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

Transcription

1 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 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/ 平成 25 年 08 月再改定 (上下巻に分離)

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

3 D CFT CMB CMB A 71 A.1 ( )

4 4 A B 77 B.1 G D D = C 79 C D 83 D E 85 E F 87

5 5 8 4 Weyl t 4 DeWitt-Schwinger δ (4) (0) = x x x x 1 δ (D) (0) = d D k = 0 4 D 4 D 8.1 D n F (QED) D 1 D δ (4) (0) = x e td x t 0 ( t + D) x e td x = 0 (Heat Kernel)

6 6 8 Euclid Wick I = d D x { 1 g t 2 C2 µνλσ + be D F µνf µν + n F j=1 i ψ j D/ψ j M 2 P 2 R + Λ } (8.1.1) D Weyl Cµνλσ 2 = R µνλσ R µνλσ 4 D 2 R µνr µν 2 + (D 1)(D 2) R2 (8.1.2) E D Euler E 4 D E D = G D 4(D 3)2 (D 1)(D 2) 2 R (8.1.3) G D = G 4 + (D 3)2 (D 4) (D 1) 2 (D 2) R2 (8.1.4) G 4 = R 2 µνλσ 4R 2 µν + R 2 Euler G D D E D G D Dirac D/ = e µα γ α D µ e α µ 4 (vierbein field) D e α µ e να = g µν e µα e µ β = δ αβ Dirac {γ α, γ β } = 2δ αβ D µ = µ ω µαβσ αβ + iea µ 1 (connection 1-form) Lorentz ω µαβ = e ν α( µ e νβ Γ λ µνe λβ ) Σ αβ = 1 4 [γα, γ β ] A.2 g µν = e 2ϕ ḡ µν = e 2ϕ (ĝe th ) µν Riegert ϕ h µν ĝ µν Euclid δ µν A µ = Z 1/2 3 A r µ, ψ j = Z 1/2 2 ψ r j, h µν = Z 1/2 h hr µν

7 8.1. D 7 QED e = Z e e r, t = Z t t r Ward-Takahashi (Z 1 = Z 2 ) Z e = Z 1/2 3 Riegert ϕ Riegert Z ϕ = 1 (8.1.5) D 4 D 4 Laurent Z 3 = 1 + x 1 D 4 + x 2 (D 4) 2 + (8.1.6) Z x 1 x 2 e r t r Euler b Euler D 4 b = 1 (4π) 2 n=1 b n (D 4) n (8.1.7) b n e r t r 4 D D 2

8 8 8 Wess-Zumino [δ ω1, δ ω2 ]Γ = 0 D 4η 1 + Dη 2 + 4(D 1)η 3 + (D 4)η 4 = 0 (8.1.8) ( A.1 ) D Weyl 2 G 4 M D = 2 R D 4 4(D 1) R2 (8.1.9) M D 2 R D E D E D = G 4 + ηm D η 2 2 R 2 D R 2 d D x gr = S (2) n S n (2) (ϕ, ḡ) = (D 2) n S n (2) (ϕ, ḡ) n! n=0 d D x ḡ { ϕ n 2 ϕ + Rϕ n + o(ϕ n ) } o(ϕ n ) ϕ n S (2) 1 Liouville-Polyakov 4 E D 4 d D x (D 4) n ge D = S n (ϕ, ḡ) (8.1.10) n=0 n!

9 8.1. D 9 S n (ϕ, ḡ) = d D x ḡ { 2ϕ n 4 ϕ + Ē4ϕ n + o(ϕ n ) } S 1 5 ( ) Riegert-Wess-Zumino S RWS 4(D 3)2 η = (D 1)(D 2) B 4 2/3 E 4 (8.1.10) G D (8.1.4) Hathrell 2 3 (e 6 r) D Weyl bg 4 + ch 2 H = R/(D 1) bg D c = (D 3)2 (D 4) b (D 2) b Laurent (8.1.7) c b Laurent c 1 = (D 3)2 D 2 b 2 = 1 2 b 2 + o(d 4) Hathrell e 6 r QED 4 2 S. Hathrell, Ann. of Phys. 142 (1982) 34.

10 10 8 G D Z ϕ = 1(8.1.5) 8.2 Laurent D 4 (counterterm) Z 3 Laurent (8.1.6) 1 4 d D x gf µν F µν = 1 4 Z 3 d D xe (D 4)ϕ FµνF r λσḡ r µλ ḡ νσ = 1 {( d D x 1 + x 1 4 D 4 + x ) 2 (D 4) + 2 ( + D 4 + x 1 + x 2 D 4 + ) F r µνf r λσḡ µλ ḡ νσ ϕfµνf r λσḡ r µλ ḡ νσ + 1 ( ) (D 4) 2 + (D 4)x 1 + x 2 + ϕ 2 F r 2 µνfλσḡ r µλ ḡ νσ } + (8.2.1) F r µν = µ A r ν ν A r µ = µ A r ν ν A r µ Laurent (8.2.1)

11 ϕf r2 µν Fµν r2 Riegert Wess-Zumino Riegert Weyl 3 D Riegert 1 t 2 d D x gc 2 µνλσ = 1 t 2 d D x ḡe (D 4)ϕ C2 µνλσ Laurent Wess- Zumino ϕ n C2 µνλσ Euler b Laurent (8.1.7) Euler (8.1.10) b d D x ge D = 1 (4π) {( b1 d D x D 4 + b 2 (D 4) + 2 )Ḡ4 ( + b 1 + b )( 2 D 4 + 2ϕ 4 ϕ + Ē4ϕ R ) 2 ( (D 4)b1 + b 2 + )( 2ϕ 2 4 ϕ + Ē4ϕ 2 + ) } + (8.2.2) G 4 Riegert-Wess-Zumino S ( ) S RWS Riegert 3 Duff, Nucl. Phys. B125 (1977) 334 Duff D 4 Weyl 1 ( R 2 )

12 12 8 Wess-Zumino S 2 b n b 1 b 1 (t r, e r ) = b 1 + b 1(t r, e r ) b 1 b 1 n 2 D ( A.2 ) Riegert Riegert d D xi ψ D/ψ { = d D x i ψγ µ µ ψ i t 4 ( ψγ µ ν ψ ν ψγµ ψ)h µν +i t2 16 ( ψγ µ ν ψ ν ψγµ ψ)h µλ h νλ + i t2 16 ψγ µνλ ψh µσ λ h νσ e ψγ µ ψa µ + et 2 ψγ µ ψa ν h µν et2 8 ψγ µ ψa ν h µλ h νλ } + o(t 3 ) γ µνλ = 1 (γ 3! µγ ν γ λ + anti-sym.) e t ψ Euclid δ µν 8.3 Riegert Wess- Zumino

13 Weyl d D x gc 2 µνλσ/t 2 { D 3 ( ) } d D x hµν 4 h µν + 2χ µ 2 D 3 χ µ D 2 D 1 χ µ µ ν χ ν χ µ = λ h λµ Euclid 2 = λ λ BRST I GF+FP = { d D xδ B c µ N µν (χ ν ζ ) 2 B ν + c ( µ A µ α ) } 2 B c µ c B µ B N µν 2 N µν = ( 2(D 3) 2 2 δ µν + D 2 ) D 2 D 1 µ ν BRST ξ µ /t c µ U(1) c δ B h µν = µ c ν + ν c µ 2 D δ µν λ c λ + tc λ λ h µν + t 2 h µλ ( ν c λ λ c ν ) + t 2 h νλ ( µ c λ λ c µ ) +, δ B A µ = µ c + t (c λ λ A µ + A λ µ c λ ) BRST δ B c µ = tc λ λ c µ,

