Gravity Waves and Linear Inflation from Axion Monodromy

1. Gravity Waves and Linear Inflation from Axion Monodromy McAllister L, Silverstein E, Westphal A arXiv: 0808.0706 [hep-th] Journal Club 2008.9.29 Reported by Hideo Kodama

2. Background

3. Flatness: |k|<0.01 Homogeneity Scalar perturbations Amplitude Spectral index: ns¼ 0.96 Adiabaticity during the period teq < t < trec Good gaussianity (Cf. CMB anomalies, cold spots, supervoid, …) Tensor/scalar ratio: r=h2/R2 <0.3. Reheating Baryon asymmetry Thermal leptogenesis ) Tr> O(109) GeV [Buchmuller et al 2005] Gravitino problem Tr < 108 GeV for m3/2>10keV [Kawasaki, Moroi 1995; Kawasaki, Takahashi, Yanagida 2006] From Observational Cosmology Recent CMB observations support predictions of the inflationary universe scenario on LSS of the Universe: Can we construct an inflation model explaining these observational features in the framework of unified theory?

4. new influm Chaotic influm Phenomenological INFLUMs • Small field model • || < Mpl • Fine-tuned flat potentials • Fine-tuned initial condition • Sensitive to quantum corrections • Examples: New influm, Racetrack model • Large field model • || À Mpl • Simple potentials such as m22 • Natural initial conditions • Ubiquitous inflation • Smooth exit to the hot FRW universe • Sensitive to SUGRA corrections • Examples: Chaotic influm • Hybrid model • Multiple (small) fields • Simple potential for the main inflaton • High temperature reheating • Sensitive to SUGRA corrections • Exmaples: hybrid influm, D3-D7 brane influm

5. String-Based Inflation Models • Natural inflation[Freese K, Frieman JA, Olinto AV 1990; +Adams FC, Bond JC 1993] • Brane inflation model[Dvali GR, Tye SHH 1999] • KKLT model: >0 vacuum by flux compactification and D/anti-D branes [Kachru S, Kallosh R, Linde A, Trivedi S 2003]． • KKLMMT model: KKLT + brane inflation [Kachru S, Kallosh R, Linde A, Maldacena J, McAllister L, Trivedi S 2003] • D3/D7-brane inflation model[Hsu JP, Kallosh R, Prokushkin S 2003] • DBI inflation model[Silverstein E, Tong D 2004] • Racetrack model[Blanco-Pillado JJ, Burgess CP, Cline JM, Escoda C, Gomez-Reino M, Kallosh R, Linde A, Quevedo F 2004] • Tachyon inflation model[Cremades D, Quevedo F, Sinha A 2005] • N-flation[Dimopoulos S, Kachru S, McGreevy J, Wacker J 2005] • Better racetrack model[Blanco-Pillado JJ, Burgess CP, Cline JM, Escoda C, Gomez-Reino M, Kallosh R, Linde A, Quevedo F 2006] • Monodromy brane inflation [Silverstein E, Westphal 2008] • Axion linear inflation [ McAllister J, Silverstein E, Westphal 2008] Cf. ''A Delicate Universe'':  problem in D3-D7 model [Baumann D, Dymarshy A, Klebanov IR, McAllister L, Steinhardt PJ 2007]

6. Eta Problem ,¿1 required! • Single inflaton slow roll model • Slow roll parameters • Potential in 4D N=1 SUGRA • Kahler potential: • Superpotential: W(z) (holomorphic) • Kahler F-term correction to the inflaton mass • -problem The eta problem may be solved if the theory has a shift symmetry Kawasaki M, Yamaguchi M, Yanagida T 2000

7. Linear Inflation

8. Basic Idea • Axion as inflaton • Flat potential protected by a shift symmetry. Cf -problem • The instanton correction produces a periodic potential. • This makes a non-negligible contribution to  leading to an uncomfortable tilt of the spectrum in the small field framework. • It is difficult to make the period À Mpl. • QCD axions in 10D superstring theories • 2 cycles  in the compactified internal space b =s B2 , c =s C2 ) axions • Coupling with D5/NS5 branes wrapping 2 cyles • DBI action ) Potential for the axion ) Monodromy unwrapping of the axion field For D5-brane wrapped on a 2-cycle  of size l(’)1/2, this gives the potential • S-duality: D5 , NS5, B2, C2

9. Observational Predictions • Slow roll inflation • Evolution equations • The slow roll conditions • The e-folding number • Perturbations • Scalar perturbation amplitude

11. Stringy Realisation

12. Conditions for Controlled Inflation • The inflaton (QCD) axion should belong to the physical spectrum. • No significant correction to the linear potential. • No significant backreaction to the warped geometry from the brane charges. • No significant correction of MPl by light particle species due to large brane charges. • No siginificant influence of the axion potential to the moduli stabilisation and vice versa.

