Light from Dark Matter

D o c t o r a l T h e s i s i n T h e o r e t i c a l P h y s i c s

Light from Dark Matterndash Hidden Dimensions Supersymmetry and Inert Higgs

Michael Gustafsson

Thesis for the degree of Doctor of Philosophy in Theoretical PhysicsDepartment of PhysicsStockholm UniversitySweden

Copyright ccopy Michael Gustafsson Stockholm 2008ISBN 978-91-7155-548-9 (pp indashxv 1ndash184)

Figure 81 91 and 92 have been adopted with authorrsquos permission fromthe sources cited in the figure captions ( ccopyPhys Rev D ccopyAstrophys J andccopyAstron Astrophys respectively)

Typeset in LATEXPrinted by Universitetsservice US AB Stockholm

Cover illustration Artistrsquos impression of two massive Kaluza-Klein parti-cles γ(1) which collide and annihilate into two gamma rays γ Under themagnifying glass it is shown how these massive particles in reality arejust ordinary photons circling a cylindrically-shaped extra dimensionccopy Andreas Hegert and Michael Gustafsson

A b s t r a c t

Recent observational achievements within cosmology and astro-physics have lead to a concordance model in which the energy con-tent in our Universe is dominated by presumably fundamentallynew and exotic ingredients ndash dark energy and dark matter To re-veal the nature of these ingredients is one of the greatest challengesin physics

The detection of a signal in gamma rays from dark matter an-nihilation would significantly contribute to revealing the nature ofdark matter This thesis presents derived imprints in gamma-rayspectra that could be expected from dark matter annihilation Inparticular dark matter particle candidates emerging in models withextra space dimensions extending the standard model to be super-symmetric and introducing an inert Higgs doublet are investigatedIn all these scenarios dark matter annihilation induces sizeable anddistinct signatures in their gamma-ray spectra The predicted sig-nals are in the form of monochromatic gamma-ray lines or a pro-nounced spectrum with a sharp cutoff at the dark matter particlersquosmass These signatures have no counterparts in the expected astro-physical background and are therefore well suited for dark mattersearches

Furthermore numerical simulations of galaxies are studied tolearn how baryons that is stars and gas affect the expected darkmatter distribution inside disk galaxies such as the Milky WayFrom regions of increased dark matter concentrations annihilationsignals are expected to be the strongest Estimations of dark matterinduced gamma-ray fluxes from such regions are presented

The types of dark matter signals presented in this thesis will besearched for with existing and future gamma-ray telescopes

Finally a claimed detection of dark matter annihilation intogamma rays is discussed and found to be unconvincing

L i s t o f A c c o m p a n y i n g P a p e r s

Paper I Cosmological Evolution of Universal Extra Dimensions

T Bringmann M Eriksson and M GustafssonPhys Rev D 68 063516 (2003)

Paper II Gamma Rays from Kaluza-Klein Dark Matter

L Bergstrom T Bringmann M Eriksson and M GustafssonPhys Rev Lett 94 131301 (2005)

Paper III Two Photon Annihilation of Kaluza-Klein Dark Matter

L Bergstrom T Bringmann M Eriksson and M GustafssonJCAP 0504 004 (2005)

Paper IV Gamma Rays from Heavy Neutralino Dark Matter

Paper V Is the Dark Matter Interpretation of the EGRET

Gamma Excess Compatible with Antiproton

Measurements

L Bergstrom J Edsjo M Gustafsson and P SalatiJCAP 0605 006 (2006)

Paper VI Baryonic Pinching of Galactic Dark Matter Haloes

M Gustafsson M Fairbairn and J Sommer-LarsenPhys Rev D 74 123522 (2006)

Paper VII Significant Gamma Lines from Inert Higgs Dark Matter

M Gustafsson E Lundstrom L Bergstrom and J EdsjoPhys Rev Lett 99 041301 (2007)

Published Proceedings Not Accompanying

Paper A Stability of Homogeneous Extra Dimensions

T Bringmann M Eriksson and M GustafssonAIP Conf Proc 736 141 (2005)

Paper B Gamma-Ray Signatures for Kaluza-Klein Dark Matter

L Bergstrom T Bringmann M Eriksson and M GustafssonAIP Conf Proc 861 814 (2006)

C o n t e n t s

Abstract iii

List of Accompanying Papers v

Preface xi

Notations and Conventions xiv

Part I Background Material and Results 1

1 The Essence of Standard Cosmology 3

11 Our Place in the Universe 312 Spacetime and Gravity 4

Special Relativity 4General Relativity 6

13 The Standard Model of Cosmology 814 Evolving Universe 915 Initial Conditions 1116 The Dark Side of the Universe 13

Dark Energy 14Dark Matter 15All Those WIMPs ndash Particle Dark Matter 17

2 Where Is the Dark Matter 19

21 Structure Formation History 1922 Halo Models from Dark Matter Simulations 2023 Adiabatic Contraction 22

A Simple Model for Adiabatic Contraction 22Modified Analytical Model 23

24 Simulation Setups 2425 Pinching of the Dark Matter Halo 2526 Testing Adiabatic Contraction Models 2727 Nonsphericity 28

Axis Ratios 29Alignments 30

28 Some Comments on Observations 3129 Tracing Dark Matter Annihilation 32

viii Contents

Indirect Dark Matter Detection 34210 Halo Substructure 36

3 Beyond the Standard Model Hidden Dimensions and More 39

31 The Need to Go Beyond the Standard Model 3932 General Features of Extra-Dimensional Scenarios 4133 Modern Extra-Dimensional Scenarios 4534 Motivations for Universal Extra Dimensions 47

4 Cosmology with Homogeneous Extra Dimensions 49

41 Why Constants Can Vary 4942 How Constant Are Constants 5043 Higher-Dimensional Friedmann Equations 5244 Static Extra Dimensions 5345 Evolution of Universal Extra Dimensions 5546 Dimensional Reduction 5647 Stabilization Mechanism 58

5 Quantum Field Theory in Universal Extra Dimensions 61

51 Compactification 6152 Kaluza-Klein Parity 6253 The Lagrangian 6354 Particle Propagators 7355 Radiative Corrections 7456 Mass Spectrum 76

6 Kaluza-Klein Dark Matter 79

61 Relic Density 7962 Direct and Indirect Detection 83

Accelerator Searches 84Direct Detection 84Indirect Detection 85

63 Gamma-Ray Signatures 86Gamma-Ray Continuum 87Gamma Line Signal 91

64 Observing the Gamma-Ray Signal 95

7 Supersymmetry and a New Gamma-Ray Signal 99

71 Supersymmetry 99Some Motivations 100The Neutralino 101

72 A Neglected Source of Gamma Rays 101Helicity Suppression for Fermion Final States 102Charged Gauge Bosons and a Final State Photon 103

Contents ix

8 Inert Higgs Dark Matter 109

81 The Inert Higgs Model 109The New Particles in the IDM 112Heavy Higgs and Electroweak Precision Bounds 113More Constraints 115

82 Inert Higgs ndash A Dark Matter Candidate 11783 Gamma Rays 119

Continuum 119Gamma-Ray Lines 119

9 Have Dark Matter Annihilations Been Observed 125

91 Dark Matter Signals 12592 The Data 12693 The Claim 12894 The Inconsistency 130

Disc Surface Mass Density 130Comparison with Antiproton Data 132

95 The Status to Date 137

10 Summary and Outlook 139

A Feynman Rules The UED model 143

A1 Field Content and Propagators 143A2 Vertex Rules 144

Bibliography 159

Part II Scientific Papers 185

P r e f a c e

This is my doctoral thesis in Theoretical Physics During my years as a PhDstudent I have been working with phenomenology This means I live in theland between pure theorists and real experimentalists ndash trying to bridge thegap between them Taking elegant theories from the theorist and makingfirm predictions that the experimentalist can detect is the aim My researcharea has mainly been dark matter searches through gamma-ray signals Theultimate aim in this field is to learn more about our Universe by revealingthe nature of the dark matter This work consists of quite diverse fieldsFrom Einsteinrsquos general relativity and the concordance model of cosmologyto quantum field theory upon which the standard model of particle physics isbuilt as well as building bridges that enable comparison of theory with exper-imental data I can therefore honestly say that there are many subjects onlytouched upon in this thesis that in themselves deserve much more attention

An Outline of the Thesis

The thesis is composed of two parts The first introduces my research fieldand reviews the models and results found in the second part The second partconsists of my published scientific papers

The organization for part one is as follows Chapter 1 introduces theessence of modern cosmology and discusses the concept of dark energy anddark matter Chapter 2 contains a general discussion of the dark matter dis-tribution properties (containing the results of Paper VI) and its relevancefor dark matter annihilation signals Why there is a need to go beyond thestandard model of particle physics is then discussed in Chapter 3 This isfollowed by a description of general aspects of higher-dimensional theoriesand the universal extra dimension (UED) model is introduced In Chapter 4a toy model for studying cosmology in a multidimensional universe is brieflyconsidered and the discussion in Paper I is expanded Chapter 5 then fo-cuses on a detailed description of the field content in the UED model whichsimultaneously gives the particle structure of the standard model After ageneral discussion of the Kaluza-Klein dark matter candidate from the UEDmodel special attention is placed on the results from Papers II-III in Chap-ter 6 This is then followed by a brief introduction to supersymmetry and theresults of Paper IV in Chapter 7 The inert Higgs model its dark mattercandidate and the signal found in Paper VII are then discussed in Chap-ter 8 Chapter 9 reviews Paper V and a claimed potential detection of adark matter annihilation signal before Chapter 10 summarizes this thesis

xii Preface

For a short laymanrsquos introduction to this thesis one can read Sections 11and 16 on cosmology (including Table 11) together with Section 31 andlarge parts of Section 32ndash33 on physics beyond the standard model Thiscan be complemented by reading the preamble to each of the chapters and thesummary in Chapter 10 ndash to comprise the main ideas of the research resultsin the accompanying papers

My Contribution to the Accompanying Papers

As obligated let me say some words on my contribution to the accompanyingscientific papers

During my work on Papers I-IV I had the privilege of closely collaborat-ing with Torsten Bringmann and Martin Eriksson This was a most demo-cratic collaboration in the sense that all of us were involved in all parts ofthe research Therefore it is in practice impossible to separate my work fromtheirs This is also reflected in the strict alphabetic ordering of author namesfor these papers If one should make one distinction in Paper III I wasmore involved in the numerical calculations than in the analytical (althoughmany discussions and crosschecks were made between the two approaches)In Paper V we scrutinized the claim of a potential dark matter detection byde Boer et al [1] Joakim Edsjo and I independently implemented the darkmatter model under study both into DarkSUSY and other utilized softwaresI did the first preliminary calculations of the correlation between gamma-rayand antiproton fluxes in this model which is our main result in the paper Iwas also directly involved in most of the other steps on the way to the finalpublication and wrote significant parts of the paper For Paper VI Mal-colm Fairbairn and I had similar ideas on how we could use Jesper-SommerLarsenrsquos galaxy simulation to study the dark matter distribution I wroteparts of the paper although not the majority Instead I did many of thefinal calculations had many of the ideas for the paper and produced all thefigures (except Figure 4) for the paper Regarding Paper VII I got involvedthrough discussions concerning technical problems that appeared in imple-menting the so-called inert Higgs model into FeynArts I found the simplicityof the inert Higgs model very intriguing and contributed many new ideas onhow to proceed with the paper performed a majority of the calculations andwrote the main part of the manuscript

Acknowledgments

Many people have influenced both directly and indirectly the outcome of thisthesis

Special thanks go to my supervisor Professor Lars Bergstrom who overthe last years has shown generous support not only financial but also for hissharing of fruitful research ideas

My warmest thanks go to Torsten Bringmann and Martin Eriksson whomade our collaboration such a rewarding and enjoyable experience both sci-entifically and personally Likewise I want to thank Malcolm Fairbairn andErik Lundstrom for our enlightening collaborations Many thanks also to mycollaborators Jesper-Sommer Larsen and Pierre Salati Not the least I wantto thank my collaborator Joakim Edsjo who has often been like a supervisorto me His efficiency and sense of responsibility are truly invaluable

Many of the people here at the physics department have had great impacton my life during my years as a graduate student and many have become myclose friends Thank you Soren Holst Joachim Sjostrand Edvard MortsellChristofer Gunnarsson Mia Schelke Alexander Sellerholm and Sara RydbeckLikewise I want to thank Kalle Jakob Fawad Jan Emil Asa Maria Johanand all other past and present corridor members for our many interestingand enjoyable discussions The long lunches and dinners movie nights funparties training sessions and the many spontaneously cheerful moments hereat Fysikum have made my life much better

I am also very grateful to my dear childhood friend Andreas Hegert forhelping me to produce the cover illustration

All those not mentioned by name here you should know who you are andhow important you have been I want you all to know that I am extremelygrateful for having had you around and for your support in all ways duringall times I love you deeply

Michael Gustafsson

Stockholm February 2008

N o t a t i o n s a n d C o n v e n t i o n s

A timelike signature (+minusminus middot middot middot ) is used for the metric except in Chapters 1and 4 where a spacelike signature (minus++ middot middot middot ) is used This reflects mychoice of following the convention of Misner Thorne and Wheeler [2] fordiscussions regarding General Relativity and Peskin and Schroeder [3] forQuantum Field Theory discussions

In a spacetime with d = 4 + n dimensions the spacetime coordinates aredenoted by x with capital Latin indices MN isin 0 1 dminus 1 if it is ahigher dimensional spacetime with n gt 0 Four-dimensional coordinates aregiven by a lower-case x with Greek indices micro ν (or lower-case Latin lettersi j for spacelike indices) Extra-dimensional coordinates are denoted byyp with p = 1 2 n That is

xmicro equiv xM (micro = M = 0 1 2 3)

xi equiv xM (i = M = 1 2 3)

yp equiv xM (p = M minus 3 = 1 2 n)

In the case of one extra dimension the notation is slightly changed so thatspacetime indices take the value 0 1 2 3 5 and the coordinate for the extradimension is denoted y (equiv y1 equiv x5) Higher-dimensional quantities such ascoordinates coupling constants and Lagrangians will frequently be denotedwith a lsquohatrsquo (as in x λ L) to distinguish them from their four-dimensionalanalogs (x λ L)

Einsteinrsquos summation convention is always implicitly understood in ex-pressions ie one sums over any two repeated indices

The notation ln is reserved for the natural logarithm (loge) whereas logis intended for the base-10 logarithm (log10)

Natural units where c = ~ = kB = 1 are used throughout this thesisexcept occasionally where ~ and c appear for clarity

Useful Conversion Factors (c = ~ = kB = 1)

1 GeVminus1 = 65822 middot 10minus25 s = 19733 middot 10minus14 cm1 GeV = 16022 middot 10minus3 erg = 17827 middot 10minus24 g = 11605 middot 1013 K1 barn (1 b) = 1012 pb = 10minus24 cm2

1 parsec (1 pc) = 32615 light yr = 20626middot105 AU = 30856 middot 1018 cm

Useful Constants and Parameters

Speed of light c equiv 299792458 middot 1010 cm sminus1

Planckrsquos constant ~ = h2π = 65821middot10minus25 GeV s~c = 197 middot 10minus14 GeV cm

Boltzmannrsquos const kB = 81674 middot 10minus14 GeV Kminus1

Newtonrsquos constant G = 66726 middot 10minus8 cm3 gminus1 sminus2

Planck mass Mpl equiv (~cG)12 = 12211 middot 1019 GeV cminus2

= 2177 middot 10minus5 gElectron mass me = 51100 middot 10minus4 GeV cminus2

Proton mass mp = 93827 middot 10minus1 GeV cminus2

Earth mass Moplus = 3352 middot 1054 GeV cminus2 = 5974 middot 1030 gSolar mass M⊙ = 1116 middot 1057 GeV cminus2 = 1989 middot 1033 gHubble constant H0 = 100h km sminus1 Mpcminus1 (h sim 07)Critical density ρc equiv 3H2

08πG= 10540h2 middot 10minus5 GeV cminus2 cmminus3

= 18791h2 middot 10minus29 g cmminus3

= 27746h2 middot 10minus7 M⊙ pcminus3

Acronyms Used in This Thesis

BBN Big Bang NucleosynthesisCDM Cold Dark MatterCERN Conseil Europeen pour la Recherche Nucleaire

(European Council for Nuclear Research)CMB Cosmic Microwave BackgroundDM Dark MatterEGRET Energetic Gamma Ray Experiment TelescopeEWPT ElectroWeak Precision TestsFCNC Flavor Changing Neutral CurrentFLRW Friedmann Lemaıtre Robertson WalkerGLAST Gamma-ray Large Area Space TelescopeIDM Inert Doublet ModelKK Kaluza-KleinLEP Large Electron-Positron ColliderLHC Large Hadron ColliderLIP Lightest Inert ParticleLKP Lightest Kaluza-Klein ParticleMSSM Minimal Supersymmetric Standard Model(s)NFW Navarro Frenk WhitePhD Doctor of PhilosophySM Standard Model (of particle physics)UED Universal Extra DimensionWIMP Weakly Interacting Massive ParticleWMAP Wilkinson Microwave Anisotropy Probe

P a r t I

Background Material and Results

C h a p t e r

The Essence ofStandard Cosmology

The Universe is a big place filled with phenomena far beyond everyday expe-rience The scientific study of the properties and evolution of our Universe asa whole is called cosmology This chapterrsquos aim is to give a primary outlineof modern cosmology present basic tools and notions and introduce the darkside of our Universe the concepts of dark energy and dark matter

11 Our Place in the Universe

For a long time Earth was believed to be in the center of the Universe Laterlowast

it was realized that the motion of the Sun planets and stars in the night skyis more simply explained by having Earth and the planets revolving aroundthe Sun instead The Sun in turn is just one among about 100 billion otherstars that orbit their mutual mass center and thereby form our own MilkyWay Galaxy In a clear night sky almost all of the shining objects we cansee by the naked eye are stars in our own Galaxy but with current telescopesit has been inferred that our observable Universe also contains the stars inhundreds of billions of other galaxies

The range of sizes and distances to different astronomical objects is hugeStarting with our closest star the Sun from which it takes the light abouteight minutes to reach us here at Earth This distance can be compared tothe distance around Earth that takes mere one-tenth of a second to travelat the speed of light Yet these distances are tiny compared to the size ofour galactic disk ndash 100 000 light-years across ndash and the distance to our nearest(large) neighbor the Andromeda galaxy ndash 2 million light-years away Still thisis nothing compared to cosmological distances Our own Milky Way belongsto a small group of some tens of galaxies the Local Group which in turn

lowast The astronomer Nicolaus Copernicus (1473-1543) was the first to formulate the helio-centric view of the solar system in a modern way

4 The Essence of Standard Cosmology Chapter 1

belongs to a supercluster the Virgo supercluster including about one hundredof such groups of clusters The superclusters are the biggest gravitationallybound systems and reach sizes up to some hundred million light-years Noclusters of superclusters are known but the existence of structures larger thansuperclusters is observed in the form of filaments of galaxy concentrationsthread-like structures with a typical length scale of up to several hundredmillion light-years which form the boundaries between seemingly large voidsin the Universe

This vast diversity of structures would make cosmology a completely in-tractable subject if no simplifying characteristic could be used Such a desiredsimplifying feature is found by considering even larger scales at which the Uni-verse is observed to be homogeneous and isotropic That is the Universe looksthe same at every point and in every direction Of course this is not true indetail but only if we view the Universe without resolving the smallest scalesand lsquosmears outrsquo and averages over cells of 108 light-years or more acrossThe hypothesis that the Universe is spatially isotropic and homogeneous atevery point is called the cosmological principle and is one of the fundamentalpillars of standard cosmology A more compact way to express the cosmologi-cal principle is to say that the Universe is spatially isotropic at every point asthis automatically implies homogeneity [4] The cosmological principle com-bined with Einsteinrsquos general theory of relativity is the foundation of moderncosmology

12 Spacetime and Gravity

Since the study of the evolutionary history of our Universe is based on Ein-steinrsquos general theory of relativity let us briefly go through the basic conceptsused in this theory and in cosmology As the name suggests general relativityis a generalization of another theory namely special relativity The specialtheory of relativity unifies space and time into a flat spacetime and the gen-eral theory of relativity in turn unifies special relativity with Newtonrsquos theoryof gravity

Special Relativity

What does it mean to unify space and time into a four-dimensional space-time theory Obviously already Newtonian mechanics involved three spatialdimensions and a time parameter so why not already call this a theory ofa four-dimensional spacetime The answer lies in which dimensions can belsquomixedrsquo in a meaningful way For example in Newtonian mechanics and ina Cartesian coordinate system defined by perpendicular directed x y and zaxes the Euclidian distance ds between two points is given by Pythagorasrsquotheorem

ds2 = dx2 + dy2 + dz2 (11)

Section 12 Spacetime and Gravity 5

However a rotation or translation into other Cartesian coordinate systems(xprime yprime zprime) could equally well be used and the distance would of course beunaltered

ds2 = dxprime2 + dyprime2 + dzprime2 (12)

This invariance illustrates that the choice of axes and labels is not importantin expressing physical distances The coordinate transformations that keepEuclidian distances intact are the same that keep Newtonrsquos laws of physicsintact and they are called the Galileo transformations The reference frameswhere the laws of physics take the same form as in a frame at rest are calledinertial frames In the Newtonian language these are the frames where thereare no external forces and particles remain at rest or in steady rectilinearmotion

The Galileo transformations do not allow for any transformations that mixspace and time on the contrary there is an absolute time that is indepen-dent of spatial coordinate choice This however is not the case in anotherclassical theory ndash electrodynamics The equations of electrodynamics are notform-invariant under Galileo transformations Instead there is another classof coordinate transformations that mix space and time and keep the lawsof electrodynamics intact This new class of transformations called Lorentztransformations leaves another interval dτ between two spacetime points in-variant This spacetime interval is given by

dτ2 = minusc2dt2 + dx2 + dy2 + dz2 (13)

where c2 is a constant conversion factor between three-dimensional Cartesianspace and time distances That is Lorentz transformations unifies space andtime into a four-dimensional spacetime (t x y z) where space and time canbe mixed as long as the interval dτ in Eq (13) is left invariant Taking thisas a fundamental property and say that all laws of physics must be invariantwith respect to transformations that leave dτ invariant is the lesson of specialrelativity

Let me set up the notation that will be used in this thesis x0 = ct x1 = xx2 = y and x3 = z The convention will also be that Greek indices run from0 to 3 so that four-vectors typically look like

(dx)α = (cdt dx dy dz) (14)

Defining the so-called Minkowski metric

ηαβ =

minus1 0 0 00 1 0 00 0 1 00 0 0 1

allows for a very compact form for the interval dτ

dτ2 =

αβ=0

ηαβdxαdxβ = ηαβdx

αdxβ (16)

In the last step the Einstein summation convention was used Repeated in-dices appearing both as subscripts and superscripts are summed over Thereis one important comment to be made regarding the sign convention on ηαβused in this thesis In Chapters 1 and 4 the sign convention of Eq (15) isadopted (as is the most common convention in the general relativity commu-nity) whereas in all other chapters ηαβ will be defined to have the oppositeoverall sign (as is the most common convention within the particle physicscommunity)

In general the allowed infinitesimal transformations in special relativityare rotations boosts and translations These form a ten-parameter non-abelian group called the Poincare group

The invariant interval (14) and thus special relativity can be deducedfrom the following two postulates [5]

1 Postulate of relativity The laws of physics have the same form inall inertial frames

2 Postulate of a universal limiting speed In every inertial framethere is a finite universal limiting speed c for all physical entities

Experimentally and in agreement with electrodynamics being the theory oflight the limiting speed c is equal to the speed of light in a vacuum Today cis defined to be equal to 299792458times108 ms Another way to formulate thesecond postulate is to say that the speed of light is finite and independent ofthe motion of its source

General Relativity

In special relativity nothing can propagate faster than the speed of light soNewtonrsquos description of gravity as an instant force acting between masseswas problematic Einsteinrsquos way of solving this problem is very elegant Fromthe observation that different bodies falling in the same gravitational fieldacquire the same acceleration he postulated

The equivalence principle There is no difference between grav-itational and inertial masses (this is called the weak equivalenceprinciple) Hence in a frame in free fall no local gravitationalforce phenomena can be detected and the situation is the sameas if no gravitational field was present Elevate this to includeall physical phenomena the results of all local experiments areconsistent with the special theory of relativity (this is called thestrong equivalence principle)

From this postulate you can derive many fundamental results of general rel-ativity For example that time goes slower in the presence of a gravitationalfield and that light-rays are bent by gravitating bodies Due to the equiva-lence between gravitational and inertial masses an elegant purely geometricalformulation of general relativity is possible All bodies in a gravitational fieldmove on straight lines called geodesics but the spacetime itself is curved andno gravitational forces exist

We saw above that intervals in a flat spacetime are expressed by meansof the Minkowski metric (Eq 15) In a similar way intervals in a curvedspacetime can be express by using a generalized metric gmicroν(x) that describesthe spacetime geometry The geometrical curvature of spacetime can be con-densed into what is called the Riemann tensor which is constructed from themetric gmicroν(x) as follows

Rαβmicroν equiv partmicroΓ

αβν minus partνΓ

αβmicro + ΓασmicroΓ

σβν minus ΓασνΓ

σβmicro (17)

Γαmicroν =1

2gαβ(partνgβmicro + partmicrogβν minus partβgmicroν) (18)

and gmicroν(x) is the inverse of the metric gmicroν(x) ie gmicroσ(x)gνσ(x) = δmicroν Having decided upon a description of gravity that is based on the idea of

a curved spacetime we need a prescription for determining the metric in thepresence of a gravitational source What is sought for is a differential equationin analogy with Newtonrsquos law for the gravitational potential

nablaφ = 4πGρ (19)

where ρ is the mass density and G Newtonrsquos constant If we want to keepmatter and energy as the gravitational source and avoid introducing anypreferred reference frame the natural source term is the energy-momentumtensor Tmicroν (where the T00 component is Newtonrsquos mass density ρ) A second-order differential operator on the metric set to be proportional to Tmicroν canbe constructed from the Riemann curvature tensor

Rmicroν minus1

2R gmicroν minus Λgmicroν =

c4Tmicroν (110)

where Rmicroν equiv Rαmicroαν and R equiv gαβRαβ These are Einsteinrsquos equations of gen-

eral relativity including a cosmological constant Λ-termdagger The left hand-sideof Eq (110) is in fact uniquely determined if it is restricted to be diver-gence free (ie local source conservation nablamicroT

microν = 0) be linear in the secondderivatives of the metric and free of higher derivatives and vanish in a flatspacetime [2] The value of the proportionality constant 8πG in Eq (110) is

dagger Independently and in the same year (1915) David Hilbert derived the same field equa-tions from the action principle (see Eq (44)) [6]

obtained from the requirement that Einsteinrsquos equations should reduce to theNewtonian Eq (19) in the weak gravitational field limit

In summary within general relativity matter in free fall moves on straightlines (geodesics) in a curved spacetime In this sense it is the spacetime thattells matter how to move Matter (ie energy and pressure) in turn is thesource of curvature ndash it tells spacetime how to curve

13 The Standard Model of Cosmology

We now want to find the spacetime geometry of our Universe as a whole iea metric solution gmicroν(x

α) to Einsteinrsquos equations Following the cosmologicalprinciple demanding a homogenous and isotropic solution you can show thatthe metric solution has to take the so-called Friedmann Lemaıtre RobertsonWalker (FLRW) form In spherical coordinates r θ φ t this metric is givenby

dτ2 = minusdt2 + a(t)2[

1 minus kr2+ r2

dθ2 + sin2 θdφ2)

where a(t) is an unconstrained time-dependent function called the scale fac-tor and k = minus1+1 0 depending on whether space is negatively curvedpositively curved or flat respectively Note that natural units where c isequal to 1 have now been adopted

To present an explicit solution for a(t) we need to further specify theenergy-momentum tensor Tmicroν in Eq (110) With the metric (111) theenergy-momentum tensor must take the form of a perfect fluid In a comovingframe ie the rest frame of the fluid the Universe looks perfectly isotropicand the energy-momentum tensor has the form

T microν = diag(ρ p p p) (112)

where ρ(t) represents the comoving energy density and p(t) the pressure ofthe fluid Einsteinrsquos equations (110) can now be summarized in the so-calledFriedmann equation

H2 equiv(

=8πGρ+ Λ

3minus k

a2(113)

dt(ρa3) = minusp d

dta3 (114)

The latter equation should be compared to the standard thermodynamicalequation expressing that the energy change in a volume V = a3 is equal tothe pressure-induced work that causes the volume change Given an equationof state p = p(ρ) Eq (114) determines ρ as a function of a Knowingρ(a) a solution a(t) to the Friedmann equation (113) can then be completely

Section 14 Evolving Universe 9

specified once boundary conditions are given This a(t) sets the dynamicalevolution of the Universe

By expressing all energy densities in units of the critical density

ρc equiv3H2

8πG (115)

the Friedmann equation can be brought into the form

1 = Ω + ΩΛ + Ωk (116)

where Ω equiv ρρc

ΩΛ equiv Λ8πGρc

and Ωk equiv minuska2 The energy density fraction Ω

is often further split into the contributions from baryonic matter Ωb (ieordinary matter) cold dark matter ΩCDM radiationrelativistic matter Ωrand potentially other forms of energy For these components the equation ofstate is specified by a proportionality constant w such that p = wρ Specifi-cally w asymp 0 for (non-relativistic) matter w = 13 for radiation and if one soprefers the cosmological constant Λ can be interpreted as an energy densitywith an equation of state w = minus1 We can explicitly see how the energy den-sity of each componentDagger depends on the scale factor by integrating Eq (114)which gives

ρi prop aminus3(1+wi) (117)

14 Evolving Universe

In 1929 Edwin Hubble presented observation that showed that the redshiftin light from distant galaxies is proportional to their distance [7] Redshiftdenoted by z is defined by

1 + z equiv λobs

λemit (118)

where λemit is the wavelength of light at emission and λobs the wavelengthat observation respectively In a static spacetime this redshift would pre-sumably be interpreted as a Doppler shift effect light emitted from an objectmoving away from you is shifted to longer wavelengths However in agree-ment with the cosmological principle the interpretation should rather be thatthe space itself is expanding As the intergalactic space is stretched so isthe wavelength of the light traveling between distant objects For a FLRWmetric the following relationship between the redshift and the scale factorholds

1 + z =a(tobs)

a(temit) (119)

The interpretation of Hubblersquos observation is therefore that our Universe isexpanding

Dagger Assuming that each energy component separately obeys local lsquoenergy conservationrsquo

Proper distance is the distance we would measure with a measuring tapebetween two space points at given cosmological time (ie

dτ ) In practicethis is not a measurable quantity and instead there are different indirect waysof measuring distances The angular distance dA is based on the flat spacetimenotion that an object of known size D which subtends a small angle δθ is ata distance dA equiv Dδθ The luminosity distance dL instead makes use of thefact that a light source appears weaker the further away it is and is definedby

dL equivradic

4πL (120)

where S is the intrinsic luminosity of the source and L the observed luminosityIn flat Minkowski spacetime these measures would give the same result butin an expanding universe they are instead related by dL = dA(1 + z)2 For anobject at a given redshift z the luminosity distance for the FLRW metric isgiven by

dL = a0(1 + z)f

dzprime

H(zprime)

f(x) equiv

sinh(x) if k = minus1x if k = 0sin(x) if k = +1

Here a0 represents the value of the scale factor today and H(z) is the Hubbleexpansion at redshift z

H(z) = H0

Ω0i (1 + z)minus3(1+wi) (122)

where H0 is the Hubble constant and Ω0i are the energy fraction in different

energy components todayIt is often convenient to define the comoving distance the distance between

two points as it would be measured at the present time This means that theactual expansion is factored out and the comoving distance stays constanteven though the Universe expands A physical distance d at redshift z corre-sponds to the comoving distance (1 + z) middot d

By measuring the energy content of the Universe at a given cosmologicaltime eg today we can by using Eq (117) derive the energy densities atother redshifts By naıvly extrapolating backwards in time we would eventu-ally reach a singularity when the scale factor a = 0 This point is sometimespopularly referred to as the Big Bang It should however be kept in mind thatany trustworthy extrapolation breaks down before this singularity is reachedndash densities and temperatures will become so high that we do not have anyadequately developed theories to proceed with the extrapolation A better(and the usual) way to use the term Big Bang is instead to let it denote the

Section 15 Initial Conditions 11

early stage of a very hot dense and rapidly expanding Universe A brieftimeline for our Universe is given in Table 11

This Big Bang theory shows remarkably good agreement with cosmologi-cal observations The most prominent observational support of the standardcosmological model comes from the agreement with the predicted abundanceof light elements formed during Big Bang nucleosynthesis (BBN) and the ex-istence of the cosmic microwave background radiation In the early Universenumerous photons which were continuously absorbed re-emitted and inter-acting constituted a hot thermal background bath for other particles Thiswas the case until the temperature eventually fell below about 3 000 K At thistemperature electrons and protons combine to form neutral hydrogen (the so-called recombination) which then allows the photons to decouple from theprimordial plasma These photons have since then streamed freely throughspace and constitute the so-called cosmic microwave background (CMB) radi-ation The CMB photons provides us today with a snapshot of the Universeat an age of about 400 000 years or equivalently how the Universe looked137 billion years ago

15 Initial Conditions

The set of initial conditions required for this remarkable agreement betweenobservation and predictions in the cosmological standard model is howeverslightly puzzling The most well-known puzzles are the flatness and horizonproblems

If the Universe did not start out exactly spatial flat the curvature tendsto become more and more prominent That means that already a very tinydeviation from flatness in the early Universe would be incompatible with theclose to flatness observed today This seemingly extreme initial fine-tuning iswhat is called the flatness problem

The horizon problem is related to how far information can have traveledat different epochs in the history of our Universe There is a maximal distancethat any particle or piece of information can have propagated since the BigBang at any given comoving time This defines what is called the particlehorizonsect

dH(t) =

dtprime

a(tprime)= a(t)

int r(t)

drprimeradic1 minus krprime2

That is in the past a much smaller fraction of the Universe was causallyconnected than today For example assuming traditional Big Bang cosmologythe full-sky CMB radiation covers about 105 patches that have never been incausal contact Despite this the temperature is the same across the whole

sect There is also the notion of event horizon in cosmology which is the largest comovingdistance from which light can ever reach the observer at any time in the future

Table 11 The History of the Universe

Time = 10minus43 s Size sim 10minus60times today Temp = 1032 K

The Planck era Quantum gravity is important current theories are in-adequate and we cannot go any further back in time

Time = 10minus35 s Size = 10minus54rarrminus26times today Temp = 1026rarr0rarr26 KInflation A conjectured period of accelerating expansion an inflaton fieldcauses the Universe to inflate and then decays into SM particles

Time = 10minus12 s Size = 10minus15times today Temp = 1015 K

Electroweak phase transition Electromagnet and weak interactions be-come distinctive interactions below this temperature

Quark-gluon phase transition Quarks and gluons become bound intoprotons and neutrons All SM particles are in thermal equilibrium

Time = 100 s Size = 10minus8times today Temp = 109 K

Primordial nucleosynthesis The Universe is cold enough for protonsand neutrons to combine and form light atomic nuclei such as HeD and Li

Time = 1012 s Size = 3 middot 10minus4times today Temp = 104 K

Matter-radiation equality Pressureless matter starts to dominate

Time = 4 times 105 yrs Size = 10minus3times today Temp = 3 times 103 K

Recombination Electrons combine with nuclei and form electrically neu-tral atoms and the Universe becomes transparent to photons The cosmicmicrowave background is a snapshot of photons from this epoch

Time = 108 yrs Size = 01times today Temp = 30 K

The dark ages Small ripples in the density of matter gradually assembleinto stars and galaxies

Dark energy The expansion of the Universe starts to accelerate A secondgeneration of stars the Sun and Earth are formed

Time = 137 times 109 yrs Size = 1times today Temp = 27 K

Today ΩΛ sim 74 ΩCDM sim 22 Ωbaryons = 4 Ωr sim 0005 Ωk sim 0

Section 16 The Dark Side of the Universe 13

sky to a precision of about 10minus5 This high homogeneity between casuallydisconnected regions is the horizon problem

An attractive but still not established potential solution to these initialcondition problems was proposed in the beginning of the 1980rsquos [8ndash10] Byletting the Universe go through a phase of accelerating expansion the particlehorizon can grow exponentially and thereby bring all observable regions intocausal contact At the same time such an inflating Universe will automaticallyflatten itself out The current paradigm is basically that such an inflatingphase is caused by a scalar field Φ dominating the energy content by itspotential V (Φ) If this inflaton field is slowly rolling in its potential ie12 φ ≪ V (Φ) the equation of state is pΦ asymp minusV (Φ) asymp minusρΦ If V (Φ) staysfairly constant for a sufficiently long time it would mimic a cosmologicalconstant domination From Eq (113) it follows that H2 asymp constant and thusthat the scale factor grows as a(t) prop eHt This will cause all normal matterfields (w gt minus13) to dilute awaydagger During this epoch the temperature dropsdrastically and the Universe super-cools due to the extensive space expansionOnce the inflaton field rolls down in the presumed minimum of its potential itwill start to oscillate and the heavy inflaton particles will decay into standardmodel particles This reheats the Universe and it evolves as in the ordinaryhot Big Bang theory with the initial conditions naturallyDagger tuned by inflationDuring inflation quantum fluctuations of the inflaton field will be stretchedand transformed into effectively classical fluctuations (see eg [13]) Whenthe inflation field later decays these fluctuations will be transformed to theprimordial power spectrum of matter density fluctuations These seeds offluctuations will then eventually grow to become the large-scale structuressuch as galaxies etc that we observe today Today the observed spectrum ofdensity fluctuations is considered to be the strongest argument for inflation

16 The Dark Side of the Universe

What we can observe of our Universe are the various types of signals thatreach us ndash light of different wavelengths neutrinos and other cosmic raysThis reveals the distribution of lsquovisiblersquo matter But how would we know ifthere is more substance in the Universe not seen by any of the above means

The answer lies in that all forms of energy produce gravitational fields(or in other words curve the surrounding spacetime) which affect both theirlocal surroundings and the Universe as a whole Perhaps surprisingly suchgravitational effects indicate that there seems to be much more out there in

dagger This would also automatically explain the absence of magnetic monopoles which couldbe expected to be copiously produced during Grand Unification symmetry breaking atsome high energy scale

Dagger A word of caution Reheating after inflation drastically increases the entropy and a verylow entropy state must have existed before inflation see eg [11 12] and referencestherein

our Universe than can be seen directly It turns out that this lsquoinvisible stuffrsquocan be divided into two categories dark energy and dark matter Introducingonly these two types of additional energy components seems to be enough toexplain a huge range of otherwise unexplained cosmological and astrophysicalobservations

Dark Energy

In 1998 both the Supernova Cosmology Project and the High-z SupernovaSearch Team presented for the first time data showing an accelerating ex-pansion of the Universe [14 15] To accomplish this result redshifts andluminosity distances to Type Ia supernovae were measured The redshift de-pendence of the expansion rate H(z) can then be deduced from Eq (121)The Type Ia supernovae data showed a late-timesect acceleration of the expan-sion of our Universe (aa gt 0) This conclusion relies on Type Ia supernovaebeing standard candles ie objects with known intrinsic luminosities whichare motivated both on empirical as well as theoreticalpara grounds

These first supernova results have been confirmed by more recent observa-tions (eg [16 17]) The interpretation of a late-time accelerated expansionof the Universe also fits well into other independent observations such as datafrom the CMB [18] and gravitational lensing (see eg [19])

These observations indicate that the Universe is dominated by an energyform that i) has a negative pressure that today has an equation of statew asymp minus1 ii) is homogeneously distributed throughout the Universe with anenergy density ρΛ asymp 10minus29 gcm3 and iii) has no significant interactions otherthan gravitational An energy source with mentioned properties could also bereferred to as vacuum energy as it can be interpreted as the energy densityof empty space itself However within quantum field theory actual estimatesof the vacuum energy are of the order of 10120 times larger than the observedvalue

The exact nature of dark energy is a matter of speculation A currentlyviable possibility is that it is the cosmological constant Λ That is the Λ termin Einsteinrsquos equation is a fundamental constant that has to be determined byobservations If the dark energy really is an energy density that is constant intime then the period when the dark energy and matter energy densities aresimilar ρΛ sim ρm is extremely short on cosmological scales (ie in redshift

sect To translate between z and t one can use H(z) = ddt

ln ( aa0

) = ddt

ln ( 11+z

) = minus11+z

para A Type Ia supernova is believed to be the explosion of a white dwarf star that hasgained mass from a companion star until reaching the so-called Chandrasekhar masslimit sim 14M⊙ (where M⊙ is the mass of the Sun) At this point the white dwarfbecomes gravitationally instable collapses and explodes as a supernova

Inclusion of broken supersymmetry could decrease this disagreement to some 1060 ordersof magnitude

range) We could wonder why we happen to be around to observe the Universejust at the moment when ρΛ sim ρm

Another proposed scenario for dark energy is to introduce a new scalarfield with properties similar to the inflaton field This type of scalar fieldsis often dubbed quintessence [20] or k-essence [21] fields These models differfrom the pure cosmological constant in that such fields can vary in time (andspace) However the fine-tuning or other problems still seems to be presentin all suggested models and no satisfactory explanation of dark energy iscurrently available

Dark Matter

The mystery of missing dark matter (in the modern sense) goes back to atleast the 1930s when Zwicky [22] pointed out that the movements of galaxiesin the Coma cluster also known as Abell 1656 indicated a mass-to-light ratioof around 400 solar masses per solar luminosity which is two orders of mag-nitude higher than in our solar neighborhood The mass of clusters can alsobe measured by other methods for example by studying gravitational lensingeffects (see eg [23] for an illuminating example) and by tracing the distri-bution of hot gas through its X-ray emission (eg [24]) Most observations oncluster scales are consistent with a matter density of Ωmatter sim 02minus 03 [25]At the same time the amount of ordinary (baryonic) matter in clusters can bemeasured by the so-called Sunayaev-Zelrsquodovich effect [26] by which the CMBgets spectrally distorted through Compton scattering on hot electrons in theclusters This as well as X-ray observations shows that only about 10 ofthe total mass in clusters is visible baryonic matter the rest is attributed todark matter

At galactic scales determination of rotation curves ie the orbital veloc-ities of stars and gas as a function of their distance from the galactic centercan be efficiently used to determine the amount of mass inside these orbits Atthese low velocities and weak gravitational fields the full machinery of gen-eral relativity is not necessary and circular velocities should be in accordancewith Newtonian dynamics

v(r) =

r (124)

where M(r) is the total mass within radius r (and spherical symmetry hasbeen assumed) If there were no matter apart from the visible galactic diskthe circular velocities of stars and gas should be falling off as 1

radicr Observa-

tions say otherwise The velocities v(r) stay approximately constant outsidethe bulk of the visible galaxy This indicates the existence of a dark (invisible)halo with M(r) prop r and thus ρDM sim 1r2 (see eg [27])

On cosmological scales the observed CMB anisotropies combined withother measurements are a powerful tool in determining the amount of dark

matter In fact without dark matter the cosmological standard model wouldfail dramatically to explain the CMB observations [18] Simultaneously thebaryon fraction is determined to be about only 4 which is in good agreementwith the value inferred independently from BBN to explain the abundanceof light elements in our Universe

Other strong support for a large amount of dark matter comes from surveysof the large-scale structures [28] and the so-called baryon acoustic peak in thepower spectrum of matter fluctuations [29] These observations show how tinybaryon density fluctuations deduced from the CMB radiation in the presenceof larger dark matter fluctuations have grown to form the large scale structureof galaxies The structures observed today would not even have had time toform from these tiny baryon density fluctuations if no extra gravitationalstructures (such as dark matter) were present

Finally recent developments in weak lensing techniques have made it pos-sible to produce rough maps of the dark matter distribution in parts of theUniverse [30]

Models that instead of the existence of dark matter suggest modificationsof Newtonrsquos dynamics (MOND) [3132] have in general problems explainingthe full range of existing data For example the so-called lsquobullet clusterrsquoobservation [33 34] rules out the simplest alternative scenarios The bulletcluster shows a snapshot of what is interpreted as a galaxy cluster lsquoshotrsquothrough another cluster (hence the name bullet) ndash and is an example where thegravitational sources are not concentrated around most of the visible matterThe interpretation is that the dark matter (and stars) in the two collidingclusters can pass through each other frictionless whereas the major part ofthe baryons ie gas will interact during the passage and therefore be haltedin the center This explains both the centrally observed concentration of X-ray-emitting hot gas and the two separate concentrations of a large amountof gravitational mass observed by lensing

In contrast to dark energy dark matter is definitely not homogeneouslydistributed at all scales throughout the Universe Dark matter is insteadcondensed around eg galaxies and galaxy clusters forming extended halosTo be able to condense in agreement with observations dark matter shouldbe almost pressureless and non-relativistic during structure formation Thistype of non-relativistic dark matter is referred to as cold dark matter

The concordance model that has emerged from observations is a Universewhere about 4 is in the form of ordinary matter (mostly baryons in the formof gas w asymp 0) and about 0005 is in visible radiation energy (mostly theCMB photons w = 13) The remaining part of our Universersquos total energybudget is dark and of an unknown nature Of the total energy roughly 74is dark energy (w sim minus1) and 22 is dark matter (w = 0) Most of the darkmatter is cold (non-relativistic) matter but there is definitely also some hotdark matter in the form of neutrinos However the hot dark matter can atmost make up a few percent [3536] Some fraction of warm dark matter ie

Dark Energy

Dark Matter

Baryonic Matter

2 Luminous (Gas amp Stars)

0005 Radiation (CMB)

2 Dark Baryons (Gas)

Figure 11 The energy budget of our Universe today Ordinary matter(luminous and dark baryonic matter) only contributes some percent whilethe dark matter and the dark energy make up the dominant part of theenergy content in the Universe The relative precisions of the quotedenergy fractions are roughly ten percent in a ΛCDM model The figure isconstructed from the data in [1837ndash39]lowastlowast

particles with almost relativistic velocities during structure formation couldalso be present This concordance scenario is often denoted the cosmologicalconstant Λ Cold Dark Matter (ΛCDM) model Figure 11 shows this energycomposition of the Universe (at redshift z = 0)

Note that the pie chart in Fig 11 do change with redshift (determined byhow different energy components evolve see Eq (117)) For example at thetime of the release of the CMB radiation the dark energy part was negligibleAt that time the radiation contribution and the matter components were ofcomparable size and together made up more or less all the energy in theUniverse

The wide range of observations presents very convincing evidence for theexistence of cold dark matter and it points towards new yet unknown exoticphysics A large part of this thesis contain our predictions within differentscenarios that could start to reveal the nature of this dark matter

All Those WIMPs ndash Particle Dark Matter

Contrary to dark energy there are many proposed candidates for the darkmatter The most studied hypothesis is dark matter in the form of some

lowastlowast The background picture in the dark energy pie chart shows the WMAP satellite imageof the CMB radiation [40] The background picture in the dark matter pie chart is aphotograph of the Bullet Cluster showing the inferred dark matter distribution (in blue)and the measured hot gas distributions (in red) [41]

yet undiscovered species of fundamental particle To have avoided detectionthey should only interact weakly with ordinary matter Furthermore theseparticles should be stable ie have a life time that is at least comparableto cosmological time scales so that they can have been around in the earlyUniverse and still be around today

One of the most attractive classes of models is that of so-called WeaklyInteracting Massive Particles ndash WIMPs One reason for the popularity of thesedark matter candidates is the lsquoWIMP miraclersquo In the very early Universeparticles with electroweak interactions are coupled to the thermal bath ofstandard model particles but at some point their interaction rate falls belowthe expansion rate of the Universe At this point the WIMPs decouple andtheir number density freezes in thereby leaving a relic abundance consistentwith the dark matter density today Although the complete analysis can becomplicated for specific models it is usually a good estimate that the relicdensity is given by [42]

ΩWIMPh2 asymp 3 middot 10minus27cmminus3sminus1

〈σv〉 (125)

where h is the Hubble constant in units of 100 km sminus1 Mpcminus1 (h is todayobserved to be 072 plusmn 003 [18]) and 〈σv〉 is the thermally averaged interac-tion rate (cross section times relative velocity of the annihilating WIMPs)This equation holds almost independently of the WIMP mass as long theWIMPs are non-relativistic at freeze-out The lsquoWIMP miraclersquo that oc-curs is that the cross section needed 〈σv〉 sim 10minus26 cm3 sminus1 is roughlywhat is expected for particle masses at the electroweak scale Typically

σv sim α2

M2WIMP

sim 10minus26 cmminus3sminus1 where α is the fine structure constant and

the WIMP mass MWIMP is taken to be about 100 GeVThere are other cold dark matter candidates that do not fall into the

WIMP dark matter category Examples are the gravitino and the axion Fora discussion of these and other types of candidates see for example [25] andreferences therein

C h a p t e r

Where Is theDark Matter

Without specifying the true nature of dark matter one can still make generalpredictions of its distribution based on existing observations general modelbuilding and numerical simulations Specifically this chapter concentrates ondiscussing the expected dark matter halos around galaxies like our own MilkyWay For dark matter in the form of self-annihilating particles the actualdistribution of its number density plays an extremely important role for theprospects of future indirect detection of these dark matter candidates Aneffective pinching and reshaping of dark matter halos caused by the centralbaryons in the galaxy or surviving small dark matter clumps can give anenormously increased potential for indirect dark matter detection

21 Structure Formation History

During the history of our Universe the mass distribution has changed dras-tically The tiny 10minus5 temperature fluctuations at the time of the CMB ra-diation reflects a Universe that was almost perfectly homogeneous in baryondensity Since then baryons and dark matter have by the influence of gravitybuilt up structures like galaxies and clusters of galaxies that we can observetoday To best describe this transition the dark matter particles should benon-relativistic (lsquocoldrsquo) and experience at most very weak interactions withordinary matter This ensures that the dark matter was pressureless andseparated from the thermal equilibrium of the baryons and the photons wellbefore recombination and could start evolving from small structure seeds ndashthese first seeds could presumably originating from quantum fluctuations inan even earlier inflationary epoch

Perturbations at the smallest length scales ndash entering the horizon priorto radiation-matter equality ndash will not be able to grow but are washed outdue to the inability of the energy-dominating radiation to cluster Laterwhen larger scales enter the horizon during matter domination dark matter

20 Where Is the Dark Matter Chapter 2

density fluctuations will grow in amplitude due to the absence of counter-balancing radiative pressure This difference in structure growth before andafter matter-radiation equality is today imprinted in the matter power spec-trum as a suppression in density fluctuations at comoving scales smaller thanroughly 1 Gpc whereas on larger scales the density power spectrum is scaleinvariant (in agreement with many inflation models) The baryons are how-ever tightly coupled to the relativistic photons also after radiation-matterequality and cannot start forming structures until after recombination Oncereleased from the photon pressure the baryons can then start to form struc-tures rapidly in the already present gravitational wells from the dark matterWithout these pre-formed potential wells the baryons would not have thetime to form the structures we can observe today This is a strong supportfor the actual existence of cold dark matter

As long as the density fluctuations in matter stay small linearized ana-lytical calculations are possible whereas once the density contrast becomesclose to unity one has to resort to numerical simulations to get reliable re-sults on the structure formation The current paradigm is that structure isformed in a hierarchal way smaller congregations form first and then mergeinto larger and larger structures These very chaotic merging processes resultin so-called violent relaxation in which the time-varying gravitational poten-tial randomizes the particle velocities The radius within which the particleshave a fairly isotropic distribution of velocities is commonly called the virialradius Within this radius virial equilibrium should approximately hold ie2Ek asymp Ep where Ek and Ep are the averaged kinetic energy and gravitationalpotential respectively

By different techniques such as those mentioned in Chapter 1 it is possibleto get some observational information on the dark matter density distribu-tion These observations are often very crude and therefore it is common touse halo profiles predicted from numerical simulations rather than deducedfrom observations In the regimes where simulations and observations canbe compared they show reasonable agreement although some tension mightpersists [25]

22 Halo Models from Dark Matter Simulations

Numerical N -body simulations of structure formation can today contain upto about 1010 particles (as eg in the lsquoMillennium simulationrsquo [43]) thatevolve under their mutual gravitational interactions in an expanding universeSuch simulations are still far from resolving the smallest structures in largerhalos Furthermore partly due to the lack of computer power many of thesehigh-resolution simulations include only gravitational interactions ie darkmatter These simulations suggest that radial density profiles of halos rangingfrom masses of 10minus6 [44] to several 1015 [45] solar masses have an almost

Section 22 Halo Models from Dark Matter Simulations 21

Table 21 Parameters for some widely used dark matter density profilemodels (see equation 21) The values of rs are for a typical Milky-Way-sized halo of mass M200 sim 1012M⊙ at redshift z = 0

Model α β γ rs [kpc]NFW 10 30 10 20Moore 15 30 15 30Kra 20 30 04 20Iso 20 20 00 4

universal formlowast A suitable parametrization for the dark matter density ρis to have two different asymptotic radial power law behaviors ie rminusγ atthe smallest radii and rminusβ at the largest radii with a transition rate α bywhich the profile interpolates between these two asymptotic powers aroundthe radius rs

ρ(r) =ρ0

(rrs)γ [1 + (rrs)α]βminusγ

It is often convenient to define a radius r200 sometimes also referred to as thevirial radius inside which the mean density is 200 times the critical densityρc The total mass enclosed is thus

M200 = 2004πr3200

3ρc (22)

For a given set of (α β γ) the density profile in Eq (21) is completely specifiedby only two parameters eg the halo mass M200 and the scale radius rs Thetwo parameters M200 and rs could in principle be independent but numericalsimulations indicate that they are correlated In that sense it is sometimesenough to specify onlyM200 for a halo (see eg the appendix of [48]) Insteadof rs the concentration parameter c200 = r200rs is also often introducedAlthough less dependent on halo size than rs c200 also varies with a tendencyto increase for smaller halo size and larger redshifts (see eg [49] and [47])

Some of the most common values of parametrization parameters (α β γ)found for dark matter halos are given in Table 21 From top to bottom thetable gives the values for the Navarro Frenk and White (NFW [50]) Mooreet al (Moore [51]) and the Kravtsov et al (Kra [52]) profile The modifiedisothermal sphere profile (Iso eg [5354]) with its constant density core isalso included

The most recent numerical simulations appear to agree on a slightly newparadigm for the dark matter density They suggest that the logarithmicslope defined as

γ(r) equiv d ln(ρ)

d ln(r) (23)

lowast The density profiles are actually not found to be fully universal as the density slope inthe center of smaller halos is in general steeper than in larger halos [46 47]

decreases continuously towards the center of the halos In accordance withthis the following so-called Einasto density profile is suggested [55 56]

ρ(r) = ρminus2 exp

minus 2

rminus2

minus 1

In this profile ρminus2 and rminus2 correspond to the density and radius whereρ prop rminus2 (Note that the logarithmic slope converges to zero when r = 0)Typically α is found to be of the order of sim 02 [56]

The dark matter profile are sometimes referred to as cored cuspy or spikeddepending on whether the density in the center scales roughly as rminusγ withγ asymp 0 γ amp 0 or γ amp 15 respectively

All the above results stem from studies of dark matter dominated systemsThis should in many respects be adequate as the dark matter makes upsim80 [18] of all the matter and therefore usually dominates the gravitationallyinduced structure formation However in the inner parts of eg galaxy halosthe baryons ie gas and stars dominate the gravitational potential andshould be of importance also for dark matter distribution

23 Adiabatic Contraction

The main difference between dark and baryonic matter is that the latterwill frequently interact and cool by dissipating energy This will cause thebaryons unless disturbed by major merges to both form disk structures andcontract considerably in the centers This behavior is indeed observed bothin simulations containing baryons and also in nature where the baryons forma disk andor bulge at the center of apparently much more extended darkmatter halos It has long been realized that this ability of baryons to sinkto the center of galaxies would create an enhanced gravitational potentialwell within which dark matter could congregate increasing the central darkmatter density This effect is commonly modeled by the use of adiabaticinvariants [53 54 57ndash66]

A Simple Model for Adiabatic Contraction

The most commonly used model suggested by Blumenthal et al [60] assumesa spherically symmetric density distribution and circular orbits of the darkmatter particles From angular momentum conservation piri = pfrf and

gravitational-centripetal force balance GM(r)r2 = p2

m2r we obtain the adiabaticinvariant

rfMf (rf ) = riMi(ri) (25)

where M(r) is the total mass inside a radius r and the lower indices i andf indicate if a quantity is initial or final respectively Splitting up the final

Section 23 Adiabatic Contraction 23

mass distribution Mf (r) into a baryonic part Mb(r) and an unknown darkmatter part MDM(r) we have Mf (r) = Mb(r) +MDM(r) Eq (25) gives

rf =riMi(ri)

Mb(rf ) +MDM(rf ) (26)

From mass conservation the non-crossing of circular orbits during contractionand a mass fraction f of the initial matter distribution in baryons we get

MDM(rf ) = Mi(ri)(1 minus f) (27)

This means From an initial mass distribution Mi of which a fraction f (iethe baryons) forms a new distribution Mb the remaining particles (ie thedark matter) would respond in a such way that orbits with initial radius riend up at a new orbital radius rf These new radii are given by Eq (26)and the dark matter mass inside these new radii is given by Eq (27)