14 14 8 δ B c = tc λ λ c, δ B c µ = B µ, δ B B µ = 0, δ B c = B, δ B B = 0 Riegert BRST δ B ϕ = tc λ λ ϕ + t D λc λ BRST I GF+FP = { d D x B µ N µν χ ν ζ 2 B µn µν B ν c µ N µν λ (δ B h νλ ) +B µ A µ α } 2 B2 c µ (δ B A µ ) B B µ 4 I GF = d D x { 1 2ζ χ µn µν χ ν + 1 } 2α ( µa µ ) 2 α = 1 ζ = 1 Feynman α = Z 3 α r ζ = Z h ζ r Feynman h r µν 4 h r µν h r µν(k)h r λσ( k) = D 2 1 2(D 3) k 4 IH µν,λσ 4 B µ det 1/2 (N µν )

15 I H µν,λσ = 1 2 (δ µλδ νσ + δ µσ δ νλ ) 1 D δ µνδ λσ I 2 H = I H Riegert Riegert Laurent (8.2.2) b 1 (4π) S 1(ϕ, ḡ) = b 1 2 (4π) 2 { d D x 2ϕ 4 ϕ + Ē4ϕ R } 2 ϕ(k)ϕ( k) = (4π)2 4b 1 1 k 4 b 1 t Riegert L 2 S 1 = b 1 { 2 (4π) 2 3 t 2 ϕ µ ν h µν + 1 } 18 t2 ( µ ν h µν ) 2, L 3 S 1 = 2b { 1 (4π) t 2 2 µ ϕ ν 2 ϕ µ λ ϕ ν λ ϕ 2 } 3 λϕ µ ν λ ϕ 2 µ ν ϕ 2 ϕ h µν, L 4 S 1 = 2b 1 (4π) 2 t2{ 2 ϕ µ ν ϕh µλ h νλ + µ ν ϕ λ σ ϕh µν h λσ } + h L 2 S 1 Ē4ϕ ( 2 R)ϕ R2 L 3 S 1 L 4 S 1 ϕ 4 ϕ

16 h Riegert Wess-Zumino Weyl h z ϕ h µν 1/(k 2 + z 2 ) 2 log z 2 Einstein 4 t r Riegert Einstein Riegert M P D = 4 2ϵ, t r = t r µ ϵ, e r = ẽ r µ ϵ, b = bµ 2ϵ t r ẽ r µ Riegert D µ ϵ Riegert I Feynman 8.1 t 2 r Riegert 2

17 t r (1) t r t 2 r (2) 8.1: Riegert t 2 r L 3 S 1 8.1(1) d D { k (2π) ϕ(k)ϕ( k) D b 1 t 2 r D 2 d D l 1 6 (4π) 2 2(D 3) (2π) D (l 2 + z 2 ) 2 {(l + k) 2 + z 2 } 2 [ 6(l 2 k 6 + l 6 k 2 ) + 24l 4 k 4 16(l k)(l 2 k 4 + l 4 k 2 ) 20(l k) 2 l 2 k 2 2(l k) 2 (l 4 + k 4 ) + 8(l k) 3 (l 2 + k 2 ) + 8(l k) D ( 36l 4 k (l k)(l 2 k 4 + l 4 k 2 ) + 40(l k) 2 l 2 k 2 3D 4(l k) 2 (l 4 + k 4 ) 16(l k) 3 (l 2 + k 2 ) 16(l k) 4 )]}. D l z 1 { } [ 2b 1 (4π) 2 k4 3 t 2 r (4π) 2 ( 1 z2 log ϵ µ + 7 )], 2 6 1/ ϵ = 1/ϵ γ + log 4π log k 2 /µ 2 (tadpole) 8.1(2) L 4 S 1 h µν 2 [ 2b 1 (4π) 2 k4 3 t 2 r (4π) 2 ( 1 z2 log ϵ µ + 7 )]. 2 12

18 18 8 Feynman t 2 r Z ϕ = ( 8.2) Weyl 2 3 b n Ḡ4 3 (1) (2) 8.2: 2 3 (background field method) t r ( nf Z t = ) t 2 r 1 3 (4π) 2 ϵ 7n F ẽ 2 r t 2 r (4π) 4 ϵ + o( t 4 r) (8.4.1) 1 Feynman t 2 r n F /80 U(1) 1/40 Riegert 1/60 199/120 t 2 re 2 r 2 Feynman ĝ µν = (e tĥ) µν Zĥ ĥ µν = Z 1/2 ĥ r ĥ µν Z tz 1/2 = 1 ĥ

19 Z h Zĥ Zĥ Z t t r β t = µ d dµ t r µ 0 = µ d dµ t = µ d dµ (Z t t r µ ϵ ) µ dz t β t = ϵ t r t r Z t dµ µd t r /dµ = β t = ϵ t r + o( t 2 r) µdẽ r /dµ = ϵẽ r + (8.4.1) β t = ( nf ) t 3 r (4π) 7n F 2 72 e 2 rt 3 r (4π) 4 + o(t5 r) Euler Ḡ4 b1 = 11n F , b 1 = n2 F ẽ 4 r 6 (4π) + o( t 2 r), 4 b2 = 2n3 F 9 ẽ 6 r (4π) 6 + o( t 4 r) (8.4.2) b 1 (11n F + 62)/360 Riegert 7/90 87/20 b 1 e 4 r b 2 e 6 r 2 3 Feynman

20 20 8 Riegert II Riegert 2 t 2 r Hathrell e 6 r Z ϕ = 1 8.3: Riegert e 4 r QED Z 3 x 1 = 8n F 3 x 2 = 32n2 F 9 ẽ 2 r (4π) 2 + 4n F ẽ 4 r (4π) 4, ẽ 6 r (4π) 6 (8.4.3) Laurent (8.2.1) Wess-Zumino e 2 r Riegert 2 e 4 r Feynman 8.3 2lp 2 (subdiagram) e 4 r 2 b 2 e 6 r 2

21 E D Laurent (8.2.2) e 6 r (5) 2 Riegert ϵ 4 (6) (7) (1) (4) e 6 r Hathrell Z ϕ = 1 e 6 r b 2 Riegert U(1) Z 3 t 2 r 8.4 Feynman (1) (2) 8.4: Z 3 t 2 r Riegert Feynman e 4 r Feynman e 6 r 2 Feynman 8.6

22 : Z 3 e 4 r Feynman 8.6: Z 3 e 6 r 2 Feynman QED (8.4.3) Z 3 Z 3 = 1 4n F 3 ( + ẽ 2 ( r 1 (4π) 2 ϵ + 8n2 F QED 2n F + 8 n 2 ) F ẽ4 r 1 27 b1 (4π) 4 ϵ n 3 ) F ẽ6 r 1 b1 (4π) 6 ϵ + 2 o(ẽ2 r t 2 r, t 4 r) β e = µ dẽ r dµ (8.4.4) Ward-Takahashi Z 1 = Z 2 e r Z e = Z 1/2 3 β e = ϵẽ r + ẽr 2 µ dz 3 Z 3 dµ (8.4.5)

23 Z 3 Z 3 = 1 + A 1 ϵ + A ( 2 ϵ b1 B1 2 ϵ + B ) 2 ϵ ẽ r A 1 = A 1,n ẽ 2n r, A 2 = A 2,n ẽ 2n r, n 1 n 3 B 1 = B 1,n ẽ 2n r, B 2 = B 2,n ẽ 2n r n 2 n 3 (8.4.4) (8.4.5) ϵ A 2,3 = 1 3 A 1,1A 1,2, B 2,3 = 1 4 A 1,1B 1,2 (8.4.6) µd b 1 /dµ = 2ϵ b B 1, B 2,3 QED β e = 4n F 3 e 3 r (4n (4π) + 2 F 8 n 2 ) F e 5 r 9 b 1 (4π) + 4 o(e3 rt 2 r) b 1 n F 24 e 5 r Λ QG e r Landau Riegert ( ) ϕf r2 µν Z 3 Z ϕ = 1 Z 3 2 e 6 r Laurent (8.2.1) Wess-Zumino ϕf r2 µν

