観測的宇宙論workshop.pptx

名古屋⼤大学  宇宙論論研究室

嵯峨  承平

(共同研究者:  市來來淨與,  杉⼭山直)

2013/12/4  観測的宇宙論論workshop

⽬目次

1.   イントロ

2.   2次摂動論論

3.   重⼒力力波(線形摂動)

4.   重⼒力力波(2次摂動)

5.   まとめ

1.  イントロ

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波}

(2

_次摂動

)

• 

⾮非ガウス性

• 

重⼒力力レンズ効果

• 

2次ドップラー効果

•

 

2次重⼒力力波

• 

磁場

Mode  coupling

(Tensor  mode)

Mode  coupling

(Vector  mode)

Non-‐‑‒linearity

Non-‐‑‒linearity

(e.g.  δ

(1)

_×v

(1)

₎

2.  2次摂動論論

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波}

(2

_次摂動

)

定式化

(

ボルツマン⽅方程式

)

ボルツマン⽅方程式

ゲージ:  Poisson  gauge

N.  Bartoro  et  al.  [1001.3957] L.  Senatore  et  al.  [0812.3652] C.  Pitrou  et  al.  [1003.0481]

G.  W.  Pettinari  et  al.  [1302.0832] M.  Beneke  et  al.  [1003.1834] A.  Naruko  et  al.  [1304.6929]

u

 2

nd

 orderではモードカップリングが起こる

スカラー + スカラー  è テンソル(重⼒力力波)

スカラー + スカラー è ベクトル(磁場)

⼀一般論論

ボルツマン⽅方程式は”Brightness”  Δを定義して多重極展開

純粋な2次:  重⼒力力

(1次)×(1次):  重⼒力力

純粋な2次:  散乱

(1次)×(1次):  散乱

n

 アインシュタイン⽅方程式(テンソルモード)

スカラーモード

1

_次

×1

_{次による寄与}

光⼦子の⾮非等⽅方圧

純粋な2次のテンソルモード

_{ニュートリノの⾮非等⽅方圧}

純粋な

2

次のテンソルモード

n

 ボルツマン⽅方程式(テンソルモード)

光⼦子の階層⽅方程式(m=±2)

ニュートリノの階層⽅方程式(m=±2)

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波}

(2

_次摂動

)

3.  重⼒力力波(線形摂動)

先⾏行行研究

純粋な2次のテンソルモードを除いた取扱いは既になされている。

A.  Mangilli  et  al.  [0805.3234]

純粋な2次のテンソルモードのニュートリノの⾮非等⽅方圧を考慮する定式化

K.  N.  Anada  et  al.  [gr-‐‑‒qc/0612013] D.  Baumann  et  al.  [hep-‐‑‒th/0703290] H.  Assadullahi  et  al.  [0907.4703]

⾮非等⽅方圧は重⼒力力波のソースとして働くので気になる。

1

st

_order

の場合

重⼒力力波の振舞

-0.2

0

0.2

0.4

0.6

0.8

1

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

h

a

without

with

S.  Weinberg.  [astro-‐‑‒ph/0306304]

J.  R.  Pritchard  and  M.  Kamionkowski.  [astro-‐‑‒ph/0412581]

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波(線形摂動)}

4.

_{重⼒力力波}

(2

_次摂動

)

0.001 0.01 0.1 1 10 100 10 100 1000 C TT l l r=0.1(without ) r=0.1(with )

1

st

_order

の場合

Ø 

⾮非等⽅方圧は放射優勢で主な影響を及ぼす(数10％ほど)

Ø 

super  horizonでは影響を及ぼさない(初期では⾮非等⽅方圧はゼロ)

Ø 

物質優勢期に⼊入ってきたモードは効かない

Ø 

2

nd

 orderでも⾮非等⽅方圧の寄与を考えることは重要。

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波(線形摂動)}

4.

_{重⼒力力波}

(2

_次摂動

)

D.  Baumann  et  al.  [hep-‐‑‒th/0703290]

スカラー1次×1次のみによる2次重⼒力力波

10

-24

10

-22

10

-20

10

-18

10

-16

10

-14

10

-12

10

-10

10

-4

10

-3

10

-2

10

-1

10

0

10

1

GW

k

z = 3400

z = 580

z = 100

z = 10

z = 0

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波(線形摂動)}

4.

