This chapter describes two problems initially studied in the 1930s, a few years

after the discovery of galaxies. In later years, they were revisited and, a posteriori,

appeared to be important precursors of the problem of dark matter, which

in the meantime had emerged clearly from the study of the rotation curves of

spiral galaxies. In reality, the early studies of clusters of galaxies

and of the vertical dynamics of the solar neighborhood had failed to provide

decisive evidence for the existence of dark matter. In fact, for both cases significant

confusion remained until the end of last century. Then, with the advent

of fresh data from the Hipparcos mission, it was proved that from the vertical

dynamics of the solar neighborhood there is no evidence for the presence

of dark matter in the disk. In turn, with the combined use of improved X-ray

observations and of gravitational lensing clusters of galaxies were eventually

shown to be associated with large amounts of dark matter, which, in contrast to the case of galaxy halos, appears to be more concentrated with respect to the distribution of visible matter. The closing dynamical section describes the properties of the self-gravitating slab model, often used to describe the local properties of galaxy disks.

The study of the universe at very short wavelengths, in the domain of X-rays

and gamma rays, started immediately after the beginning of the space age,

with the launch in 1957 of the first artificial satellite, Sputnik 1. Three extraordinary discoveries mark the birth of high energy astrophysics: X-ray binaries (and the interesting mass exchange between normal stars and compact objects), the IntraCluster Medium (a very high temperature plasma), and Gamma Ray Bursts (huge explosions occurring frequently at cosmological distances). The final section of the chapter is devoted to a brief description of the definition of fluid models. After a brief summary of the main picture, the text describes the concept of hydrostatic equilibrium, which plays a key role in the study of the hot diffuse medium in clusters of galaxies and in diagnosing the amount and distribution of dark matter in those systems. Then the opposite regime, of rotation-dominated inviscid disks, is considered, which is at the basis of the modeling of rotation curves in galaxies. Finally, the general framework of the fluid description in the presence of viscosity, which is associated with the picture of accretion disks, is presented.

The second variant of Milgrom’s theory (T1) derives the equations of motion from a Lagrangian formulation, allowing the theory to make predictions for a very general class of systems with arbitrary shapes. A number of confirmed, novel predictions follow from T1 including the “central surface density relation” (CSDR) and a prediction about the form of the vertical force law in the Milky Way. The theory also predicts a very low rate of mergers between galaxies, a prediction that may or may not be consistent with observations. However, the theory’s predictions about the kinematics of the largest bound structures in the universe, the galaxy clusters, appear to be incorrect.

The fourth variant of Milgrom’s theory (T3) is due to Justin Khoury and Lasha Berezhiani. These researchers made a change to Milgrom’s postulates in order to address the failings of theory variants T0–T2. They introduced dark matter into the theory, but postulated that, on scales corresponding to galaxies, the dark matter acts like a superfluid condensate. By adjusting the equation of state of the superfluid, they showed that the interaction of the dark matter with normal matter – via Newtonian gravity, but also via coupling of the superfluid phonons – can result in an acceleration that approximates Milgrom’s modified dynamics in the low-acceleration regime. On large scales, the dark matter would behave like the dark matter in the standard model. Although still in an early stage of development, Khoury and Berezhiani’s theory has not yet demonstrated an increase in testable content compared withT0–T2, and in fact much of the content of the earlier theory variants is lost.