### 9. COMOVING VOLUME

The *comoving volume V*_{C} 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

(27)

where *D*_{A} 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*

(28)

(Carrol, Press & Turner
1992),
where *D*_{H}^{3} is sometimes called the
*Hubble volume.* The comoving volume element and its integral
are both used frequently in predicting number counts or luminosity
densities.