**2.3. Hubble's interpretation**

We now come to one of the most remarkable episodes in all of science. Hubble's (1936b) detailed analysis of Fig. 1 is a most fascinating study of how an interpretation, without caution concerning possible systematic errors, led to a conclusion that the systematic redshift effect is probably not due to a true Friedmann-Lemaitre expansion, but rather to an unknown, then as now, unidentified principle of nature. Indeed, even in the abstract to this 1936 paper on the "Effects of Redshift on the Distribution of Nebulae" Hubble concluded: "The high density suggests that the expanding models are a forced interpretation of the data." His belief that the expansion probably is not real persisted even into his final 1953 paper which was the Darwin lecture of the RAS, given in May of the year he died in September. What were the steps leading to this conclusion that, in today's climate, seems so remarkable?

Redshifts, ubiquitous for all galaxies everywhere, decrease the received flux. The larger the redshift, the larger is the decrease. This effect is one of the two principal reasons for the observed departure of the data in Fig. 1 from the "uniform distribution" supposition. The second is the departure of the intrinsic geometry from Euclidean, measured by "curvature" in the sense introduced by Gauss (1828, 1873). Hubble considered both of these effects.

There is no redshift information in Fig. 1. Hence, a relation between redshift and apparent magnitude is required to change the N(m) relation to the more fundamental N(z) function that is needed to make the calculations. And, because the counts in Figure 1 are for field galaxies, the (m, z) relation for clusters as determined by Hubble and Humason (1931) and Humason (1936) had to be replaced by data for field galaxies. The required (m, z) relation was set out in Fig. 2 of Hubble (1936b), based on data of Hubble & Humason (1934).

Using the field galaxy (m, z) ridge-line relation of log cz = 0.2m + 0.77, the N(m) data of Fig. 1 could be changed to an N(z) relation, but only after correcting the observed magnitudes for the technical effects of redshifts, expressed as a K term, both selective and neutral (more later).

Armed now with a corrected N(z) relation, an assumption must be made concerning
the relation between "distance" and redshift so that the N(z) relation
could be changed to
an N(r) relation, assumed to be proportional to the volume contained
within "distance" r,
leading to the geometry that is either Euclidean if (vol ~
r^{3}), or non-Euclidean if otherwise.

Note the extremely complicated multiple steps and, further, the questionable
approximation of replacing distribution functions by mean values;
viz. the counts in m are
changed to N(z) and the K(z) term is changed to K(m) via an assumed
*mean* (m, z)
relation rather by using a luminosity distribution function for absolute
magnitudes that takes into account the intrinsic spread in m at a given z.

A few of the intricate details of Hubble's procedure are set out in brief in the next section. Here we only summarize his conclusions, based on his analysis of the corrections to apparent magnitudes due to the effects of redshifts.

As is now well known, if redshifts are due to a true expansion, the required correction to the observed apparent magnitudes are by two factors of (1 + z) for the so called number effect (the number of photons received per second from a receding source) plus the energy effect [each photon is degraded in energy by the redshift, again by (1 + z)]. Hubble concluded (see the next section) that if two factors of (1 + z) are applied to his Fig. 1 data, then the curvature correction needed to make the data conform to the "Uniform Distribution" condition would have to be enormous, giving a very small, high density, large curvature universe, so small and of such high density that Hubble believed that the procedure gave impossible results. He continued to write his conclusion to the end, calling into question the reality of the expansion that required the second factor of (1 + z) correction for the "number effect."

To make understandable the language of Hubble's analysis, the K
correction as used by
Hubble is the selective effect (plus the bandwidth effect)
^{(2)} of
shifting a galaxy spectrum
through the blue photographic band pass "plus" either the one or two
factors of 2.5 log
(1 + z). Hubble expressed the total effect as K = B x z.

Hubble determined B from the observations using the "departure" curve in Fig. 1. His program was then to compare this B(observed) with a theoretical B* calculated using the assumption of either a true expansion or not (either one or two factors of 1 + z), plus the selective term found by shifting an assumed galaxy spectrum through the photometric band pass (the selective part of the K term; see Humason et al. 1956, Appendix B; Oke and Sandage 1968). In what follows, the argument hinges on the comparison of B and B*.

Hubble's analysis (page 533 of the 1936b paper) of his "departure" curve
gave B =
2.94. His *calculated* K term (assuming a black body galaxy
spectrum of T = 6000° K)
was either B* = 3.0 for no expansion, or B* = 4.0 for a real expansion
(energy plus
number effect). Clearly, only the no-expansion solution fitted Hubble's
putative B = 2.94 departure data in
Fig. 1.

