Landscape, anthropic arguments and dark energy

Jaume Garriga (U. Barcelona)

SM parameters , , / , / , / , , , ,....b m S TX n n Q n n w

Probability calculus (from yesterday’s talk by Martin Kunz)

data setD

( , ) ( | ) ( ) ( | ) ( )P D X P X D P D P D X P X

modelM

prior

parametersX

likelihood

Bayesproduct rule

( ) ( ) ( ) ( )E M P D dX L X X

( )L X ( )X

Evidence for the model

A good theory should have no free parameters.Equivalently, should be calculable from the theory.( )X

Landscape: Field configuration space with a vast number of vacua, where the effective “constants” of nature take different values.

A landscape with many “vacua”

Example:

QED in 1+1 dimensions (Massive Schwinger model)

( )F qF F

2 21 1( ) ( )

2 2n F nq

q q

Electric field F dA nq F

Dual bosonized model:

( 0)F

n

Metastable “vacua”

True vacuum

m m

2 2( ) cos(2 )V q c qm

Coleman, Jackiw, Susskind 75

String theory may have many “vacua”

[4] [3]F F dA Many ingredients to play with. e.g.

Size and shape of compactifide internal space

(Fluxes)

not dynamical in 4D

[4] [4]F F F q

Membrane of charge q (2-brane)

Douglas 03,04Susskind 03

FF nq

Discretuum of vacua

Bousso+Polchinski 00

- The extra legs of a p-brane can be wraped around extra dimensions

- So they will look like a membrane (2-brane) in 4D.

- This can be done in many different ways, leading to many different types ofmembranes, coupled to different types of fluxes.

1,..., ( 100)a a aF n q a J J

1,..., ( 100)a a aF n q a J J

The 4D theory has in fact many different fluxes

21

1( ,..., ) ( )

2bare

J a aa

n n n q

# ( 1)Jvacua G

10010 ( )G googol

Bousso+Polchinski 00

Each dots represents a metastable vacuum, with a different value of (and ).

Spherical shell with a given range of

X

Toy “particle physics” Landscape:

SM + N independent scalar fields i

( )iV

i

Arkani-Hamed Dimopoulos Kachru, 05

31,..., ( 10 ) 2 vacuaNi N N

Dimensionful couplings , “scan” over a wide range of values.No need of a symmetry principle to explain their small values. There are many vacua where these parameters are small, just by chance.

hm

Dimensionless couplings (gauge couplings, etc.) are “predicted” with sharply peaked distributions. Most vacua have similar values of these couplings.

Gravitational interactions with the make the SM parameters slightly different in each vacuum.

i

Shift of paradigm in making predictions

With few vacua, a few observations determine which one is ours.

Everything else can be predicted.

“Landscape” Many vacua to scan

Many may look like our own.except for small variations of the “constants”.

Can we predict the values of such constants?

However…

Statistical approach Do all vacua carry the same weight ?What is the measure?

100'( 10 ).s

“FUNDAMENTAL THEORY”

Eternal inflation: All vacua are realized in the multiverse.

The answer may lay in cosmology.

X Set of constants characterizing a given vacuum

( )VOLP X VOLUME DISTRIBUTION

(fraction of volume occupied by vacuum of type )X

Today’s subject: Can we calculate for a given theory? ( )VOLP X

Given a referenceclass of observations {“OBS”}

( ) ( ) ( )OBS VOL OBSP X P X n XVolume distributionfor the constants

Number densityof observersX

is a useful distribution ?( )VOLP X

perhaps related to the probability for observations.

Prob. for observingthe values inthe given theory

X

( )X

• Multiverse from eternal inflation

• Probabilities

• and the Q catastrophe

• Continuum of vacua and dynamical DE

( )OBSP

( )VOLP X

Multiverse from eternal inflation:

1- Models with bubble nucleation

2- Models with quantum diffusion

dS vacua generically lead to eternal inflation

4BA e H

eff

Forbubbles of the new phase do not percolate

Eternally inflating

11

2

2

2

Be

4( 3 )1

H HtV C e

1 - Models with bubble nucleation

Tunneling uphill is also possible

1

12

Be

2

(2) (1)/ S Se Entropy difference

eff

12

31

2 1

… 4

5 13

Eventually, all vacua are occupied:

dS

AdS

space

time

1

3

152

12

3

1

22

21 3

Different “pocket universes” (Self-similar fractal)

