The Minimal Model for Dark Matter A study of the real singlet scalar dark matter model

(1)

The Minimal Model for Dark Matter

A study of the real singlet scalar dark matter model

Bram van Overeem

10222332

Report Bachelor Project Physics and Astronomy, size 15 EC Conducted between 01-12-2014 and 16-07-2015

Institute for Theoretical Physics Amsterdam & Gravition Astroparticle Physics Amsterdam Faculty of Science

University of Amsterdam

Date of submission: 16-07-2015 Supervisor: dr. F. Calore

(2)

Scientific abstract

In this thesis, I study a minimal extension of the Standard Model. I speculate on the existence of a new scalar particle X, which couples only to the Higgs field, and investigate whether it may account for the universe’s dark matter. The phenomenology of the model depends on only two parameters: the X mass, mX, and the coupling to Higgs, ρ. An expression is derived,

relating the relic density of X scalars to mX and ρ. By requiring this relic density to match

the limit on the relic density of cold dark matter as observed by the Planck experiment, it is possible to constrain the model’s parameter space. The mass range 5 GeV≤ mX ≤ 1 TeV is

scanned. It is found that away from the Higgs resonance region, where mX ≈ mh/2, a typical

value of ρ ∼ O(10−1− 10−2_{) is required for X to account for the cold dark matter. Near the}

Higgs resonance area, the coupling has to be suppressed, with a minimum of ρ ∼ 4 × 10−6. I also comment on the constraints set by direct detection and collider experiments. By combining the constraints from the most recent experimental data with a more accurate relic abundance constraint it is found that for masses 55 GeV . mX . 62.5 GeV and couplings

2.5 × 10−4_{. ρ . 10}−2 the X scalar model continues to be viable. Also, the model has yet to be ruled out for high masses mX & 1 TeV with ρ > 0.5.

(3)

Populair wetenschappelijke samenvatting

De bij ons bekende elementaire deeltjes en drie van de vier fundamentele krachten die tussen die deeltjes werken worden beschreven in het zogenaamde Standaard Model van elementaire deeltjes. Dit theoretisch model is de afgelopen decennia erg succesvol gebleken. Niet alleen heeft het Standaard Model veel experimentele waarnemingen verklaard, maar het heeft ook succesvol verschillende ont-dekkingen voorspeld. Het is inmiddels echter ook duidelijk dat het Standaard Model niet compleet is. Dit komt niet in de laatste plaats door het feit dat de deeltjes die het Standaard Model beschrijft, niet meer dan ongeveer 16 % van de materie in het heelal vormen. De overige 84 % bestaat uit onbekende, zogenaamde donkere materie. Op alle schalen in het heelal bestaat overtuigend bewijs voor de aan-wezigheid van donkere materie, maar de grote vraag blijft wat deze donkere materie nou daadwerkelijk is. ´E´en ding is zeker: het bestaat niet uit deeltjes uit het Standaard Model. Er moeten dus nieuwe, onbekende elementaire deeltjes bestaan.

In dit project wordt zo’n nieuw deeltje aan de theorie van het Standaard Model toegevoegd. Het gaat hier om de simpelst mogelijke uitbreiding van het Standaard Model, namelijk een deeltje zonder spin, dat we X noemen en dat alleen via het Higgs deeltje aan de andere deeltjes van het Standaard Model kan koppelen. Door gebruik te maken van experimentele data, wordt in dit project de mogelijkheid onderzocht dat de donkere materie uit dit deeltje bestaat. De relevante natuurkunde van dit model is slechts afhankelijk van twee parameters, namelijk de massa van het deeltje en de sterkte van de kop-peling met het Higgs deeltje. De dichtheid van het hypothetische X deeltje in het heelal is afhankelijk van deze twee parameters. Als we aannemen dat de dichtheid van X deeltjes gelijk is aan de dichtheid van donkere materie, die met data van de Planck satelliet is bepaald, kunnen we beperkingen opleggen aan de parameters van het model en informatie verkrijgen over de fundamentele eigenschappen van het deeltje.

Figure 1: De Planck satelliet

Andere experimenten, waaronder experimenten in de bekende Large Hadron Collider (LHC), leggen verdere beperkingen op aan de eigenschappen van het X deeltje. Deze beperkingen, door de meest recente experimentele data, worden ook besproken. Door de beperkte gevoe-ligheid van de experimenten kunnen niet veel zinnige uit-spraken gedaan worden over het bestaan van X deeltjes met massa’s groter dan 1000 GeV (1 GeV = 1.78 ×10−27 kg). Voor deeltjes met een lagere massa kan dit wel. Het blijkt dat het hypothetische X deeltje verantwoordelijk zou kunnen zijn voor (een deel van) de donkere materie in het heelal als het een massa tussen de 55 en 62.5 GeV heeft en de koppeling met het Higgs boson een waarde heeft van ongeveer 2.5 × 10−4 _{tot 10}−2_.

(4)

Popular abstract

The known elementary particles and three of the four fundamental forces that govern the interactions between these particles are described in the so-called Standard Model of elementary particles. During the past decades, this theory has proven to be successful. The Standard Model has explained many experimental observations, and on top of that, it successfully predicted a variety of discoveries. How-ever, by now it is also clear that the Standard Model is incomplete. The particles described by the Standard Model can only account for around 16 % of the matter in the universe. The remaining 84 % consists of unknown, dark matter. Compelling evidence for the existence of dark matter exist on every scale in the universe, but the identity of this dark matter largely remains a mystery. One thing is certain, however: it does not consist of any of the Standard Model particles. Therefore, new, unknown elementary particles must exist.

In this project such a new particle is added to the theory of the Standard Model. The new particle that is studied is the simplest possible extension of the Standard Model, namely a spinless particle that we call X, and that interacts with Standard Model particles only via the Higgs particle. In this project, the possibility that such a particle accounts for the dark matter is investigated, using experimental data. The relevant physics of this model depends on only two parameters, being the mass of the X particle and the strength of its coupling with the Higgs particle. The density of the hypothetical X particle in the universe depends on these two parameters. If we demand the density of X particles to match the density of the universe’s dark matter, which is obtained from data of the Planck satellite, we can set constraints on the parameters of the model, and obtain information about the fundamental properties of the particle.

Figure 2: The Planck satellite

Other experiments, such as experiments in the famous Large Hadron Collider (LHC), set further constraints on the properties of the X particle. These constraints, from the most recent experimental data, are also discussed. Due to the limited sensitivity of the experiments, we cannot give any useful comments on the possibility of the existence of X particles with a mass greater than 1000 GeV (1 GeV = 1.78 ×10−27 kg). We can, however, for particles with a lower mass. The hypothetical X particle could account for (a part of) the universe’s dark matter if it has a mass between 55 and 62.5 GeV and a coupling with the Higgs boson of about 2.5 × 10−4 to 10−2.

