宇宙大規模構造理論予言の現状と展望
西道 啓博 (KAVLI IPMU → 1月より基研)
Mathuriya, Bard, Mendygral+’18

He, Li, Feng+’18

Crocce & Scoccimarro ‘06

Bernardeau, Crocce & Scoccimarro ‘08

理論懇シンポジウム, 京大基研, 20 Dec 2018

宇宙の揺らぎと情報• 宇宙の揺らぎは確率場（と普通考える）• ナイーブにはオーダー 106 ~ 107 要素のデータベクトルの同時確率分布を相手にすることになる• 求めたいのは少数の宇宙論パラメタ• 主に2つのチャンネル•

Early time: 宇宙マイクロ波背景放射•

Late time: 大規模構造•

重力レンズ, 銀河分布, Lyα forest, intensity mapping, …

• 現在の標準的な理解によれば、•初期条件: インフラトンの量子揺らぎ、ほぼガウシアン•時間発展: 重力相互作用 + correction

TAKAHIRO NISHIMICHILSS REVIEW

宇宙論的統計解析:
CMBの場合X⇥(p̂) =a`m Y`m (p̂)`,mha`m a⇤`0 m0 i =``0 mm0 C`

Planck 2015 results. XIII.

‣ 宇宙背景放射の温度揺らぎマップ‣ 非常に良い近似でガウスランダム場‣ 理論、観測ともにこれを示唆

次元削減‣ a_l,mが独立なランダム変数‣ パワースペクトルC_lで尽くされる‣ 揺らぎは小さい (~10^-4)‣ 線形理論: 与えられた宇宙論パラメタに

観測データ

理論予言

統計量 X

…

対してシグナル、ノイズともすぐに計算可能‣ MCMC法によるパラメタサーチ(ベイズ推定)

ラベル

多次元宇宙論
パラメタ空間TAKAHIRO NISHIMICHILSS REVIEW

宇宙論的統計解析: LSSの場合‣ 銀河の大域的空間分布‣ ただし、δ_k1, δ_k2, …, δ_knは独立でない‣ 元をたどるとCMBと同起源‣ 球面調和関数展開 → フーリエ展開‣ 密度揺らぎδ_kがランダム変数‣ パワースペクトルが基本的な統計量(~x) =Zd3 ki~k·~xe~3(2⇡) k3 ~h ~k ~k0 i = (2⇡ 3 ) D(k + ~k 0 )P (k)SDSS BOSS final data release ©Jeremy Tinker and the SDSS-III collaboration

RESPONSE FUNCTION

TAKAHIRO NISHIMICHI

重力進化と非ガウス性/非線形性

RESPONSE FUNCTION

TAKAHIRO NISHIMICHI

重力進化と非ガウス性/非線形性

TAKAHIRO NISHIMICHIRESPONSE FUNCTION

重力進化と非ガウス性/非線形性
simulation result

linear theory

Valageas,TN, Taruya ‘13

TAKAHIRO NISHIMICHIRESPONSE FUNCTION

重力進化と非ガウス性/非線形性
simulation result

linear theory

Valageas,TN, Taruya ‘13

宇宙大規模構造の理論予言
観測データ• 将来観測からの要請

理論予言• ひとこえ、波数10h/Mpcまで 統計量 X

誤差1%以内…

• N体シミュレーション• 計算コスト•ラベル

MCMCに乗るか？

• 精度を担保できる？
• 機械学習の導入？

• できるだけ手で解きたい• 揺らぎが小さいうちは摂動展開で良さそう• より定量的な理解が必要

Schneider, Teyssier, Potter+’16

多次元宇宙論
パラメタ空間BASIC EQUATIONS FOR LARGE-SCALE STRUCTURE
Juszkiewicz ’81, Vishnac ’83, Goroff ’86,
Suto, Sasaki ’91, Makino, Sasaki, Suto ‘92•

Density and velocity kernel{Fn (q1 , . . . , qn ), Gn (q1 , . . . , qn )}continuity + Euler + Poisson eqs.

