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Yoshiharu Tanaka (YITP)
Gradient expansion approach to nonlinear superhorizon perturbations
Finnish-Japanese Workshop on Particle Cosmology@ Helsinki, 9 March, 2007
Y. Tanaka & M. Sasaki, gr-qc/0612191(to be published in PTP)Y. Tanaka & M. Sasaki, in preparation
Standard single field slow-roll inflation predicts
1NLf
Other scenarios Curvaton scenario, inhomogeneous reheating scenario, ghost inflation DBI inflation …… can make large non-Gaussianity!
Non-Gaussianity will be a smoking gun for these inflation models !
1.Introduction
Deviations from Gaussian statics of CMB anisotropy can be a powerful probe for the early universe
very small
1NLf
2gaussNLgauss f 11454 NLf (WMAP+SDSS)
Non-Gaussianity is produced by interaction of fields Thus, we need to go beyond linear theory !
Gravitational potential (which relates directly to CMB anisotropy )
We consider fluctuations whose typical scale L is larger than Hubble horizon scale, 1/H
Expand equations as a power series in ε and solve iteratively
HQQQLQ ti 1
1 HL
The solutions are effective only on superhorizon scales, but full non-linear !
ii ε= expansion parameter
We take gradient expansion approach toward non-linear theory
1H1H
Q
x
L
Previous works( Lifshitz & Khalatnikov ’60, Tomita ’72 、 ’ 75 、 Muller et al. ’89, Salopek & Bond ’90 ・・・・・ )
)( 2O
･ Most authors worked in the synchronous gauge. The gauge doesn’t fix time coordinate uniquely.
Gauge modes appear. ∙∙∙ inconvenient
･ On the other hand, there exists a convenient gauge (as uniform Hubble slicing) in which gauge invariant nonlinear scalar perturbation is conserved on superhorizonfor adiabatic case, neglecting all the spatial gradients. cf. Lyth, Malik, Sasaki ’04
Further investigations on nonlinear perturbations in gradient expansion are needed.
Correspondence to gauge-inv. linear pert. theory was unclear.
･ Scalar, vector, and tensor modes have not been identified clearly.
We formulate gradient expansion on appropriate slicing to and study the correspondence to gauge-inv. linear pert. theory.
)( 2OBut, gradient expansion on the covenient gauge, keeping second order gradients is still not formulated. is important to study non slow-roll models.
2 2k
Haz /
Slow-roll
zRQ
constRaHz
zk
22 )(
Non slow-roll
constRaHz
zk
22 )(
1/k aH ddQ
Q
terms are important to study non slow-roll models
Linear perturbation equation for curvature perturbation, R
In non slow-roll regime, R is not conserved, but enhanced, or damped on superhorizon.
