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1 KEK 2 (cosmological perturbation theory) CMB R. Durrer, The theory of CMB Anisotropies, astro-ph/ ; A. Liddle and D. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, 2000). 1 Friedmann ( ) Friedmann 100 ( ) 1 (, 2016) 2 URL: 1

1: ( ) (Sachs-Wolfe ) (CMB) CMB Planck ( 2) δt/t 10 5 2: CMB

2 1: ( ) (Sachs-Wolfe ) (CMB) CMB Planck ( 2) δt/t : CMB Planck Wilkinson Microwave Anisotropies Probe (WMAP) CMB ( 3) 2

3 Harrison- Zel dovich 3 CMB Planck 3 3: WMAP : (l <30) (30 <l<800) 3 CMB 3

4 Silk (l >800) Silk ( ) ( ) (photon diffusion) 4 1 l 800 l<800 Silk Boltzmann Boltzmann CMB CMBFAST 3 2 Friedmann ( ) Einstein Friedmann ds 2 = e 2 ˆφ ( dη 2 + dx 2), (2.1) T µ (α)ν = diag( ρ α,p α,p α,p α ) (2.2) 4 ( ) 4

5 ˆφ(η) η (conformal time) x (comoving coordinate) dτ = e ˆφdη r = e ˆφx ( ) ρ α P α α η (equation of state parameter) w α = P α ρ α (2.3) 1/3 0 µ T µ (α)ν =0 η ρ α +3 η ˆφ (ρα + P α ) = 0 (2.4) Einstein (00) M 2 Pe 2 ˆφ ( 6 2 η ˆφ +6 η ˆφ η ˆφ) ρ +3P 4Λ = 0, (2.5) 3M 2 P e 2 ˆφ η ˆφ η ˆφ + ρ +Λ = 0, (2.6) Λ ρ P ρ = α ρ α, P = α P α (2.7) (2.4) α Einstein 5

6 ρ P w = P/ρ a a = e ˆφ, η ˆφ = η a a = ah (2.8) H Hubble Einstein ( 6MP 2 a 1 η H +2H 2) = ρ 3P +4Λ=(1 3w)ρ +4Λ, (2.9) 3MP 2 H2 = ρ +Λ, (2.10) η ρ α = 3aH (ρ α + P α )= 3(1+w α ) ahρ α (2.11) Friedmann Hubble Hubble h H 0 = 100h [kms 1 Mpc 1 ] (c = h =1) H 0 = h = Mpc 1. (2.12) h =0.72 Hubble 1/H Mpc Hubble Ω α = ρ α0 3M 2 P H2 0 (2.13) ρ α0 α α (cold dark matter, CDM) c b r c b d w = 1 6

7 Ω Λ =Λ/3MP 2H2 0 Ω r = Ω γ +Ω ν = /h 2 = , (2.14) Ω b = 0.042, (2.15) Ω d = Ω c +Ω b =0.27, (2.16) Ω Λ = 0.73 (2.17) Ω r Ω γ 3 Ω ν ρ γ0 =2(π 2 /30)Tγ 4 ρ ν0 = N ν 2(π 2 /30)(7/8)T 4 ν Ω ν = ρ ν0 7 = N ν Ω γ ρ γ0 8 ( Tν T γ ) 4 =0.68 (2.18) T ν /T γ =(4/11) 1/3 N ν =3( ) Ω γ = Einstein (2.10) Ω d +Ω Λ = 1 (2.19) Ω r 1 1 r c b w r = 1 3, w c =0, w b = 0 (2.20) (2.11) ( ) a0 4 ρ r = ρ r0 =3M 2 P a H2 0 Ω r ) 3 =3M 2 ρ c = ρ c0 ( a0 a ρ b = ρ b0 ( a0 a PH 2 0Ω c ( a0 a ) 3 =3M 2 PH 2 0Ω b ( a0 a ( ) a0 4, (2.21) a ) 3, (2.22) ) 3 (2.23) 7

8 a 0 a 0 =1 a 0 =1 Einstein (2.10) Hubble H 2 = H 2 0 { Ω r ( a0 a ) 4 ( ) } a0 3 +Ωd +ΩΛ a (2.24) a 0 /a < 2 (red shift)z z +1= a 0 a (2.25) (comoving angular size distance) 5 (2.24) H = η a/a 2 dη = d = η 0 η = 1 z dz (2.26) a 0 H 0 0 Ω r (z +1) 4 +Ω d (z +1) 3 +Ω Λ z =0.1 d = 408Mpc, z =1 d = 3271Mpc, z =5 d = 7822Mpc, z dec = 1100 d dec = 13808Mpc ( ). (2.27) z dec (last scattering surface) 4 θ λ 0 /d (multipole)l (comoving wave number)k = π/λ 0 C l 5 proper motion distance angular diameter distance transverse comoving distance 8

9 0 (z =0) (z) A B 4: d dec l d = d dec l π θ = kd dec (2.28) k =0.0002Mpc 1 l 3 k =0.002Mpc 1 l 30 k =0.005Mpc 1 l 70 k =0.015Mpc 1 l 210 k =0.05Mpc 1 l 700 (2.29) p = k/a a 0 =1 k l =3 Hubble (1/H 0 ) 1/k 5000Mpc l 30 1/k 500Mpc l 700 1/k 20Mpc (super cluster of galaxies) 10 30Mpc a 4 a 3 ρ r ρ d ρ r ρ d ρ r = ρ d z eq +1= Ω d Ω r = 3333 (2.30) 9

10 ρ d =0 Einstein a η ρ r =0 a η 2 dτ = adη ( ) a τ 1/2 a τ 2/3 3 φ h µν g µν = e 2φ ḡ µν, ḡ µν =(e h ) µν = η µν + h µν + (3.1) η µν h λ λ =0 φ(η, x) = ˆφ(η)+ϕ(η, x) (3.2) ˆφ ϕ ds 2 = g µν dx µ dx ν = e 2 ˆφ(1 + 2ϕ)(η µν + h µν ) dx µ dx ν = a 2{ (1 + 2ϕ h 00 )dη 2 +2h 0i dηdx i +(δ ij +2ϕδ ij + h ij ) dx i dx j} (3.3) 6 i, j =1, 2, 3 ρ α (η, x) =ρ α (η)+δρ α (η, x), P α (η, x) =P α (η)+δp α (η, x) (3.4) 6 A B i H ij ds 2 = a 2 { (1 + 2A)dη 2 2B i dηdx i +(δ ij +2H ij ) dx i dx j} 10

