2017/09/ 20-22
研究会「生物から宇宙までの非線形現象」@ 京大人環宇宙の大規模構造の記述：
観測から理論まで
樽家篤史（基礎物理学研究所）1

阪上さんとの関わり
1995年(D1)

天体核中間発表会で阪上さんと初めて会う
カオスと量子デコヒーレンスの話に興味

1996年(D2)

基研アトム型研究員で京都滞在
インフレーション後の宇宙の再加熱期に起こるパラメーター共鳴について議論

1998年4月~1999年3月 研修員として人環の研究室に所属
日頃接する機会が増えたが、研究までにはつながらなかった
2001年

東大宇宙理論研（佐藤・須藤研） 助教
ついに共同研究の機会が訪れる（→次）2非加法エントロピー
S q = − 1 ∫ d 6τq −1[{p(x, v)} − p(x, v) ]q

q→1: Boltzmann-Gibbs− ∫ d 6τ p(x, v) ln p(x, v)

Tsallis (1988)；Tsallis, Mendes & Plastino (1998)• probabilityp ( x , v)

• escort distribution• normalized q-value
特徴

擬加法性(pseudo-additivity)

s.t.

6
d
∫ τ p ( x , v) = 1q{p ( x , v)}
Pq ( x , v) ≡
q
6
{
}
d
τ
p
(
x
,
v
)
∫

Oi q = ∫ d 6τ Oi Pq ( x , v)

S q ( A, B) = S q ( A) + S q ( B) + (1 − q) S q ( A) S q ( B)3Antonov problem
恒星系力学の古典的問題
自己重力系の熱平衡状態は安定か？
断熱壁
半径：re
質量：M=N×m

エネルギー・質量一定の下でre

エネルギー：E系のサイズが臨界半径を超えると不安定

自己重力N体系

GM 2
rcrit = 0.335(− E )

Antonov (‘62), Lynden-Bell & Wood (’68)（壁がない現実的な）
自己重力多体系には安定な熱平衡系は存在しない
4

Non-equilibrium extensionof Antonov problem

n=6

re

（非加法エントロピーの平衡状態）

恒星ポリトロープ（べき分布）の解系列
stable ?

unstable ?

n>5では臨界線を境に安定・不安定相に分離？
2002, 2003年

GRAPEによるN体シミュレーション
ポリトロープの解系列に沿って進化
密度プロファイル
ρ(r)

分布関数
f(ε)

unpublished2003, 2005年
r変分法によるポリトロープ指数の変化の解析的記述
5

ε

1 2
v + Φ ( x)2

Sardinia

Edinburgh
6

自己重力多体系の運命
e.g., Binney & Tremaine (’87, ’08)

初期

free-fall time

条件 tffrelaxation time

力学平衡(G ) 1/2 (ビリアル平衡) trelax(N/ ln N ) tff (不安定!!)

collisionlesscollisional

楕円銀河

球状星団

銀河団

熱平衡

: 質量密度N : 粒子数

以後のお話

宇宙大規模構造t↵ . tage (= 13.8 G yr)

・・・初期条件に依る・・・初期条件を忘れて行く7

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)※ 1 Mpc＝10^6 pc ~300万光年

標準的シナリオでは• 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
初期条件を忘れていないので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（最近は重力レンズ観測なども）8

A section of galaxy 3D map
120,000 galaxies

redshift

http://www.sdss.org/press-releases/astronomers-map-a-recordbreaking-1-2-million-galaxies-to-study-the-properties-of-dark-energy/
9バリオン音響振動 (BAO)(Baryon Acoustic Oscillations)• 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の音響振動)• 振動スケール は「標準ものさし」になる
10

S. Alam et al.

→ 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）
ゆらぎのパワー P (k) = 1| (k)|2 zz (=0~3)
Nk
BOSS DR12
14
L. Anderson
et al.
角径距離
スペクトル
|k|=k
P(k) [(h-1Mpc)^3]

7

音響振動スケール

BOSS DR9
宇宙膨張の変化は(SDSS-III)k [h Mpc-1 ]距離-赤方偏移関係に影響

z=1100

Alam et al. (’16)

遠方銀河

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).

10

宇宙マイクロ波背景放射

s [h 1Mpc]
Alam et al. (’16) s? [h Mpc]
s [h 1 Mpc]

150
150

80
100

40
50

0
0
50

0

40
50
100

視線方向

80
100
150
150

50

0

0

150

120
40

s2 (s , s ) [h 2 Mpc2]

0

80

100

50

s [h 1 Mpc]

50

50

40

100

100

80

100
150

120

BOSS DR12 - 0.5 < z < 0.75
150

50

100

150

1

ssk [h[h 1 Mpc]
Mpc](Redshift-Space Distortions)

銀河の特異速度場がドップラー効果を通じて赤方偏移測定
Cosmological Aに影響、銀河クラスタリングの統計的等方性が破れる
BOSS DR12 - 0.5 < z < 0.75

大スケールでは、

1

11〜1

ゆがみの強さは重力由来の密度ゆらぎの成長率に比例f (z)→ 宇宙論的スケールでの重力のテストe.g., Linder (’08); Guzzo et al. (’08);
Yamamoto et al. (’08); Percival & White (’09)

Figure 5. The measured pre-reconstruction correlation function (left) and power spectr
sight, shown for the NGC only in the redshift range 0.50 < z < 0.75. In each panel, thebest-fit model. The anisotropy of the contours seen in both plots reflects a combination
separately constrain DM (z)/rd , H(z)rd , and f 8 . The BAO ring can be seen in two
anisotropic BAO ring in the power spectrum, the right panel plots the two-dimensional pseen in this panel are analogous to the oscillations seen in the top left panel of Fig 3.

赤方偏移空間ゆがみ (RSD)DM ⇥ rd,fid /rd [Mpc]DM ⇥ rd,fid /rd [Mpc]DM ⇥ rd,fid /rd [Mpc]Measurementz = 0.38
z = 0.51
z = 0.61

redshift

1476 ± 33
1985 ± 41
2287 ± 54

Satpathy et al.
⇠(s) multipoles

Table 4. Summary table of pre-reconstruction full-shape constraints on the parameter coin the supporting papers for each of our three overlapping redshift bins

d ln D+ 線形成長因子⇥ { m (z)}d ln aスケール因子sk [hMpc]1 ]
1
[h
Mpc
s
[h
Mpc]
?
Alam et al. (’16)

100 0.150

80
40

0.2

0
0
0.0
50

0.0

0

50
0.1

100

[h Mpc]1
k s[h Mpc
]

1

40

1

視線方向

150
0.2

80
120

0.1

50 0.1100

0.1
100

50

0
0.0

0

0.0
1

50

100

80
0.3

0.1

0.4120 0.5

150
0.2

[P (k , k s)2 (sPsmooth)]/P2 ]smooth (k), s ) (k[h ,2kMpc

0.140 0.2

s [h
Mpc]
1
k
[h Mpc]5000.240 0.1 0.0

0.1

13
BOSS
DR12
NGC
0.5z<<z0.75< 0.75
BOSS
DR12- 0.5- <

80
0.3

100

0.2
150

0.2

150
0.2150

100

100
0.1
50

BOSS DR12 NGC - 0.5 < z < 0.75

150
0.2
150
0.2
150

k k [h Mpc ]BOSS DR12 - 0.5 < z < 0.75

1 galaxies
ical Analysis of BOSS1
ss [h[h 1 Mpc]
Mpc](Redshift-Space Distortions)

銀河の特異速度場がドップラー効果を通じて赤方偏移測定
logical Analysis of BOSS galaxies 13
Cosmological Aに影響、銀河クラスタリングの統計的等方性が破れる
大スケールでは、

11〜2

ゆがみの強さは重力由来の密度ゆらぎの成長率に比例f (z)→ 宇宙論的スケールでの重力のテストe.g., Linder (’08); Guzzo et al. (’08);
Yamamoto et al. (’08); Percival & White (’09)

n thedirections
perpendicularand parallel to thecorrelationline ofFigure5. Themeasured pre-reconstruction
function (left) and power spectr
ows
the data
andforthethecontours
showintheof the0.50 < z < 0.75. In each panel, thesight,
shown
NGC only
theprediction
redshift range
he AP
effect,and holds
most of theofinformation
used
to in both plots reflects a combination
best-fitmodel.
The anisotropy
the contours
seen
thecorrelation
functionDplot.(z)/rTo moreclearly show the
separately
constrainMd , H(z)rd , and f 8 . The BAO ring can be seen in two
divided by the best-fit smooth component. The wiggles

赤方偏移空間ゆがみ (RSD)b)
les

8 (z) derived

1525 ± 25
1990 ± 32
2281 ± 43
81.2 ± 2.3
87.0 ± 2.4

Grieb et al.
P (k) wedges

1501 ± 2730
DM 2010⇥ r±d,fid /rd [Mpc]37
DM 2286⇥ r±d,fid /rd [Mpc]2.4/r [Mpc]
DM82.5⇥ r±d,fidd90.2 ± 2.5

Sánchez et al.
⇠(s)wedgesMeasurement

Satpathy et al.
⇠(s) multipoles
1476 ± 33
1985 ± 41
2287 ± 54

redshiftz = 0.38
z = 0.51
z = 0.61

Table 4. Summary table of pre-reconstruction full-shape constraints on the parameter coin the supporting papers for each of our three overlapping redshift bins

⇥ rd,fid /rd , H ⇥ rd /rd,fid , and fanisotropic BAO ring in the power spectrum, the right panel plots the two-dimensional pseen in this panel are analogous to the oscillations seen in the top left panel of Fig 3.

