Transcription

1 29 7 5

2

3 i Slow roll inflation

4 ii Generation of Adiabatic perturbation via Inflation Scalar, vector, tensor decomposition of perturbation Gauge transformation of scalar perturbations Gauge invariant variables Power spectrum of scalar perturbation CMB Late stage evolution of density contrast and CMB spectrum :U = SR :U = RS

5 iii A 165 B E B 177

6

7 q i (t) i = x, y, z q i (t) q i (t) 6 q i (t), q i (t) (Lagrangian) L(q i (t), q i (t)) = T V T V V (q i ) m L(q i (t), q i (t)) = 1 m q i 2 V (q i ) 2 (action) I = t2 t 1 i dtl(q i (t), q i (t)) 1.1

8 2 1 q(t ) q(t) B q(t ) 1 A q(t ) 1 q(t ) q(t) 1.1 A t 1 B t 2 (q i (t), q i (t)) (q i (t) + δq i (t), q i (t) + δ q i (t)) δi = = = t2 t2 dtl(q i (t) + δq i (t), q i (t) + δ q i (t)) dtl(q i (t), q i (t)) t 1 t 1 t2 ( L(qi (t), q i (t)) dt δq i (t) + L(q i(t), q i (t)) ) δ q i (t) t 1 q i q i (t) [ ]t L(qi (t), q i (t)) 2 t2 ( L(qi (t), q i (t)) δq i (t) + dt δq i (t) d q i (t) t 1 t 1 q i dt L(q i (t), q i (t)) ) δq i (t) q i (t)

9 1.1 3 = t2 t 1 ( L(qi (t), q i (t)) dt d q i dt L(q i (t), q i (t)) ) δq i (t) q i (t) δq i (t 1 ) = δq i (t 2 ) = 0 δi = 0 δq i (t) L(q i (t), q i (t)) q i d L(q i (t), q i (t)) dt q i (t) = 0 (1.1) p i L q i (1.2) q i, q i q i, p i (Hamiltonian) H(q i, p i ) = i p i q i L(q i, q i ) (1.3) q i, p i q i q i + η i, p i p i + ζ i H(q i + η i, p i + ζ i ) = H(q i, p i ) + ( H η i + H ) ζ i i q i p i = ( p i q i + ζ i q i + p i η i L η i L ) η i L(q i, q i ) i q i q i = H(q i, p i ) + ( ṗ i η i + q i ζ i ) i (1.3)

10 4 1 (1.3) (1.1) (1.2) η i, ζ i H q i = ṗ i, H p i = q i (1.4) m p i = m q i H = i p 2 i 2m + V (q i) V q i = ṗ i, p i m = q i t = t + δt(t) x i = x i + δx i (t) δt, δx i t Lie δ L q i (t) q i(t) q i (t) (1.5)

11 1.1 5 t t δx δt q(t) x q(t) x q(t) x x x 1.2 t = t + δt, x = x + δx x t x q x(t) 1.2 δ L q i (t) = q i(t) q i(t ) + q i(t ) q i (t) (1.6) = q iδt + δx i (t) (1.7) δx i = 0 1.3

12 6 1 t t=t t =t t q(t) x q(t)=q(t) x qδt x x δt q(t) x q(t) x x x 1.3 t = t + δt, δx i = 0 q x(t) δt t t δt t δt q x δt t δ L q x = q x δt (1.7) I

13 1.1 7 δi = t 2 t 1 ( dt L q i(t ), dq i (t ) ) t2 dt t 1 dtl(q i (t), dq i(t) ) dt t dt = dt dt = (1 + δt)dt dt q i(t ) = q i (t) + δx i (t) dq i (t ) dt = dt dt d dt (q i(t) + δx i (t)) = (1 δt)( q i (t) + δẋ i (t)) = q i (t) δt q i (t) + δẋ i (t) δ L q i (t) = q i δt q i δṫ + δẋ i (t) dl dt = i ( L q i q i + L q i q i ) δi = = t2 t 1 t2 t 1 t2 + [ dt δtl(q i (t), q i (t)) + i dt d(δtl) dt t 1 dt i t2 dt t 1 i ( L(qi (t), q i (t)) δx i (t) + L(q i(t), q i (t)) q i q i ( L q i δt + L ) q i δt q i q i ) ( L q i δx i + L q i ( δt q i + δẋ i ) ( δt )] q i (t) + δẋ i (t)) t2 t 1 dt i L q i ( q i δt + δẋ i ) = t2 t 1 dt { d i dt ( L ) ( q i δt + δx i ) q i

14 8 1 δi δi = t2 t 1 dt d dt ( δtl + i +δt d L q i + δt dt q L q i d L δx i } i q i dt q i L ) t2 δ L q i + dt q i t 1 i ( L d L ) δ L q i q i dt q i (1.7) δi = 0 N dn/dt = 0 N δtl i L q i δ L q i (1.8) (Noether current) δt = ε δx i = 0 ε δ L q i = ε q i (1.8) N = εl + i p i q i ε = εh x δt = 0

15 1.2 9 δ x = (ε, 0, 0) δ L q = (ε, 0, 0) (1.8) N = L q x ε = εp x x x x m E p E = p2 2m E i t, p i i t ψ = 2 2m 2 2 ψ ψ E 2 = p 2 c 2 + m 2 c 4 φ 4π2 λ 2 φ = 0 or (1.9) c

16 10 1 φ m2 c 2 2 φ = 0 (1.10) 2 c 2 t λ c h/mc = c = 1 φ m 2 φ = 0 φ φ V (φ) φ dv dφ = 0 (1.11) (1.10) V = 1 2 m2 φ 2 (mass term) φ (1.11) φ = dv dφ φ(x) 4 µ φ(x) L(φ(x), µ φ(x)) x (x 0, x 1, x 2, x 3 ) = (t, x, y, z) 4 µ = 0, 1, 2, 3 I = d 4 xl(φ(x), µ φ(x)) V 4

17 V 4 4 V 4 3 φ, µ φ η(x) δi = d 4 x (L(φ + η, µ φ + µ η) L(φ, φ)) V 4 ( L = d 4 x V 4 φ η + L ) µ φ µη ( L = d 4 x V 4 φ L ) [ L ] µ η + d 4 x µ µ φ V 4 µ φ η 4 V 4 η η δi = 0 ( L 0 = d 4 x V 4 φ L ) µ η µ φ L φ L µ µ φ = 0 (1.12) µ 0 3 µ φ = φ/ x µ x µ L/ µ φ µ L µ φ = 3 µ µ=0 L µ φ

18 12 1 L = 1 2 ηαβ α φ β φ V (φ) (1.13) η αβ ( 1, 1, 1, 1) 4 4 (1.12) (1.11) π(x) L t φ(x) (1.14) x µ (x ) = x µ + δx µ (x) δx µ (x) 4 φ(x). φ (x ) = φ(x) + δφ(x) (1.15) δ L φ(x) φ (x) φ(x) (1.16) δ L φ(x) = φ (x) φ (x ) + φ (x ) φ(x) = δx α α φ (x) + δφ(x)

19 φ (x) φ(x) δx α δ L φ(x) = δx α α φ(x) + δφ(x) (1.17) δi = d 4 x L(φ (x ), µφ (x )) d 4 xl(φ(x), µ φ(x)) V 4 V 4 x x d 4 x = (x ) (x) d4 x (x )/ (x) (x ) (x) = 1 + ν δx ν µ = xν x µ x ν x ν x α x α x µ = δµ ν xν x µ 4 4 ( x ν x µ )

20 14 1 x ν x µ = δµ ν + δxν x µ x ν x µ = δµ ν δxν x µ µφ (x ) = µ φ(x) δxν x µ νφ(x) + µ δφ(x) = µ φ(x) + µ δ L φ(x) + δx ν ν µ φ(x) δi = 1 d 4 x(1 + α δx α ) c V 4 ( L(φ(x), µ φ(x)) + L φ δφ + L ) µ φ ( µδ L φ + δx ν ν µ φ) 1 d 4 xl c V 4 = 1 ( α δx α L + L c φ δφ + L ) µ φ ( µδ L φ + δx ν ν µ φ) d 4 x V 4 = 1 ( d 4 x µ δx µ L + δ L φ L ) + d 4 x L c V 4 µ φ V 4 µ φ δxν ν µ φ + 1 ( L d 4 x c V 4 φ δφ L ) µ µ φ δl φ δx µ µ L = 1 ( d 4 x µ δx µ L + δ L φ L ) + d 4 x L c V 4 µ φ V 4 µ φ δxν ν µ φ + 1 ( L d 4 x c V 4 φ δφ L ( µ µ φ δl φ δx µ L φ µφ + L = 1 ( d 4 x µ δx µ L + δ L φ L ) c V 4 µ φ + 1 ( L d 4 x c V 4 φ L ) µ δ L φ µ φ )) ν φ µ ν φ

21 δi = 1 c V 4 d 4 x µ ( δx µ L + δ L φ L ) µ φ 4 (1.18) (1.18) 4 V 4 µ N µ = 0 (1.19) N µ ( δx µ L + δ L φ L ) µ φ (1.20) (1.20) 4 (1.19) δφ(x) = 0 δx 0 = ε δx i = 0 i = 1, 2, 3 δ L φ = ε φ

22 16 1 N 00 L ( 1 = εl + ε φ φ = ε 2 ( φ 2 + ( φ) ) 2 ) + V N 0i L = ε φ i φ = ε φ i φ (1.13) L = 1 2 φ ( φ) 2 V N 00 N 0i x δx 0 = 0 δx 1 = ξ δx 2 = 0 δx 3 = 0 δ L φ = ξ 1 φ N 10 = ξ ( 1 φ φ) N 11 = ξl + ξ 1 φ L 1 φ = ξ ( 1 2 φ ( φ) 2 V + ( 1 φ) 2) N 12 = ξ 1 φ L 2 φ = ξ 1 φ 2 φ N 13 = ξ 1 φ 3 φ N 10 x (T µν )

23 ( φ 2 + ( φ) 2 ) + V φ 1 φ φ 2 φ φ 3 φ φ 1 φ 1 2 ( φ 2 ( φ) 2 ) V + ( 1 φ) 2 1 φ 2 φ 1 φ 3 φ φ 2 φ 1 φ 2 φ 1 2 ( φ 2 ( φ) 2 ) V + ( 2 φ) 2 3 φ 2 φ φ 3 φ 1 φ 3 φ 2 φ 3 φ 1 2 ( φ 2 ( φ) 2 ) V + ( 3 φ) 2 (T µν ) (1.21) ν T µν = 0 ( T µν = η µα α φη νβ β φ + η µν 1 ) 2 ηαβ α φ β φ V (φ) (1.22) φ = 0 ρ = 1 2 φ 2 + V (1.23) P = 1 2 φ 2 V (1.24) ( T µν = g µα g νβ α φ β φ + g µν 1 ) 2 gαβ α φ β φ V (φ) (1.25) g µν µ 00 (1.14) (1.13) T 00 H π φ L = 1 2 ( φ 2 + ( φ) 2 ) + V (1.26)

24 (Operator)A, B [A, B] AB BA (2.1) A, B {A, B} AB + BA (2.2) a (Hermite Conjugate a )a [a, a ] = 1 (2.3) [a, a ] = [a, a] = 0 (2.4) N a a (2.5) G > N G

25 N G > = G G > (2.6) G > < G G > = 1 (2.3),(2.4) N a, a [N, a] = Na an = a (2.7) [N, a ] = Na a N = a (2.8) G (2.6) G > < G < G a a G > = G < G a a G > a G > G 0 (2.9) N (2.6) a an G > = ag G >= Ga G > G C a (2.7) an = Na + a (Na + a) G > = Ga G > Na G > = (G 1)a G > (2.10) G > a a G > N G 1

