Lectures on Delta N formalism  (given at DPG physics school on Inflation and CMB, July 2013)

δN formalism

Misao Sasaki

Misao Sasaki

Yukawa Institute for Theoretical Physics

Yukawa Institute for Theoretical Physics

Kyoto University

Kyoto University

contents

1. Introduction

2. Linear perturbation theory

3. Linear δN formula

4. Non-linear extension

5. Non-linear ∆N formula

6. Conservation of nonlinear curvature perturbation

7. δN for ‘slowroll’ inflation

• M. Sasaki and E.D. Stewart,

A General analytic formula for the spectral index of the density perturbations produced during inflation,

Prog. Theor. Phys. 95, 71 (1996) [astro-ph/9507001].

• M. Sasaki and T. Tanaka,

Superhorizon scale dynamics of multiscalar inflation, Prog. Theor. Phys. 99, 763 (1998) [gr-qc/9801017].

• D.H. Lyth, K.A. Malik and M. Sasaki,

A General proof of the conservation of the curvature perturbation, JCAP 0505, 004 (2005) [astro-ph/0411220].

• A. Naruko and M. Sasaki,

Conservation of the nonlinear curvature perturbation in generic single-field inflation, Class. Quant. Grav. 28, 072001 (2011) [arXiv:1101.3180 [astro-ph.CO]].

• J.-O. Gong, J.-c. Hwang, W.-I. Park, M. Sasaki and Y.-S. Song, Conformal invariance of curvature perturbation,

JCAP 1109 (2011) 023 [arXiv:1107.1840 [gr-qc]].

a (highly biased) list of references

linear

quasi-nonlinear / separate universe approach

nonlinear

1. Introduction

Standard (single-field, slowroll) inflation predicts almost scale-invariant Gaussian curvature perturbations.

CMB (WMAP,PLANCK,...) is consistent with the prediction. Linear perturbation theory seems to be valid.

PLANCK 2013

PLANCK 2013

l

δ

δ

N

N

formalism for curvature perturbations

formalism for curvature perturbations

Φ=Φ_gauss+ f_NLΦ2

gauss+ ∙∙∙

Tensor perturbations (gravitational waves) have not been detected yet.

Future CMB experiments may still detect non-Gaussianity...

M

Models need to be tested.

multi-field, non-slowroll, string theory, vacuum bubbles, … -8.9< f_NL<14.3 (95%CL)

r<0.11 (95%CL) PLANCK 2013

tensor-scalar ratio:

(-)gravitational potential:

However, nature may be a bit more complicated...

PLANCK 2013

What is δN ?

•

• δN is the perturbation in # of e-folds counted

backward in time from a fixed final time t_f

•

δN is equal to conserved NL comoving curvature

perturbation R_NLon superhorizon scales at t>t_f •

• t_f should be chosen such that the evolution of the universe has become unique by that time.

• δN formula is valid independent of theory of gravity

therefore it is nonlocal in time by definition

isocurvature perturbation that persists until t=t_f must be dealt separately

“

3 types of δN

originally adiabatic end of/after inflation entropy/isocurvature → adiabatic f t = t φ₁ φ₂

2. Linear perturbation theory

(

)

( )

2 2 2

1 2

t

(1 2 )

_{ij ij} i j

ds

= − +

A dt

+

a



_

+

R

δ

+

H



_

dx dx

• propertime along xi _{= const.: d}

τ

= +_{(1 A dt)}

• curvature perturbation on Σ(t): R (3) (3) 2

4

R

a

= −

∆ R

x xi i _{= const.= const.} Σ Σ((tt)) Σ Σ((t+dtt+dt)) d dττ traceless

• expansion (Hubble parameter):

(

)

1

3 (3) 1 _t H = H − A + ∂ _ + ∆ E_ R  ɶ

( )

( )

scalar tensor 1 3 transverse-traceless (3) ij i j ij ij H E H

δ

  = ∂ ∂ −_ ∆ _   =

metric (on a spatially flat background)

• comoving slicing

₀

(

₌

( )

_{for a scalar field}

)

i

T

µ

=

φ φ

t

• flat slicing (3) (3)

₌₀

₀

2

4

R

a

= −

∆

R

⇔

R

=

• Newton (shear-free) slicing

( )

