Lemaitre and Hubble: A Tale Of Two Blunders

An essay on early determinations of the Hubble constant

Although Hubble probably wasn't aware of it at the time, when he published his 1929 paper (PNAS, 15, 168) announcing the discovery of a correlation between velocity and distance of galaxies, his paper was neither the first or even second, but rather the third to derive a value for the slope of this relation (which we call the Hubble constant). The first paper was by Lemaitre two years earlier, but Lemaitre published it in an obscure journal (1927, Annales de la Societe Scientifique de Bruxelles, A47, 49) and his derivation was not known to Hubble nor the rest of the community at the time. (The second was by Robertson in 1928 [Phil. Mag., 5, 835], also published in journal not widely read by astronomers.) Lemaitre's value for the slope was 625 km/s/Mpc, reasonably close to Hubble's value of 500 km/s/Mpc. Because Lemaitre based his result in part on data published by Hubble in 1926, it is sometimes assumed that Hubble's work was simply a repeat of Lemaitre's, using essentially the same data --
Way and Nussbaumer (2011, arXiv:1104:3031): "Two years later Hubble found the same velocity-distance relationship on observational grounds ... from practically the same observations that Lemaitre had used."
Van den Bergh (2011, arXiv:1108:0708): "1929 Hubble repeats Lemaitre's work with essentially the same data and obtains similar results"

Although these assertions sound plausible, they are actually quite wrong - while Lemaitre and Hubble did use the same published velocities and both relied on Hubble's distances to nearby, Local Group galaxies, each used a different method and different data to derive distances to more remote galaxies. Furthermore, each made a serious blunder in the analysis that greatly impacted their determination of the Hubble constant, and the fact that both arrived at a similar value is actually sheer accident. These blunders are of interest since they are part of the larger story as to why the early values of the Hubble constant were so much bigger than modern determinations (~70 km/s/Mpc).

Backdrop

To start, we need to backtrack to Hubble's monumental 1926 paper (ApJ, 64, 321) on Extragalactic Nebulae. In one section of the paper, Hubble collected togther all the galaxy distances known at the time (seven total), which were all based, directly or indirectly, on observations of Cepheid variable stars. Hubble used these distances to calibrate the absolute magnitudes of the brightest stars in galaxies. Additionally, Hubble derived the mean absolute magnitude and scatter of the galaxies themselves. Using this calibration, Hubble gave a straightforward equation (Eq. 8) to convert the apparent magnitude of a galaxy into a distance:
log D = 4.04 + 0.2 mT.              (8)
(Also of relevance, Hubble went on to estimate masses of galaxies from rotation curves and Opik's 1922 M/L calibration of the Milky Way and computed the mean mass density of baryonic stellar matter in the universe.)

Lemaitre

Lemaitre's 1927 paper is a remarkable tour de force - in today's terminology, we would say that he developed a mixed baryonic/dark energy model of an expanding universe that starts from a static Einstein universe (2/3 baryonic, 1/3 dark energy) and evolves into a pure de Sitter universe (100% dark energy). Going further, Lemaitre showed that his model predicts a linear velocity-distance relation, and he showed that one could use measurements of the slope of that relation (the Hubble constant) along with a measurement of the baryonic mass density to completely constrain the free parameters of the model. To find the value of the Hubble constant, he compiled a sample of 42 galaxies with known apparent magnitudes and velocities, used Hubble's Eq. 8 to convert magnitudes to distance, and then divided the mean velocity of all galaxies in the sample by their mean distance (with a weighting scheme not of relevance here). In essence, Lemaitre was basing his distances on Hubble's conjecture that the absolute magnitude of a galaxy, as calibrated by nearby galaxies, can be used as a standard candle. In this way Lemaitre arrived at his preferred value of 625 km/s/Mpc. By combining this value with Hubble's 1926 value for the baryonic stellar mass density, he arrived at the result that the universe has expanded by a factor 20 from its initial state.

Hubble

On the surface, Hubble's 1929 paper seems much less ambitious - it, too, derived a linear relation between velocity and distance, but not much else; further Hubble's sample included only 24 galaxies. However, there was an important difference - Hubble used brightest stars, not galaxies, as his standard candles. Thus, in going from the known distances of nearby galaxies to more remote objects, Hubble used a method that was independent of and relied on different data than that of Lemaitre. The fact that Hubble's preferred value for the slope, 500 km/s/Mpc, agreed within 25% of Lemaitre's value was not inconsequential, and would seem to validate both methods as being well-founded.

Blunders

In fact, the agreement was vacuous - Lemaitre and Hubble both committed egregious blunders that, by chance, were of nearly the same magnitude.

