Dark Matter Vienna University of Technology V2.1

Jochen Schieck and Holger Kluck Institut f¨urHochenergiephysik Nikolsdorfer Gasse 18 1050 Wien

Atominstitut derTechnischen Universit¨atWien Stadionallee 2 1020 Wien

Wintersemester 2017/18 18.12.2017 2 Contents

1 Introduction 7 1.1 What is ”Dark Matter” ...... 7 1.1.1 Dark ...... 7 1.1.2 Matter ...... 7 1.1.3 Cosmology ...... 7 1.1.4 Massive particles as origin of ”Dark Matter” ...... 7 1.1.5 ”Dark Matter” as contribution to the energy density of the universe in Cosmology ...... 7 1.2 First Indication of observing ”Dark Matter” ...... 13 1.3 Brief Introduction to Cosmology ...... 15 1.3.1 Special Relativity ...... 16 1.3.2 Diﬀerential geometry ...... 16 1.3.3 General Relativity ...... 16 1.3.4 Cosmology ...... 17 1.3.5 Decoupling of matter and radiation ...... 18 1.4 The Standard Model of Particle Physics ...... 19 1.4.1 The matter content of the Standard Model ...... 19 1.4.2 Forces within the Standard Model ...... 19 1.4.3 Shortcoming of the Standard Model ...... 20 1.4.4 Microscopic Behaviour of Gravity ...... 21

2 Evidence 23 2.1 Dynamics of Galaxies ...... 23 2.2 Gravitational Lensing ...... 24 2.2.1 Bullet Cluster ...... 31 2.3 Cosmic Microwave Background ...... 33 2.4 Primordial Nucleosynthesis (Big Bang Nucleosynthesis - BBN) ...... 35

3 Structure Evolution 39 3.1 Structure Formation ...... 39 3.1.1 The Classic Picture ...... 39 3.1.2 Structure Formation and Cosmology ...... 41 3.2 Model of the Dark Matter Halo in our Galaxy ...... 45

3 4 Unsolved Questions and Open Issues 49 4.1 Core-Cusp Problem ...... 50 4.2 Missing Satellite Problem ...... 50 4.3 ”Too big to fail” Problem ...... 51

5 Particle Character of ”Dark Matter” 55 5.1 Baryonic Dark Matter ...... 55 5.2 Weakly Interacting Particle (WIMP) ...... 56 5.2.1 The WIMP miracle ...... 56 5.2.2 Supersymmetry (SUSY) ...... 58 5.3 Sterile Neutrinos ...... 61 5.4 Asymmetric Dark Matter (ADM) ...... 63 5.5 Axions ...... 64 5.6 Alternative Theories ...... 65 5.6.1 Modiﬁed Newtonian Dynamics - MOND ...... 65

6 Orthogonal Approaches for ”Dark Matter” searches 69 6.1 The Feynman Diagram from diﬀerent directions ...... 69 6.2 Strength and Weakness of the various Approaches ...... 70

7 Indirect ”Dark Matter” detection 75 7.1 Search Strategy ...... 75 7.1.1 Expected Signal ...... 75 7.2 Instruments and Methods ...... 77 7.3 Experimental Search for ”Dark Matter” annihilation ...... 78 7.3.1 Gamma Flux from Dwarf Galaxies ...... 78 7.3.2 Neutrino ﬂux from the Galactic Center ...... 79 7.3.3 Claims for Detection of ”Dark Matter” annihilation ...... 80 7.3.4 ”Dark Matter” Signal at 130 GeV from Galactic Center ...... 80 7.3.5 Excess of Positron fraction in Cosmic Rays ...... 81

8 Direct ”Dark Matter” detection 87 8.1 Astrophysical parameters ...... 88 8.2 Signals ...... 89 8.3 WIMP-nucleus cross section ...... 90 8.3.1 Spin-independent interactions ...... 91 8.3.2 Spin-dependent interactions ...... 93 8.4 Experiments ...... 94 8.4.1 DAMA/LIBRA ...... 97 8.4.2 LUX ...... 98 8.4.3 CRESST ...... 98

9 ”Dark Matter” Production 101 9.1 Production of ”Dark Matter” in particle colliders ...... 101 9.1.1 Model dependent searches ...... 101 9.1.2 Search Strategy ...... 102 9.1.3 Eﬀective Field Theory (EFT) ...... 103

4 9.1.4 Relation to Direct and Indirect Dark Matter Detection Experiments 105 9.1.5 Results of Searches at LHC ...... 105 9.1.6 Relation between ”Dark Matter” Production and Relic Density . 106 9.1.7 Validity of EFT Approach ...... 106 9.1.8 ”Dark Matter” searches with Higgs as a mediator ...... 107

5 6 Chapter 1

Introduction

1.1 What is ”Dark Matter”

1.1.1 Dark non-luminous, i.e. electrically neutral • weak, or even less interacting with ordinary matter • stable with respect to the lifetime of the universe • 1.1.2 Matter non-baryonic • massive particle - acts gravitationally • 1.1.3 Cosmology the abundance or relic density must match the measured ”Dark Matter” density • 1.1.4 Massive particles as origin of ”Dark Matter” no evidence for the particle character of ”Dark Matter”, however, well motivated • from the particle physic point of view all observation of ”Dark Matter” are based on gravitational interactions • alternative approach: deviations from the 1/r distance relations at all scales (ie. • ”Modiﬁed Newtonian Dynamics” (MOND), not consistent with all observations)

1.1.5 ”Dark Matter” as contribution to the energy density of the universe in Cosmology (Here we just mention the basic energy and matter ingredients of our universe. A more detailed introduction to cosmology will follow later in this chapter). The information sketched below are taken from [1].

7 the Friedmann-equation describes expansion of the Universe: • a˙ 2 8 π G kc2 H2 = = ρ (1.1) a 3 − a2 with H the Hubble parameter, a being scale factor of the universe reﬂecting a comoving universe, G being the Newtonian gravitational constant, ρ the mass density and k a constant which can be identiﬁed as the curvature of the universe (k=-1,0,1). the Friedmann-equation allows to add a constant term Λ, the so called cosmological • constant, a repulsive force:

a˙ 2 8 π G kc2 Λ H2 = = ρ + (1.2) a 3 − a2 3

8 Fig. 1.1: Sketch of a 2-dim curved plane with diﬀerent values for the curvature k=1,-1,0 (Ω0 > 1, Ω0 < 1 and Ω0 = 1)in space. The triangle on the plane indicates the change of angles within a triangle (picture taken from Wikipedia).

9 using the measured Hubble parameter H we deﬁne the critical density • 2 3H 26 kg MeV 2 3 3 ρ = 10− 5200 11h H Atoms m− 5 H Atoms m− (1.3) c 8πG ≈ m3 ≈ m3 ≈ 0 − ≈ − H (with H being the Hubble Constant, h = 1 1 = 0.673(12) [2] and G being • 100kms− Mpc− the Newtonian gravitational constant.)

using the critical density ρ Eq. 1.2 can be rewritten as • c 3kc2 Λ ρ = ρ + (1.4) c 8πGa2 − 8πG or

Λ 3kc2 ρ ρ = (1.5) c − − 8πG −8πGa2 the various parts of the energy density can be normalised to the critical energy • ρ Λ 3kc2 density ρ : Ω = ,Ω = and Ω = 2 c ρc Λ 8πG ρc k 8πGa ρc 1 Ω Ω = Ω (1.6) − − Λ − k

with Ω0 = Ω + ΩΛ Ω0 = 1 + Ωk (1.7)

the curvature is consistent with a ﬂat universe k = 0 • the diﬀerent contributions to Ω are Ω = Ω +Ω +Ω and the density contribution • r lum b from the cosmological constant ΩΛ, 5 the contributions are measured to be Ωr 5 10− , the energy density contribution from radiation (can be neglected), Ω≈ × 0.01, the energy contribution from lum ≈ luminous matter, and the contribution from baryonic luminous matter, Ωb 0.05. The energy density from the cosmological constant is determined to be Ω ≈ 0.7: Λ ≈

1 = Ωr + Ωlum + Ωb + ΩΛ = 0.06 + 0.7 < 1 (1.8) i) the sum of the measured contributions is not adding up to one ii) the missing contribution acts gravitationally (not like a cosmological constant) iii) the missing gravitational part acts like non-luminous matter Dark Matter →

10 Fig. 1.2: Measurements of the energy density from the cosmological constant (ΩΛ) as a function of the total matter energy density (Ωm = Ωlum + Ωb + Ωdm). The diagonal line indicates the expectation for a ﬂat universe with k = 0 [2].

Fig. 1.3: The size of the microwave background fluctuation at the surface of last scattering. Theory (ΛCDM) predicts the size of the fluctuation to be about θ 1◦. ' Depending on the geometry this fluctuation is smaller (k < 1 and θ < 1◦), indicating that Ω < 1, larger (k > 1 and θ > 1◦), indicating that Ω > 1 or identical to one (k = 0 and θ 1◦), indicating that Ω = 1 (Picture taken from http://map.gsfc.nasa.gov/media/030639/index.html' ).

11 Fig. 1.4: Energy matter density content of our current universe.

12 1.2 First Indication of observing ”Dark Matter”

circular movement of a star around the galactic centre, the velocity can be calculated • by requiring the gravitational and the centrifugal force to be equal: G m M m v2 F = = = F (1.9) G r2 r Z GM v(r) = (1.10) r r with M being the mass within the orbit, G the gravitational constant and r as the radius example for the velocity distribution of the planes within our solar system (see • Fig. 1.5), with the mass M being the mass of the sun (Jupiter, the heaviest planet, has about 1/1000 of the solar mass) no sign for deviation from 1/r behaviour • Velocity of Planets

50 Mercury

velocity in m/s 40

Venus

Earth 30

Mars

Jupiter

10 Saturn Uranus Neptune

0 5 10 15 20 25 30 distance in AU

Fig. 1.5: Velocity distribution of the planets within our solar system with the nice v 1/r distribution. ∝ p

assuming a spherical distribution of matter with a constant density ρ, M = ρ V = • · 4 3 G ρ 4/3 π r3 ρ 3 π r , leads to v(r) = r = r G ρ 4/3 π q p 13 the measurement of spiral galaxies returns the relation v(r) constant for a test • mass m being within the spherical distribution of density ρ ∝ a measurement of the circular speed as a function of radius for the barred spiral • galaxy NGC 3198 is shown in Fig. 1.6

For large r > 10 kpc the velocity is almost constant (1 pc = 3.26 light years) • additional non-luminous gravitational matter Dark Matter • → 1985ApJ...295..305V

Fig. 1.6: The velocity distribution for the barred spiral galaxy NGC 3198. The dots from the measurement indicate a massive halo, the lines correspond to models with contributions from a halo or a disk. [3].

the relation between gravitational attraction and kinetic energy applies also to • clusters of galaxies

for a closed, bound system in equilibrium the virial theorem relates kinetic and • potential energy: < Epot >= 2 Ekin, the average potential energy is half of the average potential energy−

In 1933 Fritz Zwicky related for the ﬁrst time the observed velocity of galaxies with • 1 the gravitational potential Epot = 2 GN(N 1) with (N 1) N and N < m >= M the total massh iM of− the galaxy− cluster can be approximated− ≈ by 2 r v 2 M h i h i [4] ≈ G 1 M M

the observed ratio between mass M and luminosity L 300 L • ≈ 1L = 4πr2σT 4 L, with σ the Stefan Boltzman Constant and T the surface temperature

14 for the solar system the mass is completely dominated by baryonic matter (see • Fig. 1.5).

the galaxy cluster contains a large part of non-luminous gravitational pulling matter • Dark Matter →

Insertion Virial theorem: the virial theorem relates for a stable system on N particle the average of the total energy with the average potential energy. For simplicity the sum and time average sign over the ensemble of particles is not shown. This is not a rigid correct proof, just an idea for the proof. The complete proof will be added later.

kinetic energie for single particle: E = 1 m v2 ∂E v = 2 E • kin 2 ⇒ ∂v · 2 E = p v = d p r r d p • kin dt − dt

the ensemble of particles the G is deﬁned as G = pi ri, and for particles within • i dG P conﬁned space and restricted velocity dt = 0 2 E = r d p for G = 0 • kin − dt F = d p = ∂ U 2 E = r ∂ U = U • dt − ∂ r ⇒ kin ∂ r for a new gravitational potential: 2E = U • kin −

Dark Matter disturbing the orbit of the earth:

Distance Sun-Earth: AU = 150 109 m; diameter earth 13 106 m; π AU 2 13 106 m3 • = 106 1018 106 = 1030m3 · · · · · · · 3 27 Dark Matter density: 0.3 GeV/cm , 1 GeV = 1.8 10− kg, Dark Matter Density • 21 3 · = 1.8 10− kg/m · Dark Matter mass between sun and earth: 1.8 109 kg • · Mass sun: 2 1030kg • ·

1.3 Brief Introduction to Cosmology

Here we brieﬂy introduce the main ideas and principles of cosmology. The key relation of Dark Matter in cosmology is already discussed in section 1.1.5. For a more detailed review see [5], [6] or [7].

15 1.3.1 Special Relativity physics invariant under a Lorentz transformation, consistent with speed of light c • being constant in all coordinate systems

2 µ ν the scalar product of two vectors is calculated by s = gµν x x , with • 1 0 0 0 −0 1 0 0 g = being the metric tensor µν  0 0 1 0  0 0 0 1     the absolute distance ds2 between two space points can be expressed as ds2 = • dx2 + dy2 + dz2 c2 dt2 − 1.3.2 Diﬀerential geometry description of space can be obtained by using the distance measurement in space • the distance measurement is closely related to the metric tensor used for the scalar • product

the metric tensor contains information about the space • 1.3.3 General Relativity general relativity follows from a generalisation of the equivalence principle: you • cannot distinguish between an accelerating system and a constant gravitational force (equivalence principle: the mass which gravitationally pulls is equivalent to the mass which determines the resistance by any force)

Newtonian gravitation potential ~g(~r) = G M rˆ • − N r2 equation can be re-written as •

2Φ = 4πG ρ (1.11) ∇ N describing the relation between a potential (left hand side) and the matter / energy • (right hand side)

general relativity describes in a more general way the relation between matter and • space-time

the speed of light is constant in all systems or light always follows the shortest • distance

in the presence of gravity (or mass or, following special relativity E=mc2, energy) • space will be curved in order to allow light to follow the shortest distance in space (geodesic)

16 following Eq. 1.11 the more general relation between a curved space, playing the • role of a potential, and the energy can be written as (Einstein’s ﬁeld equation) 1 8πG R g R = T (1.12) µν − 2 µν c4 µν with R being the Ricci-tensor representing the curvature of space • µν R being the Ricci scalar • the metric tensor g describes distances and angles in space • µν

and Tµν being the energy stress tensor, with T00 being the energy density and Tii • being the pressure (e.g. from radiation) Eq. 1.12 can be even extended in a more general way, adding on the left hand side • a constant proportional to gµν: 1 8πG R g R + Λg = T (1.13) µν − 2 µν µν c4 µν with Λ the cosmological constant •

F~ = m~a (1.14) M ~g(~r) = G rˆ (1.15) − N r2 ~g dA~ = 4πG M (1.16) − N IS ~ ~g dV = 4πG ρdV (1.17) ∇ − N Z Z ~ ~g = 4πG ρ (1.18) ∇ − N 2 Φ = 4πG ρ (~g = ~ Φ) (1.19) ∇ N −∇

1.3.4 Cosmology distribution of matter and radiation in the observable universe is homogeneous and • isotropic which metric with curvature can describe space which is homogeneous and isotropic • and expands uniform in time geometry and curvature of space can be described with two cosmological parameters, • curvature k and expansion distance can be described with the Robertson-Walker-metric: • 2 2 2 dr2 2 2 2 2 ds = dt a (t) 1 kr2 r (dθ + sin θdφ ) − − (a(t) being the scaleh factor and (t, r, θ, φ) beingi the comoving coordinates)

17 with the curvature k = +1, 0, 1 • − for an isotropic energy density the energy tensor T =diag( ρ c2, p, p, p) • µν − put Robertson-Walker-metric into Einstein’s ﬁeld equation Eq. 1.13 and solve the • Equation for the (0, 0) component

a˙ 2 8πG k c2 Λc2 H2 = = ρ + (1.20) a 3 − a2 3 and for the (i, i) components together with Eq. 1.20 • a¨ 4πG Λc2 = ρc2 + 3p + (1.21) a − 3c2 3 acceleration equation describes dynamics of the expansion • the ﬁrst part of the right corresponds to the gravitational attraction, leading to a • negative acceleration (ρ and p always positive)

a suﬃcient large value of λ could counteract the gravitational attraction, leading • to a positive acceleration and an expansion of the universe

the cosmological constant Λ has a ”negative pressure” and an expansion will leading • to an increase of the internal energy (following dE + p dV = 0)

1.3.5 Decoupling of matter and radiation The temperature of the universe, and therefore the energy of the particles, cools • down with expansion

the ionisation energy of a hydrogen atom is 13.6 eV: H + γ p + e−, in thermal • equilibrium the reaction can go in both directions →

at a certain time the energy density drops below the energy necessary to ionise • hydrogen atoms, and all protons and electrons are combined to electrical neutral atoms

the universe changed from a plasma with charged particles, protons and electrons, • to a ﬂuid with neutral atoms

the universe transforms from being opaque in the plasma to being transparent, • photons decouple and can be observed as cosmic microwave background (CMB) radiation

decoupling of γ took place at a temperature of about 3000 K at a redshift z of • about z = 1100 and a time when the universe was tdec = 380000 years old H (the redshift z is related to the distance r by Hubble’s law z = c r. The redshift is λ λem determined by z = ob− ) λem

18 Fig. 1.7: Cosmic Microwave Background measured with the Planck Satelite.

1.4 The Standard Model of Particle Physics

1.4.1 The matter content of the Standard Model the Standard Model contains three families (generations) of quarks and leptons • stable baryonic matter consists of quarks of the ﬁrst family, atoms contains also • electrons

quarks and charged leptons from the second and third generation do decay into • members of the ﬁrst family with relativ short lifetimes

1.4.2 Forces within the Standard Model the Standard Model contains three different forces • – strong interaction: quarks participate in the strong interaction; the corresponding charge is the color charge and the mediator are gluons which also contain a charge and couple to each other (self coupling); quarks are bound to each other within hadron and are not freely visible (con- fined); due to the self-interaction of the gluons the strong force is a short-range force the strong interaction preserves parity (P ), charge conjugation (C) – electromagnetic interaction: all charged particles take place in the electromagnetic interaction; the force can be attractive and repulsive, depending on the charge; the mediator of the electromagnetic interaction is the photon; the range of the electromagnetic force is infinity

– weak interaction: The weak interaction acts on all particles of the Standard Model; it’s the only force which couples to the neutrinos; the mediators are the W - and Z-bosons; the range of the force is restricted by the ﬁnite mass of the exchange particles; the weak interaction violates parity (P ) and charge

19 conjugation (C) maximally and does not preserve the combined CP symmetry to a smaller extend.

The matter content and forces are summarised in Fig. 1.8. A summary of the forces can be found in table 1.1.

