3. The dark matter hypothesis

The dark matter hypothesis was essentially established in 1974 by Ostriker et al. [6], who concluded that the rotation curves of spiral galaxies could most plausibly be understood if the spiral galaxy was embedded in a giant spherical halo of invisible "dark matter." In the conventional cosmological model of spiral galaxies [9], each spiral galaxy is considered to be surrounded by a giant halo of invisible (dark) matter that provides a large contribution to the gravitational field at large distances from the center of the galaxy.

Standard model of cosmology [10] assumes that the universe is now composed of about 5% ordinary matter, 27% dark matter, and 68% dark energy, so that dark matter constitutes about 84% of the total mass, while dark energy plus dark matter constitute about 95% of the total mass-energy content of the universe. Thus, for many years, cosmologists have been confronted with the notion that 84% of the gravitational mass of the universe is dark matter.

The hypothesis of a dark matter spherical halo surrounding a spiral galaxy to account for the observed flat rotation curve of the galaxy has yet to be verified. One of the main difficulties is that the nature of the proposed dark matter is unknown.

Initially, massive compact halo objects (MACHOs), were searched for within the outer regions of galaxies, using microlensing techniques [11]. The conclusion from these observations was that at most 20% of a galactic halo consists of MACHOs, and the rest of the halo consists of nonbaryonic matter.

The only other known candidates for dark matter are the three neutrinos of the standard model (SM) of particle physics [12]. However, it was demonstrated in 1983 [13] that if dark matter consisted entirely of neutrinos, the large-scale structure of the universe would significantly differ from the observed one, since the neutrinos are relativistic particles leading to a smooth large-scale structure. Recently, Frampton [14] has suggested that the nonbaryonic component of dark matter may consist entirely of primordial intermediate mass black holes. However, this suggestion remains to be verified.

The existence of dark matter in the universe suggests that one requires new physics beyond the SM. Three such particles have been searched for without success: (1) axions, (2) weakly interacting massive particles (WIMPS), and (3) sterile neutrinos. These three particles are all hypothetical particles, some of which have been introduced into particle physics in order to resolve certain perceived problems.

The axion was postulated in 1977 by Peccei and Quinn [15] in an attempt to understand the strong CP problem in quantum chromodynamics (QCD). To date various experiments have been carried out but none have successfully identified an axion particle.

A weakly interacting massive particle (WIMP) is considered to be a new elementary particle, which only interacts via gravity and any other weak force. The basic goal of direct detection of a (WIMP) is to measure the energy deposited when it interacts with nuclei in a detector, transferring energy to nuclei. Such direct-detection experiments need to be carried out deep underground to prevent them being swamped by unwanted noise from cosmic ray particles.

The most favored (WIMP) is the lightest neutral stable particle, the neutralino, predicted by the supersymmetric (SUSY) theory of particle physics, which provides a significant relationship between elementary bosons and fermions. This relationship resolves several puzzling problems, including the hierarchy problem, for example, the extremely large difference in the strengths of the gravitational and weak interactions ≈ 10˜36. However, to date, no evidence for any SUSY particle has been found either at the large Hadron collider (LHC) in CERN or in the many underground detection laboratories. At the LHC, no previously unknown particles, which may be evidence of SUSY, have been observed since the claimed detection of the Higgs boson, so that SUSY probably does not exist. In addition, no (WIMP) has clearly been detected over several decades at any of the underground laboratories such as the large underground xenon (LUX) experiment in the Homestake Mine, Dakota.

However, there has been one claim of direct detection of dark matter from the DAMA-LIBRA experiment at the Gran Sasso laboratory [16]. This experiment has observed a possible dark matter event rate that modulates annually as the Earth travels around the Sun, while the Solar System moves within the disk of the Milky Way and hence through the hypothesized galactic dark matter halo. The count rate is expected to depend upon the relative velocity of the detector and undergoes a modulation that peaks in June, when the relative velocity is at its maximum.

This observation of the DAMA-LIBRA experiment is controversial, since it has been excluded by observations from several direct-detection experiments, including perhaps the most sensitive one, the LUX experiment. In order to test the DAMA-LIBRA claim, a more sensitive directdetection experiment, SABRE, is being undertaken with improved but similar equipment in Australia [17].

Sterile neutrinos are also hypothetical neutral particles that emerged from the development of the electroweak theory by Glashow [18, 19], who separated the neutrinos into left-handed and right-handed particles. The left-handed neutrinos interact via the left-handed weak interaction, while the right-handed neutrinos do not and only interact via gravity. The right-handed neutrinos correspond to the so-called sterile neutrinos.

The possible existence of sterile neutrinos arose in the development of the SM at a time when the neutrinos were considered to be massless. This is no longer the case so that the three "normal" neutrinos are expected to have right-handed components with the same mass as the left-handed components and hence are unsuitable as candidates for dark matter.

## 4. Milgrom MOND theory

In view of the considerable uncertainties concerning the existence and nature of the proposed dark matter, there have been several attempts to modify Newton's universal law of gravitation instead of introducing dark matter.

In 1983, Milgrom [20] developed a modification of Newtonian dynamics known as the MOND theory, as a possible alternative to dark matter. This theory is based upon describing two astronomical observations: (1) the flat rotational curves of spiral galaxies at large distances from their centers and (2) the Tully-Fisher empirical relation [21], which states that the intrinsic luminosity L (proportional to the total visible mass) of a spiral galaxy and the velocity, vf , of the matter circulating at the extremities of the galactic disks are given by:

$$L \propto \upsilon\_{f'}^{a} \tag{1}$$

where α is approximately 4.

In order to describe both the flat rotation curves of spiral galaxies and the Tully-Fisher relation, Milgrom suggested that gravity varies from the prediction of Newtonian dynamics for low accelerations. In particular, the transition from 1=r<sup>2</sup> to 1=r gravity should occur below a critical "acceleration" a<sup>0</sup> rather than beyond a critical distance r0: the former leads to the Tully-Fisher relation, while the latter leads to:

$$L \propto v\_{f'}^2 \tag{2}$$

in gross disagreement with the Tully-Fisher relation.

The modified law of gravity in terms of a<sup>0</sup> is [22]:

$$\mathbf{g} = \mathbf{G}\mathbf{M}/r^2 + (\mathbf{G}\mathbf{M}\mathbf{a}\_0)^{\frac{1}{2}}/r,\tag{3}$$

where the first term corresponds to distances for which the acceleration is ≫ a<sup>0</sup> and the second term corresponds to distances associated with the flat rotation curves, that is, with vf . Indeed the second term gives:

$$w\_f = (GMa\_0)^{\frac{1}{4}},\tag{4}$$

which, if the mass to luminosity ratio, M=L, is roughly constant for galaxies, leads to the Tully-Fisher relation.

To summarize: MOND is an empirical modification of Newton's gravitational interaction that is designed to provide agreement with two overarching observational facts: (1) the flat rotation curves of spiral galaxies and (2) the Tully-Fisher relation. It achieves this aim by causing the gravitational interaction to change from 1=r<sup>2</sup> for small distances, r, to 1=r at large galactic distances as the gravitational acceleration becomes less than a critical small acceleration <sup>a</sup><sup>0</sup> <sup>≈</sup> 1.2 � <sup>10</sup>�<sup>10</sup> m s�2.

However, MOND is incomplete in the sense that in order to be more acceptable to the overall scientific community, it needs to be related to a more general underlying theory of gravity. Just as Kepler's laws of planetary motion described mathematically but without any physical content the observed orbits of the planets, it required Newton's universal law of gravitation to understand the physics underlying Kepler's laws.
