N formalism for curvature perturbations formalism for curvature - - PowerPoint PPT Presentation

SLIDE 1

δ δ N N formalism for curvature perturbations

formalism for curvature perturbations from inflation from inflation

Yukawa Institute (YITP) Yukawa Institute (YITP) Kyoto University Kyoto University

Misao Sasaki Misao Sasaki

SLIDE 2

• 1. Introduction
• 1. Introduction
• 2. Linear perturbation theory
• 2. Linear perturbation theory
• metric perturbation & time slicing

metric perturbation & time slicing

• δ

δN formalism N formalism

• 3. Nonlinear extension on superhorizon scales
• 3. Nonlinear extension on superhorizon scales
• gradient expansion, conservation law

gradient expansion, conservation law

• local Friedmann equation

local Friedmann equation

• Δ

ΔN for slowroll inflation N for slowroll inflation

• diagrammatic method for

diagrammatic method for Δ ΔN N

• IR divergence issue

IR divergence issue

• 5. Summary
• 5. Summary
• 4. Nonlinear
• 4. Nonlinear Δ

ΔN formula N formula

SLIDE 3

• 1. Introduction
• 1. Introduction

Standard (single Standard (single-

• field, slowroll) inflation predicts scale

field, slowroll) inflation predicts scale-

• invariant

invariant Gaussian Gaussian curvature perturbations. curvature perturbations.

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

SLIDE 4

Tensor perturbations Tensor perturbations have not been detected yet. have not been detected yet.

Re Re-

• consider the dynamics on super

consider the dynamics on super-

• horizon scales

horizon scales

Ψ= Ψ=Ψ Ψgauss

gauss+ f

+ fNL

NLΨ

Ψ2

2 gauss gauss+

+ ∙∙∙ ∙∙∙ ; | ; |f fNL

NL|

| ≳ ≳ 5? 5?

So, why bother doing more research on inflation? So, why bother doing more research on inflation? PLANCK, CMBpol, PLANCK, CMBpol, … … may detect may detect non non-

• Gaussianity

Gaussianity

Because observational data does not exclude other models. Because observational data does not exclude other models.

In fact, inflation may not be so simple. In fact, inflation may not be so simple.

multi multi-

• field, non

field, non-

• slowroll, extra

slowroll, extra-

• dim

dim’ ’s, string theory s, string theory… …

T T/ /S S ~ 0.2 ~ 0.2 -

• 0.3? or smaller?

0.3? or smaller?

Nonlinear backreaction on superhorizon scales? Nonlinear backreaction on superhorizon scales?

SLIDE 5

• 2. Linear perturbation theory
• 2. Linear perturbation theory

( )

( )

2 2 2

1 2 (1 2 )

i j ij ij

t

A H ds dt a dx dx δ ⎡ ⎤ = − + + + + ⎣ ⎦ R

• propertime along

propertime along x xi

i = const.:

= const.: (1 ) d dt A τ = +

• curvature perturbation on

curvature perturbation on Σ Σ( (t t): ): Ρ

(3) (3) 2

4 R a = − Δ R

x xi

i = const.

= const. Σ Σ( (t t) ) Σ Σ( (t+dt t+dt) ) d dτ τ

• expansion (Hubble parameter):

expansion (Hubble parameter):

( )

1 3

(3)

1

t

H A H E ⎡ ⎤ = − + ∂ + Δ ⎢ ⎥ ⎣ ⎦ R %

( ) ( )

scalar tensor

transverse-trace ss le = ∂ ∂ =

ij ij i j

H E H

metric on a spatially flat background metric on a spatially flat background

Bardeen Bardeen ‘ ‘80, Mukhanov 80, Mukhanov ‘ ‘81, Kodama & MS 81, Kodama & MS ‘ ‘84, 84, … …. .

( (g g 0

0j j=0 for simplicity)

=0 for simplicity)

SLIDE 6

• comoving slicing

comoving slicing

( )

( )

= for a scalar field

i

T t

μ

φ φ =

• flat slicing

flat slicing

(3)

=0

(3) 2

4 R a = − Δ ⇔ = R R

• Newton (shear

Newton (shear-

• free) slicing

free) slicing

3 scalar traceless

1 3

( ) t ij i j ij t t

H E E E δ ⎡ ⎤ ⎡ ⎤ ∂ ≡ ∂ ∂ − Δ ∂ = ⇔ ∂ = ⇔ ⎢ ⎥ ⎣ ⎣ ⎦ = ⎦

• uniform density slicing

uniform density slicing

( )

t T ρ ρ − ≡ =

• uniform Hubble slicing

uniform Hubble slicing

Choice of time Choice of time-

• slicing

slicing ( )

