第１０回湯川財団・
木村利栄理論物理学賞
記念講演

重力構造進化の摂動計算
で探る宇宙
樽家篤史

謝辞
今回、栄えある賞をいただき誠にありがとうございます。
また今回の受賞に際し、推薦頂いた須藤 靖さん (東大理・教授)
共同研究者の方々（斎藤、高田、西道、平松 etc.）、
家族、
ならびに

研究をサポートしていただいた全ての方々

心より感謝いたします。

講演内容
宇宙の大規模構造にもとづく観測的宇宙論の舞台裏（裏事情？）
観測的宇宙論と宇宙の大規模構造受賞の対象となった研究の紹介
まとめと展望観測的宇宙論• 宇宙の成り立ち・進化を、観測に立脚してボトムアップ的に理論体系化する宇宙論の研究分野（c.f. 素粒子宇宙論）• さまざまな観測による精査を経て、無矛盾な宇宙像を築く
主導的な役割を果たしてきた観測が宇宙マイクロ波背景放射 (CMB) の非等方性
宇宙の大規模構造

今回はこちら宇宙の大規模構造
宇宙論的スケールにわたって存在する質量分布の非一様性
メガパーセク(Mpc) 〜ギガパーセク(Gpc)
標準的シナリオでは※ 1 Mpc＝10^6 pc ~300万光年• 質量分布の大半は冷たい暗黒物質（Cold Dark Matter, CDM）
• 原始密度ゆらぎを種に、宇宙膨張の影響下で重力不安定性により構造が発達・進化（→宇宙論の情報を豊富に含む）
銀河赤方偏移サーベイによる銀河の３次元地図をもとに研究が進められている（最近は重力レンズなどもある）

銀河赤方偏移サーベイ〜 大規模構造を探る窓 〜
2.5m

8.2m

光学望遠鏡で銀河１個１個を分光（スペクトル）観測

スローンデジタル

すばる望遠鏡スカイサーベイ(ハワイ)→ 銀河の赤方偏移 z を決定(ニューメキシコ)
遠方銀河z=（奥行きの ‘距離’ 指標に）

Ca,H&K OIII Hβ Na Mg

波長SDSS SkyServer/スローンデジタルスカイサーベイ(SDSS) の銀河カタログ
カタログから断面を切り取ったもの
地球
(観測者)

赤方偏移

約60億年前※ 途中段階のもの

http://www.mpa-garching.mpg.de/131601/hl201506

黄：SDSS-II main
赤：SDSS-II LRG
白：SDSS-III CMASS

スローンデジタルスカイサーベイ(SDSS) の銀河カタログ
カタログから断面を切り取ったもの
地球
(観測者)

赤方偏移

約60億年前CfA サーベイ※ 途中段階のもの

黄：SDSS-II main
1100個の明るい銀河
赤：SDSS-II
LRG
http://www.mpa-garching.mpg.de/131601/hl201506
de Lapparent, Geller &白：SDSS-III
Huchra (’86) CMASS

銀河分布の３次元地図
120,000 galaxies

赤方偏移

http://www.sdss.org/press-releases/astronomers-map-a-recordbreaking-1-2-million-galaxies-to-study-the-properties-of-dark-energy/

宇宙の進化史
宇宙マイクロ波
背景放射

宇宙の

面
ン
乱
ョ
成
散
シ
生 成
ン
終
ー
ン
合
代
最
時
グバ フレ リオ 元素
の
黒
光
バ 軽
暗
ビッ イン

成

河

形

加速膨張

銀

・

代

星

現在

初

宇宙の大規模構造

年
10
0億

年
10
億

38万年

2億

年

s
秒

1秒ec
10
0秒
1
10 年
0年

30

10

10

32

秒
se

c

(銀河・銀河団)138億年

宇宙論的情報
質量分布のランダムな空間パターンの統計的性質から• 原始密度ゆらぎの性質• 宇宙のエネルギー組成・宇宙膨張則に関する
宇宙論パラメーター
インフレーションモデルの制限
ダークエネルギーの性質（状態方程式）
ニュートリノ質量和の制限
宇宙論スケールでの重力のテスト
CMBから得られない情報も含む

imates from the the correlation function and the power spectrum.
N≡d r⟨w(r)n(r)⟩,quantify these improvements further in the following sections.
and ⟨w(r)n(r)⟩ is the expected weighted distribution of galax
We next compare the observed scatter in the best-fitting α inlocation r in the absence of clustering, and n(r) is the galaxy democks to the σ α estimated in each fit from the χ 2 (α) curve.
Fig. 7, we plot a histogram of (α − ⟨α⟩)/σ α from the mocks
The quantity δ D is the standard Dirac-δ function. We do not
d compare the result to the unit normal distribution. We findluminosity-dependent weights (as applied by Percival et al. 2cellent agreement; a Kolmogorov–Smirnov (K-S) test finds aand Reid et al. 2010), as we are only interested in the BAO, anh likelihood密度場
that the observed distribution is drawn from a unit
the overall shape of the power
1 spectrum.
n(x)
1
i
k·x
mal. Hence the Gaussian
distribution
We chose toPmodel
the expected distribution(k) =| (k)|2of galaxies us(x)probability=
1 = obtained from(k) e
Nk selected at the mean galaxy denVerror on α.
of theχ 2 statistic is an appropriate characterization
random catalogue with pointsk|k|=k

パワースペクトル

Anderson et al. (’12)

観測

SDSS-III
DR9 (BOSS)
2.6

線形理論
バリオン音響振動

k ns

105 galaxies@ z~0.5

k ns 4

As

kfs
keq

（波数）
（波数）

mh

m

m

ニュートリノによる減衰効果精密宇宙論観測の到来
望遠鏡を占有化し、これまで以上に深く広域にサーベイを行う
BOSS(~2014)

DES (2013~)

DESI(2018+)HETDEX(2015+)LSST (2022+)

eBOSS(2014~)WFIRST(2024+)EUCLID(2020)
すばる望遠鏡による宇宙論観測
FastSound(2012~2014)
SuMIRe Hyper-Suprime Cam(2014~) Prime Focus Spectrograph精密宇宙論における不安
大規模観測により観測データの統計精度は飛躍的に向上
7

5.8

14

104 galaxies
0.7 Gpc

Veff

3

L. Anderson et al.

Anderson et al. (’12)P(k) [(Mpc/h)^3]

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

Tegmark et al. (’06)

7

2.6

105 galaxies
Veff = 2.2 Gpc3

107

109 galaxies

Veff

10 Gpc3~1% precision（波数）k [h/Mpc]

k [h/Mpc]

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 clearlyvisible for k >∼ 0.09h/Mpc (vertical line).

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 clearlyvisible for k >∼ 0.09h/Mpc (vertical line).

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

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

質のよい統計データで新しい宇宙論が拓ける可能性

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

その一方、

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

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

Figure 8. The CMASS DR9 power spectra before (left) and after (right) reconstruction with the best-fit models overplotted. The vertical dotted li
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 model and dpbest-fit model calculated (including window function convolution) with no BAO. Error bars indicate Cii for the power spectrum and the rms error cfrom fitting BAO to the 600 mocks in the inset (see Section 4.2 for details).

