Transcript Solution to the IR divergence problem of interacting

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
Primordial fluctuation generated during inflation
・Influence on observables from IR divergence
- Single field case Y.U. and Takahiro Tanaka
0902.3209
・Influence on observables from IR divergence
- Multi field case Y.U. and Takahiro Tanaka
0904.4415
・Influence on observables from UV divergence
Alexei Starobinsky and Y.U.
090*.****
1
► Outline
2
1. Introduction
2. Cosmological perturbation during inflation
3. IR divergence problem - Single field 4. IR divergence problem - Multi field 5. UV divergence problem
6. Summary and Discussions
► Cosmic Microwave Background
3
WMAP 1yr/3yr/5yr…
Almost homogeneous and isotropic universe
with small inhomogeneities
1. Introduction
► CMB angular spectrum
T  T 
T
(n1 )
T
(n2 ) 4
Harmonic expansion
Ωm
ΩΛ
P
Ωb
ΩK
Primordial spectrum
← Large scale
Small scale →
1. Introduction
► Sachs-Wolfe (SW) effect
Flat plateau l  20
SW effect : Dominant effect
◆ Last Scattering surface z~1091
Inhomogeneity
gravitational potential
→ red shift → temperature
1
 T 
    LS
 T  SW 5
1. Introduction
6
► Evolution of fluctuation
Physical scale
 phys  a / k
Horizon scale
 hor  1/ H  a / a
a  e Ht
k : comoving wave number
a  t p ( p  1)
 hor  const
 hor  t
Log

Horizon reenter
phys
 hor
Horizon cross
inflation
Loga
► Adiabatic fluctuation
Log
1
 T 


 LS
 
5
 T  SW
 hor
inflation
For  phys   hor at LSS
1
 T 
 


5
 T  SW
LS
1


 hoc
LS
5
Loga
► WMAP 5yr date
Almost scale invariant, Almost Gaussian …
95％ C.L.
Pivot point
k  0.002Mpc1
3
2
k
|

|
9
k
2 

(
2
.
445

0
.
096
)

10
2 2
d ln 2
ns  1 
 0.96  0.013
d ln k
2 GW (k )
r 2
  (k )
* No running
Consistent to the prediction from
“Standard” inflation ( Single-field , Slow-roll)
9
► Beyond linear analysis
10
Within linear analysis
Observational date → Not exclude other models
More information from Non-linear effects
･ Non-Gaussianity
WMAP 5yr
 9  f NL
95％ C.L.
local
 111
151 f NL
equil
 253
→ PLANCK (2009.5)
･ Loop corrections
1. Introduction
► IR / UV divergences
◆ During inflation
Quantum fluctuation of inflaton
Quantum fluctuation of gravitational field
Ultraviolet (UV) & Inflared (IR) divergence
Regularization is necessary
Classicalization
Classical stochastic fluctuation
Observation → Clarify inflation model ??
11
► Outline
12
1. Introduction
2. Cosmological perturbation during inflation
3. IR divergence problem - Single field 4. IR divergence problem - Multi field 5. UV divergence problem
6. Summary and Discussions
Liner analysis
13
► Comoving curvature perturbation
◆ Gauge invariant quantity

 R 
s
Spatial curvature
2

Fluctuation of scalar field
(t, xi )  (~
t  t   t , xi )
 ~    Ht
~
      t
“Gauge invariant variable”
H
     

14
► Gauge invariant perturbation
Gauge invariant perturbation
Completely Gauge fixing
Equivalent
  0 /   0 “Completely gauge fixing”
H
     

Gauge invariant

Flat gauge   0

Comoving gauge   0

15
► Liner perturbation
◆ Single field inflation model
Comoving gauge
  0
GW
Non-decaying mode  (k ), h' (k )  0 as k/aH → 0
16
► Adiabatic vacuum
Positive frequency mode f.n. → Vacuum ( Fock space )
◆ Initial condition
In the distant past |η| → ∞,
⇔ k>>1 Much smaller than curvature scale
Adiabatic solution
~ Free field at flat space-time
17
► Scalar perturbation

Log
phys
 hor
Loga
hoc
e ik
k 
2k
k  k

3
2
k 3 |  k |2
1
2
  (k ) 

2 2
2 hoc
 H hoc 


2



2
H
  2
H
Almost scale invariant
18
► Chaotic inflation
19
Inflation goes on  , H 
Reheating
2
2
1  H hoc 


 
2 hoc  2 
Larger scale mode → Exit horizon earlier
→ Larger amplitude
Red tilt ns< 1
► Tensor perturbation
◆ Initial condition
In the distant past |η| → ∞,
hk ( ) 
1 1 ik
e
a 2k
Adiabatic solution
◆ Power spectrum
3
2
k
|
h
|
 H hoc 
2
k
  (k ) 
 2

