Renata Kallosh Davis, May 16, 2004 Stanford Stanford Deformation, non-commutativity and cosmological constant problem

Outline Observational data on 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 Irrational deformation (non-commutativity) parameter 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

Cmbgg OmOl CMB flat closed open How much dark energy is there?

Cmbgg OmOl CMB + LSS How much dark energy is there? WMAP + SDSS: lots flat closed open

Cmbgg OmOl CMB + LSS How much dark energy is there? flat closed open

Cmbgg OmOl CMB + LSS How much dark energy is there? flat closed open Tegmark et al, 2004

Cosmological Concordance Model n Early Universe Inflation n Near de Sitter space n 13.7 billion years ago n During 10^{-35} sec n Current Acceleration n Near de Sitter space n Now n During few billion years

DARK ENERGY n Total energy in 3d flat FRW universe O DARK n 70% of the total energy of the universe is DARK

Cosmological Constant (CC) Problem n The simplest form of dark energy: CC

String Theory and Cosmology n 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 stabilization Giddings, Kachru and Polchinski 2001 Kachru, R. K, Linde, Trivedi 2003 Kachru, R. K., Maldacena, McAllister, Linde and Trivedi 2003 Landscape Susskind Flux Vacua Douglas Volume stabilization, KKLT

The throat geometry has a highly warped region Deformed Conifold Copeland, Myers, Polchinski picture

Volume stabilization n Warped geometry of the compactified space and nonperturbative effects allows to obtain AdS space with unbroken SUSY and stabilized volume n One can uplift AdS space to a metastable dS space by adding anti-D3 brane at the tip of the conifold n Warped geometry of the compactified space and nonperturbative effects allows to obtain AdS space with unbroken SUSY and stabilized volume n 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 n Small z (resolution of conifold singularity) In our example C was 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 n Advantage: source of small parameters n 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 n Hybrid Inflation n F-term, D-term Inflation n Include Volume Stabilization: F-term for KKLT+ Shift Symmetry slightly broken by quantum corrections n Practically D-term Inflation Linde, 91 Copeland, Liddle, Lyth, Stewart, Wands; Dvali, Shafi, Shafer, 94 Binetruy, Dvali; Halyo, 96; Dvali, Tye, 99 n D3/D7 Brane Inflation as D-term Inflation Dasgupta, Herdeiro, Hirano, R.K., 2002 Hsu, R. K., Prokushkin, Burgess, Kallosh, Quevedo, 2003 Ferrara et al, 2003

Inflaton Trench SHIFT SYMMETRY The motion of branes does not destabilize the volume The motion of branes does not destabilize the volume n Supersymmetric Ground State of Branes in Stabilized Volume

Cosmology, Supersymmetry and Special Geometry Near Extremal Black Holes n In familiar case of Near Extremal Black Holes DUALITY SYMMETRY DUALITY SYMMETRY protects exact entropy formula from large quantum corrections n DUALITY SYMMETRYshift symmetry n DUALITY SYMMETRY (shift symmetry) flatness of the potential 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 Fayet- Iliopoulos 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 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 working 1) 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!

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 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 ?