観測・理論双方の系統誤差が結論に影響を与える可能性
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), and so ca
sorbed into the smooth component of the fit, and we thernot explicitly include this term in our fits.
To model the overall shape of the galaxy clusterin
spectrum we use a cubic spline (Press et al. 1992), with nin
fixed empirically at k = 0.001, and 0.02 < k < 0（その影響を考慮すべき、理論テンプレートに取り込むべき）
6.2

Fitting the power spectrum

We fit the observed redshift-space power spectrum, calculated as

理論と観測をつなぐ
線形理論

パブリックなCMBボルツマンコードによる高精度・高速計算 (camb, class,…)

質量密度ゆらぎ :mKodama &Sasaki (’84)

1m

非線形重力進化
非線形な

赤方偏移空間ゆがみ

系統的効果

銀河バイアス観測銀河の個数密度ゆらぎ :

gal

ngal
ngal

1

We
fit <thesecomplications
using
aequalmodel
involving
the amplihalo,
enhancingsmall-scalebias.
(0:380z <matter
0:474),
containing
roughly
numbers
methods (see Appendix A 1) effectively produce anderlyingpower
spectrum
inof boththree‘‘nuisanceparameters’’
Qa nlgalaxy-weighted; k" # as involving
illustrated ingalaxies,
ascomplications
illustrated
in Fig.using
2. !b;
Their
e fit
these
model
thesmoothed version of what we are plotting, with errorudeandshape,
causingbias.
mean
are [29,88],
0.235, 0.342,
and
9.redshifts
Following
we!b;
model
galaxyinbars that are smaller but highly correlated.
reeFig.
‘‘nuisanceparameters’’
Q0.421,;ourk"respectively.
# measured
as illustratednl
We also split
the samplecan
into theseven angularregions dark
Our Fourier
convention is such that the dimensionlessMultiplegalaxies
share
the
same
matter2
g. power
9.illustrated
Following[29,88],our roughly
measured
spectrum
as3, each wein Fig.
againmodel
containing
the galaxy
power " of [77] is given by "2 !k" $ 4!!k=2!"3 P!k".
alo,enhancingsmall-scale bias. 2
number ofwersamespectrumasgalaxies.
It is worth emphasizing that the LRGs
a
Qconstitutenl k
2 1a%model
these
complications
using
involving
the
2;
(4)
Pg !k#and$ uniform
Pdewiggled!k#b sample,
remarkably
cleangalaxycontaining
宇宙論パラメーター解析に
1
%
Q
k
nl
2 1 % 1:4k
; red-illustrated
(4)
P
!k#
$
P
!k#b
the
same
type
of
galaxy(luminous
early-types)
at# allasgdewiggled
nuisance
parameters’’
!b;
Q
;
k
in
線形理論
nl
"
1
%
1:4k
^
!
shifts.
Not
only
is
it
nearlycomplete
(
n!
r
"
#
1
as
men用いられた理論テンプレート
where
first
factorrightside
accounts
Following[29,88],
we
model
our
measuredtioned theabove),
but it on
is theclose
to handvolume-limited
for for the galaxy
変てこ理論
here
the
first
factor
on
the
right
hand
side
accounts
for
the
FIG.
8
(coloronline).
Constraints
on the redshz&
0:38[36,49],
i.e.,
for
our
NEAR
and
MID
slices.
nonlinear
suppressionofbaryonwiggles
and
the
last
factor(Tegmark
etal.’06)
FIG.parameters
8 (color !online).
Constraints on thpectrum
asnlinear suppressionof baryon wiggles and the last factor tortionand rテンプレート
gv . The contours show th

１０年前までは、

accounts for a combination of the nonlinear change of thetortion parameters ! and rgv . The contours
constraints from the observed LRG clustering an
constraints from the observed LRG clustethe(4)
circulardot indicating
the best-fit
values.value
The d
the
circulardot indicating
thebest-fit
SDSS LRG
thethecompletely
independent!-estimate
inferredinf
frcompletely
independent!-estimate
sissisof theWMAP3and LRG
powerpower
spectraspectra(it putsof theWMAP3and LRGhefirstfactorthehandaccounts
but, but
has hasbeenbeenplottedat rgv at$ r1).
rgv ,rgv
summarize
the processveryright
briefly here.
Thesidefirst stepis!k#; (5) foronthe
plottedon
gv $ 1).
$on
W!k#P!k#W!k#(PP Pdewiggled!k# !k#$ W!k#P!k#%%&1&1''W!k#(P!k#;(5)
nowiggle

SPECTRUM
countsIII.
forPOWER
a combinationof MEASUREMENTS
the nonlinear change of the2
globalmatterpower
spectrumshape
shape1andandscale-dependent
%
Q
k
obal
matterpower
spectrumscale-dependentnl
We measure the power spectrum of our2 various samples
P
!k#
$
P
!k#b
bias
of
the
galaxies
relative
to
the
dark
gthe PKL
dewiggled
methoddescribed in
Wedarkfollowmatter.
the ; For
as usingof thegalaxies
relativeto[28].
thematter.
For
1
%
1:4k
of [28]exactly,withsome additional
Pprocedure
!k#
we
adopt
the
prescription[88]numericaldewiggled
!k#
we
adopt
the
prescription[88]
dewiggled
improvements described in Appendix A, so we merely

to adjust the galaxy redshifts slightly to compress so-called
FIG. 8 (color online). Car suppressionofbaryonwiggles
and
the
last
factorfingers-of-god (FOGs), virialized
galaxy clusters that ap2 =2
'!k=k
#
'!k=k
#
=2
"
tortion parameters ! and rg
where
W!k#
e the line-of-sight
andP
P
!k#
denotes
the
‘‘no
here
W!k#
) e)alongand!k#denotes
the
‘‘no
pear
innonlinear
redshiftspace; wenowiggle
nowiggles foraelongatedcombinationofthechangeofthez=0.35
線形理論
do
this
with
severaldifferent
thresholds
andreturntoillustrated
how
SDSS Main from the obser
constraints
ggle’’
power
spectrumdefinedin[89]
andillustratedin
wiggle’’power
spectrumdefinedin[89]
inmatterpower
spectrumshapeandscale-dependent
this affects the results in Sec. IV F 2. The LRGs are not justg. Fig.
9.brightest
In9. otherwords,!k#
simplyweighted
the circular dot indicating t
Inclusterotherwords,Pdewiggled
Pabout!k#ofisisthemsimplyweightedgalaxies;
20%
appearaa todewiggledthegalaxies
relative
to
the
dark
matter.
For：フィッティングパラメーター
residea darkmatterhalo
withone or moreandotherthe
LRG’s.
erage
ofinthelinearpower
spectrumthewiggle-free
the completely independen
averageofthelinearpower
spectrumandwiggle-freeThewesecondstep is tothe
expandthe three-dimensional
galaxy!k#
adopt
prescription[88]
edFIG. 4 (color online). Measured
power
spectra for thesis of the
WMAP3
andfullLRG
dewiggled

nowiggle"

2

density field in N three-dimensional functions termed
LRG and main galaxy samples. Errors are uncorrelated and
PKL-eigenmodes, whose variance and covariance retain
plotteonshownrgvin, but
full window functions are
Fig. 5.has
The beensolid curves
ggledessentially all the information about the k < 0:2h=Mpc
nowigglecorrespond to the linear-theory #CDM fits to WMAP3 alone
power spectrum from the galaxy catalog. We use N $from Table 5 of [7], normalized to galaxy bias b $ 1:9 (top) and!k#引用件数$ W!k#P!k#% &1 ' W!k#(P!k#; (5)1000以上これからの精密観測では許されない

受賞の対象となった業績
宇宙の大規模構造の重力進化計算に対する✓ 摂動論的計算手法の開発・発展• 再和法による摂動計算の精度向上・高速• 修正重力理論での計算方法• 赤方偏移空間ゆがみの理論モデル• ニュートリノに質量がある場合の取り扱い✓ 銀河サーベイの理論テンプレートの作成・観測への応用

大規模構造研究の理論ツール
線形理論を越えて、冷たい暗黒物質が優勢の宇宙の構造形成を取り扱う代表的手法
TARUYA et al.
(その派生・発展版にフィッティング公式、ハローモデルなど)

宇宙論的N体シミュレーション

摂動論による解析計算

TARUYA et al.

pi = ma2

pi=
dt

PHYSICPHYSIC

dxi
dt

Gm2
a

N

j=i

xi
|xi

xj
xj |3

(i = 1, 2, · · · , N )http://www.projet-horizon.fr/

AT et al. (’12)

104

パワースペクトルの非線形重力進化103

z = 0.00

z = 0.00

N体シミュレーション
10
10
2
4

線形理論
2
Δ2 (k) )(k)Δ2=(k)k 3 P (k)/(210

2

無次元パワースペクトルz = 0.98
z = 3.05

z=03
1

z=0.98z=3.0510z=7.0210-11
10

z = 14.87

z=14.8z = 14.87

z = 3.05

弱非線形

z = 7.02

10-20
10

線形

-1
10-3

z = 14.87

Springel et al. (’05)
1.00 0.10
k [ h / Mpc ]10.001.00
100.0010.00
1k [ h / Mpc ]波数 k [h Mpc-3（強）非線形z = 0.98
z = 7.02

z = 7.02

2
0

10
10
0.10 0.01

z = 3.05

z = 0.00

-4
-2

0.01

z = 0.98]

100.00

104

パワースペクトルの非線形重力進化103

z = 0.00

z = 0.00

N体シミュレーション
10
10
2
4

線形理論
2
Δ2 (k) )(k)Δ2=(k)k 3 P (k)/(210

2

無次元パワースペクトルz = 0.98
z = 3.05z=0

3
1z = 0.98
z = 7.02

z = 7.02

20

10

z=7.0210-11
10

z = 14.87

z=14.8z = 14.87

z = 3.05

弱非線形

z = 7.0210-20
10線形理論 線形-1
10-3

z = 14.87

Springel et al. (’05)
1.00 0.10
k [ h / Mpc ]10.001.00
100.0010.00
1k [ h / Mpc ]波数 k [h Mpc-3

by T. Nishimichi（強）非線形

z=0.98z=3.0510
10
0.10 0.01

z = 3.05

z = 0.00

-4
-2

0.01

z = 0.98]

100.00大規模構造の摂動論
ゆらぎの波長 << ハッブル半径
弱い重力 → ニュートン重力冷たい暗黒物質(CDM) + バリオン

圧力ゼロの渦なし流体(single-stream approx.)