THEORY

MULTIVERSE

Structure of a “pocket” universe

Hypersurface of thermalization(infinite and spacelike)

BIG BANG in an open FRW

False vacuuminflation

Slow rollinflation

LSS

Bubble wall

FRW*

2 - Models with quantum diffusion

2( ) ( )V H q

*

Slow Rollregime

Quantum Diffusion regime

Thermalization

' ( ) ( )H H

2' ( ) ( )H H

Generic potential

/ | 'SRt H

2/ |QDt H

A. Linde , 94

3F

div J H Ft

( , )F t d Inflating volume with field value ,at proper time

( ) ( )J grad D F grad H F

Fokker-Planck equation

expansion

classical driftquantum diffusion

3D H

t

diffusion coefficient

Vanchurin, Vilenkin, Winitzky 99

t=const. snapshots

Co-moving space

Pro

per

tim

e

Still inflating

1

2

*

1 2Inflation

*

“Big bang” surface

q

Multiverse Structure of a pocket universe

Different “pocket universes”

quantum diffusion

Thermalized volume distribution ( )VOLP X

The amount of thermalized volumein vacua 1 and 2 depends crucially on how we cut it off

Co-moving space

Pro

per

tim

e

Still inflating

1

2

“Proper time” slice

“Scale factor time” slice

Problem: the total amount of thermalized volume is infinite, to compare volumes, we need a cut off.

(Linde, Linde, Mezhlumian 94)

.t const

a const

Gauge dependence: Two different gauges can give vastly different results.

• Geometrically defined (gauge independent).

• Self-consistent and of generic applicability.

• Intuitively reasonable.

A definition for

SOME BASICREQUIREMENTS

( With D. Schwartz-Perlov, A. Vilenkin and S. Winitzki hep-th/0509184 )

( )VOLP X

• Define probabilities within a given pocket

of type to find values of the constants

• Define a suitable weight factor for comparing different types of pockets

• Combine both objects to find the overall distribution

OUTLINE

( ; )P X jj X

jp

PROBABILITIES WITHIN A GIVEN POCKET

( ; )P X j

Distribution of constants inside a bubble (flat space, no gravity)

X 0R

( )P X

False Vacuum

True vacuum

intrinsic dS temperature 10R

does a random walk of step each proper time interval as the bubble wall eats false vacuum. X 1

0X R 0R

r

1/ 2 1 1/ 2 10 0( / )KX N R r r R

Kr( curvature scale)

( ) .P X const

X

Preferred tunneling direction (saddle point).

( )V ( )( ) ( ) i X xx f x e

2 200, ( )X f f x t

O(3,1) symmetry

2 2 4 1 2 2(3,1) 0 0 0 ,O XX m R R R

( ) ( ') exp( / )KX r X r r r

Consider now a tilted potential

Correlation length1 2

0( ) 1Xm R

2 2/ ( ; )( ; ) EX X I X jP X j e e

( ; )EI X j Euclidean action atconstant phase(adiabatic approximation)

(in thin wall approx.)

Kr

1Kr

r

X

Milne “multiverse”

1 1/ 20R

Infinite number of such“islands” inside each bubble?

Including gravity: one of the fields becomes inflaton for slow roll inflation inside the bubble

( ; ) 3*( ; ) ( ; )EI X jP X j e Z X j

Volume distribution right after quantumtunneling

Classical slow roll expansion factor inside the bubble,up to the time of thermalization.

* ( , )( ; ) exp

'( , )i

H XZ X j d

H X

VOLUME DISTRIBUTION (AT THERMALIZATION)WITHIN A POCKET OF TYPE j

( ; )EI X j Euclidean action for the CdL instanton

(the corresponding formula for the case of quantum diffusion will be discussed later)

PROBABILITIES FOR DIFFERENT TYPES OF POCKETS

jp

TRICKY BUSINESS: the number of pockets of any given type is infinite.

1 2

False vacuum

1 2

Intuitively, there should be more bubbles of type 2 than of type 1.

A FIRST (FAILED) ATTEMPT

Send a congruenceof co-moving observers

2 2 1 2 2

3 4

5

The result depends on initial conditions: No good.

Cjp Number of obs. who end up in pockets of type j

OUR PROPOSAL

RELATIVE NUMBER OF POCKETS OF TYPE j. COUNT THEM AT THE FUTURE BOUNDARY.

