N formalism forSuperhorizon Perturbationsand its Nonlinear Extension

Misao SasakiYITP, Kyoto University

Based onBased onMS & E.D. Stewart, PTP95(1996)71 [astro-ph/9507001]MS & E.D. Stewart, PTP95(1996)71 [astro-ph/9507001]MS & T. Tanaka, PTP99(1998)763 [gr-qc/9801017]MS & T. Tanaka, PTP99(1998)763 [gr-qc/9801017]D. Lyth, K. Malik & MS, JCAP05(2005)004 [astro-ph/0411220]D. Lyth, K. Malik & MS, JCAP05(2005)004 [astro-ph/0411220]H.C. Lee, MS, E.D. Stewart, T. Tanaka & S. Yokoyama [astro-ph/0506262]H.C. Lee, MS, E.D. Stewart, T. Tanaka & S. Yokoyama [astro-ph/0506262]A. Linde, V. Mukhanov & MS [astro-ph/0509015]A. Linde, V. Mukhanov & MS [astro-ph/0509015]

1. IntroductionStandard (single-field, slowroll) inflation predicts scale-invariant Gaussian curvature perturbations.

CMB (WMAP) is consistent with the prediction.

Linear perturbation theory is justified.

However,….

PLANCK PLANCK may detect may detect non-Gaussianitynon-Gaussianity

Inflation may be Inflation may be non-standardnon-standard multi-field, non-slowroll, extra-dim’s, …multi-field, non-slowroll, extra-dim’s, …

Revisit the dynamics on super-horizon scalesRevisit the dynamics on super-horizon scales

Backreaction from/to infrared modes (Backreaction from/to infrared modes (HH»»kk//aa)?)?

Bartolo, Kolb & Riotto ‘05, Sloth ‘05

Kolb, Matarrese & Riotto ‘05

Enhancement of tensor-to-scalar ratio?Enhancement of tensor-to-scalar ratio?

2. Cosmological perturbation theory

2 2 21 2 (1 2 ) i jij ijtA Hds dt a dx dx R

• propertime along propertime along xxi i = const.:= const.: (1 )d dtA

• curvature perturbation on curvature perturbation on ((tt):): (3) (3)

2

4R

a R

((tt))

((t+dtt+dt))

xxi i = const.= const.

dd

traceless

• expansion (Hubble parameter): expansion (Hubble parameter): 1

3

(3)

1 tH AH E R%

scalar

tensor

1

3

transverse-traceless

(3)

ij

i

i j j

j

iH E

H

metric (on a spatially flat background)

• comoving slicingcomoving slicing = for a scalar0 fieldiT t

• flat slicingflat slicing(3)

0 =0(3)

2

4R

a RR

• Newton (shear-free) slicingNewton (shear-free) slicing

scalar

10 0 0

3

(3)

t ij i j ij t tH E E E

• uniform density slicinguniform density slicing 00 tT

• uniform Hubble slicinguniform Hubble slicing

Choice of gauge (time-slicing)

1 0

3

(3)

tH AH H t E R%

matter-based gaugematter-based gauge

geometrical gaugegeometrical gauge

comoving comoving = = uniform uniform = = uniform uniform HH on superhorizon scales on superhorizon scales

2 23 8

(3)

24OH G

a R%

Hubble horizon scale on superhorizon scales

wavelength1 =

Friedmann eq. holds Friedmann eq. holds indep’t of slicing, indep’t of slicing, at leading order inat leading order in

IfIf is time-indep’t, Friedmann eq. holds up through O( is time-indep’t, Friedmann eq. holds up through O(22),),

local ‘local ‘Hubble parameterHubble parameter’ given by’ given by 23 8 2H G O %

with local ‘with local ‘curvature constantcurvature constant’ given by’ given by 2

3

(3)iK x R

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

Separate universe approach

local Friedmann eq. holds forlocal Friedmann eq. holds for adiabatic perterbations adiabatic perterbations on on comovingcomoving//uniform uniform //uniform uniform HH oror flat slices. flat slices.

0 00 08G GT

N formalism Starobinsky ’85, MS & Stewart ’96, ….

e-folding number perturbation between e-folding number perturbation between ((tt) and ) and ((ttfinfin):):

fin fin

fin

fin

fin

background

1

3

(3)2

;t t

t t

t

t

Hd Hd

E dt

N t t

Ot t

t R RR

%

NN00 ((tt,,ttfinfin))NN((tt,,ttfinfin))

((ttfinfin), ), ((ttfinfin)) ((ttfinfin))

((tt))

xxi i =const.=const.

