Big Bang Planck Big Bang 1 43 Planck Planck quantum gravity Planck Grand Unified Theories: GUTs X X W X 1 15 ev 197 Glashow Georgi 1 14 GeV 1 2

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12 Big Bang 12.1 Big Bang Big Bang 12.1 1-5 1 32 K 1 19 GeV 1-4 time after the Big Bang [ s ] 1-3 1-2 1-1 1 1 1 1 2 inflationary epoch gravity strong electromagnetic weak 1 27 K 1 14 GeV 1 15 K 1 2 GeV quark confinement neutrino decoupling nucleosynthesis photon decoupling 3K 1-4 ev 12.1: 273

274 12 Big Bang Planck Big Bang 1 43 Planck Planck quantum gravity Planck Grand Unified Theories: GUTs X X W X 1 15 ev 197 Glashow Georgi 1 14 GeV 1 27 K Big Bang 1 36 12.2 Big Bang 1 36 1 32

12.1 Big Bang 275 radius of observed universe [ m ] 1 2 Standard "Big Bang" theory 1 1-2 1-4 1-6 inflationary theory inflationary period 1-4 1-3 1-2 1-1 time after the Big Bang [ s ] 12.2: 1 32 1 6 1 12 1 GeV 1 15 K W Z 1 6 1 13 K 1GeV 1 3MeV 1 1 K e + p n + ν e (12.1) e + e + Z ν x + ν x

276 12 Big Bang 1.9 K Big Bang 1 nucleosynthesis 1 9 K D p + n D+γ (12.2) A =5 A =8 4 He 25% 4 He 4 He 3 He 5 7 Li 7 Be 4 He 75% 4 He Coulomb A =5 8 15 4 He

Thomson 12.1 Big Bang 277 4 K 1% 38 K 3 K 1 5 Cosmic Microwave Background Radiation Big Bang

278 12 Big Bang 12.2 12.2.1 E f(e) f(e) = 1 ( ) E µ exp ± 1 kt T µ (12.3) k Bolzmann ± + Fermi Bose E m p E 2 = p 2 c 2 + m 2 c 4 (12.4) N U P S N = U = P = kt dp f(e) (12.5) (2π h) 3 dp Ef(E) (12.6) (2π h) 3 S = 1 kt dp log[1 f(e) ] (12.7) (2π h) 3 [ ] U + P µn (12.8) S Boltzmann g 1 2 g =2 1 g =2 T µ (1) kt mc 2 kt µ

12.2 279 (2) kt mc 2 kt mc 2 µ Boltzmann (3) µ mc 2 kt Fermi Fermi Bose Bose.5 MeV/c 2 MeV 12.2.2 kt mc 2 kt µ E 1 E = pc dp c 3 E2 de d p (12.9) 4π N U P N = U = n(e)de = u(e)de = E 2 de 2π 2 f(e) (12.1) ( hc) 3 E 2 de 2π 2 Ef(E) (12.11) ( hc) 3 P = p(e)de = kt E 2 de 2π 2 log[1 f(e) ] (12.12) ( hc) 3 n(e) E E +de u(e) p(e) Bose E x = E/(kT ) x 2 dx e x 1 = 2!ζ(3) ζ(3) = 1.22 56 93 (12.13) x 3 dx e x 1 = π4 15 (12.14) ζ N boson = ζ(3) (kt) 3 π 2 ( hc) 3 U boson = π2 (kt) 4 3 ( hc) 3 (12.15)

28 12 Big Bang Fermi Bose x n e x 1 xn e x +1 = 2xn e 2x (12.16) 1 x n dx e x 1 x n dx e x +1 = 2x n dx e 2x 1 = 1 2 n x n ( dx e x +1 = 1 1 ) x n dx 2 n e x 1 t n dt e t 1 (12.17) (12.18) n =2 n =3 Fermi Bose N fermion = 3 4 N boson U fermion = 7 8 U boson (12.19) 12.3 Bose u(e) Fermi T =1 15 K kt = 86 GeV u ( E ) [ 1 6 fm -3 ] 7 6 5 4 3 2 T =1 15 K boson fermion 1 2 4 6 8 1 energy E [ GeV ] 12.3:

