- PDF ΔΩΡΕΑΝ Λήψη

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7 n( x) α = 0.01 n( x) α = 1 n( x) α = 3 n( x) α = 10 n( x) α = 30 ū i ( x) α = 1 ū i ( x) α = 3 ū i ( x) α = 10 ū i ( x) α = 30 δū ij ( x) α = 1 δū ij ( x) α = 3 δū ij ( x) α = 10 δū ij ( x) α = 30

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11 R µν 1 2 g µνr = 8πG N c 4 T µν Λg µν R µν R g µν G N T µν Λ g µν

12 ( ) dr ds 2 = c 2 dt 2 + α(t) kr 2 + r2 dω 2 k = 1, 0, 1 ( ) α 2 + k α α 2 = 8πG N ρ tot, 3 ρ tot, H(t) = α α, k = 0 ρ c 3H2 8πG N, ρ c Ω i ρ i Ω i ρ i ρ c, Ω = i Ω i i ρ i ρ c. Ω 1 = k H 2 α 2

13 Ω i p i = w i ρ i w i ρ i α 3(1+w i) Ω i H 2 (z) H 2 0 = [ Ω k (1 + z) 2 + i Ω i (1 + z) 3(1+w i) ] Ω k = ρ < ρ c ρ < ρ c Ω < 1 k = 1 ρ = ρ c Ω = 1 k = 0 ρ > ρ c Ω > 1 k = 1 k H 2 0 α2 0 H 0 = ( )kms 1 Mpc 1 Ω k < w de w de =

14 G a W, Z H 0 SU(2) L ( νe e ( νµ µ ( ντ τ ) ( ) u, L L d ) ( ) c, L s ) ( ) t, b L L L SU(2) L

15 d s b V ud V us V ub d = V cd V cs V cb s V td V ts V tb b SU(3) C SU(2) L U(1) Y SU(3) C SU(2) L U(1) Y SU(3) C U(1) Q SU(3) C W Z 0

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17 GN M(r) v(r) =, r G M(r) 4π ρ(r)r 2 dr 1/ r ρ(r) 1/r 2 M(r) r r 0 ρ 0 = ( r 0 kpc ) 2 3 M pc 3

18 1 dp ρ dr = a(r), a 0.6 d log ρ d log r + d log T d log r = r T ( µmp k B ) a(r), m p k B T ( )keV ( Mr ) ( ) Mpc M r

19 M r M r 10 5

20 Y lm (θ, φ) δt T (θ, φ) = + +l l=2 m= l a lm Y lm (θ, φ) a lm C l < a lm 2 > 1 2l + 1 l m= l a lm 2 C l l(l+ 1)C l /2π

21 Ω b h 2 = (68 C.L.), Ω m h 2 = (68 C.L.)

22 ρ(r) = ρ 0 (r/r) γ [1 + (r/r) α ] (β γ)/α γ = 1 α β γ

23 m i Ω ν h 2 = 3 i=1 m i 93eV m ν < 2.05eV (95 C.L.) Ω ν h Ω ν h C.L.)

24 0.01eV

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27 ρ( x, t) x t Nm ρ = δv 0 δv ρ( x, t)

28 u( x, t) P ( x, t) ρ( x, t) u( x, t) P ( x, t) F = G N m 1 m 2 r 2 ˆr m 1 m 2 ˆr r δm( r)

29 δm i ( r i ) r i δf δmδm i i = G N r r i 2 ( r r i ) δm i δf = i δf i = G N δm i δm i r r i 2 ( r r i ) δm = ρδv δf = G N ρ( r)δv i ρ( r i )δv i r r i 2 ( r r i ) f δf = δv fδv = G N ρ( r)δv i f = G N ρ( r) i ρ( r i )δv i r r i 2 ( r r i ) ρ( r i )δv i r r i 2 ( r r i ) δv i δv i dv = d 3 r r i r r f( r) = G N ρ( r) d 3 r ρ( r ) r r 2 ( r r ) r g( r) = G N d 3 r ρ( r ) r r 2 ( r r )

