WMAP Results
andINFLATIONARYUNIVERSE
Masahiro Kawasaki
Recearch Center for the Early Universe
University of Tokyo

Contents
1. Introduction
2. WMAP results
3. Testing slow-roll inflation
4. SUSY inflation modelwith Running specral index
5. Isocurvature perturbation
6. Small power at low multipoles
7. Summary

WMAP (Wilkinson Microwave Anisotropy Probe)

http://map.gsfc.nasa.gov/

WMAP Full Sky Map

History of Background Radiation

WMAP Result
Flat Universe

Wm + Wl ! 1

Scale Invariant Adiabatic Density
Perturbation
ns ! 1 "|dk|2# = kns
No Large Isocurvature Perturbation
Polarization: Anti-TE Correlation
No Tensor Mode Detected!DT DE " < 0D2hr≡ 2DR

Support for Inflation ModelAngular Power Spectrum"DT (!n) = Â Â a"mY"m(!n)" m=−"∗!a!ma!#m#$ = d!!#dmm#C!

TT correlation: adiabatic vs isocurvature102

adiabatic

10 1

10 0l(l+1)C l2π

10 -1

isocurvature
10 -2

10 -3

10 -4
1

10100

l1000

Polarization
Quadrupole
Anisotropy

hotε'

m=0v

e–ε'

Thomson
ScatteringcoldhotεScalars(Compression)

LinearPolarizationDE !−0.17(1 − cos(k̂n̂))×DhdeckQ1(hdec)

Q1 velocity of photon

Hu, White astro-ph/9706147

103Power (µK2)

102temperature

101
cross100

reion.
τ=0.1

10-1

E-pol.

10-2101l

102

103

Hu, White astro-ph/9706147

TE-Correlationisocurvatureadiabatic

Peris et al astro-ph/030225

Scalar vs Tensor Mode
Slow-roll inflation
Scalar mode

33kV
2∆2R (k) =!|R|"=k
2π 2
12π 2 Mp6 (V ! )2D2R(k0) ! 2.95 × 10−9A(k0) [k0 = 0.002Mpc−1]
Tensor mode (gravitational waves)k2V
22!|h+k | + |h×k | " =22π
3π 2 Mp4!
"2! 2∆h (k0 )Vr= 2∼∼!"1∆R (k0 )V2∆k (k) =Ratio

3

WMAP Result
Flat Universe

Wm + Wl ! 1

Scale Invariant Adiabatic Density
Perturbation
ns ! 1 "|dk|2# = kns
No Large Isocurvature Perturbation
Polarization: Anti-TE Correlation
No Tensor Mode Detected!DT DE " < 0D2hr≡ 2DR

Support for Inflation ModelWMAP Observation- cosmological parameters Parameter

WMAP

ns (k0 = 0.002 Mpc−1 )1.20+0.12−0.11

1.18+0.12−0.11

1.13 ± 0.08

r(k0 = 0.002 Mpc−1 )< 1.28/0.81/0.47< 1.14/0.53/0.37< 0.90/0.43/0.29−0.077+0.050−0.052−0.075+0.044−0.045−0.055+0.028−0.0290.71+0.10−0.11

0.73 ± 0.09

0.75+0.08
−0.09

Ωb h2

0.024 ± 0.002

0.023 ± 0.001

0.024 ± 0.001Ωm h2

0.127 ± 0.017

0.134 ± 0.006

0.134 ± 0.006

h0.78 ± 0.07

0.75+0.03−0.04

0.75 ± 0.03

τ0.22 ± 0.06

0.20 ± 0.06

0.18 ± 0.06

σ80.82+0.13−0.12

0.85 ± 0.05

0.85 ± 0.05

dns /d ln k
A(k0 = 0.002 Mpc−1 )WMAPext+2dFGRS WMAPext+2dFGRS+Lyman

Peris et al astro-ph/030225

Spectral Index Problem
ns= 1.13 ± 0.08

dnsd ln k=

at k0 = 0.002 Mpc−1+0.028−0.055−0.029

Running Spectral Index !