24 : ϕfµν 2 e6 r I e 6 r Feynman QED Riegert Feynman 8.7 (1) (2) 2 n F e 2 rϕf r2 µν 2 (3) (4) 3 ϵϕf r2 µν 3 2 Riegert ϵ (5) Z ϵϕfµν r2 Feynman 3

25 Riegert ϵ { Γ ϕaa µν (0; k, k) I = } n 2 e 6 r 1 ( ) F δµν k 2 k 9 (4π) 6 µ k ν = 0(8.4.7) ϵ Γ = d D k 1 (2π) D d D k 2 (2π) D ϕ( k 1 k 2 )A r µ(k 1 )A r ν(k 2 )Γ ϕaa µν ( k 1 k 2 ; k 1, k 2 ) (1) (2) (3) (4) (5) Riegert Feynman Feynman Feynman Riegert ϕ 3 ϕ 2 F r2 µν b ge D Laurent (8.2.2) { Γ ϕaa µν (0; k, k) II = } n 3 F 81 b 1 e 6 r 1 ( ) δµν k 2 k (4π) 6 µ k ν = 0 ϵ (1) (3) (10) (13) (14) Z 3 2 (4) (9) 8.5 Wess-Zumino

26 : ϕfµν 2 e6 r II

27 QED log(k 2 /µ 2 ) QED 2 β e /e r = y 1 /2 y 1 = 8n F 3 e 2 r (4π) 2 + 8n F e 4 r (4π) 4 y 1 e 4 r x 1(8.4.3) Riegert QED { Γ QED = 1 y ( ) 1 k 2 2 log e 4 } r 1 + x µ 2 1 ϕ + 4n F (4π) ϕ x 1 Wess-Zumino ϕ ϕ ( e 4 ) r 1 δ ϕ Γ QED = x 1 + 4n F gr F r2 (4π) µν = y 1 4 F r2 µν gr F r2 µν 8.9: ϕfµν 2 e4 r 2 ϵ k 2 (= k µ k ν δ µν ) gµν r (= e 2ϕ δ µν ) p 2 = k 2 /e 2ϕ (8.5.1)

28 28 8 { Γ QED = 1 y ( )} 1 p log gr F µ 2 µν r2 4 Wess-Zumino log n (k 2 /µ 2 ) ϕ n Fµν r2 Wess-Zumino Weyl log(k 2 /µ 2 ) Wess-Zumino ϕc 2 µνλσ β t = β 0 t 3 r (β 0 > 0) { ( )} 1 k 2 Γ W = 2β 0 ϕ + β 0 log = t 2 r 1 gr C t 2 µνλσ r2 r(p) µ 2 C r2 µνλσ { } t r (p) t 2 r(p) = 1 β 0 log(p 2 /Λ 2 QG) (8.5.2) p (8.5.1) Λ QG = µ exp{ 1/(2β 0 t 2 r)} log n (k 2 /µ 2 ) ϕ n C 2 µνλσ Euler Euler Ḡ4 b 1 Ḡ4 2 Feynman 3 8.2(2) W G (ḡ r ) = b { 1 1 d 4 x 8Ēr 1 (4π) 2 4 Ē r 4 r 1 } 4 18 R r 2

29 R 2 W G 2 Ḡ4 Riegert-Wess-Zumino b 1 S 1 W G b 1 (4π) S 1(ϕ, ḡ 2 r ) + W G (ḡ r ) = b 1 8(4π) 2 d 4 x g r E r 4 1 r 4 E r 4 (8.5.3) R 2 Riegert-Wess-Zumino 2 Polyakov 4 Riegert-Wess-Zumino L S1 = b 1 /(4π) 2 {2ϕ r 4ϕ+ } t 2 r Γ R = ( 1 a 1 t 2 r(p) + ) L S1 (ϕ, ḡ r ) [ ( ) ]} k = {1 a 1 t 2 r + 2β 0 t 4 rϕ β 0 t 4 2 r log + L µ 2 S1 (ϕ, ḡ r ) ϕ 2 r 4 ϕ t 4 r (8.2.2) b 2 t 4 r ϕ n r 4 ϕ (n 2) 8.6

30 30 8 Riegert I Λ = Λ d D x g = Λ d D xe Dϕ Riegert Λ = Z Λ Λ r = Z Λ Λr µ 2ϵ Λ r Z Λ 4 Λ r Laurent I Λ = Λ r Z Λ = 1 + u 1 D 4 + u 2 (D 4) 2 + { ( d D x 1 + u 1 D 4 + u 2 (D 4) 2 + ) e 4ϕ ( + D 4 + u 1 + u ) 2 D 4 + ϕe 4ϕ + 1 ( (D 4) 2 + (D 4)u 1 + u 2 + ) ϕ 2 e 4ϕ 2 } + (8.6.1) b 1 N e 4ϕ = n(4ϕ) n /n! Z Λ = 1 2 b1 1 ϵ 2 b2 1 1 ϵ + 2 b2 1 1 ϵ 2 +. (8.6.2)

31 n (a) n (b) n (c) 8.10: 1/b 1 1/b u 1 n (a) n (b) n (c) 8.11: ϕe 4ϕ 1/b 2 1 Feynman 8.10 (a) (b) (c) u 1 = 4/ b 1 + 4/ b 2 1 u 2 = 8/ b 2 1 ϕe 4ϕ /b 2 1 (a) (b) Laurent (8.6.1) 2 2 u 2 (c) u 1 u 2 ϕe 4ϕ

32 32 8 u n γ Λ = µ Λr d Λ r dµ µ γ Λ = 2ϵ + µ dz Λ Z Λ dµ = (8.6.3) b 1 b 2 1 Riegert δ ϕ L Λ = (4 + γ Λ )L Λ 4 Riegert e γ 0ϕ γ 0 = 4 + γ Λ γ Λ ) γ Λ = 2b 1 (1 1 4b1 4 = 4 b b b (8.6.3)

33 33 9 Planck m pl GeV Λ QG Planck Λ QG GeV Friedmann 9.1 I = d 4 x { g 1 t 2 C2 µνλσ bg 4 + M P 2 } 2 R Λ + I M M P = 1/ 8πG Planck Riegert g µν = e 2ϕ ḡ µν ḡ µν t ḡ µν = η µν + h µν + (9.1.1)

34 34 9 th µν h µν η µν = ( 1, 1, 1, 1) x µ = (η, x i ) η (conformal time) x i (comoving coordinate) Z = [dϕdhdadx]η Vol(diff.) exp {is(ϕ, ḡ) + ii(a, X, g)} S Wess-Zumino (Jacobian) Riegert- Wess-Zumino S(ϕ, ḡ) = b 1 (4π) 2 d 4 x { ḡ 2ϕ 4 ϕ + (Ḡ4 2 3 ) 2 R ϕ R } 2 b 1 Weyl N X N W N A b 1 = 1 (N X + 11 ) N W + 62N A R 2 Λ QG (β t = β 0 t 3 r, β 0 > 0) t 2 r(p) = 1 β 0 log(p 2 /Λ 2 QG) p η µν k p = k/e ϕ ( 8.5 )

35 Planck Planck Einstein m pl Λ QG Planck m x 1/m Schwarzschild r g = 2Gm x r g m Planck m pl Planck Planck l pl (= 1/m pl ) Schwarzschild 2l pl Planck Λ QG ξ Λ = 1/Λ QG ξ l pl Riegert ˆϕ(η) b 1 4π 2 4 η ˆϕ + 6M 2 Pe 2 ˆϕ ( 2 η ˆϕ + η ˆϕ η ˆϕ) = 0

36 36 9 a Hubble H a = e ˆϕ, H = ηa a 2 = ȧ a (, proper time)τ dτ = adη Hubble b 1 8π 2 (... H +7HḦ + 4Ḣ2 + 18H 2 Ḣ + 6H 4) 3M 2 P (Ḣ + 2H 2 ) = 0 (de Sitter ) H = H D, H D = 8π 2 b 1 M P = a e H Dτ π b 1 m pl (9.2.1) Planck b 1 GUT 10 H D Planck M P = GeV Planck

