## 3. Distance measures## 3.1. Distances defined by measurementIn a static Euclidean space, one can define a variety of distances according to the method of measurement, which are all equivalent. ## 3.1.1. Angular size distanceLet us consider at position since an object of projected length ## 3.1.2. Proper motion distanceThe proper motion distance is similar to the angular size distance,
except that ## 3.1.3. Parallax distanceParallax distance is similar to the proper motion distance, except
that the angle is at In the canonical case, . ## 3.1.4. Luminosity distanceSince the apparent luminosity where is the luminosity at some fiducial distance . ## 3.1.5. Proper distanceBy proper distance we mean the distance measured with a rigid ruler. ## 3.1.6. Distance by light travel timeFinally, from the time required for light to traverse a certain distance, one can define a distance by where ## 3.2. Cosmological distances## 3.2.1. General considerationsIn a static Euclidean space, which was used above when defining the distances through a measurement description, these distance measures are of course equivalent. In the general case in cosmology, where the 3-dimensional space need not be flat () but can be either positively () or negatively () curved, and where the 3-dimensional space is scaled by , not only do the distances defined above differ, but also (in the general case) . The definitions are still applicable, but different definitions will result in different distances. In reality, of course, the universe is neither perfectly
homogeneous nor perfectly isotropic, as one assumes when deriving
Eq. (1). However, as far as the usefulness of the Friedmann
equations in determining the global dynamics is concerned, this
appears to be a good approximation. (See, for example, Longair (1993)
and references therein for an interesting discussion.) The
approximation is certainly too crude when using the cosmological model
to determine distances as a function of redshift, since the angles
involved in such cases can have a scale comparable to that of the
inhomogeneities. In this paper, we assume that these inhomogeneities
can be sufficiently accurately described by the parameter
, which gives the fraction of homogeneously
distributed matter. The rest () of the matter is
distributed clumpily, where the scale of the clumpiness is For example, a halo of compact MACHO type objects around a galaxy
in a distant cluster would be counted among the homogeneously
distributed matter if one were concerned with the angular size
distance to background galaxies further away, but would be considered
clumped on scales such as those important when considering
microlensing by the compact objects themselves. Since we don't know
exactly how dark matter is distributed, different
values can be examined to get an idea as to how
this uncertainty affects whatever it is one is interested in. If one
has no selection effects, then, due to flux conservation, the
'average' distance cannot change (Weinberg 1976);
introduces an additional uncertainty when
interpreting observations. It is generally not possible to estimate
this scatter by comparing the cases and
, since, depending on the cosmological
parameters and the cosmological mass distribution, not all
combinations are self-consistent. For instance, if one looks at scales
where galaxies are compact objects, and the fraction of
due to the galaxies is We further assume that light rays from the object whose distance is to be determined propagate sufficiently far from all clumps. (See Schneider et al. (1992) - hereafter SEF - for a more thorough discussion of this point.) Compared to the perfectly homogeneous and isotropic case, the introduction of the parameter will influence the angular size and luminosity distances (as well as the proper motion and parallax distances) since these depend on angles between light rays which are influenced by the amount of matter in the beam, but not the proper distance and only negligibly the light travel time. The last two distances are discussed briefly in Sect. 3.2.2 and in Appendix B3 and B6. Since there is a simple relation between the angular size distance and the luminosity distance (Sect. 3.2.2) which also holds for the inhomogeneous case (see Appendix A), for the general case it suffices to discuss the angular size distance, which we do in Sect. 4. ## 3.2.2. Relationships between different distancesWithout derivation
where the term in parentheses takes account of, by way of
Eq. (10), the expansion of the universe. It is convenient, in
keeping with the meaning of angular size distance, to think of the
expansion of the universe changing the angle in
Eq. (13) and not Although the angle between the rays (at the source) at the time of
reception of the light is important for the luminosity distance, this
distance is From this and Eq. (17) follows the relation This means that the surface brightness of a 'standard candle'
is , a result independent of the
cosmological model parameters, including .
Of course, this applies only to the where where and The light travel time (or lookback time) between and (where ) is given by the integration of the reciprocal of Eq. (11): where the minus sign from Eq. (11) is equivalent to the
swapped limits of integration on the right-hand side so that the
integral gives instead of
, making the light travel time increase (for
) with Since the proper distance would be the same as
This gives the proper distance at the present time. Since
scales linearly with the expansion of the
universe, the proper distance at some other time can be obtained by
dividing Eq. (23) with , where
is the redshift at the corresponding time. For
homogeneous () cosmological models,
For inhomogeneous models, where this relation between global geometry and local light propagation does not exist, another approach must be used, which takes account of both the expansion of the universe as well as the local propagation of light, when calculating angle-defined distances such as the angular size distance. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |