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Non-Gaussianity of superhorizon curvature perturbations beyond δN-formalism

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### 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）

● 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)

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)

- 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

◆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)

●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

◆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)]

●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

●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

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）

- 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

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)

Einstein equation after ADM decomposition

+Uniform Hubble slicing

◆ Orders of magnitude

- Input

First-order equations

- Output

+time-orthogonal

: 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

← 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

◆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

◆Nonlinear curvature perturbation

Linear theory

Traceless

Correspondence to our notation

Constraint eq

Without non-local operators

Some integrals

Growing mode

◆Nonlinear second-order differential equation

Variable:

Integrals:

satisfies

Ricci scalar of 0th spatial metric

●Natural extension of well-known linear version

◆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)

◆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

- 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

- 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)

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