influence on observation from ir / uv divergence during inflation
DESCRIPTION
Influence on observation from IR / UV divergence during inflation. Yuko Urakawa ( Waseda univ .). Y.U. and Takahiro Tanaka 0902.3209 [ hep-th ]. Y.U. and Takahiro Tanaka 0904.4415[ hep-th ]. Alexei Starobinsky and Y.U. in preparation. Contents . - PowerPoint PPT PresentationTRANSCRIPT
Influence on observation from IR / UV divergence during inflation
Yuko Urakawa (Waseda univ.)
Y.U. and Takahiro Tanaka 0902.3209 [hep-th]Y.U. and Takahiro Tanaka 0904.4415[hep-th]
Alexei Starobinsky and Y.U. in preparation
2
Contents
・ Influence on observables from IR divergence
・ Influence on observables from IR divergence
・ Influence on observables from UV divergence
Y.U. and Takahiro Tanaka 0902.3209
Y.U. and Takahiro Tanaka 0904.4415
Alexei Starobinsky and Y.U. 090*.****
- Single field case -
- Multi field case -
Primordial fluctuation generated during inflation
3
1. Introduction
3. IR divergence problem - Single field -
► Outline
2. Cosmological perturbation during inflation
4. IR divergence problem - Multi field -
5. UV divergence problem
6. Summary and Discussions
4► Cosmic Microwave BackgroundWMAP 1yr/3yr/5yr…
1. Introduction
Almost homogeneous and isotropic universe
with small inhomogeneities
Small scale →← Large scale
5► CMB angular spectrum
ΩΛ
1. Introduction
)()( 21 nTTn
TT
Harmonic expansion
Ωm Ωb
ΩK PPrimordial spectrum
7
► Sachs-Wolfe (SW) effect Flat plateau
20l
SW effect : Dominant effect
◆ Last Scattering surface z~1091
Inhomogeneity
gravitational potential
→ red shift → temperature
LSSWT
T 51
1. Introduction
inflation
► Evolution of fluctuationPhysical scale k : comoving wave number Horizon scale
aaHhor //1
kaphys /
consthor )1( pta pHtea thor
Log
Loga
physhor
Horizon cross
Horizon reenter
► Adiabatic fluctuation
inflation Loga
hor
Log
~constant
LSSWT
T 51
LS
For at LSS
horphys hocLSSWT
T 51
51
10
► WMAP 5yr dateAlmost scale invariant, Almost Gaussian …
Consistent to the prediction from
“ Standard” inflation ( Single-field , Slow-roll)
013.096.0ln
ln12
kd
dns
92
232 10)096.0445.2(
2||
kk
1002.0 Mpck
)()(
2
2
kkr GW
95 % C.L. Pivot point
* No running
11
► Beyond linear analysis Within linear analysis
Observational date → Not exclude other models
More information from Non-linear effects
・ Non-Gaussianity
・ Loop corrections1. Introduction
WMAP 5yr 95 % C.L.
1119 localNLf 253151 equil
NLf
→ PLANCK (2009.5)
12
► IR / UV divergences◆ During inflation
Quantum fluctuation of inflaton
Quantum fluctuation of gravitational field
Classical stochastic fluctuation
Observation → Clarify inflation model ??
Classicalization
Ultraviolet (UV) & Inflared (IR) divergence Regularization is necessary
13
1. Introduction
3. IR divergence problem - Single field -
► Outline
2. Cosmological perturbation during inflation
4. IR divergence problem - Multi field -
5. UV divergence problem
6. Summary and Discussions
14
Liner analysis
15
► Comoving curvature perturbation◆ Gauge invariant quantity
),~(),( ii xtttxt
Spatial curvature 2Rs
Fluctuation of scalar field
tH ~ t ~
H
“Gauge invariant variable”
16
► Gauge invariant perturbationGauge invariant perturbation Completely Gauge fixing
Equivalent
H
Flat gauge 0
Gauge invariant
Comoving gauge
0
“Completely gauge fixing” 0/0
17
► Liner perturbation◆ Single field inflation model
Comoving gauge 0
GW
Non-decaying mode as k/aH → 0
0)('),( khk
18
► Adiabatic vacuum
Positive frequency mode f.n. → Vacuum ( Fock space )
◆ Initial conditionIn the distant past |η| → ∞,
Adiabatic solution
⇔ k>>1 Much smaller than curvature scale
~ Free field at flat space-time
19
► Scalar perturbation
ke ik
k 2
Log
Loga
phys
hor
23
kk
2
2
232
221
2||)(
hoc
hoc
k Hkk
Almost scale invariant
2HH
~constant
hoc
20
► Chaotic inflation
Reheating
Inflation goes on
H,
22
221
hoc
hoc
H
Larger scale mode → Exit horizon earlier
→ Larger amplitude
Red tilt ns< 1
21
► Tensor perturbation
◆ Initial conditionIn the distant past |η| →
∞, ik
k eka
h 211)(
Adiabatic solution
2
2
232
22
2||)(
hock Hhkk
◆ Power spectrum
Almost scale invariant , Red tilt
22
Quantum correlation
23
► Linear theory
OdxS free4 22)( mOex
)()(),( yxyxG
OGOG /11
)()()( zyx
x
y z
0)()()( zyx
0
x
y
G G
(i) Two point fn. (ii) Three point fn.
