1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a T + AT 4 3 = (nt + 13 ) AT 4 da 3 (6.5a) π 2 A = g 30 (nma 3 = ) : (84b) (6.5b) dt T = 1 + B da 1 2 + B a, B 4AT 3 3n (6.6) T a 1 n a 3 B a B(a) = B(a 0 ) B(a 0 ) = 4AT 0 3 = 4 π 2 g 3n 0 3n B 30 T 0 3 = π4 45ζ(3) g n γ 10 10 (6.7) n B (6.6) dt T + da a = 0 T 1 a (6.8)
6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2 Gρ 3c2 a 2, ρ = ρ m + ρ r + ρ Λ (6.11) ρ Λ >> ρ m,ρ r (k=0) ρ m 1/a 3 H = ȧ/a (6.11) da dt = A a a t 2/3 (6.12) a t 1/2 a e H Λt, H Λ = 8πG Λ 3 ρ Λ = 3 a H = ȧ/a a t 2/3 H = 2 1 3 t a t 1/2 H = 1 1 2 t a e H Λt H = H Λ (6.13) (6.14a) (6.14b) (6.14c) H 1 6.2 ( ) ( ) r a(t) ds 2 = 0 Z r Z dr t d H (t) = a(t) = a(t) cdt 0 1 kr 2 0 a(t ) = 2ct = ch 1 3ct = 2H 1 (6.15)
6 3 6.3 (6.11) ρ m a 3,ρ a 4,ρ Λ, (k ) a 2 (79) π 2 ( ρ r = g 30 T 4, g = g B + 7 ) B,F 8 g *1) F (6.16) (6.11) 4π H(t) = αt 2 3, α = 45 g G 1.66g1/2 (6.18) M Pl M Pl = G 1/2 = 1.22 10 19 GeV g B, g B T ( (84c) ) H = 1/2t t 1s(T 1MeV ) g = 3.36 [ ] 90 1/4 T = g 32π 3 t 1/2 1.33 1010 K G (t/sec) 1/2 1.15 MeV (t/sec) 1/2 (T 1MeV ) (6.19) T = 1MeV t(s) = 1010 K t(s) (6.20) 6.4 6.4.1 T 1MeV ( ) *2) ν + e ν + e (6.21a) ν + ν e + e + γ + γ (6.21b) ν e + n e + p (6.21c) ν e + p e + + n (6.21d) * 1) T >> 2m k k T >> 2m e 1MeV T << 1MeV g γ = 2 g = 2 + 7 8 2 3 ( 4 11 ) 4/3 = 3.36 T << 1MeV (6.17) * 2) η 6 10 10
6 4 ν = ν e,ν µ,ν τ Γ ν = n ν σ ν v ν H ( ) σ ν G 2 F T 2 3 ζ(3) n ν = g ν 4 π 2 T 3 T 3, v ν = 1 (6.22) Γ ν G 2 FTD 5 = H = αtd 2 1.66g1/2 TD 2 (6.23) M Pl T D T D 1MeV T D 0.72 MeV *3) 6.4.2 T 0.72MeV T m e /3 0.17MeV S = sv = V (ρ + P)/T P = ρ/3 4 V 3 T (ρ e + ρ e + + ρ γ) = 4 V be f ore 3 T ρ γ (6.24) a ft ρ e + ρ e + = 2 (7/8) ρ γ, ρ γ T 3 ( ) 11 4 1/3 4 (T be f ore) 3 = (T a ft ) 3 T be f ore = T a ft = 0.714T a ft (6.25) 11 T ν 1/a *4) T ν = T γ T ν = (4/11) 1/3 T γ T ν,0 = 0.714T γ,0 1.92K (6.26) (78) n ν = 110/flavor/cm 3 n ν = 330/cm 3 (6.27) * 3) [?]D.N.Schramm amd M.S.Turner: ReV. Mod. Phys.,70 (1998) 303 * 4) T T 1/a
