Non-Gaussianity of superhorizon curvature perturbations beyond δN-formalism

1. Non-Gaussianity of superhorizoncurvature perturbationsbeyond δN-formalism Resceu, University of Tokyo Yuichi Takamizu Collaborator: Shinji Mukohyama (IPMU,U of Tokyo), Misao Sasaki & Yoshiharu Tanaka (YITP,Kyoto U) Ref In progress & JCAP01 013 (2009）

2. ◆Introduction ● Inflationary scenario in the early universe • Generation of primordial density perturbations • Seedsof large scale structure of the universe • The primordial fluctuations generated from Inflation are one of the most interesting prediction of quantum theory of fundamental physics. However, we have known little information about Inflation • More accurate observations give more information about primordial fluctuations • PLANCK (2009-) ◆Non-Gaussianityfrom Inflation Will detect second order primordial perturbations Komatsu & Spergel (01)

3. ◆Introduction Bispectrum Power spectrum ●Non-Gaussianity from inflation Mainly two types of bispectrum: ■ WMAP 7-year ■Local type ・・・Multi-field, Curvaton ■Equilateral type ・・・Non-canonical kinetic term Cf) Orthogonal type:generated by higher derivative terms(Senatore etal 2010)

4. ◆Introduction • If we detect non-Gaussianity, what we can know about Inflation ? ●　Slow-rolling ? ●Single field or multi-field ? ●　Canonical kinetic term ? ●Bunch-Davies vacuum ? • Theoretical side • Standard single slow-roll scalar • Many models predicting Large Non-Gaussianity • （Multi-fields, DBI inflation and Curvaton） • Observational side PLANCK2009- Can detect within • Non-Gaussianity (nonlinearity)will be one of powerful tool to discriminate many possible inflationary models with the current and future precision observations

5. ◆Three stages for generating Non-Gaussianity ②Classical nonlinear effect ③Classical gravity effect ①Quantum nonlinear effect Ｌｏｃａｌ：Multi-field Ｅｑｕｉｌ： Non-canonical ①Before horizon crossing ②During superhorizon evolution ③ At the end of or after inflation Inflation Hot FRW ●δN-formalism (Starobinsky 85, Sasaki & Stewart 96)

6. ●Nonlinear perturbations on superhorizon scales • Spatial gradient approach : Salopek & Bond (90) • Spatial derivatives are small compared to time derivative • Expand Einstein eqs in terms of small parameter ε, and can solve them for nonlinear perturbations iteratively ◆δNformalism (Separated universe ) (Starobinsky 85, Sasaki & Stewart 96) Curvature perturbation = Fluctuations of the local e-folding number ◇Powerful tool for the estimation of NG

7. ◆Temporary violating of slow-roll condition • e.g.Double inflation, false vacuum inflation ◆δNformalism • Adiabatic (Single field) curvature perturbations on superhorizon are Constant • Independent of Gravitational theory（Lyth, Malik & Sasaki 05） • Ignore the decaying mode of curvature perturbation ◆Beyond δNformalism • is known to play a crucial role in this case • This order correction leads to a time dependence on superhorizon • Decaying modes cannot be neglected in this case • Enhancementof curvature perturbation in the linear theory [Seto et al (01), Leach et al (01)]

8. ◆Example ●Starobinsky’s model(92) ●Leach, Sasaki, Wands & Liddle (01) • Linear theory • The in the expansion • There is a stage at which slow-roll conditions are violated • Enhancement of the curvature perturbation near superhorizon Initial: Final: Growing mode decaying mode Decaying mode Growing mode • Enhancement of curvature perturbation near

9. ●Nonlinear perturbations on superhorizon scales up to Next-leading order in the expansion n 1 2 …… ∞ =Full nonlinear m 0 ・・δＮ ・・Our study 2 2nd order perturb n-th order perturb ……. Linear theory ∞ Nonlinear theory in Simple result ! Linear theory