(5)

1 Introduction

In the past decades, evidence for the existence of vast amounts of dark matter in the universe has mounted to the point that its presence seems inevitable [1]. Its nature, however, remains largely unknown. The study of Big Bang nucleosynthesis (BBN) and Cosmic Microwave Back-ground (CMB) anisotropies provides strong evidence suggesting that the constituents of dark matter are almost entirely non-baryonic [2]. Since neutrino’s cannot account for the observed dark matter [1, 3, 4], the Standard Model does not contain a suitable dark matter candidate. This leaves us with the necessity of considering theories beyond the Standard Model in order to account for the dark matter in the universe.

In this project I speculate on a minimal extension of the Standard Model and investigate whether it may explain the universe’s dark matter. The model is constructed by adding to the Standard Model a real singlet scalar X, which only couples to the Higgs field. If such a particle is stable, it could account for a dark matter density. To ensure its stability, a discrete Z2 symmetry is imposed on X. This minimal model was first considered in

cosmo-logical context by Silveira and Zee [5] and has since been extensively studied (among others [6, 7, 8, 9, 10, 11, 12, 13, 14]).

The real singlet scalar dark matter model is a simple one, but nevertheless interesting. Mini-mal models make definite predictions possible and are thus easily falsifiable. In the (probable) case that the X scalar does not account for the dark matter in the universe, studies regard-ing this minimal model can still be useful in understandregard-ing more complicated models. The discovery of the Higgs boson [15, 16, 17] and the recent progress in dark matter detection [18, 19] and CMB experiments [2] call for research to improve the constraints on the real singlet scalar dark matter model.

In this thesis I give an analysis of the real singlet scalar dark matter model. Following stan-dard procedures, I derive from Boltzmann’s equation an expression for the relic density of the X scalar in terms of the free parameters of the model. The goal is to constrain the coupling of the X scalar to the Higgs field, by using the most recent Planck limit on the relic density of cold dark matter. Besides the abundance constraint, current direct detection and collider searches provide constraints on the model’s parameter space. I comment on these constraints in order to find regions of the parameter space where the real singlet scalar dark matter model continues to be viable.

(7)

This thesis will be structured as follows. In Section 2, I discuss some general features of standard cosmology, essential for understanding what follows. Also for the sake of under-standing, Section 3 contains a brief description of the Standard Model and its deficiencies. In Section 4, I review the evidence for dark matter and the motivation for models beyond the Standard Model. In Section 5, I introduce the real singlet scalar dark matter model and identify its free parameters. The derivation of an expression to estimate the relic density of the X scalar is given in Section 6. The computation results, together with the resulting relic abundance constraint are presented in Section 7. In section 8, I comment on the parameter space constraints from direct detection and collider experiments. A discussion of the results is given in Section 9. Finally, in Section 10, I come to a conclusion.

2 Standard cosmology

Before starting, it is important to discuss some features of modern day standard cosmology, essential for understanding what follows. The currently prevailing cosmological model for the universe is the so-called Big Bang theory. According to this theory, the universe is a system evolving from an extremely dense state, that existed some 13×109 years ago. This means that the universe is expanding. The effect of the expansion of the universe is described by the scale factor, a. The ratio of the rate of change of a to its value is the Hubble parameter, H:

H(t) = ˙a(t)

a(t). (2.1)

The Hubble parameter measures the expansion rate of the universe and its present value is observed to be H0 = 67.81 ± 0.92 [2]. In practice the Hubble parameter is often expressed as

H0 = 100h km s−1 Mpc−1, (2.2)

where h is the dimensionless Hubble parameter. The time-dependence of a changes during the expansion of the universe. The time-dependence of a is determined by the universe’s energy density [39]. At early times, radiation dominates the energy density, while later matter dominates. The Friedmann equation, which models the expanding universe, defines the evolution of a and reads

˙a a + k a2 = 8πGN 3 ρtot. (2.3)

In equation 2.3 ρtot is the total energy density of the universe, GN is Newton’s constant and

k is a constant describing the spatial curvature of the universe. k can take three different values: -1, 0 and 1, corresponding to an open, flat and closed universe respectively. The universe is flat when ρtot takes the value of the critical density, ρc, which from equation 2.3

(8)

is easily seen to be

ρc≡

3H2 8πGN

= 1.054 × 10−5h2 GeV cm−3. (2.4)

It is customary to express abundances in the universe in units of ρc. Therefore, we introduce

Ωi, defined as

Ωi ≡

ρi

ρc

, (2.5)

for a substance of species i. Using data from experiments that image Cosmic Microwave Background anisotropies, it is possible to obtain accurate measurements of the abundances of different components in the universe. I will discuss this in more detail in section 4.4

3 The (incomplete) Standard Model

3.1 Introduction

The Standard Model of elementary particles reflects our best understanding of particle physics phenomena. All observed elementary particles and three of the four fundamental forces that govern them are very well described in the Standard Model. As we will see, however, the theory does not describe the whole picture. Theories beyond the Standard Model are needed. In this section, I will very briefly discuss the most important aspects of the Standard Model for the scope of this project. For a more extensive description of the Standard Model I refer to references [20, 21, 22].

3.2 Description

The Standard Model is a quantum field theory, with the gauge group SU(3) × U(2) × U(1). The elementary constituents of matter are spin 1/2 fermions and their interactions are medi-ated by integer spin gauge bosons.

The basic building blocks of matter, the fermions, can be classified into quarks and leptons. There are six flavors of quarks: up, down, strange, charm, bottom and top. They have electric charge, in units of the elementary charge e, Q = 2₃, −1₃, −1₃,2₃, −1₃ and 2₃. Quarks also have the quantum number of color, which can be of three types. The leptons are classified into three flavors of charged leptons: the electron, muon and tau, all with charge Q = −1 and corresponding uncharged neutrinos. Quarks have baryon number B = 1/3 and lepton number L = 0, while leptons have B = 0 and L = 1. Every elementary matter particle has an anti-particle with the same mass, but opposite quantum numbers. Quarks are confined in matter particles called hadrons, which are classified into baryons and mesons. Baryons are fermions composed of three quarks and mesons are bosons composed of a quark and an anti-quark.

(9)

The elementary matter particles are related in three generations. Only the particles in the first generation are stable. The second and third generation contain heavier and less stables particles.

The interactions between the fermions are mediated by spin 1 force-carrying particles, called gauge bosons. The photon mediates the electromagnetic force, eight gluons mediate the strong force and the W± and Z bosons mediate the weak interaction. The electromagnetic force has an infinite range due to the fact that the photon is massless. Although the gluons are massless as well, the range of the strong force is not infinite. This is due to the physical property of confinement [22]. As a result, the strong force has a range of about 10−13 cm. The weak interaction is governed by massive bosons, resulting in a range of about 10−16 cm.

Finally the Standard Model contains a spin 0, massive Higgs boson, responsible for the Higgs mechanism that spontaneously breaks the electroweak symmetry and provides the proper masses to the fermions and the W and Z bosons.