Juszkiewicz (’81),Vishniac (’83), Goroff et al.
(’86), Suto & Sasaki (’91), Makino, Sasaki &Suto (’92), Jain & Bertschinger (’94), ...
standard PT
| |

1

=

(1)

+ (2) + (3) + · · ·(k; t) (k ; t) = (2 )3 D (k + k ) P (|k|; t)(6)(6) andand (7),(7), we
we expandexpand thethe quantities
quantities δδ andand θθ as
as

摂動計算をちょっとだけ

(1)
(2)
(1)
(2)
δ(k;
δ(k;t)t) == δδ(1)
(k;
(k;t)
t) +
+ δδ(2)
(k;
(k;t)
t) +
+ ······ ,, θ(k;
θ(k;t)t) == θθ(1)
(k;
(k;t)
t) +
+ θθ(2)
(k;
(k;t)
t) +
+ ······ ,,rticularly
rticularly interested
interested inin thethe late-timelate-time evolution
evolution dominateddominated byby thethe growing
growing modemode22,, thtee expressed
expressed
asas
n次の摂動解初期密度場（確率場）
!! 33
デルタ関数
33
d
d
k
k
·
·
·
·
·
·
d
d
kknn
1
1
(n)
(n)
δδ (k;(k;t)t) ==δδDD(k(k −− kk12···n)Fnn(k(k11,,······ ,,kknn;; t)t)δδ00(k(k11))······δδ00(k(knn),),
12···n)F3(n−1)
3(n−1)
(2π)
(2π)
!! 33
dd kk11 ······dd33kknn
k1
(n)
(n)
摂動論カーネルθθ (k;
(k;t)
t) =
=
δ
δ
(k
(k
−
−
k
k
)G
)G
(k
(k
,
,
·
·
·
·
·
·
,
,
k
k
;
;
t)
t)
δ
δ
(k
(k
)
)
·
·
·
·
·
·
δ
δ
(k
(k
),
D
D
12···n12···n nn 11
nn
00 1F
13
00 nn),
3(n−1)
3(n−1)(2π)(2π)
パワースペクトル
k

k2

andom
andom
initialinitialdensity
field.
field.
Then,
defining
defining
theoperator ofof thethe matrix
matrix form
form (here
(here
is
(k;
t) (k density; t) = (2)3 Then,(k + k) P (|k|;thet) operatorD
k3 aa isP (k) = P (11) (k) + P (22) (k) + P (13) (k) + · · ·

⎛
⎛

⎞
⎞
dd
aa
11
初期パワースペクトル
ダイアグラム表現
⎜
⎜ da
⎟
⎟
P13
da
P Plin
P P22
P
P
P
P
⎜
⎜
⎟
⎟
(13)
(22)
(11)
P
P
"L(k)
" (k) ≡
⎜ &
⎟
⎟,,
P
(
)
)
P (
P
L(k)
≡⎜
P
0
'
'22
&
& P'
'P22 P
&
&
'
'
⎜
⎜ &P
⎟
⎟
P
22
P
0
H00
Ω
Ωm,0
ccsskkdd
ḢḢ
ccvvkk⎝⎝ 33 = H
⎠
⎠…
m,0
+
+
+
F
F
P P P
F
P
F
F
F
F
F
FP3 +
P
−
−
+ 22 +
+
+P
P0 aa
PF2
F
P
P F2 +
k
0
3
3
2
2
P
k
k
k
k
k
0
k
22 H(a)
H(a)
aa
aH
aH
da
da
H
H
aH
aH k
lin

lin

22

22

13

13

lin

lin

lin lin

2

lin

lin2

22

lin

lin

lin

lin

2

2 3

lin

lin

lin

3

3lin

lin

Figure 2: Tree-level and one-loop power spectrum.

Figure2: Tree-leveland one-loop
powerpower
spectrum.
Figure2: Tree-leveland one-loop
spectrum.
leading ordernext-to-leading
uations
uations forfor thethe kernels
kernels
F=nnk andand
G. nnForarearewrittenwrittenaswhere kF+...k G
us it willturn out toas
be important thatorder（
also for F and 1ループ）
where
k
=
k
+
.
.
.
k
.
For
us
it
will
turn
out
to
be
importantthatof also
F forandF and(線形or
F a where
similarscalingone
thek =ツリー)kholds+ .when. . k the. sum
Forofustheitarguments
will turnremains
out tofinite
be while
importantthatfor
also
1

n 2

1

2

n 2

2

先駆的研究パワースペクトルに対する解析的表式（1ループ）
PHYSICAL REVIEW LETTERSVOLUME 66, NUMBER 3

Quasinonlinear

Theory of Cosmological Self-Gravitating

21 JANUARY 1991

Systems

Yasushi Suto and Misao Sasaki

for Theoretical Physics, Kyoto University,

Uji Research Center, Yukawa Institute(Received 18 June 1990)

Suto & Sasaki (’91)
Makino, Sasaki & Suto (’92)
PHYSICAL REVIEW( D

Uji 6l I, Japan

Nonlinear eflects of self-gravitating systems in cosmology are considered on the basis of perturbation
theory. In particular, we examine several cases in which evolution of the power spectrum of density fluctuations can be analytically calculated. In some cases, nonlinearity suppresses the growth of fluctuations
relative to linear theory, and the power transfer via nonlinear mode coupling is sensitive to the specificshape of the underlying fluctuation spectrum. The result is in good agreement with recent numericalsimulations.
PACS numbers:

98.60.Mp, 05.45.+b, 98.80.Dr

The dynamics of self-gravitating systems is of fundaeffect, artificial two-body relaxation, and so on. Hence,
2
JULY 1992
NUMBER
VOLUME 46,
mentalin astrophysics.
In particular, 15 nonit is of great value to check the numerical simulation andimportancelinear growth of cosmological density Auctuations is theits validity by other analytical methods. In this paper,
first and mainin the formationand evo-

we present a quasinonlinear
perturbation analysis, following the formalism developed by Juszkiewicz
andVishniac,
to see the weakly nonlinear effect on thecosmological density fluctuations.
We found that thephysical processof nonlinear
gravitational fluctuations
Analytic approach to the perturbative expansion
lution of the large-scale structure in the Universe. If the
fieldsin cosmological density
velocity
cosmicstructure
originated from quantum flucpresentand

tuations in the inAationary epoch, their statistical properties Makino
are fairly well specified: random Gaussian density
Nobuyoshi
scale-invariant
withPhysics,
field
power Ujispectrum
Theoretical
611, Japan[P(k) ~k].
Research
YukawaInstitute
Kyoto Uniuersity,
Center,forUjiquantitatively
Hi-roshima
724, Japan describes the subUniversity, linear
Higashitheory
and Department of Physics, Hiroshima Althoughsequent evolution of Auctuations, there is no established
Sasakito follow their nonlinear evolution. ConventionMisaomethod
N-body simulations have been employed to explore
ally,
Department of Physics, Kyoto Uniuersity, Kyoto 606, Japanthe nonlinear evolution of the cosmic structure. In thisSuto' and later Suginohara et al.
numerical
Yasushi Suto approach,found that the nonlinear growth rate sensitively depends
Uji Research Center, Yukawa Institute for Theoretical Physics, Kyoto Uniuersity, Uji 611, Japanon the overall shape of the Auctuation spectrum. It is(Received 5 February 1992)
not clear, however, to what extent the simulations faithactual large-scale
structureformation
describefully areseriesinas perturbation
solved theEquations of self-gravitating systems in the Universeby expanding
thediscretization
inthe
Universe,topossibleowing
Wefields.
apare
for
and
velocity
Fourier space. The formulas for the higher-order terms
densitygivenhigher-order perturbations can be analytically integrated
for power-law-type spectra of density fluctuations. The
results clearly illustrate the role of primordial spectrum
shape on the subsequent nonlinear evolution of cosmological gravitating systems, as suggested by the earlier
numerical works.
Expand the density fluctuations
as a perturbation
series:'= 8)(k, t)+&z(k, t)+&3(k,&(k, t) —t)+where the expressions for the above perturbations can befound in Ref. 4. To second order, the spectrum reduces~摂動論は使えない！...？
CDMスペクトル