Many conclusions were made from this result, not only concerning the reality of the expansion but also concerning the consequences of a real expansion for a second-order term in the velocity-distance relation as the measurement of deceleration, the value of the space curvature, and the question of evolution of galaxy absolute magnitudes in the look-back time. Several of the direct quotes concerning these issues are of interest for the work of the present workshop.

On his page 542: "It is evident that the observed result, B = 2.94, is
accounted for if
redshifts are not velocity shifts. The comparison is based on an
effective temperature,
T_{0} of 6000°, but the uncertainties cover the range down
to about T_{0} = 5750. The
interpretation is consistent with the data [but only if] - the
*expansion and spatial curvature
are either negligible or zero*." (Emphasis added).

Concerning the redshift-distance relation; page 38 of Hubble (1937): "The inclusion of recession factors would displace all the points [in the Hubble diagram of redshift vs. apparent magnitude of great clusters his Fig. 1 of the 1937 reference] to the left [higher redshifts at a given magnitude], thus destroying the linearity of the law of redshifts". [N.b., this is not correct when the appropriate Mattig relations for the (m, z) Hubble diagram are used; see later].

For the conclusion on the reality of the expansion (Hubble 1936b, page 553):-"if redshifts are not primarily due to velocity shifts, the observable region loses much of its significance. The velocity-distance relation is linear; the distribution of nebulae is uniform; there is no evidence of expansion, no trace of curvature, no restriction of the time scale." Page 553/4: "The unexpected and truly remarkable features are introduced by the additional assumption that redshifts measure recession. The velocity-distance relation deviates from linearity by the exact amount of the postulated recession. The distribution departs from uniformity by the exact amount of the recession. The departures are compensated by curvature which is the exact equivalent of the recession. Unless the coincidences are evidence of an underlying necessary relation between the various factors, they detract materially from the plausibility of the interpretation, the small scale of the expanding model, both in space and time is a novelty, and as such will require rather decisive evidence for its acceptance."

From his Darwin lecture
(Hubble 1953):
"When no recession factors are included,
the law will represent approximately a linear relation between
red-shifts and distance.
When recession factors are included, the distance relation is expected
to be - non-linear
in the sense of accelerated expansion" [sic, not the correct sign; the
word must clearly
be *decelerated*, as he in fact wrote twice earlier in 1936 and
1937] . " - [If no recession
factor is included] the 'age of the universe' is likely to be between
3000 and 4000 million
years, and thus [again with no recession factor] comparable with the age
of rock in the crust of the Earth."

Concerning the second-order term in the velocity-distance relation: (1936b), page 546: "Since the second-order term [with recession factors included] is definitely positive, the possible models are restricted to those in which the rate of expansion has been diminishing during the past several hundred million years." And again in (1937, page 43): "The chief significance of the term for cosmological theory lies in the positive sign [of the redshift vs. distance correlation]. The rate of expansion of the universe has been slowing down, at least for the past several hundred million years. The 'age of the universe' is considerably shorter than that permitted by the linear law."

Finally, as to evolution of luminosity in the look-back time (Hubble 1936b, page 543). "As for the constancy of nebular luminosities, the question is whether or not luminosities of spirals change materially (say 10%, or 0.1 mag) during [the look-back time]. - very few students will hesitate to adopt the assumption that systematic variation in so short a [time] interval will be inappreciable." Reasons are then given in the remainder of the paragraph, none of which would pass today's referees armed with the present knowledge of population synthesis and stellar evolution.

The clearest proof that Hubble maintained these views concerning the reality of the expansion until the end is the style of argument in his 1953 Darwin Lecture, seen in particular from Fig. 1 of that lecture. No recession factor was put to the K-correction magnitudes in the abscissa of the m, z (Hubble) diagram. This diagram was the first rendering in the literature of that central cornerstone of observational cosmology using new data from the Palomar 200-inch reflector.

The several arguments set out above were of particular importance for the Palomar program on these matters that followed from 1953 through the 1980s. Reasons why Hubble's conclusions should be changed emerged slowly from this program, not only because of new observations based on photoelectric photometry, but also from a much deeper understanding of the interface between the observations and the theory (section 4).

^{2} The spectrum is stretched by
multiplying each rest wavelength by a factor of 1 + z. By
so doing, compensation must be made for the decreased bandwidth of the
stretched spectrum
over the fixed sensitivity pattern of the plate, filter, and telescope
photometric functions. I mistakenly have written
(Sandage 1995,
Lecture 2) that Hubble neglected the bandwidth
correction. However, a detailed examination of his calculated K
corrections for a blackbody with
T = 6000° K
(Hubble 1936b,
his Tables V and VI) shows that he did not neglect the bandwidth
term and that my comment in the Saas Fee Lectures is incorrect.
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