13. Specific Models I:Warped IIB CY Compactifications • Flux compactification of type IIB theory ) N=1 4D sugra with fixed complex moduli • Geometry¼ AdS5£ X5 (warped throad region) • Flux: ISD 3-form flux + SD 5-fom flux (D3 branes) • Orientifold projection ) O3/O7 branes with negative tensions to satisfy the tad pole condition • Non-perturbative effects of instantons/gaugino condensates ) Stabilisation of Kahler moduli with (light) QCD axion(s). ) adS4 SUSY vacua. • Uplifting of the vacuum energy by anti-D3 branes ) vacuum with small >0 • Z2 related 5B-anti 5B branes at the tips of the throats. ) a linear potential for the axion ) chaotic inflation

14. Physical Spectrum • Simple CY compactification of IIB ) N=2 SUGRA • Complex moduli: h2,1 (2 the vector multiplet) • double-tensor multiplet • axio-dilaton: =C0 +ie- • b2 + i c2 (the 4D part) • Kahler moduli : h1,1 (2 the hypermultiplet) • TA ;Im TA=A • Axions: h1,1 (2 the hypermultiplet) • bA, cA) GA= cA - bA • The tadpole condition requires a orientifold projection in CY flux compactifications. • H1,1= H+1,1 + H-1,1 3 (, I) • H+1,1: T • H-1,1: Gl=cl- bI In most part of the paper, a simplest case with h1,1+=2 (vL, v+) and Re TL/L4 is discussed, but an explicit model is not given.

15. Axions from string theories • Axion • Derivative coupling to other fields ) shift symmetry • Model-independent axion • B2 field in 4D spacetime: • The gauge invariance under the gauge trf  B=d L guarantees the shift symmetry. • The coupling of B2 to gauge fields can be derived from the anomaly cancelation condition • Model-dependent axions • B2 field and Cp fields in the internal space: b =s B2 , c =s Cp • The coupling of b to gauge fields can be derived from the GS counter-term • The coupling of c to gauge fields comes from the D-brane Chern-Simons term Svrcek P, Witten E: Axions in string theory, JEHP06 (2006) 051:

16. Potential corrections • Flux couplings • In general, the Chern-Simons corrections to Fp produce a mass term for the axion. • However, in type IIB flux compactification, the ISD condition guarantees the non-existence of this correction.[Grimm & Lous 2004] • problem for B • The instanton effect stabilises not the volume directly, but rather TL. • Then, the Kahler potential produces m2 of order H2, leading to the  problem, as in the standard case. • The axions of the RR-form origin, c, do not suffer from this problem.

17. Instanton effects: • In general, the non-perturbative effect due to ED1-DE3 interactions (C in W) may produce uncontrollable large corrections to W. ) It is expected that this problem does not arise if the Kahler moduli are stabilised by the gaugino condensate on D7 branes with SU(NL) SYM fields. • Corrections to the Kahler potential can be neglected if 1¿ v+¿ c gs. • ED1 correction to  • Backreactions to the moduli

18. Backreaction on the geometry • The branes in effect carry Nw» a/(2)2 units of D3-brane charge. They modify the geometry on the scales of • Hence, in order to avoid significant backreaction on compactified geomety, Nw has to satisify where R? is the curvature scale perpendicular to the branes. • From the condition, Umod¸ V ' 2.4£ 10-9 MPl4, and we obtain Cf. A successful inflation requires

19. Other conditions • Constraints from the number of light species • Effectively large D3-brane charge Nw) Nw2 light species. • For rcore¿ L’1/2, the correction to MPl by them can be neglected from the AdS/CFT correspondence (???).

20. Toy model

21. Summary KKLT + an axion inflaton with a linear potential due to 5Bs. • Nice features • Inflaton is a QCD axion field that is required to resolve the strong CP problem. • Large field chaotic inflation can be realised in the string framework avoiding the  problem. • Consistent with the present observational constraints. • Predicts a relatively large value for the tensor/scalar ratio. • Problems • It is not certain whether a good model can be actually constructed in a realistic string compactification. • Utilises anti-D3 branes that break SUSY explicitly and invalidiate the usage of the N=1 D=4 SUGRA framework as the 4D effective theory. • Warp is not taken into account in constructing 4D effective theory, which is common to most models.