1. Yukawa Institute for Theoretical Physics Kyoto University Misao Sasaki Relativity & Gravitation 100 yrs after Einstein in Prague 26 June 2012

2. 2 0. Horizon & flatness problem horizon problem conformal time is bounded from below gravity=attractive particle horizon E

3. 3 solution to the horizon problem for a sufficient lapse of time in the early universe or large enough to cover the present horizon size NB: horizon problem≠ homogeneity & isotropy problem

4. 4 flatness problem (= entropy problem) alternatively, the problem is the existence of huge entropy within the curvature radius of the universe (# of states = exp[ S ])

5. 5 solution to horizon & flatness problems spatially homogeneous scalar field: potential dominated V ~ cosmological const./vacuum energy inflation “vacuum energy” converted to radiation after sufficient lapse of time solves horizon & flatness problems simultaneously

6. 6 1 ． Slow-roll inflation and vacuum fluctuations single-field slow-roll inflation V(  )  ∙∙∙ slow variation of H inflation! metric ： field eq. ： Linde ’82,... may be realized for various potentials

7. 7 comoving scale vs Hubble horizon radius log a(t)=N log L t = t end k : comoving wavenumber inflation subhorizon superhorizon subhorizon hot bigbang

8. 8 e-folding number: N log a(t) log L L = H -1 ~ t t=t()t=t()t=t()t=t() t = t end N=N()N=N()N=N()N=N() L = H -1 ~ const redshift # of e-folds from  =   (t) until the end of inflation  determines comoving scale k

9. 9 inflaton fluctuation (vacuum fluctuations=Gaussian) rapid expansion renders oscillations frozen at k/a < H (fluctuations become “classical” on superhorizon scales ) Curvature (scalar-type) Perturbation  intuitive (non-rigorous) derivation curvature perturbation on comoving slices ∙∙∙ conserved on superhorizon scale, for purely adiabatic pertns. evaluated on ‘flat’ slice (vol of 3-metric unperturbed)

10. 10 Curvature perturbation spectrum  N - formula ~ almost scale-invariant spectrum MS & Stewart (’96) geometrical justification non-linear extension Lyth, Marik & MS (’04) Starobinsky (’85) (rigorous/1 st principle derivation by Mukhanov ’85, MS ’86)

11. 11 Tensor Perturbation canonically normalized tensor field : transverse-traceless tensor spectrum Starobinsky (’79)

12. 12 Tensor-to-scalar ratio Tensor-to-scalar ratio scalar spectrum: scalar spectrum: tensor spectrum: tensor spectrum: tensor spectral index: tensor spectral index: ··· valid for all slow-roll models with canonical kinetic term

13. 13 Comparison with observation Standard (single-field, slowroll) inflation predicts scale- invariant Gaussian curvature perturbations. CMB (WMAP) is consistent with the prediction. Linear perturbation theory seems to be valid. WMAP 7yr

14. 14 CMB constraints on inflation Komatsu et al. ‘10 scalar spectral index: n s = 0.95 ~ 0.98 tensor-to-scalar ratio: r < 0.15

15. 15 However,…. Inflation may be non-standard multi-field, non-slowroll, DBI, extra-dim’s, … Quantifying NL/NG effects is important PLANCK, … may detect non-Gaussianity B-mode (tensor) may or may not be detected. energy scale of inflation modified (quantum) gravity? NG signature? (comoving) curvature perturbation:

16. 16 2. Origin of non-Gaussianity self-interactions of inflaton/non-trivial “vacuum” nonlinearity in gravity multi-field classical physics, nonlinear coupling to gravity superhorizon scale during and after inflation quantum physics, subhorizon scale during inflation classical general relativistic effect, subhorizon scale after inflation

17. 17 Origin of NG and cosmic scales log a(t) log L t = t end k : comoving wavenumber inflation classical gravity classical/local effect quantum effect hot bigbang

18. 18 Origin １： self-interaction/non-trivial vacuum conventional self-interaction by potential is ineffective need unconventional self-interaction → non-canonical kinetic term can generate large NG Non-Gaussianity generated on subhorizon scales (quantum field theoretical) ex. chaotic inflation ∙∙∙ free field! (grav. interaction is Planck-suppressed) Maldacena (’03) extremely small!