11 ρ P T µ (α)ν = {ρ α(η, x)+p α (η, x)} u µ αu α ν + P α (η, x)(δ µ ν +Π αµ ν) (3.5) Π α µν (Πα 0ν =0) u µ α 4 g µν u µ α uν α = 1 (3.6) 4 u µ α =(1/a, 0, 0, 0) 4 u µ α uα µ = g µνu ν α (3.6) u 0 α = 1 ( 1 ϕ + 1 ) a 2 h 00, u i α = vi α a, ( u α 0 = a 1+ϕ 1 ) 2 h 00, u α i = a (vi α + h 0i ) (3.7) vα i vα i = δ ij vα j 4 (3.6) 4 T(α)0 0 = (ρ α + δρ α ), T(α)0 i = (ρ α + P α )vα i, T(α)j 0 = (ρ α + P α )(vj α + h 0j ), T(α)j i = (P α + δp α )δ i j + P α Π αi j (3.8) T µ (α)ν g µν v i = δ ij v j δ ij T 0 i = g 0λ T λi g iλ T λ0 = T i 0 v α i = i v α + v Tα i (3.9) 11

12 vi Tα (transverse) Π α ij Π αi i =0 (general coordinate transformation) δ ξ g µν = g µλ ν ξ λ + g νλ µ ξ λ δ ξ ϕ = ξ λ 1 λ ˆφ + 4 λξ λ, (3.10) δ ξ h µν = µ ξ ν + ν ξ µ 1 2 η µν λ ξ λ (3.11) 7 ξ µ = η µν ξ ν h 00 = h, (3.12) h 0i = h T i + i h, (3.13) h ij = h TT ij + (i h T j) + 1 ( 3 δ i j ijh + 1 ) 2 3 δ ij h (3.14) h T i ht i h TT ij 2 = i i ξ µ ξ 0 ξ i = ξi T + i ξ S δ ξ ϕ = ξ 0 1 η ˆφ + 4 ηξ ξ S, (3.15) δ ξ h = 3 2 ηξ ξ S, (3.16) δ ξ h = ξ 0 + η ξ S, (3.17) δ ξ h = 2 2 ξ S, (3.18) δ ξ h T i = η ξ T i, (3.19) δ ξ h T i = 2ξi T, (3.20) δ ξ h TT ij = 0 (3.21) 7 ϕ h µν δ ξ h µν = µ ξ ν + ν ξ µ 1 2 η µν λ ξ λ + ξ λ λ h µν h µλ( ν ξ λ λ ξ ν )+ 1 2 h νλ( µ ξ λ λ ξ µ )+o(h 2 ) 12

13 δ ξ T µ (α)ν = νξ λ T µ (α)λ λ ξ µ T(α)ν λ + ξλ λ T µ (α)ν δ ξ v α = η ξ S, (3.22) δ ξ vi Tα = η ξi T, (3.23) δ ξ (δρ α ) = ξ 0 η ρ α, (3.24) δ ξ (δp α ) = ξ 0 η P α, (3.25) δ ξ Π αi j = 0 (3.26) Bardeen Φ = ϕ h 1 6 h + σ η ˆφ, (3.27) Ψ = ϕ 1 2 h + σ η ˆφ + η σ (3.28) σ = h 1 η h 2. (3.29) 2 δ ξ σ = ξ 0 Bardeen (δ ξ Φ=δ ξ Ψ=0) (longitudinal gauge) (conformal Newtonian gauge) h = h =0 Bardeen Φ=ϕ + h/6 Ψ=ϕ h/2 ds 2 = a 2 [ (1 + 2Ψ) dη 2 + (1 + 2Φ) dx 2] (3.30) Ψ Newton 13

14 Υ i = h T i 1 2 ηh T i, (3.31) h TT ij (3.32) α V α = v α + 1 η h 2, (3.33) 2 D α = δρ α ρ α + ηρ α α σ 3(1 + w α ) η ˆφV ρ α = δρ α ρ α 3(1 + w α ) η ˆφ(σ + V α ), (3.34) D α = δρ α ρ α + ηρ α ρ α σ + 3(1 + w α )Φ = δρ α + 3(1 + w α )(Φ η ˆφσ), ρ α (3.35) Vi α = vi Tα ηh T i, (3.36) Ω α i = vi Tα + h T i, (3.37) Π α ij (3.38) (3.34) (3.35) η ρ α η ρ α = 3(1 + w α ) η ˆφρα D α = D α 3(1 + w α )(Φ + η ˆφV α ) Υ i + Vi α Ω α i =0 D α D α CMB 8 ρ P (2.7) 8 Θ Newton D γ /4= Θ+Φ 14

15 D D V ρd = α ρd = α ρ α D α, (3.39) ρ α D α, (3.40) (1 + w)ρv = (ρ + P )V = α = α (ρ α + P α )V α (1 + w α )ρ α V α, (3.41) P Π ij = α P α Π α ij (3.42) w = P ρ = α P α α ρ α, (3.43) D α D α D α D α V α V α w α w α D α ρ α ρ α D α δρ α (ρ α + P α )V α V ( ) ( ) P P δp = δρ + δs = c 2 sδρ + TδS (3.44) ρ S S ρ (δs =0) δρ δp (sound speed) c 2 s = P/ ρ Γ α = 1 ( ) δpα c 2 δp α P αδρ α = c2 α δρ α (3.45) α P α w α ρ α 15