d ln D+ 線形成長因子⇥ { m (z)}d ln aスケール因子reader should assume this factor implicitly in all text throughout(2016) for DR12 measurements.
this section. However, Fig. 13 and Table 9 have no such corrections
The right panel of Figure 15 compares our measurements
applied to them.
Cosmological
Analysis of BOSS galaxies 3of f 8 results those from other surveys: 2dfGRS (Percival et al.
For comparison, we focus on the DV constraints, as these pro2004b), 6dFGS (Beutler et al. 2012), GAMA (Blake et al. 2013),
vide the most information from the post-reconstruction BAO analremely strong, and nearly all observations remain consistentzon,
WiggleZ (Blake et al. 2012), VVDS (Guzzo et al. 2008), andysis and we regard the LOWZ volume as too small to obtain robust
Z 1 DV likelihood is what was used in
VIPERS (de la Torre et al. 2013), as well as the measurements from
H(z) likelihoods (the LOWZ
a cosmological constant form of dark energy. CMB measurecs (z)the SDSS-I and -II main galaxy sample (Howlett et al. 2015, MGS)
rd cosmological=dz ,the Cuesta et al. 2016a
analysis).
The consensus(1)prefrom the Wilkinson Microwave Anisotropy Probe (WMAP;
zd H(z)and the SDSS-II LRG sample (Oka et al. 2014, DR7). The meacision
on
D
from
the
combinationofthe
Cuestaetal.
(2016a)andV
ett et al. 2013), ground-based experiments such as the Atasurements plotted are conditional constraints on f 8 based on the
Gil-Marı́netal.
(2016a)results
is
1.3
per
cent
for
LOWZ
and
1.0
Cosmological
AnalysisofBOSSgalaxies
3
evaluated
at
the
drag
epoch
z
,
shortly
after
recombination,
when
Cosmology Telescope (Das et al. 2014) and the South Poledp
Planck 2015 ⇤CDM cosmological model. This can be seen as a diper centforbaryons
CMASS,after
the(seeabovescaling
byal. V
. Thee↵25
Cosmological
Analysisof BOSSgalaxies
photonsanddecouple
Aubourget2015
for consenmore
cope (George et al. 2015), and, especially, the Planck satelrect test of General Relativity. We find that our results confirm thesusDprecision
weobtain(see
Section
8.2)
is
20
per
cent
betteratVrong, andnearly all observationszon,
precise discussion). This scale appears as a localized peak in the
PlanckCollaboration
I 2015) nowremain
provideconsistentstrong constraints
validity of General Relativity. We also find reassuring consistency
lowredshift
and
the
same
at
high
redshift,
and
these
D
V constraintsZ
constant
dark energy.
CMB
correlation function or1a dampedof oscillations in the power
egicalcosmicmatterform
andofradiation
density,themeasureangular diametercs (z) theseries
between our measurements and those by different surveys.
comealmostentirely
frompost-reconstruction
BAO
analysis
30
r
=
dz
,
(1)
d
Wilkinson
Microwave(WMAP;
spectrum. Assuming
standard
matter and radiation content, theceto the surfaceof lastAnisotropy
scattering,Probeand theshape and amH(z)It is also interesting to compare this paper’s full-shape re(seethesecondcolumn
of
Fig.
11).
Our
improvement
at
low
redz
6dFGS
d
2013),
as the Ata- epoch
Planck2015
measurements
of factthe matter
anderrorbaryondensitydee of theground-based
matter powerexperiments
spectrum atsuch
the recombination
assuming
Planckcosmology
0.7
sults(Table7) withCDM
the full-shapeanalysis of the DR12 LOWZ
shiftiscompatible
with
the
that
our
in
D
is
smallerV
SDSS
MGS Pole
evaluatedtermine
at the dragepoch horizon
zd , shortly
after recombination,
when
Telescopeet al. 2014)
the lower
South
the
sound
to
0.2%.
An
anisotropic
BAO
analysis⇡gy1090.
These (Dasmeasurements
alsoandprobe
redshift matter
DR12
final
consensus
and CMASS samples, doneinFourier
space by Gil-Marı́n et al.
than the standard deviation of the mock samples (see Table 5) by
p
SDSS
DR7
photons andbaryons decouple(seeAubourgal. 2015 for and
more
orge
al. 2015),
and, especially,satel- Sachsthat
the BAO
featurein theetline-of-sightring etthrough
gravitational
lensing theandPlancktheintegrated(2016b) (scaled
by Ve↵ factors). Approximating LOWZ to
20measures
per cent, while
the resultspresented
in Cuesta et al.transverse(2016a) obPlanckagain⇤CDM
precise discussion).
This
scale appears
as aH(z)localized
peakin the angular
I 2015)
now 1967)
providestrongconstraints
WiggleZ
directions
canseparately
measureandequivalent
thecomoving
0.6 from
ellaboration(ISW; Sachs& Wolfeeffect.
Within⇤CDM, CMB
our low redshift bin and CMASS to our high redshift bin, we find atained slightly
worse precision
than thequantitycorrelation function or a damped series of oscillations in the power
matter and radiation density, the angular
diameter
BOSS
GalaxyDR12
diameterdistance
is related areto thephysicalanguDM measurement of 1.7% in the low redshift bin and 1.8% in thelone are sufficient to provide tight parameter constraints, but
the mocks.
SuchD
fluctuationsin precisionconsistent
with
those
M (z), whichsurface of last scattering, and theBOSS
shapeLyand
am-DR11 spectrum. Assuming standard matter and radiation content, the-autohigh redshift bin, which compares to 2.3% and 1.8% in Gil-Marı́nin ourmock samples.
In =
terms
the A
standard
deviation, thelarfounddiameterdistance
by DM (z)(1 +ofz)D(z) (Padmanabhan
weaken considerably when
non-zero curvature
or moreflex20Planck 2015 measurements of the matter and baryon density de- 0.5
matter power spectrum at the recombination epochet al. (2016b), respectively. Regarding H(z), our measurement of
results for
DV in Cuestaparameters
et al. (2016a)
agree withBOSS Ly-cross DR11
et consensus
al. 2008). mock
Adjustments
in cosmologicalor changes
orms of dark energy are allowed (Planck
Collaboration
XIII.
termine thethe
soundhorizonresults
to 0.2%.
An anisotropic
BAO
hese measurements also probe lower redshift matter
2.8% in both the low and high redshift bins compares to 3.8% andconsensus
presented
in Table5, fromat analysis
theextranumberof sigto
thepre-recombinationenergydensity(e.g.,
relativishereafter
Planck2015).
Supernova
measurementsoftheexthat measures
the BAO
featurein theThus,line-of-sightand transverseugh gravitational lensing and the integrated Sachs3.6% in Gil-Marı́n et al. (2016b), again showing a clear improvenificant
digitswe quote.
resultsfromthiscomparisonaretic
species)
can
alter
r
,
so
BAO
measurements
reallyconstrainonhistory
have
improved
dramaticallythankstolargegroundd
directions canseparately
measureH(z) andfrom
the comoving
angularachs & Wolfe 1967) effect. Within ⇤CDM, CMB
0.4
ment in the precision when using our new methodology. Finallyconsistentwiththeexpectation
the
tests
in
mock
catalogues
combinations
Dsurveys tothatprovide
span theredshiftrange
0.2 < z but< 0.8, imM (z)/rd , H(z)rd . An angle-averaged galaxy
diameter the
distanceDM
which
sufficienttightparameter
constraints,
our f 8 constraint of 9.5% and 8.9% in the low and high redshift
described
in(z),
Section
2.3.is related to the physical anguBAO
measurement
constrains
a combinationthat is approximately
donsiderably
local calibrator
Space
Telescopelar diameter
distance
by14
DMplots(z) =our(1 BAO-only+z)DA (z)results(Padmanabhan
whensamples,
non-zeroHubble
curvatureor moreflex- searches
bin compares to the LOWZ constraintof 12.1% and 9.6% in GilFigurein the wider conGAMA
D
(z)/r
z
M
d
xtend
the Hubble
diagramto zCollaboration⇡ 1.5, and XIII.
major efforts⇥ higher⇤measurements
et al. 2008).text
Adjustments
in cosmologicalor changes 0.3
1/3
ark
energyare allowed(Planck
2 parameters
Marı́netal.
(2016b),whichsimilarlyto
DM and H, shows a clearof other surveys
andredshift
from
the
10
6dFGS
WiggleZD.
(2)
V (z) = czDM (z)/H(z)V (z)/rd z
dependentgroupsto placemeasurements
different data of
setstheonexa common
to theDpre-recombinationenergydensity
extra relativisPlanck2015).
Supernova
improvement inthelowredshiftbin.
BOSS Ly↵ forest.
Blue,green,(e.g.,
and from
red curves/points
show DV (z),SDSS MGS
ViperszDH An(z)/rz
andidentify andmitigate thankssourcestooflarge
systematicd altertic species)r ,BAOso
BAO
reallyhavetoimproved
dramaticallyground-error (seeanisotropic
automaticallyincorporates
theredshiftsoDcan⌘ measurementsc/H(z),
dividedbyrconstrain
Additionally, we display the results based on the combinaM (z), andd D
H (z)analysis
d and with0.2
kithatet spanal. 2012;
Betoulerange
et al. 0.2
2014;
the combinations
Dthat
angle-averaged
galaxythe redshift< andz <references
0.8, im- therein).
M (z)/rd , H(z)rd . Anscalings
fit allthree
curvesontest,the samewithvisible ertion of0.3
the pre-reconstructed
power
calledAlcock-Paczynski(1979;AP)
whichplotuses
the require0.1
0.2
0.4
0.5
0.6 spectrum,
0.7
0.8bispectrum and post0.1
0.2and z ⇡ 2.5,
0.5 ror bars.constrains
1.0
2.0 thethat
BAO measurement
alinescombination
is approximately
librator samples,now
HubbleSpace
searchesmeasurements,
spanningz Telescope= 0.1 0.8
The threeshow
predictionsof a combination⇤CDM modelreconstruction BAO (from Gil-Marı́n et al. 2016a,b,c), which isment of statisticalisotropy
toconstraintheparameterzz H(z)D (z). ⇥element
Hubbleto z ⇡ 1.5,
major anefforts⇤1/3
thediagram
SN measurements
by andprovidingabsolute dis2
M
D
(z)
=
czD
(z)/H(z).
(2)
VM
tscale,groupsto place
different dataon a commonc 2016 RAS, MNRAS 000, 1–38
directmeasurementof thesets
expansion
rate H(z), andThe
localized
three-dimensional
natureofthe
BAO
featureentifyandmitigatesourcesofsystematic
error
(see
tness tosystematic
errors(see
discussion
andreferencesbeAnanisotropic
BAO ofanalysisautomaticallyincorporates
theFigure 14. The “Hubble diagram”
from15.
the worldcollection ofspectroscopic
BAO detections.
Blue,
green
points show BAO
measurementsof DVso/rd , BOSS measurements in DR11 (Alam et al. 2015b; Beutler et al. 2014a;
FigureLeft-handpanel:
Comparison
f red,8toand(z)measurements
acrossprevious
makesmeasurements
robust tomostobservational012;
etDal.
therein).
Dand
respectively,offromsources indicatedin the legend.
These BAO
can be compared
the correspondingly
colouredlines,
which represents systemM /r2014;
H /rd ,referencesd , and measurements
DirectBetoule“distance
ladder”
Hthe
thecalledAlcock-Paczynski(1979;
0 constrainp AP) test, which uses the requireSamushia
et2.5,
al.(with2014;
et0.6727).
al.(see2014)
andetbyDR12(Gil-Marı́n
al. tend
2016b,c;Chuanget al. 2016) samples. Right-hand panel: The f 8 (z) results from thispredictionsof thefiducial0.8
Planck⇤CDM⌦m =Sánchez
0.3156, h atics=
The
scalingz 2012,is arbitrary,chosen which
toetcompressthe dynamic
range
Ross
2016),
to introduce
onlyments,
now spanning
=
0.1
and
z lever⇡modelisotropy
toal.haveconstrain
theslightly
parameter
combination
nt day expansion
rate,zproviding
thebars
longestarm
sufficiently
to make error
visible on
the plot.
Foragainst
visualment
clarity,ofthestatisticalLy↵ cross-correlation
pointsbeen shifted
in redshift;auto-correlation
work
comparedwiththemeasurementsofthe
2dfGRS(Percival
etal.
2004b)
and6dFGS(Beutler et al. 2012), the GAMA (Blake et al. 2013), the WiggleZeSN
measurements
byprovidinganabsolute
dissmoothdistortions
inclustering
measurements.
Similarly,
nonplotted Freedmanat the correct effective
redshift. Measurements
shownMby(z).
open points are not incorporated in our cosmological parameter analysis because
H(z)D
MB (Riess et al. points
2011,are2016;
et al. 2012).
RSD andtheyofarethenot independentof(BlaketheBOSSmeasurements.
ectmeasurement
expansion
rate
H(z),
and
et
al.
2012),the
VVDS(Guzzoetal.
2008),
andthe
VIPERS
(de
la
Torre
et
al.
2013)
surveys, as well as the measurements from the SDSS-I andlinearevolution
andgalaxybias
areexpectedtoproducesmoothgravitational lensing measurements provide complementary The localized three-dimensional nature of the BAO featureystematic errors (see discussion andreferences
be- sample (Howlett
ratherthan
localized
distortionsofclustering.
Our
BAO
analy-IImaingalaxy
et
al.
2015,
MGS)
and
the
SDSS-II
LRG
sample
(Oka
et
al.
2014,
DR7). We have plotted conditional constraints on f 8
s of structure growth that have somewhat different paramemakes BAO measurements robust to most observational systemistance ladder” presented
measurements
ofandHthe
0 constrainsis
methods
introduceparametrized
templatestomarginalize
over
in
Table
9
denoted
as
G-M
et
al.
(2016
a+b+c).
The
assumingaPlanck⇤CDM
backgroundcosmology.
This
is
one
of
the
best
evidenceof
how
growthratemeasurements from BOSS again reaffirm the validity
nsitivity and very different systematics. Consistency of RSD
atics (see Ross et al. 2012, 2016), which tend to introduce only
pansion rate, providing
the oflongestleversetsarm
againstcombinationthese threeof results
is presented at the endsmooth distortions of observational or astrophysical origin, and reof
GeneralRelativity
inlarge
scales.
eak lensing can ofalso
test modified
gravitymodelsthisthat predictsmooth to
distortions in clustering measurements. Similarly, nonGil-Marı́n
et al.
As before,s et al. 2011, 2016;
Freedmanet (2016c).
al. 2012).
RSD and case is comparedsults are insensitive to details of these templates and to many otherent
effective
potentials
governinglight-bending
andaccelerour
full-shapecolumnofTable
7,
approximating
LOWZ
to
our
low
linearevolutionand galaxy bias are expected to produce smooth
onal lensing measurements provide complementary
analysisdetails(Vargas-Magañaet al. 2014,
evoredshift
bin and
CMASS
to these
our high
redshift bin,
where
the
volof
non-relativistic
tracers.
At
present,
structure
growthrather thanlocalized
distortionsof clustering.
Our2016).
BAO Non-linear
analyture growth thatume
havesomewhatdifferent
paramedifference factor has been taken into account. Our DM mea-lution broadens the BAO peak in the correlation function (or damps
urements
aresubstantially
lessprecisethan
expansion
historysis
methods
introduceparametrized
templatestomarginalize
overnd very differentsurement
systematics.
Consistency
ofbinRSDCOSMOLOGICALPARAMETERS
Planckof 1.7% in
the9low redshift
and 1.8% in the high redhigh-k oscillationsin the orpower
spectrum),and simulations
and2015 release (Planck Collaboration XIII 2015). We use thesmoothdistortionsof observational
astrophysicalorigin,and reurementsvs.
⇠ to1models2%),
sopredictthey
serve prishift10%
bin compares
1.5%
and
1.1%,
respectively,
in
Gil-Marı́nng can also(⇠test5 modified
gravitythat
spectra for both temperature and polarization; in detail, we
perturbationto theory
indicatethat
evolution
sultsinare
details calculationsof these templates
andto non-linear
many otherpower
2016 froma+b+c.
RegardingH(z),ouraccelermeasurementof 2.8%bothinsensitive
yivetopotentials
test departures
GRand constrain
neutrino masses
governinglight-bending
and
9.1
Data
sets
the
low
and
high
redshiftbins
compares
to
2.5%
and
1.8%
in
Gilandgalaxy
bias
can
shift
the
location
of
the
BAO
peak
at
a
level
use
the
likelihoods plik dx11dr2 HM v18 TTTEEE and lowTEB
analysis details (Vargas-Magaña et al. 2014, 2016). Non-linear evothan measuredark
parameters.
This situation
lativistic
tracers.
Atenergy
present,
these structure
growth is likely
Marı́n 2016 a+b+c. Finally our f 8 constraint of 9.5% and 8.9% inof 0.2
0.5%(Eisensteinetal.
2007b;
Padmanabhan&Whitelutionbroadens
the
BAO
peak
in
the
correlation
function(ordamps
for the high and low multipoles, respectively. We do not include
ange
in next-generation
experiments.
Observational
probesof CMASS interpretation of our results. We will
are substantiallylessexpansionthelowprecise
andhighthanredshiftbincomparesto theto
LOWZ
and
We
nowhistoryturn
cosmological2009; Seo in
et al.
et al. 2011;
Sherwin &and
Zaldarriaga
high-kOne
oscillations
the2010;powerMehta
spectrum),and simulations
energyreviewedetal.
(2006),
measurements
ofAlbrecht
9.2%
6.0%
by
Gil-Marin
2016a+b+c.
(⇠ 5 are10%
vs.
⇠by,1 e.g.,2%),
soand
they
serve
pri- Frieman,the information from the lensing of the CMB in the 4-point correuse
theconsensus
measurements,
includingoursystemMeasurements
ofindicate
theestimated
BAOthatscale
using samples
with considperturbation
theorycalculations
non-linear
evolutioncanthe improvementin Gil-Marı́n
2016a+b+c
when com-2012).
r, & Huterer(2008),
Blanchard(2010),
Astier& Pain(2012),
departures
from
GRattributeand constrain
neutrino
masseslations
pared to our measurement
to the use ofcontribution
the bispectrum,which
haserableandgalaxybias differences
can shiftmatrix,
the
ofthe
the
BAO
at a levelin location
galaxybias
thatBAO-only
sharepeakthe samevolumehaveof the CMB temperature anisotropies. We will discuss the
atic
to
the
covariancefromsurecomprehensively
dark energynot
parameters.
This situation
is likely
more
Weinberg
et al.error(2013).
Reviews fobeenby
usedin our analysis.
impactof
0.2
0.5%(Eisensteinetal.
2007b;
Padmanabhan&Whiteobtained
results
withfigures
such small
shiftsxt-generation
experiments.
Observational
probesof columns
BAO+FS
3. In
ourconsistentsubsequentandta- (Ross et al. 2014;of the recent (Planck Collaboration XLVI 2016) large-angle
more on theoriesof dark energy
andandmodifiedgravityinclude of Table
12〜1
2009;
Seo
et
al.
2010;
Mehta
et
al.
2011;
Sherwin&
Zaldarriaga
et al. 2016a).
A key elementof recent
BAO analyses
re- andresults
polarizationin §9.4.
eand,reviewede.g., Albrecht
et al.Jain(2006),
Frieman,Figure16.
Parameter
constraints
forBeutler
thelabeledowCDM
cosmologicalmodel,
comparingthe isBAO
BAO+FS
results from this paper as well as the DR12
Sami,by,& Tsujikawa(2006),&
Khoury(2010),
andis simply
bles,
the
formercase2012). Measurementsof“BAO.”the BAO scale using samples with consid-

Cosmological constraints

26

Alam et al. (’16)

S. Alam et al.

Growth of structure

f

8 (z)

distance/rd z

Geometric distances

Curvature

Hubble parameter

Dark energy EoS

Hubble parameter

Cosmological Analysis of BOSS galaxies

Curvature

Dark energy EoS

27

should assume
this factor
implicitly in all ,textZhongxu
throughout
hael Wood-Vasey, Christophe reader
Yèche, IditZehavi
Zhai , Gong(2016) for DR12 measurements.
this section. However, Fig. 13 and Table 9 have no such corrections
Zhao52,22
The right panel of Figure 15 compares our measurements
applied to them.