2
2
 2 
2
Almost scale invariant , Red tilt
20
Quantum correlation
21
► Linear theory
S free   dx 4 O
(ex) O   2  m2
(i) Two point fn.
G( x, y)   ( x) ( y)
(ii) Three point fn.
 ( x) ( y) ( z)
  ( x) ( y)  ( z)  0
Transition from y to x
x
x
G
G
y
OG  1  G  1 / O
0
y
z
22
► Non-linear theory
S  S free  Sint
S int   dx
S free   dx  O
4
 4  1
4

4!

 ← Expansion by free field
(i) Two point fn.
 ( x) ( y)
x
x
x
x
y
G
y
λ
4
O(λ0)
y
O(λ1)
y
x
O(λ2)
etc
y
23
► Non-linear theory
S  S free  Sint
S free   dx  O
4
S int   dx
4

3!
λ
3
(ii) Three point fn.  ( x) ( y) ( z)
x
x
x
λ
y
z
y
O(λ1)
z
y
z
O(λ3)
etc
24
► Summary of Interaction picture
S  S free  Sint
Propagator ↑
↑ Vertex
1. Write down all possible connected graphs
Feynman rule
2. Compute the amplitude of each graph
Loop integral
q
Fourier trans.
x
z
y
Z
k
Z
d4zG(x; z)õG(z; z)G(z; y)
k
Z
dt zGk(t x; t z) d3qõGq(t z; t z)Gk(t z; t y)
25
Non-linear perturbation
26
► Interests on Non-linear corrections
Primordial perturbation ζ
 ( x) ( y)
x
y
x
x
 ( x) ( y) ( z)
y
x
y
z
x
w
y
z
 ( x) ( y) ( z) (w)
y
z
and so on…
27
► ADM formalism
28
Comoving gauge
S = SEH + Sφ = S [ N, Ni, ζ ]
◆ Lagrange multiplier N / Ni
Hamiltonian constraint
∂L/∂N =0
Momentum constraint
∂ L / ∂ Ni = 0
eρ: scale factor
Maldacena (2002)
→
S [ N, Ni, ζ ] = S [ ζ ]
N = N[ζ]
Ni = Ni [ζ]
► Non-linear action
29
1
N  N1  N 2  ...
2!
1st order constraints
N1 , 1
1
   1   2  ...
2!
2nd order constraints
N2 , 2
(ex) 1st order constraints
 2  i i
► NGs / Loop corrections
“Quantum origin” ( Mainly until Horizon crossing)
2002 J.Maldacena
Single field with canonical kinetic term
NG → Suppressed by slow-roll parameters
2005 Seery &Lidsey
Single & Multi field(s) with non-canonical kinetic term
NG → Dependence on the evolution of sound speed
2005, 2006 S.Weinberg
2004 D.Boyanovsky
2007 M.Sloth
2007 D.Seery
2008 Y.U. & K.Maeda
and so on
Loop correction amplified at most logarithmic order
IR divergence in Loop corrections → Logarithmic
30
► IR divergence problem
31
◆ One Loop correction to power spectrum
Mass-less field ζ
k
linear
k
< ζk ζk’ >
linear
|  k linear |2  k 3
Scale-invariant
Next to leading order
q
k
Momentum ( Loop )integral
k'
∫d3q |ζq|2 = ∫ d3q /q3
Log. divergence
► Outline
32
1. Introduction
2. Cosmological perturbation during inflation
3. IR divergence problem - Single field 4. IR divergence problem - Multi field 5. UV divergence problem
6. Summary and Discussions
► Our purpose
33
Primordial perturbation
Loop integral
diverge
To extract information from loop corrections,
we need to discuss …
“ Physically reasonable regularization scheme ”
( Note )
Increasing IR corrections
Spectrum : Large Dependence on IR cut off
► IR divergence problem
34
◆ Loop corrections
Fluctuations computed by Conventional perturbation
Vertex integral  d 4 x...   dtd 3k... Diverge
Fluctuations we actually observe ex. CMB
Finite
Strategy
・ Propose “How to compute observables”
・ Prove “Regularity of observables”
Violation of Causality
35
► Non local system
36
◆ Constraint eqs.
Hamiltonian constraint
∂L/∂N =0
Momentum constraint
∂ L / ∂ Ni = 0
→
N = N[ζ]
Ni = Ni [ζ]
(N ,  ) : Solutions of Elliptic type eqs.
(ex) 1st order Hamiltonian constraint
S  S[ N   2 ..., i  i  2 ...,  ]
Non local term
► Causality
37
We can observe fluctuations within “Causal past J-(p) ”
A portion of Whole universe
η
.p
ζ(x)
Observation
x
Initial
► Violation of Causality
ζ(x)
x ∈ J-(p) affected by { J-(p) t}c
◆ Definition of fluctuation
δQ (x) = Q(x) ‐ Q
38
.p
ζ(x)
Observation
x
Q : Average value
Conventional perturbation theory
Q : Average value in whole universe
Initial
- Chaotic inflation Amplitude of ζ
39
δ2 ζ ∝ H2 / ɛ
Large scale fluctuation → Large amplitude
Q
Q on observable region
ⅹ
Q on whole universe
Large fluctuation we cannot observe
( Q - Q )2
<<
( Q - Q )2
► Violation of Causality 2
ζ(x)
40
x ∈ J-(p) affected by { J-(p) }c
◆ Gauge fixing
Completely gauge fixing at whole universe ☠ Impossible
Gauge invariant
- We can fix our gauge only within J-(p).
- Change the gauge at { J-(p) }c
→ Influence on ζ(x)
x ∈ J-(p)
Gauge degree of freedom
41
► Gauge choice
NGs / Loop corrections Computed in
Flat gauge
Comoving gauge
Maldacena (2002), Seery & Lidsey (2004) etc..
- Gauge degree of freedom
(N ,  ) : Solutions of Elliptic type eqs.
DOF in Boundary condition
42
► Boundary condition
43
Solution 1
Arbitrary integral region
Solution 2
► Scale transformation
keeping Gauge condition
Scale transformation
xi
→
～i
x
= e - f(t) xi
44
► Scale transformation
Solution 1
Solution 2
45
► Gauge condition
46
~
 ( x)   ( x)  f (t )  ....
Change homogeneous mode
Additional gauge condition
“Causal evolution”
~
 ( x) : Not affected by { J-(p) }c
(1) Observable fluctuation
~
~
 obs ( x)   ( x)   (t )
Averaged value at
＝０
J-(p)
(2) Solution of Poisson eq. ∂-2
~
 (t )   J