12.2 281 N boson =1.1 1 7 T 3 m 3 U boson =3.78 1 16 T 4 Jm 3 (12.2) T K U boson N boson = π4 3 ζ(3) kt =2.7 kt U fermion N fermion = 7 6 E x = E/(kT ) x 2 log ex e x ± 1 dx = ± U boson N boson =3.15 kt (12.21) x 2 log ex ± 1 e x dx (12.22) ± [ ] x 2 log ex ± 1 1 e x dx = ± 3 x3 log ex ± 1 e x + 1 x 3 dx 3 e x ± 1 (12.23) 1 3 P P = 1 3 U (12.24) (12.24) S = 4 3 U kt (12.25) S boson = 2π4 45 ζ(3) N boson =3.6 N boson S fermion = 7 6 S boson (12.26) - N ( kt h ) 3 S (12.27)

282 12 Big Bang 12.2.3 U boson 12.1 quarks 7 8 6 2 3 2 = 63 massive leptons 7 8 3 2 2 = 1.5 neutrinos 7 8 3 2 1 = 5.25 photon 1 2 = 2 weak bosons 3 3 = 9 gluons8 2 = 16 6 flavor 2 3 2 1 2 W ± Z 1 Higgs 8 2 U standard = 15.75 U boson (12.28) 3 4 2%

12.2 283 12.2.4 1 n(e)de = 1 1 π 2 ( hc) 3 ( ) E 2 de (12.29) E exp 1 kt u(e)de = En(E)dE = 1 1 π 2 ( hc) 3 ( E exp kt ) E 3 de (12.3) 1 E N γ = 2ζ(3) (kt)3 π 2 ( hc) 3 =2.2 1 7 T 3 m 3 (12.31) U γ = π2 (kt) 4 15 ( hc) 3 =7.56 1 16 T 4 Jm 3 (12.32) u(e) λ u(e) (12.3) E λ λ λ +dλ u(λ)dλ = 4 hc λ 5 dλ ( ) 2π hc exp 1 kt λ u(λ) 12.4 (12.33) u(λ) λ max u(λ) λ max =.21452 hc kt =2.9 1 3 K T m (12.34) λ max 5 nm T = 2.9 1 3 5. 1 7 = 58 K (12.35)

284 12 Big Bang.25.2 u ( λ ) [ ev µm -4 ].15.1.5 8 K 7 K 6 K 5 K...5 1. 1.5 wave length λ [ µm] 12.4: 12.2.5 kt mc 2 µ E E = mc 2 + p2 2m (12.36) ±1 ( ) ( ) f(e) = exp mc2 µ exp p2 (12.37) kt 2mkT Fermi Bose ( ) p 2n exp p2 dp = 2mkT = (2n 1)!! 2 n 1 π 3 2 (2mkT ) n+ 1 2 ( ) 1 π 2 3 (mkt ) 2 n =1 2 2m 3kT 2 ( ) 1 π 2 3 (mkt ) 2 n =2 2 (12.38)

12.2 285 N U N = ( ) 3 ) mkt 2 exp ( mc2 µ 2π h 2 kt (12.39) U = (mc 2 + 3 ) 2 kt N (12.4) (12.21) P f(e) 1 log [1 f(e)] f(e) P = kt N (12.41) S [ ] 5 (2πkT)3/2 S = N + log 2 (2π h) 3 N (12.42) 12.2.6 1 8 K 1 kev E mc 2 R N U matter = Nmc2 R 3 (12.43) R 3 (12.32) T 4 kt = hν = hc (12.44) λ R 4 1 m 3 1 U matter (today) = 1.67 1 27 (3. 1 8 ) 2 =1.5 1 1 Jm 3 (12.45)

