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/ 平成 25 年 08 月再改定 (上下巻に分離)
2 Planck Planck BRST Planck Λ QG Planck 10 17 GeV Planck Λ QG Friedmann CMB
3 8 5 8.1 D.............. 5 8.2................ 10 8.3..................... 12 8.4..................... 16 8.5................... 25 8.6....................... 29 9 33 9.1.................. 33 9.2............... 35 9.3.................... 41 10 47 10.1.......................... 47 10.2................... 51 10.3................ 55 11 CFT CMB 59 11.1 2.......... 59 11.2................... 61 11.3 CMB................... 67 A 71 A.1 ( )............... 71
4 A.2............... 74 B 77 B.1 G D D = 4............... 77 C 79 C.1................... 79 D 83 D.1................... 83 E 85 E.1.................... 85 F 87
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 8 Euclid Wick I = d D x { 1 g t 2 C2 µνλσ + be D + 1 4 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 + 1 2 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
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 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!
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 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)
8.2. 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) 2 + 1 2 {( b1 d D x D 4 + b 2 (D 4) + 2 )Ḡ4 ( + b 1 + b )( 2 D 4 + 2ϕ 4 ϕ + Ē4ϕ + 1 18 R ) 2 ( (D 4)b1 + b 2 + )( 2ϕ 2 4 ϕ + Ē4ϕ 2 + ) } + (8.2.2) G 4 Riegert-Wess-Zumino S 1 5.2 ( ) S RWS Riegert 3 Duff, Nucl. Phys. B125 (1977) 334 Duff D 4 Weyl 1 ( R 2 )
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
8.3. 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 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 )
8.3. 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 + 1 18 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 + 4 3 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 8 8.4 h Riegert Wess-Zumino 2 4 4 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
8.4. 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) 4 + 4 D ( 36l 4 k 4 + 24(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 8 Feynman t 2 r Z ϕ = 1 2 3 ( 8.2) Weyl 2 3 b n Ḡ4 3 (1) (2) 8.2: 2 3 (background field method) t r ( nf Z t = 1 80 + 5 ) t 2 r 1 3 (4π) 2 ϵ 7n F ẽ 2 r t 2 r 1 288 (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
8.4. 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 40 + 10 3 ) t 3 r (4π) 7n F 2 72 e 2 rt 3 r (4π) 4 + o(t5 r) Euler Ḡ4 b1 = 11n F 360 + 40 9, 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 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
8.4. 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 8.5 2 Feynman e 6 r 2 Feynman 8.6
22 8 8.5: 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 9 + 8 81 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)
8.4. 23 Z 3 Z 3 = 1 + A 1 ϵ + A ( 2 ϵ + + 1 b1 B1 2 ϵ + B ) 2 ϵ + + 2 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 1 8.5 B 1,2 8.6 2 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 8 8.7: 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 3 2 2 3 ϵϕfµν r2 Feynman 3
8.5. 25 Riegert ϵ { Γ ϕaa µν (0; k, k) I = 8 3 + 16 9 + 8 } 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 8.8 2 3 Feynman Riegert ϕ 3 ϕ 2 F r2 µν b ge D Laurent (8.2.2) { Γ ϕaa µν (0; k, k) II = 8 81 + 16 81 8 } 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 8 8.8: ϕfµν 2 e6 r II
8.5. 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 4 4 3 x 1 Wess-Zumino 4 8.9 ϕ ϕ ( e 4 ) r 1 δ ϕ Γ QED = x 1 + 4n F gr F r2 (4π) 4 4 1 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 8 { Γ QED = 1 y ( )} 1 p 2 1 2 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
8.6. 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 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)
8.6. 31 1 n (a) n (b) n (c) 8.10: 1/b 1 1/b 2 1 1 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 8.11 1/b 2 1 (a) (b) Laurent (8.6.1) 2 2 u 2 (c) u 1 u 2 ϕe 4ϕ
32 8 u n γ Λ = µ Λr d Λ r dµ µ γ Λ = 2ϵ + µ dz Λ Z Λ dµ = 4 + 8 + (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 1 + 8 b 2 1 + 20 b 3 1 + 2 (8.6.3)
