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 ...