Quantum Tunneling of Thin Wall Matthew C. Johnson, in collaboration with Anthony Aguirre

Outline Inflation and de Sitter space. Thin wall bubbles. Tunneling: the old and the new. Semi-classical calculation. Conclusions.

Inflation: The good and the not so good B The good: From generic intial conditions, you get: Flatness Homogeneity, isotropy, observed density fluctuations Magnetic monopoles Solution to the Horizon problem But, what is meant by “generic inital conditions?” Inital conditions are erased!

Initial Conditions String theory landscape: de Sitter (our universe) is a metastable state. Fluctuating from some background: how do you go from high entropy universe we observe today to the low entropy state of inflation? The Multiverse: Hopefully our universe is the most likely possibility!

De Sitter Space: Universe dominated by a substance with Scale factor is exponential! Potential dominated scalar field. Fundamental cosmological constant. ? What is it?: Our Past and Future

De Sitter Space Future null infinity Past null infinity

De Sitter Space Can never go back once outside the horizon!

Pasting Two Spacetimes Together This Talk: spherically symmetric spacetimes with two different cosmological constants. Thin Wall: Integrate and assign surface energy density k.

Thin Wall CDL Bubbles False Vacuum True Vacuum Claim: This solution has the lowest Euclidean action (is the most probable). Transition between two cosmological constants. Want this: Wall and volume energy must cancel (zero mass)! Bubbles of new phase nucleate spontaneously.

Non-Zero Mass Bubble What happens when the wall and volume energy do NOT cancel? Birkhoff’s Theorem: exterior spacetime is Schwarzschild-de Sitter. 1) Classify the wall trajectories. 2) Can massive bubbles nucleate spontaneously? New parameter, M!

Exterior Spacetime: Non-Zero Mass

Interior de Sitter

Junction Condition Allowed junctions are constant Q solutions

Potential

Allowed Spacetimes (Aguirre & Johnson gr-qc/ )

Landmarks z

z

z

z

Got Tunneling?

FGG Mechanism a) A bound solution is created in its expanding phase somehow (in the lab, quantum fluctuation, etc.) b) The bound solution grows to the turning point. c) The bound solution tunnels to an unbound solution. d) The two spacetimes become disconnected as the black hole evaporates.

Is this the only option? Still possible as Do we have to hide a universe behind the worm hole? Singularity theorems in asymptotically flat space say yes.

New Tunneling Mechanism Not possible as Goes out the cosmological horizon instead of through the worm hole! Avoids many of the problems of the FGG mechanism! (as we will see) Consistent with the singularity theorems in SdS! (Aguirre & Johnson in prep.)

Zero Mass Limit: With Worm Hole Universe “from nothing” containing a CDL bubble Undisturbed universe

Zero Mass Limit: Without Worm Hole CDL bubble

Parametrized Systems Particle in motion: The trick: Constraint

Quantization Quantize!

Gravity Metric is dynamical, but which components? 10 equations - 4 Bianchi identities = 6 DOF MatterGravity Constraints Quantize!

WKB Tunneling

Tunneling Exponent InsideOutsideWall Insideand Outside : Contributions only where or

Possibilities

Tunneling Exponent Wall CDL

Wrong Sign! Euclidean Interpolation (SdS half): another pathology No one-to-one mapping! Geometry with worm hole: Geometry without worm hole: No Problem! FGG answer: SWE yields both growing and decaying modes, need to choose by hand. But, the sign is intimately tied to the worm hole!

Formation of the Seed Entropy fluctuation produces seed:

Stability of the Seed (Aguirre & Johnson gr-qc/ ) Bound solutions are unstable. Work to 1st order in EOM: Even quantum fluctuations go non- linear! Devastating for FGG?

Putting it all Together = no worm hole = worm hole Both mechanisms are important! Is anything else important? CDL not always lowest action

Thermal Activation Direct production of a universe (50% of the time) if the instability does not interfere. 0(3) instanton exists; Euclidean action non- zero. Jaume Garriga and Ariel Megevand, Decay of de sitter vacua by thermal activation, Int. J. Theor. Phys. 43 (2004), 883–904.

Conclusions Junction condition formalism is a good classification tool. Two possible mechanisms for tunneling, two signs, two zero-mass limits. from nothing CDL One step closer to classifying all vacuum transitions.

WKB Approximation Quantize! Ansatz: To lowest order in Solution: With constraint H = E

Quantization Quantize!