Transition from y to x
24
► Non-linear theory
OdxS free4 44
int !4dxS
intSSS free
← Expansion by free field 14
x
y
λ
)()( yx (i) Two point fn.
G
x
y
x
y
x
y x
y
O(λ0) O(λ1) O(λ2)etc
25
► Non-linear theory
OdxS free4 34
int !3dxS
intSSS free λ
)()()( zyx (ii) Three point fn.
O(λ1) O(λ3) etc
x
y z
x
y z
λ
x
y z
26
► Summary of Interaction picture
intSSS free Propagator ↑ ↑ Vertex
1. Write down all possible connected graphs
2. Compute the amplitude of each graph
Feynman rule
x yzZ
d4zG(x;z)õG(z;z)G(z;y)
k k
q
ZdtzGk(tx; tz)
Zd3qõGq(tz; tz)Gk(tz; ty)
Fourier trans.
Loop integral
27
Non-linear perturbation
28
► Interests on Non-linear correctionsPrimordial perturbation ζ
)()()( zyx
)()( yx
)()()()( wzyx
x
y z
x y
x
y z
w
x
y z
x y
and so on… More i
nformation on inflation
29
Comoving gauge
► ADM formalism
S = SEH + Sφ = S [ N, Ni, ζ ]
Hamiltonian constraint ∂ L / ∂ N = 0
N = N[ζ] Momentum constraint ∂ L / ∂ Ni = 0 Ni = Ni [ζ]
→
eρ: scale factor
S [ N, Ni, ζ ] = S [ ζ ]
◆ Lagrange multiplier N / Ni Maldacena (2002)
30
► Non-linear action
...!2
121 NNN
...!2
121
1st order constraints 11,N
2nd order constraints 22 ,N
(ex) 1st order constraints
ii2
31
► NGs / Loop corrections
2002 J.Maldacena
“Quantum origin” ( Mainly until Horizon crossing)
Single field with canonical kinetic term
NG → Suppressed by slow-roll parameters
2005 Seery &Lidsey
2005, 2006 S.Weinberg Loop correction amplified at most logarithmic order
Single & Multi field(s) with non-canonical kinetic term
NG → Dependence on the evolution of sound speed
IR divergence in Loop corrections → Logarithmic 2007 M.Sloth 2007 D.Seery 2008 Y.U. & K.Maeda
2004 D.Boyanovsky
and so on
32
► IR divergence problem
∫d3q |ζq|2 = ∫ d3q /q3
< ζk ζk’ > q
k k'
Momentum ( Loop )integral
Scale-invariant
◆ One Loop correction to power spectrumMass-less field ζ
Next to leading order
Log. divergence
32|| klinearlinearlinearkkk
33
1. Introduction
3. IR divergence problem - Single field -
► Outline
2. Cosmological perturbation during inflation
4. IR divergence problem - Multi field -
5. UV divergence problem
6. Summary and Discussions
34
Primordial perturbation
► Our purpose
Loop integral
To extract information from loop corrections, we need to discuss …
diverge
“ Physically reasonable regularization scheme ”
Increasing IR corrections
Spectrum : Large Dependence on IR cut off
( Note )
35
Fluctuations computed by Conventional perturbation
Fluctuations we actually observe ex. CMB
・ Prove “Regularity of observables”
・ Propose “How to compute observables”
Strategy
► IR divergence problem
Vertex integral
◆ Loop corrections
...... 34 kdtdxd Diverge
Finite
36
Violation of Causality
37
► Non local system ◆ Constraint eqs.
: Solutions of Elliptic type eqs.