3.3 Problems of the Standard Model

The Standard Model has succeeded in explaining almost all experimental results. On top of that the theory has also accurately predicted a great deal of particle physics phenomena. However, the theory has deficiencies as well. These deficiencies are an indication for the ex-istence of physics beyond the Standard Model.

One of the deficiencies is the gauge hierarchy problem, which comes down to the question why the Higgs mass is so small. Without going into any details, quantum corrections are expected to pull up the Higgs mass to within an order of magnitude of the Planck mass, mP l= 1.22 × 1019 GeV [23]. The observed Higgs mass of about 125 GeV then requires some

very implausible fine-tuning. New physics at the weak scale would eliminate this problem.

An experimental problem is that of neutrino masses. The Standard Model predicts all neu-trinos to be massless. However, neutrino flavor oscillations have been observed, implying that at least some do have mass [24, 25]. This is direct evidence that the Standard Model is incomplete.

Without going into details, other problems are the strong CP problem and the Standard Model flavor problem, which is the question why the fermion masses are so different. The

(10)

problem of trying to comprehend the three fundamental forces as different manifestations of one underlying force, is called the grand unification problem.

Although I have been incomplete in outlining the problems of the Standard Model, it is fair to say that the model is incomplete. Physics beyond the Standard Model is needed. This notion is strengthened by the problem of dark matter. As will be discussed in the next section, the presence of dark matter requires physics beyond the Standard Model.

4 Evidence for dark matter

4.1 Introduction

Dark matter was first introduced by Fritz Zwicky [26], as early as in 1933. Zwicky measured the Doppler shifts of galaxies in the Coma cluster to derive their velocity dispersion. Using the virial theorem, he estimated the mass of the cluster. This virial mass far exceeded the mass ascribed to the luminous matter in the cluster. He concluded that the cluster must therefore contain a large amount of non-luminous (invisible) matter, to prevent the cluster from rapid expansion. He named this ’missing mass’ dunkle ’kalte’ Materie, which is the origin of the term (cold) dark matter. Since this first hint in 1933, evidence for the existence of dark matter has become stronger and is nowadays overwhelming. In this section, I will briefly discuss the most important evidence for dark matter.

4.2 Rotation curves of galaxies

The most direct and convincing evidence for the existence of dark matter comes from the rotation curves of spiral galaxies. A rotation curve is the orbital velocity, v, as a function of the distance from the galactic center, r. Spiral galaxies typically have cold neutral hydrogen in their disks that can be observed by its 21 cm line emission to distances far beyond the optical disk [27]. Rotation curves are usually obtained by using this 21 cm line. In Newtonian dynamics, using Keppler’s law, one expects the orbital velocity to be

v(r) = r

GM (r)

r , (4.1)

where, M ≡ 4πR ρ(r)r2_{dr, and ρ(r) is the mass density profile. We would thus expect that}

beyond the optical disk v(r) drops ∝ r−1/2. However, when measuring the rotation curves of spiral galaxies, in almost all cases the velocities rise linearly near the centre of the galaxy, after which they remain constant out as far as can be measured. This essentially flat behavior was first observed in the Andromeda galaxy [28] and is since then supported by systematic studies

(11)

of spiral galaxies [29]. Velocity measurements on elliptical galaxies also indicate the presence of dark matter [30]. The fact that v(r) is approximately constant implies the existence of a huge invisible spherical halo with M (r) ∝ r and ρ ∝ r−2 [31].

4.3 Galaxy cluster scale

Zwicky first noted in 1933 that galaxy clusters appear to contain large amounts of dunkle ’kalte’ Materie. Zwicky used the virial theorem to determine the mass of a cluster with the observed distribution of radial velocities. Two other independent methods for determining the mass of clusters make use of x-ray analysis and gravitational lensing. Both these methods strengthen the dark matter hypothesis.

Galaxies usually contain large amounts of hot gas, emitting X-rays [32]. If the energy and flux of the X-rays are measured and hydrostatic equilibrium is assumed, the mass distribution of the cluster can be obtained. Masses obtained using this method confirmed earlier estimates made with the virial method and suggest the presence of dark matter in galaxy clusters.

Another evident observation of dark matter is provided by gravitational lensing. Gravita-tional lensing is predicted by the theory of general relativity [33]. Intense gravitaGravita-tional fields wrap space-time, thereby changing the path of passing light. Images of background objects will distort due to the gravitational field of a cluster. This distortion can be used to derive the shape of the potential well and, thus, the mass profile of that cluster. Using gravitational lensing, one can predict (cluster) masses without relying on dynamics. This makes it a com-pletely independent method to observe and measure dark matter.

The effect of gravitational lensing can be seen at various magnitudes depending on the relative position and gravitational potential of the lensing object. Strong gravitational lensing occurs when the background object is (almost) lined up with the lensing cluster and results in an easily visible distortion of an individual light source. Measuring the distortion geometry, the mass of the cluster can be estimated. These estimates reveal vast amounts of dark matter, roughly corresponding to the virial mass estimates discussed above [34]

While strong lensing requires the light source and the lensing object to be almost lined up, weak lensing does not. The image of every background galaxy in the vicinity of a galaxy cluster will be distorted. This allows for a statistical analysis of a large number of sources to determine the cluster mass distribution. These analyses also confirm the presence of dark

(12)

matter. Moreover, advanced lensing analyses have allowed us to compose three dimensional maps of the dark matter distribution in clusters [35].

4.4 Cosmological scale and the need for non-baryonic dark matter

Despite the fact that the observations described above provide compelling evidence for the existence of dark matter, none of them allow us to calculate the total amount of dark matter present in the universe, nor do they help to pinpoint what the dark matter is made of. The analysis of Cosmic Microwave Background (CMB) can give us such information. CMB is background radiation originating from the photon decoupling in the early universe. It was discovered by Arno Penzias and Robert Wilson in 1965 [36]. The radiation is isotropic at the 10−5 level and follows very precisely the black body spectrum at a temperature of T = 2.725 K [1]. The analysis of anisotropies in CMB is a powerful tool to accurately test cosmological models and to put constraints on fundamental cosmological parameters. The Planck exper-iment is one of the experexper-iments that maps and analyses CMB anisotropies. In the Planck experiment results published in the 2015 data release, the values found for the abundance of baryons and matter in the universe are [2]

Ωbh2= 0.02226 ± 0.00023 ΩMh2 = 0.1415 ± 0.0019. (4.2)

The fact that Ωb is much smaller than ΩM means that the most significant constituent of the

matter component of the universe is non-baryonic dark matter.

Using Big Bang nucleosynthesis (BBN) one can also predict constraints on ΩB. These

pre-dictions agree with the value of ΩB obtained from CMB analysis [1].

