Internship report

Computing density of Dark Matter in Grand UnifiedTheory models

Nadège Lemarchand

Training supervisor : Yann Mambrini

Laboratoire de Physique ThéoriqueBâtiment 210

Univ. Paris-Sud 1191405 Orsay Cedex

France

May, 23 – July, 4 2014 Licence 3 et Magistère 1 de Physique Fondamentale

2

Contents

1 Resume/Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A thermal history of the Universe . . . . . . . . . . . . . . . . . . 7

3.1 A thermodynamical bath . . . . . . . . . . . . . . . . . . 73.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Recombination and decoupling of the photons . . . . . . . 123.4 What happened next ? . . . . . . . . . . . . . . . . . . . . 143.5 The neutrino decoupling . . . . . . . . . . . . . . . . . . . 15

4 The Dark Matter (original material) . . . . . . . . . . . . . . . . 164.1 Hot, warm and cold dark matter . . . . . . . . . . . . . . 164.2 Hot relic densities . . . . . . . . . . . . . . . . . . . . . . 17

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3

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Thanks

First I would like to thank my training supervisor Yann Mambrini for givingme a chance as an intern for my first internship in physics. He is interesting whenhe talks about physics and he is a nice and open-minded person who taught menew things. He was not just a researcher in physics but also has an interestingpersonnality for the less I know and so gave me the will to go on in this field. Ialso thank some PhD students like Jérémie Quevillon who worked with Yann andgave me some pieces of advice on the subject I was working on; Lucien Heurtierand Antoine Gérardin for their help with the software LaTex and others whowill recognize, those who welcomed me for lunches or coffee breaks with theirendless discussions. Thanks to my co-worker Lucile Goubayon who was in thesame office with me and who encouraged me everyday by getting up early in themorning to come. I also have to thank the Magistère for encouraging us to do aninternship and of course the laboratory for welcoming me and giving me a niceplace to work.

Premièrement, je voudrais remercier mon maître de stage Yann Mambrinipour m’avoir donné l’opportunité d’être sa stagiaire pour mon premier stage dephysique. Il parle de physique de manière intéressante, c’est une personne sym-pathique et ouverte d’esprit qui m’a appris de nouvelles choses. Ce n’est passimplement un chercheur en physique mais également une personnalité intéres-sante pour le peu que je le connaisse et il m’a donc donné l’envie de continuerdans ce domaine. Je remercie aussi quelques étudiants en thèse comme JérémieQuevillon qui travaillait avec Yann et qui m’a donné quelques conseils à proposdu sujet sur lequel je travaillais; Lucien Heurtier et Antoine Gérardin pour leuraide avec le logiciel LaTex et d’autres qui se reconnaîtront, ceux qui m’ont ac-cueillie lors des déjeuners et pauses cafés où nous avions des discussions sans fin.Merci à ma collègue et camarade Lucile Goubayon qui était dans le même bureauque moi et qui m’encourageait chaque jour en se levant tôt le matin pour venirau laboratoire. Je dois aussi remercier le Magistère pour nous pousser à faireun stage et le laboratoire bien-sûr pour m’avoir accueillie et donné un endroitagréable pour travailler.

1. RESUME/ABSTRACT 5

1 Resume/Abstract

La matière noire constitue environ 24 % de l’Univers actuel. Des indices deson existence sont donnés par les effets gravitationnels qu’elle engendre : c’estune matière qui n’émettrait pas de rayonnement électromagnétique car elle n’aencore jamais été détectée jusqu’à présent.

Après avoir retracé l’histoire primordiale de l’Univers, du bain de photonscomposé de particules relativistes du modèle standard jusqu’à l’instant du dé-couplage des photons qui donne lieu au fond diffus cosmologique, on supposeraqu’à des échelles d’énergie plus grandes qui n’ont pas encore été atteintes dansles accélérateurs, c’est-à-dire à des temps plus reculés de l’univers, des partic-ules au-delà du modèle standard existent. Nous considérerons par exemple unneutrino massif qui se comporterait de façon similaire au neutrino du modèlestandard et qui pourrait être un bon candidat matière noire.

