2 1 ds 2 = a 2 (η) ( dη 2 + γ ij dx i dx j ) (1.2) ( dt ) conformal time η η = a(t) a(t) (scale factor) t =const (3) R ijkl = K a 2 (t) (γ ikγ jl γ il

Transcription

1 1 1.1 Robertson-Walker [ ds 2 = dt 2 + a 2 dr 2 ] (t) 1 Kr 2 + r2 (dθ 2 + sin 2 θdφ 2 ) sin 2 χ = dt 2 + a 2 (t) dχ 2 + χ 2 (dθ 2 + sin 2 θdφ 2 ) = dt 2 + sinh 2 χ a 2 (t) (1 + K 4 r2 ) [d r 2 + r (dθ 2 + sin 2 θdφ 2 )] = dt 2 + a 2 (t) [dx 2 + dy 2 + dz 2 = dt 2 + a 2 (t)γ ij dx i dx j r 2r Kr 2 +1 K = 0 1 (1.1) K(xdx + ydy + ] zdz)2 1 K(x 2 + y 2 + z 2 )

2 2 1 ds 2 = a 2 (η) ( dη 2 + γ ij dx i dx j ) (1.2) ( dt ) conformal time η η = a(t) a(t) (scale factor) t =const (3) R ijkl = K a 2 (t) (γ ikγ jl γ il γ jk ) (1.3) (3) R = 6K a 2 (t) (1.4) K = ±1 a(t) R curv (t) 1 (1.1) 1.2 (χ 1, 0, 0) d(t) = χ1 0 gχχ dχ = a(t)χ 1 (1.5) v(t) d(t) = ȧ(t)χ = ȧ(t) d(t) H(t)d(t) (1.6) a(t) v = Hd a(t) H H 1 0 H 0 = 100h km/s/mpc h 0.7 1Mpc 1pc = 3.26 Mpc

3 1.3 3 null geodesic ds 2 = 0 (χ 1, 0, 0) t 1 t 0 ds 2 = dt 2 + a 2 (t)dχ 2 = 0 t0 t 1 dt χ1 a(t) = dχ = χ 1 0 t 1 + δt 1 t 0 + δt 0 t0+δt 0 t 1 +δt 1 dt a(t) = χ 1 t0 +δt 0 t 0 dt t1 +δt a(t) = 1 dt t 1 a(t) δt 0 a(t 0 ) = δt 1 a(t 1 ) λ 0 = δt 0 = a(t 0 > 1 if H > 0 (1.7) λ 1 δt 1 a(t 1 ) = z 1 λ 0 λ 1 λ 1 = a(t 0) a(t 1 ) z = a(t 0) a(t) (1.8) 1 + z a(t) z = a(t 0) a(t 1 ) a(t 1 ) ȧ(t 0) a(t 0 ) (t 0 t 1 ) ȧ(t 0 )χ = v(t 0 ) = H 0 d(t 0 ) (1.9) 1.3

4 (luminosity distance) L s energy erg/s d flux F = L s 4πd 2 Minkowski d L L s flux F 0 d 2 L L s 4πF 0 (1.10) δt s δe s (r s, 0, 0) χ r metric δt s δt 0 δe 0 δe 0 = δe s 1 + z, δt 0 = (1 + z)δt s (1.11) L s = δe s /δt s L 0 L 0 = δe 0 L s = δt 0 (1 + z) 2 (1.12) t = t 0 4π(a 0 r) 2 4π metric [ ds 2 = dt 2 + a 2 dr 2 ] (t) 1 Kr 2 + r2 }{{ dω 2 } flux F 0 = L 0 4π(a 0 r) 2 = L s 4π(a 0 r) 2 (1 + z) 2 = L s 4πd 2 L

5 1.3 5 d L = a 0 r(1 + z) (1.13) luminosity distance (angular diameter distance) D θ d A D θ (1.14) D t s r θ 2π = D 2πa s r d A = a s r = d A = a 0r 1 + z (1.15) d L (1 + z) 2 (1.16) luminosity distance (1.9) d L = z/h 0 d L = 1 [z + 1 ] H 0 2 (1 q 0)z 2 + O(z 3 ) q 0 a 0ä 0 ȧ 2 0 (1.17)

6 6 1 (deceleraton parameter) 2 (1.17) t0 t dt a(t ) z = a 0 a(t) 1 = a 0 a 0 + ȧ 0 (t t 0 ) + ä0 2 (t t 0) 2 1 = ȧ0 a 0 (t 0 t) ä0 2a 0 (t 0 t) 2 + ȧ0 a 0 2 (t 0 t) 2 + = H 0 (t 0 t) + H 2 0 ( q 0)(t 0 t) 2 + = = r 0 t0 t dr = 1 Kr 2 sin 1 r r sinh 1 r 1 [ 1 ȧ0 ] (t t 0 ) dt a 0 a 0 = r + r3 6 r r r3 6 = 1 (t 0 t) + ȧ0 (t 0 t) 2 = r ( ) a 0 2a 2 0 a 0 r = t 0 t + 1 ȧ 0 2 d L = [ z a 0 r(1 + z) = 1 ( H 0 H 0 = 1 [z + 1 ] H 0 2 (1 q 0)z 2 + (t 0 t) 2 = t 0 t + 1 a 0 2 H 0(t 0 t) q ) 0 z z 2] (1 + z) 2 2H 0 q 0 < 0 z

7 ρ(t) p(t) T0 0 = ρ, Ti 0 = 0 ( ) Tj i = pδj i ( ) (1.18) EM T µν = pg µν + (p + ρ)u µ u ν (1.19) u µ = (1, 0, 0, 0) T ν µ = diag( ρ, p, p, p) ρ p 3 R-W [ä (ȧ )2 R = 6 a + + K ] a a 2 (1.20) 00 ii (1.22) (1.21) (ȧ a )2 + K a 2 = 8πG 3 ρ (1.21) (ȧ )2 2ä a + + K = 8πGp (1.22) a a2

8 8 1 { > 0 ρ+3p < 0 ä a = 4πG (ρ + 3p) (1.23) 3 (decelerate) (accelerate) T µν ;ν = 0 ρ = 3(ρ + p)h (1.24) dρa3 dt = p da3 dt de = pdv ρ, p de = T ds pdv (+µ i dn i ) ds = 0 (1.25) P = P (ρ) (1.24) ρ a (1.21) a(t) p = wρ (w = const) (1.24) ρ a 3(1+w) (1.26) w = 1 3 ρ a 4 ρ r (massless ) w = 0 ρ a 3 ρ m (dust) w = 1 ρ = const ρ V

9 1.5 9 (ȧ a )2 + K a 2 = 8πG ρ K = 0 (1.26) 3 a(t) t 3(1+w) 2 e H V t w 1 w = 1 (1.27) (H V = 8πGρ V /3) 1.5 (1.21) H = ȧ/a K a 2 = 8πG 3 ρ H2 (Ω 1)H 2 (1.28) Ω 8πG 3H 2 ρ ρ ρ cr ρ cr = 3H2 9πG ρ cr0 = 3H2 0 8πG = h 2 g/cm 3 Ω i = ρ i Ω r, Ω m = Ω c (CDM) + p cr Ω b (baryon), Ω Λ = λ ρ V Λ Ω i 1 K 0 i Λ = 8πGρ V (1.29) Ω K K a 2 H 2 (1.30)

