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

**5.1. Method of Predicting
N(m, q_{0}, E) for an Infinitely Narrow
Luminosity Function**

The necessary apparatus is now in place to predict the expected
*N*(*m*)
relation for any assumed *q*_{0} value and luminosity
evolution rate *E*(*z*).
The *N*(*z*) relation calculated by the method of
Section 3.2 can be transformed to
*N*(*m*, *q*_{0}, *E*) using Equation
35. The conversion is
trivial if *M* is assumed to be a fixed number, <*M*>,
with no dispersion
(i.e. if the luminosity function is a spike). In this case, for
computational purposes the equations are easiest used progressively in
parametric form with the following steps, once *q*_{0} has
been fixed for a particular geometry.

- For any particular redshift
*z*, calculate*r*and*rR*_{0}from Equations 29 and 30. - Use Equations 9, 10, or 11 (depending on the value of
*k*) to calculate*V*(*z*), which aside from a normalization factor is the*N*(*z*,*q*_{0}) of Section 3.2. - For any
*z*and*q*_{0}use Equation 35 to calculate*m*for an assumed absolute magnitude <*M*>, using the*K*(*z*) and*E*(*z*) corrections. - Repeat for a variety of
*z*and*q*_{0}values, producing the predicted family of*N*(*m*,*q*_{0}) curves.

These steps are the method that was used to show numerically the
degeneracy of *N*(*m*) to go to first order in *z*
(Sandage 1961a,
his Figures 4 and 5) if *E*(*z*) = 0, despite the nondegeneracy to
*q*_{0} in
*N*(*z*). The same result that *N*(*m*) is less
sensitive to *q*_{0} than is *N*(*z*)
was shown analytically by
Robertson & Noonan
(1968),
Misner et al. (1973),
and Brown & Tinsley
(1974)
using series expansions. The
reason for the near-degeneracy of the *N*(*m*) counts to
*q*_{0} is that
although the *N*(*z*) relation is relatively sensitive to
*q*_{0}, its
dependence on *q*_{0} appears with the opposite sign from
the variation of
*m*(*z*) with *q*_{0} in Equation 35. This nearly
cancels the curvature
dependence of *N*(*m*, *q*_{0}). The calculated
*N*(*m*, *q*_{0}) curves with no *K*
correction (i.e. using the *m*_{bol} magnitude scale) are
shown in Figure 2
for *q*_{0} = 0 and *q*_{0} = 0.5. For this
idealized calculation, all galaxies
were assumed to have the same absolute magnitude of
*M*_{bol} = - 20.5; this
is a reasonable value, close to *M*_{V}^{*} for the
field luminosity function for high-surface-brightness galaxies
(Tammann et al. 1979)
in the *Revised Shapley-Ames Catalog*
(Sandage & Tammann
1981;
hereinafter RSA).

The redshift is marked along each curve in
Figure 2, showing the
different *z* values at a given
*m*_{bol} value depending on *q*_{0} given by
Equation 35. The volume ratios at various *z* values (for
*q*_{0} = 0) are listed in the table interior to the diagram.

It is therefore surprising that
Yee & Green (1987)
claim a determination of *q*_{0} from faint galaxy
counts. They find a
non-negligible dependence on *q*_{0} (their Figures 4 and
5). This result
is probably produced by the strong dependence of the *E*(*z*)
evolutionary
correction on *q*_{0} (their Table 3), presumably due to
the *q*_{0} dependence
in the conversion of *E*(*t*) to *E*(*z*), as
explained above. The
determination of *q*_{0} in this way is then not a direct
test of different
*volumes* of a non-Euclidean geometry (i.e. the direct volume test) but
rather a much more indirect route that connects secular luminosity
evolution with a *time scale* that does depend on
*H*_{0} and *q*_{0}
(Section 9).