1. MEASURES OF THE MULTIVERSE Alex Vilenkin Tufts Institute of Cosmology Stanford, March 2008

2. Salute to Andrei !

3. The measure problem i+i+ Bubbles (pocket universes) We want to find P j – probability for a randomly picked observer to be in a bubble of type j. The number of bubbles & the number of observers per bubble are infinite. Need a cutoff. Results are strongly cutoff-dependent.

4. Measure proposals Global time cutoff Garcia-Bellido, Linde & Linde (1994) Linde, Linde & Mezhlumian (1994) Pocket-based Garriga, Schwartz-Perlov, A.V. & Winitzki (2005) Easther, Lim & Martin (2005) Adjustable cutoff Linde (2007) Causal-patch Bousso (2006), Susskind (2007) We are in the process of working out the properties of different measures and their observational predictions.

5. THIS TALK: Scale-factor cutoff measure Predictions for. Contrast with pocket-based measure Based on work with Alan Guth, Andrea de Simone & Michael Salem. Work in progress…

6. t = const steady-state evolution. The distribution does not depend on the initial state (but depends on what we use as t). Garcia-Bellido, Linde & Linde (1994) Linde, Linde & Mezhlumian (1994) Linde (2007) Global time cutoff Possible choices of t : (i) proper time along geodesics orthogonal to ; (ii) scale-factor time,.

7. Volume in regions of any kind grows as Linde & Mezhlumian (1996), Guth (2001), Tegmark (2004), Bousso, Freivogel & Yang (2007) Observers who take less time to evolve are rewarded by a huge volume factor. Observers who evolve faster than us by and measure are more numerous by Driven by fastest- expanding vacuum Proper-time cutoff is ruled out. Proper time cutoff leads to “youngness paradox”

8. Scale-factor cutoff – a mild youngness bias Growth of volume: – decay rate of the slowest-decaying vacuum The probability of living at T = 2.9K is enhanced only by. Not ruled out and has interesting observational consequences.

9. Pocket-based measure – bubble abundance, Garriga, Schwartz-Perlov, A.V. & Winitzki (2005) Easther, Lim & Martin (2005) – weight factor. Sample equal comoving volumes in all bubbles (all bubble spacetimes are identical at early times). Slow-roll expansion inside the bubble Note: large inflation inside bubbles is rewarded. Similar Z-dependence for Linde’s adjustable cutoff.

10. Predictions for : Depend on the slow-roll expansion factor Z in the bubbles. Pocket-based measure favors large inflation: Scale-factor cutoff does not: (unless large Z are strongly suppressed in the landscape) Detectable negative curvature is feasible. Freivogel, Kleban, Martinez & Susskind (2006)

11. Predictions for : “Q catastrophe” Feldstein, Hall & Watari (2005) Garriga & A.V. (2006) Depend on the shape of inflaton potential. Pocket-based measure: – exponential Q-dependence Scale-factor cutoff: Mild Z-dependence no Q-catastrophe. The exact form of P(Q) is model-dependent.

12. Distribution for : standard approach A.V. (1995), Efstathiou (1995), Martel, Shapiro & Weinberg (1998). Assume in the range of interest. Assume asymptotic fraction of matter clustered in large galaxies ( ). Weinberg (1987), Linde (1987),. All constants other than are fixed. Appropriate for pocket-based measure

13. Distribution for : scale-factor cutoff Suppose observers do their measurements of at a fixed proper time after galactic halo collapse. (Allowing for chemical and biological evolution.) fraction of matter clustered in large galaxies 5 Gyr prior to the cutoff. Volume thermalized in scale factor interval da (reflects youngness bias). Proper time corresponding to scale factor change (a c /a). Cutoff at. Press-Schechter Warren et. al.

14. Once dominates, expansion accelerates, triggering scale-factor cutoff. Large values of are suppressed. De Simone, Guth, Salem & A.V. (2008).

15. Varying M and.

16. Including negative (A) Count all halos formed more than 5 Gyr before the big crunch. (B) Count all halos formed more than 5 Gyr before turnaround.

17. CONCLUSIONS Scale-factor cutoff is a promising measure proposal. Prediction for is a good fit to the data. No Q-catastrophe. Possibility of a detectable curvature. No “Boltzmann brain” problem. (Assuming that the slowest-decaying vacuum does not support BBs)