**1.3. Cosmological Parameters**

Based on (7) we can define some very
important parameters like the *Critical Density*, which is the density
necessary to obtain a flat Universe
( = 0):

(15) |

and the *Cosmological density parameter*
, which is a
unit-less measure of the density of the Universe:

(16) |

Furthermore, the constant of proportionality in Hubble's law, the
*Hubble constant*, is:

(17) |

Note that the necessity of parametrizing with *h* was due to earlier
discordant determinations of *H*_{0}. Today most studies
converge to a value of ~ 0.7 (see
section 2.4).

A convenient representation of these interrelations can be produced by re-writing Friedmann's equation, in the matter dominated era (using 10), as following:

(18) |

were the contribution to the total density parameter from the curvature and terms is:

(19) |

Note that *H*(*z*) is called *Hubble function*. It is
evident that at the present epoch we obtain from (18) that *E*(0) =
1 and thus:

(20) |

which also holds for any epoch (evaluated directly from 7). Note that we
can have a flat Universe
(_{k} = 0)
while having
_{m} <
1 (as suggested by many different observations).

**The Age of the Universe:** Using (18), evaluated at the present
epoch, we have /
*R*_{0} = *H*_{0} *E*(*z*) / (1 +
*z*) and from d*R/R*_{0} = -d*z* / (1 +
*z*)^{2} we obtain the age of the Universe:

(21) |

For example, in an *Einstein-de Sitter* universe
(_{} =
_{k} = 0) we
have:

(22) |

while for a _{} > 0 model we obtain:

(23) |

We therefore see that if
_{} > 0 we
have that the age of the Universe is
larger than what is predicted in an Einstein-de Sitter Universe.

**The
0 Universe:**
Due to the recent exciting observational
indications for a positive cosmological constant and the important
consequences that this has to our understanding of the Cosmos, we will
briefly present this model.

Originally, the cosmological
-parameter was
introduced *ad hoc* by
Einstein in his field equations in order to get a static solution
( =
= 0). From (8) he
derived *R* = (*k* /
_{c})^{1/2} and inserting this
into (7) he obtained, in a matter dominated (*p* = 0) Universe:

(24) |

where _{c}
is the critical value of
for which
=
= 0. However, it was
found that his solution was unstable and that small perturbations of the
value of _{c}
would change drastically the behaviour of *R*. From (7) we see
that if *k* 0, then
^{2} is always
nonnegative for
> 0, and thus
the universe expands for ever, while if
< 0 then
the universe
can expand and then recontract again (as in the *k* = 1,
= 0 case).

The recent SNIa observations (see
section 3.2) and the CMB
power-spectrum results (see section 3.1)
have shown that the *Standard*
Cosmological paradigm should be considered that of a flat,
_{}
0.7,
_{m}
0.3 model. Thus we will
consider such a model in the following
discussion. Evaluating (7) at the present epoch, changing successively
variables: *x* = *R*^{3/2}, *y* =
*x*(_{m}
/ _{})^{1/2}
*R*_{0}^{-3/2} and
= sinh ^{-1}
*y* and then integrating, we obtain:

(25) |

and

(26) |

It is interesting to note that in this model there is an epoch which
corresponds to a value of *R* = *R*_{I}, where the
expansion slows down and
remains in a quasi-stationary phase for some time, expanding with
> 0
thereafter (see Fig.1). At the quasi-stationary
epoch, called the inflection point, we have
= 0 and thus from
(7) by differentiation we have:

(27) |

Now from (25) and (27) we have that the age of the universe at the inflection point is:

(28) |

The Hubble function at *t*_{I} is:

so
if *t*_{0} > *t*_{I} we must have
*H*_{0} < *H*_{I}.

This is an important result because it indicates that introducing
an _{}-term, and if we
live at a time that fulfils the condition
*t*_{0} > *t*_{I}, we can increase the
age of the universe to comfortably fit
the globular cluster ages while keeping the value of
_{m} < 1 and
also a flat
(_{k} = 0)
space geometry. From (28) and (23) and for
the preferred values
_{} = 0.7 and
_{m} = 0.3
we indeed obtain *t*_{0} / *t*_{I}
1.84 (see also
Fig.1),
which implies that we live in the accelerated
phase of the Universe. Note that in order for the present time
(*t*_{0})
to be in the accelerated phase of the expansion we must have:
_{}
> 1/3.

**Importance of k and
terms in global
dynamics:** Due to the recent interest in the
> 0,

and from (20) we have

(29) |

By equating the above two terms we can find the redshift at which they have equal contributions to the dynamics of the Universe. Evidently this happens only in the very recent past:

(30) |

Observations suggest that
_{m}
0.3 and
_{}
0.7, and therefore we have
*z*_{c}
0.3, which implies that the present dynamics of the universe are
dominated by the
-term, although
for the largest part of the history of
the Universe the determining factor in shaping its dynamical evolution is
the matter content.

Similar results are found for the *k*-term in
= 0 models. In
this case we have from (29) that

and thus the redshift at which the density and curvature terms have equal impact in the global dynamics, is:

We see that as *z* increases the density term grows faster than the
curvature
term which is important only very recently. A similar line of argument
shows that also in the radiation dominated era the
and *k*
terms do not affect the dynamics of the Universe.

**Density parameter as a function of z:** From
(18) and eliminating the curvature term, using
_{k} = 1 -
_{m} -
_{}, we obtain the
time evolution of the density parameter
_{m}:

(31) |

and using (15) and (10) we have:

(32) |

It is easy to see that whatever the value of
_{m}, at
large *z* we always have
(*z*) = 1.