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 r1, we are also looking back in time to some time t1 < t0, and back to some expansion factor R1 = R(t1) that is smaller than the current value R0. Note, however, that neither r1, t1, nor R1 are directly measurable quantities. Rather, the measurable quantities are things like the redshift z; the angular diameter distance

19.

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

20.

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

21.

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)

22.

One sees in particular that dA, dM, and dL are not independent, but related by

23.

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

24.

Multiplying this equation by R0, 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,

25.

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 ks 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 dM because the extra factor of (1 + z)-1 (Equation 23) spreads the curves apart.

 Figure 5. Angular diameter distance as a function of redshift for models A-E. Objects of a small fixed z look farther away (have smaller angular diameters) in -dominated models. At larger redshifts the situation reverses. Correspondingly, a fixed angular beam subtends, at high redshift, a smaller scale for -dominated models than for open models, giving smaller cosmic microwave background fluctuations for the -dominated case.