定性的にしかシミュレーション

z=0のふるまいと一致しない！？

摂動論

当時のシミュレーションは、摂動線形理論論が適用できる領域で十分な精度・

z=1
シミュレー
ション

線形理論

摂動論

解像度がなかった
ボックスサイズ：100Mpc
粒子数： 144

3

Jain & Bertchinger (’94)

また、そんな大スケールの観測もなかったバリオン音響振動の検出！
dk k 2sin(ks )２点相関関数 ξ ( s) = ∫P( k )
2
2π
ks

2.5m
スローンデジタル
スカイサーベイ
43,000個の明るい銀河 (LRG)@ z=0.3
Typicalsize

Eisenstein et al. (’05)

バリオン音響振動• 宇宙晴れ上がり前のバリオン-光子流体の痕跡(

宇宙マイクロ波背景放射に現れる音響振動)• 振動スケール は「標準ものさし」になる→ 遠方宇宙の宇宙膨張診断（加速膨張の起源に迫る手がかり）
角径距離 zz (=0~3)z=1100

音響振動スケール

宇宙膨張の変化は距離と遠方銀河
赤方偏移の関係に影響

宇宙マイクロ波背景放射

日本でも宇宙論観測を
すばる望遠鏡を準占有化することで本格的な宇宙論のサーベイ観測を行う機運が生じた
FastSound プロジェクト (2012年~2016年)
SuMIRe プロジェクト (2014年~)
サイエンス検討会→「 観測の精度が上がるなら理論の精度も何とかしないと」
周りにも興味を持つ学生が集まりだした（当時、東大所属）

摂動論の再生

Matter Power Spectrum

3n-linear Matter Power Spectrum

3

バリオン音響振動に対する非線形進化の記述に使える！
摂動論(1-loop)
z=1

線形理論

z=6
摂動論
シミュレーション
線形理論

Jeong & Komatsu (’06)

Fig. 1.— Power spectrum at z = 1, 2, 3, 4, 5 and 6 (from top
to bottom), derived from N -body simulations (dashed lines), perturbation theory (solid lines), and linear theory (dot-dashed lines).
We plot the simulation data from 512, 256, 128, and 64 h−1 Mpc
simulations at k ≤ 0.24 h Mpc−1 , 0.24 < k ≤ 0.5 h Mpc−1 ,0.5 < k ≤ 1.4 h Mpc−1 , and 1.4 < k ≤ 5 h Mpc−1 , respectively.
Note that we did not run 64 h−1 Mpc simulations at z = 1 or 2.

これはおもしろそうだ！
Fig. 3.— Non-linearity in baryonic acoustic oscillations. All
シミュレーションの精度は？
摂動論も改良の余地あり？
the3 Power
power
spectradividedby aoscillations.
smooth powerg.f Fig.
1.—
at z =have
1, 2,in3,been
4, 5 and
6 (fromtop
3.—spectrum
Non-linearity
baryonic
acoustic
All摂動論の改良
精度の向上と適用範囲を広げたい
流体方程式にもとづく摂動計算だから流体力学を学べ
The Astrophysical Journal, 674:617–635, 2008 February 20# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.

AT & Hiramatsu (’08)

A

A CLOSURE THEORY FOR NONLINEAR EVOLUTION OF COSMOLOGICAL POWER SPECTRA
Atsushi Taruya and Takashi Hiramatsu
Research Center for the Early Universe ( RESCEU ), School of Science, University of Tokyo, Tokyo 113-0033, Japan
Received 2007 August 23; accepted 2007 November 12

ABSTRACT
We apply a nonlinear statistical method in turbulence to the cosmological perturbation theory and derive a closed
set of evolution equations for matter power spectra. The resultant closure equations consistently recover the one-loop
results of standard perturbation theory, and beyond that, it is still capable of treating the nonlinear evolution of matter
power spectra. We find the exact integral expressions for the solutions of closure equations. These analytic expressions
coincide with the renormalized one-loop results presented by Crocce and Scoccimarro apart from the vertex renormalization. By constructing the nonlinear propagator, we analytically evaluate the nonlinear matter power spectra basedon the first-order Born approximation of the integral expressions and compare it with those of the renormalized perturbation theory.
Subject headingg
s: cosmology: theory — dark matter — large-scale structure of universe
Online material: color figures

パワースペクトルの非摂動な発展方程式を導出

乱流の統計理論で知られる近似法をもとに、モーメント
連鎖方程式から閉じた方程式系を得る（完結近似）

完結近似
Closure Approximation (CLA)
従来の摂動論 P P
Plinlin

AT & Hiramatsu (’08)

lin

P22

AT & Hiramatsu,
P13 674, 617 (2008)
P13 ApJ

P22 P22

P13

(13)
P
P
P P
P
P
P0
Self-consistent treatment tocompute
P
P
P
P0
+and
=
+
both
non-linear
spectrum
F power
P P P
P
F
F
FFP
F F
F propagator
FP
P
P
PF
F
P
F
0
P0
2
2
3
k
k
k
k
k
k
(11)

P (k)

(22)

lin

lin

lin

lin

k

2

lin

lin

22

lin2

lin

lin

lin

3

3lin

lin

lin

2

2 3

lin

k

lin

+…

2: Tree-level and one-loop power spectrum.
Figure2: Tree-leveland one-loop
power
spectrum.
Figure2: Tree-leveland one-loop
power
spectrum.
Vertex function: Figure
Lowest-order
evaluation(tree
approx.)where k = k + . . . k . For us it will turn out to be important that also for F and
非摂動量（プロパゲーター）を用いて展開をk =. .+. k.when
For
us of
itusthewill
turn turnoutremains
toimportantthat
forandF andFwherea wheresimilarscaling
oneof alsothek k= +
kholds. . k. the
. sum
For
itarguments
willoutbetofinite
be while
importantthattwo-loop
alsoF forSelf-energy:
Truncationofhigher-loop
corrections
than
1

n 2

1

3

2

n 2 n 21

2

2

momenta
toscaling
infinity,
i.e.holds
F3 F
a similar
holds
when
the sumof theremains
finite finite
while while
one ofone
the of theagoes
similarscalingwhen
the sumof arguments
the arguments
remains3

書き直して、１ループレベルで打ち切る
Closedsystemk
momentagoesgoesto infinity,
i.e. i.e.
momentato infinity,

