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 Background

WMAP 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 nT

Tn

T

T 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 5

1

1. Introduction

inflation

► Evolution of fluctuation

Physical 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

z ~constan

tLS

SWT

T 5

1

LS

For 　　　　　　　 at LSS

horphys hocLSSWT

T 5

1

5

1

10

► WMAP 5yr dateAlmost scale invariant, Almost Gaussian …

Consistent to the prediction from

　 “ Standard” inflation ( Single-field , Slow-roll)

013.096.0ln

ln1

2

kd

dns

92

232 10)096.0445.2(

2

||

kk

1002.0 Mpck

)(

)(2

2

k

kr

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 perturbation

Gauge 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 condition

In the distant past |η| → ∞,

Adiabatic solution

⇔ k>>1 Much smaller than curvature scale

~ Free field at flat space-time

19

► Scalar perturbation

k

e ik

k2

Log

Loga

phys

hor

2

3

kk2

2

232

22

1

2

||)(

hoc

hoc

k Hkk

Almost scale invariant

2H

H

z ~constan

t

hoc

20

► Chaotic inflation

Reheating

Inflation goes on

H,

22

22

1

hoc

hoc

H

Larger scale mode → Exit horizon earlier

→ Larger amplitude

Red tilt ns< 1

21

► Tensor perturbation

◆ Initial condition

In the distant past |η| → ∞,

ikk e

kah

2

11)(

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 !4

dxS

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 !3

dxS

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

Z

dtzGk(tx; tz)

Z

d3qõGq(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

informatio

n on infla

tion

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|| klinearlinearlinear

kkk

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

yx

ydpJ

“Causal evolution” 　　　　　 : Not affected by { J-

(p) }c

)(~

x

)(~

)(~

)( txxobs

....)()()(~

tfxx

)(~

)(~ 3

)(xxdt

pJ Averaged value at J-(p)

48

][2 S

► Gauge invariant perturbation

0)(~

)(~ 3

)( xxdt

pJ

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

H

MN 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 expansion

Various 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

h

h

A.Starobinsky & YU (2009)

Neglig

ible

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