RenataRenata KalloshKallosh

Davis, May 16, 2004

StanfordStanford

Deformation, non-commutativity and cosmological constant

problem

OutlineOutline1. Observational data on Observational data on DARK ENERGY and INFLATION CC PROBLEM 2. String Theory- Cosmology: KKLT model of de Sitter space,Warping small parameter from deformed conifold.

Problems with warping in KKLMMT model of inflation

3. Hybrid Inflation/Acceleration in D3/D7 Brane System

4. Deformed non-linear instanton, Nekrasov-Schwarz non-commutative instanton

5. Irrational deformation (non-commutativity) parameterIrrational deformation (non-commutativity) parameter

in 6,7,8,9 space CC in 0,1,2,3 space.

Replace D0/D4by D3/D7

Non-commutative

in the space orthogonal to D3

Cosmological Constant in effective 4d

Cmbgg OmOl

How much dark energy is there?

Closed

Open

Cmbgg OmOlCMB

flatclosedopen

How much dark energy is there?

Cmbgg OmOlCMB+LSS

How much dark energy is there?

WMAP + SDSS: lots

flatclosedopen

Cmbgg OmOlCMB+LSS

How much dark energy is there?

flatclosedopen

Cmbgg OmOlCMB+LSS

How much dark energy is there?

flatclosedopen

Tegmark et al, 2004

Cosmological Concordance ModelCosmological Concordance Model Early Universe Inflation Near de Sitter spaceNear de Sitter space 13.7 billion years ago During 10^{-35} sec

Current Acceleration Near de Sitter spaceNear de Sitter space Now During few billion years

DARK ENERGYDARK ENERGY

Total energy in 3d flat FRW universe

O

70% of the total energy of the universe is DARKDARK

Cosmological Constant (CC) Problem

The simplest form of dark energy: CC

String Theory and CosmologyString Theory and Cosmology

All observations fit 4d Einstein GR: how to get this picture from the compactified fundamental 10d string theory or 11d M-theory and supergravity

How to get de Sitter or near de Sitter 4d space?How to get de Sitter or near de Sitter 4d space?

Towards cosmology in type IIB string theory

Dilaton stabilization Giddings, Kachru and Polchinski 2001

Kachru, R. K, Linde, Trivedi Kachru, R. K, Linde, Trivedi 20032003

Kachru, R. K., Maldacena, McAllister, Linde and Trivedi 2003

LandscapeLandscape Susskind Susskind Flux VacuaFlux Vacua Douglas Douglas

Volume stabilization, KKLTVolume stabilization, KKLT

The throat geometry has a highly warped region

Deformed ConifoldCopeland, Myers, Polchinski picture

Volume stabilization Warped geometry of the compactified space and

nonperturbative effects allows to obtain AdS space with unbroken SUSY and stabilized volume

One can uplift AdS space to a metastable dS space by adding anti-D3 brane at the tip of the conifold

The role of warping factor inuplifting AdS vacuum to dS

Small z (resolution of conifold singularity)

In our example C was In our example C was 1010-9-9

Small C is necessary for dialing the anti-D3 energy Small C is necessary for dialing the anti-D3 energy to AdS scale to preserve and uplift the minimum to AdS scale to preserve and uplift the minimum

The redshift in the throat plays the The redshift in the throat plays the key role inkey role in

Advantage: source of small parameters

Disadvantage: highly warped region of KS geometry corresponds to conformal coupling of the inflaton field (position of D3-brane in the throat region)

Flatness of the Inflaton Potential and of the Perturbation Spectrum Require

Few possibilities to improve the model are knownFew possibilities to improve the model are known

Supersymmetry and Inflation Hybrid Inflation

F-term, D-term Inflation

Include Volume Stabilization:F-term for KKLT+ Shift Symmetry slightly broken by quantum corrections

