表紙

http://map.gsfc.nasa.gov/m_ig/990297/990297.html

全天マップ１

T=2.725KCosmic Microwave Background

CMB

Scale factor Curvature

一様等方宇宙Standard Inflation predicts with high accuracy. 1

Hubble parameter

Density parameter

cosmological constant(dark energy)

階層

1024m

1022m

1020m

1012m

107m

1m

Earth

Solar system

galaxy

cluster

supercluster

grew out of linear perturbations under the gravity

Potential fluctuation Curvature fluctuation

Cosmological ParametersH,

ds t dt a t t d2 2 2 21 2 1 2 ( , ) ( ) ( , )x x xb g b g

Power Spectrumof Initial Fluctuation

Anisotropies in cosmicmicrowave background

Large-Scale Structures

Present Power Spectrum

Angular Power Spectrum

P k t t( , ) | ( )|0 02 k

P k t ti i( , ) | ( )| k2

Cl

Linear perturbation

A H

COBECOsmicBackgroundExplorer1993

WMAPWilkinsonMicrowaveAnisotropyProbe2003

510T

T

size 5m 、 weight 840kg

2001/6/30

2001/7/30

2001/10/1

2002/4: first full-sky map2002/10: second map

COBE の beam width は７度だった。

Full Sky Map of Cosmic Microwave Background Radiation

Temperature fluctuation is Gaussian distributed.Power spectrum determines the statistical distribution.

-200 T(μK) +200

Three dimensional spatial quantities: Fourier expansion

( , ) ( )x kkxt t e

d kz i

3

23

2bg k k k k( ) ( ) ( , )*t t P k t 3b g ( , ) ( , ) ( , )x y x y k x yt t P k t e

d ki zc h bgb g 3

32

Power Spectrum ：

Correlation Function ：

Length scale r: rk

Two dimensional angular quantities: Spherical harmonics expansion

T

Ta Ylm lm

m l

l

l

, ,b g b g

0

Angular scaleθ:

l

Angular Power Spectrum ：

Angular Correlation Function ：

a a Cl m l m l l l m m1 1 2 2 1 1 2 1 2

*

1 1,b g 2 2,b g 12

C 12b g

T

T

T

TC

lC Pl l

l1 1 2 2 12 12

0

2 1

4, , cosb g b g b g b g

Cl

( 1)

2

C

So many data points!

m0Luminosity density and average M/L of galaxies Cluster baryon fraction from X-ray emissivity and baryon density from primordial nucleosynthesis

Shape parameter of the transfer function of CDM scenario of structure formation

Many othersm0 0.3

m0 0.2 0.5

m0 0.35 0.07

m0 0.15 0.3h

m0 0.35

0 Type Ia Supernovae m-z relation

0 m01.25 0.5 0.5

log(dL)

z

d H zq

z

qa

aH

L

tM

FHG

IKJ

0

1 0 2

0 2

1

2

1

22

0

,

b g

0

m0

0K

0

SNIa+CMB+Matter density

0HCepheids H0 =75±10km/s/Mpc

SNIa H0 =71±2(stat)±6(syst)km/s/Mpc

Tully-Fisher H0 =71±3±7km/s/Mpc

Surface Brightness Fluctuation H0 =70±5±6km/s/Mpc

SNII H0 =72±9±7km/s/MpcFundamental Plane of Elliptical Galaxies H0 =82±6±9km/s/Mpc

Summary 　 H0 =72±8km/s/Mpc

HST Key Project

(Freedman et al ApJ 553(2001)47)

m0 0.3 0 0.7

m0 0 1, 0K as predicted by Inflation

Cosmic age 10 0(0.9 1.0)t H

H0 =72±8km/s/Mpc, 10 13.61 Gyr2.2 16.9H

0 11 17Gyrt centered around 　　　　　　0 13Gyrt

Observation:

0 11 14Gyrt

0 12 15Gyrt from globular cluster

from cosmological nuclear chronology

Concordance Model

Concordance Model was confirmed with high accuracy.(with the help of the HST value of Hubble parameter.)

6 ParametersNormalization of FluctuationsSpectral indexBaryon densityDark matter densityCosmological ConstantHubble parameterinSpatially Flat Universe

899 data points are fit.Approximately scale-invariant spectrum, which is predicted by standard inflation models, fits the data.But we may also find several interesting features beyond a simple power-law spectrum…

表紙

The Boltzmann equation for photon distribution in a perturbed spacetime

Collision term due to the Thomson scattering

free electron density

ds t dt a t t d2 2 2 21 2 1 2 ( , ) ( ) ( , )x x xb g b gf p x ,c h

Df

Dt

f

x

dx

dt

f

p

dp

dtC f

C f x nme e T T

e

,

8

3

2

2

0

( , , ) ( ) ( , ) ( ),k i k P

23

0

30

( , )2 1.