_{重⼒力力波}

(2

_次摂動

)

4.  重⼒力力波(2次摂動)

光⼦子の⾮非等⽅方圧

散乱項

重⼒力力項

基本的には…

Ø 

散乱項はsilk  dumpingまで優勢

Ø 

重⼒力力項はsilk  dumping後優勢だが⼤大きさはケースバイケース

⼤大きな波数(すぐにホライズンに⼊入るモード)は1st  orderのSilk  

dumpingを受け重⼒力力項が⼩小さくなる。

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波(2次摂動)}

光⼦子の⾮非等⽅方圧の時間発展①

Tight  coupling

中

6 In this subsection, we consider tensor mode (m = σ) only. It is useful to define the function Y!1,!2

!,m (ˆk1, ˆk2) as Y!1,!2 !,m (ˆk1, ˆk2) ≡ (−1)m(2" + 1) ! m1,m2 " "1 "2 " 0 0 0 # " "1 "2 " m1 m2 −m # $ 4π 2"1+ 1 Y_!∗_1,m1(ˆk1) $ 4π 2"2+ 1 Y_!∗_2,m2(ˆk2) , (19) where "1+ "2+ " must be even because of a property of the Wigner-3j symbol. Note that, for a special case

that "1 = 0 or "2 = 0, the dependence of the ˆk1 or ˆk2 are vanish as

Y!1,0

!,m(ˆk1, ˆk2) = Y!,m(ˆk1)δ!,!1 , (20)

Y0,!2

!,m(ˆk1, ˆk2) = Y!,m(ˆk2)δ!,!2 . (21)

First, we show the zeroth order of the tight-coupling solution. ∆(2,Ø) 2,σ = 20 % d3k1 (2π)3 & v_{γ 0}(1,Ø)(k1)vγ 0(1,Ø)(k2) ' Y2,σ1,1(ˆk1, ˆk2) , (22) ∆(2,Ø) !≥3,σ = 0 . (23)

It is very interesting that the anisotropic stress of photons vanish in the first-order cosmological perturbation up to zeroth order of the tight-coupling approximation, whereas the anisotropic stress of photons does not vanish in the second-order cosmological perturbation theory. The anisotropic stress of photons survive because of the scalar-mode of photon’s velocity perturbation. This result is consistent with refs. [34, 45].

Next, we consider (CPT= 2, TCA= I) and the results can be written as 9 10∆ (2,I) 2,σ = 2 ˙τc ˙χ(2,Ø) σ − % d3_k 1 (2π)3 & 9Π(1,I) γ 0 (k1)(δ_b(1,Ø)− Φ(1,Ø))(k2) ' $ 4π 5 Y2,σ∗ (ˆk1) +4% d3k1 (2π)3 & 9v(1,Ø) γ 0 (k1)vγ 0(1,I)(k2) + 8v(1,Ø)γ 0 (k1)δv_{γb 0}(1,I)(k2) ' Y2,σ1,1(ˆk1, ˆk2) +% d3k1 (2π)3 (" k1 ˙τc # ) 10δ(1,Ø) γ (k1) − 8Φ(1,Ø) * (k1)v(1,Ø)γ 0 (k2) + Y2,σ1,1(ˆk1, ˆk2) , (24) ∆(2,I) 3,σ = − " k ˙τc # √5 5 ∆ (2,Ø) 2,σ + 15 % _d3_k 1 (2π)3 & Π(1,I) γ 0 (k1)v (1,Ø) γ 0 (k2) ' Y3,σ2,1(ˆk1, ˆk2) , (25) ∆(2,I) !≥4,σ = 0 . (26)

Very interestingly, in the second order cosmological perturbation and the first order tight-coupling approxi-mation, the higher multipole " = 3 can survive because of two source terms. One is the anisotropic stress of the zeroth order tight-coupling solution, which is only streaming effect in the left-hand side of the Boltzmann equation. The other is a convolution of the first order cosmological perturbation. This term comes from the collision term. These terms do not appear in the first order cosmological perturbation and this result is only the second order effect. Note that, more higher multipole equals zero since there are no any more source.

By using the tight-coupling solution, we can find the behavior of the anisotropic stress of photons in the early time. At zeroth order of the tight-coupling parameter, Eq. (24) can be written as

∆(2,I) 2,σ = 20 % d3k1 (2π)3 & v(1,Ø)_{γ 0} (k1)v(1,Ø)γ 0 (k2) ' Y2,σ1,1(ˆk1, ˆk2) . (27)

If we adopt the adiabatic initial condition as the first-order variables [46], the velocity perturbation for photons is proportional to η. The anisotropic stress of photons is proportional to η2 _{and finally we find the spectrum}

of the second-order anisotropic stress of photons must be proportional to η4_{. In the next section, we will find}

that this estimate is correct by using a numerical calculation.

