ビッグバン宇宙論 一般相対論＋一様等方性 → ビッグバン宇宙論

Yuko Urakawa (Waseda U. ^→ U. Barcelona)

with Takahiro Tanaka (YITP) Y.U. and T. Tanaka 0902.3209[hep-th], P.T.P.122:779-803,2009 Y.U. and T. Tanaka 1007.0468[hep-th],

Y.U. and T. Tanaka 1009.2947[hep-th],

de Sitter ^{時空における}

赤外発散の問題とその周辺

ビッグバン宇宙論

一般相対論＋一様等方性 → ビッグバン宇宙論

高温・高熱 現在

構造形成

宇宙の晴れ上がり

軽元素合成

T~ 10

eV

T~ 1MeV T~ 0.1eV

?

e^-  

p

e^-  

p

宇宙背景輻射

2

宇宙背景輻射

WMAP 1yr/3yr/5yr…, Planck, ...

T~ 0.1eV   バリオンと熱平衡にあった光子 背景輻射

宇宙の初期条件

現在の宇宙の 構成要素

! δT δT "

WMAP 5yr

3

現在の宇宙の構成要素

宇宙論的パラメーター

（WMAP5yr + BAO + SN ）

Ω

= 0.0456 ± 0.0015

Ω

= 0.228 ± 0.013 Ω

= 0.726 ± 0.015

H₀ =

!a˙ a

"

0

= 70.5 ± 1.3km/s/Mpc

DM/DE

4

宇宙の初期条件

宇宙の晴れ上がり

因果的に結びつきのあるスケールを超えて等方的

●

２.７K

θ~1.7[deg]

θ

5

宇宙の初期条件

初期宇宙モデルが満たすべき条件

✔

●

✔

✔

∆²(k) = ∆²(k₀)

! k

k₀

"ns−1

∆²(k₀) = (2.445 ± 0.096) × 10⁻⁹

n_s = 0.960 ± 0.013 宇宙の晴れ上がり

6

インフレーション

¨

a = d dt

! a

H

"

> 0

物理的スケール

ホライズンスケール

λ

∝ a

宇宙の等方化

λ

λ

λ

∝ H

7

＝一定

フリードマン方程式

H

=

! a ˙ a

"

= 8πG 3 ρ

a(t) = e

インフレーションシナリオ

＝一定

◆

2φ˙² + V (φ) ! V (φ)

V (φ)

8

精密宇宙論の幕開け

T~ 10^-4eV

T~ 0.1eV

p n 晴れ上がり

T~ 1MeV 軽元素合成

ビッグバン T≤ 10⁸GeV??

インフレーション

？

高エネルギー物理

？

インフレーションモデル

ゆらぎの精密観測

T~ 10³GeV LHC

●  インフレーション宇宙

9

WHY QFT in de Sitter?

■  Inflation in early universe

■  Acceleration in current universe Two quasi-de Sitter stages

Origin of present large scale structures Quantum fluctuation of the scalar field

Solution to cosmological constant problem??

Quantum fluctuation in de Sitter spacetime

Tsamis&Woodard(96),Polyakov(07, 09),Kitazawa&Kitamoto(10)

Screen the cosmological constant Explain the smallness of C.C. ???

10

Contents of my talk

1. Introduction

2. Infrared(IR) divergence problem

4. Current status of IR issues 5. Concluding remarks

11

3. Solution to IR divergences problem

- Cosmological constant problem

Basis of cosmological perturbation

Our universe = FRW universe + Small fluctuations

G

= 8π GT

Einstein eq. covariant

DOFs in the choice of coordinates (→ Gauge DOFs) (ex) Distribution of energy density

Average

Change time coordinate t → t(x) , so that vanishes. δρ δρ x

→FRW

12

δφ = 0

h

= e

(δ

+ δγ

)

γ

δγ

= 0 ∂

δγ

= 0

■  Comoving gauge

Primordial perturbations

Linear perturbation in inflationary universe

spatial metric

s

R ! − 2e

∂

ζ

: Curvature perturbation ζ

: Gravitational waves

δγ

z := e

ζ

+ 2

ζ

− ∂

ζ = 0

δγ

+ 2

δγ

− ∂

δγ

= 0

!

