Evidences for dark matter
The idea that a part of the matter in the Universe may escape the observations because they do not emit light or are just too dim to see was already seriously considered at the beginning of the twentieth century. In 1904, Lord Kelvin hypothesized the existence of dark stars in the Milky Way possibly not bright enough to be directly observable. Considering that the stars in our galaxy are acting like molecules in a gas, Kelvin realized there was a way to calculate the total mass of luminous and non-luminous matter from the velocity dispersion of the stars . This method was, however, later reconsidered by Henri Poincaré whose final calculation showed that the observed velocity dispersion was in agreement with the mass of luminous matter . It wasn’t until the beginning of the 1930’s that a real discrepancy between the total mass and the luminous mass of an astrophysical system was measured. From this point, the observational evidences for dark matter multiplied at various astrophysical and cosmological scales.
Local Dark Matter
The first attempts to probe the existence of a population of dark astrophysical objects via their gravitational interactions with luminous matter were, in fact, not the most successful. They were, nevertheless, the first steps to great discoveries. It began with the observations of the stars in the sun vicinity (i.e. stars at a distance 0.1 – 1 kpc from the sun). In 1915, Estonian astronomer Ernst Öpik calculated the total local mass density by measuring the vertical oscillations of those stars . He eventually found that the motion of the stars could be explained by the mass of luminous matter alone and that there were no need to assume the existence of dark matter. Improved analyses were later carried out by Kapteyn  and Jeans  who found a total local density of 0.099 M pc−3 and 0.143 M pc−3. Similarly, in 1932, Jan Hendrick Oort, Kapteyn’s student, found a local density of 0.092 M pc−3 . None of them found an excessive amount of dark matter and Oort argued that by taking into account the expected number of white dwarves, his result was coherent with the contribution of ordinary matter alone.
It appears, actually, that the main source of uncertainties in the calculation of the amount of dark matter does not come from the total density measurement, which is quite coherent between the various analyses, but comes from the estimation of the amount of luminous matter. More precise estimations were recently done and the local density of dark matter is now believed to be around 0.008 M pc−3 (0.3 GeV cm−3), with large uncertainties. One can refer to the review on local dark matter density by J.I. Read for further information .
While the study of vertical oscillations of stars in the Sun’s vicinity is not the most convincing evidence of the existence of dark matter, the precise measurement of local DM density is crucial for DM direct detection (see Section 1.3.2). The first really challenging measurements came, in fact, from the observations of galaxy clusters.
One of the first striking evidences for dark matter came from the Swiss astronomer Fritz Zwicky in 1933 . At that time, Zwicky was carrying out a project in Mount Wilson concerning the measurement of galaxy cluster distances via the spectral red-shift related to the expansion of the Universe. While observing the Coma galaxy cluster, he measured the velocity of eight galaxies and deduced an approximate value of the total mass of the cluster using the virial theorem. This value was 400 times greater than the one expected by summing the masses of luminous objects. The discrepancy between the mass of luminous objects and the mass calculated via the Newtonian law of gravity lead him to suggest the existence of a non-luminous type of matter composing the cluster which he referred to as dunkle Materie or “dark matter”. His calculations should, however, be reviewed as he took a value of the Hubble constant H0 = 558 km s−1 Mpc−1. The current value of H0 is now ≈ 70 km s−1 Mpc−1, therefore the overdensity of 400 should be reduced to 50. Nevertheless, the conclusion that the majority of the matter in the cluster must be dark remains relevant.
Three years later, a similar study lead by Sinclair Smith showed that the mass of the Virgo cluster was 200 times larger than expected , giving weight to Zwicky’s hypothesis. In 1959, Kahn and Woltjer calculated the mass of the Local Group from the motion of Andromeda towards the Milky Way . They found that the Local Group was six times more massive than the observed luminous matter and suggested that the missing mass was composed of very hot gas in the intergalactic medium. At this time, it was not yet considered that dark matter cannot be composed of ordinary particles.
Figure 1.1: A purple haze shows dark matter flanking the « Bullet Cluster. » X-rays observations
of hot gas is represented in pink. Image Credit: X-ray: NASA/CXC/M.Markevitch et al. Op-tical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al. Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.
