1. Non-Gaussianities in General Single Field Inflation Xingang Chen CTP, MIT astro-ph/0507053; hep-th/0605045, with Minxin Huang, Shamit Kachru, Gary Shiu; astro-ph/0611645, with Eugene Lim, Richard Easther; and in progress; with Rachel Bean, Henry Tye, Jiajun Xu, in preparation. 陈新刚

2. Inflation Models and Observations Inflation mechanisms and models Slow-roll inflation --- using flat potential; DBI inflation --- using speed-limit in warped space; K-inflation --- inflation driven by kinetic energy. WMAP measurement on CMBR Spectral index: Running of spectral index: Tensor to scalar ratio: Non-Gaussianity:

3. Most General Non-Gaussianities in Single Field Theory Motivations  Null hypothesis on specific models; Fit or constrain parameters model-independently;  Several string models has distinctive predictions on non-Gaussianities;  Straightforward evaluation of non-Gaussianities for future models in this general class. Single field inflation:  Inflaton is responsible for density perturbations;  Lagrangian is arbitrary function of and ;  Arbitrary sound speed and (to be defined).

4. Review of several classes of models General formalism General form of non-Gaussianities Using non-G to probe new physics Outline

5. Review of Several Classes of Models 1. Slow-roll inflation (Linde 82; Albrecht & Steinhardt 82) V << 1 Slow-roll parameters: 1. Slow-roll inflation; 2. DBI inflation; 3. K-inflation

6.  dS inflation; Power-law inflation; Large field inflation; Small field inflation;  String models: Branes; Tachyons; Axions; Radions.

7. UV model (Silverstein, Tong, 03) IR model (X.C. 04) 2. DBI inflation (Silverstein, Tong & Alishahiha, 03,04; X.C. 04,05) Lagrangian

8. Multi-throat brane inflation (X.C. 04)  Antibrane-flux annihilation (Kachru, Pearson, Verlinde, 01)  Generate branes as candidate inflatons  Exit B-throat, roll through bulk, settle down in another throat  Enough warping: DBI inflation; Flat potential: slow-roll inflation.

9. Branes are moving ultra-relativistically For example, in IR DBI, Lorentz factor, In DBI inflation, potential energy dominates, despite the fact that inflatons are ultra-relativistic. Pressure and Energy

10. 3. K-inflation Lagrangian For example, Attractor solution Inflation driven by kinetic energy if Can use a hybrid field to end the inflation Not realized in string theory so far (Armendariz-Picon, Damour, Mukhanov, 99)

11. Review of several classes of models General formalism General form of non-Gaussianities Using non-G to probe new physics Outline

12. A General Formalism (Garriga & Mukhanov, 99)

13. Slow variation parameters More general than the usual slow-roll parameters  Flat potential v.s. steep potential (DBI) or no potential (k-inflation)  Non-relativistic slow-roll v.s. ultra-relativistic fast-roll Power spectrum Spectral index

14. Review of several classes of models General formalism General form of non-Gaussianities Using non-G to probe string theory Outline

15. ADM Formalism (Maldacena, 02; Seery & Lidsey 05; X.C., Huang, Kachru & Shiu, 06) Metric are Lagrangian multipliers Action

16. Decompose and expand in powers of Solve to, in order to expand the action to Plug them into the action and expand

17. The Quadratic Part Only require the variation of be slow; can be arbitrary.

18. The Cubic Part The exact cubic action for scalar perturbation

19. The 3-Point Function Define

20. The Cubic Part The exact cubic action for scalar perturbation Various contributions 1:

21. The Cubic Part The exact cubic action for scalar perturbation Various contributions 2:

22. The Cubic Part The exact cubic action for scalar perturbation Various contributions 3: This last term is absorbed by a redefinition:

23. The Cubic Part The exact cubic action for scalar perturbation Various contributions 4: Negligible, unless there are sharp features (X.C., Easther, Lim, 06) (Bean, X.C., Tye, Xu, in preparation)

24. The Cubic Part The exact cubic action for scalar perturbation Various contributions 5: Negligible, unless there are non-trivial initial conditions (X.C., Easther, Lim, in preparation)

25. The leading contributions from each terms, in absence of sharp features and non-trivial initial conditions

26. Corrections terms

27. The 3-pt function for a general single field inflation to Final Results (X.C., Huang, Kachru, Shiu, 06) Completely specified by 5 parameters:

28. Size, Shape, and Running of Non-Gaussianities Size (magnitude) of non-Gaussianities Large non-GaussianitySmall or large  WMAP’s ansatz  To compare, take equilateral limit in our results: (Note: is defined in Maldacena,02; X.C.,Huang,Kachru,Shiu,06;….; here we quote in WMAP’s convention.)