Fig. 1.8: The particle and forces of the Standard Model of particle physics (picture taken from Wikipedia).

name of the interaction relative strength range strong interaction 1 1 fm 2 electromagnetic interaction 10− inﬁnity 7 3 weak interaction 10− 10− fm 40 gravitational interaction 10− inﬁnity Table 1.1: fundamental interactions and the relative strength

1.4.3 Shortcoming of the Standard Model The Standard Model of particle physics describes well all observations at a macroscopic scale. Several astrophysical observations cannot be explained with the Standard Model:

the Standard Model does not contain gravity • the amount of CP violation in the Standard Model is not enough to explain the • baryon asymmetry observed

the Standard Model does not contain a candidate for Dark Matter • 20 1.4.4 Microscopic Behaviour of Gravity the Standard Model of Particle Physics describes only three out of four known • forces: strong, weak and electromagnetic interaction

gravitation is not part of the Standard Model • – very weak compared to other interactions: gravitational versus electromagnetic 38 interaction for a proton-antiproton pair: 10− – no impact on particle physics interaction – only attractive force known, no cancelation between positive and negative charges

which energy (or mass) scale can be related to the gravitational interaction? • consideration of dimensions • 11 m3 39 ~ c gravitational constant G = 6.67 10− = 6.71 10− (using other • · kg s2 · (GeV/c2)2 constants) ·

combine gravitational constant G with natural units in order to get value with units • mass: M = ~ c = 1.2 1019 GeV/c2 (”Planck scale”) P l G · q interpretation of Planck scale • – at the Planck scale energy quantum eﬀects are relevant for gravitational interactions – Schwarzschild radius: enough mass (or energy) is concentrated that the escape velocity exceeds the speed of light c 2 MG 2 GM ve = rs = 2 rs → c q h – Compton wavelength of a massive particle: λ = M c – at the Planck mass (or planck energy via E = m c2) the Schwarzschild radius is equal to the Compton wavelength of the massive particle 2 GM = h M = h c c2 M c → 2 G q semi-classical interpretation of escape velocity: kinetic energy 1 m v2 is equal to gravitation energy m G M 2 e rs 1 m v2 = m M G 2 e rs v = 2 MG e rs q

21 22 Chapter 2

Evidence

In this chapter we discuss the various observations of ”Dark Matter”. Please note the observation of ”Dark Matter” is closely related to a precise measurement of the other energy contributions, like baryonic matter, of the universe. For this reason we will not only discuss the observation of ”Dark Matter” but also the measurement of baryonic matter and the related missing gravitational pull.

2.1 Dynamics of Galaxies

Historically, ”Dark Matter” in the modern meaning was ﬁrst observed in the dy- • namic behaviour of galaxies in clusters, already discussed in 1.2. Two classic examples are the velocity distribution of cluster galaxies and the rotation curve of stars in individual galaxies, but F. Zwicky noted in 1933, that the velocity distribution of the galaxies in the Coma • cluster is greater than expected. Introducing the Virial theorem 5R M = v2 (2.1) 3G to astrophysics, he deduced a ratio of gravitational interacting matter to luminous matter of 400 M /L , given in terms of solar mass M and solar luminosity L [4]. This result is no single case, e.g. oberving 1000 galaxies in 16 clusters, R. Carlberg 1 et al. [8] found 295(53) hM L− where h is the Hubble constant. Therefore, most of the matter in galaxy clusters is dark. Starting in the 1970s, observations indicate that also individual galaxies contain • ”Dark Matter” These observations used individual starts or star clusters as test particle to trace the velocity curve of a given galaxy, i.e. the circular velocity around the galactic center as function of the radius of the orbit. For stars well outside the (luminous) galactic body, one expected a Keplerian behaviour, i.e. that the velocity drops with 1/√r. However, even well beyond the galactic body one observed a ﬂat rotation curve. Again, this behaviour was no single case as V. Rubin et al. [9] showed 1980: The systematic measurement of rotation curves for 21 galaxies yield no single curve that follow the expected Keplarian behaviour.

23 The observed deviation from the Keplerian velocity curves can be explained by dark • halos: As ﬁrst formulated by J. Einasto et al. [10] and J.P. Ostriker et al. [11], the ”Dark Matter” in galaxy clusters is not homogenous distributed throughout the cluster, but clustered around the individual galaxies, resulting in dark halos around the luminous part of the galaxies.

An ongoing eﬀort is the deduction of the actual distribution of ”Dark Matter” • density ρ in a galaxy from the observed rotation curve. Several distributions are discussed in literature, e.g. iso-thermal halos, tri-axial halos, etc. A popular distribution is the Navarro-Frenk-White (NFW) model [12], which can be parametrized as ρ = 0 (2.2a) ρ γ α (β γ)/α r r − R 1 + R α = 1.0, β = 3.0, γ = 1.0,R = 20 kpc (2.2b)

with the dark matter density of the halo ρ0 and a characteristic scale R. It is used also in ﬁgs. 1.6, 2.1.

All these is also true for our own galaxy, the Milky Way. The rotation curve stay flat • up to xxx, well beyond Sun’s orbit as shown in fig. 2.1. Fitting the observed curve with the known contribution of luminous matter, the ”Dark Matter” distribution could deduced, following a NFW profile. Recently, it was shown that also within the orbit of the Sun, the star orbits deviate and a contribution of ”Dark Matter” is needed, see fig. refxxx. From these measurements one can deduce the average 3 ”Dark Matter” density at the position of the Sun to ρ0 0.3 GeV/cm− [13], which may be memorized as “one (3 GeV) dark matter particle≈ in each cup of coffee”. These observation indicates that the Sun, and therefore also Earth, rotate through the dark halo of the Milky Way. Therefore, it is possible for Earth based instruments to directly search for possible particle constituents of this dark halo which we will discuss more detailed in chapter ??.

2.2 Gravitational Lensing

As already mentioned in section 1.3, the light trajectory is always a geodesic for • a given space time. As energy can curve the space time, it can also aﬀect the light trajectory. Fig. 2.2 illustrate this principle: A massive object between a light source (here a distant galaxy) and an observer can bend the light trajectory like an optical lens, hence this eﬀect is called gravitational lensing. As consequence of the bended light trajectory, the position of the distant object on the sky is changed for the observer. Consequently, the image of an extended object get distorted.

Also Newtonian mechanics predict a bending of light, however the predicted de- • flection angle is only half the angle predicted by general relativity. Therefore, the actual measurement of the deflection angle is a classical test of general relativity. The effect was observed during the total eclipse in 1919 by A. Eddington: The

24 300

250

200 −1 150 km s / 푣 100

0 0 2 4 6 8 10 12 14 16 18 20 푟 / kpc

Fig. 2.1: Rotation curve of the Milky Way for the innermost 20 kpc. Fitted to the data points are the sum (black line) of three components: A visible bulge (blue line) and disk (dashed green line), together with a dark halo based on a NFW-parametrisation (red line). Figure adapted from [14, ﬁgs. 2,5], references for the data compilation therein.

position of stars near the eclipsed Sun where changed with respect to their position in absence of the Sun, i.e. at night. The measured deﬂection angle was indeed twice the Newtonian prediction, in agreement with general relativity.

The deﬂection angleα ˆ can be calculated from the mass M of the lens and the • impact parameter ξ [15]: 4GM αˆ = . (2.3) c2ξ with the speed of light c and the gravitational constant G, see also ﬁg. 2.2.

Depending on the strength of the eﬀect, one distinguish weak gravitational lensing • without clearly visible distortion of the image [15, 17] and strong gravitational lensing with split image [18], see ﬁg. 2.3. A strong lensing event is called microlensing in case the lens is small, e.g. a massive astrophysical compact halo object (MACHO), see section 5.1.

Famous examples for strong gravitational lenses are Einstein’ Crosses like ﬁg. 2.4a • and Einstein’ Rings, like ﬁg. 2.4b. Here, the image of the source reach the observer in multiple trajectories due to a single compact gravitational lens (e.g. a compact galaxy cluster) along the line of sight.

In weak gravitational lensing, no obvious distortion of the image of an individual • source is observed. However, by observing multiple sources, one can deduce statis- tically the correlation between the images due to the gravitational lensing. Aver- aging of multiple, distant galaxies as light source, the average shape is expected to

25 Fig. 2.2: A lens of mass M will deﬂect light from a distant source by an angleα ˆ dependent on the impact parameter ξ [16].

be circular (fig. 2.5a). The averaged images of these sources in the background are elliptical distorted by gravitational lenses in the foreground (fig. 2.5b). Based on the observed ellipticities a shear map can be constructed (fig. 2.5c). The orientation of the distorted images encodes the information about the mass between source and observer (fig. 2.5e): Around a gravitational lens, i.e. a mass over density between source and observer, the minor axis of the ellipse points towards the lens. Contrary, if there is an under density between source and observer, e.g. a void, the major axis points towards it. Therefore, from the observed shear map, the distribution of mass between the source and the observer can be reconstructed (fig. 2.5d). Because one use multiple sources, more extended gravitational lenses can be studied, e.g. cosmic filaments or colliding galaxy clusters.

In both cases one can deduce the mass of the gravitational lens from the distortion • of the image. Therefore, gravitational lensing is a tool to map the distribution of gravitational interacting matter between source(s) and observer. Both eﬀects,

26 Fig. 2.3: Depending on the extend of distortion, the gravitational lensing can be classiﬁed as weak lensing with hardly visible distortion and strong lensing with split images. The intermediated case ’ﬂexion’ featured a clearly visible distortion, but no split image [19].

strong and weak gravitational lensing, shows for galaxy clusters more gravitational interacting matter than luminous matter. The diﬀerences can be explained as Dark Matter and is in agreement with the results obtained from dynamics of galaxies, e.g. from the Coma cluster.

This is not only an additional indication for Dark Matter, with gravitational lens- • ing it is also possible to map the distribution of Dark Matter in comparison with luminous matter, see e.g. ﬁg. 2.6 or the famous “Bullet cluster” (section 2.2.1).

27 (a) (b) Fig. 2.4: (a) Einstein cross G2237 + 0305: Four images of a quasar in a distant of 8 billion light years caused by strong gravitational lensing from a foreground galaxy at a distant of 400 million light years (centre). Credit: NASA, ESA, and STScI [20]. (b) Einstein ring SDSS J120540.43+491029.3: Multiple, arc-like images of a background galaxy (blue) caused by strong gravitational lensing from a foreground galaxy (centre, yellow). Credit: NASA, ESA, A. Bolton (Harvard-Smithsonian CfA) and the SLACS Team [21].

28 (a) (b)

(e) Fig. 2.5: (a) Grid of circular, distant light sources. (b) Distorted images due to gravitational lensing. (c) Shear map and (d) deduced mass distribution between sources and observer. (e) Major axes of the distorted images point toward under densities, minor axes towards over densities. For details see text. Images taken from [22].

29 (a) (b) Fig. 2.6: (a) The rich galaxy cluster Cl0024+17 features both weak and strong gravitational lensing, the latter is visible as arc like structure around the centre of the image. (b) Based on weak and strong gravitational lensing, the projected distribution of dark matter is obtained as ring like structure (blue) around the cluster centre. The ring structure may be the result of a high-speed collision of two clusters along the line-of-sight [23]. Credit: NASA, ESA, M.J. Jee and H. Ford (Johns Hopkins University)

30 2.2.1 Bullet Cluster The Bullet Cluster (1E 0657-558) is a merging galaxy cluster, consisting of two well • separated subclusters. Using weak lensing measurements, it was found that the distribution of mass within the cluster spatially coincide with the visible matter of the two subclusters (ﬁg. 2.7). However, the visible mass accounts for only . 2 % of its total mass [24], therefore the existence of dark matter is needed.

The Bullet Cluster allows also to investigate the nature of its dark matter, i.e. if it • is non-baryonic dark matter or baryonic dark matter, see section 5.1. A classic candidate for non-baryonic dark matter in galaxy clusters is the intergalactic medium, i.e. hot gas not associated with individual galaxies but with the total cluster [25].

X-ray observation of the Bullet Cluster found this gas, but it contributes only • . 15 % to the total cluster mass. Furthermore, it is located in-between the two subclusters, see ﬁg. 2.7. The location of the gas can be explained as follows: The two subcluster were interpenetrating each other. The galaxies of each subcluster where not distorted, due to the large distances between it. Contrary, the extended gas around each subcluster was separated from the galaxies by friction [24].

In the X-ray image a clear bow shock in the gas is visible. As this evoke the image • of a bullet passing through a target, the cluster was named Bullet Cluster.

However, the weak-lensing showed, that most of the cluster mass coincide with the • subclusters. Therefore, baryonic dark matter is only a minor contribution to the mass budget of the Bullet Cluster. Non-baryonic dark matter could be a solution for the missing major contribution [24].

As this potential non-baryonic mass coincide with the galaxies, it must have a low • self-interaction cross-section. Because otherwise friction would have separated it from the galaxies, similar to the intergalactic medium. This together with the observation of several other merging clusters could set limits on the self-interaction cross-section [26].

31 Fig. 2.7: Optical image of the Bullet Cluster (1E 0657-558), overlayed with an X-ray observation (red) and a lensing map (blue). For details see text. Credit: X-ray: NASA/CXC/CfA/M.Markevitch et al.; Optical: NASA/STScI; Mag- ellan/U.Arizona/D.Clowe et al.; Lensing Map: NASA/STScI; ESO WFI; Magel- lan/U.Arizona/D.Clowe et al. [24, 27].

32 2.3 Cosmic Microwave Background

Universe is matter dominated: Ω > Ω • m r a˙ 8πG 3 2/3 = ρm; ρm a− ; a t (2.4) a r 3 ∝ ∝ primordial over densities starts to collapse ⇒ Equilibrium above T 1 eV: Atom + γ nucleus + e−, see also section 1.3.5. • ≈ ↔ decouple at redshift z 1000: last scattering surface ⇒ ≈ Emitted photons redshifted to T 3 K today with a black body spectrum: cosmic • microwave background (CMB). Discovered≈ by A. Perzias and R. Wilson in 1965. Most precise measurement of the absolute, average CMB temperature by COBE: T = 2.7255 K h i 4 CMB is isotropic up to 10− strong indication for inﬂation. • ⇒ At a mK-level, the CMB is anisotropic, see ﬁg. 1.7. Decompose it in spherical • harmonics T (ˆn) T Θ = − h iY (ˆn) dΩ (2.5) lm T lm Z h i on can use the correlation function

Θ∗ Θ = δ δ C (2.6) h lm l0m0 i ll0 mm0 l the power spectrum of the Cl is shown in fig. 2.8. Before the decoupling, photons and electrons were coupled via Thompson scatter- • ing. Electrons and protons were coupled via charge conservation. – Matter got compressed during fall into gravitational well temperature increase → – Radiation pressure increase with temperature – Radiation pressure rarefied matter again temperature decrease → acoustic oscillation with wavelength λ and higher harmonics ⇒ stops at recombination ⇒ The power spectrum is sensitive to Ω ,Ω ,Ω : • T b b – first peak is sensitive to ΩT: sound horizon ∆Θ = (2.7) d where the sound horizon is the wave length of the base mode at decoupling, it depends on Ωm,ΩΛ and is 150 Mpc. The distance to the last scattering surface d depends on the curvature≈ of the universe. π ∆Θ 0.0105 rad l = 300 (2.8) ≈ ⇔ ∆Θ ≈

33 Fig. 2.8: Power spectrum of the cosmic microwave background as measured by PLANCK, the lower panel shows the residuals between measurement and model. Figure taken from [28].

34 – second peak is sensitive to Ωb: More baryons results in a stronger compression and in a higher amplitude of the odd-numbered peaks in the power spectrum. However, the rarefaction is the same, hence the amplitude of the even-numbered peaks stay the same. the ratio between even and odd peaks, e.g. between ﬁrst and second peak, ⇒ is sensitive to Ωb.

– higher peaks are sensitive to Ωm: The time of transition from the radiation dominated epoch of the universe to the matter dominated epoch depends on Ωm, e.g. for a smaller Ωm, the transition happens later. In this case the decay of the gravitational wells by radiation is stronger, which aﬀects the amplitude of all peaks.

However, the effect of ΩT,Ωb,Ωb on the power spectrum is not complete disentan- • gled1, so one has to fit the complete observed power spectrum with a cosmological model which consider the influence of all cosmological parameter. Currently the best observation of the CMB power spectrum was done with the PLANCK satellite. A fit of the ΛCMB model results in Ωb = 0.049, Ωm = 0.314 [28].

The diﬀerence Ωm Ωb 0.27 clearly indicates the existence of non-baryonic ⇒dark matter in the universe.− ≈

2.4 Primordial Nucleosynthesis (Big Bang Nucleosyn- thesis - BBN)

BBN: production of ﬁrst composite nuclei with the cool down of the universe • 6 at a temperature of about 1 GeV ( 10− s) free quarks condensed to nucleons, with • protons and neutrons ≈

all other hadrons (mesons and baryons containing heavy quarks) did already decay • at larger energies nuclei and anti-nuclei are in equilibrium with photons: p +p ¯ γ γ • ↔ the p p¯ are expected to completely annihilate to photons and the expected ratio • NB NB¯ 18 = 10− Nγ Nγ ≈ the annihilation between baryon and anti-baryons was not complete and the ob- • NB 9 NB 4 served ratio is η = 10− , with < 10− (Baryogenesis) Nγ ≈ NB¯ neutrons are more massive than protons, leading to the fact that neutrons are more • abundant than protons

in thermal equilibrium at energies between 0.1 to 10 MeV the ratio between neutrons • nn Q/T 2 and protons can be written as = e− with Q = (mn mp) c = 1.29 MeV np − 1Very instructive animations of the aﬀect of the diﬀerent parameters on the shape of the power spectrum can be found on [29].

35 at around 0.7 MeV (3 s) the reaction rate drops and the ratio freezes out to nn = np • 1.29/0.7 e− 1/6 (at that temperature≈ the reaction rate is roughly equal to the expansion rate of the universe freeze out) → around the same time the production of composed nuclei started, ie production of • deuterium (n + p D + γ + 2.2 MeV) ↔ 4 ρ(4He) all deuterium is almost transformed to He, and the ratio Yp = 0.24 • ρbaryon ≈ this ratio includes only primordial produced 4He and is not considering 4He from fusion in stars.

