General Relativity and Cosmology II
Sheet 3
Prof. Dr. F.R. Klinkhamer; Dr. S. Ertl
Institut für Theoretische Physik
WS 2012/13
Name:
Due: 12.11.12
Discussion: 13.11. 2012
..................................
Please hand in this sheet with your solutions.
(3.1) Luminosity distance versus redshift
Consider an FRW Universe (setting c=1)
2
2
2
ds = −dt + a(t)
dr2
+ r2 dΩ2
1 − kr2
The proper distance dp (z) to a radiation source emitting light, which is detected with a redshift
z, is given as
Z z
dz 0
dp (z) =
0
0 H(z )
where the Hubble parameter H ≡ aȧ is now expressed in terms of redshift z 0 . Show that if
L is the total, intrinsic power radiated by the observed object then its apparent brightness f
(measured flux in watts per unit area) is given as
f=
L
4πd2p (1
+ z)2
.
This prompts the definition of the luminosity distance dL as
dL ≡ dp (1 + z).
(3.2) Age of the FRW universe
Consider the Universe containing the three energy components: matter, radiation, vacuum
energy. Then we divide the density into its matter, radiation and vacuum components ρ =
ρm + ρr + ρvac (and wm = 0, wr = 31 , wΛ = −1) and likewise for the density parameter
Ω0 = Ωm + Ωr + Ωvac = 1. In the general case with arbitrary values of the Ω’s we can rewrite
the Friedmann equation the following way
2
a 4 8πG
a 3
ȧ
8πG
k
0
0
=
Ωr ρc
+
Ωm ρc
+ ΩΛ H02 − 2 .
a
3
a
3
a
a
1
Defining x ≡ aa0 = 1+z
and integrate from it the time it takes for the universe to expand from
a1 to a2 or from redshift z1 to z2 , obtain the age-redshift relation
Z
t(z) =
dt =
0
(P.T.O.)
t
H0−1
Z
0
1
1+z
dx
p
(1 − Ω0 ) + Ωr x−2 + Ωm x−1 + ΩΛ x2
Whats the relation for the present age of the universe?
Compute the age of the open universe (k < 0 or Ω0 < 1) but without vacuum energy (ΩΛ = 0)
and approximating Ωr ≈ 0. (Hint: Integrate the relation you get for the present age of the
Ωm
universe with the substitution x = 1−Ω
sinh2 ψ2 ).
m
What do you get if you insert h = 0.72 and Ωm = 0.3?
(3.3) Distance and redshift in the flat matter-dominated universe
The distance-redshift relationship is given by
Z 1
dx
p
d0 (z) = H0−1
1
Ω0 (x − x2 ) − ΩΛ (x − x4 ) + x2
1+z
1
where d0 (z) is the comoving distance to redshift z and x ≡ aa0 = 1+z
.
Consider a galaxy and assume that it has stayed at the same coordinate value r, i.e. it has no
peculiar velocity, then the comoving distance is equal to its present distance. The actual distance
to the galaxy at the time t1 when the light left the galaxy is
d1 (z) =
d0 (z)
.
1+z
Now, compute d1 (z) for the simplest case Ωm = 1, ΩΛ = 0, Ωr = 0 and check if the Hubble
law is valid for small z.
(P.T.O.)