*Astron. Astrophys. 318, 680-686 (1997)*
## 2. Basic theory
Considering for the moment *homogeneous*
Friedmann-Lemaître cosmological models, we can write the
familiar Robertson-Walker line element:
where the symbols are defined as follows (with the corresponding
units):
The dynamics of the universe is given by the Friedmann equations
and
where dots denote derivatives with respect to *t*, *G* is
the gravitational constant, the matter
density (this paper assumes negligible pressure),
the cosmological constant and the sign of
*k* determines the curvature of the 3-dimensional space.
Introducing the usual parameters
( and are
dimensionless and *H* has the dimension )
we can use Eq. (2) to calculate
so that
Since we can write
this is the radius of curvature of the 3-dimensional space at
time *t*. For it is convenient to
*define* the scale factor *R* to be .
In the following the index 0 will be used to denote the present
value of a given quantity, fixed, as usual, at the time
of observation.
^{2} The explicit
dependence on *t* will be dropped for brevity. Taking matter
conservation into account and using the present-day values, we have
and so from Eqs. (2), (4), (5) and (8) follows
Since below we want to discuss distances as functions of the
cosmological redshift *z*, by making use of the facts that
and that is fixed, we can use Eq. (9)
to get
where
**Note:** *Throughout this paper, the
sign should be taken to signify the *
positive*solution, except that always.*
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998
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