26 20 2 m Na m G > = (G m)a m G > m > G m G m < 0 (2.9) m = m 0 m > G m 0 Na m0 G > = (G m 0 )a m0 G > a Na m0+1 G > = (G m 0 1)a m0+1 G > a m0+1 G > N G m 0 1 (2.9) (2.9) a m0+1 G > = 0 (2.11) a m 0 G > a G > a m (2.11) a a a m0+1 G > = 0 a aa m 0 G > = Na m 0 G >= 0 a m 0 G > a m 0 G > G = m 0

27 N N 0 a N number operator N 0 > < 0 0 > = 1 (2.12) N 0 > = 0 0 > (2.13) a 0 > = 0 (2.14) (2.6) a (2.8) Na G > = (G + 1)a G > (2.15) G > a a G > N G + 1 N N n n > n a (2.10) n > a (2.15) n > N (2.3) (2.4) n n > n > = C n a n 0 >

28 22 2 < n n > = C n 2 < 0 a n a n 0 >= C n 2 < 0 a n 1 aa a n 1 0 > = C n 2 < 0 a n 1 (1 + a a)a n 1 0 > = C n 2 (< 0 a n 1 a n 1 0 > + < 0 a n 1 a aa a n 2 0 >) = C n 2 (< 0 a n 1 a n 1 0 > + < 0 a n 1 a (1 + a a)a n 2 0 >) = C n 2 (2 < 0 a n 1 a n 1 0 > + < 0 a n 1 a 2 aa n 2 0 >)... = C n 2 (n < 0 a n 1 a n 1 0 > + < 0 a n 1 a n a 0 >) = n C n 2 < 0 a n 1 a n 1 0 >... = n! C n < 0 0 >= n! C n 2 = 1 n > = 1 n! a n 0 > (2.16) {b, b } = 1 (2.17) {b, b} = {b, b } = 0 (2.18) (2.18) bb = bb = 0 b b = b b = 0 number operator N b b

29 NN = b bb b = b (1 b b)b = N b b bb = N N(N 1) = 0 (2.19) N 0 (2.17) (2.18) i ψ(t) > = H ψ(t) > (2.20) t H t = 0 ψ(0) > ψ(t) > = e iht/ ψ(0) > e iht/ e iht/ = n=0 1 ( iht n! H = H < ψ(t) = < ψ(0) e iht/ A < ψ(t) A ψ(t) > < ψ(t) A ψ(t) > = < ψ(0) e iht/ Ae iht/ ψ(0) > A A(t) A(t) e iht/ Ae iht/ )n

30 24 2 A < ψ(0) A(t) ψ(0) > A(t) da(t) dt = ih A(t) A(t)iH da(t) dt = i [A(t), H] (2.21) Maxwell rote + 1 B c t = 0 (2.22) rotb 1 E c t = 4π c j (2.23) dive = 4πρ (2.24) div B = 0 (2.25) ρ j (2.25),(2.22) 4 A ν = (φ, A) E = gradφ 1 A (2.26) c t B = rota (2.27) (2.26) (2.27) (2.23)

31 A c 2 t 2 ( 2 A 1 φ + grad c t + div A ) = 4π c j (2.28) ν A ν = 0 (2.29) (2.26) (2.27) 4 A new = A old + gradψ (2.30) φ new = φ old 1 c ψ t (2.31) ψ ψ 2 ψ = div A old diva new = 0 (2.32) 2 φ = 4πρ (2.33) (2.33) φ( x, t) = d 3 x ρ( x, t) x x (2.34) (2.34) φ = 0

32 26 2 div A = 0 (2.35) φ = 0 (2.36) A (2.28) 1 2 A c 2 t 2 2 A = 0 (2.37) E = A (2.38) c t B = rota (2.39) L A( x, t) = 1 V k r q (r) ( k, t) e r, k e i k x (2.40) V = L 3 e r, k k r A(x + L, y, z, t) = A(x, y, z, t) q (r) ( k, t) e r, k e i(kx(x+l)+kyy+kzz) = q (r) ( k, t) e r, k e i(kxx+kyy+kzz) k r

33 k x k x = 2πn x L (2.41) n x k y.k z k y = 2πn y L k z = 2πn z L (2.42) (2.43) (2.40) k = n x= n y= n z= L 0 dxe i(k x k x )x = 1 ( e 2πi(n x n i(k x k x) x ) 1 ) = 0 for k x k x = L for k x = k x n x, n x 1 V d 3 xe i( k k ) x = δ k, k (2.44) δ k, k = δ kx,k x δ k y,k y δ k z,k z (2.37) 1 2 q (r) ( k, t) c 2 t 2 + k 2 q (r) ( k, t) = 0 k ω k = ck (2.45)

34 28 2 ω k q (r) ( k, t) = q (r) e iω kt k A( x, t) = 1 V k r ( q (r) k e iω kt e i k x + q (r) k e iωkt e i k x ) e r, (2.46) k q (r) q (r) k k (2.46) (2.38),(2.39) E = B = 1 iω k V k r c e r, k 1 V k r ( (r) q e iωkt e i k x q (r) k e iωkt e i k x ) (2.47) k i ( (r) k e r, k q e iωkt e i k x q (r) k e iωkt e i k x (2.48) ) k V U = 1 d 3 x(e 2 + B 2 ) (2.49) 8π V (2.47) (2.48) d 3 xe 2 = 1 d 3 x ω k ω k V k r r c 2 e r, k e r, k ( q (r) q (r ) k e i( k+ k ) x i(ω k +ω k )t k V V k q (r) k q (r ) k e i( k+ k ) x+i(ω k +ω k )t +(q (r) q (r ) k + q (r) k q (r ) k )e i( k k ) x+i(ω k ω k )t ) k = k r ωk 2 c 2 ((q(r) q (r) k k q (r) q (r) k k e i2ω k q (r) k + q (r) q (r) k ) k q (r) k ei2ω k )

35 (2.44) ω k = ω k d 3 xb 2 = 1 d 3 x ( V k e r, k ) ( k e r, k r k )(q (r) q (r ) k e i( k+ k ) x i(ω k +ω k )t k r V V k +q (r) k q (r ) k e i( k+ k ) x+i(ω k +ω k )t +(q (r) q (r ) k + q (r) k q (r ) k )e i( k k ) x+i(ω k ω k )t ) k = k r k 2 ((q (r) q (r) k k +q (r) q (r) k k e i2ω kt + q (r) k + q (r) q (r) k ) k q (r) k ei2ω kt ) ( k e r, k ) ( k e r, k ) = k 2 e r, k = e r, k (2.45) U = ω 2 ( k (r) q k 4πc 2 k r q (r) k + q (r) q (r) ) k k (2.50) 1 M = d 3 xe 4πc B (2.51) V d 3 xe B = ω k k c e r, k ( k e r, k )((q(r) q (r ) k + q (r) k q (r ) k )δ k k, k V k r r q (r) q (r ) k e i2ωkt δ k k, k q (r) q (r ) k e i2ωkt δ k k, k ) = ω k k r c k(q (r) q (r) k + q (r) k q (r) k ) k ω k k r c ( k)(q (r) q (r) k k e i2ω kt + q (r) q (r) k k ei2ω kt ) = ω k k c k(q (r) q (r) k + q (r) k q (r) k ) k r

36 30 2 k r ω k c ( k )(q (r ) k q(r) k e i2ω k + q (r ) k q (r ) k e i2ω k ) k = k k = k ω k = ω k k, r k, r q (r) q (r) k = k q(r) k q(r) k q (r) q (r) k = k q(r) k q(r) k M = ω k k 4πc 2 k(q (r) k r q (r) k + q (r) q (r) k ) (2.52) k L r, k = ( 1) r 1 4πω k d 3 xe k B k (2.53) r = 1 r = 2 (2.52) c/ω k L = ( 1) r 1 k 4πc k(q (r) k r V q (r) k + q (r) q (r) k ) (2.54) k (2.21) (2.46) q (r), q (r) k k (2.50) U H (2.21) A( x, t) t = i V k r ( (r) ω k q e iωkt e i k x q (r) k e iωkt e i k x ) e k r, k (2.55)

37 [ (r) q, q (r ) ] k k [ (r) q, q (r )] k k = 4π c2 δ k, 2ω k δ r,r (2.56) k = [ q (r), q (r ) ] k k = 0 (2.57) [ (r) q, q (r ) k q (r ) ] k k [ (r) q, q (r ) k q (r ) ] k k [ (r) q, q (r ) k q (r )] k k [ (r) q, q (r ) k q (r )] k k = q (r) k q (r ) k q (r ) k q (r ) = q (r ) k = q (r ) k q (r) q (r ) k q (r ) k q (r) k k q (r) k q (r ) k q (r ) k k ( 4π c 2 δ k, 2ω k δ r,r + q (r k k = 4π c2 δ k, 2ω k δ r,r q (r k k ) = q (r) k q (r ) k q (r ) k q (r ) = q (r) q (r ) q (r ) k k q (r ) k k = q (r) q (r ) q (r ) k k k = 4π c2 δ k, 2ω k δ r,r q (r k k ) q (r ) k q (r) k k ( 4π c 2 q (r) q (r ) k k 2ω k ) = q (r) k q (r ) k q (r ) k q (r ) = q (r) q (r ) k q (r ) k q (r ) k k = q (r) q (r ) k q (r ) k k = 4π c2 δ k, 2ω k δ r,r q (r k k q (r ) k q (r) k k ) q (r) q (r ) k q (r ) k q (r) k k δ k, k δ r,r + q (r) q (r ) k k q (r) q (r ) k k ( 4π c2 ) = q (r) k q (r ) k q (r ) k q (r ) = q (r ) k = q (r ) k q (r) q (r ) k 2ω k q (r ) k q (r) k k q (r ) k q (r ) k k k ( 4π c2 δ k, 2ω k δ r,r + q (r k k = 4π c2 δ k, 2ω k δ r,r q (r k k ) ) δ k, k δ r,r + q (r) q (r ) k k q (r) ) ) q (r) k q (r ) ) q (r ) k q (r ) k q (r ) k q (r) k k

38 32 2 (2.21) i [ A( x, t), H] = i V k r ω k (q (r) e iωkt e i k x q (r) k e iωkt e i k x ) e k r, k (2.58) (2.55) (2.58) (2.56) (2.57) a (r), a (r) k k a (r) k a (r) k 2ωk 4π c 2 q(r) (2.59) k 2ωk 4π c 2 q(r) (2.60) k (2.56) (2.57) a (r), a (r) k k [ (r) a, a (r ) ] k k [ (r) a, a (r )] k k = δ k, k δ r,r (2.61) = [ a (r), a (r ) ] k k = 0 (2.62) a (r), a (r) k k (2.59) (2.60) (2.50) (2.61) (2.62) U = ( ω k k r a (r) a (r) k k ) (2.63) e r k number operator e r k n r, k

39 n r, k > < n r, k U n r, k > = ω k n r, k 1 ω k (2.52) M = k r ( k a (r) k a (r) k ) = k r ka (r) a (r) k (2.64) k k (2.54) L = ( 1) r k ( k k r a (r) k + 1 ) = k ( 1) r k k 2 a (r) r k a(r) k a (r) k (2.65) r = 1 r = 2 k (1.10) V (1.26) H = V [ 1 d 3 x 2c φ ( φ) ( mc ) ] 2 φ 2 (2.66)