30 r r ( ) r r 2 = 1 r r g( r) ( ) g( r) = G N d 3 r ρ( r ) 1 r r = G N d 3 r ρ( r ) r r r r Φ( r) g( r) = Φ( r) Φ( r) = G N d 3 r ρ( r ) r r g( r) = 2 Φ( r) 2 Φ( r) 2 Φ( r) = G N ( ) d 3 r ρ( r ) 2 1 r r ( ) 2 1 r r = 4πδ (3) ( r r )

31 δ (3) 2 Φ = 4πG N ρ P hi( r) ρ u M(t) = dv ρ( x, t) V dm(t) dt = V dv d ρ( x, t) dt δs u δs ds df = ρ u d S

32 F = S ρ u d S ρ u ds = S V (ρ u)dv V dv d dt ρ( x, t) = S ρ u ds = V (ρ u)dv d dt ρ + (ρ u) = 0

33 F = m α f = ρ α f grav = ρ g = ρ Φ f hyd = P f = f grav + f hyd = ρ Φ P = ρ α α = d u dt α = u t + u u u( x, t) δ u( x, t) = u( x, t) δt + t u( x, t) δx + x u( x, t) δy + y u( x, t) δz z δt 0 d u( x, t) dt = u( x, t) t + u( x, t) dx x dt u( x, t) dy + y dt u( x, t) dz + x dt u t + u u ( ) u ρ + u u = ρ Φ t P

34 P = c 2 s ρ c s 2 Φ = 4πG N ρ d dt ρ + (ρ u) = 0 ( ) u ρ + u u = ρ Φ t P P = c 2 s ρ ρ ū Φ ū = 0 ρ = 0 ρ Φ Φ = 0 ρ 0 ρ( x, t) = ρ + δρ( x, t)

35 δρ( x, t) δ( x, t) = δρ( x, t) ρ ρ( x, t) = ρ(1 + δ( x, t)) d dt ρ + (ρ u) = 0 d dt δρ + u δρ + ρ u = 0 ρ δ + ρ u = 0 δ + u = 0 u t = 1 ρ P Φ u t = c2 s δ Φ

36 ( ) u = c 2 t s 2 δ 2 Φ ( ) u = c 2 t s 2 δ 4πG N ρδ δ c 2 s 2 δ 4πG N ρδ = 0 δ( x, t) δ( k, t) δ + (c 2 sk 2 4πG N ρ)δ = 0 ω 2 = c 2 sk 2 4πG N ρ δ( k, t) = δ 0 ( k) (ωt) + δ 1 ( k) (ωt) if ω 2 > 0 δ( k, t) = δ 0 ( k) + δ 1 ( k)t if ω 2 = 0 δ( k, t) = δ 0 ( k)e ω t + δ 1 ( k)e ω t if ω 2 < 0 δ 0 ( k) δ 1 ( k) ω 2 = 0 k J = 4πGN ρ 0 c s

37 k > k J k < k J λ J = c s π G N ρ 0 λ J ū = H(t) x

38 x = 0 u = d x dt = x + u x = x + u x = 0 ρ = 0 ū = H(t) x = 3H(t) ρ + 3H(t) ρ = 0 Φ = (Ḣ + H2 ) x 2 Φ = ( Ḣ + H 2 ) x = 3(Ḣ + H2 ) Ḣ + H 2 + 4πG N 3 ρ = 0

39 ρ = ρ(1 + δ) u = ū + δ u = H x + δ u Φ = Φ + φ δ + δ u + H x δ = 0 δ u + H(δ u + x δ u) = c 2 s δ φ 2 φ = 4πG N ρδ r x r + H x r = 0 r = x α H = α α

40 δ + i k θ = 0 θ + 2H θ = i k a 2 (c2 sδ + φ) k 2 φ = 4πG N a 2 ρδ θ = δ u α ( δ + 2H δ c 2 + s k 2 ) α 2 4πG N ρ δ = 0 4πGN ρ k J = α c s k > k J k < k J

