高次元統一理論と 宇宙論

1. 高次元統一理論と宇宙論 基礎物理学研究所　小玉英雄

2. Inflation and Dark Energy • Luminosity distance-red shift relation of type Ia SNe • Accelerating expansion of the present universe [Riess et al (1998), Perlmutter et al (1999)] • CMB anisotropy observation (WMAP 2003) • Flatness of the universe • Scale-free primordial fluctuations • Dark energy and dark matter: • Confirmation of primordial inflation • Dark energy/Lambda problem • To be explained by a unified theory including gravity

3. Higher-Dimensional Unification Standard model ⇒ GUT: gauge-sector unification • αunification, hypercharge structure, neutrino mass • Baryon asymmetry, strong CP(Peccei-Quinn symmetry) GUT ⇒ SGUT: boson-fermion correspondence • Dark matter, Λ problem, hierarchy problem SGUT ⇒ Sugra GUT: inclusion of gravity • Flat inflaton potential Sugra GUT ⇒ HD Sugra GUT: matter sector unification • Generation repitition, Cabibo/neutrino mixing, CP violation • Origin of the Higgs in the adjoint representation HD Sugra GUT ⇒ Superstring/M theory • Consistency as a quantum theory, finite control parameters • No Λ freedom (M-theory)

4. Difficulties of SHD Theories Choice of theory • M-theory or 10D superstring theories • Ambiguity due to duality and branes Compactification • What determines the type of compacfication? • Moduli stabilisation • No-Go theorem against accelerating expansion • Identification of our four-dimensional universe SUSY breaking • Mechanism • Control

5. IIB SUGRA Field contents • Bosonic fields • 10D metric • Axion, dilaton: • 3-form flux: • 5-form flux: • Spinor fields

6. Calabi-Yau Compactification Product-type compactification with no flux • N=2 SUSY vacuum on Moduli • Shape moduli • Dilaton-axion moduli τ: 1 • Complex structure moduli of CY: • Size moduli • Kaehler moduli of CY: this includes the volume modulus of CY

7. 4D Effective Theory KK ansatz 4D effective action (tree) where

8. Moduli Stabilisation No scale model[Giddings, Kachru & Polchinski(2002)] • Flux compactification • Discrete set of vacua • Stabilisation of all shape moduli This privides constraints. • The size moduli is not fixed

9. KKLT Model Stabilisation of the volume modulus • When there exists a special 4-cycle Z on Y • Euclidean D3-brane wrapped on Z ⇒ • D7 brane wrapped on Z ⇒ gaugino condensate ⇒ • Combining with the no scale model ⇒Stable supersymmetric AdS vacuum with all moduli fixed Uplifting the vacuum • Adding N anti-D3 branes at the KS throat • Adjusting the number of anti-D3 branes ⇒(Metastable) dS vacuum

10. Warped Compactification Conifold compactification D3 branes ⇒ 5-form flux Klebanov-Strassler solution (2000) • Constant τ, 5-form flux and ISD 3-form flux • CY=deformed conifold regular at r=0 • Large hierarcy [Giddings, Kachru & Polchinski 2002]

11. Brane Inflation Model Brane inflation model • A pair of D3 and anti-D3 brane [Dvali & Tye 1999] ⇒ a flat inflaton potential • The slow roll condition is not satisified unless [Quevedo 2002] KKLMMT model[Kachru , Kallosh, Linde, Maldacena, McAllister & Trivedi 2003] • A mobile D3 brane on the KKLT model (the KS background) ⇒ slow roll inflation

12. Difficulties • Volume modulus – D3 brane position coupling ⇒ a large inflaton mass in general • Difficulties in the Kaeler moduli stabilisation • The volume modulus is not stabilised in a model with a single Kaeher modulus [Denef, Douglas and Florea 2004] • There exist models in which all Kaeler moduli are stabilsed by the instanton effets, but they are not generic [Denef, Douglas, Florea, Grassi & Kachru 2005] • The10D structure of the KKLT/KKLBBT model is not clear • Warped structure is not taken into account in the volume modulus statilisation argument. • Anti-D3 branes at the KS throat produce naked singularity. • In order to evade the No-Go theorem, singular objects or branes/orientifold planes with negative tension are required.

13. No-Go Theorem For any (warped) compactification with a compact closed internal space, if the strong energy condition holds in the full theory and all moludi are stabilized, no stationary accelerating expansion of the four-dimensional spacetime is allowed. • Proof For the geometry from the relation for any time-like unit vector V on X, we obtain Hence, if Y is a compact manifold without boundary, is a smooth function on Y, and the strong energy condition is satsified in the (n+4)-dimensional theory, then the strong energy condition is satisfied on X.

14. Moduli Instability in Warped Compactifications Ansatz • Metric • Form fields General solution [Kodama & Uzawa 2005] If and the metric has no null Killing where

15. Interesting Features • When the volume modulus is time-dependent, it does not factor out from the warp factor as is assumed in most effective four-dimensional theories. • The volume modulus is effectively stabilised near the KS throat. • The instability is associated with full SUSY breaking of the order • When the instability grows, it produces a Big-Rip singularity in the cosmological context. • In order to construct a realistic universe model, full 10D/11D analyses are required. It is also important to make clear how our 4D universe is identified.

16. Summary It is quite likely that the fundamental theory of nature is a higher-dimensional unified theory. However, it is still a challenging task to solve the Λ Problem and construct a natural inflationary universe model on the basis of such a theory at present.