Cosmology:
the dynamical universe
The large-scale universe
· Homogeneity
on distance scales > 100 Mpc the universe is highly isotropic and
homogeneous.
· Expansion
As time unfolds the physical distances between distant objects
in the universe increase uniformly.
Homogeneity
t’
t
· Cosmic time:
Prefered time co-ordinate at which hypersurfaces of equal time
are homogeneous
Galaxy distribution (2DF)
CMB temperature fluctuations (WMAP)
Expansion
Doppler effect:
Redshift z = ∆λ/λ:
λ0
λ + ∆λ
v
=
=1+z ≈1+
λ
λ
c
Hubble:
average galactic redshift proportional to distance
Ho d
z=
c
⇔
v = Ho d
Left: Supernova data up to z ≈ 0.15
Right: Hubble’s measurements
Ho = 73 ± 3 km/sec/Mpc
The scale factor a(t)
Equal cosmic-time distance
between points A and B:
d(t) = a(t)(rA − rB )
a
λ
1
=
=
a0
λ0
1+z
v = d˙ = Hd
⇒
H=
ȧ
a
Cosmic geometry
gµν dxµdxν = −c2dt2 + ds2,
ds2 = a2(t)
k = +1
with k = −1
k=0
dr2
!
2 dθ 2 + r 2 sin2 θ dϕ2 ,
+
r
1 − kr2
(spherical universe)
(hyperbolic universe)
(flat universe)
Dynamics of expansion
Einstein equations (c = 1)
1
Rµν − gµν R = −8πG Tµν
2
imply for cosmic geometry:
1. H 2 =
2
ȧ
a
8πGε
k
=
− 2
3
a
da3
d(εa3)
+p
=0
2.
dt
dt
In addition: equation of state ε(p).
d(εa3)
da3
+p
=0
dt
dt
1. Radiation (relativistic particle gas):
ε
p=
⇒
εr a4 = constant.
3
2. Non-relativistic matter (dust):
p=0
εma3 = constant.
⇒
3. Cosmological constant (vacuum energy):
p = −ε
⇒
εv = constant.
Critical density:
⇒
k=0
3H 2
3 ȧ2
εc =
=
.
2
8πG
8πG a
With present value H0:
εc 0 ' 5.6 GeV/m3
Friedman eqn.:
3H 2
3k
k
ε =
+
=
ε
1
+
c
8πG
8πGa2
a2 H 2
3
a0 4
a0
= εr0
+ εm0
+ εv0.
a
a
The present values of the various contribution to the energy
density can be represented as fractions of this critical density:
ε
ε
ε
2)
Ωm ≡ m 0 ,
Ωr ≡ r 0 ,
Ωv ≡ v 0 ,
Ωk ≡ −k/(a2
H
0 0
εc 0
εc 0
εc 0
→ the Friedman eqn. takes the form
4
3
2
εc
H2
a0
a0
a0
+ Ωm
+ Ωv + Ωk
.
= 2 = Ωr
εc 0
a
a
a
H0
In particular
Ωr + Ωm + Ωv + Ωk = 1.
Angular power spectrum of the CMB
Time dependence of scale factor
1. Pure radiation dominated universe:
√
1
ȧ ∝
⇒
a(t) ∝ t.
a
2. Pure matter dominated universe:
1
ȧ ∝ √
⇒
a(t) ∝ t2/3.
a
3. Pure vacuum dominated universe:
ȧ ∝ a
⇒
a(t) ∝ eH0t
4. Pure curvature dominated universe (k = −1):
ȧ = constant
⇒
a(t) ∝ t.
The age of the universe
v
u
2
a0H0 u
ȧ
a
t
H= =
Ωr + Ωm + Ωv
2
a
a
a0
a
a0
!4
a
+ Ωk
a0
!2
Then with x = a/a0:
1
1
xdx
q
t0 =
H0 0
Ωr + Ωmx + Ωv x4 + Ωk x2
Z
Standard cosmology:
Ωr < 10−4,
→
Ωm = 0.26,
t0 =
Ωv = 0.74,
1.01
= 13.5 × 109 yr
H0
Ωk = 0
Cross-over in the energy density
The energy density of matter and vacuum energy was equal:
εm = εv at scale factor given by
⇔
εm = εv
a0 3
Ωm
= Ωv ,
a
a
Ωm 1/3
x=
=
= 0.7,
a0
Ωv
t=
0.7
= 9.2 × 109 yr,
H0
or z = 0.4, at which time εr = 4 × εr 0 < 10−3.
Cross-over matter and radiation:
εm = εr
or
x=
⇔
4
a0 3
a0
Ωm
= Ωr
a
a
a
Ωr
≈ 3 × 10−4.
=
a0
Ωm
Fate of the universe
V
eff
( Ωv > 0 )
H2 Ω
0
−1
0
x
H 2 Ω +1
0
Friedmann eqn. for x = a/a0:
ẋ2 + Vef f (x) = H02 Ωk ,
where
Ωr
Ωm
2
Vef f (x) = −H02
+
Ω
x
+
v
x2
x