and Cosmology

4.4 Thermal History of the Universe
ergy density of the radiation (4.43), is a function of T
only.
The necessary condition for establishing chemical
equilibrium is the possibility for particles to be be created
and destroyed, such as in electron–positron pair
production and annihilation.
4.4.1 Expansion in the Radiation-Dominated
Phase
As mentioned above (4.28), the energy density of radiation
dominates in the early Universe, at redshifts z ≫ z eq
where
z eq = a −1
eq − 1 ≈ 23 900 Ω m h 2 , (4.54)
The radiation density behaves like ρ r ∝ T 4 , where the
constant of proportionality depends on the number of
species of relativistic particles (these are the ones for
which k B T ≫ mc 2 ). Since T ∝ 1/a and thus ρ r ∝ a −4 ,
radiation then dominates in the expansion equation
(4.18). The latter can be solved by a power law, a(t) ∝ t β ,
which after insertion into the expansion equation yields
β = 1/2 and thus
√
a ∝ t 1/2 3
, t =
32πGρ ,
t ∝ T −2
in radiation-dominated phase , (4.55)
where the constant of proportionality depends again on
the number of relativistic particle species. Since the latter
is known, assuming thermodynamical equilibrium,
the time dependence of the early expansion is uniquely
specified by (4.55). This is reasonable because for early
times neither the curvature term nor the cosmological
constant contribute significantly to the expansion
dynamics.
4.4.2 Decoupling of Neutrinos
We start our consideration of the Universe at a temperature
of T ≈ 10 12 K which corresponds to ∼ 100 MeV.
This energy can be compared to the rest mass of various
particles:
proton, m p = 938.3MeV/c 2 ,
neutron, m n = 939.6MeV/c 2 ,
electron, m e = 511 keV/c 2 ,
muon, m μ = 140 MeV/c 2 .
Protons and neutrons (i.e., the baryons) are too heavy
to be produced at the temperature considered. Thus all
baryons that exist today must have already been present
at this early time. Also, the production of muon 5 pairs,
according to the reaction γ + γ → μ + + μ − , is not efficient
because the temperature, and thus the typical
photon energy, is not sufficiently high. Hence, at the
temperature considered the following relativistic particle
species are present: electrons and positrons, photons
and neutrinos. These species contribute to the radiation
density ρ r . The mass of the neutrinos is not accurately
known, though we recently learned that they have
a small but finite rest mass. As will be explained in
Sect. 8.7, cosmology allows us to obtain a very strict
limit on the neutrino mass, which is currently below
1 eV. For the purpose of this discussion they may be
considered as massless.
In addition to relativistic particles, non-relativistic
particles also exist. These are the protons and neutrons,
and probably also the constituents of dark matter. We
assume that the latter consists of weakly interacting
massive particles (WIMPs), with rest mass larger than
∼ 100 GeV because up to these energies no WIMP candidates
have been found in terrestrial particle accelerator
laboratories. With this assumption, WIMPs are nonrelativistic
at the energies considered. Thus, like the
baryons, they virtually do not contribute to the energy
density in the early Universe.
Apart from the WIMPs, all the aforementioned particle
species are in equilibrium, e.g., by the following
reactions:
e ± + γ ↔ e ± + γ : Compton scattering,
e + +e − ↔ γ +γ : pair-production and annihilation,
ν + ν ↔ e + + e − : neutrino–antineutrino scattering,
ν + e ± ↔ ν + e ± : neutrino–electron scattering.
5 Muons are particles which behave in many respects as electrons, except
that they are much heavier. Furthermore, muons are unstable and
decay on a time-scale of ∼ 2 × 10 −6 s into an electron (or positron)
and two neutrinos.
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