The comoving volume VC is the volume measure in which
number densities of non-evolving objects locked into Hubble flow are
constant with redshift. It is the proper volume times three factors
of the relative scale factor now to then, or (1 + z)3.
Since the
derivative of comoving distance with redshift is 1 / E (z) (defined
above), the angular diameter distance converts a solid angle
d
into a proper area, and two factors of (1 + z) convert a proper area
into a comoving area, the comoving volume element in solid angle
d and redshift interval
dz is
where DA is the angular diameter distance at redshift z and E (z) is defined above (Weinberg 1972, p. 486; Peebles 1993, pp. 331-333). The comoving volume element is plotted in Figure 5. The integral of the comoving volume element from the present to redshift z gives the total comoving volume, all-sky, out to redshift z
|
Figure 5. The dimensionless comoving
volume element (1 / DH)3
(dVC / dz). The three curves are for
the three world models, ( |
(Carrol, Press & Turner 1992), where DH3 is sometimes called the Hubble volume. The comoving volume element and its integral are both used frequently in predicting number counts or luminosity densities.
) =
(1, 0), solid; (0.05, 0), dotted; and (0.2, 0.8), dashed.