Yukawa Institute for Theoretical Physics Kyoto University

1. Relativity & Gravitation 100 yrs after Einstein in Prague 26 June 2012 Inflation and Birth of Cosmological Perturbations Misao Sasaki Yukawa Institute for Theoretical Physics Kyoto University

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

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

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

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

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

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

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

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

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

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

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

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

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

18. Origin１：self-interaction/non-trivial vacuum Non-Gaussianity generated on subhorizon scales (quantum field theoretical) • conventional self-interaction by potential is ineffective Maldacena (’03) ex. chaotic inflation ∙∙∙ free field! (grav. interaction is Planck-suppressed) extremely small! • need unconventional self-interaction • → non-canonical kinetic termcangenerate large NG

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

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

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 (except for graviton/tensor-mode loops) • slow-roll inflation : dS symmetry is slightly broken G-int induces NG but suppressed by 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. rtot Origin 2：superhorizon generation • NG may appear if T mndepends nonlinearly ondf, • even if df itself is Gaussian. This effect is smallinsingle-field slow-roll model (⇔ linear approximation is valid to high accuracy) Salopek & Bond (’90) • For multi-field models, contribution to T mnfrom each field • can be highly nonlinear. NG is always of local type: x WMAP 7yr： dN formalism for this type of NG

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

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

25. f2 f1 3 types of dN end of/after inflation originally adiabatic entropy/isocurvature → adiabatic

26. Chooseflat slice at t = t1 [ SF (t1) ] and comoving (=uniform density) at t = t2 [ SC (t1) ] : NonlineardN - formula ( ‘flat’ slice: S(t) on which ) SC(t2) : comoving r (t2)=const. R(t2)=0 SF(t2) : flat SF (t1) : flat R(t1)=0

27. Nonlinear dNfor multi-component inflation : wheredf =dfFis fluctuation on initial flat slice at or after horizon-crossing. dfF may contain non-Gaussianity from subhorizon (quantum) interactions eg, in DBI inflation

28. 4．NG generation on superhorizon scales two efficient mechanisms to convert isocurvature to curvature perturbations: curvaton-type & multi-brid type • curvaton-type Lyth & Wands (’01), Moroi & Takahashi (‘01),... rcurv(f) << rtot during inflation highly nonlinear dep of drcurv on df possible rcurv(f) can dominate after inflation curvature perturbation can be highly NG

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

30. 5. Summary • inflation explains observed structure of the universe flatness:W0=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?

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

32. 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?) ...