The current universe is composed of approximately 24 % of dark matter. It issupposed to exist only because of gravitational effects observed: as it has neverbeen detected yet, it would not emit electromagnetic radiation.

After recounting the primordial history of the universe, from the primordialbath made of relativistic particles from the standard model to the time of pho-ton’s decoupling picturing the CMB, it could be supposed that at some moreenergetic scales which have not yet been reached in accelerators (in other wordsat earliest times in the universe), other particles beyond the standard model existand have not been detected yet. We will consider for example a massive neutrinowhich would behave the same way as the one from the standard model and whichcould be a good dark matter candidate.

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2 Introduction

This internship was a compulsory 3rd-year internship for the Magistere dePhysique Fondamentale at Université Paris-Sud. It took place at the Laboratoirede Physique Théorique in Orsay (UMR 8627) in the team "Particle physics". Myinternship supervisor was Yann Mambrini but I also worked with the help of thePhD student Jérémie Quevillon on "Computing density of Dark Matter in GUTmodels": the subject was at the line between particle physics and cosmology.This internship took place in the framework of the European Union FP7 ITNINVISIBLES (Marie Curie Actions, PITN- GA-2011- 289442) and and the ERCadvanced grants MassTeV. After a bibliographical work on astroparticle physicsand the thermal history of the Universe, I studied the thermodynamics of parti-cles of the standard model in the early Universe before going beyond the standardmodel by introducing another massive neutrino.

In 1933, the astrophysicist Fritz Zwicky found that the mass of Coma, acluster of galaxies, deduced from its luminosity, does not match with the grav-itational observations: they are different by a factor of 100. He was the firstto introduce the notion of Dark Matter. Then in the seventies, the astronomerVera Rubin brought another evidence of Dark Matter by comparing the rotationcurves of galaxies to the one given by the theory. Dark matter is by definitionan invisible matter because it does not emit electromagnetic radiation1. Thus,it can only be detected by its decaying, annihilation or deexcitation or by itsinteraction with ordinary matter. As the neutrino does not interact electromag-netically nor strongly it could be a good dark matter candidate but as it appearsto be too light to fill the missing mass of the galaxies as the Cowsick-McClellandbound shows, see subsection(3.5), the question of knowing what is dark matterremains open. Nonetheless, one will see that physicists have developped someideas.

1In fact the upper bound for the ratio of the electromagnetic charge of dark matter is 10−4.

3. A THERMAL HISTORY OF THE UNIVERSE 7

3 A thermal history of the Universe

3.1 A thermodynamical bath

The first seconds of the Universe are dominated by a plasma made of ele-mentary particles. The density and temperature are very high so every particlesare relativistic and the plasma can be considered as a perfect gas. Then, as theexpansion of the universe begins, the temperature and density start to decreasefollowing

T (t)R(t) = T0R0, R(t) being the scale factor of the universe (1)

Figure 1: Illustration of the expansion of the universe. The dots represent the relativisticparticles of the primordial plasma.

This equation is due to the conservation of the entropy per comoving volume aswe will see in the subsection (3.2). Indeed, the evolution of the thermal bath isvery slow (it is the first seconds of the universe) so phenomena are considered asbeing reversible, in adiabatic evolution.

When the temperature starts to decrease, the kinetic energy of a particle alsodecreases because the temperature can (under certain conditions) be viewed asits thermal agitation (Ec = 3

2kT ). Thus, the particle is no longer relativistic andso its density decreases (a we will see in Eq.16). In this way, it has less and lessinteractions with the primordial plasma: the particle decouples from the thermalbath of photons which does not possess sufficient energy to create it.

At first we will only consider the set of elementary particles from the StandardModel: photons (2 degrees of freedom because of the 2 polarizations), electrons(2 degrees of freedom because of the 2 possible values for the spin 1/2) and

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their antiparticles: positrons (2), muons (2) and antimuons (2), taus (2) andantitaus (2), neutrinos (3 because of the 3 flavors, each neutrino has only onepossible value for the spin because of their chirality) and antineutrinos (3), gluons(8 × 2 states forpolarization), W+and W−(3 for each) Higgs boson (1), quarks: down, up, top, bottom, strange, charm (2 values for spin × 3 colors for each)and antiquarks (6), and Z0 (3).

