**1.1. The hot Big Bang**

Observations of the Hubble expansion and the very isotropic microwave
background suggest that the universe has evolved from an earlier state
of high temperature and density (*hot Big Bang*) that can be reasonably
well described by Friedman-Lemaitre-Robertson-Walker cosmological
models. Further support for this view comes from observations of the
light elements D, ^{3}He, ^{4}He and ^{7}Li,
which are expected to have been
synthesised in significant quantities in nuclear reactions that set in
about 100 seconds after the Big Bang, an effect that was first
suggested by Gamow and his collaborators in the late 1940's although
their aim of explaining all elements in this way could not be
realised. The modern theory has been developed by
Peebles (1966),
Wagoner, Fowler & Hoyle
(1967)
and Yang et al. (1984),
among others, and is described by
Schramm & Wagoner (1979),
Tayler (1982),
Boesgaard & Steigman
(1985)
and Kolb & Turner
(1990).

Since the mass density of radiation and relativistic particles
varies with scale factor *R* or red-shift *z* as (1 +
*z*)^{4} and that of
non-relativistic matter only as (1 + *z*)^{3}, the
gravitating matter of
the universe was dominated by the former during the first 10^{5}
years or
so and the universe was then in a phase with significant pressure but
negligible effects from curvature or the cosmological constant (if
any), and the total mass density at any one time (essentially all
radiation and relativistic particles) is then fixed:

(1) |

The density, in turn, fixes the radiation temperature
*T*_{}
through the
equation of state which depends on the number of relativistic degrees
of freedom thermal equilibrium with photons. About ls ABB, with a
temperature of th. order of 1 Mev, we have, in comparable numbers,
photons, electrons, positron and
*N*_{} kinds of pairs
of neutrinos and
antineutrinos, all of which are relativistic, and a small sprinkling
of non-relativistic protons and neutrons, leading t the equation of
state

(2) | |

(3) |

where *g*, *g*_{i} represent statistical weight
factors and a is the usual
Stefan Boltzmann radiation density constant. The *one* in equation (3)
comes from photons, the 7/4 from electrons and positrons and the third
term from th neutrinos (and any other, hypothetical particles that
might be relativistic at temperature of a few Mev and would then act
like an additional contribution to
*N*_{}). With
*N*_{} = 3 and
*T*_{} =
*T*_{}, (1)
and (3) lead to the temperature law (with t in seconds)

(4) |

After several seconds, neutrinos have decoupled and electrons and
positrons annihilate, adding entropy to the photon gas, whereafter
*T*_{}^{3} is a factor 11/4 greater than
*T*_{}^{3}. After annihilation, nucleons and
photons are both conserved in a co-moving volume so that their ratio
=
*n*_{b} /
*n*_{}
has remained constant (a few times 10^{-10}) through the epoch of
nucleosynthesis and to this day. Since the radiation temperature and
total density are fixed functions of time, the outcome of primordial
nuclear reactions (depending on particle densities and velocities as
functions of time) can be expressed as a function of the one cosmic
parameter , apart
from physical constants which (in principle at
least) can be directly measured in the laboratory.