今後やるべき事、および 観測計画と将来の展望

BAO:

現在の理論的到達点と

今後やるべき事、および 観測計画と将来の展望

小松英一郎

テキサス大学オースティン校

名古屋大学

At

,

暗黒エネルギー

•

• ^{なぜ暗黒エネルギー}

が必要なのか？

水素・ヘリウム 暗黒物質

暗黒エネルギー

暗黒「エネルギー」

•

•

(1)

(1a

) (2)

(BAO, CMB)

が同時に説明できれば、「暗黒エネルギー」と対等 に扱って良い。

μ = 5Log 10 [D L (z)/Mpc] + 25

Wood-Vasey et al. (2007)

赤方偏移 , z

1a

w(z)=P DE (z) /ρ DE (z)

=w 0 +w a z/(1+z)

Wood-Vasey et al. (2007)

[

]

w(z)=w 0 +w a z/(1+z)

1a

赤方偏移 , z

•

標準宇宙論の枠内で 考えると・・・

•

を説明できないのは 明白。従って、暗黒 エネルギーが必要。

0.0 0.5 1.0 1.5 2.0

! _M 0.0

0.5 1.0 1.5 2.0

! "

ESSENCE+SNLS+gold

( ! _M , ! _" ) = (0.27,0.73)

! _Total =1

Wood-Vasey et al. (2007)

1a

−3 −2 −1 0 1 w ₀

−10

−5 0 5 10 15

w a

ESSENCE+SNLS+gold (w ₀ ,w _a ) = (−1,0)

•

標準宇宙論の枠内で 考えると・・・

•

真空のエネルギーで 説明できる。

•

は大きい。

Wood-Vasey et al. (2007)

w(z) = P DE (z) /ρ _DE (z) = w 0 +w a z/(1+z)

1a

D L (z) = (1+z) ² D A (z)

• ^{D A (z)}

•

0.2 2 6 1090

1a

銀河

(BAO) CMB

D L (z) D A (z)

0.02 赤方偏移 , z

D A (z) を測る

•

^d

^{D A}

できる。 なにが

d

0.2 2 6 1090

銀河

CMB

0.02

D A (galaxies)=d BAO / θ

d BAO

d CMB

D A (CMB)=d CMB / θ

θ θ

赤方偏移 , z

混乱を避けるために

• ^{D L (z)}

^{D A (z)}

従って、「共動距離」はそれぞれ

(1+z)D L (z)

(1+z)D A (z)

• ^{d CMB}

^{d BAO}

従って、「共動サイズ」はそれぞれ

(1+z CMB )d CMB

(1+z BAO )d BAO

•

^r

^{r CMB = (1+z CMB )d CMB}

r BAO = (1+z BAO )d BAO

CMB ＝標準ものさし

•

d CMB

d CMB

θ

θ~

θ

θ

θ θ

θ

θ θ

音波の地平線

• ^{d CMB}

t CMB

t CMB ~38

z CMB ~1090

•

^{t CMB}

• ^{d H (t CMB ) = a( t CMB} )*Integrate [ c dt/a(t), {t,0,t CMB }].

•

• ^{d s (t CMB ) = a( t CMB )*} Integrate[ c s (t) dt/a(t), {t,0,t CMB }],

c s (t)

-

• ^{WMAP 3}

• ^{l CMB = π/θ = πD A (z CMB )/d s (z CMB ) = 301.8 ± 1.2}

• ^CMB

^{D A (z CMB )/d s (z CMB )}

Hinshaw et al. (2007)

l CMB =301.8 ± 1.2

•

^{: l CMB =πD A (z CMB )/}

d s (z CMB )

z EQ

Ω _b h ²

•

^{: WMAP}

解析から得られる制限

(Spergel et al. 2007)

l CMB =301.8 ± ^1.2

1-Ω m -Ω Λ = 0.3040Ω m

+0.4067Ω Λ

D A (z CMB )/d s (z CMB ) から

得られるもの

0.0 0.5 1.0 1.5 2.0

! _M 0.0

0.5 1.0 1.5 2.0

! "

ESSENCE+SNLS+gold

( ! _M , ! _" ) = (0.27,0.73)

! _Total =1

10 Percival et al.

Fig. 12.—The redshift-space power spectrum recovered from the combined SDSS main galaxy and LRG sample, optimally weighted for both density changes and luminosity dependent bias (solid circles with 1-σerrors). A flatΛcosmological distance model was assumed with Ω_M = 0.24. Error bars are derived from the diagonal elements of the covariance matrix calculated from 2000 log-normal catalogues created for this cosmological distance model, but with a power spectrum amplitude and shape matched to that observed (see text for details).