41 λ J = c s α G N π ρ(t) k < k J c2 s k2 α 2 δ + 2H δ 4πG N ρδ = 0 4πG N ρ = 3H2 2 = 2 3t 2 α = (t/t 0 ) 2/3 H = 2 3t δ + 4 3t δ 2 3t 2 δ = 0 δ = t n n = 2/3 n = 1 δ(t, k) = δ 0 ( k)(t/t 0 ) 2/3 = δ 0 ( k)α r δ(t, r) = δ 0 ( k)α = d 3 k (2π) 3 ei k r δ 0 ( k)

42 α = (t/t 0 ) 1/2 H = 1 2t k < k J δ + 1 t δ 4πG N ρδ = 0 ρ 4πG N ρ 3H 2 δ + 1 t δ = 0 δ(t, k) = δ 0 ( k) + δ 1 ( k)ln(t/t 0 ) = δ 0 ( k) + δ 1 ( k)ln(α) δ + 2 δ 4πG N ρδ = 0 2 const 4πG N ρ δ + 2 δ = 0 δ = const δ = e 2Ht α 2

43 θ θ = θ + θ rot θ rot = 0 θ θ θ rot [ i k θ + 2Hθ + 1 ] α 2 (c2 s + φ) + θrot + 2Hθ rot = 0 k θ rot = 0 θ + 2Hθ + 1 α 2 (c2 s + φ) = 0 θ rot + 2H θ rot = 0 θ rot (t, k) = 1 α 2 θrot ( k) 1 α 2 δ = k 2 θ

44 θ = Hf(α) k 2 δ f(α) f(α) = d lnδ d lnα Ω 0.6 m f = Ω γ m, γ = Ω + (Ω2 ) k 2 d s 2 = ḡ µν dx µ dx ν = α(η) 2 ( dη 2 + δ ij dx i dx j ) δ ij Ḡ µν = R µν 1 2ḡµν R = 8πG N Tµν

45 H(η) = α α, H = αh, µ = ν = 0 3H 2 = 8πG N α 2 ρ = 8πG N α 2 I ρ I I ρ I µ = ν = i 2H H 2 = 8πG N α 2 P = 8πGN α 2 I P I µ µ T ν=0 ρ I + 3H( ρ I + P I ) = 0 g µν = ḡ µν + δg µν

46 δg µν ḡ µν G µν = Ḡµν + δg µν δḡ µν T µν = T µν +δt µν δg µν = 8πG N δt µν ds 2 = α(η) 2 ( (1 + 2A)dη 2 2B i dηdx i + (δ ij + h ij )dx i dx j ) B i h ij B i B = B + ˆB i ˆ i ˆBi = 0 h ij = 2Cδ ij + 2 <i j> E + 2 (i Ê j) + 2Êij <i j> E ( i j 1 3 δ ij 2 )E (i Ê j) 1 2 ( iêj + j Ê i ) i Ê i = 0 i Ê ij = 0 Êi i = 0

47 ˆBi Êi Ê ij ds 2 = α(η) 2 ( (1 + 2Φ)dη 2 + (1 2Ψ)δ ij dx i dx j + Êijdx i dx j ) Γ 0 00 = α α + Φ Γ 0 0k = Φ,k Γ 0 ij = α α δ ij Γ i 00 = Φ,i Γ i 0j = α α δi j Ψ δ i j ] [2 α (Φ + Ψ) + Ψ δ ij α Γ i kl = (Ψ,lδ i k + Ψ,kδ i l ) + Ψ,iδ kl

48 Γ α 0α = 4 α α + Φ 3Ψ Γ α iα = Φ,i 3Ψ,i Γ 0 00 = H Γ 0 0k = 0 Γ 0 ij = Hδ ij Γ i 00 = 0 Γ i 0j = Hδj i Γ i kl = 0 δγ 0 00 = Φ δγ 0 0k = Φ,k δγ 0 ij = [ 2H(Φ + Ψ) + Ψ ] δ ij δγ i 00 = Φ,i δγ i 0j = Ψ δj i δγ i kl = (Ψ,lδk i + Ψ,kδl i ) + Ψ,iδ kl R µν = Γ α νµ,α Γ α αµ,ν + Γ α αβ0 Γβ νµ Γ α νβ Γβ αµ = R µν + δγ α νµ,α δγ α αµ,ν + Γ α αβ δγβ νµ + Γ β νµδγ α αβ Γ α νβ δγβ αµ Γ β αµδγ α νβ.