2

パワース
ペクトル
プロパ
ゲーター

lim F2 ( q, q + k) / lim F2 ( q + k1 , q + k2 ) /

q2

,

2
k1-loop(2.20)
k2
lim lim
F2 ( Fq,( q q,
+q
k)+/
lim
F
(
q
+
k
,
q
+
k
)
/
,
2
1
2
2
k)
/ lim F2 ( q + k1 , q + k2 )q 2/ 2 ,
k q!1
q!1
2
q!1
lim F ( q!1q, q + k , k ) /
,
q

q!1

q!1

3

t

1

2

q2

+2

t
tʹ
k2
k 2 same order.
limtheF3momenta( q, q +k1k⇠
k22 ⇠) k/are 2of, thewhere we assumed that
1, k
lim F3 ( q, q + k1 , k2 ) q/ 2 ,Pab (k ; t , t ʹ) q!1q!1q
2.2 The bispectrum in SPT
1-loopwhere we assumed that the momenta k1 ⇠ k2 ⇠ k are of the same order.
whereassumedthethemomenta
k1the⇠ equations
k2 ⇠ k areof the andsameorder.
Let usfor thewe
momentfocusthat
only onSPT part ofof motion
postpone

t

q!1

tʹ

tʹ

(2.20)(2.20)+4

detailed
discussion
stress tensorʹ to Secs. 3 and 4. The
two- andThe
bispectrumin SPTʹ of the te↵ective
t a 2.2
tbispectrum
s1 we
s2three- t ʹtt
Thecorrelators
in SPT field are the quantities
point2.2
connectedof the stochastic
thatwill considerʹIn) moment
GLetkpaper.
| t , ttheforfocustheonly
on thepart ofarethe
equationsof motion and postpone
inthis
Fourier space,
powerand SPT
bispectrum
definedasab (usLet us forthe moment
only stresson thetensor
SPT partof the3 equationsof twomotion
postponea detailed
discussionof thefocuse↵ectiveto Secs.
and 4. The
andandthree(3)3a detailed
discussion
the
to
3 and
4.
The
and threeh (k1 , a) of
(k2of
, a)i
⌘stochastic(2⇡) D stress(k1field+ ktensor(k1 ,thea) .Secs.
(2.21)
2 ) Parepoint
connected
correlatorsthee↵ective
quantities
thatwewilltwoconsider
point
connected
correlatorsstochastic
field are are
the definedquantities
in thispaper.
In Fourier
space, ofthethepowerand bispectrum
as that we will consider
andin this paper. In Fourier space, the power- and bispectrum are defined as

これを時間発展方程式として定式化（非線形な積分微分方程式）
(3)

完結近似
Closure Approximation (CLA)
従来の摂動論 P P
Plinlin

AT & Hiramatsu (’08)

lin

P22

AT & Hiramatsu,
P13 674, 617 (2008)
P13 ApJ

P22 P22

P13

(13)
P
P
P P
P
P
P0
Self-consistent treatment tocompute
P
P
P
P0
+and
=
+
both
non-linear
spectrum
F power
P P P
P
F
F
FFP
F F
F propagator
FP
P
P
PF
F
P
F
0
P0
2
2
3
k
k
k
k
k
k
(11)

P (k)

(22)

lin

lin

lin

lin

k

2

lin

lin

22

lin2

lin

lin

lin

3

3lin

lin

lin

2

2 3

lin

k

lin

+…

2: Tree-level and one-loop power spectrum.
Figure2: Tree-leveland one-loop
power
spectrum.
Figure2: Tree-leveland one-loop
power
spectrum.
Vertex function: Figure
Lowest-order
evaluation(tree
approx.)where k = k + . . . k . For us it will turn out to be important that also for F and
非摂動量（プロパゲーター）を用いて展開をk =. .+. k.when
For
us of
itusthewill
turn turnoutremains
toimportantthat
forandF andFwherea wheresimilarscaling
oneof alsothek k= +
kholds. . k. the
. sum
For
itarguments
willoutbetofinite
be while
importantthattwo-loop
alsoF forSelf-energy:
Truncationofhigher-loop
corrections
than
1

n 2

1

3

2

n 2 n 21

2

2

momenta
toscaling
infinity,
i.e.holds
F3 F
a similar
holds
when
the sumof theremains
finite finite
while while
one ofone
the of theagoes
similarscalingwhen
the sumof arguments
the arguments
remains3

書き直して、１ループレベルで打ち切る
Closedsystemk
momentagoesgoesto infinity,
i.e. i.e.
momentato infinity,

2

パワース
ペクトル
プロパ
ゲーター

lim F2 ( q, q + k) / lim F2 ( q + k1 , q + k2 ) /

q2

,

2
k1-loop(2.20)
k2
lim lim
F2 ( Fq,( q q,
+q
k)+/
lim
F
(
q
+
k
,
q
+
k
)
/
,
2
1
2
2
k)
/ lim F2 ( q + k1 , q + k2 )q 2/ 2 ,
k q!1
q!1
2
q!1
lim F ( q!1q, q + k , k ) /
,
q

q!1

q!1

3

t

1

2

q2

+2

t
tʹ
k2
k 2 same order.
limtheF3momenta( q, q +k1k⇠
k22 ⇠) k/are 2of, thewhere we assumed that
1, k
lim F3 ( q, q + k1 , k2 ) q/ 2 ,Pab (k ; t , t ʹ) q!1q!1q
2.2 The bispectrum in SPT
1-loopwhere we assumed that the momenta k1 ⇠ k2 ⇠ k are of the same order.
whereassumedthethemomenta
k1the⇠ equations
k2 ⇠ k areof the andsameorder.
Let usfor thewe
momentfocusthat
only onSPT part ofof motion
postpone

t

q!1

tʹ

tʹ

(2.20)(2.20)+4平松尚志
detailed
discussion
stress tensorʹ to Secs. 3 and 4. The
two- andthreeThe
bispectrumin SPTʹ of the te↵ective
t a 2.2
t
s
s
t
ʹ
t
t
1
2
Thecorrelators
bispectrumin SPT field are the quantities that we will consider
point2.2
connectedof the stochastic
ʹ
G
(
k
|
t
,
t
)
for thefocustheonly
on thepart ofarethe
equationsof motion and postpone
inLet
this
paper.
In moment
Fourier space,
powerand SPT
bispectrum
defined
as
ab us

（立教大）

Let us forthe moment
only stresson thetensor
SPT partof the3 equationsof twomotion
postponea detailed
discussionof thefocuse↵ectiveto Secs.
and 4. The
andandthree(3)3a detailed
discussion
the
to
3 and
4.
The
and threeh (k1 , a) of
(k2of
, a)i
⌘stochastic(2⇡) D stress(k1field+ ktensor(k1 ,thea) .Secs.
(2.21)
2 ) Parepoint
connected
correlatorsthee↵ective
quantities
thatwewilltwoconsider
point
connected
correlatorsstochastic
field are are
the definedquantities
in thispaper.
In Fourier
space, ofthethepowerand bispectrum
as that we will consider
andin this paper. In Fourier space, the power- and bispectrum are defined as

これを時間発展方程式として定式化（非線形な積分微分方程式）(3)

これはいける！

Fig. 3. Same as figure 2, but we correct the effect of the finite volume. We truncate the expansion of equation (11) at the first term. Th
equation (13).

ボルン近似で解析計算した結果とシミュレーションを比較
z=3

従来の摂動論(1ループ）

西道啓博（カブリIPMU）
z=1

完結近似

シミュレーションの精度と収束性を入念にチェック

z=0
線形

従来の摂動論よりシミュ
レーションとよく合う！
Nishimichi, Shirata, AT,
Yahata, Saito, Suto et al. (’09)

けっこういける！

CHASING THE NONLINEAR EVOLUTION

高次のオーダーまで入れるとさらに適用範囲が広がった

NONLINEAR EVOLUTION OF BARYON ACOUSTIC . . .

PHYSICAL REVIEW D 80, 123503

解析計算

1OF MATTERCHASING THE NONLINEAR EVOLUTIONCHASINGTHE NONLINEAR...
EVOLUTION OF
PHYSICA
MATT

z =3

1

0.001
0.01

Numerical
Analytic(1st Born)
Analytic(2nd Born)
N-body0.1

k [h/Mpc]

z=1

z=3
z=1

3
2

P11(k){D(zinit)/D(z)} [Mpc/h]

2
0.1

0.01

1

z=3

3
2

P11(k){D(zinit)/D(z)} [Mpc/h]

P11(k){D(zinit)/D(z)} [Mpc/h]

3

数値計算
0.1
0.1

0.01
0.01

Numerical
Numerical
Analytic(1stBorn) Born)
Analytic(1stAnalytic(2nd Born)
Analytic(2nd Born)
N-bodyN-body0.001
0.01
1 0.01

0.001
0.01

0.1
0.1

0.01
11 0.01

kk[h/Mpc][h/Mpc]0.1

Hiramatsu&ofAT
FIG. 3 (color online). Time evolution ofFIG.
matter3 (color
power
online).
spectra,TimeP11 ðkÞ,evolutionatmatterz(’09)
¼ 3,
power
1 and
s
k evaluated[h/Mpc]vertical axis is normalized by the linear growthverticalrate,
axis DðzÞ.
is normalized
The solidbylines
the represent
linear growth
the numericalrate, DðzÞAT,
Nishimichi,
Saito&
Hiramatsu(’09)
dashed and dotted lines are the analyticdashedresultsand
includingdotted lines
the corrections
are the analytic
up to results
the firstincludin
and s

