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24 Pages

### VEU_Vilenkin

Course: GR 201207, Fall 2009
School: East Los Angeles College
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Word Count: 930

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Vilenkin MEASURES A. OF THE MULTIVERSE Alex Vilenkin Tufts Institute of Cosmology Cambridge, Dec. 2007 STRING THEORY PREDICTS MULTIPLE VACUA WITH DIFFERENT CONSTANTS OF NATURE N ~ 10 1000 Bousso &amp; Polchinski (2000) Susskind (2003) Douglas (2003) THE LANDSCAPE Eternal inflation the entire landscape will be explored. A. Vilenkin THE MEASURE PROBLEM: What is the probability for a randomly picked...

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Vilenkin MEASURES A. OF THE MULTIVERSE Alex Vilenkin Tufts Institute of Cosmology Cambridge, Dec. 2007 STRING THEORY PREDICTS MULTIPLE VACUA WITH DIFFERENT CONSTANTS OF NATURE N ~ 10 1000 Bousso & Polchinski (2000) Susskind (2003) Douglas (2003) THE LANDSCAPE Eternal inflation the entire landscape will be explored. A. Vilenkin THE MEASURE PROBLEM: What is the probability for a randomly picked observer (reference object) to be in a given type of vacuum? Assume we have a model for calculating the numbers of reference objects. Still there is a problem with infinities NOTE: A measure is needed even for predicting the CMB multipoles. PLAN Structure of the multiverse General requirements for the measure Proposals and problems The noodle measure Spacetime structure i+ i+ Bubbles (pocket universes) ? ? ? Bubbles nucleate and expand at nearly the speed of light. Terminal & recyclable bubbles Eternal geodesics Inflating spacetimes are past-incomplete. What is at the past boundary? Quantum nucleation from nothing Chaotic initial conditions Not relevant for the measure (almost). The measure problem i+ i+ Bubbles (pocket universes) The number of bubbles is infinite, even in a finite comoving volume. The number of ref. objects in each bubble is infinite. Need a cutoff. Global time measures i+ t = const Use a hypersurface t = const as a cutoff. Linde, Linde & Mezhlumian (1994) Garcia-Bellido, Linde & Linde (1994) The limit at t does not depend on the initial state. BUT: depends on what we use as t. (Most of the reference objects are near cutoff.) Requirements for the measure Measure axioms: A PA = 1, PA B = PA + PB , etc. Independence of initial conditions. Independence of time parametrization. The pocket-based measure Pj p j w j Bubble abundance Garriga, Tanaka & A.V. (1999) Garriga, Schwartz-Perlov, A.V. & Winitzki (2005) Weight factor (characterizes the number of reference objects per bubble). The bubble abundance: pj = N j / N The geodesics project all bubbles onto . Geodesic congruence Include only bubbles of projected volume bigger than . Then let 0 . p j are independent of the choice of congruence and of . independent of initial conditions: dominated by bubbles formed at late times. An equivalent prescription: Easther, Lim & Martin (2005) Calculation of pj fi -- fraction of co-moving volume in vacuum of type i dfi = Mi j f j dt M i j = i j i j r i r ij Probability per Hubble time to get to vacuum i from vacuum j. Gained from other vacua Lost to other vacua q< 0 f i (t ) f i () + si e qt + ... for bubbles of type j in parent vacuum Highest nonvanishing eigenvalue of M i j (it can be shown to be negative). si Corresponding eigenvector (it is non-degenerate). p j j s q << 1 if bubble nucleation rate is small Reduces an to eigenvalue problem The weight factor Pj p j w j Sample equal comoving volumes in all bubbles. i+ Internal FRW geometry: ds 2 = d 2 a 2 ( )(dr 2 + sinh 2 r d 2 ) =0 Sample comoving spheres: Bubble spacetimes are identical at small : a ( ) r < Same for all bubbles wj n j Pj p j n j j Z 3 j Bubble nucleation rate Slow-roll expansion inside the bubble Note: large inflation inside bubbles is rewarded. Some other proposals: A version of global time cutoff: A.V. (1994), Linde (2007) Use different cutoff times tj for different bubbles. Pj are approximately time-parameter independent. Include only observers past light cone. Pj depend on the initial state. Bousso (2006) Vanchurin (2007) Volume weighting of histories. Hawking (2006) Divergent; needs a cutoff. Hartle, Hawking & Hertog (2007) Pj = f ( Pj ) Violates additivity. Bousso (2007) The pocket-based measure satisfies all requirements. The main shortcoming of pocket-based measure: Does not account for bubble collisions. Bubbles form infinite clusters; have fractal structure. r = The cluster-based (noodle) measure Garriga, Guth & A.V. The noodle measure All clusters within the same parent vacuum are statistically equivalent. Pj p w j of type . -- probability to be in a bubble of type j in a cluster Calculate cluster abundance p with the same prescription as we used for bubbles. p j s s j assuming low bubble nucleation rate The noodle measure w j = V j n j = const Sample equal comoving spheres in all clusters (?) Spacelike version of time-parameter dependence. Sample a tube-like region around a spacelike geodesic the noodle. dVi = 1 (iV j jVi ). dl j Stationary solution: V j = j Pj ...

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