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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/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 OmOlCMB
flat
closedopen
How much dark energy is there?
Cmbgg OmOlCMB
+
LSS
How much dark energy is there?
WMAP + SDSS: lots
flat
closedopen
Cmbgg OmOlCMB
+
LSS
How much dark energy is there?
flat
closedopen
Cmbgg OmOlCMB
+
LSS
How much dark energy is there?
flat
closedopen
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 theoryTowards cosmology in type IIB string theory
Dilaton stabilization Giddings, Kachru and Polchinski 2001Dilaton stabilization Giddings, Kachru and Polchinski 2001
Kachru, R. K, Linde, Trivedi Kachru, R. K, Linde, Trivedi 20032003Kachru, 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 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
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 Holes
DUALITY SYMMETRYDUALITY SYMMETRY protects exact entropy formula from large quantum corrections
DUALITY SYMMETRYDUALITY SYMMETRY (shift symmetryshift symmetry)
protects the flatness of the potentialflatness of the potential
in 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 singularity
in 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 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
quantizedcannot 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 ?