= ∂

∂

= ∂

∂

At large scales, freezes. ^{ζ , δγ}

a = e

13

Primordial perturbations 2

h

= e

[(1 + 2ζ )δ

+ δγ

]

λ

ln λ

lna

Inflation

ζ, δγ_ij :const

λ

t

t

ζ

, δγ

∝ 1/k

ζ

, δγ

∝ e

/ √

2k

∆

=

1 ε_∗

!

2π

"

∆

=

!

2π

"

∆

=

P

Power spectrum

Scale invariant

t₁₄= t_∗

✔ No-loop corrections

● Linear theory

Beyond linear analysis

! ζζ "

✔ Gaussian

● Non-linear theory

! ζζ " , ! ζζζ " , ! ζζζζ " , · · ·

✔ Non-Gaussian

✔ Loop corrections

Sub-leading contribution

* UltraViolet divergence

* InfraRed divergence

S. Weinberg (05, 06)

Senatore & Zaldarriaga (09)

→ Break down of perturbation theory??

Leading → Consistent with CMB

15

! ζ

ζ

"

! ζ

ζ

" = | ζ

|

∝ k

Infrared(IR) divergence

Momentum ( Loop )integral Scale-invariant

■  Leading order

Logarithmic divergence

● Two point function

■  Next to leading order

k k'

q

k k'

!

d

q | ζ

|

= !

d

q/q

L

∝ ζ

ζ : mass-less field

16

Introduction of IR cutoff

log a

log k Which modes participate in loop corrections?

IR cutoff

horizon scale

Oscillation

ζ_k ∼ e^−ikη

√2k

k = aH

As inflation goes on,

more and more modes participate.

Logarithmic corrections ∝ (log a)

! ζζ... " = !

i

"

dt

d

k

...

IR divergence is physical or not?

Contents of my talk

1. Introduction

2. Infrared(IR) divergence problem

3. Solution to IR divergences problem

5. Concluding remarks

19

4. Current status of IR issues

- Cosmological constant problem

Comoving gauge

∂ L /∂ N = 0

Hamiltonian constraint

∂ L /∂ N

= 0

Momentum constraints

e

: scale factor

● Lagrange multiplier N / N

Maldacena (2002)  

ADM formalism

ds

= − N

dt

+ h

!

dx

+ N

dt " !

dx

+ N

dt "

N = N [ζ ] N

= N

[ζ ] S = S [N, N

, ζ ] = S [ζ ]

δφ = 0 h

= e

(δ

+ δγ

)

γ

δγ

= 0 ∂

δγ

= 0

20

S = S

+ S

= S [N, N

, ζ , δγ

]

DOFs in boundary conditions

δφ = 0

t ^{(ζ , δN [ζ ], N}

^{[ζ ])}

t + δt ^{ζ (t + δt, x)}

Fix the temporal gauge

Constraint eqs.

∂

= ∂

∂

∂

δN = S [ζ ] ∂

N

= S

[ζ ]

Elliptic-type eqs. → DOFs in boundary conditions Change the evolution of ^ζ

21

DOFs in boundary conditions 2

●  Hyperbolic system

t

t

Past light-cone Time evolution from t

to t

l(t)

ζ(t_f, x)

→ Bounded by

Wavelengths of fluctuation that affect k ≥ 1/l(t)

Vertexes in _Q ^(t

, x

)

ti

dt!

|x|≤l(t)

d³x

Q (t

, x

)

22

DOFs in boundary conditions 3

Time evolution from t

to t

t

t

^{ζ (t}

^{, x)}

(δN[ζ], N_i[ζ])

Boundary

b(t)

Vertexes in

|x|≤b(t)

d³x

●  Evolution of ζ

b(t) → ∞

For , IR div. appear.

→ Unbounded.