The data are correlated, and the width of the correlations is presented in Fig. 10 (the correlation between data points drops to<0.33 for

∆k >0.01hMpc⁻¹). The correlations are smaller than the oscillatory features observed in the recovered power spectrum. For comparison we plot the model power spectrum (solid line) calculated using the fitting formulae of Eisenstein & Hu (1998); Eisenstein et al. (2006), for the best fit parameters calculated by fitting the WMAP 3-year temperature and polarisation data, h= 0.73, Ω_M = 0.24, n_s = 0.96 and Ω_b/Ω_M = 0.174 (Spergel et al. 2006). The model power spectrum has been convolved with the appropriate window function to match the measured data, and the normalisation has been matched to that of the large-scale (0.01 < k < 0.06hMpc⁻¹) data. The deviation from this lowΩ_M linear power spectrum is clearly visible atk >_∼0.06hMpc⁻¹, and will be discussed further in Section 6. The solid circles with 1σerrors in the inset show the power spectrum ratioed to a smooth model (calculated using a cubic spline fit as described in Percival et al.

2006) compared to the baryon oscillations in the (WMAP 3-year parameter) model (solid line), and shows good agreement. The calculation of the matter density from these oscillations will be considered in a separate paper (Percival et al. 2006). The dashed line shows the same model without the correction for the damping effect of small-scale structure growth of Eisenstein et al. (2006). It is worth noting that this model is not a fit to the data, but a prediction from the CMB experiment.

BAO ＝標準ものさし

•

d BAO

(1+z)d BAO

Percival et al. (2006)

Okum ur a et al. (2007)

実空間 フーリエ空間

再び音波の地平線

• ^{d BAO}

t BAO

• ^{z BAO ~1080}

^{z CMB ~1090}

•

z BAO

z CMB

•

^{3ρ baryon /(4ρ photon ) =0.64(Ω b h 2} /0.022)(1090/

(1+z CMB ))

z BAO >z CMB

宙では

Ω b h ² =0.022

z BAO <z CMB (ie, d BAO >d CMB )

最新の BAO 測定結果

• z=0.2: 2dFGRS

SDSS

• z=0.35: SDSS LRG

•

^{D A (z)/d s (z BAO )}

が決まる。

Percival et al. (2007) z=0.2

z=0.35

D A (z) だけでは終わらない

• ^BAO

^{D A (z)}

各々の赤方偏移における宇宙の膨張率

H(z)

•

^BAO

^{D A (z):}

D A (z) = d s (z BAO )/θ

•

^BAO

^H(z):

H(z) = cΔz/[(1+z)d _s (z BAO )]

D A (z) と H(z) を測る

SDSS LRG

から求めた２次元の ２点相関関数

(Okumura et al. 2007) (1+z)d s (z BAO )

θ = d s (z BAO )/D A (z) cΔz/(1+z)

= d s (z BAO )H(z)

線形理論 観測データ

D V (z) = {(1+z) ² D A 2 (z)[cz/H(z)]} ^1/3

Percival et al. (2007)

2dFGRS

SDSS

SDSS LRG

(1+z)d s ( t B A O )/D V (z)

現在のデータの精度では

D A (z)

H(z)

D V (z)

Ω m =1, Ω Λ =1 Ω m =0.3, Ω Λ =0

Ω m =0.25, Ω Λ =0.75

赤方偏移 , z

CMB + BAO => 空間曲率

• ^CMB

^BAO

な距離の指標。

• ^Ia

距離の指標。

•

^CMB+BAO

空間曲率の測定に最 適。

BAO: 現在の状況

• ^BAO

^SDSS

^LRG

2dFGRS

• ^BAO

(Eisenstein et al. 2005; Percival et al. 2007)

• ^CMB

^BAO

^2%

(Spergel et al. 2007)

• ^{BAO, CMB,}

^1a

な性質に制限

(

)

BAO: 理論的挑戦

•

^,

^,

1.