4He has an binding energy of 7.07 MeV per nucleon and is much more stable than • the next heavier elements therefore the primordial production of composed nuclei basically stops with the production of 4He

4 5 some deuterium (about 10− to 10− ) escapes the production to deuterium, the • remaining fraction of deuterium strongly depends on the baryon ratio η, higher baryon ratio η leads to a smaller deuterium survival probability together with the photon density the baryon ratio can be used to calculated the • baryon density: 2 3 nbaryon = η nγ baryon = (mpc ) nbaryon = 210 20 MeV/m− [5] 3 baryon· → 210 20 MeV/m− ± Ωbaryon = = ± 3 = 0.04 0.01, and c being the critical density c 5200 1000 MeV/m− ± (see Eq.1.3). ±

36 Fig. 2.9

5 Boltzmann constant: kB = 8.6 10− eV/K 1 MeV corresponds to 1 106/k ×= 1/8.6 1011 K 1010 K × B × ≈ temperature versus time or energy:

1/2 T (t) 1010 K t − • ≈ 1s t 1/2 kT (t) 1MeV − • ≈ 1s

37 38 Chapter 3

Structure Evolution

3.1 Structure Formation

The derivation and equations follow closely the more detailed description in [5].

on very large scales ( 100 MPc) the universe is homogeneous and isotropic • ≈ at smaller distances ( 50 MPc) we start to observe voids and superclusters • ≈ how did structure form during the evolution of the universe and what is the role of • ”Dark Matter”?

key messages for ”Dark Matter” from structure formation: • – ”Dark Matter” Matter is necessary to explain current observations of matter distribution – ”Dark Matter” needs to be cold, meaning not traveling with the speed of light or close to the speed of light; neutrinos as weakly coupled particles are exclued

3.1.1 The Classic Picture classical gravitational picture for structure instabilities • gravity is an only attractive force and any small over density of matter should lead • to a gravitational collapse

define the fluctuation of a matter density ρ over the average matter densityρ ¯: • ρ ρ¯ δ = − , with δ 1 ρ¯ an upward fluctuation of the matter density can lead to an attraction of a mass m • at the edge of a sphere with density ρ =ρ ¯(1 + δ) (see Fig. 3.1)

¨ Gm∆M G 4π 3 F = m R = R2 = R2 3 R ρδ¯ , with ∆M being the mass diﬀerence with • respect to the− average mass− within the sphere with the densityρ ¯(1 + δ)

39 the relation of the radius of the sphere and the density ﬂuctuation δ is • R¨ 4πGρ¯ = δ(t) (3.1) R 3

the overall mass M of the sphere needs to be constant during the collapse : M = • 4π 3 3 ρ¯(1 + δ(t))R(t)

1/3 with R = 3M the radius can be expressed as R(t) = R (1 1 δ(t)) • 0 4πρ¯ 0 − 3 for δ 1 the second derivative can be written as R¨ 1 δ¨ • R ≈ − 3 together with Eq. 3.1 the overdensity δ evolves as δ¨ = 4πGρδ¯ • 1/tdyn 1/tdyn and the most general solution for delta can be written as δ = A1e + A2e− • 2 1/2 with t = 1 , and with ρc2 this leads to t = c dyn (4πGρ¯)1/2 ≈ dyn 4πG¯ each upward ﬂuctuation of a matter density will lead to a collapse of the sphere • with the time constant tdyn

Fig. 3.1: Sketch of the overdensity δ in a homogeneous universe with densityρ ¯

3 in our atmosphere the density of air is roughlyρ ¯ 1 kg m− , leading to tdyn = 9h, meaning in about 9h the atmosphere will collapse due≈ to gravitation

what prevents small matter density ﬂuctuations from collapsing? Radiation! • Equation of state (”Zustangsleichung”), the relation between the pressure and the • energy density, for a low-density gas of non-relativistic massive particles:

Zustandsgleichung: p V = N k T · · · 40 N µ with ρ = V· ρkT P = µ Schallgeschwindigkeit in Gasen: γ RT v = µ q

ρ – P = µ kT (µ being the mean particle mass), with the energy density being almost entirely determined by the rest mass: ρc2, leading to P kT ≈ ≈ µc2 – time to for a pressure change t for a distance R is t = R with c being pre pre cs s the speed of sound which can be determined from the equation of state cs = 1/2 dP 1/2 kT c d = c µc2 = c√ω R if the time constant related to radiation tpre is smaller than the time constant • ≈ cs needed to bring an overdensity to collapse tdyn, the pressure change can wash out the density ﬂuctuation and the equilibrium is maintained

the related length scale at which both scales are equal can be written as λJ • 2 1/2 ≈ c t c c (Jeans length) s dyn ≈ s G¯ the Jeans length scales with the sound speed in the relative medium, meaning large • sound of speed allows only large regions of overdensity to collapse without being washed out by pressure changes, smaller ones are washed out.

(compare to the equation of state of the universe which also also relates gravitational • attraction with a pressure)

3.1.2 Structure Formation and Cosmology next steps: •

i) estimate speed of sound cs for the diﬀerent epochs before and after decoupling and estimate the corresponding Jeans length ii) take expansion of the universe into account: density ﬂuctuations might be also be diluted by the expansion iii) what’s the role of ”Dark Matter”?

i) cs for the diﬀerent epochs before and after decoupling:

1/2 1 3c2 – the expansion of the universe is described by the hubble time H− = 8πG¯ ≈ 1.22tdyn (c.f. Eq. 1.3 for the hubble constant)

3 1/2 cs – the Jeans length λJ can be written as λJ = 2π 2 H – before decoupling (c.f. section 1.3.5) the elementary particles did form a single photon-baryon-electron ﬂuid

41 – the speed of sound cs is almost completely determined by the radiation of the ﬂuid c c and λ 3.0 c 1.9 1022 m and using the corresponding s √3 J H ≈ ≈ ≈ × 19 3 mass / energy density of baryon = 5 10− kg m− we get for the Jeans mass × 18 (Volume with radius λJ ) MJ 7 10 M ≈ × – structures with a mass as large as at least MJ are gravitational stable and not washed out by pressure changes – this corresponds to a mass larger than the largest supercluster observed – after decoupling the sound of speed for baryons is determined by the temper- 5 ature, c = 1.5 10− c s × – the smaller sound speed cs allows much smaller structures to develop: MJ 1 105M (smaller ones are washed out) ≈ × – time of decoupling is an important time in the structure evolution of the universe, before the decoupling only very large structure are gravitationally stable

22 5 λ before decoupling: 1.9 10 , after decoupling smaller by a factor 3 10− • J × × 3 5 3 14 M = ρ V = ρ λ (3 10− ) 10− smaller after decoupling • J J → × ≈ before decoupling 18 19 MJ 7 10 M 10 M before decoupling • ≈ × ≈ after decoupling before decoupling 14 5 MJ = MJ 10− = 10 M • ×

ii) Jeans length in an expanding universe:

– the expansion of the universe described by the hubble constant and the cor- 1 responding time H− and the Jeans time is of similar order and therefore the expansion cannot be neglected – a full relativistic calculation of the time evolution of the overdensity δ gets an ¨ ˙ 4πG additional ”Hubble friction” term: δ + 2Hδ = c2 ¯mδ – the term δ describes only ﬂuctuations of matter alone (not radiation etc)

¯m 8πGm – this leads for partly matter dominated universe Ωm = = 2 2 to critical 3c H ¨ ˙ 3 2 δ + 2Hδ = 2 ΩmH δ – in a matter dominated universe the density perturbation grows (after some time) as δ t2/3 a(t) 1 ∝ ∝ ∝ 1+z iii) the impact of cold ”Dark Matter” on structure formation

– before the decoupling the Jeans length was to large and no gravitational in- stability was build up, no structure formation

42 – after the decoupling the speed of sound and therefore the Jeans length decreased significantly and first fluctuations in the matter density δ lead to some structure – the size of the fluctuations in the CMB at decoupling originating from primor- 5 dial fluctuations is at the order of 10− 1 – for an expanding universe the density fluctuation δ 1+z leading to an increase of the density fluctuation by 103 ∝ 5 – an initial density fluctuation of 10− at z=1000 is now a 0.01 fluctuation – the observed density changes in the current universe are of order one and the 3 predicted fluctuations at decoupling time had to be of order 10− – structure formation had to be started before decoupling in an environment with a smaller speed of sound – in the baryon-radiation-plasma the speed of sound is determined from the radiation, non-radiating matter leads a smaller sound speed ”Dark Matter” →

Fig. 3.2: The jeans mass as a function of temperature. At the time of the decoupling the sound speed drops by several order of magnitudes, leading to a signiﬁcant smaller jeans mass [30].

– hot ”Dark Matter” - light relativistic particles traveling with (almost) c - decoupled from baryon-radiation plasma before decoupling - high mobility leads to wash out from initial density ﬂuctuations (diﬀusion), no structure building - (almost) massless particles like neutrinos cannot act as ”Dark Matter” candidate

43 – cold ”Dark Matter” - heavy non-relativistic particles - no interaction with radiation and decoupling from baryon-photon plasma - low mobility leads to a low speed of sound and the possibility to build up structure from initial density fluctuations - ”Dark Matter” structure grow started at about z 3600, the time when the universe goes from an radiation to a matter dominated≈ universe (or when the WIMP thermally decouples from the medium, roughly when the temperature falls below the rest mass of the WIMP) – the density fluctuation can be decomposed with a Fourier transform δ(~r) = V i~k ~r 3 12.5.(2π)HOT3 δkVERSUSe− · d kCOLD, with the mean square amplitude of the Fourier279 components defines the power spectrum: P (~k) = δ2 R h| k¯|i – the expected1 power spectrum for initial density perturbations is P (k) k, ∝ however, being.1 modified by the timeP of∝k matter-radiation equality – the spectrum.01 depends on the presence of cold or hot dark matter, hot dark matter wipes out small mass fluctuations, cold dark matter preserves them (see Fig. 3.3)..001 CDM P – for hot ”Dark.0001 Matter” the large structures would collapse first have the size of superclusters10 −5 (top down scenario) - observation only observes the collapse

of superclusters −6 only now 10 HDM – for cold ”Dark Matter” ﬁrst objects are the smallest (galaxies) followed by −7 clusters and10 superclusters (bottom up scenario). This scenario is consistent .001 .01 .1 1 10 with observations (see Fig. 3.4). k [Mpc−1]

.1 2 / 1 )

P .01 3 k ( .001

.0001

10 −5 12 14 16 18 20

log10M

Fig. 3.3:Figure The root12.5: massUpper squarepanel – massThe ﬂuctuationpower spectrumδM/Mat the time(k3Pof)1/radiation-2 shown as a function 3 ∝ of M matterk− forequalit coldy darkfor cold matterdark (solidmatter line),(solid hotline) darkand matterfor hot (dotteddark matter line). The initial power∝ spectrum(dotted line). is shownThe asinitial dashedpower linespectrum [5]. produced by ination (dashed line) is assumed to have the form P (k) k. The normalization of the power spectrum is arbitrary. Lower panel – The∝root mean square mass uctuations, 3 1/2 3 δ–Mafter/M decoupling(k P ) , are baryonsshown as falla function into ”Darkof M Matter”k− (masses halosare andin units loose energy via of M ).∝The line types are the same as in the upp∝er panel. radiation¯

44 Fig. 3.4: The distribution of galaxies as a function of redshift [31].

– at very high baryonic densities fusion can start

3.2 Model of the Dark Matter Halo in our Galaxy

the approximations used to solve the equations above are only valid for very small • density ﬂuctuation δ 1 (linear model), in the case of weak self gravitation in a given region | |

a larger density contrast δ and a strong lokal gravitational ﬁeld leads to a non-linear • regime which can no longer be analytically solved

evolution of system can be solved with N-body simulation with gravitating mass • points, calculate evolution of gravitational ﬁeld and density distribution

the simulation uses the input from a cosmological model with 5 time more ”Dark • Matter” than matter and 70% dark energy

the simulation follows a two step approach, ﬁrst a large box with ”Dark Matter” only • is simulated and in a second step a smaller box with a ”Dark Matter” halo similar to the expected halo of our galaxy is taken for further simulation (see Fig. 3.5)

besides the ”Dark Matter” velocity the ”Dark Matter” density map in the galactic • plane can be visualised (see Fig. 3.6)

3 simulation returns a ”Dark Matter” density of ρDM = 0.37 GeV/cm at the position • of the sun in our galaxy

essential input to direct Dark Matter searches •→ as simulation of the universe at diﬀerent scales and redshift is shown Fig. 3.7 •

45 Fig. 3.5: Picture of the simulated ”Dark Matter” halo used for further simulation [32].

Fig. 3.6: Density map of the ”Dark Matter” halo in the galactic plane, contours corre- 3 spond to ρDM = (0.1,0.3,1.0 and 3.0) GeV/cm [32]

46 Fig. 3.7: Result of an N-body simulation for diﬀerent redshift z ( time) and scale ( resolution) [33]. ≈ ≈

47 48 Chapter 4

Unsolved Questions and Open Issues

starting from an homogeneous universe gravity generates structures • the structure evolution in the non-linear regime with large over-densities δ can not • be calculated analytically

collisionless N-body simulations predict the evolution of structures, leading to galax- • ies inside of ”Dark Matter” halos

input to the simulation are the parameters obtained with the the Cold Dark Matter • model including the cosmological constant (ΛCDM) and the initial ﬂuctuations observed in the cosmic microwave background

no prediction of ”Dark Matter” in galaxies from ﬁrst principles • comparison of N-body simulations based on ΛCDM leads to diﬀerences with obser- • vations which are discussed below

the discussed problems appear on ”small scales”, meaning at the level of single • galaxies

currently intense discussions ongoing in the research community •

size of a halo 200 500kpc (1 pc = 3.26 LY = 3.1 1016 m) • ∼ − · distance to the centre of milky way 8 kpc • ∼ size of the universe 28 109 pc • · local group: about one part of 100 millions of the total universe, size about 3.1 Mpc •

49 4.1 Core-Cusp Problem

the density proﬁle ρ of ”Dark Matter” halos obtained with N-body simulation • indicate an inner distribution ρ rα with α = 1 [34] ∼ − following the discussion in section 1.2 a non constant ”Dark Matter” density can • lead to a change of the circular velocity

– a halo with constant ”Dark Matter” density ρ leads to a rotation curve with a linear increase 1 – a ”cuspy” halo with a ”Dark Matter” density with ρ r− leads to rotation curve increasing with the square root of the radius ∼

observational determination of α based on rotational curves result in α 0, a core • like ”Dark Matter” density distribution, which is in contradiction to∼ the results obtained from N-body simulation

more evolved N-body simulations lead to a ”Dark Matter” density profile which • can be fitted with a generic NFW [35] fit function ρ ρ = 0 (4.1) r r α 2 R 1 + R with a similar cuspy behaviour of the ”Dark Matter” profile Fig. 4.1 shows a comparison of the velocity distribution as a function of the ra- • dius from data compared to a fit with a more cuspy profile (red) using the NFW description and a core profile (blue).

possible solutions currently discussed: • – baryonic matter falls into the gravitational potential, leading to a supernova explosion which also leads to a decreased ”Dark Matter” density in the inner part of the halo, a core like density proﬁle – ”Dark Matter” only weakly interacts with ordinary matter but has strong self interaction

4.2 Missing Satellite Problem

the missing satellite problem refers to the diﬀerence to the number of predicted • cold ”Dark Matter” subhalos obtained with N-body simulations compared to the number of satellite galaxies observed in our local group.

from simulation about 4-5 times more satellite galaxies with a circular velocity • 1 v 10 20km s− is expected circ ≈ − (subhalos are rather characterised by their circular velocity vcirc than their total mass, because of problems deﬁning the subhalo with respect to the halo)

50 Fig. 4.1: [36]. Note that the visible rotational curve only corresponds to a fraction of the size of the simulated halo.

v measures the mass within the radius of the rotating subhalo • circ possible solutions currently discussed: • – processes which suppress gas accretion in these halos, keeping them ”Dark” (ultraviolet photoionisation prevents from cooling a gas ) – dwarf galaxies are to faint to be discovered (recently the SDSS has observed some ultra-faint galaxies) problem shift from a ”Dark Matter”-problem towards a problem of galaxy formation •

4.3 ”Too big to fail” Problem

massive sub-halos obtained from simulation are expected to host dwarf galaxies • comparing the mass of the observed dwarfs and the massive sub-halos from N-body • simulation leads to an discrepancy the mass of the sub-halos from simulation exceeds the mass of the observed dwarfs→ by a factor four to five ”to big too fail” - why do very massive sub-halos do not host stars? • the effect is significant and a statistical fluctuation can be excluded • similar to core-cusp problem: N-body simulation predict to much mass in the central • region

51 Fig. 4.2: ”Dark Matter” density distribution from a N-body simulation with a 1012 M CDM halo (left) . The number of subhalos exceeds the number of know Milky Way satellites (right) [36].

Deﬁnition halo: overdensity ( > 200) with respect to background, also deﬁnes the • virial radius

subhalo: self-bound object within the viral radius of a halo •

52 WHERE ARE GALACTIC SATELLITES? 7

1 Fig. 4.3: CumulativeFig. 4.— circular Properties velocity of satellite distribution systems within of satellite 200 h− systemskpc withFig. the5.— solid The same as in Figure 4, but for satellites within 1 triangles beingfrom Satellite the host from halo. MilkyTop panel: Way andThe Andromedathree dimensional and rms the ve- solid line400 h shows− kpc the from the center of a host halo. In the bottom panel locity dispersion of satellites versus maximum circular velocity we also show the cumulative velocity function for the field ha- result for halos from ΛCDM simulations [37]. 1 of the central halo. Solid and open circles denote ΛCDM and los (halos outside of 400 h− kpc spheres around seven massive CDM halos, respectively. The solid line is the line of equal satel- halos), arbitrarily scaled up by a factor of 75. The difference 1 lite rms velocity dispersion and the circular velocity of the host at large circular velocities Vcirc > 50 km s− is not statisti- halo. Middle panel: The number of satellites with circular ve- cally significant. Comparison between these two curves indi- 1 locity larger than 10 km s− versus circular velocity of the host cates that the velocity functions of isolated and satellite halos 1 halo. The solid line shows a rough approximation presented are very similar. As for the satellites within central 200h− kpc in the legend. Bottom panel: The cumulative circular velocity (Figure 4), the number of satellites in the models and in the distribution (VDF) of satellites. Solid triangles show average Local Group agree reasonably well for massive satellites with 1 VDF of Milky Way and Andromeda satellites. Open circles Vcirc > 50km s− , but disagree by a factor of ten for low mass 1 present results for the CDM simulation, while the solid curve satellites with Vcirc10 30 km s− . − represents the average VDF of satellites in the ΛCDM simulation for halos shown in the upper panels. To indicate the statistics, the scale on the right y-axis shows the total number of satellite halos in the ΛCDM simulation. Note that while 1 the numbers of massive satellites (> 50 km s− ) agrees reasonably well with observed number of satellites in the Local Group, models predict about five times more lower mass satellites with 1 Vcirc < 10 30 km s− . −

2.75 V − 1 3 velocities. The numbers of observed satellites and satel- n(> V ) = 1200 153 (h− Mpc)− , (4) 1 10 km s− lite halos cross at around Vcirc = (50 60) km s− . This means that while the abundance of− massive satellites 1 1 1 again, for R< 200 h− kpc and R< 400 h− kpc, respec- (Vcirc > 50 km s− ) reasonably agrees with what we ﬁnd tively. This approximation is formally valid for Vcirc > in the MW and Andromeda galaxies, the models predict 1 1 20 km s− , but comparisons with the higher-resolution an abundance of satellites with Vcirc > 20 km s− that is CDM simulations indicates that it likely extends to smaller approximately ﬁve times higher than that observed in the 54 Chapter 5

Particle Character of ”Dark Matter”

observations within the ΛCDM model of cosmology indicate ﬁve times more ”Dark • Matter” than baryonic matter

the Standard Model of particle physics does not oﬀer a particle as a possible ”Dark • Matter” candidate

several theories for physics beyond the Standard Model predict new new particle(s) • which could act as a candidate for ”Dark Matter”

discuss some of the most popular candidates, a ﬁrst possible separation gives • – thermally produced ”Dark Matter” (e.g. WIMPs) – non-thermally produced ”Dark Matter” (e.g. Axions)

summary of possible ”Dark Matter” candidate is shown in Fig. 5.1 •

5.1 Baryonic Dark Matter

MACHOs: MAssive Compact Halo Objects could act as a baryonic Dark Matter • candidate

MACHOs is a object of normal baryonic matter which emits little or no radiation, • like black holes, neutron stars or brown dwarfs

the gravity of a MACHO can bend light of a star and the star appears to brighter • for a observer, if observer MACHO and star in a line of sight

experimentally the census yields that MACHOs can maximally contribute 8% to • the mass of the galactic halo

a baryonic Dark Matter candidate is also in contradiction with the measured baryon • density observed in the BBN, also CMB leads to lower baryonic density

excluded as solution to the ”Dark Matter” problem • 55 Fig. 5.1: An overview of various particle ”Dark Matter” candidates as a function of mass and cross-section [38].