As an initial matter, we note that Hubble's distances to the nearby Local Group galaxies were all too small. Roughly, two effects were at work. First, the calibration of the Cepheid period-luminosity relation (from Shapley) was too faint by about 1.5 magnitudes. Second, the Mount Wilson stellar magnitude scale had a nonlinearity between the bright and faint ends. The impact of this nonlinearity is difficult to assess but might have added another 0.6 mag error to the distance modulus. These errors affected both Lemaitre and Hubble - had these been the only errors, both would have computed a value for the Hubble constant of around 200 km/s/Mpc, much closer to the modern value.

Lemaitre relied on Hubble's Eq. 8, which presumes that the mean absolute magnitude of nearby galaxies can be applied to more distant galaxies as well. However, Hubble couched this result with caveats like "extrapolation", "assumption", and "working hypothesis", implying that it was a conjecture that needed to be validated later. Lemaitre, however, ignored these caveats and adopted the equation as established fact. We now know that Hubble's conjecture was wrong. Today we would call the problem one of Malmquist bias - a sample of nearby galaxies will be dominated by the numerous, intrinsically faint objects, while at large distances, due to selection effects, one only sees (and measures) the rare but nevertheless intrinsically bright galaxies, introducing a bias. Indeed, Lemaitre's sample of more distant galaxies was intrinsically brighter by 2.3 mag in comparison to Hubble's calibrating sample of nearby galaxies. This effect accounted for Lemaitre's erroneously high value of the Hubble constant.

Hubble's brightest star method, while not very precise, was not subject to the biases encountered by Lemaitre. Further, there is a physical basis (the Eddington limit) as to why the method should work. Hubble's problem, however, was that he was unable to distinguish individual stars from HII regions in the more distant galaxies. Humason, Mayall, and Sandage (1956, AJ, 61, 97) would later show that Hubble's "brightest stars" for his most distant galaxy in the Virgo cluster were, in fact, just such HII regions, and that the brightest stars were 2 magnitudes fainter. This effect accounted for Hubble's erroneously high value. The error was nearly the same size as Lemaitre's, and also accounted for why Hubble, in a further erroneous bit of reverse engineering, concluded that his conjecture about the universality of a galaxy's absolute magnitude appeared to be correct ("... this entirely unforced agreement ...").

Who "Discovered" the "Expansion of the Universe?"

This question actually carries too much baggage - a better way to phrase it is "who discovered the linear velocity-distance relation?" One might think that Lemaitre deserves credit, but that is really not correct. Lemaitre did not plot his data in the form of a figure; had he done so, it might have looked like the figure below (left side), from Block (2011 arXiv:1106.3928). On the right is a copy of the figure presented in Hubble's paper.

Notwithstanding the suggestive straight line drawn in the left figure, Lemaitre had to assume that the linear relation, passing through the origin, is correct ab initio. He could not infer it from the data - the scatter was too large. Lemaitre was well aware of this problem and attributed it (correctly) to the fact that, as Hubble had shown, there is a large scatter in the intrinsic absolute magnitude of galaxies about the mean, and this scatter is nearly as large as the effect he was looking for, given the limited range in velocity of galaxies that was available to him. This explanation also accounts for why previous researchers, such as Lundmark and Stromberg, were unable to discern the relation from very nearly the same data. Conversely, even though Hubble had a much smaller sample of galaxies than Lemaitre and his differentiation of stars from HII regions was imperfect, his ordering of the galaxy distances was much better, and thus he was able to demonstrate the existence of a linear velocity-distance relation, passing through the origin, based purely on the data alone.

We can make this comparison more quantitative. A Spearman rank-correlation coefficient is a nonparameteric measure of the degree and significance of a monotonic correlation between two variables. The data for 42 galaxies in the Lemaitre diagram, on the left side above, have a correlation coefficient of 0.39 with a significance of 2.5 sigma - low enough that the data are only marginally inconsistent with there being NO correlation at all. The data in the Hubble diagram, on the right side, even though consisting of only 24 galaxies, have a correlation coefficient that is much higher - 0.85 - with a significance that is also greater - 5.5 sigma. If we trust in the mythical 3 sigma as being the threshold for making a discovery, credit for discovery of the velocity-distance relationship goes to Hubble.

Addendum

Some might wonder whether Hubble deserves much credit at all, given that his calibration of the velocity-distance relation was off by a factor of 7. It is important to remember, though, that, at the time, the field of observational cosmology was still in its early phase of development, when many lines of investigation would invariably lead to dead-ends, and progress often relied on a combination of empiricism and sketchy extrapolation. Measuring distances on an intergalactic scale is hard! Even today, the most accurate determination of the Hubble constant using HST involves a three-step procedure, with statistical and systematic errors creeping in at each step. The embarrassment is not that Hubble's intial value was so wrong, but rather that it remained largely uncorrected for over 25 years.