_scalar

1

0

0

0

3

(3) t

H

ij i j

δ

ij t

E

t

E

E





∂

= ∂ ∂ −

_

∆ ∂

_

=

⇔ ∂

=

⇔

=





• uniform density slicing

−

_T

0₀

≡ =

ρ ρ

( )

_t

• uniform Hubble slicing

Choice of gauge (time-slicing)

( )

1

0

3

(3) t

H

=

H t

⇔ −

H A

+ ∂



_

+ ∆

E



_

=



R



ɶ

matter-based gauge geometry-based gauge

comoving = uniform

ρ

= uniform H on superhorizon scales ┴

┴ ┴ ┴

( )

2

2

3

8

(3) 4 2

H

O

G

a

ε

π ρ

−

∆ +

R

=

ɶ

(

)

Hubble horizon scale

on superhorizon scales

wavelength

1

ε

=

≪

at leading order in ε , Friedmann equation holds

independent of time-slicing.

local ‘Hubble parameter’ given by

3

H

ɶ

2

=

8

π ρ

G

+

O

( )

ε

2

‘local’ means ‘measured on scales of Hubble horizon size’

Separate universe approach

0 0

0 8 0

G =

π

GT

further, if is time-independent,

Friedmann equation holds up through O(ε2_),

with local ‘curvature constant’ given by

( )

2

( )

3

(3)

i i

K x

= − ∆ R

x

local Friedmann eq. holds up through O(ε2_),

for adiabatic perturbations (= adiabatic limit) on comoving/uniform

ρ

/uniform H slices.

( )

2

2

3

8

(3) 4 2

H

O

G

a

ε

π ρ

−

∆ +

R

=

ɶ

( )

2

3

( )

₂ 4

8

i

K x

H

O

G

a

ε

π ρ

+

+

=

ɶ

comoving curvature perturbation R _C is conserved in the adiabatic limit:

; 2 2 2 2 2 2 2 ( ) ( ) ( ) ~ z a P O z a z H

ρ

ε

′ + ′′ + ′ = ≡ C C R R

3. Linear

δN formula

Starobinsky ’85, MS & Stewart ’96, …. e-folding number perturbation between Σ(t) and Σ(t_fin):

(

)

(

)

( )

( )

( )

fin fin fin fin background fin

1

3

(3) 2

;

t t t t t t

N t t

H d

H d

E dt

t

t

O

δ

τ

τ

ε

≡

−





=

∂

_

+ ∆

_

=

−

+





∫

∫

∫

t

R

R

R

ɶ

N N₀₀((tt,,tt_finfin))

δ

δ

NN((tt,,tt_finfin)) Σ Σ ((tt_finfin); ); RR_((tt fin fin)) Σ Σ₀₀ ((tt_finfin)₎ Σ Σ₀₀ ((tt)) xi _=const. Σ Σ ((tt); ); RR₍₍t_t)) δN=0 if both Σ(t) and Σ(t_fin) are chosen to be ‘flat’ (R=0).

By definition, δN(t; t_fin) is t-independent.

The gauge-invariant variable ‘ζ’ used in the literature

is equal to R_C on superhorizon scales (sometimes ζ = -R_C) Σ(t); R(t)=0

Σ(t_fin); R_C(t_fin)

xi _=const.

Choose Σ(t) = flat (R=0) and Σ(t_fin) = comoving:

(

;

fin

)

( )

fin

( )

C

( )

fin

N t

t

t

t

t

δ

=

R

−

R

= R

(suffix ‘C’ for comoving) curvature perturbation on comoving slice

Example: single-field slow-roll inflation

- single-field inflation, no extra degree of freedom R

C becomes constant soon after horizon-crossing (t=th):

(

h

;

fin

)

( )

fin

( )

h

N t t

t

t

δ

=

R

_C

=

R

_C log a log L L=H-1 t=t_h RC = c onst. t=t_fin inflation

Also because R_c is conserved, δN = H(t_h) δt_F→C , where δt_F→C is the time difference between the comoving and flat slices at t=t_h.