40 34 2 natural unit cgs gauss φ( x, t) = 1 V k q( k, t)e i k x (2.41) (2.43) (1.10) ω k = ( kc) 2 + (mc 2 ) 2 (2.67) q( k, t) = q k e iω kt φ( x, t) = 1 V k ( q k e iω kt+i k x + q k e iω kt i k x ) (2.66) (2.68) δ k, k H = k ω 2 k c 2 (q k q k + q k q k ) (2.69) ω 2 k k 2 c 2 (mc/ ) 2 (2.67) M ( M = d 3 x 1 c φ ) 2 φ (2.70) V natural unit cgs gauss 1/c c (2.68) M = k ω k c 2 k(q k q k + q k q k ) (2.71)

41 k k δ k, k ω k k = ω k ( k) k = ω k k = ω k k k k = φ = i [φ, H] (2.72) [ q k, q k ] a k, a k = c2 2ω k δ k, k (2.73) [q k, q k ] = [ q k, q k ] = 0 (2.74) a k a k 2ωk c 2 q k (2.75) 2ωk c 2 q (2.76) k (2.73) (2.74) a k, a k [ a k, a k ] = δ k, k (2.77) [a k, a k ] = [ a, a k k ] = 0 (2.78) a k, a k

42 36 2 H = ( ω k a a k k + 1 ) k 2 (2.79) M = k ka k a k (2.80) (2.64) k k 1 2 = 0

43 Friedmann H 2 = 8πG 3 ρ H = ȧ/a a ρ Hubble length l H = c/h Hubble length Hubble length Friedmann M H = 4π 3 l3 Hρ = c2 2G l H Hubble length Hubble length Schwarzschild 2GM H c 2 Schwarzschild Hubble length Schwarzschild

44 38 3 Schwarzschild Schwarzschild Hubble length Hubble length M H c 2 E < M H c 2 t t 2 1 M H c 2 Hubble length t age l/c = 1/H Hubble time 1 ( H = c ) 3 1 H2 H c 3 = 2GM H c 3 2M H c 2 M H 1 c 2 G Planck mass c M pl = G = g = GeV (3.1) c2 Planck

45 ρc 2 < ( )4 3 c 3 32π( c) 3 G c2 = 32π( c) 3 (M plc 2 ) 4 (3.2) π 2 15( c) 3 (k BT ) 4 k B T < M pl c 2 Planck time Reduced Planck mass t pl = sec (3.3) M pl M pl = GeV (3.4) 8π c ( ds 2 = c 2 dt 2 + a(t) 2 1 ) 1 Kr 2 dr2 + r 2 (dθ 2 + sin 2 θdφ 2 ) (3.5) a(t) (r, θ, φ) 3 3 K

46 40 3 K > K = 0 K < 0 H 2 = H 2 0 (Ω r0 (1 + z) 4 + Ω m0 (1 + z) 3 + Ω K0 (1 + z) 2 ) (3.6) H = ȧ/a t H z = a 0 /a a 0 Ω r0, Ω m0 Ω K0 ρ K = 3Kc2 8πGa 2 0 (1 + z) 2 z Ω K = ρ K ρ cr = Kc2 ȧ 2 = ȧ2 0 ȧ 2 Ω K0 (3.7) ȧ (3.7) Ω K a = (t/t 0 ) 0.5 ȧ 2 0/ȧ 2 = t/t 0 = (a/a 0 ) 2 T a = T 0 a 0 Ω K (t pl ) ( )2 kb T 0 M pl c 2 Ω K Ω K0 (3.8) 0 < Ω K0 < O(1) 62 0

47 Ω K = 0 t cdt l p (t) = a(t) 0 a (3.9) a t n ä n(n 1)t n 2 n = 1 n < 1 n > 1 (3.9) 1 l p (t) = 1 n ctn (t n+1 (t = 0) n+1 ) ct = 1 n for n < 1 = for n > 1

48 42 3 CMB 2.73K CMB CMB t i l p = ct n t 1 n i /(1 n) t i = 0 (3.7) ȧ Ω K ä a = 4πG 3 ( ρ + 3P c 2 ) (3.10) P = wρc 2 w < Slow roll inflation φ

49 ρc 2 = 1 φ 2 2c + V 2 H 2 = 8πG 3 [ 1 2c φ ] c 2 V = c [ 1 3 M pl 2 2c φ ] c 2 V (3.11) M pl c/8πg reduced Planck mass P = wρc 2 1 < w < 1/3 φ + 3H φ + dv dφ = 0 (3.12) Slow roll 1 φ 2 2c 2 φ slow roll parameters slow roll ε 1 2 M 2 pl η M pl 2 V V ( V V )2 (3.13) (3.14) V, V φ Slow roll H M V (3.15) pl 2 3H φ V (3.16) 1 2c 2 φ 2 V 1 1 V 2 2 V 9H M ( V pl 2 V )2 = 1 3 ε (3.17)

50 44 3 ε (3.15) 2HḢ = 1 3 M V φ pl 2 H 2 H 2 Ḣ = 1 6 M 2 pl V H φ 1 18 M V 2 pl 2 (3.16) H 4 Ḣ H 2 = ε (3.18) (3.16) 3H φ + 3Ḣ φ = V φ (3.12) φ 3H V φ 9H ε = 1 3 M pl 2 V V ε = 1 ( η + ε) (3.19) 3 ε, η e-folding N(t) = ln a(t end) a(t) = a(tend ) a(t) da a (3.20) t t end

51 (3.15),(3.16) (3.20) N(t) = tend t ȧ φ(tend ) a dt = φ(t) H 2 dφ = H φ φ(tend ) φ(t) V V dφ (3.21) M pl 2 ε(φ(t end )) = 1 (3.22) reheating) (event horizen) cdt l H = a(t) t a (3.23) a t n n > 1 l H = 1/(n 1)ct n < 1 c/h = ct/n l H = n c n 1 H c H

52 46 3 r H l H a t1 n t λ λ a(t λ )λ = c H(t λ ) λ λ = 1 a 0 c H 0 a 0 a(t λ )H(t λ ) a 0 H 0 = 1 a(t λ )H(t λ ) = a(t λ) a(t end ) H(t λ ) a 0 H 0 a(t end ) a 0 H 0 a(t end )T end = a 0 T 0 k B T /( )ev, k B T end GeV

53 H(t λ ) 2 = 8π 3 G V c 2 8π 3 GaT 4 end c 2 H 2 0 = 8π 3 G at 4 0 Ω γ c 2 Ω γ = h , a = 8π5 k 4 B /15c3 h 3 e-folding N(t λ/a(tλ )=c/h 0 ) = 61 (3.24) Exponential 61 horizen exit horizen reentry

54 g( x, t) g( x, t) = k ĝ k (t)e i k x (4.1) x r a(t) r = a(t) x φ 0 (t) δφ( x, t) (φ 0 + δφ) V (φ 0 + δφ) = 0 (4.2) φ 0 V (φ 0 ) = 0 δφ V (φ 0 )δφ = 0 (4.3)

55 δφ = g µν µ ν δφ = g µν µ ν δφ (4.4) = g µν ( µ ν δφ α δφγ α µν) (4.5) ds 2 = c 2 dt 2 + a(t) 2 (dx 2 + dy 2 + dz 2 ) (4.6) Γ 0 oo = 0, Γ i 00 = 0, Γ 0 ij = aȧδ ij (4.7) δ φ + 3Hδ φ 1 a 2 2 δφ + V δφ = 0 (4.8) du dt = P dv dt U = ρv V P d ( 1 dt 2 φ φ 0 δ φ + V (φ)) = ( φ φ 0 δ φ) 1 V V = a 3 d 3 x peculiar velocity 1 V dv dt = 3ȧ a + div v div v v( x, t) t dv dt

56 50 4 x(t) x(t) + d x(t) d 3 x(t) dt x(t + dt) = x(t) + v( x(t), t)dt x(t + dt) + d x(t + dt) = x(t) + d x(t) + v( x(t) + d x((t), t))dt d x(t + dt) = d x(t) + ( v( x(t) + d x(t), t) v( x(t), t)) dt ( x(t + dt)) ( x(t)) = 1 + x v x dt y v x dt z v x dt x v y dt 1 + y v y dt z v y dt x v z dt y v z dt 1 + z v z dt 1 + div vdt div v peculiar velocity v = φ 0 1 a 2 δφ ρ 0 + P 0 = 1 a 2 δφ φ 0 dv (φ 0 + δφ) dt = (V (φ 0 ) + V (φ 0 )δφ)( φ 0 + δ φ) = V φ0 + V δ φ + V δφ φ 0 φ 0 ( φ 0 + 3H φ 0 + V ) +δ φ 0 ( φ 0 + 3H φ 0 + V ) + φ 0 (δ φ + 3Hδ φ 1 a 2 δφ + V δφ) = 0

57 Massless scalar field massless free field V = 0 δφ k (t) δ φ k + 3Hδ φ k + k2 a 2 δφ k = 0 (4.9) δφ(t) δφ k δφ k = w k (t)a k + w k(t)a k (4.10) a k, a k w k δφ k ( w k = A i + k ) e i k ah (4.11) ah A Hubble horizen a/k 1/H subhorizen scale t 1 Hubble time t H(t t 1 ) 1 a(t) a 1 + ȧ 1 (t t 1 ) = a 1 (1 + H(t t 1 )) (4.12) a 1 = a(t 1 ) t 1 w k

58 52 4 k a(t)h k a 1 H k (t t 1 ) (4.13) a 1 subhorizen scale k/ah 1 w k w k k a 1 H Aei k a 1 H +i k a t 1 1 e i k a t 1 (4.14) mass less free scalar field φ( r, t) = 1 ( a p e iωpt + a V p 2ω eiω pt ) e i p r p (4.15) p V r, p r = a(t) x, p = k/a(t) ω p = p = k/a(t) subhorizen scale A = H e iδ 1 1p (4.16) ω 1p V 2ω 1p H(t t 1 ) 1 t 1 1 δ 1p = k a 1 H + k a 1 t 1 (4.17) A V = a(t) 3 V c δφ( x, t) = H 2Vc ( + i + ( k [ i + 3/2 k 1 k a(t)h k ) e i k ah iδ ip a k (4.18) ah ) e i k ah +iδip a k ]ei k x (4.19)

59 > a k 0 > = 0 (4.20) < 0 δφ( x, t) 0 > = 0 (4.21) < 0 a k = 0 < 0 δφ( x, t) 2 0 > = H2 2V c k 1 k 3 ( 1 + k2 a 2 H 2 ) (4.22) V c k (2π) 3 d3 k (4.23) Hubble horizen super horizen limit a/k 1/H k/ah 1 < 0 δφ( x, t) 2 0 > = H 2 d 3 k ( H )2 2(2π) 3 k 3 = dk 2π 0 k (4.24) g < g 2 > = 0 P g (k) dk k (4.25)

60 54 4 P φ = ( H 2π )2 (4.26) ε t /ε c t c/h ε < H H c, (Conformal time) dτ = dt a Conformal time (4.27) τ τ 1 = = t t 1 [ 1 1 a H dt a a = da a 2 H a 1 ]a a da + a 1 a 1 a a = 1 ah + 1 a 1 H 1 + = 1 ah + 1 a 1 H 1 + a 1 a a 1 d ( 1 ) da H da aȧ da a 2 H ε ( Ḣ H 2 = 1 ah + 1 ε a 1 H 1 ah + ε 1 + a 1 H 1 ) a a 1 ( da a 2 Ḣ H H 2 ε + ε ) H