6 5 6.4.3 - t eq t 0 / ρ m /ρ rad a ρ(t eq ) = ρ rad (t eq ) ρ m (t 0 )a 3 (t 0 ) = ρ m (t eq )a 3 (t eq ) (6.28) ρ rad (t 0 )a 4 (t 0 ) = ρ rad (t eq )a 4 (t eq ) (6.29) 1 + z eq = a 0 a(t eq ) = ρ m(t 0 ) ρ rad (t 0 ) = ρ m(t 0 ) = ρ cω m0 = Ω m0h 2 = 3250 (6.30) ρ γ,0 + ρ ν,0 ρ c Ω rad,0 4.14 10 5 ρ γ0 = π2 15 T 0 4 = 2.00 10 15 ev 4 = 0.2603 ev /cm 3 T =2.725 K (6.31a) ρ rad,0 = ρ γ0 + ρ ν0 = g 2 ρ γ0 = 1.68ρ γ0 = 0.4373eV /cm 3 (6.31b) Ω rad,0 = ρ rad,0 0.4373 ev /cm 3 = ρ c 1.0537 10 5 h 2 ev /cm 3 = 4.15 10 5 h 2 (6.31c) Ω m0 h 2 = 0.135 ± 0.009, Ω m0 = 0.27 ± 0.04 (6.31d) T eq = T 0 (1 + z eq ) = (2.725 ± 0.001K) 3250 8860K, or 0.76 ev (6.32) t eq = 1 H 0 1 (1 + z) 3/2 13.6Gyr 3250 3/2 74,000yrs (6.33) 6.5 (recombination=) e + p H + γ (6.34) (81a) (6.35) µ e + µ p = µ H (6.35) ( ) mt 3/2 n = g e (m µ)/t (6.36) 2π [ ][ ] gh mh 2π 3/2 n H = n e n p e B/T B = m e + m p m H = 13.6 ev (6.37) g e g p m p m e T
6 6 X e n e = n p, n B n B n B = η B n γ, η B = 2.75 10 8 (Ω B h 2 ) = 6.1 ± 0.2 10 10 (6.38a) n γ = 2 ζ(3) π 2 T 3, ζ(3) 1.202 (6.38b) g H = 4, g p = g e = 2, m H m p n H n B = 1 X [ ] e me T 3/2 n p n e Xe 2 = n B e B/T = 4 [ ] 2ζ(3) T 3/2 [ ] T 3/2 η B e B/T = 3.84η B e B/T (6.39) 2π π m e m e T = T 0 (1 + z rec ) = 2.725(1 + z rec ) X e = 0.5 (T rec ) 1 + z rec 1370, T rec = 0.323 ev = 3740 K (6.40) T = 0.308 ev X e T 0.26 ev = 3030 K X e 10 4 10 5 t dc z dc = 1100, T dc = 3000K (6.41a) t dc = H 1 0 (1 + z dc) 3/2 = 375,000 (6.41b) 6.6 6.6.1 (B) ( B) *5) B B = B B B n B η B 6 10 10 (6.42) n γ B/B B B n γ (T 10 15 GeV ) ( 1TeV ) * 5) T Λ QCD 200MeV
6 7 B L SO(10) Lepto genesis [1] 100 X B 1 (B 1 ) B 2 (B 2 ) b(b) 1 b(1 b) Γ(X B 1 ) Γ(X All) = b, Γ(X B 2 ) Γ(X All) = 1 b Γ(X B 1 ) Γ(X All) = b, Γ(X B 2 ) Γ(X All) = 1 b (6.43) Γ(X All) = Γ(X All) (6.44) X B = (b b)b 1 + {(1 b) (1 b)}b 2 = (B 1 B 2 )(b b) (6.45) B 1 B 2 b b CP CPT r Γ(B r) = r Γ(B r) = Γ(B r) (6.46) r CPT r r r Γ(r B) = Γ(r B) (6.47) r CP [2]
6 8 6.6.2 nucleo-synthesis(d.n.schramm and M.S.Turner: Rev. Mod. Phys., 70 (1998) 303) T Λ QCD 200MeV () 10MeV ( MeV/ ) p n A Z p + (A Z)n A (6.48) µ A = Zµ p + (A Z)µ n n B e µ A/T = e {Zµ p+(a Z)µ n }/T [ ] 2ζ(3) n B = n n + n p + An A n γ η B = η B A π 2 T 3 (6.49) (6.50) X A = An A n B, X p = n p n B, T (81a) X n = n n n B (6.51) n k = g k [ mt 2π X A = g A 2 A A5/2 Xp Z Xn A Z [ ] 3/2 e (µ m)/t (6.52) n B ( 2π m N T ) 3/2 e B A/[(A 1)T ] ] A 1 (6.53a) B A = Zm p + (A Z)m n m A (6.53b) [ ] A 1 n B = n γ η B (6.1 ± 0.3) 10 10 (2ζ(3)/π 2 )T 3 ( ) T 3/2 8.13 10 14 1 (6.54) MeV B A /[(A 1)T ] (T 1 0.1 MeV ) ρ rad = g π 2 30 T 4 = 3 32πG t 2 (6.55) 2 H = (2t) 1, H 2 = (8πG/3)ρ rad T 1MeV T 1 MeV (t/ ) 1/2 1010 K (t/ ) 1/2 (6.56)