Σ_F(t_h) : flat R=0, δφ=δφ_F Σ_C(t_h) : comoving

δ

t_F→C R=R_C , δφ=0

( )

(

)

( )

C fin h; fin F C F h dt t N t t H t H t d

δ

δ

δφ

φ

→ = = = − R ··· δN formula

(

)

( )

F h F C, C h i t t x t

φ

+

δ

_→ =

φ

_δφ

_{F +}

_φ

ɺ

( )

_th

_δ

_{tF→C = 0} dN = −Hdt

( )

F h dN t d

φ

δφ

=

Only the knowledge of the background evolution is necessary to calculate R_C(t_fin) .

(for slow-roll, no isocurvature perturbation)

( )

( )

( )

C fin

δ

φ

δφ

F h

δφ

F h ∂ = = ≡ ∇ ⋅ ∂

∑

R a a a a a N t N t N t

MS & Stewart ’96, MS & Tanaka ‘98

N.B. R_C is no longer conserved:

( )

F C t H 2

φ δφ

φ

⋅ = − R ɺ

ɺ ··· time-varying even on superhorizon

( )

( )

_{( )}

h F 2 ₂ 2 3 4 2 2 2 2 2 3 ₂ 4 ( ) (2 ) S H t k H P k N N

π

δφ

π

_π

_π

_φ

    = ∇ = ∇ ≥    ɺ  a a N N

φ

∂ ∇ ≡ ∂

• spectrum (for mutually independent

δφ

_Fa ₎

2 ₂ 2 _{2 2} 2 2 a a H H φ N φ N N φ = ∇ ≤ ∇ ⇒ ∇ ≥ i ɺ ɺ

tensor-to-scalar ratio

8

16

T T s S

P

n

P

≤

=

ε

• scalar spectrum:

( )

( )

2

( )

2

π

π

=

π

∇ 3 2 2 3 2 4 S k H P k N • tensor spectrum:

( )

( )

4 8 2 2

π

κ

π

=

π

3 2 2 3 2 ( ) T k H P k

• tensor spectral index: 2

1 8 2 2 2 2 2 2 2 2 2 2 _s T T S H n H H P P N N

φ

κ

φ

κ

κ

φ

ε

− = ≡ − = = ≥ = ∇ ⋅∇ ɺ ɺ ɺ ɺ φ = − = − a ∇ a dN H N dt i MS & Stewart ‘96

··· valid for any slow-roll models

1 s n

k

−

∝

∝

k

nT 8 2 G κ = π

(‘=’ for a single inflaton model)

2 s H H

ε

≡ − ɺ slow-roll parameter Einstein gravity

4. Non-linear extension

;

i

x

Q

t

Q

HQ H

G

ρ

∂

∂

∂

≪

∂

∼

∼

This is a consequence of causality:

Field equations reduce to ODE’s

Belinski et al. ’70, Tomita ’72, Salopek & Bond ’90, …

light cone

L »H-1

H-1

• On superhorizon scales, gradient expansion is valid:

metric on superhorizon scales

(

)(

)

( )

2 i

det

1,

2 2 2 i i j j ij ij

ds

dt

e

dx

dt dx

dt

O

α

_γ

_β

_β

γ

β

ε

= −

+

+

+

=

=

N

ɶ

ɶ

expansion parameter

,

i

ε

i

ε

∂ → ∂

=

( )

( ,

)

( ,

exp



_

α

t x

i



_

=

a

t

ex

p

_



ψ

t x

i

)



_

the only non-trivial assumption

e.g., choose

ψ

(t_* ,0) = 0

fiducial `background’

contains GW (~ tensor) modes

•

• gradient expansion:

•

• metric:

• Local Friedmann equation

2

8

2

( ,

)

( ,

)

(

)

3

i

G

i

H t x

ɶ

=

π

ρ

t x

+

O

ε

xi _{: comoving (Lagrangian) coordinates.}

exactly the same as the background equations.

uniform

ρ

slice = uniform Hubble slice = comoving slice

(

)

3

0

d

H

p

d

τ

ρ

+

ρ

+

=

ɶ

d

τ

= N dt : proper time along fluid flow

as in the case of linear theory

no modifications/backreaction due to super-Hubble perturbations.

(

)

(

)

( )

0

0 ;

2

;

3

t

T

u u

p g

u u

u

T

d

u

p

u

O

d

µν µ ν µν µ ν µν µ ν µ µ µ µ

ρ

α

ρ

ρ

ε

τ

=

+

+

∇

=

∂

⇒

+ ∇

+

=

∇

=

+

N

( )

normal to

const

.