61 ε = Ḣ/H2 d/dt = φ 0 d/dφ 0 ε ε = M 2 pl ( V V 2 V 2 V 3 ) V φ0 H 2 V/3 M 2 pl V φ 0 = 6 M pl 2 HḢ ε = 2H(2ε η)ε (4.28) τ Conformal time τ τ 1 = = 1 ah + 1 a 1 H 1 + ε(τ τ 1 ) τ = ε ah (4.29) w k = u k /a u k ε ( u k + k ε ) τ 2 u k = 0 (4.30) ξ = kτ u k = ξh k (ξ) d 2 H k dξ ξ dh ( k dξ + 9/4 + 3ε ) 1 ξ 2 H k = 0 (4.31) ν ν = 9/4 + 3ε 3/2 + ε Bessel J ν Neumann N ν Hankel H ν (1), H ν (2) w k

62 56 4 w k = 1 a [ α(k) ξh (1) ν (ξ) + β(k) ξh (2) ν (ξ) ] (4.32) Hubble horizen flat space time Subhorizen limit ξ = kτ k/ah 1 Hankel ξ 2 H ν (1) (ξ) H (1) ν (ξ) πξ exp [ i 2 πξ exp [ i ( ξ (2ν + 1) π )] 4 ( ξ (2ν + 1) π )] 4 (4.33) (4.34) Subhorizen limit t 1 t ξ ξ k a 1 H 1 k a 1 (t t 1 ) w k subhorizen limit w k 1 2 k [α(k) a 1 π e i a (t t 1) 1 e i( k a 1 H π (2ν+1)) a (t t +β(k) 1 ) 1 e i( k a 1 H π (2ν+1)) 1 4 π ei k Flat space time mass less scalar field β(k) = 0 (4.35) α(k) = 1 π Vc 4k e i( k a 1 H π (2ν+1)) 1 4 (4.36) V c = v/a 3 1 ω p = p = k/a 1 p r = k x w k

63 w k = 1 1 π a Vc 4k e i( k a 1 H π (2ν+1)) 1 4 ξh (1) ν (ξ) (4.37) horizen super horizen limit ξ = kτ k/ah 1 H ν (1) i Γ (ν) ( ξ π 2) ν w k H V c Γ (ν) 1 Γ (3/2) k 3/2 2ν 2 e i( k a 1 H + π (ν 1/2)) ( ah 1 2 k )ε (4.38) Γ (3/2) = π/2 Γ (ν)/γ (3/2) 1 massless P δφk (k) = ( Γ (ν) ( H )2 ( a2h )2ε k Γ (3/2))2 2ε (4.39) 2π k Super horizen limit w k ẅ k + 3Hẇ k 0 H > 0 w k = const. horizen exit k Horizen exit t a(t ) k = 1 H(t ) a H /k = 1 super horizen horizen exit P δφk (k) = ( Γ (ν) )2 ( )2 H Γ (3/2) 2π (4.40)

64 58 4 H horizen exit H(t) 2 (H + Ḣ (t t )) 2 = H 2 (1 2H ε(t t )) (a(t)/a ) 2ε e 2εH (t t ) 1 + 2H ε(t t ) H 2 a 2ε H 2 a 2ε horizen exit n n = d ln P δφ k (k) d ln k (4.41) d d ln k = φ 0 d H dφ M 2 V d pl 0 V dφ 0 d ln H 2 d ln k 2ε n 2ε massless scalar field (ε > 0) horizen exit k Scalar filed with non zero mass simplified treatment V 0 u k 3a 2 H 2 ηu 3η τ 2 u k (4.42) H k ν = 9/4 + 3ε 3η 3/2 + ε η

65 P δφk (k) k 2ε+2η (4.43) η > 0 horizen exit Generation of Adiabatic perturbation via Inflation T Inf = GeV ct pl ( Tpl T Inf )2 = cm a

66 60 4 a 0 CMB a 0 a = T Inf T 0 ( )2 Tpl T Inf ct pl = cm T Inf T δt reh δt reh = δφ φ 0 t reh g i ġ i g i δt reh δg i = g i δt reh (4.44)

67 i δg 1 = δg 2 =,,, = δg n (4.45) g 1 g 2 g n density contrast δ m, δ r ρ m = 3Hρ m ρ r = 4Hρ r ρ m δ m ρ m = ρ rδ r ρ r δ m 3 = δ r 4 (4.46) Scalar, vector, tensor decomposition of perturbation T µν g µν z Conformal time ds 2 = a 2 (τ){ (1 + 2A)dτ 2 2B i dτdx i + [(1 + 2D)δ ij + 2E ij ]dx i dx j }

68 62 4 g µν = a 2 (τ) (1 + 2A) B 1 B 2 B 3 B D + 2E 11 2E 12 2E 13 B 2 2E D + 2E 22 2E 23 B 3 2E 31 2E D + 2E 33 A lapse function B i = (B 1, B 2, B 3 ) shift vector det(g µν ) = a 8 (τ)(1 + 2A + 2D 3) g µν g µν 1 2A 6D = a 2 (τ) 1 + 6D B 1 B 2 B 3 B A + 4D + 2E E 33 2E 21 2E 31 B 2 2E A + 4D + 2E E 33 2E 32 B 3 2E 13 2E A + 4D + 2E E 22 g µν = 1 a 2 (τ) (1 2A) B 1 B 2 B 3 B 1 1 2D 2E 11 2E 21 2E 31 B 2 2E D 2E 22 2E 32 B 3 2E 13 2E D 2E 33 A, D Shift vector k B/k = B 3 = ib (B 1, B 2, 0) E ij

69 E S ij = 1 3 E V ij = i 2 E E E 0 0 E E 2 E 1 E 2 0 (4.47) (4.48) E T ij = E + E 0 E E (4.49) T µν (ρ + P )U µ U ν + P g µν + Σ µν (4.50) ρ = ρ 0 P = P 0 Σ µν = 0 U µ = dx µ /dλ = (1/a, 0, 0, 0) Σ µν v i = dxi dτ λ dλ 2 = g µν dx µ dx ν = a 2 (τ)dτ 2 { (1 + 2A) 2B i v i + [(1 + 2D)δ ij + 2E ij ]v i v j } a 2 (τ)dτ 2 (1 + 2A + 2B i v i v 2 ) dλ = a(τ) ( 1 + 2A + 2B i v i v 2 dτ = a(τ) 1 + A + B i v i 1 2 v2) dτ Conformal time A lapse function U i = dx i /dλ = v i /a, U 0 = (1 A

70 64 4 B i v i + v 2 /2)/a v i v i U µ = g µν U ν U i = a(τ)( B i + v i ) (4.51) U 0 = a(τ)(1 + A v2 ) a(τ) (4.52) Shift vector shift vector T µ ν = g να T µα T µ ν = (ρ 0 + δρ + P 0 + δp )U µ U ν + (P 0 + δp )δ µ ν + Σ µ ν (4.53) T 0 0 = (ρ 0 + δρ) (ρ 0 + P 0 )( B i v i + v 2 ) (ρ 0 + δρ)(4.54) T 0 i = (ρ 0 + P 0 )(v i B i ) (4.55) T i 0 = (ρ 0 + P 0 )v i (4.56) T i j = (P 0 + δp )δ ij + Σ i j (4.57) δρ δp k v/k = iv v = (V 1, V 2, 0) T µν = g µα T α ν T 00 = g 00 T 0 0 = a 2 (1 + 2A)(ρ 0 + δρ) = a 2 (ρ 0 + 2Aρ 0 + δρ) T 0i = g 00 T 0 i + g 0j T j i = a2 (ρ 0 + P 0 )(v i B i ) a 2 B j P 0 δ ij T ij = g i0 T 0 j + g ik T k j = a 2 [(1 + 2D)δ ik + 2E ik ][(P 0 + δp )δ k j + Σ k j ] = a 2 (P 0 δ ij + (2DP 0 + δp )δ ij + 2P 0 E ij + Σ ij ) Σ S ij = 1 3 Σ Σ Σ (4.58)

71 Σ V ij = i 2 Σ T ij = 0 0 Σ Σ 2 Σ 1 Σ 2 0 Σ + Σ 0 Σ Σ Gauge transformation of scalar perturbations (4.59) (4.60) x µ = x µ + ξ µ (4.61) ξ µ z ξ µ = (δτ, 0, 0, 0), (0, 0, 0, iδx) ξ µ = (0, δx 1, δx 2, 0) δρ ρ(t) t = t + ξ 0 (x i ) ρ( t(x i ))

72 66 4 t(x i ) = t t = t t = t ξ 0 t ρ( t(x i ) = t) = ρ(t ξ 0 (x i )) ρ(t) ξ 0 (x i ) ρ(t) (4.62) ρ < 0 ξ 0 (x i ) > 0 ρ(t) ξ 0 ρ ρ t = t + ξ 0 ξ 0 t t ξ 0 ξ 0 ξ 0 ρ Lie δ L ρ = ρ(t) ρ(t) (4.63) Lie B µν x σ = x σ + ξ σ B µν B µν ( x σ ) = xα x µ x β x ν B αβ(x σ ) (4.64)

73 x α x xα µ x µ x α x µ = δ α µ + µ ξ α (4.65) x α x µ 1 + i ξ i 0 ξ 1 0 ξ 2 0 ξ 3 = (1 µ ξ µ ) 1 ξ ξ ξ ξ 3 1 ξ 2 1 ξ 3 2 ξ 0 2 ξ ξ ξ ξ 3 2 ξ 3 3 ξ 0 3 ξ 1 3 ξ ξ ξ ξ 2 = 1 0 ξ 0 0 ξ 1 0 ξ 2 0 ξ 3 1 ξ ξ 1 1 ξ 2 1 ξ 3 2 ξ 0 2 ξ ξ 2 2 ξ 3 3 ξ 0 3 ξ 1 3 ξ ξ 3 x α x µ = δ α µ µ ξ α (4.66) Lie δ L B µν (x σ ) = B µν (x σ ) B µν (x σ ) (4.67) = ξ σ σ B µν (x σ ) µ ξ α B αν (x σ ) ν ξ β B µβ (x σ ) (4.68) ξ µ = (δτ, 0, 0, iδx) g 00

74 68 4 g 00 = g 00 0 ξ β g 0β 0 ξ α g 0α ξ σ σ g 00 = a 2 (1 + 2A) + 2δτ a 2 (1 + 2A) + 2iδx a 2 ( B 3 ) + δτ(a 2 (1 + 2A)) iδx(a 2 (1 + 2A)), 3 Ã = A δτ a a δτ = A δτ ahδτ (4.69) 0 τ, 3 x 3 0 a = a 2 H g 0i g 03 = g 03 0 ξ β g 3β 3 ξ α g 0α ξ σ σ g 03 = g 03 0 ξ 3 g 33 3 ξ 0 g 00 3 δτ = ikδτ g ij B = B + δx + kδτ (4.70) g ij = g ij i ξ β g jβ j ξ α g iα ξ σ σ g ij = g ij i ξ k g jk j ξ k g ik ξ 0 0 g ij a 2 [(1 + 2 D)δ ij + 2Ẽij] = a 2 [(1 + 2D)δ ij + 2E ij ] i ( iδx)a 2 δ j3 j ( iδx)a 2 δ i3 δτ2aa δ ij = a 2 [(1 + 2D)δ ij + 2E ij ] kδxa 2 δ i3 δ j3 kδxa 2 δ j3 δ i3 δτ2aa δ ij Ẽ = E + kδx (4.71) D = D k δx ahδτ (4.72) 3 T 00

