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Holographic Systematics of D-Brane Inflation
Liam McAllister Cornell
ISM08, Pondicherry December 9, 2008
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Building on:
S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L.M., S. Trivedi, hep-th/0308055. D. Baumann, A. Dymarsky, I. Klebanov, J. Maldacena, L.M., & A. Murugan, hep-th/0607050.D. Baumann, A. Dymarsky, I. Klebanov, L.M., & P. Steinhardt, 0705.3837.D. Baumann, A. Dymarsky, I. Klebanov, & L.M., 0706.0360.
See also:
C. Burgess, J. Cline, K. Dasgupta, & H. Firouzjahi, hep-th/0610320.O. DeWolfe, L.M., G.Shiu, & B. Underwood, hep-th/0703088.A. Krause & E. Pajer, 0705.4682.S. Panda, M. Sami, S. Tsujikawa, 0707.2848.A. Ali, R. Chingangbam, S. Panda, M. Sami, 0809.4941.
D. Baumann, A. Dymarsky, S. Kachru, I. Klebanov, & L.M., 0808.2811.
Based on:
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PlanI. Motivation and background
I. InflationII. Inflation in string theory
II. Case study: warped D-brane inflation III. Direct approach: computation of nonperturbative
superpotentialIV. Systematic method: enumeration of contributing
modesV. Pre-ConclusionsVI. Remarks on primordial tensor signatureVII.Conclusions
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Part I.
Inflation and
String Inflation
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InflationA period of accelerated expansion2 2 2 2 .Htds dt e dx H const
• Solves horizon, flatness, and monopole problems.i.e. explains why universe is so large, so flat, and so empty. A.Guth,1981.
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InflationA period of accelerated expansion2 2 2 2 .Htds dt e dx H const
• Solves horizon, flatness, and monopole problems.i.e. explains why universe is so large, so flat, and so empty.
• Predicts scalar fluctuations in CMB temperature:- approximately, but not exactly, scale-invariant
- approximately Gaussian • Predicts primordial tensor fluctuations
A.Guth,1981.
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Physics of inflation• Scalar field φ with a potential. V(φ) ~ 1016 GeV
• Potential drives acceleration.• Acceleration is prolonged if V(φ) is flat in Planck
units:
φ
V(φ)
12 ( )V
L
A.Linde,1982.A. Albrecht and P.Steinhardt,1982.
22 211 and 1
2pl plV VM MV V
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Quantum fluctuations seed structure formation
NASA/WMAP Science Team
Perturbations are small enough to permit linear analysis, for a while. This is a very clean and tractable system.
Prediction: these perturbations are approximately scale-invariant.
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Current Constraints
0.97 .015 ( 0)0.22 (95%)
sn rr
Expect dramatic improvements in these limits in next 5-10 years: Planck, SPIDER, EBEX, Clover, QUIET, BICEP, PolarBEAR,...
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From WMAP to PLANCK (2010?)
WMAP; C. Cantalupo, J. Borrill, and R. Stompor
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Goal: develop theoretical models of the early universe suitable for an era of precision
cosmology.
NASA/WMAP Science Team
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Since inflation occurred at sub-Planckian energies, why not just use effective QFT?– Consistent if one is careful. However, inflation is UV-sensitive!– So, would like toy models (at least) in which Planck-scale theory is
specified.– Moreover, rare opportunity to probe (some aspects of) Planck-
scale physics.
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Since inflation occurred at sub-Planckian energies, why not just use effective QFT?– Consistent if one is careful. However, inflation is UV-sensitive!– So, would like toy models (at least) in which Planck-scale theory is
specified.– Moreover, rare opportunity to probe (some aspects of) Planck-
scale physics.
String theory provides:• many fundamental scalar fields• considerable UV control
Hope: • use string theory to clarify UV issues in inflation• use UV-sensitive aspects of inflation to constrain string theory – or at
least, constrain string theory models of inflationWarning: many aspects of inflation are not UV-sensitive.
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Use CMB and LSS to probe fundamental physics.
Propose and constrain mechanisms for inflation in string theory.
Focus on correlated signatures (tilt; tensors; non-Gaussianity; cosmic strings; isocurvature;...).
Derive inflaton action in consistent, computable string compactifications with stabilized moduli.
Aim:
Method:
Task:
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Status Report: Inflation in String Theory• We now have
– many scenarios that are partially constructed in string theory.– a few scenarios that are complete or nearly so
• Modern models are far more realistic than those of 5-10 years ago. Moduli problem explicitly solved in a few cases, implicitly in a few more.