37 m pl = GeV H D Planck Planck τ P = 1/H D (9.2.2) δ H = H D (1 + δ) o(δ 2 )... δ +7H D δ + 15H 2 D δ + 12H 3 D δ = 0 δ = e υτ υ 4H D, ( 3 2 ± i 3 2 ( ) (power-law) ) H D Planck Λ QG 1/Λ QG Λ QG (QCD) QCD Λ QCD

38 38 9 Weyl (1/t 2 r)cµνλσ 2 Riegert Wess-Zumino b 1 ( b 1 b 1 1 a1 t 2 r + ) = b 1 B 0 (t r ) B 0 (t r ) = 1 (1 + a 1 κ t 2 r) κ κ 0 < κ 1 Riegert b 1 4π 2 B 0 4 η ˆϕ + M 2 Pe 2 ˆϕ { 6 2 η ˆϕ + 6 η ˆϕ η ˆϕ} = 0 (9.2.3) (0, 0) b 1 8π B { η ˆϕ η ˆϕ 2 2 η ˆϕ η ˆϕ} 3M 2 P e 2 ˆϕ η ˆϕ η ˆϕ + e 4 ˆϕρ = 0 (9.2.4) ρ τ d dτ t r = β( t r ) = β 0 t 3 r

39 t 2 r(τ) = τ Λ = 1/Λ QG 1 β 0 log(1/τ 2 Λ 2 QG) p 1/τ (τ > 0) t r t r (τ) B 0 Hubble b 1 8π 2 B 0(τ) (... H +7HḦ + 4Ḣ2 + 18H 2 Ḣ + 6H 4) 3M 2 P (Ḣ + 2H 2 ) = 0 (9.2.5) b 1 8π 2 B 0(τ) ( 2HḦ Ḣ2 + 6H 2 Ḣ + 3H 4) 3M 2 PH 2 + ρ = 0 (9.2.6) H H D ρ 0 H = H D B 0 Hubble H 0 < κ < 1 3 κ = 1 2 B 0 Ḧ (9.2.6)

40 40 9 ρ(τ Λ ) = 3M 2 PH 2 (τ Λ ) ρ + 4Hρ = b 1 8π 2 Ḃ0(τ) ( 2HḦ Ḣ2 + 6H 2 Ḣ + 3H 4) B 0 Planck τ P (= 1/H D ) τ Λ (= 1/Λ QG ) (e-foldings) N e = log a(τ Λ) a(τ P ) a e H Dτ N e H D Λ QG β 0 a 1 κ t r H D /Λ QG = 60 β 0 /b 1 = 0.06 a 1 /b 1 = 0.01 κ = 0.5 H D = 1 τ Λ = 60 N e = 65.0 (τ > τ Λ ) Planck M P = GeV b 1 = 10 H D = GeV Λ QG = GeV (9.2.7)

41 CMB 10 log 10 [a(τ)/a(τλ)] τ Λ log 10 (τ/τ P ) 9.1: a(τ) Planck τ P τ Λ (= 60τ P ) Friedmann 9.3 Λ QG Einstein QCD QCD Λ QCD Λ QG Riegert Λ QG Riegert Einstein Friedmann

42 ρ H Friedman H, ρ proper time,τ 9.2: Hubble H ρ H D = 1 Friedmann I low = d 4 x g {L 2 + L 4 + } Einstein L 2 = M 2 P 2 R + L M L M Einstein 1 Planck M P 4πF π Planck M P Λ QG 4 L 4 R 2, R 2 µν, R 2 µνλσ, 1 M 2 P R µν T µν M, 1 M 4 P T µν M T M µν

43 T M µν Einstein Einstein M 2 P R µν = T M µν Einstein R = 0 L 4 Euler Riemann L 4 = α (4π) 2 Rµν R µν α α E c (< Λ QG ) Einstein α α(e c ) = α(λ QG ) + ζ log(e 2 c /Λ 2 QG) (9.3.1) N X Weyl N W N A Feynman ζ = (N X + 3N W + 12N A )/120 Ricci µ R µν (= µ Tµν) M = 0 ζ Λ QG α(λ QG ) ζ (9.3.1) α(e c ) 4 Λ QG Planck

44 44 9 M 2 P (Ḣ + 2H 2 ) + α 4π 2 (... H +7HḦ + 4Ḣ2 + 12H 2 Ḣ ) = 0 (9.3.2) 3M 2 PH 2 + ρ + α 4π 2 ( 6H Ḧ + 3Ḣ2 18H 2 Ḣ ) = 0 (9.3.3) E c = 1/τ ( ) 1 α(τ) = α 0 + ζ log τ 2 Λ 2 QG α ζ α 0 log(τ 2 Λ 2 QG) α 0 = α(λ QG ) QCD τ = τ Λ H Ḣ ρ (9.3.2) Ḧ (9.3.3) α 0 = 1 ζ = 1 (9.3.2) (9.3.3) Ḣ + 2H2 = 0 3M 2 PH 2 = ρ Friedmann H Friedmann 9.2 Friedmann R 0 Friedmann

45 R = 0 R = 6Ḣ + 12H2 (9.3.2) (9.3.3) R + 3HṘ + 4π2 α M PR 2 = 0, ρ = 3MPH α (HṘ 4π + 2 H2 R 1 ) 12 R2 Planck m rsp = 8π 2 /2αM P = π/2αm pl R 0 1/m rsp Planck Friedmann R = 0 9.3: Planck ξ Λ = 1/Λ QG ( l pl ) Hubble 1/H 0 ( 5000Mpc) 1/H ξ Λ

46

47 47 10 E m pl ( ) (cosmological perturbation theory) 10.1 ( ) E δr R E2 12H 2 D (10.1.1) 1 H = H D (9.2.1) H D Planck Planck E H D Λ QG δr/r τp 0.1 δr Λ2 QG 10 5 R τλ 12HD 2 CMB 1

48 /Λ QG Planck Planck Riegert φ ϕ(η, x) = ˆϕ(η) + φ(η, x) ˆϕ(η) (9.2.3) δ ξ φ = ξ 0 η ˆϕ λξ λ, δ ξ h µν = µ ξ ν + ν ξ µ 1 2 η µν λ ξ λ h 00 = h, h 0i = h T i + i h, h ij = h TT ij + (i h T j) δ ijh + ( i j 1 ) 2 3 δ ij h

49 i j 3 2 = i i h T i h T i h TT ij ξ i = ξ T i + i ξ S δ ξ φ = ξ 0 η ˆϕ ηξ ξ S, δ ξ h = 3 2 ηξ ξ S, δ ξ h = ξ 0 + η ξ S, δ ξ h = 2 2 ξ S, δ ξ h T i = η ξ T i, δ ξ h T i = 2ξ T i, δ ξ h TT ij = 0 Bardeen Φ = φ h 1 6 h + σ η ˆϕ, σ Ψ = φ 1 2 h + σ η ˆϕ + η σ (10.1.2) σ = h 1 η h 2 2 δ ξ σ = ξ 0 (10.1.2) h = h = 0 (conformal Newtonian gauge)[ (longitudinal gauge) ] Φ = φ + h/6 Ψ = φ h/2 ds 2 = a 2 [ (1 + 2Ψ) dη 2 + (1 + 2Φ) dx 2]

50 50 10 Ψ Φ Υ i = h T i 1 2 ηh T i, h TT ij h = h = 0 h T i = 0 T Mλ λ = 0 T M0 0 = (ρ + δρ), T Mi 0 = 4 3 ρv i, T M0 i = 4 3 ρ (v i + h 0i ), T Mi j = 1 3 (ρ + δρ)δi j (10.1.3) ρ(η) (9.2.4) δρ v i δ ξ T Mµ ν = ν ξ λ T Mµ λ λξ µ T Mλ ν + ξ λ λ T Mµ ν v i = vi T + i v vi T v δ ξ (δρ) = ξ 0 η ρ, δ ξ v = η ξ S, δ ξ vi T = η ξi T