群雄割拠
ほぼ同時期に発表された摂動計算の改良方法（2006~2008年）
Renormalized Perturbation theory
R. Scoccimarro

M. Crocce

Lagrangian resummation theory
Time-RG approach

Multi-point propagator expansion
F. Bernardeau（現IAP所長）

T. Matsubara

M. Pietroni

Large-N expansion
P.Valageas

比較的早い時期に出版し、シミュレーションをもとにその有効性を示したこともあり、我々の論文も認知されるようになった

群雄割拠招へいしてもらったきっかけでほぼ同時期に発表された摂動計算の改良方法（2006~2008年）
共同研究が始まる
Renormalized Perturbation theory
R. Scoccimarro

M. Crocce

彼の提案した定式化で摂動計算
Lagrangian
resummation theoryの高速化するアイデアを思いつく
計算コードをパブリック化
Time-RG→
approach
T. Matsubara

Multi-point propagator expansion
F. Bernardeau（現IAP所長）

計算時間が劇的に短縮
M. Pietroni（数時間が数秒に）http://www2.yukawa.kyoto-u.ac.jp/~atsushi.taruya/regpt_code.html
Large-N expansion

AT, Bernardeau, Nishimichi & Codis
P.Valageas(’12)

比較的早い時期に出版し、シミュレーションをもとにその有効性を示したこともあり、我々の論文も認知されるようになったいまもうひとつの問題
重力進化の記述だけでは観測に応用できない
赤方偏移空間ゆがみ (Redshift-space distortions, RSD)
分光観測で銀河の赤方偏移を決定する際、銀河の固有運動によるドップラーシフトが混入して見かけの位置がずれる
実際の位置

観測される
見かけの位置(⌅v · ẑ)⌅s = ⌅r +ẑ ;Redshift-space distortions
tortionsa H(z)（赤方偏移空間）z-spacer-spacez-spacer-space

Kaiser Effectr-spacepeculiar velocity

large-scale coherent motion→ enhancement of clusteringk⊥ random motion
small-scale→ suppression of clustering
marro’04

divergence:ion: σv: 銀河の速度場: 観測者の視線方向観測される大規模構造

k||
μ=1

Finger-of-God
Effect
μ=0

{

v
ẑ

が非等方に
μ=0

k⊥

e.g., Scoccimarro’04

観測の解釈がより複雑に

streaming model

視線方向vel. dispersion: σv

シミュレーション：西道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.

いまもうひとつの問題
重力進化の記述だけでは観測に応用できない

赤方偏移空間ゆがみ (Redshift-spacedistortions, RSD)
銀河２点相関関数
分光観測で銀河の赤方偏移を決定する際、銀河の固有運動に
観測者の視線方向
よるドップラーシフトが混入して見かけの位置がずれる
実際の位置

観測される
見かけの位置(⌅v · ẑ)⌅s = ⌅r +ẑ ;Redshift-space distortions
tortionsa H(z)（赤方偏移空間）z-spacer-spacez-spacer-space

Kaiser Effectr-spacepeculiar velocity

large-scale coherent motion→ enhancement of clusteringk⊥ random motion
small-scale→ suppression of clustering
marro’04: 銀河の速度場: 観測者の視線方向観測される大規模構造

k||
μ=1

Finger-of-God
Effect
μ=0

{

v
ẑ

が非等方に
μ=0

k⊥

e.g., Scoccimarro’04

観測の解釈がより複雑に

streaming model

Samushia
et al.(’13)
視線方向vel. dispersion: σdivergence:ion: σvv

シミュレーション：西道

逆転の発想線形Kaiser 公式 (Kaiser ’87)(S) ~(k) = (1 + f µ ) (~k)
2

赤方偏移空間の密度場
コヒーレント
視線方向
観測者

実空間の密度場

重力由来の密度ゆらぎの成長率

な運動

f (z)
四重極非等方性

波数ベクトルと視線µ :方向の方向余弦d ln D+ 線形成長因子⇥ { m (z)}d ln aスケール因子

赤方偏移空間ゆがみから成長率パラメーター ‘ f ’ を決定することで宇宙論的スケールで重力のテストができる！
e.g., Linder (’08); Guzzo et al. (’08);Yamamoto et al. (’08); Percival & White (’09)

バリオン音響振動の観測とともに重要なサイエンスゴールに

神の指

多重極
展開

Redshift-space distortions
Redshift-space distortions(S)

P (S) (k, µ)z-space=Pr-space=even
Kaiser Effect

z=1

`=0

(k) P (µ) 赤方偏移空間
r-spacez-spacepeculiar velocityk||r-spacez-spacer-space
実空間

μ=1

large-scale coherent motionk||peculiar
velocity→ enhancement of clustering
Kaiser Effectμ=1
large-scale coherent motion
Finger-of-God Effectμ=0k⊥
small-scale random motion → enhancement of clustering→ suppression of clusteringFinger-of-God
e.g., Scoccimarro’04
Effectμ=0k⊥
small-scale random motion
streaming model→ suppression of clustering
e.g., Scoccimarro’04

減衰！ 線形理論

vel. divergence:vel. dispersion: σvvel. divergence:streaming model
vel. dispersion: σv

線形Kaiser公式の非線形拡張
減衰！
`=2版に減衰項をつけた半現象論モデルにもとづく摂動計算

何かが足りない2ループ
1ループ

赤方偏移空間歪みの理論モデル
赤方偏移空間のパワースペクトルの厳密な表式は知られていた：P (S) (k) =d3 x eik·x e ikµf

uz

{ (r) + fz uz (r)} { (r ) + f

z uz (r )}uz :規格化された視線方向の速度場

結構ややこしいuz ⌘ uz (r)x⌘ruz (r0 )r0

いろいろ考えた末、半現象論モデルを改良する方針に
長波長展開をして補正項を導出：
P(S)(k, µ) = e(kµf ⇤v )2

減衰項

P (k)

AT, Nishimichi & Saito (’10)

2f µ2 P ⇥ (k) + f 2 µ4 P⇥⇥ (k) +A(k, µ) + B(k, µ)

非線形 Kaiser 項

TNSモデルと呼ばれるようになった

高次モーメント由来の補正項

赤方偏移空間歪みの理論モデル
赤方偏移空間のパワースペクトルの厳密な表式は知られていた：P (S) (k) =d3 x eik·x e ikµf

uz

{ (r) + fz uz (r)} { (r ) + f

z uz (r )}uz :規格化された視線方向の速度場

結構ややこしいuz ⌘ uz (r)x⌘ruz (r0 )r0

いろいろ考えた末、半現象論モデルを改良する方針に
西道啓博
斎藤俊
（MPA）
長波長展開をして補正項を導出： （カブリIPMU）
AT, Nishimichi& Saito (’10)P(S)(k, µ) = e(kµf ⇤v )2

減衰項

P (k)

2f µ2 P ⇥ (k) + f 2 µ4 P⇥⇥ (k) +A(k, µ) + B(k, µ)

非線形 Kaiser 項

TNSモデルと呼ばれるようになった

高次モーメント由来の補正項

1

fiducial

赤方偏移空間歪みの理論モデル

DA,fid
DA

"

2

&1/2

µ2

,

with corrections

0.5

(26)

w/o corrections

0
0.96 0.98 1 1.02 1.04

AT, Nishimichi & Saito (’10)

最初は不満足だったが…
0.9
DA,fid
DA

%
#z=1
2

µ2

`=0

&−1/2

f

"

# %

,

0.85 線形理論
0.8

(27)

1.02

0.98

1

1

補正項あり

0.5

補正項なし

0
0.8

1.02

1.02

0.85

0.9
1

H/HfidH/Hfide the fiducial values
11
0.5
nd Hubble parameter
0.98
0.98
s. For a given set of正解
0
ift-space `power
0.98
1
1.02
0.8
0.85
0.9
0.96
1
= 2 specDA/DA,fid
H/Hff8), but we here treatn addition to the ve成長率パラメーター ‘ f ’imic a practical data
FIG. 9: Results of MCMC analysis using the model o
も正しく決定できる
um, 結構シミュレーション結果を再現
we introduce the1fiducial