75 Ãρ 0 + δ ρ = 2Aρ 0 + δρ 2δτ ρ 0 2δτaHρ 0 δτρ 0 δ ρ = δρ ρ 0δτ (4.73) T 0i a 2 (ρ 0 + P 0 )(ṽ 3 B 3 ) a 2 P 0 B3 = a 2 (ρ 0 + P 0 )(v 3 B 3 ) a 2 P 0 B 3 δx 3a 2 P 0 δτ, 3 a 2 ρ 0 Ṽ = V + δx (4.74) x x = 0 x = δx x δx δx T ij a 2 (P 0 δ ij + (2 DP 0 + δ P )δ ij + 2P 0 Ẽ ij + Σ ij ) = a 2 (P 0 δ ij + (2DP 0 + δp )δ ij + 2P 0 E ij + Σ ij ) i ξ k a 2 P 0 δ kj j ξ k a 2 P 0 δ ik ξ 0 (2aa P 0 + a 2 P 0)δ ij (2 DP 0 + δ P )δ ij + 2P 0 Ẽ ij + Σ ij = (2DP 0 + δp )δ ij + 2P 0 E ij + Σ ij kδxp 0 δ 3 i δ k 3 δ kj kδxp 0 δ 3 j δ k 3 P 0 δ ik δτ(2ahp 0 + P 0)δ ij Σ = Σ (4.75) δ P = δp δτp 0 (4.76) δφ ρ = 1/2 φ 2 + V = 1/2a 2 φ 2 + V P = 1/2 φ 2 V = 1/2a 2 φ 2 V φ = φ 0 + δφ δρ δp

76 70 4 δ φ = δφ φ 0δτ (4.77) k R (3) k R (3) k = (D 4k2 a 2 + E ) 3 (4.78) ϕ D E (4.79) Gauge invariant variables ϕ = ϕ ahδτ (4.80) Bardeen variable ζ ζ ϕ + 1 δρ (4.81) 3 ρ 0 + P 0 ρ 0 = 3aH(ρ 0 + P 0 ) Entropy perturbation P P δp P 0 ρ δρ (4.82) 0 P = P (ρ) P = dp dρ ρ

77 P = ( δp δρ dp ) δρ (4.83) dρ δp δρ = dp dρ P = 0 P 0 Velocity perturbation V s Scalar field perturbation φ V s V E k (4.84) φ δφ + 1 k Sasaki-Mukuhanov variable Q ) (B E φ 0 (4.85) k Q δφ φ 0 ah ϕ = δφ φ 0 H ϕ (4.86) threading slicing Conformal Newtonian gauge ds 2 = a 2 [ (1 + 2Ψ N )dτ 2 + (1 2Φ N )δ ij dx i dx j ] (4.87) A N Ψ N (4.88) D N Φ N (4.89) B N = E N = 0 (4.90) R (3) kn = 4k2 a 2 Φ N (4.91) slicing threading

78 72 4 Total matter gauge δφ = 0 Conformal Newtonian gauge scalar comoving gauge Slicing threading Conformal Newtonian gauge thread thread conformal Newtonian gauge δφ = 0 δx = 0 (4.92) δτ = V N k total matter gauge (4.93) Ψ T M = Ψ N V N k ah V N (4.94) k B T M = V N (4.95) D T M = Φ N ah V N (4.96) k E T M = 0 (4.97) V T M = V N (4.98) δρ T M = δρ N ρ V N 0 k δp T M = δp N P 0 V N k (4.99) (4.100) Spatially flat gauge (slicing) D = E = 0 ϕ = 0 Q = δφ (4.101) ds 2 = a 2 [ (1 + 2A)dτ 2 2B i dτdx i + δ ij dx i dx j ] (4.102)

79 g 00 = a 2 (1 + 2A) g 0i = a 2 B i g ij a 2 δ ij (4.103) g 00 = 1 a (1 2A) g 0i = Bi 2 a g ij = 1 2 a δ 2 ij Comoving gauge (slicing) T 0 i = 1 a 2 φ 0 i δφ = 0 (4.104) i δφ = 0 δφ = 0 (4.105) Q = φ 0 H ϕ (4.106) Spatially flat δφ SP F TM ϕ T M ϕ T M = H δφ SP F φ 0 = ȧδt a (4.107) δt = δφ SP F / φ 0 φ TM φ TM δφ = 0 Spatially flat δφ Spatially flat 0 δt Spatially flat by definition TM δt a = ȧδt δt < 0 a < 0 a

80 74 4 TM a a TM a/a Spatially flat gauge Q spatially flat spatially flat Γ 0 00 = 1 2 (g00 g 00,0 + g 0i (g i0,0 + g i0,0 g 00,i )) = 1 2 ( 1 a 2 (1 2A)(2aa (1 + 2A) + a 2 2A )) = a a + A Γ 0 i0 = 1 2 (g00 g 00,i + g 0j (g ji,0 + g j0,i g i0,j )) = A,i a a B i = iδ i3 (ka + a a B) Γ 0 ij = 1 2 (g00 (g 0i,j + g 0j,i g ij,0 ) + g 0k (g ki,j + g kj,i g ij,k )) = a a δ ij (B i,j + B j,i ) 2 a a Aδ ij = a a + kbδ i3δ j3 2 a a Aδ ij Γ i 00 = 1 2 (gi0 g 00,0 + g ij (2g j0,0 g 00,j )) = a a B i B i + A,i = iδ i3 ( a a B + B + ka) Γ i j0 = 1 2 (gi0 g 00,j + g ik (g kj,0 + g k0,j g j0,k )) = a a δ ij 1 2 (B i,j B j,i ) + D δ ij + E ij = a a δ ij + D δ ij + E ij

81 Γ i jk = 1 2 ( gi0 g jk,0 ) = a a B iδ jk = i a a Bδ jkδ i3 Γ µ 0µ = 4 a a + A + 3D (4.108) Γ µ iµ = A,i = ikaδ i3 (4.109) µ T ν µ = µ T ν µ + Γ µ µβ T ν β ΓµνT β µ β = 0 (4.110) T ν µ = ( g µα α φ ν φ + δ ν µ 1 ) 2 gαβ α φ β φ V (4.111) [ 1 T0 0 = 2a 2 φ V (φ 0 ) A ] a 2 φ φ 0 a 2 δφ + V δφ (4.112) T 0 i = 1 a 2 φ 0 i δφ = ik 1 a 2 φ 0δφ (4.113) T0 i = B i a 2 φ a 2 φ 0 i δφ = i a 2 (Bφ kφ 0δφ) (4.114) ( φ Tj i = δj i 2 0 2a 2 V (φ 0) + φ 0 a 2 δφ A ) a 2 φ 2 0 V δφ (4.115) δρ φ = A a 2 φ φ 0 a 2 δφ + V δφ (4.116) δp φ = φ 0 a 2 δφ A a 2 φ 2 0 V δφ (4.117) (T 0 0 ) + i T i 0 + Γ µ 0µ T Γ µ iµ T i 0 Γ 0 00T 0 0 Γ 0 0iT i 0 Γ i 00T 0 i Γ i 0jT j i = 0 0 = (ρ 0 + δρ) k a 2 (Bφ kφ 0δφ) (4 a a + A + 3D )(ρ 0 + δρ)

82 76 4 +( a a + A )(ρ 0 + δρ) 3 a a (P 0 + δp ) 3D P 0 0 = δρ k a 2 (Bφ kφ 0δφ) 3 a a (δρ + δp ) 3D (ρ 0 + P 0 ) δρ = A a 2 φ 2 0 2A a 2 φ 0φ 0 + 2a a 3 Aφ φ 0 a 2 δφ + φ 0 a 2 δφ 2a a 3 φ 0δφ +V φ 0δφ + V δφ φ 0 + 2aHφ 0 + a 2 V = 0 (4.118) 0 = δφ + 2aHδφ + (a 2 V + k 2 )δφ A φ 0 + 2a 2 AV + kbφ 0 + 3D φ 0 (4.119) spatially flat gauge T = T µ µ = ρ 0 + 3P 0 δρ + 3δP R µν = Γ α µν,α Γ α µα,ν + Γ α ασγ σ µν Γ α νσγ σ αµ (4.120) R 00 R 00 = Γ00,α α Γ0α,0 α + ΓασΓ α 00 σ Γ0σΓ α α0 σ = ( a ( a ) a + A ) + i 3 a B + B + ka + (4 a a + A ) (4 ( a a + A ) a ( a ) a + A ) + ia,3 a B + B + ka

83 ( a a + A )2 ( a a )2 = 3 a a + 3a 2 a 2 +3aHA k 2 A kahb kb T T g 00 = g 0µ T µ T g 00 = a 2 (1 + 2A)( ρ 0 δρ) a2 (1 + 2A)( ρ 0 δρ + 3P 0 + 3δP ) = a2 2 (ρ 0 + 3P 0 ) + a2 2 (δρ + 3δP ) + a2 A(ρ 0 + 3P 0 ) = a2 2 (ρ 0 + 3P 0 ) + 2φ 0δφ a 2 V δφ 2a 2 V A 00 (4.121) 3 a a + 3a 2 a 2 = ä a 1 M 2 pl a 2 2 (ρ 0 + 3P 0 ) (4.122) = 4π 3 G(ρ 0 + 3P 0 ) (4.123) 3aHA k 2 A kahb kb 3D 3 a a D = 1 M 2 pl (2φ 0δφ a 2 V δφ 2a 2 V A) (4.124) R 0i R 0i = Γ0i,α α Γ0α,i α + ΓασΓ α 0i σ ΓiσΓ α α0 σ [ ( = iδ i3 2k a a A + a a )2 ] a B + B a

84 78 4 T 0i 1 2 T g i0 = g 00 Ti 0 + g 0j T j i 1 2 T g 0i = a 2 (1 + 2A) ( 1 ) a 2 ikφ 0δφδ i3 a 2 B i P a2 B i ( ρ 0 + 3P 0 ) = ikφ 0δφδ i3 + a2 2 B i( ρ 0 + P 0 ) = ikφ 0δφδ i3 i a2 2 B( ρ 0 + P 0 )δ i3 = ikφ 0δφδ i3 + ia 2 BV δ i3 ( 2ik a a )2 a A + ia a B + i B = a 1 M 2 pl (ikφ 0δφ + ia 2 BV ) ij ( a 3 a )2 = a2 ρ M pl 2 0 (4.125) (4.122) a ( a )2 a + a = a 2 2 M 2 pl (ρ 0 P 0 ) = a2 V (4.126) M pl 2 03 B 2 a a A = 1 φ M 0δφ (4.127) pl 2 R ij R ij = Γ α ij,α Γ α iα,j + Γ σ ασγ σ ij Γ α jσγ σ αi = Γ 0 ij,0 + Γ k ij,k Γ α iα,j + Γ α α0γ 0 ij + Γ α αkγ k ij Γj0Γ 0 0i 0 ΓjkΓ 0 0j k Γj0Γ l li 0 ΓjkΓ l li k

