Astron. Astrophys. 338, 777-780 (1998)

2. Distance and volume measures in general FLRW cosmologies

We assume a homogeneous and isotropic universe described by a Robertson-Walker metric (Weinberg 1972):

or in the form

where

The relation of the measurables to the unmeasurables is (Lightman et al. 1975; Carroll, Press & Turner 1992)

where is the present time of the universe, and , and are the angular diameter distance, the proper motion distance and the luminosity distance respectively. Distances used in lensing theory are the angular diameter distances (Schneider, Ehlers & Falco 1992). From Eqs. 3and 4, one can derive an important relation

where and are the angular distances from the lens to the source and from the observer to the source, respectively.

Using the Einstein field equation, it can be shown that the relation of the comoving distance to the redshift for the general FLRW cosmologies is (see Eq. 6)

For our end, the comoving volume dV is more convenient than the traditional physical volume. Within the shell at , dV reads (Gott, Park & Lee 1989)

© European Southern Observatory (ESO) 1998

Online publication: September 17, 1998
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