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

**6.1. Local Tests for Linearity of the Redshift-Distance
Relation**

6.1.1. THE PREDICTION
If redshift is due to a real expansion of the geometrical manifold
carrying the galaxies with it, the form of the velocity field must be
isotropic and linear if space is isotropic and homogeneous. There are
many ways to prove this. One of the most straightforward is to
consider a triangle. If, upon expansion, the triangle is to be similar
to itself, it follows directly that the velocity *v*_{i} of
any number of
points along the sides of the triangle at distances *r*_{i}
from any arbitrary point must be

(40) |

Otherwise the sides of the triangle will not expand with the same
proportionality ratio. The factor *H*(*t*) sets the scale of
the velocity field at time *t*.

A linear velocity field has two fundamental properties:

- The form and the scale of the field are invariant to position, i.e.
Equation 40 holds for every point in the manifold upon transferring
the origin to it. Hence, the velocity field looks the same from every
vantage point, i.e. the ultimate democracy.
- Upon reversing direction of the velocity vectors to form a
*contracting*field, all points arrive at the origin at the same time, i.e. points twice as far away as others have twice the velocity, hence*equal travel time*to any arbitrary point. And because*all*points are equivalent, all points arrive at all other points together. There is no center of the expansion. All points*have always been*the center.

From these two properties it was early expected that the
velocity-distance relation for any reasonable model of an isotropic,
homogeneous universe (in the large) must be linear if redshift is a
true expansion of the manifold. Note also that the direct prediction
of the *m*(*z*) relation in the standard model (Equations 33
and 34), using *m* - *M* + 5 = 5 log *r*, gives *cz*
= *H*_{0}*r* in the *z*
0
limit, which
is Equation 40; this provides another way to prove that linearity is a
prediction of the standard model.

6.1.2. THE DATA FOR SMALL z For light travel times that are small compared with the age of the Universe, the World picture is nearly the same as the World map, in Milne's (1935) useful language. The light travel time can be neglected for small-enough redshifts.

In the search for the form of the expansion law, the relevant redshift
regime in which to exploit this circumstance unencumbered by the
non-simultaneity of the observations is *z*
0.2.

Four claims are extant in the literature concerning the dependence of redshifts on distance in the World map. Only one of the following is correct.

- The relation is linear everywhere, at all distances and at all
times. The standard model requires it. We argue herein that the data
demand it.
- The relation is approximately quadratic
*locally*(i.e. to*cz*4000 km s^{-1}); thereafter it becomes linear (de Vaucouleurs 1958, 1972, de Vaucouleurs & Peters 1986, their Figures 2a, b; Giraud 1985, 1986a, b, 1987). - The relation is quadratic everywhere
(Segal 1975,
1981).
- The relation is exponential as 1 +
*z*= exp(*HrR*/*c*) in the speculation of tired light [LaViolette 1986; see also Pecker & Vigier (1987) for a review].

In his discovery paper,
Hubble (1929)
claimed a linear relation only quite locally, being aware of
de Sitter's (1917)
*quadratic* prediction [cf.
Sandage (1975b)
for a history]. Hubble's final sentence of this
first paper was a cautious "and in this connection it may be
emphasized that the linear relation found in the present discussion is
a first approximation representing a restricted range in distance."
His idea evidently was that he might only be observing a linear
(tangent) approximation at very small *r* to the de Sitter
*v* ~ *r*^{2}
prediction, which might become evident at larger distances.

To test this,
Humason (1931)
began redshift measurements at Mount
Wilson of bright E galaxies in clusters and found, even at this early
date, values as large as *cz* ~ 20, 000 km s^{-1} for a
cluster in Leo. With
these new data, combined with estimates of apparent magnitude,
Hubble & Humason (1931)
extended the original Velocity-apparent magnitude
relation approximately twentyfold in distance within two years of the
initial discovery.

The slope of the redshift-magnitude relation was found to be 5 to
within the errors, proving beyond doubt that the velocity-distance law
is linear. However, in 1931 there were only 8 clusters involved in the
test. By 1936,
Hubble (1936a) and
Humason (1936)
had increased the
data base to 10 clusters reaching
*cz* = 42, 000 km s^{-1}, with the same result.
Hubble's (1953)
last discussion, in his Darwin lecture, still
had only 11 clusters, but the data now reached to *z* = 0.2, which was
the largest redshift Humason could measure, even with the early use of
the Palomar 200-inch reflector with photographic detection. The result
again proved linearity, but the data sample was still small and the
photometric measurements had been made using only photographic
techniques. The final phase of the initial proof of linearity was the
summary paper of HMS, which used new data on 18 clusters. However, the
universality of the phenomenon was more inferred than established
because there were still so few clusters.

Early claims of nonlinearity (cf.
Hawkins 1962)
used the HMS data on
*field* galaxies taken from *flux-limited* samples and failed to
correct for sample bias. Modern discussions of local nonlinearity, some of
which are referenced earlier in this section, also use field
galaxies. Because the luminosity function of field galaxies is
considerably broader than for first-ranked cluster galaxies, some
conclusions from these field galaxy studies have been criticized as
due to insufficient corrections for the selection bias. Discussions of
the bias have been given by
Teerikorpi (1975a,
b,
1984,
1987),
Bottinelli et al. (1986,
1987,
1988),
and Sandage (1988a,
b). The
principal consequences of bias neglect have been (*a*) an incorrectly
high value of the Hubble constant and (*b*) a false belief that the
consequent calculated variation of *H*_{0} with distance,
increasing outward, is real (Section 9).

Discussion of the bias effects in the flux-limited RSA catalog
became possible when the redshift coverage of that catalog became
complete, following the finalization
(Sandage 1978)
of the Humason &
Mayall (HMS) redshift program. The bias in the field galaxy sample was
shown directly
(Sandage et al. 1979),
and the apparent increase of *H*_{0}
with distance was demonstrated to be an artifact, caused by assigning
a fixed <*M*> value to all galaxies in the flux-limited
sample These
selection effects are severe enough that study of the form of the
redshift-distance relation using field galaxies is dangerous owing to
the necessary corrections that are difficult to make accurately. For
this reason, study of the *m*(*z*) relation to test for
linearity was
begun in the early 1970s using *cluster* and *group* samples. The
photometry in these studies was done by photoelectric techniques; the
results, reported in a series of papers
(Sandage 1972a,
b,
c,
1973a,
b,
1975a,
1986,
Sandage et al. 1972,
Sandage & Hardy 1973),
again confirmed the linearity of the local velocity-distance relation.

The principal conclusion of the work was that the leading term of 5
log *cz* in the theoretical prediction of Equation 34 is fully
confirmed. The standard model does, then, pass this most elementary
of its predictions.