nonlinear perturbations for cosmological scalar fields filippo vernizzi ictp, trieste...
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Nonlinear perturbations for cosmological scalar fields
Filippo VernizziICTP, Trieste
Finnish-Japanese Workshop on Particle CosmologyHelsinki, March 09, 2007
Beyond linear theory: motivations
• Nonlinear aspects:
- effect of inhomogeneities on average expansion
- inhomogeneities on super-Hubble scales (stochastic inflation)
- increase in precision of CMB data
• Non-Gaussianity
- discriminator between models of the early universe
- information on mechanism of generation of primordial perturbations
sensitive to second-order evolution
Conserved nonlinear quantities
• Salopek/Bond ‘90
• Comer/Deruelle/Langlois/Parry ‘94
• Rigopoulos/Shellard ‘03
• Lyth/Wands ‘03
• Lyth/Malik/Sasaki ‘04
Long wavelength expansion (neglect spatial gradients)
Second order perturbation
• Malik/Wands ‘02
Covariant approach
• Langlois/FV ‘05• Enqvist/Hogdahl/Nurmi/FV ‘06
Covariant approach
• Work with geometrical quantities
au 4-velocity
proper time:
world-line PguuPT baa
bba
[Ehlers, Hawking, Ellis, 60’-70’]
- perfect fluid
aau
d3
1
- volume expansion
- integrated volume expansion
31
- “time” derivative
aau
Covariant perturbations
au 4-velocity
proper time:
world-line
[Ellis/Bruni ‘89]
baa
bab uugh
projector on
• Perturbations should vanish in a homogeneous universe
• Instead of , use its spatial gradient!
• Perturbations unambiguously defined
aabb hD
• In a coordinate system:
)x,t()x,t( ii
Conservation equation
aaa D
PPD
P
babab
baa uu uL
• “Time” derivative: Lie derivative along ub
aaa DD
0
aa DP
PDPP
• Barotropic fluid
[Langlois/FV, PRL ’05, PRD ‘05]
0bab
a Tu 0 P
• Covector:
Linear theory (coordinate approach)
• Perturbed Friedmann universe
curvature perturbation
jiij dxdxtadtAds 2121 222
xi = const.
(t)
(t+dt)
d
• proper time along xi = const.: (1 )d dtA
• curvature perturbation on (t): 2
3 4
aR
Relation with linear theory
iiv'
'PP
)P(
H'
3
1
aaa D
PPD
P
[Langlois/FV, PRL ’05, PRD ‘05]
• Nonlinear equation “mimics” linear theory
H
aaa DD
dtv;alnt ii
3
1
x,ttx,t
[Wands/Malik/Lyth/Liddle ‘00][Bardeen82; Bardeen/Steinhardt/Turner ‘83]
• Reduces to linear theory
a
'aH;
dt
d'
Gauge invariant quantity
F : flat=0, =F
C : uniform densitytF→C
=0, =C
'HC
Curvature perturbation on uniform density hypersurfaces[Bardeen82; Bardeen/Steinhardt/Turner ‘83]
;tH~;t'~
ttt~t
Higher order conserved quantity
22 2
11
'
H
''
'
H'
''
H
• Gauge-invariant conserved quantity at 2nd order[Malik/Wands ‘02]
3222
2
222
1
6
11
2
2
12
1
'
H
''''
H
'
'
H
'''
'
'
H
''
H
• Gauge-invariant conserved quantity at 3rd order[Enqvist/Hogdahl/Nurmi/FV ‘06]
• and so on...
Cosmological scalar fields
• Single-field
• Scalar fields are very important in early universe models
- Perturbations generated during inflation and then constanton super-Hubble scales
log a
logℓ
L=H-1
t=tout
= const
t=tin
inflation
Cosmological scalar fields
• Single-field
• Scalar fields are very important in early universe models
- richer generation of fluctuations (adiabatic and entropy)- super-Hubble nonlinear evolution during inflation
• Multi-field
- Perturbations generated during inflation and then constanton super-Hubble scales
log a
logℓ
L=H-1
t=tout
d/dt S
t=tin
inflation
Nonlinear generalization
• Rigopoulos/Shellard/Van Tent ’05/06
Long wavelength expansion (neglect spatial gradients)
Higher order generalization
• Maldacena ‘02• FV ’04• Lyth/Rodriguez ’05 (non-Gaussianities from N-formalism)• FV/Wands ’05 (application of N)• Malik ’06
Covariant approach
• Langlois/FV ‘06
Gauge invariant quantities
F : flat
=0
=0
'H
• Curvature perturbation on uniform energy density
[Bardeen82; Bardeen/Steinhardt/Turner ‘83]
: uniform density
: uniform field=0
[Sasaki86; Mukhanov88]
'H
R
• Curvature perturbation on uniform field (comoving)
Large scale behavior
0'R
0 R
• Relativistic Poisson equation large scale equivalence
• Conserved quantities
042
2
a
G
large scales Hak
=0 : uniform density
: uniform field=0
0'
New approach [Langlois/FV, PRL ’05, PRD ‘05]
d3
1• Integrated expansion
Replaces curvature perturbation
aaa DD
• Non-perturbative generalization of
31
aaa DD
RR• Non-perturbative generalization of
aaaa DD
R
Single scalar field
)(Vaa
2
1
2
1L
= const
au
auarbitrary
ab)ba(abbaab uqPguuPT
0 aaa
a, DuDDV
aau
Single scalar field
= const
au
aau
Single-field: like a perfect fluid
)(Vaa
2
1
2
1L
PguuPT abbaab
0 ,V
abaq 0
03 ,V'H''
Single field inflation
log a
logℓ
L=H-1
t=tout
a = const.