2

1

1

3

3

H

u

n

O

n dx

dt

t

µ µ µ µ µ µ

ε

≡ ∇

= ∇

+

= −

N

⋅⋅⋅

=

ɶ

At leading order, local Hubble parameter is independent of the time slicing, as in linear theory

n

µ

u

µ t=const.

( )

0 i i

u

v

O

u

ε

≡

=

assumption uµ – nµ = O(ε) •

• energy momentum tensor:

• local Hubble parameter:

5. Nonlinear δN formula

(

)

( )

2 3 t t t a O H p a ρ ε α ψ ρ ∂ _{ } + = −∂ = −_ + ∂ _ = − + _{ } N ɺ ɶ energy conservation:

(applicable to each independent matter component)

e-folding number:

(

)

2 2 1 1 1 2 1 , ; 3 i t t i _t t t x N t t x H dt dt P

ρ

ρ

∂ ≡ = − +

∫

ɶ N

∫

where xi _{=const. is a comoving worldline.}

(

)

(

)

2 1 1 2 1 2 1 1 2 0 2 0 , ; , ; ( , ) 1 ( , ) 3 ( ) i i t t t t t x t t x N N N dt N t t P t t

ρ

ρ

ρ

≡ − = ∆ ∂ − − +

∫

(

2,

) (

1,

)

(

1, ;2

)

i i i t x t x

N

t t x

ψ

−

ψ

= ∆

where

Lyth, Malik & MS ’04 Langlois & Vernizzi ’05

This definition applies to any choice of time-slicing

To summarize:

(

) (

)

(

)

2 1 2 1 1 2 0 1 2 , , , ;

1

( , )

3

( )

i i i t t t t x t x

N

t t x

dt

N t t

P

ψ

ψ

ρ

ρ

ρ

−

= ∆

∂

= −

−

+

∫

matter geometry

relates the evolution of matter to geometry.

Here we use ∆N for general choice of slices.

N₀(t₁,t₂)

∆

N(t₁,t₂) Σ (t₂); ψ(t₂) Σ_F (t₂) Σ_F (t₁) xi _=const. Σ (t₁); ψ(t₁)

Σ_F (t): hypersurface on which

ψ

= 0 ↔ eα = a(t); ‘flat’ slice

between and

1 2 0 1 2 1 2

( , ; )i ( , ) _F ( ) _F ( )

N t t x = N t t Σ t Σ t

No need for `background’ universe

NL

δ

N - formula

Let us take slicing such that Σ(t) is ‘flat’ at t = t₁ [ Σ_F (t₁) ] and uniform density/comoving/uniform H at t = t₂ [ Σ_C (t₂) ]:

( ‘flat’ slice: Σ(t) on which

ψ

= 0 ↔ eα = a(t) )

Σ_F (t₁) : flat

Σ_C(t₂) : uniform density

_ρ

_(t2)=const.

ψ

(t₁)=0 N₀(t₂,t₁)

∆

N_C

ψ

(t₂)=0 Σ_F (t₂) : flat 1 2 1 2 ( , ; )i _C( , ; )i N t t x N t t x ∆ = ∆

(

1

,

)

0,

(

2

,

)

C

(

2

,

)

(

1

,

2

;

)

i i i i

C

t x

t x

t x

N

t t x

ψ

=

ψ

=

R

= ∆

where

∆Ν

_C is the e-folding number from Σ_F(t₂) to Σ_C(t₂):

(

)

2 2 2 2 1 1 ( ) ( 1 2 ) ( ) ( ) ( ) ( )

1

1

3

, ;

1

3

3

C F F _i F C F _i t i _t t _t C _t t t x x t t

N

dt

N t t x

dt

P

t

P

P

d

ρ

ρ

ρ

ρ

δ

ρ

ρ

Σ Σ Σ Σ Σ Σ

∂

∂

≡ ∆

= −

+

=

+

∂

−

+

+

∫

∫

∫

suffix C for comoving/uniform ρ/uniform H

indep of t₁

Σ_C(t): matter is almost homogeneous & isotropic

Then

For adiabatic case

(

p=p(ρ) ,or single-field slow-roll case),

(

)

