Σχετικά έγγραφα
m i N 1 F i = j i F ij + F x

A 1 A 2 A 3 B 1 B 2 B 3

ƒˆˆ-ˆœ œ Ÿ ˆ ˆ Š ˆˆ ƒ ˆ ˆˆ



Dissertation for the degree philosophiae doctor (PhD) at the University of Bergen

Ó³ Ÿ , º 4(153).. 449Ä471

4. Zapiši Eulerjeve dinamične enačbe za prosto osnosimetrično vrtavko. ω 2


.. ntsets ofa.. d ffeom.. orp ism.. na s.. m ooth.. man iod period I n open square. n t s e t s ofa \quad d ffeom \quad orp ism \quad na s \quad m o

ΠΕΡΙΕΧΟΜΕΝΑ ΓΕΝΙΚΗ ΘΕΩΡΙΑ ΣΧΕΤΙΚΟΤΗΤΑΣ ΒΑΡΥΤΙΚΑ ΚΥΜΑΤΑ ΣΤΟ ΚΕΝΟ ΠΑΡΑΓΩΓΗ ΒΑΡΥΤΙΚΩΝ ΚΥΜΑΤΩΝ ΑΠΟ ΠΗΓΕΣ ΑΝΙΧΝΕΥΣΗ ΒΑΡΥΤΙΚΩΝ ΚΥΜΑΤΩΝ

Theory of Cosmological Perturbations


Sur les articles de Henri Poincaré SUR LA DYNAMIQUE. Le texte fondateur de la Relativité en langage scientiþque moderne. par Anatoly A.

Theory of Cosmological Perturbations

rs r r â t át r st tíst Ó P ã t r r r â

Bogoliubov-de Gennes

Déformation et quantification par groupoïde des variétés toriques

m 1, m 2 F 12, F 21 F12 = F 21

Š Š Œ Š Œ ƒˆ. Œ. ϵ,.. ÊÏ,.. µ ±Ê



Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α



Teor imov r. ta matem. statist. Vip. 94, 2016, stor

Ax = b. 7x = 21. x = 21 7 = 3.

r r t r r t t r t P s r t r P s r s r r rs tr t r r t s ss r P s s t r t t tr r r t t r t r r t t s r t rr t Ü rs t 3 r r r 3 rträ 3 röÿ r t

Alterazioni del sistema cardiovascolare nel volo spaziale

Coupling strategies for compressible - low Mach number flows

m r = F m r = F ( r) m r = F ( v) F = F (x) m dv dt = F (x) vdv = F (x)dx d dt = dx dv dt dx = v dv dx

u(x, y) =f(x, y) Ω=(0, 1) (0, 1)

"BHFC8I7H=CB HC &CH=CB 5B8 &CA9BHIA

A Classical Perspective on Non-Diffractive Disorder

F (x) = kx. F (x )dx. F = kx. U(x) = U(0) kx2

Κβαντομηχανική Ι Λύσεις προόδου. Άσκηση 1

ˆ ˆ Œ Ÿ Š Œ ƒˆ Šˆ ˆ Ÿ ˆ ˆ Š ˆˆ ƒ ˆ ˆˆ

#%" )*& ##+," $ -,!./" %#/%0! %,!


692.66:



Φαινόμενο Unruh. Δημήτρης Μάγγος. Εθνικό Μετσόβιο Πολυτεχνείο September 26, / 20. Δημήτρης Μάγγος Φαινόμενο Unruh 1/20

T : g r i l l b a r t a s o s Α Γ Ί Α Σ Σ Ο Φ Ί Α Σ 3, Δ Ρ Α Μ Α. Δ ι α ν ο μ έ ς κ α τ ο ί κ ο ν : 1 2 : 0 0 έ ω ς 0 1 : 0 0 π μ

Conditions aux bords dans des theories conformes non unitaires

κ α ι θ έ λ ω ν α μ ά θ ω...

