and Cosmology

4.3 Consequences of the Friedmann Expansion
Fig. 4.12. A modern Hubble diagram: for several clusters of
galaxies, the K-band magnitude of the brightest cluster galaxy
is plotted versus the escape velocity, measured as redshift
z = Δλ/λ (symbols). If these galaxies all had the same luminosity,
the apparent magnitude would be a measure of
distance. For low redshifts, the curves follow the linear Hubble
law (4.9), with z ≈ v/c, whereas for higher redshifts modifications
to this law are necessary. The solid curve corresponds to
a constant galaxy luminosity at all redshifts, whereas the two
other curves take evolutionary effects of the luminosity into account
according to models of population synthesis (Sect. 3.9).
Two different epochs of star formation were assumed for these
galaxies. The diagram is based on a cosmological model with
a deceleration parameter of q 0 = 0 (see Eq. 4.33)
define the function
⎧
⎪⎨ 1/ √ K sin( √ Kx) K > 0
f K (x) = x K = 0 ,
⎪⎩
1/ √ −K sinh( √ −Kx) K < 0
where K is the curvature scalar (4.30). The comoving
radial distance x of a source at redshift z can be computed
using dx = a −1 dr =−a −1 c dt =−c da/(a 2 H).
Hence with (4.31)
x(z) = c H 0
(4.49)
∫ 1
da(c/H 0 )
× √ .
aΩm + a 2 (1 − Ω m − Ω Λ ) + a 4 Ω Λ
(1+z) −1
The angular-diameter distance is then given as
D A (z) = 1
1 + z f K [x(z)] , (4.50)
and thus can be computed for all redshifts and cosmological
parameters by (in general numerical) integration
of (4.49). The luminosity distance then follows from
(4.48). The angular-diameter distance of a source at redshift
z 2 , as measured by an observer at redshift z 1 < z 2 ,
reads
D A (z 1 , z 2 ) = 1 f K [x(z 2 ) − x(z 1 )] . (4.51)
1 + z 2
This is the distance that is required in equations
of gravitational lens theory for D ds . In particular,
D A (z 1 , z 2 ) ̸= D A (z 2 ) − D A (z 1 ).
4.3.4 Special Case: The Einstein–de Sitter Model
As a final note in this section, we will briefly examine
one particular cosmological model more closely,
namely the model with Ω Λ = 0 and vanishing curvature,
K = 0, and hence Ω m = 1. We disregard the radiation
component, which contributes to the expansion only at
very early times and thus for very small a. For a long
time, this Einstein–de Sitter (EdS) model was the preferred
model among cosmologists because inflation (see
Sect. 4.5.3) predicts K = 0 and because a finite value for
the cosmological constant was considered “unnatural”.
In fact, as late as the mid-1990s, this model was termed
the “standard model”. In the meantime we have learned
that Λ ̸= 0; thus we are not living in an EdS universe.
But there is at least one good reason to examine this
model a bit more, since the expansion equations become
much simpler for these parameters and we can
formulate simple explicit expressions for the quantities
introduced above. These then yield estimates which for
other model parameters are only possible by means of
numerical integration.
The resulting expansion equation ȧ = H 0 a −1/2 is
easily solved by making the ansatz a = (Ct) β which,
when inserted into the equation, yields the solution
( ) 3 H0 t 2/3
a(t) =
. (4.52)
2
Setting a = 1, we obtain the age of the Universe,
t 0 = 2/(3H 0 ). The same result also follows immediately
from (4.34) if the parameters of an EdS model are
inserted there. Using H 0 ≈ 70 km s −1 Mpc −1 results in
an age of about 10 Gyr, which is slightly too low to be
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