51 D = δρ ρ + ηρ ρ σ 4 η ˆϕV, V = v η h 2, V i = v T i ηh T i, Ω i = v T i + h T i Υ i V i Ω i Υ i + V i = Ω i 10.2 Riegert Einstein δγ = 1 2 = = d 4 x gt µν δg µν d 4 x { ḡ T λ λδϕ T } µν δḡ µν { d 4 x T λ λδϕ + 1 } 2 Tµ νδh ν µ T µν T µν T µν g µν = e 2ϕ ḡ µν δg µν = 2e 2ϕ ḡ µν δϕ + e 2ϕ δḡ µν T µν (g) g µν T µν (ϕ, ḡ) Riegert ḡ µν

52 52 10 T µν (ϕ, h) Minkowski η µν Riegert T µν = e 6ϕ T µν = e 6 ˆϕ(1 6φ) T µν, T µ ν = e 4ϕ µ T ν = e 4 ˆϕ(1 4φ) T µ ν h µ ν T µν = η λ(µ T λ ν) = T µν h λ (µ ˆT ν)λ T µν ˆT µν T λ λ(= η µν T µν ) = T λ λ T µν = T R µν + T W µν + T EH µν + T M µν = 0 R W EH M Riegert-Wess-Zumino Weyl Einstein δ ξ B = ξ λ λ B = ξ 0 η B σ B = B 0 σ η B 0 Riegert-Wess-Zumino b 1 b 1 B σ = 0

53 T λ λ = 0, (10.2.1) ( 1 T i 2 i 3 i j ) T 2 ij = 0 (10.2.2) (10.2.1) { ( b 1 8π B 0(τ) 2 ηφ η ˆϕ η Φ + 8 η 2 ˆϕ + 10 ) 3 2 ηφ 2 + ( 12 3η ˆϕ ˆϕ ) ( 16 η 2 η Φ ˆϕ η 4 ) Φ 3 +2 η ˆϕ η Ψ + (8 2η ˆϕ + 2 ) 3 2 ηψ 2 + (12 3η ˆϕ 10 3 ˆϕ ) η 2 η Ψ + ( η ˆϕ 2 ) } Ψ +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 } = 0 (10.2.3) (9.2.3) 4 η ˆϕ η B 0 (10.2.2) 2 { 2 t 2 r(τ) + b 1 8π 2 B 0(τ) 4 ηφ Φ 4 ηψ } 3 2 Ψ { ηφ + 4 η ˆϕ η Φ η ˆϕ η Ψ + ( η ˆϕ 8 3 η ˆϕ η ˆϕ 8 ( η ˆϕ η ˆϕ η ˆϕ ) 9 2 Φ ) } Ψ +M 2 Pe 2 ˆϕ { 2Φ 2Ψ} = 0 (10.2.4) Einstein t r 0 Riegert Φ = Ψ (= φ)

54 54 10 Einstein Einstein Φ = Ψ Φ = Ψ = φ (10.2.3) Riegert φ T µ µ tr 0 = b 1 4π 2 ( 4 η φ 2 2 η 2 φ + 4 φ ) +M 2 Pe 2 ˆϕ { 6 2 ηφ 6 2 φ + 12 η ˆϕ η φ +12 ( 2 η ˆϕ + η ˆϕ η ˆϕ) φ } j 2 T ij = 0 2 { } 3 t 2 η Υ i η 2 Υ i r(τ) b {( 1 1 8π B 0(τ) ˆϕ η ˆϕ ) ( 1 η η ˆϕ η Υ i ˆϕ η ˆϕ ) } η η ˆϕ Υ i { } +MPe 2 2 ˆϕ 1 2 ηυ i + η ˆϕΥi = 0 (10.2.5) T ij = 0 2 { 4 t 2 η h TT ij r(τ) + b 1 8π 2 B 0(τ) ηh TT ij } + 4 h TT ij {( ˆϕ η ˆϕ ) η η ˆϕ ηh 2 TT ( ˆϕ η ˆϕ ) η η ˆϕ { +MPe 2 2 ˆϕ ηh TT ij η ˆϕ η h TT ij ij + 2 h TT ij } h TT ij ( ˆϕ η ˆϕ ) η η ˆϕ } = 0 η h TT ij

55 dτ = a(τ)dη τ a(τ) = e ˆϕ(τ) Hubble H(τ) = ȧ(τ)/a(τ) 2 = a 2 ( η = a τ, ) k2 a 2 2 η = a 2 ( 2 τ + H τ ),, η 3 = a { 3 τ 3 + 3H τ 2 + ( Ḣ + 2H 2) } τ, η 4 = a { 4 τ 4 + 6H τ 3 + ( 4Ḣ + 11H2) τ 2 + ( Ḧ + 7HḢ + } 6H3) τ η ˆϕ = ah, 2 η ˆϕ = a 2 ( Ḣ + H 2), 3 η ˆϕ = a 3 ( Ḧ + 4HḢ + 2H3), 4 η ˆϕ = a 4 (... H +7HḦ + 4Ḣ2 + 18H 2 Ḣ + 6H 4) 10.3 (10.1.3) T Mλ λ = 0, T M 00 = e 4ϕ (ρ + δρ + 4ρφ), T M 0i = 4 ( 3 e4ϕ ρ v i + 1 ) 2 h 0i, T M ij = 1 3 e4ϕ (ρ + δρ + 4ρφ) δ ij (10.3.1)

56 56 10 T η ˆϕ i T 2 i0 = 0, i T 2 i0 = 0 D {( b 1 8π B 0(τ) 2 2 ˆϕ ) 2 2 η + 2 η ˆϕ η ˆϕ 3 2 ηφ 2 + ( 2 η 3 ˆϕ 4 η 2 ˆϕ η ˆϕ) η Φ ( + η ˆϕ 2 2 η ˆϕ + 2 η ˆϕ η ˆϕ ) ( 2 2 η Φ ˆϕ ) 4 η η ˆϕ Φ ( + η ˆϕ 2 η 2 ˆϕ ) 2 2 η ˆϕ η ˆϕ η Ψ + ( 2 η 3 ˆϕ η ˆϕ η ˆϕ η ˆϕ) Ψ ( + 2 η 2 ˆϕ ˆϕ ) } 2 η η ˆϕ Ψ + 2 { 4 t 2 r(τ) 3 4 Φ 4 η ˆϕ 2 η Φ + 4 } 3 4 Ψ + 4 η ˆϕ 2 η Ψ +M 2 Pe 2 ˆϕ2 2 Φ + e 4 ˆϕρD = 0 2 (10.2.3) (10.2.4) Φ Ψ D V { b 1 8π B 0(τ) ηφ t 2 r(τ) η ˆϕ 2 ηψ + ( η ˆϕ η ˆϕ η ˆϕ + 4 ( 2 2 η ˆϕ 2 3 η ˆϕ η ˆϕ + 2 { η Φ η Ψ +M 2 Pe 2 ˆϕ { 2 η Φ 2 η ˆϕΨ } 4 3 e4 ˆϕρV = 0 } ) 9 2 η Φ 4 3 ˆϕ η 2 Φ ) 9 2 η Ψ + (2 3η ˆϕ 2 3 η ˆϕ 2 ) Ψ 3 (10.2.3) (10.2.4) V }

57 T 0i = 0 Ω i 2 { } ( 2 t 2 η 2 Υ i 4 b 1 1 Υ i r(τ) 8π B 0(τ) ˆϕ η ˆϕ ) η η ˆϕ M 2 Pe 2 ˆϕ 2 Υ i 4 3 e4 ˆϕρΩ i = 0 2 Υ i Υ i 2 (10.2.5) Ω i