[Note] Geodesic equation for scalar, vector and tensor perturbations

!"#$ % &# &%#'()(! $ * +

Έργο παραγώμενο στο τοίχωμα

Jeux d inondation dans les graphes


E fficient computational tools for the statistical analysis of shape and asymmetryof 3D point sets


Περιεχόμενα. A(x 1, x 2 )

Modèles de représentation multi-résolution pour le rendu photo-réaliste de matériaux complexes

X(f) E(ft) df x[i] = 1 F. x(t) E( ft) dt X(f) = x[i] = 1 F

Ο αναλυτικός δείκτης και η χαρακτηριστική του Euler 1

Gapso t e q u t e n t a g ebra P open parenthesis N closing parenthesis fin i s a.. pheno mno nd iscovere \ centerline

ITU-R P (2009/10)

Geodesic Equations for the Wormhole Metric

On the Einstein-Euler Equations


ιανύσµατα A z A y A x 1.1 Αλγεβρικές πράξεις µεταξύ διανυσµάτων 1.2 Εσωτερικό γινόµενο δύο διανυσµάτων ca = ca x ˆx + ca y ŷ + ca z ẑ

Œ ƒ ˆ ˆˆ. Î ± É ÉÊÉ ³..., Œµ ± ˆ ˆˆ Œ ƒ ˆ ˆˆ 1051 Ð ³ Î Ö 1051 Î ± Ö É Í Ö 1059

6. ΙΑΦΟΡΙΚΗ ΑΝΑΛΥΣΗ ΤΗΣ ΡΟΗΣ

άρα : p= hf c = h λ λ= h p

Cosmology with non-minimal derivative coupling


Dark matter from Dark Energy-Baryonic Matter Couplings


f O(U) (f n ) O(Ω) f f n ; L (K) 0(n )

ITU-R P (2012/02) &' (

Ó³ Ÿ , º 4(181).. 501Ä510

Μαγνητικοί άνεμοι και απώλεια στροφορμής

ˆˆ ŸŒ ƒ ˆŸ CP- ˆŒŒ ˆˆ

Α Ρ Ι Θ Μ Ο Σ : 6.913

Πληθωριστική Κοσμολογία

36 ( ) Ω λk(k= + )-Δ <γ < (4) L (Ω) φ k λk : (-Δ) /φ γ / k=λγ k φ k { <λ λ λk (k ) D((-Δ) γ / )= {u L (Ω)stu Ω = ; (-Δ) γ / u L (Ω) = k=+ λ γ / k u φ

ΚΒΑΝΤΙΚΗ ΜΗΧΑΝΙΚΗ (ΚΕΦΑΛΑΙΟ 39 +)

ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ Ñ Ò É ÉÊÉ Ö ÒÌ ² μ, Ê

!!" #7 $39 %" (07) ..,..,.. $ 39. ) :. :, «(», «%», «%», «%» «%». & ,. ). & :..,. '.. ( () #*. );..,..'. + (# ).

ˆˆ ƒ ˆ ˆˆ.. ƒ ÏÉ,.. μ Ê μ, Œ.. Œ É Ï ²,.. ± Î ±μ

! " # $ % # "& #! $! !! % " # '! $ % !! # #!!! ) " ***

3. ΚΙΝΗΣΗ ΡΕΥΣΤΟΥ-ΕΞΙΣΩΣΗ BERNOULLI Κίνηση σωµατιδίων ρευστού

The low energy limit of the 3-flavor extended Linear Sigma Model. Jonas Schneitzer. Johann Wolfgang Goethe Universität Frankfurt am Main

PHYS606: Electrodynamics Feb. 01, Homework 1. A νµ = L ν α L µ β A αβ = L ν α L µ β A βα. = L µ β L ν α A βα = A µν (3)


Q π (/) ^ ^ ^ Η φ. <f) c>o. ^ ο. ö ê ω Q. Ο. o 'c. _o _) o U 03. ,,, ω ^ ^ -g'^ ο 0) f ο. Ε. ιη ο Φ. ο 0) κ. ο 03.,Ο. g 2< οο"" ο φ.

