### 5. PREDICTED AND OBSERVED COUNT-MAGNITUDE RELATION

The necessary apparatus is now in place to predict the expected N(m) relation for any assumed q0 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, q0, 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 q0 has been fixed for a particular geometry.

1. For any particular redshift z, calculate r and rR0 from Equations 29 and 30.

2. 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, q0) of Section 3.2.

3. For any z and q0 use Equation 35 to calculate m for an assumed absolute magnitude <M>, using the K(z) and E(z) corrections.

4. Repeat for a variety of z and q0 values, producing the predicted family of N(m, q0) 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 q0 in N(z). The same result that N(m) is less sensitive to q0 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 q0 is that although the N(z) relation is relatively sensitive to q0, its dependence on q0 appears with the opposite sign from the variation of m(z) with q0 in Equation 35. This nearly cancels the curvature dependence of N(m, q0). The calculated N(m, q0) curves with no K correction (i.e. using the mbol magnitude scale) are shown in Figure 2 for q0 = 0 and q0 = 0.5. For this idealized calculation, all galaxies were assumed to have the same absolute magnitude of Mbol = - 20.5; this is a reasonable value, close to MV* 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).

 Figure 2. Theoretical N(m, q0) relations for two values of q0 using bolometric magnitudes (i.e. no K correction has been applied) for galaxies with no spread in absolute luminosity. In practice, K corrections for each galaxy type must be applied for the particular detector band, and integrations must be performed over the luminosity functions for each morphological type and summed over the morphological mix, making the dependence on q0 smaller than in this limiting example. Redshift values are shown along each curve.

The redshift is marked along each curve in Figure 2, showing the different z values at a given mbol value depending on q0 given by Equation 35. The volume ratios at various z values (for q0 = 0) are listed in the table interior to the diagram.

It is therefore surprising that Yee & Green (1987) claim a determination of q0 from faint galaxy counts. They find a non-negligible dependence on q0 (their Figures 4 and 5). This result is probably produced by the strong dependence of the E(z) evolutionary correction on q0 (their Table 3), presumably due to the q0 dependence in the conversion of E(t) to E(z), as explained above. The determination of q0 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 H0 and q0 (Section 9).