10 10 1 Ω m + Ω r + Ω Λ }{{} λ = +Ω K = 1 (1.31) Λ 3H 2 (1.27) Ω K = Ω Λ = Ω r = 0, Ω m = 1 Einstein de Sitter a(t) t 2 3 H = 2/3t (1.21) t 0 = 2 3H 0 66 (1.32) a dt t(z) = 0 da da = dz z (1 + z)h(z) = 1 dz H 0 z (1 + z) [Ω m0 (1 + z) 3 + Ω r0 (1 + z) 4 + Ω K0 (1 + z) 2 + Ω Λ0 ] 1/2 z = 0 t 0 Ω r = 0 Ω K = 0 (1.33) t 0 = 1 H 0 0 dx (1 + z) 1 Ω m0 }{{} Ω Λ0 +Ω m0 (1 + z) 3 (1.34) Ω Λ Einstein de Sitter

11 r dr t0 dt da z = 0 1 Kr 2 t }{{} s a(t) = aȧ = dz 0 H(z)a 0 K sinh 1 ( Kr) = 1 z dz a 0 H 0 0 [Ω m0 (1 + z) 3 + Ω r0 (1 + z) 4 + Ω K0 (1 + z) 2 + Ω Λ0 ] 1/2 (1.35) t s : { 1 ΩK z } dz a 0 r = sinh H 0 ΩK 0 [Ω m0 (1 + z) 3 + Ω r0 (1 + z) 4 + Ω K0 (1 + z) 2 + Ω Λ0 ] 1/2 1 H 0 S K (z) (1.36) d L (z) = 1 H 0 S K (z)(1 + z), d A (z) = S K(z) H 0 (1 + z) (1.37) 1.6 CMB Ω r0 = Ω γ0 + Ω ν0 = Ω m0 = 0.27 a eq a = 1 z eq = T > T eq = 9197K = eV

12 z = (ρ V /ρ m0 ) = 0.47 z eq = T β 1 f BF (ω) = 1 e β(ω µ) 1 n ρ B, F n = g ρ = g d 3 q (2π) 3 f(ω) = q 2π 2 d 3 q (2π) 3 ωf(ω) = E = ω = m 2 + q 2 g 2π 2 m m q (1.38) def(e)e(e 2 m 2 ) 1 2 (1.39) def(e)e 2 (E 2 m 2 ) 1 2(1.40) x =const v x = q x /ω 2q x p = g q x>0 d 3 2q 2 x q 2 (2π) 3 ω f(ω) = g 3ω f(ω) = g 6π 2 Ξ(β, µ) Ω = T ln Ξ = pv = T V g m d 3 q (2π) 3 ln [1 e β(ω µ) ] 1 def(e)(e 2 m 2 ) 3 2(1.41) d 3 q p = ±gt ln[1 ± f(ω)] (1.42) (2π) 3 = ± g (E 2 m 2 ) 1 2 E ln [1 e β(e µ) ] de β = ± g [ (E 2 m 2 ) 3 ] 2 ln [1 e β(e µ) ] de 3β

13 = g (E 2 m 2 ) 3 e β(e µ) 2 de 3 1 e β(e µ) = (1.41) Gibbs µ = 0 G = E T S + P V = µn (1.43) s = β(ρ + p µn) (1.44) T ds = de + pdv = d(ρv ) + pdv = ρdv + V dρpdv = (ρ + p)dv + V dprho ρ = ρ(t ) T ds = ρ + p T dv + V T S V = ρ + p T = 1 T dp dt = ρ + p T dρ dt dt, S T = 1 T ( ρ + p ) = ( V T V T dρ dt + 1 T dρ dt dρ ) = 1 dt T dp dt ρ + p T 2 S = V (ρ + p) T ds = ρ + p T = ρ + p T dv V ρ + p T 2 dt + V T dv V ρ + p T 2 dt + 1 T ( dρ dt + dp ) dt dt (ρ + p)dv dt dt + V (ρ + p)dt T 2

14 14 1 = 0 OK V = a 3 d Q = de + pdv = V dρ + 3V ρ da a + 3V pda = 0 (1.45) a dρ 1 (1.24) = 3(ρ + p)da dt dt a (1.46) d Q = T ds f(ω) = e β( µ) ( mt ) 3 2 n = g e β(m µ) (1.47) 2π ρ = (m + 3 ) 2 T n (1.48) P = nt ρ (1.49) ε = ρ n = m T (1.50) ( m µ s = + 5 ) n T 2 = m µ n (1.51) T (k B T mc 2 ) µ T n B = ζ(3) π 2 gt 3, n F = 3 ζ(3) 4 π 2 gt 3 (1.52) ρ B = π2 30 gt 4, ρ F = π gt 4 (1.53) ρ n ε B = π4 30ζ(3) T 2.70T, ε F = 7π4 180ζ(3) T 3.15T(1.54) P = 1 3 ρ (1.55) s = ρ + P T = 4ρ 3T (1.56)

15 B F ζ(3) = 1 2 n 3 n=1 ρ r = π2 30 g (T )T 4, g (T ) = 0 i boson g i x 2 dx e x 1 = = j ferumion g j (1.57) s = 4π2 90 g (T )T 3 (1.58) 3 (T 100GeV) g = T i ρ r = π2 30 g (T )T 4 g (T ) = i b s = 4π2 90 g s(t )T 3 g s (T ) = i b g g s ( )4 Ti g i + T j f ( )3 Ti g i + T j f T T eq ρ tot = ρ r = π2 g 30 T 4, a(t) t 1 2 ( )4 Tj g j (1.59) T ( )4 Tj g j (1.60) T ( 1 H 2 = = 2t)2 8πG 3 ρ r = 8π3 g (T ) 90M 2 pl T 4 M pl = GeV t = 1 ( 90 ) 2 M ( pl T ) 2 32π 3 g T 2 sec (1.61) 1MeV M pl = 1 2 c 5 2 G 1 2 = G 1 2 = GeV = g

16 16 1 t pl = 1 2 G 1 2 c 5 2 = sec l pl = 1 2 G 1 2 c 3 2 = cm ρ pl = c 5 1 G 2 = g/cm 3 = M 4 pl (1.62) 1.8 Γ H (1.63) τ 2t (1.64) Γ 2 = nσc NT 3 π 2 α 2 N Γ 2 T T 2 (1.65) H T 2 T Γ 2 < H Γ H ( T α 2 ( N ) ( g ) ) 2 GeV (1.66)

17 (i) (kinetic equilibrium) (eγ eγ ) Boson Bose Fermion Fermi f(ω) = 1 e (ω µ)/t 1 (1.67) i µ i (ii) (chemical equilibrium) abc ijk µ a + µ b + µ c + = µ i + µ j + µ k + f(ω) = 1 e (ω µ i)/t 1 (iii) (Thermal equilibrium) 0 0 f(ω) = (asymmetry) µ + µ = 0 1 e ω/t 1 fermion n q n q = g 2π 2 m (1.68) q q γγ E(E 2 m 2 ) 1 [ 1 2 de e ( E µ)/t ] e ( E + µ)/t + 1