2.

3.

既存の理論の精度は、将来 のデータの精度に追いつけ るのか？

10 Percival et al.

Fig. 12.— The redshift-space power spectrum recovered from the combined SDSS main galaxy and LRG sample, optimally weighted for both density changes and luminosity dependent bias (solid circles with 1-σ errors). A flat Λ cosmological distance model was assumed with Ω

= 0.24. Error bars are derived from the diagonal elements of the covariance matrix calculated from 2000 log-normal catalogues created for this cosmological distance model, but with a power spectrum amplitude and shape matched to that observed (see text for details).

The data are correlated, and the width of the correlations is presented in Fig. 10 (the correlation between data points drops to < 0.33 for

∆k > 0.01 h Mpc

). The correlations are smaller than the oscillatory features observed in the recovered power spectrum. For comparison we plot the model power spectrum (solid line) calculated using the fitting formulae of Eisenstein & Hu (1998); Eisenstein et al. (2006), for the best fit parameters calculated by fitting the WMAP 3-year temperature and polarisation data, h = 0.73, Ω

= 0.24, n

= 0.96 and Ω

/Ω

= 0.174 (Spergel et al. 2006). The model power spectrum has been convolved with the appropriate window function to match the measured data, and the normalisation has been matched to that of the large-scale (0.01 < k < 0.06 h Mpc

) data. The deviation from this low Ω

linear power spectrum is clearly visible at k >

0.06 h Mpc

, and will be discussed further in Section 6. The solid circles with 1σ errors in the inset show the power spectrum ratioed to a smooth model (calculated using a cubic spline fit as described in Percival et al.

2006) compared to the baryon oscillations in the (WMAP 3-year parameter) model (solid line), and shows good agreement. The calculation of the matter density from these oscillations will be considered in a separate paper (Percival et al. 2006). The dashed line shows the same model without the correction for the damping effect of small-scale structure growth of Eisenstein et al. (2006). It is worth noting that this model is not a fit to the data, but a prediction from the CMB experiment.

観測データ 線形理論

この理論は信じられるか？

非線形効果を理解する

•

• ^N

•

非線形効果を理解する

•

•

⁽

¹

⁾

観測的に立証済み。

(WMAP

!)

•

•

３次の摂動論は新しい？

•

(25

)

• ¹⁹⁹⁰

•

•

^z~0

なぜ今、摂動論なのか？

•

•

(z>1)

•

BAO

Dark Energy Task Force

• ^z>1

摂動論の巻き返し

•

帰ってきてください！

時は来ました！

解くべきは３つの方程式

•

•

: rotV=0.

•

フーリエ変換すると、

•

,

– 8 –

our using θ ≡ ∇ · v , the velocity divergence field. Using equation (5) and the Friedmann equation, we write the continuity equation [Eq. (3)] and the Euler equation [Eq. (4)] in Fourier space as

δ ˙ ( k , τ ) + θ ( k , τ )

= −

! d ³ k ₁ (2π ) ³

!

d ³ k ₂ δ D ( k ₁ + k ₂ − k ) k · k ₁

k ₁ ² δ ( k ₂ , τ )θ ( k ₁ , τ ), (6) θ( ˙ k , τ ) + a ˙

a θ ( k , τ ) + 3 ˙ a ²

2a ² Ω _m (τ )δ ( k , τ )

= −

! d ³ k ₁ (2π ) ³

!

d ³ k ₂ δ D ( k ₁ + k ₂ − k ) k ² ( k ₁ · k ₂ )

2k ₁ ² k ₂ ² θ( k ₁ , τ )θ ( k ₂ , τ ),

(7) respectively.