Modified Analytical Model

In reality the process of forming the baryonic structure inside an extendedhalo is neither a fully adiabatic process nor spherically symmetric Instead itis well established that typical orbits of dark matter particles inside simulatedhalos are rather elliptical (see eg [67]) This means that M(rorbit) changesaround the orbit and M(r)r in Eq (25) is no longer an adiabatic invariantIt has therefore been pointed out by Gnedin et al [62] that Eq (25) could bemodified to try to take this into account In particular they argue that usingthe value of the mass within the average radius of a given orbit r should givebetter results

The average radius r for a particle is given by

int ra

vrdr (28)

where vr is the radial velocity ra (rp) is the aphelion (perihelion) radius andT is the radial period The ratio between r and r will change throughout thehalo but a suitable parametrization of 〈r〉 (ie r averaged over the populationof orbits at a given radius r) is a power law with two free parameters [62]

〈r〉 = r200A

The numerical simulations in [62] result in A = 085plusmn005 and w = 08plusmn002daggerDagger

dagger In [62] they used r180 instead of r200 as used here but the difference in A is very smallIn general we have A180 = A200(r200r180)1minusw which for a singular isothermal sphere

(ρ prop 1r2) implies that A180 = A200 (180200)(1minusw)2 asymp 099A200Dagger Another almost identical parametrization was used in [68] 〈r〉 = 172y082(1 +

5y)0085 where y = rrs

The modified adiabatic contraction model is given by

rf =riMi(〈ri〉)

Mb(〈rf 〉) +MDM(〈rf 〉) (210)

whereas the equation for the conservation of mass is unchanged

These two equations can now be solved for any given A and w value Thusthe model predicts the final dark matter distribution MDM(r) if one knowsthe initial mass distribution together with the final baryonic distribution in agalaxy (or some other similar system like a cluster) How well these analyticalmodels work can now be tested by running numerical simulations includingbaryons

24 Simulation Setups

In Paper VI we aimed at investigating the dark matter halos as realisticallyas possible by using numerical simulations that included both dark matterand baryons These simulations were known from previous studies to produceoverall realistic gas and star structures for spiral galaxies [69ndash71] Althoughthe numerical resolution is still far from being able to resolve many of thesmall-scale features observed in real galaxies the most important dynamicalproperties such as the creation of stable disk and bulge structures both forthe gas and star components are accomplished

Four sets of simulated galaxies were studied in Paper VI The simulationswere performed by the Hydra code [72] and an improved version of theSmoothed Particle Hydrodynamics code TreeSPH [73]sect In accordance withthe observational data the simulations were run in a ΛCDM cosmology withΩM = 03 ΩΛ = 07 H0 = 100h km sminus1Mpcminus1 = 65 km sminus1 Mpcminus1 amatter power spectrum normalized such that the present linear root meansquare amplitude of mass fluctuations inside 8hminus1 Mpc is σ8 = 10 and abaryonic fraction f set to 015 By comparing simulations with differentresolutions we could infer that the results are robust down to an inner radiusrmin of about 1 kpc The simulations were run once including only darkmatter then rerun with the improved TreeSPH code incorporating starformation stellar feedback processes radiative cooling and heating etc Thefinal results in the simulations including baryons are qualitatively similar toobserved disk and elliptical galaxies at redshift z = 0 a result that is mainlypossible by overcoming the angular momentum problem by an early epoch of

sect Hydra is a particle-particle particle-mesh code that calculates the potential amongN point masses and TreeSPH is for simulating fluid flows both with and withoutcollisionless matter Each of the simulations in Paper VI took about 1 month of CPUtime on an Itanium II 1 GHz processor

Section 25 Pinching of the Dark Matter Halo 25

Table 22 The main properties (at redshift z=0) of the benchmarkgalaxy and its dark matter halo

Simulation DM+galaxy DM only

Virial radius r200 [kpc] 209 211Total mass M200 [1011M⊙] 89 93Number of particles N200 [times105] 36 12DM particle mass mDM [106M⊙] 65 76SPH particle mass mbaryon [106M⊙] 11 Baryonic disk + bulge mass [1010M⊙] 717 Baryonic bulge-to-disk mass ratio 019

strong stellar energy feedback in the form of SNII energy being fed back tothe interstellar medium (see Paper VI and [69ndash71] for further details on thenumerical simulations)

In the following we will focus on the generic results in Paper VI Al-though four galaxies were studied I concentrate here on only one of themto exemplify the generic results [All examples will be from simulation S1

and its accompanying simulation DM1 found in Paper VI] The simulatedgalaxy resembles in many respects our own Milky Way and some of its mainproperties are found in Table 22

25 Pinching of the Dark Matter Halo

With the baryonic disks and bulges formed fully dynamically the surroundingdark matter halo response should also be realistically predicted Figure 21shows the comparison of the simulation that includes the correct fraction ofbaryons to the otherwise identical simulation with all the baryons replaced bydark matter particles It is clear how the effect of baryons ndash forming a centralgalaxy ndash is to pinch the halo and produce a much higher dark matter densityin the central part

For simulations including only dark matter the density profile in Eq (24)with a continuously decreasing slope turns out to be a good functional formThe best fit values for the two free parameters in this profile are given inTable 23 The simulation that includes baryons produces a dark matter cusp

Table 23 Best fit parameters to Eq (24) for the spherical symmetrizeddark matter halo in the simulation with only dark matter

α rminus2 [kpc] χ2dofa

0247 185 15a This χ2 fit was done with 50 bins ie 48 degrees of freedom (dof)

001 01 110

DM only sim

DM in galaxy sim

r [r200

[ρc r

Figure 21 Dark matter density for a galaxy simulation includingbaryons (solid line) compared to an identical simulation including darkmatter only (dashed line) A clear steepening in the dark matter densityof the central part has arisen due to the presence of a baryonic galaxyThe curvesrsquo parameterizations are given in Table 24 and Table 23 for thesolid curve and the dashed curve respectively The data points shownas solid and open circles are binned data directly from the simulationsThe arrows at the bottom indicate respectively the lower resolution limit(rmin = 2 kpc) and the virial radius (r200 asymp 200 kpc) These arrows alsoindicate the range within which the curves have been fit to the data

Table 24 Best fit parameters to Eq (21) for the spherical symmetrizeddark matter halos in the simulation including baryons

α β γ rs [kpc] χ2dofa

176 331 183 449 14a This χ2 fit was done with 50 bins ie 46 degrees of freedom (dof)

and was therefore better fitted with Eq (21) which allows for a steeperlogarithmic slope γ in the center The best-fit parameter values are found inTable 24 It should be realized that with four free parameters in the profile(21) there are degeneracies in the inferred parameter values (see eg [74])Although the numbers given in Table 24 give a good parametrizationpara they

para See eg [75] for a nice introduction to statistical data analysis

Section 26 Testing Adiabatic Contraction Models 27

do not necessarily represent a profile that could be extrapolated to smallerradii with confidence

Let me summarize the result of all the dark matter halos studied in Pa-per VI Simulations without baryons have a density slope continuously de-creasing towards the center with a density ρDM sim rminus13plusmn02 at about 1of r200 This is a result that lies between the NFW and the Moore profilegiven in Table 21 The central dark matter cusps in the simulations that alsocontain baryons become significantly steeper with ρDM sim rminus19plusmn02 with anindication of the inner logarithmic slope converging to roughly this value

26 Testing Adiabatic Contraction Models

The proposed adiabatic contraction models would if they included all therelevant physics be able to foresee the true dark matter density profiles fromsimulations that include only dark matter and known (ie observed) baryonicdistributions Having in disposal simulations with identical initial conditionsexcept that in one case baryons are included and in the other not we couldtest how well these adiabatic contraction models work

It turns out that the simpler contraction model by Blumenthal et al [60]significantly overestimates the contraction in the inner 10 of the virial radiusas compared to our numerical simulations see Fig 23 To continue andtest the modified adiabatic contraction proposed by Gnedin et al [62] wein addition need to first find out the averaged orbital eccentricity for thedark matter (ie determine A and w in Eq (29) for the pure dark mattersimulation) For our typical example model we found that the averagedorbital structure is well described by A = 074 and w = 069 as seen inFig 22 (similar values were found for all our simulated dark matter halos)

From these A and w values and the baryonic distribution in our corre-sponding (baryonic) galaxy simulation the final dark matter distribution isdeduced from Eq (210) and (211) Comparing the result from these twoequations with the dark matter density profile found in the actual simulationincluding baryons showed that the Gnedin et al model is a considerable im-provement compared to the Blumenthal et al model However this modelrsquosprediction also differed somewhat from the N -body simulation result that in-cluded baryons To quantify this A and w were taken as free parameters anda scan over different values was performed With optimally chosen values ofA and w (no longer necessarily describing the orbital eccentricity structure ofthe dark mater) it was always possible to obtain a good reconstruction of thedark matter density profile

Figure 23 shows the region in the (Aw)-plane that provides a good recon-struction of the dark matter halo for our illustrative benchmark simulationFrom these contour plots it follows that the fits for (Aw)=(11) ndash which cor-responds to circular orbits and therefore the original model of Blumenthal etal ndash are significantly worse than the fits for the optimal values (A sim 05 and

0 5 10 15 200

r [kpc]

ltrr200gt= A(rr200)w

065 07 075 08 08506

Figure 22 〈r〉 versus r for the halo simulation including only darkmatter The best fit (solid line) corresponding to (A w)=(074069) inEq (29) shows that the power law assumption is an excellent represen-tation of the data The large crosses represent the binned data and thesmaller horizontal lines indicate the variance for 〈r〉 in each data pointThe smaller sub-figure shows in black the 1σ (68) confidence regionwhereas the lighter gray area is the 3σ (997) confidence region in the(Aw) plane Figure from Paper VI

w sim 06) We also see that although the Gnedin et al model (marked by across in Fig 23) is a significant improvement it is not at all perfect

All of our four simulations in Paper VI showed more or less significantdeviations from the model predictions By changing the stellar feedbackstrength we could also find that this had an impact on the actual best fitvalues of (Aw) (see Paper VI for more details) This difference between(Aw) obtained directly from the relationship between 〈r〉 and r in Eq (29)and from the best fit values suggests (not surprisingly) that there is morephysics at work than can be described by a simple analysis of the dark matterorbital structure

27 Nonsphericity

We have just seen how the centrally concentrated baryons pinch the darkmatter Since the dark matter particles have very elliptical orbits and thebaryons dominate the gravitational potential in the inner few kpc it wouldbe interesting to see how the presence of the baryonic galactic disk influencesthe triaxial properties of the dark matter halo This was studied in Paper VI

Section 27 Nonsphericity 29

0 02 04 06 08 10

Figure 23 Best fit parameters for reconstructing the baryon compresseddark matter halo from its dark-matter-only halo The black area is the 1σ(68) confidence region and the larger gray is the 3σ (997) confidenceregion The (Aw) value expected from the analysis of orbit ellipticitiesas proposed by Gnedin et al [62] is marked by a cross and the originaladiabatic contraction model by Blumenthal et al [60] by a circle Figurefrom Paper VI

by relaxing the spherical symmetry assumption in the profile fitting andinstead studying the halosrsquo triaxial properties With a ellipsoidal assumptionand studying the momentum of inertia tensor Iij we determined the threeprincipal axes a b and c at different radii scales (see Paper VI for moredetails) That is we find how much we would need to stretch out the matterdistribution in three different directions to get a spherically symmetric densityprofile

Axis Ratios

Let the principal axes be ordered such that a ge b ge c and introduce theparameters e = 1 minus ba (ellipticity) and f = 1 minus ca (flatness) Figure 24shows how these quantities vary with radius We can clearly see that ellipticityand flatness differ between the simulation with only dark matter (left panel)and the simulation including the formation of a baryonic disk galaxy (rightpanel) The radius R on the horizontal axis gives the size of the elliptical shellndash that is semiaxes R (ba) middotR and (ca)R ndash inside which particles have beenused to calculate e and f

Having obtained the semiaxes we can determine whether a halo is prolatein other words shaped like a rugby ball or oblate ie flattened like a Frisbee

08DM (incl baryons)

R [kpc]10

DM (DM only)

R [kpc]

Figure 24 The triaxial parameters e = 1 minus ba (dashed line) and f =1minusca (solid line) of the dark matter halos in the simulation with only darkmatter (left panel) and the simulation including baryons (right panel)

Table 25 Values of the oblateprolate-parameter T inside R = 10 kpcfor the dark matter halo

Simulation including baryons only dark matterT value 0076 074

by the measure

T =a2 minus b2

a2 minus c2 (212)

If the halo is oblate that is a and b are of similar size and larger than c andthe measure is T lt 05 whereas if T gt 05 the halo is prolate The T valuefor the dark matter halo with and without baryons are listed in Table 25

The general result from all our four studied simulated halos is that theinclusion of the baryons causes the dark matter halo to change its shape frombeing prolate in the pure dark matter simulations into a more spherical andoblate form in simulations that include the formation of a central disk galaxyThis result agrees and compliment the studies in [76ndash78]

Alignments

Given these results of nonsphericity the obvious thing to check is whether theprincipal axes of the dark matter and the baryon distributions are alignedFigure 25 shows this alignment between the stellar disk the gaseous disk andthe dark matter lsquodiskrsquo The parameter ∆θ is the angle between each of thesevectors and a reference direction defined to correspond to the orientationvector of the gaseous disk with radius R = 10 kpc The figure shows that theorientation of the minor axes of the gas stars and dark matter is strongly

Section 28 Some Comments on Observations 31

R [kpc]

Figure 25 Diagram showing angular alignment of the gas (dotted) stars(dot-dashed) and dark matter (solid line) in our four galaxy simulationsThe vertical scale is the difference in angle between the orientation of theminor axis (around which the moment of inertia is the greatest) of thecomponent in question relative to the axis of the gas inside 10 kpc (bydefinition zero and marked with a cross) The dashed line is the darkmatter in the simulation without baryons showing that the baryonic diskis formed aligned with the plane of the original dark matter halo

correlated and they line up with each other However at 50 kpc there is aclear step in the starsrsquo alignment The reason for this discrepancy is due to amassive star concentration in a satellite galaxy outside the galactic disk

Figure 25 also shows that the orientation of the baryonic disk is rathercorrelated with the orientation of the flattest part of the dark matter halo inthe simulation without baryons The dark matter therefore seems to have arole in determining the orientation of the baryonic disk

Let me summarize the nonsphericity results for all studied halos in Pa-per VI All four galaxy simulations indicate that the inclusion of baryonssignificantly influences the dark matter halos Instead of being slightly pro-late they all became more spherical and slightly oblate with their (modest)flattening aligned with their galaxiesrsquo gas disk planes

28 Some Comments on Observations

The amount of triaxiality of dark matter halos seems to be a fairly genericprediction in the hierarchial cold dark matter model of structure formationand observational probes of halo shapes are therefore a fundamental test ofthis model Unfortunately observational determination of halo shapes is a dif-ficult task and only coarse constraints exist Probes of the Milky Way haloindicate that it should be rather spherical with f 02 and that an oblate

structure of f sim 02 might be preferable (see eg [79] and references therein)Milky-Way-sized halos formed in dissipationless simulations are usually pre-dicted to be considerably more triaxial and prolate although a large scatter isexpected [80ndash86] Including dissipational baryons in the numerical simulationand thereby converting the halo prolateness into a slightly oblate and morespherical halo might turn out to be essential to produce good agreement withobservations [77]

Having determined the ellipsoidal triaxiality of the dark matter distribu-tion we can include this information in the profile fits Including triaxialityto the radial density profile fits would not change any results (see Paper VI)This should not be surprising since the flattening of the dark matter halo isvery weak The oblate structure of the dark matter would have some mi-nor effects on the expected indirect dark matter signal [87] However thebaryonic effects found here have no indication of producing such highly disk-concentrated dark matter halo profiles as used in eg [1] to explain theexcess of diffuse gamma-rays in the EGRET data by WIMP annihilation (seeChapter 9 for more details)

Observations of presumably dark-matter-dominated systems such as lowsurface brightness dwarf galaxies indicate that dark matter halos have con-stant density cores instead of the steep cusps found in numerical simulations(see eg [88ndash93]) This could definitely be a challenge for the standard colddark matter scenario Even if baryons are included in the N -body simulationsand very explosive feedback injections are enforced it seems unlikely that itcould resolve the cusp-core problem (see eg [94] and references therein)

However several studies also demonstrate that the cusp-core discrepancynot necessarily implies a conflict Observational and data processing tech-niques in deriving the rotation curves (see eg [89ndash91]) and the neglectedcomplex effects on the gas dynamics due to the halosrsquo triaxiality proper-ties [95] indicate that there might not even be a discrepancy between ob-servation and theory One should also note that the story is actually differentfor galaxy halos where the baryons dominate the gravitational mass in theinner parts (as the halos studied in Paper VI) Here the problem of sepa-rating the dark matter component from the dominant baryonic componentallows the dark matter profile to be more cuspy without any conflict with ob-servation As adiabatic contraction increases the central dark matter densityin such a way that the dark matter density only tends to track the higherdensity baryonic component strong adiabatic contraction of the dark matterhalo in these systems should not be excluded

29 Tracing Dark Matter Annihilation

Improved knowledge about the dark matter distribution is essential for reliablepredictions of the detection prospects for many dark matter signals For anyself-annihilating dark matter particle the number of annihilations per unit

Section 29 Tracing Dark Matter Annihilation 33

time and volume element is given by

2〈σv〉 ρ

2DM(r)

where v is the relative velocity of the two annihilating particles σ is the totalcross section for annihilation and ρDM(r) the dark matter mass density at theposition r where the annihilation take place

Indirect detection of dark matter would be to detect particles producedin dark matter annihilation processes eg to find an excess in the amountof antimatter gamma rays andor neutrinos arriving at Earth [25] Sincethe expected amplitude of any expected signal depends quadratically on ρDMit seems most promising to look for regions of expected high dark matterconcentrations Unfortunately charged particles will be significantly bent bythe magnetic fields in our Galaxy and will no longer point back to their sourceOn the other hand this is not the case for neutrinos and gamma rays as theseparticles propagate more or less unaffected through our Galaxy

When looking for gamma rays towards a region of enhanced dark matterdensity the expected differential photon flux along the line of sight (los) ina given direction ψ is given by

dΦγ(ψ)

〈σtotv〉8πm2

dN effγ

dℓ(ψ)ρ2DM

(ℓ) (214)

where dN effγ dEγ is the energy differential number of photons produced per

dark matter pair annihilationExpected dark matter induced fluxes are still very uncertain Any attempt

to accurately predict such fluxes is still greatly hampered by both theoreti-cal uncertainties and lack of detailed observational data on the dark matterdistribution It can be handy to separate astrophysical quantities (ρDM) fromparticle physics properties (mDM 〈σv〉 and dNγdEγ) It is therefore conve-nient to define the dimensionless quantity [96]

〈J〉∆Ω(ψ) equiv 1

85 kpc

03 GeV cmminus3

dℓ(ψ)ρ2DM

(215)which embraces all the astrophysical uncertainties The normalization values85 kpc and 03 GeVcm

3are chosen to correspond to commonly adopted

values for the Sunrsquos galactocentric distance and the local dark matter densityrespectively For a detector of angular acceptance ∆Ω the flux thus becomes

dΦγdEγ

= 94 middot 10minus13dN eff

( 〈σtotv〉tot10minus26 cm3sminus1

middot ∆Ω 〈J〉∆Ω cmminus2sminus1TeVminus1 (216)

For example looking towards the galactic center with an angular accep-tance of ∆Ω = 10minus5 sr (which is comparable to the angular resolution of egthe HESS or GLAST telescope) it is convenient to write

∆Ω 〈J〉∆Ω(0) = 013 b sr (217)

where b = 1 if the dark matter distribution follows a NFW profile as given inTable 21 [97] On the other hand we have just seen that taking into accountthe effect of baryonic compression due to the dense stellar cluster observed toexist very near the galactic center could very well enhance the dark matterdensity significantly A simplifying and effective way to take into account suchan increase of the dark matter density is to simply allow the so-called boostfactor b to take much higher values Exactly how high this boost factor canbe in the direction of the galactic center or in other directions is still notwell understood and we will discuss this in some more detail in what follows

Indirect Dark Matter Detection

It is today impossible for galaxy simulations to get anywhere near the lengthresolution corresponding to the very center of the galaxy Despite this dif-ferent profile shapes from numerical simulations have frequently been extrap-olated into the galactic center This enables at least naıvely to predict ex-pected fluxes from dark matter annihilation in the galactic center In thespirit of comparing with the existing literature we performed the baryoniccontraction with our best fit values on A and w For typical values of theMilky Way baryon density (see Paper VI for details) and an initial Einastodark matter profile (given in Table 23) it is straightforward to apply thecontraction model The local dark matter density is here normalized to beρDM(r = 85kpc) sim 03 GeV cmminus3 Figure 26 shows both results if a 26times106

M⊙ central supermassive black hole is included in the baryonic profile and ifit is not

No attempt is made to model the complicated dynamics at subparsecscales of the galaxy other than trying to take into account the maximumdensity due to self-annihilation In other words a galactic dark matter halounperturbed by major mergers or collisions for a time scale τgal cannot containstable regions with dark matter densities larger than ρmax sim mDM〈σv〉τgalIn Fig 26 and in Table 26 it is assumed that τgal = 5times109 years and for theWIMP property a dark matter mass of mDM = 1 TeV with an annihilationcross section of 〈σv〉 = 3 times 10minus26cm3sminus1 is adopted

Table 26 shows the energy flux obtained with the values of (Aw) foundin the previous section as well the Blumenthal et al estimate This is thetotal energy luminosity not in some specific particle species and is hence theflux given in Eq (213) multiplied by 2 times the dark matter mass TheBlumenthal et al adiabatic contraction model gives fluxes far in excess of themodified contraction model Even with the modified contraction model and

r [kpc]

larr rBH

darr ρ m

baryons

Einasto rarr

(Aw)=(11)

larr (αβγ)minusprofile

(Aw)=(05106)

10minus10

10minus5

Figure 26 Diagram showing the contraction of a Einasto dark matterdensity profile given in Table 23 (dot-dashed line) by a baryon profile(dotted line) as described in Paper VI The resulting dark matter pro-files (dashed lines) are plotted for (Aw) = (1 1) and (051 06) eachsplitting into two at low radii the denser corresponding to the densityprofile achieved from a baryon profile that includes a central black holeThe extrapolated density profile from Table 24 from our simulation isshown for comparison (solid line) Remember the numerical simulation isonly robust into rmin asymp 1 kpc Also shown are the maximum density lineand the radius corresponding to the lowest stable orbit around the centralblack hole Figure adapted from Paper VI

Table 26 Luminosity in erg sminus1 from dark matter annihilation for differ-ent contraction model parameters (Aw) The initial dark matter profilesare an NFW profile given in Table 21 or the Einasto profile given inTable 23 The baryon profile includes a super massive black hole as de-scribed in the text Quoted values are for the flux from the inner 10 pcand 100 pc Note that this is the total luminosity and not of some specificparticle species A dark matter particle mass of 1 TeV is assumed

NFW (Table 21) Einasto (Table 23)A w L10pc L100pc L10pc L100pc

Initial profile rarr 39 times 1033 39 times 1034 27 times 1031 47 times 1033

1 1 28 times 1040 28 times 1040 98 times 1039 98 times 1039

the values of (Aw) found in Paper VI we find a large flux enhancementcompared to the traditional NFW profile (as given in Table 21) This staystrue also if the initial profile is the Einasto profile which does not initiallypossess a cusp at all From Table 26 the boost of the luminosity comparedto the standard NFW profile takes values of about 102 ndash 104 in the directionof the galactic center This extrapolation to very small radii neglects manypotential effects such as the scattering of dark matter particles on stars orthe effect from a supermassive black hole not exactly in the galactic centerCase studies of the dark matter in the galactic center [68 98ndash103] show thatcompared to an NFW density profile the expected flux from self-annihilationcan be boosted as much as 107 but also that the opposite effect might bepossible leading to a relative depletion of an initial dark matter cusp

210 Halo Substructure

So far the dark matter density has been described as a smooth halo profile witha peak concentration in the center Presumably the halos contain additionalstructure Numerical simulations find a large number of local dark matterconcentrations (clumps) within each halo (see eg [47 104ndash112]) Also sub-structures within substructures etc are found This should not be surprisingas the hierarchal structure formation paradigm predicts the first formed struc-tures to be numerous small dark matter halos These first formed (WIMP)halos could still be around today as clumps of about the Earth mass and withsizes similar the solar system [44 113ndash115] In the subsequent processes ac-cretion to form larger structures by the merging of smaller progenitors is notalways complete the cores of subhalos could survive as gravitationally boundsubhalos orbiting within a larger host system More than 1015 of this firstgeneration of dark halo objects could potentially be within the halo of theMilky Way [111] but gravitational disruption during the accretion process aswell as late tidal disruption from stellar encounters can significantly decreasethis number [116]

This additional substructure could be highly relevant for indirect detec-tion of dark matter As the dark matter annihilation rate into for examplegamma rays increases quadratically with the dark matter density the in-ternal substructure may enhance not only the total diffuse gamma-ray fluxcompared to the smooth halo but also individual clumps of dark mattercould be detectable with eg gamma-ray telescopes such as GLAST Theprospects for detecting these subhalos however depend strongly on the as-sumptions (see eg [117]) With surviving microhalos down to masses of10minus6 M⊙ the dark matter fraction in a galaxy could be as large as about50 of the total mass [117] Translated into an enhancement of the totaldark matter annihilation rate for whole galaxies this gives a boost factor ofa few up to perhaps some hundred [112 117 118] The local annihilationboost compared to the smooth background halo profile due to subhalos is

Section 210 Halo Substructure 37

however expected to strongly depend on the galactocentric distance In theouter regions the clumps can boost the rates by orders of magnitude Onthe other hand in the inner regions the increase in annihilation rates due toclumps might be negligible This is both because expected tidal disruptioncould have destroyed many clumps in the center and that the smooth com-ponent already give larger annihilation rates Spatial variations of the localannihilation boost mean that different dark matter signals are expected todepend on both species and energy [119] of the annihilation products Forexample positrons are most sensitive to the local boost factors Positrons arestrongly affected by magnetic fields ndash they quickly lose energy and directionalinformation ndash and become located within some kpc to their source beforediffusing outside the galactic disk and escape from the Galaxy On the con-trary gamma rays propagate almost freely in our Galaxy and are thereforeaffected also by more distant dark matter density boosts The intermediatecase are antiproton signals which like the positrons are sensitive to clumpsin all sky directions but due to their much higher mass they are less deflectedby magnetic fields and can therefore travel longer distances in the disk (seeeg [119])

C h a p t e r

Beyond theStandard ModelHidden Dimensions

and More

Why is there a need to go beyond todayrsquos standard model of particle physicsThis chapter presents motivations and introduces possible extensions whichwill be discussed further in this thesis Special focus is here put on the pos-sibility that hidden extra space dimensions could exist General features ex-pected in theories with extra space dimensions are discussed together witha short historical review The chapter concludes by giving motivations tostudy the specific model of so-called universal extra dimensions as this willbe the model with which we will start our discussions on dark matter particlephenomenology in the following chapters

31 The Need to Go Beyond the Standard Model

Quantum field theory is the framework for todayrsquos standard model (SM) ofparticle physics ndash a tool box for how to combine three major themes in modernphysics quantum theory the field concept and special relativity Includedin the SM is a description of the strong weak and electromagnetic forcesas well as all known fundamental particles The theoretical description hasbeen a great success and agrees to a tremendous precision with practicallyall experimental results up to the highest energies reached (ie some hundredGeV) However it is known that the SM is not a complete theory as it standstoday Perhaps the most fundamental drawback is that it does not include aquantum description of gravitational interactions There are also a numberof reasons directly related to particle physics for why the SM needs to beextended For example the SM does not include neutrino masses (neutrinosmasses are by now a well-accepted interpretation of the observed neutrino

40 Beyond the Standard Model Hidden Dimensions and More Chapter 3

oscillations [120ndash125]) and extreme fine-tuning is required in the Higgs sectorif no new divergence canceling physics appears at TeV energies [with no newphysics between the electroweak scale (102 GeV) and the Planck scale (1019

GeV) is usually called hierarchy problem] Physics beyond the SM is alsovery attractive for cosmology where new fields in the form of scalar fieldsdriving inflation are discussed and since the SM is incapable of explainingthe observed matter-antimatter asymmetry and the amount of dark energyThe strongest reason for new fundamental particle physics however comesperhaps from the need for a viable dark matter candidate

Despite the necessity of replacing (or extending) the SM its great successhints also that new fundamental physics could be closely tied to some of itsbasic principles such as its quantization and symmetry principles As the SMis a quantum field theory with an SU(3) times SU(2) times U(1) gauge symmetryand an SO(1 3) Lorentz symmetry it is tempting to investigate extensions ofthese symmetries

This is the idea behind grand unified theories (GUTs) where at highenergies (typically of the order of 1016 GeV) all gauge couplings have thesame strength and all the force fields are fused into a unified field This isthe case for the SU(5) GUT theory This larger symmetry group is thenthought to be spontaneously broken at the grand unification scale down tothe SU(3)times SU(2)times U(1) gauge group of the SM that we observe at todaystestable energies However the simplest SU(5) theory predicts a too-shortproton lifetime and is nowadays excluded

Another type of symmetry extension (with generators that anticommu-tate) is a spacetime symmetry that mixes bosons and fermions These are thesupersymmetric extensions of the SM where every fundamental fermion hasa bosonic superpartner of equal mass and vice versa that every fundamen-tal boson has a fermionic superpartner This symmetry must if it exists bebroken in nature today so as to give all superpartners high enough masses toexplain why they have evaded detection This possibility is further discussedin Chapter 7

Yet another possibility is a Lagrangian with extended Lorentz symmetryachieved by including extra dimensions The most obvious such extension isto let the SM have the Lorentz symmetry SO(1 3 + n) with n ge 1 an inte-ger This implies that all SM particles propagate in n extra spatial dimen-sions endowed with a flat metric These are called universal extra dimensions(UEDs) [126]lowast Other extra-dimensional scenarios also exist where all or partof the matter and SM gauge fields are confined to a (3+1)-dimensional braneon which we are assumed to live The aim of the many variations of extra-dimensional models is usually to propose different solutions or new perspec-tives to known problems in modern physics Most such scenarios are therefore

lowast Along similar lines studies of possible effects of extra dimensions felt by SM particleswere also done earlier in [127ndash129]

Section 32 General Features of Extra-Dimensional Scenarios 41

of a more phenomenological nature but a notable exception is string theoryString theory (see eg [130] for a modern introduction) certainly aims to be afundamental theory and by replacing particles with extended strings it offersa consistent quantum theory description of gravity (how much this theory isrelated to reality is however still an open question) The requirement thatthe theory should be anomaly free leads canonically to a critical value of thespacetime dimensionality In the case of supersymmetric strings the numberof dimensions must be d = 10 (or d = 11 for M-theory) [130] Performing afully consistent extra dimensional compactification within string theory thatleads to firm observational predictions at accessible energies is at the momentvery challenging Therefore fundamental string theory is not yet ready formaking unique (or well constrained) phenomenological predictions in a veryrigorous way However as string theory is perhaps the most promising can-didate for a more fundamental theory today it is of interest to try to anywayinvestigate its different aspects such as extra space dimensions from a morephenomenological perspective

Finally another approach to go beyond the SM could be to try to extendit as minimally as possible to incorporate only new physics that can addressspecific known drawbacks One such approach which extends only the scalarsector of the SM is discussed in Chapter 8 There it is shown that such anextension besides other advantages gives rise to an interesting scalar darkmatter particle candidate with striking observational consequences

32 General Features of Extra-Dimensional Scenarios

Although we are used to thinking of our world as having three spatial dimen-sions there is an intriguing possibility that space might have more dimensionsThis might at first sound like science fiction and seemingly ruled out by ob-servations but in the beginning of the 20th century Nordstrom [131] andmore prominently Kaluza [132] and Klein [133] asked whether extra dimen-sions could say something fundamental about physics To allow for an extradimension without violating the apparent observation of only three spacedimensions it was realized that the extra dimension could be curled up onsuch a small length scale that we have not yet been able to resolve the extradimension As an analogy imagine you are looking at a thin hose from a longdistance The hose then seems to be just a one-dimensional line but as youget closer you are able to resolve the thickness of the hose and you realize ithas an extended two-dimensional surface

In the original idea by Kaluza [132] and rediscovered by Klein the start-ing point was a five-dimensional spacetime with the dynamics governed bythe Einstein-Hilbert action (ie general relativity) After averaging over the(assumed static) extra dimension and retaining an ordinary four-dimensionaleffective theory the result was an action containing both Einsteinrsquos generalrelativity and Maxwellrsquos action for electro-magnetism Although it at first

seemed to be a very lsquomagicrsquo unification it can be traced back to the fact thatthe compactification of one extra dimension on a circle automatically gives aU(1) symmetry which is exactly the same key symmetry as in the electromag-netic (abelian) gauge theory There is however a flaw in the five dimensionalKaluza-Klein (KK) theory even before trying to include the weak and thestrong interactions Once matter fields are introduced and with the U(1)symmetry identified with the usual electromagnetism the electric charge andmass of a particle must be related and quantized With the quantum of chargebeing the charge of an electron all charged particles must have masses on thePlanck scale Mpl sim 1019 GeV This is not what is observed ndash all familiarcharged particles have very much smaller masses

Today much of the phenomenological studies of extra dimensions concernthe generic features that can be expected To illustrate some of these fea-tures let us take the spacetime to be a direct product of the ordinary (four-dimensional) Minkowski spacetime and n curled up flat extra dimensionsWe can then show that in the emerging effective four-dimensional theory

(1) a tower of new massive particles appears

(2) Newtonrsquos 1r law is affected at short distances

(3) fundamental coupling lsquoconstantsrsquo will vary with the volume spanned bythe extra dimensions

Feature (1) can qualitatively be understood quite easily Imagine a particlemoving in the direction of one of the extra dimensions Even if the particlersquosmovement cannot be directly observed the extra kinetic energy will still con-tribute to its total energy For an observer not aware of the extra dimensionsthis additional kinetic energy will be interpreted as a higher mass (E = mc2)for that particle compared to an identical particle that is not moving in theextra dimensions To do this more formally let us denote local coordinatesby

xM = xmicro yp (31)

where M = 0 1 3 + n micro = 0 1 2 3 and p = 1 2 n and considera scalar field Φ(x) with mass m in five dimensions Its dynamics in a flatspacetime are described by the Klein-Gordon equation

(5) +m2)

Φ(xmicro y) =(

part2t minusnabla2 minus part2

y +m2)

Φ(xmicro y) = 0 (32)

With the fifth dimension compactified on a circle with circumference 2πRie

y sim y + 2πR (33)

any function of y can be Fourier series expanded and the scalar field is de-composed as

Φ(xmicro y) =sum

Φ(n)(xmicro) eminusinRy (34)

Each Fourier component Φ(n)(x) thus separately fulfills the four-dimensionalKlein-Gordon equation

Φ(n)(xmicro) =(

part2t minusnabla2 +m2

Φ(n)(xmicro) = 0 (35)

where the masses are

m2n = m2 +

That is a single higher-dimensional field appears in the four-dimensional de-scription as an (infinite) tower of more massive KK states Φ(n) This fits wellwith the expectation that very small extra dimensions should not affect lowenergy physics (or large distances) as it would take high energies to producesuch new heavy states In general the exact structure of the KK tower willdepend on the geometry of the internal dimensions

Feature (2) that the gravitational 1r potential will be affected at smalldistances is also straightforward to realize At small distances r ≪ R thecompactification scale is of no relevance and the space should be fully (3+n)-dimensional rotation invariant At a small distance rn from a mass m thegravitational potential is thus determined by the Laplace equation nabla2V (r) =0 where nabla2 = part2

x1 + part2x2 + + part2

xn+3 The solution is

V (r) sim minusG4+nm

rn ≪ R (37)

where G4+n is the fundamental (4 + n)-dimensional gravitational constantand r2n = r2 + y2

1 + y22 + + y2

n is the radial distance The potential willthus qualitatively behave as in Eq (37) out to the compactification radiusR where the potential becomes sim G4+n

mRn+1 At larger distances space is

effectively three-dimensional and Newtonrsquos usual 1r-law is retained

V (r) sim minus G4+n

rr ≫ R (38)

At small distances the presence of the extra dimensions thus steepens thegravitational potential Such a deviation from Newtonrsquos law is conventionallyparameterized as [134]

V (r) prop 1

1 + αeminusrλ)

Experiments have today tested gravity down to sub-millimeter ranges andhave set upper limits in the (λ α)-plane At 100 microm the deviation fromNewtonrsquos law cannot be larger than about 10 ie (λ |α|) (100 microm 01)[135]

By comparing Eq (38) with Newtonrsquos law V (r) = minusGmr we find that the

ordinary Newtonrsquos constant scales with the inverse of the extra-dimensionalvolume according to

G equiv G4 prop G4+n

Rn (310)

This is exactly feature (3) for Newtonrsquos constant G ndash fundamental constantsvary with the inverse of the volume of the internal space sim Rminusn

Another instructive way of seeing the origin of the features (1)ndash(3) is towork directly with the Lagrangian [136] Take for simplicity a φ4-theory in a(4+n)-dimensional Minkowski spacetime

d4+nxradic

minusg[

2partM φ part

M φminus m2

2φ2 minus λ4+n

With the extra n dimensions compactified on an orthogonal torus with allradii equal to R the higher-dimensional real scalar field φ can be Fourierexpanded in the compactified directions as

φ(x y) =1radicVn

φ(~n)(x) exp

i~n middot ~yR

Here Vn = (2πR)n is the volume of the torus and ~n = n1 n2 nn isa vector of integers ni The coefficients φ(~n)(x) are the KK modes whichin the effective four-dimensional theory constitute the tower of more massiveparticle fields Substituting the KK mode expansion into the action (311)and integrating over the internal space we get

d4xradicminusg

partmicroφ(0))2

minus m2

2(φ(0))2

partmicroφ(~n))(

partmicroφ(~n))lowast

minusm2~nφ

(~n)φ(~n)lowast]

minus λ4

4(φ(0))4 minus λ4

4(φ(0))2

φ(~n)φ(~n)lowast +

where the dots stand for the additional terms that do not contain any zeromodes (φ(0)) of the scalar field The masses of the modes are given by

m2~n = m2 +

R2 (314)

The coupling constant λ4 of the four-dimensional theory is identified to thecoupling constant λ5 of the initial multidimensional theory by the formula

λ4 =λ4+n

Vn (315)

We thus again find that the four-dimensional coupling constant is inverselyproportional to the volume Vn spanned by the internal dimensions The sameis true for any coupling constant connected to fields in higher dimensions

Section 33 Modern Extra-Dimensional Scenarios 45

33 Modern Extra-Dimensional Scenarios

The original attempt by Kaluza and Klein of unifying general relativity andelectromagnetism the only known forces at the time by introducing a fifthdimension was intriguing After the discovery of the weak [137ndash139] andstrong forces [140141] as gauge fields it was investigated whether these twoforces could be fit into the same scheme It was found that with more extradimensions these new forces could be incorporated [142143] This developedinto a branch of supergravity in the 1970s which combined supersymmetryand general relativity into an 11 dimensional theory The 11-dimensionalspacetime was shown to be the unique number of dimensions to be able tocontain the gauge groups of the SM [143144]dagger The initial excitement over the11-dimensional supergravity waned as various shortcomings were discoveredFor example there was no natural way to get chiral fermions as needed in theSM nor did supergravity seem to be a renormalizable theory

For some time the ideas of extra dimensions then fell into slumber beforethe rise of string theory in the 1980s Due to consistency reasons all stringtheories predict the existence of new degrees of freedom that are usually takento be extra dimensions The reason for the popularity of string theory is itspotential to be the correct long time searched for quantum theory for gravityThe basic entities in string theory are one-dimensional strings instead of theusual zero-dimensional particles in quantum field theory and different oscil-lation modes of the strings correspond to different particles One advantageof having extended objects instead of point-like particles is that ultra-violetdivergences associated with the limit of zero distances get smeared out overthe length of the string This could solve the problem of unifying quantumfield theory and general relativity into a firm physical theory For superstringtheories it was shown that the number of dimensions must be 10 in orderfor the theory to be self-consistent and in M-theory the spacetime is 11 di-mensional The extra dimensions beyond the four observed which have tobe made unobservable and are commonly compactified on what is called aCalabi-Yau manifold It might also be possible that non-perturbative lowerdimensional objects called branes can host the four-dimensional world thatwe experience

With the hope that string theory will eventually turn out to be a morefundamental description of our world many string-inspired phenomenologicalscenarios have been developed For example the concept of branes in stringtheory gave room for addressing the strong hierarchy problem from a newgeometrical perspective Branes are membranes in the higher dimensionalspacetime to which open strings describing fermions and vector gauge fieldsare attached but closed strings describing gravitons are not In 1998 Arkani-Hamed Dimopoulos and Dvali (ADD) [145] proposed a string-inspired model

dagger Today many techniques exist to embed the SM gauge group in supergravity in anynumber of dimensions by eg the introduction of D-branes [130]

where all the SM particle fields are confined to a four-dimensional brane in ahigher dimensional flat spacetime Only gravity is diluted into the additionalextra dimensions and therefore the gravitational force is weakened comparedto the other known forces With the extra dimensions spanning a large enoughvolume (see Eq (310)) the gravitational scale could be brought down tothe electroweak scale ndash explaining the strong hierarchy of forces For twoextra dimensions spanning a volume Vn=2 sim (1 microm)2 the fundamental energyscale for gravity Mpl is brought down to the electroweak scale ie Mpl sim(M2

pl middot Vn)1(2+n) sim (1 TeV)2Even with small extra dimensions the strong hierarchy problem can be

addressed in a geometrical way In 1999 Randall and Sundrum proposed amodel with one extra dimension that ends at a positive and a negative tensionbrane (of which the latter is assumed to contain our visible SM) This modelis often referred to as the Randall and Sundrum I model or RS I model [146]The five-dimensional metric is not separable in this scenario but has a warpfactor eminusw|y| connecting the fifth dimension to the four others

ds2 = eminusw|y|ηmicroνdxmicrodxν + dy2 (316)

The bulk is a slice of an anti de Sitter space (AdS5) ie a slice of a space-time with constant negative curvature After integrating out the extra di-mension in this model the connection between the four-dimensional and thefive-dimensional fundamental Planck mass (that is the relationship betweenthe scale for gravity at the brane and in the bulk) is found to be

M2pl =

1 minus eminus2wL)

where L is the separation between the branes Here Mpl depends only weaklyon the size L of the extra dimension (at least in the large L limit) this isa completely different relation than found in the ADD model To addressthe strong hierarchy problem in the RS I model they looked at how the massparameters on the visible brane are related to the physical higher-dimensionalmasses In general the mass parameter m0 in the higher-dimensional theorywill correspond to a mass

m = eminuswLm0 (318)

when interpreted with the metric on our visible brane (see [146]) This meansthat with wL of about 50 it is possible to have all fundamental mass param-eters of the order of the Planck mass and by the warp factor still producemasses of the electroweak scale Instead of having large flat extra dimensionsthe large hierarchy between the electroweak and the Planck energy scales ishere induced by the large curvature of the extra dimension ie the warpfactor eminuswL In a follow-up paper [147] Randall and Sundrum demonstratedthat the metric (316) could also allow for a non-compact extra dimensionThis possibility is partly seen already in Eq (317) where it is no problem to

Section 34 Motivations for Universal Extra Dimensions 47

take the indefinitely large L limit If the curvature scale of the AdS space issmaller than a millimeter then Newtonrsquos gravitational law is retained withinexperimental uncertainties [147148] The reason why the extra dimension canbe non-compact is that the curved AdS background supports a localizationof the higher dimensional gravitons in the extra dimensions In this so-calledRS II case the hierarchy problem is not addressed

Another more recent extra-dimensional scenario is the model by DvaliGabadadze and Porrati (DGP) [149] where gravity gets modified at largedistances The action introduced is one with two gravity scales one five-dimensional bulk and one four-dimensional brane gravity scale This modelcould be used to discuss an alternative scenario for the cosmological problemof a late-time acceleration of our expanding Universe However problemssuch as violation of causality and locality make it theoretically less attractiveFor a recent review of braneworld cosmology see [150]

As mentioned above another approach for extra-dimensional phenomenol-ogy is to look at models where all SM particles can propagate in a higherdimensional space This is the case in the UED model [126] The UED modelwill be of special interest in this thesis as this model can give rise to a newdark matter candidate This dark matter candidate will not only be dis-cussed in detail but the UED model will also serve as the starting point to gothrough multidimensional cosmology the particle SM structure dark matterproperties and dark matter searches in general

34 Motivations for Universal Extra Dimensions

Even though the idea of the UED model is a conceptually simple extensionof the SM ndash basically just add extra dimensions ndash it provides a frameworkto discuss a number of open questions in modern physics Theoretical andpractical motivations to study the UED model include

bull Simplicity only 2 new parameters in its minimal version (R and Λcut)

bull A possibility to achieve electroweak symmetry breaking without anyneed to add an explicit Higgs field [151]

bull Proton stability can be achieved even with new physics coming in atlow-energy scales With the SM applicable up to an energy scale ΛSMthe proton would in general only be expected to have a lifetime of

τp sim 10minus30 years (ΛSMmp)4 (319)

where mp is the proton mass [152] In [153] it was shown that globalsymmetries within UED instead can lead to a proton lifetime of

τp sim 1035 years

500 GeV

)12(ΛcutR

where Λcut is the cut-off energy scale of the UED model This illus-trates that the UEDs model can (for relevant R Λcut values) fulfill theconstraint on proton stability of τp amp 1033 years [152154]

bull It has addressed the (unanswered) question of why we observe threeparticle generations In order to cancel gauge anomaliesDagger that appearin an even number of UEDs it has been shown in the case of two extradimensions that the number of generations must be three [155]

bull Unlike most other extra-dimensional scenarios single KK states cannotbe produced but must come in pairs This means that indirect con-straints such as those coming from electroweak precision observablesare not particularly strong As a result KK states of SM particles canbe much lighter than naıvely expected Such lsquolightrsquo new massive parti-cles should if they exist be produced as soon as the upcoming LargeHadron Collider (LHC) operates This is especially true in the region ofparameter space favored by having the dark matter in the form of KKparticles

bull Studies of the UED could lead to insights about supersymmetry whichtoday is the prime candidate for new physics at the TeV scale There aremany similarities between supersymmetry and the UED model whichhave even led some people to dub the UED scenario lsquobosonic super-symmetryrsquo [156] With two such similar models in hand it gives agreat possibility to study how to experimentally distinguish differentbut similar models In particular such studies have triggered work onhow to measure spin at LHC By studying the gamma-ray spectrumfrom annihilating dark matter particles in the UED scenario we havelearned more about similar phenomena within supersymmetry (see egPaper II versus Paper IV)

bull The UED model naturally encompasses a dark matter particle candi-date Although this was not the original motivation for the model thisis perhaps one of the most attractive reasons to study it

The following three chapters will be devoted to discussions of cosmologicalaspects of UEDs and its dark matter properties in detail

Dagger A gauge anomaly occurs when a quantum effect such as loop diagrams invalidates theclassical gauge symmetry of the theory

C h a p t e r

Cosmology withHomogeneous

Extra Dimensions

For every physical theory it is crucial that it is consistent with observationalconstraints A generic prediction in extra-dimensional models is that at leastsome of the fundamental coupling constants vary with the volume of the extra-dimensional space Due to the tight observational constraints on the potentialvariability of fundamental coupling lsquoconstantsrsquo it is necessary that the sizeof the extra dimensions stays close to perfectly static during the cosmologicalhistory of our Universe If the extension of the extra dimensions is consider-ably larger than the Planck scale it is reasonable that their dynamics shouldbe governed by classical general relativity However in general relativity it isnontrivial to obtain static extra dimensions in an expanding universe There-fore the evolution of the full spacetime in a multidimensional universe mustbe scrutinized in order to see if general relativity can provide solutions thatare consistent with current observational constraints The results presentedin this chapter coming partly from Paper I show that a homogeneous mul-tidimensional universe only can have exactly static extra dimensions if theequations of state in the internal and external space are simultaneously fine-tuned For example in the case of the UED model it is not expected that theextra dimensions stay static unless some stabilization mechanism is includedA brief discussion of the requirements of such stabilization mechanisms con-cludes this chapter

41 Why Constants Can Vary

With coupling constants defined in a higher-dimensional theory the effectivefour-dimensional coupling lsquoconstantsrsquo will vary with the size of the extra-dimensional volume In a UED scenario where all particles can propagate inthe bulk all force strengths (determined by their coupling constants) pick up

50 Cosmology with Homogeneous Extra Dimensions Chapter 4

a dependence on the internal volume This is not the case in general sincesome or all of the force carrying bosons might be confined to a membrane andtherefore insensitive to the full bulk However since gravity is associated withspacetime itself the gravitational coupling constant (ie Newtonrsquos constant)will inevitably depend on the size of the internal space

Consider specifically (4+n)-dimensional Einstein gravity in a separablespacetime M4(x)timesKn(x) with the internal space Kn compactified to form an-dimensional torus with equal radii R If the matter part is confined to ourfour dimensions then the extra spatial volume ffects only the gravitationalpart of the action

16πG4+n

d4+nxradic

minusgR (41)

where R is the Ricci scalar (calculated from the higher dimensional met-ric gMN ) and G4+n is the higher dimensional gravitational coupling con-stant By Fourier expanding the metric in KK modes with the zero (ie

y-independent) mode denoted by g(0)MN the Ricci scalar can be expanded

as R[gMN ] = R[g(0)microν ] + The missing terms represented by the dots

are the nonzero KK modes and in Section 46 it is shown that they corre-spond to new scalar fields so-called radion fields appearing in the effectivefour-dimensional theory After integrating over the internal dimensions inEq (41) the four-dimensional action takes the form

d4xradicminusg

16πGR[g(0)

microν ] +

In analogy with Eq (310) the four-dimensional Newtonrsquos constant G is givenby

G =G4+n

Vn(43)

and as before Vn =int

d4xradicminusg(n) prop (2πR)n is the volume of the internal

space and g(n) is the determinant of the metric on the internal manifold Alsoin the previous chapter we saw in the example of a φ4-theory how the volumeVn of the extra dimensions rescale higher dimensional coupling constants λ4+n

into the dynamical (ie Vn dependent) four-dimensional coupling constant

λ =λ4+n

Vn (315)

Similar relations hold also for other types of multidimensional theories

42 How Constant Are Constants

Numerous experimental and observational bounds exist on the allowed timevariation of fundamental constants and thus on the size variation of extra

Section 42 How Constant Are Constants 51

dimensions Some of these constraints are summarized below (for a morecomplete review see eg [157 158])

The constancy of Newtonrsquos constant G has been tested in the range fromlaboratory experiments to solar system and cosmological observationslowast Lab-oratory experiments have mainly focused on testing the validity of Newtonrsquos1r2 force law down to sub-millimeter distances but so far no spatial (ortemporal) variation has been detected [161] In the solar system monitor-ing of orbiting bodies such as the Moon Mercury and Venus sets an upperlimit of |∆GG| 10minus11 during the last decades of observations [157] Oncosmological scales the best limit comes perhaps from BBN which puts aconstraint of |∆GG| 02 between today and almost 14 billion years ago(ie zBBN sim 108 minus 1010) [160] The limit from BBN is derived from theeffect a change in Newtonrsquos constant has on the expansion rate of the Uni-verse and accordingly on the freeze-out temperature which would affect theabundance of light elements observed today It is worth noting that somemultidimensional models might retain the same expansion rate as in conven-tional cosmology despite an evolving gravitational constant G and thereforesome stated constraints on G might not be directly applicable [162]

The constraints on the possible variation of the electromagnetic couplingconstant or rather the fine structure constant αEM are both tight and covermuch of the cosmological history One fascinating terrestrial constraint comesfrom studies of the isotopic abundances in the Oklo uranium mine a prehis-torical natural fission reactor in central Africa that operated for a short timeabout 2 times 109 yr ago From the αEM dependence on the capture rate of neu-trons of eg 149

62Sm an upper limit of |∆αEMαEM| 10minus7 has been derived[163164] Another very suitable way of testing the constancy of the fine struc-ture constant is by analyzing light from astrophysical objects since the atomicspectra encode the atomic energy levels Analyses of different astrophysicalsources has put limits of |∆αEMαEM| 10minus3 up to a redshift z sim 4 [157]Worth noticing is that in the literature there have even been claims of anobserved variation in αEM Webb et al [165 166] and also later by Murphyet al [167] studied relative positions of absorption lines in spectra from dis-tant quasars and concluded a variation ∆αEMαEM = (minus54 plusmn 12)times 10minus6 atredshift 02 lt z lt 37 However this is inconsistent with other analyses ofquasar spectra For example Chand et al [168] and Srianand et al [169] get∆αEMαEM = (minus06 plusmn 06) times 10minus6 at redshift 04 lt z lt 23 In cosmologyCMB [170171] and BBN [160172] set the constraint |∆αEMαEM| 10minus2 atredshift zCMB sim 1000 and zBBN sim 1010 respectively

lowast Strictly speaking it makes no sense to consider variations of dimensionful constantssuch as G We should therefore give limits only on dimensionless quantities like thegravitational coupling strength between protons Gm2

p(~c) or specify which other cou-pling constants that are assumed to be truly constant The underlying reason is thatexperiments in principle can count only number of events or compare quantities withthe same dimensionality [157 159 160]

Less work has been put into constraining the weak (αw) and the strong(αs) coupling constants This is partly due to the more complex modeling forexample in the weak sector there are often degeneracies between the Yukawacouplings and the Higgs vacuum expectation value and for strong interactionsthere is the strong energy dependence on αs Nonetheless existing studies ofthe BBN ndash where a change in the weak and the strong interactions shouldhave observable effects ndash indicate no changes in αw or αs [157]

Of course we should keep in mind that there are always some underlyingassumptions in deriving constraints and that the entire cosmological historyhas not been accessible for observations (accordingly much less is known attimes between epochs of observations) Despite possible caveats it is still fairto say that there exists no firm observational indication of any variation ofany fundamental constant ranging from the earliest times of our Universe untiltoday These constraints are directly translated into the allowed variation insize of any extra dimension which coupling constants depend on Thereforein conclusion observations restrict the volume of extra spatial dimensions tobe stabilized and not vary by more than a few percentages throughout thehistory of our observable Universe

43 Higher-Dimensional Friedmann Equations

We now turn to the equations of motion that describe the evolution of space-time in a multidimensional universe Standard cosmology is well described byan isotropic and homogeneous (four-dimensional) FLRW model Any scenariowith internal spatial dimensions must therefore mimic this four-dimensionalFLRW model and at the same time be in agreement with the above constraintson the size stability of the extra dimensions The gravitational dynamics of amultidimensional cosmology will be assumed to be governed by the ordinaryEinstein-Hilbert action with n extra dimensions

d4+nxradic

minusg(

R + 2Λ + 2κ2Lmatter

where the notation κ2 = 8πG has been introduced By varying the actionwith respect to the metric we derive Einsteinrsquos field equations in d = 3+n+1dimensions

RAB minus 1

2R gAB = κ2Tmatter

AB + Λ gAB equiv κ2TAB (45)

The higher dimensional cosmological constant Λ is here taken to be a part ofthe energy momentum tensor TAB

To address the question whether it is possible to retain ordinary cos-mology together with static extra dimensions let us consider a toy-modeldagger

dagger Some similar studies of multidimensional cosmologies can be found in eg [162 173ndash181]

Section 44 Static Extra Dimensions 53

where the multidimensional metric is spatially homogeneous but has twotime-dependent scale factors a(t) and b(t)

gMN dxMdxN = gmicroν(x) dxmicrodxν + b2(x)gpq(y) dypdyq (46)

= dt2 minus a2(t)γijdxidxj minus b2(t)γpqdy

Here γij is the usual spatial part of the FLRW metric (111) for the ordinarylarge dimensions and γpq is a similar maximally symmetric metric for the inter-nal extra-dimensional space The most general form of the energy-momentumtensor consistent with the metric is in its rest frame

T00 = ρ Tij = minuspa a2γij T3+p 3+q = minuspb b2γpq (47)

This describes a homogeneous but in general anisotropic perfect fluid witha 3D pressure pa and a common pressure pb in the n directions of the extradimensions

With the above ansatz the nonzero components of the higher-dimensionalFriedmann equations (45) can be written as

b+n(nminus 1)

= κ2ρ (48a)

b+n(nminus 1)

= κ2pa (48b)

b+ (nminus 1)

n+ 2(ρminus 3pa + 2pb) (48c)

where dots (as in a) denote differentiation with respect to the cosmic time tand as usual the curvature scalars kab are +1 0minus1 depending on whetherthe ordinaryinternal spatial space is positively flat or negatively curved

With the extra dimensions exactly static (b equiv 0) the first two equations(48a48b) reduce to the ordinary Friedmann equation (111) with an effectivevacuum energy due to the internal curvature kb If also the third equation(48c) can be simultaneously satisfied this seems to be what was looked for ndasha solution to Einsteinrsquos field equations that has static extra dimensions andrecovers standard cosmology

44 Static Extra Dimensions

Since the internal curvature parameter kb is just a constant in Eq (48c)exactly static extra dimensions (b equiv 0) are only admitted if also

ρminus 3pa + 2pb equiv C

= (n+ 2)(nminus 1)kbκ2b2

= constant

stays constant This equation will severely restrict the possible solutions if wedo not allow the internal pressure to be a freely adjustable parameter

Similarly to standard cosmology the energy content will be taken to be amulticomponent perfect fluid Each matter type will be specified by a constantequation of state parameter w(i) such that

p(i)a = w(i)

a ρ(i) and p(i)b = w

(i)b ρ(i) (410)