5.2 Weakly Interacting Particle (WIMP)

weakly interacting particles as non-baryonic cold ”Dark Matter” are considered as • one of the best candidates for ”Dark Matter” - very well studied motivation originates from the fact that the relic density obtained from thermal • freeze out leads to a mass and cross section know from the electroweak theory (WIMP miracle) - more a aesthetic argument diﬀerent measurements put very strong limits on this assumption • 5.2.1 The WIMP miracle calculation of the relic density of a weakly couple particle with certain m • the relic density of a WIMP particle χ nχ can be calculated with the Boltzmann • equation: dnχ = 3Hn σ v (n2 n2 ) dt − χ − h ann i χ − eq with nχ the WIMP number density, neq the equilibrium density, H the hubble constant and σ v thermal averaged WIMP annihilation cross-section h ann i 56 the change in density with time originates from diﬀerent contributions • – the expanding universe represented by the hubble constant H dilutes the WIMP density nχ

– the annihilation of WIMPs reduces the WIMP density nχ

– the production of WIMPs from particles from the equilibrium neq enhances the WIMP density nχ, the number of produced WIMPs χ decreases exponentially mχ/T with the Boltzmann factor e− (the number of particles with enough energy to create a pair of ”Dark Matter” particles decreases)

the hubble constant 2 = 8π , with = ( ) and being the photon tempera- H 3M 2 ρ ρ ρ T T • P l ture

the massive ”Dark Matter” particle χ pair-annihilates into a pair of basically mass- • less particles, which are coupled to the cosmic plasma and in a complete thermal equilibrium

at a certain time or energy (about the mass of the particle of χ) the energy is below • the energy required to pair produce a ”Dark Matter” particle pair, the χχ-pair is no longer in equilibrium and freezes out

n the Boltzmann equation can be rewritten in terms of Y = s , with s being the • m entropy, and x = T , with m the mass of χ and T being the photon temperature, leading to: dY = 1 ds σv (Y 2 Y 2 ) dt 3H dx h i − eq the evolution and the freeze out of a hypothetical ”Dark Matter” particle χ is shown • in Fig. 5.2

the Boltzmann equation cannot be solved analytically, only numerically • interesting values: • – time of freeze out (x ): the temperature T at freeze out is T m /20 f f f ' χ – WIMP speed at freeze out vf : the speed at freeze out is vf = (3Tf /2mχ) 0.3c ' 1 2 (the energy of the gas 3kT must be equal to the kinetic energy 2 mv )

– WIMP relic density Ωχ: the asymptotic relic density can be calculated from Y at freeze out [38]: 2 Ωχh 1 0.12 σ v/c , only weak dependence (logarithmic) on mχ ' 36 2 0.1 h 10− cm i 36 2 a weak cross section of the order of 10− cm (=1 pb) can be written in natural • 2 9 2 9 2 units (0.4 mb GeV− ) 2.5 10− GeV− 10− GeV− ' · ' 2 2 within the Fermi Theorie a cross section is proportional to GF Mχ, with GF being • 5 2 the Fermi constant G = 1.2 10− GeV− F × 57 setting the thermal cross section obtained from the thermal freeze with the Boltz- • 2 2 9 2 5 2 2 χ 2 mann equation: σ = (GF Mχ) 10− GeV− = (10− GeV− ) (M ) Mχ = 9 2 → → 10− GeV − 5 2 2 (√10) GeV (10− GeV − ) ' q WIMP miracle → the cross section required to match the observed ”Dark Matter” density in the • universe leads to a cross section similar to a cross section know from weak interaction pb, assuming weak interaction (here Fermi Theory) this would lead to a required ”Dark' Matter” of (GeV) O

Fig. 5.2: The evolution of the density of a hypothetical ”Dark Matter” particle χ with mass 100 GeV. The solid line corresponds to a cross section which leads to the correct relic density, the other lines correspond to a change in cross section by 10,100 or 1000. The dashed line corresponds to particle which stays in thermal equilibrium [39].

5.2.2 Supersymmetry (SUSY) Supersymmetry (SUSY) relates bosons (particles with integer spin) and fermions • (particle with half spin): Q F ermion >= Boson > and Q Boson >= F ermion > | | | | the Algebra of Supersymmetry is the only non-trivial extension of the poincare • group [40] Open questions of particle physics • MW 17 – Hierarchy problem: why is the ratio 10− , with MW being the MP l ' electroweak scale and MP l the Planck scale, so small?

58 – Gravity: the Standard Model does not contain gravity – ”Dark Matter” candidate: the Standard Model does not contain a ”Dark Matter” candidate – Naturalness: the Higgs particle is the only fundamental scalar particle in the Standard Model radiative corrections (with fermions in the loop) are of size Λ2, with Λ being 2 2 2 the scale of new physics, leading to mH = mH,0 + (Λ ) 2 2 O with Λ being Mpl the bare Higgs mass mH,0 has to be extremely ﬁned tuned to return a Higgs mass at the weak scale, the bare Higgs mass is not at it’s natural value, the so called ”naturalness problem”

SUSY oﬀers a solution to most of the remaining problems from the Standard Model • – without SUSY the coupling constants of the three fundamental interactions do not meet at a single point, with SUSY the evolution of the coupling constants with time meet at a single point, the grand uniﬁcation scale (GUT) see Fig. 5.3 (the evolution of the coupling constants with energy originates from virtual particles in loops) – SUSY has an intrinsic relation to time-space, local invariance leads to a spin-2 particle, the graviton – the Neutralino, a mixture of the neutral supersymmetric partners of the gauge bosons, acts as a ”Dark Matter” candidate – for the radiative correction to the mass of the scalar higgs boson, besides the fermions the supersymmetric partner of the fermion contributes with the opposite sign ∆(m2 ) m m H ∝ | fermion − scalar| the minimal supersymmetric model (MSSM) extends the Standard Model, • each Standard Model particle has a supersymmetric partner, squarks and the supersymmetric partners of the gauge bosons, gauginos

within the MSSM the supersymmetric partner should have the same mass as the • Standard Model particle simple MSSM extension has additional 120 parameters ”SUSY is broken” (mass of SUSY particles is diﬀerent to mass of Standard Model particles), breaking scenario needed

mixing of neutral supersymmetric partners of gauge bosons • ˜ χ0 B W˜ χ1  3   = A ˜0 χ3 · H1  ˜0 χ4 H     2  with χ0 being the lightest neutralino

R-parity: new discrete symmetry R=( 1)3B+L+2S, which is conserved • − 59 – Standard Model particles have R=+1 and Supersymmetric Particles have R=- 1 – SUSY particles can only be produced in pairs – R-parity conservation prevents the decay of the proton (strong constraints from experiments) – lightest supersymmetric particle (LSP) can not decay into Standard Model particle

lightest supersymmetric particle (LSP), the neutralino could act as candidate for • ”Dark Matter”

predictions for the χ0 within a certain SUSY breaking scenario (cMSSM) together • with the exclusion limits from direct searches are shown in Fig. 5.4

The gravitino, the SUSY-partner of the graviton could also act as a ”Dark Matter” • candidate, interaction only via gravity - no weakly interacting particle

Fig. 5.3: Evolution of the coupling constants with Standard Model particles only (dashed) and in the MSSM (red and blue) [41]

Search for SUSY two orthogonal ways to search for SUSY at the LHC •

– indirect searches: look for deviations in precision observables, e.g. Bs µµ constraints on diﬀerent SUSY models →

60 10-40

10-41

10-42 ] -43 2 10

m -44

c 10 [

I -45 S p 10 σ 2 -46 χ minimum 10 ∆χ2 =2.30 post-LUX ∆χ2 =5.99 post-LUX -47 ∆χ2 =2.30 pre-LUX 10 ∆χ2 =5.99 pre-LUX Xenon100 225-days 90%CL LUX 85-days 90%CL 10-48 101 102 103 m 0 [GeV] χ˜1

Fig. 5.4: Prediction of the χ0 in the mass versus cross section space for a certain MSSM breaking model (cMSSM) [42], together with the existing exclusion limits from direct searches.

– direct searches: direct production of SUSY particles at colliders look for dedicated SUSY scenarios at LHC • constrain certain SUSY models with deﬁned breaking scenario, e.g. constrained • MSSM (cMSSM) with ﬁve free parameters

look for dedicated SUSY decays by reconstructing dedicated decay chains like • Fig. 5.5

Neutralinos χ can be observed as a missing transverse energy in the detector • 0 the current exclusion limits obtained for certain SUSY particles and decay scenarios • is shown in Fig. 5.6

the LSP is maximally excluded up to 300 GeV for SUSY production in diﬀerent • SUSY production channels Fig. 5.7 ≈

after LHC Run 1 SUSY is under pressure, some simple SUSY models are already • ruled out

5.3 Sterile Neutrinos

Weak interaction violates CP maximal: it couples only to left-handed particles and • right-handed anti-particles.

61 Fig. 5.5: Production and decay of a gluino-pair (left) and a stop-pair (right). Both have a neutralino χ0 in the ﬁnal state, which is seen as missing energy in the detector.

As neutrinos have only a weak charge and no electric or strong charge, only left- • handed neutrinos are observed.

Neutrinos are mass less in the standard model of particle physic. • But observed neutrino oscillation is only possible if they have mass. • Possibility: add right-handed neutrinos, so that the neutrino mass can be generated • via a Dirac mass term: m (¯ν ν +ν ¯ ν ) (5.1) L ⊂ − R L L R As right-handed neutrino would not interaction via weak, strong, or electromagnetic • interactions, it would be an sterile neutrino. Contrary, the standard left-handed neutrinos are called active neutrinos.

Sterile neutrinos may also explain why the mass of the active neutrinos is so small • compared to the mass of the other leptons via the see saw mechanism:

– The right-handed neutrinos could gain additional mass via a Majorana mass term M (¯νc ν +ν ¯ νc ) (5.2) L ⊂ − 2 R R R R with a free parameter M. – The corresponding mass matrix would be

0 m ν ν ν L (5.3) L R m M ν R – With eigenvalues of: M √M 2 + 4m2 λ = ± (5.4) ± 2 Light active neutrinos with λ m2/M. ⇒ −− ≈ Heavy sterile neutrinos with λ M. ⇒ + ≈ Sterile neutrinos are candidates for dark matter. • 62 Summary of CMS SUSY Results* in SMS framework ICHEP 2014 m(mother)-m(LSP)=200 GeV m(LSP)=0 GeV

~ ∼ 0 g → qq χ SUS 13-019 L=19.5 /fb ~ ∼ 0 g → bb χ SUS-14-011 SUS-13-019 L=19.3 19.5 /fb ~ ∼ 0 g → tt χ SUS-13-007 SUS-13-013 L=19.4 19.5 /fb ~ ~ ∼ 0 g → t(t → t χ ) SUS-13-008 SUS-13-013 L=19.5 /fb ~ ∼ ± ∼ 0 x = 0.20 g → qq(χ → Wχ ) SUS-13-013 L=19.5 /fb x = 0.50 0 gluino production ~ ~ ∼ ± ∼ g → b(b → t(χ → Wχ )) SUS-13-008 SUS-13-013 L=19.5 /fb

~ ∼ 0 q → q χ SUS-13-019 L=19.5 /fb squark ~ ∼ 0 t → t χ SUS-14-011 L=19.5 /fb ~ ∼ + ∼ 0 x = 0.25 x = 0.50 t → b(χ → Wχ ) SUS-13-011 L=19.5 /fb x = 0.75 ~ ∼ 0 ∼ 0 t → t b χ ( χ → H G) SUS-13-014 L=19.5 /fb ∼ 0 stop ~ ~ t → ( t → t χ ) Z SUS-13-024 SUS-13-004 L=19.5 /fb ~2 ~1 ∼ 01 t → ( t → t χ ) H SUS-13-024 SUS-13-004 L=19.5 /fb 2 1 1

~ ∼ 0 b → b χ SUS-13-018 L=19.4 /fb ~ ∼ 0 b → tW χ SUS-13-008 SUS-13-013 L=19.5 /fb ~ ∼ 0

sbottom b → bZ χ SUS-13-008 L=19.5 /fb

∼ 0 ∼ ± ∼ 0 ∼ 0 x = 0.05 χ χ → lll ν χ χ SUS-13-006 L=19.5 /fb x = 0.50 2 x = 0.95 ∼ +∼ - + ∼ 0∼ 0 χ χ → l l-ν ν χ χ SUS-13-006 L=19.5 /fb ∼ 0 ∼ 0 ∼ 0 ∼ 0 χ χ → Z Z χ χ SUS-14-002 L=19.5 /fb ∼ ±2∼ 02 ∼ 0 ∼ 0 χ χ → W Z χ χ 2 SUS-13-006 L=19.5 /fb CMS Preliminary ∼ 0 ∼ 0 ∼ 0 ∼ 0 χ χ → H Z χ χ SUS-14-002 L=19.5 /fb 2 2 ∼ ± ∼ 0 ∼ 0 ∼ 0 For decays with intermediate mass, χ χ → H W χ χ SUS-14-002 L=19.5 /fb ∼ 0 ∼ ±2 ∼ 0 ∼ 0 x = 0.05 χ χ → llτ ν χ χ x = 0.50 ⋅ ⋅ EWK gauginos 2 SUS-13-006 L=19.5 /fb x = 0.95 m = x m +(1-x) m ∼ 0 ∼ ± ∼ 0 ∼ 0 intermediate mother lsp χ χ → τττ ν χ χ SUS-13-006 L=19.5 /fb 2

~ ∼ 0 l → l χ SUS-13-006 L=19.5 /fb

slepton ~ g → qllν λ 122 SUS-12-027 L=9.2 /fb ~ g → qllν λ SUS-12-027 L=9.2 /fb ~ 123 g → qllν λ 233 SUS-12-027 L=9.2 /fb ~ g → qbtµ λ ' 231 SUS-12-027 L=9.2 /fb ~ g → qbtµ λ ' SUS-12-027 L=9.2 /fb ~ 233 g → qqb λ '' 113/223 EXO-12-049 L=19.5 /fb ~ g → qqq λ '' EXO-12-049 L=19.5 /fb ~ 112 g → tbs λ '' 323 SUS-13-013 L=19.5 /fb ~ g → qqqq λ '' SUS-12-027 L=9.2 /fb ~ 112 q → qllν λ 122 SUS-12-027 L=9.2 /fb ~ q → qllν λ 123 SUS-12-027 L=9.2 /fb ~ q → qllν λ

RPV SUS-12-027 L=9.2 /fb ~ 233 q → qbtµ λ ' 231 SUS-12-027 L=9.2 /fb ~ q → qbtµ λ ' SUS-12-027 L=9.2 /fb ~ 233 q → qqqq λ '' R 112 SUS-12-027 L=9.2 /fb ~ t → µ e ν t λ R 122 SUS-13-003 L=19.5 9.2 /fb ~ t → µ τν t λ SUS-12-027 L=9.2 /fb ~R 123 t → µ τν t λ R 233 SUS-13-003 L=19.5 9.2 /fb ~ t → tbtµ λ ' R 233 SUS-13-003 L=19.5 /fb 0 200 400 600 800 1000 1200 1400 1600 1800 *Observed limits, theory uncertainties not included Mass scales [GeV] Only a selection of available mass limits Probe *up to* the quoted mass limit

Fig. 5.6: Summary of the current SUSY searches at the LHC with the CMS experiment.

Sterile neutrinos can interact with matter through mass mixing, but it is suppressed • by m/M.

Claimed observation of an X-ray excess at 3.5 keV in galactic clusters could be • explained by the decay of a sterile 7 keV-neutrino [43].

5.4 Asymmetric Dark Matter (ADM)

If an asymmetry between dark matter and anti-dark matter exist, then the relict • abundance ΩDM depends on the initial ADM asymmetry and not on the thermal freeze-out: n n Ω η m , η = dm − dm (5.5) dm ∼ dm · dm dm s with the entropy density s. The mass m is related to the Ω /Ω ratio [38]: ⇒ dm b dm ηb Ωdm mdm = mp (5.6) ηdm · Ωb ·

with the proton mass mp as typical baryon.

The ration ηb/ηdm depends on the actual theory. For ηdm ηb and the experimental • values of Ω ,Ω it follows that m 5 GeV [38]. ≈ b dm dm ≈ 63 ∼ ∼± χ0-χ production 900 2 1

∼0 ∼± ∼0 ∼0 χ χ → (H χ )(W χ ) CMS Preliminary 2 1 1 1 ∼0 ∼± ∼0 ∼0 800 χ χ → (Z χ )(W χ ) 2 1 1 1 s = 8 TeV ∼0 ∼± ~ χ χ ( l , BF(l+l-)=0.5 ) ICHEP 2014 2 1 L 700 ∼+ ∼- ~ χ χ ( l , BF(l+l-)=1 ) 1 1 L LSP mass [GeV] ∼0 ∼± ∼∼ χ χ ( τντ) 600 -1 2 1 SUS-13-006 19.5 fb ∼ ∼± χ0 χ ~ + - ( lR , BF(l l )=1 ) SUS-14-002 19.5 fb-1 2 1 ∼0 χ 500 1 = m Z ∼± Observed χ +m 1 ∼0 m χ H 1 +m = m ∼0 ∼± χ Expected χ 1 m 1 = m 400 ∼± χ m 1

300

200

100

0 100 200 300 400 500 600 700 800 neutralino mass = chargino mass [GeV]

Fig. 5.7: Exclusion limit in the LSP versus Neutralino / Chargeino plane.

5.5 Axions

Axions emerge from the Peccei-Quinn solution [44] of the strong CP problem1: •

– CP violating terms are allowed in QCD but would result in an electric dipole moment of the neutron (NEDM). L 26 – But no NEDM is observed, the current limit is < 2.9 10− ecm [46]. · Fine-tuning problem! ⇒ Prevent CP violation by additional Peccei-Quinn symmetry [44], which breaks below • the scale fa.

During its evolution, the universe cools down below fa and the axion appears as • the pseudo Nambu-Goldstone-Bosone of the Peccei-Quinn symmetry.

1 The coupling of axions to matter is f − . • ∝ a As the axion radiation is an additional channel to transport energy out of stars, a • too strong coupling would result in a too fast cooling of stars and supernovae.

1A good pedagogical introduction to the strong CP problem and why axions may be a solution to it can be found in [45].

64 7 Astronomical observations set limits: fa > 10 GeV and on the axion mass ma < • 1 eV [47]. The axion is the classic example for a non-thermal produced particle candidate for⇒ dark matter.

Beside the classic axion from the Peccei-Quinn symmetry, similar light particles • exist in other theories, collectively called axion like particles (ALPs). Frequently used as examples are the KSVZ and DFSZ models [48].

Despite its small couplings, ALPs may be observable via the Primakoff effect, i.e. • the conversion of an ALP to a real photon by interacting with a virtual photon in 1 a magnetic field with a coupling G f − [48]. Aγγ ∝ A Light Shining through a Wall experiments (LSW), e.g. Any Light Particle Search • (ALPS)2 [50] or Optical Search for QED Vacuum Birefringence, Axions, and Photon Regeneration (OSQAR) [51], are based on two subsequent conversions: A light source illuminates an opaque wall in front of a light detector. In front of and behind the wall, magnetic fields are applied. In absence of ALPs, the light detector should measure no light. If ALPs exist, some photons passing through the first magnetic field in front of the wall would be converted to ALPs. Due to their low interaction rate, the ALPs can pass through the wall. Some of these ALPs would be converted back to photons in the second magnetic field behind the wall. Consequently, the detector would register light, hence the name of this experiment design.