✔
✔
✔

Crocce & Scoccimarro 06(x) = ⇥(x)/¯⇥(x) =1· v(x)PERTURBATION THEORY IS IN CRISIS
N-bodylinear theory

Blas, Garny, Konstandin ‘14

PERTURBATION THEORY IS IN CRISIS
N-bodyNext to leading orderlinear theory

Blas, Garny, Konstandin ‘14

PERTURBATION THEORY IS IN CRISIS
N-bodyNext to leading order

Next-to-next-toleading orderlinear theory

Blas, Garny, Konstandin ‘14

PERTURBATION THEORY IS IN CRISIS
N-bodyNext to leading order

Next-to-next-toleading orderlinear theory
Next-to-next-to-next-toleading order
Blas, Garny, Konstandin ‘14

PERTURBATION THEORY IS IN CRISIS
N-bodyNext to leading order

The success of the next-to-next-toleading (2 loop) order calculation justNext-to-next-toan illusion? leading orderlinear theory
Next-to-next-to-next-toleading order
Blas, Garny, Konstandin ‘14

EFFECTIVE FIELD THEORY APPROACHES
continuity + Euler + Poisson eqs.

✔

Baumann+’12, Carrasco, Herzberg, Senatore’12 …

•Neglecting the stress tensor
would be a reasonableapproximation for a CDMdominated universe at leastat early times•

EFT estimates the functional
form for the corrections
from viscosity andanisotropic stress in anempirical manner•

Introduce free parametersand determine them bysimulations✔
✔

Foreman, Perrier, Senatore’16

SYSTEM-LEVEL RESPONSE FUNCTION

Pnl (k)K(k, q) = q
Plin (q)I want to study this mode at some late time tkq1

TN, Bernardeau, Taruya ‘16

q2

what is the impact from wave mode q at the initial time t0?

q3

SYSTEM-LEVEL RESPONSE FUNCTION
TN, Bernardeau, Taruya ‘16large scale structure gravitational evolution

Input(⌦m , h, ...; z)Pnl (k)OutputSYSTEM-LEVEL RESPONSE FUNCTION
TN, Bernardeau, Taruya ‘16large scale structure gravitational evolution

Input(⌦m , h, ...; z)Plin (k)Pnl (k)To a very good approximation

OutputSYSTEM-LEVEL RESPONSE FUNCTION
TN, Bernardeau, Taruya ‘16large scale structure gravitational evolution

Input

Plin (k)Pnl (k)Pnl (k)K(k, q) = q
Plin (q)OutputSYSTEM-LEVEL RESPONSE FUNCTION
TN, Bernardeau, Taruya ‘16large scale structure gravitational evolution

Input

Plin (k)+ Plin (k)Pnl (k)Pnl (k)K(k, q) = q
Plin (q)OutputSYSTEM-LEVEL RESPONSE FUNCTION
TN, Bernardeau, Taruya ‘16large scale structure gravitational evolution

Input

Plin (k)+ Plin (k)Pnl (k)Pnl (k)K(k, q) = q
Plin (q)Output+ Pnl (k)RESPONSE FUNCTION: SIM VS PT
Rescaled quantity:

TN, Bernardeau, Taruya ‘16q ≧ 2k are shown•

SPT (2-loop) >> N-body @ high q•

This is exactly where PT breaks
down•

What N-body tells us is:“Physics at strongly nonlinear regime
does not propagate to large scales”

initial wave mode [h/Mpc]

UV SENSITIVITY? Halle, Colombi, Taruya, TN, in prep••

1D dynamics: Zel’dovich solution is theexact solution before shell crossing•

Exhibits too strong UV sensitivity•

Smoothing out shell crossing regions helps

3D LPT dynamics•PINpointing Orbit Crossing Collapsed HIerarhicalObjects (PINOCCHIO, Monaco+’02) works well

w/ Adaptive smoothing

TN, Taruya, Colombi, Halle, in prep

PRACTICAL USAGE? RECONSTRUCTION••

From the definition of a functional derivative

TN, Bernardeau, Taruya ‘17

Use this to predict Pnl for model p1 given Pnl for another model p0
RESPRESSO (Rapid and Efficient SPectrum
calculation based on RESponSe functiOn)python package

http://www-utap.phys.s.u-tokyo.ac.jp/~nishimichi/
public_codes/respresso

PT CHALLENGE
Osato, TN, Bernardeau, Taruya ‘18

繰込摂動論

標準摂動論

繰込摂動論+ 1 free paramAs (規格化)
有効場の理論ns (距離依存性)(3 free params)Ωm (物質の密度)