1 3

(3) t

H A H H t E ⎡ ⎤ ⇔ − + ∂ + = = Δ ⎢ ⎥ ⎣ ⎦ R %

matter matter-

• based slices

based slices geometrical slices geometrical slices comoving comoving = = uniform uniform ρ ρ = = uniform uniform H H on superhorizon scales

• n superhorizon scales

SLIDE 7

δ δN formalism in linear theory

N formalism in linear theory

MS & Stewart MS & Stewart ’ ’96 96

e e-

• folding number perturbation between

folding number perturbation between Σ Σ( (t t) and ) and Σ Σ( (t t fin

fin):

):

( )

( )

( ) ( )

( )

fin fin fin

fin fin background

1 3

(3) 2

;

t t t t t t

H d H d E dt O N t t t t δ τ τ ε ≡ − ⎡ ⎤ = ∂ + Δ + ⎢ = − ⎥ ⎣ ⎦

∫ ∫ ∫

t

R R R %

N N 0

0(

(t t, ,t tfin

fin)

) δ δN N( (t t, ,t tfin

fin)

) Σ Σ0

0 (

(t

tfin

fin)

)

Σ Σ0

0 (

(t

t)

)

Σ Σ ( (t tfin

fin),

), Ρ Ρ( (t t fin

fin)

) x xi

i =const.

=const. Σ Σ ( (t t), ), Ρ Ρ( (t t) ) δ δN N=0 if both =0 if both Σ Σ( (t t) and ) and Σ Σ( (t t fin

fin) are chosen to be

) are chosen to be ‘ ‘flat flat’ ’ ( (Ρ Ρ=0). =0).

SLIDE 8

Σ Σ(

(t

t), ), Ρ Ρ( (t t)=0 )=0 Σ Σ( (t tfin

fin),

), Ρ ΡC

C(

(t tfin

fin)

) x xi

i =const.

=const. Choose Choose Σ Σ( (t t) = flat ( ) = flat (Ρ Ρ=0) =0) and and Σ Σ( (t t fin

fin) = comoving:

) = comoving:

( ) ( )

fin C fin

; N t t t δ = R

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

The gauge The gauge-

• invariant variable

invariant variable ‘ ‘ζ ζ’ ’ used in the literature used in the literature is related to is related to Ρ ΡC

C as

as ζ ζ = = -

• Ρ

ΡC

C or

• r ζ

ζ = = Ρ ΡC

C

By definition, By definition, δ δN N( (t t; ; t tfin

fin) is

) is t t-

• independent

independent

SLIDE 9

Example: slow Example: slow-

• roll inflation

roll inflation

• single

single-

• field inflation, no extra degree of freedom

field inflation, no extra degree of freedom Ρ ΡC

C becomes constant soon after horizon

becomes constant soon after horizon-

• crossing (

crossing (t t= =t th

h):

):

( ) ( ) ( )

h fin fin h

; N t t t t δ = =

C C

R R

log log a a log log L L L L= =H H -

• 1

1

t t= =t th

h

Ρ ΡC

C = const.

= c

• n

s t . t t= =t tfin

fin

inflation inflation

SLIDE 10

Also Also δ δN N = = H H( (t th

h)

) δ δt tF

F→ →C C , where

, where δ δt tF

F→ →C C is the time difference

is the time difference between the comoving and flat slices at between the comoving and flat slices at t t= =t th

h.

. Σ ΣF

F(

(t

t h

h) : flat

) : flat

Σ ΣC

C(

(t th

h) : comoving

) : comoving

Ρ Ρ= 0,

= 0, δφ

δφ=

= δφ

δφF

F

δ δt tF

F→ →C C

δφ δφ= 0,

= 0, Ρ

Ρ=

= Ρ

ΡC

C

( ) ( ) ( )

C fin h fin F h

; / H t N t t t d dt δ δφ φ = = − R

··· ··· δ δN N formula formula

( )

( )

F h F C C h

,

i

t t x t φ δ φ

→

+ =

( )

F h F C

t t δφ φ δ

→

+ = & dN Hdt ⇐ = −

( )

F h

dN t d δφ φ =

Only the Only the knowledge of the background evolution knowledge of the background evolution is necessary to calculate is necessary to calculate Ρ ΡC

C(

(t tfin

fin) .

) .