58

59 59 11 CFT CMB Λ QG GeV (= GeV/3 o K) ( 9.3 ) Planck Planck (Mpc) CMB Planck Planck CMB τ i = 1/E i (E i H D ) Φ = Ψ Riegert φ 4 Riegert-

60 60 11 CFT CMB Wess-Zumino φ(τ i, x)φ(τ i, x ) = 1 4b 1 log ( m 2 x x 2) (11.1.1) b 1 Riegert-Wess-Zumino m τ i Planck m = a(τ i )H D (11.1.2) τ i r r = a(τ i ) x x (11.1.1) Planck L P = 1/H D 3 Fourier φ(x) Fourier φ(x) = φ(k) 2 d 3 k (2π) 3 φ(k)eik x φ(k) φ(k ) = φ(k) 2 (2π) 3 δ 3 (k + k ) (11.1.3) Fourier log ( m 2 x 2) = k>ϵ d 3 k 4π 2 ( ) m eik x 2 log (2π) 3 k 3 ϵ 2 e 2γ 2 k = k ϵ( 1) γ Euler Fourier δ 3 (k) (11.1.1) φ(k) 2 = π2 b 1 1 k 3

61 φ(x) P φ φ 2 (x) = dk k P φ(k) (11.1.3) φ 2 (x) = = d 3 k d 3 k (2π) 3 (2π) φ(k) 3 φ(k ) e i(k+k ) x dk k 3 k 2π 2 φ(k) 2 P φ (τ i, k) = k3 2π 2 φ(τ i, k) 2 = 1 2b 1 (11.1.4) Harrison-Zel dovich-peebles k n s 1 n s = 1 1 Υ i h ij TT 2 h ij TT 4 Weyl 2 P h (τ i, k) = k3 2π 2 h TT(τ i, k) 2 = A t A t 2 A t k n t n t = τ = τ Λ 1 n s 1

62 62 11 CFT CMB τ i Ψ = Φ = φ Φ(τ i, k) = Ψ(τ i, k) Ψ = Φ (10.2.4) Φ(τ Λ, k) + Ψ(τ Λ, k) = 0 (11.2.1) (10.2.3) (10.2.4) τ Φ Ψ 2 Φ Φ(τ Λ, k) = T Φ (τ Λ, τ i ) Φ(τ i, k) P Φ (τ Λ, k) = TΦ 2 (τ Λ, τ i )P φ (τ i, k) k a(τ) H D 2 2 k 2 /m 2 a(τ) 2 a(τ i ) = 1 Planck H D m a(τ) β 0 a 1 κ 2 τ t = H D τ

63 Φ and Ψ log 10 (τ/τ P ) τ Λ 11.1: Bardeen Φ( ) Ψ( ) Φ = Ψ(= φ) 1/ 20 k = 0.01Mpc 1 m = (= 60λ)Mpc 1 Bardeen τ Λ Φ = Ψ b 1 = 10 Planck m = Mpc 1 (e-foldings) H D /Λ QG H D /Λ QG = 60 β 0 /b 1 = 0.06 a 1 /b 1 = 0.01 κ = 0.5 N e = / 2b 1 = P φ A t = Friedmann

64 64 11 CFT CMB Bardeen Potential Φ(b 1 =10, m=0.0156) proper time, log 10 (τ/τ p ) k [Mpc -1 ] proper time τ k [Mpc -1 ] : Bardeen Φ τ = 60 Tensor Perturbation (b 1 =10, m=0.0156) k [Mpc -1 ] proper time, log 10 (τ/τ p ) 11.3: Riegert τ φ = 2 τ φ = 0 δ R = δr 12m 2 = 1 2m 2 e2φ ( 2 φ i φ i φ )

65 Fourier δ R (k) = k2 2m 2 φ NL(k) (11.2.2) Riegert φ 2 d 3 ( ) q 3 φ NL (k) = φ(k) + φ (k/2 q) φ (k/2 + q) (2π) q2 (11.2.3) k 2 φ NL (x) = φ(x) + f NL φ 2 (x) f NL 1 n s > 1 f NK 1 1/2b 1 A s ( ) Harrison-Zel dovich-peebles (n s 1) ξ Λ ξ Λ = 1/Λ QG ( L P ) Planck ξ Λ 2 t r t 2 r(k) = 1/β 0 log(k 2 /λ 2 ) ) v/ log(k 2 /λ 2 ) ( k P s (k) = A s (11.2.4) m v λ λ = a(τ i )Λ QG (11.2.5) Planck m/λ = H D /Λ QG k = λ

66 66 11 CFT CMB 9.3 k k < λ (11.2.4) P t (k) = A t ( k m ) v/ log(k 2 /λ 2 ) (11.2.6) 11.3 A t r = A t A s CMB Fourier H 2 D D(τ Λ, k) = 2 k2 e 3 H(τ Λ ) 2e 2N m Φ(τ Λ, k) 2 ρ(τ Λ ) = 3MPH 2 2 (τ Λ ) N e

67 11.3. CMB CMB P s (11.2.4) P t (11.2.6) Friedmann CMB CMBFAST WMAP wmap 5yrs acbar2008 l(l + 1)Cl/2π Multipole, l 11.4: CMB (TT ) WMAP5 ACBAR2008 r = 0.06 λ = (= m/60)mpc 1 v = EE ( ) τ e = 0.08 Ω b = Ω c = 0.20 Ω vac = H 0 = 73.1 T cmb = Y He = 0.24 [χ 2 /dof = 1.10 (2 l 1000)] λ l k l kd dec

68 68 11 CFT CMB l(l + 1)Cl/2π Multipole, l 11.5: CMB TE WMAP [χ 2 /dof = (2 l 1000)] d dec 14000Mpc l = 2, Mpc 1 λ = Mpc 1 (11.2.5) λ Λ QG GeV(9.2.7) 1 a(τ i ) = Mpc GeV /λ 4000Mpc ξ Λ = 1/Λ QG cm N e Planck

69 11.3. CMB Λ QG 3 o K (l < 100) r = 0.06 EE ( ) τ e = 0.08 TT WMAP 5 (WMAP5) ACBAR(Arcminute Cosmology Bolometer Array Receiver) 11.4 TE WMAP5 11.5

70

71 71 A A.1 ( ) Christoffel Riemann Γ λ µν = 1 2 gλσ ( µ g νσ + ν g µσ σ g µν ), R λ µσν = σ Γ λ µν ν Γ λ µσ + Γ λ ρσγ ρ µν Γ λ ρνγ ρ µσ, Ricci R µν = R λ µλν Ricci R = Rµ µ Christoffel µ A σ 1 σ m λ 1 λ n = µ A σ 1 σ m λ 1 λ n n j=1 Γ ν j µλ j A σ 1 σ m λ 1 ν j λ n + m j=1 Γ σ j µν j A σ 1 ν j σ m λ 1 λ n [ µ, ν ] A λ1 λ n = n Rµνλ j j=1 σ j A λ1 σ j λ n A D Riemann R µ νλσ + Rµ λσν + Rµ σνλ = 0, ρ R µ νλσ + λr µ νσρ + σ R µ νρλ = 0 Bianchi µ R µ λνσ = ν R λσ σ R λν µ R µ ν = ν R/2

