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

3. Distance measures

3.1. Distances defined by measurement

In 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 distance

Let us consider at position y two light rays intersecting at x with angle . If l is the distance between these light rays, it is meaningful to define the angular size distance  as

since an object of projected length l at position y will subtend an angle  (for small ) at distance  .

3.1.2. Proper motion distance

The proper motion distance is similar to the angular size distance, except that l is given by vt, where v is the tangential velocity of an object and t the time during which the proper motion is measured.

3.1.3. Parallax distance

Parallax distance is similar to the proper motion distance, except that the angle is at y instead of x, so that we have

In the canonical case, .

3.1.4. Luminosity distance

Since the apparent luminosity L of an object at distance D is proportional to , one can define the luminosity distance as

where is the luminosity at some fiducial distance .

3.1.5. Proper distance

By proper distance  we mean the distance measured with a rigid ruler.

3.1.6. Distance by light travel time

Finally, from the time required for light to traverse a certain distance, one can define a distance by

where t is the so-called look-back time.

3.2. Cosmological distances

3.2.1. General considerations

In 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 by definition of the same order of magnitude as the angles involved.

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 x, then must be .

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 distances

Without derivation ³ we now discuss some important distance measures, denoting the redshifts of the objects with the indices x and y. Due to symmetry considerations (see Appendix A)

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 l, if one identifies l as the (projected) size of an object. The angle is defined at the time when the light rays intersect the plane of the observer. Thus with the observer at defines what one normally thinks of as an angular size distance. On the other hand, and with x in general can be important in, for example, gravitational lensing. ⁴

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 not simply , since in the cosmological case the observed flux is obtained by multiplying the 'non-redshifted flux' by the factor . One factor of occurs because a given wavelength is increased by , which reduces the flux correspondingly; an additional factor of occurs because the arrival rate of photons is also decreased. Therefore, since is inversely proportional to the square root of the (observed, 'redshifted') flux the luminosity 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 . ⁵ (This result also holds for the inhomogeneous case, since Eq. (17) still holds (see Appendix A) and the additional factor due to the expansion of the universe (given by the term in parentheses in Eq. (18)) is of course present in the inhomogeneous case as well.)

Of course, this applies only to the bolometric luminosity. Observing in a finite band introduces two corrections. The so-called K -correction as it is usually defined today (see, e.g., Coleman et al. (1980) or, for an interesting and thorough discussion, Sandage (1995)) takes account of these, both of which come from the fact that the observed wavelength interval is redshifted compared to the corresponding interval on emission. This means that, first, for a flat spectrum, less radiation is observed, because the bandwidth at the observer is times larger than at the source. Second, the spectrum need not be flat, in which case additional corrections based on the shape of the spectrum have to be included. ⁶ Thus,

where m is the apparent magnitude, M the absolute magnitude, is the luminosity distance and K is the K -correction as defined in Coleman et al. (1980). Perhaps more convenient is

where N is a normalisation term: for in units of 1 pc, for in units of 1 Mpc and for in units of the Hubble length ⁷ , where

and h is the Hubble constant in units of . In practice one has to add terms to correct for various sources of extinction and consider the fact that M is the absolute magnitude of the object when the light was emitted, which of course could be different from the present M of similar objects at negligible redshift.

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 z ; thus .

Since the proper distance would be the same as were there no expansion, the former can be calculated by multiplying the integrand in Eq. (22) by . Thus

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, ⁸ the propagation of light rays is determined by the global geometry, so that there is a simple relation between and D and, thus, . This is discussed in Sect. B3. Although not 'directly' observable, the proper distance is nevertheless important in cosmological theory, since it is the basic distance of general relativity. Although not useful as a distance, the light travel time is of course important when considering evolutionary effects.

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
helpdesk.link@springer.de