6 9 : T 10 MeV, t 1 ν e + n e + p (6.57a) ν e + p e + + n (6.57b) ν e + ν e e + e + (6.57c) e + e + γ + γ (6.57d) T 0.72 MeV (6.57a) µ ν + µ n = µ e + µ p (6.58) (6.57d) µ(e + ) + µ(e ) = 0 n(e ) = n(e + ) + n p (6.59) n(e ) n(e + ) = ) µ(e + )]/T e[µ(e 1 + n p n(e + ) 1 + η B (6.60) 2µ e /T η B µ n = µ p n n n p = X n X p = e Q/T, Q = m n m p = 1.293 MeV (6.61) T Q X n = X p = 0.5 (6.62) (6.53a) A = 2 g D = 3, B D = m p + m n m D = 2.22 MeV (6.63) [ ] T 3/2 X D = 4.07 η B e 2.22/T (MeV ) 6 10 12 (6.64) m N 2 T 1 MeV, t 1 T 0.72 MeV T D n n n p = e Q/T D 1 6, X n 1 7, X p 6 7, X D 10 12 (6.65) γ + D p + n
6 10 6.1: A = 5, 8 3 T = 0.3 MeV 0.1 MeV t = 1 3 ((6.53a) η A 1 B ) 885.7 ± 0.8sec T = 0.07 MeV (B(He) = 28.3 MeV ) D + D n + 3 He, D + D p + 3 H, D + 3 He p + 4 He D + 3 H 4 He A = 5,6 ( 6.1) 6.2 (Kolb and Turner; The Early Universe) 6.2 ( ) n(p) n/p 1/7 Y 2n n + p = 2(n/p) 1 + n/p = 2/7 8/7 1 4 (6.67) 75% 25% (G.Gamow:Phys. Rev., 73 (1948) 803) (D) ( 3 He) 7 Li η B = n B /n γ D/H Ω B h 2 = 0.017 0.024 η B = 4.7 6.5 10 10 (6.68a) (6.68b)
6 11 6.2: (1 3 1 ) (D.N.Schramm and M.S.Turner: Rev. Mod. Phys., 70 (1998) 303) ( ) η 6 10 Ω B h 2 = 0.0223 ± 0.0008 Ω B = 0.043 (6.69a) η B 6.1 ± 0.3 10 10 (6.69b) - - ( ) ( ) + 6.7 () ((6.18) g ) T D Y = 2e Q/T 1 + e Q/T (6.70)
6 12 6.3: (G.Steigman; Neutrino06) D/H Y(He) 2σ WMAP N ν N ν = 2.75 ( (6.67)) (6.23)(84c) Y Y ( 1 Y ) Q T D (6.71) 2 T D T D T d = 1 g T D 6 g g = g γ + 7 8 (3g ν + g e + g e +) = 2 + 7 (3 2 + 2 + 2) = 10.75 8 (6.72b) (6.72a) N ν = 3 3+ N ν g = (7/4) N ν (6.72a) Y 0.007 N ν ( ) Ω B (= ρ B /ρ c, ρ c = 3H 2 0 /8πG) D/H Ω B 6.3 D/H WMAP CMB Ω B Ω B D/H WMAP ( 6.3 ) D/H = 2.6 ± 0.4 10 5 Y = He/H( )= 0.238 ± 0.005 N ν = 1.7 3.0 (2σ ) *6) * 6) V.Barger et al., Phys. Lett. B566 (2003) 8-18 hep-ph/0305075