18 18 1 = gt 2 [π 2 µ ( µ ) 3 ] 6π 2 T + (T m ) T ( mt ) 3 2 2g e m ( m ) T sinh (T m ) 2π T (1.69) 1.9 f(p µ, x µ ) = f(e, t) = f(p, t) Boltzmann eq. ˆL[f] = C[f] (1.70) ˆL:Liouville ˆL NR = D Dt = t + dx dt + dp dt p = t + v + F m v ˆL R = dxα dτ = p α x α + dpα dτ p α x α Γ βγp α β p γ p α = E t Hp2 E n(t) = g τ : Affine parameter E = m 2 + p 2 (1.71) d 3 p (2π) 3 f(p) 1 E L[f] = 1 E C[f]

19 dn dt + 3Hn = g d 3 p (2π) 3 C[f] (1.72) E 5 (1.71) (1.72) ψ ψ + a + b + + d i + j + + l (1.72) g d 3 p ψ (2π) 3 C[f] = E ψ dπ ψ dπ a dπ d dπ i dπ l (2π) 4 δ 4 (p ψ + p a + + p d p i p j p l ) [ M 2 f ψ f a f b f d (1 ± f i )(1 ± f j ) (1 ± f l ) ] M 2 f i f j f l (1 ± f ψ ) (1 ± f d ) (1.73) M 2 transition amplitude s matrix element T invariance M 2 = M 2 (1 ± f s) a s n >= 1 ± ns n s + 1 > induced emission Pauli blocking dπ s = g d4 (2π) 4 2πδ(p2 s m 2 s) = g d3 p 1 (2π) 3 (1.74) 2E p f s (E s ) = 1 e β(e s µ s )/T 1 1 ± f s = f s e β(es µs) (1.72) (1.73) ṅ ψ + 3Hn ψ = dπ ψ dπ a dπ d dπ i dπ l M 2 f ψ f a f d f i f l (2π) 4 δ 4 (p ψ p a + p i p l ) [ e β(e ψ µ ψ ) e β(ea µa) e β(e d µ d )

20 20 1 ] e β(e i µ i) e β(e l µ l ) (1.75) E ψ + E a + + E d = E i + + E l µ ψ + µ a + µ d = µ i + µ j + + µ l 1 ± f i Boltsmann eq. ṅ ψ + 3Hn ψ = dπ ψ dπ a dπ d dπ i dπ j dπ l (2π) 4 δ(p ψ + p a + + p d p i p l ) M 2 [f ψ f a f b f d f i f j f l ] (1.76)

21 1.10 Freeze out of pair annihiration Decoupling Freeze out of pair annihiration Decoupling ψ ψ χ χ χ χ (e.g. ) f eq χ = fχ eq 1 = e βex 1 (CPT ) ( )ψ: neutrino χ: charged lepton (1.84) n ψ + 3Hn ψ = dπ ψ dπ ψdπχdπ χ(2π) 4 δ 4 (p ψ + p ψ p χ p χ ) M 2 [f ψ f ψ (1 ± f eq χ )(1 ± f eq χ ) f eq χ f eq χ (1 ± f ψ )(1 ± f ψ)] ψ ψ kinematic equilibrium dπ (1.87) ψ ψ kinetic equilibrium dπ (1.87) dn ψ dt σ ψ ψ χ χ v = + 3Hn ψ = σ ψ ψ χ χ v (n 2 ψ n eq2 ψ ) (1.78) 1 n eq ψ dπ ψ dπ ψdπ χ Π χ (2π) 4 δ 4 (p ψ + p ψ p χ p χ ) M 2 e Eψ/T E e ψ /E σ ann v n ψ (1.77)

22 22 1 dn ψ dt + 3Hn ψ = σ ann v (n 2 ψ n eq2 ψ ) (1.79) Y ψ n ψ /s x m ψ /T (1.78) x Y eq dy ψ dx = Γ ann H [( Y ψ Y eq ) 2 1] (1.80) Γ ann n eq ψ < σ ann v > ( ) 45ξ(3) g 1 Boson Y eq (x) = 2π 4 g 3/4 Fermion 45 2π 4 (π 8 ) 1 g 2 x 2 3 e x g T m ψ T m ψ (1.81) T m ψ n eq ψ T 3 σ ann v T 1 Γ ann T 2 T, Γ ann /H < 1 hot dark matter, ν Γ ann > H freeze out n eq ψ Γ ann Γ ann < H σ ann v freeze out < σ v > CDM ( )SUSY dark matter Γ = H x x f

23 1.10 Freeze out of pair annihiration Decoupling 23 x x f Y eq x x f Y eq (x f ) (1) hot relics x f z Y Y (x ) Y eq (x f ) = g ( 1 Boson g s 3/4 Fermion s 0 = cm 3 ( ) n ψ0 = s 0 Y = g ( 1 g s 3/4 ρ ψ0 = 8.1 g ( 1 g s 3/4 Ω ψ0 = 0.15 ( g 1 g s (x f ) 3/4 ) cm 3 ) ( m ) evcm 3 1eV ) ( m ) 1eV (1.82) ) 3 Ω ν0 h 2 = 94.1eV i m νi (2) cold relics x f 3 σ ann v ( ) x = x f ( Γ ann = n eq 8π ψ σ π 2 ) 1 ann v = H = 3Mpl 2 90 g Tf 4 2 (1.83)

24 24 1 Y (x f ) = neq ψ (x f ) = s(x f ) 90 4πg s T 3 f H σ ann v = ( 45g ) 1 2 x f π g s σ ann v M pl m ψ (1.84) x f (1.92) x f = ln[0.038 g g 1/ ln{ln[0.038 g M pl m ψ σ ann v ] f 1/2 n ψ0 = g 1/2 x f σ ann v cm 3 g s M pl m ψ ρ cro = evcm 3 M pl m ψ σ ann v ]} ( ρcro = g/cm 3 1eV = g x f Ω ψ0 = m ψn ψ0 = 2.0g1/2 ρ cro g M pl σ ann v ev 1 = g1/2 x ( f σ ann v g s cm 3 /s ) 1 m ψ! (1.85) 1.11 Distribution function after decoupling Γ H: Γ < H

25 1.11 Distribution function after decoupling 25 RW u du dt = Hu p massless T dec T dec f(p, t dec ) = 1 e p/t dec 1 p(t) = p(t dec ) a(t dec) a(t) m a 3 f(p, t) = d3 n dp 3 = f(p a a dec, t dec ) = T = T dec a dec a 1 pa exp( ) 1 (1.86) T dec a dec ( ) ( f(e) exp E µ ) dec T dec (1.87) a 2 a 3 T (t) = ( a a dec ) 2 Tdec, (1.88) ( a ) 2 µ(t) = m + (µ dec m) a dec (1.89)