• Making correct predictions is still very difficult!– A key lesson across scenarios has been that solving the moduli
problem changes the scenario and can shift ns, r dramatically. So one should be skeptical of predictions for these quantities in the absence of stabilization.
• It is clear that many complete models will soon exist. At present only a few are worked out in enough detail to make correct predictions.
• This talk: one of the best-understood scenarios.
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• Successful inflation requires controlling Planck-suppressed contributions to inflaton potential.
e.g.
The eta problem
2
02P
V VM
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• Successful inflation requires controlling Planck-suppressed contributions to inflaton potential.
e.g.
The eta problem
2
02P
V VM
2 23 pH M V
2
2 13mH
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• Successful inflation requires controlling Planck-suppressed contributions to inflaton potential.
e.g.
The eta problem
2
02P
V VM
2 23 pH M V
2
2 13mH
22 2
02 3 1P
V V m HM
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The eta problem from an F-term potential
2
02P
V VM
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The eta problem from an F-term potential
2
02P
V VM
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The eta problem from an F-term potential
2
02P
V VM
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• Inflation requires controlling Planck-suppressed contributions to the inflaton potential up to dimension 6.
• Moduli stabilization almost always introduces such terms. • Knowing (and controlling) the inflaton potential therefore requires
detailed information about moduli stabilization.
Moduli stabilization and the eta problem
2
02P
V VM
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• Inflation requires controlling Planck-suppressed contributions to the inflaton potential up to dimension 6.
• Moduli stabilization almost always introduces such terms. • Knowing (and controlling) the inflaton potential therefore requires
detailed information about moduli stabilization.• ‘Detailed’ means one usually needs to know string + a corrections,
backreaction effects, etc.• Common approximations (classical, noncompact/large-volume,
probe...) are often insufficient.• String incarnation of (misnamed) supergravity eta problem
(Copeland, Liddle, Lyth, Stewart, & Wands astro-ph/9401011).
Moduli stabilization and the eta problem
2
02P
V VM
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Today’s talk • Focus on a specific, concrete model: warped D-brane
inflation. • Determine form of inflaton potential, incorporating all
relevant contributions.– approached directly, this is highly nontrivial. – We have explicitly computed a wide class of contributions.– to finish the job, we will take a different tack, inspired by
AdS/CFT, that gives the complete answer and reduces to our earlier result in the appropriate limit.
• This gives very substantial progress towards an existence proof.
• It also provides a truly general characterization of possible phenomenological models in this scenario.
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Part II.
D-brane Inflation
initiated in:
Dvali&Tye 1998Dvali,Shafi,Solganik 2001Burgess,Majumdar,Nolte,Quevedo,Rajesh,Zhang 2001
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Warped D-brane inflation
CY orientifold, with fluxes and nonperturbative W(GKP 2001, KKLT 2003)
warped throat (e.g. Klebanov-Strassler)
anti-D3-brane(3+1)d
Kachru, Kallosh, Linde, Maldacena, L.M., Trivedi, 2003
warped throat gives: weak Coulomb potential control of energy scales explicit local geometry dual CFT
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Throat GeometryFinite-length KS throat, which we approximate by the Klebanov-Tseytlin solution:
Can ignore logs and approximate as AdS5 x T1,1 :
+ angles
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Coulomb Potential
• Looks extremely flat, because interaction term is warped.• If this were the whole story, the model would be perfect
– dimensional transmutation explains smallness of slow-roll parameters
– inflation has a natural end, with the possibility of interesting topological defects from tachyon condensation: cosmic superstrings.
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Cosmic superstrings
B. Allen and E.P. Shellard
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Coulomb Potential
• Looks extremely flat, because interaction term is warped.• If this were the whole story, the model would be perfect
– dimensional transmutation explains smallness of slow-roll parameters
– inflation has a natural end, with the possibility of interesting topological defects from tachyon condensation: cosmic superstrings.
• In reality, many more important contributions to the potential!
• Origin: moduli stabilization and other `compactification effects’
• We need to understand and incorporate these effects.
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Does it work?
known in initial proposal
?
introduced by moduli stabilization
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Does it work?
known in initial proposal
One contribution to ∆V determined in 2006-2007:
D. Baumann, A. Dymarsky, I. Klebanov, J. Maldacena, L.M., & A. Murugan, hep-th/0607050.C. Burgess, J. Cline, K. Dasgupta, & H. Firouzjahi, hep-th/0610320.D. Baumann, A. Dymarsky, I. Klebanov, L.M., & P. Steinhardt, 0705.3837.A. Krause & E. Pajer, 0705.4682.D. Baumann, A. Dymarsky, I. Klebanov, & L.M., 0706.0360.
