and Cosmology

4. Cosmology I: Homogeneous Isotropic World Models
164
than baryons, a sufficient number of highly energetic
photons, with E γ ≥ E b , exist in the Wien tail of the
Planck distribution to instantly destroy newly formed D
by photo-dissociation. Only when the temperature has
decreased considerably, k B T ≪ E b , can the deuterium
abundance become appreciable. With the corresponding
balance equations we can calculate that the formation
rate exceeds the photo-dissociation rate of deuterium at
about T D ≈ 8 × 10 8 K, corresponding to t ∼ 3min.Up
to then, a fraction of the neutrons has thus decayed,
yielding a neutron–proton ratio at T D of n n /n p ≈ 1/7.
After that time, everything happens very rapidly. Owing
to the strong interaction, virtually all neutrons first
become bound in D. Once the deuterium density has
become appreciable, helium (He 4 ) forms, which is a nucleus
with high binding energy (∼ 28 MeV) which can
therefore not be destroyed by photo-dissociation. Except
for a small (but, as we will later see, very important)
remaining fraction, all deuterium is quickly transformed
into He 4 . For this reason, the dependence of helium formation
on the small binding energy of D is known as
the “bottleneck of nucleosynthesis”.
Helium Abundance. The number density of helium nuclei
can now be calculated since virtually all neutrons
present are bound in He 4 . First, n He = n n /2, since every
helium nucleus contains two neutrons. Second, the
number density of protons after the formation of helium
is n H = n p − n n , since He 4 contains an equal number of
protons and neutrons. From this, the mass fraction Y of
He 4 of the baryon density follows,
Y =
4n He
=
2n n
= 2(n n/n p )
4n He + n H n p + n n 1 + (n n /n p ) ≈ 0.25 ,
(4.61)
where in the last step we used the above ratio of
n n /n p ≈ 1/7 atT D . This consideration thus leads to
the following:
About 1/4 of the baryonic mass in the Universe
should be in the form of He 4 . This is a robust prediction
of Big Bang models, and it is in excellent
agreement with observations.
Fig. 4.13. The evolution of abundances of the light elements
formed in BBN, as a function of temperature (lower axis) and
cosmic time t (upper axis). The decrease in neutron abundance
in the first ∼ 3 min is due to neutron decay. The density of
deuterium increases steeply – linked to the steep decrease
in neutron density – and reaches a maximum at t ∼ 3 min
because then its density becomes sufficiently large for efficient
formation of He 4 to set in. Only a few deuterium nuclei do
not find a reaction partner and remain, with a mass fraction of
∼ 10 −5 . Only a few other light nuclei are formed in the Big
Bang, mainly He 3 and Li 7
The helium content in the Universe may change later
by nuclear fusion in stars, which also forms heavier
nuclei (“metals”). Observations of fairly unprocessed
material (i.e., that which has a low metal content) reveal
thatinfactY ≈ 0.25. Figure 4.13 shows the result of
a quantitative model of BBN where the mass fraction
of several species is plotted as a function of time or
temperature, respectively.
Dependence of the Primordial Abundances on the
Baryon Density. At the end of the first 3 min, the composition
of the baryonic component of the Universe is
about as follows: 25% of the baryonic mass is bound in
helium nuclei, 75% in hydrogen nuclei (i.e., protons),
with traces of D, He 3 and Li 7 . Heavier nuclei cannot
form because no stable nucleus of mass number 5 or
8 exists and thus no new, stable nuclei can be formed
in collisions of two helium nuclei or of a proton with
a helium nucleus. Collisions between three nuclei are
far too rare to contribute to nucleosynthesis. The den-