To proceed further, we assume that the universe is matter dominated, Ω _m (τ ) = 1 and a(τ ) ∝ τ ² . Of course, this assumption cannot be fully justified, as dark energy

dominates the universe at low z . Nevertheless, it has been shown that the next-to-leading order correction to P (k ) is extremely insensitive to the underlying cosmology, if one

uses the correct growth factor for δ ( k , τ ) (Bernardeau et al. 2002). Moreover, as we are

primarily interested in z ≥ 1, where the universe is still matter dominated, accuracy of our approximation is even better. (We quantify the error due to this approximation below.) To solve these coupled equations, we shall expand δ ( k , τ ) and θ ( k , τ ) perturbatively using the n-th power of linear solution, δ ₁ ( k ), as a basis:

δ ( k , τ ) =

∞

"

n =1

a ⁿ (τ )

! d ³ q ₁

(2π ) ³ · · · d ³ q n −1

(2π ) ³

×

!

d ³ q n δ D (

n

"

i =1

q _i − k )

×F n ( q ₁ , q ₂ , · · · , q _n )δ ₁ ( q ₁ ) · · · δ ₁ ( q _n ), (8)

θ( k , τ ) = −

∞

"

n =1

˙

a(τ )a ^{n −1} (τ )

! d ³ q ₁

(2π ) ³ · · · d ³ q _{n −1} (2π ) ³

×

!

d ³ q _n δ _D (

n

"

i =1

q _i − k )

×G n ( q ₁ , q ₂ , · · · , q _n )δ ₁ ( q ₁ ) · · · δ ₁ ( q _n ). (9)

δ ₁ に関してテイラー展開

• ^{δ 1}

– 8 –

our using θ ≡ ∇ · v , the velocity divergence field. Using equation (5) and the Friedmann equation, we write the continuity equation [Eq. (3)] and the Euler equation [Eq. (4)] in Fourier space as

δ ˙ ( k , τ ) + θ ( k , τ )

= −

! d ³ k ₁ (2π ) ³

!

d ³ k ₂ δ _D ( k ₁ + k ₂ − k ) k · k ₁

k ₁ ² δ ( k ₂ , τ )θ( k ₁ , τ ), (6) θ ˙ ( k , τ ) + a ˙

a θ ( k , τ ) + 3 ˙ a ²

2a ² Ω _m (τ )δ ( k , τ )

= −

! d ³ k ₁ (2π ) ³

!

d ³ k ₂ δ _D ( k ₁ + k ₂ − k ) k ² ( k ₁ · k ₂ )

2k ₁ ² k ₂ ² θ( k ₁ , τ )θ( k ₂ , τ ),

(7) respectively.

To proceed further, we assume that the universe is matter dominated, Ω _m (τ ) = 1 and a(τ ) ∝ τ ² . Of course, this assumption cannot be fully justified, as dark energy

dominates the universe at low z . Nevertheless, it has been shown that the next-to-leading order correction to P (k ) is extremely insensitive to the underlying cosmology, if one

uses the correct growth factor for δ ( k , τ ) (Bernardeau et al. 2002). Moreover, as we are

primarily interested in z ≥ 1, where the universe is still matter dominated, accuracy of our approximation is even better. (We quantify the error due to this approximation below.) To solve these coupled equations, we shall expand δ ( k , τ ) and θ( k , τ ) perturbatively using the n-th power of linear solution, δ ₁ ( k ), as a basis:

δ ( k , τ ) =

∞

"

n =1

a ⁿ (τ )

! d ³ q ₁

(2π ) ³ · · · d ³ q _{n −1} (2π ) ³

!

d ³ q _n δ _D (

n

"

i=1

q _i − k )F _n ( q ₁ , q ₂ , · · · , q _n )δ ₁ ( q ₁ ) · · · δ ₁ ( q _n ),

θ( k , τ ) = −

∞

"

n =1

˙

a(τ )a ⁿ⁻¹ (τ )

! d ³ q ₁

(2π ) ³ · · · d ³ q n −1

(2π ) ³

!

d ³ q _n δ _D (

n

"

i =1

q _i − k )G _n ( q ₁ , q ₂ , · · · , q _n )δ ₁ ( q ₁ ) · · · δ ₁ ( q _n ) Here, the functions F and G follows the following recursion relations with the trivial initial

conditions, F ₁ = G ₁ = 1. (Jain & Bertschinger 1994)

３次の項 ( δ ₁ ³ ) までキープ

• ^{δ=δ 1 +δ 2 +δ 3}

^{δ 2 =O(δ 1 2 ), δ 3 =O(δ 1 3 ).}

•

^{, P(k)=P L (k)+P 22 (k)+2P 13 (k),}

以下のようにオーダー毎に分割して書く。

Odd powers in δ 1 vanish (Gaussianity) P L

P 13 P 22 P 13

P(k): ３次の摂動論の解

• ^{F 2 (s)}

(Goroff et al. 1986)