?
introduced by moduli stabilization
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Does it work?
known in initial proposal
One contribution to ∆V determined in 2006-2007:
D. Baumann, A. Dymarsky, I. Klebanov, J. Maldacena, L.M., & A. Murugan, hep-th/0607050.C. Burgess, J. Cline, K. Dasgupta, & H. Firouzjahi, hep-th/0610320.D. Baumann, A. Dymarsky, I. Klebanov, L.M., & P. Steinhardt, 0705.3837.A. Krause & E. Pajer, 0705.4682.D. Baumann, A. Dymarsky, I. Klebanov, & L.M., 0706.0360.
Complete parameterization: D. Baumann, A. Dymarsky, S. Kachru, I. Klebanov, & L.M., 0808.2811.
?
introduced by moduli stabilization
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Part III.
The D3-Brane Potential
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D3-branes in flux compactifications
GKP solutions:
0G
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D3-branes in flux compactifications
GKP solutions:
D3-branes feel no potential in GKP solutions (‘no-scale’), butnonperturbative stabilization of Kahler moduli will spoil this.
0G
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D3-brane vacua in KKLT compactifications
ED3/D7-branes responsible for Kahler moduli stabilization
I will assume that # fermion zero modes = 2.cf. A. Uranga’s talk
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D3-brane vacua in KKLT compactifications
ED3/D7-branes responsible for Kahler moduli stabilization DeWolfe, L.M., Shiu, & Underwood, hep-th/0703088.
For generic A(y), solutions to
i.e., supersymmetric D3-brane vacua, are isolated. But where are they, and what is the potential in between?
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D3-brane vacua in KKLT compactifications
ED3/D7-branes responsible for Kahler moduli stabilization DeWolfe, L.M., Shiu, & Underwood, hep-th/0703088.
For generic A(y), solutions to
i.e., supersymmetric D3-brane vacua, are isolated. But where are they, and what is the potential in between?
We need to know A(y).
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Q. Can one compute A(y) in a special case?
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Q. Can one compute A(y) in a special case?A. Yes: Berg, Haack, & Körs were able to compute it in toroidal orientifolds. hep-th/0404087
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Q. Can one compute A(y) in a special case?A. Yes: Berg, Haack, & Körs were able to compute it in toroidal orientifolds. hep-th/0404087
In case of gaugino condensation, they compute dependence on D3 position y as threshold correction due to 3-7 strings.
D3-D7 string
D3 scalar
gluon
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Q. Can one compute A(y) in a special case?A. Yes: Berg, Haack, & Körs were able to compute it in toroidal orientifolds. hep-th/0404087
Q. What about a case directly applicable to a D3-brane in a warped throat?
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Nonperturbative Effects
• Gaugino condensation on N D7-branes wrapping a four-cycle S4
• Euclidean D3-branes wrapping a four-cycle S4
2
2
8expYM
Wg N
43exp wnpW T VS
Either case: can write 43expw
np
T VW
NS
1 12 22 ( ) ( ) i j
ijds h Y g dx dx h Y g dY dY
44
4 ( )wV d Y g h YS S
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Corrected Warped Volumes
44
4 ( )wV d Y g h YS S
1 12 22 ( ) ( ) i j
ijds h Y g dx dx h Y g dY dY
h = h(Y,X)
(probe approximation)
Including D3-brane backreaction:D3-brane position
44 4
0
0
4 40
( )
( )
( , )
( , )w
V
h X Yh h Y
V d Y g h Y d Y g h X Y
V
S S S
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D3-brane Backreaction
(10)
2 2 63( , ) 2 ( ) ( )Y bgh X Y T X Y Y
• Solve for h• Integrate over S4 to get V(X) • Read off W(X) 3 0 3 ( )exp expnp
T V T V XWN N
44 4
0
0
4 40
( )
( )
( , )
( , )w
V
h X Yh h Y
V d Y g h Y d Y g h X Y
V
S S S
Baumann, Dymarsky, Klebanov, Maldacena, L.M., & Murugan, hep-th/0607050.
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Comparison
Tree-level SUGRABackreaction on warpingGaugino or ED3Comparatively easyMore general geometriesWarping ok
One-loop open stringThreshold correctionGaugino onlyHardToroidal cases onlyUnwarped only
Closed String Giddings-Maharana; Us
Open String BHK
Perfect agreement where comparison is possible.
