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Lopez Honorez, L. (2007). Dark matter: signs and genesis (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des Sciences – Physique, Bruxelles. Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/210692/4/e07112e3-d12a-4be1-a0c8-2707de442dbb.txt
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D 03486
Université Libre de Bruxelles Faculté des Science Service de Physique Théorique
Dark Matter
SiGNS AND GENESIS
Thèse présentée en vue de l’obtention du Grade Légal de Docteur en Sciences Physique
Laura Lopez Honorez
Mai 2007
Université Libre de Bruxelleb
0033BB021 Université Libre de Bruxelles Faculté des Science Service de Physique Théorique
Dark Matter
SiGNS AND GENESIS
Thèse présentée en vue de l’obtention du Grade Légal de Docteur en Sciences Physique
Laura LoPEZ Honorez
Mai 2007
1
Contents
Table of contents iii
Abbreviation List v
Introduction 4
1 Dark matter tools 5 1.1 Friedman-Lemaître-Robertson-Walker model ...... 5 1.2 The evidence for dark matter...... 7 1.3 Thermodynamics in the Early universe...... 9 1.3.1 Boltzmann équations...... 10 1.3.2 Annihilations and relie abundance: approximate solution ...... 12 1.3.3 Coannihilation...... 15 1.4 History of the universe...... 17 1.4.1 Thermal History of the universe...... 17 1.4.2 Add-on to Big Bang Cosmology; Baryogenesis...... 25
2 Constraints on dark matter and some particle physics candidates 31 2.1 Constraints from Direct and Indirect détection ...... 31 2.1.1 Direct Détection...... 32 2.1.2 Indirect détection...... 34 2.2 Constraint from unitarity ...... 36 2.3 Hot and Cold dark matter: distinction from structure formation...... 38 2.3.1 Jeans length and free streaming length...... 38 2.3.2 The power spectrum...... 40 2.3.3 The “Standard Model” neutrino: a hot dark matter candidate .... 42 2.3.4 The WIMPs from the MSSM: popular cold dark matter candidates 44
3 The Inert Doublet Model 47 3.1 The model...... 49 3.1.1 Scalar potential ...... 49 3.1.2 Constraints...... 50 3.2 Dark Matter abundance...... 51 3.3 Inert scalars détection...... 53 3.3.1 Indirect détection...... 53 3.3.2 Direct détection...... 54 3.3.3 Détection at colliders ?...... 55 3.4 Analysis of the IDM...... 56 CONTENTS
3.4.1 Low mass régime, Mhq around Mw ...... 57 3.4.2 The high mass régime, Mfj » Mw...... 60 3.5 Concluding discussion on the IDM...... 63
4 MeV right-handed neutrîno as dark matter 69 4.1 The L-R symmetric model ...... 70 4.2 Constraints...... 73 4.2.1 Stability of the right-handed neutrinos...... 74 4.2.2 Constraint on the decay rate from INTEGRAL and COMPTEE ... 75 4.2.3 Constraints from neutrino experiments...... 77 4.3 Cosmological abundance ; low reheating température scénario...... 79 4.4 The link with Intégral ?...... 82 4.4.1 The 511 keV signal ...... 82 4.4.2 Light dark matter annihilation as a source for positrons...... 84 4.4.3 The MeV right-handed neutrino at the origin of the 511 keV signal? 86 4.5 Concluding discussion on MeV right handed neutrino ...... 87
5 Matter Genesis Mechanism 89 5.1 The Model...... 90 5.1.1 Symmetry breaking and Residual Z2 symmetry...... 91 5.1.2 Fermion masses...... 92 5.1.3 Gauge boson masses...... 93 5.2 The Matter Genesis mechanism...... 93 5.2.1 Initial B-L asymmetry...... 94 5.2.2 Annihilation of the messengers and first contribution to D asymmetry 95 5.2.3 Decay of messengers into vr...... 96 5.3 Constraints...... 97 5.4 Observational implications ?...... 99 5.5 Concluding discussion about Matter Genesis...... 100
Conclusions and perspectives 105
A Conventions 107 A.l System of units and Conversion factors...... 107 A.2 Scalar and fermion fields: normalization conventions ...... 107 A.3 Charge conjugation and Majorana fields...... 108
B Relation between B and L asymmetries 111
C Cross-sections and Unitary bounds 113 C.l 5-matrix unitarity and Optical Theorem...... 113 C.2 Partial wave expansion and Unitarity bounds...... 114 C.2.1 Partial wave unitarity...... 115 C.2.2 Unitary bounds on cross-sections...... 115 C.3 Cross-section - General expression ...... 116 C.3.1 Spin averaged cross-section...... 116 C.3.2 Thermally averaged cross-section...... 117 CONTENTS iii
D Left-Right symmetric model 119 D.l Interaction lagrangian and Feynmann rules ...... 119 D.1.1 Right-handed Charged current ...... 119 D.l.2 Z' Neutral current...... 119 D.l.3 Majorana neutrinos, tricky Feynmann rules...... 120 D.2 The V A A cross-section ...... 121 D.3 Reduced cross-sections for the MeV vr production...... 123 D.3.1 Charged current ...... 123 D.3.2 // ^ vrî^r ■ Neutral current -f Charged current...... 124
E Figures with colors 127
Bibliography 127 IV CONTENTS Abbreviation List
BBN Big Bang Nucleosynthesis BEH Brout-Englert-Higgs CDM Cold dark matter CM center of mass CMB Cosmic Wave Background DM dark matter dof degrees of freedom FWHM Full Width at Half Maximum FRLW Priedmann-Lemaître-Robertson-Walker HDM Hot dark matter IDM Inert doublet model INTEGRAL International Gamma-Ray Astrophysics Laboratory L-R Left-Right LSP Lightest Supersymmetric particle LZP Lightest particle under Zi symmetry MSSM Minimal Supersymmetric Standard Model SM Standard Model SUSY Supersymmetry vev vacuum expectation value WMAP Wilkinson Microwave Anisotropy Probe WDM Warm dark matter WIMP Weakly interacting massive particle VI Abbreviation List 1
Introduction
What is the universe made of? As of today one can not answer to this question with certainty. Nevertheless, some pièces of information can be inferred from observations. The Cosmic Microwave Background analysis, along with the study of supernova and clusters data rather converge for a matter contribution to the total energy density of the universe of ~ 30%. Yet from the confrontation of the Big Bang Nucleosynthesis (BBN) theory with the observed relative abondances of the light éléments (He, D, Li) one can deduce that ordinary matter or baryons, i.e. what we are made of, can only afford 1/6 of the matter content. That would mean that the dominant component of matter is made of some invisible and unknown material, the dark matter!
This conclusion however involves that our understanding of the laws of gravitation is correct. We are maybe misunderstanding some déviation from the theory of General Relativity. Anyway, even if it has not already been detected directly, indications of dark matter are compiling. The existence of daxk matter can be inferred from the flatness of galactic curves of spiral galaxies. On larger scales, the clusters of galaxies (~ 10^^ solar masses) also give significant signs of the presence of non luminous matter. These can be found in the analysis of their mass-to-light ratio, in the study of the X-ray emitting intracluster gas, and from the lensing of background objects. Moreover, one can not wait for baryons to découplé from radiation in order to begin the structure formation. Some other form of matter with no or at least very suppressed electromagnetic interaction is needed.
We know that dark matter should be neutral, stable and rather weakly interacting. The question that arises then is what is dark matter made of? If we limit ourselves to the Standard Model we know that the neutrinos must be part of dark matter. However, bringing together the results of the tritium beta decay experiment, the observations of neutrino oscillation and the measurements of Z boson decay at LEP, one can conclude that the “Standard Model” neutrino masses are small < 2.3 eV and that neutrinos can not account for the totality of dark matter. One is thus led to consider extensions of the Standard Model including new candidates for dark matter.
Before questioning the nature of dark matter, we also hâve to ask how should it be producedt The most commonly assumed mechanism is the freeze-out one. This mechanism supposes that the species, dark matter or not, were in thermal equilibrium in the early universe and decoupled once their interaction rate became unable to compete with the expansion rate of the universe. For the sake of illustration, let us remind to the case of 2 Introduction the “Standard Model” neutrinos. Since they axe weakly interacting, in the sense of SU(2)£^ interactions, they decoupled when the température of the universe reached T ~ 1 MeV. At that time they were relativistic. They were still relativistic at the epoch of matter- radiation equality. As a conséquence, they are Hot Dark Matter (HDM) candidates^. Notice that this is one more argument against neutrinos as a dominant contribution to dark matter. Indeed, HDM leads to a “top down” scénario for structure formation which means that larger structures formed first, and eventually fragmented in order to give rise to the smaller ones. However, the observation of galaxies at rather large redshift (2 > 3) tends to favor a “bottom up” scénario driven by Cold Dark Matter (CDM) candidates. Let us emphasize however that the subject of structure formation will not be discussed much in this work (for a recent review on the subject, see e.g. [1]).
Let us momentarily assume that the standard freeze-out mechanism is responsible for dark matter abondance. The most popular candidates behave as cold dark matter and in most cases they are cold relies. As a conséquence, it can be inferred from their évolution équations that their relie abondance scales like the inverse of their annihilation cross section, ^dm oc l/(cru), and it can be shown that one needs {av) ~ 1 pb in order to saturate the dark matter density (see section 1.3.2). Ail the ingrédients for dark matter can be recovered in the Supersymmetric extensions of the Standard Model and the neutralino, a Majorana fermion, has been adopted as a reference for dark matter with a mass in the GeV-TeV range. Let us stress however that the neutralino is not the only possibility even if we restrict ourselves to the standard freeze-out mechanism.
In this thesis we study the case of a neutral scalar as a candidate for dark matter in the framework of the so-called Inert Doublet Model (IDM). This model is rather more transparent and more easy to handle than the Supersymmetric model since it only includes one additional scalar doublet and one extra discrète symmetry compared to the Standard Model. There are two acceptable mass range in agreement with the DM matter relie density constraint not conflicting with LEP data. The first one is below the W threshold ranging from ~ 40 GeV until ~ 80 GeV while the second one is above ~ 400 GeV and is limited by the unitarity bound on the mass ~ 100 TcV.
This model incorporâtes several interesting features. As it was first pointed out by Barbiéri et al. the IDM can rather naturally include a heavy Higgs with a mass up to 500 GeV and still fulfill the LEP Electroweak Précision Test (EWPT) measurements. Several authors also studied its infiuence on the neutrino sector and the possibility for inert scalars to be at the origin of neutrino masses. We will show that this model can also hâve clear signatures in indirect détection searches even if it will eventually be detected directly with difficulty. This of course dépends on the dark matter density profile in our galaxy and, in particular, around the galactic center. Let us stress that the latter is still a source of intensive searches and debate among the experts. On more general grounds the IDM is a rather simple while complété model for dark matter. For the non initiated, it gives the occasion to get knack of dark matter models. To our eyes, the inert doublet can be considered as a new archétype for dark matter.
^Let us emphasize that the notion of Hot and Cold Dark Matter must not be confused with the notion of hot cind cold relies. Hot (Cold) Dark Matter is relativistic (non relativistic) at the time of matter radiation equality T ~ 0.4 eV. Hot (cold) relies 8U"e relativistic (non relativistic) at the time of their decoupling of the thermal bath, i.e. when Finteract ~ H. Introduction 3
There are other mechanisms for the genesis of dark matter than the standard freeze- out one. What would happen if we one consider a low reheating température? As has been already pointed out by Giudice, Kolb and Riotto, this circumstance would lead to a smaller relie density and would change the dependence of Weakly Interacting Massive Particles (WIMP) abundance on their mass and annihilation cross section. For instance, the standard freeze-out relation ÇIdm oc l/{av) can become inverted fioM oc [av) with candidates that never reach thermal equilibrium densities. Let us emphasize that the reheating température is not necessarily the température associated with the end of the period of inflation and that several reheat events can hâve occurred since then.
In this thesis, we study the case of a MeV right-handed neutrino ur as dominant compo- nent of dark matter (DM). We work in the framework of a Left-Right symmetric extension of the Standard Model and the ur is assumed to be produced in a last reheating period just before nucleosynthesis. We analyze the constraints on the model that arises from a too early decay of the neutrinos in today’s universe and from the solar neutrino experiments. The interest of that kind of dark matter candidate is that through its interactions with the baryons it could be at the origin of the 511 keV gamma-ray line which has been detected by the INTEGRAL experiment at the galactic center. Once again, one can argue that the baryon and the dark matter density profile are rather poorly known in the central régions of the galaxy. Nevertheless, we will show that DM-baryon diffusion into positrons could be more efficient than DM-DM annihilation to reproduce the morphology of the 511 keV signal inferred from INTEGRAL data.
Let us also mention that the production of the dark matter candidate can also be non thermal, resulting from the late decay of some heavy particle of the dark sector. This mechanism sounds like the leptogenesis mechanism in which some heavy Majorana particle produces a lepton asymmetry that is transmitted later to the baryon sector through non-perturbative effects. We will take advantage of this plausible parallelism between the production of ordinary and dark matter. Through the non thermal and CP violating decay of one single Majorana particle of the dark sector we generate an asymmetry in both the visible and the dark sector. The latter will be the source of ordinary and dark matter abondances, some sort of “Matter Genesis” mechemism. Indeed, ~ 0{1) could be the sign of a common origin for both dark and ordinary matter.
This beautiful idea has been already discussed by several authors, is however rather difflcult to realize in a simple way. Our “Matter Genesis” scénario takes place in a particular Left-Right symmetric framework with some of the ingrédients of the so-called “universal seesaw” model. The dark matter candidate is a right-handed neutrino again, but this time with a mass in the GeV range. The main drawback of this model is that the dark matter mass range cornes in as a constraint, not as a prédiction and that dark matter candidate must be very (very) weakly interacting.
I think that the trails that we will follow among the signs of dark matter and its possible genesis mechanism is rather cleax now. Let me rapidly overview the content of this thesis chapter by chapter. 4 Introduction
In Chapter I, we review the cosmological framework and the évidences for dark matter. We also re-derive the necessaxy tools in order to estimate the abundance of species. Armed with ail these tools, we will run through the history of the universe and look deeper into some particular events: the epoch of recombination, BBN and the baryogenesis.
In Chapter 2, we study the bounds that can be put on dark matter models from direct an indirect searches as well as from the unitarity of the S matrix. After recapitulating the basis of structure formation, we define the notion of Hot and Cold Dark Matter and investigate one typical example for each case: the neutrino and the WIMPs of the MSSM.
In Chapter 3, we investigate the Inert Doublet Model and its repercussions for direct and indirect détection searches.
In Chapter 4, we explore how a MéV right-handed neutrino can account for dark matter within a low reheating scénario. We also study the advantages of that kind of candidate in order to reproduce the INTEGRAL signal.
In Chapter 5, we look for a “Matter Genesis” mechanism explaining the rather similar relie abondances of ordinary and dark matters. 5
Chapter 1
Dark matter tools
In this chapter, we introduce the cosmological framework and study how the évolution of the universe is dictated by its constituents. We will look at the diverse hints about the existence of dark matter which hâve been put forward over the years and which are still under study today. We will finally develop the mathematical tools necessary to estimate the abondance of species and show how it models the history of the universe. The cos- mic microwave background is a precious witness of the original conditions and so is the abondance of light éléments in the universe. Along with the observation of the Hubble expansion these two subjects give solid grounds to the model of the Big Bang. They will among other things underline the necessity of a baxyon asymmetry production in the re- mote past. The latter, however, can not be explained in the framework of the big bang theory and we will bare the necessary éléments for baryogenesis to take place.
1.1 Priedman-Lemaître-Robertson-Walker model
Before to deepen the subject of dark matter, let us briefly recall the cosmological frame work given by the Priedmann-Lemaître-Robertson-Walker cosmological model (see e.g. [2], chapters 2 and 3 and [3] chapter 2). Moreover, let us emphasize that throughout this work we will use the natural System of units, with c, k, and h each equaling 1. For conversion factors in these units see table A.l in annex A.
The Cosmological Principle States that the universe is homogeneous and isotropie. This statement has been confirmed by observations at largest scales. The metric describing the geometry of spacetime and satisfying these conditions can be written in the following form: dr^ — g^iidx^dx'^ = dt^ — a{t) 4- r‘^{d6^ 4- sin^ 9d(l)) (1.1) 1 — where a{t) is the scale factor, k measures the spatial curvature, it can be négative zéro or positive corresponding to open, flat and closed Universes respectively. l = (r, 0,
The matter-energy content of the universe is described by the stress energy tensor. To satisfy the symmetries it takes the form: = diag(p, p, p, p) where p is the energy density and p is the pressure. The conservation of the energy momentum tensor implies
d =—p d . (1.2)
We can extract the évolution of the diverse fluids (matter, radiation, ...) with expansion by introducing the équation of State p = ujp in équation (1.2). After intégration, we obtain: p oc
The geometry is related to the matter-energy content of the universe by the Einstein équations. In the FLRW framework, it translates into the so-called Priedmann équations:
Stt k (1.3) 3mp| ^ à 47T , „ , (1.4) a where overdots indicates dérivatives with respect to time and mpi == 1.22 10“^^ GeV is the Plank mass. The first équation relates the expansion rate, called the Hubble parameter H = à/a, to the energy density and the spatial curvature of the universe. If we plug the solutions of the conservation équation p oc into this équation, for k = 0, we get a oc i2/3(i+w) resuit, for a radiation dominated era a oc and H = ^ and for a matter dominated era a oc and H — The second Priedmann équation relates the accélération of the expansion to the energy density and the pressure. The first équation can be re-expressed in the following form:
K n-i = (1.5) a^H^' where fi = p/pc with pc = Srrip^H^/Stt, the critical energy density accounting for a fiat universe {i.e. k — 0). Today, we hâve pc = 8.1 lO^^^fi^GcV'* and we write Hq — lOOfikms^^ Mpc~^. Observations give h w 0.7 [4].
To specify a particular epoch of the universe history, we will generally use the tem pérature which is decreasing with expansion. It can also be characterized by its redshift. Indeed, for a wavelength \emit emitted at time te and observed as Xobs at time to, the redshift parameter z is defined as: 1 + z = Xobs/^emit- In PLRW cosmology, the redshift is related to the évolution of the scale factor a{t) in the following way:
Cl{to) 1 + z = (1.6) a{te) and today’s scale factor uq is chosen to be equal to 1. Another chaxacteristic would be the âge of the universe at that period, defined as
dajt') ^ ~ Jo à(t') ■ A general expression for the Hubble rate at any time t, given équation (1.3), is:
H^(t) P(i) ^0^ Pc i p<^ i ' 1.2. The Evidence for dark matter 7
where pc = Smp^HQ/Sn is the présent critical energy density, pip are the contributions to today’s energy density from several sources such as matter, radiation,..., and tUj are their équation of State parameter. We bave used the fact that uq = 1, îli,o = Pi,o/Pc and the définition of the redshift (1.6). The âge of the universe can thus be rewritten as:
rait) t = H,0 dx- i.QX -l-3wi,o
This expression permits to test the composition of the présent universe as old globular clusters and nuclear chronology give close values of the âge of the universe: ~ 14 Gyr. It can be shown that this value can not be reached for a fiat universe filled with matter only, indeed for Qtot = ^ = — 5/i“^ Gyr. Adding a cosmological constant A, the universe can be made older:
A 6.525/1 ^Gyr 1 T v^IIa t0 ------;__ In (1.7) v7l — Oa where we took i'itot = 1 = 11m + ÎIa- For an estimate of these parameters see section 1.4.1.
1.2 The evidence for dark matter
More than one piece of evidence for non radiating matter has been put forward from scales ~ kpc to Hubble radius by the observation of diverse luminous objects. In 1930’s, Zwicky was the first to pick up evidence for dark matter from the observation of the mass-to-light ratio of the Coma Cluster of galaxies. At that time, the mass-to-light ratio for some of its spiral galaxies were available and estimated to M/L ^ 1}i{Mq/Lq)y where the 0 refers to the Sun, V refers to the visual band and h was estimated to be 5.58. The total luminosity of the Cluster was also known. Zwicky applied the virial theorem^ to the Coma cluster knowing the velocities of seven of its galaxies, gave an approximate value for the Cluster radius and determined this way the total mass of the Cluster. He obtained a mass-to-light ratio for Coma Cluster of M/L ^ bt)h{MQ/Lq)v- He made several errors in its estimate but the factor 50 between individual galactic M/L and the Cluster one remains, even considering an updated method. Zwicky concluded that a large amount of non luminous matter was necessary to accelerate the galaxies at the observed velocities.
For the sake of completeness, let us mention that one can evaluate the mean lumi nosity of galaxies £ ~ 2 ± 0.2h{LQ Mpc~^)y and define the critical mass-to-light ratio as {M/L)c = Pc/^ — 1OOO(M0/L0)v (see t.g. [6] or [5] for a dérivation and references there in). The matter density parameter can be expressed as LLm — {^/L)v/{M/L)c, the mass-to-light ratio is thus an indicator of the matter density. For clusters, we get LLm ~ 0.3 though in the solar neighborhood ÎIm ~ 0.001.
Apart from clusters mass-to-light ratio, one can also work out the total mass permitting to bind the observed X-ray emitting hot gas. Assuming hydrostatic equilibrium, one can
^The virial theorem is defined for a finite collection of point particles interacting via Newtoniain me- chanics for which positions and velocities are bounded for ail times. It guarantees, if the System is in a stationary state, that 2{Ec)t + (Ep)i = 0 where (..)( stands for time average, Ec (resp. Ep) is the total kinetic (resp. gravitational potential) energy. See e.g. [5] for a discussion. 8 1. DARK MATTER TOOLS relates the température of the gas of the Cluster mass and verify that luminous mass is not enough [7]. Moreover, the baryonic mass content of clusters is dominated by X-ray emitting intracluster gas and it was argued that the resulting estimate of the baryon mass fraction in cluster fs — M^/Mgrav should not differ from the universal one fs = ^b/^m [8]- In [9], Allen et al recently derived a matter density paxameter consistent with Qm = 0-3 from Chandra observations.
Gravitational lensing data is still another argument in favor of non luminous matter. Indeed, image distortions of background galaxies appearing on the sky map are driven by the gravitational mass distributed along the line of sight. Strong lensing effects are due to the presence of a foreground cluster generating multiple images of a single background galaxy shaped as axes and axclets. Weak lensing effects are less spectaculax giving rise to one single image. Studying these effects probes cluster masses and properties, agréé with the requirement of a dark matter component. For more informations on gravitational lensing see e.g. the following recent reviews [10, 11] and references therein.
= -21.8
Figure 1.1: Synthetic rotation curves for galaxies with magnitude (M) = —21.2. Dotted curve show disk contribution and dashed line show the halo one. Rppt is defined in [12] as the radius encircling 83% of the integrated light.
The most popular example for dark matter evidence is given by spiral galactic rotation curves, i.e. the rotational velocity of stars v plotted as a function of the radius of their galactic orbit r (see figure 1.1). Indeed, assuming the stellax System in virial equilibrium, one can get a relation between the mass inside the orbit M(r) and the velocity at radius r: _ GM{r) r If light is supposed to trace masses, this would imply that the velocity of stars decreases as v{r) oc l/^/r when they are moving beyond the point where most of the light ceases. However, the observation of the 21 cm line émission of neutral hydrogen gas situated in the outer région of the galaxies shows a rather constant dépendance of the velocity as function of r, v{r) oc est as can be seen in figure 1.1. One solution to this inconsistency could be the presence of a dark halo, both in inner and outer régions, dominating stellar dynamics at large distances, with Mhaio(^) oc r or equivalently pi,aio(’') oc 1/r^.
The Cosmic microwave background provides a test of both the matter and the baryon 1.3. THERMODYNAMICS in THE Early universe 9 densities while Big Bang nucleosynthesis put constraints on the baryon to photon ratio. As we will see in section 1.4.1, these two different probes agréé ohÇIb — 0.023 and CMB gives Üm — 0.13. The discrepancy between these two numbers advocates once more for the existence of non luminous and non baryonic type of matter which would amount for 85 % of the total matter content of the universe.
Structure formation and galax;y power spectrum further require the presence of dark matter likewise testing dark matter nature. We will deepen these subjects in section 2.3. Our understanding of the large scale structure and our knowledge of our own galaxy is however far from satisfactory. Numerical simulations suggest a wide range of dark matter density profiles (see section 2.1). Let us emphasize that currently the debate is open especially on the galactic core shape. The profile for the inner région of the Milky Way is still uncertain for both the dark matter and the baryons (see also chapter 4). The local density of dark matter piocai in onr neighborhood, situated at a distance Rq — 8.5 kpc from the galactic center, is less uncertain than the one of the galactic center. The local density piocal and the velocity dispersion of dark matter particles v — are derived from the rotation curve of our galaxy (for a discussion see [13] section 2.4). Our position in the Milky way is neverthelesfe not simplifying the measurements (it is much easier to measure rotation curves in other galaxies than in our own) and the errors following from that source of uncertainty is rather important. The local density is estimated to be Piocal — 0.3 GcV / cm^ within a factor 2 and the velocity of dark matter particles at the location of our solar System is about v ~ 22Qkm/s.
1.3 Thermodynamics in the Early universe
In order to describe évolution of the particles abundance throughout the history of the universe, we express their number density in term of their phase space distribution, f{x,p). The latter can be interpreted as the occupancy probability of quantum modes in a given région of phase space with position x and momentum p. The number density and the energy density for a particle tp is given by :
J Ui^,P)d^P, Pi> = j E{p)f^{x,p)(Pp (1.8) where counts the number of ip internai degrees of freedom (dof). In a FLRW model the phase space is homogeneous and isotropie so that: f = f{\p\,t) = f{E, t), and for particles in kinetic equilibrium, / is given by:
" e{E-ix^)IT ^ 1 which is the familiax Fermi-Dirac (+1) or Bose Einstein (-1) distributions and is the Chemical potential. When there is Chemical equilibrium, the sum of the Chemical potentials of initial and final States in any process are equals.
The Einstein équation for an homogeneous and isotropie universe implies the relation (1.2), so that the entropy density s times the comoving volume is conserved (see e.g. [2] 10 1. DARK MATTBR TOOLS and [3]): d{sa^ P + P with (1.10) dt T as a conséquence, s scales like a The entropy density is dominated by relativistic particles contributions for which p is calculated with (1.8):
TT Prel =
with g* = 9i (1.11) T j ^ 8 E i=bosoiis '' z=fermioiis where % is the number of internai degrees of freedom for species i and Tj/T is the ratio be- tween the relativistic species températures and photon température, g* counts the number of relativistic degrees of freedom^. As a conséquence, one gets with p = p/3:
s g*sT^ 45 with (1.12) E + i E ». 2=bosons ^ ^ z=fcrmions
For most of the history of the universe the ratios Tj/T are equal to one so that g^s — 9*- For example, at températures between the électron and the muon mass, (0.5 MeV < T < 100 MeV), the e^, the photons and three types of neutrinos contribute, so that g*s = (2 + 7/8(2 *2 + 3*2)) = 10.75. At earlier times (for T < m*), we hâve to add the contributions of gluons, 5 quarks and anti-quarks, p,, p,, r and f so that g*s = 91-5. Today, the électrons are non relativistic and we hâve to take into account the ratios between photon and neutrinos températures. Using Tq = 2.75 K, we get ,g*(To) = 3.36, ,g*s(îo) = 3.91 and s(To) = So = 2970 cm“^.
1.3.1 Boltzmann équations
For a collisionless gas of particles in non-relativistic kinetic theory, the évolution of the phase space density, is governed by the Boltzmann équation:
rrri ^ -1 T ^ dx d dv d L+1=0 with 1.=- + (1.13)
It asserts that the distribution fonction is constant along any trajectory in phase space. After relativistic generalization of the Liouville operator L, using FLRW metric and inte- grating over the physical 3-momentum p, Boltzmann équation translates into:
9i> d à \p\^ d _3Ô(a^n^) = 0 with = 0 (1.14) (27t)^ di^a~WdË “ dt i.e. in absence of collision the number of particles in comoving volume is constant.
^for detailed values of g, through the history of the universe, see [14] in the section 19.3.2 of the Big Bang Cosmology review 1.3. Thermodynamics in THE Early universe 11
When we take collisions into considération, the évolution équations take the form
HU] = c[/^] where C[f^] is the collision operator. Let us study the évolution of the number density of a particle ^ involved into processes of the type: 'ijj + X ^ Y where X and Y can be multiparticles States. The collision term is given by:
J = - J d^^d^xd^Y i‘^Tf)^iS‘^{Pi^+Px-PY) Ufx + \My (1.15) where px is the sum of the momenta and fx is the product of the phase space densities of particles in State X, and
d^Pa d^x = n d^a with (1.16) (27r)32Ea ■ aex
The same kind of définition hold for py, fy and d^y. The probability of transition t/j+X —>■ Y (resp. Y tp Y X) is encoded into the matrix element squared \M^j^x^y\^ (resp. |Ady_».0+xP) which is summed over initial and final spins and includes the appropriate symmetry factors l/n‘^! for identical particles in the initial and/or final States^.
Hereafter, in this section, we will assume CP (or equivalently T) invariance, which implies: \M^^x-^y\^ = and we will write the matrix element Also, we will study Systems at températures for which E — ^ T, so that we will use the Maxwell-Boltzmann approximation for the phase space densities (1.9):
MW, E _E T = -----^ P eq U{E) = e r e‘ eq ^ with = (1.17) n^, where is tp equilibrium number density^. For non relativistic particles, T m^, the equilibrium density is exponentially decreasing oc exp(—m^/T) while for relativistic particles, T » m.^, it scales like T^. Then, using energy conservation E^p + Ex = Ey, the Boltzmann équation becomes:
■ . orr^ _ eq ( ny n^Ux n-ijj Y oHn^ — 7w,+X—*Y 1 eq eq eq ^ \^Y V^X
with 7^+x^Y = J d^^d^xd^y e ^ {p^ Y px - Py) |M|f.l8) with nx = riaex = YlaeX while équivalent définitions hold for ny and Ey.
^ is related to the matrix element squared defined in Eq. C.15 in the following way :
“ ■ aea
‘‘When annihilation processes are in equilibrium, particle-antiparticle processes: ff <-> 77 occurs rapidly and Pf Y Pf = 2py = 0. Moreover, it canbe shown that n/ —is characterized by the ratio Pf IT,as a conséquence, if there are no asymmetries: pj = Pf = 0, what will be assumed at thermal equilibrium in the sequel. 12 1. DaRK MATTER TOOLS
1,3.2 Annihilations and relie abundance: approximate solution
In the sequel, we will concentrate on 2 body-2 body processes, 1 2 34, in which case the Boltzmann équations take the form:
nan4 niU2 ^ eq ni + ZHïii = {av) eq „eq ëq with {av) = ^eq^eq7l2- >34 (1.19) TI^Ua where (av) is the thermally averaged annihilation cross-section times the relative velocity V in the initial State. The rate of interactions is F = n2{av). When F
In generic scénarios, the weakly interacting massive particle (WIMP) x stayed in ther mal equilibrium with the rest of the plasma by annihilating into lighter Standard Model (SM) particles l: XX ^ Let us suppose ni = n®'’ for the period of interest^ and = n^. The Boltzmann équation takes the familiar form
-|- 3Hn^ = {aAv){n^'^ — n^) d^X ^ {(^AV)SX 2 _ y2) (1.20) dx ^ ^
with = n^/s, s is the entropy density, the dimensionless parameter x = m-^/T^ H{x) ■= H{m-^)x~'^ during radiation dominated era and we hâve used dx/dt = H x. The differential équation can be integrated over time numerically, but it is possible to get ap proximate solutions for relie density by using Lee- Weinberg standard calculation. When the expansion rate becomes larger than the annihilation rate, H > Va, the annihi lation processes freeze-out. There after, the abundance of x in a comoving volume stays constant as a conséquence of 1.14: Y{x > Xf) = Y{xf) with Xf = rny./Tf and Tf the freeze out température. The relie abundance calculation differs for hot and hot relies i.e for particles that freeze-out while being relativistic or not.