ζ(t_f, x)

Wavelengths of fluctuation that affect

23

Residual gauge modes

Single field inflation

S_φ = −1 2

! √

−g [g^µνφ_,µφ_,ν + 2V (φ)]d⁴x

−1

4∂ⁱG_i,1(x) δN1(x) = 1

ρ^!

!

ζ₁^!(x)

"

Nˇ_i,1(x) = ∂_i

! φ^{! 2}

2ρ^{! 2}∂⁻²ζ₁^!(x) − 1

ρ^!ζ₁(x)

"

−1 4

!

1 + φ^{! 2} 2ρ^{! 2}

"

∂_i∂⁻²∂^jG_j,1(x) + G_i,1(x)

∂²G_i,1(x) = 0

δN, N ˇ

= e

N

●  General solutions of

From Hamiltonian&Momentum constraints at 1st order

DOFs in → Residual gauge DOFs δN & N

24

! δN, N ˇ

"

for G

= 0

"

for G_i != 0

δx_i = −

!

dη G_i + 1 4

!

dη∂_i∂⁻²∂^jG_j + 1 4

! dη ρ^"

!

dη∂_i∂^jG_j + · · ·

(i) Scale transformation

ζ (x) → ζ ˜ (x) = ζ (x) − f (η) + · · ·

x

→ e

x

25

Residual gauge modes 2

●  Gauge transformation:

■  Time coordinate

(t, x

) → (t + δ t, x

+ δ x

) δφ = 0

Fixed by

■  Spatial coordinates

∃

Residual gauge modes

γ

δγ

= 0 ∂

δγ

= 0

(ii) Shear transformation

x

→ x

+ C

(η )x

C

= 0, C

= C

δγ

(x) → δ γ ˜

(x) = δγ

(x) − 2C

(η) + · · ·

Def. of fluctuations

x

δ Q := Q − Q ¯

Q ¯ := !

d

x Q / !

d

x

Q

δQ

Q¯

Q¯^! δQ^!

Distribution of _Q

Q = ζ , δγ

Gauge transformation : _Q ^¯ _{→ Q} ^¯

Averaged value

Gauge-inv. perturbations

Fluctuations, we observe, are gauge-invariant.

Physical DOFs

Gauge DOFs → Un-physical DOFs

{

All DOFs

27

Why don’t you perform gauge-inv. perturbations?

2. Construction of gauge-invariant variables 1. Complete gauge fixing

Y.U.&T.Tanaka(09)

Y.U.&T.Tanaka(10¹,10²)

Local gauge condition

●  Boundary conditions for ^{δN & N}

: Observable region = Causally connected region O

Y.U.&T.Tanaka(09)

Fluctuations within → Not affected by outside of _{O O} Boundary conditions at ∂ O

t

t

^{ζ (t}

^{, x)}

∂O

∂O

Blue region: _O t

Remove gauge DOFs associated with boundary cond.

28

Local gauge condition 2

●  Boundary conditions for ^{δN & N}

: Observable region = Causally connected region O

Y.U.&T.Tanaka(09)

Fluctuations within → Not affected by outside of _{O O} Boundary conditions at ∂ O

Remove gauge DOFs associated with boundary cond.

(ex) Scale transformation

ζ (x) → ζ ˜ (x) = ζ (x) − f (η) + · · ·

x

→ e

x

!

O

d

x ζ ˜ (t, x

) = 0

Fix Averaged value in O

measures deviation from local average

ζ˜

29

●  Momentum integral

Y.U.&T.Tanaka(09)

Regularization scheme

Vertex integral

t

t

^{ζ (t}

^{, x)}

Blue region: _O t

L(t)

Vertexes in ^{ζ ˜ (t}

, x

)

|x|≤L(t)

d³x

! dt !

d ³ k

1/L(t)

Effective cutoff at

t

dt

a(t) ! L_f + 1 a(t)H

30

Regularization scheme 2

●  Time integral _{log a}

log k horizon scale

k = aH Vertex integral !

dt !

d ³ k

t

Suppressed Oscillation

t = t_c 1/L(t)

Effective cutoff ^L(t) _! ^L

+ (aH )

k = 1/L_f

31

Upper bound on secular growth

Assumption

UV renormalization is safely performed.

