2018年度
東北大学天文学教室

集中講義

暗黒物質優勢宇宙の構造形成
樽家篤史（京都大学 基礎物理学研究所）

自己紹介
名前：樽家 篤史（たるや あつし）
生年：1970年
出生地：京都府 長岡京市
出生高校：高槻高校
大学・大学院：名古屋大学（CG研）

現QG研の前身

1998年学位取得後、3年間のポスドクを経て2001年〜2013年
東京大学ビッグバン宇宙国際研究センター助教
2013年〜 京都大学 基礎物理学研究所 准教授

自己紹介
専門：観測的宇宙論

PHYSICAL REVIEW D 86, 103528 (2012)TARUYA et al.

宇宙の構造形成の理論をベースに、宇宙の大規模構造など
TARUYA et al.
PHYSICAL REVIEW D 86, 103528 (2012)の観測から宇宙論的制限を得るための理論・観測的研究
TARUYA et al.
PHYSICAL REVIEW D 86, 103528 (2012)
摂動論的手法を駆使した

非線形重力進化が進んだ宇宙の大規模構造の精密理論計算とその観測的応用
TARUYA et al.
PHYSICAL REVIEW D 86, 103528 (2012)
4 A. Oka et al.

Planck Collaborati

理論テンプレート

0.8
SDSS MGS

観測的制限

観測データ

k1.5P2f8

k1.5P0

900
350
0.7
800
300
FastSound
700
250 SDSS LRG
0.6
600
200
BOSS
VIPERS
6dFGS
DR12
500
150
+SnIa
0.5
400
100
300
50
0.4
200
6dFGS 0
DR14 quasars
GAMA
Best-fitting
0.3
100
-50
WiggleZ
SDSS and REGPT calculations. In each
FIG. 9 (color online). Comparison of power spectrum results between
N-body simulations
panel, the
0
-100
0.2
0.3
0 the smooth
0.1 linear 0.2
0.3
results at z ¼ 3, 2, 1, and 0.35 are shown (from top to bottom). 0Left panel0.1shows the0.2ratio of power
spectrum
to
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
[h/Mpc]from the no-wiggle formula of thek [h/Mpc]
linearspectrum, PðkÞ=Pno$wiggle ðkÞ, where the reference spectrum Pno$wiggle ðkÞ is kcalculated

講義の目標
観測的宇宙論の基盤をなす宇宙の構造形成の理論と、観測から宇宙の成り立ち・進化を探るための方法を学ぶ
宇宙の標準モデル（ΛCDMモデル）にもとづく宇宙の構造形成
非一様宇宙の観測
非線形性が進んだ構造形成の理論的記述

概観

観測的宇宙論とは？
宇宙論
宇宙の成り立ち、進化を物理的に明らかにする学問
トップダウン
物理の基礎理論にもとづき、理論的に整合性のとれた
宇宙創成・初期宇宙のモデル・シナリオを構築
ボトムアップ
観測データにもとづき宇宙の進化を記述する理論を構築
あるいは

観測データを説明する理論を構築し、宇宙を理解する

宇宙論の観測対象

観測対象：宇宙膨張、ゆらぎの進化の情報を担う天体（現象）The Astrophysical Journal, 784:90 (27pp), 2014 April 1

Okabe et al.

銀河のクラスタリング
銀河

銀河団

小スケール
(~kpc)

