MEASURES OF THE MULTIVERSE Alex Vilenkin Tufts Institute of Cosmology Cambridge, Dec. 2007

STRING THEORY PREDICTS MULTIPLE VACUA WITH DIFFERENT CONSTANTS OF NATURE Bousso & Polchinski (2000) Susskind (2003) Douglas (2003) “THE LANDSCAPE” Eternal inflation the entire landscape will be explored.

THE MEASURE PROBLEM: What is the probability for a randomly picked observer (“reference object”) to be in a given type of vacuum? Still there is a problem with infinities Assume we have a model for calculating the numbers of reference objects. 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

i+i+ Bubbles nucleate and expand at nearly the speed of light. Terminal & recyclable bubbles Eternal geodesics Inflating spacetimes are past-incomplete. Bubbles (pocket universes) Spacetime structure ? ? ? i+i+

What is at the past boundary? Quantum nucleation from nothing Chaotic initial conditions Not relevant for the measure (almost).

i+i+ 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. Bubbles (pocket universes) The measure problem i+i+

t = const Use a hypersurface t = const as a cutoff. BUT: depends on what we use as t. Linde, Linde & Mezhlumian (1994) Garcia-Bellido, Linde & Linde (1994) The limit at does not depend on the initial state. (Most of the reference objects are near cutoff.) Global time measures i+i+

Requirements for the measure Measure axioms: Independence of initial conditions. Independence of time parametrization.

Bubble abundance Weight factor (characterizes the number of reference objects per bubble). Garriga, Schwartz-Perlov, A.V. & Winitzki (2005) The pocket-based measure Garriga, Tanaka & A.V. (1999)

Include only bubbles of projected volume bigger than. Then let. The bubble abundance: The geodesics project all bubbles onto. 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) Geodesic congruence

-- fraction of co-moving volume in vacuum of type i Gained from other vacua Lost to other vacua Probability per Hubble time to get to vacuum i from vacuum j. Highest nonvanishing eigenvalue of (it can be shown to be negative). Corresponding eigenvector (it is non-degenerate). Calculation of p j Reduces to an eigenvalue problem if bubble nucleation rate is small for bubbles of type j in parent vacuum

i+i+ Internal FRW geometry: Bubble spacetimes are identical at small : Sample comoving spheres: Same for all bubbles The weight factor Sample equal comoving volumes in all bubbles.

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: Use different cutoff times t j for different bubbles. A.V. (1994), Linde (2007) P j are approximately time-parameter independent. Include only observer’s past light cone. P j depend on the initial state. Bousso (2006) Vanchurin (2007) Volume weighting of histories. Divergent; needs a cutoff. Hawking (2006) Hartle, Hawking & Hertog (2007) 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. The cluster-based (“noodle”) measure Garriga, Guth & A.V.

-- probability to be in a bubble of type j in a cluster of type. Calculate cluster abundance with the same prescription as we used for bubbles. The noodle measure All clusters within the same parent vacuum are statistically equivalent. assuming low bubble nucleation rate

The noodle measure Stationary solution: assuming low bubble nucleation rate Spacelike version of time-parameter dependence. Sample a tube-like region around a spacelike geodesic – “the noodle”. Sample equal comoving spheres in all clusters (?) Agrees with pocket-based measure.

What is the probability for us to observe a collision with another bubble (i.e. to be in a collision-affected region)?

Reducible landscapes The landscape splits into several sectors inaccessible from one another. Probability of vacuum j in sector A: Nucleation probability with vacuum k Probability of j, calculated using the hypersurface

CONCLUSIONS We formulated requirements for the measure: Additivity; independence of initial conditions; independence of time parametrization. The noodle measure is applicable to general bubble spacetimes & satisfies all requirements. Open issues: Uniqueness? Extension to quantum diffusion Freak observers

Youngness paradox: Most observers evolve at very early cosmic times (high T CMB ), when the conditions for life are still hostile. Low probability of evolving is compensated by exponential increase in the volume. Linde & Mezhlumian (1996) Guth (2001); Tegmark (2004) t = const Global time measures i+i+ The observed (low) value of T CMB is highly unlikely.