Hot relies
For hot relies, the WIMP density today is given by its equilibrium value at the time of freeze out: T(xo) = Y^{xf) and
SQY^^{xj)mAp siw, = ------(1.21) Pc where «o is the value of the entropy density today. One can show that [2]
for fermions , Y^\xf) = Q.27%g,ff/g.s{Tf) with | (1.22) for bosons ,
® The lighter l particles usually hâve additional interactions which are more efficient to keep them in thermal equilibrium than their interaction with y. 1.3. ThERMODYNAMICS in THE EARLY UNIVERSE 13 where g counts the number of internai dof of the relie species. The dependence of Y^^{xf) (and thus in the freeze-out epoch for relativistic species thus only appeaxs through the number of relativistic dof contributing to s at that time, 5*5(Ty).
One typical example of cold relie is given by light, weakly interacting neutrinos {i.e. interacting through SU(2)x, interactions). When ~ G'pT^/{T'^/m-p\) — 1, weak interaction processes of the type vv <-» e+e“ freeze out so that ~ MeV. As a consé quence, neutrinos with masses smaller than a few MeV behave like hot relies, g^s — 10.75 and using today’s critical density and the entropy today sq (see eq.(1.5) and (1.12)), we get:
(1.23) 92eV Imposing < 1, one gets the so called Cowsik-McClelland bound [15] on neutrino masses: Ylj < 92/i^eV. Considering the case of dark matter heing exclusively made of light neutrinos and using the experimental bound on < 0.129, we get the following cos- mological bound on the neutrino masses: Ylj ^ 12 eV. For a more detailed discussion on dark matter made of neutrinos see section 2.3.3 and chapter 5.
Cold relies
For cold relies, the présent day abondance of particle species is determined by annihilations after a particle species has become non relativistic. In order to get an approximate solution for the differential équation (1.20), we use a Taylor expansion of (cr^n) as a fonction of the relative velocity v or equivalently as a fonction of the températures given that the equipartition theorem gives: {v^) oc T. It can be shown that the expansion of (cr^u) as a fonction of x takes the form®:
{(tav) = a + h{v^) + 0{v‘^) « a + 6- + O(x^). (1-24) X For the following calculations, we will write {(Jav) ~ For annihilation processes dominated by s-wave contribution n = 0 and â = a, while for those dominated by p-wave contribution n = 1 and â = 66, etc.
Let us study the physics of the équation (1.20) for cold relies. At early times, when X 1, the annihilation processes occurs rapidly enough so that Y tracks Ri constant. Once the température drops below the mass, a: > 1, F®'’ becomes exponentially (F'^'i oc e~^) suppressed and the particles becomes too diluted to keep their equilibrium density with the help of annihilations: F^ = n^{aAv) drops below H, particles découplés from the thermal bath. The température at which freeze out occurs, Tf, roughly correspond to the time when F^ and H are of the same order, generally Tf ^ m^/25. It can be shown that the freeze out point is given by: 0.0038mpi g^ rn^{av) Xf (1.25) = '"( y This équation must be iteratively solved. For ce !» 1 the évolution équation becomes:
—Xx —n—2 with (1.26) dx H{m^)
6 (see appendix C.3 for more details) 14 1. DARK MATTER TOOLS
Figure 1.2: General behavior of the relie comoving number density of cold relies F as a funetion of time ~ x depending on the annihilation eross-seetion around thermal freeze out.
where s = sx^. Integrating out from x — Xf to x = oo and using the faet that usually Y{xf) » Foo, we get Foo = (n + Using fl = soYoom^/pc and inserting ail the appropriate numerieal factors we obtain:
, l.OTlO^GeV-^ (n +l)x’l+^ O 1___ ^ {g*s/^*)mp^c7
The general trend of the relie eomoving number density as a funetion of the annihilation eross-seetion is illustrated in figure 1.2. It is sometimes useful to estimate an order of magnitude for (cru) whieh would give a relie density eonsistent with eurrent estimâtes of dark matter density, i.e = ^DMh^ ~ 0-1. Using typieal values x/ = 25 and 9* = 9*S = 100, we should hâve â ~ 10~®GeV“^ ~ 1 pb for eold dark matter annihilating espeeially trough s-wave proeesses.
One typieal example of eold relies is given by nueleons. Assuming a baryon symmetrie universe, nueleons and antinueleons remain in thermal equilibrium until Ty ~ 22 MeV, and their annihilation eross seetion â ~ m~^ is strong enough to give Fqo ~ 10“^^. Today’s observed abundanee of nucléons is however of order of tib/s ~ r]/7 ~ 10“^^. Usual scénario of eold relies lead to a real “annihilation catastrophe” for nucléons, their relie abundanee estimate being ~ 9 order of magnitude under the observed one. To avoid this, we will need to introduce a baryonic asymmetry in the past, see section 1.4.2 for more details and chapter 5 for application to baryon-dark matter correlated production mechanism.
Another typieal example of eold relies is the heavy neutrino, interacting with the other SM particles trough SU(2)l interactions. Their averaged annihilation eross-seetion is s- 1.3. Thermodynamics in THE Early universe 15
wave dominated for Dirac neutrinos and p-wave dominated for Majorana ones (their s-wave contribution is suppressed by a factor m;/m^ where rrii is the mass of light SM fermions), but for the two of them a ^ G^.m.^/27r. It can be shown, requiring ilu+û < 1,
(5) GeV
for Dirac (Majorana) neutrinos, this is the so called Lee-Weinberg bound on neutrino masses [16]. Remember that in the previous Hot relie section, we found a upper limit on SU(2)£, interacting neutrinos, as a conséquence, to be cosmologically acceptable weakly interacting neutrinos must hâve their masses greater than 2-5 GeV or smaller than 92 eV. To avoid these bounds, one can play with extentions of the standard model giving rise to different annihilation processes and/or modify the “production” mechanism. In this work, we will study Left Right symmetric extensions of the SM in which ~ MeV neutrinos in a low reheating température framework, see chapter 4, and ~ GeV non thermally produced neutrinos, see chapter 3, becomes more acceptable.
As pointed out in [17], the method we just described fails in some particular cases. We are going to study the coannihilation problem in more details in the next section.
1,3.3 Coannihilation
Let us study a set of N particles, xi> •••> Xn which are nearly degenerate in mass and which differs from SM species by a multiplicatively conserved quantum number (like for R-parity in SUSY). We will assume that their masses axe labeled in such a way that rrii < rrij for * < j, the lightest being xi- If the mass différence Aruj = mj — rui, is small enough, Le. Arrii Tf m\l2b, the particle Xi is thermally accessible to xi at the epoch of freeze out and can play a major rôle in the détermination of xi relie density. If xi is stable, it is a candidate for dark matter.
The évolution of the number density of the zth particle of the set is going to be influenced by the following processes:
XiXj^ll' Xi^^Xjl' Xj^Xill' where l and V are SM particles lighter than the xi- The abondances of the Xi is regulated by a set of N Boltzmann équations:
ûi = -SHui - '^WijVij){ninj - jW - - n^7Î^) - {a'jiVij)(njni> - n^"‘n)^,"‘)] j^iW - E - nf) - - nf)] (1.28) with (Tij = aiXiXj ^ II'), o'ij = (^{XilXjl'), 1^ji = '^{Xj ^ Xill') 16 1. DaRK MATTER TOOLS and the relative velocity Vij is defined as:
Vij — EiEi where pi is the 4-momentum of particle Xi- Since ail the \i which survive annihilation decay into the lightest one xii the relevant number density to be considered is n = 'Ym=i By summing the N équations (1.28), we obtain:
N n = -3Hn - (1.29) i,j=l 1,1'
where ternis in the second and the third line in (1.28) cancel in the sum.
If we suppose that the cross-sections aÿ are of the same order than a[p the reactions Xi^ ^ Xjl' will occur faster than the XiXj ^ ^because the equilibrium density of SM particles 1,1' lighter than XiXj is higher. Indeed, we are working around the freeze out température and as the set of x particles is supposed to be nearly degenerate, they are ail non relativistic and their abondance (equilibrium density) is exponentially suppressed compared to the SM particles ones which are still relativistic. As a resuit, the following relation holds for the reaction rates: nirna'^j riirijaij, and because it is the reaction XiXj II' that déterminés the freeze out, we can approximate the ratios rii/n by the equilibrium one The évolution équation takes finally the form:
N ^eq n = —3Hn — {aeffv){n^ — n®'*^) with {oe/fv) = i,j=l 1,1'
The freeze out point is given this time by:
\ y/9^ J ^ (1.31) If we don’t take into account the coannihilations, the relie density for the lightest x can be slightly underestimated. Even if ail the (Tjj are equals, there can be a différence in the Xi relie abondance given the change in 7/ due to the replacement of by ,9e//- We will see some effects of coannihilations in chapter 3 in studying the possibilities for a neutral scalar to be dark matter. In that case, we hâve computed the relie densities using micrOMEGAs2.0, a new package for numerical calculation of thermal relie abondances of dark matter for any model containing a discrète symmetry to differentiate the dark sector from SM species. This code uses the trick presented in this section, i.e. given that ail daxk sector particles will finally decay into the lightest one, ail the Boltzmann équation determining their abondance évolution are summed giving only one équation of the same kind than (1.30). The dark matter relie density is obtained by integrating the latter from T = 00 to today’s température T = Tq. 1.4. HISTORY OF THE UNIVERSB 17
1.4 History of the universe
In this section we will study the constraints on the matter-energy content of the universe given our understanding of its évolution. We will first go trough its thermal history ac- cording to hot Big Bang theory and dérivé constraints on its matter content, especially from the BBN and the CMB. We will go further on, by studying one of the unanswered questions in the Big Bang’s framework, Baryogenesis.
1.4,1 Thermal History of the universe
Our understanding of the universe today is based on the successful hot Big Bang theory. The first observational evidence about the expanding universe was given in 1929 by E. Hubble. He observed that galaxies were receding from us with a speed proportional to their distance, the proportionality constant accounting for the Hubble constant. This expansion does not provide enough évidences for the hot Big Bang theory. The formulation of the model really began in 1940s when Gamov and his collaborators pointed out the possible origin of light éléments abondances by synthesizing nuclei in a primeval hot and dense bath of particles. The prédiction for relie background radiation just followed Gamov’s suggestion. We saw in the previous section that species are in thermal equilibrium in the early universe as long as their interaction rate T is larger than H. Once F < H, the particles découplé. This is the trigger mechanism for Nucleosynthesis and Cosmic Microwave Background (CMB) émission.
The history of the very early universe from the Big Bang until it reached a température of ~ 10^ — 10^ GeV is still rather spéculative. Indeed that température range corresponds to the energy range accessible today in particle accelerators. Before that we hâve no evidence of the applicable physical laws and we must use extrapolations based on our current understanding of the particle physics. There should hâve been a period of inflation in order to explain for example the quasi isotropy of the Cosmic microwave background or the fiatness of the universe (for more details see e.g. [18]). There should also hâve been a sériés of spontaneous symmetry breaking. During spontaneous symmetry breaking phase transitions some gauge bosons and some particles aquire a mass through the Brout-Englert- Higgs mechanism and the gauge symmetry relevant before is broken to a lower symmetry [19, 20, 21]. In this work, we make reference to the Left-Right symmetry breaking (see chapter 4 and 5) and to the electroweak symmetry breaking (EWSB). Let us draw without being thorough a time-line for the thermal history of the universe since then:
• T ~ 10^ GeV; Electroweak Symmetry Breaking (EWSB): the Standard Model gauge group SU(2)^ x U(l)y breaks into U(l)g. •
• T ~ 0.3 GeV: QCD phase transition: quarks and gluons become confined into hadrons.
• T ~ 1 MeV: Decoupling of the neutrons.
• T ~ 100 keV: Big Bang Nucleosynthesis (BBN): protons and neutrons fuse into light éléments D,^He,'^He and ïii. 18 1. DARK MATTER TOOLS
• T ~ 1 cV: Matter-Radiation equality, structure formation begins.
• T ~ 0.4 eV: Photons decouplings, émission of the CMB.
• T ~ 10-^ eV~ 2.7K: Today.
In the following we are going to study BBN and CMB in more details and présent their prédictions about the universe content, more details can be found e.g. in [2, 3, 22, 14].
BBN
The Big Bang Nucleosynthesis (BBN) theory predicts the universal primordial abondances of light éléments, D,^He,'^He and which are fixed after the “first three minutes” of the universe. These prédictions are in good agreement with the abondances inferred from today’s observations. Here we briefly describe the BBN theory and its implications on the baxyon density and new physics.
To get these results, one should solve a set of coupled differential équations, one équa tion of the type of 1.19 for each element. It is however possible to understand qualitatively the situation. Before going through the theory and the observations, let us introduce some définitions. It is convenient to work with the mass fractions of a nucléus made of A nucléons (out of Z proton and A — Z neutrons) defined as:
Aua nw where ua is the number density of nucléons with atomic number A and the total number of nucléons un = rin + Up + '^i{^nA)i- If we suppose that for the period of interest our nucléons are nonrelativistic and in thermal abundances, ua is suppressed as usual by the Boltzmann factor. Using Chemical equilibrium relationship between the Chemical potentials of nucléons n, p and nucléus, we can re-express the argument of the exponential factor in term of the binding energy of the nucléus: Da = Zmp -|- (^ — Z)mn — tua, and the mass fraction shows the following dependence:
Xa oc 77^^“^ (—J with 77 = ——— (1-32) \ nip J
The factor 77 gives the comoving baryon number density (up to a factor 7) as well as the necessary baryon asymmetry necessary to avoid “the annihilation catastrophe” (see section 1.3.2).
At T > 1 MeV, the cosmic plasma is made of relativistic 7, e* and v, and non relativistic baryons. The number of relativistic dof is thus: 5, = 2 + 7/8(4 + 2Nu) where Ni, is the number of neutrinos flavor and the total energy density in this radiation dominated period is given by: p = 7r^/30g'*T'^. At these températures, the weak interaction rate is larger than the expansion rate and reactions such as:
71 + P + n + Ue<^p-\-e~ , n ^ p + e~ + Pe (1.33) 1.4. HISTORY OF THE UNIVBRSE 19 keep the neutrinos in thermal equilibrium.
At T ~ 1 MeV= Tf, the interactions (1.33) freeze ont, neutrinos découplé and using (1.8), we extract the neutron to proton ratio at that time: n„/np = cxp{—Q/Tf) with Q = rrin — rup = 1.293 MeV. This ratio is sensitive to ail known interactions (!): Q dépend on both strong and electromagnetic interactions, and Tf dépend on weak as well as gravitational interactions. Nucleosynthesis begins well after the température drops below the binding energy of the light éléments. Indeed as it can be seen in équation (1.32), the nucléon equilibrium abundance becomes of order of the baryon abundance when the température dépendent factor ~ exp(Syi/T) compensâtes the very small baryon to photon ratio t]. The formation of the Deuterium begins the nucleosynthesis chain and the Deuterium binding energy is Bd = 2.23 MeV.
When température falls below T^uc ~ 0-1 MeV, light éléments can form without being immediately photo-dissociated. In the meantime the neutron to proton ratio ratio dropped to 1/7 due to occasional beta decay of the neutron. Since the binding energy of '^He is larger than deuterium one, approximative^ ail the neutrons^ will form ‘^He and the final ^He abundance can be approximated by the half of neutron ones at T^uc- A rough estimate of primordial mass fraction of ‘^He, conventionally referred to as Yp, is thus given
4n, Tp = X4 = — ~ 2Xn{TNuc) ^ Ü.25, Tl}) which is in outstanding agreement with the exact solution see figure 1.3.
The précisé détermination of the abondances of the light nuclei relies on the exact nu- merical computation (Wagoner code is publicly available [23]), see figure 1.3 for the results. The light éléments abondances show an important dependence in rj. For larger baryon to photon ratio, nucleosynthesis begins earlier when the neutron to proton ratio was larger. As a resuit the ‘^He abundance is larger. D and abondances decrease with increasing value of ^i7e and r/ because they constitute the fuel in ^He production, and earlier the ^He production begins lessD and ^He are left. The primordial abondances also dépends on g» or equivalently on N^. Indeed the expan sion rate H oc ex; and the ratio drive the value of the freeze out température, as a conséquence: Tf oc gl^^. Higher 9*, implies higher Tf and larger ‘^He abondances. A maximum likelihood analysis on 77 and N y based on the observed '^He and D abondances gives 4.9 < 7710 < 7.1 and 1.8 < < 4.5 (see [14] and reference therein). The limits on N,y put bounds on additional light particle species at T ~ MeV. Additional constraints can be put on life-time and abundance of particles which are nonrel- ativistic at the epoch of BBN. For instance, their decay products can destroy light éléments by photodissociation or enhance inappropriately the baryon density leading to an overpro- duction of ^He,D,^He and ^Li. We will consider these limits in the particular framework of chapter 5.
Shortly after BBN took place, at matter radiation equality, structures begin to form and grow, and at a latter time stellar nucleosynthesis begins. These processes can alter the primordial light éléments abundance and produce heavier éléments such as C, N, O and
^No heavier éléments will form because there is no new stable nuclei with A = 5..8. In stars three *He can fuse to give but in the early universe it is not possible as the '*He abundance is far too low 20 1. DARK MATTBR TOOLS
Baryon density 0.005 0.01 0.02 0.03
Figure 1.3: The abundances of and ^ Li as predicted by the BBN. Boxes indicate the observed light element abundances (smaller boxes: 2a statistical errors; larger boxes: ±2a statisticcd and systematic errors). The narrow vertical band indicates the CMB measure of the cosmic baryon density. The curves are obtained using = 3. See [14] for more details.
Fe (referred to as metals). Trying to infer light element abundances closest to primordial ones from observations, one has to study astrophysical sites with low métal abundances, but we are left with important systematic uncertainty, above ail for ^He. Let us look over the light element abundances observations.
• Deuterium is believed not to hâve astrophysical sources, however, local interstellar D may hâve been depleted by a factor 2 or more by stellar processing. By averaging the 5 most précisé observations of D, detected in low-metalicity quasar absorption System, one gets D/H\p = (2.78 ± 0.29) 10“^, here the error is statistical only.
• ^He abundance receives a small stellar contribution which is correlated to metals production. Most available data about C,N,0 and cornes from metal-poor clouds of ionized hydrogen (HII régions), and extrapolating to zéro metalicity, one gets for primordial ^He abundance: Tp = 0.249±0.009. The error is an estirnate of systematic uncertainties as it has been shown that the determined value of Yp is highly method of analysis dépendent (see [22] for a recent review). •
• ^Li abundance is studied in metal-poor stars in the halo population (PopII) of our galaxy. Observations hâve shown that ^Li doesn’t vaxy significantly in Pop II stars with metalicities < 1/30 of the solar one. Extrapolating to zéro metalicities, one 1.4. HiSTORY OF THE UNIVERSE 21
gets for ^Li primordial values: Li/H\p = (1.7 ± 0.021q^) 10 but there remains serious systematic uncertainties.
Consistency between theory and observations leads to:
4.710“^° < r/ < 6.5 10“^'^ =» 0.017 < < 0.024 or equivalently ps = (3.2 — 4.5) 10“^^ gcm“^ at 95% CL. We took fixed by présent the CMB température (see next section). Using i'itot — 1 and h ~ 0.7, according to BBN, baryons only amounts for ~ 5% of the total matter-energy content of the universe.
CMB and Concordance model
The cosmic microwave background is one of the pillar of Big Bang Cosmology. It was first detected in the 1960’s by Penzias and Wilson. The précisé measurement of the Black body température was done by the NASA’s COBE satellite in 1992 along with the first discovery of large angle anisotropies. Since then there has been intense activity to map the sky with ground based and balloon borne measurement and last improvement with the analysis of the 3 year NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) data.
The CMB offers a look of the universe when it was about 1000 times smaller than it is today and aged of a few hundred thousand years and is a powerful probe of the early universe. Once the the température of the universe had cooled enough for the ionized plasma to combine and form neutral atoms, the universe became transparent to photons, allowing them to freestream towaxd us until today. This is called the ’last scattering surface’ or the epoch of recombination. Before the émission, collisions between électrons and photons were occurring fast enough to ensure thermal equilibrium. Therefore the CMB should hâve a black body spectrum, in agreement with COBE observations. The resulting estimate of the température T-y = 2.275 ± O.OOIAT corresponds to = 2(7r^/30)T.^ ~ 4.64 10“^'^ gcm~^. Comparing to the results for ps of the previous section we see that radiation constitutes a very small fraction of today’s total matter-energy content although CMB originates from an epoch of radiation dominated universe. This clecU-ly illustrâtes the fact that the radiation energy density {p oc a~^) redshifts faster than the matter one {p oc with the expansion of the universe.
The CMB shows tiny variations of températures across the sky 5T/T ~ 10“^ which encodes a lot of cosmological informations. In order to characterize these anisotropies one expands the CMB température in spherical harmonies:
T{9,4>) = Y.^imYr{e,
Before going further on with CMB constraints on cosmological parameters, let us stress that the quasi isotropie background radiation can not be explained in the standard Big Bang cosmology. Indeed, in this framework, causally connected régions at the time of CMB émission subtend an angle of about 1 degree in the sky today, there is thus a priori no reason for ail these régions to share the same température. One solution to this 'horizon problem’ is given by an early epoch of accelerated expansion, a period of inflation (see e.g. [18] for an introduction). Before inflation started, typical size of causally connected régions was larger than any scale of cosmological interest today.
Figure 1.4: Evolution of a given scale compared to the causal horizon (Hubble length), before, during and after inflation in physical coordinates in the upper panel and in co- moving ones in the lower panel. Concentrating on physical scales, we see that the causal horizon {du ~ ^/H) is constant during inflation while linearly evolving with time in radi ation dominated era on the other hand physical scales are exponentially increasing during inflation while evolving as during radiation era. Prom [18] and modified.
Moreover, the vacuum fluctuations of the inflaton fleld driving inflation should be at the origin of energy-density perturbations ôp at ail scales in causal contact during inflation. AU along the period of inflation, larger scales leave the horizon, the perturbations generated at these scales freeze in, and no causal physics can alter them anymore. Let us consider the scales re-entering the horizon before recombination (the smaller scales re-enter flrst) long after inflation ends, back in a radiation dominated era (see flgure 1.4). Causal physics can affect the perturbation amplitude again which evolve under competing influences of pressure of the CMB photons and gravity. This set up acoustic oscillations in the plasma which is made of a single ’baryon-photon’ fluid at that time. As a conséquence, a pattern of peaks is induced in the spectrum of the CMB anisotropies.
Assuming a Gaussian distribution (generally predicted by inflation), the anisotropies can be fully characterized by the variance of the température fleld Ci = (|a;^p). The Power spectrum, usually plotted as l{l + 1)C;, resulting from diverse experiments CMB measurements is shown in flgure 1.5. We recognize right away the peaks pattern in lower angular scales which correspond to larger l in the power spectrum (Û ~ differentiate 3 régions in these plots.
The flrst one concerns multipole modes with l < 100 corresponding to scales which were 1.4. HiSTORY OF THE UNIVERSE 23
AngularSoale Angular Scale
Figure 1.5: The figure on the left gives the angular power spectrum of the CMB from WMAP3. The solid line show the prédiction from best fitting ACDM model. The multipole axis is logarithmic. The figure on the right gives the power spectrum estimâtes from several experiments: WMAP, BOOMERANG, VSA, CBI and ACBAR. The spectrum extends up to higher multipole moments (/ ~ 2000) than the first plot and multipole axis is linear. From [14]
larger than the causal horizon at the epoch of recombination. Fluctuations of températures at these scales shows little évolution and quite a fiat spectrum, they refiects the initial conditions. It is hard to see unless the multipole axis is plotted logarithmically, see figure 1.5, plot on the left. Amplitude of anisotropies at these scales should hâve been set during infiation.
Next région concerns multipole modes with 100 < l < 1000 corresponding to the latest scales Crossing the horizon before recombinations. The perturbations associated with the ’first peak’, entered the horizon, begun to grow under the eff'ect of gravity and had just time to reach their maximum amplitude (maximal compression) at the time of recombination. It appears at the angular scale of sound horizon at last scattering surface. The 'second peak’ in the anisotropy power spectrum correspond to perturbations that entered the horizon earlier on, oscillated and reached minimal amplitude at the time of recombination. And it goes on with perturbations imprinted on scales entering the horizon earlier and earlier, odd pealcs are associated with overdensities of photons and even peaks with underdensities.
Finally, the power spectrum of fluctuations with multipole moments l > 1000 is damped due to imperfect coupling between photons and électrons at small distances. Indeed, photons travel a finite distance between two scatters which dépends on the électron number density Ue and the Thompson scattering cross-section ax- After one Hubble time H~^, they pass through a distance A ~ , any perturbation smaller than this diffusion scale is expected to be washed out.
The exact shape of the resulting power spectrum dépend on several cosmological pa- rameters, we study here the effect of some of them, for illustration see figure 1.6. First of ail the position of the first peak is in concordance with fiat universe. Going from fiat 24 1. DARK MATTER TOOLS
Figure 1.6: Changes in auisotropy spectrum as baxyon density, matter density and cos- mological constant vary. The dark solid curve correspond to a model with ÎÎa = 0-7 and ÇIm = 0.3. Prom [24]
universe to closed one for instance, a given physical scale would be projected on much smaller angular scales, the peaks pattern would therefore be shifted on laxger l. In the following, we will suppose that the total energy density equals the critical one, as WMAP3 measurements give: i\ot = 1-0031°;°]?.
Increasing baryon density has two effects: it will increase the amplitude of the oscil lations since it increases the importance of gravity compaxed to pressure and decreases the Sound speed since heavy baryons reduce the velocity of wave propagation. As a con séquence the frequency of oscillations goes down and peaks are shifted to larger l. More generally, increasing the matter density (baryons + dark matter) will increase the gravita- tional potential and boost* the anisotropies.
The Cosmological constant energy density was quite negligible at the time of recom bination so that it is only involved in late time effects. It will affect the conversion from angular scales to physical scales, the curves shift horizontally with a change in vacuum density.
The values for cosmological parameters obtained using a fit of a ACDM cosmology supposing a power law initial spectrum: {{Spk/p}'^) ~ to WMAP3 data alone, are given by [4] with uncertainties of 1er :
h = 0.73l°;°i = 0.1271S, 05^2 = 0.02231°;°°°^, n = 0.95ll°;°]^. (1.34)
Notice that inflation models usually predicts a fiat primordial power spectrum, i.e. n = 1. Since the parameter fitting has to be done in a multi parameter space, previous results dépend on the assumed prior range for each of the parameters {e.g. 0.5 < h < 1). WMAP data can test BBN prédiction for the baryon to photon ratio r? as it can be seen in figure
®Actually, increasing baxyon density will increase the height of odd peaks corresponding to higher overdensities of photons, and since photons get more easily trapped in potential wells, even peak will be lower as they correspond less underdense perturbations. This is what helps to distinguish between baryons and dark matter 1.4. HiSTORY OF THE UNIVERSE 25
1.3. The T] range for CMB values are obtained for a different assumption on the primordial spectra of density perturbations n [14] but WMAP and BBN prédictions seem nevertheless to overlap.
Several combinations of the parameters can lead to an équivalent fit of the anisotropies. In order to break the degeneracy of fitting cosmological model, it is useful to constrain the cosmological parameters with different sources of informations, such as studies on galaxy clustering, abondance of galaxy clusters, gravitational lensing. Type la supernova distances and Lyman a forest clouds. Figure 1.7, for example, shows the preferred région in — plane for complementajy data sets, best fit values giving CIm ~ 0.3 and Ha « 0.7. More detailed results are given in Spergel et al. [4]
r I M 1 T r“i i \ [ rn 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 SNe; Knop et, al. (2iXi3) No Big Bang CMB Speigei et al. (200.3) ■ Ciuste.'s: Allen et al, (2002)
Supemovae
. . T'®:»- CMB 1
. Clu4ers \ % :
\ : 1 1 1 1 1 1 1 1111111111111 ■ I II 11 111~
Figure 1.7: Preferred région in ÇIm — plane for complementary data sets. From [14]
1.4.2 Add-on to Big Bang Cosmology: Baryogenesis
As it was put forward in BBN and in cold relie abondance calculations, a baryonic asym- metry must hâve been produced in the remote past in order to avoid the 'annihilation catastrophe’. This is not the only argument in favor of the asymmetry. The observed uni verse seems to be filled with matter, while anti-matter is practically absent. Indeed, some antiprotons and positrons are observed (produced) in accelerators and cosmic rays but in either case they amounts for secondary products of particles interactions. Moreover, if any macroscopie object made of antimatter would exist, annihilation should take place 26 1. DARK MATTER TOOLS at the matter-antimatter interface and produce gamma ray émission, however, this has never been observed. There is thus no evidence for a noticeable quantity of antimatter in our neighborhood.
In 1967, Sakharov [25] pointed out the three ingrédients which must necessarily flavor the primordial soup for the asymmetry to be generated. These are Baryon number viola tion, C and CP violation and departure from thermal equilibrium. The first one is quite obvious but quite constrained by the proton lifetime. If we start from a baryon symmetric universe, rj = 0, we need baryon number violating processes in order to evolve to an asymmetric one, r/ ^ 0. Some processes must violate the charge conjugation symmetry C, otherwise the rate for the transition ^ —> /, producing an excess of baryons, would equal the one for i f one, producing antibaryon excess. CP is the composition of parity with C, and it is équiv alent to time reversion T by the CPT theorem. As a conséquence, if CP is conserved, the rate of z(rj,pi,Sj) ^ f{rj,pf,Sf) processes should equal the one of i{ri,—pi,—Si) —> f{vf, —Pf, —Sf) processes. So even if an asymmetry would hâve been produced locally in phase space, after intégration over momenta p and sommation over the spin s, it would be completely washed out. Finally, concerning thermal equilibrium, we saw that species follows Fermi Dirac or Bose Einstein distribution defined in (1.9). Moreover, the CPT theorem implies that a particle X and its antiparticle X hâve the same masses. In order to get an asymmetry between their number densities their Chemical potentials should thus be different. However, rapid annihilation of X and X into photons implies px = baryon number violating processes XX XX at equilibrium imply px = ~ 0- thermal equilibrium, the particles and antiparticles number densities are consequently equal, no asymmetry can be produced.
Sphalerons Before studying baryogenesis in the framework of Standard Model or any of its extensions, we introduce a récurrent source for B-hL violation. In models where fermions couples to gauge fields, it can be verified that the lepton and baryon number currents Ji,Jq are conserved classically, but not at quantum level [26, 27[. This is a conséquence of the so called Alder-Bell-Jackiw anomaly. Their divergences are effectively equal, = d^J^, so that the différence is anomaly free. These non-conserved currents lead to violation oi D + L symmetry in non-abelian gauge théories through non- perturbative effects {i.e. not described by Feynmann diagrams).
The vacuum structure of these théories can be viewed as a sequence of ground States labeled by an integer ncs» a Chern-Simons number, sepaxated from one another by a potential energy barrier. The anomaly relates the change in ncs to ^ change in baryon and lepton number: Ahb = AriL = NfAncs, where Nj is the number of fermions families. In the Standard Model the smallest jump is thus given by Ans — Aul — ±3.
There are two ways to transit between the vacua of the gauge theory. At zéro tempér ature, tunneling is enabled and is due to instanton configurations. The transition rate is however exponentially suppressed and can be neglected in B violating processes operating 1.4. HiSTORY OF THE UNIVERSE 27
Figure 1.8: Schematic vacuum structure for non abelian théories where the tunneling instanton path and the sphaleron path over the barrier are represented. From [28]
for baryogenesis. At finite température, sphaleron configurations (literally “ready to fall” in Greek), which are unstable solution of the équations of motion for the gauge-Higgs System, enable a transition through thermal fluctuations over the barrier. The rate of transition is sensitively higher, and it can be shown these transitions are at thermal equilibrium at températures ranging from T ~ 10^^ GeV to T^w ~ 100 GeV (see e.g. [29] and reference therein).