Cosmological
Analysis of BOSS galaxies 3of f 8 results those from other surveys: 2dfGRS (Percival et al.
For comparison, we focus on the DV constraints, as these pro2004b), 6dFGS (Beutler et al. 2012), GAMA (Blake et al. 2013),
vide
the
most
informationfrom the post-reconstruction BAO analuly 2016
remely strong, and nearly all observations remain consistentzon,
WiggleZ (Blake et al. 2012), VVDS (Guzzo et al. 2008), andysis and we regard the LOWZ volume as too small to obtain robust
Z 1 DV likelihood is what was used in
VIPERS (de la Torre et al. 2013), as well as the measurements from
H(z) likelihoods (the LOWZ
a cosmological constant form of dark energy. CMB measurecs (z)the SDSS-I and -II main galaxy sample (Howlett et al. 2015, MGS)
rd cosmological=dz ,the Cuesta et al. 2016a
analysis).
The consensus(1)prefrom the Wilkinson Microwave Anisotropy Probe (WMAP;
zd H(z)and the SDSS-II LRG sample (Oka et al. 2014, DR7). The meacision
on
D
from
the
combinationofthe
Cuestaetal.
(2016a)andV
ett et al. 2013), ground-based experiments such as the Atasurements plotted are conditional constraints on f 8 based on the
Gil-Marı́netal.
(2016a)results
is
1.3
per
cent
for
LOWZ
and
1.0
Cosmological
AnalysisofBOSSgalaxies
3
evaluated
at
the
drag
epoch
z
,
shortly
after
recombination,
when
Cosmology Telescope (Das et al. 2014) and the South Poledp
Planck 2015 ⇤CDM cosmological model. This can be seen as a diper centforbaryons
CMASS,after
the(seeabovescaling
byal. V
. Thee↵25
Cosmological
Analysisof BOSSgalaxies
photonsanddecouple
Aubourget2015
for consenmore
cope (George et al. 2015), and, especially, the Planck satelrect test of General Relativity. We find that our results confirm thesusDprecision
weobtain(see
Section
8.2)
is
20
per
cent
betteratVrong, andnearly all observationszon,
precise discussion). This scale appears as a localized peak in the
PlanckCollaboration
I 2015) nowremain
provideconsistentstrong constraints
validity of General Relativity. We also find reassuring consistency
lowredshift
and
the
same
at
high
redshift,
and
these
D
V constraintsZ
constant
dark energy.
CMB
correlation function or1a dampedof oscillations in the power
egicalcosmicmatterform
andofradiation
density,themeasureangular diametercs (z) theseries
between our measurements and those by different surveys.
comealmostentirely
frompost-reconstruction
BAO
analysis
2
30
r
=
dz
,
(1)
d
Wilkinson
Microwave(WMAP;
spectrum. Assuming
standard
matter and radiation content, theceto the surfaceof lastAnisotropy
scattering,Probeand theshape and am2 reH(z)It is also interesting to compare this paper’s full-shape(seethesecondcolumn
of
Fig.
11).
Our
improvement
at
low
redz
6dFGS
d
2013),
ground-based
experiments
such
as
the
AtaPlanck2015
measurements
of factthe matter
anderrorbaryondee of the matter power spectrum at the recombination epoch
assuming
Planckcosmology
0.7
sults (Table7)2withCDM
the full-shapeanalysis of the DR12 LOWZ
3shift iscompatible
with thethat ourin Ddensity
smaller
V is
SDSS
MGS Pole
evaluatedtermine
at the dragepoch
zd , shortly
after recombination,
when
Telescopeet al. 2014)
the lower
South
3
the
sound
horizonto
0.2%.
An
anisotropic
BAO
analysis⇡gy1090.
These (Dasmeasurements
alsoandprobe
redshift matter
DR12
final
consensus
and CMASS samples, doneinFourier
space by Gil-Marı́n et al.
than the standard deviation of the mock samples (see Table 5) by
p
SDSS
DR7
photons andbaryons decouple(seeAubourgal. 2015 for and
more
orge
al. 2015),
and, especially,thesatel- Sachsthat
the BAO
featurein theetline-of-sightring etthrough
gravitational
lensing 3
andPlancktheintegrated(2016b) (scaled
by Ve↵ factors). Approximating LOWZ to
20measures
per cent, while
the resultspresented
in Cuesta et al.transverse(2016a) obPlanckagain⇤CDM
precise discussion).
This
scale appears
as aH(z)localized
peakin the angular
I 2015)
now 1967)
providestrongconstraints
WiggleZ
directions
canseparately
measureandequivalent
thecomoving
0.6 from
ellaboration(ISW; Sachs& Wolfeeffect.
Within⇤CDM, CMB
our low redshift bin and CMASS to our high redshiftbin, we find atained slightly
worse precision
than thequantityMMcorrelation function or a damped series of oscillations in the power
matter and radiation density, the angular
diameter
BOSS
GalaxyDR12
diameterdistance
is related areto thephysicalanguDM measurementof 1.7% in the low redshift bin and 1.8% in thelone are sufficient to provide tight parameter constraints, but
the mocks.
SuchD
fluctuationsin precisionconsistent
with
those
M (z), whichM
surface of last scattering, and theBOSS
shapeLyand
am-DR11 spectrum. Assuming standard matter and radiation content, the-autohigh redshift bin, which compares to 2.3% and 1.8% in Gil-Marı́nin ourmock samples.
In =
terms
the A
standard
deviation, thelarfounddiameterdistance
by DM (z)(1 +ofz)D(z) (Padmanabhan
weaken considerably whenor moreflex20 non-zero curvature
Planck 2015 measurements of the matter and baryon density de- 0.5
matter power spectrum at the recombination epochet al. (2016b), respectively. Regarding H(z), our measurement of
results for
DV in Cuestaparameters
et al. (2016a)
agree withBOSS Ly-cross DR11
et consensus
al. 2008). mock
Adjustments
in cosmologicalor changes
orms of dark energy are allowed (Planck
Collaboration
XIII.
termine thethe
soundhorizonresults
to 0.2%.
An anisotropic
BAO
hese measurements also probe lower redshift matter
2.8% in both the low and high redshift bins compares to 3.8% andconsensus
presented
in Table5, fromat analysis
theextranumberof sigto
thepre-recombinationenergydensity(e.g.,
relativishereafter
Planck2015).
Supernova
measurementsoftheexthat measures
the BAO
featurein theThus,line-of-sightand transverseugh gravitational lensing and the integrated Sachs3.6% in Gil-Marı́nM
et al. (2016b), again showing a clear improvenificant
digitswe quote.
resultsfromthiscomparisonaretic
species)
can
alter
r
,
so
BAO
measurements
reallyconstrainonhistory
have
improved
dramaticallythankstolargegroundd
directions canseparately
measureH(z) andfrom
the comoving
angularachs & Wolfe 1967) effect. Within ⇤CDM, CMB
0.4
M
ment
in
the
precision
when using our new methodology. Finallyconsistentwiththeexpectation
the
tests
in
mock
catalogues
M
combinations
Dgalaxysurveys tothatprovide
span theredshiftrange
0.2 < z but< 0.8, imM (z)/rd , H(z)rd . An angle-averaged
diameter the
distanceDM
which
sufficienttightparameter
constraints,
our f 8 constraintof 9.5% and 8.9% in the low and high redshift
8
described
in(z),
Section
2.3.is related to the physical anguBAO
measurement
constrains
a combinationthat is approximately
donsiderably
local calibrator
Space
Telescopelar diameter
distance
by14
DMplots(z) =our(1 BAO-only+z)DA (z)8(Padmanabhan
whensamples,
non-zeroHubble
curvatureor moreflex- searches
8 to the LOWZ constraint
bin comparesof 12.1% and 9.6% in Gilresultsin the wider conGAMA
DM (z)/rd Figure
z
xtend
the
Hubblediagram
to
z
⇡
1.5,
and
major
efforts⇥⇤etal.
2008).
Adjustments
incosmologicalparameters
or changes 0.3
1/3
ark energy are allowed10(Planck Collaboration XIII.
2
Marı́netal.
(2016b),whichsimilarlyto
DM and H, shows a clear
text of other surveys
andhigherredshiftmeasurements
from
the
6dFGS
WiggleZD.
(2)
V (z) = czDM (z)/H(z)V (z)/rd z
dependentgroupsto placemeasurements
different data of
setstheonexa common
to theDpre-recombinationenergydensity
extra relativisPlanck2015).
Supernova
improvement inthelowredshiftbin.
BOSS Ly↵ forest.
Blue,green,(e.g.,
and from
red curves/points
show DV (z),SDSS MGS
ViperszDH An(z)/rz
andidentify andmitigate thankssourcestooflarge
systematicd altertic species)r ,BAOso
BAO
reallyhavetoimproved
dramaticallyground-error (seeanisotropic
automaticallyincorporates
theredshiftsoDcan⌘ measurementsc/H(z),
dividedbyrconstrain
Additionally, we display the results based on the combinaM (z), andd D
H (z)analysis
d and with0.2
kithatet spanal. 2012;
Betoulerange
et al. 0.2
2014;
the combinations
Dthat
angle-averaged
galaxythe redshift< andz <references
0.8, im- therein).
M (z)/rd , H(z)rd . Anscalings
fit allthree
curvesontest,the samewithvisible ertion of0.3
the pre-reconstructed
power
calledAlcock-Paczynski(1979;AP)
whichplotuses
the require0.1
0.2
0.4
0.5
0.6 spectrum,
0.7
0.8bispectrum and post0.1
0.2and z ⇡ 2.5,
0.5 ror bars.constrains
1.0
2.0 thethat
BAO measurement
alinescombination
is approximately
librator samples,now
HubbleSpace
searchesmeasurements,
spanningz Telescope= 0.1 0.8
The threeshow
predictionsof a combination⇤CDM modelreconstruction BAO (from Gil-Marı́n et al. 2016a,b,c), which isment of statisticalisotropy
toconstraintheparameterzz H(z)D (z). ⇥element
Hubbleto z ⇡ 1.5,
major aneffortsM⇤1/3
thediagram
SN measurements
by andprovidingabsolute dis2
M
D
(z)
=
czD
(z)/H(z).
(2)
VM
tscale,groupsto place
different dataon a commonc 2016 RAS, MNRAS 000, 1–38
directmeasurementof thesets
expansion
rate H(z), andM
M
The
localized
three-dimensional
nature of the
BAO feature
entifyandmitigatesourcesof systematicerror(seecollection betness tosystematic
errorsdiscussion
andreferencesAn
anisotropic
BAO
analysisautomaticallyincorporates
theFigure
14. The(see“Hubblediagram”
fromthe worldofspectroscopic
BAO detections.
Blue,
red, andgreen
points show BAO
measurementsof DVso/rd , BOSS measurements in DR11 (Alam et al. 2015b; Beutler et al. 2014a;
Figure15.
Left-handpanel:
Comparisonoff(z)measurements
acrossprevious
8
makes
BAO
measurements
robusttomostobservational
system012;
etDal.
therein).
Dand
respectively,offromsources indicatedin the legend.
These can be comparedto the AP)
correspondingly
colouredlines,thewhich
represents
M /r2014;
H /rd ,referencesd , and measurements
DirectBetoule“distance
ladder”
Hthe
thecalledAlcock-Paczynski(1979;
test, whichuses
require0 constrainpSamushia
et2.5,
al.(with2014;
et0.6727).
al.(see2014)
andetbyDR12(Gil-Marı́n
al. tend
2016b,c;Chuanget al. 2016) samples.KRight-hand panel: The f 8 (z) results from thispredictionsof thefiducial0.8
Planck⇤CDM⌦m =Sánchez
0.3156, h atics=
The
scalingz 2012,is arbitrary,chosen which
toetcompressthe dynamic
range
Ross
al.constrain
2016),
to introduce
onlyments,
now spanningzproviding
=
0.1
and
z lever⇡modelmentofstatisticalisotropytotheparameter
combination
nt day expansion
rate,
the
longestarm
againstsufficiently to make error bars visible on the plot. For visual clarity, the Ly↵ cross-correlation points have been shifted slightlyK
in redshift; auto-correlation
work
comparedwith themeasurementsof the 2dfGRS(Percival
etparameteral. 2004b)
and
6dFGS
e SN(Riess
measurements
byprovidingan effective
absolute
dis-Measurementssmoothdistortionsin clustering
measurements.
Similarly,non-(Beutler et al. 2012), the GAMA (Blake et al. 2013), the WiggleZ
plottedat the correct
redshift.
shownMbyopen points
are not incorporated
in our cosmologicalanalysis
because
H(z)D
(z).
MB
et al. points
2011,are2016;
Freedmanet al. 2012).
RSD andtheyofarethenot independentof(BlaketheBOSSmeasurements.
ectmeasurement
expansion
rate
H(z),
and
et
al.
2012),the
VVDS(Guzzoetal.
2008),
andthe
VIPERS
(de
la
Torre
et
al.
2013) surveys,Kas well as the measurements from the SDSS-I andlinearevolution
andgalaxybias
areexpectedtoproducesmoothgravitational lensing measurements provide complementary The localized three-dimensional nature of the BAO featureystematic errors (see discussion andreferences
be- sample (Howlett
ratherthan
localized
distortionsofclustering.
Our
BAO
analy-IImaingalaxy
et
al.
2015,
MGS)
and
the
SDSS-II
LRG
sample
(Oka
et
al.
2014,
DR7).
We have plotted conditional constraints on f 8
s of structure growth that have somewhat different paramemakes BAO measurements robust to most observational systemistance ladder” presented
measurements
ofandHthe
0 constrainsis
methods
introduceparametrized
templatestomarginalize
over
in
Table
9
denoted
as
G-M
et
al.
(2016
a+b+c).
The
assumingaPlanck⇤CDM
backgroundcosmology.
This
is
one
of
the
best
evidenceof
how
growthratemeasurementsfrom BOSS again reaffirm the validity
8atics (see Ross et al. 2012, 2016), which tend to introduce only
nsitivity and very different systematics. Consistency of RSD
pansion rate, providing
the oflongestleversetsarm
againstcombinationthese threeof results
is presented at the endsmooth distortions of observational or astrophysical origin, and re1ofGeneralRelativity
inlarge
scales.
1
eak lensing
also test modified
gravitymodelsthisthat predict
8 cansmooth to
distortions in clustering measurements. Similarly, nonof Gil-Marı́n
et al.
As before,s et al. 2011,2016;
Freedmanet (2016c).
al. 2012).
RSD and case is comparedsults are insensitive to details of thesetemplates1and to many other0ent
effective
potentials
governinglight-bending
andacceler1
ourfull-shapecolumnofTable
7,
approximating
LOWZ
to
our
low
linearevolution
andgalaxybias
areexpectedtoproducesmoothonal lensing measurements provide complementary
analysisdetails(Vargas-Magañaet al. 2014,
evo0rather
8highredshiftbin andto theseourredshift bin,wherethe volof
tracers.
At CMASS
present,
structure
growththanlocalized
distortionsof clustering.
Our2016).
BAO Non-linear
analyturenon-relativistic
growth thatume
havesomewhatdifferent
paramedifference
factor has
been taken
into account. Our DM mea-lution broadens the BAO peak in the correlation function (or damps
1
urements
aresubstantially
lessprecisethan
expansion
history1
sismethods
introduceparametrized
templatestomarginalize
over1nd very differentsurement
systematics.
Consistency
ofbinRSD
1
COSMOLOGICALPARAMETERS
PlanckXIII 2015). We use theof 1.7% in
the9low redshift
and 1.8% in the high redhigh-k oscillationsin the orpower
spectrum),and simulations
and2015 release (Planck Collaboration0smoothdistortionsofobservational
astrophysicalorigin,andreurements
(⇠
5
10%
vs.
⇠
1
2%),
so
they
serve
pri0
shift bin compares
1.5% and
1.1%,
respectively, in Gil-Marı́nng can also test modified
gravitytomodels
that
predict11spectra for both temperatureand polarization; in detail, we
perturbationto theory
indicatethat
evolution
sultsinare
details calculationsof these templates
andto non-linear
many
otherpower
2016 froma+b+c.
RegardingH(z),ouraccelermeasurementof 2.8%bothinsensitive
1
yivetopotentials
test departures
GRand constrain
neutrino masses
1
governinglight-bending
and
0peak
9.1
Data
sets
the
low
and
high
redshiftbins
compares
to
2.5%
and
1.8%
in
Gilandgalaxy
bias
can
shift
the
location
of
the
BAO
at
a
level
use
the
likelihoods
plik
dx11dr2
HM
v18
TTTEEE
and
lowTEBm0
analysis details (Vargas-Magaña et al. 2014, 2016). Non-linear
evothan measuredarkenergyparameters.
This situation
is likely
1lativistic
tracers.
At
present,
these
structure
growth1
Marı́n 2016 a+b+c. Finally our f 8 constraint of 9.5% and 8.9% inof 0.2
2 high and low multipoles,(Eisenstein
et al. 2007b;
Padmanabhan& Whitelutionbroadens
the 0.5%
BAO peakin the correlation
function(ortherespectively. We do not include 1
m interpretation
0 dampsforange
in next-generation
experiments.
Observational
probesof CMASS
are substantiallylessexpansionthelowprecise
andhighthanredshiftbincomparesto theto
LOWZ
and
We
nowhistoryturn
cosmologicalof
ourresults.
We
will
2009; Seo in
et al.
et al. 2011;
Sherwin &and
Zaldarriaga
0 of the CMB in the 4-point correhigh-kOne
oscillations
the22010;
powerMehta
spectrum),and simulations
energyreviewedetal.
(2006),
measurements
ofAlbrecht
9.2%
6.0%
by
Gil-Marin
2016a+b+c.
(⇠ 5 are10%
vs.
⇠by,1 e.g.,2%),
soand
they
serve
pri- Frieman,theinformation
fromthelensingusetheconsensus
measurements,
includingoursystemMeasurements
ofindicate
theestimated
BAOthatscale
using samples
with considperturbation
theorycalculations
non-linear
evolutioncanthe improvementin Gil-Marı́n
2016a+b+c
when com-2012).
r, & Huterer(2008),
Blanchard(2010),
Astier& Pain(2012),
departures
from
GRattributeand constrain
neutrino
masseslationsof the CMB temperature
anisotropies. We will discuss the 1
pared to our measurement
to theuse ofcontribution
the bispectrum,which
haserableandgalaxy
bias
can
shift
the
location
of
the
BAO
peak
at
a
level
differencesingalaxy
bias
that
share
the
same
volume
have
m
0
atic
error
to
the
covariancematrix,from
the
BAO-only
surecomprehensively
dark energynot
parameters.
This situation
is likely
more
Weinberg
et al. (2013).
Reviews fobeenby
usedin our analysis.
impactof 0.2
0.5%3.(Eisenstein
et al. 2007b;
PadmanabhanWhiteobtained
results
withfigures
such
small
shiftset2
al. 2014;of the recent (Planck Collaboration XLVI 2016) large-angle
xt-generation
experiments.
Observational
probesof columns
BAO+FSof TableInourconsistentsubsequent
andta-&(Ross
more on theoriesof dark energy
andandmodifiedgravityinclude
12〜2
2009;
Seo
et
al.
2010;
Mehta
et
al.
2011;
Sherwin&
Zaldarriaga
et al. 2016a).
A key elementof recent
BAO analyses
re- andresults
polarizationin §9.4.
eand,reviewede.g., Albrecht
et al.Jain(2006),
Frieman,Figure16.
Parameter
constraints
forBeutler
thelabeledowCDM
cosmologicalmodel,
comparingthe isBAO
BAO+FS
results from this paper as well as the DR12
Sami,by,& Tsujikawa(2006),&
Khoury(2010),
andis simply
bles,
the
formercase2012). Measurementsof“BAO.”the BAO scale using samples with consid-