85 = ( a a + ( a a )2) ( a ( a )2) δ ij 2A a + δ ij a a a A δ ij +k 2 Aδ i3 δ j3 + 2k a a Bδ i3δ j3 + k a a Bδ ij + kb δ i3 δ j3 T ij 1 2 T g ij = g iµ T µ j 1 2 T g ij = a 2 (P 0 + δp )δ ij a2 2 ( ρ 0 + 3P 0 δρ + 3δP )δ ij = a2 2 (ρ 0 P 0 )δ ij + a 2 V δφδ ij ( a ( a )2) a + a = a 2 2 M (ρ pl 2 0 P 0 ) (4.128) a 2 ( a ((V M ( a )2) δφ + 2D)δ pl 2 ij + 2E ij ) = 2A a + δ ij a a a A δ ij + k 2 Aδ i3 δ j3 +2k a a Bδ i3δ j3 + k a a Bδ ij + kb (4.129) δ i3 δ j3 3a 2 ( a V M ( a )2) δφ = 6 pl 2 a + A 3 a a a A +k 2 A + 5k a a B + kb (4.130) a = a 2 H, a = 2a 3 H 2 + a 3 Ḣ a ( a )2 a + a = a 2 (3H 2 + Ḣ)

86 80 4 a ( a )2 a a (4.138) H 2 = = a 2 2 M 2 pl (ρ 0 P 0 ) = a2 V (4.131) M pl 2 = a 2 ä a = a2 4π 3 G(ρ 0 + 3P 0 ) (4.132) 1 3 M 2 pl ρ 0 = 1 ( 1 ) 3 M pl 2 2a 2 φ V (4.133) 3H 2 + Ḣ = V M 2 pl (4.134) (4.140) (4.141) Ḣ = 1 φ 2 0 2a 2 M 2 pl (4.135) conformal time Ḧ = φ 0φ 0 a 3 M pl 2 = 2φ 0 aφ 0 + H φ 2 0 a 2 M pl 2 Ḣ 2HḢ (4.136) V = 1 a 2 (φ 0 + 2aHφ 0) 3 H ( k )2 a A A k a a HB k a 2 B = 1 M 2 pl ( 2φ 0δφ a 2 V δφ 2V A) (4.137)

87 HA = 1 a M φ 0δφ (4.138) pl 2 3 M 2 pl V δφ = 6(3H 2 + Ḣ)A 3H a A ( k )2 + A + 5 k a a HB + k a 2 B (4.139) δφ + 2aHδφ + (a 2 V + k 2 )δφ A φ 0 + 2AV + kbφ 0 = 0 (4.140) (4.149) A = 1 2 M 2 pl 1 δφ ah φ 0δφ = aḣ H φ 0 (4.141) (4.148) (4.150) 1 M 2 pl ( φ 0δφ a 2 + V δφ V A) = 3(3H 2 + Ḣ)A + 2k a HB (4.146) 1 M 2 pl φ 0δφ a 2 = φ 2 0 2a 2 M 2 pl 2δφ = 2Ḣ δφ φ 0 (4.145) conformal time V φ M pl 2 0 = 6aHḢ + aḧ (4.142) Ṽ M 2 pl A = (3H 2 + Ḣ)A

88 82 4 (1.152) 2 k δφ HB = 2Ḣ a φ + (6aHḢ + aḧ)δφ 0 φ 0 +2(3H 2 + Ḣ)A 2 k δφ HB = 2Ḣ a φ + aḧ δφ 2aḢ2 δφ 0 φ 0 H φ 0 k a B = Ḣ δφ H φ + a δφ ( ) Ḧ 0 φ 0 2H Ḣ2 H 2 (4.143) δφ + 2aHδφ + (a 2 V + k 2 )δφ (4.152) conformal time A = a Ḣ Hφ δφ + aḣ 0 H +a 2 Ḣ2 H 2 φ δφ a 2 0 (4.155) = A φ 0 2AV kbφ 0 (4.144) φ 0 φ 2 0 Ḧ Hφ 0 (r.h.s.) = φ 0[ a Ḣ Hφ δφ + aḣ 0 H +a 2 Ḣ2 H 2 φ δφ a 2 Ḧ 0 ( 2 a Ḣ aφ 0 [ = a 2 Ḣ ah Hφ 0 δφ Hφ 0 δφ a 2 Ḣ φ δφ (4.145) 0 φ 0 φ 2 0 δφ δφ a 2 Ḣ φ δφ] 0 ) ) ( 1 a 2 (φ 0 + 2aHφ 0) ( δφ Ḣ H φ + a δφ 0 φ 0 φ 0 2Ḣ2 φ + 0 ( Ḧ 2H Ḣ2 H 2 H 2 3 Ḧ 2 H 5Ḣ δφ )) ] δφ

89 (4.147) ( (r.h.s.) = 2a 2 3Ḣ Ḣ2 H 2 + Ḧ ) δφ H δφ + 2aHδφ + k 2 δφ +a (V (3Ḣ Ḣ2 H 2 + Ḧ H )) δφ = 0 (4.146) Klein-Golden mass term m 2 V + 2 (3Ḣ Ḣ2 H 2 + Ḧ ) H (4.147) d dt d dt ( a 3 ) Ḣ H ( a 3 ) Ḣ H = a 3 Ḧ H + 3a2 ȧḣ H a3 Ḣ2 H 2 (Ḧ ) = a 3 Ḣ2 + 3Ḣ H = 1 M 2 pl d dt ( a 3 φ2 0 H H 2 ) +a 2 [ (k a )2 Mass term δφ + 2aHδφ + V 1 a 3 M 2 pl d dt V = d [ 1 ] dφ 0 a 2 (φ 0 + 2Haφ 0) = dτ dφ 0 d dτ ( )] a 3 φ2 0 δφ = 0 (4.148) H [ 1 a 2 (φ 0 + 2Haφ 0) ]

90 84 4 m 2 = 1 a 2 [ = 1 [ φ 0 a 2 φ 0 φ 0 φ 0 2H 2 a 2 + 2aH ] + 2H 2 a 2 2 H 2 H 2 + 4H H u = aδφ u = a 2 Hδφ + aδφ φ 0 φ 0 u = aδφ + 2a 2 Hδφ + (a 2 H + 2a 3 H 2 )δφ ( u (a 2 H + 2a 3 H 2 )δφ + a 3 k a +a [ φ 0 φ 0 + 2H 2 a 2 2 H 2 H 2 + 4H H u + [k 2 φ 0 φ ah 2 H 2 0 H 2 + 4H H )2 δφ φ 0 φ 0 φ 0 φ 0 ] ] δφ = 0 ] u = 0 z φ 0 H (4.149) H = 2 φ 0 φ ahh (4.150) 0 1 z z = φ 0 φ 0 2 H H φ 0 φ H 0 H + 2H 2 H 2 (4.151) [ u + k 2 1 d 2 z ] z dτ 2 u = 0 (4.152) Spatially flat Q = δφ P aq [ P + k 2 1 d 2 z ] z dτ 2 P = 0 (4.153)

91 Power spectrum of scalar perturbation 1 d 2 z z dτ 2 (4.160) 2 3η + 9ε τ 2 ν 3 η + 3ε (4.154) 2 ( P Q (k) = (2H) 6ε 2η )2 H ( Γ (ν) ( k ) 6ε+2η 2π Γ (3/2))2 a (4.155) spatially flat Q = ϕ TM TM spatial curvature spatially flat ϕ T M = Ḣ φ 0 δφ SF (4.156) ( P ϕ (k) = (2H) 6ε 2η )2 H ( H )2 ( Γ (ν) ( k ) 6ε+2η (4.157) φ 0 2π Γ (3/2))2 a (4.144),(4.146) P ϕ (k) 1 24π 2 M 4 pl V ε ( k ) 6ε+2η 1 2aH 2 M pl 2 ( )2 H 1 2π ε (4.158) dε d ln k = 2εη + 4ε 2 (4.159) n

92 86 4 n 1 d ln P ϕ d ln k = 2η 6ε (4.160) n 1 δρ k 2 k n (4.161) n 1 2 Φ N = 4πGa 2 δρ k Φ N k 2 δρ k d ln kp Φ (k) d ln kk 3 k 4 δρ k 2 d ln kk n 1 n = 1 n = 1 n = 1 Harrison-Zel dovich spectrum Conformal Newtonin Conformal Newtonian g 00 = a 2 (1 + 2Ψ) g 0i = 0 g ij = a 2 (1 2Φ)δ ij g 00 = 1 a 2 (1 2Ψ) g0i = 0 g ij = 1 (1 + 2Φ)δij a2

93 Γ 0 00 = 1 2 g00 g 00,0 = a a + Ψ Γ 0 i0 = 1 2 g00 g 00,i = Ψ,i = ikψδ i3 Γ 0 ij = 1 2 g00 ( g ij,0 ) = ( a a (1 2Ψ N 2Φ N ) Φ N)δ ij Γ i 00 = 1 2 gii ( g 00,i ) = Ψ,i = ikψδ i3 Γ i 0j = 1 2 gii g ij,0 = ( a a Φ )δ i j Γ i jk = 1 2 gii (g ij,k + g ik,j g jk,i ) = Φ,k δ ij Φ,j δ ik + Φ,i δ jk = ikφ( δ k3 δ ij δ j3 δ ik + δ i3 δ jk ) Γi0 i = 3( a a Φ ) Γji i = 3ikΦ N δ j3 Γ0α α = 4 a a 3Φ N + Ψ N Γiα α = ( 3ikΦ N + ikψ N )δ i3 T 0 0 = ρ 0 δρ T 0 i = (ρ 0 + P 0 )v i T i 0 = (ρ 0 + P 0 )v i T i j = (P 0 + δp )δ i j + Σ i j P 0 = wρ 0 (4.162) density contrast δ δρ ρ 0 (4.163) µ T µ 0 = 0 0 = µ T µ 0 = T 0 0,0 + Γ 0 0σT σ 0 Γ σ 00T 0 σ + T i 0,i + Γ i iσt σ 0 Γ σ i0t i σ = ρ 0 δρ k(1 + w)ρ 0 V N 3(aH Φ )ρ 0 3aHδρ 3(aH Φ )P 0 3aHδP

94 88 4 ρ 0 + 3aH(1 + w)ρ 0 = 0 (4.164) δρ + k(1 + w)ρ 0 V N 3Φ N (1 + w)ρ 0 + 3aH(δρ + δp ) = 0 (4.165) δ N + k(1 + w)v N 3(1 + w)φ N 3aHwδ N + 3aH δp N ρ 0 = 0 (4.166) V Conformal Newtonian V = a 3 (1 2Φ N ) 3/2 d 3 x Φ peculiar velocity div v d(ρ 0 + δρ)v dt + (P 0 + δp ) dv dt = 0 1 V dv dt = 3ȧ a 3 Φ + div v V δ ρ + 3Hδρ 3ρ 0 (1 + w) Φ + kv N (1 + w)ρ 0 + 3HδP = 0 (4.167) (4.165)

95 = 0 T 0 i + j T j i = T 0 i,0 + Γ 0 σ0t σ i Γ σ 0iT 0 σ +T j i,j + Γ j σj T σ i Γ σ jit j σ = ((ρ 0 + P 0 )v i ) + δp,i + Σ j i,j + a a (ρ 0 + P 0 )v i + ikψ(ρ 0 + P 0 )δ i3 a a (ρ 0 + P 0 )v i +3 a a (ρ 0 + P 0 )v i δ i3 a a (ρ 0 + P 0 )v i i = 3 V N + ah(1 3w)V N + w 1 + w V N kψ N kδp + 2 (1 + w)ρ 0 3 kσ = 0 (1 + w)ρ 0 T = ρ 0 δρ + 3P 0 + 3δP ( R 00 = 3 a a )2 a + 3 k 2 Ψ N + 3Φ N + 3aH(Ψ N + Φ N ) a g 00 (T T ) = a2 2 (ρ 0 + 3P 0 + δρ + 3δP + 2Ψ(ρ 0 + 3P 0 )) (4.168) a ( a )2 a a = a2 6 M (ρ pl P 0 ) (4.169) k 2 Ψ N + 3Φ N + 3aH(Ψ N + Φ N ) = a 2 2 M (δρ + 3δP + 2Ψ(ρ pl P 0 )) (4.170) R ij R ij = ( a a + ( a a )2) ( a ( a )2) δ ij 2 a + (Ψ N + Φ N )δ ij a +k 2 (Ψ N Φ N )δ i3 δ j3 k 2 Φ N δ ij Φ N δ ij a a (Ψ N + 5Φ N )δ ij

