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Astron. Astrophys. 318, 680-686 (1997)

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2. Basic theory

Considering for the moment homogeneous Friedmann-Lemaître cosmological models, we can write the familiar Robertson-Walker line element:

[EQUATION]

where the symbols are defined as follows (with the corresponding units):

[TABLE]

The dynamics of the universe is given by the Friedmann equations

[EQUATION]

and

[EQUATION]

where dots denote derivatives with respect to t, G is the gravitational constant,  [FORMULA] the matter density (this paper assumes negligible pressure), [FORMULA] the cosmological constant and the sign of k determines the curvature of the 3-dimensional space.

Introducing the usual parameters

[EQUATION]

([FORMULA] and [FORMULA] are dimensionless and H has the dimension [FORMULA]) we can use Eq. (2) to calculate

[EQUATION]

so that

[EQUATION]

Since [FORMULA] we can write

[EQUATION]

this is the radius of curvature of the 3-dimensional space at time t. For [FORMULA] it is convenient to define the scale factor R to be [FORMULA]. In the following the index 0 will be used to denote the present value of a given quantity, fixed, as usual, at the time  [FORMULA] 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

[EQUATION]

and so from Eqs. (2), (4), (5) and (8) follows

[EQUATION]

Since below we want to discuss distances as functions of the cosmological redshift z, by making use of the facts that

[EQUATION]

and that [FORMULA] is fixed, we can use Eq. (9) to get

[EQUATION]

where

[EQUATION]

Note: Throughout this paper, the  [FORMULA]  sign should be taken to signify the positivesolution, except that [FORMULA] always.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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