Electroweak Baryogenesis AU the three Sakharov ingrédients are présent in the Stan dard Model. Indeed, departure from thermal equilibrium is provided by the electroweak pheise transition (EWPT), B violation could be provided by sphaleron processes and CP violation appears in the CKM matrix. Let us stress however that this CP violation source, which is responsible e.g. for the experimentally observed Kq, Kq mixing , is not strong enough to account for the observed baryon asymmetry. This is the first argument which demonstrate against electroweak baryogenesis. Furthermore, for a significant departure from thermal equilibrium the phase transition should be of first order [30]. This kind of transition is characterized by the coexistence of two phases for a short period of time. Bubbles, enclosing the broken phase, are nucleated through thermal fluctuations in the sea of the unbroken phase, begin to grow, collide and finally fill ail the space. Fermions outside the bubble internet with the bubble wall where CP violation affect their transmis sion and/or reflection, in addition, they are submitted to the active sphaleron processes. Particles and antiparticles diffuse inside the bubble, the combination of ail the effect men- tioned earlier leading to an excess of particles in the broken phase. In the context of the Standard Model, numerical simulations hâve shown that the asymmetry won’t be erased after the EWPT by sphalerons processes if the Higgs mass is lower than 45 GeV. This gives a second argument against electroweak baryogenesis given the présent LEP lower bound on the Higgs mass ruh > 114.4 GcV. In order to produce the observed baryonic asymmetry, we need to consider théories beyond the Standard Model.
Leptogenesis and neutrino masses from seesaw mechanism The non-perturbative effects contribute to B violating processes in electroweak baryogenesis but they can be 28 1. DaRK MATTBR TOOLS used more interestingly for another purpose. If a net B — L asymmetry was created once, sphaleron processes provide a way to redistribute it in both B and L asymmetry comparable to the B — L original one (see [31] and appendix B for more details). Leptogenesis takes advantage of this mechanism by producing first a lepton asymmetry in the early universe and convert it later in the expected baryon asymmetry. Usually the initial asymmetry results from the ont of equilibrium decay of heavy Majorana particles, Ni into leptons l and other species through CP violating interactions:
r(A^i +...) ^ T{Ni ~^r +...).
As a concrète example, inspired of [29], let us study the mechanism of leptogenesis on the basis of the Standard Model lagrangian for the lepton sector with additional terms including a right handed neutrino with a Majorana mass®:
C — Llî P Ll + Îri P Ir-\- Nrï P N R + uiLl^Ir + VuLl^Nr + -M {N^Nr + h.c.) where we omitted the family indices for the lepton fields, couplings and masses, and $ is the Standard Model Higgs doublet and:
Notice that the complex Yukawa matrix can contains the necessary CP phases to fulfill the second Sakharov condition. Once the température of the universe drop below M, the heavy neutrinos become unable to follow their equilibrium distribution. The third Sakharov condition can thus be fulfilled now. Finally, their decay through their Yukawa interactions with SM leptons lead to the génération of a lepton asymmetry. The decay width is given by:
TiVi = ~ = -^{yïyu)iiNii. (i.35)
Let us emphasize that at low energy, after the electroweak symmetry breaking the neutrino mass lagrangian can be conveniently rewritten as:
where rriD = is the Dirac neutrino mass matrix. Let us diagonalize the matrix Ad, assuming that the components of M are heavier than the Dirac masses rriD-, we get:
+ h.c.) - + h.c.) where = rriD^rn^. This “seesaw mechanism” leads to naturally smaller masses for light neutrino mass eigenstates i>i compared to the other fermions of the Standard Model
^Notice this kind of lagrangian can be obtained in the framework of a Left-Right (L-R) symmetric model (see e.g. [32], [33]) and section 4.1) after the L-R symmetry is breaking by e.g. An, the heavy scalar triplet. See aJso annex A.3 for the Majorana properties. 1.4. HiSTORY OF THE UNIVBRSE 29
given their suppression by the heavy Majorana mass M. The mass eigenstates are actually mixtures of the interaction eigenstates:
f \ _ f cosd sin9 \ l' i^L \ \ '^H J ~ \ sin0 COS0 J \ N^r J
with 6 — mn/M.
We can dérivé an estimate of the baryon asymmetry in the framework of leptogenesis. First we need to evaluate the CP asymmetry e resulting from the interférence between diagrams at tree level and one loop level. In a basis where the matrix M is diagonal and for Ml C M2M3, one gets:
where e dépend on the heavy Majorana masses Mi, i = 1,2,3 and also on the Yukawa couplings y,y or equivalently on rriD as it can be anticipated^° from (1.35). The resulting maximum baryon asymmetry is given by:
Yb ~ (1-37) 9*
where k dépend on the sphaleron processes and is evaluated in annex B in the framework of the Standard Model.
The CP asymmetry e satisfies to an upper bound, requiring , one gets an upper bound on neutrino masses mi,i < 0.1 eV. Asking for Tn,{T) < H{T) at T = Mi, one can get a lower bound on rrii^i using (1.35). In this way leptogenesis ask for light Majorana neutrino masses in the range :
10“^ eV< <0.1 eV
Models including these extra particles Nr can thus fulfill the three Sakharov conditions, give rise to lepton and baryon asymmetry, low masses for light neutrinos and even bounds on the later. In chapter 5, we will study a similar mechanism responsible for “Matterge- nesis”. The latter expression must be interpreted in the sense that the out of equilibrium decay of the Majorana particle will lead to a B — L asymmetry in the visible sector driving baryon abondance along with, this time, an asymmetry in a dark sector determining dark matter today’s abondance.
Loads of other extensions of the Standard Model and mechanisms hâve been proposed in order to get baryogenesis such as Plank scale beiryogenesis, baryogenesis in Grand Unified théories, Supersymmetry, ... and more recently Afflek-Dine mechanism. I won’t go further in these théories, many reviews on the subject can be found, see for example [28, 29, 34, 35].
It can also be diluted with new Majorana decay channel through Wn gauge bosons in a L-R symmetric framework, the value £ will then also dépend on Mi/Mw^ ratio, see [33]. 30 1. DARK MATTER TOOLS 31
Chapter 2
Constraints on dark matter and some particle physics candidates
In this chapter, we study the constraints on dark matter properties imposed by direct and indirect détection experiments and Quantum field theory. Direct searches look for WIMP interactions with ordinary matter in low background detectors and provide bounds on WIMP proton cross-section as a fonction of the WIMP mass. Indirect détection searches look for WIMP annihilation signais in our galaxy, we will concentrate on gamma ray détec tion and how dark matter galaxy distribution can influence their flow. We will eventually study how unitarity can put constraints on the dark matter mass.
Then, we will try to distinguish between dark matter candidates with the help of structure formation theory deflning hot an cold dark matter. Finally we will be ready to dive into the dark sea of the dark matter aspirants, we will rapidly study the qualities of the most popular ones, the neutrino and the SUSY candidates, and we will even tear some of them away from the race to the stars.
2.1 Constraints from Direct and Indirect détection
There are different categories of dark matter searches. For example, one can look for WIMP, or more generally, for new physics at accelerators. One typical signature of dark matter event is the observation of a large amount of missing transverse energy due to pair production of WIMP. Indeed some symmetry, such as iî-parity in SUSY, is usually invoked for the WIMP to be stable and particles of the Standard model are generally even under this symmetry. If such processes are observed, we still need to discriminate between the different available DM candidate. A lot of work has been devoted to Supersymmetry (SUSY) and discrimination between SUSY and théories with Universal Extra Dimensions (UED) théories (see for instance [36]). Note that there axe questions that colliders will never address such as dark matter velocity and distribution in our galaxy or the fraction of dark matter of DM made of Weakly Interacting massive particle (WIMP). Direct and 32 2. CONSTRAINTS ON DARK MATTER AND SOME PARTICLE PHYSICS CANDIDATES indirect détection searches provide a complément ary study. Although these two suffers of astrophysical uncertainties, they can nevertheless put constraints on the characteristics of dark matter.
2.1.1 Direct Détection
Low background experiments try to detect the scattering of a non relativistic WIMP, Corning from the galactic halo streaming through the Eaxth, and a target nuclei. Due to Earth’s rotation around the Sun, its rotation around its own axis and the Sun’s motion through the halo one can look for both annual and diurnal modulation of the event rate and event directionality. Hint on the nature of dark matter can also be obtained by measuring the energy recoil of the nucléus hit by a WIMP.
The direct détection rate of WIMP can be written as ;
where Ni is the number of nuclei of species i in the detector, is the local galactic WIMP number density, a-)^i is the WIMP-nucleus elastic cross-section, and (...) dénotés the average over V, the WIMP velocity relative to the detector. Up to now, the experiments were dealing with ~ kg-hundreds of kg of target material (see table 2.1). In order to enhance the détection rate, the next génération of experiments will consider larger mass scales (kg ton). Indeed, the WIMP interaction cross-section with ordinary matter is quite weak, as a conséquence, the détection rates are very low (typically less than one event per kg of target material per day). It is thus also quite important to détermine accurately and minimize ail sources of background. For example, the experiments take place in deep underground detectors to avoid contamination from neutron coming from cosmic rays and the detectors must be shielded from natural radioactivity.
The maximum recoil energy of the target nucléus is given by
2v'^mT
^ (1-I-mj’/m^)2 where are the WIMP mass and velocity (~ 200 km/s) and rriT is the target mass. En is expected to range between 1 to tens of keV, the experiments must thus keep a very low energy threshold. There are three ways of detecting this energy (see also table 2.1) :
1. the less sensitive one is to look for light produced in scintillation process. The energy collection efficiency is only of a few percent. This technique has been used in liquid Xe (Zeplin I) and Nal scintillators (DAMA).
2. For energy collection efficiency up to 10 percent depending on the target material, one can look for ionization signal using Si, Ge and liquid Xe.
3. Using cryogénies one can extract beat signal. This process get the higher energy collection efficiency. 2.1. CONSTRAINTS FROM DIRECT AND INDIRECT DETECTION 33
Name Location Technique Material Status DAMA Cran Sasso Light 100 kg Nal stopped LIERA Cran Sasso Light 250 kg Nal running NalAD Boulby mine Light 65 kg Nal running ZEPLIN I Boulby mine Light 65 kg Nal stopped CDMS I Stanford Heat-t-Ionization 1 kg Ce -F 0.2 kg Si stopped CDMS II Soudan mine Heat+lonization 5 kg Ce + 1 kg Si running CRESST II Cran Sasso HeatT Light 10 kg CaW04 running EDELWEISS I Modane Heat -l-Ionization 1kg Ce stopped EDELWEISS II Modane Heat -l-Ionization 10kg Ce (30kg in 3 years) running Xénon10 Cran Sasso Ionization+ Light 10kg Xe running WARP Cran Sasso lonization-F Light 2.3 liter Ar running
Table 2.1: Some of the current détection experiments
The nucléus-WIMP cross-section is calculated in three steps, first we hâve to déter mine the quark-WIMP interaction, combine them taking into account the embedding of the quarks in nucléons which are themselves part of the nucléus. One distinguishes spin dépendent and spin independent interactions. For models studied in this work, I will need to calculate scalar (i.e. spin independent) interactions and I will thus focus on the latter in the following. In this case, the coupling of the quarks to the WIMP is proportional to the quark mass, as a resuit, for the scattering of a proton for instance, we hâve to evaluate the following nucleonic matrix^:
{P\ X] = fprrip (2.1) Q
It can be shown that the fp parameter dominantly dépends on the fraction of mass of the nucléon arising from the non valence strange quark, usually written /ts-
(n|msss|n) / 6 21 \ Its fp 0.3 (2.2) V27 ^ 27-'^7 the first equality for fp is obtained neglecting the contributions from u and d quarks (see [13] for details) and the 0.3 is obtained using Jts parameter recently reevaluated in [37].
DAMA experiment reported annual modulation of their event rate at 6.3 a consistent with a WIMP candidate with ~ 52 GéV and ~ 710^® pb (central values [39]). Unfortunately, as it can be seen on figure 2.1.1, the others experiments (CDMS, CRESST, EDELWEISS and ZEPLIN I) contradict this interprétation. However, it is important to note that the rate évaluation in direct détection experiments rely on the assumption that the WIMP halo density is given by piocai = 0.3GeV/cm^, its average value in our neighbourhood. The shape of the nuclear recoil spectrum dépends on the WIMP velocity distribution usually taken to be Maxwellian with mean velocity typically of 220 km/s. Relaxing these assumptions, it is possible to accommodate the results of the different
^the q appearing in (p| 52, ui^gglp) for quark (resp. antiquark) of a given momentum k and spin s has to be interpreted as og ^u{k, s)/^/wk (resp. 6^ ''he conventions of annex A.2 34 2. CONSTRAINTS ON DARK MATTER AND SOME PARTICLE PHYSICS CANDIDATES
DATA listed top to boltom on plot CDMS (Soudan) 2005 Si (7 keV threshold) DAMA 2000 58k ka-days Nal Ann.Mod. 3sigma,w/o DAMA 1996 Hmit CRESST2004 lO.TTcg-day CaW04 Hdclwciss I final limit, 62Kg-days Ge 2000+2002+2003 limit ZBPLINII (Jan 2007) resuit CDMS Soudan 2007 projected Edelweiss 2 projection SupcrCDMMProjecied) 2-ST@Soudan SuperCDMS (Projected) 25kg (7-ST@Snolab) XENON IT (I tonne) projected sensiüvity
Figure 2.1: Current and future sensitivities of direct détection experiments, upper straight Unes correspond to current limits from CDMS, CRESST, Edelweiss and Zeplin II and filled surface shows the région of the parameter space favored by DAMA. Dashed and dotted Unes shows future reach for (SUPER)CDMS and Edelweiss and lower straight Une shows projected sensitivity for 1 ton experiments. This Ugure is obtained using the interface found at [38]. See full-color version on color pages in annex E experiments (see [7] section 5 and reference therein) at the cost of some fine tuning. In the following we will take into account the présent limit on given by CDMS, ie ayj> < 10“^ pb, waiting for the next génération experiments (~ ton) reaching pb sensitivities.
2.1.2 Indirect détection
Indirect détection searches are looking for signais of WIMP annihilation in the galactic halo. Different annihilation products can be detected including gamma rays^. The photons can be produced in two ways: monoenergetic Une or continuons signal. The Une must resuit from a loop mediated process as WIMP are usually uncharged, and don’t couple directly to photons. Its rate of production should thus be quite suppressed. On the other hand the continuons signal is produced in the cascade decay of other primaxy annihilation products.
The produced gamma ray flux from the annihilation of dark matter particles can be expressed as dNl l f O C ------2— / Pdm dl, (2.3) dÇldE dEy -'1.0.S. where tudm is the dark matter particle mass, pdm is the dark matter density profile, (crju) and dN^/dE^ are, respectively, the thermally averaged annihilation cross-section times the relative velocity v defined in (1.19) and the differential gamma spectrum per annihilation
^ Neutrinos, positrons, antiprotons and antinuclei are other potential annihilation products (for a review see [40, 41, 7]). 2.1. CONSTRAINTS FROM DIRECT AND INDIRECT DETECTION 35
Corning from the decay of annihilation products of final State i. The intégral is taken along the line of sight. The numerical factor C, = \j2 for a self conjugale dark matter candidate, otherwise C = 1/4. Indeed, for x = x, the expression {cTiv)nyn-^ gives twice the annihilation rate. This is consistent with the Boltzmann équation (1.20) driving the relie abundance: two X particles disappear for each annihilation when x = X- In the cases where X 7^ X> the expression {piv)nyn^ amounts for the annihilation rate, however, riy_ — — udm
One can separate halo model dépendance of the flux from the partiale physics dépen dance introducing the dimensionless quantity J(fi) defined as :
J{û) = (2.4)
where 1{Û) is the line of sight corresponding to the direction Ü, Rq = 8.5 kpc is the distance between Earth and the center of the Galaxy and piocai = 0.3GeV/cm^ is the dark matter density in our neighborhood. The astrophysical factor (J(fi)) is the average of J{Û) over a spherical solid angle Afl, centered on the direction of the galactic center:
(J(Af2)) = ^ [ J[Ü) dü. (2.5) J AU To compute (J(AD)), one has to choose a model for the dark matter halo density profile. According to numerical simulations of galaxy formation by cold dark matter, the halo profile can be parameterized in the following way [42]:
(/3-7)/q /fioV fl + jlh/ar P{'^) — Plocal (2.6) \ r J V 1 -I- [r/a)°‘ where Rq and piocal were defined for équation (2.4), a is some length scale. The exponents (q, f3,7) can be thought as characterizing p(r) power-law index for r = a, r » a and r a respectively. One of the most widely used profile is the Navarro, Frank and White (NFW) profile [43] which behaves as p{r) oc near the halo center. However, as dark matter distribution is poorly known in the innermost région of the galaxy, 7 is quite unconstrained and one can argue for much steeper profile like Moore et al [44], more gentle ones like Kratsov et al [45] or even fiat core with au isothermal profile. The prédictions for (J(AO)) are shown in table 2.2 for solid angles Af2 = 10”^ srad and Afl = 10“^ srad centered on the galactic center. The chosen Afl corresponds to the angular resolution of EGRET experiment [46, 47] and fortheoming GLAST experiment [48, 49] respectively.
a P 7 a (kpc) (J(lO-^sr)) (J(lO-'^sr)) Iso 2.0 2.0 0 3.5 2.46 X 10^ 2.47 X 10^ Kra 2.0 3.0 0.4 10.0 1.932 X 10^ 2.37 X 10^ NFW 1.0 3.0 1.0 20 1.21 X 10^ 1.26 X lO'^ Moore 1.5 3.0 1.5 28.0 1.60 X 10^ 1.24 X 10^
Table 2.2: Parameters of some widely used density profiles models and corresponding value of (J(10~^sr)) and (J(10“^sr)). From [42]. The density profiles as a fonction of the galactocentric distance are shown in chapter 4 figure 4.7. 36 2. CONSTRAINTS ON DARK MATTER AND SOME PARTICLE PHYSICS CANDIDATES
The astrophysical factor span 6 order of magnitude between models but it is not the only uncertainty in the computation of the gamma ray flux, we still hâve to choose the particle physics framework. In particular, our ignorance on the good dark matter candidate leads to uncertainties on its mass, annihilation cross section and annihilation channels corresponding to uncertainties on the energy scale, the normalization and the slope of the photon flux. The experiments can nevertheless put constraints on independently of the halo or particle physics model in absence of convincing signal. The annihilation rate and the gamma ray flux increases with increasing dark matter density. As conséquence, it is quite natural to look for signais coming from the galactic center where the dark matter density profile is expected to be steeper. The current experimental limit on the flux by EGRET is of ~ photons cm“^.s“^ [46, 47] and forthcoming GLAST expected sensitivity is of ~ 10“^° — 10“^^ photons cm“^.s“^ [48, 49] for Aü = 10“^ and 10~^ srad respectively. We will go deeper in this discussion in chapter 3.
Presently, several experiments daim the détection of an excess of gamma rays coming from the galactic center compared to the expected astrophysical background: INTEGRAL at 511 keV [50], EGRET in the GeV range and HESS in the TeV range [51]. WIMP with masses ~ 50-100 GéV could be responsible for the EGRET excess (see e.g. [52]), but > 20 TeV WIMP masses are needed to explain the HESS excess [53]. On the other side of the mass spectrum, INTEGRAL signal could be due to the annihilation of ~ MeV dark matter. We will study this possibility in more details in chapter 4. Obviously ail these signais can not be produced by one single WIMP. Dark matter could be made of several WIMP, but, it is not the only solution to explain these signais and a lot of work is still done to find currently unresolved astrophysical sources.
2.2 Constraint from unitarity
Partial wave unitarity of the S matrix gives rise to a model-independent constraint on the mass of a cold dark matter candidate provided that annihilations (and not coannihilations) détermine the dark matter relie abondance and that dark matter is made of one single WIMP, X (and its anti particle x if the WIMP is not self conjugate). To dérivé it, we will follow [54] which has been inspired by the initial work of Griest and Kamionkowski [55] (see also [56] sections 3.6 and 3.7). Here, I will sketch the dérivation of the unitary bounds on dark matter mass, more details can be found in appendix G.
We know that for dark matter not to overdose the uni verse < 1- Observations straighten the bound giving 0.094 < ÇloMh^ < 0.129. In the simplest cases, the relie abondance is inversely proportional to the annihilation cross-section (see section 1.3.2). Partial wave unitarity of the S matrix will give a maximum value for this cross section as a fonction of the dark matter mass. Using that limit and the limits on we will get an upper bound on dark matter mass.
As we are interested in the annihilation cross-section we will concentrate on the pro cesses in witch there are two particles in the initial State a labeled by numbers 1 and 2. In that particular case, unitarity of the S matrix leads to a powerful relation between the 2.2. CONSTRAINT FROM UNITARITY 37 total cross-section a tôt and the 2-body 2-body forward transition amplitude Maa-
^rel^ tôt ImMnn — (2.7) lÔTT^ ’ where Vy-ei is the relative velocity between particles 1 and 2 in the State a. This relation is sometimes called the the Optical Theorem, see C.l for more details. After expansion of transition amplitude, or equivalently of the S matrix as M = (1 — 5), équation (2.7) takes the following form in the CM frame:
7T <^tot ^(2i + l)2Re(/sn|(l - S^{E))\lsn), (2.8) fc2(2si + l)(2.S'2 + l) jls where E is the total energy in State a, s = si -f S2 is total spin, l is the orbital angular momentum that combines with s to give the total angular momentum j. n is any other discrète index labeling the particles. It can be shown (see C.2.1) that 2.8 implies the unitarity of the partial wave expansion of the S matrix, S^{E):
S^^E)S^{E) = 1 (2.9) if only two body channels can be reached from initial State a at energy E. Using that property, and evaluating the dark matter annihilation cross-section a a, by removing elastic cross-section CTel accounting for XX XX processes (resp. XX XX) the total cross-section with initial State a = xX (resp. xx) > we obtain the following bounds (see C.2.2):
rrtot < P(2si -f 1)(2.S2 + 1) ^ jls o-A < (2.10) k^{2si + l){2s2 + l)
The last step is to extract the bound on dark matter mass, m^. For this we hâve to connect the bound we got in équation (2.10) to the estimate of today’s cold dark matter density given by équation (1.27) which takes the form:
1.0710^ GcV-^3;/ (2.11) t/0*fnp\C7 AVrel where rripi = 1.2210^*^ GeV, 5* is the total number of effective relativistic degrees of freedom dit T — Tf, and Xf = my^/Tj where Tf is the freeze out température. Indeed Tf is typically of order of rnyl2b so that at the time of freeze out dark matter is non relativistic and we assumed in (2.11) that annihilation cross-section is s-wave dominated in which case (y cjA^rGl’ 1^0 can thus evaluate o^a 1r the lo^^ velocity limit, ■— nx^'u where Vrei = in the CM frame. Note that Vrei is not a physical velocity given that it can be as large as 2. In this limit, the s-wave dominated annihilation cross-section {i.e. j = 0) is given by:
47T ^ A^rel — mlVrel ' 38 2. CONSTRAINTS ON DARK MATTER AND SOME PARTICLE PHYSICS CANDIDATES
The relative velocity can be approximated by Vrei ~ y/^/^f using equipartition theorem. The relation (2.11) for the relie density and the typical values Xf ^25 and y/g^ ~ 10 give the following constraint on dark matter mass :
mx ^ y/^x^^ (2.12) TeV - 3 10-3 ■
For the estimate of currently given by experiments, we get the bounds < 120 TeV for self conjugale dark matter candidates such as real scalars^ and rrix < 90 TeV for the others as 0dm = 20;^ if there is no asymmetry in the dark matter sector.
2.3 Hot and Cold dark matter: distinction from structure formation
Now that some of the basic tools for the study of dark matter properties hâve been pre- sented it is time to shed light on the diverse candidates for dark matter. One subject however still need to be digged out in order to distinguish between the so-called hot and cold dark matter types: structure formation. In order to understand how dark matter can influence the growth of structures, we first hâve to define two aspects of the theory of gravitational instability.
2.3.1 Jeans length and free streaming length
The first one is the Jeans length, Aj, which séparâtes gravitationally stable modes from unstable ones. Aj can be obtained by comparing the dynamical time scale Tgrav ~ 1 /\/Gp, being the time scale for gravitational collapse, to the time scale for pressure response Tp ~ X/cs, where Cs is the sound velocity. When Tp » Tgrav, gravitational collapse can occur, the Jeans length is actually given by:
In order to study the évolution of inhomogeneities for modes inside of the Horizon at a given period one still has to take expansion into account. It can be shown that the growth of density perturbations is inhibited at ail scales and for ail kinds of matter during radiation dominated era which lasts until tgg- By contrast, during matter domination era the expansion rate of the universe is slower, and perturbations in dark matter density at scales A > Aj become Jeans unstable and begin to grow. As a resuit, structure formation begins at the epoch of matter-radiation equality. Let us however emphasize that, for baryons inhomogeneities, the story is a bit different. Indeed, baxyons stay strongly coupled to photon until the recombination and the growth of perturbations in the baryonic fluid are still inhibited by the pressure component until that period. In the following, Tgg and Trec
^notice that s-wave annihilation processes for Majorana fermions is usually helicity suppressed by a factor mi/rrix where mi is the mass of light SM fermions, so that cmnihilation is usually p-wave dominated 2.3. Hot and Cold dark matter: distinction from structure formation 39 will refer to the températures at the epoch of matter radiation equality and recombination respectively'*.
r^q can be found by equating radiation density at températures below the MeV
TT 21/4 2"»4 Pt 2 + 4 vil to the matter density pM- If ail of the matter was made of only baryons, the total matter density would be equal pM = where ms ~GeV is the baryon mass and r/ is fixed by baryogenesis. As n-y oc T^, the température at matter radiation equality pM = Pr is given by = 0.22mBr?~0.1 eV . For a matter content dominated by collisionless dark matter, pM = ^\pc{T/Tof and
T}q = Q.m^pc/T^ ~ 5.6 eV .
In addition, it can be shown that matter inhomogeneities 6m grew like the scale factor 6m oc a oc 1/T in the matter dominated era. Also, for equality température and departing from primordial fluctuations^, we don’t reach enough growth of perturbations to drive structure formation until nonlinear régime. The enhancement in this growth by a factor Teq/T^q ~ 56.4 h? is quite welcome in order to be in agreement with observations, for more details see [6] and [14] section 19. This gives one more piece of evidence in favor of a large non baryonic dark matter component in the matter content of the universe.
A second useful aspect of the theory of gravitational instabilities is the free streaming or equivalently, the collisionless-Landau damping. This refers to the suppression of per turbations by collisionless particle streaming out of overdense régions. On scales smaller than the free streaming length \ps inhomogeneities are smoothed out. The latter scale is deflned as the distance traveled in free fall by the particle once it has decoupled from the plasma, its motion being described by a{t)dr — v{t)dt :
XMt)= Jti a{t ) We are interested in evaluating Xpsii) before the beginning of structure formation, i.e. for t = teq and after some algebra, one gets [2j:
Ab5 ^(2 + \n%q/tNR)) = 0.2 Mpc (—) ('^){2 + hi{t,q/t^R)). (2.13) o-NR V 'mx / \ ^ / where the N R subscript refers to the epoch where the particle X became non relativistic i.e. T = mx/3 which is assumed to be prior to the matter radiation equality.
Let us study in more details the case of a particle X which découplés, at température Txdec, when it is still relativistic. Its abundance relative to the photon one is given by:
_ ^x^ 9*s{T) t=To ^ 9x^ riy 9y \T ) Çy 9*s{Txdec) 9*s(Txdec)
^In the standard scénario, Teq ~ 1 eV and Trec ~ 0.4 cV ® initial conditions closely related to the low multipole modes in the anisotropy spectrum of the CMB 40 2. CONSTRAINTS ON DARK MATTER AND SOME PARTICLE PHYSICS CANDIDATES
where is the number of X internai degrees of freedom multiplied by 3/4 if X is a fermion. The second equality is obtained using entropy conservation, the last equality is obtained using g-y = 2 and g*s{To) = 3.91 ~ 4. As conséquence, one can relate the X relie density to its température and its mass using Çlx — nxmx/pc with today’s value of Pc = 10.54 keV cm~^ and n-y = 422 cm“®:
üxh'^ = 20 (2.14)
As a conséquence, we obtain the following expression for the free streaming length:
4Mpc eff Afs ^ (2 + ln{teq/tNR))- (2.15)
This leads us to the description of hot and cold dark matter based on the first scale, Xd, on which structures begin to form. If Ad > Agg ~ IMpc, dark matter is still relativistic when structure formation begins, and is called Hot Dark Matter (HDM). The damping scale is larger than galaxies {Xgaiaxy ~ 10 — 10^ kpc) so that the first structure to form would rather be super clusters which would then fragment in order to give rise to smaller stellar objects. This is the “top down scénario” for structure formation. If Ad < Agg, we talk about Cold Dark Matter (CDM) being non relativistic at matter radiation equality. This time small scale structures form first, aggregating to form larger structures later, or “from bottom up” scénario for structure formation. The intermediate case Ad ~ Agg characterize Warm Dark Matter (WDM). WDM is for example a two component fermion species relativistic at decoupling but non relativistic at matter radiation equality. With (2.15), we need Tx/T ~ 0.4 or equivalently mx ~ keV using (1.21).
2.3.2 The power spectrum
The analysis of inhomogeneities in the sky map is a bit different than anisotropy analysis from the CMB. The first one constitute a 3D field whereas températures is a 2D field, we will thus work with Fourier expansion this time. For Sx — {p{x) — p)/p, the density perturbations in matter density, the power spectrum P{k) is defined as:
{SkSk>)^{27rŸP{k)S\k-k') where ôk is the Fourier transform of Sx and (..) dénotés a volume average. P{k) has dimen sion of length k^ so that it is sometimes expressed in term of the dimensionless variable A^(/c) = k^P{k)/2Tr^. Features of P{k) are encapsulated in the following expression:
P{k) oc k^T^{k) where /c” gives the supposedly power law dépendance of the primordial power spectrum and the transfer function T{k) describes the évolution of perturbation through horizon Crossing and matter/radiation transition.
If the initial conditions are settled by inflation, one expects n « 1 which refers to the so called Haxrison-Zel’dovich spectrum. In addition, T{k) ~ 1 for large scales (the 2.3. Hot and Cold dark matter; distinction from structure formation 41
One entering in the horizon after matter radiation equality) so that for n = 1 the power spectrum on these scales should show a linear dependence in k. Going to lower scales, the ones which entered the horizon during radiation dominated era, we will see a turnover in P{k) as perturbation growth was inhibited at that period. Its position gives an information on \eq which is closely correlated to the total matter density. Moreover for lower scales, the presence of baryons would lead to the of appearance of wiggles in the spectrum, for the same reason that oscillations appear on the anisotropy spectrum of the CMB. Their absence should be used to constrain the IÎs/IÎm ratio. Damping on low scales (large k) would be associated to the free streaming of light collisionless particles and constrains the mass of light neutrinos.
k [A/Mpc]
Figure 2.2: 2dFGRS galaxy power spectrum is shown by solid circles with one sigma error appearing as a shaded area. Triangle and error bars shows the SDSS power spectrum. Dotted vertical Unes indicate the range over which the best fit to ACDM model (solid curve) was evaluated. From [14].
