Introduction: Dark Matter—what we do and do not know

How come we are so convinced of the existence of Dark Matter? Due to a multitude of reasons. In fact, we only observe Dark Matter through its gravitational effects—but this has multiple consequences. First, the motions of galaxies and whole galaxy clusters are influenced by the presence of Dark Matter. A famous example is the rotational velocity of spiral galaxies, as a function of the distance from their centre, from which one can infer their mass-to-light ratio. Without the existence of Dark Matter, with only visible matter being present, this rotational velocity should steeply increase when radially departing from the centre, and reach a maximum value before falling off. However, the data tell us a different story: instead of declining, the rotational velocity remains nearly constant after reaching its maximum, down to the outer arms of the galaxy. This behaviour can be explained by the presence of an invisible form of matter: Dark Matter. Further evidence arises from other observations, such as the motions of objects of different sizes, spanning from stars (studied by Jan Oort) to whole galaxy clusters (studied by Fritz Zwicky).

Yet, this alone would probably convince no real scientist: for example, we could think of a modification of the law of gravity on such large scales, which has indeed been proposed as an alternative explanation for the observed behaviour. However, in 2006, an observation of two colliding clusters of galaxies was made public, the so-called Bullet Cluster. This provided the ultimate game changer: the reconstruction of the Dark Matter distribution within the clusters, as achieved by gravitational lensing, clearly showed that Dark Matter is strongly favoured over any alternative explanations for the observed clash of galaxies, to the point that practically no scientist had serious doubts about its existence anymore.

The most precise determination of the amount of Dark Matter in the Universe is derived from the peaks in the spatial correlations of anisotropies in the cosmic microwave background, electromagnetic radiation from which was produced when the Universe was about 380 000 years old and which has filled it ever since. The first very precise determinations of the amount of Dark Matter have correspondingly been put forward by two microwave satellites a few decades ago, the Cosmic Background Explorer (COBE) and the Wilkinson Microwave Anisotropy Probe (WMAP). Nowadays, the 2015 data set from the Planck satellite [1], along with complementary observations such as the expansion rate of the Universe, constrain the Dark Matter abundance to be ${{\rm{\Omega }}}_{\mathrm{DM}}{h}^{2}=0.1188\pm 0.0010$, a value determined with remarkable precision. This means that the energy density of the Dark Matter today, compared to the total energy density of the Universe, is given by ${{\rm{\Omega }}}_{\mathrm{DM}}=25.89\%$, where h = 0.673 is the reduced Hubble constant, i.e. H₀ in units of 100 (km s⁻¹) Mpc⁻¹. We can say that about 1/4 of our current Universe consists of Dark Matter, with the rest made up by ordinary matter (4.86%) and dark energy (69.11%). Only a small fraction ($\lt 0.5\%$) is present in the form of radiation, i.e. photons and neutrinos.

Given the existence of Dark Matter, the natural follow-up question is what it consists of. Sticking to what we know, the most generic attempt is to explain it by something that we have already detected. Good candidates for massive objects that are not bright, in the sense that they do not produce radiation detectable by optical telescopes, would for example be big gas planets similar to Jupiter. Such objects, called MACHOs (MAssive Compact Halo Objects), are just a little too small for nuclear fusion to be triggered in their centres, so they allow 'hiding' a lot of mass inside them without producing visible light. However, such objects would not be fully invisible, since they block and influence visible light after all, so that they would leave imprints on the light of stars whose line of sight the MACHOs may cross. On the other hand, while MACHOs can still make up a fraction (but not all) of the Dark Matter present in galaxies, it seems very unlikely that they could be behind the effects of Dark Matter observed in the cosmic microwave background, since at that time the Universe was still too hot for baryons to form big bound structures such as MACHOs [3]. Similar arguments can be given against other attempts for an explanation of Dark Matter by astrophysical objects. Thus, other alternatives such as relatively light black holes, produced by some unknown mechanism in the early Universe, also seem to be unlikely [5].

There is a very natural possibility to nevertheless explain Dark Matter, though, in a way that does not seem challenged by any observation: it could be an elementary particle [2]. We have found many different elementary particles in Nature, and if any of them (or one that has not been found yet) could make up the Dark Matter, we may even be able to derive testable predictions and trigger experimental searches for this particular particle. It turns out that, in order to act as Dark Matter, a particle needs to tick several boxes. First, it needs to be electrically neutral (in fact, in order to be sufficiently abundant, at most weakly interacting), as otherwise it would interact too strongly with light and thus not be dark at all. Second, it should be massive, because we have observed Dark Matter by its gravitational effects. Third, and this is a very non-trivial requirement, the Dark Matter candidate particle has to have been produced in the early Universe in the right amount to make up a sizeable fraction (ideally all) of the Dark Matter observed, and the velocity or momentum distribution $f(p,T)$ of the particles must not feature too many particles with very large momenta, as this would have prevented galaxies from forming. Finally, Dark Matter needs to be stable enough to still be around today, or at least not have disappeared in large amounts.

