Non-Gaussianities in General Single Field Inflation

1. Non-Gaussianities inGeneral Single Field Inflation Xingang Chen 陈新刚 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.

2. WMAP measurement on CMBR • Spectral index: • Running of spectral index: • Tensor to scalar ratio: • Non-Gaussianity: 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.

3. Single field inflation: • Inflaton is responsible for density perturbations; • Lagrangian is arbitrary function of and ; • Arbitrary sound speed and (to be defined). • 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. Most General Non-Gaussianities in Single Field Theory

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

5. V 1. Slow-roll inflation (Linde 82; Albrecht & Steinhardt 82) Slow-roll parameters: << 1 << 1 Review of Several Classes of Models 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) • Lagrangian 2. DBI inflation (Silverstein, Tong & Alishahiha, 03,04; X.C. 04,05)

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

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

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

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

13. Power spectrum • Spectral index • 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

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

15. are Lagrangian multipliers ADM Formalism (Maldacena, 02; Seery & Lidsey 05; X.C., Huang, Kachru & Shiu, 06) • Metric • 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. Only require the variation of be slow; can be arbitrary. The Quadratic Part

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

19. Define The 3-Point Function

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

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

29. 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) Shape of Non-Gaussianities (Babich, Creminelli, Zaldarriaga, 04; X.C., Huang, Kachru, Shiu, 06) Equilateral shape (DBI) Local shape (Slow-roll)

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

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

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

35. Constraining microscopic parameters: For example, the upper bound in the result: (Bean, X.C., Peiris, Xu, 07) • 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. • In the IR DBI model (X.C. 04,05) (Smith, Zaldarriaga, 06; Sefusatti, Komatsu, 07) • Large non-G can also be small enough to satisfy current observations

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

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

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. Consider a small but sharp step (Adams, Cresswell, Easther, 01) Sharp features in slow-roll potential • 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 Calculate the associated large non-G (X.C., Easther, Lim, 06) 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. Duality cascade can cause sharp features in warp factor (Hailu, Tye, 06) Abrupt change in sound speed Sharp features in DBI inflation (Bean, X.C., Tye, Xu, in preparation) • Associated with non-Gaussianities features, on top of the original • nearly-scale-invariant large non-G.

45. IR DBI mode predicts large, but regional, running of spectral index (X.C., 05, 06; Bean, X.C., Peiris, Xu, 07) Provides speed limit • Warped space Redshifts string scale (Randall, Sundrum, 99) • 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 Measuring Stringy Correlation functions

46. : 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). (4) (3) (2) (1) Density perturbations: 1) Field theory regime 2) Hubble-expansion-induced stringy phase Stringy phase transition – the reminder 1 (from the last week talk)

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

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 (Bean, X.C., Peiris, Xu, 07) stringy phase transition happens near largest CMB scales

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.