Properties of “Solutions” “Solution” - solve with (V F,  0 ) singular or regular Generically compact with singularity at t max   ±∞ for singular “solutions” Across reg. compact “sol’n”  P = 1 > 0    -∞ < 0    ∞, extra pass Across non-compact soln  P = ? Between  0 1 and  0 2 with  P ≠ 0, reg. sol’n  -V E E 00 VFVF Singular “solution” F T

Solution space No  =0 tunneling FF TT HM Reg. Sol’ns # = Passes P =1 Instanton V max = 0 (HM flat)

Solution space  =0 tunneling P =1 Noncompact Instanton

V F  0 Limit Stable V F = 0 False Vacuum No noncompact solution (by assumption) Flat Big dS T T F F Reg. Compact V F > 0Reg. Compact V F = 0 S I finite (S F = ∞ )   = 0

V F  0 Limit Unstable V F = 0 False Vacuum Noncompact solution exists (by assumption) Limit discontinuous - hard to perturb Noncompact V F = 0 Large singular compact F T TF F T Large Reg. Compact V F =   0 V F   interpolate as   0: S I  ∞  > 0

Summary Smooth V F  0 limit   0  stable flat space Ergotic landscape doubtful “Solution” space - rich structure