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A tractable background for computing A(y):a four-cycle embedded in the throat
42 2
1i
i
z
( ) 0if z
Ideally, embedding should be SUSY in KS throat.
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Result for a general warped throat
If N wrapped branes are embedded along
( ) 0if z
then the superpotential correction is
0
1( ) N a
K LT iKW efA zG
1
0 ( ) N
iA A f zso the superpotential is
Ganor, hep-th/9612007Baumann, Dymarsky, Klebanov, Maldacena, L.M., & Murugan, hep-th/0607050.
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Potential from the Kuperstein embedding
42 2
1i
i
z
1( )if z z S. Kuperstein, hep-th/0411097
0
1
1
3log ( , )
aKKL
i i
NTW G A e
K k z z
z
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Holomorphy + throat geometry constrain corrections to potential
332r T
42 3
1i
i
z r
3/ 2
iz 1npW z
3/ 21 ( )z r g
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Holomorphy + throat geometry constrain corrections to potential
332r T
42 3
1i
i
z r
3/ 2iz
1npW z 3/ 2
1 ( )z r g
new term in ∆V
No quadratic correction from this embedding! i.e. still cannot fine-tune inflaton mass per se
3/ 2iz
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Baumann, Dymarsky, Klebanov, L.M., & Steinhardt, 0705.3837; Baumann, Dymarsky, Klebanov, & L.M., 0706.0360. Krause & Pajer, 0705.4682.Dynamics:Panda, Sami, Tsujikawa, 0707.2848.
A Nearly Explicit Model of D-brane inflation
D7-brane embedding:
1( )if z z
∆V does not include a quadratic term. Can fine-tune only locally.
For fine-tuned parameters, an inflection point appears. Inflation can occur near this inflection point.
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Part IV.
Holographic Systematics
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How can we be more complete and systematic?
• We had to assume that the divisor bearing nonperturbative effects sags into the throat. – This is likely not generic, and it imposes strong constraints on the
parameters that only reflect our wish to compute, not the underlying physics.
• We have assumed that all other effects are negligible!– We have thereby neglected more distant divisors, bulk fluxes, distant
antibranes, etc.• Our result may be sensitive to the gluing of the throat
into the compact space.• Purely practically, while it is easy to compute the
superpotential for a general embedding in a general throat, it is slightly subtle and rather involved to read off the complete potential that results.
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A Simple Idea
The D3-brane potential comes from Φ- alone. So we are interested in the profile of Φ-.
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A Simple Idea
The D3-brane potential comes from Φ- alone. So we are interested in the profile of Φ-.
Arbitrary compactification effects can be represented by specifying boundary conditions for Φ- in the UV of the throat, i.e. by allowing arbitrary non-normalizable Φ-
profiles.
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A Simple Idea
The D3-brane potential comes from Φ- alone. So we are interested in the profile of Φ-.
Arbitrary compactification effects can be represented by specifying boundary conditions for Φ- in the UV of the throat, i.e. by allowing arbitrary non-normalizable Φ-
profiles.The warped geometry filters these effects; the leading
contributions are those that diminish least rapidly towards the IR (~ lowest few multipoles).
Normalizable Φ- profiles arise from supersymmetry-breaking at the tip and are already incorporated.
cf. DeWolfe, Kachru, & Mulligan 0801.1520
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Gauge Theory version
Arbitrary compactification effects can be represented by allowing arbitrary deformations of the Lagrangian of the dual (approximate) CFT.
Supersymmetric deformations: Aharony, Antebi, & Berkooz, hep-th/0508080
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Gauge Theory version
Arbitrary compactification effects can be represented by allowing arbitrary deformations of the Lagrangian of the dual (approximate) CFT.
These deformations incorporate couplings to bulk moduli X that get F-term vevs.
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Gauge Theory version
Arbitrary compactification effects can be represented by allowing arbitrary deformations of the Lagrangian of the dual (approximate) CFT.
These deformations incorporate couplings to bulk moduli X that get F-term vevs.
The RG flow filters these effects; the leading contributions are those that diminish least rapidly towards the IR, i.e. the most relevant contributions.
Deformations of the CFT state arise from supersymmetry-breaking at the tip and are already incorporated.
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Scale of the moduli F-terms
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Scale of the moduli F-terms
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Scale of the moduli F-terms
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MethodConsider linearized Φ- perturbations around a finite-length KS throat, which we approximate by AdS5 x T1,1.