33 9 Planck m pl 10 19 GeV Λ QG Planck Λ QG 10 17 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 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 + 1 18 R } 2 b 1 Weyl N X N W N A b 1 = 1 (N X + 11 ) 360 2 N W + 62N A + 769 180 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 )
9.2. 35 9.2 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 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 = 2.4 10 18 GeV Planck
9.2. 37 m pl = 1.2 10 19 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 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 { 2 0 2 3 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
9.2. 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 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 9.1 9.2 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 = 2.4 10 18 GeV b 1 = 10 H D = 6.7 10 18 GeV Λ QG = 1.1 10 17 GeV (9.2.7)
9.3. 41 9 CMB 10 log 10 [a(τ)/a(τλ)] 0-10 -20-30 -40-50 -60 τ Λ -70-2 -1 0 1 2 3 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 9 2.5 2 ρ H Friedman H, ρ 1.5 1 0.5 0 0 20 40 60 80 100 120 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
9.3. 43 5 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 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 0 1 + 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) 9.2 9.1 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
9.3. 45 R = 0 R = 6Ḣ + 12H2 (9.3.2) (9.3.3) R + 3HṘ + 4π2 α M PR 2 = 0, ρ = 3MPH 2 2 + 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 ) 10 59 Hubble 1/H 0 ( 5000Mpc) 1/H 0 10 59 ξ Λ
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 10 1/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 + 1 4 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) + 1 3 δ ijh + ( i j 1 ) 2 3 δ ij h
10.1. 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 + 1 4 ηξ 0 + 1 4 2 ξ S, δ ξ h = 3 2 ηξ 0 + 1 2 2 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 + 1 6 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 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
10.2. 51 D = δρ ρ + ηρ ρ σ 4 η ˆϕV, V = v + 1 2 η h 2, V i = v T i + 1 2 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 + 1 2 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 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
10.2. 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 4 3 2 2 η ˆϕ η Φ + 8 η 2 ˆϕ + 10 ) 3 2 ηφ 2 + ( 12 3η ˆϕ + 10 3 ˆϕ ) ( 16 η 2 η Φ + 3 2 ˆϕ η 4 ) 3 2 2 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 + ( 16 3 2η ˆϕ 2 ) } 3 2 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 2 4 3 2 Φ 4 ηψ 2 + 4 } 3 2 Ψ { 4 3 2 ηφ + 4 η ˆϕ η Φ + 4 3 η ˆϕ η Ψ + ( 28 3 2 η ˆϕ 8 3 η ˆϕ η ˆϕ 8 ( 4 3 2 η ˆϕ + 8 3 η ˆϕ η ˆϕ 4 9 2 ) 9 2 Φ ) } Ψ +M 2 Pe 2 ˆϕ { 2Φ 2Ψ} = 0 (10.2.4) Einstein t r 0 Riegert Φ = Ψ (= φ)
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) 2 3 2 ˆϕ η + 4 3 ˆϕ ) ( 1 η η ˆϕ η Υ i + 3 3 ˆϕ η + 8 3 2 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) 2 2 2 ηh TT ij } + 4 h TT ij {( 1 3 2 ˆϕ η + 4 3 ˆϕ ) η η ˆϕ ηh 2 TT ( + 7 3 2 ˆϕ η + 2 3 ˆϕ ) η η ˆϕ { +MPe 2 2 ˆϕ 1 2 2 ηh TT ij η ˆϕ η h TT ij ij + 2 h TT ij } + 1 2 2 h TT ij ( 1 3 3 ˆϕ η + 8 3 2 ˆϕ ) η η ˆϕ } = 0 η h TT ij
10.3. 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 10 T 00 + 3 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 + 20 3 ˆϕ ) 4 η η ˆϕ + 9 2 2 Φ ( + η ˆϕ 2 η 2 ˆϕ ) 2 2 η ˆϕ η ˆϕ + 3 2 η Ψ + ( 2 η 3 ˆϕ η ˆϕ + 4 2 2 η ˆϕ η ˆϕ) Ψ ( + 2 η 2 ˆϕ + 2 3 ˆϕ ) } 2 η η ˆϕ + 9 2 2 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) 2 2 3 3 ηφ + + 2 t 2 r(τ) + 2 3 η ˆϕ 2 ηψ + ( 10 3 2 η ˆϕ + 2 3 η ˆϕ η ˆϕ + 4 ( 2 2 η ˆϕ 2 3 η ˆϕ η ˆϕ + 2 { 4 3 2 η Φ + 4 3 2 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 }
10.3. 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) 2 3 2 ˆϕ η + 4 3 ˆϕ ) η η ˆϕ + 1 2 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
59 11 CFT CMB Λ QG 10 17 GeV 10 29 (= 10 17 GeV/3 o K) ( 9.3 ) 10 30 Planck Planck 10 59 (Mpc) CMB Planck Planck CMB 11.1 2 τ i = 1/E i (E i H D ) Φ = Ψ Riegert φ 4 Riegert-
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
11.2. 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 = 0 11.2 τ = τ Λ 1 n s 1
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
