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![Page 1: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/1.jpg)
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]
![Page 2: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/2.jpg)
1. IntroductionStandard (single-field, slowroll) inflation predicts scale-invariant Gaussian curvature perturbations.
CMB (WMAP) is consistent with the prediction.
Linear perturbation theory is justified.
![Page 3: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/3.jpg)
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?
![Page 4: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/4.jpg)
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)
![Page 5: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/5.jpg)
• 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
![Page 6: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/6.jpg)
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
![Page 7: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/7.jpg)
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).
![Page 8: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/8.jpg)
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
![Page 9: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/9.jpg)
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
![Page 10: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/10.jpg)
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
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• 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
![Page 12: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/12.jpg)
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)
![Page 13: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/13.jpg)
(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
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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.
![Page 15: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/15.jpg)
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:
![Page 16: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/16.jpg)
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)
![Page 17: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/17.jpg)
• 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.
![Page 18: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/18.jpg)
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
![Page 19: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/19.jpg)
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
![Page 20: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/20.jpg)
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
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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
![Page 22: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/22.jpg)
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.
![Page 23: N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71.](https://reader030.fdocuments.in/reader030/viewer/2022020417/56649f0e5503460f94c22b06/html5/thumbnails/23.jpg)
• 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…