Annu. Rev. Astron. Astrophys. 1992. 30:
499-542
Copyright © 1992 by . All rights reserved |

**3.3 Distance Measures**

As we look out from our self-defined position at *r* = 0 to observe some
object at a radial coordinate value *r*_{1}, we are also
looking back in
time to some time *t*_{1} < *t*_{0}, and back
to some expansion factor *R*_{1} =
R(*t*_{1}) that is smaller than the current value
*R*_{0}. Note, however, that
neither *r*_{1}, *t*_{1}, nor
*R*_{1} are directly measurable quantities. Rather, the
measurable quantities are things like the redshift *z*; the angular
diameter distance

where *D* is a known (or assumed) proper size of an object and
is its
apparent angular size; the proper motion distance

where *u* is a known (or assumed) transverse proper velocity and
is an
apparent angular motion; and the luminosity distance

where is a known (or assumed) rest-frame luminosity and is an apparent flux. The relation of the measurables to the unmeasurables turns out to be (Lightman et al 1975, Section 19.9)

One sees in particular that *d*_{A}, *d*_{M},
and *d*_{L} are not independent, but related by

independent of the dynamics of *R(t)*. This is perhaps a disappointment,
since it means that we cannot learn anything about
_{M},
_{} simply by
comparing two distance indicators of a single object. Rather, the
information about _{M},
_{} is contained in the
dependence of the
distance indicators on redshift *z*, which we now calculate.

Looking back along a light ray, *R*, *r*, and *t* are
related by the
equation for a radial, null geodesic of the Friedmann-Robertson-Walker
metric, namely

Multiplying this equation by *R*_{0}, and using Equations
22 and 16, and
the definitions of _{k}
and *z*, one obtains the integral formula for the
distance measure at redshift *z*,sub>1,

where ``sinn'' is now defined as sinh if
_{k} > 0 (open
universe) and as
sin if _{k} <
0. Remember that _{k}
is not independent, but given by
Equation 3. (Tn the flat case of
_{k} = 0, i.e.
_{tot} = 1, the sinn and
_{k}s disappear from
Equation 25, leaving only the integral.) The
integral in Equation 25 can be done analytically in the usual special
cases _{} = 0 and
_{tot} = 1 (see
Weinberg 1972 and
Kolb & Turner 1990),
but in general is straightforward to evaluate numerically. Because of
the dependence on _{k}
= 1 - _{tot} in the
sinn function, the qualitative
behavior of Equation 25 is not completely obvious by inspection. At
fixed _{tot}, or when
_{} = 0, the distance measures
all increase with
decreasing _{M}, for
all *z*. For fixed
_{M}, however, there is no
monotonicity as _{} is increased: The distance
measure will generally
increase at small redshifts, but decrease at redshifts greater than
some particular value. Figure 5 illustrates
these effects for the
specific models A-E. For clarity we plot
*d*_{} instead
of *d*_{M} because the
extra factor of (1 + z)^{-1} (Equation 23) spreads the curves
apart.