and Cosmology

4.3 Consequences of the Friedmann Expansion
Finally, it should be stressed again that (4.38) allows
all relations to be expressed as functions of a as well as
of z. For example, the age of the Universe as a function
of z is obtained by replacing the upper integration limit,
a → (1 + z) −1 , in (4.34).
The Necessity of a Big Bang. We discussed in
Sect. 4.3.1 that the scale factor must have attained the
value a = 0 at some time in the past. One gap in our argument
that inevitably led to the necessity of a Big Bang
still remains, namely the possibility that at sometime in
the past ȧ = 0 occurred, i.e., that the Universe underwent
a transition from a contracting to an expanding
state. This is possible only if Ω Λ > 1 and if the matter
density parameter is sufficiently small (see Fig. 4.7).
In this case, a attained a minimum value in the past.
This minimum value depends on both Ω m and Ω Λ .
For instance, for Ω m > 0.1, the value is a min > 0.3. But
a minimum value for a implies a maximum redshift
z max = 1/a min − 1. However, since we have observed
quasars and galaxies with z > 6 and the density parameter
is known to be Ω m > 0.1, such a model without
a Big Bang can be excluded.
4.3.3 Distances in Cosmology
In the previous sections, different distance measures
were discussed. Because of the monotonic behavior of
the corresponding functions, each of a, t,andz provide
the means to sort objects according to their distance. An
object at higher redshift z 2 is more distant than one at
z 1 < z 2 such that light from a source at z 2 may become
absorbed by gas in an object at redshift z 1 , but not vice
versa. The object at redshift z 1 is located between us and
the object at z 2 . The more distant a source is from us,
the longer the light takes to reach us, the earlier it was
emitted, the smaller a was at emission, and the larger z
is. Since z is the only observable of these parameters,
distances in extragalactic astronomy are nearly always
expressed in terms of redshift.
But how can a redshift be translated into a distance
that has the dimension of a length? Or, phrasing this
question differently, how many Megaparsecs away from
us is a source with redshift z = 2? The corresponding
answer is more complicated than the question suggests.
For very small redshifts, the local Hubble relation (4.39)
may be used, but this is valid only for z ≪ 1.
In Euclidean space, the separation between two
points is unambiguously defined, and several prescriptions
exist for measuring a distance. We will give two
examples here. A sphere of radius R situated at distance
D subtends a solid angle of ω = πR 2 /D 2 on our
sky. If the radius is known, D can be measured using this
relation. As a second example, we consider a source of
luminosity L at distance D which then has a measured
flux S = L/(4πD 2 ). Again, if the luminosity is known,
the distance can be computed from the observed flux. If
we use these two methods to determine, for example, the
distance to the Sun, we would of course obtain identical
results for the distance (within the range of accuracy),
since these two prescriptions for distance measurements
are defined to yield equal results.
In a non-Euclidean space like, for instance, our Universe
this is no longer the case. The equivalence of
different distance measures is only ensured in Euclidean
space, and we have no reason to expect this equivalence
to also hold in a curved spacetime. In cosmology,
the same measuring prescriptions as in Euclidean space
are used for defining distances, but the different definitions
lead to different results. The two most important
definitions of distance are:
• Angular-diameter distance: As above, we consider
a source of radius R observed to cover a solid angle ω.
The angular-diameter distance is defined as
√
R2 π
D A (z) = . (4.45)
ω
• Luminosity distance: We consider a source with luminosity
L and flux S and define its luminosity distance
as
√
D L (z) =
L
4π S . (4.46)
These two distances agree locally, i.e., for z ≪ 1; on
small scales, the curvature of spacetime is not noticeable.
In addition, they are unique functions of redshift.
They can be computed explicitly. However, to do this
some tools of GR are required. Since we have not discussed
GR in this book, these tools are not available to
us here. The distance–redshift relations depend on the
cosmological parameters; Fig. 4.11 shows the angulardiameter
distance for different models. For Λ = 0, the
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