RESPRESSOPT CHALLENGE 2TN, Takada, Senatore, Zaldarriaga++ in prep

• 「全部入り」のデータ•PL
ER

http://www-utap.phys.s.u-tokyo.ac.jp/~nishimichi/data/PTchallenge/

I
IMYRA
N

WHERE TO GO FOR A THEORISTLIMITATIONS OF SIMUL ATIONS / OUR KNOWLEDGE

small scalematter field~1Mpc

10~100Mpc

large scale
100Mpc ~analyticalBad

Good(?) Very Good(?)

N-body?

Very GoodGoodBad?

?

Very GoodGood?
N/A?

?
N/A?

biased tracers analyticalhalos

N-bodysubhalos

N-bodygalaxies

N-bodyHydro?
?
N/A?

Just a parameterization…

WHERE TO GO FOR A THEORISTLIMITATIONS OF SIMUL ATIONS / OUR KNOWLEDGE

small scalematter field~1Mpc

10~100Mpc

large scale
100Mpc ~analytical

Bad

Good(?) Very Good(?)

N-body?

Very GoodGoodBad?

?

Very GoodGood?
N/A?

?
N/A?

biased tracers analyticalhalos

N-bodysubhalosN-bodygalaxies

N-bodyHydro?
?
N/A?

Just a parameterization…

WHERE TO GO FOR A THEORISTLIMITATIONS OF SIMUL ATIONS / OUR KNOWLEDGE

small scalematter field~1Mpc

10~100Mpc

large scale
100Mpc ~analytical

BadGood(?) Very Good(?)

N-body?

Very GoodGoodBad?

?

Very GoodGood?
N/A?

?
N/A?

biased tracers analyticalhalos

N-bodysubhalos

N-bodygalaxies

N-bodyHydro?
?
N/A?

Just a parameterization…

シミュレーションと機械学習
観測データ• 問題設定の自由度は大きい•

どこからどこまでMLに頼るのか？

• 観測データの膨大な自由度を調査しInverseproblem切れるか？
統計量•

過学習を防ぐ機構？

•

やはり統計量を経由することで、自由度を減らす？

Forwardmodeling

宇宙論パラメタde Lapparent, Geller, Huchra+’86

最近の例
観測データ

Ravanbakhsh, Oliva, Fromenteau+’17
Mathuriya, Bard, Mendygral+’18>7,000,000 parameters69.33 Gflop / mimi-batch (mini-batch size = 1)

Inverseproblem• L=512Mpc/h, N=512^3• 12,632 simulations with COLA
• 150 as validation data, 50 as test data
• 8 x 128^3 voxels (101,056 sub-volumes)• Deep Convolutional Net

宇宙論パラメタ

最近の例
観測データ
ZA

2LPT

D3M

• Pairs of (Zel’dovich Approx.

— full N-body simulation)

Forwardmodeling• ZA as the input• N-body as the output• Perform 10,000 pairs ofparticle realizations• L = 128 Mpc/h, N = 323

宇宙論パラメタ

He, Li, Feng+’18

HSCでは観測データ• 全面的に機械学習を導入するのはやや時期尚早• 統計量ベースの解析• 従来のforward modeling + MCMCに則った解析

銀河の統計量Forwardmodeling

ここはよく分からない

ハローの統計量
Forwardmodeling

機械学習• シミュレーションで予言できることとできないことがある• できるところまでやろう• できないところは仕方ないので、それらしい処方箋（フリーパラメタ）を入れて、マージナライズする• Deep learningは不要• e.x., P(k) の Ωm 依存性

宇宙論パラメタWHAT WE WANT TO (HAVE TO) DO
Hyper Suprime Cam (HSC)