Number is infinite: count only the ones which are bigger than some small co-moving size and then let 0

Result is independent on initial conditions(number is dominated by bubbles formed at late times).

jp

Result is invariant under coordinate transformations.

(see also Easther, Lim and Martin, 05)

if Fraction of co-moving volume in vacuum of type i

ii j j

dfM f

dt i j i j i j r i

r

M Gained from other vacua

Lost to other vacua

3j i jH

Nucleation rate of bubbles oftype “i” in vacuum “j”.

( ) (0) ...qti i if t f s e

0q Highest nonvanishing eigenvalue of (it can be shown to be negative).

i jM

is Corresponding eigenvector(it is non-degenerate).

qi j i j j

j

p H s E.g., if all vacua emanate from a single false vacuum, then

i ip

It can be shown that

1,..., ( 100)a a aF n q a J J

The 4D theory has in fact many different fluxes

21

1( ,..., ) ( )

2bare

J a aa

n n n q

# ( 1)Jvacua G

10010 ( )G googol

Bousso+Polchinski 00

Each dots represents a metastable vacuum, with a different value of (and ).

Spherical shell with a given range of

X

Overall probability distribution

3( ) ( ; ) ( ; )qj jdP X p P X j Z X j dXNormalized distribution at the beginning of classical slow roll.

( ; )( ; ) EI X jqP X j e1- For models of bubble nucleation

2- For models with quantum diffusion ( ; )( ; ) (? )S X jqP X j e

ES I Entropy for quasi-de Sitter.

Ergodicity: are all microstates equally represented on ?.q

Ergodicity: It holds for co-moving observers who stay in the diffusion regime.

True also for the volume distribution F(X,t) in scale factor time.

Remains to be seen if it is true for any unbiasely selected surface withinthe diffusion regime (and up to its boundary ).

q

( , )CC

F X tdiv J

t

22 /( ) H S

CF H X e e

Stationary solution

3( , ) ( , )tSFT CF X t e F X t

An application:

P( ) and the Q catastrophe

Structure formation stops at the redshift at which starts dominating.z

High means few galaxies

Gz z

Very small is punished by phase space

Gz z Explains the time coincidence

Press-Schechter 741/3

3( ) .80obs

m G

n erfc

The case of 1- Assume in the range of interest( )VOLP const

2- Assume 1210GM M M ( )obsn fraction of matter in galaxies of mass

Vilenkin 95,Weinberg 96

density contrast on the galactic scale

mG

m

1/3

3( ) .80OBS

m G

dP erfc d

constant in the matter era

( )VOLP

4M

( )VOLP

0100 M

Anthropic range

The prediction for

.04 16y

3 3

m G

yQ

/ lnOBSdP d y

Martel et al., 98

Comparison of prediction with data

0.6 0.7 0.8 0.9 1od

0.2

0.4

0.6

0.8

1

1.2

1.4

8s8

WMAP

( )VOLP const

.72 .47 .99 1bh n w

1210GM M Inputs:

0m

0.6 0.7 0.8 0.9 1od

0.2

0.4

0.6

0.8

1

1.2

1.4

8s

0.6 0.7 0.8 0.9 1od

0.2

0.4

0.6

0.8

1

1.2

1.4

8s

0.6 0.7 0.8 0.9 1od

0.2

0.4

0.6

0.8

1

1.2

1.4

8s

8

8

8

1210GM M

1110GM M

1010GM M

Changing the mass of the relevant objects

The volume distribution for Lambda is very important

0.6 0.7 0.8 0.9 1od

0.2

0.4

0.6

0.8

1

1.2

1.48s8

WMAP

( )VOLP

misses by a mile

The case of varying and G

Linear density contrast on the galactic scaleG

Larger means less galaxies will formLarger means more galaxies will formG

1/3( , ) ( ) (.80 )OBS G VOL G GdP P erfc y d d

( , ) ( )VOL G VOL GP P

3/( )m Gy 3Gd dy

3 1/3( , ) ( ) (.80 )OBS G G VOL G GdP P d erfc y dy

same as before

Assume

The two distributions decouple.

