**2.2. Dynamics of the Friedmann model**

Let us now turn to the second assumption which determines the dynamics
of the universe. When several non interacting sources are present in
the universe, the total energy momentum tensor which appear on the right
hand side of the Einstein's equation will be the sum of the energy
momentum tensor for each of these sources. Spatial homogeneity and
isotropy imply that each *T*_{b}^{a} is diagonal
and has the form *T*_{b}^{a} = dia
[_{i}(*t*), - *P*_{i}(*t*), -
*P*_{i}(*t*), - *P*_{i}(*t*)] where
the index *i* = 1, 2,..., *N* denotes *N* different kinds
of sources (like
radiation, matter, cosmological constant etc.). Since the sources
do not interact with each other, each energy momentum tensor
must satisfy the relation
*T*_{b;a}^{a} = 0 which translates to
the condition *d*
(_{i}
*a*^{3}) = - *P*_{i} *da*^{3}. It
follows that the evolution of the energy densities of
each component is essentially dependent on the parameter
*w*_{i}
(*P*_{i} /
_{i})
which, in general, could be a function of time. Integrating *d*
(_{i}
*a*^{3}) = - *w*_{i}
_{i}
*da*^{3}, we get

(19) |

which determines the evolution of the energy density of each of the
species in terms of the functions *w*_{i}(*a*).

This description determines
(*a*) for
different sources but not *a*(*t*). To determine the
latter we can use one of the Einstein's equations:

(20) |

This equation shows that, once the evolution of the individual
components of energy density
_{i}(*a*) is known, the function
*H*(*a*) and thus the line element in equation (13)
is known. (Evaluating this equation at the present epoch one can
determine the value of *k*; hence it is not necessary to provide
this information separately.) Given *H*_{0}, the current
value of the Hubble parameter, one can construct a *critical
density*, by the definition:

(21) |

and parameterize the energy density,
_{i}(*a*_{0}), of different
components at the present epoch in terms of the critical density by
_{i}(*a*_{0})
_{i}
_{c}.
[Observations
[49,
50]
give *h* = 0.72 ± 0.03 (statistical) ± 0.07 (systematic)].
It is obvious from equation (20) that *k* = 0 corresponds to
_{tot} =
_{i}
_{i} = 1
while
_{tot} >
1 and
_{tot} < 1
correspond to *k* = ± 1. When
_{tot}
1, equation (20),
evaluated at the current epoch, gives
(*k* / *a*_{0}^{2}) =
*H*_{0}^{2}(_{tot} - 1), thereby fixing the value of
(*k* / *a*_{0}^{2}); when,
_{tot} = 1,
it is conventional to take *a*_{0} = 1 since its value can
be rescaled.