表紙
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
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