We can use the above tight-coupling solutions as the initial conditions and show the results of the spectrum of gravitational wave in the next subsection.

B. Numerical results

In this section, we show the result of our numerical calculations. To solve the cosmological perturbation up to second-order, we need the time evolution and the transfer function of the first order perturbations, Φ(1)_{(k, η),}

1

st

_order

_{のadiabatic initial}

condition

_{を考えると、}

で時間のベキの依存性は分かる。

6

In this subsection, we consider tensor mode (m = σ) only. It is useful to define the function Y!1,!2

!,m (ˆk1, ˆk2) as Y!1,!2 !,m (ˆk1, ˆk2) ≡ (−1)m(2" + 1) ! m1,m2 " "1 "2 " 0 0 0 # " "1 "2 " m1 m2 −m # $ 4π 2"1+ 1 Y!∗1,m1(ˆk1) $ 4π 2"2+ 1 Y!∗2,m2(ˆk2) , (19) where "1+ "2+ " must be even because of a property of the Wigner-3j symbol. Note that, for a special case

that "1= 0 or "2= 0, the dependence of the ˆk1 or ˆk2 are vanish as

Y!1,0

!,m(ˆk1, ˆk2) = Y!,m(ˆk1)δ!,!1 , (20) Y0,!2

!,m(ˆk1, ˆk2) = Y!,m(ˆk2)δ!,!2 . (21) First, we show the zeroth order of the tight-coupling solution.

∆(2,Ø) 2,σ = 20 % _d3_k 1 (2π)3 & vγ 0(1,Ø)(k1)v(1,Ø)γ 0 (k2) ' Y2,σ1,1(ˆk1, ˆk2) , (22) ∆(2,Ø) !≥3,σ = 0 . (23)

It is very interesting that the anisotropic stress of photons vanish in the first-order cosmological perturbation up to zeroth order of the tight-coupling approximation, whereas the anisotropic stress of photons does not vanish in the second-order cosmological perturbation theory. The anisotropic stress of photons survive because of the scalar-mode of photon’s velocity perturbation. This result is consistent with refs. [34, 45].

Next, we consider (CPT= 2, TCA= I) and the results can be written as 9 10∆ (2,I) 2,σ = 2 ˙τc ˙χ(2,Ø) σ − % _d3_k 1 (2π)3 & 9Π(1,I) γ 0 (k1)(δ(1,Ø)b − Φ(1,Ø))(k2)' $ 4π₅ Y2,σ∗ (ˆk1) +4% d3k1 (2π)3 & 9v(1,Ø) γ 0 (k1)vγ 0(1,I)(k2) + 8vγ 0(1,Ø)(k1)δvγb 0(1,I)(k2) ' Y2,σ1,1(ˆk1, ˆk2) + % d3_k 1 (2π)3 (" k1 ˙τc # ) 10δ(1,Ø) γ (k1) − 8Φ(1,Ø) * (k1)v(1,Ø)γ 0 (k2) + Y2,σ1,1(ˆk1, ˆk2) , (24) ∆(2,I) 3,σ = − " k ˙τc # √5 5 ∆ (2,Ø) 2,σ + 15 % _d3_k 1 (2π)3 & Π(1,I) γ 0 (k1)v(1,Ø)γ 0 (k2) ' Y3,σ2,1(ˆk1, ˆk2) , (25) ∆(2,I) !≥4,σ = 0 . (26)

Very interestingly, in the second order cosmological perturbation and the first order tight-coupling approxi-mation, the higher multipole " = 3 can survive because of two source terms. One is the anisotropic stress of the zeroth order tight-coupling solution, which is only streaming effect in the left-hand side of the Boltzmann equation. The other is a convolution of the first order cosmological perturbation. This term comes from the collision term. These terms do not appear in the first order cosmological perturbation and this result is only the second order effect. Note that, more higher multipole equals zero since there are no any more source.

By using the tight-coupling solution, we can find the behavior of the anisotropic stress of photons in the early time. At zeroth order of the tight-coupling parameter, Eq. (24) can be written as

∆(2,I) 2,σ = 20 % d3_k 1 (2π)3 & vγ 0(1,Ø)(k1)vγ 0(1,Ø)(k2) ' Y2,σ1,1(ˆk1, ˆk2) . (27)

If we adopt the adiabatic initial condition as the first-order variables [46], the velocity perturbation for photons is proportional to η. The anisotropic stress of photons is proportional to η2 _{and finally we find the spectrum}

of the second-order anisotropic stress of photons must be proportional to η4_{. In the next section, we will find}

that this estimate is correct by using a numerical calculation.

