When the Universe is a few seconds old, at T=1010 K, protons and neutrons are in thermal equilibrium, and their ratio is given by the Boltzmann equation:
But free neutrons undergo beta decay, which converts neutrons into protons with a half life of 617 seconds. When the universe was about four minutes old, the time the temperature had dropped to 109 K, and deuterium can form. At this point, neutron decay has rebalanced the neutron to proton ratio to n/p=0.164. So our proton/neutron mix above has changed to 1051 protons and 172 neutrons.
Now we are are ready for nucleosynthesis. At 109 K, deuterium will survive. So nuclear reactions will form deuterium, tritium, and helium:
These reactions happen so efficiently that we make as much helium as possible. If we have 172 neutrons, we can make a total of 86 4He nuclei, with 879 protons (H nuclei) left over. So the mass fraction of helium in the Universe was
Which is not too far off the observed value of the primordial
helium content of the Universe: ~ 23-24%.
A full nucleosynthesis model looks like this:
| Some of the less massive
nuclei are also produced: deuterium (2H), 3He,
7Li. The
abundances of these elements can be used to constrain the baryonic
density
of the universe at the time of BBN: higher densities detroy
deuterium, for example. From this, we can infer the baryonic density of
the universe now, since density drops as R-3.
The plot at the left shows the abundances of
these light
elements as a function of the present baryon density of the universe.
Based
on the observed abundances of these elements (red for deuterium, green
for Helium 3, black for Lithium 7), we infer that the
baryonic density of the universe is only a few percent of the critical
density:
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