Galaxies provide the data source for the computation of the power spectrum. However, they can not be thought as faithful tracers for matter. In order to test cosmological parameters, one has to study the galaxy “biasing” or in other words the relationship between galaxies and the underlying dark matter distribution: b{gŸ — Pg{k)/P{k) (see e.g. [57] for a discussion). The galaxy power spectrum resulting from the 2-degree Field (2dF) Galaxy Redshift Survey is shown in figure 2.2 and it is compared to the current data for Sloan Digital Sky Survey (SDSS) power spectrum. The 2dFGRS power spectrum is consistent with a ACDM model with VLm = 0.126 and 0,b/^m = 0.17 for h = 0.72 and a Harrison-Zel’dovich spectrum characterized by n = 1. 42 2. CONSTRAINTS ON DARK MATTER AND SOME PARTICLE PHYSICS CANDIDATES
Figure 2.3: Summary plot for Dirac neutrino relie density, the horizontal dashed line approximately corresponds to maximal value. Prom [22] and modified.
2.3.3 The “Standard Model” neutrino: a hot dark matter candidate
As we hâve seen in the previous chapter, section 1.3.2, by comparing neutrino annihilation cross section to the expansion rate of the universe, equilibrium for SU (2)^ interacting neutrinos can be maintained until températures of Ty ~ 1 MeV.
As a conséquence, neutrinos with masses lower than 1 MeV become hot relies,and their comoving number density stays constant after neutrino decoupling. In section 1.3.2, their relie density was shown to be: T.J m,, (2.16) 92eV ■ If we want to saturate the WMAP dark matter relie density, (2.16) translates into the bound Ylj < 12 eV.
Neutrinos with masses laxger than MeV behave as cold dark matter, their abondance being exponentially suppressed by the Boltzmann factor since they became non relativistic until annihilation interactions freeze in. Their relie density is inversely proportional to the neutrino annihilation cross-section ex l/(cru), see équation (1.27). We find that heavy Dirac (resp. Majorana) neutrinos saturâtes WMAP3 dark matter relie density for
Trii,. = 3 (resp.7) GeV . j
The latter corresponds this time to a lower bound on acceptable mass for dark matter made of heavy neutrinos.
The general trend for the relie neutrino density as a fonction of the mass is shown in figure 2.3. For rriu < MeV, we see that increases linearly with the mass as dictated by équation (2.16). For neutrino masses MeV< < few TeV, the cross-section shows a {av) oc ml dependence in the neutrino mass so that decreases with m^, (see equ. 2.3. Hot and Cold dark matter: distinction from structure formation 43
(1.27)), with a dip around M^/2 due to strong annihilation at Z boson pôle. Over the TeV range,it is expected that the annihilation cross-section takes unitarity bound dependence oc 1/m^ (see section 2.2) and should increase again.
Let us emphasize that light neutrinos can not account for the totality of dark matter for two main reasons (see e.g. [58] section 11 and référencés therein). The first one is associated to their hot dark matter properties: structures formed first at large scales. From the previous section the free streaming length can be derived as a function of neutrino masses: Xps — 600 {rn^/ eV) Mpc.
For ~ 0{ eV), it can be shown that structures on galactic scales (~ kpc range) appeax very late, at redshift z <1, whereas they are observed at redshift z > 3. This is the main recison why CDM is usually preferred to HDM. Let us also stress that WDM can still maJee an interesting dark matter candidate and solve some contradictions between simulations and observations for CDM driven structure formation such as the cuspiness of halos inner density profile or the number of low mass halos (see e.g. [59] and [60] for discussion and reference therein).
The second reason against dark matter neutrinos is associated to their fermion proper ties limiting the number of neutrinos in a particular quantum state in a given région. The resulting so-called Tremaine-Gunn limit [61] on the neutrino mass mj/ as a function of its velocity v„ in a galaxy of radius R is given by:
rui.
If neutrinos must provide ail the dark matter content of small gala;Xies of radius ~ kpc with Vy < lOOkm/s, their mass must exceed 100 eV, which violâtes the hot relie abondance bound.
In order to get the neutrino contribution to the dark matter content, we need to take into account the results of several experiments. The number of weakly interacting neutrinos is constrained to he 3 by the précisé measurement of the Z decay width at LEP. The observation of flavor changes in atmospheric and solar neutrinos constrains the two squared mass différences between neutrino mass eigenstates to be:
~ 2 10~^ eV^ Amg « 810“^ eV^ .
This implies that at least one mciss eigenstate has a mass exceeding 40 rneV. Finally, from tritium beta decay experiment [62], we get the bound < 2.3 eV. One can consider that for ail the three weakly interacting neutrinos, we hâve
< 2.3 eV .
From cosmology, one gets two additional constrains on the sum of the neutrino masses: CMB alone implies that Ylj < 2.6 eV, and large scale structures data implies Ylj "b ^ 0.5 eV (see e.g. [63] and reference therein). As a conclusion. Standard Model neutrinos constitute hot dark matter and they can be ruled out as dominant contribution to the total dark matter content of the universe. 44 2. CONSTRAINTS ON DARK MATTER AND SOME PARTICLE PHYSICS CANDIDATES
2.3.4 The WIMPs from the MSSM: popular cold dark matter candidates
One of the basic motivation for Supersymmetry is to solve the hierarchy problem appearing when one tries to evaluate the radiative corrections to the Higgs mass. Indeed, in the Standard Model, no symmetry protects the Higgs mass to get quantum corrections from diverging quadratically with the energy: SAI^ oc A^, where A would be some natural cut-off energy scale which can a priori be as large as the Plank scale. In SUSY, for each fermion in Nature there is a corresponding boson, with identical internai quantum numbers such as electric charge, isospin, .... As a conséquence, for each boson loop contributing to ÔM^ there is a corresponding fermion loop with opposite sign (see Feynmann rules, pl20 [64]) that has to be taken into account leading to the cancellation of the quadratic divergence.
Clecirly, with ail the extra degrees of freedom, we hâve a chance to find some neutral and quite weakly interacting candidate for dark matter. We are going to concentrate now on the MSSM, the Minimal Supersymmetric Standard Model, Minimal in the sense that it contains the minimal field content in order to hold the entire SM. Let us hâve a look to Fig.2.4 where the field content of the MSSM is presented. Sneutrinos ù and neutralinos Xn, n = 1, ..,4 seem to hâve the good properties being or containing the superpartners of neutral SM particles.
Standard Model particles and fields Supersymmetric partners Supcrfield SM particles Spin Suporpartners Spin Interaction cigenstates Mass cigenstates Symbol Namc Symbol Name Symbol Name Q 1/2 0 (|) g = d, c, 6, U, s,t quark QL, 9r squark Qi, h squark slepton slepton Ûfl 1/2 ^Jt 0 l = e,fiyT lepton II, Ir h, h dR 1/2 0 U = UeyV^yUr ncutrino û sncutrino ù sncutrino 9 gluon 9 gluino 9 gluino L 1/2 0 w± n’± M/’-boson wino 1 1 E^' €r 1/2 0 H- Higgs boson Hÿ higgsino \ XÎ2 chargino //i Ih 0 1/2 Higgs boson f>î higgsino j 1 Ih Ih 0 ih 1/2 B B-fieid B bino G“ 9 1 9 1/2 W^-deld W'O wino neutralino Wi Wi 1 Wi 1/2 m Higgs boson higgsino > X?,2,3.4 B B 1 B 1/2 HO Higgs boson iio higgsino 1
Figure 2.4: Standard Model particles and their superpartners in the MSSM. From [7].
One missing ingrédient is an additional symmetry, most often discrète, to differenti- ate the dark sector from the Standard Model. In SUSY, one already has to assume the conservation of R parity, under which SM particles are even and their superpartners are odd, in order to forbid new baryon and/or lepton number violating interactions leading for example to too rapid decay of the proton. As a resuit, sparticles can only decay in an odd number of sparticles (plus SM particles) and the Lightest Supersymmetric Particle (LSP) is stable constituting an excellent dark matter candidate.
The field content and the number of free parameters of the MSSM is imposing. One is rapidly tempted to reduce their number from more than 100 to a quantity more easy to handle. It is often assumed that, at some unification scale, there is: •
• a universal value for the gaugino (= masses
• a universal value for the scalar (sfermions and Higgs bosons) masses ttiq. 2.3. Hot and Cold dark mattbr: distinction from structure formation 45
Other parameters appearing regularly in the dark matter discussions are the Higgses vac uum expectation ratio tan f3 and the Higgs(ino) mass parameter //.
We do not want to go further on technical details about SUSY (for reviews on the subject, see e.g. [13, 22, 7]). Since ail the ingrédients for dark matter are présent, we rapidly OverView the chances of the LSP inspiring ourselves of the above-mentioned references:
• The sneutrino should hâve a mass me, > 20 TeV in order to be in agreement with direct détection searches. This seems to be well over the acceptable masses to account for dark matter relie density [65, 22]. •
• The lightest neutralino is one of the most acclaimed and studied dark matter can didate. Its mass dépends on the values of 9\v and m^, the masses Mi and M2 of the gauginos D and W^, tan (3 and fi. Let us emphasize that M\ and M2 don’t especially need to be equal and equal to mi/2 at the unification scale. In order to hâve an idea of the possible mass range, we can look at the figure 3.8 in chapter 3 where we hâve plotted the dark matter relie density as a function of the dark matter candidate mass. The blue points correspond to MSSM scénarios with the neutralino as LSP. They are obtained running renormalization group équations down to low energy scale with no assumption of gaugino and/or Higgsino universality of masses at unification scale ajid minimizing the Higgs potential keeping, the models predicting mz values in agreement with the observations. We see that the allowed mass range in agreement with WMAP dark matter relie density is around 100 GcW Chapter 3 The Inert Doublet Model What’s the nature of dark matter and which extension of the Standard Model do we hâve to consider to solve its lack of completeness^? These two questions are still unanswered nowadays. Actually, we should rather say that we are facing overmuch possibilities and it is still difEcult to décidé what’s the good answer. For the time being, the most popular extension of the Standard Model is the Supersymmetric one with the neutralino as candi date for dark matter (see e.g. [13, 7] and [6, 22] for a review). Supersymmetry however encompass a somewhat frightful number of parameters, that could be reduced under ad- ditional assumptions on symmetries of the theory, and a quite intricated new physics for non-initiated. In this chapter we will study the case of a simple extension of the Standard Model which contains one extra scalar doublet and one extra Z2 symmetry In that framework, the candidates for dark matter will be two neutral scalars axising from the extra doublet. The latter was called “Inert” doublet by Barbiéri et al in [68] as it will get no vacuum expectation value and will not hâve direct coupling to matter fields. However, as a funda- mental représentation of the SU(2)£^ gauge group, it is weakly interacting. As a resuit, the physics driving its neutral and charged components existence is quite simple to understand dealing with usual Higgs and gauge boson phenomenology. This is not the only attractive feature of the model. As was pointed out in [68], the Inert Doublet Model (IDM) naturally allows for a Higgs mass up to 500 GeV and still fulfill the LEP Electroweak Précision Test (EWPT) measurements (see aiso [69] for an older reference on that multiple-scaler models property and section 3.1.2 of the présent chapter). This can be interesting when we think to the prédiction of the Minimal Supersymmetric Standard Model (MSSM) [14] which constrains the Higgs to be quite light < 135 GeV. There is no more a wide margin between this upper bound and the LEP experimental lower bound of 114 GeV. The IDM is one of the models that offers an escape if no Higgs is observed at LHC below 135 GeV. ^such as neutrino masses, discrepancy between p — 2 theoretical and experimental estimate [66], what fixes the Higgs mass,... or other more fundamentai questions such as the hierarchy problem between electroweak scale and Plank mass, what’s the nature of daik energy, what’s responsible for inflation,... ^ see e.g. [67] for a two Higgs doublet model in which the Zz symmetry appears naturally. 48 3. The Inert Doublet Model In the présent work, we will nevertheless consider no Higgs masses over 200 GeV. This is because, for the range of scaJar mass différences that we considered, increasing the Higgs mass reduces the contributions to the annihilation cross-section at tree level. It also dramatically reduces the available parameter space in agreement with WMAP. This behavior can already be detected in relie density plot of figure 3.5 going from Mh = 120 GeV to Mh — 200 GeV. Let us emphasize however that in [70], the authors considered Inert scalaxs with masses lower the W threshold with suppressed tree-level annihilation but substantiaJ coannihilation in order to obtain candidates with the correct relie density. As it was stressed by the authors of this paper, considering dark matter with rather suppressed couplings to fermions and with a mass just below the W threshold, one can get a significant contribution to gama-lines through one-loop processes. We briefly discuss the results of [70] in the conclusion of this chapter. This model can also influence the neutrino sector. In [71], Ma added odd Majorana particles coupling to H2 giving rise to SM neutrino masses through dark sector loop contri butions. This mechanism was even further addressed in a sériés of papers [72, 73, 74[. We would like to clarify the possibilities by a renewed and systematic scan of the parameter space. Furthermore, some outcomes of the model still deserve clarification such as the conséquences for Higgs and daxk matter searches at colliders in the simple framework (yet addressed for a limited part of the parameter space in [68]) and extended “Ma”-framework. No results for these subjects however will be presented in this work, this is let for future investigations. Charmed by the simplicity and the potential repercussions on the dark, Higgs and neutrino sector, we decided to go through a more detailed and systematic study of the model [75]. In my opinion the IDM can even be seen as a more general introduction to dark matter. Dark matter candidates should be weakly interacting and unseen, on the other hand, the theory of the Standard Model works quite well but loads of uncertainties still remain on the scalar sector as the Higgs bosons has not been detected yet. Very often, dark matter models take advantage of these uncertainties intertwining the dark and scalar sector story. In this chapter, we hâve one of the most simple and rich example of that kind. Going through the IDM can help to get a more general intuition on dark matter phenomenology in any model even as complicated (to my eyes) as SUSY. One should maybe qualify it as an archétype for dark matter ? The plan for the next sections is first, the description of the model and the dérivation of the régions in the parameter space consistent with the WMAP estimate of dark matter relie density. We then test our results with direct and indirect détection searches. Direct détection searches probe the dark matter interactions with matter in low back ground detectors and indirect détection searches considered in this work probe the gamma ray flux resulting from dark matter annihilation in the galactic center. We will end with a quite large number dark matter candidates with two possible mass range: one below the W mass and the other one above 400 GeV. 3.1. The model 49 3.1 The model We consider a particulax two Higgs doublet model, in which one of the doublet, H\ plays the rôle of the standard Brout-Englert-Higgs doublet while the second one H2 will be the source for dark matter candidates. In order to guarantee the stability of the latter’s, we invoke Z2 symmetry under which ail Standard model fields are even and Hi Hi and H2 -H2. This discrète symmetry also prevents the appearance of flavor changing neutral currents in this model. We will assume that Z2 is not spontaneously broken, and that H2 does not develop a vacuum expectation value. The model is thus definitely not the {H2) —> 0 limit of a generic two Higgs model like, for instance, the Higgs sector of the MSSM. This model was yet recently approached by Ma [71], Cirelli et al [76] and Barbiéri et al [68] but their initial purpose and some of their assumptions were not exactly identical. Their conclusions seem nevertheless at first sight inconsistant as the neutral scalar reaching the dark matter WM AP abundance was found to be in the mass range of 60 to 75 GeV for Barbiéri et al [68] while for Cirelli et al [76] it was of order of 430 GeV. Let’s first go more deeply in the details of the model before trying to elucidate this incompatibility. 3,1.1 Scalar potential The most general, albeit renormalizable, potential of the model can be written £is Ar (3.1) There is an Peccei-Quinn t/(l) global symmetry if A5 = 0. This limit is however not favored by dark matter direct détection experiments (section 3.3). The vacuum expectation value of Hi, {Hi) = v/2 with V = = 248 GeV, while assuming for simplicity > 0, we hâve {H2) = 0. For the sake of completeness, let us define: / iW+ \ ( Œ+ \ where the scalars and Zf are the Goldstone bosons associated to the W an Z gauge bosons. The mass of the Higgs particle h is = -2^1 = 2Ait;2 (3.2) while the mass of the charged, and two neutral. Ho and Aq, components of the field H2 are given by 50 3. The Inert Doublet Model M^o — A*2 + (A3 + A4 + As)t;^/2 M\^ = fi2 + (A3 + A4 — \^)v^/2. (3.3) For appropriate quartic couplings, Hq or Aq is the lightest component of the H2 doublet. In the absence of any other lighter Z2-odd field, either one is a candidate for dark matter. For definiteness we choose Ho- AU our conclusions are unchanged if the dark matter candidate is Aq instead. Following [68] we parameterize the contribution from symmetry breaking to the mass of Hq by Al = (A3 + A4 + A5)/2, which is also the coupling constant between the Higgs field h and our dark matter candidate Hq. Of the seven parameters of the potential (3.1), one is known, v, and four can be related to the mass of the scalar particles. In the sequel, we take fi2 and A2 as the last two independent parameters. The later actually plays little rôle for the question of dark matter. 3.1.2 Constraints Some constraints on the potential and the scalars masses hâve to be taken into account for the analysis of the model: • Vacuum stability (at tree level) demands that [69] Al,2 > 0, Asî A3 + A4 — IA5I > —2\/ A1A2. (3.4) As a resuit, négative couplings, and Al < 0 among others, are largely excluded. • Perturbativity : Strong couplings |Aj| > 47r are excluded but intermediate cou plings, 1 < |Aj| < 47t, might be tolerated. • The mass of the charged Higgs scalar is constrained by LEP data to be larger than 79.3 GeV [14]. • We impose Mjjo 4- M^o < Mz to not be in conflict with LEP data on the decay width of the Z boson. • Electroweak Précision Tests (EWPT): The values of weak-interaction observables such as the ratio Mw/Mz, the Z decay width, etc can be affected by new physics through its contribution to the W and Z gauge bosons vacuum polarization eunplitudes. These effects can be described in term of the S, T electroweak précision parameters (see e.g. [77]). Notice that in our case the mass of the Inert scalars contribute to the S and T parameters as well as the Higgs mass^ even if we can not really qualify the Higgs boson as physics beyond the Standard Model. As pointed out in [68], with appropriate mass splittings between its components, an H2 doublet could screen the contribution to the T parameter of ^the top quark mass also contribute to the S and T but its contribution kept fixed in the following discussion 3.2. Dark Matter abundance 51 a large Higgs masses, Alh ~ 500 GeV. Indeed, the extra Inert scalars contribution to the T variable was computed in [68] to be: ~ 247r2a^2 - Mao){Mh+ - M^o) The Higgs contribution itself amounts for: 3 T{Mh) = - In Sttc^ Mz' Obviously, for cancellations to take place between these two contributions, one need H~^ to be the heaviest of the Inert scalars and this is what we will assume in the following. The effects on the S parameter is parametrically smaller for the région of parameter space satisfying the previous constraints. In ail the cases we hâve studied, we hâve checked that the contribution to the T parameter resulting from the Higgs mass and the H2 components are consistent with EWPT. Unlike Barbiéri et al, we only considered small values of the Higgs mass, Mft < 200 GeV. Given our choice of the Higgs mass Mh and of mass splittings between the Hq particles and the other components of H2, the constraints from EWPT are easily satisfied. 3.2 Dark Matter abundance The abundance of Hq h8is been estimated in [68] and in [76] for two different mass ranges. Let us stress that in either cases the authors considered the standard freeze-out mechanism and took into account coannihilation of Hq with the next-to-lightest scalar particle. The discrepancy between their prédictions for the dark matter candidate mass results from their distinct assumptions on the quartic couplings of the potential. In either cases the charged H^ scalars are supposed to be the heaviest component of the H2 doublet. However, Barbiéri et al considered a nonzero mass splitting between the neutral scalars Aq and Hq while Cirelli et al considered no mass différence^ or equivalently A5 = 0. Moreover, Barbiéri et al considered mass différences between the lightest and next-to-lightest Inert scalar of order of 0.1 AIhq, on the other hand Cirelli et al dealt with ~ ().001M//g mass splittings. These are the sources of their disagreement, the reasons why will become clear in the next sections. As in [68] and in [76], we consider a thermal production of our cold relie Hq. Also, as it can be expected from the above discussion, there are essentially two régimes, depending on the mass of Hq with respect to that of the W and Z bosons. If AIhq > 80GeV, the Hq annihilâtes essentially into Z and W boson pairs (see Figure (3.1)). For Mhq > Mh, the annihilation channel into h pairs opens (see Figure (3.2)). Otherwise, the Hq annihi- late essentially through an intermediate Higgs, provided the Higgs itself is not too heavy. Some amount of coannihilation of Hq with the next-to-lightest scalar particle Aq (resp. H'^) through a Z (resp. W^) boson can be présent [17], a feature that substantially com- plicates the détermination of the dark matter abundance (see Figure (3.3)) but which is ^ Due to this assumption, Cirelli et al ruled out the Inert doublet model as a viable dark matter model given the direct détection limits on dark matter proton scattering (see section 3.3). 52 3. The Inert Doublet Model essential in order to enlarge the viable parameter space. One last comment on the con- tributing processes, the décisive rôle played by the Higgs-dark sector interweaving appears clearly with the presence of the Xl coupling in each channels contributing to dark matter abundance calculation (see figures 3.1, 3.2 and 3.3). W+ {Z) Ho,^ W+{Z) Ho v\ h f i H-(Ao) W~ {Z) Ho..... {Z) W-{Z) Figure 3.1: Annihilation channels into gauge bosons final State with corresponding couplings. Ho h Ho h Hç, , h h y i^o Ho h Ho h Ho h Xl A2 AlAi Figure 3.2: Annihilation channels into Higgs final State. Ho. f /.....< _ Ho / ^LVf Figure 3.3: (Co)Annihilation channels into fermion anti-fermion final State. We hâve computed the relie abundance of Ho using micrOMEGAs2.0, a new and versa tile package for the numerical calculation of Dark Matter abundance from thermal freeze- out [78]. The latest implémentation of this code, which was originally developed to study Supersymmetric models, allows one to enter any model containing a discrète symmetry that guarantees the stability of the dark matter particle. The code takes advantage of the trick presented in section 1.3.3: each Z2—odd particles eventually decay into the lightest odd particle under the Z2 symmetry (LZP). Using the notations of the submentionned section we hâve in this case xi = Hq and Xi = -^0» H^ for i > 1. One is then led to deal with one unique Boltzmann équation for the LZP resulting from the sommation of the System of Boltzmann équations of ail odd species, this is eq. (1.30) or equivalently: ^ = \f^^Mpi{aeffv){Y\T) - y;2(r)), (3.5) where Y = udm/s is the comoving density of dark matter and {a^jfv) is defined in (1.30) and can be understood as an effective thermally averaged cross-section taking into account 3.3. InERT SCALARS DETECTION 53 annihilation as well as coannihilation processes. Equation (3.5) is numerically integrated from T = oo to T = To in an incredibly short time in micrOMEGAs2.0, which delivers finally today’s dark matter relie density using: = 2.72 X I08^|^y(ro). (3.6) Let us mention that micrOMEGAs2.0 itself is build upon CALCHEP, a code for computing tree-level cross-sections. 3.3 Inert scalars détection In the présent section, we will concentrate on the signais we should perceive from dark mat ter presence by studying the gamma ray flux arising from the galactic center or detecting the recoil of heavy nuclei in low background detectors. 3.3,1 Indirect détection The measurement of secondary particles coming from dark matter annihilation in the halo of the galaxy is a promising way of deciphering the nature of dark matter. As emphasized in section 2.1, this possibility dépends however not only on the properties of the dark matter particle, through its annihilation cross-sections, but also on the astrophysical assumptions made concerning the distribution of dark matter in the halo that supposedly surrounds our galaxy. The galactic center (GC) région is potentially a very attractive target for indirect détection of dark matter, in particular through gamma rays. The produced gamma ray flux from dark matter annihilation was given in équation (3.7) which becomes for a self conjugale candidate: dNl^ {aiv) 1 ^7 (3.7) (lildE E dE-y 2 where niDM is the dark matter particle mass, poM is the dark matter density profile, (erju) and dN^/dEy are, respectively, the thermally averaged annihilation cross-section times the relative velocity v and the differential gamma spectrum per annihilation coming from the decay of annihilation products of final State i. The intégral is taken along the line of sight. The models will be constrained by the existing EGRET [46, 47] experimental limit on the flux, ~ 10“® photons cm“^.s“\ and the fortheoming GLAST [48, 49] expected sensitivity, ~ photons cm“^.s“^ for Afl = 10“^ and srad respectively. The processes involved in the production of the gamma ray flux at tree level are shown in Figures (3.1) and (3.2) and in the Figure (3.3), first diagram. Moreover, VF-boson loop processes can give rise to line signais from final States 77 and Z-y. As it was put forward in [70], the latter’s can dominate the contributions to the gamma ray flux if one considéra the case of a heavy Higgs with mass^ ~ 500 GeV. Indeed in the low mass régime i.e. ® Heavy Higgs is acceptable in the IDM framework for a good choice of the Inert scalar mass [68] 54 3. The Inert Doublet Model ^ 80 GeV (see section 3.4.1), the annihilation through a Higgs can be shown to be too low on the major part of the parameter space in order to account for the dark matter relie density. Especially when Mh > 2Mw- A substantial amount of coannihilation with the next-to-lightest Inert scalar is then useful to decrease the Hq population before thermal freeze ont and enlarge the viable dark matter parameter space. In the universe today, there is no more coannihilation as ail the next-to-lightest Inert scalar decayed in Hq. Moreover, annihilation through Higgs can be suppressed compared to the loop processes for heavy Higgs mass. In the latter case, gamma-lines become the dominant feature of the gamma spectra. Our purpose for the time being is merely to prospect the IDM with regard to gamma ray indirect détection. We focus on the particle physics parameter dependence of the model and work mostly within a fixed astrophysical framework, the popular Navarro-Prank-White (NFW) halo profile [43] (see section 2.1). It should be emphasized that the (dark) matter distribution in the innermost région of the galaxy is poorly known. The dependence of the dark matter profile on the distance r from the G.C. can be parameterized as poM ^ in that région (see Eq. (2.6)), with 7 = 1 for a NFW profile. Let us stress however that the 7 power of r is not very constrained, a freedom that can give rise to very different values of the gamma ray flux. Typically a suppression or an enhancement of two orders of magnitude can be obtained if one considéra respectively a halo with a fiat core (e.g. isothermal, 7 = 0) or a deeper cusp (7 ~ 1.5) [44] coming for instance from baryonic infall. Hence, depending on the astrophysics assumptions, the estimate of gamma ray signal can vary quite strongly (see e.g. [42, 79] and Fig.3.9). To calculate the flux we hâve integrated (3.7) for a NFW profile above IGeV and around a solid angle of AU = 10“^ srad for EGRET and AU = 10~® srad for GLAST. The differential spectra of each channel are given by micrOMEGAs as well as (aju) at rest at tree level and we hâve integrated the square of the dark matter density along the line of sight. Indeed for the Higgs masses considered here, the processes at loop level don’t contribute much to the integrated flux (see [70] table II). In the framework of indirect détection searches, another interesting track to follow is the possible détection of other dark matter annihilation products such as positrons and antiprotons or even in neutrinos. The resulting signais would be constrained by the positron flux measurements by the HEAT collaboration [80], anti matter détection experiments such as the balloon-borne experiment HESS [81] or the space based PAMELA (see for example a équivalent study in the SUS Y framework in [82]) or neutrino détection experiments such as SuperKamiokande, AMANDA and ANTARES and in the future Ice Cube (see e.g. [40] for a review). This is however let for future work. 3.3.2 Direct détection A local distribution of weakly interacting dark matter could be detected [83] by measur- ing the energy deposited in a low background detector by the scattering of a dark matter 3.3. INERT SCALARS DETECTION 55 particle with a nucléus of the detector. One distinguishes spin dépendent and spin indé pendant interactions. For i/o interacting with the quarks of the nuclei, there are two spin Z h indépendant processes at tree levai, Hqq —> Aqq and H^q —> H^q (see Figure (3.4)). The experiments hâve reached such a levai of sensitivity that the Z exchange contribution is excluded by the current experimental limits [68]. Consequently, to forbid Z exchange by kinematics, the mass of the particle must be higher® than the mass of Hq by a few lüükeV. We will thus disregard the A5 ^ 0 limit. We assume that the main process here is the one with Higgs particle exchange h. Massaging a bit the expression of quarks q and Hq interactions with the Higgs boson h one gets the effective Lagrangian: Ant = - X] ~ XlvH^h ^ Cg/f = (3-8) q q h The resulting spin averaged squared transition matrix for the process Hqp —> Hç,p is given by: 2 2 P rn•fl-P Si Sf where fp ~ 0.3 is the nucleonic form factor evaluated in équation 2.2. |A4p is independent of the scattering angle or equivalently of the Mandelstam t variable. As a resuit, the differential cross-section defined in (C.17) can be easily integrated out over t. The intégral limiting values are given by ^pIq^^ (for 9cm = tt) and 0 (for 0cm = 0), we get thus: _ 1 |a^|2_< (3.9) where we took s ~ (M^^ -|- rrip) in the non relativistic limit and rripMHo nir = TUp + Mho is the reduced mass of the System. In addition, for Mfjo < M\y, we hâve neglected the one-loop exchange of two gauge bosons. For > Mw, we include this process with, following [76], (Ji/g-p ~ 4.610“^^pb. We take into account the constraints given by the CDMS [84, 85] experiment (~ 10“® pb) and the model is also compared to the next génération experiments like EDELWEISS II [86] or a ton-size experiment like ZEPLIN [87] with the valley of their sensitivities respectively around a 10 ^ and (T ~ 10 pb. 3.3.3 Détection at colliders ? Obviously, direct and indirect détection searches are not the only available tests of the IDM as a model for dark matter. We can look for production and decay of the inert particles ®100kcV corresponds to the ~ 1 TeV dark matter particles kinetic energy in our galcixy 56 3. The Inert Doublet Model tlo Ho î h Q Q 9^ >^LVf Figure 3.4: Leading channels contributing to ctho-p (direct détection). at colliders. The subject must still be digged out for the whole paxameter space but let us cite the study of Barbiéri et al [68] as an introduction to this work. For Mhq — 60 — 75 GeV with a Hq — Aq mass différence AMAq < 10 GeV, they showed that HqAo pairs must hâve been created at LEP2 through the process e'^e“ —> AqHq with a production cross-section ~ 0.2 pb for 200 GeV center-of-mass energy. The dilepton event with missing energy resulting from the decay HqZ*) HqIÎ seems to escape the existing limits from the several LEP collaborations given the low production cross-section. For LHC, we should concentrate on: PP Z\-i* ^ AqHo or pp ^ W* ^ H^Hq or H^Ao followed by the decays or HoW"^ and ^ HqZ giving rise to leptons, jets and missing transverse energy in the final State. 