26 26 1 (1.87) 1.12 decoupling ν Z 0 G F = GeV 2 Γ G 2 F T 5 H = 5.4 T 2 M pl Γ ( H = T 1.5MeV )3 (1.90) T < 1.5MeV ν decooupling (1.86) T 0.5MeV(= 511keV) e + e ( ) L e ± 9 R 8 = = a 3 (before)10.75t 3 νaa 3 = (after)2t 3 γaa T 3 ν a 3 T νb = T νa ν decouple e ± heat up T 3 νba 3 = T 3 νaa 3 T νa = ( 4 ) 1 3 Tγa (1.91) 11 ( 4 ) 1 3 T ν0 = 2.73 = 1.95K 11

27 g 0 = ( ) = 3.36 (1.92) g s0 = = (1.93) ((1.59),(1.60) ) 1.13 (recombination) 13.6eV 16 K n γ n B p + e H + γ T < m i (1.47) ( mi T ) 3 i 2 n i = g i e m i µ i T 2π i = p, e, H µ p + µ e = µ ψ n H = g ( ) H me T 3 2 n p ne e B T g p g e 2π B = m p + m e m H = 13.6eV (1.94) n p = n e n p + n h = n b n b = ηn γ (η ) X e n p /n b 1 X e X 2 e = 4 2ζ(3) ( T ) 3 2 B η e T (1.95) π m e decoupling (1.72) (1.78)

28 28 1 n n e + 3Hn e = σv (n eq e n eq p H n γ n eq H nrq γ n e n p ) σv recombination cross section p + e H + γ ( 2 σv 1 ( ) 0 ) = n σv (n eq e n eq ) p H n eq n 2 e H = ( met ) 3 2 n b σv [(1 X e ) e B T Xe 2 n b ] 2π (1.96) dx e = β(1 X e ) α (2) n b Xe 2 (1.97) dt ( me T ) 3 2 β σv e B T ionization rate 2π α (2) σv recombination rate α (2) (n = 1) n = eV γ (excited state) α (2) = 9.78 α2 ( B ) 1 2 ( B ) ln m 2 e T T (1.98) decoupling γ H γ : n e σ T = X e n b σ T σ = cm 2 Thomson n e σ T = XeΩ b h 2 ( a a 0 ) 3 cm 1

29 n e σ T H ( Ωb h 2 ) ( Ωm h 2 = 113X e ) 1 2 ( 1 + z 1000 ) 3 2 [ ( 1 + z 1000 ) ( Ωm h ) 1 ] 1 2 (1.99) X e 10 2 decoupling z dec = (WMAP7) (1.100) cf(1.99) X e = 1 n eσ e H 2 ( 0.02 ) 3 ( Ω m h z = 43 Ω b h < 1 z ) 1 3 (1.101) γ

30 ) particle horizon d h (t) t ( ) ( r H ) ds 2 = dt 2 + a 2 dr 2 (t) 1 Kr 2 = 0 (null) d H (t) }{{} rh 0 grr dr = a(t) rh 0 dr t = a(t) dt 1 Kr 2 t i a(t ) (2.1) a(t) t m d H (t) = t [ 1 1 m ( ti )1 m] = t t 1 m (m < 1 ) (2.2) 2) ( ) t H 1 H 1 = ȧ a = t m for a(t) tm (2.3) m < 1 t a(t) t m

31 2.15 ( ) 31!: 2.15 ( ) (ȧ a ) + K }{{} a 2 = 8πG 3 ρ = 8π 3M 2 ( ρ γ + ρ m + ρ pl }{{}}{{}}{{} Λ a 2 a 4 a 3 a 0 ) z > 0.47 MD RM a ρ K/a 2 K/a2 ρ a or a 2 Ω = (WMAP7+BAO+H 0) K a 2 = H2 (Ω 1) (1.28) R curv = a(t) = H0 1 Ω tot H0 1 K a Ω tot (t) 1 = 1 8πGρ / K 0 a 0 3 a ( a(t) )2 ( aeq ) ( a0 ) 2 1 a eq a m a m (2.4) a m (DM) (ΛD) Ω(t pl ) (2.5)

32 32 2 R curv (t pl ) l pl 2.16 High temperature symmetry restoration and phase transitons SU(3) SU(2) U(1) Higgs SU(2) ϕ L = 1 2 ( ϕ)2 V [ϕ] (2.6) }{{} V [ϕ] = λ 4 (ϕ2 µ2 λ )2 = g µν µ ϕ ν ϕ }{{} = ϕ a 2 ( ϕ)2 RW = λ 4 ϕ4 1 2 µ2 ϕ 2 + µ4 4λ = ϕ 2 + ( ϕ) 2 }{{} δl δϕ = 0 ϕ V [ϕ] }{{} = 0 (2.7) = λϕ 3 µ 2 ϕ ϕ σ ϕ φ ϕ = σ + φ φ = 0 (2.8)

33 2.16 High temperature symmetry restoration and phase transitons 33 δl δϕ = 0 σ + µ 2 σ λ (σ + φ) 3 = σ + µ 2 σ λ (σ 3 + 3σ 2 φ + 3σφ 2 + φ 3 ) φ(x, t) = σ (λσ 2 µ 2 )σ 3λ φ 2 σ 3 φ 3 = 0 (2.9) d 3 k 1 [ φ(x, t) = ak e ikx iωkt + a (2π 3/2 k ) e ikx+iω kt ] (2.10) 2ωk [ ak, a ] k = δ(k k ), ω k = k 2 + m 2 ϕ (m ϕ; ϕ = σ mass) φ 3 operator 3 ( coherent state coherent ) φ 2 = d 3 k 1 (2π) 3 (2 a k 2ω a k k }{{} occupation +1) (2.11) = 0 (vacuum) occupation 0 = 0 φ 2 φ 3 σ =const translational invariance respect σ = 0, ± µ λ σ = 0 σ = ± µ λ = β (β = T 1 ) (

34 34 2 ) ( ) T m ϕ β φ 2 β = (2.9) β a k a k β = 1 e βω k 1 (2.12) d 3 k 1 2 (2π) 3 2ω k e βω T 2 k 1 12 (2.13) σ (λσ 2 + λ 4 T 2 µ 2 )σ = o (2.14) T > sµ λ σ =const σ = 0 : }{{} ϕ ϕ λ 8 T 2 ϕ 2 (T 4 ϕ ) ϕ = 0 ϕ = σ = ± µ λ ( ) T 2µ λ T c H 1 c ± µ λ ( 8πG π 2 ) g Tc 4 2 ϕ = ϕ(x, t) ϕ = + µ λ µ λ ϕ = 0 (V (ϕ = 0) = µ4 ) 4λ

35 2.16 High temperature symmetry restoration and phase transitons 35 ± µ λ U(1) ( U(1) ) ( ) U(1) Π 0 (M) 1 U(1) Π 1 (M) 1 U(1) Π 2 (M) 1 SU(2) Π 3 (M) 1 Π n (M), n = 0: 1: 2: M M : SU(3) SU(2) U(1) cf ( h + ) H = h 0 H = 0 V [H] = µ 2 H + H + λ(h + H) 2 + const µ 2 < 0 L Higgs = ( σ ) a µ ig 2 W aµ 2 ig 2 1B µ V [H] + f l L ij i Hl j + f D Q ij i Hq j i,j=e,µ,τ i.j=d,s,b

