and Cosmology

4. Cosmology I: Homogeneous Isotropic World Models
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compatible with the age of the oldest star clusters. The
angular-diameter distance (4.45) in an EdS universe is
obtained by considering the Mattig relation (4.38) for
the case Ω m = 1:
D A (z) = 2c (
1
1 − 1 )
√ ,
H 0 (1 + z) 1 + z
D L (z) = 2c (
(1 + z) 1 − 1 )
√ , (4.53)
H 0 1 + z
where we used (4.48) to obtain the second relation from
the first.
4.3.5 Summary
We shall summarize the most important points of the
two preceeding lengthy sections:
• Observations are compatible with the fact that the
Universe around us is isotropic and homogeneous on
large scales. The cosmological principle postulates
this homogeneity and isotropy of the Universe.
• General Relativity allows homogeneous and
isotropic world models, the Friedmann–Lemaître
models. In the language of GR, the cosmological
principle reads as follows: “A family of solutions
of Einstein’s field equations exists such that a set
of comoving observers see the same history of the
Universe; for each of them, the Universe appears
isotropic.”
• The shape of these Friedmann–Lemaître world models
is characterized by the density parameter Ω m
and by the cosmological constant Ω Λ , the size by
the Hubble constant H 0 . The cosmological parameters
determine the expansion rate of the Universe as
a function of time.
• The scale factor a(t) of the Universe is a monotonically
increasing function from the beginning of the
Universe until now; at earlier times the Universe was
smaller, denser, and hotter. There must have been an
instant when a → 0, which is called the Big Bang.
The future of the expansion depends on Ω m and
Ω Λ .
• The expansion of the Universe causes a redshift of
photons. The more distant a source is from us, the
more its photons are redshifted.
4.4 Thermal History of the Universe
Since T ∝ (1 + z) the Universe was hotter at earlier
times. For example, at a redshift of z = 1100 the temperature
was about T ∼ 3000 K. And at an even higher
redshift, z = 10 9 ,itwasT ∼ 3 × 10 9 K, hotter than in
a stellar interior. Thus we might expect energetic processes
like nuclear fusion to have taken place in the
early Universe.
In this section we shall describe the essential processes
in the early Universe. To do so we will assume
that the laws of physics have not changed over time. This
assumption is by no means trivial – we have no guarantee
whatsoever that the cross-sections in nuclear physics
were the same 13 billion years ago as they are today.
But if they have changed in the course of time the only
chance of detecting this is through cosmology. Based
on this assumption of time-invariant physical laws, we
will study the consequences of the Big Bang model developed
in the previous section and then compare them
with observations. Only this comparison can serve as
a test of the success of the model. A few comments
should serve as preparation for the discussion in this
section.
1. Temperature and energy may be converted into
each other since k B T has the dimension of energy.
We use the electron volt (eV) to measure
temperatures and energies, with the conversion
1eV= 1.1605 × 10 4 k B K.
2. Elementary particle physics is very well understood
for energies below ∼ 1 GeV. For much higher energies
our understanding of physics is a lot less
certain. Therefore, we will begin the consideration
of the thermal history of the cosmos at energies below
1 GeV.
3. Statistical physics and thermodynamics of elementary
particles are described by quantum mechanics.
A distinction has to be made between bosons,
which are particles of integer spin (like the photon),
and fermions, particles of half-integer spin
(like, for instance, electrons, protons, or neutrinos).
4. If particles are in thermodynamical and chemical
equilibrium, their number density and their energy
distribution are specified solely by the temperature –
e.g., the Planck distribution (A.13), and thus the en-