In the Standard Model of elementary particle physics, no suitable particle exists. Out of all electrically neutral and massive particles, only neutrinos may possibly contribute to the Dark Matter, but their small mass and large velocities prevent them from being more than a small fraction of it. Fortunately, this picture changes once we extend the Standard Model by new physics. Historically, a breakthrough arose from an idea proposed by Lee and Weinberg in 1977 [6]: assuming that heavy stable neutrinos exist, they showed that such particles could easily be produced in the early Universe. At high temperatures T, when the Universe was very dense and small, such particles would be in 'thermal equilibrium' with all Standard Model particles; i.e. the Dark Matter particles would annihilate with a large rate into known matter, so large it even supersedes the expansion rate of the Universe. As the latter cools down, however, all species are diluted and, at some point, the particle density is so low that the Dark Matter particles no longer find any annihilation partners. Thus, given that they are stable, they have remained present in the Universe until today. This mechanism is called thermal freeze-out, and it can be used to produce a sizable amount of Dark Matter particles [4]. In order to meet the correct abundance, these particles typically need masses of a few 100 GeV. In modern language, particles with such properties are called WIMPs (Weakly Interacting Massive Particles). The fact that interaction strengths similar to that of the weak interaction are, by Dark Matter production, connected to the electroweak scale (where we anyway expect new physics to be present) is often dubbed the WIMP miracle.

While a WIMP may be quite a general category of particles, it is remarkable that several theories beyond the Standard Model do predict WIMP-candidates: for example, models based on supersymmetry generically feature an electrically neutral spin-1/2 Majorana fermion called the neutralino, which is in many cases stable and interacts roughly with weak-scale cross sections. Another option is theories based on extra spatial dimensions, which also predict particles with the characteristics of WIMPs. Note that, due to the details of the production process and the requirement of not producing too much Dark Matter, WIMPs typically act as what is called cold Dark Matter: they have non-relativistic speeds from the moment they are produced. Phenomenologically, WIMPs are interesting due to their potential to be discovered directly (by their interactions with nuclei), indirectly (by detecting their annihilation products), or even by producing them at particle colliders. A WIMP is a very good Dark Matter candidate indeed, with the only serious problem that, up to now, experiments have considerably narrowed down the viable parameter space for WIMPs without, however, a clear discovery (at most hints inconsistent with each other have been seen).

However, particle physicists do not run out of ideas easily. Many alternatives, non-WIMP Dark Matter candidates have been proposed. Some of these candidates have the practical problem that they are hardly testable—in the sense that, while they may exist and be the Dark Matter, they would be so feebly interacting that we may have no chance to ever detect them. While such an option is of course not logically forbidden, the only categories of particles we could possibly test are those with observable signatures. Thus, we would ideally like to link Dark Matter particles to something that we already know. In the case of WIMPs, that something is the weak interaction itself. There are, however, at least three other popular candidate particles that are very unlike WIMPs, but nevertheless connected to known sectors of physics: the axion is a very light and very feebly interacting spin-0 scalar that may be connected to the so-called strong CP problem; the spin-3/2 gravitino is the so far undiscovered supersymmetric partner of the graviton, the quantum field of gravitation; and the spin-1/2 sterile neutrino is connected to the existence of right-handed neutrino fields, which are singlets under the Standard Model gauge group and whose existence is ultimately linked to neutrinos being massive.

A sterile neutrino is an up-to-now hypothetical fermion that is a total singlet with respect to the Standard Model, i.e. it is perfectly neutral. Mathematically, it is introduced by adding so-called 'right-handed' neutrino fields to the Standard Model, which are singlets $({\bf{1}},{\bf{1}},0)$ under the full ${SU}{(3)}_{C}\times {SU}{(2)}_{L}\times U{(1)}_{Y}$ gauge group. Such a field can have a Majorana mass term, which implies that it is identical to its anti-particle. Furthermore, the mass of a sterile neutrino is not fixed by any principal requirement from the particle physics side. In particular, it is not tied to the electroweak scale in any way. However, a sterile neutrino mass eigenstate is not 100% sterile, because it mixes with ordinary (active) neutrinos. This admixture is what intimately connects sterile neutrino fields to the active neutrino sector and, as we will see, it is responsible for a lot of the important properties of sterile neutrinos.

It is this sterile neutrino as a serious candidate for Dark Matter that we will concentrate on in this book. As we will see, a sterile neutrino is a Dark Matter candidate that is somewhat peculiar, with properties very different from those of WIMPs. These new properties have a variety of consequences, which on the one hand make sterile neutrinos harder to observe, but which also connect them to other fields and ultimately yield testable predictions beyond direct detection. Also, from a cosmological point of view, sterile neutrinos do have new implications, starting from their production mechanisms generating non-thermal momentum distributions (see figure 1.1) to their non-standard predictions for the formation of structures in the Universe. We will also see how the connection of Dark Matter sterile neutrinos to active neutrinos could yield a potentially observable experimental signature of these particles, and where else they could have an influence. Hopefully—after having worked through this book—you, dearest reader, will appreciate why sterile neutrinos are very good Dark Matter candidates and what we can learn from them about one of the biggest questions of our Universe.

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