ĐỀ ÁN TỐT NGHIỆP-CÁC MÔ HÌNH VŨ TRỤ

− − −  − − −
a(t)
z
·····················································
R
µν
−
1
2
g
µν
R − Λg
µν
= 8ΠGT
µν
T
µν
= 0
R
µν
−
1
2
g
µν
R = 8ΠGT
µν
g

µν
R
µν
−
1
2
g
µν
R − Λg
µν
= 8ΠGT
µν
R
µν
R
g
µν
Λ
G
T
µν
T
µν
=







ρ
Λ
0 0 0
0 −ρ
Λ
0 0
0 0 −ρ
Λ
0
0 0 0 −ρ
Λ






ρ
Λ
= 3Λ/(8ΠG)
T
ij
= pδ
i
j
,
ρ
Λ
T
µν

R
µν
−
1
2
g
µν
R = Λg
µν
.
ds
2
= dt
2
− a
2
(t)(
dr
2
1 −kr
2
+ r
2
dθ
2
+ r
2
sin
2
θdφ

2
)
R
µν
R
µν
= R
α
µαν
,
R
ρ
σµν
=
∂
∂x
µ
Γ
ρ
νσ
−
∂
∂x
ν
Γ
ρ
µσ
+ Γ
ρ
µλ

Γ
λ
νσ
− Γ
ρ
νλ
Γ
λ
µσ
.
Γ
ν
µσ
Γ
λ
µν
=
g
λρ
2
(g
νρ,µ
+ g
µρ,ν
− g
νµ,ρ
) ,
R = g
µν
R

µν
g
µν
g
µν
=
1
g
µν
,
g
µν
=







1 0 0 0
0 −
a
2
1 −kr
2
0 0
0 0 −a
2
r

2
0
0 0 0 −a
2
r
2
sin
2
θ







Γ
1
01
= Γ
2
02
= Γ
3
03
=
˙a
a
; Γ
0

11
=
˙aa
1 −kr
2
Γ
1
11
=
kr
1 −kr
2
; Γ
2
12
= Γ
3
13
=
1
r
Γ
0
22
= ˙aar
2
; Γ
1
22
= −r(1 −kr

2
)
Γ
3
23
= cot θ; Γ
1
33
= −r(1 −kr
2
sin
2
θ)
Γ
2
33
= −sin θ cos θ .
R
00
= R
α
0α0
=
∂Γ
α
00
∂x
α
−
∂Γ

α
0α
∂x
0
+ Γ
α
αλ
Γ
λ
00
− Γ
α
0λ
Γ
λ
α0
.
R
00
= −3
¨a
a
.
R
11
= R
α
1α1
=
∂Γ

α
11
∂x
α
−
∂Γ
α
1α
∂x
1
+ Γ
α
αλ
Γ
λ
11
− Γ
α
1λ
Γ
λ
α1
.
R
11
= −3
¨a
a
,
R

22
= R
α
2α2
=
∂Γ
α
22
∂x
α
−
∂Γ
α
2α
∂x
2
+ Γ
α
αλ
Γ
λ
22
− Γ
α
2λ
Γ
λ
α2
R
22

= (¨aa + 2˙a
2
+ 2k)r
2
R
33
= R
α
3α3
=
∂Γ
α
33
∂x
α
−
∂Γ
α
3α
∂x
3
+ Γ
α
αλ
Γ
λ
33
− Γ
α
3λ

Γ
λ
α3
R
33
= (¨aa + 2˙a
2
+ 2k)r
2
sin
2
θ
R g
µν
R
µν
R =
R
00
g
00
+
R
11
g
11
+
R
22
g

22
+
R
33
g
33
= −6[
¨a
a
+ (
˙a
a
)
2
+
k
a
2
]
R
00
−
1
2
g
00
R = Λg
00
3(
˙a

a
)
2
+
3k
a
2
= Λ
R
11
−
1
2
g
11
R = Λg
11
2
¨a
a
+ (
˙a
a
)
2
+
k
a
2
= Λ

R
22
−
1
2
g
22
R = Λg
2
¨a
a
+ (
˙a
a
)
2
+
k
a
2
= Λ
R
33
−
1
2
g
33
R = Λg
33

2
¨a
a
+ (
˙a
a
)
2
+
k
a
2
= Λ
v = Hr
r(t) = a(t)r
0
r(t) a(t) r
0
r
0
˙r = Hr
H(t) =
˙r(t)
r(t)
=
˙a
a
k = 0 Λ > 0
H
2