),( N
],......,[ 22 iiNSS
Hamiltonian constraint ∂ L / ∂ N = 0
N = N[ζ] Ni = Ni [ζ]
→ Momentum constraint ∂ L / ∂ Ni = 0
(ex) 1st order Hamiltonian constraint
Non local term
38
► Causality
A portion of Whole universe
η
x
Observation
Initial
ζ(x)
p .
We can observe fluctuations within “Causal past J-(p) ”
39
δQ (x) = Q(x) ‐ Q
Q : Average value
◆ Definition of fluctuation
t
x
Observation
Initial
ζ(x)
p . ζ(x) x ∈ J-(p) affected by { J-(p) }c
Conventional perturbation theory
Q : Average value in whole universe
► Violation of Causality
40
Large scale fluctuation → Large amplitude
Q
ⅹ
Large fluctuation we cannot observe
Q on whole universe
( Q - Q )2 < < ( Q - Q )2
Q on observable region
- Chaotic inflation - δ2 ζ H∝ 2 / ɛ Amplitude of ζ
41
◆ Gauge fixing
ζ(x) x ∈ J-(p) affected by { J-(p) }c
Gauge invariant
Completely gauge fixing at whole universe ☠ Impossible
- We can fix our gauge only within J-
(p).- Change the gauge at { J-(p) }c
→ Influence on ζ(x) x ∈ J-(p)
► Violation of Causality 2
42
Gauge degree of freedom
43
► Gauge choice NGs / Loop corrections Computed in
Comoving gauge
Flat gauge
Maldacena (2002), Seery & Lidsey (2004) etc..
- Gauge degree of freedom
DOF in Boundary condition
: Solutions of Elliptic type eqs. ),( N
44
► Boundary condition
Solution 2
Solution 1Arbitrary integral region
45
► Scale transformation keeping Gauge condition
Scale transformation xi → xi = e - f(t) xi ~
46
Solution 2
Solution 1
► Scale transformation
47
► Gauge condition
Additional gauge condition
Change homogeneous mode
=0
(1) Observable fluctuation
(2) Solution of Poisson eq. ∂-2 ....||
~ 3
)(2
yxyd
pJ
“Causal evolution” : Not affected by { J-
(p) }c
)(~ x
)(~)(~)( txxobs
....)()()(~ tfxx
)(~)(~ 3)(
xxdtpJ
Averaged value at J-(p)
48
][2 S
► Gauge invariant perturbation
0)(~)(~ 3)(
xxdtpJ
ii
pJpJdSxd )(
23)(
0
∂ L / ∂ N = 0
◆ Naïve understanding
Local gauge condition
No Influence from { J-(p) }c
Fix Gauge within J-(p) → Determineζ(x) x ∈ J-(p)
Recovery of Gauge invariance
49
► Quantization
Adiabatic vacuum
◆ Initial condition
),(~ xti : Curvature at local comoving gauge
),( xti : Curvature at ordinal comoving gauge
P (k) 1 / k∝ 3
Divergent IR mode
Gauge transformation
We prove IR corrections of are regular.
)(~ x
50
► Regularization scheme
“Cancel” IR divergence
Extremely long inflation
Higher order corrections might dominate lower ones.
Validity of Perturbation ??
)()()(~ txx Effective cut off by k~ 1/Lt
Exceptional case1
HM
N pl
Lt: Scale of causally connected region
52
1. Introduction
3. IR divergence problem - Single field -
► Outline
2. Cosmological perturbation during inflation
4. IR divergence problem - Multi field -
5. UV divergence problem
6. Summary and Discussions
53
► Multi-field generalization
Background trajectory
( 2 ) ♯ ≧ 2
Gauge invariant → Still diverges
δσ (x) = δσ (x) ‐ δσLocal average = 0
~
( 1 ) ♯ = 1
δs
δσ
IR regular
◆ Local flat gauge
)~,~( s
s~
♯ : Number of IR divergent fields
✔
54
πk
δsk
IR mode < δsk δsk > 1 / k∝ 3 Highly squeezed
phase space
< O(x) O(y) O(z)… > O = δσ, δ s
~ ~
► IR divergence in Multi-field model
☠ Origin of IR divergence
◆ Squeezed wave packet
55
πk
δsk
IR mode < δsk δsk > 1 / k∝ 3 Highly squeezed
phase space
< O(x) O(y) O(z)… > O = δσ, δ s
~ ~
A portion of wave packet
► IR divergence in Multi-field model
Observable fluctuation
Prove IR regularity of observables
◆ Squeezed wave packet
56
Decoherence
| δs >
s
sOne of Wave packet
→ Realized
s
Wave packet of | δs >
► Wave packet of universe Observation time t = tf
Early stage of Inflation
Superposition of
Correlated Uncorrelated
Statistical Ensemble
Cosmic expansionVarious interactions
57
t = tf
► Parallel world
t = ti
Pick up
s
Causally disconnected universe
Our universe
In , another wave packet may be picked up. However, we cannot know what happens there.