●  Upper bound

n_c := ε⁻¹ − 1 = O(100)

Correlation fns.

Expanded by interaction picture field ^ζ

! ζ ζ ζ ... "

n: # of the included s ^ζ

■  Amplitude of ^ζ

A[ζ_n(t, xⁱ)] =

{

^{for for n < n n > n}

At least for , NO secular growth !!! n < n

Summary of the local gauge

33

Comoving gauge

δφ = 0

Local comoving gauge

!

O d³x ζ˜(t, xⁱ) = 0

etc ζ with all k

Locality

Gauge trans.

Initial condition : Adiabatic vacuum

ζ

_ζ ^˜

_{(x) = ζ}

_{(x) −} ^!

O

d

x ζ

(x) ζ ˜ (t

, x

) = ˜ ζ

(t

, x

)

δγ

= 0 ∂

δγ

= 0

!!NOTE!! Slight gauge dependence

Through initial condition that specifies the relation between the Heisenberg & interaction picture field

ζ ˜ with k ≥ 1/L(t)

Gauge-inv. perturbations

Fluctuations, we observe, are gauge-invariant.

Physical DOFs

Gauge DOFs → Un-physical DOFs

{

All DOFs

34

Why don’t you perform gauge-inv. perturbations?

2. Construction of gauge-invariant variables 1. Complete gauge fixing

Y.U.&T.Tanaka(09)

Y.U.&T.Tanaka(10¹,10²)

35

Y.U.&T.Tanaka(10)

●  Geodesic normal coordinate

Genuine gauge-inv. quantities

Gauge invariance regarding x

→ x ˜

= x

+ δ x

Scalar quantity, labeled by the gauge-invariant argument Gauge-invariant

s

R

■  3D scalar curvature

Two-point function on t:const surface

t:const

P1 P2

!

R(P

)

R(P

) " !

R

R " (l)

l: Geodesic distance between P

and P

dxⁱ

dλ² + ^sΓⁱ_jk dx^j dλ

dx^k

dλ = 0

Geodesic normal coordinate

: Global coordinates

: Geodesic normal coordinates

δγ

= 0 ∂

δγ

= 0

x

X

{

Xⁱ = dxⁱ dλ

!!

!λ=0

■  3D geodesics

dl

= e

[δ

+ δγ

] dx

dx

x

(X ) ! e

!

e

"

j

X

spatial metric

δx

:= x

− X

" !

e

"

e

#

j

− δ

$

X

large scale limit

λ = 1 λ = 0

Xⁱ t:const

| Ψ !

Initial state : Need to restrict to the physical state

37

Genuine gauge-inv. quantities 2

●  Gauge-invariant initial state

Gauge DOFs Physical DOFs

Quantized DOFs

Gauge cond.

g R(t, X ⁱ ) := ^s R(t, x ⁱ (X ))

● Initial condition in interaction picture

: Heisenberg picture field 1.

ζ

ζ

2. Positive frequency fn. for

Y.U.&T.Tanaka(10)

ζ

(t

) = ζ

[ζ

(t

)]

Necessary condition for gauge-invariance

Gauge-invariant initial state

ζ

: Interaction picture field

38

Choose initial conditions 1&2

■  One-loop corrections

Total derivatives (Divergent terms) ＋ (Regular terms)

!

R

R " #

!

d(log k )∂

(...)

1.

Heisenberg eq.

L : Derivative op.