((tt), ), ((tt))

NN=0 if both =0 if both ((tt) and ) and ((ttfinfin) are chosen to be ‘flat’ () are chosen to be ‘flat’ (=0).=0).

By definition, By definition, NNttttfinfin) is) is tt-independent.-independent.

The gauge-invariant variable ‘The gauge-invariant variable ‘’ popularly used’ popularly used

in the liturature is related to in the liturature is related to C C as as = -= -CC

((tt), ), ((tt)=0)=0

((ttfinfin), ), CC((ttfinfin))

xxi i =const.=const.

Choose Choose ((tt) = flat () = flat (=0)=0) and and ((ttfinfin) = comoving:) = comoving:

fin C fin;N t t t R

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

Example: slow-roll inflation• single-field inflation, no extra degree of freedomsingle-field inflation, no extra degree of freedom

CC becomes constant soon after horizon-crossing ( becomes constant soon after horizon-crossing (tt==tthh):):

h fin fin h;N t t t t C CR R

log log aa

log log LL

LL==HH-1-1

tt==tthh

CC = const. = const.

tt==ttfinfin

inflationinflation

Also Also NN = = HH((tthh)) ttFF→C→C , where , where ttF→CF→C is the time difference is the time difference

between the comoving and flat slices atbetween the comoving and flat slices at t t==tthh..

FF((tthh) : flat) : flat=0, =0, ==FF

CC((tthh) : comoving) : comovingttF→CF→C

=0, =0, ==CC

C fin h fin F h;/H

t N t t td dt

R

··· ··· NN formula formula

• No need to solve perturbation equations to calculate No need to solve perturbation equations to calculate CC((ttfinfin):):

Only the Only the knowledge of the background evolutionknowledge of the background evolution is is necessary.necessary.

F h F C C h, it t x t F h F C 0t t &

dN Hdt

F h

dNt

d

• Extension to a multi-component scalar:Extension to a multi-component scalar: (for slow-roll, no isocurvature perturbation)(for slow-roll, no isocurvature perturbation)

C fin F ha

aa

Nt N t

R

MS & Stewart ’96, MS & Tanaka ‘98MS & Stewart ’96, MS & Tanaka ‘98

N.B. N.B. CC is no longer constant in time is no longer constant in time::

FC 2t H

R

&

&······ time varying even on time varying even on superhorizon scalessuperhorizon scales

Extension to non-slow-roll case is also possible, if Extension to non-slow-roll case is also possible, if general slow-roll conditiongeneral slow-roll condition is satisfied at horizon-crossing. is satisfied at horizon-crossing.

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

12

2

2 2, , , ...,O O

H HO

H

& &&

=&&&&

&

hC fin F

2

22 2 2 2

2

H tt N N

R

a a

NN

Tensor-to-scalar ratio

• scalar spectrum:scalar spectrum: 2 2

3 22

3 2

4s

k HP k N

• tensor spectrum:tensor spectrum: 4

82 2

3 22

3 2( )g

k HP k

• tensor spectral index:tensor spectral index:2

2 2

2 222 2g

Hn

H H N

& &&

&

aa

dNH N

dt

g

8gg

s

Pn

P

MS & Stewart ‘96MS & Stewart ‘96

··· ··· valid for any slow-roll modelsvalid for any slow-roll models

1snk

gnk

1

82

2g

s

P

PN

82 G

(‘=’ for a single inflaton model)(‘=’ for a single inflaton model)

(a) Maybe possible in a non-slow-roll /non-Einstein model… (a) Maybe possible in a non-slow-roll /non-Einstein model…

But the scalar spectrum must be almost scale-invariant.But the scalar spectrum must be almost scale-invariant.

need bothneed both scale-invariancescale-invariance AND AND large large PPg g /P/Pss !!

(b) Suppress (b) Suppress PPss by some post-inflationary perturbation? by some post-inflationary perturbation?

This does not work becauseThis does not work because

Enhancement of the ratio Pg /Ps?

total post-inf

2 2 2 2total inf post-inf inf

infN N N

N N N N

unlessunless NNinfinf andand NNpost-infpost-inf are are strongly correlated.strongly correlated.

Need an entirely new idea, if seriously want to increase Need an entirely new idea, if seriously want to increase PPg g /P/Pss

But this brings us back to the case (a)!But this brings us back to the case (a)!

