and Cosmology

4.4 Thermal History of the Universe
it is not a recombination but rather the (first) transition to
a neutral state – however the expression recombination
has now long been established. The recombination of
electrons and nuclei is in competition with the ionization
of neutral atoms by energetic photons (photoionization),
whereas collisional ionization can be disregarded completely
since η – (4.59) – is so small. Because photons
are so much more numerous than electrons, cooling has
to proceed to well below the ionization temperature,
corresponding to the binding energy of an electron in
hydrogen, before neutral atoms become abundant. This
happens for the same reasons as apply in the context of
deuterium formation: there are plenty of ionizing photons
in the Wien tail of the Planck distribution, even if
the temperature is well below the ionization temperature.
The ionization energy of hydrogen is χ = 13.6eV,
corresponding to a temperature of T > 10 5 K, but T
has to first decrease to ∼ 3000 K before the ionization
fraction
number density of free electrons
x =
total number density of existing protons
(4.65)
falls considerably below 1, for the reason mentioned
above. At temperatures T > 10 4 Kwehavex ≈ 1, i.e.,
virtually all electrons are free. Only at z ∼ 1300 does x
deviate significantly from unity.
The onset of recombination can be described by an
equilibrium consideration which leads to the so-called
Saha equation,
( )
1 − x
kB T 3/2 ( ) χ
x 2 ≈ 3.84 η
m e c 2 exp ,
k B T
which describes the ionization fraction x as a function
of temperature. However, once recombination occurs,
the assumption of thermodynamical equilibrium is no
longer justified. This can be seen from the following
consideration.
Any recombination directly to the ground state leads
to the emission of a photon with energy E γ >χ.However,
these photons can ionize other, already recombined
(thus neutral), atoms. Because of the large cross-section
for photoionization, this happens very efficiently. Thus
for each recombination to the ground state, one neutral
atom will become ionized, yielding a vanishing net effect.
But recombination can also happen in steps, first
into an excited state and then evolving into the ground
state by radiative transitions. Each of these recombinations
will yield a Lyα photon in the transition from the
first excited state into the ground state. This Lyα photon
will then immediately excite another atom from the
ground state into the first excited state, which has an
ionization energy of only χ/4. This yields no net production
of atoms in the ground state. Since the density
of photons with E γ >χ/4 is very much larger than of
those of E γ >χ, the excited atoms are more easily ionized,
and this indeed happens. Stepwise recombination
thus also provides no route towards a lower ionization
fraction.
The processes described above cause a small distortion
of the Planck spectrum due to recombination
radiation (in the range χ ≫ k B T) which affects recombination.
One cannot get rid of these energetic
photons – in contrast to gas nebulae like HII regions,
in which the Lyα photons may escape due to the finite
geometry.
Ultimately, recombination takes place by means of
a very rare process, the two-photon decay of the first
excited level. This process is less probable than the direct
Lyα transition by a factor of ∼ 10 8 .However,it
leads to the emission of two photons, both of which
are not sufficiently energetic to excite an atom from the
ground state. This 2γ -transition is therefore a net sink
for energetic photons. 7 Taking into account all relevant
processes and using a rate equation, which describes
the evolution of the distribution of particles and photons
even in the absence of thermodynamic equilibrium,
gives for the ionization fraction in the relevant redshift
range 800 z 1200
√
x(z) = 2.4 × 10 −3 Ωm h 2 ( z
) 12.75
. (4.66)
Ω b h 2 1000
The ionization fraction is thus a very strong function of
redshift since x changes from 1 (complete ionization)
7 The recombination of hydrogen – and also that of helium which
occurred at higher redshifts – perturbed the exact Planck shape of
the photon distribution, adding to it the Lyman-alpha photons and
the photon pairs from the two-photon transition. This slight perturbation
in the CMB spectrum should in principle still be present today.
Unfortunately, it lies in a wavelength range (∼ 200 μm) where the
dust emission from the Galaxy is very strong; in addition, the wavelength
range coincides with the peak of the far-infrared background
radiation (see Sect. 9.3.1). Therefore, the detection of this spectral
distortion will be extremely difficult.
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