landscape, anthropic arguments and dark energy jaume garriga (u. barcelona)
TRANSCRIPT
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