surrounding subhalos, the contribution of the main cluster
mass distribution to subhalo masses is excluded. Third, themass density of subhalos is expected to be close to zero
outside of the tidal radius, and the measured aperture mass
corresponds to the subhalo mass itself. If the mass density
profile follows the universal NFW profile (Navarro et al. 1996,1997) without any truncation radii, the aperture mass is higher
than the spherical one (Okabe et al. 2010b). As expected fromtidal destruction, the radial profile of the projected mass issaturated outside the truncation radii, rt . We measure projectedmasses for all the candidates. Since the smoothing kernel forthe mass reconstructions gives rise to centroid uncertainties of
the candidates, we determine the central position by choosing
maximal lensing signals within a 8.′ × 8.′ box where the centeris aligned with the map peak position. For accurate massmeasurements of subhalos with a variety of sizes, it is important
to explore truncation radii where the projected mass profile issaturated. We systematically compute projected mass profiles
by changing the background annulus and then statisticallydetermining the truncation radii. Here, the inner radius changes
from 0.′ 7 to 14.′ 5 in steps of 0.′ 2 and the width is fixed at 3.′ . The
projected mass M2D is computed from saturated values, taking
into account the error covariance matrix. The measurement
method is detailed in Appendix C. The same analysis was
repeated for different background widths which showed that theresult does not significantly change. Mass measurements used a
considerably large number of source galaxies (4×103 –2×104 ).
The number is comparable or less than that for main clusters atz ∼ 0.2 (e.g., Okabe et al. 2010b) for which the backgroundnumber densities are ng ∼ 5–20 (arcmin−2 ). Less massive
subhalos which are detected inside more massive ones should
be excluded in order to avoid double-counting these subhalos.
We count the ith subhalo using two conditions of the radius
rt,i > rt,j and the subhalo mass M2D,i > M2D,j (i ̸= j ). The
number of candidates is then reduced from 49 to 39 using thisprocedure. As mentioned above, the LSS model fails to fully
explain the lensing signals of background systems, especially ongroup scales. Furthermore, since there is a possibility to detectmass structures behind the cluster, we conservatively select thecandidates hosting spectroscopically identified member galaxies
within their truncation radii as the cluster subhalos. Having
applied these limitations, 32 peaks are identified as dark matter
subhalos. Three candidates are associated with the backgroundsystems (Table 2). Four candidates have no optical counter:they are located around ∼70.′ in the south-east direction and thenorth-west direction, respectively.
These 32 subhalos are labeled by integers, in the order ofright ascension. The resulting subhalo masses, M2D , range
from ∼2 × 1012 h−1 M⊙ to ∼5 × 1013 h−1 M⊙ (Table 3).
As shown in Figure 6, the radial profiles of the projected mass
clearly show saturation at some outer radii. The subhalos arewidely distributed from the northeast to the southwest in the sky(Figure 3). Interestingly, the direction connecting between the
Coma cluster and A1367 which are parts of the Coma superclus-

マイクロ波背景放射

ライマンαの森

Figure 5. X-ray surface brightness distribution in the 0.1–2.4 keV band fromROSAT X-ray satellite. The contours of the mass map are overlaid with
FWHM = 8.′ 3, taking into account the LSS lensing model. The contour level
starts at 1σ and increases in steps of 1σ .
(A color version of this figure is available in the online journal.)

the model does not perfectly describe the full LSS lensing
effect. Three other peaks associated with the known backgroundobjects (Table 2) are detected with the above conditions. One isthe background object “I” and two peaks are around the object“F” (see Figure 3). These objects are likely to be groups because
the lensing signals are stronger than what is expected from theluminosity of a single galaxy. Furthermore, there is a possibility
that background groups are accidentally superimposed withcluster subhalos, giving a systematic bias on mass estimatesof subhalos. This point is discussed in Section 3.4.1.
Next, we measure the model-independent projected masses(Clowe et al. 2000, see also Appendix C) for shear-selectedsubhalo candidates. This measurement has several important

Ia型超新星セファイド変光星（クェーサー吸収線系）

大スケール(~Gpc)

宇宙に広がる「ゆらぎ」
銀河分布(SDSS main galaxies)

宇宙マイクロ波背景放射(Planck)

絶対温度2.7Kの黒体放射だがわずかに温度ムラがある
網目状構造の非一様な空間分布（大規模構造）「ゆらぎ」のもつ統計的性質
パワースペクトル
乱雑なパターンに隠された情報を読み解く鍵
密度
温度
ゆらぎ
ゆらぎ

平均からのズレ
平均からのズレ

波に分解
波に分解T ( , )T ( , n(x)
)
i k·x
⇥
(⇥x)== a m(Yk)me( , )
(x) =
T
T n
（フーリエ展開）,m（球面調和関数展開）
平均温度
銀河の平均
個数密度

