and Cosmology

9.4 Reionization of the Universe
This raises the question of how this reionization occurred,
in particular which process was responsible for
it. The latter question is easy to answer – reionization
must have happened by photoionization. Collisional
ionization can be ruled out because for it to be efficient
the IGM would need to be very hot, a scenario
which can be excluded due to the perfect Planck spectrum
of the CMB – the argument here is the same as
above, where we excluded the idea of a hot IGM as the
source of the CXB. Hence, the next question is where
the energetic photons that caused the photoionization of
the IGM come from.
Two kinds of sources may account for them – hot
stars or AGNs. Currently, it is not unambiguously clear
which of these is the predominant source of energetic
photons causing reionization since our current understanding
of the formation of supermassive black holes
is still insufficient. However, it is currently thought that
the main source of photoionization photons is the first
generation of hot stars.
9.4.1 The First Stars
Following on from the above arguments, understanding
reionization is thus directly linked to studying the first
generation of stars. In the present Universe star formation
occurs in galaxies; thus, one needs to examine when
the first galaxies could have formed. From the theory of
structure formation, the mass spectrum of dark matter
halos at a given redshift can be computed by means of,
e.g., the Press–Schechter model (see Sect. 7.5.2). Two
conditions need to be fulfilled for stars to form in these
halos. First, gas needs to be able to fall into the dark
halos. Since the gas has a finite temperature, pressure
forces may impede the infall into the potential well.
Second, this gas also needs to be able to cool, condensing
into clouds in which stars can then be formed. We
will now examine these two conditions.
The Jeans Mass. By means of a simple argument, we
can estimate under which conditions pressure forces are
unable to prevent the infall of gas into a potential well.
To do this, we consider a slightly overdense spherical
region of radius R whose density is only a little larger
than the mean cosmic matter density ρ. If this sphere
is homogeneously filled with baryons, the gravitational
binding energy of the gas is about
|E grav |∼ GMM b
,
R
where M and M b denote the total mass and the baryonic
mass of the sphere, respectively. The thermal energy of
the gas can be computed from the kinetic energy per
particle, multiplied by the number of particles in the
gas, or
Here,
E th ∼ c 2 s M b .
c s ≈
√
k B T b
μm p
is the speed of sound in the gas, which is about the average
speed of the gas particles, and μm p denotes the
average particle mass in the gas. For the gas to be bound
in the gravitational field, its gravitational binding energy
needs to be larger than its thermal energy, |E grav | > E th ,
which yields the condition GM> c 2 s R. Since we have
assumed an only slightly overdense region, the relation
M ∼ ρ R 3 between mass and radius of the sphere applies.
From the two latter equations, the radius can be
eliminated, yielding the condition
M >
( c
2
s
G
) 3/2
1 √ρ . (9.4)
Thus, as a result of our simple argument we find that
the mass of the halo needs to exceed a certain threshold
for gas to be able to fall in. A more accurate treatment
yields the condition
M > M J ≡ π5/2
6
( c
2
s
G
) 3/2
1 √ρ . (9.5)
In the final step we defined the Jeans mass M J , which
describes the minimum mass of a halo required for the
gravitational infall of gas. The Jeans mass depends on
the temperature of the gas, expressed through the sound
speed c s , and on the mean cosmic matter density ρ. The
latter can easily be expressed as a function of redshift,
ρ(z) = ρ 0 (1 + z) 3 .
The baryon temperature has a more complicated dependence
on redshift. For sufficiently high redshifts, the
small fraction of free electrons that remain after recombination
– the gas has a degree of ionization of ∼ 10 −4 –
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