2 1 2 1 1 2 ( , ) 2 2 1 ( , ) 1 1 , ; 3 ( ) ( ) 1 ( , ) ( , ) ln 3 ( ) ( ) i i t i t t t x i i t x N t t x dt P a t d t x t x P a t ρ ρ

ρ

ρ

ρ

ρ

ψ

ψ

ρ

ρ

∂ = − +   = − = − + _{ } + _{ }

∫

∫

6. conservation of NL curvature perturbation

2 1 2 1 2 2 1 1 ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( ) ( ) 2 2 1 1 1 3 ( ) 1 1 1 3 ( ) 3 ( ) 3 ( ) ( ) ( , ) ( , ) ln ( ) i i i i t x t x t x t x t t t t i i d P d d d P P P a t t x t x a t ρ ρ ρ ρ ρ ρ ρ ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ψ

ψ

− + = − + − + + +   = − + _{ }  

∫

∫

∫

∫

, dρ ≈ V d′ φ 2 2 3 V P V ρ + =φɺ ≈ ′ 1 3 d V d N P V ρ δρ φ δφ ρ φ ρ φ δ ρ + + = = ′ +

∫

∫

ex.: single-field slow-roll inflation

h 0 ( ) ψ δ ₌ = = R NL N t t NL ( , ) ( )

1

( )

( ,

)

3

( )

i t x i i t

d

x

t x

P

ρ ρ

ρ

ψ

ρ

ρ

≡

+

+

∫

R

non-linear generalization of

conserved ‘gauge’-invariant quantity ζ or R

c

(

ψ

and

ρ

can be evaluated on any time slice)

···slice-independent 1 2 1 2 ( , ) ( , ) 1 _{( )} 2 _{( )} 1 1 ( , ) ( , ) 3 ( ) 3 ( ) i i t x t x i i t t d d t x t x P P ρ ρ ρ ρ

ρ

ρ

ψ

ψ

ρ

ρ

ρ

ρ

+ = + + +

∫

∫

Example 2: Curvaton model

ρ

_φ=

ρ

_γ

∝

a-4 _and

ρ

χ

∝

a-3, hence

Ω

χ /

Ω

γ

∝

a Ω_χ Ω_γ Ω t

2-field model: inflaton (φ) + curvaton (χ)

2 2

1

( )

2

V

V

m

χ

φ

χ

=

+

2 2

8

3

GV

m

H

χ

π

≈

≪

•

• after inflation,

χ

begins to dominate if it does not decay. • during inflation

φ

dominates.

•

( , ) 1 ln 3 ( ) χ χ χ

ρ

ψ

ρ

  = + _{ }   R i t x t ( , ) 1 ln 4 ( ) γ γ γ

ρ

ψ

ρ

  = + _{ }   R i t x t 3( ) 4( ) ( , )t xi ( , )t xi e χ ψ e γ ψ χ γ χ γ

ρ

+

ρ

=

ρ

− R − +

ρ

− R − •

• With sudden decay approx, final curvature pert amp

ζ

is determined by 3( ) 4( )

( , )

t x

i

( , )

t x

i

e

χ ζ

e

γ ζ χ γ χ γ χ γ

ρ

+

ρ

=

ρ

− R −

+

ρ

− R −

=

ρ

+

ρ

•

• On homogeneous total density slices,

ψ

=

ζ

( A A) A C A P P ρ ζ ρ + = = +

∑

R R nonlinear version of

(

)

4( ) 3( )

1

e

γ ζ

e

χ ζ

1

χ χ − −

− Ω

R

+ Ω

R

=

Ω_χ : density fraction of

χ

at the moment of its decay MS, Valiviita & Wands (2006)

7

. NL δN for ‘slowroll’ inflation

• Nonlinear

δ

N for multi-component inflation :

(

) ( )

1 2 1 2 1 ! δ φ δφ φ δφ δφ δφ φ φ φ = + − ∂ = ∂ ∂ ∂

∑

n n A A A n A A A A A A n N N N N n _⋯ ⋯ •

• In slow-roll inflation, all decaying mode solutions of the (multi-component) inflaton field

φ

die out.

MS & Tanaka ’98, Lyth & Rodriguez ‘05

•

• If

φ

is slow rolling (or already at an attractor stage) when the scale of our interest leaves the horizon, N is only a

function of

φ

(independent of d

φ

/dt ), no matter how complicated the subsequent evolution is.

where

δφ

=

δφ

_F (on flat slice) at horizon-crossing.

