2 The Missing Mass Problem in a Nutshell

There exists overwhelming evidence for mass discrepancies in the Universe from multiple independent observations. This evidence involves the dynamics of extragalactic systems: the motions of stars and gas in galaxies and clusters of galaxies. Further evidence is provided by gravitational lensing, the temperature of hot, X-ray emitting gas in clusters of galaxies, the large scale structure of the Universe, and the gravitating mass density of the Universe itself (Figure 1View Image). For an exhaustive historical review of the problem, we refer the reader to [394Jump To The Next Citation Point].

View Image

Figure 1: Summary of the empirical roots of the missing mass problem (below line) and the generic possibilities for its solution (above line). Illustrated lines of evidence include the approximate flatness of the rotation curves of spiral galaxies, gravitational lensing in a cluster of galaxies, and the growth of large-scale structure from an initially very-nearly–homogeneous early Universe. Other historically-important lines of evidence include the Oort discrepancy, the need to stabilize galactic disks, motions of galaxies within clusters of galaxies and the hydrodynamics of hot, X-ray emitting gas therein, and the apparent excess of gravitating mass density over the mass density of baryons permitted by Big-Bang nucleosynthesis. From these many distinct problems grow several possible solutions. Generically, the observed discrepancies either imply the existence of dark matter, or the necessity to modify dynamical laws. Dark matter could, in principle, be any combination of non-luminous baryons and/or some non-baryonic form of mass-like neutrinos (hot dark matter) or some new particle, whose mass makes it dynamically cold or perhaps warm. Alternatively, the observed discrepancies might point to the need to modify the equation of gravity that is employed to infer the existence of dark matter, or perhaps some other fundamental dynamical assumption like the equivalence of inertial mass and gravitational charge. Many specific ideas of each of these types have been considered over the years. Note that none of these ideas are mutually exclusive, and that some form or the other of dark matter could happily cohabit with a modification of the gravitational law, or could even be itself the cause of an effective modification of the gravitational law. Question marks on some tree branches represent the fruit of ideas yet to be had. Perhaps these might also address the dark energy problem, with the most satisfactory result being a theory that would simultaneously explain the acceleration scale in the dark matter problem as well as the accelerating expansion of the Universe, and explain the coincidence of scales between these two problems, a coincidence exhibited in Section 4.1.

The data leave no doubt that when the law of gravity as currently known is applied to extragalactic systems, it fails if only the observed stars and gas are included as sources in the stress-energy tensor. This leads to a stark choice: either the Universe is pervaded by some unseen form of mass – dark matter – or the dynamical laws that lead to this inference require revision. Though the mass discrepancy problem is now well established [394, 465], such a dramatic assertion warrants a brief review of the evidence.

Historically, the first indications of the modern missing mass problem came in the 1930s shortly after galaxies were recognized to be extragalactic in nature. Oort [342] noted that the sum of the observed stars in the vicinity of the sun fell short of explaining the vertical motions of stars in the disk of the Milky Way. The luminous matter did not provide a sufficient restoring force for the observed stellar vertical oscillations. This became known as the Oort discrepancy. Around the same time, Zwicky [518Jump To The Next Citation Point] reported that the velocity dispersion of galaxies in clusters of galaxies was far too high for these objects to remain bound for a substantial fraction of cosmic time. The Oort discrepancy was approximately a factor of two in amplitude, and confined to the Galactic disk – it required local dark matter, not necessarily the quasi-spherical halo we now envision. It was long considered a serious problem, but has now largely (though perhaps not fully) gone away [194Jump To The Next Citation Point, 240]. The discrepancy Zwicky reported was less subtle, as the required dark mass outweighed the visible stars by a factor of at least 100. This result was apparently not taken seriously at the time.

One of the first indications of the need for dark matter in modern times came from the stability of galactic disks. Stars in spiral galaxies like the Milky Way are predominantly on approximately circular orbits, with relatively few on highly eccentric orbits [132]. The small velocity dispersion of stars relative to their circular velocities makes galactic disks dynamically cold. Early simulations [343Jump To The Next Citation Point] revealed that cold, self-gravitating disks were subject to severe instabilities. In order to prevent the rapid, self-destructive growth of these instabilities, and hence preserve the existence of spiral galaxies over a sizable fraction of a Hubble time, it was found to be necessary to embed the disk in a quasi-spherical potential well – a role that could be played by a halo of dark matter, as first proposed in 1973 by Ostriker & Peebles [343Jump To The Next Citation Point].

