Annu. Rev. Astron. Astrophys. 1988. 26:
561-630
Copyright © 1988 by Annual Reviews. All rights reserved |

**3.2. The Predicted K(z, q _{0}) Relation**

Combining Equations 9-11 with Equations 29 and 30 gives the exact
*N*(*z*, *q*_{0}) relation, calculated therefrom
directly in parametric form. A
calculation via this route shows the dependence of *N(z)* on curvature
for three *q*_{0} values in Figure 1.
The normalization of the volume is
arbitrary in this diagram; *N(z)* is proportional to *V(z)*, the
proportionality factor being the volume density of galaxies.

The lowest value of *q*_{0} shown in
Figure 1 is that which is required
by adding the luminosity density of galaxies obtained from the
observed *N(m)* data. As discussed by
Binggeli et al. (1988)
in their review of the luminosity function in this volume, this minimum
permissible value of *q*_{0} is

(31) |

which is *q*_{0} = 0.03 if *M* / *L* = 40. The
second case shown in Figure 1 is for
_{0} =
1(*q*_{0} = 1/2) required by Grand Unification. The third
case (*q*_{0} = 1) is for a highly curved Reimannian space
of curvature *c*^{2} / *R*^{2} =
*H*_{0}^{2}. If *H*_{0} = 50 km
s^{-1} Mpc^{-1}, the radius of
curvature would be *R* = 6000 Mpc, which is only 300 times the distance
to the Virgo cluster!

Figure 1a shows *N*(*z*,
*q*_{0}) vs. log *z*. Figure 1b
is the same calculation, but it is displayed in *z* rather than log
*z*. The marked
interval along the ordinate is a actor of 10 in the counts. For small
redshifts (*z* < 0.1) the slope of *N*(*z*) curves for
all reasonable *q*_{0} is
*d* log *N*(*z*)/*dz* = 3, becoming considerably
flatter for higher redshifts,
as marked along the curves in Figure 1a. For
large redshifts, the
volume becomes much smaller than the *z*^{3} Euclidean case
for *q*_{0} > 1/2,
but larger for the hyperbolic geometry of *q*_{0} <
0.5. The ratio of the
volumes encompassed from the observer to redshift *z* for
*q*_{0} = 0
compared with *q*_{0} = 0.5 is shown as a function of
*z* in the table in Figure 1a.