72 72 A δg µν = g µλ g νσ δg λσ, δ g = 1 2 gg µν δg µν, δγ λ µν = 1 2 gλσ ( µ δg νσ + ν δg µσ σ δg µν ), δr λ µσν = σ δγ λ µν ν δγ λ µσ = 1 2 gλρ{ σ µ δg νρ + σ ν δg µρ σ ρ δg µν ν µ δg σρ ν σ δg µρ + ν ρ δg µσ }, δr µν = δr λ µλν = 1 { ( )} µ λ δg λν + ν λ δg λµ 2 δg µν µ ν g λσ δg λσ 2 R λ µ σ νδg λσ + 1 ( ) R λ 2 µ δg λν + Rν λ δg λµ, δr = δg µν R µν + g µν δr µν = R µν δg µν + µ ν δg µν 2 (g µν δg µν ) δ( µ A) = µ δa, δ( µ ν A) = µ ν δa 1 2 λ A ( µ δg νλ + ν δg µλ λ δg µν ), δ( 2 A) = 2 δa δg µν µ ν A µ A ν δg µν λ A λ (g µν δg µν ) A Weyl Weyl δ ω g µν = 2ωg µν δ ω gr = (D 2)ω gr 2(D 1) g 2 ω 2 δ ω gr µνλσ R µνλσ = (D 4)ω gr µνλσ R µνλσ 8 gr µν µ ν ω, δ ω gr µν R µν = (D 4)ω gr µν R µν 2 gr 2 ω

73 A.1. ( ) 73 2(D 2) gr µν µ ν ω, δ ω gr 2 = (D 4)ω gr 2 4(D 1) gr 2 ω, δ ω g 2 R = (D 4)ω g 2 R + (D 6) g λ R λ ω 2 gr 2 ω 2(D 1) g 4 ω, δ ω gfµν F µν = (D 4)ω gf µν F µν D (8.1.8) [δ ω1, δ ω2 ]Γ = 2{4η 1 + Dη 2 + 4(D 1)η 3 + (D 4)η 4 } d D x grω [1 2 ω 2] Euler D = 2 Euler R µν = 1 2 g µνr D = 4 Euler R µλσρ R λσρ ν 2R µλνσ R λσ 2R µλ R λ ν + R µν R = 1 4 g µνg 4 g µν = e 2ϕ ḡ µν Riegert Γ λ µν = Γ λ µν + ḡ λ µ ν ϕ + ḡ λ ν µ ϕ ḡ µν λ ϕ, R λ µσν = R λ µσν + ḡ λ ν µσ ḡ λ σ µν + ḡ µσ λ ν ḡ µν λ σ +(ḡ λ νḡ µσ ḡ λ σḡ µν ) ρ ϕ ρ ϕ, R µν = R µν (D 2) { µν ḡ 2 µν ϕ + (D 2) λ ϕ λ ϕ }, R = e { 2ϕ R 2(D 1) 2 ϕ (D 1)(D 2) λ ϕ λ ϕ }

74 74 A µν = µ ν ϕ µ ϕ ν ϕ ḡ µν = (ĝe h ) µν h µν Γ λ µν = ˆΓ λ µν + ˆ (µ h λ ν) 1 2 ˆ λ h µν ˆ (µ (h 2 ) λ ν) 1 4 ˆ λ (h 2 ) µν h λ ˆ σ (µ h σ ν) hλ ˆ σ σ h µν + o(h 3 ), R = ˆR ˆR µν h µν + ˆ µ ˆ ν h µν 1 4 ˆ λ h µ ˆ ν λ h ν µ ˆR σ µλνh λ σh µν ˆ ν h ν ˆ µ λ h λµ ˆ µ (h µ ˆ ν λ h ν λ) + o(h 3 ), R µν = ˆR µν ˆR σ µλνh λ σ + ˆR λ (µh ν)λ + ˆ (µ ˆ λ h ν)λ 1 2 ˆ 2 h µν 1 2 hλ (µ ˆ 2 h ν)λ 1 2 ˆ λ h σ µ ˆ σ h νλ 1 4 ˆ µ h λ σ ˆ ν h σ λ 1 2 ˆ λ (h λ σ ˆ (µ h σ ν)) ˆ λ (h σ (µ ˆ ν) h λ σ) ˆ λ (h λ σ ˆ σ h µν ) + o(h 3 ) a (µ b ν) = (a µ b ν + a ν b µ )/2 R = ḡ µν Rµν ḡ µν = ĝ µν h µν + [ ˆ λ, ˆ ν ]h λ µ = h λ σ ˆR σ µνλ + h µσ ˆRσ ν R µν R A.2 g µν = e α µe να D α β γ δ Lorentz µ ν λ σ Einstein Lorentz {γ α, γ β } = 2η αβ Einstein e µ αγ α ψ Dirac (adjoint) Lorentz ψ = ψ γ 0 D µ = µ ω µαβσ αβ

75 A (connection 1-form)ω µ dx µ ( ) ω µαβ = e ν α µ e νβ = e ν α µ e νβ Γ λ µνe λβ Lorentz ω µαβ = ω µβα Σ αβ Lorentz [ Σ αβ, Σ γδ] = η βγ Σ αδ η αγ Σ βδ + η βδ Σ γα η δα Σ γβ [D µ, D ν ] = 1 2 ( µω ναβ ν ω µαβ + [ω µ, ω ν ] αβ ) Σ αβ = 1 2 R µναβσ αβ Lorentz Σ αβ = 0 Einstein Σ µν = e µ αe ν βσ αβ (Σ µν ) λσ = g µ λ gν σ g µ σg ν λ D µ = ν Σ αβ = 1 [ γ α, γ β] 4 Weyl Weyl δ ω g µν = 2ωg µν δ ω e µ α = ωe µ α, δ ω e µα = ωe µα, δ ω ψ = 1 D 1 D ωψ, δ ω ψ = ω 2 2 ψ δ ω ω µαβ = ( e µα e λ β e µβ e λ α) λ ω, δ ω (e µ αγ α D µ ψ) = D + 1 ωe µ 2 αγ α D µ ψ γ α Σ αβ = 1(D 2 1)γβ ( δ ω g µ ψe α γ α D µ ψ ) ( = Dω + 1 D ω D + 1 ) g µ ω ψe 2 2 αγ α D µ ψ = 0 D Weyl

76 76 A Riegert Riegert ē µα = (e 1 2 h ) µα = η µα h µα (h2 ) µα +, ē µ α = (e 1 2 h ) µ α = δ α µ 1 2 hµ α (h2 ) µ α + ē α µē να = ḡ µν ē µ αē µβ = η αβ Lorentz ω µαβ = ( ē ν a µ ē νβ Γ ) λ µνē λβ = 1 2 ( αh µβ β h µα ) 1 ( ) h λ 8 α µ h λβ h λ β µ h λα 1 ( hµλ α h λ β h µλ β h λ 1 ( ) α) + h λ 4 4 α λ h µβ h λ β λ h µα +o(h 3 )

77 77 B B.1 G D D = 4 D 4 Euler G 4 D = 4 d D x gg 4 = (D 4) n n=0 n! { d D x ĝ ϕ n Ḡ 4 + 4(D 3)ϕ n Rµν µ ν ϕ 2(D 3)ϕ n R 2 ϕ 2(D 2)(D 3)(D 4)ϕ n 2 ϕ λ ϕ λ ϕ (D 2)(D 3) 2 (D 4)ϕ n ( λ ϕ λ ϕ) 2 } (B.1.1) D M D (8.1.9) d D x gm D = D 4 d D x gr 2 4(D 1) 1 (D 4) n { = d D x ĝ (D 4)ϕ n R2 4(D 1) n! n=0 2(D 1)(D 6)ϕ n R 2 ϕ + 2(D 1)(D 2)ϕ n λ R λ ϕ +4(D 1) 2 ϕ n 4 ϕ + 8(D 1) 2 (D 4)ϕ n 2 ϕ λ ϕ λ ϕ +(D 1) 2 (D 2) 2 (D 4)ϕ n ( λ ϕ λ ϕ) 2 } (B.1.2) E D (8.1.3) (B.1.2) (D 4) n o(ϕ n+2 ) o(ϕ n+3 ) (B.1.1) E D (8.1.10) o(ϕ n+1 ) o(ϕ n+2 ) o(ϕ n+3 )