29. Shape of Non-Gaussianities (Babich, Creminelli, Zaldarriaga, 04; X.C., Huang, Kachru, Shiu, 06) Equilateral shape (DBI)Local shape (Slow-roll) Current Bound: (WMAP team; Creminelli, Senatore, Zaldarriaga, Tegmark, 06) CMB: Planck (Smith, Zaldarriaga, 06) LSS: high-z galaxy surveys: similar or better resolutions. (Sefusatti, Komatsu, 06)

30. Slow-Roll Inflation In this limit, our formulae recover the slow-roll results of Maldacena, 02; Seery & Lidsey, 05. In slow-roll inflation, the non-Gaussianity is unobservable,

31. DBI Inflation (Alishahiha, Silverstein & Tong, 04)

32. K-Inflation Another leading shape (Gruzinov, 04) Potentially observable in K-inflation Remind: Sound speed is constant, non-G does not run

33. Review of several classes of models General formalism General form of non-Gaussianities Using non-G to probe new physics Outline 1) Constraining String models; 2) Probing compactification geometry; 3) Probing sharp features; 4) Probing inflationary vacuum; 5) Measuring stringy correlation-functions.

34. Constraining String Models In GKP-type warp compactification, is restricted by the size of the throat Excessive non-Gaussianities (X.C., 05; Baumann, Mcallister, 06; Bean, Shandera, Tye & Xu, 07) In the UV DBI model (Silverstein, Tong & Alishahiha, 03,04) Viable only if Note: No comparison with data has been made. In fact, before data comparison is made, probe brane back-reaction is already too large. Require: But: (Bean, X.C., Peiris, Xu, 07) (see last week talk)

35. In the IR DBI model (X.C. 04,05)  Large non-G can also be small enough to satisfy current observations  Testable in the future experiments: In future experiments: on CMB scales, Planck can achieve on LSS scales, high-z galaxy surveys can reach similar or better resolutions. (Smith, Zaldarriaga, 06; Sefusatti, Komatsu, 07)  Constraining microscopic parameters: For example, the upper bound in the result: (Bean, X.C., Peiris, Xu, 07)

36. Probing Geometry in String Compactification Running of non-Gaussianity Shape of geometry in extra dimension (X.C. 05) Combining with the correlated feature in 2-pt function (Shiu, Underwood, 06) Radius dependence of warp factortime dependence of sound speed Scale dependence (running) of non-G

37. Probing Inflationary Vacuum General vacuum state for inflaton fluctuations: The Bunch-Davis vacuum: Consider corrections Replace one of with (X.C.,Huang,Kachru,Shiu,06) (Martin, Brandenberger, 00)

38. The size of the non-Gaussianities The shape of the non-Gaussianities  Peak in the folded triangle limit,  Divergence is artificial: if non-standard vacuum exits only up to M, divergence is replaced within

39. Probing Sharp Features Glitches in CMB power spectrum: Cosmic variance, or new physics?

40. Sharp features in slow-roll potential Consider a small but sharp step (Adams, Cresswell, Easther, 01) Without the step, with the step, Cause a dip in density perturbations with ratio: (Covi, et al, 06)

41. As the inflaton falls down the step, within results in abrupt changes in:

42. The contribution becomes important (X.C., Easther, Lim, 06) Calculate the associated large non-G Choose c and d to fit the power spectrum Predict the non-G Distinctive features: 1) localized around the location of feature; 2) characteristic oscillatory running, c.f. mild running in DBI.

43. Since running dominates, shape dependence varies a lot Experimental bound for such non-Gaussianities has not been done.

44. Sharp features in DBI inflation (Bean, X.C., Tye, Xu, in preparation) Duality cascade can cause sharp features in warp factor (Hailu, Tye, 06) Abrupt change in sound speed Associated with non-Gaussianities features, on top of the original nearly-scale-invariant large non-G.

45. In IR DBI inflation, at earlier times, i.e. larger scales, Hubble energy > redshifted string scale. (Phase transition) Not only scalar fluctuations, but also stringy fluctuations. Happens at Warped space Provides speed limit Redshifts string scale (Randall, Sundrum, 99) IR DBI mode predicts large, but regional, running of spectral index (X.C., 05, 06; Bean, X.C., Peiris, Xu, 07) Measuring Stringy Correlation functions

46. (1)(2)(3)(4) 2) Hubble-expansion-induced stringy phase1) Field theory regime Density perturbations: 1) : Field theory applies; 2) : Open string creation (Stringy quantum fluctuations); 3) : Closed string creation starts; 4) : Closed strings smooth out background (de Sitter back-reaction cuts off the throat). Stringy phase transition – the reminder 1 (from the last week talk)

47. Stringy phase transition:  Hubble scale < string scale:  Fluctuation speed < speed of light:  Density perturbations:  Spectrum index: Field theory regime Phase transition at: if Stringy phase transition – the reminder 2

48. Large non-Gaussianities are stringy near larger scales Stringy 2-pt is only estimated; but even estimation of stringy non-G is currently unavailable. Experiments ahead of string theory! Compare IR model with data stringy phase transition happens near largest CMB scales (Bean, X.C., Peiris, Xu, 07)

49. Conclusions A full non-Gaussianity in general single field inflation specified by 5 parameters; Explicit form of momentum dependence, including a few potentially observable; Recovered all previously known results, explore unknown regions. Probing new physics and string theory models, including field theoretic with strong string motivations and completely stringy physics.