Today, the most convincing evidence of the existence of dark matter is based on the observation of the Bullet Cluster (1E 0657-558), which consists in two colliding galaxy clusters at a co-moving radial distance of approximately one giga-parsec from us . Its observation in X-rays reveals hot gas, represented in pink in figure 1.1, which forms a distinct arc characteristic of a shock-wave. It is also possible to map the distribution of mass in the cluster by studying gravitational lensing. Gravitational lensing is a phenomenon related to General Relativity. The bullet cluster, thanks to its mass, deforms space-time around it. If a luminous object, such as a galaxy, is located behind the Bullet Cluster in the line of sight, the space-time deformation bends the trajectory of the light emitted by the galaxy. This effect allows us to observe the galaxy, but its image is slightly curved. The study of this curvature is key to calculate the mass distribution of the Bullet Cluster, which is represented by a purple haze in figure 1.1. One can notice that the location of mass is clearly decorrelated from the position of hot gas in the Bullet Cluster. That can be easily explained if one considers that the cluster is mainly composed of gas and dark matter. Ordinary matter, such as the gas suffers from friction due to electromagnetic interaction. Therefore, when the two clusters collided, ordinary matter tended to stay at the position of collision. This is not the case for dark matter, which is not stopped by frictions. When the two clusters collided, dark matter continued its motion by inertia. This explains why the mass, mainly carried by dark matter, is separated from the gas in the Bullet Cluster. This observation was extremely important as it challenged significantly models of modified gravity.
Another remarkable evidence that dark matter exists comes from the study of the motion of lu-minous matter in the periphery of spiral galaxies. The first obvious specimen for this kind of observations is our neighbour, the Andromeda galaxy (M 31). In 1939, American astronomer Ho- race W. Babcock obtained the spectra of M31 and deduced the rotational velocity of the different regions of the galaxy, up to ≈ 20 kpc from the center of the galaxy . He found that outer re-gions had an unexpectedly high velocity compared to the Keplerian velocity calculated from the luminous mass of the galaxy. He explained this either by the presence of poorly luminous matter in the outer region of the galaxy or by strong dust absorption. Similar unexpected results were obtained by Oort when studying galaxy NGC 3115 .
Results on the Andromeda galaxy were improved after WWII, when Oort and his team realized that German radars could be rehabilitated into radio-telescopes. They discovered that neutral hy-drogen gas emitted interesting radio waves at 21 cm wavelength. The 21-cm line is now one of the most important astrophysical probe. Oort’s student, Van de Hulst, was then able to measure the rotational velocity of hydrogen gas up to 30 kpc from the center of Andromeda galaxy, improving previous analyses .
According to the Big Bang model, the Universe starts with an extremely hot and dense state of matter. During the first seconds of the Universe, the very high temperatures prevent any group of particles from bonding. For instance, helium nuclei formed through the fusion of two protons and one or two neutrons are immediately destroyed by collisions with high energetic photons. However, as the Universe expands, the photons lose energy and are not able to break the nuclei anymore. The first light nuclei (helium-3, helium-4, lithium and beryllium), can thus be stably produced. Approximatively 20 minutes after the Big Bang, the production of light nuclei freezes-out as the temperature becomes too low for the production process to occur. The abundances of light elements have not evolved since and remain observable today. This phenomenon is called Big Bang Nucleosynthesis (BBN) and first was described by Alpher, Bethe and Gamow in 1948 . It constitutes a crucial source of information in cosmology and allows us to constrain the nature of dark matter, as we will see in section 1.2.1.
After BBN, the electrons and nuclei are still decoupled because of the photo-dissociation. Thus, photons have very short mean free paths as they scatter on electrons and nuclei and are trapped in the very hot and dense gas. It is only around 380 000 years after the Big Bang that the Universe becomes cool enough for the electrons and the nuclei to finally couple. This is the recombination. Atoms are formed and photons can eventually escape and propagate through the whole Universe. This light is still observable as a nearly-perfect black body radiation with a temperature of 2.7K, homogeneous and isotropic through the Universe. The Cosmic Microwave Background (CMB), as it is called, was first observed by Penzias and Wilson in 1965  and is now a pillar of cosmology. Three main space telescopes were launched in order to study the CMB, starting with the Cosmic Background Explorer (COBE) in 1989. It revealed tiny fluctuations in the CMB which can be explained by overdensities at the epoch of recombination . Those overdensities are believed to be quantum fluctuations appearing at the very beginning of the Universe which grew into large structures, such as clusters of galaxies and galaxies, by attracting matter thanks to their gravita-tional potential. High precision measurements of the fluctuations followed with space-telescopes WMAP  and Planck . The analysis of the angular correlation of these fluctuations is key to constraining cosmological parameters. The power spectrum associated to these correlations displays multiple peaks (see figure 1.3). While the position of the first peak reveals information on the total energy density ρtot of the Universe, the position of the second peak allows us to con-strain non-baryonic dark matter density ρD M .
Without dark matter, the matter could not have collapsed in time to form these current structures. As baryonic matter interacts through electromagnetism, the resulting pressure in the hot primordial gas slows down gravitational collapse. This is not the case for cold dark matter which can easily form clusters and then accrete ordinary matter. In order to reproduce all the features of large structures in numerical simulations, it is also needed that dark matter be non-relativistic at the time of the formation of large structures. From this result, it is possible to rule out neutrinos as dark matter candidates .