49 R 00 = 3H + 3Ψ + 2 Φ + 3H(Φ + Ψ ) R 0i =2(Ψ + HΦ),i R ij =(H + 2H 2 )δ ij + [ Ψ + 2 Ψ H(Φ + 5Ψ ) (2H + 4H 2 )(Φ + Ψ)]δ ij + (Ψ Φ),ij R µ ν = g µα R αν = (ḡ µα + δg µα )( R αν + δr αν ) = R µ ν + δg µα Rαν + ḡ µα δr αν R 0 0 =3α 2 H + α 2 [ 3Ψ 2 Φ 3H(Φ + Ψ 6H Φ)] R 0 i = 2α 2 (Ψ + HΦ),i R i 0 = R 0 i = 2α 2 (Ψ + HΦ),i R i j =α 2 (H + 2H 2 )δ i j + α 2 [ Ψ + 2 Ψ H(Φ + 5Ψ ) (2H + 4H 2 )Φ]δ i j + α 2 (Ψ Φ),ij R =R R i i =6α 2 (H + H 2 ) + α 2 [ 6Ψ (2Ψ Φ) 6H(Φ + 3Ψ ) 12(H + H 2 )Φ]

50 G 0 0 =R R = 3α 2 H 2 + α 2 [ 2 2 Ψ + 6HΨ + 6H 2 Φ] G 0 i =R 0 i = 2α 2 (Ψ + HΦ),i G i 0 =R i 0 = 2α 2 (Ψ + HΦ),i G i j =R i j 1 2 δi jr =α 2 ( 2H H 2 )δ i j + α 2 [2Ψ + 2 (Φ Ψ) + H(2Φ + 4Ψ ) + (4H + 2H 2 )Φ]δ i j + α 2 (Ψ Φ),ij. T µν =( ρ + p)ū µ ū ν + pḡ µν T µ ν =( ρ + p)ū µ ū ν + pδ µν ρ = ρ(η) p = p(η) ū µ = (ū 0, 0, 0, 0) ū µ ū µ = 1 ū µ = α( 1, 0) T µν = T µν + δt µν T µ ν =(ρ + p)u µ u ν + pδ µν

51 ρ = ρ + δρ p = p + δp u i =ū i + δu i = δu i 1 α v i v i αu i δ δρ ρ u µ =ū µ + δu µ (α 1 + δu 0, α 1 v 1, α 1 v 2, α 1 v 3 ) u µ =ū µ + δu µ ( α + δu 0, δu 1, δu 2, δu 3 ). u ν = g µν u µ u µ u µ = 1 u 0 = g µ0 u µ = α α 2 δu 0 2αA δu 0 = α 2 δu 0 2αA. δu i = u i = αb i + αv i δu 0 = 1 α A

52 ( ) ( ) ρ 0 T ν µ δρ ( ρ + p)(v i B i ) = + 0 pδj i ( ρ + p)v i δpδj i ( ) δp δtj i = δpδj i + Σ i j p p δi j + Π i j. Σ i j Πi j Σi j / p δp 1 3 δt k k Σ i j δt i j 1 3 δi jδt k k Σ i j = 0 δρ δp v Π ij v = v + ˆv i iˆv i = 0 Π ij = Π S ij + Π V ij + Π T ij, Π S ij = ( i j 1 3 δ ij 2 )Π Π V ij = (Π i,j + Π j,i ) και δ ik Π T ij,k = 0