6 13 6.8 6.1: 0.72 MeV, 0.6 10 10 K 1sec 0.07 MeV, 0.6 10 9 K 3 3250 0.76 ev, 8860K 74,000 1370 0.323 ev, 3740K 270,000 1100 0.26 ev, 3030K 37,5000.1 k = 1 Z g n ( ) = (2π) 3 f (p)d 3 p (73) Z g ρ ( ) = (2π) 3 ε f (p)d 3 p (74) P ( ) = n < pv > = g Z p 2 3 (2π) 3 3ε f (p)d3 p (75) f (p) = [exp((ε µ)/t ) ± 1] 1, ε = p 2 + m 2 (76) g g = 2 ± (FD) (BE) µ i + j k + j µ i + µ j = µ k + µ l (77) * 7) (µ ) (µ + ) µ = µ + (T m,ε m) E T m (T µ) n = g ζ(3) π 2 T 3 1 : BE (78) 3/4 : FD ζ(3) = 1.202 ρ = g π2 30 T 4 1 : BE (79) 7/8 : FD P = ρ 3 (80) * 7) e + e + 2γ ( 10 9 ) µ < 9 10 6 D.J.Fixsen et al.: Astro. Phys. J. 473 (1996), 576
6 14 BE FD T γ,0 = 2.725 ± 0.001 n γ,0 = 2 ζ(3) T 3 π 2 0 = 410.4 ± 0.5/cm3, Ω γ,0 = ρ γ,0 /ρ c = (2.471 ± 0.004) 10 5 /h 2 (T m,ε m)(be, FD) ( ) mt 3/2 n = g e (m µ)/t (81a) 2π ρ = n (m + 12 ) m v2 = n (m + 32 ) T (81b) P = nt ρ (81c) ds = dq T s = S V = ρ µn+ P T (82) (83) µ = 0 s(t 0 ) k s = ρ + P = 4 ρ T 3 T = 2π2 45 g st 3 = π4 g s 45ζ(3) n γ (84a) π 2 ρ = g 30 T 4 (84b) g s = g = g B + 7 8 g F T 1MeV (84c) g s = 2 + 7 8 3 2 ( 4 3/3 } 11) = 3.91 g = 2 + 7 8 3 2 ( ) 4 4/3 T 1MeV (84d) 11 = 3.36 ( ) 3 T0 = 7.04n γ = 2889.2 cm 3 (84e) 2.725 4: SU(3) SU(2) U(1)
6 15 g s, g T g SU(3) SU(2) U(1) g s g T m e T m e ( ) T ν = (4/11)T γ (?? ) (??) (??) s 0 (75) T S sr 3 (T R) 3 const (85) dp dt = ρ + P µn T dp = ρ + P µn dt (86) T T ds = d(ρv ) + PdV = d[(ρ + P)V ] V dp (87) ds = 1 T [ ] dt (ρ + P)V d[(ρ + P)V ] (ρ + P µn)v T 2 = d + µnv dt T T 2 (88) ( ) S = (ρ + P)V T + (89) ( (87)) T ds ds = 0 S =.1.1 K U N ( m k ) r k n I = 1 2 N k=1 m k r k 2 (90) di dt = m k r k ṙ k = r k p k (91a) d 2 [ I dt 2 = ṙ k p k + r k dp ] k = 2K + r k F k = 2K r k k U (91b) dt U(ar 1,ar 2, ) = a n U r k k U(r 1,r 2, ) = nu (92)
6 16 (91b) d 2 I = 2K nu (93) dt2 d 2 I dt 2 = 1 di t dt di t dt, < ( ) >= 1 0 t Z t 0 dt( ) (94) < K >= 1 2 n < U > (95) E = K +U (96) < K >= n 2 E, < U >= n + 2 n + 2 E (97) N T 3 2 kt = 1 2 < m kv 2 k >= < K > = n < U > = n N 2N 2 < u > (98) (98) n = 1 < K >= 1 < U >= E (99) 2 1/2 1/2 M R < u > = 3 5 GmM R = 1 2 < mv2 > (100a) < v 2 > =< v 2 r > + < v 2 θ > + < v2 φ >= 3 < v 2 r > (100b) M = M virial 5R < v2 r > G (100c) < v 2 r > < v 2 > R
17 [1] A.D.Sakharov,Pizma ZhETF,5(1967)32. [2] M.Yoshimura ;Phys. Rev. Lett., 41(1978)281