78 78 B E D = G 4 + ηm D η η = 4(D 3) 2 /(D 1)(D 2) E D G D (8.1.4) E D d D x ge D = d D x gg D G D d D x gg D = (D 4) n n=0 n! d D x ĝ {ϕ n Ē D + 4(D 3)2 D 2 ϕn 4 ϕ +4(D 3)ϕ n Rµν µ ν ϕ 4(D 3)(D2 6D + 10) ϕ n R (D 1)(D 2) 2 ϕ 2(D 3)2 (D 6) 2(D 3)(D 4)3 (D 1)(D 2) ϕn λ R λ ϕ ϕ n D 2 2 ϕ λ ϕ } λ ϕ ( = d D x ĝ{ḡ4 + (D 4) 2ϕ 4 ϕ + Ē4ϕ R ) (2ϕ 2 (D 4)2 2 4 ϕ + Ē4ϕ 2 + 6ϕ 4 ϕ + 8ϕ R µν µ ν ϕ 28 9 ϕ R 2 ϕ ϕ λ R λ ϕ 14 9 R 2 ϕ R 2 ϕ R ) ! (D 4)3( 2ϕ 3 4 ϕ + Ē4ϕ ϕ λ ϕ λ ϕ + 9ϕ 2 4 ϕ + ) } +o((d 4) 4 ) 1 1 D Ricci D = 2 d D x (D 2) n gr = n! n=0 d D x ĝ { (D 1)ϕ n 2 ϕ + Rϕ n} n = 1 Liouville-Polyakov

79 79 C C.1 D Euclid d D p = dω D = D Euclid p D 1 dp dω D, (p 2 = p µ p µ ) D 1 sin D 1 l θ l dθ l = 2πD/2 l=1 Γ ( ) D 2 d D p (2π) D p 2 p 2n (p 2 + L) = 1 Γ α (4π) D/2 ( n + D 2 p µ ) ( ) Γ α n D 2 ) L D/2+n α Γ(α) Γ ( D 2 d D p (2π) p µp D ν f(p 2 ) = 1 D δ µν d D p (2π) p µp D ν p λ p σ f(p 2 1 ) = d D p (2π) D p2 f(p 2 ), D(D + 2) (δ µνδ λσ + δ µλ δ νσ + δ µσ δ νλ ) d D p (2π) D p4 f(p 2 ) p µ

80 80 C Feynman Feynmann 1 Γ(α + β) 1 = A α Bβ Γ(α)Γ(β) 0 dx (1 x)α 1 x β 1 [(1 x)a + xb] α+β A = p 2 +z 2 B = (p + q) 2 + z 2 z 2 d D p f(p µ, q ν ) (2π) D (p 2 + z 2 ) α ((p + q) 2 + z 2 ) β = Γ(α + β) Γ(α)Γ(β) 1 0 dx(1 x) α 1 x β 1 d D p f(p µ xq µ, q ν ) (2π) D [p 2 + z 2 + x(1 x)q 2 ] α+β D = 4 2ϵ ϵ Γ(ϵ) = 1 ϵ γ + ϵ 2 ( γ 2 + π2 6 a ϵ = e ϵ ln a = 1 + ϵ ln a + o(ϵ 2 ) ) + o(ϵ 2 ), a p 2 z 2 D Euclid {γ µ, γ ν } = 2δ µν γ λ γ λ = D, γ λ γ µ γ λ = (D 2)γ µ, γ λ γ µ γ ν γ λ = (D 4)γ µ γ ν + 4δ µν, γ λ γ µν = γ λµν δ λµ γ ν + δ λν γ µ

81 C δ µν γ µν = 1 2 [γ µ, γ ν ], γ λµν = 1 3! (γ λγ µ γ ν + γ µ γ ν γ λ + γ ν γ λ γ µ γ λ γ ν γ µ γ ν γ µ γ λ γ µ γ λ γ ν )

82

83 83 D D.1 (10.2.3) (10.2.4) t r Hubble H = H D / B 0 = 1 T = b 1 B 0 t 2 r/8π 2 ( 1) k 2 /a 2 a 2... Φ Φ 36 Φ 48 Φ Ψ +14 Ψ + 36 Ψ + 48Ψ +6 ( Φ + 4 Φ Ψ 4Ψ ) = 0 (D.1.1) 4 3 Φ Φ Φ 4 3 Ψ Ψ + 8 T 2(Φ + Ψ) = 0 ( Φ + Φ Ψ Ψ) (D.1.2) f = Ψ Φ... f +7 f + 15f + 12f = 0, (... Φ ) 12 T 7 Φ 12 T Φ = f ( ) 6 T f 1 12 T f

84 84 D ( ) ( ) f = c 1 e 4τ + c 2 e τ sin 2 τ + c 3 e τ cos 2 τ ( Φ = (a 1 + c 1 )e τ + (a 2 + c 2 ) +c T c 2 3c 3 e 3 2 τ sin ) 12 T τ e 4τ 3c2 + 5c 3 + e 3 2 τ cos 14 ( ) 3 2 τ ( + (a 3 + c 3 ) ) ( 3 2 τ T T τ ) e τ (D.1.3) T = 0 (D.1.1) Φ = Ψ = ω... ω +6 ω ω 3 ω 12ω = 0 (D.1.2) 9.2 ˆϕ e τ e 4τ e 3τ/2 sin( 3τ/2) e 3τ/2 cos( 3τ/2) T = 0 (D.1.3) Φ T 1 Φ T τ

85 85 E E.1 Planck h = cm 2 g s 1 (speed of light) c = cm s 1 Newton G = cm 3 g 1 s 2 Planck m pl = g = GeV/c 2 Planck M P = GeV/c 2 Planck l pl = cm Planck t pl = s Boltzmann k B = erg K 1 (Megaparsec) 1Mpc = cm Hubble H 0 = 100h km s 1 Mpc 1 Hubble c/h 0 = 2998h 1 Mpc ( h 0.7 ) (c = h = k B = 1) 1 cm = h/gev 1 s = h/gev/c 1 g = GeV/c 2 1 erg = GeV 1 K = GeV/k B

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87 87 F J. Collins, Renormalization (Cambridge University Press, 1984). T. Muta, Foundations of Quantum Chromodynamics (World Scientific, 1987). S. Hathrell, Trace Anomalies and λϕ 4 Theory in Curved Space, Ann. of Phys. 139 (1982) 136. S. Hathrell, Trace Anomalies and QED in Curved Space, Ann. of Phys. 142 (1982) 34. K. Hamada and F. Sugino, Background-metric Independent Formulation of 4D Quantum Gravity, Nucl. Phys. B553 (1999) 283. K. Hamada, Resummation and Higher Order Renormalization in 4D Quantum Gravity, Prog. Theor. Phys. 108 (2002) 399. K. Hamada, Renormalizable 4D Quantum Gravity as a Perturbed Theory from CFT, Found. Phys. 39 (2009) 1356.

88 88 F E. Kolb and M. Turner, The Early Univrse (Westview Press,1990). A. Liddle and D. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, 2000). S. Weinberg, Cosmology (Oxford University Press, 2008) CMB R. Durrer, The Theory of CMB Anisotropies, J. Phys. Stud. 5 (2001) 177., ( K. Hamada and T. Yukawa, CMB Anisotropies Reveal Quantized Gravity, Mod. Phys. Lett. A20 (2005) 509. K. Hamada, S. Horata and T. Yukawa, Space-time Evolution and CMB Anisotropies from Quantum Gravity, Phys. Rev. D74 (2006) K. Hamada, S. Horata and T. Yukawa, From Conformal Field Theory Spectra to CMB Multipoles in Quantum Gravity Cosmology, Phys. Rev. D81 (2010) K. Hamada, S. Horata and T. Yukawa, Focus on Quantum Gravity Research (Nova Science Publisher, NY, 2006), Chap. 1 entitled by Background Free Quantum Gravity and Cosmology.