53 Π ij = 0 δg 0 0 =α 2 [ 2 2 Ψ + 6HΨ + 6H 2 Φ] = 8πG N δρ δg 0 i = 2α 2 (Ψ + HΦ),i = 8πG N ( ρ + p)v,i δg i 0 =2α 2 (Ψ + HΦ),i = 8πG N ( ρ + p)v,i δg i j =α 2 [2Ψ + 2 (Φ Ψ) + H(2Φ + 4Ψ ) + (4H + 2H 2 )Φ]δ i j + α 2 (Ψ Φ),ij = 8πG N [δpδ i j + p(π,ij 1 3 δi j 2 Π)]. 3H(Ψ + HΦ) 2 Ψ = 4πG N α 2 δρ (Ψ + HΦ),i = 4πG N α 2 ( ρ + p)v,i Ψ + H(Φ + 2Ψ ) + (2H + H 2 )Φ (Φ Ψ) = 4πG N α 2 δp ( i j 1 3 δ ij 2 )(Ψ Φ) = 8πG N α 2 p( i j 1 3 δ ij 2 )Π (Ψ Φ) ij = 8πG N α 2 pπ,ij για i j. k i k j (Ψ k Φ k ) = k ik j k 2 8πG Nα 2 pπ k για i j. k k 2 (Ψ k Φ k ) = 8πG N α 2 pπ k για k 0.

54 (Ψ Φ) = 8πG N α 2 pπ (Ψ + HΦ) = 4πG N α 2 ( ρ + p)v = 3 2 H2 (1 + w)v 2 Ψ = 4πG N α 2 ρ[δ + 3H(1 + w)v] (Ψ Φ) = 8πG N α 2 pπ Ψ + HΦ = 3 2 H2 (1 + w)v Ψ + H(Φ + 2Ψ ) + (2H + H 2 )Φ (Φ Ψ) = 4πG N α 2 δp 2 Φ = 4πG N α 2 ρ[δ + 3H(1 + w)v] = 3 2 H2 [δ + 3H(1 + w)v] Φ + HΦ = 4πG N α 2 ( ρ + p)v = 3 2 H2 (1 + w)v Φ + 3HΦ + (2H + H 2 )Φ = 3 2 H2 δp/ ρ 2 Φ = 4πG N α 2 ρ[δ + 3H(1 + w)v] = 4πG N α 2 ρ = δ + 3H(1 + w)v

55 µ T µ ν = µ T µ ν + Γ µ µαt α ν Γ α µνt µ α = 0 ν = 0 ρ + δρ + i v i ( ρ + p) + 3H( ρ + δρ) 3 ρφ + 3H( p + δp) 3 pφ ρ = 3H( ρ + p) δρ = 3H(δρ + δp) + 3Φ ( ρ + p) v ( δ p ρ ) ( ( δp v 3Φ ) + 3H δρ p ρ ) δ = 0 p ρ ν = i v + H v 3H p ρ v = δp ρ + p Φ p ρ

56 δ( k) δ b = δ e k > k J k < k J

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61 f(t, x, p) d 3 pf(t, x, p) f(t, x, p) p

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63 ( x i, p i ), i = 1...N f k = (t, x, p) = 1 N N δ D ( x x i )δ D ( p p i ) i=1 δ D x p = α 2 m d x dt α df k dt = f k t + d x dt f k x + d p dt f k p = 0 t f k = p α 2 m f k + m Φ p f k Φ 2 Φ = 4πG N ρ α ( ) d 3 pf k 1 ρ < d 3 pf k > vol = 1 f s =< f k > Φ p f k f k f 2s (t, x, p, x, p ) = f s (t, x, p)f s (t, x, p ) + f 2c (t, x, p, x, p ) t f s = p α 2 m f s + m Φ p f s + m d 3 x d 3 p Φ( x x ) p f 2c (t, x, p, x, p )

64 t f = p α 2 m f + m Φ p f 2 Φ = 4πG ( ) N ρ d 3 pf 1 α n d ( x) θ d ( x) f d (t, x, p) = n d ( x)δ D ( p θ d ( x)) f d n d θ d /m t n d = 1 mα 2 (n d θd ) t θ d = 1 2mα 2 ( θ d ) 2 mφ d 2 Φ d = 4πG N ρ (n d 1) α