36 36 2 H = iσ 2 H L = ( ) ν e + f U Q ij i Hqj i,j=u,c,t h 0 = L ( ) u, Q = d M 2 4λ, h = 0 L, (2.15) m = f h 0, M w = g 2 2 h 0, M 2 = 1 2 g g2 2 h2 2.17? (historical background) 1970 M GUT = GeV M pl = GeV 3 ρ = V [0] =const (ȧ a )2 + K a 2 8πG 3 ρ = 8πG 3 V (0) = H2 inf (2.16) a e H inft K a

37 t dt d H (t) = a(t) t i a(t ) = 1 H eh(t t i) (2.17) ρ =const ρ a n a(t) t 2 n (1.27) n = 3(1 + w) n < 2 (2.2) a(t) ρ total = ρ inf (t) + ρ m (t) + ρ γ + (2.18) ρ inf (t) ρ γ (t): Fiedmann (a 2 ) ρ inf (t) Hot Big Bang Cosmology r = 2π k 2 t k k t k ρ inf t s t f ( )

38 38 2 ρ inf ( T R ρ inf = ρ r = π2 ( 30 g TR 4 g ) 60 TR 4 (2.19) 200 g T 3 a 3 g s0 = 3.91, T CMBO = 2.735K a 0 = T ( R g ) 13 ( g ) 1 3 T R = 3.7 (2.20) a R T CMB T CMB H 1 inf L 0 H 1 inf eh inf(t f t s ) a 0 a R ( = g ) 1 ( ) 4 H 1 inf 2 e H inf (t f t s ) GeV 1 GeV ( = g ) 1 ( ) 4 H 1 inf 2 e H inf (t f t s ) Mpc (2.21) GeV ( 8πGρinf ) 1 2 H inf = L 0 500H 1 0 e N N N H inf (t f t s ) > ( 2 ln Hinf ) N min (2.22) GeV

39 (1.28) K a 2 = H2 (Ω 1) Ω(t 0 ) 1 ( )2 Ω(t s ) 1 = a(ts )H inf 1 = a 0 H 0 (L 0 H 0 ) 2 (2.23) < Ω(t 0 ) 1 < Ω(t s ) 1 (2.24) Ω(t s ) 1 Ω ρ inf (t) ρ γ 2 ρ inf (t) 3 CMB ( ) 2 1

40 ρ inf (t) ϕ V (ϕ) ϕ (i) Higgs (ii) (iii) S = d 4 x gl ϕ = d 4 x ( g 1 ) 2 gµν µ ϕ ν ϕ V [ϕ] (2.25) RW ϕ t ϕ + V (ϕ) = ϕ + 3H ϕ + V (ϕ) = 0 (2.26) ϕ = 1 µ (g µν g ν ϕ) g (2.27) (00 ) (ȧ a )2 = H 2 = ρ ϕ 3MG 2, ρ ϕ = 1 2 ϕ 2 + V (ϕ). (2.28) ρ ϕ T µν = 2 δs ( 1 ) g δg µν = µϕ ν ϕ g µν 2 gαβ α ϕ β ϕ + V [ϕ] (2.29)

41 (δg = gg µν δg µν ) (2.30) ρ ϕ = T0 0 = ϕ ( ϕ ) 2 + V [ϕ] = 1 2 ϕ 2 + V [ϕ] (2.31) ( P ϕ δj i = Tj i = δj i 1 2 ϕ ) 2 + V [ϕ] = 1 2 ϕ 2 V [ϕ] (2.32) T µν = P g µν + (ρ + P )u µ u ν u µ u ν = 1 T 0 0 = ρ T i j = P δi j u µ = (1, 0, 0, 0)cf(1.19) T µ ν = diag( ρ, P, P, P ) cf (2.26) d(p ϕ V ) = P ϕ dv (1.23) (V = a 3 ) ( ϕ ϕ + V (ϕ) ϕ)a 3 + 3a 2 ȧ( 1 2 ϕ 2 + V ) = 3a 2 ȧ( 1 2 ϕ 2 V ) ȧ a ϕ + 3H ϕ + V (ϕ) = 0 = 4πa 1 ( 1 (ρ + 3P ) = 3 6MG 2 2 ϕ 2 + V ϕ ) 2 3V 1 = (V (ϕ) ϕ 2 ) (2.33) 3M 2 G V (ϕ) > ϕ 2 = ϕ (2.26) ϕ (2.28) 1 2 ϕ 2 3H ϕ + V (ϕ) = 0 (2.34) (ȧ )2 = H 2 = V (ϕ) a 3MG 2 (2.35) ε M 2 G 2 ( V )2 (ϕ) V (ϕ) η MG 2 V (ϕ) V (ϕ) (2.36)

42 ( ) ϕ ϕ (i)new inflation GUT Coleman-Weinberg (1 loop radiative correction symmetry breaking ) [ ( ϕ V CW [ϕ] = Aϕ 4 2 ) ln v 2 1 ] + B(T )ϕ 2 + A 2 2 v4 + C(T ) (2.37) ϕ = 0 V CW [ϕ] A 2 v4 A ϕ 4 (A > 0) (2.38) A ϕ 4 Av 4 (t i ϕ i ) 3H ϕ + V CW[ϕ] = 0 H 2 1 = A 2 v4 ϕ(t) 2 = 3M 2 G [ 1 ϕ 2 i 4A ] 1 3H (t t i) (2.39) ϕ small field model ( ) (ii) large field model mass term V (ϕ) = 1 2 m2 ϕ 2

43 ϕ+ 3 H ϕ + m 2 ϕ = 0 (2.40) [ 1 ( 1 H = 3MG 2 2 ϕ m2 ϕ 2)]1/2 > mϕ 6MG friction term m (2.41) (2.42) ϕ 6M G ϕ 3H ϕ + m 2 ϕ = 0, H 2 m2 ϕ 2. (2.43) 6M 2 G ϕ(t) = 2 ϕ i 3 mm G(t t i ) (2.44) [ mϕi a(t) = a i exp (t t i ) m2 6MG 6 (t t i) 2] (2.45) [ 1 ] = a i exp (ϕ 2 i ϕ 2 (t)) 4M 2 G ϕ ϕ ϕ ȧ = H slow-roll a ϕ ϕ = 2 mm G 3 ϕ = mϕ 6MG = H ϕ = 2M G = ϕ f N = 1 4MG 2 (ϕ 2 s ϕ 2 f ) = ϕ2 s 4MG ϕ s 3.4 8πM G = 3.4M pl N 70 t = t pl = M 1 pl

44 44 2 L ϕ = 1 2 ( ϕ)2 V [ϕ] ϕ 2 M 4 pl, V [ϕ] M 4 pl V [ϕ s ] Mpl 4 ϕ s Mpl 2 /m coherent domain L ϕ 2 ϕ2 s L 2 M pl 4 m 2 L 2 M pl 4 L 1 m l pl horizon ϕ (iii) small-field inflation L ϕ = 1 2 ( ϕ)2 V [ϕ], V [ϕ] = λ 4 (ϕ2 v 2 ) 2 (2.46) ϕ ((2.6) v 2 = M2 λ = ±v ) 16 domain wall ϕ = ±v ( ) λ ϕ(x) = v tanh 2 vz (2.47) 2 1 d 0 λ v (( ϕ) 2 ( v d 0 ) 2 Vc ) 1/2 vvc, V c = λ 4 v4 ( ) 1 V 0 Hc 1 Vc 2 ( 3 )1/2 = = 3MG 2 MG V c v M G d 0 Hc 1 V V c