1
1
PC
(k)==
3
角度パワースペクトル
パワースペクトル
(2
)
2 +1

k

m

| (|ak)|m2|2

|k|=k
= m

角度サイズ
波長

2 /|⇥k| /パワースペクトル
宇宙マイクロ波背景放射

宇宙の大規模構造

Planck Collaboration: The cosmological legacy of Planck

Planck

1
C =
2 +1

m=

|a m |2 ;Planck 2018 results 1

P (k) =

1
(2 )3

|k|=k

| (k)|2

BOSS(SDSS DR9)
大角度

小角度

•空間パターンを波として調和関数展開• いろんな波の振幅をスペクトル表示
特徴的なふるまいがみえる

大スケール 1/波長 [h/Mpc] 小スケール

宇宙の構造形成

様々な天体分布やマイクロ波背景放射に見られる「ゆらぎ」とその統計的性質（非一様性・非等方性）
統一描像による見解• ゆらぎの祖先は共通• 宇宙初期の小さなゆらぎが宇宙膨張と重力の影響を受けて時間発展してきた
パワースペクトルの振る舞いに現れる「ゆらぎ」の時間進化を理論的に記述することで（構造形成の理論）、観測から宇宙論に対する様々情報を引き出せる宇宙の進化史
宇宙マイクロ波背景放射河

プランク衛星

面
銀
ン
乱
代
ョ
ジ
散
初
シ
成
イ
終
・
ー
生 成
エ
ン
最
星
レ
ン
合
ク
バ
の
代
フ
オ 素
グ
ー
光
初
ン
リ
元
ダ
ッ
イ
バ
ビ

原始ゆらぎ
の生成

重力不安定

加速膨張！
現在

大規模構造
(銀河・銀河団)

38万年

138億年

構造形成の理論
輻射〜物質・暗黒エネルギー優勢期における膨張宇宙のもとで• 宇宙初期の物質進化： 光と物質（電子）の電磁相互作用• 重力不安定性による構造進化：（一般相対論にもとづく）
地平線スケールを超えるゆらぎの進化

これらの効果を考慮した非平衡進化過程の時間発展を解けば、• 宇宙の大規模構造• 宇宙マイクロ波背景放射のゆらぎに対する定量的な予言が可能（1980年代末〜1990年代初頭に確立）線形理論のもとづく予言
様々なパラメーターに依存している（宇宙論パラメーター）
温度ゆらぎの角度パワースペクトル

質量密度ゆらぎの
パワースペクトルksmear overall
shape

bh

ns

大角度

b/

2

k

ns
As

振動の振幅
m

ns 4

As
keq

2
h
m

小角度
大スケール

mh

小スケール

標準宇宙モデル
観測との比較を経て宇宙論パラメーターが決定され、宇宙の成り立ち・進化に関する「標準モデル」が確立 →ΛCDM モデル
原子などの通常の物質（バリオン）
26%

5%

暗黒エネルギー（宇宙定数）• 宇宙の物質・エネルギー組成が確定 暗黒物質
69%
• 宇宙がほぼ平坦• 宇宙の年齢は138億年• 原始密度ゆらぎの性質はインフレーション仮説を支持（スケール不変に近い断熱ゆらぎ）

（冷たい）暗黒物質• 光も電波も出さない物質 未発見の素粒子が候補• 通常の物質とは重力を介してのみ相互作用する
銀河の回転曲線や重力レンズ現象• 構造形成の観点から「冷たい」ことが要請（速度分散が小さい）
暗黒物質の存在自体は 1934年からすでに指摘
かみのけ座銀河団の質量の運動学的見積もり

F. Zwicky暗黒エネルギー• 負の圧力を伴う未知のエネルギー体（=斥力として働く）• 宇宙を一様に覆い尽くしている（ようだ）
一般相対論で静的宇宙を実現するため、1917年にアインシュタインが導入した宇宙定数と似ている

実は不安定 → 加速膨張

Ia型超新星の観測
宇宙が加速膨張しているという事実は、Ia型超新星の光度曲線

24mag. residualfrom empty cosmology

Supernova Cosmology Project
Knop et al. (2003)ΩΜ , ΩΛ

暗0.25,0.75
0.25, 0
1, 0

20

暗い→遠い

Supernova
Cosmology
Project

22

B

空っぽ宇宙からの明るさずれ 見かけの明るさ
effective mを標準光源として使った観測から得られていた明

明るい→近い18

Calan/Tololo
& CfA
16

14
1.0
0.5

暗

ΩΜ , ΩΛ
0.25,0.75

0.0
0.5

0.25, 0
1,

明

1.0
0.0

0.2

0.4

0.6

redshift z

0.8

1.0

0

S.Paulmutter B.P. Schmidt

A.G. Riess

2011年ノーベル物理学賞

標準モデルを超える研究の進展により謎が深まってしまった
ただし、加速膨張の性質自体、よくわかっていない（宇宙定数でいいのか？）
さらに、検証が必要な仮説・仮定：✓ 宇宙は大域的に一様・等方✓ 宇宙論スケールで一般相対論が成り立つ✓ ニュートリノの質量はゼロ✓ 原始ゆらぎはガウス統計に従う
今後の大規模構造の観測から
ΛCDMモデルからのずれを突き止める