( p)
~
d x  ( x)
3
d3y
~ 2
   J  ( p)
....
| x y|
► Gauge invariant perturbation
◆ Naïve understanding
   S[ ]  
2
∂L/∂N =0
Local gauge condition
~
 (t )   J

( p)
~
d x  ( x)  0
3
0   J  ( p ) d x    J  ( p ) dS  i
3
2
i
No Influence from { J-(p) }c
Fix Gauge within J-(p) → Determineζ(x) x ∈ J-(p)
Recovery of Gauge invariance
47
► Quantization
48
◆ Initial condition

 (ti , x) : Curvature at ordinal comoving gauge
Adiabatic vacuum
P (k) ∝ 1 / k3
Divergent IR mode
Gauge transformation
~ 
 (ti , x) : Curvature at local comoving gauge
~
We prove IR corrections of  ( x) are regular.
► Regularization scheme
~
 ( x)   ( x)   (t )
49
Effective cut off by k~ 1/Lt
“Cancel” IR divergence
Lt: Scale of causally connected region
Exceptional case
Extremely long inflation
 M pl
N  
 H
1



Higher order corrections might dominate lower ones.
Validity of Perturbation ??
► Outline
51
1. Introduction
2. Cosmological perturbation during inflation
3. IR divergence problem - Single field 4. IR divergence problem - Multi field 5. UV divergence problem
6. Summary and Discussions
► Multi-field generalization
δs
◆ Local flat gauge
52
δσ
～
δσ (x) = δσ (x) ‐ δσ
Local average = 0
♯ : Number of IR divergent fields
（1） ♯ = 1
(~, ~
s)
IR regular
✔
（2） ♯ ≧ 2
~
s Gauge invariant → Still diverges
Background trajectory
► IR divergence in Multi-field model
53
◆ Squeezed wave packet
IR mode
phase space
< δsk δsk > ∝ 1 / k3
Highly squeezed
πk
δsk
＜O(x) O(y) O(z)…＞
～
～
O = δσ, δ s
☠ Origin of IR divergence
► IR divergence in Multi-field model
54
◆ Squeezed wave packet
IR mode
< δsk δsk > ∝ 1 / k3
phase space
πk
Highly squeezed
A portion of wave packet
δsk
Observable fluctuation
＜O(x) O(y) O(z)…＞
～
～
O = δσ, δ s
Prove IR regularity of observables
► Wave packet of universe
Early stage of Inflation
Superposition of | δs >
s 
55
Observation time t = tf
Statistical Ensemble
Correlated
Uncorrelated
Cosmic expansion
Various interactions
s
Decoherence
Wave packet of | δs >
s
One of Wave packet
→ Realized
► Parallel world
t = tf
56
Pick up
s
Our universe
Causally disconnected universe
t = ti
In
, another wave packet may be picked up.
However, we cannot know what happens there.
► Projection
 (s (t f )   )
P( )  exp
2