Lyth ’05, Linde, Mukhanov & MS ‘05Lyth ’05, Linde, Mukhanov & MS ‘05

Bartolo, Kolb & Riotto ’05, Sloth ‘05 Bartolo, Kolb & Riotto ’05, Sloth ‘05

4. Non-linear extension

; ix tQ Q HQ H G

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 »»HH-1-1

HH-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.

metric on superhorizon scales

2

i det 1,

2 2 2 i iij

i

j j

j

ds dt dx dt dx de t

O

N %

%

expansion parameter ,i i

curvature perturbation, ln , ;i it x ta t x :

the only non-trivial assumptionthe only non-trivial assumption

e.g., choose (t* ,0) = 0

fiducial `background’

contains GW (~ tensor) modescontains GW (~ tensor) modes

• gradient expansion:gradient expansion:

• metric:metric:

0

0 ; 23

;

t

T u u p g u u u T

dd

upu O

N

normal to const.

21 13 3

H u n

n

O

tdx dt

N

%

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

nnuu

t=const.

0

ii u

v Ou

assumptionassumption uu– – nn= = OO (())

• Energy momentum tensor:Energy momentum tensor:

• local Hubble parameter:local Hubble parameter:

(absence of vorticity mode)(absence of vorticity mode)

• Local Friedmann equation

2 28( , ) ( , )

3( )i iG

H t x t x O %

xi : comoving (Lagrangean) coordinates.

exactly the same as the background equations.exactly the same as the background equations.

uniform uniform sliceslice == uniform Hubbleuniform Hubble sliceslice == comoving comoving sliceslice

3 0 d

H pd %

d= dt : proper time along fluid flow

cf. Hirata & Seljak ‘05 Noh & Hwang ‘05

as in the case of linear theoryas in the case of linear theory

no modifications/backreaction due to super-no modifications/backreaction due to super-Hubble Hubble perturbations.perturbations.

5. Nonlinear N formula

23

tt t

aH

p aO

N& %

energy conservation:

e-folding number:

2 2

1 12 1

1, ;

3 i

t ti t

t tx

N t t x H dt dtP

N%

where xi=const. is a comoving worldline.

This definition applies to any choice of time-slicing.

2 1 2 12

1

, ; , ;( )

ln( )

i it t x t t xa t

N Na t

2 1 2 1, , , ;i i it x t x t t xN where

Lyth, Malik & MS ’04

N - formulaLet Let us take slicing such that us take slicing such that ((tt)) is is flat at flat at t t == t t11 [ [ F F ((tt11) ]) ]and and uniform densityuniform density//uniform uniform HH//comoving at comoving at t t == t t22 : :

( ‘flat’ slice: ( ‘flat’ slice: ((tt) on which ) on which = 0 = 0 ↔ ↔ ee= = aa((tt) ) ))

F F ((tt11) : flat) : flat

((tt22) : uniform density) : uniform density (t2)=const.

(t1)=0

(t1)=const.

((tt11) : uniform density) : uniform density

between an d 20 2 1

11 22 1

( )( , ) ln

( )( , ) ( ) ( ); i a t

N t ta t

N t t x t t

N (t2,t1;xi)

NNFF

C 22 2 1,, , ;ii iFt x t tx xt N

where FF is the e-folding number from FF((tt11)) to ((tt11):):

2 2

1 1

1

1

( ) ( )

( )

(

( )

)

( )

2 1

1 1, ;

3 3

13

F

iF

i

t ti t t

xt

t

t

t

tF

x

N t t x dt dtP P

dtP

Then

For slow-roll inflation In linear theory, this reduces to

1C 2 1 2 1 (( ) ; )FF Ca

aa

t N t t tt H tN

R

suffix suffix C C for for comovingcomoving//uniform uniform //uniform uniform HH

For adiabatic For adiabatic ( ( pp==pp(() ) )) case, case,

2

1

2

1

2 1

( , )2

2 1( , )1

1, ;

3 ( )

( )1( , ) ( , ) ln

3 ( ) ( )

i

i

ti t

t

t x i i

t x

N t t x dtP

a tdt x t x

P a t

NL( )

( , )( ,

)) ( )

13 (

it x i

t

i dP

t x x

non-linear generalization of conservednon-linear generalization of conserved ‘ ‘gauge’-invariant quantity gauge’-invariant quantity or or cc

( and can be evaluated on any time slice)

||

Conservation of nonlinear

······slice-independentslice-independent

6. Summary

• Superhorizon scale perturbations can never affect local (horizon-size) dynamics, hence never cause backreaction.

• There exists a non-linear generalization of which is conserved for an adiabatic perturbation on superhorizon scales.

• There exists a non-linear generalization of N formula, which may be useful in evaluating non-Gaussianity from inflation.

• It is impossible to suppress adiabatic perturbations once they are generated.

• Tensor/scalar ratio could be enhanced in a non-slow-roll/non-Einstein model, but no such model is known.

About tensor-to-scalar ratio…