amendola l. dark energy

•
•
•
•
•
•
•
R
µν
−
1
2
Rg
µν
= 8πGT
µν
gravity matter
T
ν
µ
=







ρ 0 0 0
0 −p 0 0

0 0 −p 0
0 0 0 −p







•
T
µν
˙ρ = 0
∇p = 0
ρ = nmc
2
p
i
= nmv
2
i
T
µν
= diag(ρ, p, p, p)
T
µν
,µ
= 0
ρ, p, u
µ

= dx
µ
/ds, g
µν
T
µν
= (ρ + p)u
µ
u
ν
− pg
µν
p ρ
R
µν
−
1
2
Rg
µν
= 8πGT
µν
•
E
ν
µ
E
ν
µ;ν
= 0

•
RR
µν
, R
;µν
, R
2
g
µν
,
F = F (R)
E
µν
≡ F
′
R
µν
−
1
2
F g
µν
+ g
µν
F
′
− F
′
;µ;ν
•

Λg
µν
R
µν
−
1
2
Rg
µν
− Λg
µν
= 8πGT
µν
•
•
R
µν
−
1
2
Rg
µν
= 8πGT
µν
+ Λg
µν
T
µν(Λ)
=


Λ
8π

g
µν
•
T
ν
µ(Λ)
=

Λ
8π

δ
ν
µ







ρ 0 0 0
0 −p 0 0
0 0 −p 0
0 0 0 −p








=







Λ
8π
0 0 0
0
Λ
8π
0 0
0 0
Λ
8π
0
0 0 0
Λ
8π








p
Λ
= −
Λ
8π
, ρ
Λ
=
Λ
8π
Λ > 0
•
p = wρ
w = −1
p = mv
2
≈ 0 → w = 0
p = ρ/3 → w = 1/3
g
µν
ds
2
= g
00
dt
2

+ 2g
0i
dx
i
dt − σ
ij
dx
i
dx
j
g
0i
= 0
dτ =
√
g
00
dt → g
00
= 1
ds
2
= dt
2
− σ
ij
dx
i
dx
j

ds
2
3
= σ
ij
dx
i
dx
j
|r| dx
2
+ dy
2
+ dz
2
=
dr
2
+ r
2
(dθ
2
+ sin
2
θdφ)
ds
2
3
= a
2

(t)λ
2
(r)[dr
2
+ r
2
(dθ
2
+ sin
2
θdφ)]
r
ds
2
3
= a
2
(t)[λ
′2
(r
′
)dr
′2
+ r
′2
(dθ
2
+ sin
2
θdφ)]

λ(r)
a
2
= x
2
1
+ x
2
2
+ x
2
3
+ x
2
4
x
1
= a cos χ sin θ sin φ
x
2
= a cos χ cos θ
x
3
= a cos χ sin θ cos φ
x
4
= a sin χ
x
4
dx

4
= −(x
1
dx
1
+ x
2
dx
2
+ x
3
dx
3
)
ds
2
= dx
2
1
+ dx
2
2
+ dx
2
3
+ dx
2
4
= dx
2

1
+ dx
2
2
+ dx
2
3
+
(x
1
dx
1
+ x
2
dx
2
+ x
3
dx
3
)
2
x
2
4
= a
2
(dχ
2
+ sin

2
χ(dθ
2
+ sin
2
θdφ
2
))
sin χ = r dχ = λdr
λ =
1
√
1 − r
2
a
2
= x
2
1
+ x
2
2
+ x
2
3
+ kx
2
4
ds
2

3
= a
2
(dχ
2
+ F (χ)(dθ
2
+ sin
2
θdφ
2
))
F (χ) =
sin χ k = 1
χ k = 0
sinh χ k = −1
λ =
1
√
1 − kr
2
ds
2
= dt
2
−a
2
(t)[
dr
2

1 − kr
2
+r
2
(dθ
2
+sin
2
θdφ
2
)]
•
•
ds
2
= dt
2
−a
2

dr
2
1 − kr
2
+ r
2
sin θdφ
2
+ r
2

dθ
2

k = 0 a
0
= 1
H
2
≡

˙a
a

2
=
8π
3
ρ
¨a
a
= −
4π
3
(ρ + 3p) = −
4π
3
ρ(1 + 3w)
w < −1/3
w < −1/3
→

•
H
2
≡

˙a
a

2
=
8π
3
ρ
Λ
=
Λ
3
a = a
0
e
√
Λ
3
t
•
H
2
≡

˙a

a

2
=
8π
3
(ρ
γ
+ ρ
M
+ ρ
Λ
) −
k
a
2
˙ρ
i
+ 3H(ρ
i
+ p
i
) = 0
ρ
γ
∼ a
−4
ρ
M
∼ a

−3
ρ
k
≡
k
a
2
∼ a
−2
ρ
Λ
∼ a
0
•
rad. → matter → curvature → cosm.const.
10
0
10
1
10
2
10
3
10
4
1z
0
0.2
0.4
0.6

0.8
1

MatRad 
•
•
•
E
0
=
1
2
ω
•
E
0
=

i
1
2
ω
i
k
i
= 2π/λ
i
λ
i
= L/n

i
L dn
i
= dk
i
L/2π
dk
i
E
0
=
1
2
L
3

d
3
k
(2π)
3
ω
k
ω
2
= k
2
+ m
2
/

2
k
max
ρ
vacuum
= lim
E
L
3
= 
k
4
max
16π
2
•
k
max
• k
max
E
P lanck
= 10
19
GeV
ρ
vacuum
= 10
92
g/cm

3
ρ = 3H
2
/8πG ≃ 10
−29
g/cm
3
•