A. Zwicky and Smith
The Swiss-American astronomer Fritz Zwicky is arguably the most famous and widely cited pioneer in the field of dark matter. In 1933, he studied the redshifts of various galaxy clusters, as published by Edwin Hubble and Milton Humason in 1931 , and noticed a large scatter in the apparent velocities of eight galaxies within the Coma Cluster, with differences that exceeded 2000 km/s . The fact that Coma exhibited a large velocity dispersion with respect to other clusters had already been noticed by Hubble and Humason, but Zwicky went a step further, applying the virial theorem to the cluster in order to estimate its mass.
This was not the first time that the virial theorem, borrowed from thermodynamics, was applied to astronomy; Poincare had done so more than 20 years earlier in his Leçons sur les hypothèses cosmogoniques professées à la Sorbonne . But to the best of our knowledge, Zwicky was the first to use the virial theorem to determine the mass of a galaxy cluster.
Zwicky started by estimating the total mass of Coma to be the product of the number of observed galaxies, 800, and the average mass of a galaxy, which he took to be 109 solar masses, as suggested by Hubble. He then adopted an estimate for the physical size of the system, which he took to be around 106 light-years, in order to determine the potential energy of the system. From there, he calculated the average kinetic energy and finally a velocity dispersion. He found that 800 galaxies of 109 solar masses in a sphere of 106 light-years should exhibit a velocity dispersion of 80 km/s. In contrast, the observed average velocity dispersion along the line-of-sight was approximately 1000 km/s. From this comparison, he concluded:
If this would be confirmed, we would get the surprising result that dark matter is present in much greater amount than luminous matter.
This sentence is sometimes cited in the literature as the first usage of the phrase “dark matter”. It is not, as we have seen in the previous chapter, and it is not even the first time that Zwicky used it in a publication. He had, in fact, used the same phrase in a article published earlier the same year, pertaining to the sources of cosmic rays :
According to the present estimates the average density of dark matter in our galaxy (ρg) and throughout the rest of the universe (ρu) are in the ratio ρg / ρu > 100,000.
Although he doesn't explicitly cite any article, it is obvious from this sentence that he was well aware of the work of Kapteyn, Oort and Jeans discussed in the previous chapter. His use of the term “dark matter” is, therefore, in continuity with the community of astronomers that had been studying the dynamics of stars in the local Milky Way.
In 1937, Zwicky published a new article – this time in English, in the Astrophysical Journal  – in which he refined and extended his analysis of the Coma Cluster. The purpose of this paper was to determine the mass of galaxies, and he proposed a variety of methods to attack this problem. In particular, he returned to the virial theorem approach that he had proposed in 1933, this time assuming that Coma contained 1000 galaxies within a radius of 2 × 106 light-years, and solving for the average galaxy's mass. From the observed velocity dispersion of 700 km/s, he obtained a conservative lower limit of 4.5 × 1013 M⊙ on the mass of the cluster (to be conservative, he excluded a galaxy with a recession velocity of 5100 km/s as a possible outlier), corresponding to an average mass-per-galaxy of 4.5 × 1010 M⊙. Assuming then an average absolute luminosity for cluster galaxies of 8.5 × 107 times that of the Sun, Zwicky showed that this led to a surprisingly high mass-to-light ratio of about 500.
Zwicky's work relied on Hubble's relationship between redshift and distance, and in the 1937 paper he used the results of Hubble and Humason , which pointed to a Hubble constant of H0 = 558 km/s/Mpc, with an estimated uncertainty of 10-20%. If we rescale these results adopting the modern value of H0 = 67.27 ± 0.66 , we see that Zwicky overestimated the mass-to-light ratio by a factor of ∼ 558 / 67.27 = 8.3. Despite this substantial correction, Coma's velocity dispersion still implies a very high mass-to-light ratio and points to the existence of dark matter in some form.
What did Zwicky think that the dark matter in Coma and other galaxy clusters might be? An illuminating sentence in his 1937 paper provides a rather clear answer to this question:
[In order to derive the mass of galaxies from their luminosity] we must know how much dark matter is incorporated in nebulae in the form of cool and cold stars, macroscopic and microscopic solid bodies, and gases.
Meanwhile, another estimate for the mass of a cluster of galaxies had appeared in 1936, this time from Sinclair Smith, who had studied the Virgo Cluster. Assuming that the outer galaxies were in circular motion around Virgo, Smith calculated a total mass for the cluster of 1014 M⊙. When divided by the number of observed galaxies, 500, he found an average mass per galaxy of 2 × 1011 M⊙, which he pointed out was much higher than Hubble's estimate of 109 M⊙.