世界規模で進む大規模銀河サーベイ

DES

2013~

eBOSS
DESI

4m
Dark Energy Survey

2.5m

2014~

HETDEX

extended Baryon Oscillation
Spectroscopic Survey
Dark Energy
Spectroscopic Instrument

Euclid

LSST

2018+

4m
10m

2021

2018

Hobby-Eberly Telescope Dark
Energy EXperiment

WFIRST
8.4m
2021+

1.2m

Large Synoptic Survey Telescope

2.4m
2020’sWide Field InfraRed Survey Telescopeすばる望遠鏡による宇宙論観測(口径8.2m)

SuMIRe 2014~
Subaru Measurements of Imaging and Redshifts

Prime Focus Spectrograph

HyperSuprime-Cam（超広視野多天体分光器）（超広視野カメラ）

ファイバー
システム

メトロロジ
カメラ
分光器

（弱い）重力レンズ効果
コスミックシア
手前に存在する暗黒物質の質量分布により遠方の背景銀河のイメージが歪む現象
銀河の歪み具合（楕円率）=幾何学的重み ×密度ゆらぎの振幅
ab

時間

イメージの歪みの空間相関から宇宙論的情報を引き出せる(Hirata & Seljak 2003). An advantage of this method is that it
has been applied extensively to Sloan Digital Sky Survey data,
and thus the systematics of the method are well understood(Mandelbaum et al. 2005, 2013). In this method, the shape of agalaxy image is defined as

Subaru HSC 1year result

ear shear catalogam (HSC) is a wide-field imaging camera
ter field-of-view mounted on the prime focus
ubaru telescope (Miyazaki et al. 2012, 2015,
rvey is using 300 nights of Subaru time overa multi-band wide-field imaging survey withvey consists of three layers; Wide, Deep andide layer, which is specifically designed for
ology, aims at covering 1400 square degrees
e broadbands, grizy , with a 5 point-source
ihara et al. 2018b). Since i-band images arepe measurements for weak lensing analysis,
preferentially taken when the seeing is better.
eve a median PSF FWHM of ⇠ 0.00 58 for thed to construct the HSC first-year shear catathe software pipeline used to reduce the data
et al. (2018), and particulars about the acmetry and the performance of the deblender
sing a synthetic imaging pipeline in Huang

Ellipticity of each object :1 (b/a)2e = (e1 , e2 ) =(cos 2 , sin 2 ),
1 + (b/a)2

a (1)ϕb iswhere b/a is the observed minor-to-major axis ratio andthe position angle of the major axis with respect to the equatorial coordinate system. The shear of each galaxy, (obs) , isestimated from the measured ellipticity e as follows:(z=0.625)

1
(obs)
=
1 + hmi

⇣

e
2R

⌘

c ,

(2)

where R represents the responsivity that describes the responseof our ellipticity definition to a small shear (Kaiser et al. 1995;
測光赤方偏移のBernstein & Jarvis 2002) and is given by
R=1

he2rms i .

情報を使って、(3)

Here erms is the intrinsic root mean square(RMS) ellipticity per
異なる時刻の重力component. The symbols h···i denote a weighted average where
each galaxy carries a weight w definedレンズ効果を測定
as the inverse variance(z=0.25)(クレジット：HSC Project/東京大学)

https://www.subarutelescope.org/Pressrelease/2018/09/25/j_index.html

重力レンズトモグラフィー

s of the Astronomical Society of Japan, (2014), Vol. 00, No. 0

6

Publications of the Astr

19

Table 1. Summary of properties of individual tomographic bins.⇤

bin number
1
2
3
4
All

z range
0.3 – 0.6
0.6 – 0.9
0.9 – 1.2
1.2 – 1.5
0.3 – 1.5

zmed
0.446
0.724
1.010
1.300
0.809

Ng
28426352848777
2103995
1185335
8980742ng [arcmin 2 ]
5.9
5.9
4.3
2.4
18.5

n

⇤

We show redshift ranges (z range), median redshifts (zmed ), total numbers of source galaxies (Ng ), raw nu
2013) (ng,eff ) and the mean intrinsic RMS ellipticity per component (he2rms i1/2 ), which is related to shear b
are assigned into four tomographic bins using photo-z best estimates, zbest , derived by the Ephor AB photweighted average [equation (4)]異なる赤方偏移間の相関から、ゆらぎの構造進化と宇宙膨張を制限

