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 positivesolution, except that always.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
helpdesk.link@springer.de