Without going into details it is important to mention that the study of cosmic structure for-mation also provides strong counter evidence for baryonic dark matter. A baryon-dominated universe would not have formed into the structure observed today. A baryonic universe does not allow primordial density perturbations to evolve in a way as to allow the formation of structures present today [6, 29].

4.5 WIMPS

I have been necessarily brief and incomplete in this section, but it is clear that the evidence for non-baryonic dark matter is compelling and present at all astrophysical scales. Although there might be some baryonic dark matter, the bulk of it must be non-baryonic. The only Standard

(13)

Model particles that remain as dark matter candidate are the neutrinos. For various reasons however, neutrinos cannot account for the bulk of the dark matter. Such reasons include the upper limit on neutrino masses [1] and the fact that they can not dominate the dark matter in dwarf galaxies [3]. Furthermore, neutrinos moved at relativistic speeds when structures started to form. They represent so-called hot dark matter. This impedes small structure formation, and hot dark matter can not be the dominant component of dark matter [4]. The bulk of the dark matter must be cold, i.e. moving at non-relativistic speeds when structure formation starts. Since baryons and neutrinos are not the (bulk of the) dark matter, it is necessary to postulate new particles, beyond the Standard Model. The main candidates are weakly interacting massive particles, WIMPs. These are particles interacting on the weak scale and with masses in the range where new particles are also expected in the scope of the electroweak symmetry breaking and the hierarchy problem, ∼ 102 GeV [1].

5 The real singlet scalar dark matter model

5.1 Introduction

In the previous sections it has become clear that the evidence for dark matter consisting of WIMPs is overwhelming. Since the Standard Model does not contain a suitable dark matter candidate, an extension of the model is needed in order to account for the dark matter in the universe. Here, I speculate on the existence of an additional real singlet scalar X to the Standard Model to account for the dark matter. If such a scalar is stable, it could account for a cold dark matter density. This possibility will be explored. This is a simple dark matter model, but its phenomenology is interesting. For this reason, and its predictability, the model has over the years been explored by various authors [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

5.2 The model

The model that is studied consists of the addition of a real scalar, X, to the standard theory, which transforms as a singlet under SU(3) × SU(2) × U(1), the gauge group of the Standard Model. In order to be a viable dark matter candidate, the X particle must be stable. The stability of the particle is ensured by applying a discrete reflection Z2 symmetry (X → −X)

on X [9]. This means X appears in the Lagrangian in even powers only, suggesting that its decay is forbidden. The X field can only couple to the Higgs field. Since this coupling is the only way for X to couple to ordinary matter, it naturally interacts weakly [5]. This could explain the fact that the particle has not been detected. The X scalar is a prototypical

(14)

WIMP. In the described model, the Lagrangian reads L = L_SM+1 2∂µS∂ µ_{S −} 1 2m 2_X2₋1 4ηX 4_{− ρX}2_H†_H. _(5.1)

Here LSM denotes the Standard Model Lagrangian and H the Standard Model Higgs

dou-blet. η is the self-coupling and only determines the X self-coupling strength. It is pretty much unconstrained and does not play any relevant phenomenological role in the model. ρ is the X-Higgs coupling, which controls couplings to all Standard Model fields. After spontaneous symmetry breaking, we find for the X mass, m2_X = m2+ ρv_EW2 , where vEW = 246.2 GeV [7].

The physics of the model is determined by only two parameters, mX and ρ. I take these two

to be the free parameters of the real singlet scalar model. I have taken the X mass in the range 5 GeV ≤ mX ≤ 1 TeV.

6 Relic density

6.1 Freeze-out of species

I will now consider the thermal relic density of the X scalar. The X particle in the model is stable. This means only (inverse) annihilation processes can change the number of X particles in a comoving volume1. For simplicity’s sake, only 2 ↔ 2 (inverse) annihilation processes will be considered. The most natural origin of a relic density of X scalars is the thermal relic density due to the freeze-out of equilibrium in the radiation-dominated era [6, 37]. There do exist other possibilities for the production of Xs, such as out-of-equilibrium decay of other, heavy particles.

In the very early, hot and dense universe particles were in thermal equilibrium. The produc-tion rate of particles, among which X, was equivalent to the annihilaproduc-tion rate, Γ. Suppose we would lower the temperature of the universe adiabatically below the X mass, then the particle would permanently remain in thermal equilibrium. The abundance of X would be thermally suppressed by exp(−m/T ), since the equilibrium number density is nEQ = g mT_2π

3/2 e−m/T for the region m T . The presence of X would drop to zero [38, 39]. However, the universe is not only cooling, it is also expanding at a rate given by the Hubble parameter, H. Due to this expansion, the gas of X particles will dilute. Eventually the expansion rate exceeds the annihilation rate, H Γ and the gas becomes so dilute that the X’s cannot find each other to annihilate. At this point the particle freezes-out. The number of X’s then asymptotically

1

The comoving volume is the cosmological volume with the universe’s expansion scaled out. In the comoving volume the number density of a non-evolving object in the expanding universe is constant with redshift.

(15)

approaches a constant, its relic density [38]. After the point of freeze-out, the number density simply decreases as a−3.

The picture drawn of the particle evolution is described by the Boltzmann equation. One can obtain the relic abundance of a thermally-produced dark matter candidate by integrating the Boltzmann equation. The Boltzmann equation reads [38]

dnX

dt + 3HnX = −hσvAi[(nX)

2_{− (n}eq X)

2_], _(6.1)

where nX is the number density of X particles, nEQ is the equilibrium number density and

hσv_Ai is the thermal average of the product of the annihilation cross section and the relative velocity of two annihilating X’s. Henceforth, I will simply write hσvAi as hσvi. The second

term on the left-hand side of the Boltzmann equation accounts for the dilution effect as a result of the expansion of the universe. The right-hand side includes interactions. The first term accounts for the decrease of X’s due to annihilation. The second term accounts for the creation of X’s.

By using the Boltzmann equation we can obtain the relic abundance of X. We can then demand this relic abundance to match the experimentally obtained value. By doing this, one can obtain information about the fundamental properties of the X particle.