3.4 Analysis of the IDM In this section we analyze in details the viable parameter space for the IDM to be a dark matter model. As emphasized in section 3.2, there are two qualitatively distinct régimes, depending on whether the Hq is lighter than the W and Z and/or the Higgs boson, in which case annihilation proceeds through the diagrams of Figure (3.3). At low mass, the second diagram of (3.3) may contribute to the dark matter abondance if there is a substantial number of Aq at the time of freeze-out {i.e. coannihilation). We begin the analyze with fixed Inert scalar mass différences and follow with a scan over several mass splittings. In the plots of Figure (3.5), the relie abondance of Hq dark matter particles, the gamma ray flux due to Hq annihilation at the galactic center (indirect détection) and, finally, their cross-section for elastic scattering off a proton through Higgs boson (direct détection) exchange are shown, for two particular Higgs masses (M^ = 120 GeV and 200 GéV) in the {Mh^,P2) plane. We refer to this as the low Ho mass régime. For higher Ho masses, similar plots are displayed in Figure (3.6) (for = 120 GeV), respectively for the dark matter abondance, the gamma ray flux and the cross-section for direct détection. In each of these three cases, the color gradients correspond respectively to gradients in log(DifQ/i^), log(«ï>^(cm-2s-i)) and log((THo-p(pb)). Since we plot our results in the {MhqiP2) plane, the diagonal line corresponds to 3.4. Analysis of the IDM 57 Xl — 0, i.e. to no coupling between Hq and the Higgs boson. Away from this line, Xl increases, with < 0 (resp. A/, > 0) above (resp. below) the diagonal. Also, we write AA/Aq = — Mifg and /\MHc = Mfj+ — Mfjo- In the plots of the dark matter abundance, the areas between the two dark Unes cor respond to régions of the parameter space such that 0.094 < ÇloMh^ < 0.129, the range of dark matter energy densities consistent with WMAP data. In the cross-section and gamma ray flux plots, the Unes indicate the areas of the parameter space within reach of the various experiments we take in considération. The shaded areas in the plots of Figures (3.5) and (3.6) correspond to régions that are excluded by the constraints enumerated in section 3.1.2: • Vacuum stability contributes largely to the exclusion of A^ < 0 couplings. This is the shaded area in the domains p2 > Mhq of Figures (3.5) and (3.6). • Perturbâtivity: tolerable couplings such with 1 < |Aj| < 47t correspond to the régions with horizontal Unes, the couplings |Aj| > 47t are excluded. Notice that the mass splitting relative to the p2 scale is smaller in the high mass régime (see Eq.(3.3)). For this reason, going away from the diagonal line in the plots of Figure (3.6), we run faster into the large coupling régime than in those of Figure (3.5). The régions excluded by the strong coupling constraint is of course symmetric with respect to the p2 = ^Ho axis. • Charged Higgs scalar constraint requires that > 79.3 GeV. As we fix the mass différences in our plots of Figures (3.5) and (3.6), this constraint translates in the excluded région Mhq ^ 30 GeV in the low mass régime abundance plots. • In order to avoid the conflict with LEP data on the width of the Z boson, we should impose + M^o < Mz- For our choice of mass différences in figures (3.5) and (3.6), this constraint translates into an excluded région Mhq ^ 40 GeV in the low mass régime abundance plots. • Given our choice of M^. and Inert doublet mass splittings EWPT constraint doesn’t wipe out any more space of the allowed areas. We now turn to the analysis of the model, starting with the low Hq mass régime. 3.4.1 Low mass régime, Mho around Mw The results of this section are shown in Figure (3.5). Two processes are relevant below the W, Z or h threshold: Hq annihilation through the Higgs and Hq coannihilation with Aq through Z exchange. Both give fermion-antifermion pairs, the former predominantly into bb. As the mass of Hq goes above W, Z or h threshold, Hq annihilation into WW, ZZ and hh become increasingly efficient, an effect which strongly suppresses the Hq relie density. Coannihilation into a Z may occur provided AAIAq is not too important^, roughly AM coannihilation is suppressed for our choice of AMHc- 58 3. The Inert Doublet Model logIO [A : mli=120 Q«V ; I2s10' MAO» 10 GaV ;A MHc= 50 G»V logIO |n h*l : mh=200 GaV ; 12=10 ' ;â MA0= 10 QaV ; A MHc= 50 GaV mHO (GeV) mHO (GeV) k>g10 [fluxlY(cfn'’r')] : mh=120 GeV ; 12=10'* ;A MA0= 10 GaV ;A MHc= 50 GeV logIO INuxlY(cm'^s-')] : mh=200 GeV ; 12=10'* ;A MA0= lOGeV ;A MHc= 50 GeV logIO [o^l : mh=120 GaV ; 12=10'' ;a MA0= 10 GaV ;aMHc= 50 GaV logIO : mh=200 GaV ; 12=10 * ;A MAO= 10 GaV ;A MHc= 50 GaV >4) O O '3^ ZEPUN 120 180 mHO (GeV) mHO (GeV) Figure 3.5; Prom top to bottom: Relie density, gamma indirect détection and direct dé tection contours in the (Mho,/^2) plane. Left: = 120 GeV. Right: Mh = 200 GeV. See full-color version on color pages in annex E 3.4. Analysis of the IDM 59 must be of order of T/o ~ Mhq/25. Otherwise Hq annihilate dominantly into bb. This régime is quite interesting: the interactions are either completely known {i.e. electroweak interactions), or highly sensitive to the Higgs boson mass Mh and the Hoh effective coupling Al! For the sake of illustration, we study two cases, = 120 GeV and Mh — 200 GeV to show how the prédictions change as a function of the Higgs boson mass. A few general lessons can be extracted from these two spécifie examples. Note that the mass différences AMAq and AMHc are fixed respectively to 10 and 50 GeV while AMAq is large enough to avoid the constraints from direct détection and small enough to hâve a some amount of coannihilation into a Z. We may distinguish five different régions (see Figure (3.5), top left): Région 1 is excluded by the combination of the constraints on the production of the charged Higgs and on the contribution to the Z decay width. Région 2 is excluded by the vacuum stability constraint. The latter involves the self coupling of the Higgs, Ai and thus dépends on the Higgs mass (see Eq.(3.2)): région 2 is broader for smaller Higgs mass Mh- Although région 1 is experimentally ruled out, it is interesting to understand the trends. In this région, most of the physics dépend on the Hq annihilation into two fermions through a Higgs, the abundance roughly varying as the inverse of av{HoHo —>■ ff), the product of the annihilation cross-section with the relative velocity of the Hq. With a center of mass energy y/s ~ 2Mhq, one gets : aviHQHQ ^ ff) ^,2 2 4n + where Ne = 3 for quarks and 1 for leptons. The relie abundance of Hq decreases with increasing Mhq as one goes along fi2 = Mhq- This is because when we increase Mhq-, Hq can annihilate with fermions of higher masses irif leading to higher av{HQHQ ^ ff) and lower relie densities. Comparing the Mh — 120 GeV and Mh = 200 GéV plots we observe that the gradients dépend, again, on the Higgs mass. This is simply due to the l/M^ dependence of the annihilation cross-section, i.e. smaller abundance for lower Mh- In région 2, the abundance is smaller the further one deviates from p2 = Mhq-, reflecting the dependence of the annihilation cross-section on |Al|- For Mhq < Mw (resp. Mho > M^), HqHq —> bb (resp. HqHq hh) is the dominant annihilation process. Otherwise, for Mw < Mho ^ Mh, HqHq WW, Z Z dominate. In région 3, the couplings are rather large, but nevertheless consistent with vacuum stability. Since P2 > Mhq, Mw, the dominant contribution to the cross-section is given by HqHq —> W^W~ process, large enough to bring the relie abundance far below WMAP data. Région 4 is below the W threshold. It is the only région consistent with the dark matter abundance predicted by WMAP data in the low mass régime. The process that détermines the relie abundance of Hq is again the annihilation through the Higgs. Coannihilation processes become dominant near the résonance at Maq + Mhq ~ Mz- This is the origin 60 3. The Inert Doublet Model of the dip in the Hq relie abundance around Mhq ~ 40 GeV (corresponding to M^o ~ 50 GeV for our choice of mass splittings). Similarly, annihilation near the Higgs résonance generates a second dip around Mfjg « Mh/2. There is an island consistent with WM AP data, which extends up to Mhq ~ 80 GeV, at which point WW annihilation becomes important and suppresses the relie abundance. Région 5, finally, corresponds to Mhq > Mw- Annihilation into a gauge boson pair is dominant, at least as long as Mhq < M^. Gauge boson pair production dominâtes above the Higgs threshold for 112 < Mjjo while Higgs pair production is dominant if ^2 > Mhq- In ail instances, the relie abundance is too suppressed to be consistent with WMAP for Mho ^400 GeV (see Fig.3.8 first plot) . As expected, and as revealed by visual inspection, the photon flux plots shares some of the characteristics of the abundance plots. The only new salient feature is the absence of a dip at the Z résonance. This is of course because today, in contrast with the early universe, ail Aq are gone. Existing experiments are not very constraining but future gamma ray détection experiments, such as GLAST, might severely challenge the model. As usual we should bear in mind the astrophysical uncertainties: by changing the galactic dark matter density profile, we can get higher (or smaller) fluxes (see Fig.3.9). The plot of the cross-section for direct détection is pretty transparent. Sufhces to notice (see section 3.3.3) that (Tho-p at tree level is proportional to The dominant contribution to the elastic scattering cross-section is thus zéro on the ^2 = AIho axis while it increases for larger values of \fX2 — AIhq\- The dependence on the Higgs mass (Eq. 3.9) clearly appears when comparing the ctho-p plots for = 120 GéV and Mh — 200 GeV. For Mhq > M\y, the one-loop contribution to the cross-section from W exchange is taken into account. However it does not affect much the results as it only amounts for 10“^^ pb. Unless we suppose that the mass of Hq and Aq are nearly degenerate, existing direct détection experiments are not very constraining. Fortheoming experiments however might put a dent on some of the solutions consistent with WMAP. 3.4.2 The high mass régime, Mh Mw The results of this section are shown in Figure (3.6), left column. No new annihilation channel opens if Mhq is heavier than the Higgs or gauge bosons. There are then essentially two sort of processes which control both the abundance and the gamma ray flux: the annihilation into two gauge bosons, dominant if fi2 < Mhq, and the annihilation into two Higgs, which dominâtes if ^2 > Mhq- Coannihilation plays little rôle. It affects a bit the relie abundance along the diagonal but, even so, it is not the key process to get the WMAP abundance. The abundance of dark matter is suppressed over most of the area of the plot because of large quartic coupling effects on the cross-sections. Strong couplings are excluded on a physical basis, but we found this limit nevertheless useful to understand the interplay between the varions processes. In the présent subsection, we will argue that it is possible to reach agreement with WMAP data, but only at the price of some fine tuning between 3.4. Analysis of the IDM 61 Iog10: mh=120 GeV ; 12=10 ' ;A MA0= 5 Q«V ;AMHc= 10 G«V mHO (GeV) loglO (fluxly (cm V)] ; mh=120 GeV ; 12=10 ' ;A MA0= S GeV ;A MHc= 10 GeV NPW mHO (GeV) loglO [o^ : mh=120 GeV ; I2=10 ' ;AMA0= 5 GeV ;A MHc= 10 OeV mHO (GeV) Figure 3.6: Left: same as Figure (3.5) for € [700,1400] GéV. Notice that the scale for the color gradient is however different. See full-color version on color pages in annex E 62 3. The Inert Doublet Model = ITcV. AMAo = 5GeV, AMII^ = lüGeV = ITeV. AMAo - lUGeV, AA//L = 5UGeV Figure 3.7: {crv)y^Q as a function of /i2 for Mhq = GeV and small mass splittings AMAo — 5 GéV, AMH+ = 10 GeV for the plot on the left and larger mass splittings AMAo = 10 GeV, AMH+ = 50 GeV for the plot on the right. Dashed line corresponds to annihilation into two Higgs, continuons Unes to annihilation into two gauge bosons. the different annihilation channels. Consider first the annihilation into two Higgs bosons (the dashed line in the graphs Figure (3.7)). Its cross-section is vanishing for Ax, = 0. For H2 < M^o (corresponding to Ai > 0), there is a destructive interférence between the diagrams of Figure (3.2), which is absent if jjL2 > Mhq ■ The annihilation into gauge bosons dépends on the quartic couplings between the scalars (see Figure (3.1)). Indeed, the annihilation through an intermediate Higgs is controlled by Ax,. Also, the t and u channels exchange diagrams are sensitive to the mass différences between the components of the H2 doublet, which dépend respectively on A5 for the annihilation into Z bosons and on A4 + A5 for the annihilation into W bosons. If Ax, — 0, there is no or little annihilation into a Higgs. The relevant diagrams are then the quartic vertex with two gauge bosons and the t and u channels with Aq (resp. //+) exchange. If A5 = 0 (resp. A4 -h A5 = 0) the cross-section into a Z (resp. W) boson pair is minimal and scales like ct^/Mfj^. If, for instance, A5 ^ 0 the annihilation into a Z boson pair reçoives a contribution which grows like o^^Maq — /M'^. A similar resuit holds for the annihilation into a W boson pair provided A4 + A5 7^ 0. Notice that o^{Maq — MhoŸ/M^ effects of weak isospin breaking between the components of H2 is reminiscent of what happens for the Higgs in the régime of strong Ai coupling*. To hâve an abundance of dark matter in agreement with WMAP, the mass splittings between the components of H2 must be kept relatively small. First because large mass splittings correspond to large couplings and second because the different contributions to the annihilation cross-section must be suppressed at the same location, around Ax, = 0 (ie. Mxxo ~ /i ~ Mao — Mh+ in this case). This is illustrated in Figure (3.7) for two different mass splittings. The first plot is for small mass différences (AMAq = 5 GeV and ® The so-called Goldstone boson equiv^llence theorem (see e.g. [64], section 21.2 and édso [68] annex D.l.) States that unphysical Goldstone boson eaten up by the massive gauge bosons can still control the amplitude of absorption and émission of the gauge boson in its longitudinal polarization State. As is can be seen in the scalax potential 3.1, the coupling of Ho — Ao to Zj for example is A5. 3.5. CONCLUDING DISCUSSION ON THE IDM 63 AMHc = 10 GeV), the second one for (relatively) laxger splittings {AMAq = 10 GeV and AMHc = 50 GeV). In the second case, the cross-section is too large to obtain an abundance consistent with WMAP, (crv)wMAP ~ pb. In the limit of small mass splittings and vanishing we hâve the right amount of dark matter provided Mhq ^ 400 GeV (see Fig.3.8). This régime corresponds to the narrow région around the diagonal in Figure (3.6). The abundance increases for increasing Mhq, but this can be somewhat compensated by playing with the mass splittings, which, as discussed above, tend to increase the cross-section. For instance, for AMAq = 5 GeV and AMHc = 10 GeV, we get the right relie abundance for Mhq ^ 800 GeV. There is however a limit to this trend. Indeed, as we hâve seen in section 2.2, the total annihilation cross-section of a scalar particle, like our Hq, is constrained by unitarity to be smaller than (3.10) a resuit which, in the context of dark matter relies from freeze-out, has been first put forward by Griest and Kamionkowski [55]. In the présent model, increasing the mass splitting drives the annihilation cross-section toward the strong quartic coupling régime. In our case, the bulk of the mass of the components of the H2 doublet cornes from the mass scale fi2, which is a priori arbitrary. However, the unitarity limit and WMAP data constraints, translate into the upper bound Mh^ < 120 TeV [55]. Unfortunately neither direct, nor indirect détection experiments are sensitive to the large mass région discussed in this section. Fortheoming direct détection experiments might do better, but as the plots show rather clearly, other forms of dark matter would then hâve to be introduced in order to explain the amount of dark matter that is currently observed. However, if the profile of dark matter in the galaxy is as assumed for the plot of Fig. 3.6 {i.e. NFW), future gamma detectors (GLAST) will probe most of the parameter space considered in this section, keeping in mind that we had to finely tune the mass splittings to obtain the right abundance. 3.5 Concluding discussion on the IDM The dark matter candidate of the Inert Doublet Model stands fiercely by the neutralino. The lightest stable scalar is a weakly interacting massive particle with a rich, yet simple, phenomenology and it has a true potential for being constrained by existing and fortheom ing experiments looking for dark matter. Figures 3.8 and 3.9 can be used in order to summarize of the IDM properties. In Figure 3.8, we show a scatter plot of and logiQaoM-p as a fonction of the mass of the dark matter candidate Mdm for a fair sample of models for the IDM and, for the sake of comparison, for the MSSM. In Figure 3.9, we show the log^o ^7 from the Galactic center for the same sample of models but for three different dark matter density profiles, going from flatter (isothermal profile with 7 = 0) to the most cuspy one (Moore profile with 64 3. The Inert Doublet Model Figure 3.8; Left: Relie density and Right: scattering cross-section intervening in direct détection searches, ail as a fonction of the mass of dark matter and comparison with the MSSM. For the direct détection plot, the light colors correspond to 0.01 < üoAih^ < 0-3, while the daxk colors correspond to 0.094 < < 0.129 and the two clouds of the IDM correspond to different values of Al, both in sign and amplitude. See full-color version on color pages in annex E 7 = 1.5) from top to bottom. Until now we only discussed the case of the NFW profile corresponding to the plot in the middle of Fig.3.9. In the {Mdmi ^DMh^) plane, in Fig.3.8 plot on the left , we clearly see the two régimes (low mass and high mass) of the IDM that may give rise to a relevant relie density {i.e near WMAP). The MSSM models hâve a more continuons behavior, with 0(100 GeV) dark matter masses. Let us emphasize that the IDM provide dark matter candidates with masses as small as 40 GeV and as large as 600 GeV in contrast with the MSSM more concentrated around ~ 100 GeV. For direct détection we see in Fig.3.8, plot on the right, that the low mass régime candidates should be detected by future ton sized experiments such as Zeplin. For the higher mass régime however, there is no hope for future détection in low background detector. Indeed, the WMAP requirement for dark matter relie density constraints the Al couplings to be vanishing while the same couplings drive the amplitude of the matter-i/o scattering cross-section. For indirect détection, the two mass régimes aiso clearly show up, as it C2m be seen in Fig.3.9. The IDM dark matter candidates hâve typically higher détection rates than the neutralino in SUSY models, especially at high mass. It is in particular interesting that the IDM can give the right relie abondance in a range of parameters which will be probed by GLAST for NFW dark matter profiles. GLAST will however hâve no chance to observe the gamma ray flux produced by annihilating i/o in the galactic center for flatter profiles such as the isothermaJ one (see the first plot of Fig.3.9). On the contrary for the most cuspy profile (Moore profile, last plot of Fig.3.9), the IDM is almost fully ruied out by 3.5. CONCLUDING DISCUSSION ON THE IDM 65 htb Core NFW Moore Figure 3.9: Integrated gamma ray flux from the galactic center resulting from dark matter annihilation as a function of the mass of the dark matter candidate for the same sample of models than for direct détection in Fig.3.8. Again, the light colors correspond to 0.01 < ÜDMh^ < 0.3, while the dark colors correspond to 0.094 < < 0.129. From top to bottom, the took an isothermal, NFW and Moore dark matter profile going from fiat to more cuspy profiles. See full-color version on color pages in annex E 66 3. The Inert Doublet Model IDM bcnchmark models. (In units of GeV.) Mode! mi, 771.40 771;/ + /*2 AaxlGeV 1 500 70 76 190 120 0.1 II 500 35 44 155 120 0.1 III 200 70 80 120 125 O.l IV 120 70 80 120 95 O.l IDM benchmark mode) results. Model '■«'LT" Branching ratios (%): (cm“s ') 77 Z'f l cc. 7-^r I 1.0 X 10 36 33 26 2 3 0.10 II 1.0 X 10-” 10 0 77 5 8 0.12 111 8.7 X 10“^' 2 2 81 5 9 0.12 IV 1.9 X 10 “ 0.04 0.1 85 5 10 0.11 Figure 3.10: Predicted gamma-ray spectra from the IDM benchmark model I and II as seen by GLAST (solid Unes). The predicted gamma flux is from a AQ, = lO^^srd région around the direction of the galactic center, assuming a NFW halo profile (with a boost factor as indicated in the figure) and convolved with a Gaussian 7 % energy resolution. The boxes show EGRET data and the thick line HESS data in the same sky direction and GLAST sensitivity is represented with dotted line. From [70]. EGRET observations. Let us emphasize that cuspy profiles are nevertheless not favored by observations, see e.g. [88] and reference there in. Let us stress that the plots for Indirect détection were obtained taking into account annihilation processes at three level only. In [70], the authors added the loop processes giving rise to line signais from 77 and Z7 final States in the low mass régime. The latter’s do not contribute much to the integrated flux for the Higgs masses we considered in this Work. This can be extracted from model IV of the table of results of figure 3.10. For higher Mh, their contribution of the gamma-line signais become dominant, the annihilation cross section is however suppressed by one or two orders of magnitude. For a NFW profile, they can fit EGRET data but they hâve to invoke a boost factor ~ 10“^ (assumed to arise form clumpiness), see Fig.3.10. Notice however that the IDM would typically produce stronger line signais than the MSSM in the mass range where the two models overlap (from ~ 50 GeV to ~ 80 GeV), see figure 3.11. The phenomenology of the IDM candidate for dark matter is intertwined with that of the Brout-Englert-Higgs particle, h. It could be of some interest to investigate the prospect for détection of the h and H2 components at the LHC, in the light of the constraints for dark matter discussed in the présent work. 3.5. CONCLUDING DISCUSSION ON THE IDM 67 WIMP Mass [GeV] Figure 3.11: Annihilation rates into gamma-ray Unes (upper band) and voz'^ (middle band) from the scan over the IDM parameter space with = 500 GeV, and Mu± = Mfjo -|- 120 (to fulfill EWPT) and A2 = 0.1. Importantly, one notes that the right relie density is obtained with a significant amount of coannihilations with the inert particle. For comparison the lower-right région indicate the corresponding results within MSSM as obtained with the DarkSUSY package. From [70]. 68 3. The Inert Doublet Model 69 Chapter 4 MeV right-handed neutrino as dark matter Is a right-handed neutrino with a mass in the MeV range still a viable candidate for dark matter? The answer to this question is of course model dépendent. In the présent chapter, we Work in the Left-Right (L-R) symmetric model (see section 4.1). Let us emphasize however that in this framework with the standard freeze-out mechanism MeV right-handed neutrinos can not account for the total amount of dark matter. The Lee-Weinberg bound imposes > 2 GeV for heavy^ neutrinos charged under SU(2)^. For SU(2)^ interacting neutrinos, the annihilation cross section is suppressed by a factor < 1 [14]. As a conséquence, the bound is shifted to even larger masses > 2{Mwr/Mwi^Ÿ GeV. One way out is to consider a low reheating température. Indeed, as was already men- tioned by Giudice, Kolb and Riotto in [89], a lower reheating température Tr// can lead to a smaller relie abundance. Notice that the reheating température Trh is not necessarily the température associated with the end of the period of inflation as several reheating events can hâve occurred since then. The most stringent constraint on Trr cornes from Big Bang nucleosynthesis, and in [90] it was shown that Trh can be as low as 0.7 MeV. Using this bound for the last reheating period before nucleosynthesis the authors of [89] hâve shown that the dependence of WIMP abundance on its mass and its annihilation cross section differs from the usual results. For a heavy, SU(2)^ interacting neutrinos for instance, we are used to the Cowsik-McClelland-Lee-Weinberg [15, 16] bound resulting from the constraint < 1 which excludes masses in the range 92h^ eV< < 2 GeV. Nevertheless, the cosmologically excluded région can become 33 kcV< < 5 MeV by lowering Trr to 0.7 MeV. Let us emphasize that in the latter case, the neutrinos never reached their equilibrium number density. Moreover the relie density is proportional to the neutrino annihilation cross section: oc ( ^“heavy” neutrinos refers to rrii, > 1 MeV which correspond to the freeze-out température of the SU(2)j;^ interacting neutrino, see section 1.3.2. 70 4. MeV RIGHT-HANDED NEUTRINO AS DARK MATTER The purpose of this work is not to study the details of the mechanism or the fields responsible for low reheating températures. We adopt an effective approach, in which neutrons and protons are produced during the reheating stage at the end of inflation, while right-handed neutrinos are produced afterwards by interactions between nucléons and électrons. We obtain a daxk matter candidate with a mass ~ 0( MeV) by considering the température at which the production of the vji become effective to be Tr// ~ 1 — 10 MeV. This resuit takes into account constraints on the assumed right-handed interactions. In this work, independently of the dark matter relie abondance, we derived a constraint on the Mwji/Mwi, ratio from the measurements by solar neutrino experiments. Our initial motivation for looking at dark matter made of MeV neutrinos came from the INTEGRAL experiment. A 511 keV signal [50] has been detected at the galactic center and has been identified as an electron-positron annihilation line. The source of the low energy positrons is unknown. This has prompted a number of spéculations, both on the astrophysics^ and on exotic models^ . More precisely, the scénario that inspired us proposes that light dark matter annihilation [102] is the origin for these positrons. Let us emphasize that the mass of this dark matter candidate is rather constrained, the upper limit being Mdm ^ 3 MeV [128]. Notice that a 7.5 MeV upper bound on Mdm was also derived in [129] assuming that the level of ionization of the medium in which the positrons propagate is of ~ 50%. In section 4.4, we study in more details the limits of this model. One of the characteristics of the 511 keV radiation is its close association to the bulge area of the galaxy. A natural question arises then: could the radiation (or rather the production of positrons which induces it) be linked to an interaction of dark matter with baryonic matter. It can be seen that this would lead to an émission région concentrated on the central part of the galaxy. In our case, the constraint Mdm ^ 3 MeV for annihilating dark matter translates into a constraint on the positron injection energies: Einj < 3 MeV. We corne to the conclusion that such particles interactions are possible, and could reproduce the peculiar angular distribution, but unfortunately with our right-handed neutrino we cannot reproduce the flux of the INTEGRAL signal. 4.1 The L-R symmetric model The absence of neutrino masses in the Standard Model, of a convincing baryogenesis mech anism, of dark matter candidates are some of the reasons among others why physicist look for extensions of the Standard Model. Often one invokes the existence of right-handed neutrinos to give a solution to these problems. Usually, they corne along with couplings to new gauge bosons and new scalars. One popular model that includes ail these ingrédients is the L-R symmetric model, with the gauge group SU(2)^ x SU(2)^ x U(l)g_£. It has been first introduced [130, 131, 132] because it provides an extension of the Standard Model which is initially parity symmetric, as it is the case for electromagnetic, strong and gravitational interactions. ^ see e.g. [91, 92, 93, 94, 95, 96, 97, 98, 99, 100] ® see e.g. [101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127] 4.1. The L-R symmetric model 71 In the L-R symmetric models, leptons and quarks are members of SU(2)^ and SU(2)^ doublets The right-handed neutrino that we consider as a dark matter candidate in the following is thus charged under SU(2);^. Assuming that the lagrangian is invariant under parity, the covariant dérivative can be written as: D^ = æ- ig{W^,rl - (4.2) with identical SU(2)^ and SU(2)^ gauge couplings g^ — gn = g- The matrices are the generators of SU(2)^ satisfying to the Lie algebra [r£,r^] = where a, b and c = 1,2,3, and an équivalent définition holds for r^. We will assume the usual L-R framework and give here a short description of the model. More details can be found in e.g. [133, 134, 135]. The scalar sector consists of one bidoublet^ : 01 4>2 4> = ~(l,2,2)o, 4>ï 02 which can generate Dirac masses for the fermions through Yukawa interactions of the form Fl^Fr with F = Q,L, and two additional triplets: Ax,~(1,1,3)2 and Aiî~(l,3,1)2 which are necessary in order to recover the SM gauge group after L-R symmetry breaking. The.'r advantage compared to two additional doublets charged under U(l)^_2^ is that their Yukawa couplings with leptons, of the form L'^C~^îcf2 Ll or {L R.), can generate Majorana neutrino masses and seed the seesaw mechanism assuming (A/,) {^r) {a = 1,2,3, (Tq are the Pauli matrices and C is the charge conjugation matrix, see the annex A.3). The symmetry breaking occurs in two steps: SU(2)^ X SU(2)^ X U(1)^_^-^SU(2)^ x U(1)^^U(1)q . (4.3) with Y = T3R + {B — L)/2 and Q = + t^r + {B - L)/2. Let us write ki^2 the vacuum expectation value (vev) of kl,r the vev of the triplets. Notice that the bidoublet (or a doublet) must break the SM gauge group in order to recover a relation = cosOyjMz- As a conséquence, for the scalar content considered here, we hâve to impose (A^) < (4>). The scalars give masses to ail the gauge bosons except for the photon field A^. The latter is defined as a combination of the three neutral gauge bosons W^.^, and B^- ^ a ~ (c,n,n')B-L means that a is in the représentation of dimension c of SU(3)^, of dimension n (resp. n') of SU(2)^ (resp. SU(2)^) and carry a charge B — L under U(l)3_^ 72 4. MeV right-handed neutrino as dark matter More generally, one gets the three neutral gauge bosons and through the orthogonal transformation: / ^ \ / -Su, -Su, y/cj- sj \ ( Wlz \ Z = cu, -sj/cw Su,jcu,\J Su, = sin 9.,'w Let us emphasize that the Z is an interaction eigenstate which couples to the particles of the SM in the same way as the SM Z boson. This can be seen by re-expressing the neutral gauge boson part of the covariant dérivative (4.2) in terms of Z^ and Z'^^: stKsd - + s'^B- ■ For massive neutral gauge bosons we hâve the following mass term: 4 A I ;z' ) (4.6) Z Vï _2ll , s2 ..2 fc?