Misao Sasaki YITP, Kyoto University

1. 29 June, 2009 ICG, Portsmouth Non-Gaussianity from Inflation Misao Sasaki YITP, Kyoto University

2. 2 contents 1. Inflation and curvature perturbations dN formalism 2. Origin of non-Gaussianity subhorizon or superhorizon scales 3. Non-Gaussianity from superhorizon scales two typical models: curvaton & multi-brid inflation 4. Multi-brid inflation 5. Summary

3. V(f) log L H-1 ∽a2 f hot (bigbang) universe L=H-1~ const. inflation log a t=tend 1. Inflation and curvature perturbations • single-field inflation, no other degree of freedom Linde ’82, ... slow-roll evolution

4. N : Number of e-folds of inflation • number of e-folds counted backward in time • from the end of inflation ~log(redshift) log L N=N(f) L=H-1 ~ t L=H-1~ const log a(t) inflation t=t(f) t=tend

5. Cosmological curvature perturbations - standard single-field slow-roll case - Inflaton fluctuations (vacuum fluctuations=Gaussian) Oscillation freezes out at k/a < H (~ classical Gaussian fluctuations on superhorizon scales) Curvature perturbations ∙∙∙ conserved on superhorizon scales

6. dN formalism • spectrum of Rc: ~ scale-invariant • curvature perturbation = e-folding number perturbation (dN) Starobinsky (’85) general slow-roll inflation MS & Stewart (’96) nonlinear generalization Lyth, Malik & MS (’04)

7. Theory vs Observation • Prediction of the standard single-field slow-roll inflation: almost scale-invariant Gaussian random fluctuations • perfectly consistent with CMB experiments

8. However, nature may not be so simple... • Tensor (gravitational wave) perturbations? tensor perturbations induce B-mode polarization of CMB r<0.2 (95%CL) tensor-scalar ratio: WMAP+BAO+SN (‘08) cf. chaotic inflation: r~0.15 Non-Gaussianity? Komatsu & Spergel (‘01) ( -)gravitational pot.: minus -9< fNL<111 (95%CL) WMAP (‘08) (fNL=51±30 at 1s) What wouldthe presence of non-Gaussianity mean ?

9. 2. Origin of Non-Gaussianity • three typical origins: • Self-interaction of inflaton field quantum physics, subhorizon scale during inflation • Multi-component field classical physics, nonlinear coupling to gravity superhorizon scale during and after inflation • Nonlinearity of gravity classical general relativistic effect, subhorizon scale after inflation

10. Origin of non-Gaussianity and cosmic scales k: comoving wavenumber log L classical/local NL effect classical NL gravity quantum NL effect log a(t) t=tend inflation hot Friedmann

11. Origin of NG (1):Self-interaction of inflaton field Non-Gaussianity from subhorizon scales (QFT effect) • interaction is very small for potential-type self-couplings Maldacena (’03) ex. chaotic inflation ∙∙∙ free field! (gravitational interaction is Planck-suppressed) • non-canonical kinetic term • → strong self-interaction → large non-Gaussianity

12. example : DBI inflation Silverstein & Tong (2004) kinetic term: ~ (Lorenz factor)-1 perturbation expansion: = ∝ ∝ ∝ 0 g -1 g 5 g 3 large non-Gaussianity for large g Seery & Lidsey (‘05), ... , Langlois et al. (‘08), Arroja et al. (‘09)

13. fNL for equilateral configuration is large bispectrum (3-pt function) of curvature perturbation from DBI inflation Alishahiha et al. (’04) WMAP 5yr constraint:

14. Origin of NG (２):Nonlinearity of gravity ex. post-Newton metric in harmonic coordinates Newton potential NL terms may be important after the perturbation scale re-enters the Hubble horizon Effect on CMB bispectrum seems small (but non-negligible?) Bartolo et al. (2007)

15. Origin of NG (3): Superhorizon scales Even if df is Gaussian, dT mn may be non-Gaussian due to its nonlinear dependence on df This effect is small for a single-field slow-roll model (⇔ linear approximation is extremely good) Salopek & Bond (’90), ... But it may become large for multi-field models Lyth & Rodriguez (‘05), .... Non-Gaussianity in this case is local:

16. 16 In the rest of this talk we focus on this case, ie non-Gaussianity generated on superhorizon scales

17. 3. Non-Gaussianity from superhorizon scales - two typical models for fNLlocal - • curvaton model Linde & Mukhanov (1996), Lyth & Wands (2001), Moroi & Takahashi (2001), ... • multi-brid inflation model MS (2008), Naruko & MS (2008) both may give large fNLlocal but in the case of curvaton scenario tensor-scalar ratio r will be very small.

18. linear approx. is valid for df dc contribution is small including NL corrections Curvaton model Inflation is dominated by inflaton = f Curv. perts. are dominated by curvaton = c during inflation: f dominates curvature perturbations during inflation

19. ∙∙∙fraction of c at decay time After inflation, f thermalizes, c undergoes damped oscillation. Assume c dominates final amp of curvature perturbations: fNL~ 1/q Lyth & Rodriguez (‘05) Malik & Lyth (‘06) MS, Valiviita & Wands (‘06) Large NG is possible if q<<1 But tensor mode is strongly suppressed:

20. V f N=0 inflation 4. Multi-brid inflation • hybrid inflation: • inflation ends by a sudden destabilization of vacuum • multi-brid inflation = multi-field hybrid inflation: Slow-roll eom N as a time:

21. simple 2-brid example: the case of radial inflationary orbits on (f1,f2) f2 q f N=0 f1 N=const.

22. Three types of dN ( ) indicates field perturbations f2 f1 end of inflation or phase transition originally adiabatic (standard scenario) entropy → adiabatic

23. condition to end the inflation f1, f2∙∙∙ inflaton fields (2-brid inflation) c∙∙∙ waterfall field during inflation V0 inflation ends when c

24. Simple analytically soluble model MS (’08) • exponential potential: parametrize the end of inflation • dN to 2nd order in df : • Spectrum of curvature perturbation spectral index: tensor/scalar:

25. isocurvature perturbation isocurvature contributes at 2nd order possibility of large non-Gaussianity

26. Just an example ... input parameters: outputs: indep. of waterfall c

27. WMAP 5yr Komatsu et al. ‘08 WMAP+BAO+SN WMAP present example tensor-scalar ratio r is not suppressed AND fNLlocal ~ 50 (positive and large)

28. Another example: O(2) SSB model Inflation ends at symmetry breakdown at the end of inflation f2 Alabidi & Lyth ‘06 f1

29. curvature perturbation spectrum: • spectral index: • tensor/scalar: • non-gaussianity: for Again, fNLlocal may be large and positive.

31. It seems fNL>0 always in multi-brid inflation Any good reason for

32. 33 Perhaps yes! inflaton trajectories y For ‘hyperbolic’ end of inflation condition fNL may become negative. x

33. 34 5. Summary • 3 types of non-Gaussianity 1. subhorizon ∙∙∙ quantum origin These are important 2. superhorizon ∙∙∙ classical (local) origin 3. gravitational dynamics ∙∙∙ classical origin • DBI inflation --- type 1. can be large • curvaton scenario, multi-brid inflation --- type 2. can be large. sign is important, too. but curvaton scenario predicts r≪1 if fNL is large. multi-brid inflation can give r~0.1

34. 35 Non-Gaussianity plays an important role in determining (constraining) models of inflation 4-pt function (trispectrum) may be detected in addition to 3-pt function (bispectrum) Arroja & Koyama (’08), Huang (’09), … NG may be detected in the very near future PLANCK launched a month ago (on 14 May)!