6.2 Solving the Boltzmann equation

We consider the evolution of the particle number in a comoving volume (as is customary, [1, 37, 40, 41]). This is useful, because by doing this we scale out the effect of the expansion of the universe. This is done by dividing the number density by the entropy density, s. Both the total density and the total entropy are conserved quantities and both their densities scale as a−3. We use Y as the new variable, which is defined

Y ≡ nX

s (6.2)

YEQ≡ nEQ

s . (6.3)

The entropy density is s = 2π2g∗T3/45 [38]. g∗ counts the number of effective degrees of

freedom, which is [38, 42] g∗ = X i=1,bosons gi Ti T 4 +7 8 X i=1,f ermions gi Ti T 4 , (6.4)

(16)

where T is the photon temperature. Remembering sa3 = constant, the left-hand side of the Boltzmann equation becomes [1]

dnX

dt + 3HnX = s dY

dt. (6.5)

We now have an equation depending on time. However, particle interactions will generally depend on temperature. We therefore introduce a new independent variable

x ≡ m

T, (6.6)

where m is the particle mass. In order to change from t to x, the Jacobian dx_dt = Hx is needed [39]. The Boltzmann equation then becomes

dY dx = −xλ(Y 2_{− Y}2 EQ), (6.7) with H(m) = x2H(T ) and λ = hσvis H(m). (6.8)

Multiplying the limiting forms of nEQin the non-relativistic (x 3) regime and the extremely

relativistic one (x 3) by s−1, we compute the forms of YEQ in these regions [38]

YEQ(x) = 45 2π4( π 8) 1/2 g g∗s x3/2e−x = −0.145 g g∗s x3/2e−x (x 3) (6.9) YEQ(x) = 45ζ(3) 2π4 geff g∗s = 0, 278geff g∗s (x 3). (6.10)

The effective number of degrees of freedom, geffis equal to g for bosons and 3g/4 for fermions.

Before solving equation 6.7 we discuss the qualitative behavior of the solution. As long as Γ H, λ is large and Y stays in thermal equilibrium and tracks YEQ. As the universe

expands and cools, λ decreases and eventually, Γ ≈ H. This happens when x = xf ≡ m/Tf,

where Tf is the freeze-out temperature of the particle. This is when freeze-out occurs and

from there, dY_dx becomes small. So what is expected is that for x ≤ xf, Y ' YEQ and for

x ≥ xf, Y = YEQ(xf). It is assumed that the freeze-out of X occurs when the particles are

already non-relativistic (x ≥ 3).

Unfortunately the Boltzmann equation in equation 6.7 does not have an analytic solution [42]. However, using some approximations we can obtain an accurate solution to equation 6.7. Since it is assumed that the freeze-out of X occurs when the particles are already non-relativistic, we can expand hσvi in powers of v2 [1]

(17)

where a and b are expressed in units GeV−2. Next, equation 6.7 is rewritten to obtain the differential equation for ∆ ≡ Y − YEQ, which is the departure from equilibrium,

∆0 = −Y_EQ0 − f (x)∆(2Y_EQ+ ∆), (6.12)

where the prime means _dxd and where

f (x) =r πg∗

45 mP lm(a + 6b/x)x

−2_. _(6.13)

This is the final version of the Boltzmann equation. Equation 6.12 can be solved in the two extreme regions x xf, corresponding to long before freeze-out, and x xf, corresponding

to long after freeze-out. For x xf, Y tracks YEQ closely. This means ∆ YEQ and

|∆0| −Y_EQ0 . By setting ∆0= 0, equation 6.12 can be solved:

∆ ' − Y 0 EQ f (x)(2YEQ+ ∆) ' − Y 0 EQ 2f (x)YEQ for x xf. (6.14)

Long after freeze-out, when x xf, YEQ is exponentially small compared to Y , so ∆ ≈ Y

YEQ. It is therefore safe to set YEQ = Y_EQ0 = 0 in this region and solve equation 6.12 to

obtain

∆0 = −f (x)∆2 for x xf. (6.15)

The goal is to obtain the number of X’s present today, so equation 6.15 is integrated from the point of freeze-out, x = xf, which is the lower limit of this approximation’s valid range [1], to

x = ∞. Using ∆∞ ∆xf, ∆∞ can be derived and, subsequently using ∆ ≈ Y YEQ, gives

Y∞= r 45 πg∗ 1 mP lm xf (a + 3b/xf) . (6.16)

Now Y (present) ≈ Y∞ (here it is assumed that the present photon temperature ≈ 0 GeV).

From Y∞ it is straightforward to compute the present relic density of X, ρX. This density is

given by

ρX = mnX = ms0Y∞, (6.17)

where s0 is the present entropy density, with the value s0 = 2889.2 cm−3 [1]. As mentioned

in section 2, it is conventional to write the relic density in terms of the critical density, ρc.

From equation 2.4 we have ρc= 1.054 × 10−5h2 GeV cm−3. It is then possible to express the

relic density in terms of the Hubble parameter ΩXh2≈ 1.04 × 109 GeV−1 mP l √ g∗ xf (a + 3b/xf) . (6.18)

This is the equation we were aiming for. The particle mass, m, is not explicitly present in equation 6.18. xf and the coefficients a and b however, depend on m.

(18)

Numerical solutions to the Boltzmann equation (equation 6.1) are shown in figure 3, where the abundance in a comoving volume is given as a function of x. It can be seen clearly that the larger the annihilation cross section is, the longer the particle stays in equilibrium and the smaller its relic abundance turns out to be.

Figure 3: The comoving number density of a random WIMP in the early universe as a function of x ≡ m/T , which increases with time [38, 42]. The solid curve is the equilibrium density and the dashed curves are the actual densities for different values of the annihilation cross section, hσvi.

7 Results

Starting from the Boltzmann equation 6.1, I derived equation 6.18, expressing the relic density of X in terms of the the cross section, which as we will see depends on the free parameters of the model, mX and ρ. The abundance constraint is obtained now by requiring ΩXh2

to match the cosmologically preferred value for the cold dark matter density. This value is obtained from the Planck experiment results published in the 2015 data release [2]. Planck images CMB anisotropies and this data has been used to set accurate constraints on many cosmological parameters, including the universe’s cold dark matter density. From the Planck experiment we have

(19)

In general, a WIMP freezes-out of equilibrium at a temperature that is approximately given by Tf ≈ m/20 [5, 8, 9, 38, 39, 42]. This value depends only logarithmically on variables such

as ρ and mX, and is mostly stable under variations of these free parameters [5]. We therefore

take the freeze-out criterium to be xf = 20. This value justifies the assumption that the X

scalar is non-relativistic when freezing-out.

As set forth above, the range for the X mass that is scanned in this project is 5 GeV ≤ mX ≤ 1

TeV. For this range of masses the number of effective relativistic degrees of freedom, g∗, is

computed, using equation 6.4. The result is the step function slowly varying with the X mass shown in figure 4.

Figure 4: The number of effective degrees of freedom, g∗, as a function of the scalar mass,

mX.

Depending on its mass, the X scalar can annihilate via Higgs exchange following the channels XX → f ¯f , W+W−, ZZ, hh. We use the contributions of these channels to the total anni-hilation cross section, hσvi, given by McDonald in reference [6] ([8, 9]). The contributions of the different annihilation channels are the following.