+fco + °w^r,^''w'yR. y/cl - slZ' We see that the charged SU(2)^ and SU(2)^ gauge bosons mix. We go to the physical basis through the orthogonal transformation: f Z,\^f cosC -sin^ V ^ ^ \Z2 J \ sinC cosÇ J\Z' J with approximate masses squared: {kj + kl {cl - si) 3^/ i + 4 Ml 4(24+ Ml + 2- f), (c; and the mixing angle, neglecting kl compared the vev of the other scalars, is given by tan 2^ ~ 2 y/cl- slMl/M^. The charged gauge boson mass term is given by: ^ + 4 -k,k2 'Vt \ sHwe w^) (4.7) -k,k2 ^ + 4 )' We define the mass eigenstates Wi and W2 as: / Wi \ / cos C — sin C \ V W2 / V sinC cosC /V Wr ) 4.2. CONSTRAINTS 73 and with the assumption kr » fci,2 ^ «l, we get: kik2 M^2 and tanC — The mixing angles C and ^ must be adequately suppressed to comply with experimental bounds (see section 4.2.1 and 4.2.2). Moreover the hierarchy kr » ki^2 ^ >^L implies that the SU(2)^ gauge interaction are weaher than the SU(2)^. In particular, neglecting the Q and ^ mixing angles, gauge processes for a Majorana vr reduces to interactions through the exchange of a Z' and a Wr^ which are much weaker than those among the SM fermions. For the neutrinos, we invoke the seesaw mechanism (see section 1.4.2) to get light "left-handed", and heavy "right-handed" neutrinos. The Yukawa lagrangian: -C = —ViLl^Lr — ÿiLR^LR — iX (L\C V2 ctoA^ L/, + (T <-*• R)) + h.c. with 4> = ct24>*(J2, reduces, for the neutrino sector, to the mass term: + h.c.) - + h.c.) where rtiuff — ^Xkr and ~ + Vik2)- The mass eigenstates ui and vr are mixtures of interaction States, VI cos 6 — sin6 \ h i>L \ (4.8) sin0 cos 9 ) \ ) where the mixing angle is given by tan 6^ ~ )^^^ for one génération. It should be empheisized that it is possible in the L-R framework to hâve a right-handed neutrino in the MeV range, while having right-handed gauge bosons masses in the TeV range. Compared to the case where heavy neutrino and right-handed gauge bosons m8isses are alike, this scénario does not seem as "natural" in the sense that it supposes tiny Yukawa couplings, but is still viable at this point. 4.2 Constraints Given the nature of our ~ McV Majorana neutrino, we hâve to take into account constraints Corning from two different origins. First of ail, we would like to hâve a stable dark matter candidate. As a conséquence it should hâve a lifetime at least of the order of the âge of the universe ~ 1.4 • 10^*^ years ~ 4.5 • lO^^s. The decay modes and the resulting limits on the mixing angles 6 and ( are studied in section 4.2.1. More stringent constraint on these mixing angles are derived in section 4.2.2 from the contribution of the heavy neutrino decay to the 511 keV and the MeV range gamma ray flux. In addition, our candidate interacts with SM model fermion in a very similar way than the SU(2)£^ interacting neutrinos. In particulax, the process i/R + n (^~^A) sr +p (^A) can contaminate the results of the solar neutrino experiments. In section 4.2.3, we dérivé bounds on the ratio or more precisely on Mw2/X'Iwi, arising mostly from the Chlorine experiments, for a ~ McV SU(2)^ interacting neutrino. 74 4. MbV right-handed neutrino as dark matter Figure 4.1: The diagrams (a) and (b) give the leading contributions to the decay channel i>H ^ è + e + vi, the diagram (c) to uh —> 3i^i and the diagrams (d) and (e) to uh —> i^i+7- The l in diagrams (d) and (e) corresponds to the leptons e, fj, and r. When fermion Unes are not represented with axrows, it is because the two fermion flows hâve to be taken into account. 4.2.1 Stability of the right-handed neutrinos The diagrams contributing to the decay channels of a MeV Majorana neutrino ë + e + ui —> 3^1 I/; +7 (4.9) are shown in figure 4.1. For the decay into e + ë + i^i, at first order in the mixing angles, we hâve three contributions. The effective lagrangian describing the interaction in the diagram 4.1 (a) is: -C(a) = -4^ + h.C. with — and is a Majorana neutrino. For the first term the decay proceeds via the ul — vr mixing in vr while for the second term it is via the Wr — Wr mixing in Wi. The effective lagrangian describing the interaction in the diagram 4.1 (b) is: £(b) = -2^ 47^((-1 + 24)^ + 2s^i7)t/'e + h.c. The resulting contribution from figures 4.1 (a) and (b) to the decay width to lowest order in 0,C, are: 2G? uu^eeui - 192^3 (0^ + + 450^) with 5 = 1/4 + 2s^ — 4- (4.10) 4.2. CONSTRAINTS 75 For the decay into Si'i, we hâve only one contribution to first order in the mixing angles through the i/^ — i^R mixing in uh- The effective lagrangian describing the interaction in the diagram 4.1 (c) is: £(c) = -2^ + h.c. The resulting contribution to the decay width at lowest order in 9 is: û2(~<2 were we hâve taken into account the 3 neutrino families. The calculation of the radiative decay vr ^il contribution is a little bit more in- volved. It is rather easy to see without going through the details of the calculation^ that there will be two separate leading contributions, one through the vl — mixing and the other one through the Wr — Wr mixing. The effective lagrangian describing the interaction can be written: ^ a where the index a = 1, 2,3 for the 3 lepton families which can appear in the loop process, 111 lu] and Tia includes the two previously mentioned contributions and involves the 75 matrix (the notation refers to [136] eq.(lO)). The resulting is dominated by the contribution oc C, originating from the Wr — Wr mixing: ^ua-^ui^l ^ (4-12) where mi is the mass of the heaviest intermediate charged lepton coupled to uh- One can see that the most stringent constraint on the mixing angles cornes from the radiative decay. The requirement > tu implies very suppressed mixings, (4.13) For a 4 MéV heavy neutrino it translates into the constraint 0 < 10 ® and C < 10 When these requirements are fulfilled, we hâve vi ~ ur so that one of the three light neutrinos is almost massless ~ 0, ~ vr and Wi ~ Wr, W2 — Wr. As a conséquence in the following we will use indifferently Mw^ r or Mh/j 2 and ur^r or ui^r- 4.2.2 Constraint on the decay rate from INTEGRAL and COMPTEE In [103], Picciotto and Pospelov hâve pointed out that the decay of a heavy neutrino into positrons and photons can dramatically contribute to the 511 keV line émission and to the diffuse gamma ray background. The resulting flux of photon will be too large at the ^Details of the calculation of l*® found in e.g. [136], [133] section 10.2 and 10.3 or [137]. 76 4. MeV right-handed neutrino as dark matter galactic center to be compatible with the observations of INTEGRAL and COMPTEL if we only impose the above mentioned constraints on the mixing angles, eq. (4.13). The expected gamma ray flux resulting from the neutrino decay in a solid angle Afl around the line of sight (l.o.s.) with direction tp can be deflned as = —-3- / f dspDM 47t Ji.o.s. where poM is the dark matter density profile, N-y = \ when P = and = 2x^ when r = where give the percentage of produced positrons that effectively® contribute to the 511 keV line (~ 30%). For rather steep dark matter density profiles in the central part of the galaxy^, we get in first approximation: ,3-7 J_ Plaçai ds pdm 47T J/AQ /Jl.o.i 3-7 using pDAiir) = Piocal{r/Ro) with piocal = 0.3 GeVcm ^ and Rq = 8.5 kpc, and assuming that the dominant contribution gamma ray flux cornes from an émission région with radius ~ Re at the galactic center. For the neutrino decay into positron, we hâve to compare the resulting {xp, Afî) = 07511 to the total flux measured by the INTEGRAL experiment at 511 keV, ^intégral — 10“^ cm“^ We get for® Re 1 kpc and 7 = 1: 07511 /MeV\ ~ 3.3 • 10^®s • ri/n^eevL (4.14) (pINTEGRAL V J This resuit is in good agreement with [103]. From the radiative neutrino decay, we get an additional photon line at an energy mj/^/2. We hâve to compare the resulting flux (0, Afî) = 0.y MeV to the COMPTEL measurements of the diffuse photon flux in the inner région of the Milky Way. The latter can be estimated as (Pgomptel ~ 0.016 (£^-),/MeV)~*^ ® Aü cm“^ from [128]. As a conséquence, we get for Re ~ 1 kpc and 7 = 1: 07 MeV MeV\°'^ ~ 1.3 • 10^*5 • rVR->'yi/L (4.15) (pCOMPTEL mVH J In order to avoid that the gamma ray flux resulting from the decay of heavy neutrino significantly exceed the experimental data today, we need the ratios (4.14) and (4.15) to be smaller than one. This constraint imposes to suppress the decay rates by an additional ® It has been observed that, in the interstellar medium of the galaxy, only ~ 3% of the nonrelativistic positrons annihilate directly with électrons to produce two photons of 511 keV. The others form first a positronium bound State with an électron. 25% of the time, they constitute para-positronium which decay into two photons and contribute to the 511 keV gamma ray line while 75% of the time they constitute ortho-positronium which decay into tree photons. As a resuit, the effective contribution of the low energy positrons to the two gamma line émission x-, ~ 0.03 + 0.97 * 0.25 ~ 30% (see e.g. [128] and references there in). ^like e.g. Moore and NEW profile, see section 2.1 ®A spherically symmetric émission région with a radius of Re ~ 1 kpc at the galactic center extends approximatively on ~ 8° in the sky and corresponds to a solid angle of Afl ~ 0.02 srad. 4.2. CONSTRAINTS 77 factor 10 ® — 10 compaxed to the stability constraints of eq. (4.13) or, equivalently, to suppress the mixing angle 6 and C by an additional factor of 10“®/^ and 10~^ respectively. Moreover, the heavy neutrino that decayed into photons and positrons ail over the matter dominated era of the Universe may hâve contributed to the diffuse gamma ray background. However, constraining the vh contribution with experimental data does not seem to provide a better sensitivity to the parameter of our model when < 1 [103] (see also [138]). Notice that we could also take the argument the other way around. Could the decay of a heavy neutrino be responsible for the 511 keV line émission observed by INTEGRAL at the galactic center? Light decaying dark matter scénario was already proposed in e.g. [105, 103, 138] but the likelihood analysis of [139] tends to rule out that kind of models: They seem to be incompatible with the morphology of the 511 keV émission région. We are currently verifying if their daim is founded or not in our framework. As a conclusion for this section, we hâve imposed that the neutrino decay into positrons and photons do not contribute to the photon flux at galactic center beyond the observation of INTEGRAL and COMPTEL experiments. This leads to more stringent constraints on the mixing angle than the stability bounds of eq. (4.13). For a ~ 4 MeV neutrino, we hâve 0< 10-13 and C< 10-13. 4.2.3 Constraints from neutrino experiments Figure 4.2; (a) Left(Right) charged current driving ^ ^ ei+and iyR+^ M sr + ^A.(b) Process producing the positron for the 511 keV INTEGRAL signal In our scénario, it turns out that the most stringent constraint on Mwn/Mwi cornes from solar neutrino Chemical experiments. As can be seen in figure 4.2, the positron production reaction -, Wh - . ^R-\-p----- > CR + n needed to explain the INTEGRAL signal, and the heavy neutrino capture reaction ur +^-i a Br +'^ A including ur + u gr + p are closely related due to the Majorana nature of the neutrino. As emphasized in the introduction, the INTEGRAL diffuse gamma-ray data is compatible only with positron 78 4. MeV right-handed neutrino as dark matter injection energies lower than 3 MeV in a neutral medium [128]. As a resuit, we consider right-handed neutrinos with a mass® lower than 5 MeV. Therefore, only the Gallium (Ga) experiments (Sage, Gallex, GNO) and Chlorine (Cl) experiments (Homestake) can be afFected. The energy threshold in water Cerenkov experiments such as SuperKamiokande and SNO is higher than this value. The observed rates in Ga and Cl experiments are [140, 141, 142, 143, 144, 145, 146] (/)(T|Ga = 68.1 ±3.85 SNU ; (/)cr|ci = 2.56 ± 0.16(stat) ± 0.16(syst) SNU, with 1 SNU (Solar Neutrino Unit) = 10”^® capture/atom/s. As these values are in good agreement with the results of Cerenkov water experiments (Kamiokande, SuperKamiokande and SNO), the possible right-handed contribution could at most fit inside the rate uncer- tainty. The total rate uncertainty is actually larger than the experimental error quoted above, because one should add the uncertainties coming from the predicted rates with and without oscillation based on our knowledge of the solar model, the capture cross sections and the results of SNO. It turns out that the total uncertainty at 1 cr amounts to 0.44 SNU for the Cl experiment and 6.6 SNU for Ga experiments [137]. (MeV) (a) (b) Figure 4.3: {a)Mwn/Mw[^ lower bound from the constraint (Pror < 0.44 SNU for Cl and (pR(^R ^ 6.6 SNU for Ga the solar neutrino Chemical experiment; (b) Maximal cross section aR allowed by solar neutrino data. Assuming equality between left and right-handed couplings, one can see that the rela tion between the left and right-handed neutrino scattering cross sections ctr^r = (r{vR,Ln ^r,lp) is quite simple (for a re-derivation of the cross section for the capture of a left or right-handed neutrino by a nucléus see annex D.2). The only différences résidé in the G constants and in the velocities \vi,\, o-R = ÜL- (4.16) ^Notice that 1.29 MeV of the 5 MeV neutrino mass fills in the mass différence between the initial proton and the final neutron in the positron production reaction va + p —^ ën -I- n needed to explain the INTEGRAL signal. 4.3. COSMOLOGICAL ABUNDANCE : LOW REHEATING TEMPERATURE SCENARIO 79 F;4MeV) cm^) ^Wa(io-4b from [147] from [148] 1 5.21 9.830.10^ 2 3.70.10^ 3.972. 3 1.02.10^ 9.905.10^ 4 2.23.10^ 2.129.10^ 5 5.38.10^ 4.380.103 6 1.44.10^ 8.434.103 7 4.62.10^ 1.530.10^ Table 4.1: Values of the neutrino absorption cross sections on Chlorine and Gallium as a function of neutrino energy in MeV. The cross sections are given in units of cm^. The second column refers to the cross sections on Chlorine obtained in [147] and the third column give the best-estimate cross sections on ^^Ga obtained in [148]. More informations can be found on the home page of John Bahcall: http://www.sns.ias.edu/~jnb/ If the local vr density is set to piocal = 0-3 GeV/cm^, we hâve a vr flux (j)R — Vi^j^piocai/fnuR, so that 2 (Gr /MeV o-L 4>r(^r = 0-9 10^ SNU. (4.17) \Gl V 10 ^°pb As emphasized at the beginning of this section, the later contribution to the total 4>a must be smaller than 0.44 SNU for the Cl experiment and 6.6 SNU for Ga experiments. As shown in fig. 4.3, the most constraining experiment is Chlorine. The values for were taken from table 4.1. For rjiuj^ ~ MeV we obtain Mr/Mr > 10 — 20, which corresponds to very small cross sections, typically gr < 10“^^ pb. As a resuit, the ur positron production cross section is limited in the same range. We find that it reaches a maximum of 2.4 10“^^ pb for ~ 4.5 MeV. It is interesting to note that we hâve a constraint on dark matter from neutrino experiments. This constraint is more stringent than direct accelerator limits, although less stringent than the much more spéculative astrophysical bounds [14]. Our value relies on the assumption that the local dark matter density is around the usually quoted value of 0.3 GeV/cm^. If, for some reason, this density is well below this value, the constraint could fall behind other accelerator bounds on 4.3 Cosmological abundance : low reheating température scénario As emphasized in the introduction, in the standard thermal freeze-out scénario, the so- called Lee-Weinberg bound forbids a neutrino with a mass in the MeV range and a cross section of the order of weak interactions to be a cold dark matter relie. For a Majorana 80 4. MeV right-handed neutrino as dark matter neutrino, its mass should be higher than 5 GeV, in order to hâve a relie density fit/ < 1. A crude évaluation for the cold dark matter relie density is given by 3 X 10~^^em^ (4.18) (av) For right-handed neutrinos, the Lee-Weinberg limit is even further inereased by a faetor Gr/GI ~ 10^ - 1.610^ We are thus led to eonsider a low reheating température seenaxio for our dark matter eandidate. Indeed, it has been notieed [89] that a deerease in the reheating température Trh can reduee the dark matter abundanee. Usually, Trr is supposed to be high, but the only serious eonstraint is that the standard big-bang nueleosynthesis (BBN) seenario should not be spoiled, leading to Trr ^0.7 MeV. We do not eonsider the decay of the field responsible for the reheating and we work with pre-BBN degrees of freedom to produce our dark matter neutrinos. We adopt an effective approach, in which neutrons and protons are produced during the reheating stage at the end of inflation, right-handed neutrinos are produced afterwards by interactions between nucléons and électrons. In the following Trr will refer to the température at which dark matter production processes become effective; let us stress one last time that it is supposed to hâve taken place well after inflation. The right-handed neutrino relie abondance is obtained by solving numerically the Boltz mann équation for the vr abondance after reheating. That équation can be derived from eq. (1.18) (see aiso [149, 150]). Using the variable z = rriu^/T, we hâve for z > Ziî/f: H{m^j^)s{z) dY^ = _.y(pe->ny) Y Y V. Vp Yn ■ Z dz y;\ ■ y2 ^ U - 1 (4.19) yeq ; where Yi = rii/s is the comoving density for the species i. Remember that below T ~ 100 MeV, the number of relativistic d.o.f appearing in the entropy s(z) is 5*5 = 10.75 while nowadays it is ,g*s = 3.91 (see section 1.3). logio(^) Figure 4.4: For = 4 MeV and Mwh/^Wl ~ 20, we plotted the logio(7/ MeV*) as a fonction of logio('2) with 2 = log]^o(7^^^~*”^V MeV^) and logio(7^”^~"^‘^V MeV^) merge in the dashed line, logio(7^^^~“^‘"V MeV"^) is represented with a continuons line. We see that ^ fQj. températures smaller than ~ O (10) MeV. The numerical computation was done re-expressing the 7 production rates as a fonction of the reduced cross-section à and the dimensionless variable x = slrnf,^^, with s the center 4.3. COSMOLOGICAL ABUNDANCB : LOW REHEATING TEMPERATURE SCENARIO 81 (a) (b) Figure 4.5: (a) Relie density of right-handed neutrino in the plane, (b) Solu tions of the Boltzmann équation for = 4 MeV (in units of the critical density today). of mass energy squared, not the entropy [151]: ™4 /-oo ^(x,+X2^u+y) ^ / dxà^Xx+X2^u+Y){x) Ki{zs/x) Xnxin where â = 8/s[(pi.p2)^ — (4.20) and Ki{y) is the first Bessel fonction following the conventions of [152]. This is valid for two body scattering. For more details on the reduced cross section appearing in our computation see annex D.3. In figure 4.4, we see that ^ r^ive^nu)_^ryine^pu)^ for températures smaller than ~ 0(10) MeV. This is due to the 7 dependence in the equilibrium number densities. Indeed, ^{Xi+Xi^v+Y) = nfnl^{a^x^^x2^u+x)v) and for these températures the equilibrium number densities are such'^° that n%^ 3> n^. Also, we suppose that the baryons decoupled long before this last period of reheat- ing, as a conséquence their comoving number densities appearing in eq. (4.19) differs from the equilibrium one. Assuming that the relation between neutron densities YnlYp = exp(—Q/r) with Q = mn — rrip — 1.293 MeV is valid for T ~ Trh (see section 1.4.1), we hâve: y y e-Q/T + i^ e+Q/T^l' Yb ~ 10“^^ is the relie baryon abondance. The final right-handed neutrino relie density is = P^vK^-y{2.iK'^'nivn/Pc- The results are shown on fig. 4.5 where we see that for ~ O(MeV), we can reach WMAP values Nevertheless, let us stress that the cross section iT(ee-.i/i/) is roughly two orders of magnitude smaller than and C(pe->ni^) in the non relativistic limit. 82 4. MeV right-handed neutrino as dark matter ~ 0.2) for Trjj ~ 1 — 10 MeV, which is just above the BBN limit T ~ 0.7 MeV. We can also see that the low reheating températures, the relie density is proportional to the production cross section, oc {av) in contrast with eq. (4.18) but in agreement with the results of [89]. Indeed, in fig. 4.5 (b), for a given neutrino mass and reheating température, the relie density decreases when the charged gauge boson mass ratio is increased from 10 to 20. This is illustrated by the gap of the relie 0,^^ between the dotted and the continuons curve. Moreover for a given neutrino mass and relie density, a higher reheating température is allowed when is increased. This is illustrated by the gap of Trh (the température at which the production of vr is “switched on”) between the dashed and the continuons curve. 4.4 The link with Intégral ? The gamma excess at 511 keV coming from the galactic bulge observed by INTEGRAL suggests electron-positron annihilations at rest. Also the positrons hâve to be produced with a low energy {Einj < 10 MeV) in order to stay in the bulge. Different scénarios hâve been proposed to explain the positron production, but there is no preferred explanation. Here, we consider a neutrino-baryon (ur-D) scattering through right-handed charged cur- rent interaction. Before going through the specificities of light dark matter models let us first review the characteristics of the 511 keV signal. 4.4.1 The 511 keV signal The INTEGRAL (International Gamma-Ray Astrophysics Laboratory) experiment study the celestial gamma-ray sources in the energy range 15 kéV to 10 MeV. In the following, we concentrate on the sky map of the 511 keV gamma ray line émission obtained from the data accumulated by the SPI spectrometer aboard the INTEGRAL satellite. Notice that the angular resolution of the SPI spectrometer is rather low (~ 3° Full Width at Half Maximum (FWHM)). As it can be seen in figure 4.6 the signal appears to be concentrated in the Galactic bulge region’^^. The shape of the émission région in figures (a) and (b) of 4.6 is nevertheless rather different. More quantitatively, in [50], Knôdlseder et al. characterized the morphology of the 511 keV line émission with a 2d angular Gaussian surface brightness distribution of 8.1° ± 0.9° FWHM extension in longitude {l) and 7.2° ± 0.9° FWHM extension in latitude {b). The total flux was estimated to be of (1.09 ± 0.04).10~^ ph cm“^s“^ and the émission was associated to the bulge région with no evidence for a contribution from the Galactic disk. The bulge-to-disc ratio flux was estimated to be in the range 1-3. These are the data on which was based our analysis for the MeV right-handed neutrino model in [125]. The Galactic bulge refers to a région with a dense population of stars around the Galactic center of spireil galaxies. For the Milky Way it extends from the Galactic center up to the galactocentric distance R = -V y'^ ~ 2.5 kpc and the height above the galactic plane z ~ 2.5 kpc. For more details on the galactic structure of the Milky Way see e.g. [154] 4.4. The link with Intégral ? 83 (a) (b) Figure 4.6: Model independent sky map of the 511 kéV gamma ray line émission obtained with the December 10, 2004 public INTEGRAL data release and additional data from the INTEGRAL Science Working Team for fig. (a) [50] and with the April 20, 2006 public INTEGRAL data release for fig. (b) [153]. The cleaned data set consist of 6821 pointed observations with a total exposure of 15 10® s and 18101 pointed observations with a total exposure of 36 10® s respectively. The contours indicate intensity levels of 10~^, 10“® and 10'“^ ph cm~^ s“^ srad“^(from the center outwards). The images are obtained with an image deconvolution algorithm based on an itérative procedure which was stopped after itération 17 and 19 respectively. With the resulting images one can recover the main chaxacteristics of 511 keV flux presented in [50] and [153]. Let us stress however that faint diffuse émission from the Galactic disk can’t be recovered at that point. For more details on the data analysis see the quoted references. See full-color version on color pages in annex E In the last data analysis presented in [153], by Weidenspointner et al. a significant émission is detected outside the bulge. Adding a disk component significantly improves the fits and there axe hints at a possible halo like émission region^^. The bulge-to-disc ratio flux is found to be in the range 0.4-1.4 which is lower than the préviens estimation. Let us emphasize however that for a “bulge only” model the authors of [153] obtains a total flux of (0.96 ±0.06). 10“® ph cm“^s“^ in agreement with the previous one. They also modeled the bulge émission using this time an ellipsoidal distribution with a Gaussian radial profile in longitude and latitude. They found a FWHM extension in longitude (/) of 6.5°lo'9° and 5.1°to8° in latitude (b) which is rather smaller than the ~ 8° FWHM extension of Knôdlseder et ai. The Galactic disk is made of a population of stars of different âges with a quite lower density than the bulge. It spreads out from the Galactic center to ~ 0(10 kpc). Let us aJso emphasize that the Galactic disk does not show a spherically symmetric structure. By contrast, in first approximation, the bulge can be considered as axisymmetric, even if it should rather be considered as bar shaped (see e.g. [155, 156]). For more detaûls on the galactic structure of the Milky Way see e.g. [154]. 84 4. MeV right-handed neutrino as dark matter O >a O § Q 3 'bO O log(iî (kpc)) \og{R (kpc)) Figure 4.7: Left: Some widely used dark matter density profiles pdm paxameterized by the function (2.6) and table 2.2. The fiatter ones, such as the Isothermal profile, are favored by observations (see e.g. [88] and reference there in). Right: Baryonic density profiles for the Bulge component of the Milky Way. In the figure presented here, we assumed a spherically symmetric halo (see équation (4.24)). Notice that the baryonic density profiles ps axe less extended in the galactocentric distance R than the halo dark matter profiles pdm- 4.4.2 Light dark matter annihilation as a source for positrons Several astrophysical sources hâve been proposed to explain the low energy positrons. The main difficulty seems to be to account for the morphology of the signal and for the large bulge-to-disk ratio of the émission. We will not go deeper in the details for the astrophysical sources^^. Light daxk matter annihilation XX ^ e'*'e” (4.21) has also been proposed as being at the origin of these positrons [102]. In that case there are two sources of uncertainties, the particle physics candidate and the dark halo one. Indeed, the expected gamma ray flux in a solid angle ACl around the line of sight (l.o.s.) with direction -0 is given by <^(^,Afî)/ dû f ds pIm (4.22) JAQ Jl.o.s. where C is a factor which dépend on the nature of the daxk matter candidate (see section 2.1), (cru) is the thermal average of the cross section for the process (4.21) multiplied by the relative velocity and give the percentage of produced positrons that effectively contribute to the 511 keV line (~ 30%). In [139], the authors hâve tried to constrain the daxk halo of the Milky Way assuming that the 511 kéV signal résulta from (4.22) testing separately a (av) dominated by the ‘^(see e.g. [91, 92, 93, 94, 95, 96, 97, 98, 99, 100]) 4.4. The link with Intégral ? 85 s-wave or p-wave contributions {a and b ternis in (1.24)). They concluded that a light (~ MeV) dark matter candidate annihilating essentially through s-wave process today with a {Mdm/ MeV)^(2.6 ± 0.12).10“^° cm^s~^ can be the source of the 511 kéV line émission and that the best fitting dark matter density profile is the NFW one (see figure 4.7 and section 2.1). Let us stress however that the dark matter density profiles extend well over the bulge région so that there can be a significant contribution to the total flux from the outermost région. The trick which “saves” light dark matter annihilation scénario is that the intensity of the émission outside the bulge région is so low compared to the bulge one that it is currently difficult to discriminate it from instrumental background. Let us emphasize that the mass of the light dark matter in the annihilation scénario is quite constrained. Moreover, these constraints can be translated in rather model indepen- dent bounds on the positrons injection energy that we hâve already taken into account in the choice of our neutrino mass range. Lets us rapidly review the constraints. The adjective “light” refers to particles with as mass Mdm ^ 100 MeV, i.e. particles which axe lighter than the most preferred WIMP, with masses in the GeV-TeV range (see section 2.3.4 and 3.5). This upper bound on the mass is chosen in order to account for the low energy positron production through annihilation without an excessive production of gamma rays at the galactic center’, in order not to contradict with the observations of OSSE, COMPTEE an EGRET (see e.g. [157, 158] and reference there in). However, it was shown in [159] (see also [124]) that internai Bremsstrahlung radiation (arising from Feynmann diagrams XX e'^e~'y, not from propagation in the medium) would conflict with COMPTEE and EGRET observations unless the injection energy of the positrons (= Mdm) is < 20 MeV. In addition, it was shown in [160], that the light dark matter candidate will remain confined to the galactic center in the presence of the micro-Gauss magnetic field if Mdm iS 10 MeV (with net displacements < 100 pc, much less than the émission région of ~ 0(1 kpc) size). The last constraint and the most stringent one cornes from inflight annihilation of energetic positrons in the interstellax medium. The resulting contribution to the galactic diffuse gamma ray flux is compatible with the data for Mdm ^ 3 [128] (resp. 7.5 MeV [129]) for the dark matter propagating in a neutral (resp. a partially ionized) medium. Taking this constraint into account, we hâve restricted the study of the ail chapter to MeV neutrinos with mass < 5 MeV. 86 4. MeV right-handed neutrino as dark matter 4.4.3 The MeV right-handed neutrino at the origin of the 511 keV sig nal? What would be gained with a neutrino-baxyon [ur-B) scattering as a source for low energy positrons conapared to the light dark matter annihilation scénario ? The answer is rather clear looking at figure 4.7 when one realizes that in our case the 511 keV gamma ray flux dépends on / PbPdm in contrast with the f ppj^ dependence of (4.22). With neutrino- baryon scattering the resulting 511 keV signal naturally reproduces the angular distribution of the INTEGRAL distribution as the baryonic density profile ps strongly decreases at the outskirts of the bulge région. This is rather useful especially when one tries to reproduce the morphology of the 511 kéV signal in agreement with the last data release. More precisely the expected gamma ray flux in a sohd angle Afî around the line of sight (l.o.s.) with direction tp, is defined in om case as: pB{r) = po,B (4.24) 0,.b) where the parameters Rt, Rq,b,0b and jb are defined in the table 4.2. The density of the bulge is proportional to for R < Rq, proportional to R~^^ for Rq R Rt and softly truncated at R = Rt- Moreover, the value po,s — 30 GeV/cm“^ can be derived from [161] for Dehnen & Biimey profile but it can be shown [137] that it is consistent with the other profiles by studying the galaotic rotation curve [162] at the bulge scale. Ail this discussion on the baryon-light dark matter scattering is quite instructive, un- fortunately, the magnitude of the measured flux ~ 10~^ photons cm“^ s~^ cannot be re- produced in our model. Taking into account the neutrino experiment constraints on the cross section, we get a maximum flux ~ 10“^^ photons cm“^ s“^, far too low to match the data! This contribution is suppressed by the choice of profiles and by the Majorana nature of un in the annihilation cross section. 4.5. CONCLUDING DISCUSSION ON MeV RIGHT HANDED NEUTRINO 87 Ro,b (kpc) Rt (kpc) 0B 7b Reference Dehnen & Binney (DB) 1 1.9 1.8 1.8 [161] Gerhard & Bissantz (GB) 0.1 2.8 1.8 0 [163] Gaussian(G) - V2 0 0 see e.^.[164] Table 4.2: Paxameters of some bulge density profiles, see équation (4.24) and figure 4.7. 