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)