16 Γ P Γ=δP c 2 sδρ (3.46) δp δρ c 2 s = ηp η ρ = α η P α (3.47) α η ρ α c 2 s α c 2 α 4 ( ) 4.1 Einstein δi = 1 d 4 x gt µν δg µν 2 = 1 d 4 x ḡ { 2 2 T λ λδφ + T } µν δḡ µν { = d 4 x T λ λδφ + 1 } 2 Tµ νδh ν µ = 0 (4.1) T µν T µν T µν g µν = e 2φ ḡ µν δg µν =2e 2φ ḡ µν δφ + e 2φ δḡ µν (4.2) T µν (g) g µν T µν (φ, ḡ) ḡ µν T µν (ϕ, h) η µν 16

17 T µν = e 6φ T µν = e 6 ˆφ(1 6ϕ) T µν, (4.3) T µ ν = e 4φ T µ ν = e 4 ˆφ(1 4ϕ) T µ ν (4.4) h µ ν T µν = η λ(µ T λ ν) (4.5) = T µν h λ (µ ˆT ν)λ (4.6) ˆT µν T µν T λ λ (= ηµν T µν )= T λ λ Einstein T µν = T EH µν + T Λ µν + T M µν = 0 (4.7) T M µν = α T (α) µν δ ξt µν = 0 Einstein (3.8) T Mλ λ = e 4 ˆφ { ρ +3P δρ +3δP +4( ρ +3P )ϕ}, T M 00 = e 4 ˆφ(ρ + δρ +4ρϕ), T M 0i = e 4 ˆφ(ρ ( + P ) v i + 1 ) 2 h 0i, T M ij = e 4 ˆφ {(P + δp +4Pϕ)δ ij + P Π ij } (4.8) δρ δp (2.7) v i (3.42) V 17

18 ( Π ij = i j + 1 ) 2 3 δ ij Π S + (i Π V j) +Π T ij (4.9) Π V i Π T ij Π α ij Einstein h µν T EH µν = M 2 Pe 2φ { 2 µ ν φ 2 µ φ ν φ + η µν ( 2 φ λ φ λ φ ) (µ χ ν) h µν 2h λ (µ ν) λ φ +2h λ (µ ν)φ λ φ 2 (µ h λ ν) λ φ + λ h µν λ φ ( )} 1 +η µν 2 λχ λ +2h λσ λ σ φ + h λσ λ φ σ φ +2χ λ λ φ (4.10) χ µ = λ h λ µ = λ λ = η φ T EHλ λ = M 2 Pe 2φ{ 6 φ 6 λ φ λ φ + λ χ λ +6h λσ λ σ φ +6χ λ λ φ +6h λσ λ φ σ φ } (4.11) φ ˆφ ϕ ϕ h = h = h T i =0 Einstein { T EHλ λ = MP 2 ˆφ e2 6 η 2 ˆφ +6 η ˆφ η ˆφ + 12( 2 η ˆφ + η ˆφ η ˆφ)ϕ +6 ηϕ ϕ +12 η ˆφ η ϕ + ηh h +6 η ˆφ η h +6( η 2 ˆφ } + η ˆφ η ˆφ)h { T EH 00 = MP 2 ˆφ e2 3 η ˆφ η ˆφ 6 η ˆφ η ˆφϕ 6 η ˆφ η ϕ +2 2 ϕ (4.12) 18

19 3 η ˆφ η ˆφh η ˆφ η h + 1 } 3 2 h (4.13) { T EH 0i = MP 2 ˆφ e2 2 η i ϕ 2 η ˆφ i ϕ η i h + η ˆφ i h h T i +( η 2 ˆφ } T η ˆφ η ˆφ)h i (4.14) { [ T EH ij = MP 2 ˆφ e2 2 i j ϕ + δ ij 2 2 η ˆφ + η ˆφ η ˆφ +2 2 η ϕ 2 2 ϕ +2 η ˆφ η ϕ + ( 4 η 2 ˆφ ] 1 +2 η ˆφ η ˆφ) ϕ 3 i j h [ 1 +δ ij 3 2 η h h ˆφ η η h + ( 2 η 2 ˆφ ] + η ˆφ η ˆφ) h + η (i h T j) +2 η ˆφ (i h T j) η htt ij h TT ij η ˆφ η h TT ij } (4.15) ϕ = (3Φ + Ψ)/4 h = 3(Φ Ψ)/2 T Λ µν = Λe 4 ˆφ(1 + 4ϕ)η µν (4.16) e 4 ˆφT λ λ = 0, ( T i i 3 i j ) T 2 ij = 0, ( T η ˆφ i ) T 2 i0 = 0, 4 ˆφ e 4 ˆφ e 4 ˆφ i e T 2 i0 = 0 (4.17) 19

20 M 2 Pe 2 ˆφ { 6 2 ηφ+18 η ˆφ η Φ 4 2 Φ 6 η ˆφ η Ψ + ( 12 2 η ˆφ +12 η ˆφ η ˆφ 2 2 ) Ψ } +(3c 2 s 1)ρ { D + 3(1 + w) η ˆφV } +3wρΓ +(3w 1)ρ(3Φ + Ψ) 4Λ(3Φ + Ψ) = 0 (4.18) (3.46) δp P Γ+c 2 sδρ Φ Ψ M 2 Pe 2 ˆφ( 2 2 )(Φ+Ψ)+2P Π S = 0 (4.19) Poisson M 2 Pe 2 ˆφ2 2 Φ+ρD = 0 (4.20) M 2 Pe 2 ˆφ { 2 η Φ 2 η ˆφΨ } (1 + w)ρv = 0 (4.21) Einstein (2.5) (2.6) Π S =0 Φ Ψ D V

21 4 ˆφ j e T 2 ij = 0, (4.22) e 4 ˆφT 0i = 0 (4.23) Ω i { } MPe 2 2 ˆφ 1 2 ηυ i + η ˆφΥi P ΠV i = 0 (4.24) 1 2 M 2 Pe 2 ˆφ 2 Υ i (1 + w)ρω i =0. (4.25) Einstein (2.5) (2.6) Π V i =0 (4.24) (4.25) Ω i { MPe 2 2 ˆφ ηh TT ij η ˆφ η h TT ij e 4 ˆφT ij =0 + 1 } 2 2 h TT ij + P Π T ij = 0 (4.26) Π T ij =0 4.2 Einstein α Einstein µ T µ (α)ν = 1 ( ) µ gt µ 1 g (α)ν + 2 ( νg µλ ) g λσ T µ (α)σ = 0 (4.27) Einstein 21

