MEASURES OF THE
MULTIVERSE
Alex Vilenkin
Tufts Institute of Cosmology
Cambridge, Dec. 2007
STRING THEORY PREDICTS MULTIPLE VACUA WITH
DIFFERENT CONSTANTS OF NATURE
1000
N ~ 10
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?
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+
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.
i+
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
f i -- fraction of co-moving volume in vacuum of type i
df i
 Mi j f j
dt
M i j  i j   i j   r i
r
Gained from
other vacua
Lost to
other vacua
q 0
f i (t )  f i ()  si e qt  ...
p j   j s
ij
for bubbles of type j
in parent vacuum 
si
Probability per Hubble time to get
to vacuum i from vacuum j.
Highest nonvanishing
eigenvalue of M i j
(it can be shown to be negative).
Corresponding eigenvector
(it is non-degenerate).
q  1
Reduces to an eigenvalue problem
if bubble nucleation
rate is small
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
Bubble spacetimes are identical
a ( )  
at small  :
Sample comoving spheres:
r 
wj  n j
Same for all bubbles
Pj  p j n j  j Z
Bubble
nucleation
rate
3
j
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 observer’s 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
Sample equal comoving
spheres in all clusters (?)
  const
Spacelike version of time-parameter dependence.
Sample a tube-like region around a spacelike geodesic – “the noodle”.
dVi
  1  iV j   jVi .
dl
j
Pj   j s
Stationary solution:
assuming low bubble
nucleation rate
Agrees with pocket-based measure.
V j   j 
What is the probability for us to observe a collision
with another bubble (i.e. to be in a collision-affected region)?
Pcoll ~   1.
Reducible landscapes
The landscape splits into several sectors inaccessible from one another.
Probability of vacuum
j in sector A:
PjA   P P
k A
(k )
nucl
Nucleation probability
with vacuum k
k
(k )
j
Probability of j,
calculated using
the hypersurface  k
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?
Freak observers
Extension to quantum diffusion
Global time measures
i+
t = const
Youngness paradox:
Most observers evolve at very early cosmic times (high TCMB), when
the conditions for life are still hostile. Low probability of evolving
is compensated by exponential increase in the volume.
The observed (low) value of TCMB is highly unlikely.
Linde & Mezhlumian (1996)
Guth (2001); Tegmark (2004)