Vishniac (1983); Fry (1984); Goroff et al. (1986); Suto&Sasaki (1991);

Makino et al. (1992); Jain&Bertschinger (1994); Scoccimarro&Frieman (1996)

– 10 – where

P ₂₂ (k ) = 2

! d ³ q

(2π ) ³ P L (q )P L (| k − q |) "

F ₂ ^{( s )} ( q , k − q ) # 2

, (16)

2P ₁₃ (k ) = 2π k ²

252 P L (k )

! _∞

0

dq

(2π ) ³ P L (q )

×

$

100 q ²

k ² − 158 + 12 k ²

q ² − 42 q ⁴ k ⁴

+ 3

k ⁵ q ³ (q ² − k ² ) ³ (2k ² + 7q ² ) ln

% k + q

|k − q |

& '

, (17)

where P _L (k ) stands for the linear power spectrum. While F ₂ ^{( s )} ( k ₁ , k ₂ ) should be modified for different cosmological models, the difference vanishes when k ₁ # k ₂ . The biggest correction comes from the configurations with k ₁ ⊥ k ₂ , for which

[F ₂ ^{( s )} (ΛCDM)/F ₂ ^{( s )} (EdS)] ² % 1.006 and ! 1.001 at z = 0 and z ≥ 1, respectively. Here,

F ₂ ^{( s )} (EdS) is given by equation (13), while F ₂ ^{( s )} (ΛCDM) contains corrections due to Ω _m '= 1 and Ω _Λ '= 0 (Matsubara 1995; Scoccimarro et al. 1998), and we used Ω _m = 0.27 and

Ω _Λ = 0.73 at present. The information about different background cosmology is thus almost entirely encoded in the linear growth factor. We extend the results obtained above to

arbitrary cosmological models by simply replacing a(τ ) in equation (15) with an appropriate linear growth factor, D(z ),

P δδ (k, z ) = D ² (z )P L (k ) + D ⁴ (z )[2P ₁₃ (k ) + P ₂₂ (k )]. (18) We shall use equation (16)–(18) to compute P (k, z ).

2.2. Non-linear Halo Power Spectrum : Bias in 3rd order PT

In this section, we review the 3rd-order PT calculation as the next-to-leading

order correction to the halo power spectrum. We will closely follow the calculation of (McDonald 2006). In the last section, we reviewed the 3rd-order calculation of matter

power spectrum. Here, the basic assumptions and equations are the same previous section,

but to get the analytic formula for the halo power spectrum, we need one more assumption,

３次の摂動論 vs N 体計算 Jeong & Komatsu (2006)

BAO: 非線形効果 Jeong & Komatsu (2006)

3rd-order PT Simulation

Linear theory

P. McDonald (2006) の引用

“...this perturbative approach to the galaxy power spectrum (including beyond-linear corrections)

has not to my knowledge actually been used to interpret real data. However, between

improvements in perturbation theory and the need to interpret increasingly precise

observations, the time for this kind of approach

may have arrived (Jeong & Komatsu, 2006).”

でも、銀河は？

•

•

•

局在銀河形成仮定

•

•

P g (k)=b 1 2 P(k)

b 1

•

•

^:

我々が興味あるスケールでは成立するとみなす。

δ _g を δ でテイラー展開する

δ g (x) = c 1 δ(x) + c 2 δ ² (x) + c 3 δ ³ (x) + O(δ ⁴ ) + ε(x)

ここで

δ

ε

: <δ(x)ε(x)>=0.