(20)

XX → f f : m2_W πg2 λ2_fρ2 [(4m2 X− m2h)2+ m2hΓ2h] 1 − m 2 f m2 X !3/2 . (7.2) XX → W+W−: 2 " 1 +1 2 1 −2m 2 X m2 W 2# ρ2m4_W 8πm2 X[(4m2X − m2h)2+ m2hΓ2h] × 1 −m 2 W m2 X 1/2 . (7.3) XX → ZZ: 2 " 1 +1 2 1 −2m 2 X m2_Z 2# ρ2m4_Z 16πm2_X[(4m2_X− m2 h)2+ m2hΓ2h] × 1 − m 2 Z m2_X 1/2 . (7.4) XX → hh: ρ2 64πm2 X 1 − m 2 h m2 X 1/2 . (7.5)

Here λf is the fermion Yukawa coupling and its value is λ = mf/v, where v = 250 GeV and

mf is the fermion mass. Γh is the Higgs decay width, which has the value Γh = 4.03 × 10−3

GeV. We are scanning the region wherein mX ≥ 5 GeV > mb, the bottom quark mass. In

this region we make the approximation mf = 5 GeV, taking together all fermions except for

the top quark [5]. The top quark channel is taken into account separately. In figure 5 the annihilation cross section of the X scalar is shown as a function of mX for different values of ρ.

In figure 5 it is shown that from mX = 5 GeV on, the annihilation cross section initially

increases with increasing mX. In the range of small scalar mass, mX < mW, the b¯b

chan-nel dominates the annihilation cross section [5, 10]. When the X mass reaches the value 2mX ≈ mh, the annihilation cross section rapidly rises about 9 orders of magnitude. This is

due to the so-called Higgs resonance, coming from the appearance of the Higgs propagator in the f ¯f annihilation process (equation 7.2). After reaching a peak at mX = mh, hσvi

decreases rapidly with increasing mX. When mX ≈ mW, the W+W− channel opens and the

annihilation cross section abruptly rises about 2 orders of magnitude. For mX > mW, the

annihilation cross section decreases with increasing mX. After the W+W−channel opens the

X scalar will mostly annihilate into a W+W− pair. Additional, subdominant contributions come from the ZZ (for mX > mZ), the hh (for mX > mh) and the t¯t (for mX > mt) final

states.

Now that we computed g∗ and hσvi, it is possible to calculate the relic density, using equation

6.18. Note that the annihilation cross section does not depend on v, so we can set hσvi = a in equation 6.18. For the same values of ρ as used in figure 5, the relic density of the X

(21)

Figure 5: The product of the annihilation cross section and relative velocity, hσvi, as a function of the scalar mass, mX, for ρ = 10, 1, 0.1, 0.01.

scalar is shown as a function of its mass in figure 6. It is demonstrated in equation 6.18 that ΩXh2 ∝ hσvi−1. It is therefore seen clearly in figure 6 that the change of the relic density

with mX is contrary to the change of hσvi with mX. This results in a strong suppression of

the relic density around mX ≈ mh. The opening of the W+W− channel is also clearly seen.

The opening of this channel results in a significantly smaller relic density for mX & mW.

Furthermore, equations 7.2 - 7.5 show that hσvi ∝ ρ2. It is therefore seen in figure 6 that the higher ρ, the lower ΩXh2.

Since hσvi ∝ ρ2, we can say hσvi ≡ ρ2hσvi_m. Requiring ΩXh2 to take the value of 0.1186 ±

0.0020, experimentally obtained by the Planck experiment, an expression relating the cou-pling, ρ, to the scalar mass, mX, can be obtained. By doing this it is possible to constrain

the mX − ρ parameter space of the model. From equation 6.18 the relation is obtained

ρ ≈ 1.04 × 10 9 _GeV−1 mP l √ g∗ xf (0.1186 ± 0.0020)hσvim 1/2 . (7.6)

The result is shown in figure 7. In other words, figure 7 shows the constraint that the Planck data sets on the mX − ρ parameter space.

(22)

(a) ρ = 0.01 (b) ρ = 0.1

(c) ρ = 1 (d) ρ = 10

Figure 6: The relic abundance of the X scalar in terms of the Hubble parameter, ΩXh2, as a

function of the scalar mass, mX, for (a): ρ = 0.01, (b): ρ = 0.1, (c): ρ = 1 and (d): ρ = 10.

The red horizontal lines (so close together you can hardly tell them apart) denote the Planck limit’s upper and lower bounds, ΩXh2 = 0.1206 and ΩXh2 = 0.1166 respectively.

For 5 GeV ≤ mX ≤ 1 TeV and away from the Higgs resonance region, the abundance

con-straint requires ρ to take a typical value of ρ ∼ O(10−1− 10−2). The scalar X model explains the presence of dark matter naturally. Obtaining the correct relic density of dark matter does not require fine-tuning the parameters [7, 10]. In the Higgs resonance region, where 2mX ≈ mh, the annihilation occurs very efficiently and the annihilation rate is enhanced. In

this region ρ is suppressed down to a level of up to O(10−6), with a minimum of ρ ∼ 4 × 10−6. X scalars with small masses need larger values of ρ to have a relic density that is compatible with the experimentally observed value. The result in figure 7 is the viable (mX− ρ)

param-eter space of the real singlet scalar model, after using Planck experiment data to invoke the abundance constraint. The region below the line is excluded because points in this region result in an overabundance of X dark matter. The line denotes the viable parameter space for a model in which X account for all cold dark matter, ΩX = ΩCDM. The region of the

(23)

Figure 7: The coupling, ρ, as a function of the scalar mass, mX, for ΩXh2 = 0.1186 ± 0.0020.

parameter space above the black line is viable if the X scalar does not account for all observed cold dark matter or in case the X scalar can also be produced non-thermally. In other words, we computed a lower limit in the parameter space.

8 Further constraints on the parameter space: LHC and direct

detection

In addition to the CMB analysis of the Planck experiment, other experimental methods can set constraints on the viable parameter space of the real singlet scalar dark matter model. In this section, I will comment on the consequences of recent LHC and direct detection data for the parameter space of the model.

The ρX2H†H coupling in the Lagrangian (equation 5.1) allows for a contribution to the in-visible Higgs decay. In case mX . mh/2, the Higgs decay channel h → XX is open, and

contributes to the invisible Higgs width, Γinv [11, 12]. The Large Hadron Collider (LHC)

experiments put limits on Γinv. Since the contribution to Γinv depends on the parameters

of the model, limits on Γinv bring along constraints on the parameter space for X masses

< mh/2. In this low mass region, the invisible width constraint forbids couplings larger than

ρ ∼ 0.02 − 0.03 [11, 43]. In combination with the abundance constraint, LHC data excludes X masses . 53 − 56 GeV, according to the analyses in references [11, 12, 13].

(24)

usu-ally by measuring the recoil energy of the nucleon in the interaction [1]. These measurements can limit the scattering cross section of the dark matter particle with the nucleon. This in turn limits the parameter space of the dark matter model. In reference [11], the most recent XENON100 data is used [18]. Just like the LHC data, XENON100 also excludes X masses smaller than 53 GeV. This means that the real singlet scalar dark matter model cannot explain the experimental data pointing to masses ∼ 10 GeV (coming from the DAMA and CoGeNT experiments [44]) [11]. XENON100 allows for a range of masses 53 GeV . mX . mh/2,

with couplings ρ ∼ 10−2− 10−3.5_{. It furthermore excludes a range where m}

X is larger than

mh/2. For mX & mW, XENON100 data leaves the parameter space largely unconstrained

[11]. However, reference [11] does predict near future experiments to rule out most of the remaining relevant parameter space of the model for mX > mh/2.