In the next section, we will see more specifically what are the possible candidates for dark matter.
Dark matter candidates
Numerous models have been built over the past decades in order to describe the nature of dark matter. As it would be too long and tedious to describe them all, I will focus only on two types of model and present a few examples at the end of this section. The first model is the most natural approach as it postulates that dark matter consists in feebly luminous astrophysical ob-jects, named MACHOs. This model is, however, severely constrained. The second one, the WIMP hypothesis, is one of the most popular models in particle physics. In this model, dark matter is described as a weak interacting massive particle (WIMP) beyond the Standard Model. It will be at the center of the rest of my Ph.D. thesis.
Massive Astrophysical Compact Halo Objects (MACHOs)
One of the most immediate answers to the question of the nature of dark matter is that it should be composed of ordinary matter too dim to be observed. Compact astrophysical objects such as brown dwarves, red dwarves, white dwarves, neutron stars or black holes are very difficult to ob-serve via their emission of light and could be excellent candidates for dark matter. Those kinds of objects are commonly named Massive Astrophysical Compact Halo Objects (MACHOs). Several evidences are, however, suggesting that MACHOs could only compose a small fraction of dark matter.
The first evidence involves micro-gravitational lensing. This phenomenon occurs when a massive object lies in the line of sight of a star. If the massive object is compact enough, its gravitational field will have for effect to enhance the apparent luminosity of the star behind it. In 1986 Bo-hdan Paczynski´ proposed a method to detect MACHOs in the halo of the Milky Way , which was, one year later, used with more detailed calculations in the Ph.D. thesis of Robert Nemirof . The method involves the observation of a nearby galaxy, such as the Magellanic Clouds. If the halo of the Milky Way is entirely composed of MACHOs, they calculated that at any time, any star in the Magellanic Clouds would have a probability of about one out of a million to be magnified by the gravitational field of a MACHO. If a large amount of stars are monitored in the Magellanic Clouds, it would therefore be possible to estimate the number of MACHOs in the Milky Way halo. In addition, it would give limits on the mass of the compact objects as the du-ration of a microlensing event is a function of the mass t ∼ 130days × MM . From this relation, one can deduce that with such a method, only MACHOs with masses ranging from ∼ 10−7 M to ∼ 102 M would be easily observable, which corresponds to times of observation ranging from a couple of hours to a few years.
A project, simply called MACHO, was dedicated to this task. In 2000, after 5.7 years of observa-tions of the Large Magellanic Cloud (LMC), using the 1.27-meter telescope at Mount Stromlo Ob-servatory, the MACHO Collaboration published their results. From the monitoring of 40 million stars in the LMC, only between 14 and 17 candidate microlensing events were identified. They concluded that between 8% and 50% of the mass the Milky Way’s halo consisted of MACHOs . Six years later, a similar project, EROS (Experience pour la Recherche d’Objets Sombres), pub-lished the results of 6.7 years of monitoring of both Magellanic Clouds and showed that MACHOs could not make more than 8% of the halo .
The second evidence showing that MACHOs can only account for a small fraction of dark matter comes from cosmology. As seen in section 1.1.4, the study of the CMB shows that there is, in mass, five times more non-baryonic dark matter than baryonic matter in the Universe. As MACHOs count as baryonic matter, this would leave a large fraction of non-baryonic dark matter whose nature remains unknown and would also severely constrain the density of MACHOs. It is also possible to draw constraints on baryonic matter density from BBN. By observing the abundance of light elements in the Universe, one can deduce the values of cosmological parameters such as Ωb . In 1973, Reeves et al., managed to calculate an upper limit on the baryon density parameter Ωb < 0.1 from the deuterium abundance fraction D/H . Deuterium is a good indicator, as it is believed to be only produced during BBN and not in stellar processes. Several studies measured more precisely the deuterium abundance and showed that Ωb ≈ 0.02 with 10% precision, which is coherent with CMB analyses [31–35]. Such a baryonic density leaves little room for MACHO dark matter.
Recently, there has been a renewed interest for MACHOs with the observation of black hole merg-ers by gravitational-wave interferometers LIGO and VIRGO . The unexpectedly high masses of the observed black holes may suggest that they were not created by the gravitational collapse of a star but were produced at the very beginning of the Universe, during inflation, from small overdensities. Primordial black holes could make a credible candidate for dark matter as they evade cosmological constraints and also micro-lensing constraints, depending on their mass dis-The hypothesis which draws the most the attention in particle physics remains, nevertheless, that dark matter consists of weakly interacting massive particles.
