Perturbative vs Non- Perturbative Effects for Moduli Stabilisation and Cosmology J. Conlon, FQ [hep-ph] M. Cicoli, J. Conlon, FQ, [hep-th] + to appear J. Conlon, R. Kallosh, A. Linde, FQ, to appear F. Quevedo, Cambridge. MPI 2007.

Moduli Stabilisation and Supersymmetry Breaking

KKLT Scenario Type IIB String on Calabi-Yau orientifold Turn on Fluxes ∫ a F 3 = n a ∫ b H 3 = m b Superpotential W = ∫ G 3 Λ Ω, G 3 = F 3 –iS H 3 Scalar Potential: V= e K |D a W| 2 Minimum D a W = 0 Fixes U a and S T moduli unfixed: No-Scale models Size of cycle a = U a GKP …GKP, KKLT, …

To fix Kähler moduli: Non-perturbative D7 effects Fluxes Non-perturbative Volume SUSY AdS minimum (W 0 << 1)

ALL MODULI STABILISED !?

Exponentially Large Volumes

Argument: In general: Then: Usually V 0 dominates but V 0 =0 (no-scale ) Dominant term is V J unless W 0 <<1 (KKLT)

Exponentially Large Volumes Perturbative (alpha’) corrections to K At least two Kähler moduli (h 21 >h 11 >1) Example : BBCQ, CQS (2005)

String scale: M s 2 =M 2 Planck /V W 0 ~1-10 Exponentially large volumen !!! a s ~2π/g s N Robustness: CQS, C (2005), BHP (2007)

General Conditions for LARGE Volume h 12 > h 11 At least one blow-up mode (point-like singularity) Blow-up mode fixed by non-perturbative effects, volume by alpha’ corrections For N small blow-up modes, there are L=h 11 -N small -1 flat directions at tree-level These directions are usually lifted by perturbative effects L moduli lighter than volume!

String Loop Corrections Berg-Haack-KorsBerg-Haack-Pajer Leading order correction for V is alpha’ but string loops relevant to fix other moduli

Example: K3 Fibrations K3 Fibration with 2 Kahler moduli [1,1,2,2,6] model No LARGE volume

K3 Fibration with 3 Kahler moduli (one blow up) tau3 small, tau 1 and tau 2 LARGE

PHENOMENOLOGY (LARGE Volume)

The Standard Model in the CY

Standard Model on D7 Branes Solve hierarchy problem M string = GeV! W 0 ~1 (no fine tuning) Kahler potential for chiral matter computed Conlon, Cremades, FQ (2006)

Chiral Matter on D7 Branes Soft SUSY Breaking terms Universality!Universality! No extra CP violation!No extra CP violation! M i = m 3/2 / log (Mp/m 3/2 )M i = m 3/2 / log (Mp/m 3/2 ) String scale GeVString scale GeV Solves hierarchy problem!Solves hierarchy problem! Conlon et al. Simplest case λ=1/3

Stringy source of universality (approximate) CP Violation Physical phases vanish ! Also: Anomaly mediation suppressed !!! Approximate no-scale + locality See: Conlon [hep-th]

COSMOLOGY (Inflation, Cosmological moduli problem, etc.)

KAHLER MODULI INFLATION Candidate Inflatons: Closed string modes  Kahler Moduli (size/axion)  Small modulus inflaton

Kähler Moduli Inflation Calabi-Yau: 21 > 11 h 21 >h 11 >2 volume τnτn V Conlon-FQBond-Kofman-Prokushkin Small field inflation No fine-tuning!! 0.960<n<0.967

Open Questions Loop effects tend to spoil flatness Non tensor perturbations (r=T/S<<<1)? Tension phenomenology vs cosmology Gravitino mass 1 TeV/Gravitino mass >> 1 TeV ?? (string scale GeV/ string scale ~ GUT scale)

Possible ways out Low scale inflation? Runaway before reheating? Conlon, Kallosh, Linde, FQ Ross et al, …

Runaway + Oscillations after inflation

No Overshooting problem!!!

New Candidate Inflatons L=h 11 -N small -1 `almost’ flat directions are light Mass < Volume modulus mass=V -3/2 ~H Recall η ~ mass 2 /H Cicoli et al.

After Inflation

Physics of Moduli Fields Dilaton and Complex Structure Small (heavy) Kahler moduli Large (light) Kahler modulus Moduli masses:

Physical Fields Decay Rates

Summary

Other Cosmological Implications Cosmological moduli problem Observational implications of light volume modulus? U,S: trapped at their minimum T: except for volume, heavy ad decay fast ! (No CMP nor gravitino overproduction) Volume: (mass MeV) CMP Gamma rays, e + -e - J.Conlon, FQ DCQR, BKN

The 511 keV Line

INTEGRAL/ SPI 511 keV line Light Modulus χ: Dark matter? Mass 1MeV, coupling to electrons dominant 511 keV from volume modulus decay? (prediction!)

Intensity INTEGRAL

General Picture N small heavy Kahler moduli (cosmological harmless) Volume modulus (CMP keV line+ predicted monochromatic line) L lighter moduli (candidate inflatons, CMP, new monochromatic lines!! (observable?)) CMP (late low energy inflation?)

Solution of CMP? Thermal Inflation Lyth+Stewart (1995) Number of e-folds N~10 dilutes moduli

Conclusions Non-perturbative effects fix only a fraction of Kahler moduli (N small ) Alpha’ effects fix volume to exponentially large values Loop corrections needed to fix the rest of Kahler moduli Interesting cosmology (gravitino mass problem + overshooting problem can be handled, still some tuning) Astrophysics (forest of monochromatic lines)