Perhaps the most persuasive piece of evidence was then provided, notably through the seminal works of Bosma and Rubin, by establishing that the rotation curves of spiral galaxies are approximately flat [67Jump To The Next Citation Point, 370Jump To The Next Citation Point]. A system obeying Newton’s law of gravity should have a rotation curve that, like the Solar system, declines in a Keplerian manner once the bulk of the mass is enclosed: Vc ∝ r−1∕2. Instead, observations indicated that spiral galaxy rotation curves tended to remain approximately flat with increasing radius: Vc ∼ constant. This was shown to happen over and over and over again [370] with the approximate flatness of the rotation curve persisting to the largest radii observable [67], well beyond where the details of each galaxy’s mass distribution mattered, so that Keplerian behavior should have been observed. Again, a quasi-spherical halo of dark matter as proposed by Ostriker and Peebles was implicated.

Other types of galaxies exhibit mass discrepancies as well. Perhaps most notable are the dwarf spheroidal galaxies that are satellites of the Milky Way [427Jump To The Next Citation Point, 477Jump To The Next Citation Point] and of Andromeda [217]. These satellites are tiny by galaxy standards, possessing only millions, or in the case of the ultrafaint dwarfs, thousands, of individual stars. They are close enough that the line-of-sight velocities of individual stars can be measured, providing for a precise measurement of the system’s velocity dispersion. The mass inferred from these motions (roughly, M ∼ rσ2∕G) greatly exceeds the mass visible in luminous stars. Indeed, these dim satellite galaxies exhibit some of the largest mass discrepancies observed. In contrast, bright giant elliptical galaxies (often composed of much more than the 11 ∼ 10 stars of the Milky Way) exhibit remarkably modest and hard to detect mass discrepancies [367Jump To The Next Citation Point]. Thus, it is inferred that fainter galaxies are progressively more dark-matter dominated than bright ones. However, as we shall expand on in Section 4.3, the primary correlation is not with luminosity, but with surface brightness: the lower the surface brightness of a system, the larger its mass discrepancy [279Jump To The Next Citation Point].

On larger scales, groups and clusters of galaxies also show mass discrepancies, just as individual galaxies do. One of the earliest lines of evidence comes from the “timing argument” in the Local Group [213]. Presumably the material that was to become the Milky Way and Andromeda (M31) was initially expanding apart with the general Hubble expansion. Currently they are approaching one another at −1 ∼ 100 km s. In order for the Milky Way and M31 to have overcome the initial expansion and fallen back towards one another, there must be a greater-than-average gravitating mass between the two. To arrive at their present separation with the observed blueshifted line of sight velocity after a Hubble time requires a dynamical mass-to-light ratio M ∕L > 80. This greatly exceeds the mass-to-light ratio of the stars themselves, which is of order unity in Solar units [42Jump To The Next Citation Point] (the Sun is a fairly average star, so averaged over many stars each Solar mass produces roughly one Solar luminosity).

Rich clusters of galaxies are rare structures containing dozens or even hundreds of bright galaxies. These objects exhibit mass discrepancies in several distinct ways. Measurements of the redshifts of individual cluster members give velocity dispersions in the vicinity of 1,000 km s−1 typically implying dynamical mass-to-light ratios in excess of 100 [24Jump To The Next Citation Point]. The actual mass discrepancy is not this large, as most of the detected baryonic mass in clusters is in a diffuse intracluster gas rather than in the stars in the galaxies (something Zwicky was not aware of back in 1933). This gas is heated to the virial temperature and emits X-rays. Mapping the temperature and emission of this X-ray gas provides another probe of the cluster mass through the equation of hydrostatic equilibrium. In order to hold the gas in the clusters at the observed temperatures, the dark matter must outweigh the gas by a factor of ∼ 8 [175Jump To The Next Citation Point]. Furthermore, some clusters are observed to gravitationally lens background galaxies (Figure 1View Image). Once again, mass above and beyond that observed is required to explain this phenomenon [227]. Thus, three independent methods all imply the need for about the same amount of dark matter in clusters of galaxies.