Cosmological constraints

ABSTRACT

Hubble parameter

Hubble parameter

Dark energy EoS

f

8 (z)

distance/rd z

ABSTRACTWe
present
cosmologicalresults from the final galaxy clustering data set of the Baryon OscilABSTRACT
26 S. Alamal.
Alamal.Baryon(’16) OscilWe etpresent
cosmological results from the final galaxy clustering data
set ofet
lation Spectroscopic
Survey,part clusteringof the Sloan
DigitalSky
SurveyIII.
OurthecombinedgalaxyWe present cosmologicalresults
from
the
final
galaxy
data
set
of
the
BaryonOscillation Spectroscopic
Survey,
part
of
the
Sloan
DigitalSky
SurveyIII.
Our
combinedgalaxyGeometric
distances
Growthofstructurelation Spectroscopic
Survey,part oftheSky
SurveyIII.overOur
combinedgalaxysamplecomprises
1.2Sloanmillionmassivegalaxies
ananeffectiveareaof93299329degdegandandsamplecomprises
1.2Digitalmillionmassivegalaxies
overeffective
area ofvol-volsample comprisesume
1.2 ofmilliongalaxies
overthreean effective
area
ofoverlapping
9329 degredshift
and vol18.7
Gpc, divided
intopartially
overlapping
slicescentredat at
effectiveumemassiveof 18.7
Gpc, divided
into
three
partially
redshift
slices centredeffective
ume of 18.7 Gpcredshifts, dividedredshifts
into threepartially
overlapping
slicescentreddiameterat effective
0.38,
0.51,
and
We measuretheangulardistanceandHubble0.38,
0.51,0.61.
and 0.61.
Weredshiftmeasuretheangulardiameterdistance DD and
Hubbleredshifts 0.38, 0.51,
and
0.61.
We
measuretheangulardiameterdistanceDandHubblefromthe baryon
acoustic
oscillation(BAO)method
method after
after applying
parameterparameter
H fromHthe
baryonacoustic
oscillation(BAO)
applyingreconstrucreconstrucparameter H from the baryon
acoustic
oscillation(BAO)
methodafter
applying
reconstruction to non-linear
reduce non-linear
effects
on BAOthe BAO
feature.
Usingthethe anisotropic
anisotropic clusteringof thetion to reduce
effectsonthefeature.
Using
clusteringof thetion to reduce non-linear
effectson
the
BAO
feature.
Using
theanisotropic
clusteringofthepre-reconstruction
density
field,
wemeasurethe
product
D
H
from
the
Alcock-Paczynski
pre-reconstruction
densitythe
field,
we measure
the theproduct
D H from the Alcock-Paczynski
pre-reconstruction
density (AP)field,effectwe measureproduct
D H quantified
fromAlcock-Paczynski
and the growthof structure,by f (z), from redshift-space distortions(AP)effectthe growthof structure,by f (z),from redshift-space distortions(AP) effect and thegrowthof andstructure,quantified
by f measurements(z),quantified
from redshift-space(RSD).
We combine
individual
presented indistortions
seven companion papers into a set(RSD).
We
combineindividual
measurements
presented
in seven
papersa set(RSD). We combine
individual
measurements
in sevenobtaining
companion
papersinto
settighter andof consensus
values presented
and likelihoods,
constraints
that acompanion
aremore intorobustof consensus values
and likelihoods,
obtaining
constraints
that
are tighter
andmore
of consensus
valuesandanylikelihoods,
constraints
thatrobust
are tighterand morerobustthan thosefromone
method;obtaining
in particular,theAPmeasurementfrom sub-BAO
scalesCosmological
AnalysisofBOSSgalaxies
27
than those from thanany onemethod;inparticular,
thepost-reconstruction
APinmeasurement
fromAP
sub-BAO
scales
those
from
any
one
method;particular,themeasurement
frombetween
sub-BAO
scalessharpens
constraints
from
BAO
by
breaking
degeneracy
D andSummaryof
BAO/RSDmeasurements
sharpens constraints
from post-reconstruction
BAO by2015
breaking
degeneracy
D degeneracy
andH.
Combinedfrom
with post-reconstruction
Planckcosmicmicrowave
backgroundmeasurements,
our distancesharpens
constraints
BAO bybetween
breaking
D and
Planck+ BAObetween+FS
H. Combined with Planckscale
2015measurements
cosmic microwave
backgroundmeasurements,
ourdistance
simultaneously
imply
curvature⌦ = 0.0003± 0.0026 and oura darkenH. CombinedwithPlanck2015 ⌦
cosmicmicrowave
backgroundmeasurements,
distance
scale measurements•simultaneously
imply
curvature=
0.0003±0.0026andadark
enCurvatureergy equationsimultaneouslyof state parameterimply
w = curvature
1.01 ± 0.06,
in strongaffirmationof theandspatially
flat enmeasurements⌦the= 0.0003± 0.0026
a darkergy equation of scalestate parameter
w
=
1.01
±
0.06,
in
strongaffirmationofspatiallyflat
cold dark matter model with a cosmological constant (⇤CDM). Our RSD measurements of•Dark
energyergyequationof state EoS
parameter= 1.01Our±consistent
0.06,
strong affirmationof combined
the spatiallyflat
cold dark matter model withconstantw(⇤CDM).
RSD inmeasurementsof Whenf a, cosmologicalat 6 per cent precision,
are similarly
with this model.
with
cold
darksupernova
mattermodel
witha cosmologicalconstantOurmeasurements
f , at 6 per centprecision,
are similarly
Whencombinedwith
Ia data,consistent
we findHwith= this67.3model.
± 1.0km
s (⇤CDM).
Mpc evenfor RSD
our most
general dark of
(+SNe
data)
•
Hubbleparameter
f we, findat 6energy
per=cent
similarly
consistent
withgeneral
thismodel.
combinedwithsupernova Ia data,H
67.3precision,±in1.0km sare
Mpc
even
for
our most
dark
model,
tensionwithsome direct
measurements.
Addingextra Whenrelativistic
species as(at
z=0)
energy model, insupernova
tension withmeasurements.
Addingas
Iasomedata,
we findH = the67.3± 1.0extrakm
Mpctospecies
for ±
our
a degreeofdirect
freedomloosens
constraint
onlysrelativisticslightly,Heven= 67.81.2mostkm s general
Mpc dark.
a degree of freedom
loosens
theconstraint
only
slightly,
to
H
=
67.8
±
1.2
km
s
Mpc
.
energymodel,
in tension
withdirectmeasurements.
extra±relativistic
species, as
Assuming
flat⇤CDM
wesomefindΛCDM⌦= 0.310± 0.005 andAdding
H = 67.6
0.5 km s Mpc
consistent
with
Planckmodel
Assuming flat ⇤CDM
weCurvaturefind⌦ find= a0.310± 0.005and67.6on± the
0.5neutrino
kmtos H Mpc, ± 1.2
CurvatureDark
energyEoSkm s Mpc 1 .
andwe
95% upperof H
0.16=eV/c
mass
sum.
a degreeof freedom
loosens
thelimit
constraint
only
slightly,= 67.8and we find a 95% upper
limit
of 0.16 eV/c
onlocalthe neutrino
massmeasurementsum.
but
tensionwithlensing
Assuming
flat⇤CDM
we find⌦ Hubble= 0.310± 0.005andH &=weak
67.6±
0.5 km s Mpc 1 ,Keywords:
cosmology:
observations,distance
scale,
large-scale
structure
Key words: cosmology:
observations,distance
scale,
large-scale
and we finda 95% upper
limit
of 0.16eV/c structureon the neutrino mass sum.

Large-scale structure as selfgravitating collisionless system
Main
ingredients

Cold dark matter (CDM)(invisible matter component)(Newtonian)Gravity & Cosmic expansion

Cosmological many-body system
pi
=
dt

Gm
a

2

N

j=i

xi
|xi

xj
xj |3

t↵ . tage (= 13.8 G yr)

2 dxi
pi = madta(t) : cosmic scale factor(expansion of the Universe)
13〜1N(i = 1, 2, · · · , N )m : mass of CDM particle
G : Newton const.

∂∂ done in Fourier∂ ∂ 2[46]).∂ ∂1
4π
but can be systematicallyspace(e.g.,
+
2H
ϵ
ψ
=
−
ϵ
ψ
+
2H
ψ
,
ϵijkijk
ϵpqr ψp,j+ 2H−
ρm ψk,r , (4.2
ijk j,k−p,ki,p ψj,q 2
2
2

∂t
∂t
∂t ∂t
∂t ∂t
2
3
! ∂2
"
!
"
2
∂ k . The right-hand-side∂of∂ equations represent the nwhere ψj,k =+∂ψ
the
above
j /∂q
2H
ϵ
ψ
=
−
ϵ
ψ
+
2H
ψp,k ,(4.2
ijk j,kijk p,j2
2
4.3 source(Eulerian)
Perturbation
theory∂t∂t have to be evaluated by∂t order-by-order∂tterms,
which
calculation. Once w
whereψj,k = ∂ψjsolutions/∂qk . Theforright-hand-side
the aboveequationsrepresent
perturbative
longitudinalofandtransverse
modes(i.e., ψtheand ϵik,k non-line
sourcewhichhave to construct
be evaluatedbydisplacement
order-by-order
calculation.
Once
we getfinal terms,
step isto explicitly
the
field
itself. This
is nottrivt
CollisionlessBoltzmannequation(Vlasov-Poisson
system)perturbative
for longitudinal
and transverse
modes(i.e., ψk,k and ϵijk ψj,k )but can besolutions
systematicallydone in Fourier
space (e.g.,[46]).

Large-scale structure as selfgravitating collisionless system

final step is to explicitly# construct the displacement field$ itself. This is not trivial at a
Cold
dark
matter(CDM)∂ space∂Ψ(e.g.,∂ [46]).
but canbe systematically∂done inp Fourier
Main+Perturbation−(invisiblemf (x, component)p) = 0,matter2
4.3(Eulerian)theory(Newtonian)∂t ma ∂x∂x ∂pingredients& equation:
supplemented
with Gravity
the Poisson
Cosmic expansion
4.3 (Eulerian)
Perturbation
theory$%Collisionless Boltzmann equation#(Vlasov-Poisson
system)m
3∇ Ψ(x) = 4π G adp f (x, p) − ρm .
3
Collisionless Boltzmann #equation (Vlasov-Poissona$ system)
2

2

∂
p ∂
Cosmological
Vlasov-Poisson
system
Cosmologicalmany-bodysystem ∂Ψt↵ ∂. tage (= 13.8 G yr)+−m $f (x, p) = 0,
Here, m is the mass of CDM(+baryon)particle.
#2∂ ∂t p ma∂ ∂x ∂Ψ ∂∂x ∂p
Vlasov equation+−mf (x, p)N= 0,
N the Poisson2
2withsupplemented
equation:
∂x
∂x ∂p
pi
Gm
xi∂t xma
j
2 dx
#
$
%i
=
p
=
ma
i
3
supplemented
withthe |xPoissondt
aapproximationi 2 xj |equation: 2 m dt 3
Single-stream=p)1,$−2, ρ· m· · ,.N )Poisson equation∇ Ψ(x) = 4π G# a %d p f(i(x,j=im a3 3∇ Ψ(x) = 4π G a
d: massp f (x,ofp)CDM− ρmparticle.
3m&a particle.
Here,m is: themassof CDMcosmicfactor (+baryon)
Single-stream
flowscalea(t)32Ansatz2'f (x, p) = n a {1 + δ (x)} δ p − m a v(x) .

