**1.3. Cosmic Timescales**

Of major importance for our subject is the temperature behaviour of the various timescales involved in the physics of the expansion. We distinguish the macroscopic timescale related to the expansion and the various microscopic timescales related to the particle reaction rates.

**1.3.1. The expansion timescale**

As discussed before, in the early universe the two dominant terms of
equation (2) yield the simple relation
(*R*^{0}/*R*)^{2} =
8*G*
/ 3 or

(5) |

The energy density is then dominated by all the relativistic
particles *i* (*kT* >> *Mc*^{2}) for which
_{i}
*g*_{i}*T*_{i}^{4} where
*g*_{i} is the statistical
multiplicity factor (*g*_{i} = 2 for the photons). A
*demographical factor*
*g*^{*} is usually introduced to represent the
effect of all the particle species on the mass-density of the early
universe;

(6) |

where
*T*_{} is the temperature of the photon gas. For reasons to
be discussed shortly, the various relativistic gases may not be at the
same temperature. fact will influence their density contribution and
will appear in the expression for *g*^{*} in the
following way

(7) |

The factor 7/8 reflects the difference of the statistics for bosons,
(*b*), and fermions, (*f*). At one MeV for instance, the
*standard*
demography consists of electrons and positrons, three types of
neutrinos and their antineutrinos (in the standard model of elementary
particles) and the photons. This gives
*g*^{*} = 9.75.

Thus we get the cosmic expansion timescale as (see fig. 1)

(8) |

**1.3.2. The reaction timescales**

For the reaction timescale, let us consider a reaction of the type
*A* + *B*
*C* + *D*. For instance the capture of a neutrino by neutron
to give an electron and a proton.

The capture cross-sections
are function of the
energy with a given
power (usually positive). The probability of one capture event is
given by the value of the product of the cross-section times the
relative velocity:
<*v*>,
averaged over the velocity distribution pf particles at temperature
*T*. This average value will be proportional to the strength of
the interaction, *G*, times a certain power *m* of the
temperature. For the weak interaction implied in the neutrino capture
reactions, *m* = 2 and the Fermi constant *G*_{F}
appears squared <*v*> (weak)
*G*_{F}^{2}*T*^{2}.

The probability of reaction per unit volume is proportional to
<*v*> times
the number-density of capturing particles per
unit volume *n*(*T*) which, in the expanding universe, is
*n*(*T*)
*T*^{3}. Thus the lifetime
*t*(reac) for a given neutrino (in our example) to interact with
neutrons (the inverse of the reaction probability) is given by (see
fig. 1):

(9) |

For other reactions the temperature negative exponent will be (*m* + 3)
which, for all physical processes of importance here, will always be
larger than the exponent value of 2 characteristic of the expansion
timescale. The result is that the reaction timescales are always
shorter than the expansion timescale in the very early universe. The
two curves meet (fig 1) 1 at a certain
temperature, called the
*decoupling temperature* *T*_{d} which is a function
of many parameters such
as the ratio of the coupling constants (*G*), the demography of the
universe (through *g*^{*}) and other factors
influencing the
cross-sections, including its energy dependance (through the power
*m*).

One very important consequence of this comparative behaviour of the
time scale is that all the physical processes are in *statistical
equilibrium* at early moments of the expansion. The relative abundances
of the reacting particles are then given by the law of mass-action,
such as the law of Boltzman, and are independent of the past
situations. After decoupling, on the contrary, the processes occur in
a state of disequilibrium and the abundances reflect the past history.

For the weak interactions the decoupling temperature is given by equating eqs. (8) and (9):

(10) |