22 D α V α Ω α i 1 µ T µ (α)0 ρ α = 0, (4.28) 1 µ T µ (α)i (1 + w α )ρ α = 0 (4.29) 0 η D α +3 ( c 2 α w α ) η ˆφD α +(1 + w α ) 2 V α +3w α η ˆφΓ α = 0 (4.30) i i / 2 η V α + ( 1 3cα) 2 η ˆφV α +Ψ 3c 2 αφ + c2 α D α + w α [Γ α 2 ] 1+w α 1+w α 3 ΠSα = 0 (4.31) i η Ω α i + ( 1 3c 2 α) η ˆφΩ α i + w α 2(1 + w α ) 2 Π Vα i = 0 (4.32) (2.4) η w α = 3(1 + w α ) ( c 2 α w α) η ˆφ (4.33) Einstein 22

23 α D V w c 2 s ρ 1 µ T Mµ 0 =0 (1 + w) 1 ρ 1 µ T Mµ i =0 D α D α = D α +3(1+w α ) ( Φ+ η ˆφV α ) (4.34) V α (4.31) c2 α η V α + η ˆφV α +Ψ+ D α 1+w α + w α [Γ α 2 ] 1+w α 3 ΠSα =0. (4.35) D α (4.30) Einstein (2.5) (2.6) (4.21) η D α 3w α η ˆφD α +(1+w α ) 2 V α +2w α η ˆφΓ α + 3 (1 + w 2MP 2 α )(1+w)ρa 2 (V V α ) = 0 (4.36) ρ w V D α Ψ Φ D D Γ Π S Ψ(η, x) = [d 3 k]ψ(η, k)e ik x (5.1) [d 3 k]= 1 d 3 k (5.2) (2π) 3 k

24 k = k V V (η, x) = ( [d 3 k] 1 ) V (η, k)e ik x (5.3) k V (η, k) V i Ω i Υ i h TT ij V i (η, x) = [d 3 k]v i (η, k)e ik x, (5.4) h TT ij (η, x) = [d 3 k]h TT ij (η, k)eik x (5.5) Λ Π ij CMB Integrated Sachs-Wolfe Π ij (Γ α =0) Λ=Π α ij =Γα =0 k 2 Φ= a2 ρ 2MP 2 α D α α = a2 ρ 2MP 2 α {D α +3(1+w α ) (Ψ+aH V α )}, (5.6) α k Φ= Ψ, (5.7) η D α +3 ( c 2 α w α) ahd α = (1 + w α ) kv α, (5.8) η V α + ( ) 1 3c 2 α ahv α = k ( Ψ 3c 2 α Φ) + c2 α 1+w α kd α (5.9) η Υ i +2aHΥ i =0, (5.10) η Ω α i + ( 1 3c 2 α) ahω α i =0, (5.11) 24

25 2 ηh TT ij +2aH η h TT ij + k 2 h TT ij = 0 (5.12) 6 Planck P d = δp d =0 ρ = ρ r + ρ d δρ = δρ r + δρ d, (6.1) P = P r = 1 3 ρ r, δp = δp r = 1 3 δρ r (6.2) c 2 s = ηp η ρ = ρ d (6.3) 4 ρ r TδS = δp c 2 sδρ = 1 3 δρ r 1 3 = 1 3 ρ d 1+ 3 ρ d 4 ρ r ρ d 4 ( 3 4 (δρ r + δρ d ) ρ r δρ ) d ρ d δρ r ρ r (6.4) 25

26 δs =0 δρ r = 4 δρ d (6.5) ρ r 3 ρ d D r =(4/3)D d V r = V d D r =(4/3)D d CDM CDM (6.5) δρ r = 4 δρ c = 4 ( δρ b = δρ ) (6.6) ρ r 3 ρ c 3 ρ b ρ k k p = k/a a 0 =1 k 26

27 k =0.0002Mpc 1 1/a =1+z = Mpc 1 τ dτ = adη (5.10) Υ i +2HΥ i = 0 (7.1) τ H e 2Hτ H Ω α i (5.11) c2 α 1/3 c 2 α < 1/3 CMB ḧ TT ij +3HḣTT ij + k2 a 2 htt ij = 0 (7.2) k/a ( ) k/a ḣtt ij =0 Hubble H k/a a/k 1/H a/k 1/H 27

28 (super-horizon) (sub-horizon) CMB (1/k) Mpc Hubble 1/H 0 = 4164Mpc a/k > 1/H CMB 7.2 x = kη (0 <x< ) (7.3) ( ) a η (η 2 ) ah = η a/a =1/η (2/η) a k > 1 H = x<1(x<2) (7.4) x 1 (2) 1 (2) 1 2 x x 1 28

29 1/k super horizon > x =2 > x =1 eq > sub horizon > dec 5: 1/k (k = ah) CMB l =2, 3 5 1/k k =0.002Mpc 1 (l 30) k =0.005Mpc 1 (l 70) k = 0.015Mpc 1 (l 210) k = 0.05Mpc 1 (l 700) l π/θ = kd dec CMB (l 30) l 210 ah = η a/a η Υ i +2aHΥ i = η (a 2 Υ i )=0, (7.5) η Ω α i +(1 3c 2 α)ahω α i η (a 1 3c2 α Ω α i ) = 0 (7.6) c α Υ i a 2, Ω α i a 3c2 α 1 (7.7) 29