(expansionof the(+baryon)
Universe) particle.
condition)
G : mNewtonD const.
Here,(initial
m is themass of CDM

(4.2

(4.2

13〜2 first velocity moments of Eq. (4.23) yielWith this ansatz, taking the zeroth andTheoretical tools of LSS
cold dark matter (CDM)(Newtonian)TARUYA et al.
gravity & cosmic expansion

CosmologicalN-body simulationpi = ma2pi=
dt

statistical properties
of LSSPHYSICAL REVIEW D

Perturbation theory (PT)TARUYA et al.

(based on fluid approx.)

PHYSICAL REVIEW D

dxi
dt

Gm2
a

N

j=i

xi
|xi

xj
xj |3

(i = 1, 2, · · · , N )AT et al. (’12)

http://www.projet-horizon.fr/

FIG. 9 (color online). Comparison of power spectrum results between N-body simulations andcalculation
On top of the gravitational evolution,(redshiftresults at z ¼ 3, observational
2, 1, and 0.35 are shown (from top effects
to bottom). Left panel
shows the ratio of power spectrum t
ðkÞ, where the reference spectrum P
ðkÞ is calculated from the no-wiggle fo
spectrum, PðkÞ=Ptransferfunction
in
Ref.
[47].
Solid
lines
are
the
results,
while
dottedlines represent the linear theory predspace distortions, galaxy bias, …plots) thealso
need to be consideredresults normalized by the no-wiggle spectrum, idifference between N-body and
REGPT

no$wiggle

no$wiggle

FIG. 9 (color online). Comparison of power spectrum results between N-body simulations and REGPT calculation
REGPT
results at z ¼ 3, 2, 1, and 0.35 are shown (from top to bottom). Left panel shows the ratio of power spectrum t
REGPTspectrum Pno$wiggle ðkÞ is calculated from the no-wiggle fo
spectrum, PðkÞ=Pno$wiggle ðkÞ, where the reference
14PðkÞ'=PðkÞ.
In
eachpanel,
theverticalarrowsrespectively
indicate
maximum
belo
RegPT
no$wiggle
transfer function in Ref. [47]. Solid lines are the REGPT results,
while dottedlinestherepresent
the wavenumber
linear theory predRegime of our interest
Most of interesting cosmological information (BAO, RSD,
signature of massive neutrinos, …) lies at k<0.2-0.3 h/Mpc
Weakly nonlinear regime
Dimensionless
power spectrum※ Based on linear theory

Nonlinear

Weakly nonlinear

z=0.5
1

2
3
Linear theory

LinearN-body simulations
by T. Nishimichi

15Perturbation theory (PT): reloaded
Single-stream approx. of Vlasov-Poisson system

CDM + baryon → pressureless & irrotational fluid
Basic
eqs.

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

Standard PT ( 1=1)

1 + 2 + 3 + ···Recent progress• Improving accuracy by resummation or renormalized PT treatment• Higher-order calculation & fast PT code (RegPT)

2-loop (next-to-nextto leading order)• Incorporating other systematics (massive ν, modified gravity, halo bias,…)16

Performance of resummed PT
fast resummed PT code (http://ascl.net/1404.012) Lbox = 2, 048 h

RegPT including 2-loop (next-to-next-to-leading) order

TARUYA et al.

RECT AND FAST CALCULATION OF REGULARIZED . . .

Power spectrum

Mpc
3# of particles：1, 024
1

PHYSICAL REVIEW D 86,
60103528 (2012)# of runs：
PHYSICALREVIEW D 86, 103528 (201
cosmology：wmap5

Correlation function

RegPT
LinearRegPT
Linear

G. 10 (color online). Comparison of two-point correlation function
N-bodyNishimichi
and REGPT resultsat z('12)¼ 3, 2, 1, and 0.
AT,between
Bernardeau,& Codisom bottom to top). In each panel, magenta solid, and black dotted lines represent the prediction from REGPT and linear theo17
lculations,FIG.
respectively.
Left panel
focuseson thespectrum
behavior
aroundbaryonacoustic
peakandinREGPT
linearcalculations.
scales, while
right
panel
9 (color online).
Comparisonof power
results
betweenN-bodysimulations
In eachpanel,
the shows t

Cosmic propagators
Propagator should carry information on
non-linear evolution & statistical properties
Crocce & Scoccimarro (’06)

Evolved (non-linear) density fieldm (k; t)0 (k )

Initial density fieldD (kk ) (1)(k;(k;t) t) Propagator

Ensemble w.r.t randomness of initial condition

Contain statistical information on full-nonlinear evolution(Non-linear extension of Green’s function)

18

Multi-point propagators
Bernardeau, Crocce & Scoccimarro (’08)
integrated PT
Matsubara (’11)

As a natural generalization,nm (k; t)0 (k1 ) · · ·0 (kn )

Multi-point propagator= (2 )3(1 n) D (k

k)

(n)

(k1 , · · · , kn ; t)With this multi-point prop.
• Building blocks of a new perturbative theory (PT) expansion
Γ-expansion or Wiener-Hermite expansion• A good convergence of PT expansion is expected(c.f. standard PT)
19

(t)

Γ(t) (q 1 , · · · , q t ; η) (t)= Γac1 ···ct (q 1 , · · · , q t ; η)uc1 · · · uct(t) aΓa (q 1 , · · · , q t ; η) = Γac1 ···ct (q 1 , · · · , q t ; η)uc1 · · · uct
(12)
$
%$
%$
%
&
%$
%$
%
&
##r $
!For
" "spectrum,
power
P
(k;
η)
=
P
(k;
η),
! the matter
+
s
s
+
t
t
+
r
r
+
s
s
+
t
t
+
r
11
3′ ′′3 ′′ 3 ′′3 ′′ForΨthepower
P (k; η) = P11 (k; η), r!s!t!
)Ψ1bPower(kb2(k)Ψ(kspectrum) spectrum,d3r qd′13.q.′1.Initiald. .3.qd′s3 d
Ψa1matter(k)Ψ
r!s!t!d3 qd13.q.1. d. .3.qdr3 q
qpower. .spectrum.qdt qt
a (k
2c)Ψ
c2(k
2 )= =
sqd1 .q.1. d
r
s
t
r
s
t
!
)r,s,tr,s,t
*2
)
*2
3
!
)
*
3 d) q(1) *2(2)2 η)P (k; tη) =(1)×δ
Γ ×δ
(k;(k
η)
P
(k)q−′+q2′)dδ q
P+′′0 (q)P(|k
′ Γ ′ (q,′′ k ′′− q;
t
0−
(2)
3
(k
−
q
(k
+
q
−
q
)
δ
+3q
+−q0+
)− q|)
−
q
)
δ
(k
+
q
−
q
)
δtD03(kq′′1...tq1...r)
1...r1...r
1...r
P (k; η) = Γ (k;Dη)D1 P10 (k)+ 2 1...s1...sD 3(2π)(q,1...sk −1...tq; 1...tη) D (kP(q)Pq|)
D2 Γ 2 1...s
1...t
0 (|k
! 6 ′3 (2π)(s+t)(s+t) ′
(r+s)′*2 ′′′ ′′
(r+s)
′
!
×Γ×Γ
(q
,
.
. 1. ., ,. q(q.3.rpd,, qq)r1,,qq. .′1)., ,. q.(3).′s,)qΓ′sb) Γb (−q(−q. . , −q
. .′′t,)q′′t )1 6.1., ,.dqa a
1 , . .1 .,*,.2−qs , qs1,,qq+ 6d′′ pd
Γ (p, q, k − p η)
− q;Pη)(p)PP0 (p)P(q)P−0 (|k− p − q|) ′′ (13)
0(|k
6 (3)
(t+r)
′ 0
′ −′ q|)′′+ ·′′ · ·(t+r)′′ (2π)′′Γ
′′ (p, q, k − p − q; t
′
+
6
(q)P
p
0
0
× Γ×
, . 1. ., ,.6−q) .1. ). P). t′′ ).
(−q. . , −q
. . , −qP10)(q. 1. ). P
. .0.(qPr0)(qPr 0) (qP10)(q. 1. ). P
. .0.(qPs0)(qPs 0) (qP10 (q. .0.(qPt0 (q
1 , . .1., ,. −qr ) Pr0)(qc Γc (−q1(2π)
t , −qt , −q(59)(59)(p) (p) (p)1-loop2-loopwith.
initial P(k)
q
-q
with
Γ(p)Γ = Γ=1 Γ.1 P(k)
q
-q
This
is diagrammatically
represented
in Fig.
it runs
numberof linesconnecteachThissumsumis diagrammatically
represented
in Fig.
13. 13.
WeWe
see seethatthat
it runs
overoverthethenumberthatthatconnecteach
pof-plines= theof thediagram(withconstraint
at mostof theindicesis zero,otherwise
we
would
have
sidesideof thediagram(withtheconstraint
thatthatat mostoneoneof theindicesr,otherwise
we+
would
have
+
2
+ sr,6ors torist zero,...
k-k TheThek leading-k(tree)kcontribution-kobtained
k for fora disconnected
diagram).
leadingorder(tree)
contribution
is thenobtained= t1,=-kt 0=(plus
0 (pluscyclica disconnected
diagram).
orderis thenr =r s==s 1,
cyclick-q
k-p-q-(k-p-q)
permutations),
to one-loop
corrections
squarebrackets)wehavepermutations),
up up
to one-loop
corrections(in (insquarebrackets)we -(k-q)
have

Bispectrum(2)(1)(1) (1)
) =
2 Γ(k
) Γ(k
P10 )P(k10)P+ cyc.
B(kB(k2 Γ(2), k12,)kΓ2(1)(k20 )(k+2 )cyc.
1 , k12, k23,)k3=
1(k1 )(kΓ1 ) Γ(k2 )(kP20)(k
&
&
'
'
(2)
(2)
(2) (2) (k + q, −q)Γq Γ(kq, q)Γ
(qk−− q)P− q|)P+ q|)P+ +8 8d3 qdΓ3(2)−1q,−q)Γ
(k2 +2q, −q)Γ(2) (q−−k−1q|)P+2q|)P
1 , −k1(k1 ,k2 −2q)P
0 (|k01(|k0 (|k02(|k0 (q)0 (q)
& &
( (
3
(3)
(2)
(1)
(2)
(1)
q Γ(−k(−k+−q)Γ
q, −q)Γ
q, q)Γ
− q|)P(k+3 )cyc.
+ cyc.
+ +6 6d3 qdΓ(3)(k2(k−2q,−q)Γ
(k3(k)P30)P(|k02(|k−2q|)P. . (60)(60)
3 , −k0 (q)P0)
3 , −k
2 +2q,
0 (q)P
0 (k3

k3

k1

k3

k1

k3

k1

information
about
initialpower
spectrumNotehavingresummed
multi-point
propagators information
about
thetheinitialpower
spectrum
P0 ,Pit0 , isit isNotethatthat
havingresummed
thethemulti-point
propagators
=
B(k
1
,
k
2
,
k
3
)
2
+
6
+
cyc.
+ 8 advantageous
advantageous
to simultaneously
measurepower
specmeans
manyof theone-loopcorrections
in standard
to simultaneously
measurethethepower
specmeans
thatthat
manyof theone-loopcorrections
in standard(p)trum
higher-order
spectraat large
scalesincludealready
encodednumberandandhigher-order
spectraat large
scalesandandincludePTPT
arearealready
encodedin Γin(p)Γ andandthusthus
thethenumberof of trum
+parameter...parameter
relationships
when
cosmologicalone-loop
diagrams
is reduced.
power
spectrum thesekpower2
kthese2 relationships
when
doing
cosmologicalone-loop
diagrams
is reduced.
ForForthethespectrumkdoing2tree
1-loopestimation.
have
insteadof twodiagrams,
bispectrum20estimation.
we we
have
oneoneinsteadof twodiagrams,
for forthethebispectrum
As a preliminary application of these results, we comwe have two instead of the four in standard PT [21].

p11= 3Generic property of propagators
(1)

(1)

Crocce & Scoccimarro ’06, Bernardeau et al. ’08

FIG. 7. This figure illustrates the effect of the time-ordering exchanges (thick double arrow lines). Through such exchanges, thecomplete set of diagrams that correspond to a given fpij g can be explored. Successive time exchanges can, however, lead to identical
diagrams (e.g. left and right diagrams in each panel). The unordered time integration then leads to a multiplicity factor for each
2 is2obtained each time two-loop lines are exchanged. There
2 diagramdiagram. Forkthe top+panel (corresponding
to p12 ¼ 3),2the sames2v
either by theare p12 ! of such possible exchanges.
n For the1 bottom paneln(corresponding tovp11 ¼ 3), the samevdiagram can be obtained
2exchange of loop lines, or by the exchange of the initial and final times of each of their loops (double arrow dashed line). There are thus
p11 !2p11 of such possible exchanges.

(n)

(n)

(n) /2 k
(n)
k
F (k
,
·
·
·
k )
tree e tree e(k) PHYSICALREVIEW D 78, 103521 (2008)
with !
defined in Eq. (19). This is a truly remarkable

(1)

RO

LTIPOINT

opagator

e two-pointr theory andfields. The
d in detail in
otivate theirdefinition inwith respect; si Þ þ . . .

(24)r correction
near regime.
e-loop’’ dian in Fig. 3.
is expected
e correspon-

ð2Þ
abc;treeresult. It shows that the whole effect of loop summation is
encoded in the value of k3 in exactly the same way as forthe two-point propagator Gab .
We now compare this result to measurements in numerical simulations, which will be described in detail in
Sec. VI. As done for the two-point propagator (seeFig. 4), we test for the Gaussian decay in the high-k limit
by plotting in Fig. 8 equilateral configurations
2log!ð2Þ
1 ðk; k; kÞ vs k , for which Eq. (37) predicts a straightline with known slope. We do so for three different redshifts, z ¼ 0, 0.5, 1, finding very good agreement in allcases with the predictions of Eq. (37), shown by solid lines.
This validates our resummation scheme.
Equation (37) and its generalization to other multipoint
propagators have important implications for the power
spectrum and higher-order statistics, that we discuss inSec. VII. We note also that a second, faster method to
perform the loop resummation is discussed inAppendix B. We now consider the extension of theseresults to arbitrary multipoint propagators.
FIG. 4 (color online). The large-k limit of the two-point den-

21

(2)

/2 ;

=

dq
P (q)
6

(k1 , k2 , k3 )Equilateral

FIG. 8 (color online).