4 (2 ) 2 1

kd kC

We consider temperature fluctuation averaged over photon energy in Fourier and multipole spaces.

direction vector of photon

T

T

T

T ki , , , , , , ,k k k

kc h b g b g :conformal time

Boltzmann equation

collision term

directionally averaged

Baryon (electron) velocity

LNM

OQP ik P i Vb b g 0 2 2

1

10( )

Euler equation for baryons

Va

aV k

RV V R

p pb b bb

b

b

d i,

3

4

Metric perturbation generated during inflation

:Poisson equation , k

a

k

a

H2

2

2

2

23

2

Boltzmann eq. can be transformed to an integral equation.

zb gb g

b gm r b g

, ,

( ) ( ) ( )

0

00

00

k

i V e e e dbik

ax ne e T

conformal time

Optical depth

zb gb g

b gm r b g

, ,

( ) ( ) ( )

0

00

00

k

i V e e e dbik

( ) ( ) z zd ax n de e T

0 0

If we treat the decoupling to occur instantaneously at ,

1

now

Last scattering surface Propagation

e

v e

d

d

( )

( )( ) ( )

b g

b g

, , ,0 0 00

00k kb g b gb gbg b gb g b g zi V e e db d

ik ikd

d

e ( )

d

d 0

manyscattering

no scattering

Visibility function

In reality, decoupling requires finite time and the LSS has a finite thickness. Short-wave fluctuations that oscillate many times during itdamped by a factor with corresponding to 0.1deg. e k kDb g2 Mpck hD

10 1

Observable quantity

on Last scattering surface

Integrated Sachs-Wolfe effect

, ,0

1

4

1

30

2 00kb g bg b gb g b g b g

FHG

IKJ zi V e e e db d

ik k k ikd D

d

： Temperature fluctuations

： Doppler effect

： Gravitational Redshift Sachs-Wolfe effect

small scale

Large scale

0

1

4 d

i Vb d bg1

3 dbg

They can be calculated from the Boltzman/Euler/Poisson eqs., if the initial condition of k,tiand cosmological parameters are given.

We need to calculate and at the Last scattering surface when photons and baryons are decoupled.

Vb dbg 0 dbgBehavior of photon-baryon fluid in the tight coupling regime 　　 Small scales ：　　　　　　　　 below sound horizon (Jeans scale) Oscillatory 　 （　　　　　　 is the sound speed. ）　 Large scales ：　　　　　　　

c H a ks 1

k 0

c Rs2 1 3 3 b g 0 const

, , ,0 0 00

2 00k kb g b gb gbg b gb g b g b g zi V e e e db d

ik k k ikd D

d

Specifically they are given by the solution of the following eqn.

source term is given bymetric perturbation.

0 02 2

0

2

1 1 3

R

R

a

ak c

R

R

a

a

kFs ( )

Inflation

Initial condition of is also given by generated during inflation (if adiabatic fluc.)

0 k

LSS

Θ ～ π/l

d

r

Observer

～２ π/k

図のような幾何学的関係からフーリエ空間の量が multipole 　空間の角度パワースペクトル　　に関係づけられる。

C

, , ( ) , ( )0 0k kb g b g i Pll

ll

C d k

ll l

4 2 2 1

3

3

0

2

2

z( )

,

( )

kb g

l ～ kd にピーク

Fourier modes are related with angular multipolesas depicted in the figure.

大スケールでほぼ一定

小スケールで振動

一般相対論

的重力赤方偏

移

流体力学的揺らぎ

Sound horizon at LSS corresponds to about 1 degree,which explains the location ofthe peak

180200

hydorodynamical

Gravitational

The shape of the angular power spectrum depends on

（ spectral index 　　 etc ） as well as the values of cosmological parameters.（　　　　　 corresponds to the scale-invariant primordial fluctuasion. ）

42( , ) | ( ) | sni iP k t t Ak k

sn

1sn

Increasing baryon density relatively lowers radiation pressure,which results in higher peak.Decreasing Ω （ open Universe ） makes opening angle smallerso that the multipole l at the peak is shifted to a larger value.Smaller Hubble parameter means more distant LSS with enhanced early ISW effect.Λalso makes LSS more distant, shifting the peak toward right with enhanced Late ISW effect.

Thick line

2

1, 0

1, 0.5

0.01b

n h

h

Old standard CDMmodel.

1 0.5 0.30.05

0.03

0.01

0.3

0.5

0.7

0.7

0.3 0

表紙