and Cosmology

4.3 Consequences of the Friedmann Expansion
lecting the vacuum energy density. However, there
are very strong observational indications that in fact
Λ>0.
• Models with Ω m + Ω Λ = 1, i.e., K = 0. Such flat
models are preferred by the so-called inflationary
models, which we will briefly discuss further below.
A special case is the Einstein–de Sitter model, Ω m = 1,
Ω Λ = 0. For this model, t 0 = 2/(3H 0 ) ≈ 6.7 h −1 ×
10 9 yr.
For many world models, t 0 is larger than the age of
the oldest globular clusters, so they are compatible with
this age determination. The Einstein–de Sitter model,
however, is compatible with stellar ages only if H 0 is
very small, considerably smaller than the value of H 0
derived from the HST Key Project discussed in Sect. 3.6.
Hence, this model is ruled out by observations.
It is believed that the values of the cosmological
parameters are now quite well known. We list them here
for later reference without any further discussion. Their
determination will be described in the course of this
chapter and in Chap. 8. The values are approximately
Ω m ∼ 0.3 ; Ω Λ ∼ 0.7 ; h ∼ 0.7 . (4.35)
4.3.2 Redshift
The Hubble law describes a relation between the redshift,
or the radial component of the relative velocity,
and the distance of an object from us. Furthermore,
(4.6) specifies that any observer is experiencing a local
Hubble law with an expansion rate H(t) which depends
on the cosmic epoch. We will now derive a relation
between the redshift of a source, which is directly observable,
and the cosmic time t or the scaling factor a(t),
respectively, at which the source emitted the light we
receive today.
To do this, we consider a light ray that reaches us today.
Along this light ray we imagine fictitious comoving
observers. The light ray is parametrized by the cosmic
time t, and is supposed to have been emitted by the
source at epoch t e . Two comoving observers along the
light ray with separation dr from each other see their
relative motion due to the cosmic expansion according
to (4.6), dv = H(t) dr, and they measure it as a redshift
of light, dλ/λ = dz = dv/c. It takes a time dt = dr/c for
the light to travel from one observer to the other. Furthermore,
from the definition of the Hubble parameter,
ȧ = da/dt = Ha, we obtain the relation dt = da/(Ha).
Combining these relations, we find
dλ
λ = dv
c = H c
dr = H dt =
da
a . (4.36)
The relation dλ/λ = da/a is now easily integrated since
the equation dλ/da = λ/a obviously has the solution
λ = Ca, where C is a constant. That constant is determined
by the wavelength λ obs of the light as observed
today (i.e., at a = 1), so that
λ(a) = a λ obs (4.37)
(see Fig. 4.10). The wavelength at emission was therefore
λ e = a(t e )λ obs . On the other hand, the redshift z
is defined as (1 + z) = λ obs /λ e . From this, we finally
obtain the relation
1 + z = 1 a
(4.38)
between the observable z and the scale factor a which is
linked via (4.34) to the cosmic time. The same relation
can also be derived by considering light rays in GR.
The relation between redshift and the scale factor is of
immense importance for cosmology because, for most
sources, redshift is the only distance information that
we are able to measure. If the scale factor is a monotonic
function of time, i.e., if the right-hand side of
(4.31) is different from zero for all a ∈[0, 1], then z is
also a monotonic function of t. In this case, which corresponds
to the Universe we happen to live in, a, t,and
z are equally good measures of the distance of a source
from us.
Fig. 4.10. Due to cosmic expansion, photons are redshifted,
i.e., their wavelength, as measured by a comoving observer,
increases with the scale factor a. This sketch visualizes this
effect as a standing wave in an expanding box
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