58
If finite
← Finite
► Projection
α
σ
s
s
2
2))((exp)(
ftsP
< P(α) O(x) O(y) … >
Proof of IR regularity
Our “Observables”
≩
Actual observable correlation fn.
⊋
59
π
δs
< P(α) O(x) O(y) O(z)…. > O = δσ, δ s ~ ~
After decoherence, a portion of wave packet contributes
Momentum integrals Regular
Temporal integral Logarithmic secular evolution
► Regularization scheme
phase space
◆ Loop integrals ...... 34 kdtdxd
60
1. Introduction
3. IR divergence problem - Single field -
► Outline
2. Cosmological perturbation during inflation
4. IR divergence problem - Multi field -
5. UV divergence problem
6. Summary and Discussions
61
► UV regularization ζ : curvature perturbation / hij : GW
Diverge in x → y limit
◆ Adiabatic regularization
UV mode ~ Solution in adiabatic approximation
Parker & Fulling (1974)
)(|||| 42)(2 kOsbkad
k
Regular ☠ Divergent
Adiabatic expansion
62
► Influence from Adiabatic reg. Parker (2007), Parker et.al. (2008/2009)
×( Slow-roll parameter )
×( Slow-roll parameter )
Amplitudes of ζ / GW suppressed by subtraction terms
at horizon crossing time
Single field inflation
63
► No Influence from Adiabatic reg.
Exact solution Solution in Adiabatic approx.
Constant value Decay
Super horizon limit
2
2)(
|)(||)(|
k
sbkad
2
2)(
|)(||)(|
k
sbkad
hh
A.Starobinsky & YU (2009)
Negligible
64
► Summary - IR regularization -
Comoving gauge
Flat gauge
+ Local gauge G :“ Causality” is preserved
◆ Single field case
NGs/Loop corrections are free from IR divergence ( except for models with extremely long durations )
◆ Multi field case Gauge fixing is not enough to discuss observables To consider them, we need to consider “decoherence”.
We cannot deny the existence of secular evolution.
65
► Summary - UV regularization -
◆ Adiabatic regularization Regularize UV divergence
We should introduce subtraction terms for all modes
Subtraction term decays during cosmic evolution
→ No-influence on observables
, which appears in the coincidence limit
However…
66
- Supplement -
67
τ = τi
3. Regularization scheme ~ Multi field ~
► Decoherence process
Initial state : adiabatic vacuum | 0 >ad
Correlated
Superposition of | >
| 0 >ad = ∫d | >< |0 >ad
Include the contribution from all wave packets
ad < 0 | ζ(x1) ζ(x2) ζ(x3) … |0 >ad Overestimation
68
@ Causally connected region
Expansion by GF , GD , G+, G -
Evolution of < in | ** | in >
time
GR : Regular in IR limit
GF GD G+ or G-
x
・x’
≠0, Finite value
CTP
Expansion by GR
◆ Closed Time Path
◆ Expansion by Retarded Green f.n.
► Expansion by Retarded Green fn.
69
(Ex.) N = 4
R
[ Detailed exp. ] ► Expansion by Retarded Green fn.
)(x =
・・・
+ R
= R+ + R R
= + R + R R + ・・・
70
► Expansion by Retarded Green fn.2 ◆ Contraction
'kk
k R'k
Contraction
k k'R
71
IR regular ??
3. Proof of IR regularity[ Detailed exp. ]
= ∑ ( IR regular functions GRm) × a< 0 |P(α)ζI ζI … ζI | 0 >a
~ Eigenetate for ζI with finite wave packet
FiniteFinite region
FiniteInfinite region→ ∞
・ Without P (α)
・ With P (α)
► Mode expansion
IIId 1
I
aIIIIIIaII dd 0''.....0'
aIIIIIIaII Pdd 0''.....)(0'
72
Stochastic inflation → Decoherence
Necessity to consider Local quantity ( |x| < L )Lyth (2007)
Local quantity Cut off only for external momentum
→ Introduction of IR Cut off 1/L
Bartolo et. al (2008)
Riotto & Sloth (2008)
Enqvist et.al. (2008) k < kc Stochastic fluctuation
Neglecting a part of quantum fluctation
Include the artificial cut-off scale
Under-estimation of IR corrections
→ Doubtful
► Recent topics