2
57
s 



σ
Proof of IR regularity
Our “Observables”
α
＜P(α) O(x) O(y) …＞
≩
⊋
Actual observable correlation fn.
If finite
← Finite
s
► Regularization scheme
＜ P(α) O(x) O(y) O(z)….＞
phase space
58
～
～
O = δσ, δ s
π
δs
After decoherence, a portion of wave packet contributes
◆ Loop integrals
 d 4 x...   dtd 3k...
Momentum integrals Regular
Temporal integral
Logarithmic secular evolution
► Outline
59
1. Introduction
2. Cosmological perturbation during inflation
3. IR divergence problem - Single field 4. IR divergence problem - Multi field 5. UV divergence problem
6. Summary and Discussions
► UV regularization
ζ : curvature perturbation / hij : GW
Diverge in x → y limit
◆ Adiabatic regularization
Parker & Fulling (1974)
UV mode ~ Solution in adiabatic approximation
Adiabatic expansion
|  k | | 
2
( ad )
k
2
|
sb
☠ Divergent
4
 O(k )
Regular
60
► Influence from Adiabatic reg.
Parker (2007), Parker et.al. (2008/2009)
Single field inflation
×( Slow-roll parameter )
×( Slow-roll parameter )
Amplitudes of ζ / GW suppressed by subtraction terms
at horizon crossing time
61
► No Influence from Adiabatic reg.
A.Starobinsky & YU (2009)
Exact solution
Solution in Adiabatic approx.
Super horizon limit
Constant value
Decay
|  ( ad ) k ( ) |2 sb

2
|  k ( ) |
| h ( ad ) k ( ) |2 sb

2
| hk ( ) |
62
► Summary - IR regularization ◆ Single field case
Flat gauge
Comoving gauge
+ Local gauge G ：“Causality” is preserved
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.
63
► Summary - UV regularization ◆ Adiabatic regularization
Regularize UV divergence
, which appears in the coincidence limit
We should introduce subtraction terms for all modes
However…
Subtraction term decays during cosmic evolution
→ No-influence on observables
64
- Supplement -
65
► Decoherence process
 
τ = τi
66

Correlated
>
Superposition of |
Initial state : adiabatic vacuum | 0 >ad
| 0 >ad = ∫d
|
><
|0 >ad
Include the contribution from all wave packets
ad <
0 | ζ(x1) ζ(x2) ζ(x3) … |0 >ad
Overestimation
3. Regularization scheme ~ Multi field ~
► Expansion by Retarded Green fn.
67
◆ Closed Time Path
Evolution of < in | ** | in >
GF
time
GD
G+ or G－
CTP
Expansion by GF , GD , G+, G－
◆ Expansion by Retarded Green f.n.
x
@ Causally connected region
・x’
≠0, Finite value
GR : Regular in IR limit
Expansion by GR
► Expansion by Retarded Green
fn.
[ Detailed exp. ]
R
(Ex.) N = 4
 (x)
=
+
R
=
+
R
+
R
R
R
+
R
R
・
・
・
=
+
+
・・・
68
► Expansion by Retarded Green fn.2
◆ Contraction
 k k '
 k'
k
R
Contraction
k
R
k'
69
► Mode expansion
=
3. Proof of IR regularity
70
[ Detailed exp. ]
∑ ( IR regular functions GRm) × a< 0 |P(α)ζI ζI … ζI | 0 >a
IR regular ??
1   d I  I  I
I
~
Eigenetate for ζI with finite wave packet
・Without P (α)
 d  d '
I
I
a
0 I

I
Infinite region
 I ..... I  'I  ' I 0
a
→∞
Finite
・With P (α)
 d  d '
I
I
a
Finite region
0 I

I
P( )  I ..... I  ' I
 'I 0
Finite
a
► Recent topics
Lyth (2007)
71
Necessity to consider Local quantity ( |x| < L )
→ Introduction of IR Cut off 1/L
Local quantity
Bartolo et. al (2008)
Riotto & Sloth (2008)
Enqvist et.al. (2008)
Cut off only for external momentum
Stochastic inflation → Decoherence
k < kc Stochastic fluctuation
Neglecting a part of quantum fluctation
Include the artificial cut-off scale
Under-estimation of IR corrections
→ Doubtful