Table 2. Comparison of lensing catalog properties of KiDS-450 (Hildebrandt et al. 2017
paper) used for cosmic shear analyses.⇤survey catalog
KiDS-450
DES Y1HSC Y1

area [deg2 ]
450
1321
137

⇤

No. of galaxies
14.6M
26M
9.0M

ng,e↵ [arcmin 2
6.85
5.14
16.5

We compare the survey area, the number of galaxies after cuts for cosmic shear analysis, the effective nu
tomographic analysis.

ℓ

1
2
|
|
C
=
e
ℓsurvey boundaries, non-uniform
from bright star masks,
surveyℓm which is related∑2ℓ
1 m=−ℓdepths, and non-uniform galaxy
shape+
weights.
The observedC
is the pse(obs)`

shear field is given by the weighted sum of shear values overgalaxies in each sky pixel as(obs)e( θ )⃗ =(✓) = W (✓) (true) (✓),∑ℓ,mMultipole(ℓ ∼ π/θ)eℓm Yℓm( (5)θ )⃗where W (✓) represents the survey window defined as the sumof shear weights in each pixel. When a sky position ✓ is outside
the survey area or masked due to a bright star, W (✓) is set tozero. We define a rectangular-shape region enclosing each of

directly measure`2 /2⇡ is a conve
trum. The sum i
In order to remo
tations of indivi
spectrum N` , an
10000 Monte C
tions to estimate

Hikage et al. (arXiv:1809.091

Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0

shear can tightly constrain a combination of cosmological parameters S8 (↵) ⌘ 8 (⌦m /0.3)↵ , which we adopt to quantify
cosmological constraints from the HSC first year data. By carrying out a linear fit of the logarithm of the posterior samplesof ⌦m and 8 , we find that the tightest constraints for S8 are
obtained with ↵ = 0.45. However, the previous studies by
DES (Troxel et al. 2017) and KiDS (Hildebrandt et al. 2017;
Köhlinger et al. 2017) have presented constraints on S8 with
↵ = 0.5. To present best constraints as well as constraints thatcan be directly compared with these previous cosmic shear results, in this paper we present our results of S8 both for ↵ = 0.45
and ↵ = 0.5.
In Figure 5, we show our marginalized constraints in ⌦m 8 and ⌦m -S8 (↵ = 0.45) planes. As expected, there is nostrong correlation between ⌦m and S8 . We find S8 (↵ = 0.45) =+0.0860.800+0.029
0.028 and ⌦m = 0.162 0.044 . Our HSC first-year cosmic shear analysis places a 3.6% fractional constraint on S8 ,d in the ⌦m -S8 (↵ = 0.45) plane (right), where S8 (↵) ⌘ 8 (⌦m /0.3)↵ , in the fiducialcomparison, we plot cosmic shear results from KiDS-450 with correlation function (CF)
) (Köhlinger et al. 2017) and DES Y1 (Troxel et al. 2018) with the same set of cosmologicalnshaw et al. 2013) (yellow) and Planck 2015 CMB constraints without CMB lensing (Planck

which is compa
and KiDS (Hildslightly degrade↵ = 0.5. We coS8 (↵ = 0.5) pl(Troxel et al. 20
ent methods, corand quadratic es
the plotted resu
logical paramete
different from t
For the KiDS rethe literature bu
et al. 2018). Fig
their 1- errorsthat there is no stained by these i

Fig. 6. The 68% credible interval on S8 (↵ = 0.5) from the HSC first-year data in the ⇤CDM model analized posterior contours in the ⌦m - 8 plane (left) and in the ⌦m -S8 (↵ = 0.45) plane (right), where S8 (↵) ⌘ 8 (⌦m /0.3)↵ , in the fiducial
el. Both 68% and 95% credible levels are shown. For comparison, we plot cosmic shear results from KiDS-450 with correlation function (CF)
Hildebrandt et al. 2017) and with quadratic estimators (QE) (Köhlinger et al. 2017) and DES Y1 (Troxel et al. 2018) with the same set of cosmologicaland priors as adopted in this paper, as well as WMAP9 (Hinshaw et al. 2013) (yellow) and Planck 2015 CMB constraints without CMB lensing (Planckn et al. 2016) (purple).