We can use the above tight-coupling solutions as the initial conditions and show the results of the spectrum of gravitational wave in the next subsection.

B. Numerical results

0

_{次のTCAが破れて1次のTCAに移る}

次に効いてくるのが、メトリックの項

10-36 10-34 10-32 10-30 10-28 10-26 10-24 10-22 10-20 10-18 10-16 10-14 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 P a k = 10-3 k = 10-2 k = 10-1 k = 100 k = 101

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波(2次摂動)}

13/20

Silk  dumping

TCAが破綻する。

この時点でソースは散乱項と重⼒力力

項が同じくらい

è 

ソースもSilk  dumpingしてる

ので2次の⾮非等⽅方圧にも効かず

1次と同様Silk  dumping。

※ただし、重⼒力力優勢になるとそれ

に⽀支えられる

光⼦子の⾮非等⽅方圧の時間発展②

10-36 10-34 10-32 10-30 10-28 10-26 10-24 10-22 10-20 10-18 10-16 10-14 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 P a k = 10-3 k = 10-2 k = 10-1 k = 100 k = 101

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波(2次摂動)}

ソースに依存

(

波数に依存

)

(a)Streaming

(b)

⽀支えられる

光⼦子の⾮非等⽅方圧の時間発展③

(b)  遅くhorizonに⼊入るモード

w 

ソースが⼤大きいまま残る

w 

ソースに⽀支えられstreamingしない

(a)  早くhorizonに⼊入るモード

w 

ソースも早くにhorizonに⼊入る

w 

そもそも光⼦子のΔ

_l

が早くにdecay

w 

重⼒力力項も効かずにstreamingする

10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 106 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Source a reference: large k, small k, large k,

ソースに依存

(

波数に依存

)

10-36 10-34 10-32 10-30 10-28 10-26 10-24 10-22 10-20 10-18 10-16 10-14 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 P a k = 10-3 k = 10-2 k = 10-1 k = 100 k = 101

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波(2次摂動)}

ソースのベキ

最初はソースのベキによって作られる。

重⼒力力波が作り出す

(

光⼦子と同じベキ

)

ニュートリノの⾮非等⽅方圧①

10

-34

10

-32

10

-30

10

-28

10

-26

10

-24

10

-22

10

-20

10

-18

10

-16

10

-14

10

-8

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

P

a

k = 10

-3

k = 10

-2

k = 10

-1

k = 10

0

k = 10

1

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波(2次摂動)}

ソースが早くにhorizonに⼊入ってstreamingして落落ちてい

るため2nd  orderでもstreamingする。

ニュートリノの⾮非等⽅方圧②

10

-34

10

-32

10

-30

10

-28

10

-26

10

-24

10

-22

10

-20

10

-18

10

-16

10

-14

10

-8

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

P

a

k = 10

-3

k = 10

-2

k = 10

-1

k = 10

0

k = 10

1

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波(2次摂動)}

重⼒力力波に与える影響

1次×1次  再掲(パワースペクトル)

10

-32

10

-30

10

-28

10

-26

10

-24

10

-22

10

-20

10

-18

10

-16

10

-14

10

-4

10

-3

10

-2

10

-1

10

0

10

1

P

k

z = 3400

z = 580

z = 100

z = 10

z = 0

ソースが作る

ソースが⽀支える

ソースを失う

1.

_イントロ

2. 2

_{次摂動論論}

3.

_{重⼒力力波}

(

_線形摂動

)

4.

_{重⼒力力波(2次摂動)}

10-32 10-30 10-28 10-26 10-24 10-22 10-20 10-18 10-16 10-14 P z=3400 z=580 z=100 z=10 z=0 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 10-4 10-3 10-2 10-1 100 101 P /P0 k

結果(preliminary)

10-32 10-30 10-28 10-26 10-24 10-22 10-20 10-18 10-16 10-14 P z=3400 z=580 z=100 z=10 z=0 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 10-4 10-3 10-2 10-1 100 101 P /P0 k

ü 

small  kで初期  è  2次のソースが⽣生きている

ü 

large  k  è  free  streamingしているが重⼒力力波に寄与(1次×1次は死ぬ)

ü 

初期には光⼦子は散乱項のソースのためニュートリノより⼤大きな寄与

5.  まとめ

Ø 

super  horizonスケールでも振幅を増やす働き  (1次とは異異なる)

Ø 

1次×1次  が切切れたあとでも⾮非等⽅方圧が重⼒力力波に寄与

Ø 

正味30％ほどの増幅

k

Ω

_GW

~∼30％

~∼20-‐‑‒30％

スカラー1次×1次