This permits the equation TA

0A = 0 to be integrated to

ρ(i) prop aminus3(1+w(i)a )bminusn(1+w

) (411)

The total energy ρ and pressures pa and pb are the sum of the individual

matter components ie ρ =sum

i ρ(i) and pab =

i w(i)abρ

(i) With a multi-

component fluid allowed in Eq (49) [that is C equiv sum

i C(i) where C(i) equiv

ρ(i) minus 3p(i)a + 2p

(i)b ] we could in principle imagine a cancellation of the time

dependency of individual C(i) such that C still is time independent Howeverin the case of static extra dimensions this requires that canceling terms have

the same w(i)a and therefore it is convenient to instead define this as one

matter component with the given w(i)a and then adopt an effective w

With this nomenclature each matter component (i) must separately fulfillEq (49) to admit static extra dimensions

ρ(i)(

1 minus 3w(i)a + 2w

equiv C(i) = constant (412)

From Eq (411) and (412) it is clear that static internal dimensions in

an evolving universe (a 6= 0) requires that either w(i)a = minus1 so that ρ(i) is

constant or that the equations of state fulfill (1 minus 3w(i)a + 2w

(i)b ) = 0 Thus

in summary

In a homogeneous non-empty and evolving multidimensional cos-mology with n exactly static extra dimensions the equation of statefor each perfect fluid must fulfill either

I w(i)a = minus1 or II w

(i)b =

3w(i)a minus 1

2 (413)

Note that a matter component that fulfills case I (and not II) impliesthat the internal space must be curved since it implies C 6= 0 in Eq (49)Similarly it is not possible to have a multidimensional cosmological constant

(w(i)a = w

(i)b = minus1) together with flat extra dimensions (kb = 0)

To actually determine the expected equations of state in a multidimen-sional scenario we have to further specify the model In the next section wetake a closer look at the scenario of UED and KK states as the dark matter

Section 45 Evolution of Universal Extra Dimensions 55

45 Evolution of Universal Extra Dimensions

In the UED model introduced in the previous chapter all the SM particlesare allowed to propagate in the extra dimensions Momentum in the directionof the compactified dimensions gives rise to massive KK states in the effectivefour-dimensional theory Furthermore if the compactification scale is in theTeV range then the lightest KK particle (LKP) of the photon turns out to bea good dark matter candidate The exact particle field content of the UEDmodel and the properties of the dark matter candidate are not important atthis point and further discussion on these matters will be postponed to thefollowing chapters However from the mere fact that the KK dark matter par-ticles gain their effective four-dimensional masses from their own momentumin the extra dimensions it is possible to predict the pressure

The classical pressure in a direction xA is defined as the momentum fluxthrough hypersurfaces of constant xA In case of isotropy in the 3 ordinarydimensions and also ndash but separately ndash in the n extra dimensions we find that(see Paper I)

3pa + npb = ρminuslang

For the SM particles with no momentum in the extra dimensions there is nocontribution to the pressure in the direction of the extra dimensions (pb = 0)whereas for KK states with momentum in the extra dimensions we canalways ignore any SM mass m compared to their total energy E sim TeV (iemomentum in the extra dimensions) Thus with KK states being the darkmatter the energy in the universe will not only be dominated by relativisticmatter (m2E ≪ ρ) during the early epoch of radiation domination but alsoduring matter domination in the form of LKPs Thus Eq (414) gives theequation of states

3 wb = 0 (4D Radiation dominated) (415a)

wa = 0 wb =1

n(4D Matter dominated) (415b)

During four-dimensional radiation domination the requirement II in (413)for static extra dimensions is actually satisfied whereas this is clearly not thecase during what looks like matter domination from a four-dimensional pointof view

With exactly static extra dimensions being ruled out a numerical evolu-tion of the field equations (48) was performed in Paper I to test if nearlystatic extra dimensions could be found The freeze-out of the LKPs takes placeduring four-dimensional radiation domination and a tiny amount of the en-ergy density in the universe is deposited in the form of the LKPs (which muchlater will dominate the energy during matter domination) Consequently theinitial condition for the numerical evolution is a tiny amount of LKPs with

10 15 2000

Figure 41 Evolution of the ordinary and internal scale factors (a andb respectively) as well as the fractional LKP energy density (ρDMρ) forn =1 (thin) 2 (medium) and 7 (thick) extra dimensions The evolutionis within a model with Λ = ka = kb = 0 and initially ρDMρ = 10minus7During a long period of radiation domination the extra dimensions staynearly static and a evolves in accordance with standard cosmology WhenρDMρ gt 01 however neither a nor b show the desired behavior a b andt are dimensionless arbitrarily scaled to unity as their initial conditionsFigure from Paper I

stable extra dimensions in the regime of radiation domination However notsurprisingly as soon as the relative amount of energy in dark matter becomessignificant (sim10) the size of the extra dimensions start escalating as is shownin Fig 41 Such an evolution would severely violate the constraints on bothαEM and G presented in Section 42 In conclusion not even approximatelystatic extra dimensions can be found in this setup and therefore some extramechanism is needed in order to stabilize the extra dimensions and reproducestandard cosmology

46 Dimensional Reduction

To study dynamical stabilization mechanisms it is more practical to considerthe equations of motion after dimensional reduction of the action In thissection such a dimensional reduction is performed before we return to thestabilization of internal spaces in the next section

Integrating over the internal dimensions in Eq (44) with the metric

Section 46 Dimensional Reduction 57

ansatz (46) gives (see eg [182 183])

d4xradicminusgbn

R + bminus2R + n(nminus 1)bminus2partmicrobpartmicrob+

2Λ + 2κ2Lmatter

where κ2 equiv κ2int

dnyradicminusg and R and R are the Ricci scalars constructed

from gmicroν and gpq respectively With the assumed metric the internal Ricci

scalar can also be written as R = n(nminus 1)kb A conformal transformation tothe new metric gmicroν = bngmicroν takes the action to the standard Einstein-Hilbertform (so-called Einstein frame) ie the four-dimensional Ricci scalar appearswith no multiplicative scalar field [182]

d4xradicminusg

2κ2R[gmicroν ] minus

2partmicroΦpart

microΦ minus Veff(Φ)

This is our dimensionally reduced action It gives ordinary four-dimensionalgeneral relativity coupled to a new scalar field (the radion field)

Φ equivradic

n(n+ 2)

2κ2ln b (418)

with an effective potential

Veff(Φ) = minus R2κ2

eminusradic

2(n+2)n

κΦ +1

Λ minus κ2Lmatter

eminusradic

2nn+2 κΦ (419)

Although the equations of motion for the metric gmicroν and Φ derived from thisnew action can be directly translated to the higher-dimensional Friedmannequations (48) the equation of motion for the radion field

Φ = minus part

partΦVeff (420)

where = 1radicminusgpartmicroradicminusgpartmicro now has a much more intuitive interpretation

regarding when stable extra dimensions are expected As the internal scalefactor depends only on t we have Φ = Φ + 3HΦ We have thus foundthat the scalar field Φ has the same equation of motion as a classical particlemoving in a potential Veff with a friction term given by three times the Hubbleexpansion rate H If the effective potential Veff(Φ) has a stationary minimumthen the radion field and thus the extra dimension has a static and stablesolution

Before we continue the discussion in the next section on how to obtainstable extra dimensions it is interesting to note a subtlety regarding con-formal transformations From the higher-dimensional perspective the naıve

guess would be that the four-dimensional metric part gmicroν with the scale factora(t) describes the effective four-dimensional space However after the con-formal transformation into the Einstein frame (ie the action written as inEq (417)) suggests that it should rather be the effective four-dimensionalmetric gmicroν with the scale factor a(t) = a(t)bn2(t) that describes the physi-cal four-dimensional space The problem of which of the conformally relatedframes should be regarded as the physical one or if they both are physicallyequivalent is to some extent still debated [184ndash186] However for the dis-cussion in this chapter the distinction between a(t) and a(t) is not of anyimportance because the tight observational constraints on the variation ofcoupling constants imply that the extra dimensions can always be assumedto be almost static In that case the different frames are in practice identical

47 Stabilization Mechanism

Let us briefly investigate what is needed to stabilize the extra dimensionsThe effective potential for Φ in Eq (419) can be rewritten as

Veff(b(Φ)) = minus R2κ2

bminus(n+2) +sum

ρ(i)0 aminus3(1+w(i)

a )bminusn2 (1minus3w(i)

a +2w(i)b

) (421)

This expression follows from the definition in Eq (418) that Lmatter = minusρ =minussumi ρ

(i) and that each fluid component dependency on a = abminusn2 and Φis given by

ρ(i) = ρ(i)0 aminus3(1+w(i)

a )bminusn(1+w(i)b

) =κ2

κ2ρ(i)0 aminus3(1+w(i)

a )bn2(1+3w(i)a minus2w

where the four-dimensional densities ρ(i)0 are defined by ρ

(i)0 equiv ρ

(i)0 Vn equiv

ρ(i)0 κ2κ2

We can now easily verify that partmicroΦ equiv 0 in Eq (420) implies the samecondition (49) as obtained earlier for exactly static extra dimensions How-ever in the perspective of the radion field in its effective potential it is alsoobvious that even the static extra-dimensional solution is only stable if thepotential has a stationary minimum In fact the static solution found earlier(for a radiation-dominated universe with wa = 13 and wb = 0) is not reallya stable minimum since the effective potential is totally flat (when kb = 0)During matter domination in the UED model with wa = 0 and wb = 1nthe radion field is of course not in a stationary minimum either In the caseof a higher dimensional cosmological constant wa = wb = minus1 there couldin principle be a stationary minimum if ρΛ(a) lt 0 and R lt 0 but then theeffective four-dimensional vacuum energy from the cosmological constant andthe internal curvature contribution will sum up to be negative This wouldgive an anti de Sitter space which is not what is observed

Section 47 Stabilization Mechanism 59

At this point let us ad hoc introduce a stabilization mechanism in theform of a background potential V bg(Φ) for the radion field If this potentialhas a minimum say at Φ0 the contribution to the potential is to a first-orderexpansion around this minimum given by

Vbg(Φ) ≃ Vbg(Φ0) +m2

2(Φ minus Φ0)

2 where m2 equiv part2Vbg

part2Φ

The minima of the total potential are found from V primetot = V prime

eff(Φ)+V primebg(Φ) = 0

and can implicitly be written as

Φmin ≃ Φ0+(1 minus 3wa + 2wb)

2 + nρ(a bmin)minus

2nκ2bminus(n+2)min (423)

Here the sum over different fluid components has been suppressed for brevityand only one (dominant) contributor ρ(a bmin) is included Due to the cou-pling to the matter density ρ(a b) the minimum Φmin will thus in general betime dependent For small shifts of the minimum it can be expressed as

∆Φmin ≃ (1 minus 3wa + 2wb)

2 + ntimes ∆ρ(a bmin) (424)

Thus a change in b(Φ) of less than 1 between today and BBN correspondingto ∆ρ sim 1019 eV4 would only require the mass in the stabilization potentialto be m amp 10minus16 eV ndash a very small mass indeedDagger

In fact since a light mass radion field would mediate a new long-range(fifth) force with the strength of about that of gravity sub-millimeter tests ofNewtonrsquos law impose a lower mass bound of m amp 10minus3 eV [187] (see also [188]for more cosmological aspects on radions in a UED scenario)

Still only a fairly shallow stabilization potential is needed to achieve ap-proximately static extra dimensions for a radion field that tracks its potentialminimum The stabilization mechanism could be a combination of differentperfect fluids with tuned w

(i)a and w

(i)b to produce a potential minimum or one

of many proposed mechanisms Examples are the Freund-Rubin mechanismwith gauge-fields wrapped around the extra dimensions [189] the Goldberger-Wise mechanism with bulk fields interacting with branes [190] the Casimireffect in the extra dimensions [191] quantum corrections to the effective po-tential [192] or potentially something string theory related (like the Kachru-Kallosh-Linde-Trivedi (KKLT) model with nonperturbative effects and fluxesthat stabilizes a warped geometry [193]) Whichever the mechanism mightbe it seems necessary to introduce an additional ingredient to stabilize extradimensions In the following discussions of the UED model it will be implic-itly assumed that the extra dimensions are stabilized and that the mechanismitself does not induce further consequences at the level of accuracy considered

Dagger In general a small ∆bb shift requires m amp

radic2(1minus3w

(i)a +2w

(2+n)κ2∆ρ(i)

C h a p t e r

Quantum Field Theoryin Universal Extra

Dimensions

We now turn to the theoretical framework for the UED model Proposedby Appelquist Cheng and Dobrescu [126] in 2001 the UED model consistsof the standard model in a higher dimensional spacetime As all standardmodel particles are allowed to propagate into the higher-dimensional bulk inthis model this means that when it is reduced to a four-dimensional effectivetheory every particle will be accompanied by a Kaluza-Klein tower of iden-tical but increasingly more massive copies Conservation of momentum inthe direction of the extra space dimensions imply that heavier Kaluza-Kleinstates can only be produced in pairs This chapter develops the effective four-dimensional framework and will simultaneously give an overview of the fieldstructure of the standard model of particle physics

51 Compactification

The simplest reasonable extension of spacetime is to add one extra flat dimen-sion compactified on a circle S1 (or a flat torus in more extra dimensions) withradius R However such a compactification of a higher-dimensional versionof the SM would not only give new massive KK particles but also unwantedlight scalar degrees of freedom and fermions with the wrong chirality

The extra scalar degrees of freedom that appear at low energies for afour-dimensional observer are simply the fifth component of any higher di-mensional vector fields which transform as scalars under four-dimensionalLorentz transformations Such massless scalar fields (interacting with usualgauge strengths) are not observed and are strongly ruled out by the extralsquofifthrsquo force interaction they would give rise to [194]

In the SM the fermions are chiral meaning that the fermions in the SU(2)doublet are left-handed whereas the singlets are right-handed (and the oppo-

62 Quantum Field Theory in Universal Extra Dimensions Chapter 5

site handedness for anti-fermions) With one extra dimension fermions canbe represented by four-component spinors but the zero modes will consist ofboth left-handed and right-handed fermions Technically the reason behindthis is that the four-dimensional chiral operator is now a part of the higherdimensional Dirac algebra and higher-dimensional Lorentz transformationswill in general mix spinors of different chirality

If the extra dimensions instead form an orbifold then the above problemscan be avoided With one extra dimension the simplest example is an S1Z2

orbifold where Z2 is the reflection symmetry y rarr minusy (y being the coordinateof the extra dimension) Fields can be set to be even or odd under this Z2

symmetry which allows us to remove unwanted scalar and fermionic degreesof freedom and thereby reproduce the particle content of the SM From aquantum field theory point of view the orbifold can be viewed as projectingthe circular extra dimension onto a line segment of length πR stretching be-tween the two fixpoints at y = 0 and y = πR Fields say Φ are then givenNeumann (or Dirichlet) boundary conditions part5Φ = 0 (or Φ = 0) so thatthey become even (or odd) functions along the y-direction

52 Kaluza-Klein Parity

The circular compactification S1 breaks global Lorentz invariance but localinvariance is preserved By Noetherrsquos theorem local translational invariancecorresponds to momentum conservation along the extra dimension A setof suitable base functions are thus ei

nRy which are the eigenstates of the

momentum operator iparty (the integer n is a conserved quantity called the KKnumber) For example a scalar field Φ is expanded as

Φ(xmicro y) =

infinsum

n=minusinfinΦ(n)(xmicro)ei

nRy (51)

We might expect that the fifth component of the momentum should be aconserved quantity in UED However the orbifold compactification S1Z2with its fixpoints breaks translational symmetry along the extra dimensionand KK number is no longer a conserved quantity However as long as thefixpoints are identical there is a remnant of translation invariance namelytranslation of πR which takes one fixpoint to the other Rearranging thesum in Eq (51) into terms that are eigenstates to the orbifold operator PZ2

(ie Φ(y) is even or odd) and simultaneously KK mass eigenstates (iemomentum squared or part2

y eigenstates) gives

Φ(xmicro y) = Φ(0)even(x

micro) +

infinsum

Φ(n)even cos

(n)odd sin

R (52)

Section 53 The Lagrangian 63

Depending on KK level the terms behave as follows under the translationy rarr y + πR

Rrarr cos

n(y + πR)

R= (minus1)n cos

R (53a)

Rrarr sin

n(y + πR)

R= (minus1)n sin

R (53b)

Hence a Lagrangian invariant under the πR translation can only containterms that separately sum their total KK level to an even number In otherwords every term in the UED Lagrangian must have (minus1)ntot = 1 where ntot

is the sum of all the KK levels in a particular term In an interaction termsplit ntot into ingoing and outgoing particles such that ntot = nin +nout andwe have (minus1)nin = (minus1)nout This is known as the conservation of KK parity(minus1)n and will be essential in our discussion of KK particles as a dark mattercandidate

53 The Lagrangian

In the UED model all the SM particles with its three families of fermionsforce carrying gauge bosons and one Higgs boson are allowed to propagatein the extra S1Z2 dimension In such a higher dimensional Lagrangian thegauge Yukawa and quartic-Higgs couplings have negative mass dimensionsand the model is non-renormalizable [3] Therefore should the UED model beviewed as an effective theory applicable only below some high-energy cutoffscale Λcut With the compactification scale 1R distinctly below the cutoffΛcut a finite number of KK states appears in the effective four-dimensionaltheory If only KK states up to Λcut are considered the UED model is fromthis perspective a perfectly valid four-dimensional field theory In the minimalsetup all coupling strengths in the UED model are fixed by the measured four-dimensional SM couplings and the only new parameters are the cutoff andthe compactification scale Λcut and R respectively

For later chapters mainly the electroweak part of the Lagrangian is rel-evant Therefore the fermions the U(1) times SU(2) gauge bosons the Higgsdoublet and their interactions will be discussed in some detail whereas theQuantum Chromo Dynamics (QCD) sector described by the SU(3) gaugegroup is left outlowast Furthermore with the expectation that much of the inter-esting phenomenology can be captured independently of the number of extraspace dimensions we will in the following only consider the addition of oneextra space dimension

lowast The reason for the QCD sector being of little importance in this thesis is because itcontains neither any dark matter candidate nor do the gluons (ie gauge bosons ofQCD) interact directly with photons (which means they are not relevant for the gamma-ray yield calculations made in Paper IIIII)

The UED Lagrangian under consideration will be split into the followingparts

LUED = LBosons + LHiggs + LFermions (54)

In general the inclusion of gauge fixing terms will additionally result in aLghost-term including only unphysical ghost fields In what follows all theseparts are reviewed separately

Gauge Bosons

The first four components of the SU(2) and U(1) gauge fields denoted ArM

and BM respectively must be even under the orbifold projection to retain theordinary four-dimensional gauge fields as zero modes in their Fourier series

PZ2 Armicro(x

micro y) = Armicro(xmicrominusy) (55)

A four-dimensional gauge transformation of such a gauge field is given by

Armicro(xmicrominusy) = Armicro(x

micro y) rarr Armicro(xmicro y) +

gpartmicroα

r(xmicro y) + f rstAsmicroαt (56)

where f rst is the structure constant defining the lie algebra for the genera-tors [σr σs] = if rstσt and g is a five-dimensional coupling constant FromEq (56) we read off that the function α(xmicro y) has to be even under Z2Therefore partyα

r(xmicro y) is odd and hence

Ar5(xmicro y) = minusAr5(xmicrominusy) (57)

to keep five-dimensional gauge invariance The Fourier expansions of thegauge fields in the extra dimension are then

Aimicro(x) =1radic2πR

Aimicro(0)

(x) +1radicπR

infinsum

Aimicro(n)

(xmicro) cosny

R (58a)

Ai5(x) =1radicπR

infinsum

Ai5(n)

(x) sinny

R (58b)

Note that the five-dimensional gauge invariance automatically implies theabsence of the unwanted zero mode scalars Ar5 and B5 in the four-dimensionaltheory

The kinetic term for the gauge fields in the Lagrangian reads

Lgauge = minus1

MN minus 1

MNF rMN (59)

with the U(1) and SU(2) field strength tensors given by

FMN = partMBN minus partNBM (510a)

= partMAiNminus partNA

+ gǫijkAjMAk

N (510b)

Inserting the field expansions (58) and integrating out the extra dimensionresults in the SM Lagrangian accompanied by a KK tower of more massivestates For example the U(1) part of Lgauge becomes

gauge sup minus1

int 2πR

dy FMNFMN

= minus1

partmicroB(0)ν minus partνB

(0)micro

partmicroB(0)ν minus partνB(0)micro)

minus1

infinsum

partmicroB(n)ν minus partνB

(n)micro

partmicroB(n)ν minus partνB(n)micro)

infinsum

partmicroB(n)5 +

RB(n)micro

partmicroB(n)5 +

RB(n)micro

and with a similar expression for the kinetic part of the SU(2) gauge fieldsAt each KK level a massive vector field B(n) with mass n

R appears The

scalars B(n)5 are however not physical because by a gauge transformation

Bmicro rarr B(n)micro minus (Rn)partmicroB

(n)5 the scalar field terms partmicroB

(n)5 can be removed

This actually becomes apparent already from a naıve counting of degrees offreedom A massive four-dimensional vector field has three degrees of freedomwhich is the same as the number of polarization directions for a masslessfive-dimensional gauge field Once the Higgs mechanism with electroweaksymmetry breaking is added (discussed in the next section) the additionalGoldstone scalar fields will form linear combinations with the vector fieldrsquosfifth component to form both physical as well as unphysical scalar fields

For the non-abelian gauge fields there will also be cubic and quartic in-teraction terms and we can identify the ordinary four-dimensional SU(2)coupling constant with

g equiv 1radic2πR

g (512)

This mapping between the four-dimensional couplings in the SM and the five-dimensional couplings is general and will hold for all coupling constants

The Higgs Sector

The electroweak masses of the SM gauge fields are generated by the usualHiggs mechanismdagger The Higgs field is a complex SU(2) doublet that is ascalar under Lorentz transformations

φ equiv 1radic2

χ2 + iχ1

H minus iχ3

χplusmn equiv 1radic2(χ1 ∓ iχ2) (513)

dagger Sometimes also called the Brout-Englert-Higgs mechanism Higgs-Kibble mechanism orAnderson-Higgs mechanism

To have the SM Higgs boson as the zero mode the expansion must be evenunder the Z2 orbifold

φ(x) =1radic2πR

φ(0)(x) +1radicπR

infinsum

φ(n)(x) cosny

R (514)

To make the H field electromagnetically neutral the hypercharge is set toY = 12 for the Higgs doublet and its covariant derivative is

DM = partM minus igArM

minus iY gYBM (515)

where gY and g are the higher dimensional U(1) and the SU(2) couplingconstants respectively and σr are the usual Pauli matrices

0 11 0

0 minusii 0

1 00 minus1

The full Higgs Lagrangian is written as

LHiggs = (DMφ)dagger(DMφ) minus V (φ) (517)

where the potential V (φ) is such that spontaneous symmetry breaking occursThat is

V (φ) = micro2φdaggerφ+ λ(φdaggerφ)2 (518)

where the values of the parameters are such that minusmicro2 λ gt 0 This lsquoMexicanhatrsquo potential has a (degenerate) minimum at

|φ|2 =minusmicro2

2λ2equiv v2

2 (519)

By choosing any specific point in the minimum as the vacuum state aroundwhich the physical fields then are expanded the symmetry is said to be spon-taneously broken With the vacuum expectation value chosen to lie along the

real axis 〈φ〉 = 1radic2

the Higgs field H should be replaced by h + v so

that h is a perturbation around the vacuum and hence represents the Higgsparticle field

Electroweak gauge boson mass terms will now emerge from the kinematicpart of Eq (517) As in the standard Glashow-Weinberg-Salam electroweaktheory three of the gauge bosons become massive

WplusmnM

= 1radic2

A1M∓ iA2

with mass mW =gv

2 (520a)

ZM = cWA3Mminus sWBM with mass mZ =

(520b)

and one orthogonal to Z is massless

AM = sWA3M

+ cWBM (521)

The Weinberg angle θW that appears above is (at tree level) given by

sW equiv sin θW =gY

g2 + g2Y

cW equiv cos θW =g

g2 + g2Y

This relates the electromagnetic and weak coupling constants in such a waythat the gauge field Amicro becomes the photon with its usual coupling to theelectric charge e = sW g = cWgY

In the quadratic part of the kinetic Higgs term

L(kin)Higgs sup

partMh)2

partMχ3 minusmZZM

)2+∣

∣partMχ+ minusmWW

there are unwanted cross-terms that mix vector fields and scalar fields (egZmicropartmicroχ

3) This is the same type of unwanted terms that appeared in theLagrangian of Eq (511) To remove these unwanted terms properly fivedimensional gauge-fixing terms are added to the Lagrangian

Lgaugefix = minus1

GY)2 minus

Gi)2 (524)

Gi =1radicξ

partmicroAimicro minus ξ(

minusmWχi + part5A

(525a)

GY =1radicξ

partmicroBmicro minus ξ(

sWmWχ3 + part5B

(525b)

which is a generalization of the Rξ gauge [3] These are manifestly five-dimensional Lorentz breaking terms However this should not be worryingsince in the effective theory we are restricted to four-dimensional Poincaretransformations anyway and the above expressions are still manifestly four-dimensionally Lorentz invariant If we add up the scalar contributions fromthe gauge bosons in Eq (59) the kinetic part of the Higgs sector in Eq (517)and the gauge fixing terms in Eq (524) and then integrate over the internaldimensions the result is

L(kin)scalar =

infinsum

partmicroh(n)partmicroh(n) minusM

2h(n)2

partmicroG(n)0 partmicroG

(n)0 minus ξM

partmicroG(n)+ partmicroG

(n)minus minus ξM

(n)+ G

(n)minus

infinsum

partmicroa(n)0 partmicroa

(n)0 minusM

2a(n)0

partmicroa(n)+ partmicroa

(n)minus minusM

2a(n)+ a

(n)minus

partmicroA(n)5 partmicroA

(n)5 minus ξ

The above appearing mass eigenstates in the four-dimensional theory aregiven by

a0(n)=M (n)

χ3(n)+

Z(n)5 (527a)

aplusmn(n)

=M (n)

χplusmn(n)+

Wplusmn5

(n) (527b)

G0(n)=

χ3(n) minus M (n)

Z5(n) (527c)

Gplusmn(n)=

χplusmn(n) minus M (n)

Wplusmn5

(n) (527d)

M(n)X equiv

M (n) =n

R (528)

Now let us count the number of physical degrees of freedom in the bosonicsector At KK zero-level we recover the SM with one physical Higgs field h(1)

together with the three massive gauge fields Z(0) Wplusmn(0) that have eaten the

three degrees of freedom from the three unphysical Goldstone bosons G(0)0plusmn

At each higher KK mode there are four additional scalar degrees of freedomcoming from the fifth components of the gauge bosons These scalars formtogether with four KK mode scalars from the Higgs doublet four physical

scalars h(n) a(n)0 and a

(n)plusmn and four unphysical Goldstone bosons A

(n)5 G

and G(n)plusmn that have lost their physical degrees of freedoms to the massive KK

vector modes A(n)micro Z

(n)micro and W

plusmn (n)micro respectively

Ghosts

In the case of non-Abelian vector fields or when Abelian vector fields ac-quire masses by spontaneous symmetry breaking the gauge-fixing terms inEq (524) are accompanied by an extra Faddeev-Popov ghost Lagrangianterm [3] These ghost fields can be interpreted as negative degrees of freedomthat serve to cancel the effects of unphysical timelike and longitudinal polar-ization states of gauge bosons The ghost Lagrangian is determined from the

gauge fixing terms in Eq (524) as

Lghost = minusca δGa

δαbcb (529)

where a b isin i = 1 2 3 Y The ghost fields ca are complex anticommutingLorentz scalars The bar in the expression denotes Hermitian conjugationThe ghosts are set to be even under the Z2 orbifold The functional derivativesin Eq (529) are found by studying infinitesimal gauge transformations

gpartMα

i + ǫijkAjMαk (530)

δBM =1

partMαY (531)

δφ =

iαiσi

φ equiv 1radic2

δχ2 + iδχ1

δH minus iδχ3

δχ1 =1

α1H minus α2χ3 + α3χ2 + αY χ2]

(533a)

δχ2 =1

α1χ3 + α2H minus α3χ1 minus αY χ1]

(533b)

δχ3 =1

minus α1χ2 + α2χ1 + α3H minus αYH]

(533c)

The ghost Lagrangian then becomes after the ghost fields have been rescaledaccording to ca rarr (ga

radicξ)12ca

minus caδGaδαb

cb = ca[

minus part2δab minus ξ(Mab minus part25δab)]

gǫijkδkbδia(ξpart5Aj5 minus partmicroAjmicro) minus ξ

cb (534)

h minusχ3 χ2 χ2

χ3 h minusχ1 minusχ1

minusχ2 χ1 h minushχ2 minusχ1 minush h

g2 0 0 00 g2 0 00 0 g2 minusggY

0 0 minusggY g2Y

Integrating out the extra dimensions the kinetic part of the ghost La-grangian becomes

L(kin)ghost =

infinsum

minuspart2δab minus ξMab(n)

where the mass matrix Mab(n)now is given by

M (n) =

20 0 0

0 M(n)W

0 0 M(n)W

2minus 1

4v2ggY

0 0 minus 14v

2ggY14v

The same rotation as in Eq (520)-(521) diagonalizes this matrix and theghosts mass eigenstates become cplusmn equiv 1radic

2(c1 ∓ c2) cZ and cγ with the cor-

responding massesradicξM

radicξM

(n)Z and

radicξM (n)

γDagger As for the Goldstone

bosons the gauge dependent masses indicate the unphysical nature of thesefields

Fermions

Fermions can appear both as singlets and as components of doublets underSU(2) transformations Furthermore in the massless limit fermions have adefinite chirality in even spacetime dimensions In the SM all fermions in

doublets ψ(0)d have from observation negative chirality whereas all singlets

ψ(0)s have positive chirality

γ5ψ(0)d = ψ

(0)d and γ5ψ(0)

s = minusψ(0)s (537)

where γ5 equiv iΓ0Γ1Γ2Γ3 is the four-dimensional chirality operator constructedfrom the generators of the Clifford algebra The d-dimensional Clifford algebrareads

ΓM ΓN = 2ηMN (538)

For an even number of spacetime dimensions (d = 2k + 2) the fundamentalrepresentation of the Clifford algebrarsquos generators are 2k+1 times 2k+1-matriceswhereas for odd spacetime dimensions (d = 2k+3) they are the same matricesas for the case of one less spacetime dimension and with the addition of

Γ2k+2 equiv minusi1+kΓ0Γ1 Γ2k+1 (539)

Note that our four-dimensional chirality operator γ5 is the same as iΓ4 in fivedimensions ie the chirality operator γ5 is a part of the Clifford algebrarsquosgenerators in five dimensions

For an even number of spacetime dimensions we can always reduce theDirac representation into two inequivalent Weyl representations characterizedby their spinorsrsquo chirality For odd spacetime dimensions this is not possiblesince the ΓM matrices then form an irreducible Dirac representation and they

Dagger Note that c+ and cminus are not each otherrsquos Hermitian conjugates

mix under Lorentz transformations However we can still always artificiallysplit up spinors with respect to their four-dimensional chirality operator Thatis

ψ = PRψ + PLψ equiv ψR + ψL (540)

where PRL are the chirality projection operators

PRL equiv 1

1 plusmn γ5)

= PRL PLPR = PRPL = 0 (541)

Five-dimensional Lorentz transformations mix ψR and ψL but if restricted toonly four-dimensional Lorentz transformations as in the effective KK theorythey do not mix Therefore it makes sense to assign different orbifold projec-tions depending on a fermionrsquos four-dimensional chirality when constructingan effective four-dimensional model To recover the chiral structure of the SMthe fermion fields should thus be assigned the following orbifold properties

PZ2 ψd(y) = minusγ5ψd(minusy) and PZ2 ψs(y) = γ5ψs(minusy) (542a)

With these orbifold projections the doublets and singlets have the followingexpansions in KK modes

ψd =1radic2πR

ψ(0)dL

+1radicπR

infinsum

ψ(n)dL

(543a)

ψs =1radic2πR

ψ(0)sR

+1radicπR

infinsum

ψ(n)sR

R+ ψ(n)

(543b)

where (as wanted) the zero mode doublets are left-handed and the zero modesinglets are right-handed

The fermion Lagrangian has the following structure

Lfermion = i(

ψdU ψdD)

ψdUψdD

+ iψs 6Dψs + L(Yukawa)fermion (544)

where U and D denote up-type (T3 = +12) and down-type (T3 = minus12)fermions in the SU(2) doublet (where T3 is the eigenvalue to the SU(2) gen-erator σ3 operator) respectively and the covariant derivative is

6D equiv ΓM

partM minus igArM

minus iY gYBM

where Y isin YdU YdD Ys is the hypercharge of the fermion in question Notethat the Ar

Mterm is absent for the singlet part since ψs does not transform

under SU(2) rotationsThe fermions have Yukawa-couplings to the Higgs field

L(Yukawa)fermion = minusλD

(ψdU ψdD) middot φ]

ψsD minus λU[

ψsU + hc (546)

from which after the spontaneous symmetry breaking in the Higgs sectorthe fermions get their electroweak fermions masses The conjugate of theHiggs doublet is defined by φa equiv ǫabφ

daggerb The electroweak masses will become

EW= (λUD v)

radic2

When several generations of quarks are present there can be couplingsthat mix quark generations It is still always possible to diagonalize the Higgscouplings which is also the base that diagonalizes the mass matrix Howeverthis diagonalization causes complications in the gauge couplings regardingquarks and will lead to the need for introducing a quark-mixing matrix V ij Let

ψidU = (ψudU ψcdU ψ

tdU) and ψidD = (ψudD ψ

cdD ψ

tdD) (547)

denote the up- and down-type quarks for the three families in the mass basisThese mass eigenstates ψ (ie the base that diagonalize the Higgs sector) arerelated to the interaction eigenstates ψprime by unitary matrices UU and UD

ψprimeidU = U ijU ψ

jdU and ψprimei

dD = U ijDψjdD (548)

This difference between ψprime and ψ will appear only in the interaction with WplusmnThe covariant derivative in the mass eigenstate base becomes [3]

6Dij equiv ΓM

δijpartM minus ieAMQδij minus i

ZM(σ3 minus s2WQ)δij

minusi gradic2(W+σ+ +Wminusσminus)V ij

where V equiv U daggerUUD is known as the Cabbibo-Kobayashi-Maskawa (CKM) mixing

matrix σplusmn = (σ1plusmn iσ2) are the usual step operators in SU(2) and the chargeQ is related to the weak isocharge and the hypercharge by Q = T3 + Y Forthe terms containing the photon and the Z boson the unitary matrices UD

and UU effectively disappear because they appear only in the combinationU daggerU = 13times3 In the absence of right-handed neutrinos (ψsU ) the CKMmatrix can also be made to vanish for the couplings to the Wplusmn and thereplacement Vij rarr δij should be done in the leptonic sector [3]

Let us now finally integrate out the extra dimension and see how the KKmasses arise from the kinetic term

int 2πR

iψdΓMpartMψd + iψsΓ

MpartMψs)

= iψ(0)γmicropartmicroψ(0)

infinsum

ψ(n)d

iγmicropartmicro minus n

ψ(n)d + ψ(n)

iγmicropartmicro +n

ψ(n)s

ψ(0) equiv ψ(0)dL

+ ψ(0)sR ψ

(n)d equiv ψ

+ ψ(n)dR ψ(n)

s equiv ψ(n)sL

+ ψ(n)sR (551)

Section 54 Particle Propagators 73

Hence for each SM fermion ψ(0) there are two fermions at each KK level ψ(n)s

and ψ(n)d (this can also be seen directly from Eq (543)) Note that the singlet

fields get the lsquowrongrsquo sign for their KK masses Adding the Yukawa massesmEW to the KK masses the mass matrix becomes

ψ(n)d ψ

R+ δm

(n)d mEW

mEW minus[ n

R+ δm

ψ(n)d

ψ(n)s

where δm(n)d and δm

(n)s are additional radiative loop corrections that appear

(these corrections will be further discussed in Section 55) By a rotationand a correction for the lsquowrongrsquo sign of the singletsrsquo mass terms the followingmass eigenstates are found

ξ(n)d

ξ(n)s

1 00 minusγ5

cosα(n) sinα(n)

minus sinα(n) cosα(n)

ψ(n)d

ψ(n)s

where the mixing angle is

tan 2α(n) =2mEW

2 nR + δm(n)d + δm

The physical masses for these states are

m(n)ds = plusmn1

(n)d minus δm(n)

(n)d + δm

(n)s )]2

+m2EW (555)

For all fermions except for the top quark the mixing angles α(n) are veryclose to zero

54 Particle Propagators

The five-dimensional propagators get modified by the orbifold compactifica-tion There are two differences compared to an infinite Lorentz invariantspacetime The first is trivial and comes from the compactification itselfwhich implies that the momentum in the extra dimension is quantized and isgiven by py = nR where n takes only integer values The other is related tothe reflection symmetry of the orbifold boundaries which breaks momentumconservation in the direction of the extra dimension and therefore allows fora sign flip of py in the propagator

By the introduction of unconstrained auxiliary fields the momentum spacepropagators can be found [195 196] For example a scalar field Φ can beexpressed in terms of an unconstrained auxiliary field χ as

Φ(xmicro y) =1

2(χ(xmicro y) plusmn χ(xmicrominusy)) (556)

where the plusmn-sign depends on whether the scalar is constrained to be evenor odd under the orbifold transformation Note that since the unconstrainedfields contain both y and minusy the propagator 〈Φ(xmicro y)Φ(xprimemicro yprime)〉 will (after aFourier transformation into momentum space) depend on py minus pprimey as well aspy + pprimey The five-dimensional propagator for a massless scalar field becomes[195196]

Φ(pmicro py)Φlowast(pprimemicro y)

δpypprimey plusmn δminuspypprimey

p2 minus p2y

In the effective four-dimensional theory the propagators for the KK particlestake the usual form (see also Appendix A)

Φ(pmicro)Φlowast(pprimemicro)

p2 minusm2 (558)

where m is the mass of the KK particle

55 Radiative Corrections

A main characteristic of KK theories is that at each KK level the particlestates are almost degenerate in mass This is true even after generation ofelectroweak masses mEW at least as long as the compactification scale 1Ris much larger than mEW The mass degeneracy implies that all momentum-conserving decays are close to threshold and radiative corrections will deter-mine if certain decay channels are open or not For cosmology these loopcorrections are essential in that they determine which of the fields is the light-est KK particle (LKP) and hence if the model has a natural dark mattercandidate

For example in the massless limit of five-dimensional QED the reaction

e(1) rarr e(0) + γ(1) (559)

is exactly marginal at three level After inclusion of the electroweak electron

mass me the reaction becomes barely forbidden as M(1)e equiv

1R2+m2

1R + me Radiative mass corrections are however naıvely expected to belarger than mEW because they are generically of the order of α sim 10minus2which is much larger than mEWm

(n) (which ranges from 10minus12 for electronsto 10minus2 for the top quarks at the first KK level) The study in [196] of one-loop radiative corrections show that radiative corrections generically open updecay channels like the one in Eq (559) so that all first-level KK modes candecay into the LKP and SM particles

Radiative corrections to masses arise from Feynman loop diagrams con-tributing to the two-point correlation functions These contributions can inthe UED model be artificially split into three types (1) five-dimensionalLorentz invariant loops (2) winding modes and (3) orbifold contributions

Section 55 Radiative Corrections 75

xmicro

Figure 51 Winding modes around the compactified extra dimensiongive Lorentz violating loop corrections to Kaluza-Klein masses

I will only briefly outline what goes into the calculation of the radiativemass corrections (the interested reader is referred to [196] and referencestherein) The five-dimensional dispersion relation reads

pmicropmicro = m2 + Z5p2y equiv m2

phys + Z5Zm2(n) (560)

where Z and Z5 are potential radiative quantum corrections In the SM the(divergent) quantum correction Z is absorbed into the physically observedmassesmphys Hence any extra mass corrections to the KK masses must comefrom extra contributions to Z5 In general both Z and Z5 receive (divergent)loop radiative quantum corrections but in a five-dimensional Lorentz invari-ant theory they are protected to stay equal Z = Z5 to preserve the usualdispersion relation At short distances (away from the orbifold fixpoints)the compactification is actually not perceptible and localized loops shouldtherefore preserve Lorentz invariance In fact all five-dimensional Lorentzinvariance preserving self-energy contributions can therefore be absorbed intothe renormalization of the zero mode mass mrarr mphysZ so these do not giveany additional radiative mass correction to the KK modes

Although local Lorentz invariance still holds under a S1 compactificationglobal Lorentz invariance is broken Such non-local effects appear in thoseFeynman diagrams that have an internal loop that winds around the compactdimension as shown in Fig 51 The radiative corrections from such windingpropagators can be isolated by the following procedure Because of the S1

compactification the momenta py in the compact dimension are quantizedin units of 1R and therefore the phase space integral over internal loopmomenta reduces to a sum over KK levels This sum can be translated intoa sum over net winding n around the compactified dimension where the firstterm with n = 0 exactly corresponds to an uncompactified five-dimensionaltheory The non-winding loops (n = 0) can be dealt with in the same way asdescribed in the previous paragraph The sum of the remaining winding modes(n 6= 0) turns out to only give finite and well-defined radiative corrections toeach KK mass

Finally the third kind of radiative correction appears due to the orb-ifold compactification S1Z2 This is a local effect caused by the orbifoldrsquosfixpoints that break translational invariance in the y-direction As shownin Section 54 this leads to modified five-dimensional propagators Thus wemust redo the two previously described calculations but now instead with thecorrect orbifold propagators The finite contribution stated in the previousparagraph remains the same (although divided by two because the orbifoldhas projected out half of the states) but also new logarithmically divergentterms localized at the orbifold fixpoints appear This means that counterterms should be included at the boundaries to cancel these divergences andthat these calculations strictly speaking determine only the running behav-ior In order to keep unknown contributions at the cutoff scale Λcut undercontrol they are assumed to be small at that high energy cutoff scale ndash a self-consistent assumption since the boundary terms are generated only by loopcorrections If large boundary terms were present they could induce mixingbetween different KK modes [196]

56 Mass Spectrum

Including these radiative corrections we can calculate the mass spectra of theKK particles In most cases the electroweak mass can be ignored with theprominent exceptions of the heavy Higgs and gauge bosons and the top quarkIn the case of fermions these radiative corrections were already included inthe fermionic mass matrix Eq (552) Similarly this can be done also forthe bosons Let us take a closer look at the cosmologically important massmatrix for the neutral gauge bosons In the B A3 basis including radiativecorrections δmB(n) and δm

A3(n) the mass matrix reads(

)2+ δm2

B(n) + 14v

minus 14v

)2+ δm2

A3(n) + 14v

By diagonalizing this matrix the physical KK photon mass and Z-boson masscan be found The nth KK level lsquoWeinbergrsquo angle for the diagonalizing rotationis given by

tan 2θ(n) =v2ggY

δm2A3(n) minus δm2

B(n) + v2

4 (g2 minus g2Y)] (562)

For small compactifications scales this Weinberg angle is driven to zero sincegenerally δm2

A3(n) minus δm2B(n) ≫ v2ggY For example for Rminus1 amp 500 GeV and

ΛcutR amp 20 θ(n) is less than 10minus2 Therefore the KK photon γ(n) is verywell approximated by the weak hypercharge gauge boson B(n) and these twostates are often used interchangeably

Figure 52 shows the spectrum for the first KK excitations of all SM par-ticles both at (a) tree level and (b) including one-loop radiative corrections

Section 56 Mass Spectrum 77

(a) Tree level

γgWplusmnaplusmnZa0H0

bdbsτdτsντ

(b) One-loop level

γaplusmna0H0

ZW plusmn

τdντ

bs bdts

Figure 52 The mass spectrum of the first Kaluza-Klein states at (a) treelevel and (b) including one-loop radiative corrections A compactificationradius of Rminus1 = 500 GeV Higgs mass of 200 GeV and cutoff of Λcut =20Rminus1 have been used The first column shows the Higgs and gaugebosons The second column shows the quark SU(2) doublets (Q) and thesinglets (ud) for the two first families In the last column this is repeatedfor the third family taking into account the larger electroweak masses ofthe tau lepton (τsd ντ ) the bottom quark (bsd) and the top quark (tsd)

We find that the lightest SM KK particle is the first KK level photon γ(1)Since unbroken KK parity (minus1)(n) guarantees the LKP to be stable it pro-vides a possible dark matter candidate Due to the loop corrections the massdegeneracy will be lifted enough so that all other first KK level states willpromptly cascade down to the γ(1)

Instead of the minimal UED model ndash defined by using the described one-loop expressions with boundary terms negligible at the cutoff scale Λcut (un-less specified otherwise Λcut = 20Rminus1) ndash you could take a more phenomeno-logical approach and treat all the KK mass corrections as independent inputto the theory (see eg [197]) With such an approach other KK parti-cles than the γ(1) can become the LKP With the γ(1) not being the LKPthe LKP still has to be electrically neutral to be a good dark matter candi-date [198199] Therefore the only other potentially attractive LKP optionsare h(1) Z(1) ν(1) g(1) and if going beyond SM particles also the first KKlevel of the graviton At least naıvely some of these can be directly excludedthe high electroweak Higgs mass naturally excludes the Higgs particles thegluons are color charged and thus excluded by their strong interactions [199]the Z(1) is expected to be heavier than γ(1) since usually both electroweakand one-loop order correction contribute to the mass matrix in a way that

the lighter state is almost B(1) (ie γ(1)) When it comes to the graviton itis interesting to note that for Rminus1 800 GeV the first KK graviton G(1) inthe minimal UED model becomes lighter than the γ(1) This result is underthe reasonable assumption that the radiative mass corrections to the gravitonis very small since it only couples gravitationally and therefore the mass ofG(1) is well approximated by Rminus1 The G(1) will be a superweakly interact-ing particle as it interacts only gravitationally Therefore it is hard if notimpossible to detect it in conventional dark matter searches However thepresence of G(1) could cause effects both on the BBN and the CMB radiationpredictions [200] (see also [201ndash203]) In the case of a KK neutrino thermalrelic calculations show that the favored mass range is between 08 TeV and13 TeV [197] This is in strong conflict with direct detection searches thatexclude ν(1) masses below 1000 TeV [204205] [206207]

For the remaining chapters γ(1) asymp B(1) will be taken to be the LKP

C h a p t e r

Kaluza-KleinDark Matter

Numerous phenomenological constraints must be fulfilled in order to makea particle a viable dark matter candidate In the UED model it turns outthat the first Kaluza-Klein excitation of the photon γ(1) is an excellent darkmatter candidate If γ(1) (asymp B(1)) is the lightest Kaluza-Klein state and itsmass is about 1 TeV then this candidate can make up all the dark matterIt is therefore of importance to find direct and indirect ways to detect sucha candidate The properties of this Kaluza-Klein dark matter particle arediscussed and observational constraints are reviewed The chapter concludesby discussing the results of Paper II and Paper III where characteristicindirect detection signals were predicted from pair-annihilation of these darkmatter particles So-called final state radiation producing very high energygamma rays and a monocromatic gamma-ray line with an energy equal tothe mass of the dark matter particle produce distinctive signatures in thegamma spectrum

61 Relic Density

In the early Universe the temperature T was higher than the compactifica-tion scale (T gt Rminus1 sim 1 TeV) and KK particles were freely created andannihilated and kept in thermal and chemical equilibrium by processes like

X(n)i X

(n)j harr xk xl (61a)

X(n)i xj harr X

(n)k xl (61b)

X(m)i rarr X

(n)j xk (61c)

where X(n)i is the nth-level excitation of a SM particle xi (i j k l = 1 2 )

Processes of the type (61a) are usually referred to as annihilation when i = jand coannihilation when i 6= j (in fact coannihilation processes usually refer

80 Kaluza-Klein Dark Matter Chapter 6

to all pair annihilations that not exclusively include the dark matter candidateunder consideration)

As the Universe expands the temperature drops and the production of KKstates soon becomes energetically suppressed leading to their number densitydropping exponentially This happens first for the heavier KK particles whichby inelastic scattering with numerous SM particles (Eq (61b)) and decays(Eq (61c)) transform into lighter KK particles and SM particles Due toconservation of KK parity (minus1)n (n being the KK level) the lightest of theKK particles is not allowed to decay and can therefore only be destroyed bypairwise annihilation (or potentially by coannihilation if there is a sufficientamount of other KK particles around) Hence at some point the LKPs anni-hilation rate cannot keep up with the expansion rate the LKP density leavesits chemical equilibrium value and their comoving number density freezesWhat is left is a thermal remnant of non-relativistic LKPs that act as colddark matter particles That only the LKPs contribute to the dark mattercontent today is of course only valid under the assumption that all level-1KK states are not exactly degenerate in mass and can decay into the LKPGenerically this is the case for typical mass splittings as in eg the minimalUED model at the one-loop level

As stated in the previous chapter the lightest of the KK particles is in theminimal UED model the first KK excitation of the photon γ(1) which to agood approximation is equal toB(1) In practice there is a negligible differencebetween using γ(1) and B(1) as the LKP (See Section 56) Henceforth I switchnotation from γ(1) to B(1) for the KK dark matter particle as this is the stateactually used in the calculations

Quantitatively the number density of a dark matter particle is describedby the Boltzmann equation [208209]

dt+ 3Hn = minus〈σeffv〉

n2 minus (neq)2]

where n =sum

ni is the total number density including both the LKP andheavier states that will decay into the LKP H is the Hubble expansion rateand neq the sum of the chemical equilibrium number densities for the KKparticles which in the non-relativistic limit are given by

≃ gi

(minusmX

where gi is the internal number of degrees of freedom for the particle in ques-tion Finally the effective annihilation (including coannihilation) cross sectiontimes the relative velocity (or more precisely the Moslashller velocity [210]) of theannihilating particles 〈σeffv〉 is given by

〈σeffv〉 =sum

〈σijvij〉neqi

neq(64)

Section 61 Relic Density 81

where the thermal average is taken for the quantity within brackets 〈〉In general a lower effective cross section means that the comoving number

density freezes out earlier when the density is higher and therefore leaves alarger relic abundance today A high effective cross section on the other handmeans that the particles under consideration stay in chemical equilibriumlonger and the number density gets suppressed The exact freeze-out tem-perature and relic densities are determined by solving Eq (62) but a roughestimate is that the freeze-out occurs when the annihilation rate Γ = n〈σv〉fall below the Hubble expansion H As mentioned in Chapter 1 a rule ofthumb is that the relic abundance is given by

〈σeffv〉 (65)

At freeze-out typically around T sim mWIMP25 there are roughly 1010 SMparticles per KK excitation that by reaction (61b) can keep the relativeabundance among KK states in chemical equilibrium (ninj = neq

i neqj prop

exp (minus∆MT )) Coannihilation is therefore only of importance when themass difference between the LKP and the other KK states is smaller or com-parable to the freeze-out temperature Since the masses of the KK states arequasi-degenerated by nature we could expect that coannihilation are espe-cially important for the UED model In general the presence of coannihila-tions can both increase and decrease relic densities In the case of UED wecould also expect that second-level KK states could significantly affect therelic density if the appear in s-channel resonances see [211212]

To see how differences in the KK mass spectrum affect the relic densitylet us take a look at three illustrative examples These cases set the massrange for which B(1) is expected to make up the observed dark matter

First consider a case where we artificially allow only for self-annihilationThis would mimic the case when all other KK states are significantly heavierthan the LKP The effective cross section in Eq (64) then reduces to σeff =σB(1)B(1) The relic density dependence on the compactification scale orequivalently the LKP mass is shown by the dotted line in Fig 61 In thiscase the B(1) mass should be in the range 700 GeV to 850 GeV to coincidewith the measured dark matter density

Imagine as a second case that coannihilation with the strongly interactingKK quarks and gluons are important This can potentially cause a strongenhancement of the effective cross section if these other states are not muchheavier than the LKP Physically this represents the case when other KKparticles deplete the LKP density by keeping it longer in chemical equilibriumthrough coannihilations Thus this effect will allow for increased B(1) massesAs an illustrative example take the mass spectrum to be that of the minimalUED model except that the KK gluon mass mg(1) is treated a free parameterFor a mass difference ∆g(1) equiv (mg(1) minusmB(1))mB(1) as small as 2 an allowed

relic density can be obtained for B(1) masses up to about 25 TeV (see the

Rminus1[TeV]

05 1 15 2 25 3

B(1)B(1)

B(1) g(1) ∆g(1) = 2

Figure 61 Relic density of the lightest Kaluza-Klein state B(1) as afunction of the inverse compactification radius Rminus1 The dotted line isthe result from considering B(1)B(1) annihilation only The dashed line isthe when adding coannihilation with the Kaluza-Klein gluon g(1) using asmall mass split to the B(1) ∆g(1) equiv (mg(1) minus mB(1))mB(1) = 002 Thesolid line is from a full calculation in the minimal UED model where allcoannihilations have been taken into account The gray horizontal banddenotes the preferred region for the dark matter relic density 0094 ltΩCDMh2 lt 0129 Figure constructed from results in [213]

dashed line in Fig 61) The small distortion in the curve around 23 TeVis due to the details concerning the change of the total number of effectivelymassless degrees of freedom that affects the Hubble expansion during freeze-out (see [213])

As a final case let the mass spectrum be that of the minimal UED modeland include all coannihilation effects with all the first level KK states Theminimal UED model is here set to have a cutoff scale equal to Λcut = 20Rminus1The relic density result is shown as the solid line in Fig 61 At first sight thedisplayed result might look surprising since in the previous example includingcoannihilations with quarks and gluons reduced the relic density Howeverin the minimal UED scenario the KK quarks and gluons get large radiativecorrections and therefore they are more than 15 heavier than B(1) and arenot very important in the coannihilation processes Here a different effectbecomes important Imagine that the coannihilation cross section σB(1)X(1)

with a state X(1) of similar mass is negligibly small Then the effective cross

Section 62 Direct and Indirect Detection 83

section in Eq (64) for the two species in the limit of mass degenerationtends towards

σeff asymp σB(1)B(1)g2B(1)(gX(1) + gB(1)) + σX(1)X(1)g2

X(1)(gX(1) + gB(1)) (66)

If σX(1)X(1) is small then σeff may be smaller than the self-annihilation crosssection σB(1)B(1) This is exactly what happens in the minimal UED modelthe mass splittings to the leptons are small (about 1 to the SU(2)-singletsand 3 to the SU(2)-doublets) Contrary to the case of coannihilating KKquarks and gluons (and the usual case in supersymmetric models with theneutralino as dark matter) the coannihilation processes are here weak (oractually of similar strength as the self-annihilation) and the result is thereforean increase of the relic density The physical understanding of this situation isthat the two species quasi-independentlylowast freeze-out followed by the heavierstate decaying thus enhancing the LKP relic density

In summary relic density calculations [197213214] show that if the KKphoton B(1) is to make up the observed amount of dark matter it must havea mass roughly in the range from 500 GeV up to a few TeV

While freeze-out is the process of leaving chemical equilibrium it is notthe end of the B(1) interacting with the much more abundant SM particlesElastic and inelastic scattering keeps the LKPs in thermal equilibrium untilthe temperature is roughly somewhere between 10 MeV and a few hundredMeV (see eg [205 215 216]) The kinetic decoupling that occurs at thistemperature sets a distance scale below which dark matter density fluctua-tions get washed out In other words there is a cutoff in the matter powerspectrum at small scales This means that there is a lower limit on the sizeof the smallest protohalos created whose density perturbations later go non-linear at a redshift between 40 and 80 The consequence is that the smallestprotohalos (or dark matter clumps) should not be less massive than about10minus3 to 103 Earth masses Whether these smallest clumps of dark matterhave survived until today depends critically on to which extent these struc-tures are tidally disrupted through encounters with eg stars gas disks andother dark matter halos [217ndash219]

62 Direct and Indirect Detection

With the B(1) as a dark matter candidate the experimental exclusion limitson its properties and its prospect for detection should be investigated

lowast Species never really freeze-out completely independently of each other because of thepresence of the processes (61b) that usually efficiently transform different species intoeach other

Accelerator Searches

High-energy colliders are in principle able to produce heavy particles belowtheir operating center of mass energy The absence of a direct discovery of anynon-SM particle therefore sets upper limits on production cross sections whichin the case of UED translates into a lower limit on KK masses Because of KKparity conservation KK states can only be produced in pairs For the LargeElectron-Positron Collider (LEP) at CERN this means that only masses lessthan ECM sim 100 GeV could be reached The circular proton-antiprotonTevatron accelerator at the Fermi laboratory can reach much higher energiesas it is running at a center of mass energy of ECM sim 2 TeV Non-directdetection (ie no excess of unexpected missing energy events) in the Tevatronsets an upper limit on one extra-dimensional radius of R (03 TeV)minus1 [126220ndash222] The future LHC experiment will be able to probe KK masses up toabout 15 TeV [205] Suggested future linear electron-positron colliders couldsignificantly improve measuring particle properties such as masses couplingsand spins of new particles discovered at the LHC [156223] but will probablynot be a discovery machine for UED unless they are able to probe energiesabove sim 15 TeV

Physics beyond the SM can manifest itself not only via direct productionbut also indirectly by its influence on other observables such as magneticmoments rare decays or precision electroweak data [205] For a sim 100 GeVHiggs mass electroweak precision data limits 1R to be amp 800 GeV whichweakens to about 300 GeV for a sim 1 TeV Higgs Strong indirect constraintsalso come from data related to the strongly suppressed decay b rarr sγ whichis less dependent on the Higgs mass and gives 1R amp 600 GeV [224]