Peris et al astro-ph/030225

Slow-roll Infaltion
WMAP resultをよく説明するinflation model?
Inflation

Scalar field = Inflaton

Slow-rolling inflatonφ̇2 ! Vφ̈ ! V!

Inflation

Slow-roll Parameters!
ηξ≡

Mp22!

VV!

!

"2"!!

Mp2 VV! ! !!! "≡ Mp4 V VV2≡!!1η!1VφInflationns − 1 = −6! + 2η
dnsd ln k= 16!η − 24! − 2ξ2

Testing Inflation Models

Kinney et al hep-ph/0305130

Hubble slow-roll parameters Peris et al astro-ph/030225
"22 !
!
Mp
H!H = !
!H ≡2H! !! "
ηH = η − !
2 H
ηH ≡Mp H!

λH≡! !−1 !+1(H)H2 !
(2Mp ) H ! ddφ!+1Inflationary flow equations ( N : e-fold)d!HdNd(! λH )dN

= 2!H (ηH − !H )= [(# − 1)ηH − #!H ](# λH ) + (#+1 λH )

Choose Initial values at random fromN= [40, 70]!H= [0, 0.8]
ηH
= [−0.5, 0.5]
3

λH

M +1

λH
Observables= [−0.025, 0.025]···= 0r= 16!H [1 + 2C(!H − ηH )]ns − 1 = (2ηH − 4!H )[1 − (3 − 5C)!H /4]
dnsd ln k−(3 − 5C)!2H + (3 − C)ξH /2!
"dns
1= − 1−!
C=0.0806
dNH

Case A: negative curvature

η<0V = Λ4 [1 − (φ/µ)p ]V = Λ [1 + α ln(φ/µ)]4new inflation

Case B: small positive curvature0 ≤ η ≤ 2"

V = Λ4 (φ/µ)p

V = Λ4 exp(φ/µ)

chaotic inflation

Case D: Large positive curvatureη > 3"
V = Λ [1 + (φ/µ) ]4p

Case C: Intermediate positive curvature

2! < η ≤ 3!

However, each model is not aloways realized
since the end of inflation is set by hand
Realistic Model Building Necessary

Slow-roll

Nealy Flat potential

Keep flat against radiative corrections

Supersymmetry(SUSY )SUSY inflation model with running
MK, Yamaguchi, Yokoyama hep-ph/03004126
spectral index
ns= 1.13 ± 0.08

dnsd ln k=

at k0 = 0.002 Mpc+0.028−0.055−0.029

これを現実的なモデル説明したい
Hybrid inflation model in supergravity
running spectral index
However, number of e-fold is insufficientanother inflation が必要−1

Hybrid inflation in supergravity
superpotential

W = λS Ψ̄Ψ − µ S
2R-charge S(+2) Ψ̄Ψ(0)U(1) gauge int S(0) Ψ̄(−1) Ψ(+1)

Köhler potential
K = |S|2 + |Ψ|2 + |Ψ̄|2(minimal)
scalar potential

Potential√σ ≡ ReS/ 2

σ > σc ≡√2µ/λ ⇒ Ψ = Ψ̄ = 0

flat potential-2-3

ln V -4
-50

0.2
0.1

inflation
V (σ) ! µ4!

λσ
1 4
1 + 2 ln+ σ
8π
σc82

0.1
0.2
00.3

σ"-0.10.4

ψ

0.5 -0.2-2-3

ln V-4

0.5-50.2

one-loopcorrection

0.4
0.3

0.1

SUGRA

0.20ψ0.1-0.1-0.2 0σ

e-fold, spectral index!
σ > σd ≡ λ/(2π)

GUGRA effectone-loop correction! σi! σd 228π4π
N=dσσdσ !
32λλσcσd σ

σ < σdns − 1 = 2σ 2 − σd4 /σ 2dnsd ln k

= −(σd8 /σ 4 + 4σd4 + 3σ 4 )
2µ∆R (k0 ) = √3π!