Sloan Digital Sky Survey

弱重力レンズ: convergence κ (Oguri+’17)

銀河分布: overdensity δg<κ κ>
銀河バイアス無し —> Hikage+’18<δg κ><δg δg>バイアスと宇宙論の縮退は解けるか？
—> Miyatake, TN+ in prepDARK EMULATOR: WHAT IT CAN DO
OVERVIEW

Halo mass function

Halo-Matter Cross CF

TN+’18Halo-Halo Auto CF

1 curve ~ 100 mili secs on a typical laptop computerDARK EMULATOR: WHAT IT CAN DO

TN+’18

L ARGE SC ALES
Large-scale bias

Scale-dependent bias around BAODARK EMULATOR: WHAT IT CAN DO
Mass profile of halos

SMALL SC ALES

Halo exclusion effectSplashback radiusc.f., Sugiura san &Hamabata san’s Talk

TN+’18

Concentration-mass relation

SMALL-SCALE UNCERTAINTIESPut here (as) many (asyou want) nuisanceparameters to accountfor unknowns
Galaxy-galaxy lensing

Galaxy (projected) clustering

TN+’18SMALL-SCALE UNCERTAINTIESPut here (as) many (asyou want) nuisanceparameters to accountfor unknowns
Galaxy-galaxy lensing

Typical halo mass

Galaxy (projected) clustering

TN+’18SMALL-SCALE UNCERTAINTIESPut here (as) many (asyou want) nuisanceparameters to accountfor unknowns
Galaxy-galaxy lensing

Galaxy (projected) clustering

Typical halo mass
Environment
Distribution of galaxies around cluster center

TN+’18SMALL-SCALE UNCERTAINTIESPut here (as) many (asyou want) nuisanceparameters to accountfor unknowns
Galaxy-galaxy lensing

Galaxy (projected) clustering

Typical halo mass
Environment
Distribution of galaxies around cluster center(Theory)

Large-scale bias

TN+’18SMALL-SCALE UNCERTAINTIES

TN+’18

Put here (as) many (asyou want) nuisanceparameters to accountfor unknowns
Galaxy-galaxy lensing

Galaxy (projected) clustering

Typical halo mass
Environment
Distribution of galaxies around cluster center(Theory)

Large-scale bias

Cosmology!

ns

DARK QUEST SIMULATIONS

0.96

TN+’18

DQ 1: to be made public after HSC
cosmology analyses (~early next year)

0.94
0.92
−0.8
−0.85
−0.9
−0.95
−1−1.05−1.1−1.15•

Ensemble of N=2048^3 simsw•

2 base resolutions••

L = 1 Gpc/h and 2 Gpc/h

100+1 6D-wCDM models•

28 HR (14 LR) fiducial runs

• 1 0.13
0.02150.0220.02250.023 0.11 0.115 0.12 0.125
0.65 100
0.7 0.75
2.6 2.8 3
run0.55
at every
LHD0.8sampleωbωcΩΛ
ln(Circles: 100 parameter sets to be covered
Centered at Planck 2015ωb = Ωbh2: ±5%ωc = Ωch2: ±10%
ΩΛ: ±20%ln(1010 As): ±20%
ns: ±5%w: ±20%•density on 1024^3 grid points•

Rockstar halos + postprocess•

Subhalos excluded•

Spherical density profile (40 binsfrom 10kpc/h to 5Mpc/h)

LATIN HYPERCUBE DESIGN
σ8•Curse of dimensionality••

Regular lattice is not
tractable in high dimensions

Latin hypercube•

Definition: each sample
point is the only one both on
the row and the column•

Can tell if one input
parameter has the dominanteffectΩm

LATIN HYPERCUBE DESIGN
σ8•Curse of dimensionality••

Regular lattice is not
tractable in high dimensions

Latin hypercube•

Definition: each sample
point is the only one both on
the row and the column•

Can tell if one input
parameter has the dominant
effectΩm

LATIN HYPERCUBE DESIGN
σ8•Curse of dimensionality••

Regular lattice is not
tractable in high dimensions

Latin hypercube•

Definition: each sample
point is the only one both on
the row and the column•

Can tell if one input
parameter has the dominant
effectΩm

LATIN HYPERCUBE DESIGN
DiagonalOptimalRandom•

LHD is not unique••Some look nicer, others bad.

