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

5. PREDICTED AND OBSERVED COUNT-MAGNITUDE RELATION

5.1. Method of Predicting N(m, q₀, E) for an Infinitely Narrow Luminosity Function

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 m_bol 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 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).

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 m_bol 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).