אוניברסיטת בן-גוריון

1. Ram Brustein אוניברסיטת בן-גוריון Models of Modular Inflation with I. Ben-Dayan, S. de Alwis To appear ============ Based on: hep-th/0408160 with S. de Alwis, P. Martens hep-th/0205042 with S. de Alwis, E. Novak • Introduction: The case for closed string moduli as inflatons • Cosmological stabilization of moduli • Designing inflationary potentials (SUGRA, moduli) • The CMB as a probe of model parameters

2. Stabilizing closed string moduli • Any attempt to create a deSitter like phase will induce a potential for moduli • A competition on converting potential energy to kinetic energy, moduli win, and block any form of inflation • Inflation is only ~ 1/100 worth of tuning away! • (but … see later)

3. Generic properties of moduli potentials: • The landscape allows fine tuning • Outer region stabilization possible • Small +ve or vanishing CC possible • Steep potentials • Runaway potentials towards decompactification/weak coupling • A “mini landscape” near every stable mininum + additional spurious minima and saddles

4. Cosmological stability hep-th/0408160 • The overshoot problem

5. r r r Proposed resolution:role of other sources • The 3 phases of evolution • Potential push: jump • Kinetic : glide • Radiation/other sources : parachute opens Previously: Barreiro et al: tracking Huey et al, specific temp. couplings • Inflation is only ~ 1/100 worth of tuning away!

6. Example: different phases == potential == kinetic == radiation

7. Example: trapped field == potential == kinetic == radiation

8. Using cosmological stabilization for designing models of inflation: • Allows Inflation far from final resting place • Allows outer region stabilization • Helps inflation from features near the final resting place

9. 2 -1 f/mp -4 -2 0 2 4 • –wall thickness in space (D/d)2 ~ L4 Inflation d H > 1 D > mp H2~1/3 L4/mp2 D (My) preferred models of inflation: small field models • “Topological inflation”: inflation off a flat feature Guendelman, Vilenkin, Linde Enough inflation V’’/V<1/50

10. Results and Conclusions: preview • Possible to design fine-tuned models in SUGRA and for string moduli • Small field models strongly favored • Outer region models strongly disfavored • Specific small field models • Minimal number of e-folds • Negligible amount of gravity waves: all models ruled out if any detected in the foreseeable future  Predictions for future CMB experiments

11. Designing flat features for inflation • Can be done in SUGRA • “Can” be done with steep exponentials alone • Can (??) be done with additional (???) ingredients (adding Dbar, const. to potential see however ….. ) • Lots of fine tuning, not very satisfactory • Amount of tuning reduces significantly towards the central region

12. Designing flat features for inflation in SUGRA Take the simplest Kahler potential and superpotential Always a good approximation when expanding in a small region (f < 1) For the purpose of finding local properties V can be treated as a polynomial

13. Design a maximum with small curvature with polynomial eqs. Needs to be tuned for inflation

14. Design a wide (symmetric) plateau with polynomial eqs. In practice creates two minima @ +y,-y (*) A simple solution: b2=0, b4=0, b1=1,b3=h/6, b5 determined approximately by (*)

15. Designing flat features for inflation in SUGRA A numerical example: The potential is not sensitive to small changes in coefficients Including adding small higher order terms, inflation is indeed 1/100 of tuning away b2=0, b4=0, b1=1,b3=h/6, Need 5 parameters: V’(0)=0,V(0)=1,V’’/V=h DTW(-y), DTW(+y) = 0 b5 y4(y2+5) + y2+1=0 h =6 b1 b3 – 2(b0)2

16. Designing flat features for inflation for string moduli: Why creating a flat feature is not so easy • An example of a steep superpotential • An example of Kahler potential Similar in spirit to the discussion of stabilization

17. Why creating a flat feature is not so easy (cont.) • extrema • min: WT = 0 • max: WTT = 0 • distance DT Example: 2 exponentials WT = 0  WTT = 0   For (a2-a1)<<a1,a2

18. Amount of tuning For (a2-a1)<<a1,a2 To get h ~ 1/100 need tuning of coefficients @ 1/100 x 1/(aT)2 The closer the maximum is to the central area the less tuning. Recall: we need to tune at least 5 parameters

19. Designing flat features with exponential superpotentials Trick: compare exponentials to polynomials by expanding about T = T2 Linear equations for the coefficients of Need N > K+1 (K=5N=7!) unless linearly dependent

20. Numerical examples 7 (!) exponentials + tuning ~ h (aT)2 = UGLY

21. Lessons for models of inflation • Push inflationary region towards the central region • Consequences: • High scale for inflation • Higher order terms are important, not simply quadratic maximum

22. Phenomenological consequences • Push inflationary region towards the central region • Consequences: • High scale for inflation • Higher order terms are important, not simply quadratic maximum

23. Models of inflation: Background de Sitter phase r + p << r  H ~ const. Parametrize the deviation from constant H by the value of the field Or by the number of e-folds Inflation ends when e = 1

24. Models of inflation:Perturbations • Spectrum of scalar perturbations • Spectrum of tensor perturbations Spectral indices r =C2Tensor/ C2Scalar (quadropole !?) Tensor to scalar ratio (many definitions) r is determined by PT/PR , background cosmology, & other effects r ~ 10 e (“current canonical” r =16 e) CMB observables determined by quantities ~ 50 efolds before the end of inflation

25. Wmapping Inflationary Physics W. H. Kinney, E. W. Kolb, A. Melchiorri, A. Riotto,hep-ph/0305130 See also Boubekeur & Lyth

26. Simple example:

27. For example: The “minimal” model: Quadratic maximum End of inflation determined by higher order terms Sufficient inflation Qu. fluct. not too large Minimal tuning  minimal inflation, N-efolds ~ 60  “largish scale of inflation” H/mp~1/100

28. The “minimal” model: Quadratic maximum End of inflation determined by higher order terms Unobservable!

29. WMAP 1/2 < 25|V’’/V| < 1  Expect for the whole class of models  Detecting any component of GW in the foreseeable future will rule out this whole class of models !

30. Summary and Conclusions • Stabilization of closed string moduli is key • Inflation likely to occur near the central region • Will be hard to find a specific string realization • Specific class of small field models • Specific predictions for future CMB experiments