(

δφ

_F may contain non-gaussianity from subhorizon interactions) eg, DBI inflation

example:

multi-brid

inflation

0

φ





=

_

_



∑



exp

_A

(

_A

)

A

V

V

u

1 3 1

φ

φ

φ

φ

φ

∂ ∂ _′ = = = = ∂ ∂ 2 1 - ( ) A A A A A A d d V V u dN H dt H V • slow-roll eom: 1 2

1

2

_,

( )

A A A

L

_φ

g

µν _µ

φ

_ν

φ

V

φ

=

= −

∑

∂

∂

−

_V φφφφ inflation 1 2

,

) +

( ,

φ φ

⋅⋅⋅

φ

_n

χ

MS MS ‘‘0808

= −

dN

Hdt

N as a time variable: 2 2 2 2 3 1 8

κ

κ

₌

π

=₌ − ₌ Pl H V G M inflaton

(

)

A A A A A

dq

d

q

u

φ

φ

≡

_′

1 cos , 2 sin , q = q θ q = q θ θ = const. •

• transformation of field variables:

1

ln

_A

d

q

dN

=

1

1

1

3

(

)

(

)

A A A A A A A

d

V

d

u

dN

V

u

dN

φ

φ

φ

φ

φ

∂

_′

=

=

⇒

=

′

∂

set set 2 A

;

1

A A A

q

=

qn

∑

n

=

d

ln

q

₁

_,

dn

A

₀

dN

=

dN

=

angular coordinates n_A are conserved..

q q₂₂ q q₁₁

θ

θ

N N=0=0 q q 1, 2

(

q q

)

N N=const.=const.

trajectories are radial in space

trajectories are radial in space

1 2

(

, )

( , )

( , )

q q N

N

N q

N

θ

θ

φ φ

=

↔

=

=

1 , 0 ln d q d dN dN θ = = _{( , ) ln ln ( )} f N q θ = q − q θ f f

( )

q

=

q

θ

Assume that inflation ends at

g

₁2

φ

₁2

+

g

₂2

φ

₂2

=

σ

2 1 f 2 f 1 2 , cos , , sin g g

σ

σ

φ

=

γ

φ

=

γ

Parametrize orbits by an angle at the end of inflation

2 2 2 1 2 / , A mA A q = eφ q = q + q

and the universe is thermalized instantaneously.

1 1 2 2 1,f 1 2,f 2 2 2 f 2 2 1 2 / / / ln ln ( ) ln m m m m e e N q q e e φ φ φ φ

θ

 +  = − = _{ } +  

(

2 2 2 2

)

2 2 2 2 0 1 1 2 2 1 2

χ

4

λ

σ

φ

φ

λ

χ

  = + + _ − _   V g g realized by For exponential pot.:

0 φ 0 φ = _{ } = _{ } 

∑

 

∑

 exp _A( _A) exp _{A A} A A V V u V m φ φ φ = = ′ ( ) A A A A A A A dq d d q u m

(

1 2

)

2 2 1 ₁ 1₁ 22 2 2 _{2 2}

1

2

/ / cos / sin /

,

ln

m m g m g m

e

e

N

N

e

e

φ φ σ γ σ γ

φ φ



+



=

=

_

_

+





where

γ

=

γ

(

φ

₁,

φ

₂)

(

1 1

,

2 2

)

(

1

,

2

)

N

N

N

δ

=

φ δφ φ δφ

+

+

−

φ φ

• δN valid to full nonlinear order is simply given by

1 1 2 2 1 2 1 1 2 2

cos

sin

ln

q

q

m

m

g m

g m

φ

φ

σ

γ σ

γ





=

−

=

−









This determines

γ

in terms of

φ

₁ &

φ

₂ . (∙∙∙ const of motion)

1 2 f 0

1 2

exp m cos m sin

V V g g

σ

σ

γ

γ

  = _ + _  

• To be precise, one has to add a correction term to adjust the energy density difference at the end of inflation

f 1 2 0 1 2 1 4 ln 4 cos sin c V m m N V g g

σ

γ

γ

    = _{ } = _ + _    

(assuming instantaneous thermalization)