4.5 Concluding discussion on MeV right handed neutrino We hâve studied the possibility of a MeV right-handed Majorana neutrino as dark matter candidate, in a Left-Right symmetric framework. The stability constraints can be ful- filled thanks to very suppressed left-right mixings for both bosons and neutrinos. The discrepancy between the produced gamma ray fiux through right-handed neutrino decay and INTEGRAL-COMPTEL observations can be avoided by imposing even more strin- gent bounds on the mixing angles. The solar neutrino Chemical experiments give the most stringent constraint on the right-handed interaction cross section, which appeaxs to be very suppressed. Such a low cross section also excludes the standard freeze-out scénario. The WM AP relie abundance for the right-handed neutrino can still be achieved, at the cost of a very low reheating température scénario (Trh ~ C>(10MeV)). Finally, we hâve shown that the INTEGRAL 511 kéV line signal morphology can be in good agreement with a dark matter-baryon interaction. In the spécifie model we hâve considered, the right-handed neutrino is a viable dark matter candidate, but the cross section is too small to produce the right flux of photons. We point out that the careful comparison of solax neutrino experiments does and will provide stringent constraints on this and similar dark matter candidates. [• » •• h *1 t * <_• t ) t /• 1 f f . ê \Y .4 89 Chapter 5 Matter Genesis Mechanism What is hidden beneath the word matter ? In cosmology matter refers to two a priori very different kind of particles: baxyons and daxk matter. First of ail, excluding neutrinos, dark matter seems to be absent in the Standard Model. Then there is the question of the production mechanism. In the simplest scénarios, dark matter abundance is driven by the freeze-out mechanism. If the same mechanism was at the origin of baryons, we would hâve faced the “annihilation catastrophe” and baryon abundance would hâve been rather negligible. We need to invoke a mechanism generating a baryon asymmetry well before the beginning of the exponential decrease of the baryon density in order to account for the observed baryon density. Hence, CIdm and ils seems to be two quantities fixed independently. If the standard scénario is correct, why axe the dark matter and baryon densities so close today ^dm/^b ~ 6? Moreover, although this ratio is constant today, it is not necessarily the case for ail the history of the universe. How to explain that we finally get Üdm/^b ~ 0{1) ? This looks like one of the so called “coincidence problem” that one can face trying to understand what rules our Universe. Let us emphasize that in the ratio ^dm/^b one has to understand both the particle number density ratio noM/nB and the particle mass ratio Mdm/Mb- In the présent chapter, we propose to give the same origin to baryon and dark matter number densities, the two of them resulting from an asymmetry in the dark and the visible sector generated through one single mechanism, a sort of “Matter Genesis”. Then, the mass of dark matter particles will corne in as a constraint. In a sense, this scénario is similar to leptogenesis which fixes the ratio between the number densities of one component of dark matter (in the form of neutrinos) and the one of baryons. Today, the ratio of their abundances is 4 < Üb/^u ^ 50, where the lower bound cornes from large scales structure formation (mjy < 0.5 eV) while the upper bounds cornes from neutrino oscillations {rriy > 0.04 eV), see section 2.3.3. This is surprising, since leptogenesis has nothing to say about the baryon to neutrino mass ratio. Yet the ratio of baryon to neutrino energy densities are almost similar. 90 5. Matter Genesis Mechanism The main idea of “Matter Genesis” goes back to old works of Barr et al [165] and Kaplan [166] and more recent inputs of Kuzmin [167] and Kitano and Low [168, 169]. For other proposais a bit further from the scénario we présent here, see e.g. [170, 171]. Let us outline the version of [168] that inspired us: By necessity, there is a dark sector and it is composed of a set of new particles. The visible sector, which consists of, among other things, baryons, and the dark sector communicate with each other but the interactions are suppressed at low energies. The lightest of these particles is protected from decay by a Z2 symmetry, analogous to R-parity. This lightest particle cannot be produced thermally in the Universe. If it were, the tiny asymmetry in the dark sector would be drowned by numbers. This last condition motivâtes the introduction of a particle in the dark sector that we call the messenger particle. This particle is strongly interacting and in thermal equilibrium in the early universe. Because it is strongly interacting, it stays in thermal equilibrium even when it becomes non-relativistic and that messengers and their antiparticles begin to annihilate. The situation in the dark sector at this point is like that for ordinary baryons in the visible sector. Baryons and messengers both survive to annihilation thanks to a tiny asymmetry in their respective sector. In the visible sector, neutrons decay into protons and the chain ends. In the dark sector, the messengers decay into the lightest stable particle, that should better be electrically neutral. Our Matter Genesis scénario takes place in a particular L-R symmetric model including “universal seesaw” ingrédients [172] and a Z2 symmetry discriminating dark sector from visible one [173]. Let us emphasize however that in this case the only fermions that won’t get their masses from a seesaw mechanism will be the neutrinos (and also the d quarks) by opposition to what was done initially in universal seesaw models [172]. The mechanism responsible for Matter Genesis is leptogenesis and dark matter will be made of light, Mdm ~ few GeV, right-handed Majorana neutrinos. The model is very constrained and, we agréé, not the nicest model one would dream of. However we believe that there are some lessons to be drawn from it. As we shall discuss, the main drawback of this model and its siblings, will be that, at the end of the day, it does not look very natural. Then, the dark matter mass range cornes in as a constraint, not as a prédiction. Finally, the kind of dark matter of the type we consider would escape ail attempts at détection. The messenger particle could be observed in high energy colliders, since it is a strongly interacting particle, similar to a (very very) heavy quark. 5.1 The Model We hâve chosen to concentrate on a spécifie L-R symmetric extension of the Standard Model that was proposed many years ago in [172] as an alternative to the SM way of giving mass to the quarks and leptons and is known in the literature as the "universal seesaw model". The gauge group is SU(2)j^ x SU(2)^ x U(l)^_£,, left and right-handed quarks Qr,l and leptons Lr^l are respectively SU(2)^ and SU(2)^ doublets as in section 4.1. The Higgs sector however will be slightly different, we consider two Brout-Englert- 5.1. The Model 91 Higgs doublets instead of one bidoublet^: XL ~ (1,2, l)i and XR ~ (1, l,2)i. To give mass to the quarks and leptons, one introduces a set of SU(2) singlet Weyl fermions and a Majorana fermion N: ~ (3,1,1)4/3 .D ~ (3,1, l)-2/3 ^^~(l,l,l)-2 A^~(l,l,l)o- Note the unusual B — L charge assignment of these fields. The Higgs bosons, for instance, hâve a non-zero B — L charge, and there is a completely neutral field N. The latter will play the rôle of the heavy Majorana particle, analogous to the heavy right-handed Majorana neutrinos in standard leptogenesis scénarios. This model looks nice but, unfortunately, we will need to complicate it a bit further. In particular we need to implement a discrète symmetry to protect the dark sector. We follow in that an old proposai of Babu et al [173]. First we add two Higgs scalars in the adjoint représentation of the SU(2): Ai ~ (1,3,1)2 Aiî~(l,l,3)2 which will give masses to neutrinos. Then we impose the following Z4 symmetry; Dl —Dl Qr iQr Lr -* —ILr Xr —* -iXR ^R ~^R Nr -Nr, ail other fields transforming trivially under Z4. The first effect of this symmetry is to forbid a Dirac mass term for the D field and Yukawa couplings to the N (would be neutrino Dirac mass terms usually leading to a seesaw origin for light neutrino masses, see 1.4.2). The allowed Yukawa couplings and mass terms then take the form = VciQlXlDr + VuQlXlUr + VbLlXl^r + XLi,C ^i(T2 (Ta^L, ^L + MuÜlUr + MeËrEr + + {L^ R) + h.c. (5.1) where a = 1,2,3 and 5.1.1 Symmetry breaking and Residual Z2 symmetry The interesting things corne with symmetry breaJcing. We consider the following breaking pattern: SU(2)^ X SU(2)^ X U(l)g_i XZ4 -^SU(2)i x U(l)5. XZ2 -^U(1)q XZ2 ^ with a ~ {c,n,n')B-L means that a is in the représentation of dimension c of SU(3)^, of dimension n (resp. n') of SU(2)^ (resp. SU(2)^) and carry a charge B — L under U(l)g_^ 92 5. Matter Genesis Mechanism with V — T3R + (B — L)/2 and Q — t^l + t^r -\-{B - L)/2. Let us write vl,r the vacuum expectation value (vev) of xl.r and the vev of the triplets. For reasons that will become clear after, we want the doublet and not the triplet to break the L-R symmetry, we hâve thus vr » kr. Moreover, the hierarchy Mwn 3> Mwi with^ Mw^ — Cw^z is obtained for vr » vl and vr » kr (see section 5.1.3). We stay with a residual Z2 symmetry which will play the rôle of the R-parity in SUS Y. The fields dR, VR, Dr, Xr, Nr (5.2) axe ail odd under Z2 and belong to the dark sector of our model. 5.1.2 Fermion masses The neutrino fields axe ail pure Majorana A K R vIur + A' K R v^VR + Mn N^^N. The up-like quarks and charged leptons get their mass from mixing with the heavy Dirac singlets through; where f — e,u and F = U, E thus foUowing the usual "universal seesaw" pattern. The twist is in the down-like quark sector. Because there is no Dirac mass term for the D field, mixing is maximal {dRÙR) ^ ) ( Dfl ) ^ dRÜR + ydVR ÜRdR = Vd '^L^Rd'R + Vd t^R D'rD'r, and the rôle of the "light" and "heavy" right-handed down-like fields axe so to speak exchanged. The D' = -D^)^ paxticle couples to SU(2)^ gauge bosons through the originally written-down dR member of a SU(2)^ doublet. It is odd under the Z2 symmetry, it is thus a member of the dark sector. It will be our strongly interacting messenger paxticle and it is supposed to be lighter than the singlet fermions. The ur get their mass from the vev of the SU(2)^ adjoint scalar field. It can thus easily be the lightest paxticle under the remnant Z2 symmetry (LZP) choosing (A^) C {xr) ^^nd our candidate for daxk matter as it is weakly interacting and neutral. In the sequel, we assume that rrii,^ «C M^. (The mass of the U and E axe not very much constrained. We will only request that the E, U disappeax before the electroweak phase transition.) is the cosines of the Weinberg angle 5.2. The Matter Gbnesis mechanism 93 5.1.3 Gauge boson masses By studying the kinetic terms of the Higgs sector lagrangian, we can rapidly see that the R their masses are given by The two massive neutral gauge bosons X\ and X2 (notations from section 4.1) mixes and parameterizing their mass term as: we hâve the following mass matrix A4: ^i/4 + «i —Â-A M = si{v}./i+^.) a^(^^f,/4+K|)+c£(u|/4+K^) (^w ®ty) \^u(c^ where and are the sinus and the cosines of the Weinberg angle and the second equality cornes from the assumed hierarchy vr^ vl with vr » kr and vr kr- In tbis approximation, after diagonalization of the mass matrix, we get Ml 9^/^w ^i/2 and Ml g^cHicl - si)vy2 with a mixing angle C ~ Vr{cI, - sl)/vy%. 5.2 The Matter Genesis mechanism Steps for Matter Genesis are summarized in figure 5.1. It begins with the out-of-equilibrium decay of the Majorana dark sector fermion N producing & B — L asymmetry in both the dark and the visible sectors of the same magnitude, but with opposite B — L charges. For what concerns the dark sector, the strongly interacting D' particles caxry the asymmetry and when T M ri they annihilate very efficiently leaving a tiny D' asymmetry momen- tarily out of reach of the visible sector. In the meantime, the visible B ~ L asymmetry initially carried by the SU(2) singlets U and E passes to SM fermions u and e through their respective decay. Then B + L violating processes, active imtil Tewi the electroweak symmetry breaking température, fixes the amount of baryon asymmetry with visible sector B — L asymmetry origin. On the top of it, the D' particles eventually decay after Tew into i/R and SM fermions giving a second contribution to the baxyon asymmetry and fixing dark matter number density. Let us emphasize that D' fermions must decay at T < Tew» otherwise the B — L asymmetry so-released would washed out the asymmetry présent in the visible sector and we would not be able to account for baryon density. We now go through a more detailed analysis of this succession of events. 94 5. Matter Genesis Mechanism 1. B-L asynunetry T N vis dark Ndec n = - n T D (D) II E (U E ) B-L B-L asym 2. Annihilation n - n = n D D D A D U E 3. U, e decay D II e Ist baryonic asymmetry T vis EW n = Cn B B-L D 1 » t / / > T 4. D decay 1 -► 'x U /e ' V ' Ddec -1 —->- n = n dm v„ fin ^ vis U t ...... ► n = C n + n B-L Figure 5.1: Steps of the Matter Genesis scénario. Darker (Dark blue) color refers to the dark sector while lighter (pink) color refers to the visible one and D = D' defined in section 5.1.2. 5.2.1 Initial B-L asynametry For definiteness, we assume that the out-of-equilibrium, CP violating N decay takes place after left-right symmetry breaking. The abundance of N^s could be thermal or they could be created during reheating after inflation. Note that these flelds are odd under the Z2 symmetry and are thus the grandfather of our dark matter particles. The decay process is supposed to be dictated by higher scale interactions but we can parameterize it by dimension six eff'ective operators like ■^NEDU + h.c., where the D particle is the mass eigenstate, odd under the Z2 symmetry (since there should be no confusion at this point, we drop the prime on the D). Assuming CP violation, these decay processes may sequestrate a, B — L asymmetry between the dark and visible sectors in a similar way that the right handed Majorana neutrino generate a B — L asymmetry through it is CP violating out-of-equilibrium decay in the simplifled model of leptogenesis studied in section 1.4.2: ms ,D -L — —— —QB-LiP'D — 'Rd)) where — ri[) = ni; — nu = ng — ue = sn^, with ^ — i^N-^EUD ~ ^N-*EDu)/^N- 5.2. The Matter Genesis mechanism 95 5.2.2 Annihilation of the messengers and first contribution to B asym- metry After séquestration of a B — L asymmetry in the dark sector, the Universe contains U, E and D particles on top of the usual Standard Model fermions. In the visible sector, the E and U are in thermal and Chemical equilibrium with the Standard Model fermions, and ail together they carry a Z2-even B — L asymmetry. Eventually, we will require the E and U disappeaxs through annihilation and decay be- fore the electroweak phase transition, leaving only SM degrees of freedom behind. As in standard leptogenesis scénarios, baryon number violating processes that are in equilibrium give birth to a non-zero baryon asymmetry ub = C ti'q-i = —Cq^_j^{ri£) — tiq). (5-3) The constant of proportionality C must be calculated in the standard way [31]. In annex B, we evaluated C in the Standard Model framework. Here the only change in the basic relations appears in the computation of the total electromagnetic charge of the universe Q, the D particle contributing for q^no- Before the electroweak symmetry breaking, using the relation between the diverse number densities and the Higgs and lepton doublet ones, no and ni^, given the unchanged Gauge, Yukawa and Sphaleron processes at equilibrium (after L-R symmetry breaking), the équation (B.2) becomes: 8 Q = {2Nf + l)no - -NfULi^ + q^no where Nf is the number of families. Re-expressing the D contribution in terms of the B — L asymmetry, the electromagnetic neutrality condition (B.3) becomes: 8 qD — NfTlL^ + -ns-L 2 2Nf + l \ 3 ^B-L As a resuit, for three families the relation between the visible sector contribution to B asymmetry and the initial B — L asymmetry is given by: /28 Q q^ \ nB = CriBtL with V79 79çf_J’ where 28/79 is the usual SM contribution. As q^-L — —2/3 and q^ — —1/3, the constant of proportionality C equals 25/79 in the présent model. In the dark sector, the messenger particles D carry a Z2-odd B — L asymmetry. They are heavy, Md oa vr, strongly interacting particles and when the température of the universe drops below their mass, they annihilate into light quarks but a small asymmetry survives — nf) k,ud ^ en/v. It is crucial to require that there are essentially no vr in the universe at this level since we want to obtain a relation between the baryon asymmetry and the density of 96 5. Matter Genesis Mechanism Figure 5.2: Decay of messengers into SM fermions. Left: dominant D decay channel through the exchange of a Wr, Right: F = U,E decay channel through Yukawa interaction into f = u,e. dark matter. As we will see in section 5.3, this condition constrains the scale of left-right symmetry breaking. It is also crucial that the messenger particles are strongly interanting so as to leave only the asymmetry as a remnant. 5.2.3 Decay of messengers into ur The dominant D decay channel (see figure 5.2) is D U + e + R with the decay rate: ____^ M% (5.4) 2®7T^ If the messenger particles were to decay before the electroweak phase transition, baryon number violating processes in equilibrium would completely erase the asymmetry (5.3). Indeed the vr carry no B — L charge in our framework and ail the B — L that was sequestrated in the dark sector is released in the u and e degrees of freedom. If D decay takes place after electroweak symmetry breaking, the final B asymmetry is given by (5.3) plus the contribution from the D decay into baxyons = C u'rLl + Qb D ^ + 9b) '’^d- (5.5) ^B-L The density of dark matter is simply equal to RDM = Rvr = no- Taking the ratio we obtain VLb ms rriB — {~^Qb-l + Qb) 0.5 Üdm Ol/R R^UR which implies that ^ 3 GeV. As expected, the mass of the dark matter particle is of order of the proton mass. This scénario is quite involved. The main element is that & B — L asymmetry is sequestrated in a sector insensitive to iî + L violating processes, at least as long as they are active, and is eventually released. In the présent model, this is possible thanks to an exact discrète symmetry which differentiate the dark and the visible sector. 5.3. CONSTRAINTS 97 5.3 Constraints There are several constraints to put on scales and couplings for the above scénario to work. They are summarized in the présent section. Ufl/TeV Figure 5.3: \og{vji/TeV) as a function of log(yd)- The different constraints : (5.6) line, (5.7) dashed, (5.9) dot-dashed (the excluded région is under these Unes) and (5.8) dotted (the excluded région is over this line). The allowed région is shadowed (in green). 1. Decay before BBN: The messenger particles D hâve to decay after EW symme- try breaJking to protect the baryon number from erasure. Moreover, since the D decay products contributes to the baryon number, D decay should take place before nucleosynthesis i.e. td < tsBN ~ 10^^ eV“^ where td is defined in (5.4). As a conséquence, the first constraint imposes that Vd^R ^ 10-21 TeV, (5.6) where is the D Yukawa couphng. This is quite a nasty constraint, since it require a rather small Yukawa coupling to be satisfied. 2. Decay after D annihilation: In order to get a ratio of baryon and dark matter number density of 0(1), we require D to decay after the completion oî D — D annihilation. This implies, using for the température of annihilation interactions freeze-out = Mn/xf {xf = 0(20)) : IQi^ Vd^'^R TeV . (5.7) 98 5. Matter Genesis Mechanism 3. Only D (no D): the D asymmetry produced in N decay must be larger than the D — D relie from freeze-out. In other words: > nÿ“{x,). Assuming that ail the D asymmetry contribute to dajk matter, one can evaluate riDM{xf) by blue-shifting today’s dark matter abundance until T^°. The D OT D rehc number density at that time can be estimated using the Boltzmann équation approximate solution (1.27) for s-wave annihilation process (n = 0) through strong SM model interactions with â ~ 0.2/Md [168]. As a resuit, we obtain following bound: VdVR < xio^ TeV. (5.8) V J 4. No thermal i/r: the abundance ur produced after reheating at Trr must be neg- ligible compared to the abimdance from D decay. The ur can be produced in the early uni verse through thermal scattering in the plasma around the reheating. Under assumption of instantaneous reheating (see e.g. [174] for the gravitino case), their production is driven by the following Boltzmann équation: where the light subscript refers to the species lighter than the reheating tempéra ture. Assuming that gauge interactions dominâtes^ the contributions to the cross section, we hâve (av) ~ g^T'^fm^. Neglecting the friction term and sup- posing very fast thermalization (in a radiation dominated era), we get: ~ This thermaUy produced number density must be smaller than the originating from D asymmetry which is about the baryon asymmetry ub/s t]/7 (see section 1.4.1). For Trr > Md, we get: hfvR >1Q23 TeV. (5.9) Ail together, these constraints yield a parameters space reduced to 10^ TeV< VR < 10^' TeV and 10"^ < Vd < 10“®. (5.10) This région is showed in Figure 5.3. There is a small but non-vanishing région where ail the constraints can be met. In particular, the D messenger particles axe rather light compared to the scale of left-right symmetry brealcing, with a mass in the TeV range: 1 TeV < Md < 10® TeV . This resuit is consistent with the results of Kitano and Low [168, 169]. Let us emphasize that U and E decay through their Yukawa interaction with the SM fermions: Fr —> fiXL with F = U, E and f = u,e. On the other hand, D decay through gauge interactions (see Figure 5.2). As a conséquence: ylMp IL ~ ~ 327T td y}Mp ® or equivalently we neglect the vr production via Yukawa interactions compared to gauge interactions 5.4. Observational implications ? 99 Figure 5.4: i/r annihilation through Z' into SM fermions /. with Mp » Md = Udi^R and for yj > y^, we hâve tf < W~^y^rD- This is quite compatible with U and E decay into SM paxticles earlier than the D one, before electroweak symmetry breaking. 5.4 Observational implications ? Our dark matter candidate is, by construction, rather light ~ GeV and abundant. Its interaction cross-section is, by necessity, very small. This is essentially because our right- handed neutrinos must be non-thermal relies, with nearly the same number density as baryons. We had to pay a heavy price to achieve this resuit. First, the discrète symmetry of our model is not particulaxly natural. Second, the Yukawa coupling of the messenger particle is quite small. Last, the mass of the dark matter candidate is fixed by hand. On the observational side, we expect our right-handed neutrinos to be présent in the core of the Galaxy where they could annihilate with each other producing for example a heavy Z' boson, or be coannihilated with right-handed quarks or leptons. Let us compute the annihilation cross section through Z' exchange represented in Figure 5.4, inspiring ourselves of the calculation we did in a slightly different L-R symmetric model in annex D.1.2. The Gauge boson mass ratio is given this time by Mz>/Mwr — Cwly/c^ — , the annihilation process is thus described by the following lagrangian: f Z^ f where '(pf = Îr+Îl (/ refers to any SM fermion) and the couphngs are defined in (D.2). Going to the non relativistic limit, one gets for the annihilation cross section multiplied by the relative velocity v: ^^0 N, ~ mj/ni, (Jv{uRVR ff) ~ ------a 4 4 ------4 Stt where Ne = 1 for leptons and Ne = 3 for quarks, the 1/4 factor cornes from the average over internai degrees of freedom of the transition matrix and m/ is the fermion mass. Unfortunately the cross-section is way too small, av < 10“^^ pb, to give any observable signal. Indeed, by way of comparison, the cross-section needed to reach the sensitivity of INTEGRAL signais would be C>(10 — 100 pb) for a dark matter candidate with mass of 0{ GeV) (see e.g. previous chapter or [102] for more details about the INTEGRAL signal and it is corrélation with light dark matter annihilation). We expect this conclusion to 100 5. Matter Genesis Mechanism be generic for daxk matter candidates related to the baryon asymmetry of the Universe as their thermally produced relie density must be suppressed compared to the dark sector asymmetry, although we hâve no general proof. Our dark matter candidate and the messenger hâve otherwise similar chaxacteristics as in the model discussed in [169]. In particular, baring other explanations, light right- handed neutrinos might be of interest to explain the apparent suppression in the power spectrum on small scales as it can behave as warm dark matter (see section 2.3.1). With realistic values for the D decay life time, using the relation (11) of [169], we get a dark matter free-steaming length ~ 1 — 10“^ Mpe for the upper part of the mass D spectrum (~ 10® — 10® TeV). Dark matter associated to the rest of the mass D spectrum tends to hâve smaller free-streaming length. The only hope to detect something in our model is by the production at a collider of the strongly interacting messenger particle, analog to a very heavy quark. Our messenger has a mass range between 1 TeV and 10® TeV, corresponding to a life time between 10^ s and 10“^® s. As already underlined in [168] such a particle could be produced at the LHC at least at the very lower part of this mass range. 5.5 Concluding discussion about Matter Genesis We hâve discussed a mechanism of Matter Genesis, based on a particular (and, we hâve to recognise, quite intricated) Left-Right symmetric extension of the Standard Model. At L-R symmetry breaking a Z2 symmetry remains to differentiate dark sector from visible one. The lightest particle odd under the Z2 symmetry (LZP) is a ~ 3 GeV right handed Majorana neutrino and it is our candidate for dark matter. In order to relate its current density to the matter one, we need its abundance to be driven by an asymmetry in the daxk sector. The latter has to be generated at some stage in the history of the Universe together with an asymmetry of the same amplitude in the visible sector which seeds the baryon asymmetry. As a conséquence, we need our LZP to be very weakly interacting which implies that it will escape to both direct and indirect détection seaxches. We expect this feature to be generic of dark matter candidates related to the baryon asymmetry. As one single mechanism is reponsible for the génération of visible and dark asymmetry, we refers to it as the Matter Genesis mechanism. The idea, which has been already proposed by several authors, is quite attractive. However we found it quite difficult to realize in practice. Although one should perhaps not try to draw a general conclusion from our model, the introduction of realistic gauge and Yukawa couplings shows that such a scénario is doomed to be very constrained. To ensure the survival of asymmetry, our ur is non thermally produced through the decay of a strongly interacting daxk sector candidate. The later has to hâve a rather long lifetime, which gives us a chance to detect it at LHC, provided that it is not too heavy. Another property of the model is that the daxk matter candidate can behave as warm daxk 5.5. CONCLUDING DISCUSSION ABOUT MATTER GENESIS 101 matter which could be of interest for structure formation. The main drawback of the model is that the mass of the dark matter candidate has to be fixed by hand. For the sake of completeness, let us mention one attempt which could confront this difficulty. This mechanism could arise in the context of scalax-tensor théories of gravity coupled to matter. Since the mass of the matter fields dépends generically on the vev of a scalar field ip, the presence of matter induces an effective potential for if. For concreteness, suppose that the coupling of V {(p) = mBe°“^nB + moMe^^'^noM, with a,P>0. Then ^dm/^b = 103 Conclusions and Perspectives In this thesis we hâve seen that hints for daxk matter are compelling. The success of Big Bang Nucleosynthesis (BBN) combined with the detailed analysis of the small imperfections of the Cosmic Microwave Background blackbody spectrum lead to the conclusion that most of the matter content of our universe is made of some non-baryonic material. We reviewed the so-called freeze-out mechanism which may settle the relie density of the species in the framework of the standard Big Bang model. We also examined principally two methods of détection of daxk matter, direct and indirect seaxches. Although the latter is suffering of astrophysics uncertainties, it gives a source of information and constraints on daxk matter properties which are complementaxy to colhder seaxches. Incidentally, we looked at the signais which, up to now, can not be explained by astrophysical sources or as resulting from interactions of Standard Model paxticles. Let us stress that the Standard Model on its own is unable to provide enough aspirants for the rôle of dark matter. As a conséquence, one has to dig into the tremendous domain of physics “Beyond the Standard Model” in order to hâve a chance to elucidate the problem of the missing mass. In this work in paxticulax, we hâve considered the Inert Doublet Model (IDM) which includes an additional Higgs doublet, enclosing two neutral scalars candidates for dark matter. We hâve also analyzed two different Left-Right symmetric extensions of the Standard Model with the daxk matter made of right-handed neutrinos. A rather detailed and systematic analysis of the IDM was caxried out assuming the standard freeze-out mechanism. We showed that the IDM offers an interesting alternative to Supersymmetry and its dark matter candidate, the neutralino. We recovered the results for the daxk matter in the IDM of Barbiéri et al. and Cirelli et al. which a priori did not seem to match. This is because the IDM provides daxk matter candidates in two rather separate mass ranges, one between 40 and 80 GéV, the other one between 400 GeV and 1 TeV. The physics driving the existence of daxk matter in these régions of the parameter space is quite different. One clear common feature is the dependence of the inert doublet phenomenology on its coupling to the Brout-Englert-Higgs paxticle and on the mass of the latter. We hâve also investigated the prospects for direct and indirect détection seaxches. Con- cerning direct détection seaxches, the low mass régime candidates should be detected with the futures ton sized experiments while the high mass régime will stay out of reach. We hâve emphasized that to avoid confiieting with current status of direct détection experi ments, we hâve to impose a non zéro mass splitting between the two neutral components 104 Conclusions and Perspectives of the inert doublet. For indirect détection searches we looked at the gamma-ray flux generated at the galactic center by dark matter annihilation. Whatever the daxk matter density profile assumed, we hâve corne to the conclusion that the inert scalars hâve typi- cally higher détection rates than the neutralino in SUSY models, especially at high mass and for relatively standard Higgs masses [M^ ~ 120 or 200 GeV). Moreover, the IDM could be probed by the future GLAST experiment. On the top of that, Gustafsson et al. recently shown that in the low mass range, considering a rather heavy Higgs, the IDM should be the source of particularly significant gamma-line signais arising at the one-loop level. Some work on the IDM is let for future investigations. The prospects for détection at colliders of the rather intertwined inert doublet and Higgs particles should be done. Other indirect détection channels, such as neutrinos, positrons and antiprotons should be studied. Moreover, the IDM can hâve repercussions on the neutrino sector, a more systematic analysis of this subject should be carried out. In this thesis, we hâve also addressed a low reheating température scénario for the genesis of dark matter in the form of MéV right-handed neutrinos. We worked in a Left- Hight symmetric extension of the Standard model. We hâve asked for the stabihty and a suppressed contribution of the neutrino decay to the gamma-ray observations of INTE GRAL and COMPTEL experiments. It turns out that in order to fulfil these constraints the mbdng angles of the neutrinos and of the gauge bosons must be very suppressed (~ 10“^^ and 10“^®). In addition, we hâve obtained a limit on the gauge boson mass ratio > 10 — 20 from solar neutrino experiments assuming that the mass of our right-handed neutrino is < 5 MeV. This restriction on the mass of our dark matter candidate cornes from our attempt to explain the excess of 511 keV signal observed by INTEGRAL which appears to concentrate on the galactic bulge région of the Milky Way. Looking at the shape of the dark matter and baryon density profiles at the galactic center, one can conclude that a dark matter- baryon diffusion into low energy positrons could more easily reproduce the morphology of the 511 keV signal than the already proposed scénarios of light dark matter annihilation. However, the bound on the SU(2);j gauge bosons implies a far too small interaction cross section, and consequently too small 511 keV gamma-ray flux to match the INTEGRAL data. Nevertheless, one possibility is still open in the same framework: we still hâve to test the abihty of a right-handed decaying neutrino to bear the excess of 511 keV photons at the galactic center. Finally, prompted by the possibihty to explain the baryon and dark matter rather similar abundances by one single “Matter Genesis” mechanism, we studied a non-thermal production mechanism for dark matter. The framework is again a Left-Right symmetric model but more heavy particles are involved in order give masses to ail the fermions of the Standard Model (a part for the light neutrino and the d quark) through a “universal seesaw” mechanism. The dark matter candidate this time is a ~ 3 GeV right-handed neutrino produced through the late decay of the dark sector messenger, some heavy right handed d-type quark D (with a mass Md ^ 1 TeV). This particle has to be strongly interacting in order for his abondance to be rapidly Conclusions and Perspectives 105 reduced to the asymmetry in the eaxly universe. It also has to be relatively stable to decay after the B+L violating processes fall ont of thermal equilibrium in order to protect the baryon asymmetry from erasure. Finally, it should decay before nucleosynthesis. These tree constraints are rather restrictive and impose a large Left-Right symmetry breahing scale {vji > lO^TeV). As a conséquence the right-handed neutrino interactions are very very suppressed (cru < pb!) and the dark matter particle will not be détectable. One possibility to test the model is to look for the strongly interacting D at colüders. Notice that, unfortunately, the mass of the dark matter candidate has still to be fixed by hand. The conclusion that we draw from “Matter Genesis” mechanism is that, beside its attractive motivation, it is quite difhcult to realize and to test. This is the end of this thesis but not of the intensive theoretical as well as experimental research on dark matter. Loads of questions on the subject are still open. Waiting for a direct identification of dark matter, indirect détection searches, despite their dependence on the astrophysics, can help to put rather restrictive bounds on the properties of dark matter. To conclude, in this thesis we hâve studied tree different extensions of the Standard Model with tree different dark matter candidates and production mechanisms. This is maybe the occasion to insist on the fact that Supersymmetry and its neutralino are not the only way out and that other models and candidates should be taken into account for future prospects. 