The determination of distances is one of the most important problems in extragalactic astronomy and cosmology. Distances between two objects X and Y depend on their redshifts and , the Hubble constant , the cosmological constant , the density parameter and the inhomogeneity parameter . 1 Usually, smaller distances are determined by the traditional 'distance ladder' technique and larger distances are calculated from the redshift, assuming some cosmological model. Since the redshift is for most purposes exactly measurable, knowledge of or assumptions about two of the factors (a) Hubble constant, (b) other cosmological parameters and (c) 'astronomical distance' (i.e. ultimately tied in to the local distance scale) determines the third. In this paper we discuss distances given the Hubble constant , the redshifts and and the cosmological parameters , and . Traditionally, a simple cosmological model is often assumed for ease of calculation, although the distances thus obtained, and results which depend on them, might be false if the assumed cosmological model does not appropriately describe our universe. A general method allows one to look at cosmological models whether or not they are easy-to-calculate special cases and offers the possibility of determining cosmological distances which are important for other astrophysical topics once the correct cosmological model is known.
We stress the fact that the inhomogeneity can be as important as the other cosmological parameters, both in the field of more traditional cosmology and in the case of gravitational lensing, where, e.g. in the case of the time delay between the different images of a multiply imaged source, the inhomogeneity cannot be neglected in a thorough analysis (Kayser & Refsdal 1983). For an example involving a more traditional cosmological test, Perlmutter et al. (1995) (see also Goobar & Perlmutter (1995)) discuss using supernovae with -0.5 to determine ; for z near the top of this range or larger, the uncertainty due to our ignorance of is comparable with the other uncertainties of the method.
The plan of this paper is as follows. In Sect. 2the basics of Friedmann-Lemaître cosmology are briefly discussed; this also serves to define our terms, which is important since various conflicting notational schemes are in use. (For a more thorough discussion using a similar notation see, e.g., Feige (1992).) Sect. 3defines the various distances used in cosmology. In Sect. 4our new differential equation is derived. Similar efforts in the literature are briefly discussed. Sect. 5briefly describes our numerical implementation and gives the details on how to obtain the source code for use as a 'black box' (which however can be opened) for use in cosmology and extragalactic astronomy. The symmetry properties of the angular size distance, analytic solutions and methods of calculating the volume element are addressed in three appendices.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998