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AdS/CFT on the Brane --- Holographic Cosmology Talks... · Gradient Expansion Method The idea, the...
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AdS/CFT on the Brane
--- Holographic Cosmology ---
Jiro Soda 早田次郎Kyoto University
5th Aegeon Summer School, Islands of Milos, Greece, 25 Sep. 2009
• Braneworld Effective Action at Low Energies and AdS/CFT correspondence, Phys.Rev. D66, 043526 (2002);
• Radion and Holographic Brane Gravity, Phys.Rev. D66, 083506 (2002);
based on works with Sugumi Kanno
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Goal of this talk As you have already learned, AdS/CFT has been applied to various phenomena on various scales
QCD Condensed matter Cosmology Black holes
Here, I would like to discuss braneworld cosmology from a view of AdS/CFT. Namely, I will explain AdS/braneworld correspondence.
Admittedly, the role of AdS/CFT in braneworld has not been fully understood. I just mention what I understand.
Goal of this talk is to convince you that AdS/CFT is related to brane cosmology. In doing so, we show how the cosmological evolution on the brane is affected by the bulk geometry.
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Introduction to higher dimensions
4-dimensions
Extra-dimensions
Kaluza-Klein mechanism
Both matter and gravity live in the higher dimensions
Extra-dimensions are compactified to the Planck scale
Traditional idea
Hence, our universe is essentially 4-dimensions.
How to reconcile extra-dimensions with our experience?
Unified theory such as string theory predicts that our space-time is more than 4-dimensions.
This is an attractive idea, but not a unique one.
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Braneworld
4-d (mem)brane is embedded in higher-dimensions
Matter on the brane Gravity in the bulk
4-dimensions Bulk spacetime
graviton
e, p, photon etc.
Akama (1982) , Rubakov & Shaposhnikov (1983) Horava & Witten (1996), Arkani-Hamed et al (1998)
Closed string ~ graviton
Open string ~ matter
D-brane
compactification scale > Planck scale It could be 1mm!
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Warped compactification: Randall-Sundrum models (RS)
5-d AdS spacetime
From string cosmology point of view, the warped compactification is more interesting and has relevance to AdS/CFT.
Our universe
The spacetime is shrinking towards infinity. Hence, volume is finite although an extra-dimension is not compact.
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Issue in the cosmology side
Conventional picture of the evolution of cosmological structure.
Observed by WMAP
Fluctuations are generated by inflation
How the evolution of cosmological structure is modified in the braneworld picture?
? Effects of the bulk geometry
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Plan of my talk
1. RSII single brane models 2. Geometrical Holography 3. RSII via AdS/CFT 4. Gradient expansion method 5. Effective equations and CFT 6. RSI two-brane models 7. Effective equations and radion 8. Conclusion
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RSII: Single-brane Model Randall & Sundrum (1999)
: 5-d metric
: 4-d induced metric
: 5-d Ricci scalar
5d
Bulk Curvature scale
Brane tension Coupling constant
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Extrinsic curvature
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Geometrical Holography 4-d components of 5-d Einstein tensor
``electric” part of Weyl tensor
junction condition
Shiromizu-Maeda-Sasaki equation
4-d Einstein tensor
high energy corrections
bulk effects
Shiromizu, Maeda and Sasaki (2000)
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Effective Friedman equation
Homogeneous Cosmology
Dark Radiation
Unknown!
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AdS/CFT correspondence
This part contains brane tension, 4-d gravity and higher curvature terms!
Newtonian Gravity
Gubser (2001)
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AdS/CFT correspondence
Geometrical holography
How are these two pictures related?
What is the precise meaning of AdS/CFT on the brane?
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How can the high energy effect be recovered?
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Strategy to reveal the relation
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Since the brane is not located at the boundary of AdS, CFT is a cut off theory which is a vague concept.
Geometrical holography is a kind of cheating because equations of motion have never been solved.
We now solve bulk geometry using the technique common in AdS/CFT and derive the low energy effectve theory without resorting to AdS/CFT correspondence.
The method we use is the gradient expansion method.
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Basic Eqations
Evolution Equation
Hamiltonian Constraint
Momentum Constraint
Extrinsic curvature
Junction condition
symmetry energy momentum tensor of matter
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Energy density on the brane << brane tension
One can solve bulk Eqs. by expanding fields in the order of
We assume:
Gradient Expansion Kanno & Soda (2002)
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Gradient Expansion Method
The idea, the so-called gradient expansion method, presented in this talk goes back to Lifshitz and Khalatnikov 1963, subsequently it has been developed by Tomita 1975, and later by Salopek 1992, Comer et al. 1994, Soda et al. 1995, and many others. There, inflationary universe has been envisaged. In the context of braneworld, a relation to AdS/CFT correspondence has been pointed out by Gubser 2001. The gradient expansion method has been applied to braneworld cosmology by Kanno and Soda 2002.
A Historical Remark
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First order Solutions
constant of integration
Hence, can be interpreted as energy momentum tensor of CFT.
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Effective Eqs. on the Brane
Junction condition
Regularity requires
Einstein equation on the brane
If black holes exists in the bulk, we need to include dark radiation. This direction is related to the recent hydrodynamics calculations.
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Second order corrections By solving Hamiltonian constraint and integrating evolution equation, we have
where we have solved momentum constraint which yields three constant of integration. One of which is a tensor field satisfying a conservation law
Other two parameters are coeffcients of tensors
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Trace anomaly!
: const. of integration
CFT = const. of integration
Bulk gravitational waves
Junction condition
gives
Effective equation on the brane and CFT
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Projected Weyl tensor
Trace anomaly
Thus, one can reveal the relation between the geometrical holography and AdS/CFT correspondence.
AdS/CFT can be related to geometrical holography
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Cosmological Perturbation Theory
The energy momentum tensor can be related to the effective action by
Correlation function can be calculated as
Then cosmological perturbation theory can be
CFT generates a nonlocal effect on cosmological fluctuations. Detailed analysis requires a numerical work.
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RSI: Two-brane Model Randall & Sundrum (1999)
: 5-d metric
: 4-d metric
: 5-d Ricci scalar
Brane tension Coupling constant
RS fine tuning
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Linear analysis
Geometrical holography
How are these two pictures related?
Brans-Dicke Theory?
Is there any AdS/CFT description of RSI?
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The constant of integrations can be fixed by boundary conditions.
Einstein Eq. on the positive tension brane
Einstein Eq. on the negative tension brane
Gradient Expansion Method
Radion
Warp factor
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Scalar-Tensor Theory Brans-Dicke Theory
Eq. of motion for the radion
Eq. of motion for the metric
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In the single brane limit , vanish!
Geometrical holography & Brans-Dicke Theory
Two equivalent representations!
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Effective action for two-brane system Kanno & Soda (2002)
conformally invariant scalar
Einstein frame potential
The brane is inflating in spite of the steep potential!
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Warped brane Inflation
Kachuru et al (KKLMMT 2003)
KSスロート
D3ブレーン 反D3ブレーン
インフラトン
: warp factor
フラックス付き カラビ・ヤオ
eta problem
The previous result shows the difficulty in brane inflation.
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Summary
The leading correction of brane cosmology comes from CFT!
How to formulate AdS-BH/braneworld correspondence?
AdS/CFT may play a more important role in brane cosmology.
What I do not understand is
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We have shown the equivalence between geometrical holography and AdS/CFT correspondence.
Two brane system can be described by conformaly invariant radion action. In this sense, AdS/CFT correspondence is still relevant.