超弦理論とインフレーション

1. 超弦理論とインフレーション 向山信治　（東大理）第１回「超弦理論と宇宙」研究会＠尾道

2. Three mysteries: Inflation, Dark Energy & Dark Matter String Theory? The Cosmic Uroboros by Sheldon Glashow

3. String theory until 2002 Bad thing • No 4-dimensional de Sitter solution with stabilized moduli. • No-go theorem! • Contradict with inflation and dark energy? • No way to reconcile with cosmology??? Three mysteries: Inflation, Dark Energy & Dark Matter The Cosmic Uroboros does not close? String Theory? ?

4. Recent Progress In 2003, a 4-dimensional de Sitter solution was finally found! Kachru, Kallosh, Linde and Trivedi (KKLT) In the previous no-go theorem, non-perturbative effects and branes were not taken into account. dS 4D AdS 6D Compactification (GPK) Volume stabilization (KKLT) NS-NS Fluxes R-R Warped throat （KS) Anti-D-branes (KKLT)

5. 3 models of stringy brane inflation • Wrapped DBI inflation arXiv:0708.4285 [hep-th] with T.Kobayashi and S.Kinoshita • Conformal rapid-roll inflation arXiv:0709.1952 [hep-th] with L.Kofman • Chaotic brane inflation work in progress with L.Kofman

6. Conformal Rapid-roll Inflation arXiv:0709.1952 [hep-th] with L.Kofman

7. The KKLMMT modelKachru, Kallosh, Linde, Maldacena, McAllister, and Trivedi 2003 = brane inflation (Dvali&Tye 1998) in KKLT The inflaton gets the mass D3 inflaton anti-D3 However, inflation with almost scale-invariant spectrum of perturbations requires This can be achieved by considering f-dependent W and fine-tuning it.

8. KKLMMT fine-tuning • mf2 = 2H2 would stop inflation. • This is due to the conformal coupling -Rf2/12 . • However, people have not yet looked at modification of Einstein equation. • We should take it into account![Kofman&Mukohyama 2007]

9. Scalar field with non-minimal coupling to curvature Looks like an additional mass term… Actually, there are more terms!

10. Note that this is NOT a slow roll! Inflation without the KKLMMT fine-tuning! RAPID ROLL INFLATION!!!

11. Attractor behavior de Sitter attractor !!!

12. Inflation with V(f) Starting with: (3) (1) (2) Goal: (1’) (3’) (2’) Definitions:

13. Consistency condition I Starting with: (1) Goal: (1’) (3’) (2’) Definitions: (1) to (1’)

14. Consistency condition II Starting with: (2) Goal: (1’) (3’) (2’) Definitions: (2) to (2’)

15. Consistency condition III Starting with: Goal: (1’) (3’) (2’) Definitions: (3) to (3’)

16. Condition for inflation with conformal coupling • Usual slow roll condition+ additional conditions • The 3rd condition is not satisfied by power-law potentials. • The D/anti-D potential satisfies it!

17. Chaotic Brane Inflation Work in progress with L.Kofman

18. Chaotic Inflation driven by brane motion Large motion of anti-D3-brane In 4D, V~lf4 in progress, with L.Kofman 4D universe 6D space Anti-D3-brane Distance to the bottom = Inflaton

19. Phase portrait for an anti-D3-branewithout non-rel. approximation Chaotic Inflation!

20. Length of the throat Can we have ? For stack of anti-D3s, the answer is NO. Better for stack of wrapped D5s and D7s.

21. String theorists & Cosmologists must talk to each other! The Cosmic Uroboros by Sheldon Glashow

22. 質問 • Warped conifold内のコンパクトな4-cycleで、角度方向に対して安定なものは？ • D5とD3のbound state（橋本さん＆今村さん＠昨日の飲み会）で、D5の枚数をどこまで増やせるか？ • Conformal coupling (m2=2H2)を、対称性（AdS?)でprotectすることは可能か？

24. Wrapped DBI inflation arXiv:0708.4285 [hep-th] with T.Kobayashi and S.Kinoshita

25. The KKLMMT modelKachru, Kallosh, Linde, Maldacena, McAllister, and Trivedi 2003 = brane inflation (Dvali&Tye 1998) in KKLT The inflaton gets the mass D3 inflaton anti-D3 However, inflation with almost scale-invariant spectrum of perturbations requires This can be achieved by considering f-dependent W and fine-tuning it.

26. KKLMMT fine-tuning • mf2 = 2H2 would stop inflation. • This is based on dynamics of a scalar with canonical kinetic term . • However, the brane position is described by nonlinear DBI kinetic action. • We should take it into account! [Silberstein&Tong 2003]

27. DBI inflation: model description • Mobile D3-brane with relativistic speed • Action • Energy density & Pressure DBI part r : radial position of the brane

28. DBI inflation: good things • A kind of k-inflation with general sound speed:a new model of stringy inflation! • No need for slow-roll:a remedy to the η problem(KKLMMT fine-tuning)? • Large non-Gaussianity: signature of stringy inflation?

29. DBI inflation: bad thing Baumann&Mcllister 2006; Lidsey&Huston 2007 • (UV) DBI inflation with large non-Gaussianity seems inconsistent with WMAP data. • Can be consistent only in the limit when it goes back to the slow-roll KKLMMT inflation. • The reason: large but not too large |fNL| (say, 20<|fNL|<300) large r large Df/MPl i) large Df long throat large V6 ii) not large MPl not large V6 confliction

30. Useful equations to derive constraints

31. Our attempt: wrapped DBI Kobayashi, Mukohyama and Kinoshita 2007 • The essential reason for the inconsistency of DBI inflation with WMAP data:large Df long throat • For D3, this is inevitable: • For a wrapped D5 or D7, we can getlarger Df from the same throat! • This significantly ameliorates the confliction! large volume of 2- or 4-cycle

32. More stringent bound on wrapped DBI inflation • Wrapped DBI inflation with large non-Gaussianity still requires a long throat. (Not as long as for D3 but still long.) • Known Calabi-Yau manifold cannot sustain such a long throat, (let alone for D3). long throat large background chargelarge Euler number of CY, exceeding the known maximal valuec = 1820448 [Klemm,Lian,Roan&Yau 1997]

33. Summary of wrapped DBI inflation • Wrapping D5 or D7 over a cycle changes the relation between the brane position and the inflaton field. • This significantly ameliorates the confliction between (UV) DBI inflation and WMAP data. • However, successful wrapped (UV) DBI inflation requires Euler number larger than the known maximal value.

35. e-foldings & mass hierarchy • e-foldings: a f ~ const.N ~ ln ( fi / fe ) • Mass hierarchy a la Randall-Sundrum :M / Mpl ~ e-N • Enough inflation vs TeV gravity: N ~ 62 + ln ( M / 1016GeV ) M ~ TeVThese conditions are equivalent!

36. Summary of conformal rapid-roll inflation • Conformal coupling does NOT necessarily spoil inflation. • Brane / anti-brane inflation may work without severe fine-tuning. • E-foldings & mass hierarchy are related. • Modulated reheating can generate scale-invariant density perturbation.

37. Summary of this talk • It seems that we can really enjoy cosmology in the framework of string theory. • Model I: Wrapping D5 or D7 over a cycle ameliorates the confliction between the DBI inflation and WMAP data.However, a successful DBI inflation requires large Euler number of CY manifold. • Model II: Conformal rapid-roll inflation is possiblewithout the KKLLMT fine-tuning. The mass hierarchy and e-foldings are related! • Model III: Chaotic brane inflation may be possible with wrapped D7.