L F [ζ

] = 0

ρ:e-folding

(C2)

Gauge-invariant initial state 2

homogeneous solution

39

(1 + ε) ∂

ζ

− x

∂

ζ

+ εζ

+ ... = − (∂

+ 3/2) ζ

ζ (t

) = ζ [ζ

(t

)]

ζ = Σ

a

F [ζ

] + L

S L ζ = S [ζ ]

Conditions on → (C1) a

2. Positive frequency fn. for ^ζ

■  Leading order

Bunch-Davies vacuum (C1), (C2) OK!

a

Remarks

O (ε

)

●  Slow-roll approximation

■  Higher orders

Adiabatic vacuum & ^ζ

^(t

^{) = ζ}

^(t

⁾

→ (C1), (C2) are not satisfied

●  Canonical commutation relations

Choosing the appropriate

Commutation relations can be compatible with the gauge-invariance condition

40

Contents of my talk

1. Introduction

2. Infrared(IR) divergence problem

5. Concluding remarks

41

3. Solution to IR divergences problem 4. Current status of IR issues

- Cosmological constant problem

Comment on multi-filed models

Inflation with multi light scalar fields

φ₁ φ₂ V (φ₁, φ2)

entropy mode

adiabatic mode entropy mode: gauge invariant

Regularization of IR divergence in entropy mode?

Origins of IR divergences

Single field (Adiabatic)

Multi field (Isocurvature) Momentum

integral

Time integral

Gauge artifacts

Graviton

Cosmological constant problem

ρ

= ρ

= !

d³k (2π)^3/2

1 2

√ k

+ m

"

k

: Cutoff scale

If cosmological constant is vacuum energy Λ

Zero point oscillation

k

= M

! 10

GeV

k

= M

! 10

GeV

ρ

! 10

× ρ

ρ

! 10

× ρ

For For

Why cosmological constant takes such a small, but non-vanishing value?

44

Screening of cosmological constant

Tsamis&Woodard(96,97)

Graviton loops in de Sitter

Screen the cosmological constant

+ + . . .

! h

" =

●  Two-loop tadpole diagram of _h

g

= a

(η) [η

+ h

] Perturbation

= A(η )η

+ B (η )t

t

t

= (∂

)

Assume homogeneity & isotropy

a(η ) = − 1/(H

η )

Dynamical sol. to C.C. problem?

Tsamis&Woodard(96,97)

Secular growth from IR contributions of graviton Dynamically screen C.C. ?

Back-reaction on expansion rate

●  Objection to Tsamis&Woodard

Screening of C.C. is gauge artifact?

Garriga&Tanaka(07)

H

= H

!

1 − 1 24π

"

H

M

#

log a + ...

$

Unruh(98)

-H

is not gauge-invariant.

- Scalar curvature R is not affected by graviton loops

and remains constant.

Other IR issues

- Re-summation can cure IR singularity?

= + + + ...

Generate effective mass Singular behavior in mass-less limit

→ Break-down of perturbation theory?

C. Burges et al.(09,10)

- Effects of decoherence can cure IR singularity?

Y.U.&T.Tanaka(09)

Momentum integral can be regularized. Time integral?

- Stability of de Sitter spacetime

Analytic continuation from Euclidean S

Polyakov(07, 09), Marolf and Morrison(09,10)

1. Origin of Infrared divergence in single field models 2. Two ways of regularization

Summary of my work

→ Presence of non-local gauge DOFs

→ Gauge inv. perturbations (1) Gauge fixing

(2) Construct genuine gauge inv. variables Momentum integrals are regularized.

No secular growth for n < n

= O (100) a. Geodesic normal coordinate

b. Gauge-inv. initial vacuum

48

Origins of IR divergences

Single field (Adiabatic)

Multi field (Isocurvature) Momentum

integral

Time integral

Gauge artifacts

Absence of decoherence

Y.U.&T.T(09)

Graviton

?

?

Supplement

DOFs in boundary conditions

f₁(x) := 1 4

!

dη∂ⁱG_i,1(x)

! ζ

− "

+ 2(ln z)

!

ζ

− "

− ∂

!

ζ

− "

f

f

f

= 0

+ 1

2ρ^!∂i∂j∂^kGk,1 + ...

= 0

δγ

+ 2(ln a)

δγ

− ∂

δγ

z :=e^ρφ^! ρ^!