( ) nV

By analogy with , they assume ( ) .VOLP const

( ) .VOL G GP

which pushes to very large density contrast,

(Graesser,Hsu,Jenkins,Wise 04)

1/ 2G

5

5**( ) 10OBS G

cut off

P

Needs to be very small. In fact, it looks like it needs to be finely-tuned

4( )OBS G G GdP d

* Observed density contrastin our local universe

Anthropic upper limit *10cut off

(Chaotic inflation)

1210

(Tegmark+Rees)

THE LARGE Q CATASTROPHE

( ) .VOLP const the choice

is not really justified in the case of the inflaton coupling: The expansion factor depends exponentially on

3( )Z

Applying our general recipe for the volume distribution, we find instead:

2/( 2)( ) exp[ ]nVOLP C

4/( 2)( ) exp[ ' ]nVOLP Q C Q SMALL Q CATASTROPHE !!

HOWEVER…

Small Q catastrophe can be avoided altogether in curvaton (or in inhomogeneous reheating) models, where Q is not correlated with the duration of slow roll inflation.

J.G., A. Vilenkin, 05

The Curvaton Web (A. Linde and V. Mukhanov, 05)

2 2 21( )

2V A m A H 2 2m H

A A

A HQ

A A

You are here4

22

HA

m

Adiabatic Gaussian

Perturbations are not produced byinflaton, but by another field called curvaton.

1 1( ) N NVOL A A AdP Q A dA Q dQ

With N curvatons

J.G., A. Vilenkin, 05

Continuum of vacua and dynamical dark energy.

eff ( )V s 2Ps H M

Slow-roll condition

1- Field starts at rest, while H is large.2- Comes to dominate the energy density, and drives accelerated expansion.

3- Picks up speed as H^2 falls below s /M_p .4-Slips down into negative values of the potential.

5- Negative potential energy eventually turns the expansion into contraction.6- Local universe undergoes a big crunch.

3/ 2

2sP

tM s

We are doomed, on a timescale

10st H if

20 Ps H M

(we need )20 Ps H M

In the absence of ad-hoc adjustments, we should expect

hence 20 Ps H M 1

0st H

In this case 1p

w

(Like

Models with several “moduli”

Both and become random variables ( )eff aV | ( ) |a as V

( , )prior effP s

If prior favours small s 1p

w

sIf prior favours large Anthropic selection determines

J.G.+Vilenkin 03Dimopoulos+Thomas 03

20 Ps H M 0st t

Leads to observable signatures: ( )w w zKallosh,Kratotchvil, Linde,Linder+Schmakova 03Dimopoulos+Thomas 03J.G., Pogosian, Vachaspati 03

Both types of models can be constructed J.G.+Linde+Vilenkin 03

Doomsday model

a(t)

0H t

today doomsday

203 P

ss

H M

1s

2

34s

acce

lera

tion

Forecasting doomsday

z

w(z)

3

2

4

s= 1

w= dark energy equation of state parameter=p

Time variation of w may lead to observable Consequences on:

• CMB angular spectrum

• dimming of supernovae

• matter power spectrum

• matter/CMB cross-correlation

J.G.,Pogosian,Vachaspati 03

Substantial variation at low redshifts

time to big crunchs=4 14 Gyrs=3 17 Gyr s=2 22 Gyrs=1 45 Gyr

• Can we tell the value of s ?

• Can we tell doomsday from constant w ?

1

01

0

( ) D

D

da w aw

da

<w> h s=0 -1.00 .72 s=1 -0.94 .69 s=2 -0.81 .66s=3 -0.66 .62

These four models have almost degenerate CMB angular spectra

For each <w>, we must adjust h to reproduce the peak structure

CMB (fast…)

Similar ISW too !

Cross correlation CMB/LSS for doomsday (vs. constant w)

s=3

W=-.66

Breaks degeneracy between doomsday and constant w Models.

Kallosh,Kratochvil,Linde,Linder,Shmakova, 2003

Planck+Snap

30 Gyr

40 Gyr

SUMMARY:

1- We discussed a proposal for the volume distribution of the constants in the multiverse

2- It is based on its geometrical building blocks, the pocket universes (rather than some ad-hoc volume cut-off or any other artificial construction).

3- are determined by counting pockets at future infinity.

4- Analytic estimates for remain a challenge. Is it really ergodic in the case of quantum diffusion?

5- Application to models with variable Lambda and Q. Curvaton (or inhomogeneous reheating) seem to be favoured with respect to inflaton generated density perturbations.

6- A continuum of vacua may give rise to dynamical dark energy.

3( ) ( ; ) ( ; )j j qdP X p P X j Z X j dX

jp

qP