The ﬁrst conversion, photons to ALPs, can happen also in the Sun. The CERN • Axion Solar Telescope (CAST) [52], a so called helioscope, is searching for an ALP signal from the sun since 2002 [53], by pointing a light detector behind an opaque shield and a magnetic ﬁeld at the position of the Sun. The signal would be an ALP from the Sun which converts back to a photon.

So far, no ALPs were observed [48] and limits could be set on the parameter space, • see ﬁg. 5.8.

5.6 Alternative Theories

5.6.1 Modified Newtonian Dynamics - MOND The modified Newtonian dynamics (MOND) [54] based on the assumption that • the observed non-Keplerian rotation curves of galaxies indicates no missing mass. Instead, it proposes a modification of Newton’s second law.

MOND states that Newton’s second law was never tested at acceleration values as • low as the acceleration of stars in the outskirt area of a galaxy. Therefore, it seems

2Starting as ALPS I in 2007, now in its second stage ALPS II [49].

65 10-6 LSW VMB (OSQAR) (PVLAS)

-8 ) 10 -1 Telescopes

Sun | (GeV 10-10 Helioscopes (CAST)

A γγ Horizontal Branch Stars HESS Fermi SN 1987A 10-12

Haloscopes (ADMX and others) Axion Coupling |G 10-14 KSVZ

DFSZ 10-16 10-10 10-8 10-6 10-4 10-2 100 Axion Mass mA (eV) Fig. 5.8: Parameter space for ALP’s two-photon interactions, i.e.ALP-photon coupling GAγγ over axion mass mA. The yellow diagonal regions indicate the parameter space favored by the KSVZ and DFSZ models. The other regions are excluded via, among other experimental techniques, astronomical observations (gray), helioscopes like CAST (blue) and light shinning through walls experiments (LSW) (orange). Figure taken from [48] and references therein.

valid to propose a generalization [54]:

F~ = ~a µ(a/a ) m (5.7) · 0 · The function µ(x) is only deﬁned via its limits [54]

1 for x 1 µ(x) , (5.8) ≈ x for x 1 ( a possible function which fulﬁlls these constrains is µ(x) = x/√1 + x2 [55]. The behaviour of MOND at these limits is as follows:

– at a a the dynamic is Newtonian: F = m a. 0 · – at a a the dynamic is modified: F = m a2/a . 0 · 0 10 2 MOND claims to explain most galactic rotation curves with a0 2 10− ms− • [54]. ≈ · We can compare this value with the acceleration of a star in the outskirt of a galaxy: • for e.g.NGC 3198 we find (fig. 1.6) that at r 20 kpc3 the circular velocity is v 1 ≈ 2 11 ≈2 150 kms− , i.e. the centrifugal acceleration of the star is a = v /r 9.47 10− ms− . According to MOND, the dynamics of this star should therefore≈ be modified.·

3One parsec (pc) is roughly 3.26 light years, one light year is around 9.46 1015 m. · 66 14 2 Laboratory experiments at Earth can measure accelerations as small as 5 10− ms− • [56]. However, these measurements happen against the background acceleration· of the Earth and can therefore not directly contradicting MOND [56].

However, MOND faces diﬃculties to explain strong and weak gravitational lensing: •

– strong lensing indicates missing mass also where a a0, i.e. where MOND behaves by definition Newtonian. need dark matter to explain the missing mass ⇒ – MOND can not explain weak lensing observation where the peak of the total mass does not coincide with the peak of baryonic mass, e.g. the Bullet Cluster in section 2.2.1. need of non-baryonic dark matter ⇒ These obstacles may be solved by even more generalized modifications, e.g. Modified • Gravity (MOG) which introduce a position dependent gravitational coupling to explain the Bullet Cluster [57].

67 68 Chapter 6

Orthogonal Approaches for ”Dark Matter” searches

6.1 The Feynman Diagram from diﬀerent directions

reminder: ﬁve time more ”Dark Matter” than ordinary matter, however, evidence • for ”Dark Matter” only from gravitational interaction

expectation: additional (weak) interactions beyond gravity between ”Dark Matter” • and Standard Model particles

aim: detection of ”Dark Matter” through non-gravitational interaction • no conclusive prediction of a ”Dark Matter” particle candidate and size and type • of interaction

diversity of ”Dark Matter” particle candidates with broad range of predictions • need diﬀerent approaches with diﬀerent experimental reach → compelling solution for ”Dark Matter” problem requires coherent answer from dif- • ferent approaches

orthogonal approaches: • i) direct detection: elastic ”Dark Matter” nucleon scattering ii) indirect detection: annihilation of ”Dark Matter” particles into Standard Model particles iii) production: annihilation of Standard Model particles into ”Dark Matter” particles iv) (Astrophysical probes:) non-gravitation interaction of ”Dark Matter” particles on astrophysical scales

see Fig: 6.1 for approach i) to iii), diﬀerent experimental approaches can be see • as realisation of same Feynman diagram with space, time and direction being ex- changed

69 in case of discovery: characterisation of ”Dark Matter” particle and interaction • requires also diﬀerent approaches

Fig. 6.1: Feynman diagram with the possible interaction for studying non-gravitational interaction between matter and ”Dark Matter”. To describe the various approaches the arrow indicating the evolution of time and space needs to be rotated. Fig. taken from [58]

6.2 Strength and Weakness of the various Approaches

diﬀerent approaches have diﬀerent strength and weaknesses • i) direct detection: - very promising candidate to detect thermally produced WIMPs - better sensitivity for spin-independent interaction (scalar (χχff) and vec- µ tor (χγ χfγµf)) compared to spin-dependent interaction (axial-vector µ 5 5 (χγ γ χfγµγ f)) - low sensitivity for velocity dependent interactions due to very small ”Dark Matter” speed - less sensitive in the low-mass region due to very small energy deposition (asymmetric ”Dark Matter”) - direct detection experiments also searching for axion in the µeV to meV region ii) indirect detection: - identical process responsible for ”Dark Matter” annihilation at the thermal freeze out - ”Dark Matter” particles are not directly observed

70 - search for decays in all possible Standard Model particles, leptons, quarks and gauge bosons - strong line of sight dependence on astrophysical models describing the ”Dark Matter” density and propagation model for charged products - high sensitivity towards high mass ”Dark Matter” particle candidates - almost no sensitivity for asymmetric ”Dark Matter” scenarios (anti-”Dark Matter” already completely annihilated) iii) production: - production of ”Dark Matter” particles in the annihilation of Standard Model particles in accelerator experiments like LHC - good sensitivity for low mass ”Dark Matter” particles - no sensitivity for life time of ”Dark Matter” candidate, decay after 100 ns not visible - hadron colliders (like LHC) have very little sensitivity for ”Dark Matter” candidates coupling to leptons - diﬀerent detection scenarios - a) direct production of two ”Dark Matter” particles leaving the detector without signal (”missing Energy”) less sensitive to the underlying ”Dark Matter” model allowing to put direct limits on the mass and interaction - b) production of non-Standard Model particles decaying into ”Dark Matter” candidates typical scenario for LSP in SUSY type models, signature: hadronic activity and ”missing energy” initial production rate of non-”Dark Matter” particles insensitive on ”Dark Matter” properties iv) (Astrophysical probes:) - N-body simulation based on ΛCDM assumption describe structure of universe reasonable well - several detailed observations indicate diﬀerences between simulation and observation (discussed in detail in Section 4) like ”core versus cusp” or ”too big to fail” problem - possible solution strongly self-interacting ”Dark Matter” (SIDM) (strong with respect to gravitation) σDM 2 - current limit M < 0.47cm /g, corresponding to σDM < 1 b for M = 1 GeV - ”warmth” of ”Dark Matter” candidate from structure formation - hot ”Dark Matter” strongly disfavoured (see Section 3) - impossible to identify non-gravitational interaction of ”Dark Matter” con- clusively with astrophysical probes only, no precision measurements possible - conclusion of non-gravitational interaction of ”Dark Matter” from astrophysical probes are already discussed in not further discussed in Section 3 and Section 4 and will not be discussed in the subsequent chapters

71 the complementarity for the diﬀerent approaches and their strength is shown in • Fig. 6.2

assuming the thermally produced WIMP case: • – an observation in a green band indicates that the cross-section measured cannot explain the observed ”Dark Matter” relic density and additional ”Dark Matter” candidates are required – an observation in the yellow band indicates that the ”Dark Matter” relic density is too large and additional annihilation channels are needed to explain it

Fig. 6.2: Reach for the diﬀerent approaches for diﬀerent masses. The cross-section is normalised to the cross-section required for a thermally produced WIMP Picture. The black line corresponds to the sensitivity from direct detection, blue indirect detection and red production with colliders. Figure is taken from [59].

speciﬁc SUSY with R-parity violation models have ”Dark Matter” candidates (LSP) • variation of SUSY parameters lead to diﬀerent predictions of mass and cross-section • for ”Dark Matter” candidates (see Fig. 6.3)

even after a discovery we need orthogonal approaches for a coherent characterisation • of the ”Dark Matter” particle and it’s role within a post-Standard Model of Particle Physics

(possible) remaining questions after a ”Dark Matter” discovery: is it the only ”Dark • Matter” particle? Relation between production, elastic scattering and annihilation?

72 7 10−

9 10−

11 (pb) 10− SI σ ·

R 13 10−

15 10−

10 17 − 102 103 0 m(χ˜1) (GeV) Fig. 6.3: Model independent scan of SUSY parameter space and the impact on the mass of the ”Dark Matter” candidate χ and the spin-indepedent WIMP-nucleon cross section. Each point corresponds to one possible concrete model. ”Dark Matter” candidates can be discovered with direct detection (green), indirect detection (blue), both (red) or can only be reached by the LHC (grey). R = Ωχ . Figure taken from [59]. ΩDM

73 74 Chapter 7

Indirect ”Dark Matter” detection

7.1 Search Strategy

basic idea: ”Dark Matter” particles annihilate in a dense environment to Standard • Model particles

all Standard Model particles can act as a decay product, including gammas and • neutrinos (neutral particles will not be deﬂected in a interstellar magnetic ﬁeld)

annihilation process used for indirect ”Dark Matter” detection identical to the pro- • cess responsible for the ”Dark Matter” relic density produced during WIMP freeze out hint for size (or benchmark process) for expected cross section → indirect ”Dark Matter” searches mainly focus on the detection of WIMP-like ”Dark • Matter” particles in the GeV to TeV mass region as expected from freeze out

7.1.1 Expected Signal in general the expected signal can be subdivided into two parts: a particle physics • contribution P and a astrophysical contribution J

for gammas and neutrinos undisturbed traveling through space can be assumed • dR the expected diﬀerential gamma rate can be written dt dA dE = P J(∆Ω) (see [60] • for more details) ·

Particle Physics Part P

i σann v dNγ P = h 2 i BRi 2m dEi • χ · i i dNγ with theP diﬀerential gamma yield per annihilation, BRi the corresponding dEi branching fraction of the decay, σ v the thermally averaged cross-section and h ann i mχ the WIMP mass the produced Standard Model particles could be charged (fermions) or electrically • neutral (photons or neutrinos)

75 neutral particles are not deflected in any magnetic field (see above) • for charged particles possible interactions with the interstellar medium and propa- • gation effects need to be taken into account

gammas could produced by several sources • – no direct production since ”Dark Matter” by definition not couples electro- magnetically, only production via loops reduced cross section → – final state radiation (FSR) from charged particles in the final state – hadronization of quarks with π0 in the final state mainly decaying into gammas (see Fig. 7.1)

dNγ the diﬀerential energy spectrum of the photons dE depends on the production • process

photons from π0-decays have a deeply falling energy spectrum peaking towards • lower energies

photons from ﬁnal state radiation have a ﬂatter spectrum with a sharp cut of a the • maximum energy of Eγ = mχ, the probability for emitting a photon is proportional to the squared of the charge of the fermion, charged leptons have the highest FSR probability

”Dark Matter” annihilation in the t-channel with a charged squark also virtual • internal bremsstrahlung is possible (see Fig. 7.3)

Astrophysical Contribution J

∞ 2 J(∆Ω) = dl dΩ ρχ(l) • ∆Ω l=0 R R with ρχ being the ”Dark Matter” density along the line of sight, ∆Ω the solid angle • element

the signal size depends on the ”Dark Matter”-density ρχ squared, which is predicted • by N-body simulations (see Fig. 7.4):

– simulated ”Dark Matter” halo density proﬁle – the expected ”Dark Matter” sub-halo structure – ”Dark Matter” velocity distribution

diﬀerent ”Dark Matter” sources can be used for ”Dark Matter” searches • – Milky Way galactic center (GC): brightest source for ”Dark Matter” detection large uncertainties of density in the inner halo proﬁle (cusp versus core) large number of astrophysical foregrounds in the line of sight towards the GC

76 Fig. 7.1: Annihilation of a ”Dark Matter”-pair into a pair of vector boson or quarks with possible photons, neutrinos or anti-matter in the the ﬁnal state [61].

– Milky way dwarf spheroidal galaxies (dSphs): high ”Dark Matter” density expected from measured velocity distribution dSphs are outside the galactic plane and almost no astrophysical foreground contribution expected less dependent on the ”Dark Matter” halo density since the overall halo size matches the detector resolution (see Fig. 7.5) – Sun: large ”Dark Matter” density in the center of the sun due to gravitational attraction expected no gamma measurement possible, but signal from ”Dark Matter” decay into neutrinos

for charged leptons the propagation within the galactic medium and the energy loss • has to be taken into account

7.2 Instruments and Methods

diﬀerent signals for detecting ”Dark Matter” decay products • – Photons: detection via pair production in satellites (e.g. Fermi-LAT) or ground based air Cherenkov telescopes (e.g. MAGIC, HESS, CTA - see Fig. 7.6) – Charged particles: satellite or ISS based detector (e.g. AMS-02 - see Fig. 7.7)

77 10 x dN/d 1 2 x

10•1

10•2

10•3 χχ →bb 10•4 χχ →tt χχ →µ+µ• χχ →τ+τ• •5 + • 10 χχ →W W χχ →ZZ 10•6 10•2 10•1 1 x = E’/m χ

Fig. 7.2: The differential gamma flux from different decay channels as a function of x= Eγ , mχ weighted with x2 [62].

– Neutrinos: large area detectors detecting neutrinos via leptons produced in a charged interaction

7.3 Experimental Search for ”Dark Matter” annihilation

7.3.1 Gamma Flux from Dwarf Galaxies search of gammas from Dwarf Spheroidal Galaxies with the MAGIC (ICT) telescope • Dwarf Spheroidal Galaxies (dSphs) have a very high ”Dark Matter” density • M

dSphs Segue 1 has L 3400 • ≈ no contamination of gamma flux from other astrophysical sources • no significant gamma flux observed, set a limit on ”Dark Matter” annihilation • cross-section

interpretation for several diﬀerent gamma production scenarios leading to diﬀerent • i spectral shapes dNγ dEi summary of limit obtained is shown in Fig. 7.8 • 24 3 1 strongest limit at mχ about 500 GeV with σann v 1.2 10− cm s− , above the • expected cross-section for thermally producedh WIMPsi ≈ ·

78 10 x τ+τ•γ + •

dN/d µ µ γ

x FSR

1 VIB

10•1

10•2

10•3 10•2 10•1 1 x = E’/m χ

Fig. 7.3: The diﬀerential gamma ﬂux for virtual internal bremsstrahlung in the t-channel annihilation [62].

Distance to Segue 1 about 23 2 kpc • ± MAGIC observation time about 160h • typical J values: • 19 2 5 – MAGIC dwarf: J = 1.1 10 GeV cm− = [62] · 22 2 5 – Fermi-LAT: J = 13.9 10 GeV cm− [66] · 22 2 5 – ANTARES: J 1 10 GeV cm− [67] ≈ ·

7.3.2 Neutrino ﬂux from the Galactic Center the highest ”Dark Matter” density and therefore ﬂux of decay products is expected • from the GC

to overcome astrophysical background contributions the neutrino rate instead of • the gamma rate is observed

only muon neutrinos νµ are reconstructed via the weak interaction and production • of muons, producing Cherenkov light

predict diﬀerential energy spectrum of neutrinos • no excess of neutrinos observed - set a limit see Fig. 7.9 • 79 Fig. 7.4: Angular distribution of an expected gamma-ray signal obtained with N-body simulation of a Milky-way sized halo using the simulated J-value [63]

+ 22 3 1 limit for σ v in ”Dark Matter” annihilation in a τ τ −-pair about 10− cm s− • h ann i ∼

7.3.3 Claims for Detection of ”Dark Matter” annihilation several possible ”Dark Matter” annihilation studied with a possible signal from • ”Dark Matter” annihilation the interpretation as a clear signal for ”Dark Matter” annihilation is not possible • because of to little statistics or too large astrophysical systematic uncertainties if the observed signal is interpreted as a ”Dark Matter” signal the gamma ﬂux is • above the prediction expected from a generic ”Dark Matter” model

7.3.4 ”Dark Matter” Signal at 130 GeV from Galactic Center measurement of gamma ﬂux from the galactic center • expect large gamma rate from ”Dark Matter” annihilation • 3 ”Dark Matter” particles travel at non-relativistic velocities (v 10− c) leading to • a monochromatic gamma line ≈ large astrophysical uncertainties from modelling the ”Dark Matter” distribution • and the gamma dispersion

claim for excess with a local signiﬁcance of 4.5 σ obtained from external scientists • using publicly available data