•

^x

•

Gaztanaga & Fry (1993); McDonald (2006)

銀河のパワースペクトル

•

b 1 , b 2 , N

c 1 , c 2 , c 3 , ε

•

b 1 , b 2 , N

P g (k)

McDonald (2006)

ミレニアム “ 銀河 ” シミュレーション

•

•

(Springel

et al. 2005)

より作られたカタログを使う。

• ^MPA

: De Lucia & Blaizot (2007)

• ^Durham

: Croton et al. (2006)

摂動論 vs MPA 銀河

• ^{k max}

の

P(k)

•

も

k max

•

バイアスモデル は、良く合う！

Jeong & Komatsu (2007)

BAO: 非線形バイアス

• ^BAO

アスが重要なのは 明白。

•

(500 Mpc) ³

大スケールの

P g (k)

Jeong & Komatsu (2007)

銀河の質量依存性

•

バイアスが大きい。

•

も良く合っている。

•

も摂動論は使える！

Jeong & Komatsu (2007)

あまり感動しませんよね？

•

:

• “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” - John von Neumann

•

P g (k)

D A (z)

H(z)

D A (z) を P g (k) から求める

•

３次の摂動論を用いて、

正しい

D A (z)

“

”

•

^z=1

Jeong & Komatsu (2007)

D A /D A (input) D A /D A (input)

D A /D A (input)

D A /D A (input) D A /D A (input) D A /D A (input) 1σ

1σ

1σ

現在の到達点

• ^z>2

^BAO

摂動論を用いて理解できた。

•

•

P g (k)

D A (z)

今後やるべき事

• ^z=1

•

「くりこみ摂動論」

Crocce&Scoccimarro;

Matarrese&Pietroni; Velageas; Taruya; Matsubara.

•

•

３点相関関数（バイスペクトル）の摂動論計算。

３点相関関数

•

b 1

b 2

Q _g (k ₁ ,k ₂ ,k ₃ )=(1/b ₁ )[Q _m (k ₁ ,k ₂ ,k ₃ )+b ₂ ]

Q _m

•

2dFGRS

(Verde et al. 2002): z=0.17

b ₁ =1.04 ± 0.11; b ₂ =-0.054 ± 0.08

•

(Sefusatti & Komatsu 2007)

•

最も難しい問題

•

P g (k)

「赤方偏移空間の歪み。」

•

^H(z)

•

•

^z~3

赤方偏移空間の歪み

•( 左 ) コヒーレントな速度場 => 視線方向の相関の上昇 –“Kaiser” 効果

•( 右 ) ビリアル的ランダム運動 => 視線方向の相関の減少

–“Finger-of-God” 効果

赤方偏移空間の歪み

赤方偏移空間の摂動論

•

Kaiser

•

z=3

N

•

小さく抑えられてい る。

=> Finger-of-

God

•

Finger-of- God

• P g (k)/(1+k para 2 σ ² )

•

赤方偏移空間の摂動論

観測計画（地上）

•

[“low-z” = z<1; “mid-z” = 1<z<2; “high-z” = z>2]

• Wiggle-Z (Australia): AAT/AAOmega,

, low-z

• FastSound (Japan): Subaru/FMOS, 2008, mid-z (Hα)

• BOSS (USA): SDSS-III, 2009, low-z (LRG);high-z (LyαF)

• HETDEX (USA): HET/VIRUS, 2011, high-z (LyαE)

• WFMOS (Japan+?): >2011, low-z (OII); high-z (LBG)

•

• SPACE (Europe): >2015, all-sky, z~1 (Hα)

• ADEPT (USA): >2017, all-sky, z~1 (Hα)

• CIP (USA): >2017, 140 deg ² , 3<z<6 (Hα)

•

Dark Energy Task Force

“Stage IV”

• ^{Stage IV}

現在の制限よりも１０倍以上良い、という条件

観測計画（宇宙）

将来の展望

•

•

• ^BAO

• ^BAO

•

•

•

^{Stage IV}

•

•

^WMAP

•

将来の展望

WMAP: 使用前と使用後

• ^BAO

Hinshaw et al. (2003)

国産 BAO 衛星計画？

•

(>2017)

• ^JDEM

²⁰⁰⁸

• ^{SNAP (}

⁺

) vs ADEPT (BAO) vs CIP (BAO) vs ...

•

(>2015)

• ^{DUNE (}

⁺

) vs SPACE (BAO) vs ...

•

まとめ（１／３）

•

• ^BAO

^{D A (z)}

^H(z)

^2dFGRS

や

SDSS

•

まとめ（２／３）

•

• ^z~1

• ^z<2

•

•

•

• ^H(z)

まとめ（３／３）

•

•

^BAO

•

•

•