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Path Integral Approach to Volume-Weighted Eternal Inflation

Path Integral Approach to Volume-Weighted Eternal Inflation. Steven Gratton, Institute of Astronomy, Cambridge. 10 th July 2009. Inflation. Successes Homogeneity A fraction more time before the big bang Inhomogeneity Small quantum fluctutations Problems Initial conditions

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Path Integral Approach to Volume-Weighted Eternal Inflation

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  1. Path Integral Approach to Volume-Weighted Eternal Inflation Steven Gratton, Institute of Astronomy, Cambridge. 10th July 2009

  2. Inflation • Successes • Homogeneity • A fraction more time before the big bang • Inhomogeneity • Small quantum fluctutations • Problems • Initial conditions • How did inflation start?

  3. Stochastic Eternal Inflation • Successes • Homogeneity • Lots more time before the big bang • Inhomogeneity • Large quantum fluctutations • Problems • Initial conditions • Who cares?

  4. Volume-Weighting • Basically “rewards” histories that inflate a lot • Perhaps related to “spatial averaging” • Perhaps related to the anthropic principle: more inflation might lead to more final volume and so to more observers

  5. The Youngness Paradox (1) • “Normal” version: • Universes that have evolved less will have formed later, so there should be more of them • A “typical” low-energy observer should be the first one in a pocket, and so say see as hot a CMB as possible

  6. The Youngness Paradox (2) • “Extreme” version: • Should crazy things have happened in our past, with the skewed statistics dominating over normal dynamics? • A “typical” low-energy observer should see Planck-scale inflation just to the past! • So no nice density perturbations…

  7. Stochastic Inflation • We start with the slow-roll equation… • …and add noise:

  8. Trying to solve this directly is the Langevin approach • Cf. Heisenberg approach to QM • Can also convert this into a PDE for the probability density; the Fokker-Planck approach • Cf. Schrodinger approach to QM

  9. New Approach: Path Integral, cf Feynman • Consider a general path . • After a short time the field will be at . • According to the Langevin equation, the field “should” change by with a spread of

  10. So, the probability of this segment of history occurring is: • Multiplying for all the segments and taking the limit we obtain:

  11. Probability of a history… with where

  12. Addition of the Volume-Weighting Term • Just need to multiply each trajectory by its volume, namely: • So, putting it together we obtain…

  13. ...The Measure for Volume-Weighted Stochastic Inflation with where Now we can treat vol. weighting seriously!

  14. Simplification to Quartic Case • If the potential is a change of variables simplifies the lagrangian to where

  15. Get Approximate Results via Saddle Point Histories U R P I F As the time interval increases, such solutions stop existing!

  16. R Where do we have to go? Into the complex plane! R 0 0 P I F 0

  17. R Loitering and Attractors, orWhy Eternal Inflation Forgets R 0 0 P I F 0

  18. And Inflation Ends Classically! t

  19. Conclusions • There exists a powerful path integral formulation of stochastic eternal inflation, • in which volume-weighting can naturally be incorporated and taken seriously. • Complex saddle points emerge, that are vital in allowing one to see • eternal inflation depends on initial conditions (weakly) • that the “extreme” youngness paradox doesn’t occur (…very hard with Langevin/Fokker-Planck)

  20. Reflection: A democracy of measures? • As inflation ends in the same way basically independently of final time, we’ve seen that the much-maligned proper time volume weighting can be used to answer at least this question unambiguously… • Perhaps the “measure problem” is as much a problem with specific questions as for the weighting schemes themselves!

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