96 90 4 ( a ( a )2) a + a = a 2 2 M (ρ pl 2 0 P 0 ) (4.171) i = j = 2 ( a ( a )2) 2 a + (Ψ N + Φ N ) k 2 Φ N Φ N a a a (Ψ N + 5Φ N ) i = j = 3 = a2 2 M (δρ δp + 2 pl 2 3 Σ 2Φ(ρ 0 P 0 )) ( a k 2 ( a )2) (Ψ N Φ N ) 2 a + (Ψ N + Φ N ) k 2 Φ N Φ N a a a (Ψ N + 5Φ N) = a2 2 M (δρ δp 4 pl 2 3 Σ 2Φ(ρ 0 P 0 )) k 2 (Ψ N Φ N ) = a2 Σ (4.172) M pl 2 Σ Ψ N = Φ N R 30 2ikΦ N + 2 a a ikψ N = 1 M 2 pl (ρ 0 + P 0 )a 2 iv (4.173) Φ N + ahψ N = a 2 2 M 2 pl ρ 0 (1 + w) V k (4.174) 3k 2 Φ N + k 2 (Ψ N Φ N ) 3Φ N 3aH(Ψ N + 5Φ N ) = a2 2 M [6(ρ pl 2 0 P 0 )Ψ N + 3(δρ δp )] (00) (30)

97 k 2 Φ N = a2 2 M ρ pl 2 0 [δ N + 3aH(1 + w) V N k ] (4.175) Bardeen Bardeen conformal time (4.166) ζ = Φ N + δ N 1 + w = k 3 V N ah P 1 + w ρ 0 P = 0 conformal time w = 0 (4.168) ζ = k 3 [ kw ( δ N + 3aH(1 + w) V N 1 + w k ) ahv N + kψ N ] (4.175) ζ = k [ k w ( 2 k 2 ) ] w 3 a 2 H 2 Φ N ahv N + kφ N = ahζ + 1 ( 3 k2 Φ N w k 2 ) w a 2 H 2 (4.176) (4.175) Φ N = 3 2 ( ah k )2 [ δ N + 3aH(1 + w) V N k ah/k 1 Bardeen ζ + 2H ζ = 0 P 0 = wρ 0 H = 2/(3(1 + w)t) ζ = t n n = 0, (3w 1)/3(1 + w) 1 < w < 1/3 ζ const. ]

98 CMB CMB CMB (last scattering surface) 0.01 COBE CMB CMB CMB Sachs-Wolfe Spatially flat TM Bardeen TM Newtonian TM TM δ = δ N + 3aH(1 + w) V N k (4.177) δp = V N δp N + 3aHw(1 + w)ρ 0 k V N w ρ 0 k (4.178) V = V N (4.179) TM

99 δ = 3aHwδ (1 + w)kv (4.180) V = δp ahv + + kψ N (1 + w)ρ 0 (4.181) k 2 Φ N = 4πGa 2 ρ 0 δ (4.182) Ψ N = Φ N (4.183) ϕ = Φ N ah k V (4.184) ζ = Φ N ah k V + 1 δ w (4.185) k 2 δ = 2 3 a 2 H 2 Φ (4.186) TM δ Φ N ζ ϕ TM (4.180) conformal time t (4.182) δ TM (4.184) Φ w HΦ = 3 (1 + w)hϕ (4.187) 2 2 ϕ = constant (4.188) w = 1 H = constant Φ = Φ 0 e Ht a t 2 3(1+w)

100 94 4 Φ N decay Φ N w 1 3(1 + w) t Φ N = 1 t ϕ (4.189) Φ N t 5+3w 3(1+w) 3(1 + w) Φ N = 5 + 3w ϕ (4.190) radiation dominant w = 1/3 matter dominant w = 0 Φ N = 2 3 ϕ (4.191) Φ N = 3 5 ϕ (4.192) COBE DMR Q rms = 17.1 ± 1.5µK (4.193) δt T = (4.194) COBE COBE Sachs-Wolfe CMB

101 Sachs-Wolfe Newtonian ds 2 = (1 + 2Ψ)dt 2 + a 2 (1 2Φ)δ ij dx i dx j (4.195) (A.47) hν d x0 dλ (4.196) ε = hν P 0 = η 00 P 0 = ε P 0 = xµ x 0 P µ g 00 = (1 + 2Ψ) = xµ x 0 x ν x 0 η µν g 0i = 0 = xµ x 0 x ν x i η µν g ij = a 2 (1 2Φ)δ ij = xµ x i x ν x j η µν P 0 x 0 x = (1 + Ψ), x0 x 0 x i = 0, i x 0 = 0 x i x j = a(1 Φ)δ ij (4.197) P 0 = (1 + Ψ)ε (4.198) x 1 (A.47)

102 96 4 d(1 + Ψ)ε dx 0 = 1 2 g P α P β αβ,0 P 0 = 1 (g 00,0 P 0 (P 1 ) 2 ) + g 11,0 2 P 0 = 1 P 0 ( Ψ(P 0 ) 2 a 2 Φ(P 1 ) 2 + H(1 2Φ)a 2 (P 1 ) 2 ) 0 = g µν P µ P ν = (1 + 2Ψ)(P 0 ) 2 + a 2 (1 2Φ)(P 1 ) 2 a 2 (P 1 ) 2 d(1 + Ψ)ε dx 0 = ( Ψ + Φ)P 0 + H(1 + 2Ψ)P 0 P 0 = g 00 P 0 = (1 2Ψ)P 0 = (1 Ψ)ε 1 dε ε dt 1 daε aε dt = dψ dt + Ψ t + Φ t H = dφ dt + 2 Φ t (4.199) Ψ = Φ Sachs-Wolfe m m d v dt = m Φ dε dt = v Φ = dφ dt + Φ t ε = v 2 /2

103 CMB Sachs-Wolfe CMB intrinsic Θ 0 (t ls, x ls, n) (last scattering surface) (δt/t ) jour t ls x ls CMB n ( δt T ) jour = δ(aε) aε t0 Φ( x, t) = Φ( x ls, t ls ) Φ( 0, t 0 ) + 2 t ls t dt (4.200) Sachs-Wolfe CMB CMB

104 98 4 Integrated Sachs-Wolfe Intrinsic δ N r δn m N Conformal Newtonian δ N r 4 = δn m 3 (4.201) Θ = δt/t CMB intrinsic Θ0 N (t ls, x ls, n) = δn r 4 = δn m 3 (4.202) TM (w = 0) δ m = δ N m + 3 ah k V (4.203) V + ahv = d dt (av ) = kφ N (4.204) Φ N (4.190) Φ N (4.204) Φ N V = t k a Φ N = 2 k 3 ah Φ N (4.205) Ω m = 1 a t 2/3, t = 2/3H (4.202)

105 Θ0 N (t ls, x ls ) = 1 (δ m 3 ah ) 3 k V = 1 3 δ m(t ls, x ls ) 2 3 Φ N (t ls, x ls ) 2 3 Φ N(t ls, x ls ) (4.206) intrinsic ( δt T ) obs = 1 3 Φ N (t ls, x ls ) (4.207) (4.192) CMB ( δt T ) obs = 1 5 ϕ(t ls, x ls ) (4.208) (4.208) (4.158) CMB (4.194) V 1/4 ε 1/4 = GeV (4.209) CMB (4.186) (4.191) k 2 δ = 2 5 a 2 H 2 ϕ (4.210) density contrast

106 100 4 ( k )4 P δ = δh 2 (4.211) ah δh P R (4.212) δ H k = a 0 H 0 COBE COBE normalization n = 1 δ H (k pivot ) = (4.213) k pivot = 7.5a 0 H 0 (4.214) Late stage evolution of density contrast and CMB spectrum (4.204) Φ N CMB CMB z = 1100 z = 0 Sachs-Wolfe Integrated Sachs-Wolfe Ω m = 1 (ah/k 1)

107 (4.175) k 2 Φ = 3 2 a2 H 2 δ (4.215) Density contrast (4.166) peculiar velocity (4.168) δ + kv = 0 V + ahv kψ = 0 Newtonian δ + ahδ 3 2 a2 H 2 δ = 0 Ω m = 1 δ + 2H δ 3 2 H2 δ = 0 (4.216) δ + 4 3t δ 2 3t 2 δ = 0 δ t 1 δ t 2/3 a (4.217) (4.215) k 2 Φ = 1 2 M ρ pl 2 0 a 2 δ ρ 0 a 3 = const (4.218) ρ 0 a 3 = const

108 102 4 density contrast Φ (4.215) ρ 0 δ Φ CMB l Integrated Sachs Wolfe L = M [ pl 2 ( E +2 + ( E + ) 2 ) + ( E 2 + ( E ] ) 2 )(4.219) Ψ + 2 M pl E +, Ψ 2 M pl E L = 1 2 [( Ψ + ) 2 + ( Ψ + ) 2 ] [( Ψ ) 2 + ( Ψ ) 2 ] (4.220) Ψ +, Ψ mass less free scalar field P T (k) = 2 1 ( Γ (ν) ( H )2 ( ah 2 M pl 2 Γ (3/2))2 2π k )2ε

109 k 2ε (4.221) r P T /P δ 12.5ε (4.222) r V 1/4 = r 1/4 GeV (4.223) mass less scalar field exact ḧ + 3Hḣ + k2 a 2 h = 0 (4.224) H > 0 h = const Conformal time h + 2aHh + k 2 h = 0 (4.225) Conformal time η

110 104 4 Conformal time t e Conformal time η e (4.28) a ε H I η q H I η (4.226) H I (4.236) α = kη Conformal time a = 1 k 1 ε Hα (4.227) H I a Conformal time η energy equipartition t eq a eq a(t) = a eq ( t t eq )1/2 η η e = = t dt t e a = 2t eq a 2 (a(t) a(t e )) = 2t ( e teq ) eq a 2 (a(t) a(t e )) eq t e 2t e a(t e ) 2 (a(t) a(t e)) 1 2ε k 2 H I αea(t) ε k α e

111 H I = ȧ a (t e) = 1 2t e (4.228) a(t e ) = 1 k 1 ε H I α e (4.229) a(t) = 1 k (α (2 ε)α e ) 1 2ε H I α 2 e (4.230) Conformal time t eq t 1/2 t 2/3 t eq a(t) = a eq ( t t eq )2/3 (4.231) η η eq = 3t eq (a 1/2 a 1/2 eq ) = 3 2 a 3/2 eq a(t) = 1 1 k ε H I α 2 e 1 H 2 I a2 e a 1/2 eq (a 1/2 a 1/2 eq ) (2α + α eq 3(2 ε)α e ) 2 α eq (2 ε)α e (4.232) (4.30) V = 0 ξ = α u k = ξhk (ξ) d 2 H k dξ ξ dh ( k dξ + 9/4 + 3ε ) 1 ξ 2 H k = 0 (4.233) h u h = u/a (4.36)

