**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 *t*_{exp}' (for *g*^{*'}
larger than *g*^{*}) is shifted below the standard
one.

Consider next the timescale of weak interactions, eq.(9). The two
curves (*t*_{exp} and *t*_{reac}) meet around
a temperature of 1 MeV (called the
decoupling temperature *T*_{d}). 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 *T*_{d} 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 *T*^{4}
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
*T*^{4}(*i*) / *T*^{4}(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.