45 ϕ ( ) xy wallϕ = 0 ϕ(x, t c ) λ 2 v2 z k 0 z V [ϕ] V c 1 2 µ2 ϕ 2, µ 2 = λv 2 µ 2 Hc 2 slow-roll [ µ 2 ] ϕ(x, t) = kz exp (t t c ) 3H c a(t) = a c e Hc(t tc) ϕ = ϕ Z (t) z (t) = k 1 ϕ exp [ H2 ] (t t c ) 3H c L(t) = a(t)z (t) = a c k 1 exp [(1 µ2 ) ] H c (t t c ) 3H 2 c z slow-roll inflation ϕ H ϕ ϕ = 0 m ϕ ( k = 0 ) ϕ + 3H ϕ + m 2 ϕ = 0 ϕ [ d 1 dt 2 ϕ ] 2 m2 ϕ 2 }{{} = 3H ϕ 2 }{{} = ρ ϕ = ρ ϕ ( ) (2.48)

46 46 2 d dt ρ ϕ = 3H }{{} ρ ϕ (2.49) ρ ϕ a 3 ϕ g2 2 ϕ2 χ 2 χ χ + (g 2 ϕ 2 + m 2 χ)χ = 0 (2.50) ϕ fϕ ψψ Yukawa coupling Γ ϕ = f 2 dρ ϕ dt dρ r dt 8π m = 3Hρ ϕ Γ ϕ ρ ϕ (2.51) = 4Hρ r + Γ ϕ ρ ϕ (2.52) decay product thermalize t Γ 1 ϕ H 2 = 1 3MG 2 (ρ ϕ + ρ r ) ( a(t) ρ ϕ (t) = ρ ϕ (t f ) a(t f ) ρ r (t) = Γ ϕ t t f ) 3 e Γ ϕ(t t f ) (2.53) ( a(t) ) 4 ρ ϕ (τ)dτ (2.54) a(t f )

47 H 2 = ρ r 3M 2 G = 1 π 2 g 3MG 2 30 T R 4 }{{} = a t 1/2 ( 1 = 2t)2 ( )2 Γϕ 2 1 ( 200 ) 2 ( 1 T R 0.3 MG Γ ϕ ) 4 ( Γ ) 1 ϕ 2 GeV (2.55) g g 10 5 GeV

48 Robertson Walker ds 2 = dt 2 + a 2 (t)dx 2 φ a(t) 1 a(t) = e Ht (H:const) gd [ 1 S = 4 xl φ = dtd 3 xa 3 (t) 2 φ2 1 ] 2a 2 ( φ)2 V (φ) (3.1) V [φ] = 1 2 m2 φ 2 π(x, t) φ(x, t) = π(x, t) = δs δ ϕ = a3 (t) φ(x, t) (3.2) [φ(x, t), π(x, t)] = iδ 3 (x x ) (3.3) d 3 k (2π) 3/2 ( ak φ k (t)e ikx + a k φ k(t)e ikx) (3.4)

49 d 3 k (2π) 3/2 ˆφ k(t)e ikx ( ˆφ k (t) = a k φ k (t) + a k φ k(t) ) ( + m 2 )φ = 0 k [ d 2 dt 2 + 3ȧ d a dt + k2 a 2 + m2] φ k (t) = 0 (3.5) 2 (ϕ 1, ϕ 2 ) = i {ϕ 1 (x) t ϕ 2(x) ( t ϕ 1 (x))ϕ 2(x)}a 3 (t)d 3 x i ϕ 1 t ϕ 2a 3 (t)d 3 x φ (φ k (t)e ikx, φ k (t)e ik x ) = (2π) 3 δ 3 (k k ) φ k (t) φ k(t) φ k (t)φ k(t) = i a 3 (t) (3.6) (3.3) [ ak, a ] k = δ(k k ) (3.7) [a k, a k ] = [ a k, ] a k = 0 [ϕ(x, t), π(x, t)] = = = d 3 k d 3 k (2π) 3/2 (2π) 3/2 a3 (t) [ ak φ k e ikx + a k ψ ke ilx, a k φ k e ik x + a k φ x ] k e ik d 3 k d 3 k (2π) 3/2 (2π) 3/2 a3 (t){[a k, a k ] φ k φ k eikx ik x + [ a k, a ] k φ k φ k e ikx+ik x } d 3 k (2π) 3/2 a3 (t)(φ k φ k φ k φ k)e ik(x x ) = iδ(x x )

50 50 3 Minkowski φ k (t) = e iω kt 2ωk, ω k = k 2 + m 2 (3.8) i t φ k = ω k φ k positive frequency mode ȧ a = H =const,a = eht (3.4) [ d 2 dt 2 + 3H d dt + k2 e 2Ht + m2] φ k (t) = 0 (3.9) ds 2 = a 2 (η)( dη 2 + dx 2 ) = dt 2 + a 2 (t)dx 2 conformal time η t d dt = 1 d a dη = Hη d dη ( d 2 dη 2 2 η dt a(t) = 1 He Ht = 1 (< 0) (3.10) Ha d dη + k2 + m2 ) H 2 η 2 φ k (η) = 0 (3.11) III,p.161 ( η) 3 2 H ν (1) ( kη) ( η)h ν (2) ( kη), ν = 3 [1 4m2 ] 2 9H (3.6)(conformal time φ k (η)φ k (η) φ k (η)φ k (η) = i a 2 ) 1 (η) 1 2

51 φ k (η) = π 4 H( η) 3 2 H (1) ν ( kη) (3.12) kη = k Ha kη 1 (1) (2) Hν (z) φ k (η) Hη 2k e 2 2ν+1 πz ei(±z 4 π) (1) (2) 2ν+1 2ν+1 ikη i 4 π e ikη i 4 π = 2ka(t) dt = a(η)dη a =const φ k (η) = e ikη 2ka = k e i a t 2 k a a 3 2 = e ik physt 2k phys a 3 2 = φ k (t) (3.13) Minkowski (3.8) (3.6) H ν (1) ( kη) ( ) η = MG 2 V V = m2 3H 2 1 massless 2 φ 2 (x, t) = ( H 2π )2 Ht (3.14)

52 52 3 ( ) m 2 (3.12) ν = 3 2 π φ k (t) = 4 H( η) 3 2 H (1) 3 ( kη) = ih (1 + ikη) 2 2k 3 e ikη (3.15) ϕ 2 (x, t) 0 φ 2 (x, t) 0 = φ k (t) = d 3 k (2π) 3 φ k(t) 2 (3.16) ih ( 1 i k ) e i k Ha ih [ ( a + O ( k 2k 3 Ha 2k 3 Ha )2)] (3.17) k k 3 φ k (t) 2 H2 2k 3 (3.18) d3 k (2π) 3 (3.14) (3.18) (3.16) (super horizon modes) φ 2 (x, t) = He Ht H φ k (t) 2 d3 k He Ht (2π) 3 = H 2 4πk 2 dk ( H )2 H 2k 3 (2π) 3 = Ht 2π (3.19) t = 0 k = Ha cut off H 1,1step ± H 2π (2 t )