112 106 4 h k = 1 1 π a Vc 4k e i( k a 1 H π (2ν+1)) 1 4 ξh (1) ν (ξ) (4.234) Conformal time η R = t 0 dt a = 2t eqa a 2 eq (4.235) ah = a 1 t eq 2t eq t = a2 eq 2t eq 1 a = 1 η R η R h + 2 η R h + k 2 h = 0 h = u/a u ξ R k η R u + k 2 u = 0 d 2 u dξr 2 + u = 0 (4.236) Conformal time η M = t 0 dt a = 3t eq a 1/2 (4.237) a 3/2 eq ah = 2a 3/2 eq 3t eq a 1/2 = 2 η M h + 4 η M h + k 2 h = 0

113 h = u/a ξ M = k η M d 2 u dξ 2 M ( ) ξm 2 u = 0 (4.238) ν u = ξχ d 2 u ( dξ ν2 1/4 ) ξ 2 u = 0 (4.239) d 2 χ dξ ( ) ξ χ + 1 ν2 ξ 2 χ = 0 (4.240) ν = ±1/2 ν = ±3/2 h k = 1 ξ(aj1/2 (ξ) + BJ 1/2 (ξ)) = 1 2 (A sin ξ + B cos ξ) a a π = 2t eq 2 ( a 2 k A sin ξ R + B cos ξ ) R eq π ξ R ξ R = 1 2ε H I αe 2 k 2 π ( A sin ξ R + B cos ξ ) R ξ R ξ R (4.241) ξ R = ka 2t eq a 2 eq = ka 1 H I a 2 e = a(1 2ε) H I k α2 e = α (2 ε)α (4.242) e t 0 ξ 0 B = 0 (4.245) (4.245) A

114 108 4 H V c = 1 2ε H I αe 2 k Γ (ν) 1 Γ (3/2) k 3/2 2ν 2 e i( k a 1 H + π (ν 1/2)) ( a 1 2 e H I k 2 π Asin ξ R,e (4.243) ξ R,e ξ R,e Conformal time ξ 1 h const ξ > 1 ξ 1 )ε h 1 ξ 1 a (4.244) h = a 1 ξ(cj3/2 (ξ) + DJ 3/2 (ξ)) = 1 2 [ ( sin ξ ) ( C cos ξ D a π ξ = 9t2 eq 2 a 3 k 2 eq π 1 ξ 2 M sin ξ + cos ξ )] ξ [ ( sin ξm ) C cos ξ M D ξ M (4.245) ( sin ξ M + cos ξ M ξ M )] (4.246) ξ M = 3 1 ka 1/2 1/2 1 2ε eq a 2 H I k 2 HI 2 αe 2 = α + α eq (2 ε)α e (4.247) energy equipartition ξ 1 D = 0 Energy equipartition t = t eq (4.252) C

115 ξ 1 h const ξ 1 h 1 ξ 2 1 a (4.248) α e 1 α eq 1 ξ M α α e 1 α eq > 1 ξ R α ξ M α + α eq /2

116 A, B [A, B] = ic < ( A)2 > (< B >) 2 < C > 2 (5.1)

117 A = A < A > B = B < B > t < A + t < A, B > B, A + t < B, A > B > 0 < B, B > < A, B > 2 t < A, B > 2 t+ < A, A > 0 t D = < A, B > 4 < A, A >< B, B > < A, B > 2 0 (5.2) < A, B > 2 4 < A, A >< B, B > (5.3) A, B { A, B} < A, B > = 1 2 < [ A, B] > +1 < { A, B} > 2 < [ A, B] > = < [ B, A] >= < [ A, B] > < { A, B} > = < { B, A} >=< { A, B} > < [ A, B] > < { A, B} > < A 2 >< B 2 > 1 < [ A, B] > + < { A, B} > 2 4 = 1 4 < [ A, B] > 2 + < { A, B} > 2 1 < [ A, B] > 2 4 [ A, B] = ic

118 112 5 A, B e λa Be λa = B + λ[a, B] + λ2 2! f(λ) = e λa Be λa λ3 [A, [A, B]] + [A, [A, [A, B]]]+,,, 3! (5.4) f(λ) λ f (λ) = Ae λa Be λa e λa Be λa A = [A, f(λ)] f (λ) = Af (λ) f (λ)a = [A, [A, f(λ)]] f (λ) = [A, [A, [A, f(λ)]]] f(λ) λ = 0 f(λ) = f(0) + f (0)λ + 1 2! f (0)λ ! f (0)λ = B + [A, B]λ + 1 2! [A, [A, B]]λ ! [A, [A, [A, B]]]λ A, B [A, B] = C C C λ2 λ(a+b)+ e 2 C = e λa e λb (5.5) l.h.s. = 1 + λ(a + B) + λ2 2 C + 1 (λ 2 (A 2 + AB + BA + B 2 ) + λ 3 C(A + B) + λ4 2! 4 C2) + 1 3! (λ3 (A 3 + AAB + ABA + BAA + ABB + BAB + BBA + B 3 ) + O(λ 4 )) = 1 + λ(a + B) + λ2 2 (C + A2 + B 2 + 2AB C) + 1 3! λ3 (A 3 + B 3 + 3AAB 3CA + 3ABB 3CB + 3C(A + B))...

119 = 1 + λ(a + B) + λ2 2 (A2 + B 2 + 2AB) + 1 3! λ3 (A 3 + B 3 + 3AAB + 3ABB)... r.h.s = (1 + λa + λ2 2! A2 + λ3 3! A3...) (1 + λb + λ2 2! B2 + λ3 3! B3...) = 1 + λ(a + B) λ2 2 (A2 + B 2 + 2AB) + λ3 3! (A3 + B 3 + 3AAB + 3ABB)... Baker-Hausdorff B(t) [B(t ), B(t )] = C(C C ) T [ t ] T exp dt B(t ) 0 ( t = exp dt B(t ) t 0 t dt 0 ) dt [B(t ), B(t )] N N t t/n N [ t ] T exp dt B(t ) 0 [ t = T exp dt B(t ) + (N 1) t (N 1) t (N 2) t (5.6) dt B(t )... + t k = k t (1/2) t t 0 ] dt B(t ) l.h.s. = T exp [ t(b(t N ) + B(t N 1 )... + B(t 1 ))] = e tb(t N ) e tb(t N 1)...e tb(t 1) (5.7) (5.18) e tb(t N ) e tb(t N 1) ] = exp [ t(b(t N ) + B(t N 1 )) + t2 2 [B(t N ), B(t N 1 )] [ t ] T exp dt B(t ) = exp( t(b(t N ) + B(t N 1 )... + B(t 1 )) 0 + t2 2 {[B(t N ), B(t N 1 )] + [B(t N ) + B(t N 1 ), B(t N 2 )] +...

120 [B(t N ) + B(t N 1 ) B(t 2 ), B(t 1 )]}) = exp( t(b(t N ) + B(t N 1 )... + B(t 1 )) + t2 2 {[B(t N ), B(t N 1 ) + B(t N 2 ) B(t 1 )] +[B(t N 1 ), B(t N 2 ) B(t 1 )] [B(t 2 ), B(t 1 )]}) θ < θ > θ H Ψ S (t) i Ψ S(t) t = HΨ S (t) (5.8) H 0 = p2 2 + Ω2 2 q2 (5.9) Ω p.q [q, p] = i a a

121 a = a = 1 (Ωq + ip) (5.10) 2 Ω 1 (Ωq ip) (5.11) 2 Ω [a, a ] = 1 (5.12) H 0 = Ω(a a ) (5.13) ω n = Ω(n + 1/2) Ψ S,n (t) = n > e iω nt = 1 n! (a ) n 0 > e iω nt p = i q ( Ω ) 1 2 ξ q 2 2 H 0 = Ω 4 ξ 2 + Ωξ2 (5.14) a = ξ 1 2 ξ (5.15) a = ξ ξ (5.16)

122 116 5 a a = ξ (5.17) n = 0 u 0 (ξ) = ( 2 π )1/4 e ξ2 (5.18) n u n (ξ) = 1 ( ξ 1 n! 2 )n ( 2 ξ π )1/4 e ξ2 (5.19) q = i p ( )1/2 1 η p 2 Ω H 0 = ( Ω η 2 + η2) (5.20) a = i 1 iη 2 η (5.21) a = i 1 + iη 2 η (5.22) a + a = i η (5.23) u 0 (η) = ( 2 π )1/4 e η2 (5.24)

123 u n (η) = 1 ( )n (i) n 1 ( 2 )1/4 n! 2 η η e η2 (5.25) π e ikq = e i p q = e 2iξη ξ η ˆf(η) = 1 π dξf(ξ)e 2iηξ (5.26) dη ˆf(η) 2 = dξ f(ξ) 2 (5.27) u 0 (ξ) 1 π ( 2 )1/4 dξ e ξ2 e 2iξη = π ( 2 π )1/4 e η2 = u 0 (η) < q2 > < p 2 > 2 (5.28) < ξ 2 > = ( 2 dξxi 2 u 0 (ξ) 2 )1/2 = dξξ 2 e 2ξ2 = 1 π 4 < q2 > = ( 2 )1/2 < ξ2 > = Ω 2Ω

124 < q2 > < p 2 > = 2 hattori_cosmo < η 2 > = 1 4 < p2 > = Ω 2 < m H 0 n > = ω n < m n >= ω m < m n > < m n > = δ mn (5.29) dq q >< q = I (5.30) n >< n = I (5.31) n I ψ S (q, t) Ψ S (t) > = dq q >< q Ψ S (t) > ψ S (q, t) = < q Ψ S (t) > (5.32) u n (q)

125 u n (q) = < q n > (5.33) u n(q )u n (q) n = < q n >< n q >=< q q > n = δ(q q) (5.34) u n (q) f(q) u n (q) f(q) = n c n u n (q) (5.35) u n (q) f(q) = n dq f(q )u n(q )u n (q) = dq f(q ) n u n(q )u n (q) f(q) V i Ψ S(t) > t = (H 0 + V ) Ψ S (t) > (5.36) t 0 t U(t, t 0 ) Ψ S (t) > = U(t, t 0 ) Ψ S (t 0 ) > (5.37) i U(t, t 0) t = HU(t, t 0 ) (5.38) < Ψ S (t) Ψ S (t) > = < Ψ S (t 0 ) Ψ S (t 0 ) >

126 120 5 U (t, t 0 ) = U 1 (t, t 0 ) U (t, t 0 ) i U (t, t 0 ) t = U (t, t 0 )H (5.39) U(t 0, t 0 ) = 1 t1 ( i ) t ( i )2 t U(t, t 0 ) = 1 + dt 1 H(t 1 ) + dt 1 dt 2 H(t 1 )H(t 2 ) t 0 t 0 t 0 ( i )3 t t1 t2 + dt 1 dt 2 dt 3 H(t 1 )H(t 2 )H(t 3 )+,,(5.40),, t 0 t 0 t 0 step function U(t, t 0 ) = n=0 ( i θ(τ) = 1 for τ 0 )n t t t 0 = 0 for τ < 0 t 0... t dt 1 dt 2,,, dt n θ(t 1 t 2 )θ(t 2 t 3 )...θ(t n 1 t n )H(t 1 )...H(t (5.41) n t 0 U(t, t 0 ) = 0 1 n! ( ih(t t0 ) )n ( = exp i H(t t 0) T ) (5.42) T [H(t 1 )H(t 2 )...H(t n )] = p Θ(t p1, t p2,..., t pn )H(t p1 )H(t p2 )...H(t pn ) (5.43) Θ(t p1, t p2,..., t pn )H(t p1 ) = θ(t p1 t p2 )θ(t p2 t p3 )...θ(t pn 1 t pn ) (5.44) t 1, t 2,..., t n p