In general, many other modes are turned on, but at the linear level they do not couple to D3-branes!
CFT version: interested only in operators that generate a potential on the Coulomb branch.
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Linearization around ISD compactifications
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Linearization around ISD compactifications
So, solve the Laplace equation in AdS5 x T1,1.
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Solution: Kim, Romans, & van Nieuwenhuizen, 1985. Gubser, 1998. Ceresole, Dall’Agata, D’Auria, Ferrara, 1999.
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Solution: Kim, Romans, & van Nieuwenhuizen, 1985. Gubser, 1998. Ceresole, Dall’Agata, D’Auria, Ferrara, 1999.
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Simple case: one mode
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Simple case: one mode
0, 1c c
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What are the lowest modes?
∆=3/2 chiral mode
∆=2 nonchiral mode
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Gauge Theory Description
Klebanov & Witten, hep-th/9807080Klebanov-Witten CFT:SU(N) x SU(N) gauge groupSU(2) x SU(2) x U(1)R global symmetrybifundamentals Ai, Bi
ultraviolet deformation:
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Gauge Theory Description
Klebanov & Witten, hep-th/9807080Klebanov-Witten CFT:SU(N) x SU(N) gauge groupSU(2) x SU(2) x U(1)R global symmetrybifundamentals Ai, Bi
ultraviolet deformation:
contributingchiral operators:
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Gauge Theory Description
Klebanov & Witten, hep-th/9807080Klebanov-Witten CFT:SU(N) x SU(N) gauge groupSU(2) x SU(2) x U(1)R global symmetrybifundamentals Ai, Bi
ultraviolet deformation:
contributingchiral operators:
most relevant chiral operators:
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Discrete symmetries can forbid chiral perturbations
Moreover, the following nonchiral operators are in the supermultiplets of global symmetry currents, and have protected dimension, ∆=2:
If the compactification preserves a discrete symmetry that forbids the leading chiral perturbation O3/2, the dominant compactification effect comes from O2.
e.g., forbids
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Scenario I:Inflection point inflation again!
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Scenario II:wherein the inflaton mass admits fine-tuning
(This is the leading term if discrete symmetries forbid the chiral perturbation.)
Phenomenology: Firouzjahi & Tye, hep-th/0501009
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Part IV.
Pre-Conclusions
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Conclusions, Part I: Method• We have an easy and completely systematic approach to
computing the inflaton potential in warped brane inflation.• Method: consider generic perturbation of ultraviolet region,
sourced by supersymmetry-breaking compactification effects, and focus on most relevant terms.
• Equivalently, perturb dual CFT Lagrangian by most relevant operators.
• Our approach reproduces, extends, and simplifies the results of direct computation of W from wrapped D7-branes.
• We reliably capture totally general effects of compactification, provided that the D3-brane is far from the top and far from the tip.
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Conclusions, Part II: Implications
• Two scenarios: inflection point inflation is most generic possibility. discrete symmetry leads instead to terms that allow
direct fine-tuning of the inflaton mass.
• Study of multifield effects is now straightforward.• Moreover, since we know the structure of the most general
potential for this system, one could try to understand generic predictions.
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Part V.
Primordial Gravitational Waves
from String Inflation
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Vacuum Fluctuations: Tensors
ij ij ij ijg h h e h e
23
3
2( ) ( ) ( ) Th k h k k kk P
May be visible through induced curl of CMB photons’ polarization (B-mode): SPIDER, Clover, QUIET, BICEP, EBEX, PolarBEAR,...
T
S
r PPtensor-to-scalar ratio, a measure of the primordial tensor signal
22
2SH H
P2
2
82T
P
HM
P
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Polarization seen by WMAP
NASA/WMAP Science Team
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E-mode vs. B-mode
Image: Seljak and Zaldarriaga
(curl-free) (curl)
Primordial tensors induce curl of CMB photons’ polarization (B-mode).
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Lyth Bound
Threshold for detection: 210r 310 ?r
next decade
ultimate
2
0.009P MAX
rM
OBSERVABLE TENSORS REQUIRE TRANS-PLANCKIAN FIELD VARIATIONWHICH REQUIRESULTRAVIOLET INPUT.
D.H. Lyth, 1996
2
2
8CMB
eff P MAX
rN M
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Planckian distances
Tensors in CMB
String theory
Lyth bound
geometry of compact space
constraints/predictions
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• Small-field inflation requires controlling Planck-suppressed corrections up to dimension 6
• Large-field inflation requires control of Planck-
suppressed contributions of arbitrary dimension!