Direct Detection

Direct detection experiments are based on the idea of observing elastic scat-tering of dark matter particles that pass through the Earthrsquos orbit Thesearched signal is that a heavy WIMP depositing recoil energy to a target nu-cleus There are three common techniques to measure this recoil energy (andmany experiments actually combine them) (i) Ionization in semiconductortargets like germanium (Ge) [206 225 226] silicon (Si) [206 226] or xenon(Xe) [207] the recoil energy can cause the surrounding atoms to ionize anddrift in an applied electric field out to surrounding detectors (ii) Scintillationfor example sodium iodide (NaI) crystals [227ndash229] or liquidgas Xenon (Xe)scintillators [207 230] can produce fluorescence light when a WIMP interac-tion occurs This fluorescence light is then detected by surrounding photondetectors (iii) Phonon production cryogenic (ie low temperature) crystalslike germanium and silicon detectors as in [206225226] look for the phonon(vibration) modes produced by the impulse transfer due to WIMP scattering

To achieve the required sensitivity for such rare scattering events a lowbackground is necessary Operating instruments are therefore well shielded

and often placed in underground environments Features that are searchedfor are the recoil energy spectral shape the directionality of the nuclear recoiland possible time modulations The time modulation in the absolute detectionrate is expected due to the Earthrsquos spin and velocity through the dark matterhalo (see Section 91 for a short comment on the DAMA experimentrsquos claimof such a detected annual modulation)

The elastic scattering of a WIMP can be separated into spin-independentand spin-dependent contributions The spin-independent scattering can takeplace coherently with all the nucleons in a nucleus leading to a cross sectionproportional to the square of the nuclei mass [42] Due to the available phasespace there is an additional factor proportional to the square of the reducednuclei massmr = mDMmN(mN+mDM) [wheremN andmDM are the nuclei anddark matter particle mass respectively] The present best limits on the spin-independent cross section comes from XENON [207] and is for WIMP massesaround 1 TeV of the order of σSI 10minus6 pb per nucleon (that is per proton orneutron respectively) whereas for WIMP masses of 100 GeV it is somewhatbetter σSI 10minus7 pb This translates roughly to mB(1) amp 05 TeV [204231]when assuming a mass shift of about 2 to the KK quarks (the limits getsweakened for increased mass shifts)

The spin-dependent cross-section limits are far weaker and in the sameWIMP mass range they are at present roughly σSD 01minus1 pb These limitsare set by CDMS (WIMP-neutron cross section) [226] and NaIAD (WIMP-proton cross section) [229]

Indirect Detection

Indirect searches aim at detecting the products of dark matter particle an-nihilation The most promising astrophysical indirect signals seem to comefrom excesses of gamma-rays neutrinos or anti-matter As the annihilationrate is proportional to the number density squared nearby regions of expectedenhanced dark matter densities are the obvious targets for studies

WIMPs could lose kinetic energy through scattering in celestial bodieslike the Sun or the Earth and become gravitationally trapped The concen-tration of WIMPs then builds up until an equilibrium between annihilationand capture rate is obtained In the case of B(1) dark matter equilibriumis expected inside the Sun but not in the center of the Earth The onlyparticles with a low enough cross section to directly escape the inner regionsof the Sun and Earth where the annihilations rate into SM particles is thehighest are neutrinos Current experiments are not sensitive enough to putany relevant limits and at least kilometer size detectors such as the IceCubeexperiment under construction will be needed [204231232] Although onlythe neutrinos can escape the inner parts of a star it has been proposed thatin optimistic scenarios the extra energy source from dark matter annihilationin the interior of stars and white dwarfs could change their temperatures in a

detectable way [233ndash235]

Another way to reveal the existence of particle dark matter would be tostudy the composition of cosmic rays and discover products from dark matterannihilations Unfortunately charged products are deflected in the galacticmagnetic fields and information about their origin is lost However darkmatter annihilation yields equal amount of matter and antimatter whereasanti-matter in conventionally produced cosmic rays is expected to be relativelyless abundant (this is because anti-matter is only produced in secondary pro-cesses where primary cosmic-ray nuclei ndash presumably produced in supernovashock fronts ndash collide with the interstellar gas) A detected excess in the anti-matter abundance in cosmic rays could therefore be a sign of dark matterannihilation

In this manner the positron spectrum can be searched for dark mattersignals In the UED model 20 of the B(1) annihilations are into monochro-matic electron-positron pairs and 40 into muons or tau pairs Muons andtaus can subsequently also decay into energetic electrons and positrons Asizable positron flux from KK dark matter with a much harder energy spec-trum than the expected background could therefore be looked for [231236]No such convincing signal has been seen but the High-Energy AntimatterTelescope (HEAT) [237] covering energies up to a few 10 GeV has reporteda potential small excess in the cosmic positron fraction around 7-10 GeV (seeeg [238] for a dark matter interpretation including KK states of the HEATdata)

In the UED scenario the large branching ratio into leptonic states makesthe expected antiprotons yield ndash mainly produced from the quark-antiquarkfinal states ndash relatively low compared to the positron signal Today an-tiproton observations do not provide any significant constraints on the UEDmodel [239 240] However the PAMELA (Payload for Antimatter MatterExploration and Light-nuclei Astrophysics) [241] satellite already in orbitmight find a convincing energy signature that deviates from the convention-ally expected antiproton or positron spectrum as it will collect more statisticsas well as probe higher energies (up to some 100 GeV)

63 Gamma-Ray Signatures

In addition to the above-mentioned ways to detect WIMP dark matter thereis the signal from annihilation into gamma rays In general this signal hasmany advantages (i) gamma rays point directly back to their sources (ii) atthese energies the photons basically propagate through the galactic interstellarmedium without distortion [242] [243244] (iii) they often produce character-istic spectral features that differ from conventional backgrounds (iv) existingtechniques for space and large ground-based telescopes allow to study gammarays in a large energy range ndash up to tens of TeV

Dark matter annihilations into photons can produce both a continuum of

Section 63 Gamma-Ray Signatures 87

gamma-ray energies as well as monochromatic line signals when γγ γZ orγh are the final states

Gamma-Ray Continuum

At tree level with all first KK levels degenerated in mass B(1) dark matterannihilates into charged lepton pairs (59) quark pairs (35) neutrinos(4) charged (1) and neutral (05) gauge bosons and the Higgs boson(05)

The fraction of B(1)B(1) that annihilates into quark pairs will in a sub-sequent process of quark fragmentation produce gamma-ray photons mainlythrough the decay of neutral π0 mesons

B(1)B(1) rarr qq rarr π0 + rarr γγ + (67)

These chain-processes result in differential photon multiplicities ie the num-ber density of photons produced per annihilation and is commonly obtainedfrom phenomenological models of hadronization In Paper III we use aparametrization of the differential photon multiplicity dN q

γdEγ publishedin [245] for a center of mass energy of 1 TeV This parametrization was basedon the Monte Carlo code Pythia [246] which is based on the so-called Lundmodel for quark hadronization Since Pythia are able to reproduce experi-mental data well and dN q

γdEγ is fairly scale invariant at testable high ener-gies we judge it to be reliable to use Pythia results up to TeV energies

The massive Higgs and gauge bosons can also decay into quarks that pro-duce photons in their hadronization process For B(1) dark matter this givesa negligible contribution due to the small branching ratios into these particles

The remaining and majority part of the B(1) dark matter annihilationsresult in charged lepton pairs electrons (e) muons (micro) and tau (τ) pairs eachwith a branching ratio of about 20dagger The only lepton heavy enough to decayinto hadrons is the τ Hence τ pairs generate quarks that as described abovesubsequently produce gamma rays in the process of quark fragmentationThe photon multiplicity dN τ

γ dxγ for this process is taken from reference

[245] Due to the high branching ratio into τ -pairs for B(1) annihilation thiscontribution is very significant at the highest photon energies (see Fig 63 atthe end of this section)

An even more important contribution to the gamma-ray spectrum at thehighest energies comes from final state radiationDagger This process is a three-body final state where one of the charged final state particles radiates a

dagger This is radically different from the neutralino dark matter candidate in supersymmetrictheories where annihilation into light fermions are helicity suppressed (see the discussionin Section 72)

Dagger Pythia partly takes final state radiation into account but certainly not as specificallyas in Paper II and Paper IV (see also [247]) This is particularly true for internalbremsstrahlung from charged gauge bosons treated in Chapter 7 and in Paper IV

ℓ+ℓ+ℓ+

ℓminusℓminus

ℓ(1)

γγγ

Figure 62 Tree-level diagrams contributing to final state radiation inlepton pair production B(1)B(1) rarr ℓ+ℓminusγ Figure from Paper II

photon The relevant Feynman diagrams for B(1) annihilation with a finalstate photon is depicted in Fig 62 In principle diagrams with an s-channelHiggs bosons also exists These diagram can safely be neglected since theHiggs boson couple very weakly to light leptons (prop mℓ) and are typically farenough from resonance Typically the galactic velocity scale for WIMPs is10minus3c and the cross section can be calculated in the zero velocity limit ofB(1) As found in Paper II the differential photon multiplicity can be wellapproximated by

dN ℓγ

dxequiv d(σℓ+ℓminusγv)dx

σℓ+ℓminusv≃ α

(x2 minus 2x+ 2)

m2B(1)

(1 minus x)

where x equiv EγmB(1) and mℓ is the mass of the lepton species in considerationThe factor απ arises due to the extra vertex and the phase space difference

between two- and three-body final states The large logarithm ln(m2B(1)m

appears due to a collinear divergence behavior of quantum electrodynamicsThis effect is easy to see from the kinematics Consider the propagator of thelepton that emits a photon in the first (or third) diagram of Fig 62 Denoting

the outgoing photon [lepton] momentum by kmicro = (Ek ~k) [pmicro = (Ep ~p)] thedenominator of the propagator is

(p+ k)2 minusm2ℓ = 2p middot k = 2Ek(Ep minus |~p| cos θ) (69)

where θ is the angle between the outgoing photon and lepton This expressionshows that for a highly relativistic lepton (|~p| rarr Ep) and a collinear (θ rarr 0)outgoing photon the lepton propagator diverges meaning that leptons tendto rapidly lose their energy by emission of forward-directed photons

Let us be slightly more quantitative and investigate the cross section Innext to lowest order cos θ rarr 1+θ22 and |~p| rarr Ep(1minusm2

ℓ2E2p) and the de-

nominator of the propagator becomes EkEp(θ2 +m2

ℓE2p) The photon vertex

itself gives in this limit a contribution u(p)γmicrou(p+ k) where the approxima-tion consists of treating the virtual almost on shell electron with momentump+ k as a real incoming electronsect Squaring this part of the amplitude and

taking the spin sum leads tosum

|ǫmicrou(p)γmicrou(p+ k)|2 = minusTr[(6p+ 6k +mℓ)γmicro(6p+mℓ)γmicro]

= 8(p+ k)micropmicro asymp 8(EkEp minus ~k~p) asymp 8EkEp(1 minus cos θ) asymp 8EkEpθ2

where the standard notation that u(p) and u equiv udaggerγ0 are dirac spinors and ǫmicrois the photon polarization In the second line the lepton mass is set to zeromℓ = 0 The final ingredient including θ is the phase space factor

d3k Inthe small θ limit we have d3k = 2πpperpdkdkperp rarr 2πEkθdEkdθ and thereforethe photon multiplicity should scale as

dN ℓγ

dxpropint θmax

dθθ3

θ2 +m2ℓE

]2 asymp 1

2ln(E2

pm2ℓ) (610)

where only the leading logarithm is kept and θmax is an arbitrary upperlimit for which the used collinear approximations hold In the colinear limitenergy conservation in the vertex of the radiating final state photon givesEp =

radics2 minus Eγ (

radics being the center of mass energy) which for incoming

non-relativistic (dark matter) particles reduces to Ep asymp mWIMP(1minusx) This isqualitatively the result we obtained for the UED model above I would like tostress that these arguments are very general Thus for any heavy dark mattercandidate with unsuppressed couplings to fermions high-energy gamma raysas in Eq (68) are expected This means that a wide class of dark matterparticles should by annihilation produce very hard gamma spectra with asharp edge feature with the flux dropping abruptly at an energy equal to thedark matter mass

The general behavior of final state radiation from dark matter annihila-tions was later studied also in [247] where they further stress that annihilationinto any charged particles X and X together with a final state radiated pho-ton takes a universal form

dσ(χχ rarr XXγ)

dxasymp αQ2

πFX(x) ln

s(1 minus x)

σ(χχ rarr XX) (611)

where QX and mX are the electric charge and the mass of the X particlerespectively The splitting function F depends only on the spin of the Xparticles When X is a fermion

Ff(x) =1 + (1 minus x)2

x (612)

sect I am definitely a bit sloppy here A full kinematically and gauge invariant calculation(including all contributing Feynman diagrams) would however give the same resultIf we so wish we could be a bit more correct and imagine the electron-positron pair(momentum p1 = p + k and p2) to be produced directly from a scalar interactionwhere the p1 particle radiates a photon with momentum k The exact spinor part of theamplitude then becomes ǫmicrou(p)γmicro(6p1)u(p2) which again leads to minusTr[(6p2 6p1γmicro 6pγmicro 6p1] propθ2 in lowest order in θ

effγdx

x equiv EγmB(1)

Figure 63 The total number of photons per B(1)B(1) annihilation (solidline) multiplied by x2 = (Eγm

B(1))2 Also shown is what quark frag-

mentation alone would give (dashed line) and adding to that τ leptonproduction and decay (dotted line) Here a B(1) mass of 08 TeV and a5 mass split to the other particles first Kaluza-Klein level are assumed ndashthe result is however quite insensitive to these parameters Figure fromPaper II

whereas if X is a scalar particle

Fs(x) =1 minus x

x (613)

If X is a W boson the Goldstone boson equivalence theorem implies thatFW (x) asymp Fs(x) [248] Unfortunately a sharp endpoint is not obvious inthe scalar or W boson case According to Eq (613) limxrarr1 Fs(x) = 0 andthe flux near the endpoint might instead be dominated by model-dependentnon-collinear contributions [247] (see Section 72 and Paper IV for internalbremsstrahlung in the case of neutralino annihilations)

Including also this final state radiation the total observable gamma spec-trum per B(1)B(1) annihilation is finally given by

dN effγ dx equiv

κidNiγdx (614)

where the sum is over all processes that contribute to primary or secondarygamma rays and κi are the corresponding branching ratios The result isshown as the solid line in Fig 63

There are also other sources of photon production accompanying the LKPannihilation For example the induced high-energy leptons will Comptonscatter on the CMB photons and starlight Although these processes producegamma rays they are expected to give small fluxes [249] Another source ofphoton fluxes emerge when light leptons lose energy by synchrotron radiationin magnetic fields [249] With moderate although uncertain assumptions forthe magnetic fields in the galactic center the synchrotron radiation could givea significant flux of photons both at radio [250] and X-ray [251] wavelengths

Gamma Line Signal

Since the annihilating dark matter particles are highly non-relativistic theprocesses B(1)B(1) rarr γγ B(1)B(1) rarr Zγ and B(1)B(1) rarr Hγ result inalmost mono-energetic gamma-ray lines with energies Eγ = m

B(1) Eγ =mB(1)(1 minus m2

B(1)) and Eγ = mB(1)(1 minus m2h4m

2B(1)) respectively With

the expectation that dark matter being electrically neutral these annihila-tion processes are bound to be loop suppressed (since photons couple only toelectric charge) On the other hand such gamma-ray lines would constitutea lsquosmoking gunrsquo signature for dark matter annihilations if they were to beobserved since it is hard to imagine any astrophysical background that couldmimic such a spectral feature

In the case of B(1) dark matter with unsuppressed couplings to fermionsit could be expect that loops with fermions should dominate the cross sectionThis is a naıve expectation from the tree-level result of a 95 branching ratiointo charged fermions (but this expectation should be true if no particulardestructive interference or symmetry suppressions are expected to occur atloop-level)

In Paper III we calculated the line signal B(1)B(1) rarr γγ The purefermionic contributions give in total 2times12 different diagramspara for each chargedSM fermion that contributes to B(1)B(1) rarr γγ see Fig 64 The calculation ofthe Feynman amplitude of B(1)B(1) rarr γγ within the QED sector is describedin detail in Paper III The basic steps in obtaining the analytical result areas follows

1 The total amplitude

M = ǫmicro1

1 (p1)ǫmicro2

2 (p2)ǫmicro3

3 (p3)ǫmicro4

4 (p4)Mmicro1micro2micro3micro4(p1 p2 p3 p4) (615)

is formally written down from the Feynman rules given in Appendix A

2 Charge invariance of the in and out states and a relative sign betweenvector and axial couplings

CψγmicroψCminus1 = minusψγmicroψ Cψγmicroγ5ψCminus1 = ψγmicroγ5ψ (616)

para Remember at a given KK level the fermionic field content is doubled as compared tothe SM

ψ(0)ψ(0)

ξ(1)ds

Figure 64 Fermion box contributions to B(1)B(1) rarr γγ including thefirst KK levels Not shown are the additional nine diagrams that areobtained by crossing external momenta Figure from Paper III

means that an odd number of axial couplings in the Feynman ampli-tudes must automatically vanish (ie no γ5 in the trace) Splitting theamplitude into a contribution that contains only vector-like couplingsMv and terms that contains only axial vector couplings Ma is thereforepossible In this specific case where the axial part has equal strength asthe vector part in couplings between B(1) and fermions (see Appendix Aor Paper III) we also have Ma = Mv and the full amplitude can bewritten as

Mmicro1micro2micro3micro4 = minus2iαemαYQ2(Y 2

s + Y 2d )Mmicro1micro2micro3micro4

v (617)

where αEM equiv e24π αY equiv g2Y4π and Q and Y are the electric and

hypercharge respectively of the KK fermions in the loop

3 From momentum conservation in the zero velocity limit of the B(1)s (p =p1 = p2 = (mB(1) 0)) and transversality of the polarization vectors wecan decompose the amplitude into the following Lorentz structure

Mmicro1micro2micro3micro4v =

m4B(1)

pmicro1

3 pmicro2

4 pmicro3pmicro4 +B1

m2B(1)

gmicro1micro2pmicro3pmicro4

m2B(1)

gmicro1micro3pmicro2

4 pmicro4 +B3

m2B(1)

4 pmicro3 +B4

m2B(1)

3 pmicro4

m2B(1)

3 pmicro3 +B6

m2B(1)

3 pmicro2

+ C1 gmicro1micro2gmicro3micro4 + C2 g

micro1micro3gmicro2micro4 + C3 gmicro1micro4gmicro2micro3 (618)

4 The expressions for the coefficients A B and C are determined byidentification of the amplitude found in step 1 above However byBose symmetry and gauge invariance of the external photons the wholeamplitude can be expressed by means of only three coefficients whichare chosen to be B1 B2 and B6

5 These coefficients B1 B2 and B6 are linear combinations of tensorintegrals

D0DmicroDmicroν DmicroνρDmicroνρσ (k1 k2 k3m1m2m3m4)

1 qmicro qmicroqν qmicroqνqρ qmicroqνqρqσ(

q21 minusm21

q22 minusm22

q23 minusm23

q24 minusm24

) (619)

q1 = q q2 = q+k1 q3 = q+k1 +k2 q4 = q+k1 +k2 +k3 (620)

All of these can in turn be reduced to scalar loop integrals [252] forwhich closed expressions exist [253] However for incoming particleswith identical momenta the original reduction procedure [252] of Pas-sarino and Veltman breaks down and we therefore used the LERGprogram [254] which has implemented an extended Passarino-Veltmanscheme to cope with such a case

6 With the first-level fermions degenerate in mass we finally found

(σv)γγ =α2

EMg4eff

144πm2B(1)

3 |B1|2 + 12 |B2|2 + 4 |B6|2 minus 4Re [B1 (Blowast2 +Blowast

(621)where

g2eff equiv

Q2(Y 2s + Y 2

d ) =52

9 (622)

The sum runs over all charged SM fermions and the analytical expres-sions of B1 B2 and B6 can be found in the appendix of Paper III

Figure 65 shows the annihilation rate (σv)γγ as a function of the mass shiftbetween the B(1) and the KK fermions

In addition to the fermion box diagrams there will be a large number ofFeynman diagrams once SU(2) vectors and scalar fields are included Thereare 22 new diagram types that are not related by any obvious symmetry andthey are shown in Fig 66 Obviously it would be a tedious task to ana-lytically calculate all these contributions by hand Instead we took anotherapproach and implemented the necessary Feynman rules into the FeynArts

software [255] FeynArts produces a formal amplitude of all contributing di-agrams that can then be numerically evaluated with the FormCalc [256]package (which in turn uses the Form code [257] and LoopTools [256] toevaluate tensor structures and momentum integrals) This numerical methodcould also be used to check our analytical result of the fermion loop contribu-tion Adding the Feynman diagrams including bosons as internal propagatorsit was numerically found that they make only up a slight percentage of the

101 102 105 11 12

mξ(1)mB(1)

(σv) γγ

[10minus

Figure 65 The annihilation rate into two photons as a function of themass shift between the B(1) and first Kaluza-Klein level fermions ξ(1) Thisis for m

B(1) = 08 TeV but the dependence on the B(1) mass is given bythe scaling (σv)γγ prop mminus2

B(1) A convenient conversion is σv = 10minus4 pb =

10minus4c pb asymp 3 middot 10minus30 cm3 sminus1 Figure from Paper III

total cross section This is in agreement with the naıve expectation previ-ously mentioned that the fermionic contribution should dominate HigherKK levels also contributes however the larger KK masses in their propaga-tors suppress these contributions By adding second-level fermions we couldconfirm that our previous result only changed by a few percent In conclu-sion the analytical expression (621) is a good approximation for two photonproduction and (σv)γγ sim few times 10minus30 cm3s (1 TeVmB(1))2

The two other processes that can give mono-energetic photons B(1)B(1) rarrZγ and B(1)B(1) rarr Hγ have so far not been fully investigated The fermioniccontribution to B(1)B(1) rarr Zγ can at this stage easily be calculated The onlydifference to the two photon case is here that the Z boson has both a vectorand axial part in its coupling to fermions From the values of these couplingswe have analytically and numerically found that this Zγ line should have across section of about 10 compared to the γγ line

The Hγ line could also enhance the gamma line signal If the Higgs massis very heavy it could also potentially be resolved as an additional line atenergy Eγ = mB(1)(1 minus m2

h4m2B(1))) The contributing diagrams can have

significantly different structure than in the B(1)B(1) rarr γγ process Although

In supersymmetry the Hγ line is inevitably very week as its forbidden in the limit ofzero velocity annihilating neutralinos

Section 64 Observing the Gamma-Ray Signal 95

aG aGB (1)

Hψ(0)

WB (1)

aGB (1)

Figure 66 The 22 different types of bosonic diagrams in addition tothe fermion loop type in Fig 64 that contribute to B(1)B(1) rarr γγ

it is not expected to give any particularly strong signallowastlowast an accurate analysisof this processes has not yet been carried out

64 Observing the Gamma-Ray Signal

The characteristic dark matter signature in the gamma-ray spectrum ndash a hardspectrum with a sharp drop in the flux and potentially even a visible gammaline at an energy equal to the mass of the B(1) ndash is something that can besearched for in many experiments Due to the large uncertainties in the darkmatter density distribution the expected absolute flux from different sourcessuch as the galactic center small dark matter clumps and satellite galaxies

lowastlowast Estimates including only fermion propagators indicate that these contributions are notvery significant

as well as the diffuse extragalactic is still very difficult to predict with anycertainty

The velocity scale for cold dark matter particles in our Galaxy halo isof the order of v sim 10minus3c For an ideal observation of the monochromaticgamma line this would lead to a relative smearing in energy of sim 10minus3 dueto the Doppler shift This narrow line is much too narrow to be fully resolvedwith the energy resolution of current gamma-ray telescopes To compare atheoretically predicted spectrum to experimental data the predicted signalshould first be convolved with the detectorrsquos response The actual detectorresponse is often unique for each detector and is often rather complicatedTo roughly take the convolution into account it is reasonable to use a simpleconvolutionsmearing of the theoretical energy spectrum before comparingwith published data For a Gaussian convolution function with an energyresolution σ(Eprime) the predicted experimental flux is given by

dΦexp

int infin

minusinfindEprime dΦtheory

Eprime

eminus(EprimeminusE)2(2σ2)

radic2πσEprime

As an illustrative example let us compare our theoretically predicted spec-trum with the TeV gamma-ray signal observed from the direction of thegalactic center as observed by the air Cerenkov telescopes HESS [258]Magic [259] VERITAS [260] and CANGAROO [261] The nature of thissource is partly still unknown Because the signal does not show any appar-ent time variation is located in a direction where a high dark matter densityconcentration could be expected and is observed to be a hard spectrum upto high gamma-ray energies (ie the flux does not drop much faster thanEminus2) it has been discussed if the gamma flux could be due to dark matterannihilations (see eg [262] and references therein)

In Fig 67 the predicted gamma spectrum from annihilating B(1)s withmasses of 08 TeV smeared with an energy resolution of σ = 015Eprime is showntogether with the latest HESS data Annihilation of such low mass darkmatter particles can certainly not explain the whole range of data Howeverit is interesting to note that the flux comes out to be of the right order ofmagnitude for reasonable assumptions about the dark matter density distri-bution For the flux prediction an angular acceptance of ∆Ω = 10minus5 sr and aboost factor b asymp 200 to the NFW density profile were used in the flux equation(216) With a better understanding of backgrounds and more statistics itmight be possible to extract such a dark matter contribution especially sincethere is a sharp cutoff signature in annihilations spectrum to look for Whenthe first data were presented from the HESS collaboration it was noticedthat the spectral energy distribution of gamma rays showed a very similarhard spectrum as predicted from final state radiation from light leptons Topoint this out we suggested in Paper II a hypothetical case with M

B(1) sim 10TeV A good match to the 2003 year data (solid boxes in Fig 67) [263] wasthen found Later observations during 2004 [258] did not match the prediction

Eγ [TeV]

E2 γdΦγdEγ

[mminus

2sminus

01 1 10

10minus9

10minus8

10minus7

Figure 67 The HESS data (open boxes 2003 data [263] solid tri-angles 2004 data [258]) compared to the gamma-ray flux expected froma region of 10minus5 sr encompassing the galactic center for a B(1) mass of08 TeV a 5 mass splitting at the first Kaluza-Klein level and a boostfactor b sim 200 (dotted line) The solid line corresponds to a hypothetical9 TeV WIMP with similar artificial couplings a total annihilation rategiven by the WMAP relic density bound and a boost factor of around1000 Both signals have been smeared to simulate an energy resolution of15 appropriate for the HESS telescope

for such a heavy B(1) particle neither was any cutoff at the highest energiesfound Although it is in principle possible to reconstruct the observed spectralshape with dark matter particles of tens of TeV as illustrated in [262] withnon-minimal supersymmetry models this does not work for the most simpleand common models of dark matter [258 262] With more statistics in the2004 data a search for any significant hidden dark matter signal on top of asimple power law astrophysical background was performed but no significanttypical dark matter component could be found [258] When it comes to sug-gested astrophysical explanations of the observed TeV signal from the galacticcenter the main proposed sources are particle accelerations in the SagittariusA supernova remnant processes in the vicinity of the supermassive black holeSagittarius Alowast and the detected nearby pulsar wind nebula (see [264] andreferences therein)

When it comes to experimental searches for the monochromatic gammaline signal from B(1) annihilation a much better energy resolution wouldbe needed Nevertheless with an energy resolution close to the mentionednatural Doppler width of 10minus3 the line could definitely be detectable In

Eγ [GeV]

dΦγdEγ

[10minus

minus2sminus

minus1]

780 790 800 810

Figure 68 The gamma line signal as expected from B(1) dark matterannihilations The line-signal is superimposed on the continuous gamma-ray flux (solid line) and is roughly as it would be resolved by a detectorwith a Gaussian energy resolution of 1 (dashed) 05 (dotted) and025 Ersquo (dash-dotted) respectively The actual line width of the signal isabout 10minus3 with a peak value of 15 middot 10minus7 mminus2 sminus1 TeVminus1 The examplehere is for a B(1) mass m

B(1) = 08 TeV and a mass shift mξ(1)mB(1) =

105 An angular acceptance of ∆Ω = 10minus5 sr and a boost factor ofb = 100 to a NFW profiles were assumed Figure from Paper III

Fig 68 the expected gamma-ray spectrum around a 08 TeV mass B(1) isshown for three different detector resolutions an energy resolution better than1 would be needed to resolve the line signal Typically the energy resolutionof todayrsquos detectors is a factor of ten larger

C h a p t e r

Supersymmetryand a New

Gamma-Ray Signal

The most well known dark matter candidate is the neutralino a WIMP thatappears in supersymmetry theories Although the detection signals for thisdark matter candidate have been well studied the internal bremsstrahlungcontribution to the gamma-ray spectrum for typical heavy neutralinos haspreviously not been investigated In our study in Paper IV we found thatinternal bremsstrahlung produce a pronounced signature in the gamma-rayspectrum in the form of a very sharp cutoff or even a peak at the highestenergies This signal can definitely have a positive impact on the neutralinodetection prospects as it not only possesses a striking signature but also sig-nificantly enhances the expected total gamma-ray flux at the highest energiesWith the energy resolution of current detectors this signal can even dominatethe monochromatic gamma-ray lines (γγ and Zγ) that previously been shownto provide exceptional strong signals for heavy neutralinos

71 Supersymmetry

Supersymmetry relates fermionic and bosonic fields and is thus a symmetrymixing half-integer and integer spins The generators of such a symmetry mustcarry a half integer spin and commute with the Hamiltonian These generatorstransform non-trivially under Lorentz transformations and the internal sym-metries interact non-trivially with the spacetime Poincare symmetry Thismight seem to violate a no-go theorem by Coleman and Mandula statingthat any symmetry group of a consistent four-dimensional quantum field the-ory can only be a direct product of internal symmetry groups and the Poincaregroup (otherwise the scattering-matrix is identically equal to 1 and no scat-tering is allowed) However the new feature of having anticommuting spin

100 Supersymmetry and a New Gamma-Ray Signal Chapter 7

half generators for a symmetry turn out to allow for a nontrivial extensionof the Poincare algebra The details of the construction of a supersymmetrictheory are beyond the scope of this thesis and only some basic facts and themost important motivations for supersymmetry will be mentioned here

Since it is not possible to relate the fermions and bosons within the SMeach SM fermion (boson) is instead given new bosonic (fermionic) supersym-metric partner particles The nomenclature for superparners is to add a prefixlsquosrsquo to the corresponding fermion name (eg the superpartner to the electron iscalled the selectron) whereas superpartners to bosons get their suffix changedto lsquoinorsquo (eg the superpartner to the photon is called the photino)

Superpartners inherit mass and quantum numbers from the SM particlesOnly the spin differs Since none of these new partners have been observedsupersymmetry must be broken and all supersymmetric particles must haveobtained masses above current experimental limits The actual breaking ofsupersymmetry might introduce many new unknown parameters Explicitsymmetry breaking terms can introduce more than 100 new parameters [265]In specific constrained models where the supersymmetry is broken sponta-neously the number of parameters is usually much smaller For example inthe minimal supergravity (mSUGRA) model [266] the number of supersym-metry parameters is reduced to only five specified at a high energy grandunification scale

In the following only minimal supersymmetric standard models (MSSM)are considered They are minimal in the sense that they contain a minimalnumber of particles the SM fields (now with two Higgs doubletslowast) and onesupersymmetric partner to each of these

Some Motivations

A main motivation for having a supersymmetric theory other than for themathematical elegance of a symmetry relating fermions and bosons is thatit presents a solution to the fine-tuning problem within the SM Within theSM the Higgs mass mh gets radiative corrections that diverge linearly withany regulating ultraviolet cutoff energy With the cutoff of the order of thePlanck scale (1019 GeV) the required fine-tuning is of some 17 orders ofmagnitude to reconcile the Higgs mass mh with the indirectly measured valuemh sim 100 GeV This naturalness problem (or fine-tuning problem) of the SMis elegantly solved within supersymmetry by the fact that supersymmetricpartners exactly cancel these divergent quantum contribution to the Higgsmass Even if supersymmetry is broken there is no need for extreme fine-tuning at least not as long as the breaking scale is low and the supersymmetricpartners have masses not much higher than the Higgs mass

lowast This is a type II two Higgs doublet model (to be discussed more in Section 81) andis needed in supersymmetry to generate masses to both the up- and down-type quarks[267 268]

Section 72 A Neglected Source of Gamma Rays 101

A second motivation for supersymmetry is that the required spontaneoussymmetry breaking of the electroweak unification can be obtained throughradiative quantum corrections where the quadratic Higgs mass parameter isdriven to a negative value

A third intriguing attraction has been that the running of the three gaugecoupling constants of the electromagnetic weak and strong forces are modi-fied in such a way that they all become equal within a unification scheme atan energy of about 1016 GeV That this force strength unification occurs atan energy scale significantly below the Plank scale and where the theory stillis perturbatively reliable is far from trivial

Finally supersymmetry can provide dark matter candidates Usually thisis the case in models with an additional symmetry called R-parity which is animposed conserved multiplicative quantity Every SM particle is given posi-tive R-parity whereas all supersymmetric particles are given negative This isvery important for cosmology because R-parity guarantees that the lightestsupersymmetric particle is stable since it cannot decay into any lighter statehaving negative R-parity The introduction of such a symmetry can be fur-ther motivated as it automatically forbids interactions within supersymmetrythat otherwise would lead to proton lifetimes much shorter than experimentallimits

The Neutralino

In many models the lightest stable supersymmetric particle is the lightestneutralino henceforth just lsquothe neutralinorsquo It is a spin-12 Majorana particleand a linear combination of the gauginos and the Higgsinos

χ equiv χ01 = N11B +N12W

3 +N13H01 +N14H

02 (71)

With R-parity conserved the neutralino is stable and a very good dark mattercandidate This is the most studied dark matter candidate and there aremany previous studies on its direct and indirect detection possibilities (seeeg [25 42] and references therein) We will here focus on a new type ofgamma-ray signature first discussed in Paper IV

72 A Neglected Source of Gamma Rays

Previous studies of the gamma-ray spectrum from neutralino annihilationshave mainly focused on the continuum spectrum arising from the fragmen-tation of produced quarks and τ -leptons and the second order loop-inducedγγ and Zγ line signals [96 269 270] For high neutralino masses the almostmonochromatic γγ and Zγ photon lines can be exceptionally strong withbranching ratios that reach percent level despite the naıve expectation of be-ing two to three orders of magnitude smaller The origin of this enhancement

is likely due to nonperturbative binding energy effects in the special situa-tion of very small velocities large dark matter masses as well as small massdifferences between the neutralino and the lightest chargino [271272]

The contribution from radiative processes ie processes with one ad-ditional photon in the final state should naıvely have a cross section twoorders of magnitude larger than the loop-suppressed monochromatic gammalines since they are one order lower in the fine structure constant αem Asinvestigated in Paper IV internal bremsstrahlung in the production ofcharged gauge bosons from annihilating heavy neutralinos results in high-energy gamma rays with a clearly distinguishable signature This is partlyreminiscent of the case of KK dark matter where final state radiation in anni-hilation processes with charged lepton final states dominates the gamma-rayspectrum at the highest energies

Helicity Suppression for Fermion Final States

In Paper II we found that final state radiation from light leptons producedin B(1) annihilation gave an interesting signature in the gamma-ray spectrumNeutralino annihilations into only two light fermion pairs have an exception-ally strong suppression and typical branching ratios into electrons are oftenquoted to be only of the order of 10minus5 The reason for this so-called helicitysuppression can be understood as follows The neutralinos are self-conjugate(Majorana) fermions obeying the Pauli principle and must therefore form anantisymmetric wave function In the case of zero relative velocitydagger the spatialpart of the two particlesrsquo wave function is symmetric ndash ie an orbital angularmomentum L = 0 state (s-wave) ndash and the spin part must form an antisym-metric (S = 0) singlet state Thus the incoming state has zero total angu-lar momentum The contributing neutralino annihilation processes conservechiralityDagger so that massless (or highly relativistic) fermions and antifermionscome with opposite helicities (ie handedness) Therefore the spin projectionin the outgoing direction is one which precludes s-wave annihilationsect Theconclusion must be that the annihilation cross section into monochromaticmassless fermions is zero for massive fermions it is instead proportional tom2fm

2χ An interesting way to circumvent this behavior of helicity suppres-

dagger Typical dark matter halo velocities are v sim 10minus3 and p-wave annihilations would besuppressed by v2 sim 10minus6

Dagger Both the Z-fermion-antifermion and fermion-sfermion-gaugino vertices conserve helicityContributions from Higgs-boson exchange from Higgsino-sfermion-fermion Yukawa in-teractions and from sfermion mixing violate chirality conservation but they all includean explicit factor of the fermion mass mf [273]

sect Orbital angular momentum of the outgoing fermions can never cancel the spin com-ponent in the direction of the outgoing particles (this is clear because orbital angularmomentum of two particles can never have an angular momentum component in thesame plane as their momentum vectors lie in) Therefore the total angular momentummust be nonzero in contradiction to the initial state of zero angular momentum

χminus1

WminusWminusWminus Wminus

Figure 71 Contributions to χχ rarr W +Wminusγ for a pure Higgsino-likeneutralino (crossing fermion lines are not shown) Figure from Paper IV

sion is to have a photon accompanying the final state fermions [274] Thisopen up the possibility of a significant photon and fermion spectrum wherethe first-order corrected cross section can be many orders of magnitude largerthan the tree-level result I will not pursue this interesting possibility here(see however [274] and the recent work of [275]) but instead discuss internalbremsstrahlung when Wplusmn gauge bosons are produced by annihilating heavyHiggsinos [Paper IV]

Charged Gauge Bosons and a Final State Photon

In order not to overclose the Universe a TeV-mass neutralino must in generalhave a very large Higgsino fractionpara Zh (Zh equiv |N13|2 + |N14|2) ensuring asignificant cross section into massive gauge bosons A pure bino state witha TeV mass on the other hand does not couple to W at all in lowest orderThis usually excludes TeV binos as they would freeze-out too early and over-produce the amount of dark matter Let us therefore focus on a Higgsino-likeneutralino with N11 asymp N12 asymp 0 and N13 asymp plusmnN14 The annihilation rate intocharged gauge bosons often dominates and radiation of a final state photonshould be of great interest to investigate

For a pure Higgsino the potential s-channel exchanges of Z and Higgsbosons vanish and the only Feynman diagrams contributing to the W+Wminusγfinal states are shown in Fig 71

For the analytical calculation of these Feynman diagrams there is onetechnicality worth noticing Due to the Majorana nature of the neutralinosthe Feynman diagrams can have crossing fermion lines and special care mustbe taken to deal with the spinor indices correctly Proper Feynman ruleshave been developed (see eg [276]) which also have been implemented indifferent numerical code packages (eg FeynArts and FormCalc [277]) Formanual calculations a practical simplifying technique can be adopted in thelimit of zero relative velocity the two ingoing annihilating neutralinos must

para This is the case if the usual GUT condition M1 sim M22 is imposed otherwise a heavywino would also be acceptable For a pure wino the results are identical to what isfound for the anti-symmetric N13 = minusN14 Higgsinos considered here apart from amultiplicative factor of 16 in all cross sections

form a 1S0 state (as explained above) and the sum over all allowed spin stateconfigurations of the two incoming Majorana particles can be replaced by theprojector [278]

P1S0equiv minus 1radic

2γ5 (mχ minus 6p) (72)

where p is the momentum of one of the incoming neutralinos P1S0is sim-

ply inserted in front of the gamma-matrices originating from the Majoranafermion line and then the trace is taken over the spinor indices All analyticalcalculations in Paper IV were performed both by this technique of using theP1S0

projector operator and direct calculations by explicitly including all thediagrams with their crossing fermion lines The calculations were also furtherchecked by numerical calculations with the FeynArtsFormCalc numericalpackage

The analytical result is rather lengthy but up to zeroth order in ǫ equivmWmχ and retaining a leading logarithmic term the resulting photon mul-tiplicity is given by

dxequivd(σv)WWγdx

(σv)WW≃ αem

4(1 minus x+ x2)2 ln(2ǫ)

(1 minus x)x

minus 2(4 minus 12x+ 19x2 minus 22x3 + 20x4 minus 10x5 + 2x6)

(2 minus x)2(1 minus x)x

+2(8 minus 24x+ 42x2 minus 37x3 + 16x4 minus 3x5) ln(1 minus x)

+ δ2(

2x(2 minus (2 minus x)x)

(2 minus x)2(1 minus x)+

8(1 minus x) ln(1 minus x)

(2 minus x)3

+ δ4(

x(xminus 1)

(2 minus x)2+

(xminus 1)(2 minus 2x+ x2) ln(1 minus x)

(2 minus x)3

where x equiv Eγmχ and δ equiv (mχplusmn1minusmχ)mW with mχplusmn

1denoting the chargino

massFigure 72 shows the photon multiplicity together with a concrete realized

minimal supersymmetric model example as specified in Table 71Two different effects can be singled out to cause increased photon fluxes

at the highest energies The first occurs for large mass shifts δ between theneutralino and the chargino whereby the last two terms in Eq (73) dominateThese terms originate from the longitudinal polarization modes of the chargedgauge bosons Such polarization modes are not possible for a 1S0 state with

The MSSM parameters specify the input to DarkSUSY [279] M2 micro mA and mf

are the mass scales for the gauginos Higgsinos supersymmetry scalars and fermionsrespectively Af (= At = Ab) is the trilinear soft symmetry breaking parameter andtan β = vuvrmd is the ratio of vacuum expectation values of the two neutral Higgsdoublet All values are directly given at the weak energy scale For more details of this7-parameter MSSM see [280]

0 05 1001

x equiv Eγmχ

dNW γdx

Figure 72 The photon multiplicity for the radiative process χχ rarrW +Wminusγ The dots represent the minimal supersymmetric model givenin Table 71 as computed with the FormCalc package [256] for a relativeneutralino velocity of 10minus3 The thick solid line shows the full analyticalresult for the pure Higgsino limit of the same model but with zero relativeneutralino velocity Also shown as dashed and dotted lines are two pureHiggsino models with a lightest neutralino (chargino) mass of 10 TeV (10TeV) and 15 TeV (25 TeV) respectively Figure from Paper IV

Table 71 MSSM parameters for the example model shown in Fig 72-73 and the resulting neutralino mass (mχ) chargino mass (m

χplusmn1

) Hig-

gsino fraction (Zh) branching ratio into W pairs (Wplusmn) and neutralinorelic density (Ωχh2) as calculated with DarkSUSY [281] and micrOMEGAs

[282] Masses are given in units of TeV

M2 micro mA mf Af tanβ mχ mχplusmn1

Zh Wplusmn Ωχh2

32 15 32 32 00 100 150 151 092 039 012

only two vector particles (remember that the initial state must be in thisstate due to the Majorana nature of the low velocity neutralino) but whenan additional photon is added to the final state this channel opens up andenhances the photon flux at high energies Typically MSSM models arehowever not expected to have very large mass shifts δ The other effect ison the other hand dominated by transversely polarized photons For heavyneutralino masses the W bosons can be treated as light and the cross sectionis thus expected to be enhanced in a similar way to the infrared divergence thatappears in QED when low-energy photons are radiated away For kinematical

01 05 1 201

Eγ [TeV]

d(vσ) γdEγ

[10minus

2sminus

minus1]

06 1 22

Eγ [TeV]d(vσ) γdEγ

[10minus

2sminus

minus1]

Figure 73 Left panel The total differential photon distributionfrom χχ annihilations (solid line) for the minimal supersymmetric modelof Table 71 Also shown separately is the contribution from inter-nal bremsstrahlung χχ rarr W +Wminusγ (dashed) and the fragmentation ofmainly the W and Z bosons together with the χχ rarr γγ Zγ lines (dot-ted) Right panel A zoom in of the same spectra as it would approxi-mately appear in a detector with a relative energy resolution of 15 percentFigures from Paper IV

reasons each low energy W boson is automatically accompanied by a highenergy photon The resulting peak in the spectrum at the highest energiesis hence an amusing reflection of QED infrared behavior also for W bosonsThe two different effects are illustrated in Fig 72 by the dotted and dashedcurves respectively

In addition to the internal bremsstrahlung discussed above secondarygamma rays are produced in the fragmentation of theW pairs mainly throughthe production and subsequent decay of neutral pions Similarly productionof Z-bosons (or quarks) results in secondary gamma rays altogether produc-ing a continuum of photons dominating the gamma flux at lower energiesPrevious studies have also shown that there are strong line signals from thedirect annihilation of a neutralino pair into γγ [269] and Zγ [270] Due to thehigh mass of the neutralino (as studied here) the two lines cannot be resolvedbut effectively add to each other at an energy almost equal to the neutralinomass Adding all contributions and using the model of Table 71 the totalspectrum is shown in the left panel of Fig 73

The practical importance of internal bremsstrahlung is even clearer whentaking into account an energy resolution of about 15 which is a typicalvalue for current atmospheric Cerenkov telescopes in that energy range Theresult is a smeared spectrum as shown in the right panel of Fig 73 We can

Eγ [TeV]

0minus

3sminus

01 05 1 2

Figure 74 Comparing the shape of the total gamma-ray spectrum thatcan be expected from B(1) Kaluza-Klein (dashed line) and neutralino (solidline) dark matter annihilations as seen by a detector with an energyresolution of 15 In both cases the dark matter particle has a mass of15 TeV and both spectra are normalized to have a total annihilation crosssection of 〈σv〉0 = 3 middot 10minus26cm3sminus1 Figure from Paper B

see that the contribution from the internal bremsstrahlung enhances the fluxin the peak at the highest energies by a factor of about two The signal is alsodramatically increased by almost a factor of 10 at slightly lower energiesthereby filling out the previous lsquodiprsquo just below the peak This extra flux athigh energies improves the potential to detect a gamma-ray signal It is worthpointing out that this example model is neither tuned to give the most extremeenhancements nor is it only pure Higgsinos with W final states that shouldhave significant contributions from this type of internal bremsstrahlung Onthe contrary this type of internal bremsstrahlung can contribute 10 timesmore to high-energy photon flux than the gamma-ray line see [275] for aextensive scan of MSSM parameters

As in the case of the KK dark matter candidate B(1) the internal radiationof a photon in the neutralino annihilation case also gives a very characteristicsignature in the form of a very sharp cutoff in the gamma-ray spectrum atan energy equal to the neutralino mass This is a promising signal to searchfor and with current energy resolution and with enough statistics the shapeof the gamma-ray spectra could even provide a way to distinguish betweendifferent dark matter candidates Figure 74 illustrates this by comparing thegamma-ray spectrum of a 15 TeV neutralino (specified in Table 71) and a15 TeV B(1) KK dark matter candidate

C h a p t e r

Inert HiggsDark Matter

A possible and economical way to incorporate new phenomenology into thestandard model would be to enlarge its Higgs sector One of the most minimalway to do this which simultaneously gives rise to a dark matter candidate isthe so-called inert doublet model (or inert Higgs model) obtained by adding asecond scalar Higgs doublet with no direct coupling to fermions The lightestof the new appearing inert Higgs particles could if its mass is between 40 and80 GeV give the correct cosmic abundance of cold dark matter One way tounambiguously confirm the existence of particle dark matter and determineits mass would be to detect its annihilation into monochromatic gamma raysby current or upcoming telescopes In Paper VII we showed that for theinert Higgs dark matter candidate the annihilation signal into such monochro-matic γγ and Zγ final states is exceptionally strong The energy range andrates for these gamma-ray line signals therefore make them ideal to searchfor with upcoming telescopes such as the GLAST satellite This chapter re-views the inert Higgs dark matter candidate and discusses the origin of thesecharacteristic gamma line signals

81 The Inert Higgs Model

Let us start by shortly reviewing why there is a need for a Higgs sector in thefirst place In the SM of particle physics it is not allowed to have any explicitgauge boson or fermion mass terms since that would spoil the underlyingSU(2)timesU(1) gauge invariance and lead to a non-renormalizable theorylowast Tocircumvent this we start with a fully gauge invariant theory ndash with no gaugefield or fermion mass terms ndash and adds couplings to a complex Lorentz scalar

lowast A fundamentally non-renormalizable theory would lack predicability as the canonicallyappearing divergences from quantum corrections can not be cured by the renormalizationprocedure of absorbing them into a finite number of measurable quantities

110 Inert Higgs Dark Matter Chapter 8

SU(2) doublet φ which spontaneously develops a non-vanishing vacuum ex-pectation value and thereby breaks the full SU(2)timesU(1) gauge structure andgenerates particle masses This is exactly what was technically described inSection 53 of the UED model (but of course now without the complicationsof having an extra spatial dimension) The Higgs Lagrangian written down inEq (517) with the potential (518) is in four dimensions the most generalsetup we can have with one Higgs field φ As explained in Section 53 thisHiggs field has four scalar degrees of freedom Three of the degrees of freedomare absorbed by the new polarization modes of the now massive gauge bosonsand thus only one degree of freedom is left as a physical particle h (the Higgsparticle) The mass of the Higgs particle h is a completely free parameter inthe SM (which can be measured and constrained)

Another way to express the need for the Higgs particle is that withoutit certain cross sections would grow with the center of mass energy (denotedby Ecm) beyond the unitarity limit for large enough Ecm An example isthe process f f rarrW+Wminus into longitudinally polarized Wplusmn (those W rsquos thatarose by the Higgs mechanism) If the Higgs particle is not included the onlycontributing Feynman diagrams are from s-channel gauge bosons and t- oru-channel fermions which give rise to a term that grows as m2

fE2cm This is

the piece that is exactly canceled by the s-channel Higgs boson that couplesproportionally to mf This should make it clear that a physical Higgs or asimilar scalar interaction must be included to have a sensible theory

So why is there only one Higgs doublet in the standard model The in-clusion of just one Higgs doublet is the most economical way of introducingmasses into the SM but in principle nothing forbids models with more com-plicated Higgs sectors This might at first sound as a deviation from theprinciple of Occamrsquos razor but we will soon see that such extensions can bemotivated by its potential to address several shortcomings of the SM and stillsatisfy theoretical and existing experimental constraints

A minimal extension of the SM Higgs sector would be to instead have twoHiggs doublets H1 and H2 The Lagrangian for such so-called two-Higgs-doublet models can formally be written as

|DmicroH1|2 + |DmicroH2|2 minus V (H1 H2) (81)

The most general gauge invariant renormalizable potential V (H1 H2) thatis also invariant under the discrete Z2 symmetry

H2 rarr minusH2 and H1 rarr H1 (82)

can be written as

V (H1 H2) = micro21 |H1|2 + micro2

2 |H2|2 + λ1 |H1|4 + λ2 |H2|4 (83)

+λ3 |H1|2 |H2|2 + λ4|Hdagger1H2|2 + λ5Re[(H

dagger1H2)

Section 81 The Inert Higgs Model 111

where micro2i λi are real parametersdagger The latter constraint of a discrete Z2

symmetry is related to the experimental necessity to diminish flavor-changingneutral currents (FCNCs) and CP-violations in the Higgs sector [139267268283]

The experimental limits on FCNCs are very strong and come from egstudies of the neutral K0 meson K0 is a bound state (containing a downquark and a strange anti-quark) that would if FCNCs were mediated by s-channel Z or Higgs bosons at tree level rapidly oscillate into its antiparticleK0 or decay directly into lepton pairs These processes are so rare that theyare only expected to be compatible with loop-level suppressed reactions (or insome other way protected as for example pushing the FCNC mediator to veryhigh masses) In the SM FCNCs are naturally suppressed as they are for-bidden at tree levelDagger Technically this comes about since the diagonalizationof the mass matrix automatically also flavor diagonalizes the Higgs-fermioncouplings as well as the fermion couplings to the photon and the neutral Zgauge boson This lack of FCNCs would in general no longer be true in theHiggs sector once additional scalar doublets are included that have Yukawacouplings to fermions In a theorem by Glashow and Weinberg [284] it wasshown that FCNCs mediated by Higgs bosons will be absent if all fermionswith the same electric charges do not couple to more than one Higgs doubletAdopting this approach to suppress FCNCs the scalar couplings to fermionsare constrained but not unique To specify a model it is practical to imposesa discrete symmetry Any such discrete symmetry must necessarily be of theform H2 rarr minusH2 and H1 rarr H1 (or vice versa) [267] The Z2 symmetry canthen be used to design which Yukawa couplings are allowed or not This isexactly the Z2 symmetry that was already incorporated in the potential givenin Eq (83)

In what is called a Type I two-Higgs-doublet model the fermions coupleonly to the first Higgs doubletH1 and there are no couplings between fermionsand H2 This is the same as saying that the Lagrangian is kept invariantunder the Z2 symmetry that takes H2 rarr minusH2 and leaves all other fieldsunchanged This is the type of model we will be interested in here A Type IImodel is when the down-type fermions only couple directly to H1 and up-typefermions only couple directly to H2 (corresponding to the appropriate choicefor the Z2 transformation of the right-handed fermion fields uR rarr minusuR)The minimal supersymmetric models belongs to this Type II class of modelsOther choices where quarks and leptons are treated in some asymmetrical way

dagger An additional term is actually possible but can always be eliminated by redefining thephases of the scalars [267 268]

Dagger Actually even the loop-level FCNCs in the SM need to be suppressed In 1970 GlashowIliopoulos and Maiani realized that this could be achieved if quarks come in doubletsfor each generation (the GIM mechanism) [139] Their work was before the detection ofthe charm quark and therefore predicted this new quark to be the doublet companionfor the already known strange quark

could in principle also be possibleComing back to the Type I model where only Yukawa terms involving H1

are allowed This means that H1 must develop a nonzero vacuum expectationvalue v 6= 0 to generate fermion and gauge masses In other words thepotential must have a minimum for H1 6= 0 For H2 there are however twochoices for its potential Either H2 also develops a vacuum expectation valuevH2

6= 0 by spontaneous symmetry breaking and the potential has a globalminimum for H2 6= 0 or the Z2 symmetry H2 rarr minusH2 is unbroken and thepotential has a global minimum at H2 = 0 (Note that in general this lattercase is not the vH2

rarr 0 limit of the vH26= 0 case)

The model that contains our dark matter candidate is the latter of thetwo Type I two-Higgs-doublet models that have vH2

= 0 In other wordsthis is an ordinary two-Higgs-doublet model with the H2 rarr minusH2 symmetryunbroken The H1 field is identified as essentially the SM Higgs doublet ndash itgets a vacuum expectation value and gives masses to the W Z and fermionsexactly as in the SM On the other hand the H2 does not get any vacuumexpectation value and does not couple directly to fermions This H2 will becalled the inert Higgs doublet and the model the inert doublet model (IDM)sect

The origin of this IDM goes back to at least the 1970s [285] when thedifferent possibilities for the two-Higgs-doublet models were first investigatedThe IDM has recently received much new interest Besides providing a darkmatter candidate [286 287] this type of model has the potential to allowfor a high Higgs mass [286] generate light neutrinos and leptogenesis (seeeg [288] and references therein) as well as break electroweak symmetryradiatively [289]

The New Particles in the IDM

Let us set up some notation and at the same time present how many freeparameters and physical fields this inert doublet model contains The twoHiggs doublets will be parameterized according to

H1 =1radic2

G+radic

2v + h+ iG0

H2 =1radic2

H0 + iA0

where Gplusmn Hplusmn are complex scalar fields while h G0 H0 and A0 are realpara

We can always use the freedom of SU(2)timesU(1) rotations to get the vacuum

sect The name inert Higgs doublet might be found misleading as it is neither completelyinert (since it has ordinary gauge interactions) nor contributes to the Higgs mechanismto generate masses The name dark scalar doublet has later been proposed but we willhere stick to the nomenclature used in Paper VII and call it the inert Higgs or inertscalar

para I have here slightly changed the notation for the scalar fields compared to Section 53

expectation value v for H1 to be real valued and in the lower component of thedoublet G+ and G0 constitute Goldstone fields that in unitarity gauge canbe fixed to zero After giving mass to the gauge bosons five out of the originaleight degrees of freedom in H1 and H2 remain Besides the SM Higgs particle(h) the physical states derived from the inert doubletH2 are thus two chargedstates (Hplusmn) and two neutral one CP-even (H0) and one CP-odd (A0) Theh field in the H1 doublet will be referred to as the SM Higgs particle andthe particle fields in H2 as the inert Higgs particles The corresponding (treelevel) masses are given by

m2h = minus2micro2

m2H0 = micro2

2 + (λ3 + λ4 + λ5)v2

m2A0 = micro2

2 + (λ3 + λ4 minus λ5)v2

m2Hplusmn = micro2

2 + λ3v2 (86)

Measurements of the gauge boson masses determine v = 175 GeV and we areleft with only 6 free parameters in the model A convenient choice is to workwith mhmH0 mA0 mH+ micro2 and λ2

Heavy Higgs and Electroweak Precision Bounds

One of the original motivations for the IDM was that it could incorporate aheavy SM Higgs particle Let us briefly review why this might be of interestand how this is possible in contrast to the SM and the MSSM