30 Υ i Ω α i c2 α < 1/3 x (q =1) (q =2) ah = q/x 2 x htt ij +2 q x xh TT ij h TT ij h TT ij = + h TT ij = 0 (7.8) = e ij x 1/2 q J 1/2 q (x) e ij const. 1 a for x 1 (super-horizon) for x>1 (sub-horizon) (7.9) 8 (CDM) x = kη 8.1 CDM ρ r ρ c (8.1) Friedmann 3M 2 PH 2 = ρ ρ r Poisson (5.6) ρ c Ψ 3 { 1 D r +4 (Ψ+ 1 )} 2 x 2 x V r 30 (8.2)

31 (5.7) a η ah = η a/a =1/η a 2 ρ r /2MP 2 =(3/2)(aH)2 =3/2η 2 w r = c 2 r =1/3 CDM w c = c 2 c =0 x D r V r =0, (8.3) x V r =2Ψ+ 1 4 Dr (8.4) x D c + V c =0, (8.5) x V c + 1 x V c = Ψ (8.6) (8.2) (8.3) (8.4) (x 2 +6) x 2 Dr + 12 x xd r (x2 6)D r = 0 (8.7) { ( ) x D r = A cos 2 ( )} { ( ) 3 x x 3 x sin +B sin x ( )} x cos 3 (8.8) x 0 B =0 { ( ) x D r = A cos 2 ( )} 3 x 3 x sin, (8.9) 3 V r = 3 4 xd r = A 3 { x 2 ( ) 6 x 3 4 sin + 2 ( )} x 3 3x 2 x cos, (8.10) ( 1 Ψ = 3D r + 12 ) x 2 x V r (8.11) (x 1) Ψ = Ψ i 1 30 Ψ ix 2 +, (8.12) D r = 6Ψ i 1 3 Ψ ix 2 +, (8.13) V r = 1 2 Ψ ix + (8.14) 31

32 Bardeen Ψ i = A/6 k x 2 Ψ D r D r D r = 2 3 Ψ ix 2 (8.15) D CDM V c Ψ (8.6) D c V c (8.5) D c (x =0) = 3 4 Dr (x =0), V c (x =0) = V r (x = 0) (8.16) Ψ =Ψ i + CDM D c = 9 8 Ψ i 1 4 Ψ ix 2 +, (8.17) V c = 1 2 Ψ ix +. (8.18) D c (x 1) D r V r Bardeen Ψ= 3 2x 2 Dr (8.19) 1/x 2 Ψ 0 CDM D c log x, (8.20) V c 1 x (8.21) D c (Mezaros ) 32

33 8.2 ρ r ρ c Friedmann 3M 2 P H2 = ρ c a η 2 ah =2/η Poisson (5.6) k 2 Ψ=(a 2 /2MP)ρ 2 c D c =(6/η 2 )D c Bardeen CDM CDM w c = c 2 c =0 CDM ( (x ) x 2 V c + 4x + 72 x (x )Ψ = 6D c + 36 x V c, (8.22) x D c + V c =0, (8.23) x V c + 2 x V c = Ψ (8.24) ) ( x V c V c = V 0 x + V 1 x x 2 ) V c = 0 (8.25) (8.26) x 0 V 1 =0 Ψ = Ψ i, (8.27) D c = 5Ψ i 1 6 Ψ ix 2, (8.28) V c = 1 3 Ψ ix, (8.29) Ψ i =3V 0 w r = c 2 r =1/3 x D r V r =0, (8.30) x V r =2Ψ+ 1 4 Dr (8.31) 33

34 2 xv r V r =2 x Ψ (8.32) Bardeen D r ( ) ( ) x x V r = A sin + B cos, (8.33) 3 3 D r = 4A ( ) x cos 4B ( ) x sin 8Ψ i (8.34) A B x 0 V r = V c D r = (4/3)D c B =0 A =Ψ i / 3 D r = 8Ψ i + 4 ( ) x 3 Ψ i cos, (8.35) 3 V r = Ψ ( ) i x sin (8.36) 3 3 (x 0) D r ( 20/3)Ψ i Bardeen Bardeen CMB Bardeen Sachs-Wolfe (11.15) T/T 10 5 Bardeen (x 2) D c D r (x 2) CDM D c x 2 D r 34

35 CDM V c x V r x D r x 0 (x 0) l<200 Ψ D r (η = η eq ) D r = 6Ψ i D r =( 20/3)Ψ i 1 (first acoustic peak) CMB 11 Sachs-Wolfe (11.12) CMB T T (η 0) 1 4 Dr (η dec ) + 2Ψ(η dec )= 1 3 Ψ i cos (c s x dec ) (8.37) (8.27) (8.35) c s = c r =1/ 3 c s x dec = c s kη dec =0,π,2π, k 1peak = π/r s r s = c s η dec (2.28) l 1peak k 1peak d dec = π(η 0 η dec ) c s η dec = π c s ( zdec +1 1 ). (8.38) z dec = 1100 l 174 c s < 1/ 3 l 35

36 9 σ T = (8π/3)α 2 /m 2 e (α = e2 /4π 1/137) Boltzmann w b,c 2 b 1 10 w γ = w ν = c 2 γ = c 2 ν =1/3, w c = w b = c 2 c = c2 b = 0 (9.1) P γ = ρ γ /3 P ν = ρ ν /3 P c = P b =0 Poisson 2MP 2 k 2 { a Ψ=ρ 2 c D c +3 (Ψ+aH V c )} { + ρ ν D ν +4 (Ψ+aH V ν )} k k +ρ γ {D γ +4 (Ψ+aH V γ )} { ( + ρ b D b +3 Ψ+aH V b )} (9.2) k k η D c = kv c, (9.3) η V c + ahv c = kψ, (9.4) 10 ρ b = nm P b = nt b m 1GeV n ( 1/a 3 ) T b ( 1/a) w b = P b /ρ b = T b /m c 2 b = ηp b / η ρ b =4T b /3m z eq z dec T b m w b = c 2 b =0 36