Starobinsky Starobinsky ‘ ‘85 85

SLIDE 11

• δ

δ N N for a for a multi

multi-

• component scalar:

component scalar:

(for slowroll inflation) (for slowroll inflation)

( ) ( )

C fin F h a a a

N t N t δ δφ φ ∂ = = ∂

∑

R

N.B. N.B. Ρ ΡC

C (=

(=ζ ζ) is no longer constant in time ) is no longer constant in time: :

( )

MS & Stewart MS & Stewart ’ ’96 96

F C 2

t H φ δφ φ ⋅ = − R & &

··· ··· time varying even on time varying even on superhorizon scales superhorizon scales

Further extension to non Further extension to non-

• slowroll case is possible, if

slowroll case is possible, if general slow general slow-

• roll condition

roll condition is satisfied at horizon is satisfied at horizon-

• crossing.

crossing.

Lee, MS, Stewart, Tanaka & Yokoyama Lee, MS, Stewart, Tanaka & Yokoyama ‘ ‘05 05

( ) ( ) ( )

1 2

2 2 2

, , , ..., O O O H H H φ φ φ ξ ξ ξ ξ φ φ = = = & & & & & & = & &

( ) ( ) ( )

h C fin F

2

2 2 2 2 2 2

H t t N N δφ π = ∇ = ∇ R

a a

N N φ ∂ ∇ ≡ ∂

SLIDE 12

• 3. Nonlinear extension
• 3. Nonlinear extension

;

i

x t

Q Q HQ H Gρ

∂ ∂ ∂ ∂

= : :

This is a consequence of causality:

Field equations reduce to ODE’s

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

light cone

L L »

»H

H -

• 1

1

H H -

• 1

1

• On superhorizon scales, gradient expansion is valid:

On superhorizon scales, gradient expansion is valid:

• At lowest order, no signal propagates in spatial directions.

At lowest order, no signal propagates in spatial directions.

SLIDE 13

metric on superhorizon scales metric on superhorizon scales

( )( )

( )

2 i

det 1,

2 2 2 i i j j j ij i

ds dt dx e dt dx dt O

α

β β γ β ε γ = − + = = + + N % %

expansion parameter ,

i i

ε ε ∂ → ∂ =

( )

( )

( )

curvature perturbation , ln , ;

i i

t x t a t x α ψ ψ = + :

the only non the only non-

• trivial assumption

trivial assumption e.g., choose ψ (t* ,0) = 0 fiducial ` background’ contains GW (~ tensor) modes contains GW (~ tensor) modes

• gradient expansion:

gradient expansion:

• metric:

metric:

SLIDE 14

( )

( )

( )

0 ;

2

3 ;

t

T u u p g u u u T d d u p u O

μ μ μ μν μ ν μν μ ν μν μ ν μ

ρ ρ ρ τ α ε = + + ∇ = ⇒ + + = ∂ ∇ ∇ = + N

( )

normal to const.

2

1 1 3 3 H n u n O t dx dt

μ μ μ μ μ μ

ε ≡ ∇ = ∇ = + ⋅⋅⋅ = −N %

At leading order, local Hubble parameter on any slicing is At leading order, local Hubble parameter on any slicing is equivalent to equivalent to expansion rate of matter flow. expansion rate of matter flow.

n n μ

μ

u u μ

μ

t= const.

( )

i i

u v O u ε ≡ = u u μ

μ –

– n n μ

μ =

= O O( (ε ε) )

assumption: assumption:

• Energy momentum tensor:

Energy momentum tensor:

• Local Hubble parameter:

Local Hubble parameter: (absence of vorticity mode) (absence of vorticity mode) 1 3 H u μ

μ

≡ ∇ % So, hereafter, we redefine So, hereafter, we redefine H H to be to be ~ ~

SLIDE 15

Local Friedmann equation Local Friedmann equation

2 2

8 ( , ) ( , ) 3 ( )

i i

G H t x t x O ε π ρ = + %

xi : comoving (Lagrangean) coordinates.

uniform uniform ρ

ρ slice

slice = = uniform Hubble uniform Hubble slice slice = = comoving slice comoving slice

( )

3 d H p d ρ ρ τ + + = %

• cf. Hirata & Seljak ‘05

Noh & Hwang ‘05

as in the case of linear theory as in the case of linear theory dτ = dt : proper time along matter flow

no modifications/backreaction due to super no modifications/backreaction due to super-

• Hubble

Hubble perturbations. perturbations. exactly the same as the background equations. exactly the same as the background equations.

“ “separate universe separate universe” ”

SLIDE 16

• 4. Nonlinear
• 4. Nonlinear Δ

ΔN N formula

formula

( )

( ) ( )

2 2

3

t t t

a H p a O O ρ ε α ψ ρ ε ∂ ⎛ ⎞ = −∂ = − + ∂ = − ⎜ ⎟ + + ⎝ ⎠ + N & % energy conservation: (applicable to each independent matter component)

e-folding number:

( )

2 2 1 1

2 1

1 , ; 3

i

t t i t t t x

N t t x H dt dt P ρ ρ ∂ ≡ = − +

∫ ∫

N %

where xi= const. is a comoving worldline. This definition applies to any choice of time-slicing.