Figure 2: Table of the Standard Model particles taken from [9].

Bosons and fermions are particles which have properties of symmetry or an-tisymmetry: the wave function of a system remains identical under the exchangeof two same bosons, whereas its sign is changed in the case of two same fermions.Thus, the state of a system made of identical bosons is totally symmetric whereasit is totally antisymmetric if made of identical fermions. In fact, every particle Ais a boson (-) or a fermion (+) distributed by the Fermi-Dirac or Bose-Einsteinstatistic:

fA(p) =gA

e(E−µA)/kT ± 1(2)

with E2 = m2c4 + p2c2 and the chemical potential µA of every particle is null.The chemical potential can be seen as the enery given to the system by one moleof the particle when temperature and pressure are constants. Thus the energy

3. A THERMAL HISTORY OF THE UNIVERSE 9

density is given by:

ρA(T ) =1

h3

∫E(p)fA(p)d3p =

1

(2π~)3

∫E(p)fA(p)d3p (3)

While the temperature still very high, we are at the relativistic limit so E ' pc,then:

ρA(T ) =gA2π2

(kT )4

(~c)3

∫ ∞0

x3

ex ± 1dx (4)

where x = pckT and gA is the internal degree of freedom of the particle A (see

above). By introducing a new system of units where ~ = k = c = 1, theexpression for the energy density becomes:

ρA(T ) =gA2π2

T 4

∫ ∞0

x3

ex ± 1dx (5)

So one just has to multiply by k4

(~c)3 to get the formula with the common units.Thus one then can write:

ρA(T ) =π2

30gAT

4 for bosons (-) (6)

ρA(T ) =7

8

π2

30gAT

4 for fermions (+) (7)

In the non-relativistic approximation, which is valid for lower temperatures (whenthe mass of the particle is no longer negligible compared to the temperature ofthe thermal bath), the Fermi-Dirac distribution becomes the Maxwell-Boltzmannone:

fMBA = e−E/kT

Then, by expanding at the second order in x, the energy density becomes:

ρA(T ) =gA2π2

T 4ae−a∫ ∞0

dxx2e−x22a , (8)

Where a is given by a = mT (the unit system is still ~ = k = c = 1) and

ρA(T ) = gA(mAT

2π2)3/2mAe

−mA/T . (9)

If one compares the non-relativistic energy density of a particle to the relativisticone, one can see that the former is negligible face to the later so it does not haveto be integrated in the calculation of the total density of the thermal bath.

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The effective degrees of freedom geff is obtained by comparing the total energydensity to the energy density of a boson with only one degree of freedom so onecan interpret it as being the numbers of degrees of freedom of a gas of relativisticbosons with one degree of freedom (the plasma, whose temperature is T). Withthe previous approximations

geffT4 ≈

∑bosons

gbTb4 +

7

8

∑fermions

gfTf4

so:geff (T ) ≈

∑bosons

gb(TbT

)4 +7

8

∑fermions

gf (TfT

)4, (10)

As we consider a thermal bath, every particles are in thermal equilibrium sothe temperature is uniform and equal to the temperature of the photons (Tb =Tf = T ). So for example, at a temperature of 1 TeV when every particles arerelativistic, one has

geff ≈ (3+3+3+8×2+2+1)+7

8(3×2+3×2×2+6×2×2×3) = 106.75. (11)

Then, as the Universe expands the temperature starts to decrease and particlesstart to decouple from the bath one by one, from the more massive to the lighterones, which means they are no longer in interaction with the plasma. The firstparticles to decouple are the heaviest ones because they are the first to becomenon-relativistic particles (when the temperature starts to decrease, the kineticenergy decreases (kBT) so the heavier the particle is, the lower its velocity willbe because kinetic energy is by definition 1

2mv2). In first approximation, we will

consider that a particle of mass M decouples from the bath when the temperatureof the bath becomes lower than its mass as depicted in Fig.3.