In addition to the abundant evidence for mass discrepancies in the dynamics of extragalactic systems, there are also strong motivations for dark matter in cosmology. Two observations are particularly important: (i) the small baryonic mass density Ωb inferred from Big-Bang nucleosynthesis (BBN) (and from the measured Hubble parameter), and (ii) the growth of large scale structure by a factor of 5 ∼ 10 from the surface of last scattering of the cosmic microwave background at redshift z ∼ 1000 until present-day z = 0, implying Ωm > Ωb. Together, these observations imply not only the need for dark matter, but for some exotic new form of non-baryonic cold dark matter. Indeed, observational estimates of the gravitating mass density of the Universe Ωm, measured, for instance, from peculiar galaxy (or large-scale) velocity fields, have, for several decades, persistently returned values in the range 1∕4 < Ωm < 1 ∕3 [116Jump To The Next Citation Point]. While shy of the value needed for a flat Universe, this mass density is well in excess of the baryon density inferred from BBN. The observed abundances of the light isotopes deuterium, helium, and lithium are consistent with having been produced in the first few minutes after the Big Bang if the baryon density is just a few percent of the critical value: Ωb < 0.05 [480Jump To The Next Citation Point, 107Jump To The Next Citation Point]. Thus, Ωm > Ωb. Consequently, we do not just need dark matter, we need the dark matter to be non-baryonic.

Another early Universe constraint is provided by the Cosmic Microwave Background (CMB). The small (microKelvin) amplitude of the temperature fluctuations at the time of baryon-photon decoupling (z ∼ 1000) indicates that the Universe was initially very homogeneous, roughly to one part in 105. The Universe today (z = 0) is very inhomogeneous, at least on “small” scales of less than ∼ 100 Mpc (∼ 3 × 108 ly), with huge density contrasts between planets, stars, galaxies, clusters, and empty intergalactic space. The only attractive long-range force acting on the entire Universe, that can make such structures, is gravity. In a rich-get-richer while the poor-get-poorer process, the small initial over-densities attract more mass and grow into structures like galaxies while under-dense regions become less dense, leading to voids. The catch is that gravity is rather weak, so this process takes a long time. If the baryon density from BBN is all we have to work with, we can only obtain a growth factor of 2 ∼ 10 in a Hubble time [424], orders of magnitude short of the observed 5 10. The solution is to boost the growth rate with extra invisible mass displaying larger density fluctuations: dark matter. In order not to make the same mark on the CMB that baryons would, this dark matter must not interact with the photons. So, in effect, the density fluctuations in the dark matter can already be very large at the epoch of baryon-photon decoupling, especially if the dark matter is cold (i.e., with effectively zero Jeans length). The baryons fall into the already deep dark matter potential wells only after that, once released from their electromagnetic link to the photon bath. Before decoupling, the fluctuations in the baryon-photon fluid did not grow but were oscillating in the form of acoustic waves, maintaining the same amplitude as when they entered the horizon; actually they were even slightly diffusion-damped. In principle, at baryon-photon decoupling, CMB fluctuations on smaller angular scales, having entered the horizon earlier, would have been damped with respect to those on larger scales (Silk damping). Nevertheless, the presence of decoupled non-baryonic dark matter would provide a net forcing term countering the damping of the oscillations at recombination, meaning that the second and third acoustic peaks of the CMB could then be of equal amplitude rather than exhibiting a damping tail. The actual observation of a high third-peak in the CMB angular power spectrum is another piece of compelling evidence for non-baryonic dark matter (see, e.g., [229Jump To The Next Citation Point]). Both BBN and the CMB thus drive us to consider a form of mass that is non-baryonic and which does not interact electromagnetically. Moreover, in order to form structure (see Section 3.2), the mass must be dynamically cold (i.e., moving much slower than the speed of light when it decouples from the photon bath), and is known as cold dark matter (CDM).

Now, in addition to CDM, modern cosmology also requires something even more mysterious, dubbed dark energy. The fact that the baryon fraction in clusters of galaxies was such that Ωm was implied to be much smaller than 1 – the value needed for a flat Euclidean Universe favored by inflationary models – , as well as tensions between the measured Hubble parameter and independent estimates of the age of the Universe, led Ostriker & Steinhardt [344Jump To The Next Citation Point] to propose in 1995 a “concordance model of cosmology” or ΛCDM model, where a cosmological constant Λ – supposed to represent vacuum energy or dark energy – provided the major contribution to the Universe’s energy density. Three years later, the observations of SNIa [351Jump To The Next Citation Point, 365Jump To The Next Citation Point] indicating late-time acceleration of the Universe’s expansion, led most people to accept this model. This concordance model has since been refined and calibrated through subsequent large-scale observations of the CMB and of the matter power spectrum, to lead to the favored cosmological model prevailing today (see Section 3). However, as we shall see, curious coincidences of scales between the dark matter and dark energy sectors (see Section 4.1) have prompted the question of whether these two sectors are really physically independent, and the existence of dark energy itself has led to a renewed interest in modified gravity theories as a possible alternative to this exotic fluid [100Jump To The Next Citation Point].

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