σd4
σ+ σ3"−1ns > 1
ns < 1k = 0.002 Mpc

λ = 0.43
k = 0.002 Mpc−1−1

k = 0.002 Mpc−1

ns ! 1.1 dns /d ln k ! −0.042N ! 10.3

Result
SUSY hybrid inflationでrunning spectralindexを説明できる可能性
しかし、十分な膨張ができずスケールの対応関係がうまくいかない
Another Inflationで問題を回避できる

Hybrid-New inflation modelIzawa MK Yanagida (1997)
MK, Sugiyama, Yanagida (1998)
MK, Yangida (1999)

Isocurvature Perturbations
adiabtic vs isocurvatureρadaibatic

ρ

x

Smγ=

δ(nm /nγ )nm /nγ=δρm3 δργ
ρm − 4 ργ = 0

isocurvaturex

Smγ != 0

One Inflaton

Thermal Radiation, CDM,
baryon number
Adiabatic Perturbation

In general
Inflaton +other scalars

Adiabatic + isocurvaturePerturbation

たとえばAxion = CDMδργ + δρa ! 0 δρa /ρa "= 0isocurvatureもう一つの例Curvaton φ

1 2 2V (φ) = m φ2During InflationHδφ !
2π
isocurvature
If Curvaton-dominated
Universe
curvatonthermalRadiation, CDM,
baryon number

adiabaticδρφ
δφ!2ρφφρ
inflaton

curvatonいろいろな場合

Hamaguch,MK,Moroi,Takahashi (2003)

Radiation, CDM,
baryon number
If some components aregenerated non-thermallycurvatonΨ∝S

adiabatic +isocurvaturecorrelated isocurvature

multi-curvaton fields

adiabatic+ correlated isocurvature+ uncorrelated isocurvatute

CMB anisotropies
adiabatic + generic isocurvature
CMB angular power spectrum

C! = N (C!ad + 2κcorr C!corr + κ2iso C!iso )κcorr
κ2iso(δφA )
κm

=
==!
!

[Ψ!

A(δφA ) 2A

[Ψ(δφA ) ]2(δφ[Ψ(δφA ) ]2 κm A!

A](δφA )
κm

A
(δφ )
Smr A
Ψ(δφA )[Ψ(δφA ) ]2)2

10

κ m= 1

10 1

κcorr = κiso = κm

κ m= 010 02πl(l+1)C l

S = κm Ψ2

κ m= 310 -1

κ m = 10

10 -2

κm= ∞

10 1

κ m= 010 -3
10 0

κ m = -1

10l

100

1000

10 -1

κ m = -32π1l(l+1)Cl

10 -4[!(! + 1)C! /2π]!=10 = 1

κ m = -1010 -2

κm= - ∞
10 -3

10 -4

1

10l

100

1000

Constraint from WMAP
0.7

0.7

0.6

0.6

99%

0.5

κ iso

95%

0.5

0.4

0.4

0.3

0.3

0.2

0.2

68%
0.1

0.1

00-0.3

-0.2

-0.1

00.1

0.2

0.3

κ corrHamaguch,MK,Moroi,Takahashi(2003)

Contribution of isocurvature < 30%Conslusion
Contribution of isocurvature < 30%
Light scalar fieldが存在する理論に強い制限を与える
Axion
Curvaton
Affleck-Dine Field
String Moduli......

baryogenesis

Small power at Low mulipole

Cosmic Variance or Existence of Cutoff ?

Cutoff
spectrumContaldi, Peloso, Kofman
Linde astro-ph/0303636

Summary
WMAPの結果は単純なインフレーション宇宙モデルの予言と良くあっている
統計的信頼度はまだないが単純なモデルからのずれがみられる？
インフレーションモデルを特定する上での手がかり？
素粒子モデルに対する制限