3.2 Entropy

The entropy density of a particle A is given by the thermodynamics:

sA =ρA + PA

T(12)

where the pressure P can be obtained the same way as ρ. Then:

sA =gA2π2

T 3

∫ ∞0

x2dxy + x2/3y

ey ± 1(13)

3. A THERMAL HISTORY OF THE UNIVERSE 11

106 107 108 109 1010 1011 1012

10

100

50

20

30

15

150

70

T HeVL

geff

Figure 3: Effective degrees of freedom of the primordial plasma as a function of the temper-ature.

where y = E/kT . Thus, at the relativistic limit the plasma can be characterizedby its entropy density

s ≈ 2π2

45gsT

3, (14)

withgs(T ) =

∑bosons

gb(TbT

)3 +7

8

∑fermions

gf (TfT

)3. (15)

For the same reasons as in (10), only relativistic particles participate to theentropy density. While all relativistic particles are still in the thermal bath, onehas Tb = Tf = T and so gs ≈ geff . The entropy of the universe is given byS = sR3 and is a constant (at this moment the universe is the thermal bath),which explains the relation (1). Later we will see the link between changes oftemperature in the thermal bath and the constancy of the entropy.Sometimes, as it is the case for neutrinos, particles do not exactly decouple fromthe plasma when the temperature gets equal to their mass. In fact, one hasto check when the interaction rate Γ = n < σv >, where n is the density ofthe particle, σ is the average cross section and v the velocity of the particlereaches the expansion rate of the universe : H(t)=dR(t)

dt1R . Indeed, by comparing

these two rates, one compares if the expansion of the Universe is faster thanthe collisions between particles or not as they are homogen to the inverse of a

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time. When Γ gets lower than H, this means that there are no longer interactionsenough to keep the equilibrium between particles and the rest of the plasma. Forexample, neutrinos decouple before the temperature gets lower than their mass.Indeed, as the universe expands, the temperature and the density n of a particledecrease so there will have less and less interactions between the particle andthe rest of the plasma (this is the reason why they are said decoupled). In fact,because of the decrease of the temperature,the relativistic density of a particleA

nA = gAζ(3)

π2T 3 (bosons),

nA = gA3

4

ζ(3)

π2T 3 (fermions), (16)

becomesnA = gA(

mAT

2π)3/2e

µA−mAT , (17)

when T ≈ mA, so one can clearly see that the density decreases with the temper-ature. If Γ ≤H for this temperature, we talk about kinetic or thermal decoupling.Meanwhile, if Γ ≤H for a different temperature, it might be because of a chemicaldecoupling which is the case for light and still relativistic particles. In this case,it is the annihilation cross section which is too small to maintain the particlein equilibrium with the plasma as the chemical equilibrium is made throughinteractions between the plasma and the particle (thanks to interactions, thetemperature of the particle can relax to the temperature of the plasma).In what follows the two different cases of the photon (massless) and the neutrinoare considered.

3.3 Recombination and decoupling of the photons

Several reactions take place in the primordial plasma before every particlesand especially photons decouple. For example, approximately 380 000 yearsafter the Big Bang, the recombination of atoms takes place, which means thefirst atoms of hydrogen are created. Before this recombination, other reactionstook place so protons were already created (from quarks during the formation ofthe hadrons, epoch called BBN for Big Band Nucleosynthesis). At that moment,the plasma was barely composed of photons, electrons, protons and hydrogenatoms in formation

p + e ←→ H + γ (18)

3. A THERMAL HISTORY OF THE UNIVERSE 13

Except for the photons, particles are no longer relativistic so their density are

nA = gA(mAT

2π)3/2e

µA−mAT .

According to the chemical equilibrium (18), µe + µp = µH so considering mH ≈mp one has

nH =gHgegp

npne(2π

meT)3/2e

BHT . (19)

Where BH is the binding energy of the hydrogen defined by BH = me + mp −mH = 13.6 eV and ge=gp=2 (spin 1/2), gH=4 (two spin 1/2). The ionizationfraction can be defined as Xe = ne

nBwith nB being the density of baryons nB =

np + nH . Now one has to satisfy the neutrality of the universe so ne = npand Xe = ne

nB. Moreover nB = nγηΩBh

2 (η = 2.68 × 10−8,ΩB = 0.044, h =

0.71 and nγ = 2ζ(3)T 3

π2 because photons are still relativistic) so Xe is determinedby the following equation

1−Xe = X2e

2.68× 10−8 × 4√

2ζ(3)√π

(T

me)3/2e

BHT ΩBh

2. (20)

The plot of Xe as a function of the temperature is presented in Fig.4.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

T HeVL

XeHTL

Figure 4: Ionisation rate Xe =nenB

as a function of the temperature T.