( ) ( )

2 1 2 1

2 1

, ; , ;

( ) ln ( )

i i

t t x t t x

a t N N a t ⎛ ⎞ Δ ≡ − ⎜ ⎟ ⎝ ⎠

( ) ( ) ( )

2 1 2 1

, , , ;

i i i

t x t x t t x

N ψ ψ − = Δ

where

SLIDE 17

Δ ΔN N -

• formula

formula

Let Let us take slicing such that

us take slicing such that Σ

Σ(

(t

t) is flat at

) is flat at t

t = = t t1

1 [

[ Σ ΣF

F (

(t

t 1

1) ]

) ] and uniform density/uniform and uniform density/uniform H

H/

/ comoving at

comoving at t t = = t t2

2 [

[ Σ ΣC

C (

(t

t 1

1) ]

) ] :

: ( ( ‘ ‘flat flat’ ’ slice: slice: Σ Σ( (t t) on which ) on which ψ ψ = 0 = 0 ↔ ↔ e eα

α =

= a a( (t t) ) ) ) Σ ΣF

F (

(t

t 1

1) : flat

) : flat

Σ ΣC

C(

(t

t 2

2) : uniform density

) : uniform density

ρ (t2)= const.

• ψ (t1)= 0

ρ (t1)= const. Σ ΣC

C(

(t

t 1

1) : uniform density

) : uniform density

1 2 2 1 1 2

between and ( ) ( ( ) ) ( , ) ln ( )

C C

a t N t t a t t t ⎛ ⎞ = ⎜ ⎟ Σ Σ ⎝ ⎠

N (t2,t1;xi) Δ ΔN N F

F

Lyth & Wands Lyth & Wands ‘ ‘03, Malik, Lyth & MS 03, Malik, Lyth & MS ‘ ‘04, 04, Lyth & Rodriguez Lyth & Rodriguez ‘ ‘05, Langlois & Vernizzi 05, Langlois & Vernizzi ‘ ‘05 05

2 1 2 1

( , ; ) ( , )

i F

N t t x N t t N = + Δ

SLIDE 18

( ) ( ) ( )

C

2 2 1

, , ,

i i i

F

t x t t x x

N ψ ψ ψ − Δ = =

where ΔΝ ΔΝF

F is equal to e-folding number from Σ

ΣF

F(

(t t1

1)

) to Σ ΣC

C(

(t t1

1):

):

2 1 1 1 1 2

( ( ) ( ) ( ) ( ) ) ( )

1 1 3 3 1 3

C C i F C C F i

t t t t F x t t t x t t

N dt dt P P dt P ρ ρ ρ ρ ρ ρ

Σ Σ Σ Σ Σ Σ

∂ ∂ Δ = − + + + ∂ = − +

∫ ∫ ∫

Then For slow-roll inflation in linear theory, this reduces to

( ) ( )

2 C 1 2 1 2 1

( ) ( ) ) ; (

F

a a a C F C

t t N t t H t t N t ψ δ δ δφ φ

→

⎡ ⎤ ∂ = ⎢ ⎥ ∂ ⎣ ⎦ ≡ = =

∑

R

suffix suffix C C for for comoving comoving/ / uniform uniform ρ

ρ/

/ uniform uniform H

H

SLIDE 19

Δ ΔN for

N for ‘ ‘slowroll slowroll’ ’ inflation inflation

• Nonlinear ΔN for multi-component inflation :

( ) ( )

1 2 1 2

1 ! φ δφ φ δφ δφ δφ φ φ φ Δ = + − ∂ = ∂ ∂ ∂

∑

n n

A A A n A A A A A A n

N N N N n L L

• In slowroll inflation, all decaying mode solutions of the

In slowroll inflation, all decaying mode solutions of the (multi (multi-

• component) inflaton field

component) inflaton field φ φ die out. die out.

MS & Tanaka MS & Tanaka ’ ’98, Lyth & Rodriguez 98, Lyth & Rodriguez ‘ ‘05 05

• If

If φ φ is is slow rolling when the scale of our interest leaves the slow rolling when the scale of our interest leaves the horizon horizon, , N N is only a function of is only a function of φ φ ( (indep indep’ ’t t of

• f d

dφ φ/ /dt dt, apart , apart from trivial dep. on time from trivial dep. on time t tfin

fin from which

from which N N is measured), no is measured), no matter how complicated the subsequent evolution would be. matter how complicated the subsequent evolution would be. where where δφ δφ = =δφ δφF

F (on flat slice) at horizon

(on flat slice) at horizon-

• crossing.

crossing. ( (δφ δφF

F may contain non

may contain non-

• gaussianity from subhorizon interactions)

gaussianity from subhorizon interactions)

• cf. Maldacena
• cf. Maldacena ‘

‘03, Weinberg 03, Weinberg ’ ’05, ... 05, ...

