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MEASURES OF THE MULTIVERSE
Alex Vilenkin
Tufts Institute of Cosmology
Stanford, March 2008
Salute to Andrei !
The measure problemi+
Bubbles(pocket universes)
We want to find Pj – 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.
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 ofdifferent measures and their observational predictions.
THIS TALK:
Scale-factor cutoff measure
Predictions for .
Contrast with pocket-based measure
Based on work with Alan Guth,Andrea de Simone & Michael Salem.
,,Q
Work in progress…
t = const
steady-state evolution.
The distribution does not depend on the initial state(but depends on what we use as t).
t
Garcia-Bellido, Linde & Linde (1994)Linde, Linde & Mezhlumian (1994) Linde (2007)
Global time cutoff
t Possible choices of t :
(i) proper time along geodesics orthogonal to ;(ii) scale-factor time, .at
Volume in regions of any kind grows as
Linde & Mezhlumian (1996),Guth (2001), Tegmark (2004),Bousso, Freivogel & Yang (2007)
.~~ , max Plj MHeV
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
Gyr 1
)10exp() exp( 60
2.9KCMBT
Driven by fastest-expanding vacuum
Proper-time cutoff is ruled out.
Proper time cutoff leadsto “youngness paradox”
Scale-factor cutoff – a mild youngness bias
.3 , aV jGrowth of volume:
min)3( – decay rate of the slowest-decaying vacuum
The probability of living at T = 2.9K
is enhanced only by . 2.1/ 30 TT
Not ruled out and has interesting observational consequences.
Pocket-based measure
jjj wpP
jp – 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).
jw
3jj ZP
Slow-roll expansion inside the bubble
Note: large inflation inside bubbles is rewarded.
Similar Z-dependence for Linde’s adjustable cutoff.
Predictions for : Depend on the slow-roll expansion factor Z in the bubbles.
Pocket-based measure favors large inflation:
3jj ZP .1
Scale-factor cutoff does not:
1.-3 ,3 jj ZP
(unless large Z are strongly suppressed in the landscape)
Detectable negative curvature is feasible.
Freivogel, Kleban, Martinez & Susskind (2006)
Predictions for : Q
“Q catastrophe”Feldstein, Hall & Watari (2005)Garriga & A.V. (2006)
Depend on the shape of inflaton potential.
Pocket-based measure:
3)( ZQP – exponential Q-dependence
Scale-factor cutoff:
Mild Z-dependence no Q-catastrophe.
The exact form of P(Q) is model-dependent.
Distribution for : standard approach
A.V. (1995), Efstathiou (1995),Martel, Shapiro & Weinberg (1998).
Assume
)( )()( )()( selecprior fPP
constP prior )()( in the range of interest.
Assume )()(selecf asymptotic fraction of matter
clustered in large galaxies ( ).
Weinberg (1987), Linde (1987),
)(logd
dP
*
1012MM .
All constants other than are fixed.
Appropriate forpocket-based measure
0
Distribution for : scale-factor cutoff
Suppose observers do their measurements of at a fixedproper time after galactic halo collapse.
Gyr 5
(Allowing for chemical and biological evolution.)
)(P fraction of matter clustered in large galaxies 5 Gyr prior to the cutoff.
])/([ )( 1 aafadaP c
ac
Volume thermalized in scale factor interval da(reflects youngness bias).
Proper time corresponding to scalefactor change (ac /a).
3 ,)]2/3sinh([)( 3/21 HHa
Cutoff at .
Press-SchechterWarren et. al.
caa
)(logd
dP
*
Gyr 5
0
Once dominates, expansion accelerates, triggering scale-factor cutoff. Large values of are suppressed.
De Simone, Guth, Salem & A.V. (2008)
1012MM .
Varying M and
10 ,10 ,10 101112 MM .
Gyr 8 ,5 ,2
Including negative
ddP
ddP
*
*
(A) Count all halos formed more than 5 Gyr before the big crunch.
(B) Count all halos formed more than 5 Gyr before turnaround.
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)