37 η D ν = 4 3 kv ν, (9.5) η V ν = 2kΨ+ 1 4 kdν, (9.6) η D γ = 4 3 kv γ, (9.7) η V γ = 2kΨ+ 1 4 kdγ 1 ( V γ V b), η T (9.8) η D b = kv b, (9.9) η V b + ahv b = kψ+ 1 4 ρ ( γ V γ V b) η T 3 ρ b (9.10) η T = η (z >z Ω b h (1+z eq) 1/2 1+z eq) aσ T n e 10 1 Ω b η (z h (1+z) 3/2 eq >z>z dec ) (9.11) n e ρ b /m(m 1GeV) Ω b 0.04 Hubble h =0.72 z eq = 3333 z dec = 1100 η T η η T 0 V r = V b D c (0) = D b (0) = 3 4 Dγ (0) = 3 4 Dν (0), V c (0) = V b (0) = V γ (0) = V ν (0) (9.12) D b (x) 3 4 Dγ (x) (9.13) (9.7) (9.9) η (D b 3D γ /4) = k(v b V γ ) 37

38 V b = V r ρ = ρ γ + ρ b, P = P γ + P b = 1 3 ρ γ (9.14) w = P ρ = ρ, b ρ γ (9.15) c 2 s = ηp η ρ = ρ b 4 ρ γ (9.16) D = 1 ( ργ D γ + ρ b D b), ρ (9.17) V = 1 { (ργ + P γ ) V γ + ρ b V b} ρ + P (9.18) α = bγ Poisson (5.6) α = c, ν, bγ 2MP 2 k 2 { a Ψ = ρ 2 c D c +3 (Ψ+aH V c )} { + ρ ν D ν +4 k { ( +ρ D + 3(1 + w) Ψ+aH V )} k (Ψ+aH V ν )} k (9.19) (9.2) (5.7) (5.8) (5.9) η D +3 ( c 2 s w ) ahd = (1 + w)kv, (9.20) η V + ( 1 3c 2 s ) ( ) ahv = k 1+3c 2 c 2 s Ψ+ s kd (9.21) 1+w 38

39 6: D γ ( ) CDM D c ( ) Bardeen Φ( ) z (z 10 3 ) Harrison-Zel dovich Φ k 1 D c z eq D γ Φ 7 (9.7) (9.9) η D ρ η D = (1 + w)ρkv +3waHρD ahρ r D γ (9.22) V b V γ η (D b 3D γ /4) 0 D b 3D γ /4 D = 3(1 + w)d γ /4 ρ γ D γ 3c 2 sρd (9.20) (9.8) (9.10) (1 + w)ρ η V 39

7: V b ( ) CDM V c ( ) 6 Φ( ) Φ z eq z eq (1 + w)ρ η V =(1+w)ρ { ( 1 3c 2 s ) ahv + ( 1+3c 2 s ) kψ } + 1 3 kρ rd γ (9.23) (9.21) 6 9 CDM D c V c D ν V ν D V (9.3) (9.4) (9.5) (9.6) (9.

40 7: V b ( ) CDM V c ( ) 6 Φ( ) Φ z eq z eq (1 + w)ρ η V =(1+w)ρ { ( 1 3c 2 s ) ahv + ( 1+3c 2 s ) kψ } kρ rd γ (9.23) (9.21) 6 9 CDM D c V c D ν V ν D V (9.3) (9.4) (9.5) (9.6) (9.20) (9.21) Bardeen Φ(= Ψ) Poisson (9.19) Friedman 2 Ω b =0.042 Ω d =0.27 Ω r = Ω γ = Ω Λ =0.73 h =0.72 Harrison-Zel dovich Ψ i = 1 (8.1 ) D γ = 4(1 + w)d/4 V b = V Sachs-Wolfe 40

41 8: Sachs-Wolfe D γ /4+2Ψ 6 (z 10 3 ) cos [(8.37) ] 0.02Mpc 1 CMB 1 D γ /4+2Ψ (9.20) (9.21) (9.16) c s (η dec )= 1 3 ( ) = 1+ 3Ω b a dec 4Ω γ a ( 1+ 3Ω b 1 4Ω γ z dec +1 ) (9.24) 2 c s = (8.38) 11 Hu-Sugiyam, ApJ, 444 (1995) 489 k 3/2 (Θ 0 +Ψ) Fourier k 3/2 41

42 9: Bardeen Φ( ) Sachs-Wolfe D γ /4 + 2Ψ( ) V b ( ) (z 10 3 ) l 1peak π ( zdec +1 1 ) = 220 (9.25) c s (η dec ) 10 (Sachs-Wolfe ) CDM 42

43 CDM η D c,b = kv c,b, (10.1) η V c,b + ahv c,b = kψ (10.2) V c,b 2 η Dc,b + ah η D c,b = k 2 Ψ (10.3) CDM 2 η ( D c D b) + ah η ( D c D b) = 0 (10.4) η D c = η D b CDM CDM η D c (η dec ) > 0 η D b (η dec ) 0 CDM CDM CDM 11 Sachs-Wolfe i f F µν A µν ψ F µν = Re(A µν e iψ ) A µν e iψ 43

44 10: i f ds 2 = a 2 dσ 2 12 dσ 2 = G µν dx µ dx ν =(η µν + H µν ) dx µ dx ν (11.1) dσ 2 (affine parameter) λ n µ = dxµ dλ, G µνn µ n ν =0, dn µ dλ + Γ µ αβ (G)nα n β = 0 (11.2) ψ (x µ + n µ λ) =ψ (x µ ) n µ ψ x µ = 0 (11.3) K ψ x µ = KG µνn ν (11.4) 12 ḡ µν dx µ dx ν G µν (g µν = a 2 G µν ) 44