19. 19 1a. Non-canonical kinetic term: DBI inflation kinetic term ： Silverstein & Tong (2004) ~ (Lorenz factor) -1 perturbation expansion 0 = ∝ ∝  3  3+2 large NG for large 

20. 20 Bi-spectrum (3pt function) in DBI inflation f NL large for equilateral configuration WMAP 7yr ： Alishahiha et al. (’04)

21. 21 1b. Non-trivial vacuum de Sitter spacetime = maximally symmetric (same degrees of sym as Poincare (Minkowski) sym) gravitational interaction (G-int) is negligible in vacuum slow-roll inflation : dS symmetry is slightly broken G-int induces NG but suppressed by (except for graviton/tensor-mode loops) But large NG is possible if the initial state (or state at horizon crossing) does NOT respect dS symmetry (eg, initial state ≠ Bunch-Davies vacuum) various types of NG : scale-dependent, oscillating, featured, folded... Chen et al. (’08), Flauger et al. (’10),...

22. 22 templates for primordial bispectra squeezed type (Komatsu&Spergel 2001) local in real space (f NL =constant) max for squeezed triangles: k<<k’,k’’ equilateral type (Creminelli et al 2005) peaks for k 1 ~k 2 ~k 3 orthogonal type (Senatore et al 2009) k k’’ k” k1k1 k2k2 k3k3

23. 23 Origin 2 ： superhorizon generation NG may appear if  T   depends nonlinearly on , even if  itself is Gaussian. This effect is small in single-field slow-roll model ( ⇔ linear approximation is valid to high accuracy) For multi-field models, contribution to T  from each field can be highly nonlinear. NG is always of local type: Salopek & Bond (’90)  N formalism for this type of NG x  tot WMAP 7yr ：

24. 24 Origin 3 ： nonlinearity in gravity ex. post-Newtonian metric in asymptotically flat space NL (post-Newton) terms Newton potential important when scales have re-entered Hubble horizon effect on CMB bispectrum may not be negligible Pitrou et al. (2010) (for both squeezed and equilateral types) (in both local and nonlocal forms) distinguishable from NL matter dynamics? ?

25. 25 3.  N formalism  N is the perturbation in # of e-folds counted backward in time from a fixed final time t f  N is equal to conserved NL comoving curvature perturbation on superhorizon scales at t>t f t f should be chosen such that the evolution of the universe has become unique by that time.  N is valid independent of gravity theory What is  N? therefore it is nonlocal in time by definition =adiabatic limit

26. 26 3 types of  N originally adiabatic end of/after inflation entropy/isocurvature → adiabatic 11 22

27. 27 Nonlinear  N - formula Choose flat slice at t = t 1 [  F ( t 1 ) ] and comoving (=uniform density) at t = t 2 [  C ( t 1 ) ] : ( ‘flat’ slice :   ( t ) on which )  F ( t 1 ) : flat  C ( t 2 ) : comoving  ( t 2 )=const. R ( t 1 )=0  F ( t 2 ) : flat R ( t 2 )=0

28. 28 Nonlinear  N for multi-component inflation : where  =  F is fluctuation on initial flat slice at or after horizon-crossing.  F may contain non-Gaussianity from subhorizon (quantum) interactions eg, in DBI inflation

29. 29 4 ． NG generation on superhorizon scales curvaton-type Lyth & Wands (’01), Moroi & Takahashi (‘01),... two efficient mechanisms to convert isocurvature to curvature perturbations:  curv (  ) <<  tot during inflation highly nonlinear dep of  curv on  possible  curv (  ) can dominate after inflation curvature perturbation can be highly NG curvaton-type & multi-brid type

30. 30 multi-brid inflation MS (’08), Naruko & MS (’08),... sudden change/transition in the trajectory  tensor-scalar ratio r may be large in multi-brid models, while it is always small in curvaton-type if NG is large. curvature of this surface determines sign of f NL

31. 31 5. Summary inflation explains observed structure of the universe flatness:  0 =1 to good accuracy curvature perturbation spectrum almost scale-invariant almost Gaussian inflation also predicts scale-invariant tensor spectrum will be detected soon if tensor-scalar ratio r>0.1 any new/additional features?

32. 32 3 origins of NG in curvature perturbation multi-field model: origin 2. DBI-type model: origin 1. 1. subhorizon ∙∙∙ quantum origin 2. superhorizon ∙∙∙ classical (local) origin 3. NL gravity ∙∙∙ late time classical dynamics may be large may be large: In curvaton-type models r ≪ 1. Multi-brid model may give r ~0.1. NG from inflation need to be quantified non BD vacuum: origin 1. any type of may be large non-Gaussianities

33. 33 Identifying properties of non-Gaussianity is extremely important for understanding physics of the early universe not only bispectrum(3-pt function) but also trispectrum or higher order n-pt functions may become important. Confirmation of primordial NG? PLANCK (February 2013?)...