We notice that for T ≈ 0.3 eV the plasma becomes dominated by hydrogen andthere are no electrons free left (Xe ≈ 0) to interact with photons. Precisely, the

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recombination time is defined by the moment when there is still only 10 percentof the electrons. Thus the recombination temperature is Trec = 0,296 eV.

3.4 What happened next ?

As there are not enough electrons for photons to interact with in the plasma,they will soon decouple to be free. The interaction rate of the photons interactingwith electrons (through the Thomson scattering process) is given by:

Γγ = σT cne with σT = 2500GeV −2 and ne = XenB. (21)

Furthermore the Hubble horizon (the expansion rate of the universe) is deducedfrom the Friedmann’s equation for a matter dominated universe (which is thecase at 0.3 eV)

H(T ) = 2.13× 10−42h√

Ω0M (

T

2.37× 10−4)3/2 GeV (22)

with Ω0M = 0.3 being the energy density ratio of the matter today. Comparing

these two quantities we get the temperature when photons decouple from theplasma and so become free to wander in the universe giving us this beautifulpicture of that moment we call "The last scattering surface" better known asCMB (Cosmic Microwave Background)(Fig.5).

Figure 5: The Cosmic Microwave Background observed by the satellite PLANCK taken from[10].

3. A THERMAL HISTORY OF THE UNIVERSE 15

It is the last time the Thomson scattering process can happen, after that thedensity of electrons is too low because of the formation of hydrogen and asatoms are electronically neutral, they do not interact with photons. Solving theequation

2500ne(T ) = 2.13× 10−42h√

Ω0M (

T

2.37× 10−4)3/2, (23)

we get Tdec=0.234 eV which means that the last scattering surface happenedright after the recombination. The relation between time and temperature is

tr ≈ 4.9× 10−6(1GeVT

)21

geffseconds, (24)

in a radiation dominated universe which means T > 1 eV and

tm ≈ 0.09× (1GeVT

)3/2 seconds, (25)

in a matter dominated universe which means T < 1 eV. Thus you notice thatthe last scattering happened 380 000 years after the Big Bang.

3.5 The neutrino decoupling

As we discussed above, the neutrino decouples before the temperature gets equalto its mass which means it is still relativistic at its decoupling time. The inter-action rate of the reactions eν → eν and νν → ee is given by:

Γν ≈ nνG2F

8π× 9.93× T 2, (26)

with GF = 1.1664× 10−5Gev−2 and nν =3

4

ζ(3)

π2gνT

3.

Moreover, in a radiation dominated universe (which is still the case as we will see)the Hubble constant is H(T ) = 1.66

√geffT

2/Mp with Mp=1/√G = 1.2211 ×

1019 GeV being the Planck mass. In first approximation one has

G2FT

5 ≈ T 2/Mp for ∼ 1 MeV.

At this temperature, geff=10.75 so finally T decν =3.57 MeV.After decoupling, neutrinos have no longer interactions with the plasma but keepthe same temperature while the temperature of the photons does not change, as

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they behave the same way (they are relativistic). Thanks to the conservation ofthe entropy, one can deduce the temperature of the neutrinos nowadays. Afterthe neutrino decoupling, the only particles left in the plasma are the electronswhich decouple at T=511 keV. The conservation of the entropy gives

gbefores (T beforeγ )3R3 = gafters (T afterγ )3R3,

with gbefores = 2 + 4× 7

8=

11

2and gafters = 2.