Fig. 5. Marginalized posterior contours in the ⌦m - 8 plane (left) and in the ⌦m -S8 (↵ = 0.45) plan
⇤CDM model. Both 68% and 95% credible levels are shown. For comparison, we plot cosmic she
estimators (Hildebrandt et al. 2017) and with quadratic estimators (QE) (Köhlinger et al. 2017) and DES
parameters and priors as adopted in this paper, as well as WMAP9 (Hinshaw et al. 2013) (yellow) and P
Collaboration et al. 2016) (purple).

Hikage et al. (arXiv:1809.091
which is comparable to the results of DES (Troxel et al. 2017)
and KiDS (Hildebrandt et al. 2017). For comparison, we find aal paantify

宇宙論的制限Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0

year data in the ⇤CDM model as well as from several literature.

銀河赤方偏移サーベイ
赤方偏移を使うと奥行きの情報が得られる
移

120,000 galaxies→大規模構造の３次元情報

偏

方

赤

http://www.sdss.org/press-releases/astronomers-map-a-recordbreaking-1-2-million-galaxies-to-study-the-properties-of-dark-energy/バリオン音響振動 (BAO)• 原始バリオン-光子流体の音響振動スケール (~150Mpc)(

Figure 15. As Figure 15, but for the DR11 LOWZ correlation function
CMBの音響ピークのスケール)
transformed as defined by Eq. 46 with a = 0.39 and b = 0.04. As before,• 標準ものさしとして遠方銀河までの距離測定に使える(理論プライヤー)

these error bars are nearly independent, with a worst case of 12 per centand an r.m.s. of 3.4 per cent in the off-diagonal elements of the reduced
covariance matrix.

→加速膨張のプローブ
DR9

7

L. Anderson et al.

DR10P(k) [(h-1Mpc)^3]

14

DR11

BOSS DR9(SDSS-III)k [h Mpc-1 ]FIG. 4: Measured power spectra for the full LRG and main galaxy samples. Errors are uncorrelated and full window functions are shown

Anderson et al. (’13)

バリオン音響振動 (BAO)• 原始バリオン-光子流体の音響振動スケール (~150Mpc)(

CMBの音響ピークのスケール)• 標準ものさしとして遠方銀河までの距離測定に使える→加速膨張のプローブ(理論プライヤー)
Angular diameter
distance

z 音響振動のRedshiftz (=0~3)

Redshift z=1100

スケール

cosmic expansion
changes the distances !! distant galaxies

cosmic microwave backgroundsurements. Section 7n the ⇤CDM modeld possible deviations
iscuss our results in

Figure 1. The number density of CMASS DR11 galaxies in redshift binsof z = 0.01 in northern and southern Galactic hemispheres, computed
assuming our fiducial cosmology.

赤方偏移空間ゆがみ

n of a cosmologicals into comoving
銀河disdopt a spatially-flat
0.7銀河
for this purpose. r
llow Anderson et al.
rs relative to a model, for which the BAO

BOSS DR11, CMASS samples
700,000gals @ 0.43<z<0.7
観測者の
視線方向

観測者の視線方向

uses a dedicated 2.5rform spectroscopice using a now-retired
バリオン音響l. 2006)
that imaged
al. 1996)
to a limitピーク(尾根)
on et al. 2013) is theSamushia et al.(’14)

重力理論のプローブ
カイザー
公式(Kaiser ’87)(S)2(k) = (1 + f µk )線形成長因子(k) ;d ln D+f⌘d ln aスケール因子重力由来のゆらぎの成長を表すパラメーター（赤方偏移と）重力理論に応じて f の値は変わりうる

