表紙
T=2.725K
Cosmic Microwave Background
CMB
Scale factor
Curvature
Hubble parameter
 Density parameter
cosmological constant
(dark energy)
Standard Inflation predicts
     1 with high accuracy.
一様等方宇宙
cluster
1022m
階層
1020m
1012m
Solar system
galaxy
107m
1024m
supercluster
1m
Earth
grew out of linear perturbations under the gravity
Linear perturbation
b
g
b
g
ds   1  2 ( x , t ) dt  a (t ) 1  2 ( x , t ) dx
2
2
2
Potential fluctuation
Power Spectrum
of Initial Fluctuation
P( k , ti )  | k (ti )|2
   A   H
Cosmological
Parameters
H, , ,...
2
Curvature fluctuation
Large-Scale Structures
Present Power Spectrum
P( k , t0 )  | k (t0 )|2
Anisotropies in cosmic
microwave background
Angular Power Spectrum
Cl
T
T
 10
5
COBE
COsmic
Background
Explorer
1993
WMAP
Wilkinson
Microwave
Anisotropy
Probe
2003
2001/6/30
2001/7/30
2002/4: first full-sky map
2002/10: second map
2001/10/1
size 5m、weight 840kg
COBEのbeam widthは７度だった。
Full Sky Map of Cosmic Microwave Background Radiation
-200
T(μK)
+200
Temperature fluctuation is Gaussian distributed.
Power spectrum determines the statistical distribution.
Three dimensional spatial quantities: Fourier expansion
3

ikx d k
 ( x , t )   k (t )e
Length scale r: r 
2
k
z
bg
3
2
b g
 k (t ) k  * (t )  P( k , t ) 3 k  k 
Power Spectrum：
c hz
d 3k
b
g
Correlation Function：  ( x , t ) ( y, t )   x  y  P( k , t )e
3
ik x  y
b2 g
Two dimensional angular quantities: Spherical harmonics expansion
T
T
bg

l
bg
 ,     almYlm  , 
l  0 m l
Angular scaleθ:
al1m1 al*2m2  Cl1 l1l2  m1m2
Angular Power Spectrum： Cl
bg
Angular Correlation Function： C  12
T
T
1

2
2
12
l
l 0
b , g

b , g
1
2l  1
 , g b
 ,  g Cb
 g

C Pb
cos g
b
T
T
4
1


l
l
12
12
1
2
2
(  1)C
2
So many data points!
m0
Luminosity density and average M/L of galaxies
m0  0.2  0.5
Cluster baryon fraction from X-ray emissivity and
baryon density from primordial nucleosynthesis
m0  0.35  0.07
Shape parameter of the transfer function of CDM
scenario of structure formation   m0h  0.15  0.3
m0
Many others
0.35
m0
0.3
0
dL
q0
Type Ia Supernovae m-z relation
Fz  1  q z IJ,
H G
H 2
K
a
1

 b
  2 g
aH
2
1
0
0
2
M
2
log(dL)

t0
z
0 1.25m0  0.5  0.5
SNIa+CMB
+Matter density
0
K 0
0
m0
H0
HST Key Project
Cepheids 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/Mpc
Fundamental Plane of Elliptical Galaxies
H0 =82±6±9km/s/Mpc
Summary H0 =72±8km/s/Mpc
(Freedman et al ApJ 553(2001)47)
m0
0.3
0
0.7
m0  0  1, K  0
Concordance Model
as predicted by Inflation
Cosmic age t0  (0.9  1.0) H
1
0
1
0
H0 =72±8km/s/Mpc, H  12.2 13.6 16.9Gyr
t0  11 17Gyr
centered around
t0  13Gyr
Observation:
t0  11 14Gyr from globular cluster
t0  12 15Gyr from cosmological nuclear chronology
Concordance Model was confirmed with high accuracy.
(with the help of the HST value of Hubble parameter.)
6 Parameters
Normalization of
Fluctuations
Spectral index
Baryon density
Dark matter density
Cosmological Constant
Hubble parameter
in
Spatially 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
c h
b
g


The Boltzmann equation for photon distribution f p , x
in a perturbed spacetime ds2   1  2 ( x, t ) dt 2  a 2 (t ) 1  2 ( x, t ) dx 2
Df
f dx 
f dp 
 
 
 C f Collision term due to
Dt x dt
p dt
the Thomson scattering
b
g
8 2
C f  xe ne T ,  T 
3me2
 free electron density
We consider temperature fluctuation averaged over photon energy
in Fourier and multipole spaces.
T i
T
k 
 :conformal time
 , , k 
 , , k    , , k ,  
c
T
hTb g b g
k
direction vector of photon
 d k
2 1
C 
 ( , k ,  )   (i)  ( , k ) P (  ),
0 (2 )3
4