(t) = (
˙a
a
)
2
=
Λ
3
H(t)

Λ
3
˙a
a
H(t)
ln a = Ht
a(t) = exp(Ht) = exp(

Λ
3
t).
a(t)
v = Hr
∗
ds
2
= dt
2
− a
2

(t)(
dr
2
1 −kr
2
+ r
2
d
2
θ + r
2
sin
2
θd
2
φ).
T
µν
d
2
x
i
ds
2
+ Γ
i
µν
dx
µ
ds

dx
ν
ds
= 0
dr dθ dφ dx
i
ds dt
dx
i
ds
Γ
i
00
dx
0
ds
dx
0
ds
Γ
i
00
θ φ
T
µν
= (p + ρ)u
µ
u
ν
− pg

µν
.
p ρ u
µ
T
µν
;µ
= 0.
T
µν
;α
= T
µν
,α
+ Γ
µ
αρ
T
ρν
+ Γ
ν
αρ
T
ρµ
.
R
µν
−
1
2

g
µν
= 8ΠGT
µν
.
ρ
ρ = ρ
m
+ ρ
rad
+ ρ
vac
.
ρ
vac
=
Λ
8πG
.
(
˙a
a
)
2
+
k
a
2
=
8πG

3
ρ.
2
¨a
a
+ (
˙a
a
)
2
+
k
a
2
= −8πGp.
ρ =
3
8πG
[(
˙a
a
)
2
+
k
a
2
]
p = −
1

8πG
[2
¨a
a
+ (
˙a
a
)
2
+
k
a
2
]
=⇒ ˙ρ =
3
8πG
[2
˙a
a
¨aa −(˙a)
2
a
2
−
2k ˙a
a
3
] =
3

4πG
a˙a¨a − ˙a
3
− k ˙a
a
3
=⇒ 3a
2
˙a(p + ρ) + a
3
˙ρ = 3a
2
˙a[−
1
8πG
(
2¨a
a
+ (
˙a
a
)
2
+
k
a
2
) +
3
8πG

((
˙a
a
)
2
+
k
a
2
)]
+a
3
3
4πG
a˙a¨a − (˙a)
3
− k ˙a
a
3
=⇒ 3a
2
˙a(p + ρ) + a
3
˙ρ = 3a
2
˙a
1
4πG
((
˙a

a
)
2
+
k
a
2
) −
3a
2
˙a
8πG
2¨a
a
+
3
4πG
(a˙a¨a − ˙a
3
− k ˙a)
=⇒ 3a
2
˙a(p + ρ) + a
3
˙ρ = 0
=⇒
d
dt
[a
3

(ρ + p)] 3a
2
˙a(ρ + p) + a
3
( ˙p + ˙ρ) ˙pa
3
˙pa
3
=
d
dt
(pa
3
) +
d
dt
(ρa
3
) = p
d
dt
a
3
+ ˙pa
3
+
d
dt
(ρa
3

)
⇐⇒
d
dt
(ρa
3
) = −p
d
dt
a
3
.
da
3
−p
d
dt
a
3
p
ρ
p = αρ
α
p ρ
α =
1
3
=⇒ p =
ρ
3

α = 0 =⇒ p = 0
v << c mc
2
p = 0
α = −1 =⇒ p = −ρ
ρ
ρ
a(t)
d
dt
(ρa
3
) = −αρ
d
dt
a
3
⇐⇒ ˙ρa
3
= −(1 + α)ρ
d
dt
a
3
⇐⇒
˙ρ
ρ
= −(1 + α)
3˙a
a

= −3(1 + α)
˙a
a
lnρ = −3(1 + α)lna + const
⇐⇒ log
a
ρ = −3(1 + α) + con st.
ρ = const · a
−3(1+α)
.
α
ρ
α =
1
3
ρ ∼
1
a
4
α = 0
ρ ∼
1
a
3
a(t) a
3
(t)
a
−3
(t)

α = −1
ρ ∼ const
¨a
a
=
−4πG
3
(ρ + 3p)
a ∼ t
β
=⇒ ¨a ∼ t
β−2
=⇒
¨a
a
∼ t
−2
(3p + ρ) = (1 + 3α)ρ ∼ a
−3(1+α)
∼ t
−3β(1+α)
.
=⇒ −2 = −3β(1 + α)
=⇒ β =
2
3(1 + α)
.
a(t) ∼ t
2
3(1 + α)