286 12 Big Bang 2.7 K (12.32) U γ (today) = 7.56 1 16 2.7 4 =4. 1 14 Jm 3 (12.46) 4 R R 3 R 4 f U matter = f 3 U matter (today) U γ = f 4 U γ (today) (12.47) U matter = U γ f = 4. 1 14 1.5 1 1 =3 1 4 (12.48) T = 3K 3 1 4 =15 K (12.49)

12.3 287 12.3 Big Bang BBN Big Bang Nucleosynthesis Primordial Nucleosynthesis primordial Big Bang D 3 He 4 He 7 Li 4 He/H.8 7 Li/H 1 1 9 (1) (2) η = N baryon N γ 5 1 1 (12.5) p + p D+e + + ν e B/A 12.3.1 t<1s p + e n + ν e n + e + p + ν e (12.51) n p + e + ν e (12.52)

288 12 Big Bang kt > MeV Boltzmann Boltzmann n p exp ( mc2 kt ) mc 2 =(m n m p )c 2 =1.293 MeV (12.53) kt n/p 1 n N n t 1s kt < mc 2 (12.51) mc 2 n p = exp fr freeze-out ( mc2 kt fr ) 1 6 (12.54) T fr (12.51) Fermi 1.9 K 2.7 K t>1s (12.52) D = 2 H p + n D+γ (12.55) B(D) = 2.23 MeV kt.1 MeV 1 τ n = 882 s

12.3 289 n p 1 7 (12.56) 12.3.2 (12.55) 4 He D+n 3 H+γ 3 H+p 4 He + γ D+p 3 He + γ 3 He + n 4 He + γ (12.57) D+D 3 H+p 3 H+D 4 He + n D+D 3 He + n 3 He + D 4 He + p (12.58) 4 He 2p +2n 4 He + γ (12.59) 1:7 4 He 1:12 4 He Y p = 4 4+12 =.25 (12.6) Big Bang kt 1 MeV NSE nuclear statistical equilibrium i fraction X i = A i N i N baryon = A i (N i /N H ) 1+ i A i (N i /N H ) (12.61)

29 12 Big Bang A i N i N baryon X i = g i [ ζ(3) A i 1 π (1 A i)/2 2 (3A i 5)/2 ] A 5/2 i ( kt ) 3(Ai 1)/2 m N c 2 η A i 1 X Z i p B i i m N ( ) (12.62) X A Bi i Z i n exp kt i Γ i Λ i i t dn i dt = 3HN i + Λ i Γ i N i (12.63) H =ȧ(t)/a(t) Hubble i i n p D 3 H 3 He 4 He 6, 7, 8 Li 7, 9 Be 16 O (12.52) (12.57) (12.58) A =7 3 He + 4 He 7 Be + γ 7 Be + n 7 Li + p (12.64) 7 Li + p 2 4 He Coulomb 5 8 4 He 5 He 5 He 4 He 4 He 8 Be 8 Be 4 He 12.5 [3] Big Bang t 18 s 1 6 9 7 Be 53.29 3 H 12.33 7 Be + e 7 Li + ν e 3 H 3 He + e + ν e (12.65)

12.3 291 Big Bang D, 3 He, 4 He, 7 Li 12.3.3 Big Bang η 12.6 [2] D D+D D 4 He 7 Li 7 Be 7 Li Helium-4 Big Bang 4 He 4 He HII 4 He Big Bang 4 He Y p =.238 ±.2 (stat) ±.5 (sys) (12.66) HII Lithium-7 7 Li 1/3 Li Li Fe Li Big Bang Li ( ) Li/H p = 1.23 ±.6 +.68+.56.32 1 1 (12.67) +.56 Li Li Li-Fe 6 Li D Big Bang Big Bang D QAS quaser absorption system D QAS D/H =(3. ±.4 (stat)) 1 5 (12.68)

