Landscape anthropic arguments and dark energy Jaume Garriga

Landscape, anthropic arguments and dark energy Jaume Garriga (U. Barcelona)

Probability calculus (from yesterday’s talk by Martin Kunz) Bayes product rule likelihood prior Evidence for the model A good theory should have no free parameters. Equivalently, should be calculable from theory. 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) Metastable “vacua” True vacuum Dual bosonized model: Coleman, Jackiw, Susskind 75

String theory may have many “vacua” Bousso+Polchinski 00 Douglas 03, 04 Susskind 03 (Fluxes) Many ingredients to play with. e. g. Size and shape of compactifide internal space Membrane of charge q (2 -brane) not dynamical in 4 D Discretuum of vacua - The extra legs of a p-brane can be wraped around extra dimensions - So they will look like a membrane (2 -brane) in 4 D. - This can be done in many different ways, leading to many different types of membranes, coupled to different types of fluxes.

Spherical shell with a given range of Bousso+Polchinski 00 The 4 D theory has in fact many different fluxes Each dots represents a metastable vacuum, with a different value of (and ).

Toy “particle physics” Landscape: SM + N independent scalar fields Gravitational interactions with the make the SM parameters slightly different in each vacuum. Arkani-Hamed Dimopoulos Kachru, 05 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. Dimensionless couplings (gauge couplings, etc. ) are “predicted” with sharply peaked distributions. Most vacua have similar values of these couplings.

Shift of paradigm in making predictions With few vacua, a few observations determine which one is ours. Everything else can be predicted. “FUNDAMENTAL THEORY” “Landscape” However… 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? Statistical approach Do all vacua carry the same weight ? What is the measure?

The answer may lay in cosmology. Eternal inflation: All vacua are realized in the multiverse. Set of constants characterizing a given vacuum VOLUME DISTRIBUTION (fraction of volume occupied by vacuum of type Today’s subject: Can we calculate ) for a given theory?

is a useful distribution ? perhaps related to the probability for observations. Given a reference class of observations {“OBS”} Prob. for observing the values in the given theory Volume distribution for the constants Number density of observers

• Multiverse from eternal inflation • Probabilities • and the Q catastrophe • Continuum of vacua and dynamical DE

Multiverse from eternal inflation: 1 - Models with bubble nucleation 2 - Models with quantum diffusion

1 - Models with bubble nucleation d. S vacua generically lead to eternal inflation For bubbles of the new phase do not percolate 1 1 2 Eternally inflating 2 2

Tunneling uphill is also possible 1 Entropy difference 2

Eventually, all vacua are occupied: 1 2 2 1 … 3 2 time 1 Ad. S 3 5 5 1 3 d. S 1 space 4 1

Different “pocket universes” (Self-similar fractal) 1 2 3 THEORY 2 1 3 1 2 MULTIVERSE 2

Structure of a “pocket” universe LSS FRW Slow roll inflation False vacuum inflation Hypersurface of thermalization (infinite and spacelike) BIG BANG in an open FRW Bubble wall

2 - Models with quantum diffusion Generic potential Quantum Diffusion regime Thermalization Slow Roll regime

A. Linde , 94

Inflating volume with field value at proper time , Fokker-Planck equation expansion quantum diffusion coefficient classical drift

1 Inflation 2 t=const. snapshots Proper time Vanchurin, Vilenkin, Winitzky 99 1 2 Still inflating Co-moving space “Big bang” surface

Different “pocket universes” quantum diffusion Multiverse Structure of a pocket universe

Thermalized volume distribution

Problem: the total amount of thermalized volume is infinite, to compare volumes, we need a cut off. The amount of thermalized volume in vacua 1 and 2 depends crucially on how we cut it off (Linde, Mezhlumian 94) Proper time “Proper time” slice 1 2 Still inflating “Scale factor time” slice Co-moving space Gauge dependence: Two different gauges can give vastly different results.

A definition for SOME BASIC REQUIREMENTS • Geometrically defined (gauge independent). • Self-consistent and of generic applicability. • Intuitively reasonable. ( With D. Schwartz-Perlov, A. Vilenkin and S. Winitzki hep-th/0509184 )

OUTLINE • 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

PROBABILITIES WITHIN A GIVEN POCKET

Distribution of constants inside a bubble (flat space, no gravity) True vacuum False Vacuum intrinsic d. S temperature does a random walk of step as the bubble wall eats false vacuum. each proper time interval ( curvature scale)

Consider now a tilted potential Preferred tunneling direction (saddle point). O(3, 1) symmetry (in thin wall approx. ) Correlation length Euclidean action at constant phase (adiabatic approximation)

Milne “multiverse” Infinite number of such “islands” inside each bubble

Including gravity: one of the fields becomes inflaton for slow roll inflation inside the bubble VOLUME DISTRIBUTION (AT THERMALIZATION) WITHIN A POCKET OF TYPE Volume distribution right after quantum tunneling Classical slow roll expansion factor inside the bubble, up to the time of thermalization. Euclidean action for the Cd. L instanton (the corresponding formula for the case of quantum diffusion will be discussed later)

PROBABILITIES FOR DIFFERENT TYPES OF POCKETS TRICKY BUSINESS: the number of pockets of any given type is infinite.