Practically D-term Inflation

Linde, 91

Copeland, Liddle, Lyth, Stewart, Wands;Dvali, Shafi, Shafer, 94

Binetruy, Dvali; Halyo, 96; Dvali, Tye, 99

D3/D7 Brane Inflation as D-term InflationDasgupta, Herdeiro, Hirano, R.K.,

2002

Hsu, R. K., Prokushkin, 2003-2004Hsu, R. K., Prokushkin, 2003-2004

Burgess, Kallosh, Quevedo, 2003

Ferrara et al, 2003Ferrara et al, 2003

Inflaton TrenchInflaton Trench

SHIFT SYMMETRYSHIFT SYMMETRY The motion of branes does not destabilize the volumeThe motion of branes does not destabilize the volume

Supersymmetric Ground State of Branes in Stabilized VolumeSupersymmetric Ground State of Branes in Stabilized Volume

Cosmology, Supersymmetry and Cosmology, Supersymmetry and Special GeometrySpecial Geometry

In familiar case of Near Extremal Black HolesNear Extremal Black HolesDUALITY SYMMETRYDUALITY SYMMETRY protects exact entropy

formula from large quantum corrections

DUALITY SYMMETRYDUALITY SYMMETRY (shift symmetryshift symmetry) protects the flatness of the potentialflatness of the potentialin D3/D7 inflation model from large quantum

corrections

The Potential of the Hybrid D3/D7 Inflation Model

is a hypermultipletis a hypermultiplet

is an FI triplet: resolution of the singularityis an FI triplet: resolution of the singularity

Same Potential without Fayet-Iliopoulos term

Flat direction corresponding to the singularityFlat direction corresponding to the singularityin the moduli space of instantons in D3/D7in the moduli space of instantons in D3/D7

D3/D7 BRANE INFLATION D3/D7 BRANE INFLATION MODELMODEL

The mass of D3-D7 strings (hypers) is split due to the presence of the

deformed flux on D7

De Sitter stage- Waterfall- Ground State

DeDe Sitter: Inflation or current accelerationSitter: Inflation or current acceleration

Ground state: D3/D7 bound stateGround state: D3/D7 bound state

Higgs branch: non-commutative instantonsHiggs branch: non-commutative instantons

NS non-commutative instantons:Higgs branch, bound state of D0/D4

D3 can move away from D7 when the deformationparameter vanishes, the moduli space is singular:

there is no de Sitter space

Resolution of singularity of the moduli space of

instantons in D3/D7 Higgs branch

requires that the Coulomb branch has a non-vanishing D-term potential

Deformation-non-commutativity-resolution of singularity

de Sitter space

DBI kappa-symmetric action and non-linear deformed instantons

Seiberg,Witten, 99; Marino, Minassian, Moore, Strominger, 99

D3/D7 bound state and unbroken supersymmetry

Deformed flux on the world-volume Non-linear deformed instanton

Bergshoeff, R. K., Ortin, Papadopoulos, 97

D-term volume stabilization

2 possibilities to make this mechanism working1) Place D7 in highly warped region of space

Instead of anti-D3 add D7 with flux. The D-term potential depends on the ASD deformed flux and volume modulus

Burgess, R. K., Quevedo

2) Use deformation: irrational quantized

cannot be gauged away into

Deformation parameter (non-commutativity)is not quantized, it can be small!

DiscussionDiscussion In the context of non-commutative instantons (Nekrasov-Schwarz, 1998) and

Dirac-Born-Infeld non-linear instantons (Seiberg-Witten, 1999) FI terms are necessary to make the Abelian instantons non-singular.

It is tempting to speculate that in D3/D7 cosmological model with volume stabilization mechanism there is an explanation of the non-vanishing effective cosmological constant

Non-commutativity parameter (FI term in effective theory) is needed to remove the instanton moduli space singularity in the description of the supersymmetric D3/D7 bound state when D3 has dissolved into D7.

The same cosmological model must have a non-supersymmetric de Sitter stage when D3 is separated from D7

Hopefully, with the further development of the theory we will

find an answer to this question

Can we measure the non-commutativity parameters of the internal space

by looking at the sky ?