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Non-Gaussianity of superhorizon curvature perturbations beyond 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 ). ◆ Introduction.

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non gaussianity of superhorizon curvature perturbations beyond n formalism

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)




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


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


Equil: Non-canonical

①Before horizon crossing

②During superhorizon evolution

③ At the end of or after inflation




(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


  • 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



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


1 2 …… ∞

=Full nonlinear




・・Our study


2nd order perturb

n-th order perturb




Nonlinear theory in

Simple result !

Linear theory


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)


●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


◆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)


◆ Basic assumptions

Einstein equation after ADM decomposition

+Uniform Hubble slicing

◆ Orders of magnitude

  • Input

First-order equations

  • Output



●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

●General solution

← Gravitational wave


  • 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



◆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


: Ricci scalar of 0th spatial metric


◆Nonlinear curvature perturbation

Linear theory


Correspondence to our notation

Constraint eq

Without non-local operators

Some integrals


Decaying mode

Growing mode

◆Nonlinear second-order differential equation




Ricci scalar of 0th spatial metric

●Natural extension of well-known linear version


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)


◆Matched Nonlinear sol to linear sol

Approximate Linear sol around horizon crossing @

● Final result

Enhancement in Linear theory


Nonlinear effect

●In Fourier space, calculate Bispectrum as


◆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


  • There is a stage at which slow-roll conditions are violated

●Linear analysis

(Starobinsky’s model)


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)