The large-k limit of the three-point

Origin of Exp. damping

A 2% Distance to z = 0.35 : Meth

P0 (k)For Gaussian initial condition,m (k; t) 0 (k ) =(1)(k; t)

0 (k) 0 (k )initial power spectrum

Cross correlation between initial & evolved density fields
A 2% Distance to z = 0.35 : Methods and Data

initial3

evolved (by Zel’dovich approx)
150MpcPadmanabhan
et al. (’12)

Initial structure becomes blurred by the local cosmic flow

Figure 1. A pictoral explanation of how density-field reconstruction can improve the acoustic scale measure
show a thin slice of a simulated cosmological density field. (top left) In the early universe, the initial densities ar
the acoustic feature with a ring of 150 Mpc radius from the central points. A Gaussian with the same rms widthof the black points from the centroid of the blue points is shown in the inset. (top right) We evolve the particle
by the Zel’dovich approximation (Zel’dovich 1970). The red circle shows the initial radius of the ring, centered o
the blue points. The large-scale velocity field has caused the black points to spread out; this causes the acoust
The inset shows22the current rms radius of the black points relative to the centroid of the blue points (solid line

origin of Gaussian damping in propagatorfinite-mode sampling also affects the calculation of thetwo-point correlation function. We thus correct it by subtracting and adding the extrapolated linear density field as^ % !^ lin ðrÞ þ !lin ðrÞ, where !^ lin is the correlation func!ðrÞ
tion estimated from the Gaussian density field, and !lin is
the linear-theory prediction of two-point correlation
function.

Born approximation in the improved PT are all positiveand mostly the smooth function of k. Further, the higherorder corrections in the improved PT have a remarkablescale-dependent property compared to those in the standard PT; their contributions are well localized around somecharacteristic wave numbers, and they are shifted to thehigher k modes as increasing the order of PT. These trends
clearly indicate that the improved PT with closure approximation has a better convergence property. Qualitative beNishimichi,
Codis
haviors ofAT,
the Bernardeau,higher-order correctionsquite resemble
the (’12)
predictions of RPT by Crocce and Scoccimarro [34].
AT et al. (’09)
Now, let us focus on the behavior of BAOs, anddiscuss how the convergence properties seen in Fig. 4
affect the predictions of BAO features. In Fig. 5, adopting
the WMAP3 cosmological parameters, we plot the ratio
PðkÞ=Pno-wiggle ðkÞ, where the function Pno-wiggle ðkÞ is theWhy improved PT works well?
B. Results in real space
1. Power spectrum
Before addressing a quantitative comparison between
the N-body simulation and improved PT, we first discuss
the convergence properties of the improved PT, and consider how well the calculation based on the improved PT
does improve the prediction compared to the standard PT.

• All corrections becomecomparable at low-z.
• Positivity is not guaranteed.

Corrections are positive & localized,
shifted to higher-k for higher-loop

Improved PT(RegPT) [ (1) ]2 P0Standard PT

negative

Linear1-looppositive

2-loopnegative

positive23

positive

FIG. 4 (color online). Convergence properties of standard PT (left) and improved PT (right) expansions in the matter power

3-loop : source of trouble
Further including 3-loop (i.e., next-to-next-to-next-to-leading order),PT calculations start to get worse !!
z=1.75N-body simulations
Standard PT 2-loop
Standard PT 3-loop
Linearz=0.35next-to-next-to-next-to-leading
order (3-loop)
Blas et al. (’14)
24〜13-loop : source of trouble
Further including 3-loop (i.e., next-to-next-to-next-to-leading order),PT calculations start to get worse !!
z=1.75N-body simulations
PT 2-loop
COSMIC PROPAGATORSStandard
AT TWO-LOOP
ORDER
Standard PT 3-loop
LinearPn-loop (k)d ln q Kn-loop (k, q) P0 (q)PHYSICAL REVIEW D 89, 023502 (2014)

A large UVcontribution !!

z=0.35next-to-next-to-next-to-leading
order (3-loop) Bernardeau, AT & Nishimichi (’14)
FIG.et11al.
The shape
Does this really happen
in(color
realonline).
universe? of the kernel functions
Blas
(’14)
FIG. 10 (color online). Regular parts of the density propagator
Reg Gp−loop ðkÞ at one-, two-, and three-loop order with, respec1þ

P0 ðqÞK 1-loop ðk; qÞ (blue solid line), P0 ðqÞK 2-loop ðk; qÞ (green
24〜2dashedline) for k ¼ 0.1 h=Mpc and P0 ðqÞK 3-loop ðk; qÞ (red dot-Nature of nonlinear mode-coupling
How the small-scale fluctuations affect the evolutionof large-scale modes ? (or vice versa)
How the small disturbance added in initial power spectrum cancontribute to each Fourier mode in final power spectrum ?
P0 (k)

P0fid (k) + P0 (k)Pnl (k)P0fid (k)

fid
Pnl
(k)

InitialN-body

Final
PT

25〜1

Nature of nonlinear mode-coupling
How the small-scale fluctuations affect the evolutionof large-scale modes ? (or vice versa)
How the small disturbance added in initial power spectrum cancontribute to each Fourier mode in final power
spectrum ?
ResponsefunctionP0 (k)

P0fid (k) + P0 (k)Pnl (k) =P0fid (k)Final (nonlinear)

Pnl (k)d ln q K(k, q) P
(q)
0
fid
Pnl (k)N-bodyinitial (linear)

Initial

Final
PT

25〜2

scenarios. Here we wish to introduce a well-define
ynamics of the cosmic fluid
function and investigate it at fully nonlinear lest when gravity only is atconsider the nonlinear power spectrum as a fu
methods has been heightnl
nl
of
the
linearpower
spectrum,
i.e.,
P
=
P
[P
he baryon acoustic oscilladefine the kernel function as its functional deriv
g of galaxies at late times
Nishimichi, Bernardeau & AT (’16)
nledictions of the nonlinearδP(k; z)
Response of power spectrum at k
K(k, q; z) = q lin.
ally important.
to a small initial variation at q
δP (q; z)ely that mode couplings beWe omit the explicit dependence on z from the aridable. It makes PT results
Even for The
low-knormalization
modes,
inwhat
follows.
for K is cho
n a controlled manner. We
nl a large UVStandard PT
gets
that a small variation
in P
is related to that oh couplings with the use of! (q-modes):contribution
n[15], defined as the linear negative
nl
lin
δP
(k)
=
d
ln
q
K(k,
q)δP
(q).
2-loop>1-loop>N-bodyh respect to initial perturectrum at wave-mode q. In
ns Ref. [3] showed progresThis relation provides us a simple way to measure
In otherfunction as increasing thenel function
fromwords,
simulations. In order to do so
low-k
mode inwith
simulation
at a regularizationpare two initialconditions
small modulatio
positivescheme
is UV-insensitive

A measurement result

protected against small-scale uncertainty
26

near mode at k = 0.161 h Mpc 1 indicated byarrow. The filled (open) symbols show L9-N9lines depict L9-N8, while the big hatched symscales are L9-N10. Positive (negative) values
as the upward (downward) triangles or the solids.

function in the high q region. This strongly suggests thattem, that
this anomaly is genuinely non-perturbative.
Displacem
We propose an e↵ective description of this observed
pected to
behavior. As illustrated in Fig. 4 it can be modeled with
spectra. T
a Lorentzian:driving th
FIG. 3: Rescaled response
function,
T
(k,
q)
⌘
[K(k,
q)
⇥
⇤
cluster of
1
lin
loop
T e↵. (k,q)(k)].
= T 1 PT(k,q) + T 2 loop (k,q) shown by
K lin (k, q)]/[qP
calculationsarelines,
geneities w
1 + (q/q0 )2
whereas the symbols are L9-N9 (see legend for detail).
The
dependen(4)1nonlinear characterizedwave-modebybin
is fixed at k =0.161wave
h Mpc
homogenea time-dependent
criticalmode, (verNishimichi,
Bernardeau
&
AT
(’16
&’17
in
prep.)
2
It is howtical arrow).
is taken where
into accountthegrowth
analyticalq0 (z) Binning= 0.3D+ (z)h/Mpc,
D+ is the tolinearevaluate t
factor,and the prefactor
is determined by fitting tocalculations
consistently
to the0.3simulations.
nlSumma
the data. Note that, as it can be checked in Fig. 4, q0 issurement
independent of k preserving the k dependence of the re0
pendencesponsefunctionat thesmall scale
limit. Thisat
dependence
We thenrescaletheresponsefunctionvarious redspectrum
is in full agreement with PT predictions.
shifts as T(k, q) = [K(k, q) K lin (k, q)]/[qP lin (k)], where
surement
lin
K is the linear contribution, and plot them in Fig. simulation
3.
The resul
They are compared with the one-loop PT calculation
resolution(solid), which is time-independent with this normalizathat meastion. The simulation data indeed shows little time deopment opendence at q . k in remarkable agreementwith theThe res
1next andone-loop calculation, reproducing /the1 expectedqdepen2+ (q/q0 )with mea
dence [44], as well as the change of sign between large andwide rang
transfers
small scales. The small but non-negligible z-dependencepressed c
at k ⇠ q is further reproduced by the two-loop calculaat late timtion (see the figure legend). Note that at the wave-mode
tail with1
k plotted here (i.e., 0.161 h Mpc ), the two-loop SPTThese r
first give iprediction for the nonlinear power spectrum agrees withmological
simulations within 1% at z & 1 and the agreement gets
most of t
worse at lower redshift reaching to ⇠ 5% at z = 0 (seenals the v
order. It
e.g., [10]).
FIG. 4: Response function divided by the two-loop PT at the1
regularize
modes k ,shownin the legend.
plot data points
At27〜1
q &three
0.3waveh Mpc
however,the Wemeasured
response
also markonly at q
2k for definiteness. The over-plotted solid lines

Refined measurement

Response of power spectrum at k
to a small initial variation at q

P (k; z)K(k, q; z) = q
P (q; z)1,400 simulations
2-loop SPT

onse function predicted by PT (un-binned) up
solid) and two-loop (thick solid) order at k =t z = 1. Dashed (dotted) lines show each of theLorentzian

UV suppression is seen at various knear mode at k = 0.161 h Mpc 1 indicated byarrow. The filled (open) symbols show L9-N9lines depict L9-N8, while the big hatched symscales are L9-N10. Positive (negative) values
as the upward (downward) triangles or the solids.

function in the high q region. This strongly suggests thattem, that
this anomaly is genuinely non-perturbative.
Displacem
We propose an e↵ective description of this observed
pected to
behavior. As illustrated in Fig. 4 it can be modeled with
spectra. T
a Lorentzian:driving th
FIG. 3: Rescaled response
function,
T
(k,
q)
⌘
[K(k,
q)
⇥
⇤
cluster of
1
lin
loop
T e↵. (k,q)(k)].
= T 1 PT(k,q) + T 2 loop (k,q) shown by
K lin (k, q)]/[qP
calculationsarelines,
geneities w
1 + (q/q0 )2
whereas the symbols are L9-N9 (see legend for detail).
The
dependen(4)1nonlinear characterizedwave-modebybin
is fixed at k =0.161wave
h Mpc
homogenea time-dependent
criticalmode, (verNishimichi,
Bernardeau
&
AT
(’16
&’17
in
prep.)
2
It is howtical arrow).
is taken where
into accountthegrowth
analyticalq0 (z) Binning= 0.3D+ (z)h/Mpc,
D+ is the tolinearevaluate t
factor,and the prefactor
is determined by fitting tocalculations
consistently
to the0.3simulations.
nlSumma
the data. Note that, as it can be checked in Fig. 4, q0 issurement
independent of k preserving the k dependence of the re0
pendencesponsefunctionat thesmall scale
limit. Thisat
dependence
We thenrescaletheresponsefunctionvarious redspectrum
is in full agreement with PT predictions.
shifts as T(k, q) = [K(k, q) K lin (k, q)]/[qP lin (k)], where
surement
lin
K is the linear contribution, and plot them in Fig. simulation
3.
The resul
They are compared with the one-loop PT calculation
resolution(solid), which is time-independent with this normalizathat meastion. The simulation data indeed shows little time deopment opendence at q . k in remarkable agreementwith theThe res
1next andone-loop calculation, reproducing /the1 expectedqdepen2+ (q/q0 )with mea
dence [44], as well as the change of sign between large andwide rang
transfers
small scales. The small but non-negligible z-dependencepressed c
at k ⇠ q is further reproduced by the two-loop calculaat late timtion (see the figure legend). Note that at the wave-mode
tail with1
k plotted here (i.e., 0.161 h Mpc ), the two-loop SPTThese r
first give iprediction for the nonlinear power spectrum agrees withmological
simulations within 1% at z & 1 and the agreement gets
most of t
worse at lower redshift reaching to ⇠ 5% at z = 0 (seenals the v
order. It
e.g., [10]).
FIG. 4: Response function divided by the two-loop PT at the1
regularize
modes k ,shownin the legend.
plot data points
At27〜2
q &three
0.3waveh Mpc
however,the Wemeasured
response
also markonly at q
2k for definiteness. The over-plotted solid lines

Refined measurement

Response of power spectrum at k
to a small initial variation at q

P (k; z)K(k, q; z) = q
P (q; z)1,400 simulations
2-loop SPT

onse function predicted by PT (un-binned) up
solid) and two-loop (thick solid) order at k =t z = 1. Dashed (dotted) lines show each of theLorentzian

UV suppression is seen at various kWhat’swrong?
Short summary• Higher-order mode-coupling gets a larger UV contribution
Blas, Garny & Konstandin (’14), Bernardeau, AT & Nishimichi (’14)

However !

• In simulation, actual UV contribution is suppressed
Nishimichi, Bernardeau & AT (’16, ’17 in prep.)

Most likely

Breakdown of single-stream PT treatment(even at large scales)

What is a role of small-scale dynamics ?
Is there a way to go beyond single-stream PT ?

