3.4. Evidence from primordial nucleosynthesis
Most of the story is illustrated in fig. 1. The curve t(exp) is plotted with the g* value corresponding to the three standard families of physics (g* = 9.75). Adding new families would increase the value of g*, hence increase the total density through eq. (6), and consequently increase the rate of expansion through eq. (5). The resulting curve texp' (for g*' larger than g*) is shifted below the standard one.
Consider next the timescale of weak interactions, eq.(9). The two curves (texp and treac) meet around a temperature of 1 MeV (called the decoupling temperature Td). Below this temperature, the reaction rate is too slow to follow the expansion rate and the equilibrium is lost. After this time, the neutron essentially freely decays (lifetime of about one thousand sec, more about this later). Around T = 0.1 MeV, (one hundred seconds later) the deuteron manages to resist photodisintegration. Essentially all the surviving neutrons are then captured by protons and transformed gradually in (mostly) helium-4(BBN).
In a nutshell, the junction of the timescales curves fixes the n/p ratio at decoupling (upper part of fig. 1). Very few neutrons decay before BBN, and the rest results in helium. Thus, assuming the existence of new families results in an increase of g*), which increases Td and hence the abundance of He.
The estimates of the helium cosmic abundance after BB (obtained from observations of galaxies with very low metal abundance) will be given later. The data is best reproduced in the calculations if we assume three families with neutrinos of masses less than 0.5 MeV (to insure that they are relativistic and weigh a full T4 term in the density balance of eq. (6)). With the uncertainties on the data, it is possible to include one more family, perhaps even two, but certainly not more. It is on the basis of these arguments that BBN did make its successful prediction on the limitation of the number of families of elementary particles (3).
As discussed before, assume, for instance, that this decoupling occurs before the muon anti-muon annihilation around one hundred MeV. The released energy would be shared amongst the electrons and left-handed neutrinos (all these particles seeing their temperature go up by a factor of 1.3) but not with the i particle. In consequence, the i particles should be included in the expression of g* weighted with a factor of T4(i) / T4(photons) = 1/2.95. A similar computation could be made for particles decoupling before the nucleon masses; their contribution would be correspondingly smaller. Big-bang nucleosynthesis specifies that the value of g* is somewhere between 9 and 13. Although this range limits the number of particles interacting with the standard Fermi interaction, it clearly does not preclude the existence of a large number of other species, provided their interaction strength is weak enough not to contribute to the g* in an exaggerate way.