11.2. 63 0.25 Φ and Ψ 0.15 0.0008 0.0004 0.05 0-0.0004 1.74 1.76 1.78-0.05-3 -2-1 0 1 log 10 (τ/τ P ) τ Λ 11.1: Bardeen Φ( ) Ψ( ) Φ = Ψ(= φ) 1/ 20 k = 0.01Mpc 1 m = 0.0156 (= 60λ)Mpc 1 Bardeen τ Λ Φ = Ψ b 1 = 10 Planck m = 0.0156Mpc 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 = 65.0 1/ 2b 1 = P φ 11.1 11.2 A t = 10 5 11.3 Friedmann
64 11 CFT CMB Bardeen Potential Φ(b 1 =10, m=0.0156) 0.20 0.10 0.00 0.20 0.10 0.00 5 10-4 3 10-4 1 10-4 -2-1 0 proper time, log 10 (τ/τ p ) 1 10-3 k [Mpc -1 ] 10-2 58.0 58.5 59.0 proper time τ 59.5 60.0 10-3 k [Mpc -1 ] 10-2 11.2: Bardeen Φ τ = 60 Tensor Perturbation (b 1 =10, m=0.0156) 2 10-5 +1 10-5 1 10-5 -2 5 10-6 10-3 k [Mpc -1 ] 10-2 -1 0 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 )
11.2. 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) 3 4 + 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 ( 10 10 ) 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 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
11.3. CMB 67 11.3 CMB P s (11.2.4) P t (11.2.6) Friedmann CMB CMBFAST WMAP 6000 4000 wmap 5yrs acbar2008 l(l + 1)Cl/2π 2000 0 1 10 100 500 1500 Multipole, l 11.4: CMB (TT ) WMAP5 ACBAR2008 r = 0.06 λ = 0.00026 (= m/60)mpc 1 v = 0.00002 EE ( ) τ e = 0.08 Ω b = 0.043 Ω c = 0.20 Ω vac = 0.757 H 0 = 73.1 T cmb = 2.726 Y He = 0.24 [χ 2 /dof = 1.10 (2 l 1000)] λ l k l kd dec
68 11 CFT CMB 200 100 l(l + 1)Cl/2π 0-100 -200-300 -400 20 10 0-10 -20 1 10 100 1 10 100 500 700 900 Multipole, l 11.5: CMB TE WMAP5 11.4 [χ 2 /dof = 0.977 (2 l 1000)] d dec 14000Mpc l = 2, 3 0.0002Mpc 1 λ = 0.00026Mpc 1 (11.2.5) λ Λ QG 1.1 10 17 GeV(9.2.7) 1 a(τ i ) = 0.00026Mpc 1 1.1 10 17 GeV 1.5 10 59 1/λ 4000Mpc ξ Λ = 1/Λ QG 2 10 31 cm N e Planck
11.3. CMB 69 10 30 Λ QG 3 o K 10 29 10 59 (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
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 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 + 1 2 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
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 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 + 1 2 ˆ (µ (h 2 ) λ ν) 1 4 ˆ λ (h 2 ) µν h λ ˆ σ (µ h σ ν) + 1 2 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 + 1 2 ˆR σ µλνh λ σh µν + 1 2 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)) + 1 2 ˆ λ (h σ (µ ˆ ν) h λ σ) + 1 2 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 + 1 2 ω µαβσ αβ
A.2. 75 (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 A Riegert Riegert ē µα = (e 1 2 h ) µα = η µα + 1 2 h µα + 1 8 (h2 ) µα +, ē µ α = (e 1 2 h ) µ α = δ α µ 1 2 hµ α + 1 8 (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 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 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 + 1 18 R ) 2 + 1 (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 + 8 9 ϕ λ R λ ϕ 14 9 R 2 ϕ + 1 9 R 2 ϕ + 5 54 R ) 2 + 1 3! (D 4)3( 2ϕ 3 4 ϕ + Ē4ϕ 3 6 2 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 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 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
C.1. 81 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 )
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 14... Φ 36 Φ 48 Φ + 2... Ψ +14 Ψ + 36 Ψ + 48Ψ +6 ( Φ + 4 Φ Ψ 4Ψ ) = 0 (D.1.1) 4 3 Φ + 16 3 Φ + 20 3 Φ 4 3 Ψ + 4 3 Ψ + 8 T 2(Φ + Ψ) = 0 ( Φ + Φ Ψ Ψ) (D.1.2) f = Ψ Φ... f +7 f + 15f + 12f = 0, (... Φ 1 + 7 ) 12 T 7 Φ 12 T Φ = f ( 1 + 1 ) 6 T f 1 12 T f
84 D ( ) ( ) f = c 1 e 4τ + c 2 e 3 3 2 τ sin 2 τ + c 3 e 3 3 2 τ cos 2 τ ( Φ = (a 1 + c 1 )e τ + (a 2 + c 2 ) +c 1 360 7T 1800 + 5c 2 3c 3 e 3 2 τ sin 14 1 7 ) 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 1 + 7 12 T τ ) e τ (D.1.3) T = 0 (D.1.1) Φ = Ψ = ω... ω +6 ω... +8 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 1 7 12 T τ 11.1 11.2
85 E E.1 Planck h = 1.055 10 27 cm 2 g s 1 (speed of light) c = 2.998 10 10 cm s 1 Newton G = 6.672 10 8 cm 3 g 1 s 2 Planck m pl = 2.177 10 5 g = 1.221 10 19 GeV/c 2 Planck M P = 2.436 10 18 GeV/c 2 Planck l pl = 1.616 10 33 cm Planck t pl = 5.390 10 44 s Boltzmann k B = 1.381 10 16 erg K 1 (Megaparsec) 1Mpc = 3.086 10 24 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 = 5.068 10 13 h/gev 1 s = 1.519 10 24 h/gev/c 1 g = 5.608 10 23 GeV/c 2 1 erg = 6.242 10 2 GeV 1 K = 8.618 10 14 GeV/k B
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