● General solution of

−1

4∂ⁱG_i,1(x) δN₁(x) = 1

ρ^!

!

ζ₁^!(x)

"

Nˇ_i,1(x) = ∂_i

! φ^{! 2}

2ρ^{! 2} ∂⁻²ζ₁^!(x) − 1

ρ^!ζ₁(x)

"

−1 4

!

1 + φ^{! 2} 2ρ^{! 2}

"

∂_i∂⁻²∂^jG_j,1(x) + G_i,1(x)

∂²G_i,1(x) = 0

δN, N ˇ

= e

N

● Modification of EOM

51

Residual gauge modes 2

Comoving gauge δφ = 0 ^h

= e

!

e

]

δγ

= 0 ∂

δγ

= 0

Vector fns.

One-loop diagrams

gR(t, Xⁱ) :=^sR(t, xⁱ(X)) = Σ^∞_n=0δxⁱ¹δxⁱ²...δxⁱⁿ

n! ∂_i1∂_i2...∂_in^sR(X) δxⁱ := xⁱ − Xⁱ " !

e^−ζ "

e^−δγ/2#i

j − δⁱ_j$ X^j

s

R = e

!

e

"

∂

∂

ζ + ...

!

R

R " = !

R

R

"

● Expansion by interaction picture field

+ !

R

R

" + !

R

R

" + !

R

R

"

tree

1-loop

sR3(X) ^sR₁(Y)

contract

sR2(Y)

sR2(X)

contract

Possibly divergent terms

● Momentum integral

IR divergences are at most logarithmic.

If all propagators are multiplied by derivatives,

sR3(X) ^sR₁(Y)

∂ⁱ, ∂_t

∂ⁱ, ∂_t

sR2(X) ^sR2(Y)

momentum integrals no longer diverge.

(∂ψ)

ψ

g

R

= ∂

∂

ψ

ψ

∂ψ

(∂ψ)

g

R

! ψ∂ψ

g

R

! (∂ψ)

∂ = ∂

or ∂

IR IR

Neglect the terms that do not yield divergences.

More about secular growth

loga

logk

k = aH

t = t_c

Region I Region 2

Contributions from region I

{a_iH_iL(t)}!

H_i M_plε^1/2

"n

!

M_plε^1/2

"

{

ζ

Contributions from region II

For , suppression is not enough to eliminate the contributions from the distant past.

n > n

m² ≤ 0

(δ σ, δs) ˜

δ σ ˜ (x) = δσ(x) − δ σ ¯ (t)

(i) # = 1 (ii) # ≥ 2

多成分系への拡張

●  ^{局所的ゲージ}

＝０

#: 赤外発散する場の数

赤外発散なし

δs

赤外発散が残る

背景の軌跡

δs δσ

φ₁ φ₂

ゲージ自由度でなはく物理的な自由度

δ s ¯

多成分系におけるゆらぎの古典化

宇宙の波動関数

の波束

固有状態の重ね合わせ 統計集団

デコヒーレンス

: 固有状態

Ψ 相関あり Ψ 相関なし

我々の宇宙で実現されるのは一つの波束  デコヒーレンス→ “ 物理的な効果”

| Ψ ! = !

δs¯

C (δ s) ¯ | δ s ¯ ! | Ψ ˜ ! | δ s ¯ !

| δ s ¯ ! δ s ¯ δ s ¯

パラレルワールド

t = t

t = t

我々の宇宙

因果的に結びつきのない宇宙

別の宇宙で起こることを知る由はない

我々の宇宙で選択された波束のみ考えたい ピックアップ ^Ψ

δ s ¯

デコヒーレンス

粗視化を用いずにデコヒーレンスの効果を考慮

δ s ¯

P(α) = exp

!

−(δs(t¯ _f) − α)²

∆²

"

! P (α)δσ(x

)δσ (x

) · · · "

≩ ⊋

◆

α Ψ

∆

“観測量”となるゆらぎ

実際の観測量

正則性を示す

観測量は正則

◆