Nuclei can form in abundant amounts as soon as the temperature of the Universe has cooled down to temperatures corresponding to their binding energy, and the number densities of protons and neutrons are sufficiently high. For a (non-relativistic) species with mass number A and charge number Z [such a species will be called A(Z), and contains Z protons and A − Z neutrons], the equilibrium number density can be obtained from Eq. (A):
The chemical potential 
is related to those of protons and neutrons as
which allows us to rewrite Eq. (3.194) as
Writing 
) and 
in terms of the proton and neutron mass densities given by Eq. (3.189), respectively, and defining the nucleon mass 
we obtain
where
is the binding energy of the species A(Z). Next we define the ‘mass fraction’ or ‘abundance’ of nucleus A as
Here 
is the number density of baryons in the Universe, with the summation over all nuclear species so that 
. Substituting Eq. (3.197) in Eq. (3.199) we obtain
where 
is the present-day baryon-to-photon ratio. Since 
[see Eq. (3.130)], and 
we have
where 
is the present-day baryon density in terms of the critical density for closure. Eq. (3.200) reveals that species A(Z), with A > 1, can only be produced in appreciable amounts once the temperature has dropped to a value 
given by
The binding energies of the lightest nuclei, such as deuterium and helium, are all of the order of a few MeV, corresponding to a temperatures of a few × 10^10 K. However, because of the small number of baryons per photon (10^−10 ≲ η ≲ 10^−9), or, in other words, the high entropy per baryon, their synthesis has to wait until the Universe has cooled down to temperatures of the order of (1 → 3) × 10^9 K.
At such low temperatures, however, the number densities of protons and neutrons are already much too low to form heavy elements by direct many-body reactions, such as 2n + 2p → ^4He. Therefore, nucleosynthesis must proceed through a chain of two-body reactions. The dominant reactions in this chain are:
(10)
Here the notation X(a, b)Y indicates a reaction of the form X + a → Y + b. Since the cross-sections for almost all these reactions are accurately known, the reaction network can be integrated numerically to compute the final abundances of all elements.
Note that the reaction network does not produce any elements heavier than lithium. This is a consequence of the fact that there are no stable nuclei with atomic weight 5 or 8. Since direct many-body reactions at earlier epoch are very inefficient in producing heavy elements, we can conclude that elements heavier than lithium are not produced by primordial nucleosynthesis. Indeed, as we will see in Chapter 10, heavy elements can be synthesized in stars where the density of helium is so high that a short-lived ⁸Be, formed through ⁴He–⁴He collisions, can quickly capture another ⁴He to form a stable carbon nucleus (¹²C), thus allowing further nuclear reactions to proceed.
Inspection of the reaction network of primordial nucleosynthesis given above reveals that it can only proceed if the first step, the production of deuterium, is sufficiently efficient. Since deuterium has the lowest binding energy of all nuclei in the network, its production serves as a “bottleneck” to get nucleosynthesis started. The production of deuterium 
has a rate per free neutron given by
which is much larger than the expansion rate 
. Therefore, for temperatures T ≳ 5 × 108 K, deuterium nuclei are always produced with the equilibrium abundance:
From this we see that large amounts of deuterium are only produced once the temperature drops to 
. This occurs when the Universe is about 100 seconds old, and signals the onset of primordial nucleosynthesis. The subsequent reaction chain proceeds very quickly, because at T ≃ TD all nuclei heavier than deuterium can possess high equilibrium abundances. However, nuclei heavier than helium are still rare because of the instability of nuclei with A = 5 and A = 8, and because the temperature is already too low to effectively overcome the large Coulomb barrier in reactions like 4He (3H,
)7Li and4He(3He
7
Be. As a result, almost all free neutrons existing at the onset of nucleosynthesis will be bound into 4He, the most tightly bound species with A < 5. The mass fraction of ^4He can therefore be approximately written as
where 
is the neutron-to-proton ratio at 