80 M. ACKERMANN et al. PHYSICAL REVIEW D 89, 042001 (2014)

FIG. 1 (color online). Known dwarf spheroidal satellite galaxies of the Milky Way overlaid on a Hammer-Aitoff projection of a 4-year LAT counts map (E>1 GeV). The 15 dwarf galaxies included in the combined analysis are shown as filled circles, while additional Fig.dwarf 7.5: galaxies Distribution are shown as open circles. of observed dwarf galaxies of our local Milky Way [64]

radius of 25 dwarf spheroidal satellite galaxies Fig. 1. The P7REP_CLEAN_V15 instrument response functions CLEAN event class was chosen to minimize particle back- (IRFs) corresponding to the LAT data set selected above. grounds while preserving effective area. At high Galactic When performing the Bayesian analysis in Sec. VII,we reanalyseslatitudes in the energy of data range from by 1Fermi-LAT to 500 GeV, the particle collaborationutilize the same [66] LAT combined data set but follow with different improved data data • reconstructionbackground contamination and in thegamma CLEAN class dispersion is ∼30% modelpreparation reduce and background the modeling signal procedures, significantly which to less of the extragalactic diffuse γ-ray background [28], while are described in that section. thanbetween 3.3 500σ MeV, corresponding and 1 GeV the particle to backgroundglobal significance of 1.5 σ is comparable to systematic uncertainties in the diffuse III. MAXIMUM LIKELIHOOD ANALYSIS Galactic γ-ray emission. Studies of the extragalactic γ-ray Limited γ-ray statistics and the strong dependence of the signalbackground width at energies significant greater than 500 smaller GeV suggest than that at monochromatic signal convoluted with detector LAT performance on event energy and incident direction • these energies the fractional residual particle background is resolution motivate the use of a maximum likelihood-based analysis greater than at lower energies [30]. To reduce γ-ray con- to optimize the sensitivity to faint γ-ray sources. We define tamination from the bright limb of the Earth, we reject events the standard LAT binned Poisson likelihood, interpretingwith zenith angles the larger measurement than 100° and events ascollected an upper limit on the thermally averaged cross- during time periods when the magnitude of rocking angle Y nk −λ • λ e k sectionof the LAT wasσann greaterv thanleads 52°. to a value below the valueL expectedðμ; θjDÞ¼ fork a; WIMP(1) consistent n ! withWe thecreateh relic14°×14 densityi° regions-of-interest (ROIs) by bin- k k ning the LAT data surrounding each of the 25 dwarf D galaxies into 0.1° pixels and into 24 logarithmically-spaced as a function of the photon data, , a set of signal pair produced gammas are expect to be loop suppressed,μ the measuredθ branching bins of energy from 500 MeV to 500 GeV. We model the parameters, , and a set of nuisance parameters, . The fractiondiffuse background (assuming with a structured a thermally Galactic γ-ray averaged emis- number cross-section of observed counts from in each freeze energy and out) spatial to bin, two gam- D massion model is between (gll_iem_v05.fit 5%) and to an 10%, isotropic higher contribution thanindexed expected by k, depends from on the ”Dark data, nk ¼ Matter”nkð Þ, while models the (or from extragalactic γ rays and charged particle contamina- model-predicted counts depend on the input parameters, 1 λ ¼ λ ðμ; θÞ. This likelihood function encapsulates infor- thetion annihilation (iso_clean_v05.txt). crossWe build section a model ofσ pointlikeann v isk largerk than expected γ-ray background sources within 15° of each dwarfh galaxy i mation about the observed counts, instrument performance, exposure, and background fluxes. However, this likelihood beginning with the second LAT source catalog (2FGL) “ ” [27]. We then follow a procedure similar to that of the function is formed globally (i.e., by tying source spectra 2FGL to find additional candidate pointlike background across all energy bins simultaneously) and is thus neces- sources by creating a residual test statistic map with sarily dependent on the spectral model assumed for the 7.3.5pointlike Excess[27]. No new of sources Positron are found within fraction 1° of source in of interest. Cosmic To mitigate thisRays spectral dependence, it is any dwarf galaxy and the additional candidate sources have common to independently fit a spectral model in each chargeda negligible particlesimpact on our dwarf from galaxy ”Dark search. WeMatter” use the decaysenergy bin, willj (i.e.,propagate to create a spectral to energy the distribution earth • for a source) [31]. This expands the global parameters μ and θ into sets of independent parameters fμjg and fθjg. signal1http://fermi.gsfc.nas from ”Darka.gov/ssc/data/access/lat/BackgroundModels Matter” decays mustLikewise, differ the from likelihood cosmic function rays in Eq. (1) can be • .html. reformulated as a “bin-by-bin” likelihood function, requires spectral feature which is clearly different from the background • 042001-4 energy of particle from ”Dark Matter” decay cannot exceed mass of the ”Dark • Matter” particle

anti-particles suppressed in cosmic rays and could originate from ”Dark Matter” • pair annihilation to pair of Standard Model particles

increase of anti-particles with energy followed by an abrupt decrease around the • ”Dark Matter” rest mass would be a good sign for ”Dark Matter”-annihilation

no astrophysical explantation for an antiparticle source could explain abrupt de- • crease

81 Fig. 7.6: Sketch of a telescope of the Cherenkov telescope array (CTA). Photons hit- ting the atmosphere will shower and produce Cherenkov light being detected by the telescope [63].

e+ Fig. 7.12 shows the measured positron fraction + , increasing from 0.05 at 10 e +e− • GeV to 0.15 between 200 and 350 GeV measured with the AMS-02 detector

cosmic ray propagation models for our galaxy predict much lower positron fraction • the cross-section required to explain positron fraction excess with ”Dark Matter” • annihilation is too large compared to the expected cross-section from WIMP freeze out

additional gamma signal from compton scattering and synchrotron radiation ex- • pected, but not observed

large positron fraction could also originate from primary positrons produced by • pulsars

no clear drop of positron fraction observed • AMS-02 can separate positron and electron up to an energy of about 1 TeV • measurements from proton / anti-proton ratio are not consistent with expectations •

82 week ending PRL 113, 121101 (2014) PHYSICAL REVIEW LETTERS 19 SEPTEMBER 2014

magnet and the tracker. This ensures that most of the secondary particles produced in the TRD and in the upper TOF planes are swept away and do not enter into the ECAL. Events with large angle scattering are also rejected by a quality cut on the measurement of the trajectory using the tracker. The matching of the ECAL energy, E, and the momentum measured with the tracker, p, greatly improves the proton rejection. To differentiate between e and protons in the TRD, signals from the 20 layers of proportional tubes are combined in a TRD estimator formed from the ratio of the log-likelihood probability of the e hypothesis to that of the proton hypothesis in each layer. The proton rejection power of the TRD estimator at 90% e efficiency measured on orbit is 103 to 104 [2]. To cleanly identify electrons and positrons in the ECAL, an estimator, based on a boosted decision tree algorithm Fig. 7.7: SketchFIG. of 1 the (color). AMS-02 A 369 detector GeV positronoperated event at the as internationalmeasured by the space station[12], is constructed using the 3D shower shape in the (ISS). The chargeAMS of detectorthe particle on isthe determined ISS in the by bending measuring (y-z the) plane. curvature Tracker with aECAL. strong The proton rejection power of the ECAL estimator planes 1 to 9 measure the particle charge, sign, and momentum. reaches 104 when combined with the energy-momentum permanent magnet [65]. The TRD identifies the particle as e . The TOF measures the matching requirement E=p > 0.75 [2]. absolute charge value to be one and ensures that the particle is The entire detector has been extensively calibrated in a downward going. The RICH independently measures the charge test beam at CERN with eþ and e− from 10 to 290 GeV=c, and velocity. The ECAL measures the 3D shower profile, 400 = π e with protons at 180 and GeV c, and with from 10 independently identifies the particle as an , and measures its 180 = energy. A positron is identified by (1) positive rigidity in the to GeV c which produce transition radiation equiv- 1 2 = tracker, (2) an e signal in the TRD, (3) an e signal in the alent to protons up to . TeV c. In total, measurements ECAL, and (4) the matching of the ECAL shower energy and axis with 18 different energies and particles at 2000 positions with the momentum measured with the tracker and magnet. Note: were performed [2]. the 3D ECAL has nine superlayers along the z axis with fibers in Data sample and analysis procedure.—Over 41 billion alternating directions. In the (y-z) plane the wider rectangles events have been analyzed following the general procedure display the width of the shower in five superlayers and the presented in [2]. Optimization of all reconstruction algo- narrower rectangles display the energy deposition per layer in the other four alternating superlayers. The shower axis is defined rithms was performed using the test beam data. Several from the 3D shower shape. 83 corrections are applied to the data to ensure long term stability of the absolute scales in the constantly varying on- orbit environment. These corrections are performed using Č erenkov detector (RICH) [10]; and an electromagnetic specific samples of particles, predominantly protons. They calorimeter (ECAL) [11]. The figure also shows a high include off-line calibrations of the amplitude response of energy positron of 369 GeV recorded by AMS. AMS TRD, TOF, tracker, and ECAL electronic channels. These operates without interruption on the ISS and is monitored calibrations are performed every 1=4 of an orbit with the continuously from the ground. The timing, location, and attitude of AMS are deter- exception of the alignment of the outer tracker planes 1 and mined by a combination of global positioning system units 9 which is performed every two minutes. The stability of affixed to AMS and to the ISS. The AMS coordinate the electronics response is ensured by onboard calibrations system is concentric with the center of the magnet. The of all channels every half-orbit (∼46 min). The corrections x axis is parallel to the main component of the magnetic also include the alignment of all the AMS detectors and the field and the z axis points vertically. The (y-z) plane is temperature correction of the magnetic field strength. the bending plane. The maximum detectable rigidity over Monte Carlo simulated events are produced using a tracker planes 1 to 9, a lever arm of 3 m, is ∼2 TV. dedicated program developed by AMS based on GEANT- Detector performance, described in detail in [2,4], is steady 4.9.4 [13]. This program simulates electromagnetic and over time. hadronic interactions of particles in the materials of AMS Three main detectors provide clean and redundant and generates detector responses. The digitization of the identification of positrons and electrons with independent signals, including those of the trigger, is simulated accord- suppression of the proton background. These are the TRD ing to the measured characteristics of the electronics. The (above the magnet), the ECAL (below the magnet), and digitized signals then undergo the same reconstruction as the tracker. The TRD and the ECAL are separated by the used for the data. The Monte Carlo samples used in the

121101-3 •20 ] 10 •1 bb s χχ → 3 χχ →tt • 10•21 χχ →µ+µ

> [cm + •

v χχ →τ τ

σ + •

< χχ →W W 10•22 χχ →ZZ

10•23

10•24

10•25

10•26 102 103 104 m χ [GeV]

Fig. 7.8: Limit for ”Dark Matter” particles decaying with a gamma in the ﬁnal state obtained from observing a dSphs Segue 1 [62].

-20 ) 10 -1 .s 3 IceCube-DeepCore 79 2010-2011 10-21 IceCube 59 2009-2010 v> (cm A σ < 10-22 ANTARES 2007-2012

10-23

10-24 MAGIC 2011-2013

10-25 Fermi-LAT 2008-2014 natural scale

10-26 10 102 103 104 MWIMP (GeV)

Figure 11. The 90% C.L. upper limit on the WIMP velocity averaged self-annihilation cross-section, Fig. 7.9: Exclusion< σAv >, as limit a function for of the the WIMP thermally mass in the produced range 25 GeV cross-section< MWIMP < 10 TeVσann for thev self-obtained by + h i ANTARES,annihilation MAGIC channel and WIMP FERMI-LAT WIMP τ τ (using− for ANTARES dSphs). 2007-2012 The (red) dashed with QFit line and corresponds ΛFit to results combined. This is compared to→ the limits from IceCube 59 2009-2010 [47] for the Virgo cluster the expected(black), cross-section Fermi-LAT 2008-2014 from [48 WIMP] for the combined relic density analysis of expectations. 15 satellite galaxies (green)The shaded and areas MAGIC 2011-2013 [49] for Segue 1 (purple), and the IceCube-DeepCore 79 2010-2011 sensitivity [46] correspondfor to the signal GC (blue) interpretation is also shown. Interpreting for ”Dark observed Matter” electron/positron annihilation excesses as dark from matter Pamela, self- Fermi and HESSannihilations, [67]. the orange (PAMELA) and green (PAMELA, Fermi-LAT and H.E.S.S.) ellipses have been obtained [50]. The dashed line indicates the natural scale for which a WIMP is a thermal relic of the early Universe [51].

84 ray limits from Fermi-LAT [48], and MAGIC [49]. ANTARES 2007-2012 data provides the + best upper limit at 90% C.L. on < σAv > for the channel τ τ − from a neutrino telescope. Furthermore, the interpretation [50] of the PAMELA excess as a dark matter self-annihilation signature, after being constrained by Fermi-LAT and H.E.S.S., is rejected at 90% C.L.. All the results are summarised in Tables 1+2 and 3+4, where for each WIMP mass and channel the values of the optimised angular separation, Ψ, the 90% C.L. sensitivity,

Φνµ+¯νµ , computed from the background without signal expectation, the 90% C.L. upper limits, Φ , the acceptance, A¯ (M ) T , and the 90% C.L. upper limits on <σ v > νµ+¯νµ eff WIMP × eff A are presented. In these tables, the results shown in Figures 8 and 10 are highlighted in bold. To evaluate the influence of the dark matter halo profile used in the computation of the < σAv > upper limits, different profiles have been tried. The Einasto profile [53], favoured by recent dark matter-only simulations, is given by:

ρ(r) = ρ exp (2/α)[(r/r )α 1] (5.5) s {− s − } where rs = 21.7 kpc and α = 0.17. The isothermal proﬁle [54] is given by:

– 14 – 70 (a) P7CLEAN R3 1D Eγ = 130.0 GeV 60 nsig = 24.8 evts nbkg = 298.2 evts 50 σ Γ slocal = 4.5 bkg = 2.78 40 30

Events / 5.0 GeV 20 10 0 )

σ 4 Energy (GeV) 2 0 -2 Resid. ( -4 40 60 80 100 120 140 160 180 200 220 Energy (GeV)

Fig. 7.10: Initial diﬀerential gamma distribution obtained with the Fermi-LAT satellite with a 4.5 σ claim for an gamma excess at about 130 GeV [66].

70 (c) P7_REP_CLEAN R3 2D E = 133.0 GeV 60 γ n = 17.8 evts n = 276.2 evts 50 sig bkg s = 3.3 σ Γ = 2.76 40 local bkg 30

Events / 5.0 GeV 20 10 0 )

σ 4 Energy (GeV) 2 0 -2 Resid. ( -4 60 80 100 120 140 160 180 200 220 Energy (GeV)

Fig. 7.11: Diﬀerential gamma distribution obtained with the Fermi-LAT satellite after reprocessing and applying an improved model for the dispersion of the gammas [66].

85 week ending PRL 113, 121101 (2014) PHYSICAL REVIEW LETTERS 19 SEPTEMBER 2014

0.3 0.002 AMS-02 PAMELA Fermi (a) AMS-01 HEAT -1 0.001 TS93 CAPRICE94 0.2 0

Data

Slope [GeV ] -0.001 . Fit c log(E/E0 ) 0.1 -0.002 Positron Fraction 10 102 0.2 0 (b) 100 200 300 400 500 Energy [GeV] 0.1 FIG. 3 (color). Thee+ positron fraction above 10 GeV, where it Fig. 7.12: Positron fraction + as a function of particle energy measured with the e +e− Data AMS-02 experimentbegins to[65]. increase. The present measurement extends the energy range to 500 GeV and demonstrates that, above ∼200 GeV, the Minimal Model Positron Fraction positron fraction is no longer increasing. Measurements from 0 PAMELA [21] (the horizontal blue line is their lower limit), 1 10 102 103 Fermi-LAT [22], and other experiments [17–20] are also shown. Energy [GeV]

FIG. 4 (color). (a) The slope of the positron fraction vs energy −γ − −γ −E=E Φe− ¼ Ce− E e þ CsE s e s ; ð2Þ over the entire energy range (the values of the slope below 4 GeV are off scale). The line is a logarithmic fit to the data above (with E in GeV). A fit of this model to the data with their 30 GeV. (b) The positron fraction measured by AMS and the fit of total errors (the quadratic sum of the statistical and a minimal model (solid curve, see text) and the 68% C.L. range of the fit parameters (shaded). For this fit, both the data and the systematic errors) in the energy range from 1 to model are integrated over the bin width. The error bars are the 500 GeV yields a χ2=d:f: ¼ 36.4=58 and the cutoff 1=E ¼ 1 84 0 58 −1 quadratic sum of the statistical and systematic uncertainties. parameter s . . TeV with the other Horizontally, the points are placed at the center of each bin. parameters having similar values to those in [2], C þ =C − ¼ 0 091 0 001 C =C − ¼ 0 0061 0 0009 e e . . , s e . . , The new measurement shows a previously unobserved γ − − γ þ ¼ −0 56 0 03 γ − − γ ¼ 0 72 0 04 e e . . , and e s . . . behavior of the positron fraction. The origin of this (The same model with no exponential cutoff parameter, behavior can only be ascertained by continuing to collect 1=E i.e., s set to 0, is excluded at the 99.9% C.L. when fit to data up to the TeV region and by measuring the antiproton the data.) The resulting fit is shown in Fig. 4(b) as a solid 86 to proton ratio to high energies. These are among the main curve together with the 68% C.L. range of the fit param- goals of AMS. eters. No fine structures are observed in the data. In our In conclusion, the 10.9 × 106 primary positron and previous Letter, we reported that solar modulation has no electron events collected by AMS on the ISS show that, observable effect on our measured positron fraction, and above ∼200 GeV, the positron fraction no longer exhibits this continues to be the case. an increase with energy. This is a major change in the An analysis of the arrival directions of positrons and behavior of the positron fraction. electrons was presented in [2]. The same analysis was performed including the additional data. The positron to We thank former NASA Administrator Daniel S. Goldin electron ratio remains consistent with isotropy; the upper for his dedication to the legacy of the ISS as a scientific limit on the amplitude of the dipole anisotropy is δ ≤ 0.030 laboratory and his decision for NASA to fly AMS as a DOE at the 95% C. L. for energies above 16 GeV. payload. We also acknowledge the continuous support of Following the publication of our first Letter [2], there the NASA leadership including Charles Bolden, William have been many interesting interpretations [3] with two Gerstenmeier, and Mark Sistilli. AMS is a U.S. DOE popular classes. In the first, the excess of eþ comes from sponsored international collaboration. We are grateful for pulsars. In this case, after flattening out with energy, the the support of Jim Siegrist, Michael Salamon, Dennis positron fraction will begin to slowly decrease and a dipole Kovar, Robin Staffin, Saul Gonzalez, and John O’Fallon anisotropy should be observed. In the second, the shape of of the DOE. We also acknowledge the continuous support the positron fraction is due to dark matter collisions. In this from M.I.T. and its School of Science, Michael Sipser, case, after flattening out, the fraction will decrease rapidly Marc Kastner, Ernest Moniz, Edmund Bertschinger, and with energy due to the finite and specific mass of the dark Richard Milner. We acknowledge support from: CAS, matter particle, and no dipole anisotropy will be observed. NNSF, MOST, NLAA, and the provincial governments Over its lifetime, AMS will reach a dipole anisotropy of Shandong, Jiangsu, and Guangdong, China; CNRS, sensitivity of δ ≃ 0.01 at the 95% C.L. IN2P3, CNES, Enigmass, and the ANR, France, and

121101-7 Chapter 8

Direct ”Dark Matter” detection

Earth is located in the galactic dark halo (see ??), hence Earth based experiments • can search for scattering of Dark Matter particles on standard model particles. Here, we assume a WIMP as Dark Matter candidate and consider only interaction with quarks. As no free quarks exists, the observable eﬀect would be a recoiling nucleus in the detector target. For a direct Dark Matter detection, one need:

– a theoretical prediction of the observation, – a experimental set-up which is in principle able to observer those effects, – a model of the background, i.e. other particles which can mimic the expected Dark Matter effect in the detector, – finally, a comparison of the experimental data with the predications will tell if a significant effect is observed.

The following simpliﬁed equation is useful to motivate the quantities and units used • in this section: N R = Φ σ T (8.1) χ · χN · m Here, R is the rate of recoiling nuclei per target mass m. The number of nuclei in the target is related to the atomic weight A via NT = mNA/A with Avogadro’s constant N . The ﬂux of incident WIMPs is given as Φ = n v with the the A χ χ · χN particle density nχ = ρ0/mχ and the velocity of the WIMP relative to the target nucleus vχN. Finally σχN is the cross section for WIMP-nucleus-scattering, its unit 2 28 2 is usually barn: 1 b = 100 fm = 10− m .

3 WIMPs are non-relativistic today (v /c 10− ), therefore the energy of the recoil- • χ ≈ ing nucleus ER can be given as [68]:

2 2 µ vχN mχmN ER = (1 cos θ), µ = , (8.2) mN − mχ + mN

with the scattering angle θ. Consequently, the transferred momentum is q2 = 2mχER [13].