1 1 2 2 1 1 2 2 2 2 2 2 1 2 / / cos / sin / ln m m c g m g m e e N N e e φ φ σ γ σ γ  ₊  = _{ } + +   where

However, this correction is negligible

1

,

2 Pl

1

m m

≪

M

=

• δN to 2 order in δφ : 2 2 1 2 1 1 2 2 2 1 1 2 3 1 1 2 2 1 1 2 2 2 2 cos sin ( )

cos sin ( cos sin )

g g g m m N m g m g m g g m g δφ γ δφ γ δφ δφ δ γ γ σ γ γ + − = + + +

• comoving curvature perturbation spectrum 2 2 2 2 2 1 2 2 1 1 2 2 2 γ γ γ γ π = +   = _{ } + _{ } P_{( )} cos sin ( cos sin ) S k Ha g g H k m g m g 2 2 1 2 1 ( ) s n = − m + m 2 1 1 2 2 2 2 2 2 1 2 8 γ γ γ γ + = = + S P P ( ) ( cos sin ) ( ) cos sin T k m g m g r k g g • cf: single-field case f f mN N m

φ φ

φ

= +

φ

⇔ = − No non-Gaussianity if δφ is Gaussian spectral index: tensor/scalar: 2 2 local 1 , 8 , 0 s NL n = − m r = m f =

1 1 2 2 1 2 2 1

1 1 2 2 2 1 1 2

,

cos sin sin cos

cos sin cos sin

L g g g g N S m g m g m g m g δφ γ δφ γ δφ γ δφ γ δ γ γ γ γ + − ≡ ≡ + − Let

“true” entropy perturbation

2 2 2 local _{1 2 2 1 1 2} 2 2 2 2 2 1 2 1 1 2 2 5 6 ( cos sin )

( cos sin ) cos sin

NL g g m g m g f g g m g m g γ γ σ γ γ γ γ − = + +

(

)

local 2 3 5 NL L L N N f N S

δ

=

δ

+

δ

+ 2 0 for 2 A B AB L H N S δ δφ δφ δ π   ⋅ = _{=  }  

(

)

local 1 2 1 2

for

/

,

~ ( ),

,

~ ( ).

NL

f

=

O gm

σ

m m

O m

g g

O g

practically any non-Gaussianity is possible

linear entropy perturbation contributes at 2nd _order

local

0 ( . .,N B f_NL > )

• example of parameters

model parameters: outputs: 2 2 1 ~ .0 005 , 2 ~ .0 035 m m

(

)

2 4 9 5 3H =

σ

4

λ

~ .1 5 10 × − ⇔ PR( ) ~ .k 2 5 10× − 2 2 1 2 1 ( ) ~ .0 96 s n = − m + m 2 1 8 ~ .0 04 r ≈ m 2 local ₂ 1 4 1 5 40 6 ~ / NL gm g f m

σ

λ

≈ 1 2 18 1= M_Pl = (8

π

G)− / = 2 43 10 GeV. × 1cos 2 sin m

γ

≫ m

γ

assume 2 1 2 4 10 / ~

σ

λ

_× − 2 2 2 1 2 g = g ≡ g independent of waterfall field

Planck 2013 constraint on r & n

_s

example

f_NLlocal _{can be ~ 10}

8. Conformal frame (in)dependence

why bother?

-In cosmology, we encounter various frames of the metric which are conformally equivalent.

But it is often said that there exists a unique physical frame

on which we should consider actual ‘physics.’

They are mathematically equivalent, so one can work in any frame as long as mathematical manipulations are concerned.

Einstein frame, Jordan frame, string frame, ...

How does physics depend/not depend

on choice of conformal frames?

• Einstein frame

“gravitational” part : R+L(

φ

) ~ minimal coupling between g and

φ

matter part: G(

φ

)L(

ψ

, A,…)

ψ

: fermion, A : vector, ...