t=tin
inflation
03
23
a
,a D
V
0 aa R
• Generalized nonlinear Poisson equation 0aD
Two-field linear perturbation
s
• Global field rotation: adiabatic and entropy perturbations
[Gordon et al00; Nibbelink/van Tent01]
Adiabatic
Entropy
sincos cossin s
'
'
tg
= 0
= 0
= 0
iiii '''q
Total momentum is the gradient of a scalar
sincos
'
HR
s'
'H'
2 0R
Evolution of perturbations
• Curvature perturbation sourced by entropy field
[Gordon/Wands/Bassett/Maartens00]
033 2 sV'sH''s ss,
• Entropy field perturbation evolves independently
arbitrary !
Two scalar fields
),(Vaa
aa
2
1
2
1
2
1L
= const
au
= const au
aaa DDq
ab)ba(abbaab uqPguuPT
[Langlois/FV ‘06]
Covariant approach for two fields
aaa sincos
aaas cossin
• Local redefinition: adiabatic and entropy covectors:
• Adiabatic and entropy angle:
22
spacetime-dependent angle
aaaa DDDq
• Total momentum:
bbaa hq
tg
Total momentum may not be the gradient of a scalar
(Nonlinear) homogeneous-like evolution equations
• Rotation of Klein-Gordon equations:
aaaa
a, ssVH
13
aaaa
as, ssV
1
1st order 2nd order
1st order 2nd order
03 ,V'H''
0'
V' s,
• Linear equations:
(Nonlinear) linear-like evolution equations
• From spatial gradient of Klein-Gordon equations:
as,as,c
cca
acc
aas,
,aa
sVsVss
VV
31
33
as,,bc
bcca
acc
aass,a,a
VVhs
ssVsVs
1
21 2
Adiabatic:
Entropy:
Adiabatic and entropy large scale evolution
• Entropy field perturbation
• Curvature perturbation: sourced by entropy field
033 2 ass,aa sVss
aa s 2
033 2 sV'sH''s ss,
s'
H''
2
• Linear equations
0 aRa aaD
aR
Second order expansion
2cossin
''s
's
's'
ss iii
'ss'
2
1sincos
iiii V'
s'
'
1
s's'ssV iii 2
1
• Entropy:
Vector term
• Adiabatic:
iii Vs''q • Total momentum cannot be the gradient of a scalar
= 0
= 0
= 0
Vector term
• On large scales:
iiii V''
RRR 1
• Second order
iiV
H
22
R
Adiabatic and entropy large scale evolution
ii
ss,sss,
,ss,
VHsVV
ssHV
ssVsHs
223
22
6952
1
2
3233
ii,
ss, VV
sVsH 222
2
242
• Entropy field perturbation evolves independently
• Curvature perturbation sourced by 1st and 2nd order entropy field
03 ii HV'V 3
1
aVi
• Nonlocal term quickly decays in an expanding universe:
(see ex. Lidsey/Seery/Sloth)
Conclusions
• New approach to cosmological perturbations
- nonlinear and covariant (geometrical formulation)
- exact at all scales, mimics the linear theory, easily expandable
• Nonlinear cosmological scalar fields
- single field: perfect fluid
- two fields: entropy components evolves independently
- on large scales closed equations with curvature perturbations
- comoving hypersurface uniform density hypersurface
- difference decays in expanding universe
F : flat
=0
=0 : uniform density
: uniform field=0
R
Q
02
2
23
RR'R
aH
'Haln
dt
d
Mukhanov equation quantization
Quantized variable [Pitrou/Uzan, ‘07]
aa DR
dDD aaa R
0RRR
a
aa DDelnd
d 2
3
• At linear order converges to the “correct” variable to quantize
• Nonlinear analog of R