α =
1
3
a(t) ∼
√
t
α = 0
a(t) ∼ t
2/3
α = −1
a(t) ∼ e
Ht
a(t
0
)
˙a > 0
ds
2
= dt
2
− a
2
(t)(
dr
2
1 −kr
2
+ r
2
dθ

2
+ r
2
sin
2
θdφ
2
)
ρ(t)
t = t
0
r
i
, θ
i
, ϕ
i
r
i
, θ
i
, ϕ
i
a(t) t
1
a(t
1
) t
2
a(t

2
)
a(t
1
)
a(t
2
)
v = Hd
H(t) =
˙a
a
t
1
d
1
= a(t
1
)s
t
2
d
2
= a(t
2
)s
v =
d
2
− d

1
t
2
− t
1
=
a(t
2
) −a(t
1
)
t
2
− t
1
s t
2
−→ t
1
v =
˙a
a
(as) = Hd H
H
0
= H(t
0
) t
0
H

0
= h ·100(kms
−1
Mpc
−1
)
h ±
p ∼ 0 ρ > 0
a(t) t
0
a(t) = e
Ht
= e
Ht
0
e
H(t−t
0
)
a(t) = a(t
0
)[1 +
1
1!
d
dt
e
H(t−t
0
)

|
t=t
0
(t −t
0
) +
1
2!
d
2
dt
2
e
H(t−t
0
)
|
t=t
0
(t −t
0
)
2
+ ···]
= a(t
0
)[1 + H
0
(t −t
0

) +
1
2
(
˙
H + H
2
)|
t=t
0
(t −t
0
)
2
+ ···]
= a(t
0
)[1 + H
0
(t −t
0
) +
1
2
¨a
a
(t −t
0
)
2

+ ···]
H =
˙a
a
˙
H =
¨a
a
− H
2
a(t) = a(t
0
)[1 + H
0
(t −t
0
) −
1
2
q
0
H
2
0
(t −t
0
)
2
+ ···]
q

0
q
0
= −
¨a
aH
2
0
ρ
crit
=
3H
2
8πG
ρ
0
crit
= 1, 9 ·10
−32
h
2
.kg.cm
−3
k
H
2
a
2
+ 1 =
ρ

3H
2
8πG
= Ω
Ω =
ρ
ρ
crit
k
H
2
a
2
= Ω − 1
Ω
Ω > 1, k > 0
Ω < 1, k < 0
Ω = 1, k = 0
Ω Ω Ω
0
t −→ 0
p = αρ
q
0
= −
¨a
aH
2
0
= −

4πG
3H
2
0
(ρ + 3p) = −
4πG
3H
2
0
ρ(1 + 3α)
=
1
2
(1 +
k
a
2
H
2
0
)(1 + 3α) =
1
2
(1 + Ω
0
− 1)(1 + 3α)
q
0
=
Ω

0
2
(1 + 3α).
α = −1 q
0
< 0
ρ
Λ
a(t)
t a
ρ
a(t) t −→ ∞
ρ
m
ρ
vac
Λ = 0
ρ
vac
= 0
ρ
m
= ρ
0
a
3
0
a
3
(t)

ρ
0
a
0
t
0
˙a
2
(t) =
8πGρ
0
a
3
0
3a(t)
− k.
ρ
0
= 0 a(t) k = −1
a
Milne
(t) = t
=⇒
ρ
0
> 0 t a ∼ t
2/3
t k
k = 0
a(t) = a

0
(
t
t
0
)
2/3
t
ρ < 0 k = −1 ˙a
2
a(t)
k
a(t) ∼ t
k = 1 ρ > 0 ˙a
2
> 0 a(t)
a
crit
=
8πGρ
0
a
3
0
3
¨a ≤ 0 a
Λ = 0
˙a
2
(t) =

8πGρ
0
a
3
0
3a(t)
− k +
Λa
2
(t)
3
Ω
M
=
8πGρ
0
3H
2
0
ρ
0
t
0
Ω
M
=
ρ
0
ρ
0

crit
ρ
0
crit
Ω
Λ
=
Λ
3H
2
0
a(t)
Λ a(t) Λ
Λ < 0 a(t)
a(t) a
crit
¨a < 0
Λ > 0
• k = 0 −1 a(t)
• k = 1
Λ ˙a ¨a 0
Λ