In the SM the Higgs boson acquires an ultraviolet divergent contributionfrom loop corrections which will be at least of the same size as the en-ergy scale where potential new physics comes in to cancel divergences Withno such new physics coming in at TeV energies a tremendous fine-tuningis required to keep the Higgs mass below the upper limit of 144 GeV (95confidence level) determined by electroweak precision tests (EWPT) [290]lowastlowastLow-energy supersymmetry provides such new divergence-canceling physicsand this is one of the strongest reasons to expect that physics beyond theSM will be found by the LHC at CERN However in the MSSM the lightestHiggs particle is naturally constrained to be lighter than sim135 GeV [152] andsome amount of fine-tuning [291] is actually already claimed to be needed tofulfill the experimental lower bound of roughly 100 GeV from direct Higgssearches [152 290] This suggestive tension has motivated several studies on

and Appendix A discussing the UED model This is for consistency with the notationof Paper VII and our implementation of the IDM into FeynArts [255] The translationbetween the notations is trivial G0 equiv minusχ3(0) Gplusmn equiv plusmniχplusmn(0) and vIDM equiv vUED

radic2

If both doublets develop vacuum expectation values the physical fields will be linearcombinations with contributions from both H1 and H2 [267]

lowastlowast The central value for the SM of the Higgs mass is 76 GeV from EWPT alone and there isa lower limit of 114 GeV from direct searches Note that if direct searches are includedthe upper mass limit on the Higgs increases to 182 GeV according to [290]

the theoretical possibilities to allow for large Higgs masses both within super-symmetry and other extensions of the SM (see eg [286291] and referencestherein) In [286] it was shown that the IDM can allow for a heavy SM-likeHiggs (ie h) This was a basic motivation for the model as it meant thatthe need for divergence canceling physics could be pushed beyond the reach ofthe upcoming LHC accelerator without any need for fine-tuning While thisargument of less fine-tuning (or improved naturalness) [286] has been dis-puted [292] the mere fact that the IDM allows for the SM Higgs mass to bepushed up to about 500 GeV is interesting in itself as it might provide a cleardistinction from the SM and the MSSM Higgs (as well as having an impacton the expected gamma-ray spectrum from annihilation of H0s as discussedin Section 82)

To allow for a heavy SM-like Higgs the upper mass limit of about 144GeV from electroweak precision tests must be avoided The so-called Peskin-Takeuchi parameters denoted S T and U are measurable quantities con-structed to parameterize contributions (including beyond SM physics) to elec-troweak radiative corrections such as the loop-diagram induced contributionto self-energies of the photon Z boson and W boson and the Weinberg an-gle [293] These S T and U parameters are defined such that they vanishfor a reference point in the SM (ie a specific value for the top-quark andHiggs masses) Deviations from zero would then signal the existence of newphysics or set a limit on the Higgs mass when the SM is assumed Insteadof the T parameter the ρ parameter is sometimes used which is defined asρ = m2

cos θw(mW ))

A deviation of ρ from 1 measures how quan-tum corrections alter the tree level SM link between the W and Z bosonmasses [267 293] In fact in most cases T represents just the shift of the ρparameter ∆ρ equiv ρminus 1 = αT

Electroweak precision measurements of the S T and U parameters limitthe Higgs boson mass A heavy h of a few hundred GeV would produce a toosmall value for the observable T whereas the S and U parameters are lesssensitive to the Higgs mass [286] see Fig 81 However a heavy Higgs canbe consistent with the electroweak precision tests if new physics produce acompensating positive ∆T For a mh = 400minus600 GeV the compensation ∆Tmust be ∆T asymp 025 plusmn 01 to bring the value back near the central measuredpoint and within the experimental limits [286] In [286] it was found thatneither the S nor the U parameter is affected much by the extra contributionfrom the IDM but that the T parameter is approximatelydaggerdagger shifted accordingto

∆T asymp 1

24π2αv2(mH+ minusmA0) (mH+ minusmH0) (87)

We thus see that a heavy SM Higgs that usually produces too negative valuesof ∆T can be compensated for by the proper choice of masses for the inert

daggerdagger This approximation is within a few percent accuracy for1 le mHplusmnmH0 mHplusmnmA0 mA0mH0 le 3

mt= 1727 plusmn 29 GeV

mh= 1141000 GeV

minus04minus04

minus02

Figure 81 Dependence of the S T parameters on the Higgs mass (mh)within the standard model The thick black band marks mh = 400 minus600 GeV The top quark mass mt range within the experimental boundsFigure adapted from [286]

Higgs particles For example for a mh = 500 GeV the required compensationis ∆T asymp025plusmn01 and the masses of the inert scalar masses in Eq (87) shouldsatisfy

(mH+ minusmA0) (mH+ minusmH0) = M2 M = 120+20minus30GeV (88)

This means that a Higgs mass mh of up to about 500 GeV can be allowed inthe IDM if only the inert Higgs masses are such that they fulfill Eq (88)

As the Higgs mass is increased the quartic scalar interactions becomestronger and the maximal scale at which perturbation theory can be useddecreases To have a natural perturbative theory up to say 15 TeV (whichis about the highest new energy scale we can have without fine-tuning theHiggs mass [286]) the Higgs mass cannot be heavier than about mh = 600GeV [286]

More Constraints

There are several other constraints that must be imposed besides the elec-troweak precision measurements bounds discussed above The following con-straints on the six free parameters are also used (which are the same con-straints as used in Paper VII)

bull Theoretically the potential needs to be bounded from below in order tohave a stable vacuum which requires

λ12 gt 0

λ3 λ3 + λ4 minus |λ5| gt minus2radic

λ1λ2 (89)

bull To trust perturbation theory calculations at least up to an energy scaleof some TeV the couplings strengths cannot be allowed to become toolarge Here and in Paper VII we followed the constraints found in [286]which can be summarized as a rule of thumb in that no couplingsshould become larger than λi sim 1 (see [286] for more details)

bull In order not to be in conflict with the observed decay width of theZ boson we should impose that mH0 + mA0 amp mZ (see Paper VIIand [294])

bull No full analysis of the IDM has been done with respect to existingcollider data from the LEP and the Tevatron experiments Howevercomparison with similar analyses of supersymmetry enable at least somecoarse bounds to be found The summed mass of H0 and A0 shouldbe greater than about 130 GeV [295] or the mass split must be lessthan roughly 10 GeV [286 295] Similarly the mass of the chargedHiggs scalars Hplusmn is constrained by LEP data to be above about 80GeV [295296]

bull To explain the dark matter by the lightest inert particle (LIP) its relicabundance should fall in the range 0094 lt ΩCDMh

2 lt 0129 See Sec-tion 82 for more details

bull Direct detection searches of dark matter set limits on scattering crosssections with nucleons At tree level there are two spin-independentinteractions whereby H0 could deposit kinetic energy to a nuclei q in

direct search detectors H0qZminusrarr A0q and H0q

hminusrarr H0q The formerprocess with a Z exchange is very strong and is forbidden by currentexperiment limits [286 296 297] However this process becomes kine-matically forbidden if the mass splitting is more than a few 100 keV astypically the kinetic energy of the dark matter candidate H0 would thennot be enough to produce an A0 (this thus excludes the λ5 rarr 0 limitsee Eq 86) With this process kinematically excluded the signals fromHiggs-mediated scattering is roughly two orders of magnitude below anycurrent limits The next generation of detectors could potentially reachthis sensitivity [296]

Also naturalness could be imposed ie parameters should not be tunedto extreme precision In Paper VII we applied the naturalness constraintsfound in [286] but to be less restrictive we relaxed their parameter bounds

Section 82 Inert Higgs ndash A Dark Matter Candidate 117

by a factor two because of the somewhat arbitrariness in defining natural-ness (this constraint is not crucial for any of the general results here or inPaper VII)

When it comes to the upcoming LHC experiment the IDM should be seenin the form of both missing transverse energy and an increased width of theSM Higgs [286294295]

82 Inert Higgs ndash A Dark Matter Candidate

The existence of the unbroken Z2 symmetry in the IDM where the inert Higgsdoublet are attributed negative Z2-parity and all SM particles have positiveparity means that none of the inert Higgs particles can directly decay intoonly SM particles The lightest inert particle (LIP) is therefore a (cosmolog-ically) stable particle which is a first necessity for a dark matter candidateFurthermore the LIP should be electric and color neutral to not violate anyof the strict bounds on charged dark matter [198199] The only choices for aninert Higgs dark matter particle are therefore H0 or A0 Although the rolesof H0 and A0 are interchangeable for all the results let us for definitenesschoose H0 as the LIP

The next crucial step is to see if this H0 candidate can give the rightrelic density to constitute the dark matter In [296] it was shown that H0

can constitute all the dark matter if its mass is roughly 10 minus 80 GeV (orabove 500 GeV if parameters are particularly fine-tuned) However this studywas made for SM Higgs masses of 120 and 200 GeV which although givinghigher gamma rates deviates from one of the motivation for the model ndash araised Higgs mass [286] The setup we had in Paper VII is based on a 500GeV SM Higgs H0 relic density calculations were therefore performed Thiswas done by implementing the proper Feynman rules from the Lagrangian inEq (81) into the Feynman diagram calculator FormCalc [256] Cross-sectioncalculations with FormCalc were then interfaced with the DarkSUSY [298] relicdensity calculator This allowed us to accurately calculate the relic densityfor any given choice of IDM parameters after imposing existing experimentalconstraints The correct relic density is still roughly obtained for masses inthe range of 40 minus 80 GeV and we will next see why this result is almostindependent of the SM Higgs mass

Typically the relic density is governed by the cross section for annihilatingtwo H0 For masses mH0 above the SM Higgs mh (gt mZmW ) the annihila-tion channels are given by the diagrams in Fig 82 If mH0 is above the Wmass then the cross sections from the middle row diagrams dominate Thesediagrams produce very large annihilation cross sections and therefore the H0

relic density becomes too small to constitute the dark matter For massesbelow the W mass only the diagrams in the bottom line will contribute (asthe heavier W and Z bosons are generically not energetically allowed to beproduced during freeze-out) For these lsquolowrsquo H0 masses the tree-level anni-

Wminus(Z)Wminus(Z)

Wminus(Z)

W +(Z)W +(Z)

W +(Z)Hplusmn(A0)

Z(Wplusmn)A0(Hplusmn)

f(fplusmn)f(fplusmn)

f(νf )f(νf )

λL λ2L λLλ1

g2 g2 λLg

λLyf g2

Figure 82 Feynman diagrams for different annihilation (and coanni-hilation) channels for the inert Higgs H0 The two top rows show thecontributing diagrams into standard model Higgs h and WplusmnZ respec-tively The bottom row shows the H0 annihilation channel into fermions(left diagram) and its possible coannihilations processes (right diagram)Displayed under each diagram is the total coupling strength from thevertex factors where g is the gauge coupling strength (g sim gY sim e)λL = (λ3 + λ4 + λ5)2 and yf = mfv In general the H0 mass is belowthe gauge bosonsrsquo otherwise the annihilation strength into W and Z istoo large (if no fine-tuned cancellation occurs) for H0 to constitute thedark matter

hilation rates are small especially for high SM Higgs masses and you couldtend to assume that the relic density would be far too high However coan-nihilations with the next-to-lightest inert scalar allow us to reach the correctrelic abundance (the right-hand-side diagram in the bottom row of Fig 82)It is mainly this coannihilation process which regulates the relic density andthis process is completely independent of the SM Higgs mass

This is an interesting aspect of the IDM ndash that the H0 mass generically hasto be just below the charged gauge boson mass because the relatively strongcoupling to W+Wminus would otherwise give a too low relic density to accountfor the dark matter ndash and as we next will see this will also affect the indirectdetection signal from gamma rays

Section 83 Gamma Rays 119

83 Gamma Rays

Continuum

The dark matter particle in this model is thus the H0 with a mass belowmW This means that only annihilations into fermions lighter than mH0 areaccessible at tree level and the only contributing Feynman diagram is thebottom left one of Fig 82 The annihilation rate is calculated to be

vrelσff =Ncπα

sin4 θWm4W

(1 minus 4m2f

s )32(m2H0 minus micro2

(sminusm2h)

2 +m2hΓ

where Nc is a color factor (which equals 1 for leptons and 3 for quarks)radics

is the center of mass energy α the fine-structure constant mW the W bosonmass θW the weak mixing angle Γh the decay width of h and mf the finalstate fermion mass

The heaviest kinematically allowed fermion state will dominate the tree-level annihilation channels since the cross section is proportional to m2

f Hence in our case of interest the bottom quark final states are the most im-portant process at tree level and with some contributions from charm quarksand τ pairs Although the running of lepton masses can be safely neglectedthe QCD strong interaction corrections to quark masses might be substantialand we therefore take the leading order correction into account by adjustingthe running quark masses [267299] to their values at the energy scale of thephysical process (sim 2mH0) Quark pairs will as already described in the caseof the KK and supersymmetric dark matter hadronize and produce gammarays with a continuum of energies Because of the much harder gamma spec-trum from the decay of τ -leptons these could also contribute significantly atthe highest energies despite their much lower branching ratio Pythia (ver-sion 64) [300] was used to calculate the photon spectrum in the process ofhadronization

Gamma-Ray Lines

As has been said the H0 couplings are relatively strong to W+Wminus (ieordinary gauge couplings) which forces the mass of H0 to be below mW if it isto explain the dark matter Virtual gauge bosons close to threshold could onthe other hand significantly enhance loop processes producing monochromaticphotons (see Fig 83) In Paper VII we showed that this is indeed correctand found lsquosmoking gunrsquo line signals for the H0 dark matter from the finalstates γγ and when kinematically allowed Zγ This in combination withsmall tree-level annihilation rates into fermions makes the gamma lines amost promising indirect detection signal

Let us see what these important line signals from direct annihilation ofH0 pairs into γγ and Zγ look like First of all these spectral lines would show

hh+ symmetry relatedand non-W contributions

g4 g4 g4

λLg3 λLg3

Figure 83 Typical contributing Feynman diagrams for the annihilationprocess H0H0 rarr γγ Due to unsuppressed couplings to Wplusmn virtual Wplusmn

in the intermediate states are expected to give the largest contribution tothis process

up as characteristic dark matter fingerprints at the energies mH0 and mH0minusm2

Z4mH0 respectively The Zγ line might not be strictly monochromatic due

to the Breit-Wigner width of the Z mass but can still be strongly peaked Thepotential third gamma line from hγ is forbidden for identical scalar particleannihilation as in the IDM due to gauge invariance

We could also note that when the branching ratio into Zγ becomes largethe subsequent decay of the Z boson significantly contributes to the continuumgamma-ray spectrum The full one-loop Feynman amplitudes were calculatedby using the numerical FormCalc package [256] ndash after the Feynman rules forthe IDM had been derived and implemented

To show the strength of the gamma-ray lines and the continuum spec-trum for different parameter choices four IDM benchmark models are de-fined shown in Table 81 The two models III and IV have a low Higgs massand could therefore be directly comparable to the relic density calculationsdone in [296] Annihilation rates branching ratios and relic densities for thesemodels are given in Table 82 As an illustrative example Fig 84 shows thepredicted gamma spectrum for model I

The spectral shape with its characteristic peaks in the hitherto unexploredenergy range between 30 and 100 GeV is ideal to search for with the GLAST

x = EγmH0

x2dNγdx

10minus4

10minus2

001 01 1

τ+τminus

Figure 84 The total differential photon distribution from annihilationsof an inert Higgs dark matter particle (solid line) Shown separately arethe contributions from H0H0 rarr bb (dashed line) τ+τminus (dash-dotted line)and Zγ (dotted line) This is for the benchmark model I in Table 81Figure from Paper VII

experiment [301] In Fig 85 this is illustrated by showing the predicted fluxesfrom a ∆Ω = 10minus3sr region around the direction of the galactic center togetherwith existing observations in the same sky direction In this figure a standardNFW density profile as specified in Table 21 and with a normalization densityof 03 GeVcm

3at 85 kpc is the underlying assumption for the dark matter

halo for our Galaxy With the notation of Eq (215) this correspond toJ times ∆Ω sim 1 for ∆Ω = 10minus3 sr Processes such as adiabatic compressionwhich we discussed in Chapter 2 could very well enhance the dark matterdensity significantly near the galactic center Therefore the predicted fluxcompared to a pure NFW profile could very well be scaled up by a largelsquoboost factorrsquo The boost factors used for the shown signals are also displayedin Fig 85 Since the continuum part of the expected spectrum is withinthe energy range covered by EGRET satellite there is an upper limit on

Table 81 IDM benchmark models (In units of GeV)

Model mh mH0 mA0 mHplusmn micro2 λ2times1 GeVI 500 70 76 190 120 01II 500 50 585 170 120 01III 200 70 80 120 125 01IV 120 70 80 120 95 01

EGRETDW=210-3

HESSDW=10-5

GLAST sensitivity

50 GeV boost ~104

70 GeV boost ~100

IDM NFW DW~10-3 ΣEΓ=7lo

2 γΦγ

minus2sminus

log(Eγ [GeV])

-10-1 0 1 2 3 4

Figure 85 Predicted gamma-ray spectra from the inert Higgs bench-mark models I and II as seen by GLAST (solid lines) The predictedgamma flux is from a ∆Ω = 10minus3 sr region around the direction of thegalactic center assuming an NFW halo profile (with boost factors as indi-cated in the figure) and convolved with a 7 Gaussian energy resolutionThe boxes show EGRET data (which set an upper limit for the contin-uum signal) and the thick line HESS data in the same sky directionThe GLAST sensitivity (dotted line) is here defined as 10 detected eventswithin an effective exposure of 1 m2yr within a relative energy range ofplusmn7 Figure from Paper VII

the allowed flux in the continuum part of our spectrum The EGRET dataare taken from [97] For example for benchmark model II we find that anoptimistic but not necessarily unrealistic [101] boost of 104 could be allowedIn that case there would be a spectacular γγ line signal waiting for GLASTHowever to enable detection boost factors of such magnitudes are not at allnecessary For H0 masses closer to the W threshold the γγ annihilation ratesbecome even higher and in addition Zγ production becomes important In

Table 82 IDM benchmark model results

Model vσvrarr0tot Branching ratios [] ΩCDMh

[cm3sminus1] γγ Zγ bb cc τ+τminus

I 16 times 10minus28 36 33 26 2 3 010II 82 times 10minus29 29 06 60 4 7 010III 87 times 10minus27 2 2 81 5 9 012IV 19 times 10minus26 004 01 85 5 10 011

fact these signals would potentially be visible even without any boost at all(especially if the background is low as might be the case if the EGRET signalis a galactic off-center source as indicated in [302]) Also shown in Fig 85 arethe data from the currently operating air Cerenkov telescope HESS [258]The HESS data are within a solid angle of only ∆Ω = 10minus5 sr but sincethe gamma-ray flux is dominated by a point source in the galactic center alarger solid angle would not affect the total flux much Future air Cerenkovtelescopes with lower energy thresholds and much larger effective area thanGLAST are planned and will once operating be able to cover the entireregion of interest for this dark matter candidate

To go beyond just a few example models we performed in Paper VII asystematic scan over the parameters in the IDM for a SM higgs mass mh =500 GeV and calculated the cross section into gamma lines The constraintsmentioned in Section 81 allowed us to scan the full parameter space for darkmatter masses below the W threshold of 80 GeV The dependence on mHplusmn

and λ2 is small and we chose to set these equal to mH0 + 120 GeV (tofulfill precision tests) and 01 respectively Importantly we note that theright relic density is obtained with a significant amount of early Universecoannihilations with the inert A0 particle The resulting annihilation ratesinto γγ and Zγ are shown in Fig 86 The lower and upper mH0 mass boundscome from the accelerator constraints and the effect on the relic density by theopening of theW+Wminus annihilation channel respectively For comparison thesame figure also shows the corresponding annihilation rates for the neutralino(χ) within MSSM The large lower-right region is the union of the range ofcross sections covered by the annihilation rates 2σvγγ and σvZγ as obtainedwith a large number of scans within generous MSSM parameter bounds withthe DarkSUSY package [298] The stronger line signal and smaller spreadin the predicted IDM flux are caused by the allowed unsuppressed couplingto W pairs that appear as virtual particles in contributing Feynman loopdiagrams In the MSSM on the other hand high γγ and Zγ rates are harderto achieve [301303ndash305] at least while still satisfying both relic density andLEP constraints for the masses of interest here

The IDMrsquos true strength lies in its simplicity and its interesting phe-nomenology The lightest new particle in the model typically gives a WIMPdark matter candidate once coannihilations are included and the model al-lows a SM Higgs mass of up to at least a few hundred GeV without con-tradicting LEP precision tests These are two typical features of the modelbut the IDM also shows the typical dark matter properties of having weakinteractions and electroweak masses The main reasons why this scalar darkmatter model gives such particularly strong gamma lines are that (1) Thedark matter mass is just below the kinematic threshold for W production inthe zero velocity limit (2) The dark matter candidate almost decouples fromfermions (ie couples only via SM Higgs exchange) while still having ordi-nary gauge couplings to the gauge bosons In fact these two properties by

WIMP Mass [GeV]

H0 H0 rarr Zγ

H0 H0 rarr γγ

rarr Zγ γγ

40 50 60 70 80

Figure 86 Annihilation strengths into gamma-ray lines 2vσγγ (upperband) and vσZγ (middle band) from the scan over the IDM parameterspace For comparison the lower-right region indicates the correspondingresults within the minimal supersymmetric standard model as obtainedwith the DarkSUSY package [298] This lower region is the union of Nγvσfrom χχ rarr γγ and χχ rarr Zγ Figure from Paper VII

themselves could define a more general class of models for which the IDM isan attractive archetype because of its simplicity with only six free parameters(including the SM Higgs mass)

C h a p t e r

Have Dark MatterAnnihilations Been

Observed

Over the past several years observed anomalies in the spectra from cosmicphotons and anti-particles have been suggested to originate from dark matterannihilations One strongly promoted claim of a dark matter annihilationsignal is based on the anomaly that the EGRET experiment found in the dif-fuse galactic gamma-ray emission For gamma-ray energies above roughly 1GeV the data seems to show in all sky directions an excess of flux comparedto what is conventionally expected It has been realized that this excessesin the spectrum could be due to dark matter annihilations De Boer andcollaborators [1 306ndash308] have therefore proposed a dark matter distributionin our Galaxy to explain this observed gamma-ray anomaly Internal con-sistency of such a dark matter explanation must however be investigatedGenerically the same physical process producing the diffuse gamma rays alsoproduces antiprotons In Paper V we therefore studied this proposed darkmatter model to see if it is compatible with measured antiproton fluxes Usingcurrent and generally employed propagation models for the antiprotons weshowed that this dark matter explanation is excluded by a wide margin whenchecked against measured antiproton fluxes

91 Dark Matter Signals

Observations that have been proposed to be the product of dark matter an-nihilations include the cosmic positron spectrum measured by HEAT the511 keV emission from the galactic Bulge measured by INTEGRAL the mi-crowave excess from the galactic Center observed by WMAP and the diffusegalactic and extragalactic gamma-ray spectra measured by EGRET All ofthese potential dark matter signals are still very speculative For a recentreview and references see eg [309] There has also been a claim of a direct

126 Have Dark Matter Annihilations Been Observed Chapter 9

detection signal of dark matter by the DAMA collaboration [227 228] butthis result is very controversial as other similar experiments have not beenable to reproduce their result [206207225226229230]

It is beyond the scope of this thesis to go through all these potential hintsof a dark matter signal in detail We will only focus on scrutinizing (as inPaper V) the perhaps most strongly promoted claim in the last few years ndashthat the GeV anomaly in the diffuse galactic gamma-ray spectrum measuredby the EGRET satellite could be well explained by a signal from WIMP darkmatter annihilations

92 The Data

Between the years 1991 and 2000 the Energetic Gamma Ray Emission Tele-scope EGRET [310] onboard the Compton gamma ray observatory tookdata During this period it made an all-sky survey of the gamma-ray fluxdistribution for energies mainly between 003 and 10 GeV

Diffuse emission from the Milky Way completely dominates the gamma-ray sky The main part of the emission originates from interactions of cosmicrays (mostly protons and electrons) with the gas and radiation fields in theinterstellar medium Any calculation of the galactic diffuse emission is there-fore primarily dependent on the understanding of the cosmic-ray spectra andinterstellar gas and radiation fields throughout our Galaxy Cosmic rays arebelieved to originate mainly from acceleration processes in supernovae andpropagate through large parts of the Galaxy whereas the radiation fieldsmainly come from the CMB and photons from stars inside the Galaxy Thephysical processes involved in the cosmic-ray interactions which produce thegamma rays are mainly the production and subsequent decay of π0 inverseCompton scattering and bremsstrahlung

The first detailed analysis of the diffuse gamma rays was done by Hunteret al [311] (using EGRET data in the galactic plane latitudes |b| le 10 ingalactic coordinates) The main assumptions in their analysis were that thecosmic rays are of galactic origin that there exists a correlation between theinterstellar matter density and the cosmic-ray density and that the cosmic-ray spectra throughout our Galaxy are the same as measured in the solarvicinity Their result confirmed that the agreement between the EGRETobserved diffuse gamma rays and the expectations are overall good Howeverat energies above 1 GeV the measured emission showed an excess over theexpected spectrum This excess is known as the EGRET lsquoGeV anomalyrsquo

Later Strong Moskalenko and Reimer [312ndash315] developed a numericalcode GALPROP [316] for calculating the cosmic-ray propagation and diffusegamma-ray emission in our Galaxy Their code includes observational data onthe interstellar matter and a physical model for cosmic-ray propagation Themodel parameters are constrained by the different existing observations suchas cosmic-ray data on BC (ie the Boron to Carbon ratio which relates

Section 92 The Data 127

bremss πο

Energy [GeV]

E2times

minus2sminus

minus1]

10minus7

10minus6

10minus5

10minus4

10minus3 10minus2 10minus1 1 10 102 103

bremssπο

Energy [GeV]

10minus7

10minus6

10minus5

10minus4

Figure 91 Gamma-ray spectrum models compared to data Left

panel The conventional model for the gamma-ray spectrum for the in-ner galactic disk The model components are π0-decay (dots red) inversecompton (dashes green) bremsstrahlung (dash-dot cyan) extragalacticbackground (thin solid black) total (thick solid blue) EGRET datared vertical bars COMPTEL data green vertical bars Right panelThe same but for the optimized model fitting the observed gamma-rayspectrum Figures adopted from [315]

secondary to primary cosmic rays) This makes it possible to derive a dif-fuse gamma-ray spectrum in all sky directions In the lsquoconventional scenariorsquoin [315] the existence of the EGRET GeV anomaly was confirmed Howeverby allowing for a spatial variation of the electron and proton injection spectrait was pointed out that an lsquooptimized scenariorsquo gives a good description of thediffuse gamma-ray sky [315] To explain the GeV anomaly this optimizedmodel allows for a deviation of the cosmic-ray spectrum (within observationaluncertainties) from what is measured in the solar vicinity The electron in-jection spectrum is made slightly harder with a drastic drop at 30 GeV andat the same time normalized upward with a factor of about 5 compared tothe measured spectrum in the solar vicinity The proton injection spectrumis also made harder and the normalization is increased by a factor 18 at100 GeV The derived spectra in the conventional and the optimized modelcompared to observational data are shown in Fig 91

The origin of the potential GeV anomaly is still a matter of debate Thereare mainly three proposed explanations of its origin (i) it is of conventionalastrophysical origin like in the mentioned optimized cosmic-ray model or dueto unresolved conventional sources (ii) it is an instrumental artefact due touncertainties in the instrument calibration or (iii) it is caused by dark matter

annihilationsThat the GeV anomaly could be due to a systematic instrumental artefact

has recently been discussed by Stecker et al [317] They argue that the lackof spatial structure in the excess related to the galactic plane galactic centeranti-center or halo indicates that the GeV anomaly above sim1 GeV is morelikely due to a systematic error in the EGRET calibration Although not atall in contradiction with a calibration problem it is notably that in a recentreanalysis [318] of the EGRET instrument response the GeV anomaly wasfound to be even larger This reanalysis was done by modifying the GLASTsimulation software to model the EGRET instrument and indicated thatpreviously unaccounted instrumental effects mistakenly lead to the rejectionof some gamma-ray events

The alternative explanation that the GeV anomaly in all sky directions isa result of dark matter annihilations has been promoted in a series of papersby de Boer et al eg [1306ndash308] The idea to use the gamma-ray excess as adark matter annihilations signal has a long history (at least [97305319320])but de Boer et al have extended this idea to claim that all the diffuse galacticgamma rays detected above 1 GeV by EGRET irrespective of the directionhas a sizeable dark matter contribution

93 The Claim

Specific supersymmetric models have been proposed as examples of viablecandidates that can explain the EGRET GeV anomaly [307] The precisechoice of dark matter candidate is in itself not crucial as long as its darkmatter particles are non-relativistic have a mass between 50 and 100 GeV andannihilate primarily into quarks that then produce photons in their processof hadronization In these cases the predicted gamma-ray spectrum has theright shape to be added to the lsquoconventionalrsquo cosmic-ray model in [315] inorder to match the the GeV anomaly see Fig 92

The price to pay is however a rather peculiar dark matter halo of theMilky Way containing massive disk concentrated rings of dark matter besidesthe customary smooth halo The dark matter distribution de Boer et alpropose is a profile with 18 free parameters With the given proposal abest fit to the EGRET data is performed This is possible to do becausegamma rays have the advantage of pointing back directly to their sourcesin the Galaxy and since the gamma-ray spectral shape from dark matterannihilations is presumed to be known (and distinct from the conventionalbackground) The sky-projected dark matter distribution can therefore beextracted from the EGRET observations The deduced dark matter profilein [1] has the following main ingredients

bull a triaxial smooth halo in the form of a modified isothermal sphere butsomewhat flattened in the direction of the Earth and in the z-direction(ie the height above the galactic plane)

Section 93 The Claim 129

EGRETbackgroundsignal

Dark MatterPion decayInverse ComptonBremsstrahlung

χ2 356χ2 (bg only) 17887

Energy [GeV]

E2times

minus2sminus

minus1]

10minus5

10minus4

10minus1 1 10 102

Figure 92 Fit of the shapes of background and dark matter annihilationsignal to the EGRET data in the inner part of the galactic disk Thelight shaded (yellow) areas indicate the background using the shape ofthe conventional GALPROP model [315] while the dark shaded (red)areas are the signal contribution from dark matter annihilation for a 60GeV WIMP mass The reduced χ2 [75] for the background only and thecorresponding fit including dark matter is indicated in the figure Notethe smaller error bars in this figure compared to in Fig 91 ndash this is dueto the disagreement in how to take into account systematic and correlatederrors (see Paper V for details) Figure adopted from [1]

bull an inner ring at about 415 kpc with a density falling off as ρ sim eminus|z|σz1 where σz1 = 017 kpc and

bull an outer ring at about 129 kpc with a density falling off as ρ sim eminus|z|σz2 where σz2 = 17 kpc

Fig 93 shows this dark matter profile The strong concentration of darkmatter to the disk (upper panel) as well as the ring structure of the model(lower panel) is clearly seen

A 50-100 GeV dark matter candidate with a distribution as describedconstitutes the claimed explanation of the EGRET GeV anomaly In addi-tion which will become important later this model also has to boost thepredicted gamma-ray flux in all sky directions by a considerable lsquoboost fac-torrsquo of around 60 With these ingredients a good all sky fit to the gamma-rayspectra as in Fig 92 is obtained

minus20 minus10 0 10 20minus20

minus10

Dark matter density ρ [MSun

x [kpc]

minus5 0 50

z [kpc]

OuterSunInnerCenter

minus10

Surface density Σ08 kpc

x [kpc]

minus20 minus10 0 10 200

x [kpc]

kpc [M

Figure 93 The dark matter distribution in the halo model of de Boer et

al [1] Upper panel The concentration of dark matter along the galacticdisk The right figure displays the density dependence as a function of thevertical distance from the galactic plane ndash at the position of the outerring (dottedgreen) solar system (solidblack) inner ring (dashedred)and galactic center (dash-dottedblue) Lower panel The dark mattersurface mass density within 08 kpc from the galactic disk The Earthrsquoslocation is marked with a times-sign Figure from Paper V

94 The Inconsistency

Even though the dark matter halo profile by de Boer et al explains theEGRET data very well it is of great importance to check its validity withother observational data

Disc Surface Mass Density

Note that the distribution of the dark matter in this model seems very closelycorrelated to the observed baryon distribution in the Milky Way ndash containinga thin and a thick disk and a central bulge (see eg [321]) Since the darkhalo is much more massive than the baryonic one one of the first things weshould investigate is whether there is room to place as much unseen matter

Section 94 The Inconsistency 131

Table 91 Derived local surface densities Σ|z| within heights |z| com-pared to the amount of dark matter in the model of de Boer et al [1]The amount of dark matter exceeds the allowed span for unidentifiedgravitational matter in the inner part of the galactic disk (ie aroundz = 0) [323324]

surface density dynamical identified unidentified DM in [1]M⊙pc2 M⊙pc2 M⊙pc2 M⊙pc2

Σ50 pc 9 ndash 11 sim 9 0 ndash 2 45Σ350 pc 36 ndash 48 sim 34 2 ndash 14 19Σ800 pc 59 ndash 71 sim 46 13 ndash 25 29Σ1100 pc 58 ndash 80 sim 49 9 ndash 32 35

in the vicinity of the disk as in the model by de Boer et al

Observations of the dynamics and density distribution of stars in the diskgive a measure of the gravitational pull perpendicular to the galactic planeThis can be translated into an allowed disk surface mass density (a methodpioneered in [322]) Observational data from the local surroundings in thegalactic disk sets fairly good limits on the disk surface mass density at thesolar system location [323] Observations are well described by a smooth darkmatter halo and a disk of identified matter (mainly containing stars whiteand brown dwarfs and interstellar matter in form of cold and hot gases)Therefore there is little room for a concentration of dark matter in the disk

Table 91 shows the observed local surface mass density in both identifiedcomponents and the total dynamical mass within several heights Their dif-ferences give an estimate of the allowed amount of dark matter in the localdisk ndash the result is an exclusion of such strong concentrations of unidenti-fieddark matter as used in the model of [1] to explain the EGRET data Forexample these observations give room for only about 001M⊙pc3 in uniden-tified matter which should be compared to the dark matter density of 005M⊙pc3 in the model of de Boer et al [1] We should keep in mind that theestimates of the possible amount of dark matter are somewhat uncertain andthat the disk models also have uncertainties of the order of 10 in their starplus dwarf components and uncertainties as large as about 30 in their gascomponents Also the de Boer et al halo model could easily be modified togive a lower disk surface mass density at the solar vicinity However such amodification just to circumvent this problem seems fine-tuned The model asit now stands already has made fine-tuning modifications ie the rings areconstructed so that they can be very massive while keeping the local densitylow Figure 93 clearly shows that our Sun is already located in a region withrelatively low disk mass surface density

The dark matter distribution does not at all resemble what would be

expected from dissipationless cold dark matter The distribution should bemuch more isotropic than that of the baryonic disk material which supposedlyforms dissipatively with energy loss but very little angular momentum loss[325] (see also Chapter 2)

Comparison with Antiproton Data

Any model based on dark matter annihilations into quark-antiquark jets in-evitably also predicts a primary flux of antiprotons (and an equal amount ofprotonslowast) from the same jets As discussed in Section 92 the propagationmodels of the antiprotons (ie cosmic rays) are observationally constrainedto enable reasonably reliably predict antiproton fluxes at Earth To find outwhat the antiproton flux would be in the model proposed by de Boer et alwe calculated the expected antiproton fluxes in detail in paper Paper V

Calculating the Antiproton Flux

There is no need to repeat here the details of the procedure to calculate theantiproton flux which can be found in Paper V The main point is thatwe followed as closely as possible how de Boer et al found the necessaryannihilation rates to explain the EGRET data on gamma rays and thencalculated the antiproton flux based on these same annihilation rates

For the background gamma flux we used as de Boer et al both theconventional diffuse gamma-ray background and the optimized backgroundshown in Fig 91

For predicting the signals we used DarkSUSY [281] to calculate the darkmatter annihilation cross sections gamma-ray and antiproton yields Wenormalized our boost factors to fit the diffuse gamma-ray data from the innerregion of our Galaxy from where most observational data exist The darkmatter profile was fixed and defined by the 18 parameters found in the deBoer et al paper [1] A least χ2-fit [75] was made to the EGRET data in 8energy bins in the energy range 007 to 10 GeV

By this procedure we were able to reproduce the result in [1] That iswe find a good fit to the EGRET data for WIMP masses between roughly 50and 100 GeV and that the required boost factors can be less than the orderof 100 Our best fit χ2-values for different dark matter masses are shownin Fig 94 for both the conventional (triangles) and the optimized (circles)diffuse gamma-ray background

Note that the fits with the optimized background never get very bad forhigher masses as there is no real need for a signal with this model Oneshould also note that we used relative errors of only 7 for the gamma fluxesalthough the overall uncertainty is often quoted to be 10-15 [315326] The

lowast The protons produced in dark matter annihilations would be totally swamped by themuch larger proton flux from conventional sources

Fit with optimized background

Fit with standard background

Neutralino mass (GeV)

Figure 94 The best χ2 for a fit of background and a dark mattersignal to the EGRET data for different dark matter masses The fit isto eight energy bins where the two free parameters are the normalizationof the dark matter and the background spectrum (ie 8-2=6 degrees offreedom) Figure from Paper V

true errors are still under debate and we chose to followed de Boer et alusing their estimate of 7 for the relative errors in our χ2-fits Our resultsare however not sensitive to this choice (apart from the actual χ2 valuesof course) The optimized background model produce in this case a reducedχ2 sim 226 which corresponds to a probability of P sim 01 (P-value) thatthe data would give this or a worse (ie greater) χ2 value if the hypothesiswere correct [75] If instead conventionallarger uncertainties of sim 15 forEGRETrsquos observed gamma fluxes were adopted the reduced χ2 was decreasedto sim 56 and a P-value of P = 56

Boost factors were determined model-by-model This means for each su-persymmetric model we demanded it to give an optimized fit to the gamma-ray spectrum which thus gave us an optimized boost factor for each modelBased on the determined boost factors the antiproton fluxes could be directlycalculated for each model Sine the boost factor is assumed to be indepen-dent of location in the Galaxy the same boost factor could be used for theantiproton flux as that found for the gamma rays To be concrete the analysisin Paper V was done within the MSSM but as mentioned the correlationbetween gamma rays and antiprotons is a generic feature and the results aremore general We used DarkSUSY [281] to calculate the antiproton fluxes fora generous set of supersymmetry models with the halo profile of de Boer et

BESS 98 1σ

BESS 98 2σ

T = 040-056 GeV

lower boundary incl uncert

χ2 lt 10

10 lt χ2 lt 25

25 lt χ2 lt 60

60 lt χ2

Boost factors lt 100

Figure 95 The antiproton fluxes boosted with the same boost factoras found for the gamma rays compared to the measured BESS data Thesolid line indicate how far down we could shift the models by choosingan extreme minimal propagation model (see Section 94) Figure fromPaper V

al Once we had the calculated antiproton fluxes at hand we could compareit with antiproton measurements

We chose to primarily compare antiproton data in the energy bin at 040ndash056 GeV and using BESS (Balloon-borne Experiment with a SuperconductingSpectrometer) data from 1998 [327] The reason for using the BESS 98 datais because the solar modulation parameter is estimated to be relatively low(φF = 610 MV) at this time and the low-energy bin correspond to an energyrange where the signal is expected to be relatively high compared to thebackground

Figure 95 shows (using the conventional background) the antiproton fluxwhen enlarged with the boost factor found from the fit to the EGRET data

Models with the correct mass ie low χ2 clearly overproduce antiprotonsIn the figure we have imposed a cut on the boost factor to only allow modelswith reasonably low boost factors To be conservative we have allowed theboost factor to be as high as 100 which is higher than expected from recentanalyses (see eg [116 328]) It is fairly evident that all the models withgood fits to the EGRET data give far too high antiproton fluxes We findthat low-mass models (masses less than 100 GeV) overproduce antiprotons by

a factor of around ten Higher-mass models (above a few hundred GeV) havea lower antiproton rate so the overproduction is slightly less However theyhardly give any improvements to the fits to the gamma-ray spectrum

Other dark matter candidates like KK dark matter would also give asimilar behavior since the gamma rays and antiprotons are so correlatedHowever for eg KK dark matter in the UED model one would not improvethe fits to EGRET data as only heavier models are favored by the relic densityconstraint For the IDM the boost factor would have to be very large in allsky directions at least as long as its gamma-ray continuum part is stronglysuppressed by having only heavy Higgs coupling to quarks In fact sinceantiprotons and gamma rays are so strongly correlated in general our resultsshould be valid for any typical WIMP

Antiproton Propagation Uncertainties

We could be worried about the well-known fact that the antiproton flux fromdark matter annihilations is usually beset with large uncertainties relatingto unknown diffusion parameters combined with uncertainties in the halodistribution In [329] it is pointed out that the estimated flux may vary byalmost a factor of 10 up or down for models that predict the correct cosmic-ray features However the results of such a large uncertainty are only validfor a relatively smooth halo profile where much of the annihilation occursaway from the disk and propagation properties are less constrained

The main reason for the large uncertainties found in [329] is a degeneracy(for the secondary signal) between the height of the diffusion box and thediffusion parameter If we increase the height of the diffusion box we wouldget a larger secondary signal because cosmic rays can propagate longer inthe diffusion box before escaping This can be counterbalanced by increasingthe diffusion coefficient to make the cosmic rays diffuse away faster fromthe galactic disk Hence for the secondary signal which originates in thegalactic disk we can get acceptable fits by changing these parameters Forthe dark matter in a smooth halo the effect of these changes is differentIf we increases the height of the diffusion box we also increase the volumein which annihilations occur and the total flux increases more than can becounterbalanced by an increase in the diffusion coefficient This is howevernot true to the same extent in the de Boer et al profile where most of thedark matter is concentrated to the disk

To investigate this effect on the antiproton flux from variations in thepropagation models we recalculated the expected antiproton fluxes with thepropagation code in [329] For illustration let us look at a supersymmetricconfiguration for which the agreement with the EGRET data is good ndash a re-duced χ2 of roughly 36 a neutralino mass of 501 GeV and a derived boostfactor of 69 By varying the propagation parameters to be as extreme as al-lowed from other cosmic-ray data (details on what this correspond to can be

Figure 96 A supersymmetric model that provides a good fit to theEGRET data has been selected and its antiproton yield has been care-fully derived It is featured by the red solid line in the case of the mediancosmic-ray configuration Predictions spread over the yellow band as thecosmic-ray propagation parameters are varied from the minimal to maxi-mal configurations (see Table 2 in Paper V) The long-dashed black curveis calculated with DarkSUSY for a standard set of propagation parame-ters [330] The narrow green band stands for the conventional secondarycomponent As is evident from this figure the antiproton fluxes for thisexample model clearly overshoots the data Figure from Paper V

found in Paper V) we get the range of allowed predicted antiproton fluxesIn Fig 96 the yellow band delimits the whole range of extreme propagationmodel configurations This gives an indication on how well the flux of neu-tralino induced antiprotons can be derived in the case of the de Boer et aldark matter distribution The red solid curve is a median cosmic-ray propaga-tion model configuration which could be compared to the longndashdashed blackcurve computed with the DarkSUSY package dagger The conventional secondary

Section 95 The Status to Date 137

background producing antiprotons is indicated as the narrow green band asit was derived in [331] from the observed BC ratio For maximal cosmic-rayconfiguration the dark matter induced antiproton flux is observed to increaseby a factor of 25 and for the minimal configuration a decrease of a factorof 26 The total uncertainty in the expected dark matter induced antiprotonflux corresponds therefore to an overall factor of only sim 65 to be comparedto a factor of sim 50 in the case of an NFW dark matter halo

Even after these propagation uncertainties are included the yellow un-certainty band is at least an order of magnitude above the secondary greencomponent This was for one example model but this argument can be mademore general In Fig 95 a solid line represents how far down we would shiftthe antiproton fluxes by going to the extreme minimal propagation modelsAs can be seen the antiprotons are still overproduced by a factor of 2 to 10for the models with good fits to EGRET data It is therefore difficult to seehow this dark matter interpretation of the EGRET data could be compatiblewith the antiproton measurements

Our conclusion is therefore that the proposal of de Boer et al [1] is notviable at least not without further fine-tuning of the model or by significantchanges in generally employed propagation models

95 The Status to Date

Currently the uncertainties regarding the data of the EGRET GeV anomalycan still be debated In any case the dark matter model proposed by de Boeret al to explain the potential GeV anomaly has a problem of severe overpro-duction of antiprotons With the usual cosmic propagation models there doesnot seem to be a way out of this problem However attempts with anisotropicdiffusion models have been proposed by de Boer et al as a way to circumventthese antiproton constraints These models feature anisotropic galactic windswhich transport charged particles to outer space and places our solar systemin an underdense region with overdense clouds and magnetic walls causingslow diffusion In such a scenario it is claimed that the antiproton flux dueto dark matter annihilations could very well be decreased by an order of mag-nitude [308 332] (see also [333 334]) This seems somewhat contrived at themoment especially since the dark matter distribution itself is not standardThere are of course also other ways to tune the model to reduce the antipro-ton flux For example if we take away the inner ring the antiproton fluxesgoes down a factor of 20 This indicates that fine-tuning the distribution evenmore by having the high dark matter density as far away as possible fromour solar system could potentially reduce the antiproton significantly with-out reducing the gamma-ray flux Remember though that the disk density(although notably still quite uncertain) has already been tuned to have a dip

dagger Because diffusive reacceleration is not included in DarkSUSY the flux falls more steeplyclose to the neutralino mass than in the median model of [329]

in order not to be in large conflict with stellar motion measurements in thesolar neighborhood All such attempts to avoid the antiproton contradictionshould be judged against optimized cosmic-ray propagation models that alsohave been shown to enable an explanation of the GeV anomaly without theneed of dark matter The actual density profiles of the rings with exponentialfall-off away from the disk is also not what is expect from WIMP dark matteralthough mergers of dwarf galaxies could leave some traces of minor ring-likedark matter structures in the galactic plane [308]Dagger In fact the de Boer etal model seems most likely to be a fit of the baryonic matter distribution ofour Galaxy and not the dark matter density This is not at all to say thatthere cannot be any hidden dark matter annihilation signal in the gamma-raysky instead we have shown that standard propagation models and antiprotonmeasurements constrain the possibilities for an all-sky WIMP dark mattersignal in gamma rays

What can be said for certain is that to date there is no dark matterannihilation signal established beyond reasonable doubt

Many open questions especially regarding the EGRET GeV anomaly willpresumably be resolved once the GLAST satellite has data of the gamma-raysky For example we will then know more about the actual disk concentrationof the gamma-ray distribution if the GeV anomaly persists and how thespectrum is continued up to higher energies Before GLAST the PAMELAsatellite [241] will collect data on eg antiprotons and positrons which couldeven further enhance the understanding of the cosmic-ray sky and potentialdark matter signals in it

Dagger It has been put forward that rotation curve [1] and gas flaring [335] data support theexistence of a very massive dark matter ring at a galactocentric distance of about 10-20kpc However this is controversial and is eg not supported by the derived rotationcurve in [336] and the recent analysis in [337]

C h a p t e r

Summary and Outlook

The need to explain a wide range of cosmological and astrophysical phenomenahas compelled physicists to introduce the concept of dark matter Once darkmatter together with dark energy (in the simplest form of a cosmologicalconstant) is adopted conventional laws of physics give a remarkably gooddescription of a plethora of otherwise unexplained cosmological observationsHowever what this dark matter is made of remains one of the greatest puzzlesin modern physics

Any experimental signals that could help to reveal the nature of dark mat-ter are therefore sought This could either be in the form of an observationaldiscovery of an unmistakable feature or the detection of several different kindsof signals that all can be explained by the same dark matter model

In this thesis I have presented the background materials and research re-sults regarding three different types of dark matter candidates These candi-dates all belong to the class of weakly interacting massive particles and arethe lightest Kaluza-Klein particle γ(1) the lightest neutralino within super-symmetry χ and the lightest inert Higgs H0 They could be characterized asdark matter archetypes for a spin 1 12 and 0 particle respectively

The first dark matter candidate that was studied originates from the fasci-nating possibility that our world possesses more dimensions than the observedfour spacetime dimensions After a discussion of multidimensional universes ndashwhere it was concluded that a stabilization mechanism for extra dimensions isessential ndash the particle content within the particular model of universal extradimensions (UED) was investigated The cosmologically most relevant aspectof this UED model is that it naturally gives rise to a dark matter candidateγ(1) (asymp B(1)) This candidate is a massive Kaluza-Klein state of an ordinaryphoton ndash ie a photon with momentum in the direction of an extra dimen-sion In this thesis prospects for indirect dark matter detection by gammarays from annihilating γ(1) particles were explored It was discovered that byinternal bremsstrahlung from charged final state fermions very high-energygamma rays collinear with the fermions are frequently produced An expectedsignature from γ(1) annihilations is therefore a gamma-ray energy spectrum

140 Summary and Outlook Chapter 10

that is very prominent at high energies and has a characteristic sharp cutoffat the energy equal to the dark matter particlersquos mass As a by-product it wasrealized that this is a quite generic feature for many dark matter candidatesThe amplitude of a potentially even more characteristic signal a monochro-matic gamma line was also calculated Although in principle a very strikingsignal its feebleness seems to indicate that a new generation of detectors areneeded for a possible detection

The second dark matter candidate studied was the neutralino appearingfrom a supersymmetric extension of the standard model Also here it wasfound that internal bremsstrahlung in connection to neutralino annihilationcan give characteristic signatures in the gamma-ray spectrum The reason forthese gamma-ray signals is somewhat different from that of the UED modelHigh-mass neutralinos annihilating into Wplusmn can give rise to a phenomenonsimilar to the infrared divergence in quantum electrodynamics that signifi-cantly enhances the cross section into low-energy Wplusmn bosons and high-energyphotons Another possible boosting effect is that the (helicity) suppression ofneutralinos annihilating into light fermions will no longer be present if a pho-ton accompanies the final state fermions For neutralino annihilations boththese effects result in the same type of characteristic signature a pronouncedhigh energy gamma-ray spectrum with a sharp cutoff at the energy equal tothe neutralino mass

The third candidate arises from considering a minimal extension of thestandard model by including an additional Higgs doublet With an unbro-ken Z2 symmetry motivated by experimental constraints on neutral flavorchanging currents an inert Higgs emerges that constitutes a good dark mat-ter candidate This dark matter candidate turns out to have the potentialto produce a lsquosmoking gunrsquo signal in the form of a strong monochromaticgamma-ray line in combination with a low gamma-ray continuum Such atremendous signal could be waiting just around the corner and be detectedby the GLAST satellite to be launched later this year

The anticipated absolute fluxes of gamma rays from dark matter anni-hilations and thus the detection prospects of predicted signals are still ac-companied by large uncertainties This is because of the large uncertaintyin the dark matter density distribution To learn more about the expecteddark matter density distribution we used numerical N -bodyhydrodynamicalsimulations The effect of baryons ie the ordinary matter particles consti-tuting the gas and stars in galaxies upon the dark matter was found to besignificant In the center of the galaxy the dark matter gets pinched and theoverall halo changes shape from somewhat prolate to more oblate The pinch-ing of dark matter in the center of galaxies could cause a boost of dark matterannihilations which might be needed for detection of annihilation signals

The claimed observation of a dark matter annihilation signal by de Boer etal was also scrutinized It was concluded that to date there is no convincingobservational evidence for dark matter particle annihilations

If dark matter consists of massive and (weakly) interacting particles thereis a chance that in the near future some or many of the existing and upcomingexperiments will start to reveal the nature of the dark matter The types ofsignal presented in this thesis have the potential to contribute significantly tothis process as they might be detected both with the GLAST satellite and withexisting and upcoming ground based air Cerenkov telescopes Other exper-iments ndash like PAMELA measuring anti-particle fluxes ndash will simultaneouslycontinue to search the sky for dark matter signals Once the Large HadronCollider at CERN is running if not earlier many theories beyond standardmodel physics will be experimentally scrutinized to see whether they give afair description of how nature behaves and whether they are able to explaindark matter

A p p e n d i x

Feynman RulesThe UED model

This Appendix collects a complete list of Feynman rules for all physical fieldsand their electroweak interactions in the five-dimensional UED model The ex-tra dimension is compactified on a S1Z2 orbifold (the fieldsrsquo orbifold bound-ary conditions are specified in Chapter 5) All interaction terms are located inthe bulk and are KK number conserving In principle radiative corrections atthe orbifold fixpoints could give rise to interactions that violate KK numberconservation [231] These types of interactions are loop-suppressed and it isself-consistent to assume they are small they will therefore not be consideredhere

A1 Field Content and Propagators

In the four-dimensional theory the mass eigenstates of vector fields are A(0) micro

Z(0)micro and W(0)microplusmn at the SM level whereas their KK level excitations are

B(n)micro A3 (n)micro and W(n)microplusmn (n ge 1) (for the heavy KK masses the lsquoWeinbergrsquo

angle is taken to be zero as it is essentially driven to zero) Depending onthe gauge there will also be an unphysical ghost field c associated to everyvector field From the Higgs sector there is one physical scalar h(0) present

at the SM-level and three physical scalars h(n) a(n)0 and a

(n)plusmn at each KK

level Depending on the gauge the Higgs sector also contains the unphysical

Goldstone bosons χ3 (0) and χplusmn (0) at SM-level and G(n)0 G

(n)plusmn and A

(n)5 at

each KK level These generate the longitudinal spin modes for the massive

vector fields Finally for every SM fermion ξ(0)s there are two towers of KK

fermions ξ(n)s and ξ

(n)d ndash except for the neutrinos which only appear as a

component in the SU(2) doublet both at zero and higher KK levels

Following the conventions in [3] the propagator for an internal particle is

144 Feynman Rules The UED model Appendix A

given by

for scalars i

q2 minusm2 + iǫ (A1)

for fermions i(6q +m)

for vectors minusi

q2 minusm2 + iǫ

ηmicroν minus qmicroqν

q2 minus ξm2(1 minus ξ)

where ǫ is a small positive auxiliary parameter which is allowed to tend to zeroafter potential integrations over q and ξ is a gauge parameter (ξ = 0 beingthe Landau gauge and ξ = 1 being the Feynman-rsquot Hooft gauge) The samepropagators apply also for the unphysical ghosts and Goldstone bosons butwith the masses replaced by

radicξmV (where mV is the mass of the associated

vector boson)In Chapter 5 mass eigenstates were expressed in the fields appearing di-

rectly in the Lagrangian see Eq (520) (521) (527) and (553) Sometimesthe inverse of these relationships are also convenient to have at hand

= swAM + cwZM (A4a)

BM = cwAM minus swZM (A4b)

Z(n)5 =

a(n)0 minus M (n)

G(n)0 (A5a)

χ3 (n) =M (n)

a(n)0 +

G(n)0 (A5b)

Wplusmn (n)5 =

a(n)plusmn minus M (n)

G(n)plusmn (A5c)

χplusmn (n) =M (n)

a(n)plusmn +

G(n)plusmn (A5d)

ψ(n)s = sinα(n)ξ

(n)d minus cosα(n)γ5ξ(n)s (A6a)

ψ(n)d = cosα(n)ξ

(n)d + sinα(n)γ5ξ(n)s (A6b)

A2 Vertex Rules

All vertex rules can be expressed in terms of five independent quantities egthe electron charge e (= minus|e|) the Weinberg angle θw the W gauge bosonmass mW the Higgs mass mh and the compactification size of the extra

Section A2 Vertex Rules 145

dimension R To shorten some of the vertex-rule expressions the followingshorthand notations will be frequently used

cw equiv cos(θw) (A7a)

sw equiv sin(θw) (A7b)

M (1) equiv 1R (A7c)

M(1)X equiv

M (1)2 +m2X (A7d)

together with the quantities

mZ = mWcw (A8a)

gY = ecw (A8b)

g = esw (A8c)

λ =g2m2

In addition all the fermion Yukawa couplings are free parameters ie thefermion masses mξ In the case of charged gauge boson interactions there arealso additional independent parameters in the Cabibbo-Kobayahi-Maskawa(CKM) Vij matrixlowast whose elements contain information on the strengths offlavor-changing interactions All momenta are ingoing in the vertex rules

All vertex rules in the UED model for the physical fields up to the firstKK level will now followdagger Additional vertex rules including unphysicalGoldstone and ghost fields are displayed if they were explicitly used in ournumerical calculation of the one-loop process B(1)B(1) rarr γγ Zγ which wasdone in the Feynman-rsquot Hooft gauge (ξ rarr 1) in Paper III In unitarity gauge(ξ rarr infin) all such unphysical fields disappear

Vector-Vector-Vector Vertices

These couplings between tree vector fields originating from the cubic termsin gauge fields that appear in the field strength part of the Lagrangian (59)The following four-dimensional Feynman vertex rules are derived

lowast In the SM Vij is a 3times3 complex unitary matrix Unitarity (9 conditions) and the factthat each quark field can absorb a relative phase (5 parameters reduction) leaves only2 times 32 minus 9 minus 5 = 4 free parameters in the CKM matrix

dagger Some of these vertex rules can also be found in [214 338 339] Note A few typos wereidentified in [214 338 339] [Personal communication with Graham Kribs and TorstenBringmann]

V micro2

q3V ν3

q1V ρ1 igV1V2V3

(q1 minus q2)ν middot ηρmicro

+(q2 minus q3)ρηmicroν + (q3 minus q1)

microηρν]

gA(0)W

(10)+ W

(10)minus

= minuse (A9)

gZ(0)W

(01)+ W

(01)minus

= minuscw g (A10)

(1)3 W

(01)+ W

(10)minus

= minusg (A11)

Vector-Vector-Scalar Vertices

Couplings between two gauge fields and one scalar field appear both in thefield strength part of the Lagrangian (59) and the kinetic part of the HiggsLagrangian (517) The following four-dimensional Feynman vertex rules arederived

V micro1

S igV1V2S middot ηmicroν

gZ(0)Z(0)h(0) =g

cwmZ (A12)