10. System : Y.T and S. Mukohyama JCAP01 (2009） • Single scalar field with general potential & kinetic term including K- inflation & DBI etc ◆Scalar field in a perfect fluid form ●The difference between a perfect fluid and a scalar field The propagation speed of perturbation（speed of sound） For a perfect fluid（adiabatic）

11. ●ＡＤＭ decomposition • Gauge degree of freedom ①Hamiltonian & Momentum constraint eqs EOM: Two constraint equations • EOMfor : dynamical equations In order to express as a set of first-order diff eqs, Introduce the extrinsic curvature ② Evolution eqs for ③ Energy momentum conservation eqs • Conservation law Further decompose, ② Four Evolution eqs • Spatial metric • Extrinsic curvature ← Traceless part

12. ◆Under general cond Amplitude of decaying mode : decaying mode of density perturbation ●Gradient expansion Small parameter: ◆Backgroundis the flat FLRWuniverse ◆ Basic assumptions ◆Since the flat FLRWbackground is recovered in the limit ●Assume a stronger condition • It can simplify our analysis • Absence of any decaying at leading order • Can be justified in most inflationary model with (Hamazaki 08)

13. ◆ Basic assumptions Einstein equation after ADM decomposition +Uniform Hubble slicing ◆ Orders of magnitude • Input First-order equations • Output +time-orthogonal

14. ●General solution : Ricci scalar of 0th spatial metric ●Density perturbation & scalar fluctuation • Densityperturb is determined by 0th Ricciscalar Scalar decaying mode ●Curvature perturbation Constant (δN ) • Its time evolution is affectedby the effect of p (speed of soundetc), throughsolution of χ

15. ●General solution ← Gravitational wave where • Growing GW (Even if spacetime is flat , 2nd order of GW is generated by scalar modes ) ●The number of degrees of freedom • The `constant` of integration depend only on the spatial coordinates and satisfy constraint eq Scalar 1(G) + 1(D) & Tensor 1(G) + 1(D)=(6-1)+(6-1)-3- 3 gauge remain under

16. ◆Nonlinear curvature perturbation Focus on onlyScalar-type mode (e.g. curvature perturbation) Complete gauge fixing Definenonlinear curvature perturbation Uniform Hubble & time-orthogonal gauge Gauge transformation Comoving & time-orthogonal gauge In this case, the same as linear ! ●Comoving curvature perturbation Time dependence δN : Ricci scalar of 0th spatial metric

17. ◆Nonlinear curvature perturbation Linear theory Traceless Correspondence to our notation Constraint eq Without non-local operators Some integrals

18. Decaying mode Growing mode ◆Nonlinear second-order differential equation Variable: Integrals: satisfies Ricci scalar of 0th spatial metric ●Natural extension of well-known linear version

19. Our nonlinear sol (Grad.Exp) ◆Matching conditions n-th order perturb. sol In order to determine the initial cond. Interest k after Horizon crossing following 2-order diff. eq Value & derivative can determine that uniquely ● Final result (at late times)

20. ◆Matched Nonlinear sol to linear sol Approximate Linear sol around horizon crossing @ ● Final result Enhancement in Linear theory δN Nonlinear effect ●In Fourier space, calculate Bispectrum as

21. ◆Summary & Remarks • We develop a theory of non-linear cosmological perturbations on superhorizon scales for a scalar field with a general potential & kinetic terms • We employ the ADM formalism and the spatial gradient expansion approach to obtain general solutions valid up through second-order • This Formulation can be applied to k-inflation and DBI inflation to investigate superhorizon evolution of non-Gaussianitybeyond δN-formalism • Show thesimple 2nd order diff eqfornonlinear variable: • We formulate a general method to match a n-th order perturbative sol • Calculate the bispectrumusing the solution including the nonlinear terms • Can applied to Non-Gaussianity in temporary violating of slow-roll cond • Extension to the models of multi-scalar field

22. ◆Application • There is a stage at which slow-roll conditions are violated ●Linear analysis （Starobinsky’s model） ・・crossing Can be amplified if This mode is the most enhanced (fast rolling) Violation of Slow-roll cond ●go up to nonlinear analysis ! Leach, Sasaki, Wands & Liddle (01)