After the 70s, it was becoming clear that dark matter consists of exotic kinds of particles. Several models in particle physics were proposed in order to find a candidate for dark matter particles. Those candidates had features in common and a category of particles emerged from those mod-els: the WIMPS, standing for Weakly Interacting Massive Particles . The WIMPS consist in particles heavy enough (m 1 −100 keV) to be non-relativistic and in thermal equilibrium at the beginning of the Universe. As the Universe expands, those particles have the particular feature of leaving thermal equilibrium and ceasing to annihilate at a co-moving density still observable at present time. This is done via the freeze-out mechanism described below.
At thermal equilibrium, dark matter particles annihilate into Standard Model (ordinary) parti-cles, and conversely, Standard Model particles annihilate into dark matter particles.
Both processes equilibrate so that dark matter keeps an equilibrium density n = ne q (T) which decreases with temperature (steps (1) and (2) in figure 1.4). However, when the expansion rate of the Universe becomes as large as the annihilation rate of dark matter, this decrease stops. Dark matter density becomes too small for annihilation to occur and the dark matter co-moving density “freezes-out” (step (3)). This remaining density is named the relic density.
Examples of particle candidates
One of the most studied WIMPS in the literature is the neutralino, which will be described in details in chapter 2 on supersymmetry. However, other models of particle physics exhibit some WIMPS. This is the case of extra-dimension theories in which new spatial dimensions are intro-duced in addition to the three dimensions that we all know. Those extra-dimensions are usually compact so that they remain unnoticed at large scale. They can, however, have some importance at a particle level. In particular, they may resolve the hierarchy problem , which will be de-scribed in section 2.2.3. Moreover, to the particles of the Standard Model are attributed some modes in relation to the extra-dimension. Those modes are assimilated to particles commonly named Kaluza-Klein (KK) particles. A KK-parity may preserve the lightest KK-particle from de-cay, which would make it a viable candidate for dark matter.
Table of contents :
1 The quest for dark matter particles
1.1 Evidences for dark matter
1.1.1 Local Dark Matter
1.1.2 Galaxy clusters
1.1.3 Spiral galaxies
1.1.4 Cosmological Scale
1.2 Dark matter candidates
1.2.1 Massive Astrophysical Compact Halo Objects (MACHOs)
1.2.2 The WIMPS
1.2.3 Examples of particle candidates
1.3 Dark matter particle detection
1.3.1 Indirect detection
1.3.2 Direct detection
2.1 The Standard Model
2.1.1 Particle content
2.1.2 Symmetry groups
2.1.3 Standard Model Lagrangian
2.1.4 Beyond the Standard Model
2.2 The supersymmetric hypothesis
2.2.1 Grand Unified Theory
2.2.2 Supergravity and theory of everything
2.2.3 Hierarchy problem
2.2.4 R-Parity and dark matter
2.3 Supersymmetric Lagrangian
2.3.1 Chiral supermultiplet
2.3.2 Gauge supermuliplet
2.3.3 Gauge interactions
2.4 The Minimal Supersymmetric Standard Model
2.4.1 Higgs sector
2.4.2 Sfermion sector
2.4.3 Gaugino-Higgsino sector
2.4.4 Constrained Models
2.4.5 Extensions of the MSSM
2.5 SUSY searches at colliders
2.5.1 Electron/positron Colliders
2.5.2 Hadron Colliders
II Astrophysical and nuclear uncertainties of dark matter direct an indirect detection constraints
3 Robustness of dark matter constraints and interplay with collider searches for New Physics
3.1 Objectives of the analysis
3.2.1 MSSM Scans
3.2.2 Dark matter constraints
3.2.3 Collider constraints
3.3.1 Relic density constraints
3.3.2 Indirect detection constraints
3.3.3 Direct detection constraints
3.3.4 Combined dark matter constraints
3.3.5 Collider and Dark Matter constraints
4 New extensions and features in Superiso Relic
4.1 SuperIso Relic
4.2 Direct detection
4.2.2 Scattering cross sections
4.2.3 Uncertainties on the nucleon and nuclear form factors
4.2.4 Experimental limits
4.3 Indirect detection
4.3.1 Fluxes at production
4.3.2 Constraints from Fermi-LAT dwarf spheroïdal galaxies
4.3.3 Constraints from AMS-02 antiprotons
III Relic density in alternative cosmological scenario
5 Dark Matter Casts Light on the Early Universe
5.2 Relic density calculation
5.3 Cosmological scenarios
5.3.1 Decaying primordial scalar field
5.4 New physics scenarios
5.4.1 Benchmark Point A
5.4.2 Benchmark Point B
5.4.3 Sample of pMSSM19 Points
5.5.1 Decaying primordial scalar field