(0)+ W

(0)minus h(0) = gmW (A13)

gZ(0)W

(1)plusmn a

(1)∓

= plusmnig mZ

gZ(0)A3 (1)h(1) = gmZ (A15)

gZ(0)B(1)h(1) = minusgY mZ (A16)

(0)plusmn A3 (1)a

(1)∓

= ∓ig mW

(0)plusmn B(1)a

(1)∓

= ∓igY mW

(0)plusmn W

(1)∓ a

= plusmnig mW

(0)plusmn W

(1)∓ h(1) = gmW (A20)

gA3 (1)A3 (1)h(0) = gmW (A21)

gA3 (1)B(1)h(0) = minusgY mW (A22)

gB(1)B(1)h(0) = gs2wc2w

mW (A23)

(1)+ W

In unitarity gauge (ξ rarr infin) unphysical scalar fields are not present How-ever in a general gauge there are also additional vertices including unphysicalscalar fields I here choose to include only those vertex rules that were explic-itly used in Paper III to calculate the one-loop process B(1)B(1) rarr γγ in theFeynamn-rsquot Hooft gauge (ξ rarr 1)

gA(0)W

(0)plusmn χ∓ (0) = ∓iemW (A25)

gZ(0)W

(0)plusmn χ∓ (0) = plusmnigs2wmZ (A26)

gA(0)W

(1)plusmn G

(1)∓

= ∓ieM (1)W (A27)

(0)plusmn B(1)G

(1)∓

= ∓igY mW

(1)plusmn B(1)χ∓ (0) = ∓igY mW (A29)

Vector-Scalar-Scalar Vertices

Couplings between one gauge field and two scalar fields appear both in thefield strength part of the Lagrangian (59) and the kinetic part of the HiggsLagrangian (517) The following four-dimensional Feynman vertex rules arederived

V micro

igV S1S2 middot (q1 minus q2)micro

gA(0)a

(1)+ a

(1)minus

= e (A30)

gZ(0)a

(1)+ a

(1)minus

2(gcw minus gY sw)

M (1)2

2 + gcwmW

2 (A31)

gZ(0)h(1)a

= minusi g

(0)plusmn a∓(1)a

= ∓g2

M (1)2

M(1)W M

1 minus 2m2

(0)plusmn a∓(1)h(1) = i

gA3 (1)a

(1)0 h(0) = i

gA3 (1)G

(1)0 h(0) = i

gB(1)a

(1)0 h(0) = minusi gY

(1)plusmn a

(1)∓ h(0) = i

In unitarity gauge (ξ rarr infin) unphysical scalar fields are not present How-ever in a general gauge there are also additional vertices including unphysicalscalar fields I here choose to include only those vertex rules that were explic-itly used in Paper III to calculate the one-loop process B(1)B(1) rarr γγ in theFeynamn-rsquot tHooft gauge (ξ rarr 1)

gA(0)χ+ (0)χminus (0) = e (A39)

gZ(0)χ+ (0)χminus (0) = gcw2

minus gY

gZ(0)χ3 (0)h(0) = ig

2cw(A41)

(0)plusmn χ∓ (0)h(0) = i

2(A42)

(0)plusmn χ∓ (0)χ3 (0) = ∓g

2(A43)

gA(0)G

(1)+ G

(1)minus

= e (A44)

gB(1)χplusmn(0)a

(1)∓

= plusmngY

gB(1)χplusmn(0)G

(1)∓

= plusmngY

gB(1)χ3 (0)h(1) = minusi gY

2(A47)

gB(1)G

(1)plusmn G

(1)∓ h(0) = i

Scalar-Scalar-Scalar Vertices

Couplings between tree scalar fields appear in the kinetic and potential inthe Higgs Lagrangian (517) The following four-dimensional Feynman vertexrules are derived

S3 igS1S2S3

gh(0)h(01)h(01) = minus3

gh(0)a

(1)+ a

(1)minus

= minusgmW

M (1)2

gh(0)a

(1)0 a

= minus g

M (1)2

In unitarity gauge (ξ rarr infin) unphysical scalar fields are not present How-ever in a general gauge there are also additional vertices including unphysicalscalar fields I here choose to include only those vertex rules that were ex-plicitly used in Paper III to calculate the one-loop B(1)B(1) rarr γγ process(which was done numerically in the Feynamn-rsquot Hooft gauge (ξ rarr 1)

gh(0)χ+ (0)χminus (0) = minusg2

gh(0)χ3 (0)χ3 (0) = minusg2

gh(0)a

(1)plusmn G

(1)∓

2M (1)

1 minus m2h

gh(0)G

(1)plusmn G

(1)∓

= minusg2mW

2 (A56)

Fermion-Fermion-Vector Vertices

Couplings between two fermions and the gauge fields appear in the covariantderivative of the fermion fields see the Lagrangian in (544) The notationbelow is that indices i and j indicate which SM generation a fermion belongsto and Vij is the Cabibbo-Kobayahi-Maskawa matrix In the case of leptonsVij simply reduces to δij (and remember that there are no singlet neutrinosQU = YsU = 0 for neutrinos) Furthermore ξ = UD denotes the mass eigen-states for up (T3 = +12) and down (T3 = minus12) type quarks respectivelyElectric charge is defined as usual as Q equiv T3+Yd = Ys With these additionalnotations the following four-dimensional Feynman vertex rules are derived

V micro iγmicrogV ξ1ξ2

gA(0)ξ(0)ξ(0) = Qe (A57)

gA(0)ξ

(1)sdξ(1)sd

= Qe (A58)

gZ(0)ξ(0)ξ(0) = (T3gcw minus YdgY sw)PL minus YsgY swPR (A59)

gZ(0)ξ

(1)dξ(1)d

T3 cos2 α(1) minus Yss2w

gZ(0)ξ

(1)s ξ

T3 sin2 α(1) minus Yss2w

gZ(0)ξ

(1)s ξ

cwT3 sinα(1) cosα(1)γ5 (A62)

(1)3 ξ(0)ξ

= T3g cosα(1)PL (A63)

(1)3 ξ(0)ξ

= minusT3g sinα(1)PL (A64)

gB(1)ξ(0)ξ

= YsgY sinα(1)PR + YdgY cosα(1)PL (A65)

gB(1)ξ(0)ξ

= minusYsgY cosα(1)PR minus YdgY sinα(1)PL (A66)

(0)+ U

(0)i D

=gradic2VijPL (A67)

(0)+ U

(1)diD

=gradic2Vij cosα

cosα(1)Dj

(0)+ U

(1)siD

=gradic2Vij sinα

sinα(1)Dj

(0)+ U

(1)diD

=gradic2Vij cosα

sinα(1)Djγ5 (A70)

(0)+ U

(1)siD

=gradic2Vij sinα

cosα(1)Djγ5 (A71)

(1)+ U

(0)i D

=gradic2Vij cosα

(1)DjPL (A72)

(1)+ U

(0)i D

= minus gradic2Vij sinα

(1)DjPL (A73)

(1)+ U

(1)diD

=gradic2Vij cosα

(1)UiPL (A74)

(1)+ U

(1)siD

(1)UiPL (A75)

The remaining vertex rules are given by gV ξ1ξ2 = glowastV daggerξ2ξ1

Fermion-Fermion-Scalar Vertices

Couplings between two fermions and a scalar originate from the covariantderivative in Eq (544) and the Yukawa couplings (546) The same notationas for the lsquofermion-fermion-vectorrsquo couplings are used The following four-dimensional Feynman vertex rules are derived

S igSξ1ξ2

gh(0)ξ(0)ξ(0) = minusg mξ

gh(0)ξ

(1)dξ(1)d

= minusg mξ

sinα(1) cosα(1) (A77)

gh(0)ξ

(1)s ξ

= minusg mξ

gh(0)ξ

(1)dξ(1)s

= minusg mξ

1 minus 2 cos2 α(1))

γ5 (A79)

gh(1)ξ(0)ξ

= minusg mξ

sinα(1)PR + cosα(1)PL

gh(1)ξ(0)ξ

= gmξ

cosα(1)PR minus sinα(1)PL

ga(1)0 ξ(0)ξ

(T3gcw minus YdgY sw) cosα(1)PR + YdgY sw sinα(1)PL

minusigT3mξ

sinα(1)PR minus cosα(1)PL

ga(1)0 ξ(0)ξ

(T3gcw minus YdgY sw) sinα(1)PR + YdgY sw cosα(1)PL

+igT3mξ

ga(1)+ U

(0)i D

= minusi gradic2

sinα(1)DjPR minus mUi

cosα(1)DjPL

+igradic2

Vij cosα(1)DjPR (A84)

ga(1)+ U

(0)i D

= igradic2

cosα(1)DjPR minus mUi

sinα(1)DjPL

+igradic2

Vij sinα(1)DjPR (A85)

ga(1)+ U

(1)diD

= minusi gradic2

cosα(1)UiPR minus mUi

sinα(1)UiPL

minusi gradic2

Vij cosα(1)UiPL (A86)

ga(1)+ U

(1)siD

= igradic2

sinα(1)UiPR minus mUi

cosα(1)UiPL

minusi gradic2

Vij sinα(1)UiPL (A87)

The remaining symmetry related fermion-fermion-scalar vertex rules arefound using gSξ1ξ2 = glowast

Sdaggerξ2ξ1for scalar couplings whereas for all pseudo-scalar

coupling parts (ie the part of gSξ1ξ2 that include a γ5) pick up an additional

minus sign gSξ1ξ2 = minusglowastSdaggerξ2ξ1

(this follows from the relation (ψ1γ5ψ2)

dagger =

minusψ2γ5ψ1 for these interaction terms in the Lagrangian)

In unitarity gauge (ξ rarr infin) unphysical scalar fields are not present How-ever in a general gauge there are also additional vertices including unphysicalscalar fields Although we in Paper III calculated the one-loopB(1)B(1) rarr γγprocess in the Feynamn-rsquot Hooft gauge (ξ rarr 1) no fermion-fermion-scalar ver-texes come in at loop order for this process

Ghost-Ghost-Vector Vertices

In unitarity gauge (ξ rarr infin) all ghosts disappear from the theory For othergauges one can derive the vertex rules from the ghost Lagrangian in Eq (534)I here choose to include only the (one) vertex rules that was explicitly used inPaper III to calculate the one-loop process B(1)B(1) rarr γγ in the Feynamn-rsquot Hooft gauge (ξ rarr 1)

A(1) micro

c∓(1)

c∓(1) q

= plusmnie qmicro

Ghost-Ghost-Scalar Vertices

As for the above ghost-ghost-vector couplings only the (one) ghost-ghost-scalar Feynman rule explicitly needed in our calculation in Paper III of theprocess B(1)B(1) rarr γγ is listed

c∓(1)

= minusi g2mWξ

Vector-Vector-Vector-Vector Vertices

Couplings between four vector fields originate from the gauge field strengthpart in the Lagrangian (59) The following four-dimensional Feynman vertexrules are derived

V micro1

igV1V2V3V4 middot(

2ηmicroνηρσ minus ηmicroρηνσ minus ηmicroσηνρ)

(0)minus W

(0)+ W

= g2 (A88)

(1)minus W

(1)+ W

2g2 (A89)

(10)minus W

(01)+ W

= g2 (A90)

(10)minus W

(01)minus W

(10)+ W

= g2 (A91)

gZ(0)Z(0)W

(0)minus W

= minusg2c2w (A92)

gZ(0)Z(0)W

(1)minus W

= minusg2c2w (A93)

gA(0)A(0)W

(0)+ W

(0)minus

= minuse2 (A94)

gA(0)A(0)W

(1)+ W

(1)minus

= minuse2 (A95)

gZ(0)A(0)W

(0)+ W

(0)minus

= minusegcw (A96)

gZ(0)A(0)W

(1)+ W

(1)minus

= minusegcw (A97)

(1)3 A

(1)3 W

(1)+ W

(1)minus

= minus3

2g2 (A98)

(1)3 A

(1)3 W

(0)+ W

(0)minus

= minusg2 (A99)

(1)3 Z(0)W

(01)+ W

(10)minus

= minusg2cw (A100)

Vector-Vector-Scalar-Scalar Vertices

Couplings between two vector and two scalar fields originate both from thegauge field strength term (59) and the kinetic Higgs term (517) of the La-grangian The following four-dimensional Feynman vertex rules are derived

V micro1

iηmicroνgV1V2S1S2

gZ(0)Z(0)h(0)h(0) =g2

2c2w(A101)

(0)plusmn W

(0)∓ h(0)h(0) =

2(A102)

gB(1)B(1)h(1)h(1) =3g2

4(A103)

gB(1)B(1)a

(1)0 a

M (1)2

2 (A104)

(1)3 A

(1)3 h(1)h(1) =

4(A105)

(1)3 A

(1)3 a

(1)0 a

M (1)2

2 (A106)

(1)plusmn W

(1)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A107)

gB(1)B(1)a

(1)+ a

(1)minus

M (1)2

2 (A108)

(1)3 A

(1)3 a

(1)plusmn a

(1)∓

2+ 13m2

2 (A109)

(1)plusmn W

(1)∓ h(1)h(1) =

4(A110)

(1)plusmn W

(1)∓ a

(1)0 a

2+ 13m2

2 (A111)

(1)∓ W

(1)plusmn a

(1)∓

2minus 13m2

2 (A112)

gB(1)A

(1)3 h(1)h(1) = minus3gY g

4(A113)

gB(1)A

(1)3 a

(1)0 a

= minus3gY g

M (1)2

2 (A114)

gB(1)A

(1)3 a

(1)plusmn a

(1)∓

=3gY g

M (1)2

2 (A115)

gB(1)W

(1)plusmn h(1)a

(1)∓

= ∓i3gY g

(A116)

gB(1)W

(1)plusmn a

(1)0 a

(1)∓

= minus3gY g

M (1)2

M(1)W M

(A117)

(1)3 W

(1)plusmn a

(1)0 a

(1)∓

= minusg2

M(1)W M

(A118)

gB(1)B(1)h(0)h(0) =g2

2(A119)

gZ(0)Z(0)h(1)h(1) =g2

2c2w(A120)

(1)3 A

(1)3 h(0)h(0) =

2(A121)

gZ(0)A

(1)3 h(0)h(1) =

2cw(A122)

gZ(0)Z(0)a

(1)0 a

M (1)2

2 (A123)

(0)plusmn W

(0)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A124)

gA(0)A(0)a

(1)plusmn a

(1)∓

= 2e2 (A125)

gZ(0)Z(0)a

(1)plusmn a

(1)∓

4c4wm2W

+ (c2w minus s2w)2M (1)2

2M(1)W

2 (A126)

(01)+ W

(01)minus h(10)h(10) =

2(A127)

(01)+ W

(10)minus h(01)h(10) =

2(A128)

(0)plusmn W

(0)∓ a

(1)0 a

2+ 3m2

2 (A129)

(0)plusmn W

(0)∓ a

(1)plusmn a

(1)∓

2 (A130)

gB(1)A

(1)3 h(0)h(0) = minusgY g

2(A131)

gB(1)Z(0)h(1)h(0) =egY

2cw(A132)

gA(0)Z(0)a

(1)plusmn a

(1)∓

2c2wm2W

+ (c2w minus s2w)M (1)2

2 (A133)

gA(0)W

(01)plusmn h(10)a

(1)∓

= ∓i e2

(A134)

gB(1)W

(0)plusmn h(0)a

(1)∓

= ∓i e2

(A135)

gA(0)W

(0)plusmn a

(1)0 a

(1)∓

= minus e2

mW +M(1)W

M(1)W M

(A136)

gZ(0)W

(01)plusmn h(10)a

(1)∓

= plusmni e2

(A137)

gZ(0)W

(0)plusmn a

(1)0 a

(1)∓

2s2wcw

s2wM(1)2 minus 2c2wm

M(1)W M

(A138)

gA(0)A(0)G

(1)+ G

(1)minus

= 2e2 (A139)

gB(1)B(1)χ

(0)+ χ

(0)minus

2(A140)

gB(1)B(1)G

(1)+ G

(1)minus

2 (A141)

gB(1)B(1)G

(1)∓ a

(1)plusmn

mWM(1)

2 (A142)

gB(1)B(1)χ3(0)χ3(0) =g2

2(A143)

gB(1)B(1)G

(1)0 G

2 (A144)

gB(1)B(1)G

(1)0 a

mZM(1)

2 (A145)

gB(1)A(0)G

(1)plusmn χ

(0)∓

= plusmniegY

(A146)

gB(1)A(0)a

(1)plusmn χ

(0)∓

= plusmniegY

(A147)

(1)plusmn A(0)h(0)G

(1)∓

= ∓i eg2

(A148)

Scalar-Scalar-Scalar-Scalar Vertices

Couplings between four scalar fields originate from the kinetic Higgs term(517) of the Lagrangian The following four-dimensional Feynman vertexrules are derived

igS1S2S3S4

gh(0)h(0)h(0)h(0) = minus6λ (A149)

gh(1)h(1)h(1)h(1) = minus9λ (A150)

ga(1)0 a

(1)0 a

= minus3g2c2wm2ZM (1)2 + 9λM (1)4

4 (A151)

gh(1)h(1)a

(1)0 a

= minusg2m2

Zc2w + 6λM (1)2

2M(1)Z

2 (A152)

gh(1)h(1)a

(1)+ a

(1)minus

= minusg2m2

W+ 6λM (1)2

2M(1)W

2 (A153)

ga(1)0 a

(1)0 a

(1)+ a

(1)minus

= minusg2m2

WM (1)2(1 + c4w)c4w + 6λM (1)4

2M(1)W

2 (A154)

ga(1)+ a

(1)+ a

(1)minus a

(1)minus

= minus2g2m2WM (1)2 + 6λM (1)4

2 (A155)

gh(1)h(1)h(0)h(0) = minusλ (A156)

gh(0)h(0)a

(1)0 a

= minusg2m2

Zc2w + 4λM (1)2

2M(1)Z

2 (A157)

gh(0)h(0)a

(1)+ a

(1)minus

= minusg2m2

W+ 4λM (1)2

2M(1)W

2 (A158)

In unitarity gauge (ξ rarr infin) unphysical scalar fields are not present How-ever in a general gauge there are also additional vertices including unphysicalscalar fields Although we in Paper III calculated the one-loopB(1)B(1) rarr γγprocess in the Feynamn-rsquot Hooft gauge (ξ rarr 1) no scalar-scalar-scalar-scalarvertexes come in at loop order for this process

A Short Note on Conventions in FeynArts

Unfortunately there is no consensus in sign conventions in the field theoryliterature In our actual implementation of vertex rules into the FeynArts

package [255] which we used in the numerical calculations in Paper III weadopted the same convention as for the preimplemented vertex rules for theSM particles in FeynArts This requires a minor change for the zero modeGoldstone bosons vertex rules compared to those given in this AppendixFeynman rules in the FeynArts convention would be obtained from this Ap-

pendix if we changes the overall sign for each χ30(0)

and multiply by plusmni for

each χplusmn(0)that appears in the vertex rule

B i b l i o g r a p h y

[1] W de Boer C Sander V Zhukov A V Gladyshev and D IKazakov EGRET Excess of Diffuse Galactic Gamma Rays as Tracerof Dark Matter Astron Astrophys 444 (2005) 51[astro-ph0508617]

[2] C Misner K Thorne and J Wheeler Gravitation San Francisco USFreeman 1973

[3] M E Peskin and D V Schroeder An Introduction to Quantum FieldTheory Boulder US Westview 1995

[4] S Weinberg Gravitation and Cosmology Principles and Applicationsof the General Theory of Relativity New York US Wiley 1972

[5] J D Jackson Classical Electrodynamics New York US Wiley 1999

[6] J A Peacock Cosmological Physics Cambridge UK Univ Pr 1999

[7] E Hubble A Relation Between Distance and Radial Velocity AmongExtra-Galactic Nebulae Proc Nat Acad Sci 15 (1929) 168ndash173

[8] A H Guth The Inflationary Universe A Possible Solution to theHorizon and Flatness Problems Phys Rev D23 (1981) 347ndash356

[9] A D Linde A New Inflationary Universe Scenario A PossibleSolution of the Horizon Flatness Homogeneity Isotropy andPrimordial Monopole Problems Phys Lett B108 (1982) 389ndash393

[10] A Albrecht and P J Steinhardt Cosmology for Grand UnifiedTheories with Radiatively Induced Symmetry Breaking Phys RevLett 48 (1982) 1220ndash1223

[11] R Penrose The Road to Reality A Complete Guide to the Laws of theUniverse London Gt Brit Cape 2004

[12] S M Carroll Is our Universe Natural Nature 440 (2006) 1132ndash1136[hep-th0512148]

[13] A R Liddle and D H Lyth Cosmological Inflation and Large-ScaleStructure Cambridge US Cambridge UP 2000

[14] Supernova Search Team Collaboration A G Riess et alObservational Evidence from Supernovae for an Accelerating Universeand a Cosmological Constant Astron J 116 (1998) 1009ndash1038[astro-ph9805201]

160 Bibliography

[15] Supernova Cosmology Project Collaboration S Perlmutteret al Measurements of Omega and Lambda from 42 High-RedshiftSupernovae Astrophys J 517 (1999) 565ndash586 [astro-ph9812133]

[16] Supernova Search Team Collaboration A G Riess et al Type IaSupernova Discoveries at z gt 1 from the Hubble Space TelescopeEvidence for Past Deceleration and Constraints on Dark EnergyEvolution Astrophys J 607 (2004) 665ndash687 [astro-ph0402512]

[17] SNLS Collaboration P Astier et al The Supernova Legacy SurveyMeasurement of OmegaM OmegaΛ and w from the First Year DataSet Astron Astrophys 447 (2006) 31ndash48 [astro-ph0510447]

[18] WMAP Collaboration D N Spergel et al Wilkinson MicrowaveAnisotropy Probe (WMAP) Three Yea Results Implications forCosmology Astrophys J Suppl 170 (2007) 377 [astro-ph0603449]

[19] M Jarvis B Jain G Bernstein and D Dolney Dark EnergyConstraints from the CTIO Lensing Survey Astrophys J 644 (2006)71ndash79 [astro-ph0502243]

[20] I Zlatev L-M Wang and P J Steinhardt Quintessence CosmicCoincidence and the Cosmological Constant Phys Rev Lett 82

(1999) 896ndash899 [astro-ph9807002]

[21] C Armendariz-Picon V F Mukhanov and P J Steinhardt ADynamical Solution to the Problem of a Small Cosmological Constantand Late-Time Cosmic Acceleration Phys Rev Lett 85 (2000)4438ndash4441 [astro-ph0004134]

[22] F Zwicky Spectral Displacement of Extra Galactic Nebulae HelvPhys Acta 6 (1933) 110ndash127

[23] J A Tyson G P Kochanski and I P DellrsquoAntonio Detailed MassMap of CL0024+1654 from Strong Lensing Astrophys J 498 (1998)L107 [astro-ph9801193]

[24] S W Allen A C Fabian R W Schmidt and H EbelingCosmological Constraints from the Local X-ray Luminosity Function ofthe Most X-ray Luminous Galaxy Clusters Mon Not Roy AstronSoc 342 (2003) 287 [astro-ph0208394]

[25] G Bertone D Hooper and J Silk Particle Dark Matter EvidenceCandidates and Constraints Phys Rept 405 (2005) 279ndash390[hep-ph0404175]

[26] J E Carlstrom G P Holder and E D Reese Cosmology with theSunyaev-Zelrsquodovich Effect Ann Rev Astron Astrophys 40 (2002)643ndash680 [astro-ph0208192]

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[27] M Persic P Salucci and F Stel The Universal Rotation Curve ofSpiral Galaxies 1 The Dark Matter Connection Mon Not RoyAstron Soc 281 (1996) 27 [astro-ph9506004]

[28] 2dFGRS Collaboration S Cole et al The 2dF Galaxy RedshiftSurvey Power-spectrum Analysis of the Final Dataset andCosmological Implications Mon Not Roy Astron Soc 362 (2005)505ndash534 [astro-ph0501174]

[29] SDSS Collaboration D J Eisenstein et al Detection of the BaryonAcoustic Peak in the Large-Scale Correlation Function of SDSSLuminous Red Galaxies Astrophys J 633 (2005) 560ndash574[astro-ph0501171]

[30] R Massey et al Dark Matter Maps Reveal Cosmic ScaffoldingNature 445 (2007) 286 [astro-ph0701594]

[31] M Milgrom A Modification of the Newtonian Dynamics as a PossibleAlternative to the Hidden mass Hypothesis Astrophys J 270 (1983)365ndash370

[32] J D Bekenstein Relativistic Gravitation Theory for the MONDParadigm Phys Rev D70 (2004) 083509 [astro-ph0403694]

[33] M Markevitch et al Direct Constraints on the Dark MatterSelf-Interaction Cross-Section from the Merging Cluster 1E0657-56Astrophys J 606 (2004) 819ndash824 [astro-ph0309303]

[34] D Clowe et al A Direct Empirical Proof of the Existence of DarkMatter Astrophys J 648 (2006) L109ndashL113 [astro-ph0608407]

[35] J Lesgourgues and S Pastor Massive Neutrinos and CosmologyPhys Rept 429 (2006) 307ndash379 [astro-ph0603494]

[36] S Hannestad Primordial Neutrinos Ann Rev Nucl Part Sci 56

(2006) 137ndash161 [hep-ph0602058]

[37] M Tegmark et al Cosmological Constraints from the SDSS LuminousRed Galaxies Phys Rev D74 (2006) 123507 [astro-ph0608632]

[38] M Fukugita and P J E Peebles The Cosmic Energy InventoryAstrophys J 616 (2004) 643ndash668 [astro-ph0406095]

[39] J N Bregman The Search for the Missing Baryons at Low RedshiftAnn Rev Astron Astrophys 45 (2007) 221ndash259 [arXiv07061787]

[40] httpmapgsfcnasagovm orhtml Credit NASAWMAPScience Team

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[41] httpchandraharvardeduphoto20061e0657 Credit X-rayNASACXCCfA [33] Optical NASASTScIMagellanUArizona [34] Lensing Map NASASTScI ESO WFIMagellanUArizona [34]

[42] G Jungman M Kamionkowski and K Griest Supersymmetric DarkMatter Phys Rept 267 (1996) 195ndash373 [hep-ph9506380]

[43] V Springel et al Simulating the Joint Evolution of QuasarsGalaxies and Their Large-Scale Distribution Nature 435 (2005)629ndash636 [astro-ph0504097]

[44] J Diemand B Moore and J Stadel Earth-Mass Dark-Matter Haloesas the First Structures in the Early Universe Nature 433 (2005)389ndash391 [astro-ph0501589]

[45] J Diemand B Moore and J Stadel Convergence and Scatter ofCluster Density Profiles Mon Not Roy Astron Soc 353 (2004) 624[astro-ph0402267]

[46] Y P Jing and Y Suto Density Profiles of Dark Matter Halo Are NotUniversal Astrophys J 529 (2000) L69ndash72 [astro-ph9909478]

[47] J Diemand M Kuhlen and P Madau Formation and Evolution ofGalaxy Dark Matter Halos and Their Substructure Astrophys J 667

(2007) 859 [astro-ph0703337]

[48] J F Navarro C S Frenk and S D M White A Universal DensityProfile from Hierarchical Clustering Astrophys J 490 (1997) 493ndash508[astro-ph9611107]

[49] R H Wechsler J S Bullock J R Primack A V Kravtsov andA Dekel Concentrations of Dark Halos from Their AssemblyHistories Astrophys J 568 (2002) 52ndash70 [astro-ph0108151]

[50] J F Navarro C S Frenk and S D M White The Structure of ColdDark Matter Halos Astrophys J 462 (1996) 563ndash575[astro-ph9508025]

[51] B Moore T Quinn F Governato J Stadel and G Lake ColdCollapse and the Core Catastrophe Mon Not Roy Astron Soc 310

(1999) 1147ndash1152 [astro-ph9903164]

[52] A V Kravtsov A A Klypin J S Bullock and J R Primack TheCores of Dark Matter Dominated Galaxies Theory Vs ObservationsAstrophys J 502 (1998) 48 [astro-ph9708176]

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(2004) 1039 [astro-ph0311231]

[57] O J Eggen D Lynden-Bell and A R Sandage Evidence from theMotions of Old Stars That the Galaxy Collapsed Astrophys J 136

(1962) 748

[58] P Young Numerical Models of Star Clusters with a Central BlackHole I ndash Adiabatic Models Astrophys J 242 (1980) 1243

[59] Y B Zeldovich A A Klypin M Y Khlopov and V M ChechetkinAstrophysical Constraints on the Mass of Heavy Stable NeutralLeptons Sov J Nucl Phys 31 (1980) 664ndash669

[60] G R Blumenthal S M Faber R Flores and J R PrimackContraction of Dark Matter Galactic Halos Due to Baryonic InfallAstrophys J 301 (1986) 27

[61] B S Ryden and J E Gunn Contraction of Dark Matter GalacticHalos Due to Baryonic Infall Astrophys J 318 (1987) 15

[62] O Y Gnedin A V Kravtsov A A Klypin and D Nagai Responseof Dark Matter Halos to Condensation of Baryons CosmologicalSimulations and Improved Adiabatic Contraction Model Astrophys J616 (2004) 16ndash26 [astro-ph0406247]

[63] V F Cardone and M Sereno Modelling the Milky Way ThroughAdiabatic Compression of Cold Dark Matter Halo Astron Astrophys438 (2005) 545ndash556 [astro-ph0501567]

[64] J A Sellwood and S S McGaugh The Compression of Dark MatterHalos by Baryonic Infall Astrophys J 634 (2005) 70ndash76[astro-ph0507589]

[65] E Vasiliev A Simple Analytical Model for Dark Matter Phase-SpaceDistribution and Adiabatic Contraction JETP Letters 84 (2006) 45[astro-ph0601669]

[66] A A Dutton F C van den Bosch A Dekel and S Courteau ARevised Model for the Formation of Disk Galaxies Quiet History LowSpin and Dark-Halo Expansion Astrophys J 654 (2006) 27ndash52[astro-ph0604553]

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[68] F Prada A Klypin J Flix M Martinez and E Simonneau DarkMatter Annihilation in the Milky Way Galaxy Effects of BaryonicCompression Phys Rev Lett 93 (2004) 241301 [astro-ph0401512]

[69] J Sommer-Larsen M Gotz and L Portinari Galaxy FormationCDM Feedback and the Hubble Sequence Astrophys J 596 (2003)47ndash66 [astro-ph0204366]

[70] J Sommer-Larsen Where are the ldquoMissingrdquo Galactic BaryonsAstrophys J 644 (2006) L1ndashL4 [astro-ph0602595]

[71] L Portinari and J Sommer-Larsen The Tully-Fisher Relation and ItsEvolution with Redshift in Cosmological Simulations of Disc GalaxyFormation Mon Not Roy Astron Soc 375 (2007) 913ndash924[astro-ph0606531]

[72] H M P Couchman F R Pearce and P A Thomas Hydra CodeRelease astro-ph9603116

[73] L Hernquist and N Katz Treesph A Unification of SPH with theHierarchical Tree Method Astrophys J Suppl 70 (1989) 419ndash446

[74] A Klypin A V Kravtsov J Bullock and J Primack Resolving theStructure of Cold Dark Matter Halos Astrophys J 554 (2001)903ndash915 [astro-ph0006343]

[75] G Cowan Statistical Data Analysis Oxford UK Clarendon 1998

[76] J Dubinski The Effect of Dissipation on the Shapes of Dark HalosAstrophys J 431 (1994) 617ndash624 [astro-ph9309001]

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[78] I Berentzen and I Shlosman Growing Live Disks WithinCosmologically Assembling Asymmetric Halos Washing Out the HaloProlateness Astrophys J 648 (2006) 807ndash819 [astro-ph0603487]

[79] O Gerhard Mass Distribution and Bulge Formation in the Milky WayGalaxy astro-ph0608343

[80] C S Frenk S D M White D M and G Efstathiou The First DarkMicrohalos Astrophys J 327 (1988) 507

[81] J Dubinski and R G Carlberg The Structure of Cold Dark MatterHalos Astrophys J 378 (1991) 496

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[82] M S Warren P J Quinn J K Salmon and W H Zurek DarkHalos Formed Via Dissipationless Collapse 1 Shapes and Alignmentof Angular Momentum Astrophys J 399 (1992) 405

[83] P A Thomas J M Colberg H M P Couchman G P EfstathiouC J Frenk A R Jenkins A H Nelson R M Hutchings J APeacock F R Pearce and S D M White Further Evidence for aVariable Fine-Structure Constant from KeckHIRES QSO AbsorptionSpectra Mon Not Roy Astron Soc 296 (1998) 1061

[84] Y P Jing and Y Suto Triaxial Modeling of Halo Density Profileswith High- resolution N-body Simulations Astrophys J 574 (2002)538 [astro-ph0202064]

[85] J Bailin and M Steinmetz Internal and External Alignment of theShapes and Angular Momenta of LCDM Halos Astrophys J 627

(2005) 647ndash665 [astro-ph0408163]

[86] P Bett et al The Spin and Shape of Dark Matter Haloes in theMillennium Simulation of a LambdaCDM Universe Mon Not RoyAstron Soc 376 (2007) 215ndash232 [astro-ph0608607]

[87] E Athanassoula F S Ling and E Nezri Halo Geometry and DarkMatter Annihilation Signal Phys Rev D72 (2005) 083503[astro-ph0504631]

[88] B Moore Evidence Against Dissipationless Dark Matter fromObservations of Galaxy Haloes Nature 370 (1994) 629

[89] F C van den Bosch and R A Swaters Dwarf Galaxy RotationCurves and the Core Problem of Dark Matter Halos Mon Not RoyAstron Soc 325 (2001) 1017 [astro-ph0006048]

[90] W J G de Blok and A Bosma H Alpha Rotation Curves of LowSurface Brightness Galaxies Astron Astrophys 385 (2002) 816[astro-ph0201276]

[91] K Spekkens and R Giovanelli The CuspCore Problem in GalacticHalos Long-Slit Spectra for a Large Dwarf Galaxy Sample AstrophysJ 129 (2005) 2119ndash2137 [astro-ph0502166]

[92] G Gentile P Salucci U Klein D Vergani and P Kalberla TheCored Distribution of Dark Matter in Spiral Galaxies Mon Not RoyAstron Soc 351 (2004) 903 [astro-ph0403154]

[93] G Gentile P Salucci U Klein and G L Granato NGC 3741 DarkHalo Profile from the Most Extended Rotation Curve Mon Not RoyAstron Soc 375 (2007) 199ndash212 [astro-ph0611355]

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[95] E Hayashi J F Navarro A Jenkins C S Frenk C Power S D MWhite V Springel J Stadel and J W T Quinn Disk GalaxyRotation Curves in Triaxial CDM Halos astro-ph0408132

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[97] A Cesarini F Fucito A Lionetto A Morselli and P Ullio TheGalactic Center as a Dark Matter Gamma-Ray Source AstropartPhys 21 (2004) 267ndash285 [astro-ph0305075]

[98] P Gondolo and J Silk Dark Matter Annihilation at the GalacticCenter Phys Rev Lett 83 (1999) 1719ndash1722 [astro-ph9906391]

[99] P Ullio H Zhao and M Kamionkowski A Dark-Matter Spike at theGalactic Center Phys Rev D64 (2001) 043504 [astro-ph0101481]

[100] D Merritt Evolution of the Dark Matter Distribution at the GalacticCenter Phys Rev Lett 92 (2004) 201304 [astro-ph0311594]

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[103] D Merritt S Harfst and G Bertone Collisionally Regenerated DarkMatter Structures in Galactic Nuclei Phys Rev D75 (2007) 043517[astro-ph0610425]

[104] B Moore et al Dark Matter Substructure Within Galactic HalosAstrophys J 524 (1999) L19ndashL22

[105] A A Klypin A V Kravtsov O Valenzuela and F Prada Where Arethe Missing Galactic Satellites Astrophys J 522 (1999) 82ndash92[astro-ph9901240]

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[118] L E Strigari S M Koushiappas J S Bullock and M KaplinghatPrecise Constraints on the Dark Matter Content of Milky Way DwarfGalaxies for Gamma-Ray Experiments Phys Rev D75 (2007) 083526[astro-ph0611925]

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[140] O W Greenberg Spin and Unitary Spin Independence in a ParaquarkModel of Baryons and Mesons Phys Rev Lett 13 (1964) 598ndash602

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[153] T Appelquist B A Dobrescu E Ponton and H-U Yee ProtonStability in Six Dimensions Phys Rev Lett 87 (2001) 181802[hep-ph0107056]

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[163] I J D Craig and J C Brown Why Measure Astrophysical X-raySpectra Nature 264 (1976) 340

[164] C R Gould E I Sharapov and S K Lamoreaux Time-Variability ofalpha from Realistic Models of Oklo Reactors Phys Rev C74 (2006)024607 [nucl-ex0701019]

[165] J K Webb V V Flambaum C W Churchill M J Drinkwater andJ D Barrow Evidence for Time Variation of the Fine StructureConstant Phys Rev Lett 82 (1999) 884ndash887 [astro-ph9803165]

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[167] M T Murphy J K Webb and V V Flambaum Further Evidence fora Variable Fine-Structure Constant from KeckHIRES QSOAbsorption Spectra Mon Not Roy Astron Soc 345 (2003) 609[astro-ph0306483]

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[170] S Hannestad Possible Constraints on the Time Variation of the FineStructure Constant from Cosmic Microwave Background Data PhysRev D60 (1999) 023515 [astro-ph9810102]

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[175] K-i Maeda Stability and Attractor in Kaluza-Klein Cosmology 1Class Quant Grav 3 (1986) 233

[176] K-i Maeda Stability and Attractor in a Higher DimensionalCosmology 2 Class Quant Grav 3 (1986) 651

[177] U Gunther and A Zhuk Multidimensional Perfect Fluid Cosmologywith Stable Compactified Internal Dimensions Class Quant Grav 15

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[179] U Gunther and A Zhuk A Note on Dynamical Stabilization ofInternal Spaces in Multidimensional Cosmology Class Quant Grav18 (2001) 1441ndash1460 [hep-ph0006283]

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[181] U Gunther A Starobinsky and A Zhuk MultidimensionalCosmological Models Cosmological and Astrophysical ImplicationsPhys Rev D69 (2004) 044003 [hep-ph0306191]

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[202] S Matsumoto J Sato M Senami and M Yamanaka SolvingCosmological Problem in Universal Extra Dimension Models byIntroducing Dirac Neutrino Phys Lett B647 (2007) 466ndash471[hep-ph0607331]

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[209] J Edsjo and P Gondolo Neutralino Relic Density IncludingCoannihilations Phys Rev D56 (1997) 1879ndash1894 [hep-ph9704361]

[210] P Gondolo and G Gelmini Cosmic Abundances of Stable ParticlesImproved Analysis Nucl Phys B360 (1991) 145ndash179

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[220] T G Rizzo Probes of Universal Extra Dimensions at Colliders PhysRev D64 (2001) 095010 [hep-ph0106336]

[221] C Macesanu C D McMullen and S Nandi Collider Implications ofUniversal Extra Dimensions Phys Rev D66 (2002) 015009[hep-ph0201300]

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radics = 18-TeV PhD

thesis Yale University (FERMILAB-THESIS-2005-69) 2005FERMILAB-THESIS-2005-69

[223] E A Baltz M Battaglia M E Peskin and T WizanskyDetermination of Dark Matter Properties at High-Energy CollidersPhys Rev D74 (2006) 103521 [hep-ph0602187]

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[227] DAMA Collaboration R Bernabei et al Search for WIMP AnnualModulation Signature Results from DAMANaI-3 and DAMANaI-4and the Global Combined Analysis Phys Lett B480 (2000) 23ndash31

[228] R Bernabei et al DAMANaI Results astro-ph0405282

[229] UK Dark Matter Collaboration G J Alner et al Limits onWIMP Cross-Sections from the NAIAD Experiment at the BoulbyUnderground Laboratory Phys Lett B616 (2005) 17ndash24[hep-ex0504031]

[230] G J Alner et al First Limits on WIMP Nuclear Recoil Signals inZEPLIN-II A Two Phase Xenon Detector for Dark Matter DetectionAstropart Phys 28 (2007) 287ndash302 [astro-ph0701858]

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[232] D Hooper and G D Kribs Probing Kaluza-Klein Dark Matter withNeutrino Telescopes Phys Rev D67 (2003) 055003 [hep-ph0208261]

[233] I V Moskalenko and L L Wai Dark Matter Burners Astrophys J659 (2007) L29ndashL32 [astro-ph0702654]

[234] G Bertone and M Fairbairn Compact Stars as Dark Matter ProbesarXiv07091485

[235] M Fairbairn P Scott and J Edsjo The Zero Age Main Sequence ofWIMP burners arXiv07103396

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[237] S Coutu et al Cosmic-ray Positrons Are There Primary SourcesAstropart Phys 11 (1999) 429ndash435 [astro-ph9902162]

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[243] F W Stecker M A Malkan and S T Scully Intergalactic PhotonSpectra from the Far IR to the UV Lyman Limit for 0 iexcl z iexcl 6 and theOptical Depth of the Universe to High Energy Gamma-RaysAstrophys J 648 (2006) 774 [astro-ph0510449]

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[261] CANGAROO-II Collaboration K Tsuchiya et al Detection ofSub-TeV Gamma-Rays from the Galactic Center Direction byCANGAROO-II Astrophys J 606 (2004) L115ndashL118[astro-ph0403592]

[262] S Profumo TeV Gamma-Rays and the Largest Masses andAnnihilation Cross Sections of Neutralino Dark Matter Phys RevD72 (2005) 103521 [astro-ph0508628]

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[265] S Dimopoulos and D W Sutter The Supersymmetric FlavorProblem Nucl Phys B452 (1995) 496ndash512 [hep-ph9504415]

[266] A H Chamseddine R Arnowitt and P Nath Locally SupersymmetricGrand Unification Phys Rev Lett 49 (1982) 970

[267] J F Gunion H E Haber G L Kane and S Dawson The HiggsHunterrsquos Guide New York US Addision-Wesley 1990

[268] J F Gunion H E Haber G L Kane and S Dawson Errata forlsquoThe Higgs Hunterrsquos Guidersquo hep-ph9302272

[269] L Bergstrom and P Ullio Full One-Loop Calculation of NeutralinoAnnihilation Into Two Photons Nucl Phys B504 (1997) 27ndash44[hep-ph9706232]

[270] P Ullio and L Bergstrom Neutralino Annihilation Into a Photon anda Z Boson Phys Rev D57 (1998) 1962ndash1971 [hep-ph9707333]

[271] J Hisano S Matsumoto and M M Nojiri Explosive Dark MatterAnnihilation Phys Rev Lett 92 (2004) 031303 [hep-ph0307216]

[272] J Hisano S Matsumoto M M Nojiri and O Saito Non-PerturbativeEffect on Dark Matter Annihilation and Gamma Ray Signature fromGalactic Center Phys Rev D71 (2005) 063528 [hep-ph0412403]

[273] M Drees and M M Nojiri The Neutralino Relic Density in MinimalN=1 Supergravity Phys Rev D47 (1993) 376ndash408 [hep-ph9207234]

[274] L Bergstrom Radiative Processes in Dark Matter PhotinoAnnihilation Phys Lett B225 (1989) 372

[275] T Bringmann L Bergstrom and J Edsjo New Gamma-RayContributions to Supersymmetric Dark Matter AnnihilationarXiv07103169

[276] H E Haber and G L Kane The Search for Supersymmetry ProbingPhysics Beyond the Standard Model Phys Rept 117 (1985) 75ndash263

[277] T Hahn and C Schappacher The Implementation of the MinimalSupersymmetric Standard Model in FeynArts and FormCalc ComputPhys Commun 143 (2002) 54ndash68 [hep-ph0105349]

[278] P Ullio Indirest Detection of Neutralino Dark Matter PhD thesisStockholm University 1999

180 Bibliography

[279] J Edsjo Aspects of Neutrino Detection of Neutralino Dark MatterPhD thesis Stockholm University 1997

[280] L Bergstrom and P Gondolo Limits on Direct Detection ofNeutralino Dark Matter from brarr s Gamma Decays Astropart Phys5 (1996) 263ndash278 [hep-ph9510252]

[281] P Gondolo et al DarkSUSY Computing Supersymmetric DarkMatter Properties Numerically JCAP 0407 (2004) 008[astro-ph0406204]

[282] G Belanger F Boudjema A Pukhov and A SemenovmicrOMEGAs A Program for Calculating the Relic Density in theMSSM Comput Phys Commun 149 (2002) 103ndash120[hep-ph0112278]

[283] B Grzadkowski J F Gunion and J Kalinowski Finding theCP-violating Higgs Bosons at e+ e- Colliders Phys Rev D60 (1999)075011 [hep-ph9902308]

[284] S L Glashow and S Weinberg Natural Conservation Laws forNeutral Currents Phys Rev D15 (1977) no 7 1958ndash1965

[285] N G Deshpande and E Ma Pattern of Symmetry Breaking with TwoHiggs Doublets Phys Rev D18 (1978) 2574

[286] R Barbieri L J Hall and V S Rychkov Improved Naturalness witha Heavy Higgs An Alternative Road to LHC Physics Phys Rev D74

(2006) 015007 [hep-ph0603188]

[287] E Ma Verifiable Radiative Seesaw Mechanism of Neutrino Mass andDark Matter Phys Rev D73 (2006) 077301 [hep-ph0601225]

[288] E Ma and U Sarkar Revelations of the E(6U)(1)(N) ModelTwo-Loop Neutrino Mass and Dark Matter Phys Lett B653 (2007)288ndash291 [arXiv07050074]

[289] T Hambye and M H G Tytgat Electroweak Symmetry BreakingInduced by Dark Matter Phys Lett B659 (2008) 651ndash655[arXiv07070633]

[290] LEP Electroweak Working Group Collaboration webpagehttplepewwgwebcernchLEPEWWG

[291] R Barbieri L J Hall Y Nomura and V S Rychkov SupersymmetryWithout a Light Higgs Boson Phys Rev D75 (2007) 035007[hep-ph0607332]

Bibliography 181

[292] J A Casas J R Espinosa and I Hidalgo Expectations for LHC fromNaturalness Modified Vs SM Higgs sector Nucl Phys B777 (2007)226ndash252 [hep-ph0607279]

[293] J L Hewett The Standard Model and Why We Believe It Lecturesgiven at TASI 97 Supersymmetry Supergravity and SupercollidersBoulder 1997 (1997) 3ndash83 [hep-ph9810316]

[294] Q-H Cao E Ma and G Rajasekaran Observing the Dark ScalarDoublet and Its Impact on the Standard-Model Higgs Boson atColliders Phys Rev D76 (2007) 095011 [arXiv07082939]

[295] A Pierce and J Thaler Natural Dark Matter From an UnnaturalHiggs Boson and New Colored Particles at the TeV Scale JHEP 08

(2007) 026 [hep-ph0703056]

[296] L Lopez Honorez E Nezri J F Oliver and M H G Tytgat TheInert Doublet Model An Archetype for Dark Matter JCAP 0702

(2007) 028 [hep-ph0612275]

[297] D Majumdar and A Ghosal Dark Matter Candidate in a HeavyHiggs Model - Direct Detection Rates hep-ph0607067

[299] A Djouadi The Anatomy of Electro-Weak Symmetry Breaking I TheHiggs Boson in the Standard Model hep-ph0503172

[300] T Sjostrand S Mrenna and P Skands PYTHIA 64 Physics andManual JHEP 05 (2006) 026 [hep-ph0603175]

[301] G Zaharijas and D Hooper Challenges in Detecting Gamma-Raysfrom Dark Matter Annihilations in the Galactic Center Phys RevD73 (2006) 103501 [astro-ph0603540]

[302] D Hooper and B L Dingus Limits on Supersymmetric Dark Matterfrom EGRET Observations of the Galactic Center Region Phys RevD70 (2004) 113007 [astro-ph0210617]

[303] L Bergstrom and P Ullio Full One-Loop Calculation of NeutralinoAnnihilation into Two Photons Nucl Phys B504 (1997) 27ndash44[hep-ph9706232]

[304] P Ullio and L Bergstrom Neutralino Annihilation into a Photon anda Z Boson Phys Rev D57 (1998) 1962ndash1971 [hep-ph9707333]

182 Bibliography

[306] W de Boer Evidence for Dark Matter Annihilation from GalacticGamma Rays New Astron Rev 49 (2005) 213ndash231[hep-ph0408166]

[307] W de Boer C Sander V Zhukov A V Gladyshev and D IKazakov The Supersymmetric Interpretation of the EGRET Excess ofDiffuse Galactic Gamma Rays Phys Lett B636 (2006) 13ndash19[hep-ph0511154]

[308] W de Boer et al The Dark Connection Between the Canis MajorDwarf the Monoceros Ring the Gas Flaring the Rotation Curve andthe EGRET Excess of Diffuse Galactic Gamma RaysarXiv07105106 Proceeding of 15th International Conference onSupersymmetry and the Unification of Fundamental Interactions(SUSY07) Karlsruhe Germany 26 Jul - 1 Aug 2007

[309] D Hooper Indirect Searches For Dark Matter Signals Hints andOtherwise arXiv07102062

[310] EGRET Collaboration webpagehttpcosscgsfcnasagovdocscgroegret

[311] S D Hunter et al EGRET Observations of the Diffuse Gamma-RayEmission from the Galactic Plane Astrophys J 481 (1997) 205ndash240

[312] I V Moskalenko and A W Strong Production and Propagation ofCosmic-Ray Positrons and Electrons Astrophys J 493 (1998)694ndash707 [astro-ph9710124]

[313] A W Strong and I V Moskalenko Propagation of Cosmic-RayNucleons in the Galaxy Astrophys J 509 (1998) 212ndash228[astro-ph9807150]

[314] A W Strong I V Moskalenko and O Reimer Diffuse ContinuumGamma Rays from the Galaxy Astrophys J 537 (2000) 763ndash784[astro-ph9811296]

[315] A W Strong I V Moskalenko and O Reimer Diffuse GalacticContinuum Gamma Rays A Model Compatible with EGRET Data andCosmic-Ray Measurements Astrophys J 613 (2004) 962ndash976[astro-ph0406254] Erratum-ibid54111092000

[316] GALPROP Collaboration A W Strong and I V Moskalenkohttpgalpropstanfordeduweb galpropgalprop homehtml

Bibliography 183

[317] F W Stecker S D Hunter and D A Kniffen The Likely Cause ofthe EGRET GeV Anomaly and its Implications Astropart Phys 29

(2008) 25ndash29 [arXiv07054311]

[318] B M Baughman W B Atwood R P Johnson T A Porter andM Ziegler A Fresh Look at Diffuse Gamma-ray Emission from theInner Galaxy arXiv07060503

[319] P Gondolo WIMP Annihilations and the Galactic Gamma-Ray Halo Prepared for 2nd International Conference on Dark Matter in Astroand Particle Physics (DARK98) Heidelberg Germany 20-25 Jul 1998

[320] D D Dixon et al Evidence for a Galactic Gamma Ray Halo NewAstron 3 (1998) 539 [astro-ph9803237]

[321] SDSS Collaboration M Juric et al The Milky Way Tomographywith SDSS I Stellar Number Density Distribution Astrophys J 673

(2008) 864 [astro-ph0510520]

[322] J N Bahcall The Distribution of Stars Perpendicular to GalacticDisk Astrophys J 276 (1984) 156

[323] J Holmberg and C Flynn The Local Surface Density of Disc MatterMapped by Hipparcos Mon Not Roy Astron Soc 352 (2004) 440[astro-ph0405155]

[324] J Holmberg and C Flynn The Local Density of Matter Mapped byHipparcos Mon Not Roy Astron Soc 313 (2000) 209ndash216[astro-ph9812404]

[325] S M Fall and G Efstathiou Formation and Rotation of DiscGalaxies with Haloes Mon Not Roy Astron Soc 193 (1980) 189

[326] EGRET Collaboration J A Esposito et al In-Flight Calibration ofEGRET on the Compton Gamma-Ray Observatory Astropart J 123

(1999) 203

[327] BESS Collaboration T Maeno et al Successive Measurements ofCosmic-Ray Antiproton Spectrum in a Positive Phase of the SolarCycle Astropart Phys 16 (2001) 121ndash128 [astro-ph0010381]

[328] L Pieri E Branchini and S Hofmann Difficulty of DetectingMinihalos Via Gamma Rays from Dark Matter Annihilation PhysRev Lett 95 (2005) 211301 [astro-ph0505356]

[329] F Donato N Fornengo D Maurin and P Salati Antiprotons inCosmic Rays from Neutralino Annihilation Phys Rev D69 (2004)063501 [astro-ph0306207]

184 Bibliography

[330] J Edsjo M Schelke and P Ullio Direct Versus Indirect Detection inmSUGRA with Self- Consistent Halo Models JCAP 0409 (2004) 004[astro-ph0405414]

[331] F Donato et al Antiprotons from Spallation of Cosmic Rays onInterstellar Matter Astrophys J 536 (2001) 172ndash184[astro-ph0103150]

[332] W de Boer I Gebauer C Sander M Weber and V Zhukov Is theDark Matter Interpretation of the EGRET Gamma Ray ExcessCompatible with Antiproton Measurements AIP Conf Proc 903

(2007) 607ndash612 [astro-ph0612462]

[333] X-J Bi J Zhang Q Yuan J-L Zhang and H Zhao The DiffuseGalactic Gamma-Rays from Dark Matter Annihilationastro-ph0611783

[334] X-J Bi J Zhang and Q Yuan Diffuse Γ-Rays and p Flux from DarkMatter Annihilation ndash A Model for Consistent Results with EGRETand Cosmic Ray Data arXiv07124038

[335] P M W Kalberla L Dedes J Kerp and U Haud Dark Matter inthe Milky Way II the HI Gas Distribution as a Tracer of theGravitational Potential Astron Astrophys 619 (2007) 511ndash527[arXiv07043925]

[336] S Demers and P Battinelli C Stars as Kinematic Probes of the MilkyWay Disk from 9 to 15 kpc Astron Astrophys 473 (2007) 143ndash148

[337] X X Xue et al The Milky Wayrsquos Rotation Curve to 60 kpc and anEstimate of the Dark Matter Halo Mass from Kinematics of 2500SDSS Blue Horizontal Branch Stars arXiv08011232

[338] M Eriksson The Dark Side of the Universe PhD thesis StockholmUniversity 2005

[339] T Bringmann Cosmological Aspects of Universal Extra DimensionsPhD thesis Stockholm University 2005

P a r t I I

Scientific Papers

P a p e r I

T Bringmann M Eriksson and M Gustafsson Cosmological Evolution of Universal

Extra Dimensions Phys Rev D 68 063516 (2003)ccopy 2003 The American Physical Society (Reprinted with permission)

P a p e r I I

L Bergstrom T Bringmann M Eriksson and M Gustafsson Gamma Rays from

Kaluza-Klein Dark Matter Phys Rev Lett 94 131301 (2005)ccopy 2005 The American Physical Society (Reprinted with permission)

P a p e r I I I

L Bergstrom T Bringmann M Eriksson and M Gustafsson Two Photon Anni-

hilation of Kaluza-Klein Dark Matter JCAP 0504 004 (2005)ccopy 2005 IOP Publishing Ltd and SISSA (Reprinted with permission)

P a p e r I V

Heavy Neutralino Dark Matter Phys Rev Lett 95 241301 (2005)ccopy 2005 The American Physical Society (Reprinted with permission)

P a p e r V

L Bergstrom J Edsjo M Gustafsson and P Salati Is the Dark Matter Interpre-

tation of the EGRET Gamma Excess Compatible with Antiproton MeasurementsJCAP 0605 006 (2006)ccopy 2006 IOP Publishing Ltd and SISSA (Reprinted with permission)

P a p e r V I

M Gustafsson M Fairbairn and J Sommer-Larsen Baryonic Pinching of Galactic

Dark Matter Haloes Phys Rev D 74 123522 (2006)ccopy 2006 The American Physical Society (Reprinted with permission)

P a p e r V I I

M Gustafsson E Lundstrom L Bergstrom and J Edsjo Significant Gamma Lines

from Inert Higgs Dark Matter Phys Rev Lett 99 041301 (2007)ccopy 2007 The American Physical Society (Reprinted with permission)

Abstract

List of Accompanying Papers

Preface

Part I Background Material and Results

The Essence of Standard Cosmology

Our Place in the Universe

Spacetime and Gravity

Special Relativity

General Relativity

The Standard Model of Cosmology

Evolving Universe

Initial Conditions

The Dark Side of the Universe

Dark Energy

Dark Matter

All Those WIMPs -- Particle Dark Matter

Where Is the Dark Matter

Structure Formation History

Halo Models from Dark Matter Simulations

Adiabatic Contraction



Simulation Setups

Pinching of the Dark Matter Halo

Testing Adiabatic Contraction Models

Nonsphericity

Axis Ratios

Alignments

Some Comments on Observations

Tracing Dark Matter Annihilation


Halo Substructure

Beyond the Standard Model Hidden Dimensions and More

The Need to Go Beyond the Standard Model

General Features of Extra-Dimensional Scenarios

Modern Extra-Dimensional Scenarios

Motivations for Universal Extra Dimensions

Cosmology with Homogeneous Extra Dimensions

Why Constants Can Vary

How Constant Are Constants

Higher-Dimensional Friedmann Equations

Static Extra Dimensions

Evolution of Universal Extra Dimensions

Dimensional Reduction

Stabilization Mechanism

Quantum Field Theory in Universal Extra Dimensions

Compactification

Kaluza-Klein Parity

The Lagrangian

Particle Propagators

Radiative Corrections

Mass Spectrum

Kaluza-Klein Dark Matter

Relic Density

Direct and Indirect Detection


Direct Detection

Indirect Detection

Gamma-Ray Signatures

Gamma-Ray Continuum

Gamma Line Signal

Observing the Gamma-Ray Signal

Supersymmetry and a New Gamma-Ray Signal

Supersymmetry

Some Motivations

The Neutralino

A Neglected Source of Gamma Rays



Inert Higgs Dark Matter

The Inert Higgs Model



More Constraints

Inert Higgs -- A Dark Matter Candidate

Gamma Rays

Continuum

Gamma-Ray Lines

Have Dark Matter Annihilations Been Observed

Dark Matter Signals

The Data

The Claim

The Inconsistency



The Status to Date

Summary and Outlook

Feynman Rules The UED model

Field Content and Propagators

Vertex Rules

Bibliography

Part II Scientific Papers

Thesis for the degree of Doctor of Philosophy in Theoretical PhysicsDepartment of PhysicsStockholm UniversitySweden