赤方偏移空間歪みの理論モデル

DA,fid
DA

"

2

&1/2

µ2

,

with corrections

0.5

(26)

w/o corrections

0
0.96 0.98 1 1.02 1.04

AT, Nishimichi & Saito (’10)

最初は不満足だったが…
0.9
DA,fid
DA

%
#z=1
2

µ2

`=0

&−1/2

f

"

# %

,

0.85 線形理論
0.8

(27)

1.02

0.98

1

1

補正項あり

0.5

補正項なし

0
0.8

1.02

1.02

0.85

0.9
1

H/HfidH/Hfid補正項入り摂動計算（２ループ）
e the fiducial
values1
1
0.5
nd Hubble parameter
0.98
0.98
s. For a given set of正解
0
ift-space `power
0.98
1
1.02
0.8
0.85
0.9
0.96
1
= 2 specDA/DA,fid
H/Hff8), but we here treatn addition to the ve成長率パラメーター ‘ f ’imic a practical data
FIG. 9: Results of MCMC analysis using the model o
も正しく決定できる
um, 結構シミュレーション結果を再現
we introduce the観測への応用
観測グループでも関心が集まり、モデルの有効性が検証され、Cosmological parameters

やがて実データへと応用され始めた

密度ゆらぎの成長率
f 8

0.7

0.6

TNSモデルが応用
SDSS MGSSDSS LRG

VIPERS
WiggleZ

0.5

Planck 2015 XIII

0.4
6DFGS
0.3

BOSS CMASSBOSS LOWZ

0.2

0.1

0.2

0.3

0.4

PlanckΛCDM

最近のSDSS BOSS DR12でも使われている

0.5

z
赤方偏移

0.6

0.7

0.8

0.9

Beutler, Seo, Saito et al. (’16)

観測への応用
観測グループでも関心が集まり、モデルの有効性が検証され、Cosmological parameters

やがて実データへと応用され始めた

密度ゆらぎの成長率
f 8

0.7

0.6

TNSモデルが応用
SDSS MGSSDSS LRG

VIPERS
WiggleZ

0.5

Planck 2015 XIII

0.4
6DFGS
0.3

BOSS CMASSBOSS LOWZ

0.2

0.1

0.2

0.3

0.4

PlanckΛCDM

最近のSDSS BOSS DR12でも使われている

0.5

z
赤方偏移

0.6

0.7

0.8

0.9

Beutler, Seo, Saito et al. (’16)

=4DA (z)/DA, CDM (z)H(z)/H CDM (z)=2

Downloaded from http://mnras.oxfordjournals.org/ at Kyoto University Library on March 3, 2014

Oka et al.

Planck ΛCDM

Downloaded Downloaded
from http://mnras.oxfordjournals.org/
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from http://mnras.oxfordjournals.org/
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Downloaded from http://mnras.oxfordjournals.org/ at Kyoto University Library on November 26, 2015