Ó³ Ÿ , º 1(199).. 66Ä79 .. Ê 1. Œμ ±μ ± μ Ê É Ò Ê É É ³. Œ.. μ³μ μ μ, Œμ ±

ΟΡΘΟΔΙΑΓΩΝΙΑ ΤΕΤΡΑΠΛΕΥΡΑ

Inflation and Reheating in Spontaneously Generated Gravity

Analysis of a discrete element method and coupling with a compressible fluid flow method

ITU-R P ITU-R P (ITU-R 204/3 ( )

Κβαντική Φυσική Ι. Ενότητα 12: Θεωρήματα Ehrenfest-Parity- -Μέση τιμή τελεστή. Ανδρέας Τερζής Σχολή Θετικών Επιστημών Τμήμα Φυσικής

C M. V n: n =, (D): V 0,M : V M P = ρ ρ V V. = ρ

-! " #!$ %& ' %( #! )! ' 2003

υναµικό Coulomb - Λύση της εξίσωσης του Schrödinger

Œˆ ˆ ƒ ˆŸ Ÿ ˆ ˆ Ÿ Œˆ ˆ

Ορίζουμε την τυπική πολυδιάστατη κανονική, σαν την κατανομή του τυχαίου (,, T ( ) μεταξύ τους ανεξάρτητα. Τότε

Mesh Parameterization: Theory and Practice

Transcript:

f H f H ψ

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

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

( ) dr ds 2 = c 2 dt 2 + α(t) 2 2 1 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

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 = (67.8 0.9)kms 1 Mpc 1 Ω k < 0.005 w de w de = 1.006 0.045

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

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

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 = 4.5 10 2 ( r 0 kpc ) 2 3 M pc 3

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 2 1.5 k B T (1.3 1.8)keV ( Mr ) ( ) Mpc 10 14 M r

M r M r 10 5

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

Ω b h 2 = 0.02230 0.00014 (68 C.L.), Ω m h 2 = 0.14170 0.00097 (68 C.L.)

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

m i Ω ν h 2 = 3 i=1 m i 93eV m ν < 2.05eV (95 C.L.) Ω ν h 2 0.07 Ω ν h 2 0.006795 C.L.)

0.01eV

ρ( x, t) x t Nm ρ = δv 0 δv ρ( x, t)

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)

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 )

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 )

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

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

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

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)

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

( ) 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

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

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

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

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

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)

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

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

θ = Hf(α) k 2 δ f(α) f(α) = d lnδ d lnα Ω 0.6 m f = Ω γ m, γ = 6 11 + 15 2057 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

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

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

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

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.

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 0 0 + R i i =6α 2 (H + H 2 ) + α 2 [ 6Ψ + 2 2 (2Ψ Φ) 6H(Φ + 3Ψ ) 12(H + H 2 )Φ]

G 0 0 =R 0 0 1 2 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

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

( ) ( ) 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

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 + 1 3 2 (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.

(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 + 1 2 2 (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

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 + 1 + 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

δ( k) 76 24 δ b = δ e k > k J k < k J

δ b δ dm

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

ħ

( 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 )

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

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 2 2 + 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

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

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 )

f W ħ f W f W = e σ2 x 2 2 + 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

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

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]

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

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 )

ħ

n( x)

n( x)

n( x)

n( x)

n( x)

θ θ rot ū i ( x) α = 1

θ θ rot ū i ( x) α = 3

θ θ rot ū i ( x) α = 10

θ θ rot ū i ( x) α = 30

δū ij ( x) ū k k ( x) σ ij

δū ij ( x) ū k k ( x) σ ij

δū ij ( x) ū k k ( x) σ ij

δū ij ( x) ū k k ( x) σ ij

http://www.esa.int/spaceinimages/images/2013/03/planck_cmb γ