カイザー公式は重力理論にかかわらず成り立つ
宇宙論的スケールで相対論のテストに使える
加速膨張の起源を探るヒントにもなる
e.g., Linder (’08); Guzzo et al. (’08);Yamamoto et al. (’08); Percival & White (’09)

are geometric
2
rence
for =
high
nd NdofNb Baryon
8. Theacoustic
last twooscillation
columns(BAO)
give measurementsforparameterpin the nonlinear evolue Plik likeli- and largely una↵ected by uncertainties( 2 = 3.1), tion of the matter density field and other systematic errors that
may a↵ect other types of astrophysical data. As in PCP13, wes.
use BAO as a primary astrophysical dataset to breaknk allother pa- therefore
cosmology
SPT cosmology
the predicted parameter degeneracies from CMB measurements.
2
2
N
N
PTE version of figure 15 from
Figure
14
shows
an
of
the
secondp updatedinmin
all scales (see PCP13. The plot shows the acoustic-scale distance ratio
6.1 compen2.91
DV137.4(z)/rdrag2.31
measured9.85
from a0.08
number of large-scale strucower
1.4
2.38
70.3
2.27
3.38
0.64
cally, ns is in- ture surveys with e↵ective redshift z, divided by the mean
7.3
1.96
61.4 1.37ratio in8.21
the base0.15⇤CDM cosmology using Planckodel andAs is acoustic-scale
cale power to TT+lowP+lensing. Here rdrag is the comoving sound horizon atd ` = 20 (see the end of the baryon drag epoch and DV is a combination of theH(z),nge in ns , fol- angular diameter distance DA (z) and Hubble parameter
0.82⌧d As e 1.10
fallsSDSSSDSS MGS"#1/3
cz
ng tail due to MGS WiggleZ
2 2
D
(z)
=
(1
+
z)
D
(z)
.
(25)
0.7
V
A
gle ✓D (whichH(z)2⌧As and1.05
As e
DES (D) figure show the ±1 and ±20.6 ranges SDSS LRG
ith these cos- The grey bandsin MtheBOSS
er than in the allowed by Planck in the base ⇤CDM cosmology.

銀河サーベイからの制限

8

赤方偏移空間ゆがみによる
Planck Collaboration: The
構造成長率の測定

1.00

f(D/rdrag )/(D/rdrag )Planck

バリオン音響振動による宇宙論的距離の測定

BOSS Ly0.95

BOSS
DR12
6DFGS

0.5

VIPERS

0.4
6dFGS

0.90

FastSound(DM )SDSS quasars

DR14 LRG

DR12

6dFGS
+SnIa

Planck ΛCDM

0.3

Planck ΛCDM

GAMA

DR14 quasars
WiggleZ

0.2
0.5

1.0

1.5

z

2.0

2.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

z

Planck 2018 VI.

Fig. 11. Acoustic-scale distance measurements divided by Fig.
the 20. Constraints on the growth rate of fluctuations, f 8 , as afunction of redshift, compared to the predictions of the ⇤CDM
corresponding mean distance ratio from Planck TT,TE,EE
model constrained by Planck (from Planck Collaboration VI

精密科学化する宇宙論
大規模かつ質の高い観測データからこれまで以上の統計精度で大規模構造の様子が幅広い赤方偏移でわかる (→ 精密宇宙論)
7

7

5.8

104 galaxies

k [h/Mpc]

14

L. Anderson et al.

Anderson et al. (’12)P(k) [(Mpc/h)^3]

P(k) [(Mpc/h)^3]

Tegmark et al. (’06)

2.6

105 galaxies

107

109 galaxies~1% precisionk [h/Mpc]

FIG. 4: Measured power spectra for the full LRG and main galaxy samples. Errors are uncorrelated and full window functions are shown

FIG. 4: Measured power spectra for the full LRG and main galaxy samples. Errors are uncorrelated and full window functions are shownin Figure 5. The solid curves correspond to the linear theory ΛCDM fits to WMAP3 alone from Table 5 of [7], normalized to galaxy bias
b = 1.9 (top) and b = 1.1 (bottom) relative to the z = 0 matter power. The dashed curves include the nonlinear correction of [29] for
A = 1.4, with Qnl = 30 for the LRGs and Qnl = 4.6 for the main galaxies; see equation (4). The onset of nonlinear corrections is clearly> 0.09h/Mpc (vertical line).
visible for k ∼in Figure 5. The solid curves correspond to the linear theory ΛCDM fits to WMAP3 alone from Table 5 of [7], normalized to galaxy bias
b = 1.9 (top) and b = 1.1 (bottom) relative to the z = 0 matter power. The dashed curves include the nonlinear correction of [29] for
A = 1.4, with Qnl = 30 for the LRGs and Qnl = 4.6 for the main galaxies; see equation (4). The onset of nonlinear corrections is clearly> 0.09h/Mpc (vertical line).
visible for k ∼• 観測と比較するための理論の精度向上

Our Fourier convention is such that the dimensionless
power ∆2 of [77] is given by ∆2 (k) = 4π(k/2π)3 P (k).