53 Hubble timeh 1 H 1 δφ H 2π (3.1) (3.4) ˆφ k (t) S = [ 1 dtd 3 ka 3 (t) 2 ˆφ k (t) 2 k2 2a 2 ˆφ k(t) m2 ˆφ k (t) 2] ˆφ k (t) ˆπ k (t) = a 3 (t) ˆφ k (t) (3.17) φ k φ k(t) ˆφ k (t) φ k (t) ( a k a ) k ˆπ k (t) a 3 (t) φ k (t) ( a k a ) k operator decaying mode( ) ˆφ k ˆπ k 3.21 ds 2 = dt 2 + a 2 (t)dx 2 = a 2 (η)( dη 2 + dx 2 ) (3.20) ds 2 = (1 + 2A)dt 2 ab j dtdx j + a 2 (δ ij + 2H L δ ij + 2H T ij )dx i dx j i, j = 1, 2, 3 (3.21) 3 B j = j + B j, j Bj = 0 (3.22) rotation free divergence free ( p.225) H T ij H T ij = ( i j δ ij 3 2) H T + i HT j + j HT i + H T T ij (3.23)

54 54 3 j HT j = 0, J H T T ij = 0, trh T T ij = 0 H T T :transverce-traceless 3 ( ) ( 1 ) ( 2 ) A, B, H L, H T : B j, H T j : ( ) H T T ij : ds 2 = (1 + 2A)dt 2 a j Bdtdx j ( +a [δ 2 ij + 2H L δ ij + 2 i j δ ] ij 3 2) H T dx i dx j (3.24) }{{} i, j δ ij A B H L H T : (Lapse) : (shift) : : RW ( ) A, B x µ A, B, H L, H T x µ Ā, B, H L, H T

55 x 0 = x 0 + δx 0 = x 0 + T x i = x i + δx i = x i + i L (3.25) δx i i L ḡ µν ( x) = xα x β x µ x ν g αβ x ḡ µν ḡ µν (x) = xα x µ x β x ν g αβ(x δx) = g µν (x) (δx α ),µ g αν (x) (δx β ),ν g µβ (x) g µν,λ δx λ (3.26) (3.25) T, L 1 g 00 = g 00 2T g 00 + ġ 00 T + g 00,i L,i }{{} 2 Ā = A T ḡ 20 = g 20 2 T g 00 2 Lg22 a 2 B = a 2 B + 2 T a 2 2 L B = B 1 a T + a L ḡ 22 = g 22 ( i L),2 g i2 ( i L),2 g 2i g 22,0 T = g Lg 22 T g 22,0 [ a H L + 2 ( ] 3 2) HT [ = a H L + 2 ( ] 3 2) H T 2a 2 2 2L 2aȧT ḡ 11 + ḡ 22 + ḡ 33 H L = H L 2 L HT 3

56 56 3 ḡ 23 = g 23 ( 3 L),2 g 33 ( 2 L),3 g 22 = g La 2 2a HT = 2a H T 2a L H T = H T L Ā = A T B = B 1 a T + a L H L = H L L HT (3.27) H T = H T L (3.27) 4 ( ) generatort, L 2 Φ A = A (ab) a 2 (ḦT + 2HḢT ) (3.28) Φ H = H L 2 3 H T ȧb aȧḣt (3.29) L, T T µν = pg µν + (ρ + p)u µ u ν, g µν u µ u ν = 1 (3.30) ρ ρ + δρ p p + δp ( 1 A, ) i a 2 V u µ (i, j δ ij ) g µν u µ u ν = 1 g µν u µ g ν = (1 + 2A)(1 A) 2 +

57 = 1 + O(2 ) u 0 = g 0ν u ν = (1 + 2A)(1 A) = (1 + A) u i = g iν u ν = g i0 u 0 + g ij u j = a i B(1 A) i V = i (V + ab) u µ = ( 1 A, i (V + ab)) δt 0 0 = δρ ρδ δ δρ ρ δt j 0 = (ρ + p) j a V δtj 0 = (ρ + p) j (V + ab) a δtj i = p [π L δ ij + 1 ( 2 ) ] a 2 i j δ ij π T 3 (3.31) ( ) π T 0 π L = δρ ρ nonminimal coupling π T = 0 δ, V, π L, π T ϕ(x, t) = ϕ 0 (t) + δϕ(x, t) ϕ( x) = ϕ(x) ϕ(x) = ϕ(x δx) = ϕ(t T, x i i L) δϕ = δϕ ϕ 0 T = ϕ 0 (t T ) + δϕ(t T, x i i L) = ϕ 0 (t) ϕ 0 (t)t + δϕ(x) = ϕ 0 (t) + δϕ(x)

58 58 3 ϕ δϕ (ab a 2 Ḣ T ) ϕ 0 (3.32) T, L H T = 0 L = H T B = 0 T (3.28) A = Φ A (3.29) H L = Φ H 2 ds 2 = (1 + 2Φ A )dt 2 + a 2 (1 + 2Φ H )dx 2 (3.33) (longitudinal) ϕ = δϕ δ v 3.22 δg 0 0 = 8πGδT 0 0 δg µ ν = 8πGδT µ ν (3.34) 6H 2 Φ A 6H Φ H a 2 Φ H = 8πGρδ (3.35) δg j 0 = 8πGδT j 0 δg i j = 8πGδT i j [ ( 2 H 2 + ä a 1 ( 2 a 2 i j δ ij 3 HΦ A + Φ H = 4πG(ρ + p)v (3.36) ] a 2 (Φ A + Φ H ) [π L δ ij + 1 ( 2 a 2 i j δ ij ) Φ A + 2H Φ A 6H Φ H 2 Φ H ) (Φ A + Φ H ) = 8πGp 3 δ ij ) π T ] (3.37)

59 (3.37) (3.35)+6H (3.36) Φ A + Φ H = 0 (3.38) +2 2 a 2 Φ H = 8πGρδ 24πG(ρ + p)hv 2 a 2 Φ H = 4πGρ, δ + 3(1 + w)hv Poisson 2 a 2 Φ A = +4πGρ (3.39) δt µ ν;µ = 0 0 i 3Hw = (1 + w) 2 V (3.40) a2 V = 1 ρ + p (pπ L G 2 ρδ + c 2 sρ ) + Φ A (3.41) c 2 s = dp dρ (3.42) ṗ = 3(1 + w)hρ (3.36) (3.38) Φ H + HΦ H = 4πG(ρ + p)v = 3 2 H2 (1 + w)v 3 (1 + w)hυ 2 (3.43) Υ HV Υ (1 + w)hυ = HΦ H + Γ 1 p (δp c2 sδρ) = 0 H 1 + w (c2 s + wγ ) (3.44) p(ρ) =