87 In general the cross section depends on the recoil energy ER and the kinetic energy • 2 of the incident WIMPs Eχ = mχvχN/2. Also vχN is generally not constant, but weighted by a distribution f(~vχN). Consequently, the event rate per unit target mass is modiﬁed to [68]:

vesc ∞ ρ dσ R = dE 0 v f(~v ) χN (E , v )d3~v . (8.3) R m m χN χN dE R χN χN ZET N χ Zvmin R

Here vmin = vmin(ER) is the minimal velocity needed to produce an recoiling nucleus with energy ER, ET is the minimal measurable energy, i.e. the detector threshold, and a WIMP with vesc can escape the gravitation well of the Galaxy. One can divide the parameters in three groups: •

– Astrophysical parameters (ρ0, vχN, vesc) depends on the used halo model and are ﬁxed by astrophysical observations. To compare the results of several direct detection experiments, usually a standard model is used, see section 8.1.

– Detector parameters (A, m, ET) are chosen via the material selection for the target and the experimental set-up.

– Particle parameters (mχ, σχN) depends on the actual model for the Dark Mat- ter particle and its interaction with the nucleus. As result of a direct detection experiment, limits or detection claims are given for σχN(mχ). To compare the results of experiments with diﬀerent target materials, σχN is often scaled to σχn, the WIMP-nucleon-cross section. For spin-independent scattering as discussed in section 8.3.1, experiments could exclude cross sections as low as 10 σ < 7.6 10− pb, see also ﬁg. 8.4. χn · For such low cross section, the expected event rate is in the order of 1 event per ton • per year. To obtain a high statistic of potential scatterings, the exposure, i.e. target mass m time observation time, has to be large. Currently experiments with a target mass in the multi-ton range are planned, e.g. the LZ experiment [69]. Additionally, one aims for low detector thresholds ET.

8.1 Astrophysical parameters

The velocity distribution f(~vχN) is given by the model of the Dark Halo. As stan- • dard model the isotropic isothermal sphere with a WIMP density distribution of 2 3 ρ(r)/ρ 1/r , ρ 0.3 GeVcm− is used [13]. 0 ∝ 0 ≈

WIMPs in the isothermal sphere have a Gaussian velocity distribution f(~vχ) with • respect to the galactic rest frame [68]:

2 1 ~vχ 3 f(~vχ) = exp | | ; σ = v0 (8.4) √2πσ − 2σ2 2 r 1 which is truncated at vesc 544 kms− [13], and the dispersion σ is given by the ≈ 1 local circular speed v 220 kms− [13]. 0 ≈ 88 ae r ouae ihapro foeya.Hwvr ntenrhr eipeetemxmmis maximum the hemisphere northern the on recoil Because However, the year. atmosphere. hence the one and of of flux, muon thickness period 28 detector. the the June a winter, Matter on on with during Dark depends contract modulated which the and are flux in summer rate, muon during recoils produce the expand nuclear and atmosphere on spallation cause nuclear the depends cause can rate can neutrons it recoil muon-induced detector, The Matter this Dark Finally, a near material neutrons. through pass muon a If Signals 8.2 scattering WIMP-nucleon for section cross a cases all In Ge. and of Xe nuclei target for and rate Event 8.1: Fig. σ 1 • • • neapeaemo nue etos omcry rdc iheeg un nteatmosphere. the in muons high-energy produce rays cosmic neutrons: induced muon are example An χ n at’ ri a euti ouae inlwt eido n year one of period a with with signal correlated modulated are a which in processes result depends other and can Although properties orbit WIMP trajectory. the Earth’s Earth’s from independent on are only modulation the of phase on year Each substitutes nfis prxmto,teeetrate event the approximation, first In is velocity. Earth minimal later, the year in a resulting orbit, Half its coordinates. of side galactic opposite in the velocity on maximal its in results the get To h oaino h u rudteGlx ( Galaxy the around Sun the orbit of Earth’s rotation from: the ( contributions Sun has it the coordinates, around galactic the in velocity Earth’s h bouevlecnb prxmt ih[13]: with approximate be can value absolute The 10 = th − [70]. 8 bi sue.Fgr dpe rm[68]. from adapted Figure assumed. is pb relative f (

t log₁₀(dR/dE) / keV⁻¹ kg⁻¹ d⁻¹ ~v 0 R -6 -5 -4 -3 χ ≈ | ) 04 08 100 80 60 40 20 0 ~v sfnto fterci energy recoil the of function as ~v E Ge → ue2 June Earth ≈ | pe fWMswt epc otetre nucleus target the to respect with WIMPs of speed Xe f 4 kms 244 ,tepcla oino h u nteGlx ( Galaxy the in Sun the of motion peculiar the ), ( ~v χ nd N at oe ndrcino h aatcrtto,which rotation, galactic the of direction in moves Earth , = ) − f 1 ( 5kms 15 + ~v χ + 89 R ~v E 6] Here, [68]. ) spootoa to proportional is − 1 ~v cos S , r , E σ | ~v R χn year 1 S m m m o ieetWM masses WIMP diﬀerent for , =10⁻⁸pb r 2 ≈ | χ χ χ π =100 keV =200 keV = 50keV ~v ( 2 kms 220 E E t R − = /keV | t ~v ~v 0 E S ) , | r h eidand period The . − + 1 . [13]. ) ~v S , p f + ( ~v ~v S χ ~v , p N Earth ,and ), 1 ,one ), the , (8.5) m χ is , χ χ χ g χ χ

H q H q Z q g q (a) q χ (b) q (c) q Fig. 8.2: Example Feynman diagrams contributing to: spin-independent WIMP-quark scattering (a) and WIMP-gluon scattering (b), spin-dependent WIMP-quark scattering (c). Based on [71].

usually have not the correct phase t0 which is also related to the movement of the Sun. Therefore, an observation of such a signal is widely considered as a strong indication for elastic scattering of WIMPs from the galactic halo oﬀ target nuclei in the detector.

The diﬀerential rate dR/dER is described by an feature less, exponential falling • spectrum [68]: dR 2µ2v2 ER/Ec 0 e− ,Ec = , (8.6) dER ≈ mN see also ﬁg. 8.1.

– Via the reduced mass µ, the slope of the spectrum is sensitive to the WIMP mass mχ. Therefore, if WIMP induced nuclear recoils are observed, mχ can be deduced from dR/dRR. – Due to the falling slope, a low detection threshold is advantaged to get a high rate. However, at low recoil energies also background from ambient radioactivity and noise of the detector are present which can mimic an exponential falling spectrum.

8.3 WIMP-nucleus cross section

The construction of the diﬀerential WIMP-nucleus cross section dσ/dER occurs on • three diﬀerent scales [13, 68]:

– On the most microscopic scale, the WIMP interacts with the elementary particles of the standard model. In context of direct searches for Dark Matter, mainly the interactions with quarks are considered. – The eﬀective WIMP-nucleon interaction is the next level, it depends on the quark content per nucleon, i.e. per proton and per neutron. – Finally, the WIMP-nucleus cross section is constructed by coherently adding the contributions from the individual nucleons per nucleus.

For the lightest SUSY neutralino as WIMP, the coupling to quarks can be split into • two contributions:

90 (a) (b)

Fig. 8.3: In ﬁrst order Born approximation, the form factor F (q) (a) for spin-independent elastic WIMP-nucleus scattering is the Fourier transformed of the nuclear density ρ(r) (b).

– coupling to the quark spin via axial-vector interactions, called spin-dependent (SD) interactions. – coupling to the quark mass via scalar interactions, called spin-independent (SI) interactions.

Consequently, the diﬀerential cross section is usually split [68] dσ dσ dσ = SD + SI (8.7) dER dER dER and each of the summands can be written as [68]: dσ m x = N 2( ) x = SD SI (8.8) 2 2 σ0,xFx q , , dER 2µ v Here, the form factor F contains the dependence on the transferred momentum 2 q = 2mNER and σ0 describe the case q = 0. As we will see later, for most nuclei the SI interaction is dominant over the SD • interaction.

8.3.1 Spin-independent interactions In case of q = 0: • – On the nuclear scale the cross section depends on the mass of the proton group and the mass of the neutron group. Because mass is a scalar quantity, these are just the number of protons Z times the proton mass mp and the number

91 of neutrons A Z times the neutron mass mn. Here, the masses mp, mn are part of the respective− coupling constants f p, f n [68, 71]: 4µ2 σ = (f pZ + f n(A Z))2 . (8.9) 0,SI π − – The coupling constants can be calculated on the nucleonic level via [68, 71]: f x αs 2 αs = q x + x q x = n p (8.10) fTq fTG , , mx mq 27 mq q=uX,d,s q=cX,b,t f x = 1 f x . (8.11) TG − Tq q=uX,d,s The ﬁrst summand described the coupling of the WIMP to the light quarks (u, d, s). The second summand described the coupling to the gluons via heavy quarks (c, b, t) at loop level. The contribution of quark type q with mass mq to the nucleon mass is given by x which are measured parameters with a fTq precision of 20 % ... 50 % [68]. ≈ s – On the fundamental level, one has to calculate the WIMP-quark couplings αq for scalar interaction by evaluating the corresponding Feynman diagrams [71]. Examples for the coupling to the light quarks and to the gluons are shown in ﬁgs. 8.2a, 8.2b.

For q > 0, the coherence loss has to be taken into account by considering the form • factor FSI(q). In ﬁrst order Born approximation, F is the Fourier transformation of the “scattering centres” [13]. As the scalar interaction couples on the mass, FSI is the Fourier transformation of the nuclear density ρ(r). Several parametrizations of ρ(r) and F (q) are available. Most often direct searches for Dark Matter use the Helm form factor2 [13, 72]:

j1(qrn) (qs)2/2 F (q) = 3 e− . (8.12) qrn

1/3 Here rn is an effective nuclear radius which can be fitted to data as rn = 2.47 fmA + 0.83 fm and s = 0.9 fm is the skin thickness over which ρ(r) falls to zero [13], see fig. 8.3.

Important things to know about the SI cross section: • – In most models f n f p, i.e. the cross section scales like A2 [13, 71]. ≈ – To aim for high event rates, use heavy targets to take advantage from the A2 scaling. – But reject target materials which have a minimum of F 2(q) in the expected q(mχ)-range. – In most models, a potential WIMP signal should feature the A2 scaling. Hence, one can cross check the nature of the signal by using diﬀerent targets.

2 With the spherical Bessel function of the ﬁrst kind j1.

92 – One can compare results from experiments with diﬀerent targets via the WIMP- 2 nucleon cross section, i.e. by the scaled WIMP-nucleus cross section σ0,SI/A [13]. – The uncertainty on the diﬀerential WIMP-nucleus cross section can be di- vided in the uncertainty due to the chosen model and in uncertainties on the experimental parameters, e.g. , x , and ( ). mq fTq F q – The results of experiments directly searching for the spin-independent scattering of WIMPs is ambiguous (see also section 8.4): Most experiments found no indication of dark matter and set limits on the spin-independent cross section. However, the DAMA/LIBRA experiment claimed to observe an annual modulated signal as predicted for the scattering of WIMPs from the dark halo of the Galaxy in Earth based detectors. – The currently leading limit on spin-independent interaction is set by the LUX 10 2 experiment: σ 7.6 10− pb for m = 33 GeV/c [73]. 0,SI ≤ · χ

8.3.2 Spin-dependent interactions

The parametrization of dσSD/dER is similar to the one of dσSI/dER. However, the • spin of the proton group and the spin of the neutron group are calculated diﬀerently, because the spin is a vector quantity.

In case of q = 0: • – On the nuclear scale, the cross section is given as [68, 71]:

32µ2 (J + 1)J σ = G2 (ap S + an S )2 , (8.13) 0,SD π F h pi h ni J 2

with Fermi’s constant GF, the total nuclear spin J, the expectation values of the spin of the proton group Sp and neutron group Sn , and the related coupling constants ap, an respectively.h i The expectationh i values have to be calculated in a appropriated nuclear model. For example, in the single-shell particle model Sp , Sn are equal the total spin of the single unpaired proton, neutron respectivelyh i h [68,i 71, 13]. – The coupling constants can be calculated on the nucleonic level via [68, 71]:

a x αq x a = ∆q , x = n, p. (8.14) √2GF q=uX,d,s The contribution of quark type q to the nucleon spin is given by the measured x parameters ∆q . a – On the fundamental level, one has to calculate the WIMP-quark couplings αq for axial-vector interaction by evaluating the corresponding Feynman diagrams [71]. An example for the coupling to the light quarks are shown in ﬁg. 8.2c.

93 Like for the spin-independent case, the coherence loss has to be taken into account • for q > 0 by considering the form factor FSD(q). The calculation is not as straight forward as in the spin-independent case: The spin-dependent interaction couples to the spin of the proton, neutron group, which is in the shell model equal to the spin of the single unpaired proton, neutron respectively. In this model, FSD(q) can be interpreted as the Fourier transformed of a thin shell that approximate the single outer shell nucleon. However, FSD(q) used in actual experiments is based on more precise calculations which taken into account the contributions from all nucleons [71, 13].

Important things to know about the SD cross section: • – To be sensitive to spin-dependent interaction, the target nucleus needs an unpaired nucleon, e.g. 19F [13]. – There is no scaling with A2 nor is there a scaling with J 2 [71]. – Consequently, for A & 20, the spin-independent contribution to the scattering cross section is dominant over the spin-dependent contribution [13, 68]. – So far no indication for spin-dependent interactions are observed. Conse- quently, direct searches can set limits on the cross section. Assuming a pure coupling to neutrons (an = 1, ap = 0), the leading limit is set by the XENON100 4 2 experiment: σ0,SD 3.5 10− pb at 45 GeV/c [74]. Assuming a pure proton coupling (an =≤ 0, ap· = 1), the best limit from a direct search is the 3 2 one of COUPP σ0,SD 3 10− pb at 30 GeV/c [75, 76]. However, for pure proton coupling, the leading≤ · limit is set by an indirect search3 based on the SuperKamiokande experiment [76].

8.4 Experiments

10 WIMP induced nuclear recoils are a rare process (σ . 10− pb). As discussed in section 8.2, a successful observation of this process would feature an exponential falling spectrum in recoil energy and an annual modulation with a maximum around June 2nd. This signal would be an excess on top of the background, i.e. physical processes which can lead to similar features and consequently can mimic the signal. Classic examples are neutrons which scatter also in target nuclei and can consequently lead to nuclear recoils similar to the WIMP induced ones. Besides mimicking the signal, background can reduce the sensitivity of an experiment also by increasing its dead time. This is the time during which no new measurement is possible because the previous measurement is processed. For example, a gamma-ray may deposit energy in the target via Compton scattering on the electrons of a atom,

3Indirect limits on the WIMP-proton scattering cross section can be derived by searching for neutrinos from WIMP self-annihilation in the Sun: If a WIMP from the dark halo of the Galaxy scatters on a proton in the Sun it may loss enough energy so that its velocity fells below the escape velocity of the Sun and it get captured. After reaching an equilibrium between WIMP capturing and self-annihilation, the ﬂux of neutrinos is proportional to the scattering cross section and not to the self-annihilation cross section [71].

94 0 10 http://cedar.berkeley.edu/plotter Gaitskell,Mandic,Filippini,Speller,Wang

DAMIC 2013

DAMA 2010 -5 CDMSlite 2013 10 CDMS (Si) 2013 CRESST-II 2014 SuperCDSM 2014 Cross-section [pb]

LUX2013

-10 CMSSM fit 10 Strege et al. 2014

0 1 2 3 10 10 10 10 WIMP Mass [GeV/c] Fig. 8.4: Parameter space for spin-independent WIMP-nucleon scattering, i.e. WIMP- nucleon cross section σχn over WIMP mass mχ. The red and green contours show the parameter regions favoured by a dark matter interpretation of the DAMA/LIBRA and CDMS (Si) observations. The experiments DAMIC, CDMSlite, CRESST-II, Super- CDMS, and LUX found no indication for dark matter and set exclusion limits, i.e. exclude the parameter space above the given line. These limits constrained also the parameter region favoured by the constrained minimal super-symmetric standard model (CMSSM). Figure created with the DMTOOLS, references in [77].

95 causing the detector to start a measurement. The Compton scattering is diﬀerent from the searched nuclear recoils, and may be identiﬁed and reject in data analysis after the measurement. However, the detection of the gamma-ray add dead time to the experiment, because during the measurement and processing of the Compton scattering the detector is not sensitive to a potential WIMP-induced nuclear recoil. In summary, a good direct search for dark matter needs: A large exposure 1000 kgyr to compensate for the expected low cross section of • WIMP-nucleus scattering.≈

A low detection threshold . 1 keV, because of the exponential falling recoil spec- • trum.

A low background. The best strategy to achieve this depends on the type and the • source of the background. Common backgrounds are [78]:

– contamination of the detector material itself with radioactive nuclei, e.g. from the U/Th decay chains. To reduce it, only materials with very low contamination, called radiopure materials, are selected for the detector construction. – ambient radioactivity from contamination in the surrounding of the detector, e.g. U/Th in the concrete of the laboratory floor. It is reduced by placing shields around the experiment, e.g. Pb against gamma-rays, or polyethylene against neutrons. – radioactivity produced by cosmic rays in the detector itself or near-by, e.g. muons can produce neutrons via nuclear spallation, hadrons cause cosmo- genic activation of the detector materials, and gamma-rays can be produced as bremsstrahlung. To reduce significant the flux of cosmic rays it is necessary to place the experiment at underground laboratories, where the rock overburden act as shield. The shielding power of the rock overburden is usually given as the thickness of a water column of equivalent shielding power, abbreviated as meter water equivalent (mwe). Usually, deep underground laboratories have a shielding power of several thousand mwe.

The possibility to identify and reject remaining backgrounds, for examples see the • description of the LUX and CRESST experiments below. Currently, several experiments are searching for events induced by scattering of dark matter particles, an overview is given in [76]. Some selected results are shown in ﬁg. 8.4. In the following we will discuss some of the experiments, the technique they use and the results they obtained. As examples for the experiments which obtained null results and set exclusion limits, we select LUX [73] and CRESST [79] which set the leading limits for the lowest cross section and the lowest recoil energy respectively. As example for a potential WIMP observation4 we will discuss the result of DAMA/LIBRA [84, 85].

4There are three more experiments that observed an excess of measured events which may be interpreted as potential WIMP signal: CoGeNT [80], an older result of CRESST [81], and CDMS (Si) [82]. However, more recent analysis indicates that the results of CoGeNT and CRESST can be probably explained by analysis artefacts [83] and background [79], respectively. Concerning the CDMS (Si) excess, the CDMS collaboration does not believe it is signiﬁcant enough to claim a discovery [82].

96 2-6 keV DAMA/NaI ≈ 100 kg DAMA/LIBRA ≈ 250 kg (0.29 ton×yr) (0.87 ton×yr) Residuals (cpd/kg/keV)

Time (day)

Fig. 8.5: The combined data set of DAMA/LIBRA and its predecessor DAMA/NaI contains an annual modulated signal with a maximum around June 2nd. Shown is the residual between the measured event rate and the average as function of time since start of the measurement. Figure taken from [85].

8.4.1 DAMA/LIBRA

DAMA/LIBRA [84, 85] is located at the LNGS underground laboratory in Italy • under 1400 m of rock overburden, which has a shielding power of 3600 mwe.

As target, scintillating NaI crystals are used with a total mass of 240 kg. • ≈ The scintillation light is registered via photomultiplier tubes (PMTs) attached to • each crystal.

Using 1170 kgyr of data, DAMA/LIBRA claimed the observation of a dark matter • induced signal with a signiﬁcance of 8.9σ. Most prominently, this signal featured a sinusoidal modulation with a period of 1 year and a phase equivalent to June 2nd (ﬁg. 8.5), as expected from the interaction of the detector with dark matter particles from the galactic dark halo.