Copyright ccopy Michael Gustafsson Stockholm 2008ISBN 978-91-7155-548-9 (pp indashxv 1ndash184)

Figure 81 91 and 92 have been adopted with authorrsquos permission fromthe sources cited in the figure captions ( ccopyPhys Rev D ccopyAstrophys J andccopyAstron Astrophys respectively)

Typeset in LATEXPrinted by Universitetsservice US AB Stockholm

Cover illustration Artistrsquos impression of two massive Kaluza-Klein parti-cles γ(1) which collide and annihilate into two gamma rays γ Under themagnifying glass it is shown how these massive particles in reality arejust ordinary photons circling a cylindrically-shaped extra dimensionccopy Andreas Hegert and Michael Gustafsson

A b s t r a c t

Measurements

C o n t e n t s

Abstract iii

Preface xi

viii Contents

Contents ix

Bibliography 159

P r e f a c e

xii Preface

Acknowledgments

Michael Gustafsson

P a r t I

C h a p t e r

Special Relativity

ds2 = dx2 + dy2 + dz2 (11)

ηαβ =

minus1 0 0 00 1 0 00 0 1 00 0 0 1

dτ2 =

αβ=0

αdxβ (16)

General Relativity

σβmicro (17)

Γαmicroν =1

Rmicroν minus1

c4Tmicroν (110)

1 minus kr2+ r2

dθ2 + sin2 θdφ2)

H2 equiv(

=8πGρ+ Λ

3minus k

a2(113)

dt(ρa3) = minusp d

dta3 (114)

ρc equiv3H2

8πG (115)

1 = Ω + ΩΛ + Ωk (116)

1 + z equiv λobs

λemit (118)

1 + z =a(tobs)

a(temit) (119)

dL equivradic

4πL (120)

dL = a0(1 + z)f

dzprime

H(zprime)

f(x) equiv

H(z) = H0

Ω0i (1 + z)minus3(1+wi) (122)

dH(t) =

dtprime

a(tprime)= a(t)

int r(t)

Dark Energy

ln ( aa0

) = ddt

ln ( 11+z

) = minus11+z

Dark Matter

v(r) =

r (124)

radicr Observa-

Dark Energy

Dark Matter

Baryonic Matter

〈σv〉 (125)

σv sim α2

M2WIMP

C h a p t e r

ρ(r) =ρ0

M200 = 2004πr3200

3ρc (22)

d ln(r) (23)

minus 2

rminus2

minus 1

rf =riMi(ri)

int ra

vrdr (28)

〈r〉 = r200A

rf =riMi(〈ri〉)

Mb(〈rf 〉) +MDM(〈rf 〉) (210)

001 01 110

DM only sim

DM in galaxy sim

r [r200

[ρc r

0 5 10 15 200

r [kpc]

065 07 075 08 08506

27 Nonsphericity

0 02 04 06 08 10

Axis Ratios

08DM (incl baryons)

R [kpc]10

DM (DM only)

R [kpc]

by the measure

T =a2 minus b2

a2 minus c2 (212)

Alignments

R [kpc]

2〈σv〉 ρ

2DM(r)

dΦγ(ψ)

〈σtotv〉8πm2

dN effγ

dℓ(ψ)ρ2DM

(ℓ) (214)

85 kpc

03 GeV cmminus3

dℓ(ψ)ρ2DM

dΦγdEγ

∆Ω 〈J〉∆Ω(0) = 013 b sr (217)

r [kpc]

larr rBH

darr ρ m

baryons

Einasto rarr

(Aw)=(11)

(Aw)=(05106)

10minus10

10minus5

C h a p t e r

and More

xM = xmicro yp (31)

(5) +m2)

Φ(xmicro y) =(

y +m2)

y sim y + 2πR (33)

Φ(xmicro y) =sum

Φ(n)(xmicro) =(

Φ(n)(xmicro) = 0 (35)

m2n = m2 +

rn ≪ R (37)

1 + y22 + + y2

rr ≫ R (38)

V (r) prop 1

1 + αeminusrλ)

Rn (310)

d4+nxradic

minusg[

2partM φ part

M φminus m2

2φ2 minus λ4+n

φ(x y) =1radicVn

φ(~n)(x) exp

i~n middot ~yR

d4xradicminusg

partmicroφ(0))2

minus m2

2(φ(0))2

partmicroφ(~n))(

minusm2~nφ

(~n)φ(~n)lowast]

minus λ4

4(φ(0))4 minus λ4

4(φ(0))2

m2~n = m2 +

R2 (314)

λ4 =λ4+n

Vn (315)

M2pl =

1 minus eminus2wL)

τp sim 1035 years

500 GeV

)12(ΛcutR

C h a p t e r

Extra Dimensions

16πG4+n

d4+nxradic

minusgR (41)

d4xradicminusg

16πGR[g(0)

microν ] +

G =G4+n

Vn(43)

λ =λ4+n

Vn (315)

d4+nxradic

minusg(

RAB minus 1

2R gAB = κ2Tmatter

b+n(nminus 1)

= κ2ρ (48a)

b+n(nminus 1)

= κ2pa (48b)

b+ (nminus 1)

= constant

p(i)a = w(i)

(i)b ρ(i) (410)

) (411)

i ρ(i) and pab =

i w(i)abρ

(i) With a multi-

ρ(i)(

1 minus 3w(i)a + 2w

(i)b ) = 0 Thus

in summary

(i)b =

3w(i)a minus 1

2 (413)

(w(i)a = w

wa = 0 wb =1

10 15 2000

d4xradicminusgbn

2Λ + 2κ2Lmatter

d4xradicminusg

2partmicroΦpart

Φ equivradic

n(n+ 2)

2κ2ln b (418)

eminusradic

2(n+2)n

κΦ +1

Λ minus κ2Lmatter

eminusradic

2nn+2 κΦ (419)

Φ = minus part

partΦVeff (420)

bminus(n+2) +sum

a +2w(i)b

) (421)

a )bminusn(1+w(i)b

) =κ2

(i)0 equiv ρ

(i)0 Vn equiv

ρ(i)0 κ2κ2

2(Φ minus Φ0)

part2Φ

(i)a and w

radic2(1minus3w

(i)a +2w

(2+n)κ2∆ρ(i)

C h a p t e r

Dimensions

51 Compactification

Φ(xmicro y) =

infinsum

nRy (51)

micro) +

infinsum

Φ(n)even cos

(n)odd sin

R (52)

Rrarr cos

n(y + πR)

R= (minus1)n cos

R (53a)

Rrarr sin

n(y + πR)

R= (minus1)n sin

R (53b)

53 The Lagrangian

Gauge Bosons

PZ2 Armicro(x

gpartmicroα

Aimicro(0)

(x) +1radicπR

infinsum

Aimicro(n)

(xmicro) cosny

R (58a)

Ai5(x) =1radicπR

infinsum

Ai5(n)

(x) sinny

R (58b)

Lgauge = minus1

MN minus 1

MNF rMN (59)

+ gǫijkAjMAk

N (510b)

gauge sup minus1

int 2πR

dy FMNFMN

= minus1

(0)micro

minus1

infinsum

(n)micro

infinsum

partmicroB(n)5 +

RB(n)micro

partmicroB(n)5 +

RB(n)micro

R appears The

(n)5 can be removed

g equiv 1radic2πR

g (512)

The Higgs Sector

φ equiv 1radic2

χ2 + iχ1

H minus iχ3

φ(x) =1radic2πR

φ(0)(x) +1radicπR

infinsum

φ(n)(x) cosny

R (514)

minus iY gYBM (515)

0 11 0

0 minusii 0

1 00 minus1

|φ|2 =minusmicro2

2λ2equiv v2

2 (519)

WplusmnM

= 1radic2

A1M∓ iA2

with mass mW =gv

2 (520a)

(520b)

AM = sWA3M

+ cWBM (521)

g2 + g2Y

cW equiv cos θW =g

g2 + g2Y

L(kin)Higgs sup

partMh)2

partMχ3 minusmZZM

)2+∣

Lgaugefix = minus1

GY)2 minus

Gi)2 (524)

Gi =1radicξ

minusmWχi + part5A

(525a)

GY =1radicξ

sWmWχ3 + part5B

(525b)

L(kin)scalar =

infinsum

2h(n)2

(n)0 minus ξM

(n)minus minus ξM

(n)+ G

(n)minus

infinsum

(n)0 minusM

2a(n)0

(n)minus minusM

2a(n)+ a

(n)minus

(n)5 minus ξ

a0(n)=M (n)

χ3(n)+

Z(n)5 (527a)

aplusmn(n)

=M (n)

χplusmn(n)+

Wplusmn5

(n) (527b)

G0(n)=

χ3(n) minus M (n)

Z5(n) (527c)

Gplusmn(n)=

Wplusmn5

(n) (527d)

M(n)X equiv

M (n) =n

R (528)

(n)5 G

(n)micro and W

Ghosts

δαbcb (529)

gpartMα

δBM =1

partMαY (531)

δφ =

iαiσi

φ equiv 1radic2

δχ2 + iδχ1

δH minus iδχ3

δχ1 =1

(533a)

δχ2 =1

(533b)

δχ3 =1

(533c)

radicξ)12ca

minus caδGaδαb

cb = ca[

cb (534)

h minusχ3 χ2 χ2

g2 0 0 00 g2 0 00 0 g2 minusggY

0 0 minusggY g2Y

L(kin)ghost =

infinsum

M (n) =

20 0 0

0 M(n)W

0 0 M(n)W

2minus 1

4v2ggY

0 0 minus 14v

2ggY14v

radicξM

(n)Z and

radicξM (n)

Fermions

γ5ψ(0)d = ψ

(0)d and γ5ψ(0)

ΓM ΓN = 2ηMN (538)

PRL equiv 1

1 plusmn γ5)

ψd =1radic2πR

ψ(0)dL

+1radicπR

infinsum

ψ(n)dL

(543a)

ψs =1radic2πR

ψ(0)sR

+1radicπR

infinsum

ψ(n)sR

R+ ψ(n)

(543b)

Lfermion = i(

ψdU ψdD)

ψdUψdD

6D equiv ΓM

partM minus igArM

minus iY gYBM

ψsD minus λU[

ψsU + hc (546)

EW= (λUD v)

radic2

cdD ψ

tdD) (547)

jdU and ψprimei

6Dij equiv ΓM

int 2πR

MpartMψs)

infinsum

ψ(n)d

ψ(n)d + ψ(n)

ψ(n)s

ψ(0) equiv ψ(0)dL

+ ψ(0)sR ψ

(n)d equiv ψ

+ ψ(n)dR ψ(n)

s equiv ψ(n)sL

+ ψ(n)sR (551)

ψ(n)d ψ

R+ δm

(n)d mEW

mEW minus[ n

R+ δm

ψ(n)d

ψ(n)s

ξ(n)d

ξ(n)s

1 00 minusγ5

cosα(n) sinα(n)

ψ(n)d

ψ(n)s

tan 2α(n) =2mEW

m(n)ds = plusmn1

(n)d minus δm(n)

(n)d + δm

(n)s )]2

+m2EW (555)

Φ(xmicro y) =1

p2 minus p2y

p2 minusm2 (558)

e(1) rarr e(0) + γ(1) (559)

1R2+m2

xmicro

phys + Z5Zm2(n) (560)

56 Mass Spectrum

)2+ δm2

B(n) + 14v

minus 14v

)2+ δm2

A3(n) + 14v

tan 2θ(n) =v2ggY

B(n) + v2

4 (g2 minus g2Y)] (562)

(a) Tree level

bdbsτdτsντ

(b) One-loop level

γaplusmna0H0

ZW plusmn

τdντ

bs bdts

C h a p t e r

61 Relic Density

X(n)i X

X(n)i xj harr X

(n)k xl (61b)

X(m)i rarr X

(n)j xk (61c)

n2 minus (neq)2]

where n =sum

≃ gi

(minusmX

〈σeffv〉 =sum

〈σijvij〉neqi

neq(64)

〈σeffv〉 (65)

i neqj prop

Rminus1[TeV]

05 1 15 2 25 3

B(1)B(1)

B(1) g(1) ∆g(1) = 2

X(1)(gX(1) + gB(1)) (66)

Direct Detection

Indirect Detection

Gamma-Ray Continuum

ℓ+ℓ+ℓ+

ℓminusℓminus

ℓ(1)

γγγ

dN ℓγ

(x2 minus 2x+ 2)

m2B(1)

(1 minus x)

dN ℓγ

dxpropint θmax

dθθ3

θ2 +m2ℓE

]2 asymp 1

2ln(E2

pm2ℓ) (610)

radics2 minus Eγ (

dσ(χχ rarr XXγ)

dxasymp αQ2

πFX(x) ln

s(1 minus x)

Ff(x) =1 + (1 minus x)2

x (612)

effγdx

x equiv EγmB(1)

Fs(x) =1 minus x

x (613)

dN effγ dx equiv

κidNiγdx (614)

Gamma Line Signal

M = ǫmicro1

1 (p1)ǫmicro2

2 (p2)ǫmicro3

3 (p3)ǫmicro4

ψ(0)ψ(0)

ξ(1)ds

v (617)

m4B(1)

pmicro1

3 pmicro2

m2B(1)

4 pmicro4 +B3

m2B(1)

4 pmicro3 +B4

m2B(1)

3 pmicro4

m2B(1)

3 pmicro3 +B6

m2B(1)

3 pmicro2

q21 minusm21

q22 minusm22

q23 minusm23

q24 minusm24

) (619)

q1 = q q2 = q+k1 q3 = q+k1 +k2 q4 = q+k1 +k2 +k3 (620)

(σv)γγ =α2

EMg4eff

144πm2B(1)

(621)where

g2eff equiv

Q2(Y 2s + Y 2

d ) =52

9 (622)

101 102 105 11 12

mξ(1)mB(1)

(σv) γγ

[10minus

aG aGB (1)

Hψ(0)

WB (1)

aGB (1)

dΦexp

int infin

Eprime

radic2πσEprime

Eγ [TeV]

E2 γdΦγdEγ

[mminus

2sminus

01 1 10

10minus9

10minus8

10minus7

Eγ [GeV]

dΦγdEγ

[10minus

minus2sminus

minus1]

780 790 800 810

C h a p t e r

Gamma-Ray Signal

71 Supersymmetry

Some Motivations

The Neutralino

3 +N13H01 +N14H

02 (71)

χminus1

dxequivd(σv)WWγdx

(σv)WW≃ αem

(1 minus x)x

+ δ2(

(2 minus x)3

+ δ4(

x(xminus 1)

(2 minus x)2+

(2 minus x)3

0 05 1001

x equiv Eγmχ

dNW γdx

χplusmn1

) Hig-

Zh Wplusmn Ωχh2

32 15 32 32 00 100 150 151 092 039 012

01 05 1 201

Eγ [TeV]

d(vσ) γdEγ

[10minus

2sminus

minus1]

06 1 22

[10minus

2sminus

minus1]

Eγ [TeV]

0minus

3sminus

01 05 1 2

C h a p t e r

fE2cm This is

can be written as

2 |H2|2 + λ1 |H1|4 + λ2 |H2|4 (83)

dagger1H2)

H1 =1radic2

G+radic

2v + h+ iG0

H2 =1radic2

H0 + iA0

m2h = minus2micro2

m2H0 = micro2

2 + (λ3 + λ4 + λ5)v2

m2A0 = micro2

2 + (λ3 + λ4 minus λ5)v2

m2Hplusmn = micro2

2 + λ3v2 (86)

radic2

cos θw(mW ))

∆T asymp 1

mh= 1141000 GeV

minus04minus04

minus02

More Constraints

λ12 gt 0

λ1λ2 (89)

Wminus(Z)Wminus(Z)

Wminus(Z)

W +(Z)W +(Z)

W +(Z)Hplusmn(A0)

f(νf )f(νf )

λL λ2L λLλ1

g2 g2 λLg

λLyf g2

83 Gamma Rays

Continuum

vrelσff =Ncπα

sin4 θWm4W

(1 minus 4m2f

(sminusm2h)

2 +m2hΓ

Gamma-Ray Lines

g4 g4 g4

λLg3 λLg3

x = EγmH0

x2dNγdx

10minus4

10minus2

001 01 1

τ+τminus

EGRETDW=210-3

HESSDW=10-5

GLAST sensitivity

50 GeV boost ~104

70 GeV boost ~100

2 γΦγ

minus2sminus

log(Eγ [GeV])

-10-1 0 1 2 3 4

WIMP Mass [GeV]

H0 H0 rarr Zγ

H0 H0 rarr γγ

rarr Zγ γγ

40 50 60 70 80

C h a p t e r

Observed

92 The Data

bremss πο

Energy [GeV]

E2times

minus2sminus

minus1]

10minus7

10minus6

10minus5

10minus4

bremssπο

Energy [GeV]

10minus7

10minus6

10minus5

10minus4

93 The Claim

χ2 356χ2 (bg only) 17887

Energy [GeV]

E2times

minus2sminus

minus1]

10minus5

10minus4

10minus1 1 10 102

minus10

x [kpc]

minus5 0 50

z [kpc]

OuterSunInnerCenter

minus10

x [kpc]

kpc [M

BESS 98 1σ

BESS 98 2σ

T = 040-056 GeV

χ2 lt 10

10 lt χ2 lt 25

25 lt χ2 lt 60

60 lt χ2

C h a p t e r

Summary and Outlook

A p p e n d i x

(n)plusmn and A

(n)5 at

given by

for scalars i

for vectors minusi

q2 minusm2 + iǫ

= swAM + cwZM (A4a)

Z(n)5 =

a(n)0 minus M (n)

G(n)0 (A5a)

χ3 (n) =M (n)

a(n)0 +

G(n)0 (A5b)

Wplusmn (n)5 =

G(n)plusmn (A5c)

χplusmn (n) =M (n)

a(n)plusmn +

G(n)plusmn (A5d)

ψ(n)s = sinα(n)ξ

ψ(n)d = cosα(n)ξ

A2 Vertex Rules

M(1)X equiv

M (1)2 +m2X (A7d)

mZ = mWcw (A8a)

gY = ecw (A8b)

g = esw (A8c)

λ =g2m2

V micro2

q3V ν3

q1V ρ1 igV1V2V3

microηρν]

gA(0)W

(10)+ W

(10)minus

= minuse (A9)

gZ(0)W

(01)+ W

(01)minus

= minuscw g (A10)

(1)3 W

(01)+ W

(10)minus

= minusg (A11)

V micro1

gZ(0)Z(0)h(0) =g

cwmZ (A12)

(0)+ W

gZ(0)W

(1)plusmn a

(1)∓

= plusmnig mZ

gZ(0)A3 (1)h(1) = gmZ (A15)

(0)plusmn A3 (1)a

(1)∓

= ∓ig mW

(0)plusmn B(1)a

(1)∓

= ∓igY mW

(0)plusmn W

(1)∓ a

= plusmnig mW

(0)plusmn W

(1)∓ h(1) = gmW (A20)

gA3 (1)A3 (1)h(0) = gmW (A21)

gB(1)B(1)h(0) = gs2wc2w

mW (A23)

(1)+ W

gA(0)W

gZ(0)W

gA(0)W

(1)plusmn G

(1)∓

= ∓ieM (1)W (A27)

(0)plusmn B(1)G

(1)∓

= ∓igY mW

(1)plusmn B(1)χ∓ (0) = ∓igY mW (A29)

V micro

gA(0)a

(1)+ a

(1)minus

= e (A30)

gZ(0)a

(1)+ a

(1)minus

2(gcw minus gY sw)

M (1)2

2 + gcwmW

2 (A31)

gZ(0)h(1)a

= minusi g

(0)plusmn a∓(1)a

= ∓g2

M (1)2

M(1)W M

1 minus 2m2

(0)plusmn a∓(1)h(1) = i

gA3 (1)a

(1)0 h(0) = i

gA3 (1)G

(1)0 h(0) = i

gB(1)a

(1)plusmn a

(1)∓ h(0) = i

gA(0)χ+ (0)χminus (0) = e (A39)

minus gY

gZ(0)χ3 (0)h(0) = ig

2cw(A41)

(0)plusmn χ∓ (0)h(0) = i

2(A42)

(0)plusmn χ∓ (0)χ3 (0) = ∓g

2(A43)

gA(0)G

(1)+ G

(1)minus

= e (A44)

gB(1)χplusmn(0)a

(1)∓

= plusmngY

gB(1)χplusmn(0)G

(1)∓

= plusmngY

2(A47)

gB(1)G

(1)plusmn G

(1)∓ h(0) = i

S3 igS1S2S3

gh(0)h(01)h(01) = minus3

gh(0)a

(1)+ a

(1)minus

= minusgmW

M (1)2

gh(0)a

(1)0 a

= minus g

M (1)2

gh(0)χ3 (0)χ3 (0) = minusg2

gh(0)a

(1)plusmn G

(1)∓

2M (1)

1 minus m2h

gh(0)G

(1)plusmn G

(1)∓

= minusg2mW

2 (A56)

gA(0)ξ(0)ξ(0) = Qe (A57)

gA(0)ξ

(1)sdξ(1)sd

= Qe (A58)

gZ(0)ξ

(1)dξ(1)d

gZ(0)ξ

(1)s ξ

gZ(0)ξ

(1)s ξ

(1)3 ξ(0)ξ

gB(1)ξ(0)ξ

(0)+ U

(0)i D

=gradic2VijPL (A67)

(0)+ U

(1)diD

=gradic2Vij cosα

cosα(1)Dj

(0)+ U

(1)siD

=gradic2Vij sinα

sinα(1)Dj

(0)+ U

(1)diD

=gradic2Vij cosα

sinα(1)Djγ5 (A70)

(0)+ U

(1)siD

=gradic2Vij sinα

cosα(1)Djγ5 (A71)

(1)+ U

(0)i D

=gradic2Vij cosα

(1)DjPL (A72)

(1)+ U

(0)i D

(1)DjPL (A73)

(1)+ U

(1)diD

=gradic2Vij cosα

(1)UiPL (A74)

(1)+ U

(1)siD

(1)UiPL (A75)

S igSξ1ξ2

gh(0)ξ

(1)dξ(1)d

= minusg mξ

gh(0)ξ

(1)s ξ

= minusg mξ

gh(0)ξ

(1)dξ(1)s

= minusg mξ

γ5 (A79)

gh(1)ξ(0)ξ

= minusg mξ

gh(1)ξ(0)ξ

= gmξ

ga(1)0 ξ(0)ξ

minusigT3mξ

ga(1)0 ξ(0)ξ

+igT3mξ

ga(1)+ U

(0)i D

= minusi gradic2

cosα(1)DjPL

+igradic2

ga(1)+ U

(0)i D

= igradic2

sinα(1)DjPL

+igradic2

ga(1)+ U

(1)diD

= minusi gradic2

sinα(1)UiPL

minusi gradic2

ga(1)+ U

(1)siD

= igradic2

cosα(1)UiPL

minusi gradic2

dagger =

A(1) micro

c∓(1)

c∓(1) q

= plusmnie qmicro

c∓(1)

= minusi g2mWξ

V micro1

igV1V2V3V4 middot(

(0)minus W

(0)+ W

= g2 (A88)

(1)minus W

(1)+ W

2g2 (A89)

(10)minus W

(01)+ W

= g2 (A90)

(10)minus W

(01)minus W

(10)+ W

= g2 (A91)

gZ(0)Z(0)W

(0)minus W

= minusg2c2w (A92)

gZ(0)Z(0)W

(1)minus W

= minusg2c2w (A93)

gA(0)A(0)W

(0)+ W

(0)minus

= minuse2 (A94)

gA(0)A(0)W

(1)+ W

(1)minus

= minuse2 (A95)

gZ(0)A(0)W

(0)+ W

(0)minus

= minusegcw (A96)

gZ(0)A(0)W

(1)+ W

(1)minus

= minusegcw (A97)

(1)3 A

(1)3 W

(1)+ W

(1)minus

= minus3

2g2 (A98)

(1)3 A

(1)3 W

(0)+ W

(0)minus

= minusg2 (A99)

(1)3 Z(0)W

(01)+ W

(10)minus

= minusg2cw (A100)

V micro1

iηmicroνgV1V2S1S2

gZ(0)Z(0)h(0)h(0) =g2

2c2w(A101)

(0)plusmn W

(0)∓ h(0)h(0) =

2(A102)

gB(1)B(1)h(1)h(1) =3g2

4(A103)

gB(1)B(1)a

(1)0 a

M (1)2

2 (A104)

(1)3 A

(1)3 h(1)h(1) =

4(A105)

(1)3 A

(1)3 a

(1)0 a

M (1)2

2 (A106)

(1)plusmn W

(1)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A107)

gB(1)B(1)a

(1)+ a

(1)minus

M (1)2

2 (A108)

(1)3 A

(1)3 a

(1)plusmn a

(1)∓

2+ 13m2

2 (A109)

(1)plusmn W

(1)∓ h(1)h(1) =

4(A110)

(1)plusmn W

(1)∓ a

(1)0 a

2+ 13m2

2 (A111)

(1)∓ W

(1)plusmn a

(1)∓

2minus 13m2

2 (A112)

gB(1)A

(1)3 h(1)h(1) = minus3gY g

4(A113)

gB(1)A

(1)3 a

(1)0 a

= minus3gY g

M (1)2

2 (A114)

gB(1)A

(1)3 a

(1)plusmn a

(1)∓

=3gY g

M (1)2

2 (A115)

gB(1)W

(1)plusmn h(1)a

(1)∓

= ∓i3gY g

(A116)

gB(1)W

(1)plusmn a

(1)0 a

(1)∓

= minus3gY g

M (1)2

M(1)W M

(A117)

(1)3 W

(1)plusmn a

(1)0 a

(1)∓

= minusg2

M(1)W M

(A118)

gB(1)B(1)h(0)h(0) =g2

2(A119)

gZ(0)Z(0)h(1)h(1) =g2

2c2w(A120)

(1)3 A

(1)3 h(0)h(0) =

2(A121)

gZ(0)A

(1)3 h(0)h(1) =

2cw(A122)

gZ(0)Z(0)a

(1)0 a

M (1)2

2 (A123)

(0)plusmn W

(0)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A124)

gA(0)A(0)a

(1)plusmn a

(1)∓

= 2e2 (A125)

gZ(0)Z(0)a

(1)plusmn a

(1)∓

4c4wm2W

2M(1)W

2 (A126)

(01)+ W

(01)minus h(10)h(10) =

2(A127)

(01)+ W

(10)minus h(01)h(10) =

2(A128)

(0)plusmn W

(0)∓ a

(1)0 a

2+ 3m2

2 (A129)

(0)plusmn W

(0)∓ a

(1)plusmn a

(1)∓

2 (A130)

gB(1)A

2(A131)

gB(1)Z(0)h(1)h(0) =egY

2cw(A132)

gA(0)Z(0)a

(1)plusmn a

(1)∓

2c2wm2W

2 (A133)

gA(0)W

(01)plusmn h(10)a

(1)∓

= ∓i e2

(A134)

gB(1)W

(0)plusmn h(0)a

(1)∓

= ∓i e2

(A135)

gA(0)W

(0)plusmn a

(1)0 a

(1)∓

= minus e2

mW +M(1)W

M(1)W M

(A136)

gZ(0)W

(01)plusmn h(10)a

(1)∓

= plusmni e2

(A137)

gZ(0)W

(0)plusmn a

(1)0 a

(1)∓

2s2wcw

M(1)W M

(A138)

gA(0)A(0)G

(1)+ G

(1)minus

= 2e2 (A139)

gB(1)B(1)χ

(0)+ χ

(0)minus

2(A140)

gB(1)B(1)G

(1)+ G

(1)minus

2 (A141)

gB(1)B(1)G

(1)∓ a

(1)plusmn

mWM(1)

2 (A142)

gB(1)B(1)χ3(0)χ3(0) =g2

2(A143)

gB(1)B(1)G

(1)0 G

2 (A144)

gB(1)B(1)G

(1)0 a

mZM(1)

2 (A145)

gB(1)A(0)G

(1)plusmn χ

(0)∓

= plusmniegY

(A146)

gB(1)A(0)a

(1)plusmn χ

(0)∓

= plusmniegY

(A147)

(1)plusmn A(0)h(0)G

(1)∓

= ∓i eg2

(A148)

igS1S2S3S4

gh(0)h(0)h(0)h(0) = minus6λ (A149)

gh(1)h(1)h(1)h(1) = minus9λ (A150)

ga(1)0 a

(1)0 a

= minus3g2c2wm2ZM (1)2 + 9λM (1)4

4 (A151)

gh(1)h(1)a

(1)0 a

= minusg2m2

Zc2w + 6λM (1)2

2M(1)Z

2 (A152)

gh(1)h(1)a

(1)+ a

(1)minus

= minusg2m2

W+ 6λM (1)2

2M(1)W

2 (A153)

ga(1)0 a

(1)0 a

(1)+ a

(1)minus

= minusg2m2

WM (1)2(1 + c4w)c4w + 6λM (1)4

2M(1)W

2 (A154)

ga(1)+ a

(1)+ a

(1)minus a

(1)minus

= minus2g2m2WM (1)2 + 6λM (1)4

2 (A155)

gh(1)h(1)h(0)h(0) = minusλ (A156)

gh(0)h(0)a

(1)0 a

= minusg2m2

Zc2w + 4λM (1)2

2M(1)Z

2 (A157)

gh(0)h(0)a

(1)+ a

(1)minus

= minusg2m2

W+ 4λM (1)2

2M(1)W

2 (A158)

160 Bibliography

(1999) 896ndash899 [astro-ph9807002]

Bibliography 161

(2006) 137ndash161 [hep-ph0602058]

162 Bibliography

(2007) 859 [astro-ph0703337]

(1999) 1147ndash1152 [astro-ph9903164]

Bibliography 163

(2004) 1039 [astro-ph0311231]

(1962) 748

164 Bibliography

Bibliography 165

(2005) 647ndash665 [astro-ph0408163]

166 Bibliography

Bibliography 167

(2007) 262 [astro-ph0611370]

(2003) 103003 [astro-ph0301551]

(2004) L23 [astro-ph0309621]

168 Bibliography

(2005) 081802 [hep-ex0411038]

Bibliography 169

(1987) 1087ndash1170

(1998) 263ndash272 [hep-ph9803315]

170 Bibliography

(2000) 096006 [hep-ph0006238]

(2003) 403 [hep-ph0205340]

Bibliography 171

172 Bibliography

(1998) 2025ndash2035 [gr-qc9804018]

(2002) 024036 [hep-th0110149]

Bibliography 173

(2004) 004 [hep-ph0310258]

174 Bibliography

Bibliography 175

(2007) 191ndash198 [astro-ph0608495]

radics = 18-TeV PhD

176 Bibliography

Bibliography 177

(1977) 1519

(2006) 123515 [astro-ph0607327]

178 Bibliography

(1979) 151

Bibliography 179

180 Bibliography

(2006) 015007 [hep-ph0603188]

Bibliography 181

(2007) 026 [hep-ph0703056]

(2007) 028 [hep-ph0612275]

182 Bibliography

Bibliography 183

(2008) 25ndash29 [arXiv07054311]

(2008) 864 [astro-ph0510520]

(1999) 203

184 Bibliography

(2007) 607ndash612 [astro-ph0612462]

P a r t I I

Scientific Papers

P a p e r I

P a p e r I I

P a p e r I I I

P a p e r I V

P a p e r V

P a p e r V I

P a p e r V I I

Abstract

Preface



Special Relativity

General Relativity


Evolving Universe

Initial Conditions


Dark Energy

Dark Matter








Simulation Setups



Nonsphericity

Axis Ratios

Alignments




Halo Substructure















Compactification

Kaluza-Klein Parity

The Lagrangian



Mass Spectrum


Relic Density



Direct Detection

Indirect Detection


Gamma-Ray Continuum

Gamma Line Signal



Supersymmetry

Some Motivations

The Neutralino








More Constraints


Gamma Rays

Continuum

Gamma-Ray Lines


Dark Matter Signals

The Data

The Claim

The Inconsistency



The Status to Date

Summary and Outlook



Vertex Rules

Bibliography


A b s t r a c t

Measurements

C o n t e n t s

Abstract iii

Preface xi

viii Contents

Contents ix

Bibliography 159

P r e f a c e

xii Preface

Acknowledgments

Michael Gustafsson

P a r t I

C h a p t e r

Special Relativity

ds2 = dx2 + dy2 + dz2 (11)

ηαβ =

minus1 0 0 00 1 0 00 0 1 00 0 0 1

dτ2 =

αβ=0

αdxβ (16)

General Relativity

σβmicro (17)

Γαmicroν =1

Rmicroν minus1

c4Tmicroν (110)

1 minus kr2+ r2

dθ2 + sin2 θdφ2)

H2 equiv(

=8πGρ+ Λ

3minus k

a2(113)

dt(ρa3) = minusp d

dta3 (114)

ρc equiv3H2

8πG (115)

1 = Ω + ΩΛ + Ωk (116)

1 + z equiv λobs

λemit (118)

1 + z =a(tobs)

a(temit) (119)

dL equivradic

4πL (120)

dL = a0(1 + z)f

dzprime

H(zprime)

f(x) equiv

H(z) = H0

Ω0i (1 + z)minus3(1+wi) (122)

dH(t) =

dtprime

a(tprime)= a(t)

int r(t)

Dark Energy

ln ( aa0

) = ddt

ln ( 11+z

) = minus11+z

Dark Matter

v(r) =

r (124)

radicr Observa-

Dark Energy

Dark Matter

Baryonic Matter

〈σv〉 (125)

σv sim α2

M2WIMP

C h a p t e r

ρ(r) =ρ0

M200 = 2004πr3200

3ρc (22)

d ln(r) (23)

minus 2

rminus2

minus 1

rf =riMi(ri)

int ra

vrdr (28)

〈r〉 = r200A

rf =riMi(〈ri〉)

Mb(〈rf 〉) +MDM(〈rf 〉) (210)

001 01 110

DM only sim

DM in galaxy sim

r [r200

[ρc r

0 5 10 15 200

r [kpc]

065 07 075 08 08506

27 Nonsphericity

0 02 04 06 08 10

Axis Ratios

08DM (incl baryons)

R [kpc]10

DM (DM only)

R [kpc]

by the measure

T =a2 minus b2

a2 minus c2 (212)

Alignments

R [kpc]

2〈σv〉 ρ

2DM(r)

dΦγ(ψ)

〈σtotv〉8πm2

dN effγ

dℓ(ψ)ρ2DM

(ℓ) (214)

85 kpc

03 GeV cmminus3

dℓ(ψ)ρ2DM

dΦγdEγ

∆Ω 〈J〉∆Ω(0) = 013 b sr (217)

r [kpc]

larr rBH

darr ρ m

baryons

Einasto rarr

(Aw)=(11)

(Aw)=(05106)

10minus10

10minus5

C h a p t e r

and More

xM = xmicro yp (31)

(5) +m2)

Φ(xmicro y) =(

y +m2)

y sim y + 2πR (33)

Φ(xmicro y) =sum

Φ(n)(xmicro) =(

Φ(n)(xmicro) = 0 (35)

m2n = m2 +

rn ≪ R (37)

1 + y22 + + y2

rr ≫ R (38)

V (r) prop 1

1 + αeminusrλ)

Rn (310)

d4+nxradic

minusg[

2partM φ part

M φminus m2

2φ2 minus λ4+n

φ(x y) =1radicVn

φ(~n)(x) exp

i~n middot ~yR

d4xradicminusg

partmicroφ(0))2

minus m2

2(φ(0))2

partmicroφ(~n))(

minusm2~nφ

(~n)φ(~n)lowast]

minus λ4

4(φ(0))4 minus λ4

4(φ(0))2

m2~n = m2 +

R2 (314)

λ4 =λ4+n

Vn (315)

M2pl =

1 minus eminus2wL)

τp sim 1035 years

500 GeV

)12(ΛcutR

C h a p t e r

Extra Dimensions

16πG4+n

d4+nxradic

minusgR (41)

d4xradicminusg

16πGR[g(0)

microν ] +

G =G4+n

Vn(43)

λ =λ4+n

Vn (315)

d4+nxradic

minusg(

RAB minus 1

2R gAB = κ2Tmatter

b+n(nminus 1)

= κ2ρ (48a)

b+n(nminus 1)

= κ2pa (48b)

b+ (nminus 1)

= constant

p(i)a = w(i)

(i)b ρ(i) (410)

) (411)

i ρ(i) and pab =

i w(i)abρ

(i) With a multi-

ρ(i)(

1 minus 3w(i)a + 2w

(i)b ) = 0 Thus

in summary

(i)b =

3w(i)a minus 1

2 (413)

(w(i)a = w

wa = 0 wb =1

10 15 2000

d4xradicminusgbn

2Λ + 2κ2Lmatter

d4xradicminusg

2partmicroΦpart

Φ equivradic

n(n+ 2)

2κ2ln b (418)

eminusradic

2(n+2)n

κΦ +1

Λ minus κ2Lmatter

eminusradic

2nn+2 κΦ (419)

Φ = minus part

partΦVeff (420)

bminus(n+2) +sum

a +2w(i)b

) (421)

a )bminusn(1+w(i)b

) =κ2

(i)0 equiv ρ

(i)0 Vn equiv

ρ(i)0 κ2κ2

2(Φ minus Φ0)

part2Φ

(i)a and w

radic2(1minus3w

(i)a +2w

(2+n)κ2∆ρ(i)

C h a p t e r

Dimensions

51 Compactification

Φ(xmicro y) =

infinsum

nRy (51)

micro) +

infinsum

Φ(n)even cos

(n)odd sin

R (52)

Rrarr cos

n(y + πR)

R= (minus1)n cos

R (53a)

Rrarr sin

n(y + πR)

R= (minus1)n sin

R (53b)

53 The Lagrangian

Gauge Bosons

PZ2 Armicro(x

gpartmicroα

Aimicro(0)

(x) +1radicπR

infinsum

Aimicro(n)

(xmicro) cosny

R (58a)

Ai5(x) =1radicπR

infinsum

Ai5(n)

(x) sinny

R (58b)

Lgauge = minus1

MN minus 1

MNF rMN (59)

+ gǫijkAjMAk

N (510b)

gauge sup minus1

int 2πR

dy FMNFMN

= minus1

(0)micro

minus1

infinsum

(n)micro

infinsum

partmicroB(n)5 +

RB(n)micro

partmicroB(n)5 +

RB(n)micro

R appears The

(n)5 can be removed

g equiv 1radic2πR

g (512)

The Higgs Sector

φ equiv 1radic2

χ2 + iχ1

H minus iχ3

φ(x) =1radic2πR

φ(0)(x) +1radicπR

infinsum

φ(n)(x) cosny

R (514)

minus iY gYBM (515)

0 11 0

0 minusii 0

1 00 minus1

|φ|2 =minusmicro2

2λ2equiv v2

2 (519)

WplusmnM

= 1radic2

A1M∓ iA2

with mass mW =gv

2 (520a)

(520b)

AM = sWA3M

+ cWBM (521)

g2 + g2Y

cW equiv cos θW =g

g2 + g2Y

L(kin)Higgs sup

partMh)2

partMχ3 minusmZZM

)2+∣

Lgaugefix = minus1

GY)2 minus

Gi)2 (524)

Gi =1radicξ

minusmWχi + part5A

(525a)

GY =1radicξ

sWmWχ3 + part5B

(525b)

L(kin)scalar =

infinsum

2h(n)2

(n)0 minus ξM

(n)minus minus ξM

(n)+ G

(n)minus

infinsum

(n)0 minusM

2a(n)0

(n)minus minusM

2a(n)+ a

(n)minus

(n)5 minus ξ

a0(n)=M (n)

χ3(n)+

Z(n)5 (527a)

aplusmn(n)

=M (n)

χplusmn(n)+

Wplusmn5

(n) (527b)

G0(n)=

χ3(n) minus M (n)

Z5(n) (527c)

Gplusmn(n)=

Wplusmn5

(n) (527d)

M(n)X equiv

M (n) =n

R (528)

(n)5 G

(n)micro and W

Ghosts

δαbcb (529)

gpartMα

δBM =1

partMαY (531)

δφ =

iαiσi

φ equiv 1radic2

δχ2 + iδχ1

δH minus iδχ3

δχ1 =1

(533a)

δχ2 =1

(533b)

δχ3 =1

(533c)

radicξ)12ca

minus caδGaδαb

cb = ca[

cb (534)

h minusχ3 χ2 χ2

g2 0 0 00 g2 0 00 0 g2 minusggY

0 0 minusggY g2Y

L(kin)ghost =

infinsum

M (n) =

20 0 0

0 M(n)W

0 0 M(n)W

2minus 1

4v2ggY

0 0 minus 14v

2ggY14v

radicξM

(n)Z and

radicξM (n)

Fermions

γ5ψ(0)d = ψ

(0)d and γ5ψ(0)

ΓM ΓN = 2ηMN (538)

PRL equiv 1

1 plusmn γ5)

ψd =1radic2πR

ψ(0)dL

+1radicπR

infinsum

ψ(n)dL

(543a)

ψs =1radic2πR

ψ(0)sR

+1radicπR

infinsum

ψ(n)sR

R+ ψ(n)

(543b)

Lfermion = i(

ψdU ψdD)

ψdUψdD

6D equiv ΓM

partM minus igArM

minus iY gYBM

ψsD minus λU[

ψsU + hc (546)

EW= (λUD v)

radic2

cdD ψ

tdD) (547)

jdU and ψprimei

6Dij equiv ΓM

int 2πR

MpartMψs)

infinsum

ψ(n)d

ψ(n)d + ψ(n)

ψ(n)s

ψ(0) equiv ψ(0)dL

+ ψ(0)sR ψ

(n)d equiv ψ

+ ψ(n)dR ψ(n)

s equiv ψ(n)sL

+ ψ(n)sR (551)

ψ(n)d ψ

R+ δm

(n)d mEW

mEW minus[ n

R+ δm

ψ(n)d

ψ(n)s

ξ(n)d

ξ(n)s

1 00 minusγ5

cosα(n) sinα(n)

ψ(n)d

ψ(n)s

tan 2α(n) =2mEW

m(n)ds = plusmn1

(n)d minus δm(n)

(n)d + δm

(n)s )]2

+m2EW (555)

Φ(xmicro y) =1

p2 minus p2y

p2 minusm2 (558)

e(1) rarr e(0) + γ(1) (559)

1R2+m2

xmicro

phys + Z5Zm2(n) (560)

56 Mass Spectrum

)2+ δm2

B(n) + 14v

minus 14v

)2+ δm2

A3(n) + 14v

tan 2θ(n) =v2ggY

B(n) + v2

4 (g2 minus g2Y)] (562)

(a) Tree level

bdbsτdτsντ

(b) One-loop level

γaplusmna0H0

ZW plusmn

τdντ

bs bdts

C h a p t e r

61 Relic Density

X(n)i X

X(n)i xj harr X

(n)k xl (61b)

X(m)i rarr X

(n)j xk (61c)

n2 minus (neq)2]

where n =sum

≃ gi

(minusmX

〈σeffv〉 =sum

〈σijvij〉neqi

neq(64)

〈σeffv〉 (65)

i neqj prop

Rminus1[TeV]

05 1 15 2 25 3

B(1)B(1)

B(1) g(1) ∆g(1) = 2

X(1)(gX(1) + gB(1)) (66)

Direct Detection

Indirect Detection

Gamma-Ray Continuum

ℓ+ℓ+ℓ+

ℓminusℓminus

ℓ(1)

γγγ

dN ℓγ

(x2 minus 2x+ 2)

m2B(1)

(1 minus x)

dN ℓγ

dxpropint θmax

dθθ3

θ2 +m2ℓE

]2 asymp 1

2ln(E2

pm2ℓ) (610)

radics2 minus Eγ (

dσ(χχ rarr XXγ)

dxasymp αQ2

πFX(x) ln

s(1 minus x)

Ff(x) =1 + (1 minus x)2

x (612)

effγdx

x equiv EγmB(1)

Fs(x) =1 minus x

x (613)

dN effγ dx equiv

κidNiγdx (614)

Gamma Line Signal

M = ǫmicro1

1 (p1)ǫmicro2

2 (p2)ǫmicro3

3 (p3)ǫmicro4

ψ(0)ψ(0)

ξ(1)ds

v (617)

m4B(1)

pmicro1

3 pmicro2

m2B(1)

4 pmicro4 +B3

m2B(1)

4 pmicro3 +B4

m2B(1)

3 pmicro4

m2B(1)

3 pmicro3 +B6

m2B(1)

3 pmicro2

q21 minusm21

q22 minusm22

q23 minusm23

q24 minusm24

) (619)

q1 = q q2 = q+k1 q3 = q+k1 +k2 q4 = q+k1 +k2 +k3 (620)

(σv)γγ =α2

EMg4eff

144πm2B(1)

(621)where

g2eff equiv

Q2(Y 2s + Y 2

d ) =52

9 (622)

101 102 105 11 12

mξ(1)mB(1)

(σv) γγ

[10minus

aG aGB (1)

Hψ(0)

WB (1)

aGB (1)

dΦexp

int infin

Eprime

radic2πσEprime

Eγ [TeV]

E2 γdΦγdEγ

[mminus

2sminus

01 1 10

10minus9

10minus8

10minus7

Eγ [GeV]

dΦγdEγ

[10minus

minus2sminus

minus1]

780 790 800 810

C h a p t e r

Gamma-Ray Signal

71 Supersymmetry

Some Motivations

The Neutralino

3 +N13H01 +N14H

02 (71)

χminus1

dxequivd(σv)WWγdx

(σv)WW≃ αem

(1 minus x)x

+ δ2(

(2 minus x)3

+ δ4(

x(xminus 1)

(2 minus x)2+

(2 minus x)3

0 05 1001

x equiv Eγmχ

dNW γdx

χplusmn1

) Hig-

Zh Wplusmn Ωχh2

32 15 32 32 00 100 150 151 092 039 012

01 05 1 201

Eγ [TeV]

d(vσ) γdEγ

[10minus

2sminus

minus1]

06 1 22

[10minus

2sminus

minus1]

Eγ [TeV]

0minus

3sminus

01 05 1 2

C h a p t e r

fE2cm This is

can be written as

2 |H2|2 + λ1 |H1|4 + λ2 |H2|4 (83)

dagger1H2)

H1 =1radic2

G+radic

2v + h+ iG0

H2 =1radic2

H0 + iA0

m2h = minus2micro2

m2H0 = micro2

2 + (λ3 + λ4 + λ5)v2

m2A0 = micro2

2 + (λ3 + λ4 minus λ5)v2

m2Hplusmn = micro2

2 + λ3v2 (86)

radic2

cos θw(mW ))

∆T asymp 1

mh= 1141000 GeV

minus04minus04

minus02

More Constraints

λ12 gt 0

λ1λ2 (89)

Wminus(Z)Wminus(Z)

Wminus(Z)

W +(Z)W +(Z)

W +(Z)Hplusmn(A0)

f(νf )f(νf )

λL λ2L λLλ1

g2 g2 λLg

λLyf g2

83 Gamma Rays

Continuum

vrelσff =Ncπα

sin4 θWm4W

(1 minus 4m2f

(sminusm2h)

2 +m2hΓ

Gamma-Ray Lines

g4 g4 g4

λLg3 λLg3

x = EγmH0

x2dNγdx

10minus4

10minus2

001 01 1

τ+τminus

EGRETDW=210-3

HESSDW=10-5

GLAST sensitivity

50 GeV boost ~104

70 GeV boost ~100

2 γΦγ

minus2sminus

log(Eγ [GeV])

-10-1 0 1 2 3 4

WIMP Mass [GeV]

H0 H0 rarr Zγ

H0 H0 rarr γγ

rarr Zγ γγ

40 50 60 70 80

C h a p t e r

Observed

92 The Data

bremss πο

Energy [GeV]

E2times

minus2sminus

minus1]

10minus7

10minus6

10minus5

10minus4

bremssπο

Energy [GeV]

10minus7

10minus6

10minus5

10minus4

93 The Claim

χ2 356χ2 (bg only) 17887

Energy [GeV]

E2times

minus2sminus

minus1]

10minus5

10minus4

10minus1 1 10 102

minus10

x [kpc]

minus5 0 50

z [kpc]

OuterSunInnerCenter

minus10

x [kpc]

kpc [M

BESS 98 1σ

BESS 98 2σ

T = 040-056 GeV

χ2 lt 10

10 lt χ2 lt 25

25 lt χ2 lt 60

60 lt χ2

C h a p t e r

Summary and Outlook

A p p e n d i x

(n)plusmn and A

(n)5 at

given by

for scalars i

for vectors minusi

q2 minusm2 + iǫ

= swAM + cwZM (A4a)

Z(n)5 =

a(n)0 minus M (n)

G(n)0 (A5a)

χ3 (n) =M (n)

a(n)0 +

G(n)0 (A5b)

Wplusmn (n)5 =

G(n)plusmn (A5c)

χplusmn (n) =M (n)

a(n)plusmn +

G(n)plusmn (A5d)

ψ(n)s = sinα(n)ξ

ψ(n)d = cosα(n)ξ

A2 Vertex Rules

M(1)X equiv

M (1)2 +m2X (A7d)

mZ = mWcw (A8a)

gY = ecw (A8b)

g = esw (A8c)

λ =g2m2

V micro2

q3V ν3

q1V ρ1 igV1V2V3

microηρν]

gA(0)W

(10)+ W

(10)minus

= minuse (A9)

gZ(0)W

(01)+ W

(01)minus

= minuscw g (A10)

(1)3 W

(01)+ W

(10)minus

= minusg (A11)

V micro1

gZ(0)Z(0)h(0) =g

cwmZ (A12)

(0)+ W

gZ(0)W

(1)plusmn a

(1)∓

= plusmnig mZ

gZ(0)A3 (1)h(1) = gmZ (A15)

(0)plusmn A3 (1)a

(1)∓

= ∓ig mW

(0)plusmn B(1)a

(1)∓

= ∓igY mW

(0)plusmn W

(1)∓ a

= plusmnig mW

(0)plusmn W

(1)∓ h(1) = gmW (A20)

gA3 (1)A3 (1)h(0) = gmW (A21)

gB(1)B(1)h(0) = gs2wc2w

mW (A23)

(1)+ W

gA(0)W

gZ(0)W

gA(0)W

(1)plusmn G

(1)∓

= ∓ieM (1)W (A27)

(0)plusmn B(1)G

(1)∓

= ∓igY mW

(1)plusmn B(1)χ∓ (0) = ∓igY mW (A29)

V micro

gA(0)a

(1)+ a

(1)minus

= e (A30)

gZ(0)a

(1)+ a

(1)minus

2(gcw minus gY sw)

M (1)2

2 + gcwmW

2 (A31)

gZ(0)h(1)a

= minusi g

(0)plusmn a∓(1)a

= ∓g2

M (1)2

M(1)W M

1 minus 2m2

(0)plusmn a∓(1)h(1) = i

gA3 (1)a

(1)0 h(0) = i

gA3 (1)G

(1)0 h(0) = i

gB(1)a

(1)plusmn a

(1)∓ h(0) = i

gA(0)χ+ (0)χminus (0) = e (A39)

minus gY

gZ(0)χ3 (0)h(0) = ig

2cw(A41)

(0)plusmn χ∓ (0)h(0) = i

2(A42)

(0)plusmn χ∓ (0)χ3 (0) = ∓g

2(A43)

gA(0)G

(1)+ G

(1)minus

= e (A44)

gB(1)χplusmn(0)a

(1)∓

= plusmngY

gB(1)χplusmn(0)G

(1)∓

= plusmngY

2(A47)

gB(1)G

(1)plusmn G

(1)∓ h(0) = i

S3 igS1S2S3

gh(0)h(01)h(01) = minus3

gh(0)a

(1)+ a

(1)minus

= minusgmW

M (1)2

gh(0)a

(1)0 a

= minus g

M (1)2

gh(0)χ3 (0)χ3 (0) = minusg2

gh(0)a

(1)plusmn G

(1)∓

2M (1)

1 minus m2h

gh(0)G

(1)plusmn G

(1)∓

= minusg2mW

2 (A56)

gA(0)ξ(0)ξ(0) = Qe (A57)

gA(0)ξ

(1)sdξ(1)sd

= Qe (A58)

gZ(0)ξ

(1)dξ(1)d

gZ(0)ξ

(1)s ξ

gZ(0)ξ

(1)s ξ

(1)3 ξ(0)ξ

gB(1)ξ(0)ξ

(0)+ U

(0)i D

=gradic2VijPL (A67)

(0)+ U

(1)diD

=gradic2Vij cosα

cosα(1)Dj

(0)+ U

(1)siD

=gradic2Vij sinα

sinα(1)Dj

(0)+ U

(1)diD

=gradic2Vij cosα

sinα(1)Djγ5 (A70)

(0)+ U

(1)siD

=gradic2Vij sinα

cosα(1)Djγ5 (A71)

(1)+ U

(0)i D

=gradic2Vij cosα

(1)DjPL (A72)

(1)+ U

(0)i D

(1)DjPL (A73)

(1)+ U

(1)diD

=gradic2Vij cosα

(1)UiPL (A74)

(1)+ U

(1)siD

(1)UiPL (A75)

S igSξ1ξ2

gh(0)ξ

(1)dξ(1)d

= minusg mξ

gh(0)ξ

(1)s ξ

= minusg mξ

gh(0)ξ

(1)dξ(1)s

= minusg mξ

γ5 (A79)

gh(1)ξ(0)ξ

= minusg mξ

gh(1)ξ(0)ξ

= gmξ

ga(1)0 ξ(0)ξ

minusigT3mξ

ga(1)0 ξ(0)ξ

+igT3mξ

ga(1)+ U

(0)i D

= minusi gradic2

cosα(1)DjPL

+igradic2

ga(1)+ U

(0)i D

= igradic2

sinα(1)DjPL

+igradic2

ga(1)+ U

(1)diD

= minusi gradic2

sinα(1)UiPL

minusi gradic2

ga(1)+ U

(1)siD

= igradic2

cosα(1)UiPL

minusi gradic2

dagger =

A(1) micro

c∓(1)

c∓(1) q

= plusmnie qmicro

c∓(1)

= minusi g2mWξ

V micro1

igV1V2V3V4 middot(

(0)minus W

(0)+ W

= g2 (A88)

(1)minus W

(1)+ W

2g2 (A89)

(10)minus W

(01)+ W

= g2 (A90)

(10)minus W

(01)minus W

(10)+ W

= g2 (A91)

gZ(0)Z(0)W

(0)minus W

= minusg2c2w (A92)

gZ(0)Z(0)W

(1)minus W

= minusg2c2w (A93)

gA(0)A(0)W

(0)+ W

(0)minus

= minuse2 (A94)

gA(0)A(0)W

(1)+ W

(1)minus

= minuse2 (A95)

gZ(0)A(0)W

(0)+ W

(0)minus

= minusegcw (A96)

gZ(0)A(0)W

(1)+ W

(1)minus

= minusegcw (A97)

(1)3 A

(1)3 W

(1)+ W

(1)minus

= minus3

2g2 (A98)

(1)3 A

(1)3 W

(0)+ W

(0)minus

= minusg2 (A99)

(1)3 Z(0)W

(01)+ W

(10)minus

= minusg2cw (A100)

V micro1

iηmicroνgV1V2S1S2

gZ(0)Z(0)h(0)h(0) =g2

2c2w(A101)

(0)plusmn W

(0)∓ h(0)h(0) =

2(A102)

gB(1)B(1)h(1)h(1) =3g2

4(A103)

gB(1)B(1)a

(1)0 a

M (1)2

2 (A104)

(1)3 A

(1)3 h(1)h(1) =

4(A105)

(1)3 A

(1)3 a

(1)0 a

M (1)2

2 (A106)

(1)plusmn W

(1)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A107)

gB(1)B(1)a

(1)+ a

(1)minus

M (1)2

2 (A108)

(1)3 A

(1)3 a

(1)plusmn a

(1)∓

2+ 13m2

2 (A109)

(1)plusmn W

(1)∓ h(1)h(1) =

4(A110)

(1)plusmn W

(1)∓ a

(1)0 a

2+ 13m2

2 (A111)

(1)∓ W

(1)plusmn a

(1)∓

2minus 13m2

2 (A112)

gB(1)A

(1)3 h(1)h(1) = minus3gY g

4(A113)

gB(1)A

(1)3 a

(1)0 a

= minus3gY g

M (1)2

2 (A114)

gB(1)A

(1)3 a

(1)plusmn a

(1)∓

=3gY g

M (1)2

2 (A115)

gB(1)W

(1)plusmn h(1)a

(1)∓

= ∓i3gY g

(A116)

gB(1)W

(1)plusmn a

(1)0 a

(1)∓

= minus3gY g

M (1)2

M(1)W M

(A117)