These predictions are largely supported by reference [12], using data from the more recent LUX experiment [19]. It should be mentioned that there is a difference in the methods used in reference [11] and [12]. Where the first method assumes that the area above the line in figure 7 corresponds to the scenario in which X makes up only part of the cold dark matter, the second method assumes that for points in this region, X scalars still make up all of the cold dark matter. Another mechanism, like non-thermal production, has accounted for the presence of an abundance of X’s compatible with the observed cold dark matter. This strengthens the direct detection constraints [12]. Due to limits of the sensitivity of the LUX experiment, direct detection has yet to rule out the region with mX > 1 TeV and ρ > 0.5

[12]. Reference [13] draws a more conservative constraint from the analysis of LUX data and concludes that singlet scalar dark matter is still a viable possibility for mX & 100 GeV. Just

as reference [11], both reference [12] and [13] agree on a small viable parameter space for the real singlet scalar model. Taking direct detection, indirect detection2 _{and the relic abundance}

constraint together, the X scalar model is viable only for 56 GeV . mX . 63 GeV and

2.5 × 10−4. ρ . 5 × 10−3 [12].

9 Discussion

As set forth above, the singlet scalar dark matter model has been studied extensively over the years. The results of computing the abundance constraint in this project are shown in figure 7 and are comparable to the results of multiple earlier results. There are, however, certain differences.

2_{Indirect detection is the experimental method that uses the observation of radiation produced in} annihi-lations of dark matter [1].

(25)

First of all, almost all previous research on the model has been done while the Higgs mass, mh, was still unknown. The model strongly depends on the value of mh. Therefore, the

discovery of the Higgs boson and accurate estimates of its mass by the LHC [15, 16, 17], take away one free parameter in the model. Additionally, with the 2015 Planck data release constraints on cosmological parameters, among which ΩCDM, have become more stringent.

These experimental efforts make the abundance constraint derived in this project more accu-rate.

However, we have to continue to be realistic; it is important to state that the methods and approximations used in this project carry along inaccuracies in the resulting abundance con-straint. The most important inaccuracy comes from the calculation of the annihilation cross section. Here, I use the contributions shown in 7.2 - 7.5. These contributions do not give a very accurate approximation of the annihilation cross section near the thresholds for produc-ing boson pairs and near the Higgs resonance region [11, 14, 40, 45]. The resultproduc-ing dramatic decrease in the Higgs resonance region shown in figures 6 and 7 in the relic density and cou-pling respectively, should actually be not nearly as deep and narrow. It can be seen from figure 7 that the coupling decreases as far as ρ ∼ 5×10−6. Recent results, using more accurate approximations of the annihilation cross section, in references [11] and [12], show dips only to ρ ∼ 5 × 10−4, a difference of two orders of magnitude with the results presented in this thesis. This inaccuracy is especially important, since it occurs in a region that is still promising in the sense that is has not yet been ruled out by experimental efforts.

The inaccuracy in the Higgs resonance region is increased by the assumption that xf = 20.

Although this is a fairly good assumption for most of the mass range I scanned, xf deviates

from this value, especially in the Higgs resonance region and near the W threshold [9]. Instead of assuming xf to be 20, one should find the iterative solution of the equation

xf = ln  c(c + 2) r 45 8 g 2π3 mP l(a + 6b/xf) g1/2∗ x1/2_f  , (9.1)

where c is a constant of order unity [1]. A last remark has to be made on equation 6.4, used for the calculation of g∗. Instead of using this equation, most authors prefer the computation

of g∗ presented in reference [40], which gives a more accurate result.

As is shown in this project, the singlet scalar dark matter model is an excellent example of the predictiveness and falsifiability of a simple model. As shown in sections 7 and 8, the

(26)

parameter space of the model becomes heavily constrained and the model is therefore highly predictive. Also, section 8 illustrates that the model is a clear example of how different dark matter detection methods, including direct detection and collider searches, can effectively complement each other.

10 Conclusion

In the roughly 80 years that have past since Fritz Zwicky first introduced dark matter, the evidence for its existence has become compelling. At the same time, it has become clear that the Standard Model of particle physics is incomplete. To account for the universe’s dark matter, particles beyond the Standard Model must be considered. In this project I speculated on a minimal extension of the Standard Model, namely the addition of a real singlet scalar X, with minimal coupling. The stability of the scalar particle is ensured by applying a discrete reflection symmetry Z2 on X. The phenomenology of this model is determined by only two

parameters: the coupling to Higgs, ρ, and the X mass, mX. Using experimental data, I have

set constraints on the parameter space of the scalar X model.

Starting from Boltzmann’s equation an expression relating the relic density of X to ρ and mX is derived. This derivation is done by using standard procedures, making the resulting

expression in equation 6.18 a standard estimate. By requiring the relic abundance of X to match the relic density of cold dark matter observed by the Planck experiment, the abun-dance constraint on the model’s parameter space is computed for the X mass range 5 GeV ≤ m_X ≤ 1 TeV. Away from the Higgs resonance region, where m_X ≈ m_h/2, a typical value of ρ ∼ O(10−1− 10−2) is required for the relic density of X to meet the Planck limit. In the Higgs resonance region, ρ is suppressed down to a value of up to ρ ∼ 5 × 10−6.

Besides computing the abundance constraint, I have commented on the constraints from direct detection (LUX and XENON100) and LHC searches. It is found that an interesting region in parameter space lies near the Higgs resonance. In this region especially, the annihilation cross section I computed, does not give an accurate estimate. By combining a more accurate abundance constraint and constraints from direct detection and collider experiments, it is found that for masses 55 GeV . mX . 62.5 GeV and couplings 2.5 × 10−4 . ρ . 10−2 the

model continues to be viable. For high masses mX & 1 TeV with ρ & 0.5, the model has yet

(27)

Acknowledgement

I would like to sincerely thank my supervisor dr. Francesca Calore, for her guidance, support and flexibility.

(28)

11 Bibliography

[1] Bertone, G., Hooper, D. & Silk, J. Particle Dark Matter: Evidence, Candidates and Constraints, Phys. Rept., 2005, 405, 279.

[2] Ade, P.A.R., Aghanim, N., Arnaud, M. et al. 2015, Planck 2015 results. XIII. Cosmo-logical parameters, arXiv:1502.01589v2.

[3] Tremaine, S. & Gunn, J.E., Dynamical Role of Light Neutral Leptons in Cosmology, Phys. Rev. Lett., 1979, 42, 407.