However, DAMA/LIBRA can not determine if these modulated events are caused • by recoiling nuclei, as expected for a standard WIMP scattering, or by recoiling electrons, e.g. by Compton scattering of gamma-rays from background.

Furthermore, to be compatible with the null-results, e.g. of xenon-based LUX, the • hypothetical dark matter particle observed by DAMA/LIBRA is most probably not a standard WIMP, e.g. it may predominately scatter on electrons instead of quarks (leptophilic dark matter [86]). Even with such exotic dark matter candidates, it is doubtful if the DAMA/LIBRA claim will be compatible with the null-results [87].

Consequently, the discovery claim is not generally accepted by the physics commu- • nity.

97 8.4.2 LUX LUX is located at the Sanford Underground laboratory in the US, which has a • shielding power of 4300 mwe.

In its latest publication [73], 250 kg of liquid xenon (LXe) as target were used. For • the successor LZ a multi-ton target mass is planned [69]. LXe has several technical advantages:

– Radioactive contaminations can be easily removed. – It is rather simple to increase the target mass by adding more LXe, whereas experiments like DAMA/LIBRA or CRESST need to manufacture more target crystals. – By dividing the LXe into an outer and an inner volume, the outer volume can shield the inner one which act as target.

LUX can distinguish nuclear recoils from electron recoils by measuring two signals: • scintillation and ionisation. An interaction in the LXe cause the ionisation of the Xe and the emission of scintillation light. An electric ﬁeld extract the produced free electrons out of the LXe in a volume of gaseous Xe where they cause electroluminescence. Both signals, i.e. the prompt scintillation light and the delayed light from electroluminescence, are detected by PMTs. The ratio of ionisation over scintillation depends on the recoiling particle, i.e. recoiling nucleus or recoiling electron.

All observed events can be explained by background. • For SI interactions, LUX set the leading exclusion limit on the WIMP-nucleon cross • 10 2 section: < 7.6 10− pb for a WIMP mass of 33 GeV/c [73]. · 8.4.3 CRESST Like DAMA/LIBRA, CRESST is also located at the LNGS underground laboratory. •

It uses cryogenic crystals of CaWO4 at a temperature of 10 mK. The actual • number and mass of target crystals depends on the measuring≈ campaign, the latest result [79] is based on a single crystal with 250 g. ≈ CRESST can distinguish nuclear and electron recoils by measuring the phonon • excitation of the crystal lattice and the emitted scintillation light. The phonons are detected by transition edge sensors (TES) evaporated on the CaWO4 target. A TES is a thin ﬁlm of W stabilized at a temperature in the transition from superconducting to normal conducting phase. If the temperature of the TES increases due to phonon absorption, it results in an measurable increase of the resistance. The scintillation light is absorbed by a cryogenic light detector which is read-out by a second TES. The ration of phonon signal over scintillation signal depends on the recoiling particle, it is highest for recoiling electrons and decrease with the mass of the recoiling nuclei.

98 CRESST is sensitive to a wide mass range of potential WIMPs because of its multi- • element target. Due to the A2 scaling most of the SI interactions are expected to 2 happen on the W nuclei. However, for WIMPs with mχ . 5 GeV/c the energy of the recoiling W is below the detection threshold. Whereas the scattering on the lighter Ca and O is still above the detection threshold [81, 79].

The latest published result used a detector with a very low detection threshold of • 2 600 eV and stayed sensitive for WIMP-oxygen scattering down to mχ 1 GeV/c [79].≈ ≈

All observed events can be explained by background, and CRESST could set the • leading SI low-mass limit below 3 GeV/c2, see ﬁg. 8.4.

99 100 Chapter 9

”Dark Matter” Production

It is expected that ”Dark Matter” particles weakly couple to standard model particles and for this reason ”Dark Matter” should be able to be produced in high energetic particle collisions and is called ”Dark Matter” production. This fact is used to search for ”Dark Matter” in particle colliders like the LHC at CERN. The energy of the collider needs to be at least high enough to pair produce possible massive ”Dark Matter” particles.

9.1 Production of ”Dark Matter” in particle colliders

”Dark Matter” particles do not interact via electromagnetic interaction (Dark!); • particle physics detectors detect particles via the electromagnetic interaction

”Dark Matter” particle do not leave any signature in the detector, but carry energy • and momentum

clear ”Dark Matter” signature missing energy E • → T two ways to search for ”Dark Matter” E signatures in an experiment • T i) model dependent: look for dedicated decay chains with ”Dark Matter” like particles in the decay (e.g. production of neutralinos) ii) model independent: annihilation of pair of standard model particles to a pair of ”Dark Matter” particle (see Fig. 9.1)

limited sensitivity on lifetime of ”Dark Matter” candidate, ”Dark Matter” candidate • needs to be stable on time scale of the universe 1010 y 3 1017 s ≈ ×7 particle detectors are about 3 m, which corresponds to 10− s (assuming speed of light) 1024 orders of magnitude diﬀerence →

9.1.1 Model dependent searches ”Dark Matter” can be produced as part of a decay chain (see e.g. Fig. 5.5 for • SUSY)

101 Eﬀective Field Theory Propagator γ

q¯ χ

Î χ¯

q Õ

Fig. 9.1: Feynman diagram of ”Dark Matter” particle pair production in a four-point interaction (eﬀective theory) (left) and within a simpliﬁed model (right) [88].

LSP could act as ”Dark Matter” candidate • searches are based on missing transverse energy ET , interpretation in the context • of a certain model

no signal, exclusion of certain ”Dark Matter” candidates within a certain model • (e.g. SUSY neutralino)

9.1.2 Search Strategy ”Dark Matter” do not leave any signature in the detector - how do we know that a • ”Dark Matter” was produced

production of standard model particle at the interaction needed, in order to • – standard model create signal in detector which allows to trigger on the event – visible energy recoil with respect to the energy carried away by the ”Dark Matter” particles

initial state radiation of standard model particle (photon or boson) provides neces- • sary signal see Fig. 9.1

neutrinos do not interact with detectors and leave similar signal like ”Dark Matter” • particles irreducible background which needs to be well understood events with→ Z0 νν decays build largest background contribution → from comparison between expected and observed number of events a signal or a • maximal possible cross-section for ”Dark Matter” production can be determined

102 as an example the number of selected events as a function of missing transverse • energy is shown in Fig. 9.2, γ + Z0 νν build the largest contribution to the background →

ATLAS Data Data 2011 ( s =7 TeV) -1 γ+Z(→ νν ) 3 ∫ L dt = 20.3 fb s = 8 TeV 2 ATLAS Z(→ νν )+γ 10 γ+W(→lν) 10 W/Z+γ W/Z+jet,top,diboson •1 W/Z+jet γ → L dt = 4.6 fb top, γ+jet, multi•jet, diboson 2 +Z( ll) ∫ 10 γ+jet 10 Total background

Events / GeV ADD NLO, M =1.0 TeV, n=2 uncertainty WIMP, D5, mD=10 GeV, M=400 GeV χ * Events / 100 GeV 10 1

•1 1 10

10•2 150 200 250 300 350 400 450 500 550 1.5 1 10•3 0.5

Data/Bkg 150 200 250 300 350 400 450 500 150 200 250 300 350 400 450 500 550 miss miss ET [GeV] ET [GeV]

Fig. 9.2: Number of selected events as a function of missing transverse energy. Events are selected by looking for a high energetic photon and missing transverse energy. The left Figure is from [88] and the right Figure from [89] also showing the number expected events for a certain model.

9.1.3 Effective Field Theory (EFT) estimate expected production cross section of Dark Matter particles in high ener- • getic collisions what happen in the circle of the left diagram of Fig. 9.1? • need for general language to allow possibility to compare results from different • approaches results from dedicated models (SUSY) are useful, but difficult to translate to other • applications EFT is based on the following assumption: we do not care which physics (model) • is responsible for the interaction but we assume it’s very heavy compared to the experiments we perform describe interaction as a contact interaction between ”Dark Matter” particles and • standard model particles identical to the Fermi Theory as a low energy approximation of the electroweak • theory for small energy transfer and large masses of the mediator the propagator can be • replaced by a effective coupling constant: 2 2 g q2 0 g → (9.1) 2 2 2 (q + MMediator) −−−→ MMediator

103 Name Description Operator Spin Dependent (SD) or Spin Independent (SI) 3 scalar D1 mqχχ¯ qq/M¯ SI µ ∗ 2 vector D5 χγ¯ χqγ¯ µq/M SI µ 5 5 ∗ 2 axial-vector D8 χγ¯ γ χqγ¯ µγ q/M SD µν 2∗ tensor D9 χσ¯ χqσ¯ µνq/M SD ∗ Table 9.1: Operators coupling fermion ”Dark Matter” candidates to hadronic matter (see e.g. [90])

the contact interaction can be written in a general formχ ¯Γχq¯Γq/M n with Γ being • a 4 4 matrix ∗ × a list with the most commonly used operators is shown in Tab.: 9.1 •

the eﬀective operators can be used to predict a cross section as function of the • suppression scale M and the ”Dark Matter” mass mχ ∗ in case of no signal the parameter range for a certain operator can be excluded (see • Fig. 9.3)

± σ ATLAS observed limit ( 1 theo) 1100 expected limit EFT model, D5 operator ± σ [GeV] expected 1 * 1000 ± σ -1 expected 2 M s = 8 TeV, ∫Ldt = 20.3 fb truncated, coupling=1 900 truncated, max coupling 800 700 600 90% CL limit on 500 400 300

1 10 102 103 mχ [GeV]

Fig. 9.3: Exclusion of the parameter range M and mχ for the eﬀective operator D5 for data selected with an isolated photon and large∗ missing energy. The parameter space above the red line is excluded [88]. .

104 9.1.4 Relation to Direct and Indirect Dark Matter Detection Experiments results interpreted in the context of EFT can be related to the direct ”Dark Matter” • experiments

the EFT interaction has to be convoluted with nuclear matrix element for a nucleon • at a low momentum interaction

not all EFT operators contribute to ”Dark Matter” direct detection in the limit of • low momentum transfer, some operators are velocity suppressed and are expected 3 to be 10− ≈

the limits obtained in the mχ versus M plane can be translated into the mχ versus • ∗ σSD/SI plane as a function of M and the reduced mass of the ”Dark Matter”- mχmN ∗ nucleon system µχ = (see [90]) mχ+mN

9.1.5 Results of Searches at LHC at the LHC ”Dark Matter” is searched for using diﬀerent particles radiating in the • initial state

the probability (cross section) depends on the radiated particle leading to a diﬀerent • sensitivity

0 searches are performed with: photons, gluons (jets) and weak bosons Z or W ± • in the case of gluons ans missing transverse energy only about 1/3 radiates exactly • one gluon, about 2/3 radiate at least two gluons

the results will be interpreted in terms of EFT operators (see Section 9.1.3), mainly • subdivided in spin-independent (SI) and spin-dependent (SD) interactions

Fig. 9.4 and Fig. 9.5 summarise the results for the searches for ”Dark Matter” at • the LHC

no signal for ”Dark Matter” production observed in LHC • similar sensitivity to ”Dark Matter” for diﬀerent search channels (photon, jet or • W ±)

same experimental data provide diﬀerent sensitivity for diﬀerent operators (or spin- • dependent versus spin-independent)

sensitivity of spin-independent search channels weaker (exception D11) • reduced sensitivity to scalar operator since proportional to initial quark mass; search • for events with large quark mass (top-quark)

105 3 1 10 102 1031 10 102 10

10-30 10-30 90% CL, Spin Independent 90% CL, Spin Dependent -1 10-31 CMS (19.5 fb , 8 TeV), D5 10-31 CMS (19.5 fb-1, 8 TeV), D8 ATLAS (10.5 fb-1, 8 TeV), D5 ATLAS (10.5 fb-1, 8 TeV), D8 -32 -32 10 CMS (5.0 fb-1, 7 TeV), D5 10 CMS (5.0 fb-1, 7 TeV), D8 -1 -33 Mono-jet ATLAS (4.7 fb , 7 TeV), D5 -33 Mono-jet ATLAS (4.7 fb-1, 7 TeV), D8 ] 10 -1 10 2 CMS (19.5 fb , 8 TeV), D11 ATLAS (4.7 fb-1, 7 TeV), D9 -1 10-34 ATLAS (10.5 fb , 8 TeV), D1110 -34 CDF ATLAS (4.7 fb-1, 7 TeV), D11 SIMPLE 2012 -35 -35 - 10 CDF 10 IceCube W+W -36 LUX -36 PICASSO 10 SuperCDMS 10 XENON 100 (Neutron) 10-37 10-37 10-38 10-38 10-39 10-39 10-40 10-40 10-41 10-41 -42 -42

-Nucleon Cross Section [cm 10 10 χ 10-43 10-43 10-44 10-44 10-45 10-45 10-46 10-46 1 10 102 1031 10 102 103 Mχ [GeV] Mχ [GeV]

Fig. 9.4: Limits on spin independent (left) and spin dependent (right) χ-nucleon interaction as a function of mχ obtained with the LHC experiments for events with a mono-jet an missing energy [91] .

sensitivity worse than direct detection experiments in the mass region above 5-10 • GeV

sensitivity better than direct detection experiments in the mass region below 5-10 • GeV, depends on interpretation (operator) and direct detection experiments catch up

9.1.6 Relation between ”Dark Matter” Production and Relic Density the limits obtained can be compared to the ”Dark Matter” relic density observed • in cosmology and to indirect ”Dark Matter” experiments

above 10-100 GeV (depending on the model) the measured thermal cross-section, • assuming a similar mechanism as the production at the LHC, would be too large and additional annihilation mechanism are needed (see Fig. 9.6); the observed ”Dark Matter” density would be too large

9.1.7 Validity of EFT Approach the EFT interpretation assumes a heavy mediator being more massive than the • momentum transfer of the interaction MMediator > Q (see Eq. 9.1)

for MMediator > Q the contact interaction must be resolved and replaced by a UV- • complete theory

106 3 1 10 102 103 1 10 102 10

10-30 10-30 90% CL, Spin Independent 90% CL, Spin Dependent -31 -31 10 10 CMS (19.6 fb-1, 8 TeV), D8 CMS (19.6 fb-1, 8 TeV), D5 -32 -32 CMS (5.0 fb-1, 7 TeV), D8 10 -1 10 CMS (5.0 fb , 7 TeV), D5 ATLAS (4.6 fb-1, 7 TeV), D8 -33 Mono-photon -33 Mono-photon ] 10 -1 10 2 ATLAS (4.6 fb , 7 TeV), D5 ATLAS (4.6 fb-1, 7 TeV), D9 -34 -34 -1 SIMPLE 2012 10 ATLAS (4.6 fb , 7 TeV), D110 - -35 -35 IceCube W+W 10 LUX 10 PICASSO -36 -36 10 SuperCDMS 10 XENON 100 (Neutron) 10-37 10-37 10-38 10-38 10-39 10-39 10-40 10-40 10-41 10-41 -42 -42

-Nucleon Cross Section [cm 10 10 χ 10-43 10-43 10-44 10-44 10-45 10-45 10-46 10-46 1 10 102 103 1 10 102 103 Mχ [GeV] Mχ [GeV]

Fig. 9.5: Limits on spin independent (left) and spin dependent (right) χ-nucleon interaction as a function of mχ obtained with the LHC experiments for events with a mono-photon an missing energy [91] .

the product of the coupling to the standard model particles g and to the ”Dark • SM Matter” particles gχ is assumed to be between 0 < gSM gχ < 4π the width Γ has an impact of the result using an UV-complete theory • Mediator diﬀerent ranges of validity • – for large mediator mass the limit obtained on Λ = M /√gSM gχ corresponds to the limit obtained with the EFT ∗

– at MMediator 2 mχ a resonant behaviour for ”Dark Matter” production ex- ≈ 1 pected (σ (E M )2+Γ2/4 ) leading to an improved limt on Λ ∼ − Mediator – for smaller mediator masses the cross section is suppressed and the limit obtained with the EFT approach is overestimating the limit obtained with a UV-complete theory – Fig. 9.7 shows limit obtained with a UV-complete theory as a function of MMediator

9.1.8 ”Dark Matter” searches with Higgs as a mediator the standard model does not contain a candidate for a ”Dark Matter” particle • no constrained on character of mediating particle • the mediator between the standard model particle and the ”Dark Matter” particle • could be a standard model particle, like the Z0 or the Higgs

107 •1 /s] •18 ATLAS 95% CL s=8 TeV, 20.3 fb 3

•19 2 × ( Fermi•LAT dSphs ( χχ ) → uu, 4 years)

10 Majorana 2 × (HESS 2011 ( χχ ) → qq, Einasto profile)

> [cm •20 Majorana 10 2 × (HESS 2011 ( χχ ) → qq, NFW profile) rel Majorana •21 µ

v D5: γχ χqγ q → (χχ) 10 µ qq µ Dirac •22 D8: γχ γ 5χqγ γ 5q → (χχ) µ Dirac → 10 χχ •23 truncated, coupling = 1

σ 10 truncated, max coupling < 10•24 thermal relic 10•25 10•26 10•27 10•28 10•29 10•30 10•31 1 10 102 103

WIMP mass mχ [GeV]

Fig. 9.6: The limits for ”Dark Matter” production compared to indirect ”Dark Matter” detection experiments and the observed relic density [92] .

a measurement of the branching fraction of the corresponding particle into invisible • particles can be interpreted as a decay into ”Dark Matter”

0 the decay of a Z -boson into ”Dark Matter” particles into mχ mit mχ < 45 GeV • excluded

the interaction between the standard model and the ”Dark Matter” particles could • take place via a Higgs (”Higgs portal”)

the limit on the branching ratio to two invisible particle interpreted for diﬀerent • characters of ”Dark Matter” is summarised in Fig. 9.8

108 3000 2 mχ=500 GeV/c , Γ=M/3 CMS Preliminary 2 mχ=500 GeV/c , Γ=M/10 s = 8 TeV 2

[GeV] Γ π 2500 mχ=500 GeV/c , =M/8

Λ 2 -1 mχ=50 GeV/c , Γ=M/3 ∫L dt = 19.5 fb 2 Γ mχ=50 GeV/c , =M/10 2 mχ=50 GeV/c , Γ=M/8π 2000

1500 90% CL limit on 90% CL 1000

χγ χ γ µ ( µ )(q q) 500 Λ2

0 10-1 1 10 Mediator Mass M [TeV/c2]

Fig. 9.7: Upper limit on interaction scale Λ as a function of the mediator mass [91] . ]

2 -37 10 Higgs-portal Model ATLAS -38 10 s = 7 TeV, ∫ Ldt=4.5 fb-1 10-39 s = 8 TeV, ∫ Ldt=20.3 fb-1 10-40 ZH → ℓℓ + inv. 10-41 10-42 10-43 10-44 10-45 10-46 -47

Nucleon cross section [cm 10

− -48 10 DAMA/LIBRA 3σ CRESST 2σ -49 CDMS 95% CL CoGeNT

DM 10 XENON10 XENON100 -50 LUX ATLAS, scalar DM 10 ATLAS, vector DM ATLAS, fermion DM 10-51 1 10 102 103 DM Mass [GeV]

Fig. 9.8: Limits on ”Dark Matter”-nucleon interaction as a function of ”Dark Matter” mass for the mediator being a Higgs. The limit is obtained for the ”Dark Matter”-particle being a vector, scalar or a fermion particle [93].

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