SLIDE 20

Diagrammatic method for nonlinear Diagrammatic method for nonlinear Δ

Δ N N

1 2 1 2 1 2 1 2

; ! ζ δφ δφ δφ φ φ φ =Δ = ≡ ∂ ∂ ∂

∑

A A An n A A An n

n A A A A A A n

N D N N N n

L L

L L

2 3

1 2 1 3 ( ) ( ) ( ) ( ) ! ( ) !

A AB A AB ABC ABC c

x y G N N N N x y G N G y N x y x ζ ζ = − + − + − +L ( ) ( ) ( ) ( )

A B AB

x y h G x y δφ δφ φ = −

• connected

connected n n-

• pt function of

pt function of ζ ζ: : x x y y A A A A 2 2-

• pt function

pt function x x y y A A B B B B A A ‘ ‘basic basic’ ’ 2 2-

• pt function:

pt function:

field space metric field space metric

+ + ··· ··· + +

δφ δφ is assumed to be Gaussian is assumed to be Gaussian for non for non-

• Gaussian

Gaussian δφ δφ, there will be basic , there will be basic n n-

• pt functions

pt functions

2! 2! 1 1 3! 3! 1 1

+ +

x x y y A A BC BC BC BC A A

Byrnes, Koyama, MS & Wands Byrnes, Koyama, MS & Wands ‘ ‘07 07

SLIDE 21

2

perm. perm. 1 2 ( ) ( ) ( ) ( ) ( ! ) ( ) ( ) ( ) ( ) ( )

A B AB AB CA BC A BC c ABC

x y z G x y G y z G x y G y z G z x N N N G x y G y N N N z N N N ζ ζ ζ = − − + + − − − + − − + +L

+ perm. + perm. x x y y z z A A B B B B C C A A C C

+ + + +

x x y y z z A A B B B B A A x x y y z z A A B C B C B C B C A A

+ + ··· ···

+ perm. + perm. 2! x 2! x 3 3-

• pt function

pt function 2! 2! 1 1 2! 2! 1 1

SLIDE 22

IR divergence problem IR divergence problem

x x y y z z A A B B B B C C A A C C

( ) ( ) ( )

AB CA BC

G x y G y N z G x N N z − − −

Loop diagrams like Loop diagrams like in the in the m m -

• pt function give rise to

pt function give rise to IR divergence IR divergence in in the ( the (m m -

• 1)

1)-

• spectrum if

spectrum if P P( (k k) )~k ~k n

n− −4 4 with

with n n ≤ ≤ 1 1. . eg eg, the above diagram gives , the above diagram gives

3 3 1 2 3 1 2 3 1 2

( , , ) ( ) (| ( | ) (| ) ) | P k k k k k k P k p k p p d p P P δ + + + −

∫

:

Is this IR cutoff physically observable? Is this IR cutoff physically observable? (real space 3 (real space 3-

• pt fcn is perfectly regular if

pt fcn is perfectly regular if G G0

0(

(x x) is regular.) ) is regular.)

Boubekeur & Lyth Boubekeur & Lyth ‘ ‘05 05

cutoff cutoff-

• dependent!

dependent!

SLIDE 23

• 8. Summary
• 8. Summary

Superhorizon scale perturbations can Superhorizon scale perturbations can never affect local never affect local (horizon (horizon-

• size) dynamics

size) dynamics, hence never cause backreaction. , hence never cause backreaction. nonlinearity on superhorizon scales are always nonlinearity on superhorizon scales are always local local. .

However, However, nonlocal nonlinearity (non nonlocal nonlinearity (non-

• Gaussianity)

Gaussianity) may may appear due to quantum interactions on subhorizon scales. appear due to quantum interactions on subhorizon scales.

• cf. Weinberg
• cf. Weinberg ‘

‘06 06

There exists a There exists a nonlinear generalization of nonlinear generalization of δ

δ N N formula

formula which which is useful in evaluating is useful in evaluating non non-

• Gaussianity

Gaussianity from inflation. from inflation.

diagrammatic method diagrammatic method can by systematically applied. can by systematically applied. IR divergence IR divergence from loop diagrams needs further from loop diagrams needs further consideration. consideration.