[4] Bond, J.R., Efstathiou, G. & Silk, J. Massive Neutrino’s and the Large-Scale Structure of the Universe, Phys. Rev. Lett., 1980, 45.

[5] Silveira, V. & Zee, A. Scalar Phantoms, Phys. Lett., 1985, 161B, 136.

[6] McDonald, J. Gauge singlet scalars as cold dark matter, Phys. Rev. D, 1994, 50, 3637. [7] Burgess, C.P., Pospelov, M. & ter Veldhuis, T. The Minimal Model of nonbaryonic dark

matter: a singlet scalar, Nucl. Phys. B, 2001 619, 709.

[8] Guo, W. & Wu, Y. The real singlet scalar dark matter model, J. High Energy Phys., 2010, 1010, 83.

[9] Biswas, A. & Majumdar, D. Real gauge singlet scalar extension of the Standard Model: A possible candidate for cold dark matter, Pramana J. Phys., 2012, 80, 539.

[10] Yaguna, C.E. Gamma rays from the annihilation of singlet dark matter, J. Cosm. As-tropart. Phys., 2009, 3, 3.

[11] Cline, J.M., Kainulainen, K., Scot, P. & Weniger, C. Update on scalar singlet dark matter, Phys. Rev. D, 2014, 88.

[12] Feng, L., Profumo, S. & Ubaldi, L. Closing in on singlet scalar dark matter: LUX, invisible Higgs decay and gamma-ray lines, J. High Energy Phys., 2015, 3, 45.

[13] De Simone, A., Giudice, G.F. & Strumia, A. Benchmarks for Dark Matter Searches at the LHC, J. High Energy Phys., 2014, 6, 81.

[14] Cline, J.M. & Kainulainen, K. Electroweak baryogenesis and dark matter from a singlet Higgs, J. Cosm. Astropart. Phys., 2013, 1, 12.

[15] CMS Collaboration Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B, 2012, 716, 30.

(29)

[16] ATLAS Collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B, 2012, 716, 1.

[17] ATLAS and CMS Collaborations Combined Measurement of the Higgs Boson Mass in pp Collisions at √s = 7 and 8 TeV with the ATLAS and CMS Experiments, Phys. Rev. Lett., 2015, 114, 191803.

[18] Aprile, E., Alfonsi, M., Arisak, K. et al. Dark Matter Results form 225 Live Days of XENON100 Data, Phys. Rev. Lett., 2012, 109.

[19] Akerib, D.S., Araujo, H.M., Bai, X. et al. First results from the LUX dark matter exper-iment at the Stanford Underground Research Facility, Phys. Rev. Lett., 2014, 112. [20] Halzen, F. & Martin, A.D., Quarks and Leptons, John Wiley & Sons, 1984.

[21] Thomson, M., Modern Particle Physics, Cambridge University Press, Cambridge, 2013. [22] Herrero, M.J. 1998 The Standard Model, lectures presented at the NATO ASI 98 School, Techniques and Concepts of High Energy Physics, St. Croix, Virgin Islands, June 18-29, 1998.

[23] Feng, J.L. Dark Matter Candidates from Particle Physics and Methods of Detection, Annu. Rev. Astrom. Astrophys., 2010, 48:495.

[24] Ahmad, Q.R., Allen, R.C., Andersen, T.C. et al. Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory, Phys. Rev. Lett., 2002, 89, 011301.

[25] Fukuda, Y., Hayakawa, T., Ichihara, E. et a. Evidence for Oscillation of Atmospheric Neutrinos, Phys. Rev. Lett., 1998, 81, 1562.

[26] Zwicky, F. Die Rotverschiebung von extragalaktischen Nebeln, Helv. Phys. Acta., 1933, 6, 110.

[27] Murayama, H. 2006 Physics Beyond the Standard Model and Dark Matter, Lectures presented at Les Houches Summer School, Session 86, Particle Physics and Cosmology: the Fabric of Spacetime, Les Houches, France, July 31 - August 25.

[28] Babcock, H.W. The rotation of the Andromeda Nebula, Lick Observatory Bulletin, 1939, 19, 41.

[29] Raffelt, G.G. 1997, Dark Matter: Motivation, Candidates and Searches, Lectures pre-sented at European School of High-Energy Physics, Menstrup, Denmark, June 1997.

(30)

[30] Boriello, A. Salucci, P. & Danese, L. The fundamental plane of ellipticals, Mon. Not. Roy. Astron. Soc., 2003, 341, 1109.

[31] Ostriker, J.P. & Peebles, P.J.E. A Numerical Study of the Stability of Flattened Galaxies: or, can Cold Galaxies Survive?, Astrophys. J., 1973, 186, 467.

[32] Sarazin, C.L. X-ray emission from clusters of galaxies, Rev. Mod. Phys., 1986, 58, 1. [33] Bartelmann, M. & Narayan, R. 2008, Lectures on Gravitational Lensing,

arXiv:astro-ph/9606001v2.

[34] Bartelmann, M. & Narayan, R. Formation of Structure in the Universe, ed. by Dekel, A. & Ostriker, J.P., Cambridge University Press, Cambridge, 1995

[35] Kitching, T., Massey, R. & Richard, J. The dark matter of gravitational lensing, Rep. Prog. Phys. 2010, 73, 086901.

[36] Penzias, A.A. & Wilson, R.W. A Measurement of Excess Antenna Temperature at 4080 Mc/s, Astrophys. J., 1965, 142, 419.

[37] Gelmini, P. & Gondolo, G. Particle Dark Matter: Observations, Models and Searches, ed. by Bertone, G., Cambridge University Press, Cambridge, 2010.

[38] Kolb, E.W. & Turner, M.S. The Early Universe, Westview Press, 1994. [39] Dodelson, S. Modern Cosmology Academic Press, 2003.

[40] Gelmini, P. & Gondolo, G. Cosmic abundances of stable particles: improved analysis, Nucl. Phys. 1991, B360, 145.

[41] Olive, K.A., Srednicki, M. & Watkins, R. Calculations of Relic Densities in the Early Universe, Nucl. Phys., 1988, B310, 693.

[42] Jungman, G., Kamionkowski, M. & Griest, K. Supersymmetric dark matter, Phys. Rep., 1996, 267, 195.

[43] Belanger, G., Dumont, B., Ellwanger, U., Gunion, J.F. & Kraml, S. Global fit to Higgs signal strenghts and couplings and implications for extended Higgs sectors, Phys. Rev. D, 2013, 88.

[44] Andreas, S., Arina, C., Hambye, T., Ling, F-S. & Tytgat, M.H.G. A light scalar WIMPS through the Higgs portal and CoGeNT, Phys. Rev. D 2010, 82.

[45] Griest, K. & Seckel, D. Three exceptions in the calculation of relic abundances, Phys. Rev. D, 1991, 43, 3191.

The Minimal Model for Dark Matter A study of the real singlet scalar dark matter model