Deformation, non-commutativity and cosmological constant problem Renata Kallosh Stanford Davis, May 16, 2004 Outline 1. 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) parameter in 6,7,8,9 space CC in 0,1,2,3 space. Replace D0/D4 by 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

How much dark energy is there? ed os cl t fl a e n op CMB Cmbgg OmOl How much dark energy is there?

+ ed os cl t fl a e n op CMB Cmbgg OmOl

LSS WMAP + SDSS: lots How much dark energy is there? + LSS ed

os cl t fl a e n op CMB Cmbgg OmOl How much dark energy is there? +

LSS Tegmark et al, 2004 ed os cl t fl a e n op CMB

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

Current Acceleration Near de Sitter space Now During few billion years DARK ENERGY

Total energy in 3d flat FRW universe O 70% of the total energy of the universe is DARK Cosmological Constant (CC) Problem

The simplest form of dark energy: CC String 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?

Towards cosmology in type IIB string theory Dilaton Dilaton stabilization stabilization Giddings, Giddings, Kachru Kachru and and Polchinski Polchinski 2001 2001

Volume stabilization, KKLT Landscape Susskind Kachru, Kachru, R. R. K, K, Linde, Linde, Trivedi Trivedi 2003

2003 Flux Vacua Douglas Kachru, R. K., Maldacena, McAllister, Linde and Trivedi 2003 Deformed Conifold Copeland, Myers, Polchinski picture

The throat geometry has a highly warped region 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 in uplifting AdS vacuum to dS Small z (resolution of conifold singularity) In our example C was 10-9 Small C is necessary for dialing the anti-D3 energy to AdS scale to preserve and uplift the minimum

The redshift in the throat plays the key 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 known Supersymmetry and Inflation Hybrid Inflation

F-term, D-term Inflation D3/D7 Brane Inflation as D-term Inflation Linde, 91 Copeland, Liddle, Lyth, Stewart, Wands; Dvali, Shafi, Shafer, 94 Binetruy, Dvali; Halyo, 96; Dvali, Tye, 99 Dasgupta, Herdeiro, Hirano, R.K.,

2002 Burgess, Kallosh, Quevedo, 2003 Include Volume Stabilization: F-term for KKLT+ Shift Symmetry Ferrara et al, 2003 slightly broken by quantum corrections Hsu, R. K., Prokushkin, 2003-2004

Practically D-term Inflation Inflaton Trench Supersymmetric Ground State of Branes in Stabilized Volume SHIFT SYMMETRY The motion of branes does not destabilize the volume Cosmology, Supersymmetry and

Special Geometry In familiar case of Near Extremal Black Holes DUALITY SYMMETRY protects exact entropy formula from large quantum corrections DUALITY SYMMETRY (shift symmetry) symmetry protects the flatness of the potential in D3/D7 inflation model from large quantum corrections

The Potential of the Hybrid D3/D7 Inflation Model is a hypermultiplet is an FI triplet: resolution of the singularity Same Potential without FayetIliopoulos term Flat direction corresponding to the singularity in the moduli space of instantons in D3/D7

D3/D7 BRANE INFLATION MODEL 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 De Sitter: Inflation or current acceleration Ground state: D3/D7 bound state

Higgs branch: non-commutative instantons NS non-commutative instantons: Higgs branch, bound state of D0/D4 D3 can move away from D7 when the deformation parameter 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 Bergshoeff, R. K., Ortin,

Papadopoulos, 97 Deformed flux on the world-volume Non-linear deformed instanton D-term volume stabilization Instead of anti-D3 add D7 with flux. The D-term potential depends on the ASD deformed flux and volume modulus 2 possibilities to make this mechanism working 1) Place D7 in highly warped region of space 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! Discussion 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 nonvanishing 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 nonsupersymmetric de Sitter stage when D3 is separated from D7 Can we measure the non-commutativity parameters of the internal space

by looking at the sky ? Hopefully, with the further development of the theory we will find an answer to this question