Multi-stream flows(formation/merger of halos)
28

Suto et al. (2016)

Fig. 2.— Halo I’s color contrast images of the phase space density (left) and th
dashed), radial (black solid) and tangential (red dotted) velocity dispersions, for z
image for z = 0, the self-similar solution (Filmore & Goldreich 1984; Bertschinger

ng
tsnd
w
aite
cs
er
ihe
ncch
re-

sheet. Thus, solving Vlasov-Poisson system with cold initialcondition is mathematically equivalent to solving the equation of motion for each mass sheet:

1D cosmology

dx
v
= ,
Simplification
may help us to understand what’s going on
dt
a
dv
1
+ H v = − ∇x φ,
dt
a
x
2
2
∇x φ(x) = 4π Gρ a δ(x),

(7)

(8)

(9)

where the density field δ. To deal with this system, especially
Force / (# of sheets at RHS) - (# of sheets at LHS)
for the dynamics after shell-crossing, it would be essential
to introduce the Lagrangian coordinate, q, and to express• Generic features of nonlinear mode-coupling :the comoving position and the peculiar velocity of the mass
Response function
element as x(q, t) and v(q, t). Assuming the uniform density• Perturbative description beyond shell-crossing: Post-collapse PT
in the Lagrangian coordinate, the mass conservation implies
Learn something in simple 1D„ cosmology
«−1
∂x
29

19 91MNRAS.249. .2413

Mon. Not. R. astr. Soc. (1991) 249, 241-247

Sakagami & Gouda (’91)

On the collective relaxation in self-gravitating stellar systems
Masa-aki Sakagami
Faculty of Education, Fukui University, Bunkyo 3-9-1, Fukui 910, Japan

269

Naoteru Gouda
Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606, Japan

Yamashiro, Gouda & Sakagami (’92)

Progress of Theoretical Physics, Vol. 88,No. 2, August 1992
Accepted 1990 October 16. Received 1990 September 12; in original form 1988 July 29

Origin of Core-Halo Structure
in One-Dimensional Self-Gravitating System

SUMMARY
We study the applicability of the method by the geodesic deviations to relaxation
processes in stellar systems. Using the one-dimensional self-gravitating Af-sheet
YAMASHIRO,GOUDA and Masa-aki SAKAGAMI*
model, we investigate the relationship of the value Toshinobuof the curvaturein the equation Naoteru
ofthe geodesic deviations to two relaxation time-scales observed in numerical
Department
ofexist.
Physics,
Kyoto University, Kyoto 606-01
simulations. The expected naive correspondence between
them does not
It*
Departmentof
Education,
Fukui University, Fukui 910
follows that the rapid time-scale of the relaxation, which was suggested by Gurzadyan& Sawidy, has nothing to do with the evolution of macroscopic observables.

(Received March 23, 1992)
1

INTRODUCTION

scale shorter than rb, the evolution of the self-gravitating NTheberelaxation
self-gravitating systems is examined by using one-dimensional numeribody systems candescribed processby theof collisionlesscal simulation.
We get the asymptotic distribution function which disagrees with that proposed by
Boltzmann equation(Vlasov equation),

Real stellar systems, globular clusters and elliptical galaxies,
Lynden-Bell. Our distribution function has two peaks in low and high energy regions and a valley
can be approximately regarded as systems oí N gravitating
dt/(x, v) + v • dj{\,v) mediumdx (j> ' dj(\,energyv) = 0. region. This(1.1)
inthecharacteristic core-halo structure in phase space has been
bodies. Due to the fact that gravitation is a long-range attracin encounter,many simulations.
Vire clarify the dynamical mechanism which generates this 'core-halo'
In the case observedof the binary^(x, v) was recognizedtive force, it is well known that Debye screening is absent in
structure.
The essenceof thisis that the elements of the system are accelerated (oras a slow-varying
mean gravitational
potential
andmechanism
we dealt
gravitating systems as distinct from a plasma. This remarkdecelerated)
effectively
byHowever,
the evolving gravitational potential in a specific energy region,
with only relaxation
due to very
two-body
collisions.
able nature results in the rich features of their evolution,
633
since the gravitation
is long
does not cause thealthough it makes their behaviour very complicated. The
reflecting
the range
initialandconditions.
Debye screening, the potential (f is strongly affected bygravity plays an essential role in the formation of structure incollectivea system and varies rapidly, except in thestellar systems. Hence,in orderoftoTheoretical
discuss theirPhysics,
structureVol. 81,
Progress
No.3,motion
Marchof 1989
case of the system in a near-stationary state. Thus the energy
formation, it is very important to understand the general§ 1. Introductionof each star along its orbit is no longer conserved and thefeatures of the evolution of the self-gravitating systems. For
phase densityCosmological/, which obeys the Vlasov equation (1.1), is
Growthof states
One-Dimensional
example, we Non-Linear
must clasify their (quasi)
stationaryandSomestellarsystems,suchas elliptical galaxies, seem to have alreadyviolently changed.
Theseclasses
processesof
cause
the relaxationofinvestigate relaxation mechanisms to such states.
Density
Fluctuation
andCatastrophe
Theorymacroscopic
quantities,
known
as thefinal
violent
relaxation
There are two well-known mechanisms of relaxation insettleddown
to theirequilibrium
configuration. This is strongly suggested from(Lynden-Bell 1967).
self-gravitating Af-body systems. One is the relaxation due tothe fact
that mostgalaxiesshow the universal luminosity profile which is
The qualitative
discussion
on the ellipticalviolent relaxation
sugbinary encounters whichNhas
been considered
detailTakashi
byaoteruGOUDA inand
NAKAMURA
gests that characterized
its time-scale is of thebysameorderawof magnitude
asR1!4-l.*} Moreover,most giant elliptical galaxies have little
Chandrasekhar (Chandrasekhar 1960). Each encounterthat of the motion of each star, i.e. the free-fall time,causes a deflection of the star from its original direction ofrotation. Department
Physics,
Kyoto
Kyoto and606 random stellar velocity. The velocity dispersion of the stars in the
motion and an exchangeof energyof
betweenthe twostars University,
rf = (4jrGpav)-1/2.
(1.2)
innerTsukuba
few kiloparsecs
is also universallycorrelated with luminosity according to thetaking part in theencounter. Laboratory
These effects arefor
accumulatedas
National
High Energy
Physics,
305
Although there are many numerical2 simulations for thethe star successively undergoes encounters with other stars.
)
Faber-Jackson
law.
Therefore,
there
must
be a common relaxation mechanism
violent relaxations (van Albada 1982; Hohl & Feix 1967;
The relaxation time for the binary encounters, rb, represents(Received
July 21, 1988)
guarantees
this 1987)
universality.
Severne &which
Luwel 1986;
Reidl Jr & Miller
it seems that The ordinary relaxation mechanism due to thea time-scale within which the star has considerably
deviated30
a few analytic
treatmentsoftheseprocesses
havebeen carfrom the original orbit and forgotten its initial state. It is well
two-body
gravitational
interaction(binary encounter) has been the only wellWethat
have
the relaxation
non-linear
growthof densityried
fluctuations
a one-dimensional
self- be typicalout. The in
Lynden-Bellapproach might
known
theexaminedcorrespondingtime
for elliptical**

Downloaded froma

Gouda & Nakamura (’89)

stream motion around t
In
one-dimensional
case,
the
so-called
Zel’dovich approxima2.2 Initialcondition andpre-collapse dynamics
Initial conditionand pre-collapse
dynamics
Zel’dovichtion givesInanone-dimensional
exact solutioncase,for thetheso-called
dynamics
of massapprox
sheet
tionthegives
an exact
solution forthe dynamics
of mass s
one-dimensional
so-called
approximae-dimensionalIncase,the so-called
Zel’dovich
approximabefore case,shell-crossing.
TheZel’dovich
Zel’dovich
solution
also
provides3.1 peP
3.1 Post-collapse
beforeshell-crossing.
The ofZel’dovich
solution
also prois
tion
gives ansolutionformassthe colddynamics
mass sheet
gives an exact
solution
foraexact
thedynamics
ofsheet
naturalbasis fortheinitialcondition.
The solution
a naturalbasis
forsolution
the cold also
initialcondition.
TheThesolutib
Theprovides
basic
formalism
tobeforeThe
Zel’doviche shell-crossing.
Theshell-crossing.
Zel’dovich
also
providesgiven bysolution
by(Zel’dovich
follows
Startingthnaturalbasis forthegivencold The
initialcondition.
solutionis ’70) withural basis forathe
cold initialcondition.
solutionis Thefollows.
dD+ (τ
) we(τprewefirstfollow
the
dD
)firs
given
by
by
x(q; τ ) = x(q;q + τψ(q)(τ ), D+ (τv(q;τ ) v(q;= ψ(q).+ (20)Exact) = qD++ψ(q)),: linear
τ
)
=
ψ(q)
.
growth factordτ dτ
Zel’dovich
solution.
Th
Zel’dov
single-stream
dD+ (τ )dD+ (τ )thefunction
D+τ.isthelinear
growthfactorsatisfyingτ ) = q + ψ(q)
D+ (τ
v(q;
=
ψ(q)
(20)
x(q;
τ )),= q +Here,ψ(q)τ )D), thev(q;
)=
ψ(q)
(20)
shell-crossing,weswitchshell-cdisplacement
fieldfactor
+ (τ
Here,
functionD
linear. growthsatis+ is :thesolutiondτdτ
the following
equation:compute the backreactio
compu
the following
equation:
the functionHere,D+ isthethefunction
linearfactorsatisfying
D+» is 2thelinearsatisfying» 2 growth–growth– factor
anexplicit functionalforan expPhase-space
d
3
d
3
2
2
ollowing equation:
the following equation:− Ωm,0) H0Da(τ
)) =
D+0.
(τ
) =shell-crossing
0.
− Ωm,0(21)
theregion.
+ (τ
the
she
2 H02 a(τ2
Shellcrossing
dτ
dτ
2–
» 2–
exerted at each position
3 v
exertedd322− Ωm,0 H0 a(τ ) −D+Ω
(τm,0
)Note
=H0.
(21)
)Note
Dterms(τ ) in=
0.the cosmic
(21)
terms
of the cosmic
timet,(21)
Eq. is(21)is red
+that
0 a(τ
that
in
of
time
t,
Eq.
reducedfrom(19).
Integrati
2
from
E
2
dτ
2
to
the
standard
form
of
the
linearevolution
equation:to the standard form of the linear evolution
equation:tain
tain the correction
of th
th
»
–
that in termsNote
ofx thecosmic
time
t,
Eq.
(21)
is
reduced22that in »termsof theis reduced
xd t,32Eq.
Eq. (16). Further
d cosmic timeΩ–m,0(21)H0from2 xfromin
E
d
d
3
Ω
H
+
2H(t)
−
D
(t)
=
0.
m,0
0
+
e standard formofthelinearevolution
equation:
tofield
the standard form
ofdtthelinearevolution
equation:2+ 2H(t)
−
(22)
dt
2 a3D
(t)
+ (t) = 0.
Density23–
dt
dt2 – 2 a (t)» 2d3d Ωm,0 H02 d
1 The shell-crossing1 point
3=Ω0.
m,0 H0+ 2H(t) −
D
(t)
(22)
The
+
+
2H(t)
−
D
(t)
=
0.
(22)
+
3
δ dt
Solution
is
exact
until
2
3
2
dt a (t)dt
2 MNRASa (t) 000, 1–21 (2015)
from Lagrangian tofrom
EuleriL

1D Zel’dovich solution

0.04

0.04

0.04

0.02

0.02

0.02

0.00

0.00

0.00

!0.02

!0.02

!0.02

!0.04

0.0

!0.04

!0.04

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

10

10

10

8

8

8

6

6

6

4

4

4

2

2

0

0

0.2

0.4

0.6

0.8

1.0

0.4

0.6

0.8

1.0

shell crossing

MNRAS 000, 1–21 (2015)
20

AS 000, 1–21MNRAS(2015)
000, 1–21 x(2015)
x
0.0

0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

31

0.4

x

0.6

0.8

1.0

>
; |Q|
<not
Qc ,interested in th
>
are
n
o
>
>
>
>
−2 + B(q0 ; τ ) Q −andC(q0 ; τ ) Q3
subsequent section, but>>:
rized in Eqs. (53) and (5; |Q| < Qc ,F (q0 , τ ) = −ψ(q0 ) D+ (τ ), (65) for inner part, toget

Post-collapse PT:beyond shell-crossing
andvelocity

velocity

velocity

velocity

andC2.are defined by Eqs
where the quantities A, B, andF (q0 , τ ) = −ψ(q0 ) D+ (τ ),(30). Note that in (41)deriving Eq. (39), we have assum
bc
AT & Colombi
systemfollows
Zel’dovich
solution(’17)
at |Q| > Q
where the quantities A, B, and C are thedefinedby Eqs.
(28)–
resultant
expressions
are written
in terms of th
Breakdown
ofthatZel’dovich
solution(30). Notethat simulationin deriving Eq. (39), thewehave
assumedCold collapse
in 1-D
quantities characterizing
densityVelocity
peak at andpositionposi
b c . Since the3.3.1
the system follows Zel’dovich solution at |Q| > Q
the shell-crossing time τ , Eq. (39) is still applicable
the resultant expressions are written in terms of the local 0 Let us first consider the ocases in which there appear other shell-crossing reg
quantities characterizing the density peak atbposition q0 and(ii). In this case, the cor|Q| > Qc .
the shell-crossing time τ0 , Eq. (39) is still applicable to the
3 2cases in which there appear other shell-crossing regions at
∆v
(Q;
τ,
τ
b
)
=
−
H0
out
c
bc .
|Q| > Q2
flow to theformationof haloflow
3.3multi-stream
Corrections
Zel’dovichshell crossing

Providedposition
the explicit expressionposition
for the force in multi-×
3.3 Corrections to the Zel’dovich
flowwe now compute the corrections to the Zelregion,Computingback-reaction to the Zel’dovich
flow:
Lagrangian
flow based on the formal solution
Eqs.
Recallinginthe
fact(16)
thatanτb
Provided the explicit expression for the force in multi-stream
which givethe approximate
expression
the
above
are atperfo
1. Expanddisplacement
aroundshell-crossing
q0integrals: relevant
region, thewe now
compute the fieldcorrections
to theZel’dovich point,
valued region:
Appendix D [see Eqs. (D
3
flow
based
on
the
formalsolutionin
Eqs.
andZ (17),x(q; ) A(q0 ; ) B(q0 ; )(q q0 ) +C(q(16)
;
)(q
q
)
+
·
·
·
0
τ0
the ′first term, we obtainwhich give the approximate expression relevant at the multi-′∆v(Q; τ, τq ) =dτ F (x(Q,τ the)), short period afteforvaluedregion:2. Compute force F (x(q; )) = x (x(q; )) at multi-stream
region
τq
Z τ
Z τ
Z τ
′
′
′
dτ ∆v(Q; τ ′ , τq ).
∆v(Q; τ, τq ) =dτ F (x(Q, τ )), ∆x(Q; τ, τq ) = (42)dτ ′ a(τ ′ ) J (Q; q0 ,τqτqτbc (Q)
Z τ"ȷfunction
of
Q=q-q
0 on
up
to
7th
order
′polynomial′
Noticethat
dependingthepositioninLagrangianκ∆x(Q; τ, τq ) =dτ ∆v(Q; τ , τq ).
(43)
≃
a(τ
)
T
Q
+
−
32 of our interest, the expression of the
0 force is diﬀereτq8
position

position

10

A. Taruya and S. Colombi

10

A. Taruya and S. Colombi

10

A. Taruya and S. Colombi

10

A. Taruya and S. Colombi

10

A. Taruya and S. Colombi

10

A. Taruya and S. Colombi

Post-collapse PT: single cluster
10

A. Taruya and S. Colombi

AT & Colombi (‘17)

Post-collapse PT basically fails after next shell-crossing, but it still
gives
Simulation
and10S. Colombi
A. Taruya
and10S.reasonable
ColombiA. Taruya and10S.prediction
ColombiA. Taruya and10S.for
ColombiA.density
Taruya and10S.profiles
ColombiA. Taruya and S. Colombi

andS. Colombi
Colombind S.