60 (3.43) (3.44) d dt (ΦΥ H) = Hc2 s 1 + w = + 2c2 sh 3(1 + w) superhorizon k ah Φ H Υ (3.43) Φ H = Υ = Φ H + 2 a 2 H 2 Φ H (3.45) 2 3(1 + w) (Φ H + H 1 ΦH ) ζ R c =const. = c 1 (3.46) k ah Bardeen ζ comoving curvature perturbation (3.46) Φ H Φ H = c 1 ( 1 H a t const a(t )dt ) (3.47) ( ) t Φ H = c 1 Ḣ a + H2 a(t )dt c 1 H a [ 3 = c 1 2 (1 + w)h2 + H 2] 1 t a(t )dt c 1 H ζ = Φ H + = 3 2 H(1 + w)c H 1 a = 3 2 H(1 + w)c H 1 a a t ( a(t )dt c 1 H 1 H a t a(t )dt HΦ H 2 3(1 + w) (Φ H + H 1 ΦH ) t a(t )dt )

61 ζ = c 1 ( 1 H a t ) a(t )dt H t + c 1 a(t )dt = c 1 (3.48) a (3.47) H a const. (3.43)(3.44) Φ H (3.47) 1 w = p ρ = 2 const. a(t) t 3(1+w) Φ H = c 1 [1 + ] 1 2 = 3(1 + w) 2 3 c 1 RD 3 5 c 1 MD (3.49) Φ H 9/10 R c = ζ 3.23 (2.29) ( 1 ) T µν = µ ϕ ν ϕ g µν 2 gαβ α ϕ β ϕ + V [ϕ] (2.29) longitudinal gauge (ϕ = ϕ 0 + δϕ) δt0 0 = ρ ϕ δ ϕ = ϕ 0 δϕ ϕ 2 0Φ A + V [ϕ 0 ]δϕ (3.50) δtj i = p ϕ δπ Lϕ δj i = ( ϕ 0 δϕ ϕ 2 0Φ A V [ϕ]δϕ) δj i (3.51)

62 62 3 (3.36) (3.52) δt j 0 = (p ϕ + ρ ϕ ) j a 2 V = + ϕ j ϕ a 2 = 1 a 2 ϕ j δϕ (3.52) (3.50) (3.52) g jµ T µ0 = a 2 ϕ j ϕ Φ H + HΦ H = 4πG ϕδϕ (3.53) Υ ϕ = HV ϕ = H ϕδϕ = Ḣ δϕ (3.54) ρ ϕ + p ϕ ϕ ρ ϕ ϕ = ρ ϕ δ ϕ + 3(ρ ϕ + p ϕ )HV ϕ }{{} ϕ 2 = ϕ δϕ ϕδϕ ϕ 2 Φ A (3.55) 2 a 2 Φ H = 4πGρ ϕ ϕ Ṙc ( Ṙ c = Φ H Υ = Φ Ḣ ) H ϕ δϕ = HΦ H 4πG ϕδϕ Ḣδϕ ϕ = 4πG( ϕ δϕ ϕδϕ + ϕ 2 Φ H ) (3.56) H δϕ + H ϕδϕ δ ϕ 2 = Ḣ ϕ ρ H 2H ϕ ϕ = + 2 4πGa 2 ϕ 2 2 Φ H = 3(1 + w) a 2 H 2 Φ H (3.57) (3.45) c 2 s = 1 R c 2 ϕ 2 V [ϕ] w 1 2

63 R c = Φ H Υ ϕ = Hδϕ ϕ (3.58) R c k 2 (3.18) ( ) Ḣ 2 ( ) Ḣ 2 R ck 2 = δϕ k 2 H 2 = ϕ ϕ 2k 3 (3.59) 4πk3 (2π) 3 r = 2π k 2 2 R ( Ḣ 2 R(r) = ϕ ) 2 H 2 4πk 3 ( H 2 ) 2 2k 3 (2π) 3 = 2π ϕ (3.60) 3H ϕ = V (ϕ) ( 3H 2 2 R = 2πV (ϕ) )2 = V (ϕ) 3 12π 2 M 6 G V (ϕ) 2 = V (ϕ) 24π 2 εm 4 G = H2 8π 2 εm 2 G (3.61) ε 2 R (r = 2π k ) kn 1 n n 1 = d ln 2 R (r) d ln k = 3M 2 G ( V V )2 + 2MG 2 V V 6ε + 2η (3.62) n = 1 Harrison-Zel dovich running α dn d ln k = 16εη 24ε2 2ξ (3.63) ξ MG 4 V (ϕ)v (ϕ) V 2 (ϕ) R c RD MD

64 64 3 Φ H = 2 3 R c RD 3 5 R c MD k (3.39) k 2 a 2 Φ H = 4πGρ = 3 2 H2 = 2 ( k )2 Φ H 3 Ha k CMB 3.24 (3.21)(3.23) H T T ij h ij ds 2 = dt 2 + a 2 (t)(δ ij + 2h ij )dx i dx j (3.64) ij δ ij h ij = 0 2 +, h ij h ij = d 3 k [ ] h (2π) 3/2 +k (t)e+ ij (ˆk) + h k (t)e ij (ˆk) e ikx (3.65) e A ij (ˆk)e A ij ( ˆk) = δ AA (3.66) ˆk x > 0 e + ij (ˆk) = 1 (û i û j ˆv iˆv j ) e ij (ˆk) = 1 (û iˆv j + ˆv i û j ) 2 2 ˆk, û, ˆv 3 x < 0 x > 0 ˆk ˆk

65 S = 1 16πG [ 1 gd 4 xr 1 d 3 ka 3 (t) 8πG (2π) 3 2 ḣ+k (t) 2 + k2 a 2 h +k (t) ḣ k (t) 2 + k2 a 2 h k (t) 2 ] (3.67) massless scalar 2 φ A k (t) = 1 h 8πG Ak (t) ( d 2 dt 2 + 3H d dt + k2 ) a 2 φ A k = 0 a(t) = e Ht φ A k (t) 2 = H2 2k 3 k ah H 1 δφ = H 2π 8πGH h = = H +, 2π 2πM G ( H (ϕ(f)) )2 h 2 (f) 2 h ij h ij = 4 = V [ϕ] 2πM G 3π 2 MG h (3.68) a(t) t p 1 3p ( h(f, a) a(t) 2p p J 3p 1 2(1 p) 1 p k ) a(t)h(t) k = 2πfa 0 (3.69)

66 66 3 ω dρ GW d ln f = ω2 32πG h2 t in (f) ω = H (t in (f)), h = h inf dρ GW (f) d ln f = H2 (t in )) 32πG h2 inf = 1 24 ρ cr (t in (f)) 2 h (f) Ω GW (f, t in (f)) = h (f) (3.70) ρ GW a 4 ρ cr = ρ tot w = p/ρ ρ tot a 3(1+w) Ω GW (f, t 0 ) (3.61) (3.68) r = 2 h 2 = 16ε (3.71) R n t = d ln 2 h d ln k = d ln 2 h d ln ah d ln 2 h d ln a 1 d ln 2 h H dt = 2 H (ln H) 1 H (ln V ) = V ϕ HV = V 2 3H 2 V ( V = MG 2 )2 = 2ε (3.72) V CMB