Moreover T beforeγ = Tν so Tν = (4/11)1/3Tγ . Thanks to this result T 0ν = 1.95K

and one can deduce an upper bound for the mass of the neutrinos. Indeed, thedensity ratio of neutrinos nowadays is

Ω0ν =

ρ0νρ0c

with ρ0ν = mνn0ν = mν × 2× 3

4

ζ(3)

π2(T 0ν )3,

because the particle is still relativistic when it decouples from the thermal bath.Moreover ρ0c =

√38πMpH

2 = 10−5h2 Gev/cm3 so

Ω0νh

2 ≈ mν

92 eV,

Now as Ω0M ∼ 0.3 so Ω0

ν ≤ 0.3 and h ≈ 0.71 then

mν ≤ 14 eV.

This upper bound for a stable light neutrino species is called the Cowsik-McClellandbound we can find in [1].

4 The Dark Matter (original material)

4.1 Hot, warm and cold dark matter

Today, the two remaining hot species are the photon and the neutrino becauseof their lightness. This very lightness is another reason to exclude the neutrinoas a viable Dark Matter candidate. Dark matter is a matter which is not visiblebecause it does not interact electromagnetically (in fact, it has to interact veryweakly; indeed researchers look for radiation rays as we will see later). It issupposed to exist only by its gravitational effects. As the neutrino is uncharged,

4. THE DARK MATTER (ORIGINAL MATERIAL) 17

its does not interact electromagnetically nor strongly what makes it a good can-didate for dark matter, at first. In fact, hot species, in other words relativisticspecies, cannot be dark matter candidates because of the structure formation.Relativistic particles are too fast to have shaped structures like galaxies in thepast. However, the neutrino of the standard model is the best candidate for aspecies called "hot dark matter" which cannot constitute the whole dark mat-ter by itself but can participate in a mixed dark matter. Furthermore, as theuniverse is made of 24 percent of dark matter, neutrinos would have to be muchmore heavier to be the dark matter (mν ∼ 10 eV as we saw above). Thus, thestructure formation could be explained by warm or cold dark matter. Nonethe-less, if the whole dark matter were cold, there would be more satellite galaxiesorbiting around the galaxies we observe. Actually another candidate is warmdark matter which has properties between hot and cold dark matter.

4.2 Hot relic densities

The calculation of the neutrino relic density can be generalized to otherspecies which decouple from the thermal bath when they are still relativistic.In fact:

Ω0A =

ρAρc

with ρc = H2 38πG and ρA = mAna(nA = gA

34ζ(3)π2 T

3 for relativistic fermions).

Ω0 =g × g0sgs(xf )

8mT 30 ζ(3)

3πH20M

2p

(27)

with g=gA for a boson and g=34gA for a fermion and xf = m

Tfis the time of

freeze out, i.e. when Γ(Tf ) = H(Tf ).The first idea is to introduce a new particle of matter which does not belongto the Standard Model. For example, a massive neutrino ν ′ of 1 TeV whichdecouples very early from the thermal bath. Indeed, one can increase m byincreasing gs(xf ).First, one can compute the temperature of this neutrino nowadays, as we havedone for the neutrino decoupling. As we suppose that the massive neutrinodecouples when it is relativistic, its decoupling temperature has to be higherthan 1 TeV. The first particle to decouple after this neutrino is the quark topwhen T=mtop=173.34 GeV. The quark top is no longer relativistic and so "gives"

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its entropy to the rest of the plasma when it decays. The calculation is alike tothe one made with the neutrinos from the standard model

gbefores = geff = 106.75 and gafters = 2,

after every particles has decoupled from the plasma and there are only photonsleft. So Tν′ = 0.72 K.

Supposing that the massive neutrino annihilates into the bath in two photonsthanks to the exchange of a massive gauge boson Z’, the interaction rate is givenby Γ ≈ g4T 5/M4

Z′ and then the decoupling temperature characterized by Γ ≈ His

T ≈ 0.1(MZ′

1 TeV)4/3 MeV.

As the massive neutrino decouples when it is relativistic, one has T≥1 TeV soMZ′ ≥ 1.8× 105 GeV. If one compares the interaction rate with the relation (26)one gets a new GF ′ = 1.1× 10−12 GeV−2 (see Fig. 6).

Figure 6: Illustration of an effective coupling with a virtual gauge boson Z’.

