and Cosmology

9. The Universe at High Redshift
384
provide a thermal coupling of the baryons to the cosmic
background radiation, by means of Compton scattering.
This is the case for redshifts z z t , where
(
Ωb h 2 ) 2/5
z t ≈ 140
;
0.022
hence, T b (z) ≈ T(z) = T 0 (1 + z) for z z t . For smaller
redshifts, the density of photons gets too small to maintain
this coupling, and baryons start to adiabatically cool
down by the expansion, so that for z z t we obtain
approximately T b ∝ ρ 2/3
b
∝ (1 + z) 2 .
From this temperature dependence, the Jeans mass
can then be calculated as a function of redshift. For
z t z 1000, M J is independent of z because c s ∝
T 1/2 ∝ (1 + z) 1/2 and ρ ∝ (1 + z) 3 , and its value is
(
M J = 1.35 × 10 5 Ωm h 2 ) −1/2
M ⊙ , (9.6)
0.15
whereas for z z t we obtain, with T b ≃ 1.7 × 10 −2
(1 + z) 2 K,
) −1/2
(
M J = 5.7 × 10 3 Ωm h 2
0.15
(
Ωb h 2 ) −3/5 ( ) 1 + z 3/2
×
M ⊙ . (9.7)
0.022 10
Only in recent years has it been discovered that
molecular hydrogen represents an extremely important
component in cooling processes. Despite its very small
transition probability, H 2 dominates the cooling rate of
primordial gas at temperatures below T ∼ 10 4 K–see
Fig. 9.29 – where the precise value of this temperature
depends on the abundance of H 2 .
By means of H 2 , the gas can cool in halos with a temperature
exceeding about T vir 3000 K, corresponding
to a mass of M 10 4 M ⊙ ; the exact values depend on
the redshift. In these halos, stars may then be able to
form. However, these stars will certainly be different
from those known to us, because they do not contain
any metals. Therefore, the opacity of the stellar
plasma is much lower. Such stars, which at the same
mass presumably have a much higher temperature and
luminosity (and thus a shorter lifetime), are called Population
III stars. Due to their high temperature they are
much more efficient sources of ionizing photons than
stars with “normal” metallicity.
Cooling of the Gas. The Jeans criterion is a necessary
condition for the formation of proto-galaxies. In order
to form stars, the gas in the halos needs to be able to
cool further. Here, we are dealing with the particular situation
of the first galaxies, whose gas is metal-free, so
metal lines cannot contribute to the cooling. This means
that cooling can only happen via hydrogen and helium.
Since the first excited state of hydrogen has a high energy
(that of the Lyα transition, thus E ∼ 10.2 eV), this
cooling is efficient only above T 2 × 10 4 K. However,
the halos which form at high redshift have low mass,
so that their virial temperature is considerably below
this energy scale. Therefore, atomic hydrogen is a very
inefficient coolant for these first halos, insufficient to
initiate the formation of stars. Furthermore, helium is
of no help in this context, since its excitation temperature
is even higher than that of hydrogen. The problem
resulting from these arguments has become even worse
given the WMAP discovery of a higher reionization
redshift than previously estimated.
Fig. 9.29. Cooling rate as a function of the temperature for
a gas consisting of atomic and molecular hydrogen (with
0.1% abundance) and of helium. The solid curve describes
the cooling by atomic gas, the dashed curve that by molecular
hydrogen; thus, the latter is extremely important at temperatures
below ∼ 10 4 K. At considerably lower temperatures the
gas cannot cool, hence no star formation will take place