• Detectable tensors require super-Planckian displacements, so observations will distinguish these cases.
2 262
1
P
V m HM O
The challenge:
4
0
p
pp
V
Reasonable to enumerate and fine-tunethese terms.
Symmetry clearly essential.
PM
PM
2
0.01P
rM
Lyth 1996
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Computing the Field Range
rmax5
2 2 2Xdr r ds
warped cone over X5
1 12 22 ( ) ( ) i j
ijds h Y g dx dx h Y g dY dY
rR
2
3
5
4 2Vol( )4 ( ')s XR g a Gubser
5 5AdS ×X
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Compactification Volume determines the 4d Planck Mass
MP2
V6w
102
V6w d6y g h(y)(V6
w )throat (V6w )bulk
102 1
2 (2 )7gs
2(a ')4
max 45 5
6 5
4 2 4 2 215 max max2
( ) ( ) Vol( )
Vol( ) 2 ( )
rw
throat
s
RV dr r d h r X dr rr
X R r g r a
4 2 22 max
210
2 ( )sP
g rM a
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Compactification Constraint on the Field Variation
+
1/T3 (2 )3gs(a ')
2
Canonical Inflaton Field2 2 2
3 3 maxT r T r
4d Planck Mass4 2 2
2 max210
2 ( )sP
g rM a
2
4
pM
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Resulting field range, for any scenario involving a D3-brane
in a throat:2
4
PM
N = the number of colors in the dual gauge theory
D. Baumann and L.M., hep-th/0610285.
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Resulting field range, for any scenario involving a D3-brane
in a throat:2
4
PM
N = the number of colors in the dual gauge theory
D. Baumann and L.M., hep-th/0610285.
4.009 .01r
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However, large field ranges, and hence tensor signals, are possible in string inflation – just not in warped D3-brane inflation.
Mechanism:Linear Inflation from Axion Monodromy
L.M., Silverstein,& Westphal, 0808.0706
see also:
Silverstein & Westphal, 0803.3085.
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Natural Inflation in string theory?
Axion shift symmetry protects inflaton potential.
4 cosV f Freese, Frieman, & Olinto, 1990: f > MP
Our idea: recycle a single axion N times.
Banks, Dine, Fox, & Gorbatov, hep-th/0303252:
f > MP not attainable in string theory? (cf. also Svrcek & Witten, hep-th/0605206)
Dimopoulos, Kachru, McGreevy, & Wacker, hep-th/0507205
“N-flation”: use N ~103 axions at once, as a collective excitation.
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Axion Inflation from Wrapped Fivebranes
CY orientifold, with fluxes and nonperturbative W(GKP 2001, KKLT 2003)
warped throat
D5-brane/NS5-brane
(3+1)d
warped throat gives:control of energy scales
2CS2B
S
Σ
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Axion monodromy from wrapped fivebranes
2 1 4 2 35 35 3 22
detd ebgs
V G B L b a a
S
• Fivebrane contribution not periodic: as axion shifts by a period, potential undergoes a monodromy
• This unwraps the axion circle and provides a linear potential over an a priori unlimited field range.
• In practice, controllable over large (>> MP) but finite range.
cf. inflation from D-brane monodromy Silverstein&Westphal, 0803.3085.
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It Works • Backreaction on local geometry can be made small.• Checked consistency with moduli stabilization (KKLT scenario).• Renormalization of Planck mass from many light species is small. For very reasonable parameter choices, the instanton contributions
to the effective action can be neglected, and the additional problems above are addressed.
• Assumptions Summarized:
– NS5-brane wrapping a curve Σ that is in a warped region and stabilized at finite volume.
– Remaining moduli stabilized as in KKLT, by fluxes and strong gauge dynamics (ED3 stabilization may be manageable as well).
– Tadpole canceled by e.g. anti-NS5-brane on distant cycle homologous to Σ
• Reasonable to expect more efficient/minimal realizations.• Core idea is very simple: the axionic shift symmetry protects the
inflaton potential, just as in Natural Inflation.
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Conclusions• We can construct string inflation models that are
realistic, reasonably explicit, and falsifiable.• Well-understood class of small-field hybrid models:
warped D3-brane inflation. Computable local model with systematic incorporation of
compactification effects from moduli stabilization. Signatures include cosmic strings but not tensors.
• New class of large-field models: axion monodromy. Axion shift symmetry naturally controls contributions to
the inflaton action. Robust tensor prediction r=.07 can be falsified in 5-10
years.