superhorizon scales:
jiij
ii
kk dxdxxttadtdxdtds ~),()(2)( 42222
Assumptions
)(~ Oij )( Oi
1~det ij
Stress-energy tensor
))(2(2
1
VgT
Cf. Lyth, Malik & Sasaki ’04
2. Gradient expansion for a single-scalar system
1HL
Fixing ,1H 0 L limit
For simplicity,
)(),(~ 32 OO iij
As ε→ 0, locally observable universe becomes homogeneous and isotropic universe
for local Friedmann eq. to hold
On uniform Hubble slicing = uniform
which fixes the time coordinate uniquely, so time dependent gauge modes do not appear
Einstein equations yield
)(1),(,)(~ 222 OOOA ij
Cf. Shibata & Sasaki ’99
)0( Hi3
K
a
aH
ijijij AaK
K~
342
)()0()0( t
....),( )2()0( ixt )()( nOn
Basic equations
Klein Gordon equation on uniform Hubble slicing with )( 3 Oi
....),( )2()0( ixt
....),( )2()0( ixt
)()( nOn
Basic equations (continued)
)(/)1(3/6 4 Oaa
)(~2~ 4 OAijijt
ijijij AaK
K~
342
Einstein equations on uniform Hubble slicing with
)(][2]~[~~
8
1~~~ 4)2(
)0(
)0()2()2(2)0(
25
Od
dVaRDD kl
klji
ij
)(8)~
(~ 5
)2()0(66 OAD iij
j
)(]][3
][[1~
3~ 4
42
ORR
aA
a
aA ij
ijijijt
)( 3 Oi
Hamiltonian constraint
Momentum constraint
Evolution equationsijij a ~42
1~det ij
)](3[ tHK ))(2(83
0 )2()0()2(
)0(
)0()2(2)2()2(
2
Vd
dVG
K
Solution represented by four arbitrary spatial dependent scalars and tensors
)()(,)(,)()(,)( 2)2()2(
0)0()0( OxDxCOxfxL kk
ijk
ijk
)(][3
][1
)( 24442)2( OfLR
fffLR
LaxF kl
klij
iji
ij
)()(48
][)( 2
)0(4
)0(2
4)0(
)2(
OtLa
fLRfxC kl
kli
satisfy Friedmann equation)(,)( )0( tta
Momentum constraint
Gravitational waves should not contribute to R.H.S. of this constraint.
can be decomposed to longitudinal part and Transverse-Traceless part uniquely
ijA~6
(Gravitational waves)
Mode identification (scalar and tensor modes; no vector for a scalar)
(Cf. York 1972)
)(8)~
(~ 5
)2()0(66 OAD iij
j
GWs are conformally invariant, determined non-locally and can be generated by nonlinear interactions of only scalar modes
Counting the physical degrees of freedom
Counting d.o.f. contained in four arbitrary scalars and tensors )(),(),(),( )2()2()0()0(
kkij
kij
k xDxCxfxL)()0(
kxL
)()0(k
ij xf
)()2(k
ij xC
1
5
5
Total: 9 d.o.f.
)()0(kxL
)()0(k
ij xf
)()2(k
ij xC
Counting the physical d.o.f.
Scalar field : growing mode 1 + decaying mode 1 = 2 d.o.f.
)()0(k
ij xf
Thus, : 1 (scalar growing mode)
2 (GW)=5 – 3 (constraints)
)()2(kxD
GW from metric : 2 d.o.f.GW from extrinsic curvature : 2 d.o.f.
Total: 6 d.o.f.
Remaining 3 d.o.f. are spatial gauge freedom: they are contained in
1
Momentum constraints relate to :
)()2(kxD : 1 (scalar decaying mode)
)()2(k
ij xC )()2(kxD
2 (GW)=5 – 3 (spatial gauge)
3
is the nonlinear generalization of gauge inv. linear scalar curvature perturbation
In Starobinsky model (’93),
2 LMSeCf. Lyth, Malik & Sasaki ’04
)(V
Hteta )(
0
slow-roll
Friction-dominated…. Non slow-roll period
⇒ later, slow-roll again
analytic solution in )( 0O
-10 -5 5 10 15
4500
4750
5000
5250
5500
410
A
A
Non slow-roll period
t
ofO )( 2
0t
terms decay during slow-roll, but may become constant even on superhorizon scales if non slow-roll
)( 2O
If terms were constant at horizon crossing, the curvature perturbation would change from its value at horizon crossing on superhorizon scales, because of terms’ decay at late times.
)( 2O
)( 2O
• We obtained the general solution to for the metric, scalar field, and especially the nonlinear scalar curvature perturbation
on uniform Hubble slice with for a single-scalar system.
• We identified the scalar and tensor modes in the general solution to in gradient expansion .
• GWs are conformally invariant, and can be generated by nonlinear interactions of only scalar modes.
• Issues:
Calculation of non-Gaussianity generated in non slow-roll model.
Extension to multi-scalar fields.
3. Summary
)( 3 Oi
)( 2O
)( 2O