65 u d = θ d /m t n d = 1 α 2 (n d u d ) t u d = 1 2α 2 ( u d ) u d Φ d 2 Φ d = 4πG N ρ (n d 1) α u d = 0 σ x σ p x p f( x, p) = d 3 x d 3 p [ (2πσ x σ p ) 3 ( x x ) 2 2σx 2 ( p p ) 2 ] 2σp 2 f( x, p ) f = e σ2 x σ2 p 2 2 pf ( ( 2 )(AB) = [ ( 2 )(A)] 2 ) [ ( 2 )(B)]. p t f = α 2 m f σ2 p α 2 m p f + m Φ (σ 2 x ) p f x p

66 α ħ2 iħ t ψ = 2α 2 m 2 ψ + mφψ 2 Φ = 4πG N ρ ( ψ 2 1 ) α ψ( x) = n( x)e iθ( x)/ħ n( x) < n > vol = 1 θ( x) ħ 2 t n = 1 mα 2 (n θ) t θ = 1 2mα 2 ( θ) 2 mφ + ħ2 2 n 2α 2 m n 2 Φ = 4πG N ρ (n 1) α u = /m t n = 1 α 2 (n u) t u = 1 ( 2α 2 ( u ) u Φ + ħ2 2α 2 m 2 ) n 2 n u =0 ħ n ψ

67 j n u t n = 1 α 2 j t j = 1 2α 2 j i j i n n (Φ ħ2 2 ) n 2α 2 m 2 n ψ( x) f W ( x, p) f W ( x, p) = d 3 x (πħ) 3 e2 i ħ p x ψ( x x)ψ ( x + x) f W ( x, pr) t f W = p α 2 m f W + i ħ d 3 x (πħ) 3 e2 i ħ p x m[φ( x + x) Φ( x x)]ψ( x x)ψ ( x + x) t f W = p α 2 m f W + mφ 2 ( ħ ħ 2 [ ] p 2 2 = 2α 2 m + mφ ħ p ) f W ( ħ 2 ( p p ) f W ħ 2 t (f W f) ħ2 24 x i xj Φ pi pj p f W + O(ħ 4 )

68 f W ħ f W f W = e σ2 x σ2 p 2 2 pf W σ x σ p ħ/2 t fw = p α 2 m f W σ2 p α 2 m p fw + m Φe σ2 x 2 ħ (ħ p ) 2 f W ψ H ( x, p) = d 3 yk H ( x, y, p)ψ( y) exp K H ( x, y, p) = [ ( x y)2 4σ 2 x i ħ p ( y 1 2 x)] (2πħ) 3/2 (2πσ 2 x) 3/4 f H = Ψ H 2 f H = f W, για σ x σ p = ħ/2 f W

69 t ( f W f) ħ2 24 x i xj Φ pi pj p fw + O(ħ 4, ħ 2 σ 2 x) σ x σ p ħ/2 σ x x typ σ p p typ x typ p typ M (0) = M (1) i M (2) ij d 3 pf H =m 1 d 3 pp i f H =m 2 d 3 pp i p j f H n( x) u i ( x) σu ij ( x) f H ψ [ ] ( x y) n( x) = d 3 2 y 2σx 2 ψ( y) 2 ū i ( x) = 1 [ ] ( x y) n( x) ħ d 3 2 y 2σx 2 I[ψ,i ψ]( y) [ ] M ij( x) 2 = ħ2 4σx 2 n( x)δ ij + ħ2 ( x y) d 3 2 y 2 2σx 2 R[ψ,i ψ,j ψ,ij ψ]( y) M ij 2 δū ij ( x) = ( x) ū i ( x)ū j ( x) n( x) R I

70 δ lin (α, q) = D(α) ( πq L ) α = 1 n d (α, q) = [1 δ lin (α, q)] 1 Ψ (q) = q + Ψ(a, q) Ψ d (a, q) = D(α) L π sin(πq L ), x θ d = u d (q) = α 3 H(α) α Ψ d (a, q) n d ψ ini = n d (a ini, x) [iθ d (a ini, x)/ħ]

71 θ n n α = 0.01, α = 1, α = 3, α = 10 α = 30 ū i ( x) ū i ( x) θ θ rot

72 δū ij ( x) δū k k σ ij δū ij ( x) = 1 3 δūk k ( x)δ ij + σ ij S = d 3 xd 3 pf H (f H )

73 ħ