45 (10) s ψ = ψ dx µ ψ s = x µ ds x µ uµ s (11.5) u µ = dx µ /ds 4 g µν u µ u ν = 1 E f E i = ν f ν i = ψ s f ψ s i = (G µνnµ uν ) f (G µν n µ u ν ) i (11.6) n 0 =1 n i n i =1 4 u µ =(1/a, 0, 0, 0) E f /E i = a i /a f n µ =(1, n)+δn µ dδn µ ( dλ = (α H µ β) + 1 ) 2 µ H αβ n α n β (11.7) d(h µ β nβ ) dλ = dhµ β d λ nβ = dxα dλ αh µ β nβ =( α H µ β )nα n β (11.8) δn µ f i = H µ β nβ f i f i dλ( µ H αβ )n α n β (11.9) T 0 T = T 0 /a + δt T f = a ( i 1+ δt f δt ) i = a ( i 1+ 1 ) δρ γ f (11.10) T i a f T f T i a f 4 ρ γ i ρ γ T 4 (11.9) 0 (11.10) (11.6) E f = a { [ i 1+ ϕ 1 ] f } E i a f 2 h 00 (v i + h 0i ) n i + δn 0 i 45

46 = T [ f 1 T i 1 4 { = T f T i 1 δρ γ ρ γ ϕ h 00 +(v i + h 0i ) n i δn 0 [ 1 4 Dγ + i V b n i +Ψ Φ+Ω b in i ] f i f + dλ ( η Ψ η Φ+ η Υ i n i η h TT ij ni n j) i } ] f i (11.11) V b E f /E i = T f /T i {} 1 Bardeen [] i=η dec f=η 0 Ψ(η 0 ) n i i V b (η 0 ) CMB ( T/T)(η 0 )= (1/4)D γ (η 0 ) 13 Sachs-Wolfe ( ) T T S ( ) T V ( ) T T T (η 0, x 0 )= + +, T T T ( ) T S { 1 = T 4 Dγ + i V b n i +Ψ Φ} (η dec, x dec ) η0 + ( T T ( T T η dec dη ( η Ψ η Φ) (η, x(η)), (11.12) ) V =Ω b i (η dec, x dec ) n i + η0 dη η Υ i (η, x(η)) n i, (11.13) η dec ) T η0 = dη η h TT ij (η, x(η)) n i n j (11.14) η dec x(η) =x 0 +(η η 0 )n l <10 (8.35) (8.36) 13 CMB CMB 46

47 x 1 D γ = ( 20/3)Ψ i V γ =(1/3)xΨ i 0 V b V γ Bardeen Ψ(= Φ) Ψ(η dec )=Ψ i ( T T ) S (η 0, x 0 ) 1 3 Ψ(η dec, x dec) (11.15) Ordinary Sachs-Wolfe (11.12) Integrated Sachs-Wolfe 12 CMB 12.1 k (transfer function)t F =Ψ, D, F (η f,k)=t F F (η i,k) (12.1) CMB Sachs-Wolfe η dec Bardeen Integrated Sachs-Wolfe Bardeen CMB 47

48 Υ i (7.7) Ω b i c 2 s < 1/3 (7.7) F F (η, k)(5.1) 2 F (η, k)f (η, k ) = F (η, k) 2 k 3 (2π) 3 δ 3 (k + k ) (12.2) F P F (η, k) = 1 2π 2 F (η, k) 2 (12.3) 2 F (η, x)f (η, x) = [d 3 k][d 3 k ] f(η, k)f (η, k ) e i(k+k ) x = = 0 d 3 k 1 (2π) 3 k F (η, 3 k) 2 dk k P F (η, k). (12.4) Bardeen Ψ i Ψ h TT ij 14 P s (k) = 1 ( ) ns 1 k 2π Ψ(η i,k) 2 = A 2 s, (12.5) m 14 m (pivot) Planck 48

49 P t (k) = 1 ( ) nt k 2π 2 htt (η i,k) 2 = A t (12.6) m h TT (12.28) r = A t (12.7) A s 12.2 CMB CMB T T (x 0, n,η 0 )= l l=0 m= l a lm (x 0 )Y lm (n) (12.8) a lm a lm =0 C l = a lm 2 a lm a l m = C lδ ll δ mm (12.9) CMB 2 m m= l Y lm (n)y lm(n )= 1 4π (2l +1)P l(n n ) (12.10) n n = cos θ CMB 2 T C(θ) = T (x 0, n,η 0 ) T T (x 0, n,η 0 ) = a lm a l m Y lm(n)yl m (n ) l,l,m,m = 1 (2l +1)C l P l (cos θ) (12.11) 4π l 49

50 CMB TT TT l<50 ( ) CMB (11.12) Ψ=Φ { T 1 T (x 0, n,η 0 )= 4 Dγ + i V b n +2Ψ} i (η dec, x dec ). (12.12) x dec = x 0 (η 0 η dec )n { T 1 T (k, n,η 0) = 4 Dγ iˆk nv +2Ψ} b (k,η dec ) e ik n(η 0 η dec ) { 1 = 4 Dγ +2Ψ+ V b } k η (k,η dec ) e ik nη) η=η0 η dec (12.13) ˆk = k/k =(θ k,ϕ k ) ordinary Sachs-Wolfe (l <30) (11.15) CMB Bardeen (12.13) T T (k, n,η 0) 1 3 Ψ(k,η dec)e ik n(η 0 η dec ) (12.14) C(θ) 1 d 3 k d 3 k T C(θ) = (2π) 6 k 3 k 3 T (k, n,η 0) T T (k, n,η 0 ) = 1 (2π) 6 1 (2π) 3 e i(k+k ) x 0 d 3 k d 3 k ) x 0 1 k 3 k 3 ei(k+k 9 Ψ(k,η dec)ψ(k,η dec ) e ik n(η 0 η dec ) e ik n (η 0 η dec ) d 3 k 1 k 3 9 Ψ(k,η dec) 2 e ik n(η 0 η dec ) e ik n (η 0 η dec ) (12.15) 50