White2008),
andweT will
investigate the robustness of our model in
OWSP
3 M O D E L L I N GP T
HE E
MR
U LT
I PEOCLE RA
following reason. The BAO scales and RSD (roughly speaking,
the
ously
through
RSDs
andAP
effect in an unbiased manner. For thisP OW E R S P E C Tdetail
R A in Section 4. The model presented here is based on the perturquadrupole-to-monopole ratio, P0 /P2 ), respectively, well constrainInthis section,
we modelling
briefly review
thetheoreticalmodelofthe multipurpose,a propertheshapeand theamplitudeof thebationtheorycalculation,
andofwe
willseparately
give
prescription
In this section, we briefly review the theoretical model of the multiDA2 /H and fσ 8 free from bias parameters. The degeneracy between
pole
power
spectraused
inOur
theisgoalcosmologicalanalysis.
Our goal is toanisotropic
power analysis.
spectrum
rather(e.g. Padmanabhan
&
on in
how
to computethe multipole
spectra.
pole power spectra usedthe cosmologicalispower
to crucial
in
DA and H is broken by both amplitude and shape information
constrain
thegeometricallinearrate
and geometricalfactors
simultaneWhite
2008),
and wegrowthwill
investigate
the robustnessof ourmodel inconstrain the linear growthrateandfactors
simultanethe anisotropic power spectrum, which, respectively, depends
on RSDsdetailously throughandAPeffectinanunbiasedmanner.
For
thisously throughRSDs
andmodelAP effectin anhere
unbiasedmanner.
thisin Section
4. Thepresented
is basedon theFor
perturpurpose,&a proper modelling
of
the
shape
and
the
amplitudeoftheb02 H /DA2 and b02 DA H (Percival et al. 2007, 2010; Padmanabhanpurpose,a propermodellingthewillshape
and the give
amplitudeof thebationtheorycalculation,
andofweseparately
prescription
anisotropic power spectrum
is rather crucial (e.g.
Padmanabhan& non-linear gravitational
3.1 Redshift-spacedistortions
andWhite 2008; Anderson et al. 2012).
anisotropic
power
spectrum
rather(e.g. Padmanabhan &to computethe multipole
power
spectra.
White 2008), and weon
willhow
investigate
the robustnessof ourismodelin crucial
growthAlthough we have already presented our main results, itdetail
would
White
and
we
will investigate
in Section 4. The
model2008),
presented
here
is basedon the pertur- the robustness of our model inbe still worthwhile mentioning how robust our modelling isbation
againsttheory calculation,
and
we
will
separately
give
prescription
detail inand
Section
4. The model
presented
here ison theperturRSDs
gravitational
clustering
involve,inbasednature,non-linear
on
how
to
computethemultipole
power
spectra.
different set-ups as a check. Here we go through a similar study
bationtheory calculation, andseparately
give
prescription
andand weit andiswillquite
essentialgravitational
to takea proper3.1 non-GaussianRedshift-spaceeffects,
distortions
non-linear
to what we have done for the mock catalogue. Namely, we comon how toofcompute
thea multipole
power spectra.
account
these forrobust cosmologicalanalysis beyond thegrowthpare constraints using several slightly different set-ups which are
scales.
Since wegravitational
are interested in a large-scale anisotropic
3.1 Redshift-space linear
distortions
and non-linear
summarized in Table 1. The labels of the set-up in Table
1 aregrowthRSDs
andgravitational
clusteringin nature,should
non-linear
clustering at moderately highredshift,involve,
the PT approachworksummarized as follows.
andnon-Gaussian
effects,
and
it
is
quite
essential
totakea
proper2012
well,
and a percent-level
precisionis achievable
PT calculation
RSDs and gravitational
involve,
indistortions
nature,non-linear
3.1 clustering
Redshift-spaceandnon-linearwith
gravitational−1analysis beyond theandnon-Gaussian
effects,
and
it
is
quite
essential
totakea
properaccount
of
these
for
a
robustcosmologicalinweakly non-linear regime k ! 0.2 (h Mpc ).
(i) Planck-z03. The fiducial model described in the above. The
growthaccount of these forlinear
a robust
cosmological analysis
thewe RSDs.
arebeyondinterested
a large-scale
redshift at which we evaluate the model power spectra is z linear= 0.3.
Let scales.
us first Since
considerIt is wellinknown
that theanisotropic
clustering
scales. Since we are interested in a large-scale anisotropic
RSDs
andgravitational
clustering
involve,innature,non-linear
clusteringatmoderately
highredshift,thePT
approachshould(ii) Planck-z035. Same cosmological model as Planck-z03
but at moderately
statistics
in redshift
spaceshould
are influenced
by the two effects,worktheclusteringhigh redshift,
the PT approachwork
andnon-Gaussian
effects,
and
it
is
quite
essential
totakea
proper
well,
and
a
per
cent-level
precisionisachievablewithPT
calculation
and a per cent-level
precisionachievable
withWhile
PT calculation
we evaluate the model power spectra at z = 0.35, whichwell,is norKaiser
and isFoG
effects.
the former comes from the coherent−1).regime
in weakly non-linearin
regimek !of0.2(h Mpcfor
account
thesearobustbeyondthe).amplitude,
weaklynon-lineark!
0.2clustering(h Mpc−1analysis
mally quoted as the effective redshift in the FKP-type measurement
motion of galaxies and enhancescosmologicalthethe latterLet us first consider
RSDs.
It
is
well
known
that
the
clustering
linearscales.
Since
weare virializedinterested
a large-scale(Percival et al. 2010).
Let
us first
considerIt is wellinknown
that theanisotropic
clustering
mainlyattributed
to RSDs.
thestatistics in redshiftisspace
are influenced
by thetwo
effects, the random motion of galaxies
clusteringmoderately
highredshift,the PTby
approachshould(iii) WMAP5. We assume the cosmological parameters favoured
statistics
inredshiftareinfluenced
the twosignificantly
effects,work
theKaiser and FoG effects.
Whileinthe
comes
from
the
coherent
sitting
aatformer
halo
and spacesuppresses
the power
spectrum
well,
and
athe
per
cent-level
precisionisachievablePTthe
calculationof galaxies andenhancesclustering
amplitude,
thethelatter
= 0.7,
by the WMAP 5-year result, "m = 0.28, "b = 0.046, hmotion
Kaiser
and
FoG
While
formercomes
fromcoherent
along
the
line
ofeffects.
sight.
Strictly
speaking,
thesewitheffects
cannotbe−1
ismainlyattributedtothevirialized
randommotionofgalaxies).
inweaklynon-linear
regime
k
!
0.2
(h
Mpc
ns = 0.96, σ 8 = 0.8, and z = 0.3, for computing the model spectra.
motionofgalaxiesandenhancestheclustering
amplitude,thelatterseparately,
and throughthe higher order corrections, a tightsitting in a halo andtreatedsuppresses
the power spectrum
significantly(iv) Planck-noAB. Same as Planck-z03, but with the Aalong
andtheB line of sight.
Let
us
first
consider
RSDs.
It
isbewellknownfields
that theismainlyattributedtothevirialized
randommotionofclusteringgalaxies
correlation
betweendensityvelocity
still
plays
an
Strictly speaking,thesetheeffects
cannotandcorrection terms in the RSD model (8) dropped out.
statistics
inredshiftspace
areinfluenced
by
the
two
effects,
thetreated separately, andthroughthehigherorder
corrections,atightsitting
in
a
halo
and
suppresses
the
power
spectrumsignificantly
important role on the scales of our interest. In the present paper,correlation
the density
andvelocity
fieldsWhilestill playsan(v) Planck-cbias. Same as Planck-z03, but with the galaxy
bias between along
KaiserandFoG
theformerthecannot
coherent
theline
ofeffects.
sight.
Strictly
speaking,
these
effects
beamongseveralproposedmodelstoaccountcomesforthefrom
non-linear
RSDs
important
role
on
the
scalesofourinterest.
In
thepresent
paper,(14) being a constant, i.e. b(k) = b0 .
motionof galaxies
andenhancestheclustering
amplitude,
the2011),treatedseparately,
andthehigherorder&
corrections,a lattertight(Matsubara
2008a;for
Reid& White2011;
SeljakMcDonald
among several proposedmodelsto accountthe throughnon-linear
RSDs
ismainlyattributedtothevirialized
randommotionofgalaxies2
correlation
betweendensityand velocity
fields2012
still plays
anWhite the2011;
Seljak &the
McDonald2011),
we &adopt
model
given
by Taruya
et al. (2010,
hereafter
TNS
As is seen from Table 1, Planck-z03 gives the smallest χ(Matsubara, while 2008a; Reid
we adopt the modelsitting
given
by
Taruyaetal.
(2010,
hereafter
TNS
in
a
halo
and
suppresses
the
power
spectrumsignificantly
importantmodel): role on the scales of our interest. In the present paper,the difference is small. The constraints on (f, DA , H) are all consismodel):alongline proposedof sight. Strictly
speaking,
these
cannotbeamongtheseveralmodels toaccount fortheeffects
non-linear
RSDs
tent with each other, excepting Planck-cbias. The bias parametersssµ)separately,= 2008a;
DFoG (kµfσv&) WhiteP σ(k,
treatedandthe2011;higherorder&corrections,a tight
P (k,(Matsubara
ReidthroughSeljakMcDonald 2011),v)
赤方偏移
are in fact more important than the others in order to well fit
to µ)the= DFoG (kµfcorrelation
betweenthedensityandvelocity
fieldshereafter
still
plays
an
we
adopt
model
given
by
Taruyaet al.of(2010,
TNS
monopole. Comparison between WMAP5 and Planck-z03 shows × [PKaiser
Figure8.f ) Comparisons
ourµ;previous
works(k, µ;+ theA(k,f ) +ofB(k,µ;results
ff)],
(8)
(k,
) +withA(k,thoseµ; f ) +B(k, µ;f )],and (8)× µ;[PKaiserimportantrole onscalesofrateour(top),interest.
In diameterthe present
paper,
model):
model
predictions.
Thethelinear
growthangulardistancethat our constraints are not sensitive to choice of the underlying
where σ v is a nuisance parameter, which is related to the oneamongmodelstoasaccountfor
non-linear(middle),andHubbleparameter(bottom)a whichfunction
redshiftarecosmology for the model power spectrum. We thus conclude
that velocity
is function
aproposed
nuisance
parameter,
isoftherelatedto shown.
theRSDs
onewhere
σseveralvThedimensional
dispersion.
DFoG (kµfσsv ) charac−1
(k,
µ)
=
D
(kµf
σ
)
P
=
0.175
(h
Mpc
)
in
filled(red;
colours
are
We
plot
our
results
with
k
FoG
v
max
(Matsubara
2008a;
Reid
&
White
2011;
Seljak&McDonald2011),
our results are robust against such systematics.
terizes the suppressionof the power spectrum
the FoG effect,The
for function DFoG (kµfσ v ) characdimensional
velocitybydispersion.
fortheonlineversion)circles
as well as
results withwhich we adopt theavailable
Gaussian
form:
we
adopt
the
model
given
by
Taruyaetoural.aggressive(2010,
hereafter
TNS
terizesthesuppression
of
the
power
spectrum
by
the
FoG
−1
(k,
µ;
f
)
+
A(k,
µ;
f
)
+
B(k,
µ;
feffect,)], for(8)
×[P
Kaiser
k2max
=
0.205
(h
Mpc
)
in
open
(magenta)
circlestocaveatthesystematic
model):
which
we
adopt
the
Gaussian
form:
(x)
=
exp(−x).
(9)D
FoG
5.2 Comparison with previous works
due to non-linear RSDs. For comparison, we also display an open (purple)

パワースペクトルから加速膨張と重力理論の同時検証を初めて行う (Oka et al. ’14)

予想以上に使われた
わかりやすくて手軽さ・汎用性が受けた（計算コードも公開）
引用数：129 (ADS)• 当時ポスドクでバークレーにいた斎藤くんが、SDSS BOSSチームに関わりデータ解析に用いたことで認知度があがった
Beutler, Saito et al. (’14), Beutler, Seo, Saito et al. (’16)• 西道くんがシミュレーションをもとにモデルの詳細に検討を進めたおかげで有効性・実用性を示せた
Nishimichi & Taruya (’11), Ishikawa, Totani, Nishimichi et al. (’14)

若手の活躍と健闘のおかげ！

摂動論が広げる可能性
精密宇宙論の時代に入り、銀河サーベイから統計精度の高い
大スケールにわたる観測データが得られるようになってきた… 摂動論にもとづく理論計算が本質的役割を果たす時代
紹介しきれなかった他の研究• ニュートリノの影響を取り入れた摂動論テンプレート→観測データへ応用、 ニュートリノ質量和の制限
Saito, Takada & AT (’08, ’09, ‘11)

• 修正重力理論における摂動論 → f(R)重力理論の観測的制限
Koyama, AT & Hiramatsu (’09) AT, Nishimichi, Bernardeau & Koyama (’14)
AT, Koyama, Hiramatsu & Oka (’14) Song, AT, et al. (’15)

日本の観測プロジェクトへの活用に向けてさらに発展

課題と展望観測的応用だけでなく、基礎研究も不可欠•重力進化摂動計算の再考・リノベーション
さらに高次に行くと流体方程式にもとづく摂動計算が破綻（UV問題）
有効理論的アプローチが必要？
より基礎的な記述へ（Vlasove方程式）•銀河バイアスのよりよい理解
質量密度ゆらぎ銀河の個数密度ゆらぎ‘nuisance parameters’ をこえた理解が可能か？

共同研究者のみなさま、今まで楽しく研究ができました。ありがとうございました。そしてこれからもよろしくお願いします。