0

3
 (0 , k )
2 1
2
.
directionally averaged
Boltzmann equation
L
M
N
b g
   ik           0   
1
 2 P2 (  )  iVb
10
collision term    axe ne T
conformal time
O
P
Q
Baryon (electron) velocity
Euler equation for baryons
 b    3 b
a


Vb  Vb  k 
V  Vb , R 

a
R
pb  p
4
d
i
Metric perturbation generated during inflation
k2
k2
3H 2 
   ,
  2 
:Poisson equation
2
a
a
2 
Boltzmann eq. can be transformed to an integral equation.
b  gb, , k g
 zm
    iV  ()e
0
0
0
0
b
 ( )
b
g re
    e
 ( )
b gd
ik   0
b  gb, , k g
 zm
    iV  ()e
0
0
0
 () 
0
b
z
0

 ( )
 ( )d  
b
z
0

g re
    e
 ( )
b gd
ik   0
axe ne T d  Optical depth
If we treat the decoupling to occur instantaneously at
b g
e  ( )      d
Visibility function
e  ( )
1
no scattering
many
scattering
b g
v ()   ()e  ( )      d
  d ,
d
0

now
b g b gb
gb g
  ,  0 , k    0 , k   0    iVb  d e b g
ik  d  0
Last scattering surface
b



e


b
g
z
0
ik   0
d
Propagation
gd
In reality, decoupling requires finite time and the LSS has a finite
thickness. Short-wave fluctuations
that oscillate many times during it
2
damped by a factor e  bk k D gwith k D  10h1Mpc corresponding to 0.1deg.
Observable quantity
F
1 
b
, , k g
G
H4 
0
0 
Ib g
JK
2
1
ik b
 d  0 g  b
k kD g
 iVb    d e
e

3

on Last scattering surface

Integrated SachsWolfe effect
b



e


b
g
z
0
d
1 
d  ： Temperature fluctuations
4 
ik   0
gd
small scale
b g：Doppler effect
iVb  d
b g：Gravitational Redshift Sachs-Wolfe effect
1
 d
3
Large scale
They can be calculated from the Boltzman/Euler/Poisson eqs., if the initial
condition of  k,ti and cosmological parameters are given.
b gd



e


b
gb g
g
z
We need to calculate  b
and V b
at the Last scattering surface
 g
 g
b g b gb
b
kk g
b
g
  ,  0 , k    0 , k   0    iVb  d e
e

ik  d  0
0
d
b
2
D
0
ik   0
d
d
when photons and baryons are decoupled.
Behavior of photon-baryon fluid in the tight coupling regime
Small scales： cs H 1 a k below sound horizon (Jeans scale)
3 3Rgis the sound speed.）
Oscillatory
（ cs2 1 b
Large scales： k 
 0
 0    const
Specifically they are given by the solution of the following eqn.
2
R
a
R
a
k


 0 
 0  k 2 cs2 0    
     F ()
1 R a
1 R a
3
source term is given by
metric perturbation.
Initial condition of  0 is also given by k
generated during inflation (if adiabatic fluc.)
Inflation
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 C に関係づけられる。
Fourier modes are related
with angular multipoles
as depicted in the figure.
r ～２π/k
LSS
d
Θ～π/l
b
b g
g
  ,  0 , k   ( i ) l  l  0 , k Pl (  )
l
l～kdにピーク
Observer
b g
Cl
d k  l 0 , k

4
(2 ) 3 (2l  1) 2
z
3
2
Sound horizon at LSS
corresponds to about 1 degree,
which explains the location of
the peak
 180
 
 200


小スケールで振動
Gravitational
重一
力般
赤相
方対
偏論
移的
流体力学的揺らぎ
大スケールで
ほぼ一定
hydorodynamical
The shape of the angular power
spectrum depends on
P(k , ti )  |  k (ti ) |2  Ak ns 4
（spectral index ns etc）as well as the
values of cosmological parameters.
（ ns  1 corresponds to the scaleinvariant primordial fluctuasion.）
Increasing baryon density relatively lowers radiation pressure,
which results in higher peak.
Decreasing Ω（open Universe）makes opening angle smaller
so 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
  1,   0
n  1, h  0.5
1 0.5 0.3
0.05
0.03
0.01
b h2  0.01
Old standard CDM
model.
0.7
0.3
0.5
0.7
0.3
0
表紙

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