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

A FIRST (FAILED) ATTEMPT 2 2 3 1 2 2 4 5 Send a congruence of co-moving observers Number of obs. who end up in pockets of type j The result depends on initial conditions: No good.

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 Result is independent on initial conditions (number is dominated by bubbles formed at late times). Result is invariant under coordinate transformations. (see also Easther, Lim and Martin, 05)

Fraction of co-moving volume in vacuum of type i Gained from other vacua Lost to other vacua Nucleation rate of bubbles of type “i” in vacuum “j”. Highest nonvanishing eigenvalue of (it can be shown to be negative). Corresponding eigenvector (it is non-degenerate). It can be shown that E. g. , if all vacua emanate from a single false vacuum, then

Spherical shell with a given range of Bousso+Polchinski 00 The 4 D theory has in fact many different fluxes Each dots represents a metastable vacuum, with a different value of (and ).

Overall probability distribution Normalized distribution at the beginning of classical slow roll. 1 - For models of bubble nucleation 2 - For models with quantum diffusion Entropy for quasi-de Sitter. Ergodicity: are all microstates equally represented on ? .

Ergodicity: It holds for co-moving observers who stay in the diffusion regime. Stationary solution 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 within the diffusion regime (and up to its boundary ).

An application: P( ) and the Q catastrophe

The case of L 1 - Assume in the range of interest Vilenkin 95, Weinberg 96 Martel et al. , 98 fraction of matter in galaxies of mass 2 - Assume High L means few galaxies Structure formation stops at the redshift range at which L starts. Anthropic dominating. Press-Schechter 74 constant in the matter era density contrast on the galactic scale The prediction for L: Very small L is punished by phase space Explains the time coincidence 2 s

Comparison of prediction with data 2 s WMAP Inputs: 1 s

Changing the mass of the relevant objects

The volume distribution for Lambda is very important WMAP 2 s 1 s misses by a mile

The case of varying L and Linear density contrast dr/r on the galactic scale Larger L means less galaxies will form Larger means more galaxies will form Assume same as before The two distributions decouple.

THE LARGE Q CATASTROPHE (Graesser, Hsu, Jenkins, Wise 04) (Chaotic inflation) Needs to be very small. In fact, it looks like it needs to be finely-tuned By analogy with , they assume which pushes to very large density contrast, Observed density contrast in our local universe (Tegmark+Rees) Anthropic upper limit

HOWEVER… the choice is not really justified in the case of the inflaton coupling: The expansion factor depends exponentially on Applying our general recipe for the volume distribution, we find instead: SMALL Q CATASTROPHE !! 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) Perturbations are not produced by inflaton, but by another field called curvaton. You are here Adiabatic Gaussian With N curvatons J. G. , A. Vilenkin, 05

Continuum of vacua and dynamical dark energy.

Slow-roll condition (we need ) In the absence of ad-hoc adjustments, we should expect 1 - Field starts at rest, while H is large. 2 - Comes to dominate the energyhence density, and drives accelerated expansion. 3 - Picks up speed as H^2 falls below s /M_p. 4 -Slips down into negative In this case values of the potential. (Like L) 5 - Negative potential energy eventually turns the expansion into contraction. 6 - Local universe undergoes a big crunch. We are doomed, on a timescale if

Models with several “moduli” Both and become random variables J. G. +Vilenkin 03 Dimopoulos+Thomas 03 If prior favours small If prior favours large Anthropic selection determines Leads to observable signatures: Kallosh, Kratotchvil, Linder+Schmakova 03 Dimopoulos+Thomas 03 J. G. , Pogosian, Vachaspati 03 Both types of models can be constructed J. G. +Linde+Vilenkin 03

ce ler ati on Doomsday model ac a(t) 2 3 today doomsday

Forecasting doomsday J. G. , Pogosian, Vachaspati 03 w= dark energy equation of state parameter= Time variation of w may lead to observable Consequences on: w(z) • CMB angular spectrum 4 3 s L 2 =1 • dimming of supernovae • matter power spectrum z • matter/CMB cross-correlation Substantial variation at low redshifts s=4 s=3 s=2 s=1 time to big crunch 14 Gyr 17 Gyr 22 Gyr 45 Gyr • Can we tell the value of s ? • Can we tell doomsday from constant w ?

CMB (fast…) These four models have almost degenerate CMB angular spectra <w> s=0 s=1 s=2 s=3 -1. 00 -0. 94 -0. 81 -0. 66 h. 72. 69. 66. 62 Similar ISW too ! For each <w>, we must adjust h to reproduce the peak structure

Cross correlation CMB/LSS for doomsday (vs. constant w) L s=3 W=-. 66 Breaks degeneracy between doomsday and constant w Models.

40 Gyr 30 Gyr Planck+Snap Kallosh, Kratochvil, Linder, Shmakova, 2003

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.