3.4 Comoving Density of Objects

One is sometimes able to count objects (e.g. galaxies) in observable volume elements, that is, per solid angle d and per redshift interval dz. If the objects counted can be identified with objects of a known comoving density (e.g. galaxies today), then one has in effect another distance measure in the relation between comoving density and redshift, and another opportunity to learn about M, . The comoving volume element of the Friedmann-Robertson-Walker metric is

26.

Notice that the volume element is not simply a function of dM, or of dM and z, but has an additional dependence on k. This shows that number counts fundamentally probe a different aspect of the universe's geometry than do the distance measures of Equation 22.

Equation 26 has the consequence that, given a population of objects of constant (or calibratable) density and determinable distance measures, one can in principle directly measure k (or tot) and determine whether the universe is open or closed, in an almost model-independent fashion: One ``simply'' determines (e.g. along a pencil beam) whether the volume V scales as dM3, or whether it shows evidence of the denominator in Equation 26. If dM, the proper motion distance, were directly accessible to measurement, this test could be performed without measuring any redshifts! Unfortunately, dM is the least accessible of distance measures. Using dL or dA instead, the test requires that redshifts be known, or estimated from a model of the sources (as in Sandage 1988).

More model-dependently, one can calculate from Equations 26 and 25 the dependence of dV on z, and use observed number counts to constrain the values of M, tot (Loh 1986). Figure 6 shows how the comoving volume element dV / dz d varies with z for the five models A-E. Notice that at modest redshifts (e.g. z = 1/2) the fractional variation among the models is significantly larger in Figure 6 than for the other distance measures in Figure 5. This is an attractive feature of number count tests, but (as we will see) it must be weighed against their susceptibility to evolutionary and selection effects. At redshifts 1 + z ~ 2, the models with significant have volumes-per-redshift larger than the open models by a factor ~ 2, and larger than the flat M = 1 model by a factor ~ 4. This results in -models becoming fashionable whenever excess counts of high-redshift objects are claimed to exist (see Section 4.3 below).

 Figure 6. Volume derivative as a function of redshift for models A-E. At redshifts z < 3 there is a clean, and quite large, separation between the -dominated and open models. Number counts would be a powerful test, were it not for a morass of evolutionary and selection effects.

Equation 26 can be integrated analytically to give the comoving volume out to a distance dM,

27.

where ``sinn'' is as defined after Equation 25.