Phase-space

Shell crossing

Next crossing

Zel’dovich
Post-collapse PT

Density profile

Figure 2. Snapshots of phase-space structure (upper inset) and density profile (lower
Figureinset)
2. forSnapshots
the single-clusterof phase-spaceformation
structurein Einstein(upper inset) and density profile (lower inset) for the single-cluster
de Sitter universe. For the initial density contrast given in Eq. (68), results of N -bodyde Sitter
simulations
universe.are
Fordepicted
the initial
as density
red lines,
contrast
while thegiven in Eq. (68), results of N -body simulations are depicted
analytic results with Zel’dovich solution are shown in green dotted lines. The blueanalytic
solid lines
results
are thewithprediction
Zel’dovich
withsolutionbasic post-collapse
are shown in green dotted lines. The blue solid lines are the prediction w
Figure2. Snapshots of phase-space structure (upper inset) and density profile (lower
Figureinset)
2. Snapshots
for the single-clusterof phase-spaceformation
structurein Einstein(upper inset) and density profile (lower inset) for the single-clusterPT
treatment.
PT
treatment.
de Sitter universe. For the initial density contrast given in Eq. (68), results of Nde-body
Sittersimulations
universe. For
are thedepicted
initialasdensity
red lines,
contrast
while given
the in Eq. (68), results of N -body simulations are depicted
analytic results with Zel’dovich solution are shown in green dotted lines. The blue
analytic
solid lines
results
are withthe prediction
Zel’dovichwith
solutionbasicarepost-collapse
shown in green dotted lines. The blue solid lines are the prediction w
PT treatment.
PT treatment.

Figure 2. Snapshots of phase-space structure (upper inset) and density profile (lower inset) for the single-cluster formation in Einsteinde Sitter universe. For the initial density contrast given in Eq. (68), results of N -body simulations are depicted as red lines, while theanalytic results with Zel’dovich solution are shown in green dotted lines. The blue solid lines are the prediction with basic post-collapse
Figure2. Snapshots of phase-space structure (upper inset) and density profile (lower
Figureinset)2. for
Snapshots
the single-clusterof phase-spaceformation
structurein Einstein(upper inset) and density profile (lower inset) for the single-cluster
PT treatment.

Of course, this does not guarantee the accuracy of powerde Sitter universe. For the initial density contrast given in Eq. (68), results of N-bodyde Sitter
simulations
universe.are
Fordepicted
the initial
as density
red lines,contrast
while thegiven in Eq. (68), results of N-body simulations are depicted
spectrumprediction
atsmall
scales(→next
slide)
analytic results with Zel’dovich solution are shown in green dotted lines. The blueanalytic
solid lines
results
are thewithprediction
Zel’dovichwith
solutionbasic arepost-collapse
shown in green dottedlines. The blue solid lines are the prediction w
PT treatment.

PT treatment.
33

Let us now consider a more relevanwith random initial conditions. Althou
setup in 1D, a relevant initial conditio
3D case may be given by the Gaussi
with the initial power spectrum:
Planck ΛCDM
2k
P1D (k) =P3D (k)2π

Post-collapse PT: ΛCDM
(Dimensionless) power spectrum
Linear

z=0

with
P3D
L = 1,
000being
Mpc the matter power sp# ofparticles (sheets): 2 transfer
wecomputed
with the⇥ 105 functi# of runs：50
MNRAS 000, 1–21Phase-space(2015)

z=0

Simulation
Zel’dovich
Post-collapse PT
AT & Colombi (‘17)
34Post-collapse PT: ΛCDM
Adaptive smoothing(Dimensionless) power spectrum
Linearz=0applied to initial density peaks(with filter scales determinedby first-barrier crossing)
Phase-space

Simulation
Zel’dovich
Post-collapse PT

AT & Colombi (‘17)
35Implication to 3D
Combination of the two methods are rather crucial:

PT scheme beyond shell crossing & Coarse-graining(post-collapse PT)(adaptive smoothing)

But, idea & technique are promising and can be extended to 3D

Issues to be addressed• Accurate pre-collapse description✓ Zel’dovich approx. is inaccurate✓ Various topologies of shell crossing• Tractable analytical calculation of
statistical quantities
36〜1

Implication to 3D
Combination of the two methods are rather crucial:

PT scheme beyond shell crossing & Coarse-graining(post-collapse PT)(adaptive smoothing)

But, idea & technique are promising and can be extended to 3D

Issues to be addressed• Accurate pre-collapse description✓ Zel’dovich approx. is inaccurate✓ Various topologies of shell crossing• Tractable analytical calculation of
statistical quantities
36〜2

N-body3rd
Lagrangian PT⇢(r) / r 1.7

Zel’dovich
Moutarde et al. (’91)

a)a. 32 + one level dynamic adaptive refinementb)State-of-the-art cosmological Vlasov code

The Astrophysical Journal, 762:116 (18pp), 2013 January 10⃝
C 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

doi:10.1088/0004-637X/762/2/116c)d)

DIRECT INTEGRATION OF THE COLLISIONLESS BOLTZMANN EQUATION
IN SIX-DIMENSIONAL PHASE SPACE: SELF-GRAVITATING SYSTEMS

2013

b. 323 + two level dynamic adaptive refinement

Kohji Yoshikawa1 , Naoki Yoshida2,3 , and Masayuki Umemura1

1 Center for Computational Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305–8577, Japan; kohji@ccs.tsukuba.ac.jp
Figure 14. The initial conditions for the “ripple-wave” test problem
2of Physics,
Theof Tokyo,113-0033,
Japan
Mon. Not. R. Astron. Soc. 000,Department
000–000 (0000)
Printed
12University
January 2015
(MN Tokyo
LATEX style
file v2.2)(cf. Sec. 4.2). Shown are
the particle locations (panel a), the density
3 Kavli Institute for the Physics and Mathematics of the Universe,
The Universityof Tokyo,
Kashiwa,
Chiba
277-8583,
field using
the tetrahedralphase spaceelements(panelb), using Japan
Received 2012 June 18; accepted 2012 November
23; published2012
December
20 elements (panel
tri-linear elements(panel c) andusing
tri-quadraticd). The linear elements are discontinuous at element boundaries,
while the quadratic is continuous.
c. 5123 N-body64^6

An adaptively refined phase-space
ABSTRACTelement method forcosmologicalsimulations
andcollisionlessdynamics
We present
a scheme for numericalsimulations ofcollisionlessself-gravitating
systems which directly integrates thetri-quadratic reconstructed from N-body 323

Vlasov–Poisson equations in six-dimensional phase space. Using the results from a suite of large-scale numericalsimulations, we demonstratethat the present scheme
can simulate collisionless self-gravitating systems properly.
?12
Oliver
Hahn
and
Raul
E.
Angulo†The integration
scheme
is
based
on
the
positive
flux
conservation
method recently developed in plasma physics.
1 Department of Physics, ETH Zurich, CH-8093 Zürich, Switzerland
2
We testthewaves
accuracy
ourcodeby performing
severaltest1,calculations,
Centrode Estudiosof
Fı́sicadel Cosmos
de Aragón, PlazaSan Juan
Planta-2, 44001, including
Teruel, Spain.the stability of King spheres, theA test case:sine(phasedespace
evolution)
gravitational instability, and the Landau damping. We show that the mass and the energy are accurately conserved for
all thesubmitted
test cases
we study. The results are in good agreement with linear theory predictions and/or analytic solutions.
to MNRAS Jan. 8, 2015
The distribution function keeps the property of positivity and remains non-oscillatory. The largest simulations are
run on 646 grids. The computation speed scales well with the number of processors, and thus our code performs
tri-quadratic 32 self-consistent
ColDICE: a parallel
Vlasov-Poisson solver using
movingadaptive
Figure17. The ripple
wave collapsesimplicial
test with dynamic adaptive
efficiently on massively parallelABSTRACT
supercomputers.
Dark matter numerical simulations and the N -body method are essential for understanding refinement. The 323 runs use the same initial conditions as in Fig. 15,
tessellation
Figure16.Universe.
Comparison
betweenthea reconstructionof thehow
structureforms–andevolvesinnumericalthe
However,
discretenatureof Ntri-body tri-quadratic elements and one (top, panel a), and two (middle, panel
Key words: galaxies: kinematics
anddynamics
methods:b) of dynamic adaptive refinement. The bottom panel shows thequadratic density field from the 322 standard N -body run (topv1 [astro-ph.CO] 8 Jan 2015

Cold initial condition

2016

3

simulations can a↵ect its accuracy when modelling collisionless systems.

2016

andthethe gravitational
self-consistent evolution
thecold
tri-quadratic
We introduce a new approachhalf-panel)
to simulate
evolutionof ofcollisionless solution of3 a high-resolution N -body run using 512 particles at theOnline-only material: color figures
same 256 PM force resolution. On clearly sees how adding moreelements (bottom half-panel). One clearly sees that N -body particle
3

a,b,c,⇤ “Lagrangian
fluids by solving the Vlasov-Poisson equations in terms of adaptively refineableThierry Sousbiea, Stéphane Colombi
supporting points approaches the high-resolution N -body solution.

noise significantly perturbs the solution, in particular, caustics arephase space elements”. Thesenot
geometricalpersistent. elements are piecewise smooth maps between Still, the top two panels have significantly fewer degrees of freedom
a Institut
three-dimensional Lagrangian
spaced’Astrophysiqueand six-dimensional
phase
and apthanbd
theN -bodyrun.
de Paris,Eulerian
CNRS UMR
7095spaceand UPMC,
98bis,
Arago,
F-75014
Paris, Franceb Departmentfunction.
proximate the continuum structure of the distribution
Theyallow for dynamicalof Physics,
The Universityof Tokyo, Tokyo 113-0033, Japanc Research
adaptive splitting to accurately
follow
the onevolutioneveninregionsofverystrongmixing.
movethesmoothgravitational
potential
calculated
fromthe113-0033, Japanusing
refinement
Figure17,
comparingonce
againstCenter
forin
the
Early
Universe,
Schoolof moreScience,
The
Universityof Tokyo,Tokyo
1. INTRODUCTION3
toy problems but let the images speak for themselves
The elements thus permit a deterministic
non-linearof
self-gravitating
the 512
particle
high-res Ndescription-bodysolutionat the sameforce coldoftheseoverall
distributionoftheparticles.
Despitethese
attractive
and perform a quantitative convergence study of refinement
and collisionless fluids in the continuous
limit.
resolution. We
only consider the tri-quadratic elements in thisfeatures,the major
disadvantageof theSCF
methodis itsGravitational interaction is one of the mostimportantphysicalin thenext section,
where we apply the Lagrangian element
We discussin detailvarious one-,and thethree-dimensional
test
problems
whichdemoncase,twoalthoughlinearelements also
perform
reasonably
well.
3 We show that our method has method to cosmological structure formation.
strate the ofcorrectness
and performanceof ourmethod.
inflexibility
that32thebasis
set must
so that the lowest
We started
withthe same
initialconditions
as inbethe chosen
fixedprocesses in the dynamics and the formation
astrophysseveral advantages
standard
Nshown-bodyinalgorithms
by nowi)global
explicitly
tracking
thethe systems underresolution
testterms
Figure 15, but
employed
the force of
orderreproducethestructureical objects, such as star clusters, galaxies,
and the comparedlarge- tofine-grained distribution function,ii) naturally
representing
caustics,
providing an
refinement
criterion witha thresholdof 0.1 toiii)
dynamically
Abstractthesefield
investigation(Weinberg
1999).
otherwords,
methodscale structure of the universe. Stars anddark matterarbitrarily
regularindensity
thatis defined
everywherein space,iv) Ingivingdirectlya 5theASCF
split
elements
if required(the resultsusingvelocityrefinement
FIRST APPLICATION: COSMOLOGICAL
arehowevernotsignificantlydi↵erent).
The
solutionallowing
canbeappliedonly
tothesymmetric
gravitational
collapseorOFtoAfollowing
smooth andResolving
regulargravitational
potential
field,
thus
eliminating
the
need
for
any
type
of
self-gravitating systems are essentially collisionless,exceptfornumerically Vlasov-Poisson equations for initially cold systemsSIMULATION
can be reduced
the UNIVERSE
evolution
WARM DMfor oneadditional
level
of refinement
isthe
shown
in the top panel,force softening.
thesecularevolutionofcollisionlesssystems.
a few cases, such as globular clusters and ad-hocstars aroundsupermasofathree-dimensional
sheet
evolving
insix-dimensional
phase-space.
We
describeapublicparallelnumerical
al- to athe one forlevelsthe middle
and the reference
Finally, we illustrate the feasibilityof twousing
ourin method
forpanel,cosmologicalstudies by We now apply our Lagrangian phase space element methodTheultimateapproachfornumerical
simulations
ofthesive black holes. The dynamics of the collisionlesssystemsiscosmologicalproblem.
Wesimulatethegravitational
evolutionN-bodysolutionatthebottom.
Rather
strikingly,
thesolutionsgorithm
consistingin
representing
thephase-space
sheet
with
a
conforming,
self-adaptive
simplicial
tessellationofsimulating structure formation in a warm dark mattercosmology. We show that spurious
37
ofbea L=20Mpc/hcube in a universe where dark matter isquickly
converge
to thereferencesolutionin theexact
shapecollisionlessself-gravitating
systemswould
directintedescribed by the collisionless Boltzmanncollisionality
equation which
orand
the large-scale
Vlasov
discreteness
noise
of
N
-body
methods
arebothstrongly
the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six- and four-

3 Sep 2015

Cold initial condition

Analytic treatment helps to understandwhat is going on in Vlasov simulations
and position of caustics. Already with one additional level, themade of warm particles of mass m= 250 eV, leading to a

Approaching shell-crossing in 3D
W/ S. Saga & S. Colombi (in progress)

In 3D, even the description of pre-collapse phase is non-trivial
Lagrangian PT treatment is the only way to analytically capture
the shell-crossing
5th LPT
4th LPT
3rd LPT
2nd LPT
1st LPT

⇢(r)
⇢

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r
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a = 0.03
LPT 5th order
LPT fastest 5th
LPT fastest 6th
LPT fastest 7th
LPT fastest 8th
LPT fastest 9th
LPT fastest 10th

by S. Saga

Approaching shell-crossing in 3D
���

y
r
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in

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W/ S. Saga & S. Colombi (in progress)

m
i
l
e
r
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Lagrangian PT treatment is the only way to analytically capture

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the shell-crossing-���Further pushing higher(y, z)calculation= (0, 0)order
may help ?
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x

⇢(r)
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5th���LPT
4th LPT
3rd LPT
2nd LPT
1st LPT������(y, z)=(0, 0)

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Shell crossing !
-���Simulation Ng = 528
Simulation Ng = 128

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r
38〜2

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Summary
宇宙論的（無衝突）自己重力多体系としての宇宙の大規模構造
その理論的記述をめぐる進展と混迷、あるいは（観測の理論テンプレート）

摂動論的計算手法の再生と受難• くりこみ・再和法の発展・観測的応用• UV問題とその起源、応答関数による定量化• 単一流近似を超える取り扱いと課題 → 今後に期待（ポストコラプス摂動論）

大規模構造は、今後も理論・観測ともに目が離せない39