Two typical frames in scalar-tensor theory

• Jordan(-Brans-Dicke) frame

“gravitational” part : F(

φ

)R+L(

φ

)

matter part: L(

ψ

, A,…) ~ minimal coupling with g matter assumed to be universally coupled with g

∙∙∙ for baryons, experimentally consistent

φ

φ

φ

φ

++++ g

if non-universal coupling: ( ( ); , , . A A A A A G

φ

L Q Q

ψ

A ⇒

∑

) = ⋅⋅⋅

conformal transformation

2 g_µν → ɶg_µν = Ω g_µν 2 2 ( 1) 2 ( 4) R → R = − _R − D − _ Ω − D − gµν ∂ Ω∂ Ω µ ν _ Ω Ω     Ω □ ɶ

• metric and scalar curvature

• matter fields ( for D = 4 )

(

)

(D 2) / 2 2

φ

_→

φ

_{= Ω}− −

φ

_{= Ω}−

φ

ɶ

(

)

(D 1) / 2 -3/ 2

ψ

_→

ψ

_{= Ω}− −

ψ

_{= Ω}

ψ

ɶ

(

)

(D 4) / 2 A_µ → Aɶ_µ = Ω− − A_µ =A_µ scalar vector fermion

cosmological perturbations

Makino & MS ‘91, Komatsu &Futamase ’99,...

• tensor-type perturbation

Definition of h_ij is apparently Ω-independent.

0 ij j jh h j ∂ = =

(

)

(

)

2 2 2 2 2 ( ) ( ) i j ij ij j ij ij i ds dt a t dx dx a d x h d h dx

δ

η

η

δ

= − + +   = _− + + _

(

)

2 2 2 2 2 2 ( ) ( ) _{ij ij} i j ds ds x aµ

η

d

η

δ

h dx dx = Ω   = Ω _− + + _ ɶ

• vector-type perturbation

(

)

2 2 2

2B_j j _{ij i}H _{j j i} i j

ds = a _−d

η

+ dx d

η

+

δ

+ ∂ + ∂ H dx dx _

Definitions of B_j and H_j are aslo Ω-independent.

0 j j jB jH ∂ = ∂ =

(

)

2 2 2 2 2 2 2 _j j _{ij i j j i} i j ds ds a d

η

B dx d

η

δ

∂ H + ∂ H dx dx = Ω   = Ω _− + + + _ ɶ

(spatial) tensor & vector are conformal frame-independent

This means in particular P_T(k) formula (~H2_{) from inflation}

• scalar-type perturbation

(

)

2 2 2 ( ) (1 2 ) 2 (1 2 ) 2 j i j j i j ij ds a d dx d dx x A d B E

η

η

η

δ

 = _− + +  + + + ∂ _ ∂ ∂ R

Definitions of B and E are Ω-independent.

( )

, 0

( )

1 ( , ) i i t x t 

ω

t x  Ω = Ω _ + _

(

)

2 2 2 2 2 2 (1 2 ) 2 (1 2 ) 2 j i j i j i j j ds ds a d dx d d A B E x dx

η

η

δ

= Ω  = Ω _− + +  + + _ ∂ ∂ ∂ + R ɶ , A → A +

ω

R → R + ω

For scalar-tensor theory with

1 1

( ) ( , ) ,

2 f R K X X 2

L =

φ

+

φ

≡ − gµν ∂ ∂_µ

φ φ

_ν

The important, curvature perturbation R

c , conserved on

superhorizon scales, is defined on comoving hypersurfaces.

R

c

= R

δφ=0

is Ω-independent!

Nevertheless, if we have

Ω = Ω

( )

φ

1 c H da a d

δφ

δφ

φ

φ

≡ − = − R R R ɺ uniform

φ

(

δφ

= 0) frame-independent

( )

φ

Ω = Ω

R

c is conformal-frame independent in the adiabatic limit

⇔if not in the adiabatic limit, the notion of adiabatic perturbation depends on choice of conformal frames

generalization to NL perturbation

Generalization is straightforward for perturbations on superhorizon scales

δN formalism:

Gong, Hwang, Park, Song & MS ‘11

R

c(tf) = δN between the final comoving surface (t=tf)

and an initial flat surface

although the number of e-folds N depends on conformal frames, δN is frame-independent in the adiabatic limit

White, Minamitsuji & MS ‘13 ....

9. Summary

• There exists a NL generalization of comoving

curvature perturbation R

_C

which is conserved for an

adiabatic perturbation on superhorizon scales.

• There exists a NL generalization of

δ

N formula,

which may be useful in evaluating non-Gaussianity

from inflation.

• (NL)

δ

N formula is independent of conformal frames if