One more to add: space filling property•

It’s all about how to cover your space•

The closest neighbor should be far (=maximin distance design)

Minimize

Taken from FIFA webpage

with some large r

DIMENSION REDUCTION•

•

•

•

Mass function

fittingf ( ) = A[ha+ b] exp•

Sheth-Tormen type functional form•b and c from Tinker•(A, a) at 21 redshifts = 42 component vector -> 6 PCsc i2

Halo-matter cross correlation•(r-bin, n-bin, z-bin) = (66, 13, 21)•

18,018 components -> 5 PCs

Halo auto correlation•(r-bin, n1-bin, n2-bin, z-bin) = (21, 8, 8, 21)•28,224 components -> 8 PCs

Propagator•

fitting

24G(k) = [c0 + c2 k + c4 k ] exp⇥⇤

k 2 d2 /2

3 parameters at 21 redshifts = 63 components -> 3PCskκThe posterior distribution (45.34) is given by))' −11&tN
P (tN +1 | tN ) ∝ exp − tN tN +1 CN +1
tN +2GAUSSIAN PROCESS•

Gaussian Process•works as a non-parametric
We can evaluate the mean and standard deviation of theor classifierof tNregressor+1 by brute-force inversion of C N +1 . There is a mo•

is a prior in a function space•

Specified by the mean function
and covariance function (kernel)

Prior

for the predictive distribution, however, which is useful• toCanbe at``trained’’by points
tuningarebe madea number of newon the basis o−1−1
N . WecanwriteCintermsofCandCNN +1N using ththe hyperparametersequations (Barnett, 1979):)*M mC−1=N +1mT mPosterior

where+,−1
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521
T −1m = κ − k CN k
You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.
m = −m C−1
N k

A=1, σ=0.01

A=1, σ=1
45.4: Examples of covariance functions

M =C−1N +1mmT .
mmatrix for the vector tN +1 ≡ (t1 , . . . , tN +1 )T. We define submatrices of CN +1
this matrix into equation (45.36) we
as follows:⎡⎡When we⎤⎡substitute⎤⎤

A=B=1, Γ=4, σ=1⎢⎣ CN⎢
CN +1 ≡ ⎢⎣&kT

The posterior distribution (45.34) iswheregiven by))**&'1
Analytic expression for P (tN +1 | tN ) ∝ exp − tN tN +1 C−1 tN .
N +1 t2N +1

the posterior-⎦⎣ k ⎦⎥1(tN +1 − t̂N +1⎥⎥ . P (tN +1 | tN ) =(45.35)
exp −⎦Z2σt̂2'& 'N+1κt̂N +1 = kT C−1N tN(45.36)
2σt̂= κ − kT C−1
N k.
N+1

GAUSSIAN PROCESS

Posterior (before optimization)

Posterior (after optimization)

GP = 隠れ層のノード数∞•We can estimate the uncertainty from GP••But we cannot always trust it

Performance depends on the choice of the kernel

ACCURACY: HALO MASS FUNCTION
Example plot at z = 0.55

Spread in HMF among the 100 models

Gaussian Process Regression

Upper: Model fitting w/ Sheth-Tormentype function (2 free parameters)

Training set

Lower: Compress the 42 (=2 x 21redshifts) coefficients into 6 PCs

Validation set

Red shades: scatter of 28 fiducial runs

SUMMARY• 摂動展開に基づく揺らぎの発展•限界が見えてきた: shell crossing•

シミュレーションとのハイブリッド: response function approach• N体シミュレーション + 機械学習••

従来のforward modeling + MCMCに代わる方法論•

２点統計以上の情報を取り出す可能性•

まだまだアイディア段階だが、いろいろな応用がありうる

Emulator•

効率的なパラメタサンプリング + 次元削減 + ガウス過程•

HSCのデータ解析へ実用段階に