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

**4.1. The Predicted Hubble Diagram With No Luminosity
Evolution**

The *N*(*z*, *q*_{0}) volume-redshift relation of
the last section is very
difficult to apply in practice because complete galaxy counts in
*redshift space* require redshift measurements of every galaxy (of all
types and surface brightness) in the volume, or else a correction for
sample incompleteness in a redshift survey that is complete to a given
magnitude limit
(Loh & Spillar 1986,
Loh 1986).
These corrections must
be highly precise if the small differences in the *N*(*m*,
*q*_{0}) curves in
Figure 1 are to be measured. For this reason, the value of
*q*_{0} via this
route is quite uncertain at the moment.

The easier test observationally is the count-*magnitude* relation,
*N*(*m*, *q*_{0}), used in its most elementary
form by Hubble (1936b),
following the theory set out by
Hubble & Tolman (1935)
for *N(r)*. We
now cast their discussion into modern form by using the closed
equation for the apparent magnitude-redshift relation via the Mattig
equations and thereby changing *N*(*z*, *q*_{0})
into the *N*(*m*, *q*_{0}) count-magnitude
prediction.

The apparent bolometric flux *f*_{b} received at Earth from
a galaxy
receding with redshift *z* whose absolute flux (at the source) is
*F*_{b} was shown by
Robertson (1938)
(after some debate) to be

(32) |

For an appreciation of this equation, consider a sphere of interval
radius *l* (Equations 12, 13) centered on the source, over which
the flux
of a light pulse is spread at the time of light reception,
*t*_{0}, at the Earth. The area of this sphere is not
4*l*^{2} if the
geometry is non-Euclidean but is, rather,
4(*R*_{0}*r*)^{2}, where
*R*_{0}*r* = *R*_{0}sin *l* /*R*
using
Equation 12 for *k* = + 1. As in the case of the spherical cap of
Equation 5, this area is smaller than
4*l*^{2} owing to
the spatial curvature if *k* = + 1, or larger if *k* = - 1:

The difference in the area compared with the Euclidean case is
accounted for in Equation 32 by the
(*R*_{0}*r*)^{2} factor rather than by
simply using the interval distance *l*, which would be incorrect. The
(1 + *z*)^{2} term accounts for the energy depletion and
dilution factors of
the radiation due to the redshift. One factor arises because each
photon is decreased in energy by (1 + *z*), and hence the entire
ensemble
is depleted by the same factor. The second factor of (1 + *z*) is
present
if the redshift is due to true expansion. It is caused by the
increased path length, with the consequent decrease in the energy
*density*. If the Universe is *not* expanding, the second (1 +
*z*) factor
would not be present, a crucial point for the surface brightness test
discussed in Section 8.

Converting Equation 32 into magnitudes and using Equation 30 for
(*R*_{0}*r*)^{2} gives the theoretical
*m*(*z*, *q*_{0}) equation for the Hubble
diagram in terms of the *bolometric* magnitude:

(33) |

where the constant *C* is
2.5 log 4 + 5 log
*c*/*H*_{0}. *Note that the factor
(1 + z)^{2} of Equation 32 is incorporated in Equation 33
as part of the
theory.* Some earlier writers, following Hubble, included the
-5 log(1 +

Series expansion of Equation 33 gives the well-known equation (e.g. Robertson 1955, McVittie 1956)

(34) |

used by Humason et al.
(1956;
hereinafter HMS) in their early analysis
of cluster data. Although adequate to *z* ~ 0.3, the deviations of
Equation 34 from Equation 33 for larger *z* become inadequately large
(cf. Mattig 1958,
his Figure 1).