Much like Zwicky, whose 1933 work he cites, Smith considers this high value for the mass-per-galaxy implied by his calculations to be a problem, in particular in light of its incompatibility with Hubble's estimate. He also acknowledges, however, that both could be correct, and that:
the difference represents internebular material, either uniformly distributed or in the form of great clouds of low luminosity surrounding the [galaxies].
In his famous book The Realm of Nebulae, Hubble cites the work of Smith (and not that of Zwicky), and clearly states that he considers the discrepancy between the masses of galaxies inferred from the dynamics of clusters and those from the rotation of galaxies to be “real and important”. And although he argued that this problem might be solved, or at least diminished, by observing that the former were likely upper limits, while the latter lower limits, he acknowledged that this argument was not entirely satisfactory. A confusing situation had indeed arisen.
B. A Confusing Situation
There was no shortage of reasons for astronomers to be skeptical of the findings of Zwicky and Smith. The assumption that Virgo was a system in equilibrium, made by Smith, was questioned by Zwicky himself in his 1937 paper. In 1940, Erik Holmberg – who will appear again in this review as a pioneer of numerical simulations – described some of the concerns of the community regarding the work of Zwicky and Smith :
It does not seem to be possible to accept the high velocities [in the Virgo and Coma cluster] as belonging to permanent cluster members, unless we suppose that a great amount of mass – the greater part of the total mass of the cluster – is contributed by dark material distributed among the cluster members – an unlikely assumption.
Holmberg argued instead that these galaxies were probably “temporary” members of the cluster, i.e. galaxies on hyperbolic orbits that had fallen into the gravitational potential of the cluster, but were not bound to it. In 1954, Martin Schwarzschild  – son of the famous Karl Schwarzchild who had made important contributions to general relativity – attempted to get rid of “interlopers”, and inferred a smaller radial velocity dispersion of 630 km/s. By adopting an updated Hubble parameter, and an average luminosity-per-galaxy of 5 × 108 L⊙, he obtained the “bewildering high” mass-to-light ratio of 800. The distance, mass, luminosity, and mass-to-light ratio of the galaxies and clusters of galaxies compiled by Schwarzschild are shown in Fig. 1.
Figure 1. A snapshot of the dark matter problem in the 1950s: the distance, mass, luminosity, and mass-to-light ratio of several galaxies and clusters of galaxies, as compiled by M. Schwarzschild in 1954 .
By the late 1950s, a number of other articles had been published on the mass-to-light ratios of galaxy clusters. Victor Ambartsumian rejected the possibility that dark matter existed in clusters and argued instead that they are unstable and rapidly expanding systems, to which the virial theorem cannot be applied. It was soon realised, however, (e.g. Burbidge and Burbidge  and Limber ) that this interpretation was in tension with the estimated age of the galaxies (requiring clusters that were younger than the galaxies they contained), and with that of the Universe (the clusters should have evaporated long ago). In August of 1961, a conference on the instability of systems of galaxies was held in Santa Barbara, and included as participants some of the most important astrophysicists active in that field of research. Jerzy Neyman, Thornton Page and Elizabeth Scott summarized the discussions that took place around the mass discrepancy in galaxy clusters as follows:
Several possible explanations of this mass discrepancy were discussed at the Conference [..]. Many of those present consider that it might be real and due to invisible inter-galactic material in the clusters, totalling 90 to 99% of their mass. If these possibilities are excluded, however, the discrepancy in mass indicates positive total energy and instability of the system involved.
The overall situation was that of a community that was struggling to find a unified solution to a variety of problems. The dark matter hypothesis was not commonly accepted, nor was it disregarded. Instead, there was a consensus that more information would be needed in order to understand these systems.
In addition to the question of whether the dynamics of galaxy clusters required the presence of dark matter, astronomers around this time began to be increasingly willing to contemplate what this dark matter might be made of. Herbert Rood  (later confirmed by Simon White ) studied the relaxation process of galaxy clusters and argued that the mass responsible for their high mass-to-light ratios must to be found within the intergalactic space, and not in the galaxies themselves.
Arno Penzias searched for free hydrogen in the Pegasus I cluster and set an upper limit of a tenth of its virial mass . Neville Woolf suggested in 1967 that the gas could be ionised, and used radio, visible and X-ray observations to set limits on it . Turnrose and Rood discussed the problems of this hypothesis in Ref. , and in 1971 Meekins et al.  obtained observational evidence for X-ray emission that limited the amount of hot intracluster gas to be less than 2% of that required for gravitational binding.
With gas ruled out as an explanation for the “missing mass” in galaxy clusters, scientists began to explore more or less exotic possibilities, including massive collapsed objects , HI snowballs , and M8 dwarf stars . As we will discuss in Chapter V, these possibilities – and others like them – were eventually ruled out by measurements of the primordial light element abundances, which instead favor a non-baryonic nature for the dark matter.