Thanks to (27) one can deduce the number of degrees of freedom at the decou-pling time which is roughly the number of particles of the new model. With sucha high mass, one gets a number of degrees of freedom of order 1012 which is fartoo much. Thus, we have to consider lighter particles to have fewer degrees offreedom in order to build a likely model. However, the temperature of freeze-outhas to remain higher than ∼ 2 TeV for several reasons: first because the typicalGUT-like models where the number of degrees of freedom is higher than 106.75(which corresponds to the Standard Model) are broken at energetic scales (∼ 100TeV) and secondly, the decaying of such a particle through the massive gaugeboson would have been discovered in the LHC, where the energies involved arein this order. Thus, to check if such a particle is likely to exist, one can compute

4. THE DARK MATTER (ORIGINAL MATERIAL) 19

the temperature of the freeze-out as a function of the gauge boson mass MZ′ , aswe show in Fig.7.

Figure 7: Evolution of the temperature as a function of the gauge boson mass for differentvalues of the coupling constant gν′ .

Reminding that T has to be higher than 1 TeV, one notices that it is possiblefor the phenomenon to happen because the temperature is still below the massof the gauge boson justifying an effective approach "a la neutrino" analogous tothe neutrino decoupling where Tµ < MZ . Moreover, we would like a mass of thegauge boson around 1015 GeV to match with the GUT scale. For MZ′ ∼ 1015

GeV, one gets a temperature of decoupling Tf ∼ 1013 GeV which could staybelow the temperature of reheating. In fact the temperature of reheating isdetermined the same way as before, by comparing H to the interaction rate ofa particle called inflaton which dominates the universe right before the thermalbath we study here ; this period is called inflation. Then TRH ∼

√MpmI with

mI ∼ 1015 GeV mass of the inflaton, TRH ∼ 1017 GeV. Then, one can deducethanks to (24) the final number of degrees of freedom of a model with a neutrinolike this. The plot of gs(xf ) as a function of the neutrino’s mass is representedin Fig.8.

So for example, if one dilutes by a factor of around 100 the gs of the standardmodel one can get a higher mass for the neutrino in the order of keV, whereasfor the standard model, the neutrino mass cannot excess 14 eV with the goodrelic abundance (Ω0

M ∼ 0.3). In February 2014, the authors of [5] have extracteda monochromatic X-ray signal of 3.5 keV from observations of galaxy clusters by

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2.× 10-6 4.× 10-6 6.× 10-6 8.× 10-6 0.00001

0

1000

2000

3000

4000

5000

6000

7000

Mν '(GeV)

gseff(T

ν')

Figure 8: Evolution of the number of degrees of freedom as a function of the heavy neutrino’smass.

XMM-Newton. It could be explained by the scenario above: a neutrino of 7 keVannihilating into two photons of 3.5 keV each in our case (or, in other scenarios,decaying into one photon and one neutrino of the standard model of 3.5 keVeach). Now, for mν=7 keV you have gs(xf ) ∼ 5000. If one takes a model wherethere are two degrees of freedom for the bosons and 4 degrees of freedom for thefermions, with that value for gs one gets the possible numbers of fermions andbosons in our new model ; as presented in Fig.9.Thus, such a neutrino could be a dark matter candidate.

5. CONCLUSION 21

0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

nf

nb

Figure 9: Possible numbers of bosons and fermions in a model where a Majorana neutrino of7 keV has been added.

5 Conclusion

The observations of the Universe tell us that there is among other things anotherkind of matter that we barely do not know anything about: the dark matter.Thus, the current Standard Model is incomplete and many other models areproposed which can describe the whole constitution of the universe. Thanks todata on the early universe we can deduce some characteristics of the dark matterand then suggest some dark matter candidates as we did in this work. Theobservations can then confirm the theory and refine its characteristics.

During this internship I have discovered the approach of a theoretical physicistin his researches. I have also discovered through my internship supervisor thework of a theoretical researcher and I liked it: he seems passionate by his field,has some freedom in his work and has the opportunity to travel to give andsee conferences. Far from discouraging me, this internship validated my taste intheoretical physics.

22 CONTENTS

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24 BIBLIOGRAPHY

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