51 2 (12.2) e ik y =4π l l=0 m= l i l j l (ky)y lm (ˆk)Y lm (ŷ) (12.16) ŷ ˆk C(θ) = (4π)2 d 3 k 1 (2π) 3 k 3 3 Ψ(k,η 2 dec) j l (kd dec )j l (kd dec ) i l l l l,l =0 m= l Y lm (ˆk)Y lm(n) l m = l Y l m (ˆk)Y l m (n ) (12.17) d dec = η 0 η dec k d 3 k = k 2 dkdω k dω k = dθ k dϕ k dω k Ylm(ˆk)Y l m (ˆk) =δ ll δ mm (12.18) (12.10) C(θ) = 1 4π l=0(2l +1)P l (cos θ) 2 π dk k 1 9 Ψ(k,η dec) 2 j 2 l (kd dec) (12.19) (12.11) C l l kd dec (2.28) Ordinary Sachs-Wolfe Ψ T Ψ 1 Ψ(η dec ) 2 Ψ i 2 =2π 2 P s Cl osw dk 1 = 4π 0 k 9 P s(k)jl 2 (kd dec ) = 2π 2 A s 1 (md dec ) ns 1 9 Γ(3 n s )Γ ( l ns 2 ( ) ( 2 3 ns Γ 2 2 n s 2 Γ l + 5 ns 2 2 Harrison-Zel dovich (n s =1) l(l +1)C osw l 2π 51 = A s 9 ) ) (12.20) (12.21)

52 P s (k) (12.13) C(θ) = 1 (2π) 6 d 3 k d 3 k { ( ) x 0 D γ ) k 3 k 3 ei(k+k 4 +2Ψ { (D γ ) 4 +2Ψ (k,η dec )+ V b (k,η dec ) k η (k,η dec )+ V b } (k,η dec ) η k } e ik nη e ik n η η=η = η 0 η dec (12.22) 2 (12.2) k C(θ) = (4π) 2 (2π) 3 d 3 k k 3 {( D γ ) 4 +2Ψ {( D γ ) 4 +2Ψ i l l l l,l =0 m= l Y lm (ˆk)Y lm(n) l m = l Y (k,η dec )j l (kη)+ V b (k,η dec ) η j l (kη) k l m (ˆk)Y l m (n ) (k,η dec )j l (kη)+ V b (k,η dec ) η j l (kη)} k η=η 0 η dec (12.23) k d 3 k = k 2 dkdω k dω k = dθ k dϕ k (12.18) (12.10) C l = 2 π {( dk D γ ) } k 4 +2Ψ (k,η dec )j l (kd dec )+V b (k,η dec )j l 2 (kd dec ) (12.24) j l (x) = xj l (x) η i η dec 52 }

53 T γ T b T Ψ ( D γ ) 4 +2Ψ (k,η dec ) = 1 4 T γd γ (k,η i )+2T Ψ Ψ(k,η i ) = ( 3 ) 2 T γ +2T Ψ Ψ i (k), V b (k,η dec ) = T b V b 1 (k,η i )=T b 2 kη iψ i (k) (12.25) (x = kη 0) D γ 6Ψ i Ψ Ψ i V b (= V γ ) kη i Ψ i /2 CMB Ψ dk C l =4π {( 3 ) } k 2 T 1 2Ps γ +2T Ψ j l (kd dec )+T b 2 kη ij l (kd dec) (k) (12.26) (11.14) 1 d 3 k η0 η0 C(θ) = dη dη e ik n(η0 η) e ik n (η 0 η ) (2π) 3 k 3 η dec η dec η h TT ij (η, k) η htt lm (η, k) n i n j n l n m (12.27) 2 h TT ij (η, k)h TT lm (η, k) { = h TT (η, k)h TT (η, k) δ il δ jm + δ im δ jl δ ij δ lm + 1 k 2 (δ ijk l k m + δ lm k i k j δ il k j k m δ im k l k j δ jl k i k m δ jm k l k i ) + 1 k 4 k ik j k l k m } (12.28) C l = 2 dk η0 dη π k η h TT (η, k) j l(k(η 0 η)) 2 (l + 2)! η dec k 2 (η 0 η) 2 (l 2)! (12.29) 53

54 11: TT 3 Ω b Ω c Ω Λ Hubble h n s (optical depth)τ [E. Martinez-Gonzalez, astroph/ ] η h TT = 0 η dec η 0 (l <50) (x = kη 2) h TT 1/a η h TT ahh TT =( 2/η)h TT a η 2 54

55 η0 η dec dη η h TT j l(k(η 0 η)) k 2 (η 0 η) 2 j l(kη 0 )) k 2 η 2 0 η0 η=2/k 2dη h TT (12.30) η h TT 1/η 2 ( 2dη/η)h TT = h TT (η 0 ) h TT (η =2/k) η 0 h TT (η 0 ) η 0 = d dec l<50 C l l<50 2 π dk k htt (η =2/k, k) 2 j2 l (kd dec) k 4 d 4 dec (l + 2)! (l 2)! (12.31) 1 (η =2/k) h TT (η =2/k, k) 2 =2π 2 P t (k) (l + 2)! dk C l l<50 4π (l 2)! 0 k = 2π 2 A t (l + 2)! (md dec ) nt (l 2)! jl 2(kd dec) P k 4 d 4 t (k) dec Γ(6 n t )Γ ( l 2+ nt nt Γ 2 ( 7 2 n t) Γ ( l +4 n t 2 ) )(12.32) n t =0 l(l +1)C l /2π A t 8l(l +1)/15(l 2)(l +3) l =2 TT r TT CMB E B 12 TT TE EE 55

56 12: (r = 0.01) TT TE EE BB( ) [E. Martinez-Gonzalez, astroph/ ] CMB BB BB EE (l <10) (optical depth)τ τ 0.1 τ r B 12( ) 56

2 Planck Planck BRST Planck Λ QG Planck GeV Planck Λ QG Friedmann CMB

2 Planck Planck BRST Planck Λ QG Planck GeV Planck Λ QG Friedmann CMB 量子重力理論と宇宙論 (下巻) くりこみ理論と初期宇宙論浜田賢二高エネルギー加速器研究機構 (KEK) 素粒子原子核研究所 http://research.kek.jp/people/hamada/ 量子重力の世界は霧に包まれた距離感のない幽玄の世界にたとえることができる深い霧が晴れて時空が現れる国宝松林図屏風 (長谷川等伯筆) 平成 20 年 11 月初版/平成 21 年 09 月改定/