107 Appendix A Conventions A.l System of units and Conversion factors Throughout this work we use the natural System of units, with c, k, and h eadi equaling 1. Conversion factors for these units are shown in table A.l which is borrowed from ref. [58]. Table A.l: Conversion factors for natural units. S ^ cm 4 K eV amu erg g S ^ 1 0.334x10-1'^ 0.764x10-1^ 0.658x10-1® 0.707x10-'!4 1.055x10-2’' 1.173xl0-4« 2.998 xl0i° 1 0.229 1.973x10-® 2.118x10-14 O.lOlxlO-ii" 0.352x10-®’' K 1.310x101^ 4.369 1 0.862x10-4 0.962x10-1® 1.381x10-1® 1.537x10-®’’ eV 1.519x101^ 0.507x10^ 1.160x104 1 1.074x10-1' 1.602x10-12 1.783x10-®® amu 1.415x10^4 0.472x1014 1.081x101'! 0.931 xl0« 1 1.492x10-® 1.661 xlO-^'* erg 0.948x10^'-' 0.316x101' 0.724x10^^* 0.624x101'! 0.670x10® 1 1.113x10-21 g 0.852 x1Q4« 2.843x10'!^ 0.651 xlO'ii' 0.561x10®® 0.602 xl0®4 0.899x1021 1 cross-sections can be expressed in diverse unity, most often, it is expressed in picobarns pb and centimeters. The Conversion factors going from one to another is Ipb = 10“^®cm^. More generally in this work we will use the following units: 1 pb= 10-3® cm2= 0.33410-^® cm3s-^= 2.569 10”® GeV-^ . A.2 Scalar and fermion fields: normalization conventions Throughout this work we use the following convention for the plane wave expansions of the field operators normalization: 1 N{k) = (A.l) 108 A. Conventions where Wk = yW+m? is the energy associated to a state of momentum k and mass m. The the plane wave expansions of the complex scalar field operator, solution of the Klein Gordon équation of motion: (9^ + - 0 derived from the lagrangian: C = dp(j^d^ — m^4>^ (t> = ^ (A.2) with the commutation relations: k — k'^ and 6^, ht = (ji — k'^ . (A.3) The Dirac équation is given by: {ip — = 0 and is derived from the lagrangian: jC = ■0(i P The Dirac field ^ can be expanded as: tp{x,t) = J (fikN{k) + bl (A.4) with the anti-commutation relations We hâve chosen the foUowing normalization for the u and v spinors: {P)us' ip)Vs' (p) = 2WpÔss' ■ (A.6) which implies using Dirac équations that: üs{p)us'{p) = 2môss' and Vs{p)Vs>{p) = -2môss' Ÿ^Us{p)Us{p) = {ip + m) and - m). (A.7) 5 5 A.3 Charge conjugation and Majorana fields First we define the charge conjugate oî ip as 'tp‘^ = C'ijF where the charge conjugation matrix satisfies: = C"\ C'^ = -C and (A.8) A.3. Charge conjugation and Majorana fiblds 109 and U and v spinors for either Dirac or Majorana fermions are related by: U ^k, s'j = Cv'^ (je, s'j equivalently ü ^k, sj = —v^ ^k, s'j and u v (A.9) With these properties, we can obtain the following useful relations: V2OU1 = —viOcU2 and Ü2OU1 = —viOcV2 where Oc = CO^C^ (A.10) The Majorana fermion is self conjugate: . The équation of motion for Majorana ôelds is the same than the Dirac équation, the lagrangian however takes a factor 1/2: C = {i ^—m)'^. (A.11) Zi For the Majorana field can thus be expanded as [176, 177]: (x, t) = Ç y* d^kNik) (aj; ^u (k, s) + at v (k, s) . (A.12) Notice that it is always possible to construct Majorana spinors from Dirac ones. Before doing so let’s remember some misleading property of the combination of parity and charge conjugation^: ir)L = Lr = {Ripy = (^r)^ Let’s fix thus the following notation once for ail: i’L = ipR = {-iPrY- If we define the following Majorana spinor: = ipR + tl^R, =» = and = ■0R the Majorana lagrangian (A.11) and the Majorana équation becomes in term of iI)r: __ __ C = ipRipxl)R-—{iliRtp% + h.c.) and i p tpR - mrp% ^ 0. One last tricky thing is the définition of the charge conjugate of a bidoublet made of Diran fields such as the one introduced in the Standard Model: (A.13) where (A.14) and it is inserted in the définition of the in order to get an opposite isospin for the components of the bidoublet under Charge conjugation. 1 In this annex t/j refers to Dirac fields and to Majorana ones 111 Appendix B Relation between B and L asymmetries In this annex, we study the emergence of Baryon and Lepton asymmetry in the electroweak plasma before the electroweak phase transition considering an initial B — L asymmetry and sphaleron processes at thermal equilibrium. This calculation was first done by Harvey and Turner in 1990 [31]. Let us suppose in addition that only Standard Model particles participâtes to the equilibrium processes, including the SU(2)j^ fermion doublets Ll,Ql and singlets UR,dR,eR, the gluons, the two chaxged gauge bosons W^, the two neutral ones (one for SU{2)l, the other for C/(l)y), and finally the two scalars from the Higgs doublet: et h~. As it can be worked out from the définition of the number density for a relativistic specie x defined in (1.8), a non zéro asymmetry nx — rix is closely related to the value of the Chemical potential fix- Supposing (Sjjbx 1 with (3 = 1/T, one gets: gT^ Pnx + O for fermions , (B.l) 20iix + O (/3/ix)" for bosons . Additional relations between Chemical potentials of the diverse species can be obtained by equating the sum of the Chemical potentials of the incoming paxticles to the sum of the Chemical potentials of the outgoing ones for processes at equilibrium. Simplifying assumptions and observations: 1. We suppose that the universe is in a singlet state of the Standard Model Gauge group SU(3)^ X SU(2)^ X U(l)y. The color singlet condition implies that red, green and blue quarks hâve identical Chemical potentials gr = fJ-v = and = 0 for gluons. Weak isospin singlet condition imphes = 0. 2. Neutral gauge boson number is not conserved in the interactions, as a conséquence they hâve vanishing Chemical potentials. 112 B. Relation between B and L asymmetries 3. We suppose that Cabibbo mixings between quark families axe strong enough in order to keep = IJ'qLR, where i is the family index i = and q = u,d type quarks. Non zéro neutrino masses lead to possibly rapid flavor-mixing interactions for leptons so that we take fip — where l = e,u. L Nine Chemical potentials still hâve to be determined and many processes at thermal equilibrium must be taken into account. Prom now on will stand for the asymmetry between x particles and their charge conjugate x in order to lighten the mathematical expressions. Gauge interactions give the following relations: W- ^ h~hP W- ^ dLUL w <-»• i/ieL M- ~ “Mo Mul “ l^dij — MQi l^ei, — — MLx, Tl— — TIq “ ^di, ~ '^Ql ^ejr, — Prom Yukawa interactions we get: <-> ulur didR glcr Mufi P'Ql Mo MQl Mo Mefl Mo UuR = nq^ + no/2 Ud^ = nq^ - no/2 n^R = - no/2 and from B + L violating Sphalerons processes we get: |0) ïlfJli'^LdLdLl'Lh + NfULL = 0 where |0) refers to the vacuum. Restricting ourselves to the SM particles, the total elec- tromagnetic charge of the uni verse is given by: 2 1 Q = NfNc[-{nuj. + Uur) - -{udj^ + UdR)] - Nf{ne^ + ng^) - 2nw- ~ (B.2) where A^g = 3 is the number of colors. Rewriting ail the number density asymmetries of Standard Model particles in terms of no et and asking for electromagnetic neutrality of the universe Q = 0, we get: {2Nf + l)no^ (B.3) Baryon and Lepton asymmetry ur and ur written in terms of uq et give: ns = NfNc^iuuR + UuR + Ud^^ + nd^)) = ^r , 13 1 =» ur-l = Nf [ —+ 2^0 riL =Nf{nei^ + ritR + = ■^/(3?^LI, “ km) The neutrality condition (B.3) yields to the following relation between any initial B — L asymmetry and Baryon and Lepton asymmetry: . 21V/ + 1 ur = kur-l, ni = (1 — a) ns-L with A = 4------(B.4) 22Nf + 13 Por three families, Nf = 3, so that ur = 28/79ur-r. These relations are valid for températures higher than the electroweak phase transition but below 10^^ GéV (see section 1.4.2). 113 Appendix C Cross-sections and Unitary bonnds We are going to get unitary bounds on annihilation cross-sections on which are based the unitary bounds on dark matter mass derived in section 2.2. The following is inspired from [56] sections 3.6 and 3.7 and [54], for more details on the details of calculation the reader is invited to look at these references. C.l S'-matrix unitarity and Optical Theorem General principles of relativistic quantum mechanics implies the unitarity of scattering matrix éléments Sa/s describing the transition between two multi-particle States a and /3. As a conséquence: J- Jd7(a|5t|7)(7|5|^) = ô{a - 0) (C.l) with f d'y = J2ai,ni,a2,n2,... I d^pid?p2-- with ai being the spin projection on Iz (or helicity for massless particles) and rii any other discrète indices labeling the particle. Lorentz invariance of the theory allow us to write : Sa/3 = (al-S'l/3) = 5(a - P) - 2mô^{pa - Pp)Ma/3. (C.2) Using définition (C.2) in équation (C.l), we obtain the following condition on the transition amplitude ImMaa ^ J^7<^^(Pa -P'r)\Mya\^. (C.3) In particular, if a is a two-particle State (7 can be any number-particle State) : 1“^- = (C-4) where Uq is the relative velocity in State a and aa is the total cross-section in this State : 114 C. Cross-sections and Unitary rounds UaEiE2 -- ^(pi.p2)^ - rnlml, = j7) = jd'y5'^{pa-p-y)\M^o.Ÿ, (C.5) 1 and 2 label the particles of the State a. The unitary condition in this form States a particular relation between the total cross-section from an initial 2-body State to ail possible final States and the 2-body 2-body forward transition amplitude. This condition is usually expressed in term of the scattering amplitude /(q —^ 7). In the center of mass (CM) frame, we define for the two body scattering (7 = 1'-!- 2' is a 2-paxticle State): f(a ->7) = -^]lyE[E!,E,E2M. da \f{a 7)|2 dü, with Pi - {Ei,k),p2 -- {E2,-k),p[ = {Ei,k'),p2 - {E2,-k'),E = Ei + E2 and dü. = sm.6d6d(j). The unitary condition (2.7) takes the form: Im/(a ^ a) = (C.6) 47T and it is called the Optical Theorem. Sometimes, this name is given to the unitary condition (C.4) (see for instance [54]) but let us stress that there is an important différence between the relations (C.4) and (C.6). The two of them résulta from unitarity of the 5-matrix éléments but, the first one is concerned by 12 —> ail final states processes whereas the second one is only valid for 1 2 —> 1' 2' processes in the CM frame. C.2 Partial wave expansion and Unitarity bounds In order to use properly the relation C.4, we expand the scattering amplitude in term of partial waves, i.e. in term of states labeled by the total momentum p — P\ + P2, the total energy E, total spin s with z-component p,, the orbital angular momentum l with z-component m and total angular momentum j and its z-component cr. Here we expand M in the CM frame for a 2-body 2-body scattering: E2 —k'ai^n'-Jicri —kcT2n = EE E jer Ismii l's'm'u,' \k\EiE2\k'\E[E!2 X Cs'^s'^W, a'ia2)Cis{ja\ m'p')Cs^s^{s, p; aia2)Cis{ja; mp) (C.7) where = {Vs'n'\M^\lsn), Cs■^s■2{s,p^■,c^\cr2) is a Clebsh-Gordan coefficients that combines the two spins si and S2 with z-component a\ and 02 to give a total spin s with z-component p, and Yf^{k) is a spherical harmonie function, k — p\ = —p2- C.2. Partial wave expansion and Unitarity rounds 115 C.2.1 Partial wave unitarity We can get the total cross-section by integrating (C.7) over the final momentum k', sum- ming over final spins and averaging over initial ones and finally summing over ail two-body channels : ^ On the other hand, the unitary condition on the 5-matrix (2.7) give us the expression of the total cross-section (from n to ail possible final States) ; ^ A;2(2si + 1)(2s2 4-1) l)2Re(lsn|(l - S^)\lsn). (C.9) Comparing the two expressions (C.8) and (C.9) for the cross section , we find that the matrix S^{E) is unitary i/only two body channels can be readied from the species content n at energy E : (Zsn|(l — 5-^)^(l — S^)\lsn) = 2Re(/sn|(l — 5-^)|Zsn) =^> S^^E)S^(E) - 1. If channels involving three or more particles are open at this energy then the cross- section atot — Yln' —> n' = 2-body, E) -|- a{n —> n", E), where n" labels the “more than 2”-body channels, and : n" \ \ / jig C.2.2 Unitary bounds on cross-sections We would like to bound the annihilation cross-section of dark matter, so that we are interested in XX, or XX scattering processes, excluding the elastic ones such that XX —> XX or XX XX. The elastic contributions are given by : 7T <^el — (2i + l)|(Z's'n|(l - S^)\lsn)\‘^. (C.IO) A;2(2si + l)(2s2 + l) jVs'ls We subtract this to the total cross-section (C.8) and get the inelastic contribution : ^ A;2(2si + 1)(2s2 + 1) jls 1 - |(Zsn|(l - S^)|Zsn|2) - ^ |(ZVn|(l - S^')|Zsn|' . (C.ll) As S^^E)S^(E) = 1, implies |(Zsn|(l — 5-^)|Zsn)p < 1 using f cZ7(Zsn|5-^^|7)(7|5-^|Zsn) > |(Zsn|(l — iS'-’)|Zsn)p, it is easy to obtain the foUowing bounds on (C.8) and (C.ll) : <^tot ^ Ew + i). A;2(2si-f-l)(2s2 + l) ^ jls 116 C. Cross-sections and Unitary rounds 7T ^inel ^ E<2j + !)■ (C.12) fc2(25i + l)(2S2 + l) jls C.3 Cross-section - General expression Throughout this work, we hâve to evaluate cross-sections for diverse processes. Let’s define once for ail our convention for the transition matrices and their relation to the 5-matrix. Using the normalization conventions of annex A.2, following nearly [14], we can also write the relation between 5-matrix to the transition matrix 7^^ as: S'/3a = where the factors Ni — 1/-\/{2'ïï)^2Ei are the particle wave functions normalizations defined in A.2 and dérivés from Feynmann rules (see e.g. [2]). C.3.1 Spin averaged cross-section The spin averaged differential cross-section is defined as: (C.14) 4\/(Pi-P2)^ - rn\ml with n 7 E E (C.15) 0 aea Sa,a€a si,,be0 where we recognize at the denominator part of the définition of the relative velocity Ua (see Eq. C.5) the numbers 1 and 2 labeling the particles in the initial State a, \A4\^ corresponds to the square of the transition matrix iTpa averaged over initial spins and summed over final ones and includes the appropriâtes symmetry factors 1/nJ^! for njj identical particles in the final State and pa counts the number of spin degrees of freedom in State a. Also d?Pb = JJ with d<ï>(, (C.16) (27t)32£;6 be/3 where the square of the normalization factors of the final State are hidden. For a 2 2 scattering process, going to Mandelstam variable s, t and n, it can be shown that the differential cross-section takes the simple form: du 1 \MŸ (C.17) dt where p\CM is the momentum amplitude of particle 1 in the center of mass frame p\qj^ — C.3. Cross-section - General expression 117 The relative velocity v = Ua defined in Eq. C.5 and it can be shown that in the center of mass frame: vcM E1CME2CM = y/{pi-P2)'^ -mlml = picMV^- As a conséquence, the relative velocity of two annihilating dark matter candidates (see P DM- / DM- / p' k' Figure C.l: Dark matter annihilation process into two fermions figure C.l) in the center-of-mass frame is given by VCM = 4picm/\/s = 2uicm (C.18) where we used Ewm = E2CM = and V\CM = Picm/Eicm- For t = {p - A:)^, one has : dt - svcm/^ y/l - m^/M|,^(l - Ucm/4) d cos 9cm svcm(^ ^/l - dcos 9cm where the second equality is valid in a non relativistic limit and 9cm is the angle between the momenta p and k in the center-of-mass frame. In this particular case, (C.17) can thus be rewritten (forgetting about the CM subscript and using s = 4M|)^/(1 — u^/4) ~ 4M|,J^^): yl vrij/M^M av{DMDM //) JlMl ^ dcos 9. (C.19) C.3.2 Thermally averaged cross-section In this work we regularly need to handle the thermally averaged cross-section times the relative velocity v in the initial State (cru) which was defined in order to get an approximate solution to the Boltzmann équation in 1.3.2. For the process 12 34, Eq. 1.19 gives: ^1+^2 e V (27r)^f(5'^(pi-f-p2 — P3 ~P4) |A4|'' where n^2 equilibrimn number densities of particles 1 and 2, d$i..4 are defined in (C.16), and eq 9a f n, e V d'^pa (227t)3 J 1 \M\ = ^^gi92\M\ (C.20) n where a = |12) is the initial State. Massaging a bit this expression and using the quantity we defined previously in this section, we get: 1 1 91 (au) = y e "r d^pi y e "T d^p2 au(12 —> 34) n n)^! 27T^ 118 C. Cross-sections and Unitary bounds where a is the integrate spin averaged differential cross-section defined in (C.14) and v = Ua is the relative velocity. Typically, one can Taylor expand the annihilation cross-section as a function of v: a A = àv~^ -|- -I- 0{v^) where à and b don’t dépend on v. As a conséquence, the expression for (ctav) can be expanded as; {cTAv)v^Q = ^ -I- l{v^) + = a 4- h{v^) + ... (C.21) where the second equality uses the conventions of section 1.3.2. As a resuit, for selfcon- jugate annihilating particles, the first term in the {(Tav) expansion for vanishing relative velocities is the half of the first term in the Taylor expansion, and this is valid at eanh order of the expansion: a = Ô/2 and 6 = 6/2 for selfconjugate annihilating particles. The equipartition theorem gives for a given particle: (l/2maU„) = 3/2kT and in our conventions of annex A, k — 1. As a conséquence {vf) — 3T/m\, and in the center of mass frame, the relative velocity between two annihilating particles^, using (C.18), is given by (forgetting about the CM subscript): (u^) — 6T/mi — 6/a;. As a resuit, we can rewrite (C.21): 0 {aAv)v^o = a -f 6- -I-... (C.22) X where ci a is the annihilation cross-section and x — m\/T. ^Obviously, they hâve identical masses mi = m2 119 Appendix D Left-Right symmetric model In this annex we refer to the L-R symmetric model presented in 4.1. D.l Interaction lagrangian and Feynmann rules D.l.l Right-handed Charged current The Wr gauge bosons interactions with fermions axe similar to the Wi ones, for ql — 9r = 9i the only diflFerence is in the projector : = 9 [Jw^R^, + h.c.^ with f.e. ^(1 + 7^)0e ^ rC.c. _ _9__j] ^Wr where ^ . This gives for example, for a lepton-neutrino interaction the following effective lagrangian : ^Wtff = + 7^)V^ei0e2 7Q(l + 7^)V’i^2- D.1.2 Z' Neutral current In the L-R symmetric models two neutral heavy gauge bosons appeax, one is the usual Z boson, the other one is called Z'. In the case where the scalar in the adjoint représentation of SU(2)jj has a vev larger than the bi-doublet and the scalar in the adjoint représentation of SU(2)^, we hâve : to^r — cos(20m/)/(2cos^ We can describe its interactions with fermions with : rN.c. L-z> 120 D. Left-Right symmetric model where Qy ^ = —4Q sin^ d\v + sin^ Ow + 2T3^ cos^ 6w sin^ 9w + cos^ 9w (D-2) As a resuit, we obtain the following effective lagrangian for Z' neutral current: 2 rN.c. _ /t ^Z'eff - (D.3) ^cos 9wmz' This gives for example, for a lepton- Right-handed Majorana neutrino interaction: -N.C. Gr -Z'eff •0v7“(l ® V)V’e- 4\/2 cos^ 9w D.1.3 Majorana neutrinos, tricky Feynmann rules Suppose that the neutrino i/ is a Majorana fermion, + ^R- Let’s study in more details the neutral current : with F“ = 7“(1 + 7®) emphasizing that for Majorana neutrinos, it can take many forms: = VR')^VR = ^^,(F“ + F“)^„ (D.4) Notice that in this case F“ + F“ = 27“7^. As a conséquence, it can be considered as well as a particle or anti-particle or semi particle semi anti-particle current. For the sake of illustration this let’s consider the following interaction e(p)e(p') —> i^R{k)vR{k') through Z' exchange. Prom the previous section we know, it is described by the lagrangian : ^ ^ 4v2 cos2 9w which can be written with indifférence, using (D.4), as for example: û ' ^ + 5a ®7^)V’e- (D.5) ■' ^ 4 V 2 cos^ 9w Now we hâve to dérivé the transition matrix for the e{p)e{p') —> i'R{k)vR{k') process. Let us emphasize that for an interaction with a Z boson (or a neutral scalar), we hâve to consider the two possible current interacting with the neutral boson. This can be seen in the proper calculation of the S fi matrix as the Majorana fermion operator contains both particle création and particle destruction operator, and a (see (A.12)). Another way is to look at the contributing Feynmann diagrams of figure D.l. As a conséquence we obtain the following transition amplitude: ftfc27°7^r;fe/V7a(flv ^ + pf ®7®)up. 4v 2 cos2 9w D.2. The u A-> e~ A CROSS-SBCTioN 121 VR <^R Sfi oc (^'iî^'fl|...^',/(A:)7a75^'^(/c')|ee) Sji ou {ujîi'ii\...'^„{k')ja'yb^u{k)\ee) Ûk'ïalbVk' ÜknalbVk Figure D.l: Feynmann diagrams illustrating the two possible current Majorana neutrino current interacting with the neutral boson. Feynmann rules are derived using lagrangian (D.5). The üfc27®7^Ufc/ = üfc(r“ + r“)ufc/ contribution antually cornes from ilfc7“7^î^fc' ~ taking into account the two possible current flow using the properties (A.10), the minus sign is due to the dissimilax fermion order in the two terms (see e.g. [176, 177] or compute properly the Sfi matrix). Let us thus stress that in the transition matrix F“ + F“ appears even if a factor 1/2 is présent in the lagrajigian and it is due to the two possible current flow. We can conclude that the matrix element {fl'^,^\i) for Majorana neutrinos ^1/ is twice as much as it is for {f\'ipv'y°‘'y^ipu\i) with the Dirac neutrinos ipu- Another important Feynmann rule is, when you hâve two identical particles in the flnal State (as it is the case when you hâve two Majorana neutrinos in the final State) you hâve to divide by two the intégrations over the angles (or over t) to calculate a cross-section. If not it is as if you considered the situation with 6 angle different than tv + 9 angle. But for two identical particles in the final State these two situations are indistinguishable. D.2 The V ^ e A cross-section We re-derive here the cross-section for the capture of a left or right-handed neutrino by a nucléus of mass number A and atomic number Z: UA ^ e~A. (D.6) The effective lagrangian describing the interaction through the exchange of a Wx, or a FF/î is give by: r j(h)jQ(0 with jo,(i) =ipe'7°‘{l±Ÿ)i}u and ± 9Al^)'>Pi where ja{i) is the leptonic current and is the hadronic current with / and F corre- sponding to initial and final nucléus and G/^/2 = /SM'^. Using the définition of C.3.1, the spin averaged cross-section is given by: a - FiEe.Z) {2Trf2Ef Yli[m\vi-v,\ 2 ^ E spins 122 D. Left-Right symmetric model where = 1/2 Es; E^.s^..^). the average bearing upon the spins of the initial nucléus, the neutrino beam is supposed to be polarized. F{Eei Z) tahe into aecount the increase (resp. decrease) of the probability of the presence of an électron (resp. a positron) near a nucléus of charge Z compared to the probability of the presence of a free électron (resp. positron) at that point: F{Ee,Z) |’î/’e,Z=oP Notice that for /3 disintegrations Z = 1 (n p) and F{Ee,Z) —> 1. The relevant transition matrix squaxed for the Wl or Wr mediated interaction is given by: Y, \Tfi? ^ 2,2{\{l)\^ + g'j,\a)\^)mi{mF + Eea^)E,Ee spins where we already took into account that in the laboratory, we hâve Ei = mj, Ep — mp + Egx where Eex is the excitation energy of the final nucléus and the ~ in the Espms définition cornes from the fact the we already neglected the term oc Qu-Qe which vanishes after the intégration over the électron momentum. We define = —9a/9v and |(1)P and |(cr)p are the squaxed of the so-called reduced matrix éléments between the initial and final nucléus States. |(1)|^ is the Fermi matrix and |(o’)p is the Gamow-Teller matrix contributing to the tansitions where initial and final nucléus spins are identical and different respectively. The resulting spin averaged cross-section is given by (7 = l{GgvfapeEe^-~^ Q{E) TT \Vu\ where o = |(l)p + p^|(cj)p, 0(F1) = &{Ee -t- {mp -b Eex) — mi — E^). The factor {Ggv)‘^a contains ail the information about the nuclei, it is usually [178] reexpressed in terms of’^ , >2 _ 27t^ ln(2)m“® _ 27t^ ln(2)m“® 2Jy -f 1 ^ ~ ^ 2Ji +1 • This is because the /ti/2 faetor is measured in laboratories for the capture processes (see e.g. [179] for the ^^Cl among others). As a conséquence, the cross-section is given by: A7r^\n{2)aZm-\,G{Ee,Z)^,^, PeF{Ee,Z) a —771---- N------[—-j where G{Ee,Z) [jh/2)ji-^Jf \Vu\ 2'ïïaZEe ^ The lifetime of the initial nucléus with nuclear spin Jj is given by {t\/2 = half life time= ln(2).r) : = ^m®(GFpv)V(AS/i, Z) (D.7) where Gf — Gl is the Fermi constant and 1 roo roo fiAEfi, Z) = —, dp^pl / dp.pU(Ee -E,- AEfi)F(Ee, Z) frie Jo Jo AEfi/m.e „ 1 dxx{x^-l)i {AEfi/me - xfF{x, Z) (D.8) where we took m„ = ~ 0. We are awaxe that in our model •'he second contribution is quite negligible. D.3. ReDUCED cross-sections for THE MeV Vr production 123 1.20610-^2 ^ where <70 (D.9) V sec >jf with We = Ee/rue. Notice that for Eg S> mg, G{Eg,Z) ~ (27raZ) ^ and for Eg ~ rug, G{Eg,Z)c^l. D.3 Reduced cross-sections for the MeV ur production The reduced cross-section is related to the usual un-averaged one by : à = 8/s[(pi.p2)^ - - Sp^Qj^a so that : dà 1 dt Stts E ITfiP- sptns Note that there is only a sum over the spins for initial and final particles, no averages !!! D.3.1 Charged current = ^9V '0eT“(l + l^p7a(l + 9A'y^)i’n + h.c. where gy = cos 9g = 0.9741 et = —gAldv = 1.2695 • Reduced cross-section for pn —> urc is (no contributions to Boltzmann équations because it cornes with factors ngqn^g and n% is negligible) : {{'^ + 9a) [{rnl + ml - t - s){ml + ml - t - s) +{ml + ml~ t){ml -f - t)] -\-2g'A [{ml + ml-t- s){ml + ml~t- s) -{ml + ml~ t){ml + ml~ t)] 4-2(1 - gA)mnmp{s -ml~ m^)| t± = ^ {(m^ - 4- mlf ((s 4- m-p - — 4smp)^'^^ ± ((s 4- - ml)^ - 4sml)^^^ 124 D. Left-Right symmetric model • Reduced cross-section for pe tivr is : ô- = dt + g2) [{s - ml - ml){s - ml - ml) + {ml + ml-t- s){ml -|- nip - t - s)] [(« - "in - -ml- ml) -{ml +ml-t-s){ml + ml-t- s)] -2(1 - 9A)mnmp{ml -h - t)| 1 r/ 2 2 2 I 2\2 t± = -{{mp-m^-m^ + mj - j^((s -I- rrip - ml)'^ - Asm-l)^^^ ± ((s -I- ml - mlf - ^sml • Reduced cross-section for en —* urp is : êr = dt {{^+gA)[iml + ml-t-s){ml + ml-t-s) +{ml + ml- s){ml + ml- s)] +2g'^ [{ml + ml-t- s){ml + ml - t - s) -{ml + ml- s){ml -f - s)] -2(1 - 9A)mnmp{ml -f - t)| t± = —{{ml-ml m;2 +, ml)^2\2 ((s + ml- mlf - ^sml) ± [{s + - mlf ^smlŸ'" D.3.2 // l'Ri'R : Neutral current + Charged current For W exchange : Gr Gw = '0e7“(l + l )'^u t^i/7a(l + 7 )V’e For Z' exchange (no vectorial current for Majorana fermions!) : 'N.C. _ Gr I _5 C V’i/7“(1 + l^)i’u {fnoc{9v^ + 9a ^Ÿ)tpf) ^ 4\/2 cos^ 6w f=e,UL,p,n where 9y^ = — 4Q sin^ Ow + sin^ 6w + 2T/^ cos^ 6w 9^^ — — 2T^ sin^ 0\v + 2T^^ cos^ 9w tIlr are the SU{2)r^r isospin eigenvalues for the fermion /. The reduced cross-section is given in this case by : dt 2 Stts spins D.3. ReDUCED cross-sections for THE MeV Ur production 125 even if there is no averages over the spin. It is because there is two identical particles in the final state. • Reduced cross-section for ee —> i/rur is : â = f dt {32 [(mg -f t -I- [m\ -|- - tŸ - 2m^(s - 2mg)] lOTT s + [{9v^ + 9Ÿ) (("^e +ml-t-s)^ COS^ t/\\r +{ml + ml- tŸ - 2m1(s - 2mg)) +2(gy^ - 9A^)m^(s - 6m^)] + cos2g^^ [(5v + 9a) (~(^e +m^-t-sŸ- (m^ + ~ tf -t-2m^(s - 2ml)) ~ ^(9v “ 9a)t^I{s ~ 6m^)] } t±^ml + ml-^^ 2^~ ^e) In this cross-section the terni in Qyg'^ is negligible in the non relativistic limit and = 4 sin^ 9w - gA = - 29w • Reduced cross-section for pp vrvr and nn —> vrï'r are negligible due to the baryonic asymmetry which suppress the amount of p and n • • Reduced cross-section for —> vrvr is negligible because it is proportional to ml in the non relativistic régime. 127 Appendix E Figures with colors DATA listed top to boltom on plot CDMS (Soudan3 2005 Si (7 keV threshold) DAMA 200058k ke-days Nal Ann.Mod. 3sigma,w/o DAMA 1996 limit CRESST2004 10.7\g-éay CaW04 Edelweiss I final limiC 62 Icg-days Ge 2000+2002+2003 linut ZEPUN n fJan 2007) resuit CDMS Soudan 2007 projected Edelweiss 2 projection SuperCDMS (wjected) 2-ST@Soudan SupeiCDMS ^jecled) 25kg (7-ST@Snolab) XQ40N1T (1 tonne) projected sensitivity OTO2OI0I3MI WIMP Mass [GeV] Figure E.l: Current and future sensitivities of direct détection experiments, upper straight Unes correspond to current limits from CDMS, CRESST, Edelweiss and Zeplin II and filled surface shows the région of the parameter space favored by DAMA. Dashed and dotted Unes shows future reach for (SUPER) CDMS and Edelweiss and lower straight line shows projected sensitivity for 1 ton experiments. This figure is obtained using the interface found at [38] (see section 2.1, figure 2.1.1). détection Figure (see 128 M2(GeV) M,{GeV) M2(GeV) 180 200 section IogIO IogIO loglO E.2: GUST EGRET fOo„pl [f1uil7(cnf*r')] {n contours h*] : : mhs120 mhs120 3.4.1, From GoV Q«V : mhs120 ; ; figure I2sl0'' I2s10 mHO mHO in top QoV 100 ' ;A the ; (GeV) (GeV) hUOs I2s1(r MAOs EQRET to 120 3.5). ’ 10 10 ;â (M//o,/X2) 0*VMHcs GeV MAO» bottom; ;A lOOoV MHcs 50 50 ;A 0«V GeV MHc= NFW plane. Relie 50 QoV ■10 -6 density, Left; O ? « Mh IogIO logIO iogIO gamma inuxl inh*] — Y ; : (cm^r')] mhs200 mhs200 120 indirect G«V ; GeV mh«200 GeV. ; E. ; i2s10 I2s10*' mHO mHO mHO QoV ‘’ F ;A ; Right: (GeV) (GeV) (GeV) MA0> igures BslO'* MAOs détection 10 10 ; A Q*V GeV MAOs ; a ; a Mh 10 MH MH with QoV c os : 50 50 ; A G*V — GeV MHc= and 200 50 colors QoV direct GéV -6 -4 ■2 0 2 -8 E. Figures with colors 129 logIO (n h*] : mhs120 Q«V ; I2s10' MAOs 5 GtV MHc» tO Q«V 1300 700 mHO (GeV) “dm (GeV) loglO [fluxl Y (cm'V')| : mhs120 QaV ; 12=10 * ;â MA0= 5 GeV ;A MHc= 10 QeV NFW 2HOM i~.13 MS5M 400 600 BOO 1000 1200 1400 mHO (GeV) (GeV loglO (Op^ : mhs120 OeV ; I2s10'' ;a MA0= 5 QtV ;A MHca 10 QeV Moore mHO (GeV) “dm (GeV) Figure E.3: Left: same as Figure (E.2) for M/fg, /X2 G [700,1400] GeV. Notice that the scale for the color gradient is however different (see section 3.4.2, figure 3.6). Right: Integrated gamma ray fiux from the galactic center resulting from dark matter annihilation as a fonction of the mass of the dark matter candidate for the same sample of models than for direct détection in Fig.E.4. Again, the light colors correspond to 0.01 < < 0.3, while the dark colors correspond to 0.094 < < 0.129. From top to bottom, the took an isothermal, NFW and Moore dark matter profile going from fiat to more cuspy profiles (see section 3.5, figure 3.9). 130 E. Figures with colors Figure E.4: Left: Relie density and Right: scattering cross-section intervening in direct détection searches, ail as a function of the mass of dark matter and compaxison with the MSSM. For the direct détection plot, the light colors correspond to 0.01 < < 0.3, while the dark colors correspond to 0.094 < < 0.129 and the two clouds of the IDM correspond to different values of A^,, both in sign and amplitude (see section 3.5, figure 3.8). Figure E.5: Model independent sky map of the 511 keV gamma ray hne émission obtained with the December 10, 2004 public INTEGRAL data release and additional data from the INTEGRAL Science Working Team for fig. (a) [50] and with the April 20, 2006 pubhc INTEGRAL data release for fig. (b) [153]. The cleaned data set consist of 6821 pointed observations with a total exposure of 15 10® s and 18101 pointed observations with a total exposure of 36 10® s respectively. The contours indicate intensity levels of 10“^, 10“® and 10~'^ ph cm“^s“^ srad“^(from the center outwards). The images are obtained with an image deconvolution algorithm based on an itérative procedure which was stopped after itération 17 and 19 respectively. With the resulting images one can recover the main characteristics of 511 keV fiux presented in [50] and [153]. Let us stress however that faint diffuse émission from the Galactic disk can’t be recovered at that point. 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