Before using these measurements to constrain cosmological models, one faces important issues regarding theirinterpretation, related to evolution, nonlinearities andsystematics.

B.

Clustering evolution

The standard theoretical expectation is for matterclustering to grow over time and for bias (the relative clustering of galaxies and matter) to decrease overtime [78–80] for a given class of galaxies. Bias is also

Our Fourier convention is such that the dimensionless
power ∆2 of [77] is given by ∆2 (k) = 4π(k/2π)3 P (k).

Before using these measurements to constrain cosmological models, one faces important issues regarding theirinterpretation, related to evolution, nonlinearities andsystematics.

B.

Clustering evolution

The standard theoretical expectation is for matterclustering to grow over time and for bias (the relative clustering of galaxies and matter) to decrease overtime [78–80] for a given class of galaxies. Bias is also

理論的（観測由来の系統的効果も含めて）
課題 （非ガウス性が発達した）• 観測量から有意な情報を取り出す方法論の開発

Figure 8. The CMASS DR9 power spectra before (left) and after (right) reconstruction with the best-fit models overplotted. The vertical do
the range of scales fitted (0.02 < k < 0.3 h Mpc 1 ), and the inset shows the BAO within this k-range,
determined by dividing both modelp
best-fit model calculated (including window function convolution) with no BAO. Error bars indicate Cii for the power spectrum and the rms
from fitting BAO to the 600 mocks in the inset (see Section 4.2 for details).

an estimate of the “redshift-space” power, binned into bins in k ofwidth 0.04 h Mpc 1 .

the power of the window function is smooth), andsorbed into the smooth component of the fit, and w
not explicitly include this term in our fits.

パワースペクトルの非線形進化
104

104

103

103

m (x)

(x)

m

2

10

10

-1

この他にも10-2• 銀河バイアス
10• 赤方偏移空間歪み
-3

10-4
0.01

2

重力非線形10
性の影響 1002
32(k)k P (k)/(2Δ2 (k) )(k)Δ=

10

Δ2 (k)

10
10

3
1

1

など

1
=
V
2
4

z = 0.00
ik·x

(k) e

z = 0.00
1

P (k) =
z = 0.98

z = 0.98|k|=k
z = 3.05kz=0N-bodyLinearz=0.98z=3.05z = 7.02

10

z = 14.87

z=7.02z = 0.98
z = 7.02

z=14.8z = 14.87

z = 3.05

Weakly nonlinearz = 7.02

10-20
10

-3
10-1

-4
10-2
10
0.10 0.01

z = 3.05

Fully Nonlinearz = 0.00

2
0

10-11
10

Nk

| (k)|2

z = 14.87

LinearSpringel et al. (’05)
1.00 0.10
k [ h / Mpc ]10.00 1.00
100.0010.00
1k [ h / Mpc ]k [h Mpc

100.00]

10-3

Figure 9: The powerFigure
spectrum
9: ofThethepowerdark spectrum
matter distributionof the dark
in matter
the Millennium
distribution
Simulationin the Millennium
at various Simulation at various
どこまで正確に大規模構造の統計的性質を定量化できるか？
epochs (blue lines). The
epochsgray (bluelinesshow theThepowergray spectrum
lines showpredicted
the power
forspectrum
linear growth,predicted
whileforthelineardashedgrowth, while the dashed10lines).
-4

講義の内容構造形成の理論を通して、宇宙論のプローブとしての宇宙の大規模構造の成り立ち・進化を理解する
１・オーバービュー
２・フリードマン宇宙モデル
３・構造形成の線形理論
４・非一様宇宙の観測
５・非線形構造形成

講義ノートと資料など
以下のサイトを参照
http://www2.yukawa.kyoto-u.ac.jp/~atsushi.taruya/Lecture2019_Tohoku/index.html