12.3 293 D/H =(1.5±.1 (stat)) 1 5 D D/H < 9.7 1 5 12.6 Helium-3 3 He HII Big Bang 3 He Big Bang η η 12.6 4 He D 7 Li 2.6 η 1 6.2 η 1 = η 1 1 (12.69) 1 9 Big Bang η (12.69) Ω b =3.65 1 3 h 2 η 1 (12.7).95 Ω b h 2.23 (12.71) Ω b 1 Ω lum.24 h 1 (12.72) Ω lum Ω b 1 6 K X Ω matter.3 Dark Matter Big Bang Big Bang η η

294 12 Big Bang 12.4 12.4.1 1 Hubble Big Bang George Gamow 9 Big Bang Gamow Ralph Alpher Robert Herman 5K 1965 Bell Arno Penzias Robert Wilson 4 GHz 7.35 cm 3.5 ±1. K Gamow, Alpher, Herman Planck Big Bang 1 5 3 K Planck T I ν = 2hν3 1 c 2 ( hν exp kt ) 1 (12.73) T =3K Planck ν 5 GHz λ 2mm COBE Cosmic Background Explorer Satellite FIRAS Far-Infrared Absolute Spectrophotometer 1 4 1 GHz 1 GHz 3 COBE FIRAS T γ N γ U γ T γ = 2.725 ±.2 K (95% CL) N γ = 2 ζ(3) π 2 Tγ 3 U γ = π2 15 T γ 4 411 cm 3 4.64 1 34 gcm 3.26 ev cm 3 (12.74)

12.4 295 12.4.2 Penzias Wilson 1% isotropic unpolarized 1 5 anisotropic T T (θ,φ) = a lm Y lm (θ, φ) (12.75) lm θ 1/l l a lm 2 (12.76) 4π l =1 m T T =1.23 1 3 ( l = 1 ) (12.77) Doppler T β = v/c Doppler T (θ) = T ( ) 1 β 2 1 β cos θ = T 1+βcos θ + β2 2 cos2θ + O(β3 ) (12.78) (11.2 ±.1) h ( 7.22 ±.8) v = 371 ±.5 kms 1 (12.79) Local Group of galaxies v LG = 627 ± 22 km s 1 (12.8) Q rms Q 2 rms T 2 γ = 1 4π a 2m 2 (12.81) m

296 12 Big Bang µk 1 6 K 4 µk Q rms 28 µk (12.82) COBE l >2 BOOMERanG MAXIMA-1 DASI CBI 1 5 12.7 [4] [ ( +1)C /2 π ] 1/2 [ µ K] 1 7 5 3 2 Angular Scale [Degrees] 1 5 2 1 5 2 1.5.2.1.5 n=1 H=5 CDM+1%B COBE QMASK FIRS MAX TEN MAXIMA IAC Pyth HACME Pyth V SP MSAM ARGO SK IAB TOCO97 QMAP TOCO98 ARCHEOPS BOOM-LDB CAT OVRO WD DASI SuZIE BIMA CBI VSA 1 2 5 1 2 5 1 1 5 eff Ned Wright - 1 Nov 22 12.7: l 12.7 l 2

BOOMERanG 12.4 297 Ω tot =1.2 ±.6 (12.83) Ω tot =1 Ω b h 2 =.22 ±.4 (12.84) Big Bang Big Bang

298 12 Big Bang 12.5 12 1. 2. K. Hagiwara et al., Phys. Rev. D66, 11-1 (22) (Particle Data Group) http://pdg.lbl.gov/22/bigbangnucrpp.pdf http://pdg.lbl.gov/22/microwaverpp.pdf 3. A Knowledgebase for Extragalactic Astronomy and Cosmology http://nedwww.ipac.caltech.edu/level5/ 4. Ned Wright scosmology Tutorial http://www.astro.ucla.edu/ wright/ 5. G. Gamow, Phys. Rev. 7 (1946) 572, R. Alpher, H. Bethe and G. Gamow, Phys. Rev. 73 (1948) 83 R. Alpher and R.C. Herman, Phys. Rev. 75 (1949) 189 6. Big-bang nucleosynthesis enters the precision era, Rev. Mod. Phys. 7 (1998) 33 7. COBE: http://space.gsfc.nasa.gov/astro/cobe/