(1)3 W

(1)plusmn a

(1)0 a

(1)∓

= minusg2

M(1)W M

(A118)

gB(1)B(1)h(0)h(0) =g2

2(A119)

gZ(0)Z(0)h(1)h(1) =g2

2c2w(A120)

(1)3 A

(1)3 h(0)h(0) =

2(A121)

gZ(0)A

(1)3 h(0)h(1) =

2cw(A122)

gZ(0)Z(0)a

(1)0 a

M (1)2

2 (A123)

(0)plusmn W

(0)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A124)

gA(0)A(0)a

(1)plusmn a

(1)∓

= 2e2 (A125)

gZ(0)Z(0)a

(1)plusmn a

(1)∓

4c4wm2W

2M(1)W

2 (A126)

(01)+ W

(01)minus h(10)h(10) =

2(A127)

(01)+ W

(10)minus h(01)h(10) =

2(A128)

(0)plusmn W

(0)∓ a

(1)0 a

2+ 3m2

2 (A129)

(0)plusmn W

(0)∓ a

(1)plusmn a

(1)∓

2 (A130)

gB(1)A

2(A131)

gB(1)Z(0)h(1)h(0) =egY

2cw(A132)

gA(0)Z(0)a

(1)plusmn a

(1)∓

2c2wm2W

2 (A133)

gA(0)W

(01)plusmn h(10)a

(1)∓

= ∓i e2

(A134)

gB(1)W

(0)plusmn h(0)a

(1)∓

= ∓i e2

(A135)

gA(0)W

(0)plusmn a

(1)0 a

(1)∓

= minus e2

mW +M(1)W

M(1)W M

(A136)

gZ(0)W

(01)plusmn h(10)a

(1)∓

= plusmni e2

(A137)

gZ(0)W

(0)plusmn a

(1)0 a

(1)∓

2s2wcw

M(1)W M

(A138)

gA(0)A(0)G

(1)+ G

(1)minus

= 2e2 (A139)

gB(1)B(1)χ

(0)+ χ

(0)minus

2(A140)

gB(1)B(1)G

(1)+ G

(1)minus

2 (A141)

gB(1)B(1)G

(1)∓ a

(1)plusmn

mWM(1)

2 (A142)

gB(1)B(1)χ3(0)χ3(0) =g2

2(A143)

gB(1)B(1)G

(1)0 G

2 (A144)

gB(1)B(1)G

(1)0 a

mZM(1)

2 (A145)

gB(1)A(0)G

(1)plusmn χ

(0)∓

= plusmniegY

(A146)

gB(1)A(0)a

(1)plusmn χ

(0)∓

= plusmniegY

(A147)

(1)plusmn A(0)h(0)G

(1)∓

= ∓i eg2

(A148)

igS1S2S3S4

gh(0)h(0)h(0)h(0) = minus6λ (A149)

gh(1)h(1)h(1)h(1) = minus9λ (A150)

ga(1)0 a

(1)0 a

= minus3g2c2wm2ZM (1)2 + 9λM (1)4

4 (A151)

gh(1)h(1)a

(1)0 a

= minusg2m2

Zc2w + 6λM (1)2

2M(1)Z

2 (A152)

gh(1)h(1)a

(1)+ a

(1)minus

= minusg2m2

W+ 6λM (1)2

2M(1)W

2 (A153)

ga(1)0 a

(1)0 a

(1)+ a

(1)minus

= minusg2m2

WM (1)2(1 + c4w)c4w + 6λM (1)4

2M(1)W

2 (A154)

ga(1)+ a

(1)+ a

(1)minus a

(1)minus

= minus2g2m2WM (1)2 + 6λM (1)4

2 (A155)

gh(1)h(1)h(0)h(0) = minusλ (A156)

gh(0)h(0)a

(1)0 a

= minusg2m2

Zc2w + 4λM (1)2

2M(1)Z

2 (A157)

gh(0)h(0)a

(1)+ a

(1)minus

= minusg2m2

W+ 4λM (1)2

2M(1)W

2 (A158)

160 Bibliography

(1999) 896ndash899 [astro-ph9807002]

Bibliography 161

(2006) 137ndash161 [hep-ph0602058]

162 Bibliography

(2007) 859 [astro-ph0703337]

(1999) 1147ndash1152 [astro-ph9903164]

Bibliography 163

(2004) 1039 [astro-ph0311231]

(1962) 748

164 Bibliography

Bibliography 165

(2005) 647ndash665 [astro-ph0408163]

166 Bibliography

Bibliography 167

(2007) 262 [astro-ph0611370]

(2003) 103003 [astro-ph0301551]

(2004) L23 [astro-ph0309621]

168 Bibliography

(2005) 081802 [hep-ex0411038]

Bibliography 169

(1987) 1087ndash1170

(1998) 263ndash272 [hep-ph9803315]

170 Bibliography

(2000) 096006 [hep-ph0006238]

(2003) 403 [hep-ph0205340]

Bibliography 171

172 Bibliography

(1998) 2025ndash2035 [gr-qc9804018]

(2002) 024036 [hep-th0110149]

Bibliography 173

(2004) 004 [hep-ph0310258]

174 Bibliography

Bibliography 175

(2007) 191ndash198 [astro-ph0608495]

radics = 18-TeV PhD

176 Bibliography

Bibliography 177

(1977) 1519

(2006) 123515 [astro-ph0607327]

178 Bibliography

(1979) 151

Bibliography 179

180 Bibliography

(2006) 015007 [hep-ph0603188]

Bibliography 181

(2007) 026 [hep-ph0703056]

(2007) 028 [hep-ph0612275]

182 Bibliography

Bibliography 183

(2008) 25ndash29 [arXiv07054311]

(2008) 864 [astro-ph0510520]

(1999) 203

184 Bibliography

(2007) 607ndash612 [astro-ph0612462]

P a r t I I

Scientific Papers

P a p e r I

P a p e r I I

P a p e r I I I

P a p e r I V

P a p e r V

P a p e r V I

P a p e r V I I

Abstract

Preface



Special Relativity

General Relativity


Evolving Universe

Initial Conditions


Dark Energy

Dark Matter








Simulation Setups



Nonsphericity

Axis Ratios

Alignments




Halo Substructure















Compactification

Kaluza-Klein Parity

The Lagrangian



Mass Spectrum


Relic Density



Direct Detection

Indirect Detection


Gamma-Ray Continuum

Gamma Line Signal



Supersymmetry

Some Motivations

The Neutralino








More Constraints


Gamma Rays

Continuum

Gamma-Ray Lines


Dark Matter Signals

The Data

The Claim

The Inconsistency



The Status to Date

Summary and Outlook



Vertex Rules

Bibliography


Measurements

C o n t e n t s

Abstract iii

Preface xi

viii Contents

Contents ix

Bibliography 159

P r e f a c e

xii Preface

Acknowledgments

Michael Gustafsson

P a r t I

C h a p t e r

Special Relativity

ds2 = dx2 + dy2 + dz2 (11)

ηαβ =

minus1 0 0 00 1 0 00 0 1 00 0 0 1

dτ2 =

αβ=0

αdxβ (16)

General Relativity

σβmicro (17)

Γαmicroν =1

Rmicroν minus1

c4Tmicroν (110)

1 minus kr2+ r2

dθ2 + sin2 θdφ2)

H2 equiv(

=8πGρ+ Λ

3minus k

a2(113)

dt(ρa3) = minusp d

dta3 (114)

ρc equiv3H2

8πG (115)

1 = Ω + ΩΛ + Ωk (116)

1 + z equiv λobs

λemit (118)

1 + z =a(tobs)

a(temit) (119)

dL equivradic

4πL (120)

dL = a0(1 + z)f

dzprime

H(zprime)

f(x) equiv

H(z) = H0

Ω0i (1 + z)minus3(1+wi) (122)

dH(t) =

dtprime

a(tprime)= a(t)

int r(t)

Dark Energy

ln ( aa0

) = ddt

ln ( 11+z

) = minus11+z

Dark Matter

v(r) =

r (124)

radicr Observa-

Dark Energy

Dark Matter

Baryonic Matter

〈σv〉 (125)

σv sim α2

M2WIMP

C h a p t e r

ρ(r) =ρ0

M200 = 2004πr3200

3ρc (22)

d ln(r) (23)

minus 2

rminus2

minus 1

rf =riMi(ri)

int ra

vrdr (28)

〈r〉 = r200A

rf =riMi(〈ri〉)

Mb(〈rf 〉) +MDM(〈rf 〉) (210)

001 01 110

DM only sim

DM in galaxy sim

r [r200

[ρc r

0 5 10 15 200

r [kpc]

065 07 075 08 08506

27 Nonsphericity

0 02 04 06 08 10

Axis Ratios

08DM (incl baryons)

R [kpc]10

DM (DM only)

R [kpc]

by the measure

T =a2 minus b2

a2 minus c2 (212)

Alignments

R [kpc]

2〈σv〉 ρ

2DM(r)

dΦγ(ψ)

〈σtotv〉8πm2

dN effγ

dℓ(ψ)ρ2DM

(ℓ) (214)

85 kpc

03 GeV cmminus3

dℓ(ψ)ρ2DM

dΦγdEγ

∆Ω 〈J〉∆Ω(0) = 013 b sr (217)

r [kpc]

larr rBH

darr ρ m

baryons

Einasto rarr

(Aw)=(11)

(Aw)=(05106)

10minus10

10minus5

C h a p t e r

and More

xM = xmicro yp (31)

(5) +m2)

Φ(xmicro y) =(

y +m2)

y sim y + 2πR (33)

Φ(xmicro y) =sum

Φ(n)(xmicro) =(

Φ(n)(xmicro) = 0 (35)

m2n = m2 +

rn ≪ R (37)

1 + y22 + + y2

rr ≫ R (38)

V (r) prop 1

1 + αeminusrλ)

Rn (310)

d4+nxradic

minusg[

2partM φ part

M φminus m2

2φ2 minus λ4+n

φ(x y) =1radicVn

φ(~n)(x) exp

i~n middot ~yR

d4xradicminusg

partmicroφ(0))2

minus m2

2(φ(0))2

partmicroφ(~n))(

minusm2~nφ

(~n)φ(~n)lowast]

minus λ4

4(φ(0))4 minus λ4

4(φ(0))2

m2~n = m2 +

R2 (314)

λ4 =λ4+n

Vn (315)

M2pl =

1 minus eminus2wL)

τp sim 1035 years

500 GeV

)12(ΛcutR

C h a p t e r

Extra Dimensions

16πG4+n

d4+nxradic

minusgR (41)

d4xradicminusg

16πGR[g(0)

microν ] +

G =G4+n

Vn(43)

λ =λ4+n

Vn (315)

d4+nxradic

minusg(

RAB minus 1

2R gAB = κ2Tmatter

b+n(nminus 1)

= κ2ρ (48a)

b+n(nminus 1)

= κ2pa (48b)

b+ (nminus 1)

= constant

p(i)a = w(i)

(i)b ρ(i) (410)

) (411)

i ρ(i) and pab =

i w(i)abρ

(i) With a multi-

ρ(i)(

1 minus 3w(i)a + 2w

(i)b ) = 0 Thus

in summary

(i)b =

3w(i)a minus 1

2 (413)

(w(i)a = w

wa = 0 wb =1

10 15 2000

d4xradicminusgbn

2Λ + 2κ2Lmatter

d4xradicminusg

2partmicroΦpart

Φ equivradic

n(n+ 2)

2κ2ln b (418)

eminusradic

2(n+2)n

κΦ +1

Λ minus κ2Lmatter

eminusradic

2nn+2 κΦ (419)

Φ = minus part

partΦVeff (420)

bminus(n+2) +sum

a +2w(i)b

) (421)

a )bminusn(1+w(i)b

) =κ2

(i)0 equiv ρ

(i)0 Vn equiv

ρ(i)0 κ2κ2

2(Φ minus Φ0)

part2Φ

(i)a and w

radic2(1minus3w

(i)a +2w

(2+n)κ2∆ρ(i)

C h a p t e r

Dimensions

51 Compactification

Φ(xmicro y) =

infinsum

nRy (51)

micro) +

infinsum

Φ(n)even cos

(n)odd sin

R (52)

Rrarr cos

n(y + πR)

R= (minus1)n cos

R (53a)

Rrarr sin

n(y + πR)

R= (minus1)n sin

R (53b)

53 The Lagrangian

Gauge Bosons

PZ2 Armicro(x

gpartmicroα

Aimicro(0)

(x) +1radicπR

infinsum

Aimicro(n)

(xmicro) cosny

R (58a)

Ai5(x) =1radicπR

infinsum

Ai5(n)

(x) sinny

R (58b)

Lgauge = minus1

MN minus 1

MNF rMN (59)

+ gǫijkAjMAk

N (510b)

gauge sup minus1

int 2πR

dy FMNFMN

= minus1

(0)micro

minus1

infinsum

(n)micro

infinsum

partmicroB(n)5 +

RB(n)micro

partmicroB(n)5 +

RB(n)micro

R appears The

(n)5 can be removed

g equiv 1radic2πR

g (512)

The Higgs Sector

φ equiv 1radic2

χ2 + iχ1

H minus iχ3

φ(x) =1radic2πR

φ(0)(x) +1radicπR

infinsum

φ(n)(x) cosny

R (514)

minus iY gYBM (515)

0 11 0

0 minusii 0

1 00 minus1

|φ|2 =minusmicro2

2λ2equiv v2

2 (519)

WplusmnM

= 1radic2

A1M∓ iA2

with mass mW =gv

2 (520a)

(520b)

AM = sWA3M

+ cWBM (521)

g2 + g2Y

cW equiv cos θW =g

g2 + g2Y

L(kin)Higgs sup

partMh)2

partMχ3 minusmZZM

)2+∣

Lgaugefix = minus1

GY)2 minus

Gi)2 (524)

Gi =1radicξ

minusmWχi + part5A

(525a)

GY =1radicξ

sWmWχ3 + part5B

(525b)

L(kin)scalar =

infinsum

2h(n)2

(n)0 minus ξM

(n)minus minus ξM

(n)+ G

(n)minus

infinsum

(n)0 minusM

2a(n)0

(n)minus minusM

2a(n)+ a

(n)minus

(n)5 minus ξ

a0(n)=M (n)

χ3(n)+

Z(n)5 (527a)

aplusmn(n)

=M (n)

χplusmn(n)+

Wplusmn5

(n) (527b)

G0(n)=

χ3(n) minus M (n)

Z5(n) (527c)

Gplusmn(n)=

Wplusmn5

(n) (527d)

M(n)X equiv

M (n) =n

R (528)

(n)5 G

(n)micro and W

Ghosts

δαbcb (529)

gpartMα

δBM =1

partMαY (531)

δφ =

iαiσi

φ equiv 1radic2

δχ2 + iδχ1

δH minus iδχ3

δχ1 =1

(533a)

δχ2 =1

(533b)

δχ3 =1

(533c)

radicξ)12ca

minus caδGaδαb

cb = ca[

cb (534)

h minusχ3 χ2 χ2

g2 0 0 00 g2 0 00 0 g2 minusggY

0 0 minusggY g2Y

L(kin)ghost =

infinsum

M (n) =

20 0 0

0 M(n)W

0 0 M(n)W

2minus 1

4v2ggY

0 0 minus 14v

2ggY14v

radicξM

(n)Z and

radicξM (n)

Fermions

γ5ψ(0)d = ψ

(0)d and γ5ψ(0)

ΓM ΓN = 2ηMN (538)

PRL equiv 1

1 plusmn γ5)

ψd =1radic2πR

ψ(0)dL

+1radicπR

infinsum

ψ(n)dL

(543a)

ψs =1radic2πR

ψ(0)sR

+1radicπR

infinsum

ψ(n)sR

R+ ψ(n)

(543b)

Lfermion = i(

ψdU ψdD)

ψdUψdD

6D equiv ΓM

partM minus igArM

minus iY gYBM

ψsD minus λU[

ψsU + hc (546)

EW= (λUD v)

radic2

cdD ψ

tdD) (547)

jdU and ψprimei

6Dij equiv ΓM

int 2πR

MpartMψs)

infinsum

ψ(n)d

ψ(n)d + ψ(n)

ψ(n)s

ψ(0) equiv ψ(0)dL

+ ψ(0)sR ψ

(n)d equiv ψ

+ ψ(n)dR ψ(n)

s equiv ψ(n)sL

+ ψ(n)sR (551)

ψ(n)d ψ

R+ δm

(n)d mEW

mEW minus[ n

R+ δm

ψ(n)d

ψ(n)s

ξ(n)d

ξ(n)s

1 00 minusγ5

cosα(n) sinα(n)

ψ(n)d

ψ(n)s

tan 2α(n) =2mEW

m(n)ds = plusmn1

(n)d minus δm(n)

(n)d + δm

(n)s )]2

+m2EW (555)

Φ(xmicro y) =1

p2 minus p2y

p2 minusm2 (558)

e(1) rarr e(0) + γ(1) (559)

1R2+m2

xmicro

phys + Z5Zm2(n) (560)

56 Mass Spectrum

)2+ δm2

B(n) + 14v

minus 14v

)2+ δm2

A3(n) + 14v

tan 2θ(n) =v2ggY

B(n) + v2

4 (g2 minus g2Y)] (562)

(a) Tree level

bdbsτdτsντ

(b) One-loop level

γaplusmna0H0

ZW plusmn

τdντ

bs bdts

C h a p t e r

61 Relic Density

X(n)i X

X(n)i xj harr X

(n)k xl (61b)

X(m)i rarr X

(n)j xk (61c)

n2 minus (neq)2]

where n =sum

≃ gi

(minusmX

〈σeffv〉 =sum

〈σijvij〉neqi

neq(64)

〈σeffv〉 (65)

i neqj prop

Rminus1[TeV]

05 1 15 2 25 3

B(1)B(1)

B(1) g(1) ∆g(1) = 2

X(1)(gX(1) + gB(1)) (66)

Direct Detection

Indirect Detection

Gamma-Ray Continuum

ℓ+ℓ+ℓ+

ℓminusℓminus

ℓ(1)

γγγ

dN ℓγ

(x2 minus 2x+ 2)

m2B(1)

(1 minus x)

dN ℓγ

dxpropint θmax

dθθ3

θ2 +m2ℓE

]2 asymp 1

2ln(E2

pm2ℓ) (610)

radics2 minus Eγ (

dσ(χχ rarr XXγ)

dxasymp αQ2

πFX(x) ln

s(1 minus x)

Ff(x) =1 + (1 minus x)2

x (612)

effγdx

x equiv EγmB(1)

Fs(x) =1 minus x

x (613)

dN effγ dx equiv

κidNiγdx (614)

Gamma Line Signal

M = ǫmicro1

1 (p1)ǫmicro2

2 (p2)ǫmicro3

3 (p3)ǫmicro4

ψ(0)ψ(0)

ξ(1)ds

v (617)

m4B(1)

pmicro1

3 pmicro2

m2B(1)

4 pmicro4 +B3

m2B(1)

4 pmicro3 +B4

m2B(1)

3 pmicro4

m2B(1)

3 pmicro3 +B6

m2B(1)

3 pmicro2

q21 minusm21

q22 minusm22

q23 minusm23

q24 minusm24

) (619)

q1 = q q2 = q+k1 q3 = q+k1 +k2 q4 = q+k1 +k2 +k3 (620)

(σv)γγ =α2

EMg4eff

144πm2B(1)

(621)where

g2eff equiv

Q2(Y 2s + Y 2

d ) =52

9 (622)

101 102 105 11 12

mξ(1)mB(1)

(σv) γγ

[10minus

aG aGB (1)

Hψ(0)

WB (1)

aGB (1)

dΦexp

int infin

Eprime

radic2πσEprime

Eγ [TeV]

E2 γdΦγdEγ

[mminus

2sminus

01 1 10

10minus9

10minus8

10minus7

Eγ [GeV]

dΦγdEγ

[10minus

minus2sminus

minus1]

780 790 800 810

C h a p t e r

Gamma-Ray Signal

71 Supersymmetry

Some Motivations

The Neutralino

3 +N13H01 +N14H

02 (71)

χminus1

dxequivd(σv)WWγdx

(σv)WW≃ αem

(1 minus x)x

+ δ2(

(2 minus x)3

+ δ4(

x(xminus 1)

(2 minus x)2+

(2 minus x)3

0 05 1001

x equiv Eγmχ

dNW γdx

χplusmn1

) Hig-

Zh Wplusmn Ωχh2

32 15 32 32 00 100 150 151 092 039 012

01 05 1 201

Eγ [TeV]

d(vσ) γdEγ

[10minus

2sminus

minus1]

06 1 22

[10minus

2sminus

minus1]

Eγ [TeV]

0minus

3sminus

01 05 1 2

C h a p t e r

fE2cm This is

can be written as

2 |H2|2 + λ1 |H1|4 + λ2 |H2|4 (83)

dagger1H2)

H1 =1radic2

G+radic

2v + h+ iG0

H2 =1radic2

H0 + iA0

m2h = minus2micro2

m2H0 = micro2

2 + (λ3 + λ4 + λ5)v2

m2A0 = micro2

2 + (λ3 + λ4 minus λ5)v2

m2Hplusmn = micro2

2 + λ3v2 (86)

radic2

cos θw(mW ))

∆T asymp 1

mh= 1141000 GeV

minus04minus04

minus02

More Constraints

λ12 gt 0

λ1λ2 (89)

Wminus(Z)Wminus(Z)

Wminus(Z)

W +(Z)W +(Z)

W +(Z)Hplusmn(A0)

f(νf )f(νf )

λL λ2L λLλ1

g2 g2 λLg

λLyf g2

83 Gamma Rays

Continuum

vrelσff =Ncπα

sin4 θWm4W

(1 minus 4m2f

(sminusm2h)

2 +m2hΓ

Gamma-Ray Lines

g4 g4 g4

λLg3 λLg3

x = EγmH0

x2dNγdx

10minus4

10minus2

001 01 1

τ+τminus

EGRETDW=210-3

HESSDW=10-5

GLAST sensitivity

50 GeV boost ~104

70 GeV boost ~100

2 γΦγ

minus2sminus

log(Eγ [GeV])

-10-1 0 1 2 3 4

WIMP Mass [GeV]

H0 H0 rarr Zγ

H0 H0 rarr γγ

rarr Zγ γγ

40 50 60 70 80

C h a p t e r

Observed

92 The Data

bremss πο

Energy [GeV]

E2times

minus2sminus

minus1]

10minus7

10minus6

10minus5

10minus4

bremssπο

Energy [GeV]

10minus7

10minus6

10minus5

10minus4

93 The Claim

χ2 356χ2 (bg only) 17887

Energy [GeV]

E2times

minus2sminus

minus1]

10minus5

10minus4

10minus1 1 10 102

minus10

x [kpc]

minus5 0 50

z [kpc]

OuterSunInnerCenter

minus10

x [kpc]

kpc [M

BESS 98 1σ

BESS 98 2σ

T = 040-056 GeV

χ2 lt 10

10 lt χ2 lt 25

25 lt χ2 lt 60

60 lt χ2

C h a p t e r

Summary and Outlook

A p p e n d i x

(n)plusmn and A

(n)5 at

given by

for scalars i

for vectors minusi

q2 minusm2 + iǫ

= swAM + cwZM (A4a)

Z(n)5 =

a(n)0 minus M (n)

G(n)0 (A5a)

χ3 (n) =M (n)

a(n)0 +

G(n)0 (A5b)

Wplusmn (n)5 =

G(n)plusmn (A5c)

χplusmn (n) =M (n)

a(n)plusmn +

G(n)plusmn (A5d)

ψ(n)s = sinα(n)ξ

ψ(n)d = cosα(n)ξ

A2 Vertex Rules

M(1)X equiv

M (1)2 +m2X (A7d)

mZ = mWcw (A8a)

gY = ecw (A8b)

g = esw (A8c)

λ =g2m2

V micro2

q3V ν3

q1V ρ1 igV1V2V3

microηρν]

gA(0)W

(10)+ W

(10)minus

= minuse (A9)

gZ(0)W

(01)+ W

(01)minus

= minuscw g (A10)

(1)3 W

(01)+ W

(10)minus

= minusg (A11)

V micro1

gZ(0)Z(0)h(0) =g

cwmZ (A12)

(0)+ W

gZ(0)W

(1)plusmn a

(1)∓

= plusmnig mZ

gZ(0)A3 (1)h(1) = gmZ (A15)

(0)plusmn A3 (1)a

(1)∓

= ∓ig mW

(0)plusmn B(1)a

(1)∓

= ∓igY mW

(0)plusmn W

(1)∓ a

= plusmnig mW

(0)plusmn W

(1)∓ h(1) = gmW (A20)

gA3 (1)A3 (1)h(0) = gmW (A21)

gB(1)B(1)h(0) = gs2wc2w

mW (A23)

(1)+ W

gA(0)W

gZ(0)W

gA(0)W

(1)plusmn G

(1)∓

= ∓ieM (1)W (A27)

(0)plusmn B(1)G

(1)∓

= ∓igY mW

(1)plusmn B(1)χ∓ (0) = ∓igY mW (A29)

V micro

gA(0)a

(1)+ a

(1)minus

= e (A30)

gZ(0)a

(1)+ a

(1)minus

2(gcw minus gY sw)

M (1)2

2 + gcwmW

2 (A31)

gZ(0)h(1)a

= minusi g

(0)plusmn a∓(1)a

= ∓g2

M (1)2

M(1)W M

1 minus 2m2

(0)plusmn a∓(1)h(1) = i

gA3 (1)a

(1)0 h(0) = i

gA3 (1)G

(1)0 h(0) = i

gB(1)a

(1)plusmn a

(1)∓ h(0) = i

gA(0)χ+ (0)χminus (0) = e (A39)

minus gY

gZ(0)χ3 (0)h(0) = ig

2cw(A41)

(0)plusmn χ∓ (0)h(0) = i

2(A42)

(0)plusmn χ∓ (0)χ3 (0) = ∓g

2(A43)

gA(0)G

(1)+ G

(1)minus

= e (A44)

gB(1)χplusmn(0)a

(1)∓

= plusmngY

gB(1)χplusmn(0)G

(1)∓

= plusmngY

2(A47)

gB(1)G

(1)plusmn G

(1)∓ h(0) = i

S3 igS1S2S3

gh(0)h(01)h(01) = minus3

gh(0)a

(1)+ a

(1)minus

= minusgmW

M (1)2

gh(0)a

(1)0 a

= minus g

M (1)2

gh(0)χ3 (0)χ3 (0) = minusg2

gh(0)a

(1)plusmn G

(1)∓

2M (1)

1 minus m2h

gh(0)G

(1)plusmn G

(1)∓

= minusg2mW

2 (A56)

gA(0)ξ(0)ξ(0) = Qe (A57)

gA(0)ξ

(1)sdξ(1)sd

= Qe (A58)

gZ(0)ξ

(1)dξ(1)d

gZ(0)ξ

(1)s ξ

gZ(0)ξ

(1)s ξ

(1)3 ξ(0)ξ

gB(1)ξ(0)ξ

(0)+ U

(0)i D

=gradic2VijPL (A67)

(0)+ U

(1)diD

=gradic2Vij cosα

cosα(1)Dj

(0)+ U

(1)siD

=gradic2Vij sinα

sinα(1)Dj

(0)+ U

(1)diD

=gradic2Vij cosα

sinα(1)Djγ5 (A70)

(0)+ U

(1)siD

=gradic2Vij sinα

cosα(1)Djγ5 (A71)

(1)+ U

(0)i D

=gradic2Vij cosα

(1)DjPL (A72)

(1)+ U

(0)i D

(1)DjPL (A73)

(1)+ U

(1)diD

=gradic2Vij cosα

(1)UiPL (A74)

(1)+ U

(1)siD

(1)UiPL (A75)

S igSξ1ξ2

gh(0)ξ

(1)dξ(1)d

= minusg mξ

gh(0)ξ

(1)s ξ

= minusg mξ

gh(0)ξ

(1)dξ(1)s

= minusg mξ

γ5 (A79)

gh(1)ξ(0)ξ

= minusg mξ

gh(1)ξ(0)ξ

= gmξ

ga(1)0 ξ(0)ξ

minusigT3mξ

ga(1)0 ξ(0)ξ

+igT3mξ

ga(1)+ U

(0)i D

= minusi gradic2

cosα(1)DjPL

+igradic2

ga(1)+ U

(0)i D

= igradic2

sinα(1)DjPL

+igradic2

ga(1)+ U

(1)diD

= minusi gradic2

sinα(1)UiPL

minusi gradic2

ga(1)+ U

(1)siD

= igradic2

cosα(1)UiPL

minusi gradic2

dagger =

A(1) micro

c∓(1)

c∓(1) q

= plusmnie qmicro

c∓(1)

= minusi g2mWξ

V micro1

igV1V2V3V4 middot(

(0)minus W

(0)+ W

= g2 (A88)

(1)minus W

(1)+ W

2g2 (A89)

(10)minus W

(01)+ W

= g2 (A90)

(10)minus W

(01)minus W

(10)+ W

= g2 (A91)

gZ(0)Z(0)W

(0)minus W

= minusg2c2w (A92)

gZ(0)Z(0)W

(1)minus W

= minusg2c2w (A93)

gA(0)A(0)W

(0)+ W

(0)minus

= minuse2 (A94)

gA(0)A(0)W

(1)+ W

(1)minus

= minuse2 (A95)

gZ(0)A(0)W

(0)+ W

(0)minus

= minusegcw (A96)

gZ(0)A(0)W

(1)+ W

(1)minus

= minusegcw (A97)

(1)3 A

(1)3 W

(1)+ W

(1)minus

= minus3

2g2 (A98)

(1)3 A

(1)3 W

(0)+ W

(0)minus

= minusg2 (A99)

(1)3 Z(0)W

(01)+ W

(10)minus

= minusg2cw (A100)

V micro1

iηmicroνgV1V2S1S2

gZ(0)Z(0)h(0)h(0) =g2

2c2w(A101)

(0)plusmn W

(0)∓ h(0)h(0) =

2(A102)

gB(1)B(1)h(1)h(1) =3g2

4(A103)

gB(1)B(1)a

(1)0 a

M (1)2

2 (A104)

(1)3 A

(1)3 h(1)h(1) =

4(A105)

(1)3 A

(1)3 a

(1)0 a

M (1)2

2 (A106)

(1)plusmn W

(1)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A107)

gB(1)B(1)a

(1)+ a

(1)minus

M (1)2

2 (A108)

(1)3 A

(1)3 a

(1)plusmn a

(1)∓

2+ 13m2

2 (A109)

(1)plusmn W

(1)∓ h(1)h(1) =

4(A110)

(1)plusmn W

(1)∓ a

(1)0 a

2+ 13m2

2 (A111)

(1)∓ W

(1)plusmn a

(1)∓

2minus 13m2

2 (A112)

gB(1)A

(1)3 h(1)h(1) = minus3gY g

4(A113)

gB(1)A

(1)3 a

(1)0 a

= minus3gY g

M (1)2

2 (A114)

gB(1)A

(1)3 a

(1)plusmn a

(1)∓

=3gY g

M (1)2

2 (A115)

gB(1)W

(1)plusmn h(1)a

(1)∓

= ∓i3gY g

(A116)

gB(1)W

(1)plusmn a

(1)0 a

(1)∓

= minus3gY g

M (1)2

M(1)W M

(A117)

(1)3 W

(1)plusmn a

(1)0 a

(1)∓

= minusg2

M(1)W M

(A118)

gB(1)B(1)h(0)h(0) =g2

2(A119)

gZ(0)Z(0)h(1)h(1) =g2

2c2w(A120)

(1)3 A

(1)3 h(0)h(0) =

2(A121)

gZ(0)A

(1)3 h(0)h(1) =

2cw(A122)

gZ(0)Z(0)a

(1)0 a

M (1)2

2 (A123)

(0)plusmn W

(0)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A124)

gA(0)A(0)a

(1)plusmn a

(1)∓

= 2e2 (A125)

gZ(0)Z(0)a

(1)plusmn a

(1)∓

4c4wm2W

2M(1)W

2 (A126)

(01)+ W

(01)minus h(10)h(10) =

2(A127)

(01)+ W

(10)minus h(01)h(10) =

2(A128)

(0)plusmn W

(0)∓ a

(1)0 a

2+ 3m2

2 (A129)

(0)plusmn W

(0)∓ a

(1)plusmn a

(1)∓

2 (A130)

gB(1)A

2(A131)

gB(1)Z(0)h(1)h(0) =egY

2cw(A132)

gA(0)Z(0)a

(1)plusmn a

(1)∓

2c2wm2W

2 (A133)

gA(0)W

(01)plusmn h(10)a

(1)∓

= ∓i e2

(A134)

gB(1)W

(0)plusmn h(0)a

(1)∓

= ∓i e2

(A135)

gA(0)W

(0)plusmn a

(1)0 a

(1)∓

= minus e2

mW +M(1)W

M(1)W M

(A136)

gZ(0)W

(01)plusmn h(10)a

(1)∓

= plusmni e2

(A137)

gZ(0)W

(0)plusmn a

(1)0 a

(1)∓

2s2wcw

M(1)W M

(A138)

gA(0)A(0)G

(1)+ G

(1)minus

= 2e2 (A139)

gB(1)B(1)χ

(0)+ χ

(0)minus

2(A140)

gB(1)B(1)G

(1)+ G

(1)minus

2 (A141)

gB(1)B(1)G

(1)∓ a

(1)plusmn

mWM(1)

2 (A142)

gB(1)B(1)χ3(0)χ3(0) =g2

2(A143)

gB(1)B(1)G

(1)0 G

2 (A144)

gB(1)B(1)G

(1)0 a

mZM(1)

2 (A145)

gB(1)A(0)G

(1)plusmn χ

(0)∓

= plusmniegY

(A146)

gB(1)A(0)a

(1)plusmn χ

(0)∓

= plusmniegY

(A147)

(1)plusmn A(0)h(0)G

(1)∓

= ∓i eg2

(A148)

igS1S2S3S4

gh(0)h(0)h(0)h(0) = minus6λ (A149)

gh(1)h(1)h(1)h(1) = minus9λ (A150)

ga(1)0 a

(1)0 a

= minus3g2c2wm2ZM (1)2 + 9λM (1)4

4 (A151)

gh(1)h(1)a

(1)0 a

= minusg2m2

Zc2w + 6λM (1)2

2M(1)Z

2 (A152)

gh(1)h(1)a

(1)+ a

(1)minus

= minusg2m2

W+ 6λM (1)2

2M(1)W

2 (A153)

ga(1)0 a

(1)0 a

(1)+ a

(1)minus

= minusg2m2

WM (1)2(1 + c4w)c4w + 6λM (1)4

2M(1)W

2 (A154)

ga(1)+ a

(1)+ a

(1)minus a

(1)minus

= minus2g2m2WM (1)2 + 6λM (1)4

2 (A155)

gh(1)h(1)h(0)h(0) = minusλ (A156)

gh(0)h(0)a

(1)0 a

= minusg2m2

Zc2w + 4λM (1)2

2M(1)Z

2 (A157)

gh(0)h(0)a

(1)+ a

(1)minus

= minusg2m2

W+ 4λM (1)2

2M(1)W

2 (A158)

160 Bibliography

(1999) 896ndash899 [astro-ph9807002]

Bibliography 161

(2006) 137ndash161 [hep-ph0602058]

162 Bibliography

(2007) 859 [astro-ph0703337]

(1999) 1147ndash1152 [astro-ph9903164]

Bibliography 163

(2004) 1039 [astro-ph0311231]

(1962) 748

164 Bibliography

Bibliography 165

(2005) 647ndash665 [astro-ph0408163]

166 Bibliography

Bibliography 167

(2007) 262 [astro-ph0611370]

(2003) 103003 [astro-ph0301551]

(2004) L23 [astro-ph0309621]

168 Bibliography

(2005) 081802 [hep-ex0411038]

Bibliography 169

(1987) 1087ndash1170

(1998) 263ndash272 [hep-ph9803315]

170 Bibliography

(2000) 096006 [hep-ph0006238]

(2003) 403 [hep-ph0205340]

Bibliography 171

172 Bibliography

(1998) 2025ndash2035 [gr-qc9804018]

(2002) 024036 [hep-th0110149]

Bibliography 173

(2004) 004 [hep-ph0310258]

174 Bibliography

Bibliography 175

(2007) 191ndash198 [astro-ph0608495]

radics = 18-TeV PhD

176 Bibliography

Bibliography 177

(1977) 1519

(2006) 123515 [astro-ph0607327]

178 Bibliography

(1979) 151

Bibliography 179

180 Bibliography

(2006) 015007 [hep-ph0603188]

Bibliography 181

(2007) 026 [hep-ph0703056]

(2007) 028 [hep-ph0612275]

182 Bibliography

Bibliography 183

(2008) 25ndash29 [arXiv07054311]

(2008) 864 [astro-ph0510520]

(1999) 203

184 Bibliography

(2007) 607ndash612 [astro-ph0612462]

P a r t I I

Scientific Papers

P a p e r I

P a p e r I I

P a p e r I I I

P a p e r I V

P a p e r V

P a p e r V I

P a p e r V I I

Abstract

Preface



Special Relativity

General Relativity


Evolving Universe

Initial Conditions


Dark Energy

Dark Matter








Simulation Setups



Nonsphericity

Axis Ratios

Alignments




Halo Substructure















Compactification

Kaluza-Klein Parity

The Lagrangian



Mass Spectrum


Relic Density



Direct Detection

Indirect Detection


Gamma-Ray Continuum

Gamma Line Signal



Supersymmetry

Some Motivations

The Neutralino








More Constraints


Gamma Rays

Continuum

Gamma-Ray Lines


Dark Matter Signals

The Data

The Claim

The Inconsistency



The Status to Date

Summary and Outlook



Vertex Rules

Bibliography


C o n t e n t s

Abstract iii

Preface xi

viii Contents

Contents ix

Bibliography 159

P r e f a c e

xii Preface

Acknowledgments

Michael Gustafsson

P a r t I

C h a p t e r

Special Relativity

ds2 = dx2 + dy2 + dz2 (11)

ηαβ =

minus1 0 0 00 1 0 00 0 1 00 0 0 1

dτ2 =

αβ=0

αdxβ (16)

General Relativity

σβmicro (17)

Γαmicroν =1

Rmicroν minus1

c4Tmicroν (110)

1 minus kr2+ r2

dθ2 + sin2 θdφ2)

H2 equiv(

=8πGρ+ Λ

3minus k

a2(113)

dt(ρa3) = minusp d

dta3 (114)

ρc equiv3H2

8πG (115)

1 = Ω + ΩΛ + Ωk (116)

1 + z equiv λobs

λemit (118)

1 + z =a(tobs)

a(temit) (119)

dL equivradic

4πL (120)

dL = a0(1 + z)f

dzprime

H(zprime)

f(x) equiv

H(z) = H0

Ω0i (1 + z)minus3(1+wi) (122)

dH(t) =

dtprime

a(tprime)= a(t)

int r(t)

Dark Energy

ln ( aa0

) = ddt

ln ( 11+z

) = minus11+z

Dark Matter

v(r) =

r (124)

radicr Observa-

Dark Energy

Dark Matter

Baryonic Matter

〈σv〉 (125)

σv sim α2

M2WIMP

C h a p t e r

ρ(r) =ρ0

M200 = 2004πr3200

3ρc (22)

d ln(r) (23)

minus 2

rminus2

minus 1

rf =riMi(ri)

int ra

vrdr (28)

〈r〉 = r200A

rf =riMi(〈ri〉)

Mb(〈rf 〉) +MDM(〈rf 〉) (210)

001 01 110

DM only sim

DM in galaxy sim

r [r200

[ρc r

0 5 10 15 200

r [kpc]

065 07 075 08 08506

27 Nonsphericity

0 02 04 06 08 10

Axis Ratios

08DM (incl baryons)

R [kpc]10

DM (DM only)

R [kpc]

by the measure

T =a2 minus b2

a2 minus c2 (212)

Alignments

R [kpc]

2〈σv〉 ρ

2DM(r)

dΦγ(ψ)

〈σtotv〉8πm2

dN effγ

dℓ(ψ)ρ2DM

(ℓ) (214)

85 kpc

03 GeV cmminus3

dℓ(ψ)ρ2DM

dΦγdEγ

∆Ω 〈J〉∆Ω(0) = 013 b sr (217)

r [kpc]

larr rBH

darr ρ m

baryons

Einasto rarr

(Aw)=(11)

(Aw)=(05106)

10minus10

10minus5

C h a p t e r

and More

xM = xmicro yp (31)

(5) +m2)

Φ(xmicro y) =(

y +m2)

y sim y + 2πR (33)

Φ(xmicro y) =sum

Φ(n)(xmicro) =(

Φ(n)(xmicro) = 0 (35)

m2n = m2 +

rn ≪ R (37)

1 + y22 + + y2

rr ≫ R (38)

V (r) prop 1

1 + αeminusrλ)

Rn (310)

d4+nxradic

minusg[

2partM φ part

M φminus m2

2φ2 minus λ4+n

φ(x y) =1radicVn

φ(~n)(x) exp

i~n middot ~yR

d4xradicminusg

partmicroφ(0))2

minus m2

2(φ(0))2

partmicroφ(~n))(

minusm2~nφ

(~n)φ(~n)lowast]

minus λ4

4(φ(0))4 minus λ4

4(φ(0))2

m2~n = m2 +

R2 (314)

λ4 =λ4+n

Vn (315)

M2pl =

1 minus eminus2wL)

τp sim 1035 years

500 GeV

)12(ΛcutR

C h a p t e r

Extra Dimensions

16πG4+n

d4+nxradic

minusgR (41)

d4xradicminusg

16πGR[g(0)

microν ] +

G =G4+n

Vn(43)

λ =λ4+n

Vn (315)

d4+nxradic

minusg(

RAB minus 1

2R gAB = κ2Tmatter

b+n(nminus 1)

= κ2ρ (48a)

b+n(nminus 1)

= κ2pa (48b)

b+ (nminus 1)

= constant

p(i)a = w(i)

(i)b ρ(i) (410)

) (411)

i ρ(i) and pab =

i w(i)abρ

(i) With a multi-

ρ(i)(

1 minus 3w(i)a + 2w

(i)b ) = 0 Thus

in summary

(i)b =

3w(i)a minus 1

2 (413)

(w(i)a = w

wa = 0 wb =1

10 15 2000

d4xradicminusgbn

2Λ + 2κ2Lmatter

d4xradicminusg

2partmicroΦpart

Φ equivradic

n(n+ 2)

2κ2ln b (418)

eminusradic

2(n+2)n

κΦ +1

Λ minus κ2Lmatter

eminusradic

2nn+2 κΦ (419)

Φ = minus part

partΦVeff (420)

bminus(n+2) +sum

a +2w(i)b

) (421)

a )bminusn(1+w(i)b

) =κ2

(i)0 equiv ρ

(i)0 Vn equiv

ρ(i)0 κ2κ2

2(Φ minus Φ0)

part2Φ

(i)a and w

radic2(1minus3w

(i)a +2w

(2+n)κ2∆ρ(i)

C h a p t e r

Dimensions

51 Compactification

Φ(xmicro y) =

infinsum

nRy (51)

micro) +

infinsum

Φ(n)even cos

(n)odd sin

R (52)

Rrarr cos

n(y + πR)

R= (minus1)n cos

R (53a)

Rrarr sin

n(y + πR)

R= (minus1)n sin

R (53b)

53 The Lagrangian

Gauge Bosons

PZ2 Armicro(x

gpartmicroα

Aimicro(0)

(x) +1radicπR

infinsum

Aimicro(n)

(xmicro) cosny

R (58a)

Ai5(x) =1radicπR

infinsum

Ai5(n)

(x) sinny

R (58b)

Lgauge = minus1

MN minus 1

MNF rMN (59)

+ gǫijkAjMAk

N (510b)

gauge sup minus1

int 2πR

dy FMNFMN

= minus1

(0)micro

minus1

infinsum

(n)micro

infinsum

partmicroB(n)5 +

RB(n)micro

partmicroB(n)5 +

RB(n)micro

R appears The

(n)5 can be removed

g equiv 1radic2πR

g (512)

The Higgs Sector

φ equiv 1radic2

χ2 + iχ1

H minus iχ3

φ(x) =1radic2πR

φ(0)(x) +1radicπR

infinsum

φ(n)(x) cosny

R (514)

minus iY gYBM (515)

0 11 0

0 minusii 0

1 00 minus1

|φ|2 =minusmicro2

2λ2equiv v2

2 (519)

WplusmnM

= 1radic2

A1M∓ iA2

with mass mW =gv

2 (520a)

(520b)

AM = sWA3M

+ cWBM (521)

g2 + g2Y

cW equiv cos θW =g

g2 + g2Y

L(kin)Higgs sup

partMh)2

partMχ3 minusmZZM

)2+∣

Lgaugefix = minus1

GY)2 minus

Gi)2 (524)

Gi =1radicξ

minusmWχi + part5A

(525a)

GY =1radicξ

sWmWχ3 + part5B

(525b)

L(kin)scalar =

infinsum

2h(n)2

(n)0 minus ξM

(n)minus minus ξM

(n)+ G

(n)minus

infinsum

(n)0 minusM

2a(n)0

(n)minus minusM

2a(n)+ a

(n)minus

(n)5 minus ξ

a0(n)=M (n)

χ3(n)+

Z(n)5 (527a)

aplusmn(n)

=M (n)

χplusmn(n)+

Wplusmn5

(n) (527b)

G0(n)=

χ3(n) minus M (n)

Z5(n) (527c)

Gplusmn(n)=

Wplusmn5

(n) (527d)

M(n)X equiv

M (n) =n

R (528)

(n)5 G

(n)micro and W

Ghosts

δαbcb (529)

gpartMα

δBM =1

partMαY (531)

δφ =

iαiσi

φ equiv 1radic2

δχ2 + iδχ1

δH minus iδχ3

δχ1 =1

(533a)

δχ2 =1

(533b)

δχ3 =1

(533c)

radicξ)12ca

minus caδGaδαb

cb = ca[

cb (534)

h minusχ3 χ2 χ2

g2 0 0 00 g2 0 00 0 g2 minusggY

0 0 minusggY g2Y

L(kin)ghost =

infinsum

M (n) =

20 0 0

0 M(n)W

0 0 M(n)W

2minus 1

4v2ggY

0 0 minus 14v

2ggY14v

radicξM

(n)Z and

radicξM (n)

Fermions

γ5ψ(0)d = ψ

(0)d and γ5ψ(0)

ΓM ΓN = 2ηMN (538)

PRL equiv 1

1 plusmn γ5)

ψd =1radic2πR

ψ(0)dL

+1radicπR

infinsum

ψ(n)dL

(543a)

ψs =1radic2πR

ψ(0)sR

+1radicπR

infinsum

ψ(n)sR

R+ ψ(n)

(543b)

Lfermion = i(

ψdU ψdD)

ψdUψdD

6D equiv ΓM

partM minus igArM

minus iY gYBM

ψsD minus λU[

ψsU + hc (546)

EW= (λUD v)

radic2

cdD ψ

tdD) (547)

jdU and ψprimei

6Dij equiv ΓM

int 2πR

MpartMψs)

infinsum

ψ(n)d

ψ(n)d + ψ(n)

ψ(n)s

ψ(0) equiv ψ(0)dL

+ ψ(0)sR ψ

(n)d equiv ψ

+ ψ(n)dR ψ(n)

s equiv ψ(n)sL

+ ψ(n)sR (551)

ψ(n)d ψ

R+ δm

(n)d mEW

mEW minus[ n

R+ δm

ψ(n)d

ψ(n)s

ξ(n)d

ξ(n)s

1 00 minusγ5

cosα(n) sinα(n)

ψ(n)d

ψ(n)s

tan 2α(n) =2mEW

m(n)ds = plusmn1

(n)d minus δm(n)

(n)d + δm

(n)s )]2

+m2EW (555)

Φ(xmicro y) =1

p2 minus p2y

p2 minusm2 (558)

e(1) rarr e(0) + γ(1) (559)

1R2+m2

xmicro

phys + Z5Zm2(n) (560)

56 Mass Spectrum

)2+ δm2

B(n) + 14v

minus 14v

)2+ δm2

A3(n) + 14v

tan 2θ(n) =v2ggY

B(n) + v2

4 (g2 minus g2Y)] (562)

(a) Tree level

bdbsτdτsντ

(b) One-loop level

γaplusmna0H0

ZW plusmn

τdντ

bs bdts

C h a p t e r

61 Relic Density

X(n)i X

X(n)i xj harr X

(n)k xl (61b)

X(m)i rarr X

(n)j xk (61c)

n2 minus (neq)2]

where n =sum

≃ gi

(minusmX

〈σeffv〉 =sum

〈σijvij〉neqi

neq(64)

〈σeffv〉 (65)

i neqj prop

Rminus1[TeV]

05 1 15 2 25 3

B(1)B(1)

B(1) g(1) ∆g(1) = 2

X(1)(gX(1) + gB(1)) (66)

Direct Detection

Indirect Detection

Gamma-Ray Continuum

ℓ+ℓ+ℓ+

ℓminusℓminus

ℓ(1)

γγγ

dN ℓγ

(x2 minus 2x+ 2)

m2B(1)

(1 minus x)

dN ℓγ

dxpropint θmax

dθθ3

θ2 +m2ℓE

]2 asymp 1

2ln(E2

pm2ℓ) (610)

radics2 minus Eγ (

dσ(χχ rarr XXγ)

dxasymp αQ2

πFX(x) ln

s(1 minus x)

Ff(x) =1 + (1 minus x)2

x (612)

effγdx

x equiv EγmB(1)

Fs(x) =1 minus x

x (613)

dN effγ dx equiv

κidNiγdx (614)

Gamma Line Signal

M = ǫmicro1

1 (p1)ǫmicro2

2 (p2)ǫmicro3

3 (p3)ǫmicro4

ψ(0)ψ(0)

ξ(1)ds

v (617)

m4B(1)

pmicro1

3 pmicro2

m2B(1)

4 pmicro4 +B3

m2B(1)

4 pmicro3 +B4

m2B(1)

3 pmicro4

m2B(1)

3 pmicro3 +B6

m2B(1)

3 pmicro2

q21 minusm21

q22 minusm22

q23 minusm23

q24 minusm24

) (619)

q1 = q q2 = q+k1 q3 = q+k1 +k2 q4 = q+k1 +k2 +k3 (620)

(σv)γγ =α2

EMg4eff

144πm2B(1)

(621)where

g2eff equiv

Q2(Y 2s + Y 2

d ) =52

9 (622)

101 102 105 11 12

mξ(1)mB(1)

(σv) γγ

[10minus

aG aGB (1)

Hψ(0)

WB (1)

aGB (1)

dΦexp

int infin

Eprime

radic2πσEprime

Eγ [TeV]

E2 γdΦγdEγ

[mminus

2sminus

01 1 10

10minus9

10minus8

10minus7

Eγ [GeV]

dΦγdEγ

[10minus

minus2sminus

minus1]

780 790 800 810

C h a p t e r

Gamma-Ray Signal

71 Supersymmetry

Some Motivations

The Neutralino

3 +N13H01 +N14H

02 (71)

χminus1

dxequivd(σv)WWγdx

(σv)WW≃ αem

(1 minus x)x

+ δ2(

(2 minus x)3

+ δ4(

x(xminus 1)

(2 minus x)2+

(2 minus x)3

0 05 1001

x equiv Eγmχ

dNW γdx

χplusmn1

) Hig-

Zh Wplusmn Ωχh2

32 15 32 32 00 100 150 151 092 039 012

01 05 1 201

Eγ [TeV]

d(vσ) γdEγ

[10minus

2sminus

minus1]

06 1 22

[10minus

2sminus

minus1]

Eγ [TeV]

0minus

3sminus

01 05 1 2

C h a p t e r

fE2cm This is

can be written as

2 |H2|2 + λ1 |H1|4 + λ2 |H2|4 (83)

dagger1H2)

H1 =1radic2

G+radic

2v + h+ iG0

H2 =1radic2

H0 + iA0

m2h = minus2micro2

m2H0 = micro2

2 + (λ3 + λ4 + λ5)v2

m2A0 = micro2

2 + (λ3 + λ4 minus λ5)v2

m2Hplusmn = micro2

2 + λ3v2 (86)

radic2

cos θw(mW ))

∆T asymp 1

mh= 1141000 GeV

minus04minus04

minus02

More Constraints

λ12 gt 0

λ1λ2 (89)

Wminus(Z)Wminus(Z)

Wminus(Z)

W +(Z)W +(Z)

W +(Z)Hplusmn(A0)

f(νf )f(νf )

λL λ2L λLλ1

g2 g2 λLg

λLyf g2

83 Gamma Rays

Continuum

vrelσff =Ncπα

sin4 θWm4W

(1 minus 4m2f

(sminusm2h)

2 +m2hΓ

Gamma-Ray Lines

g4 g4 g4

λLg3 λLg3

x = EγmH0

x2dNγdx

10minus4

10minus2

001 01 1

τ+τminus

EGRETDW=210-3

HESSDW=10-5

GLAST sensitivity

50 GeV boost ~104

70 GeV boost ~100

2 γΦγ

minus2sminus

log(Eγ [GeV])

-10-1 0 1 2 3 4

WIMP Mass [GeV]

H0 H0 rarr Zγ

H0 H0 rarr γγ

rarr Zγ γγ

40 50 60 70 80

C h a p t e r

Observed

92 The Data

bremss πο

Energy [GeV]

E2times

minus2sminus

minus1]

10minus7

10minus6

10minus5

10minus4

bremssπο

Energy [GeV]

10minus7

10minus6

10minus5

10minus4

93 The Claim

χ2 356χ2 (bg only) 17887

Energy [GeV]

E2times

minus2sminus

minus1]

10minus5

10minus4

10minus1 1 10 102

minus10

x [kpc]

minus5 0 50

z [kpc]

OuterSunInnerCenter

minus10

x [kpc]

kpc [M

BESS 98 1σ

BESS 98 2σ

T = 040-056 GeV

χ2 lt 10

10 lt χ2 lt 25

25 lt χ2 lt 60

60 lt χ2

C h a p t e r

Summary and Outlook

A p p e n d i x

(n)plusmn and A

(n)5 at

given by

for scalars i

for vectors minusi

q2 minusm2 + iǫ

= swAM + cwZM (A4a)

Z(n)5 =

a(n)0 minus M (n)

G(n)0 (A5a)

χ3 (n) =M (n)

a(n)0 +

G(n)0 (A5b)

Wplusmn (n)5 =

G(n)plusmn (A5c)

χplusmn (n) =M (n)

a(n)plusmn +

G(n)plusmn (A5d)

ψ(n)s = sinα(n)ξ

ψ(n)d = cosα(n)ξ

A2 Vertex Rules

M(1)X equiv

M (1)2 +m2X (A7d)

mZ = mWcw (A8a)

gY = ecw (A8b)

g = esw (A8c)

λ =g2m2

V micro2

q3V ν3

q1V ρ1 igV1V2V3

microηρν]

gA(0)W

(10)+ W

(10)minus

= minuse (A9)

gZ(0)W

(01)+ W

(01)minus

= minuscw g (A10)

(1)3 W

(01)+ W

(10)minus

= minusg (A11)

V micro1

gZ(0)Z(0)h(0) =g

cwmZ (A12)

(0)+ W

gZ(0)W

(1)plusmn a

(1)∓

= plusmnig mZ

gZ(0)A3 (1)h(1) = gmZ (A15)

(0)plusmn A3 (1)a

(1)∓

= ∓ig mW

(0)plusmn B(1)a

(1)∓

= ∓igY mW

(0)plusmn W

(1)∓ a

= plusmnig mW

(0)plusmn W

(1)∓ h(1) = gmW (A20)

gA3 (1)A3 (1)h(0) = gmW (A21)

gB(1)B(1)h(0) = gs2wc2w

mW (A23)

(1)+ W

gA(0)W

gZ(0)W

gA(0)W

(1)plusmn G

(1)∓

= ∓ieM (1)W (A27)

(0)plusmn B(1)G

(1)∓

= ∓igY mW

(1)plusmn B(1)χ∓ (0) = ∓igY mW (A29)

V micro

gA(0)a

(1)+ a

(1)minus

= e (A30)

gZ(0)a

(1)+ a

(1)minus

2(gcw minus gY sw)

M (1)2

2 + gcwmW

2 (A31)

gZ(0)h(1)a

= minusi g

(0)plusmn a∓(1)a

= ∓g2

M (1)2

M(1)W M

1 minus 2m2

(0)plusmn a∓(1)h(1) = i

gA3 (1)a

(1)0 h(0) = i

gA3 (1)G

(1)0 h(0) = i

gB(1)a

(1)plusmn a

(1)∓ h(0) = i

gA(0)χ+ (0)χminus (0) = e (A39)

minus gY

gZ(0)χ3 (0)h(0) = ig

2cw(A41)

(0)plusmn χ∓ (0)h(0) = i

2(A42)

(0)plusmn χ∓ (0)χ3 (0) = ∓g

2(A43)

gA(0)G

(1)+ G

(1)minus

= e (A44)

gB(1)χplusmn(0)a

(1)∓

= plusmngY

gB(1)χplusmn(0)G

(1)∓

= plusmngY

2(A47)

gB(1)G

(1)plusmn G

(1)∓ h(0) = i

S3 igS1S2S3

gh(0)h(01)h(01) = minus3

gh(0)a

(1)+ a

(1)minus

= minusgmW

M (1)2

gh(0)a

(1)0 a

= minus g

M (1)2

gh(0)χ3 (0)χ3 (0) = minusg2

gh(0)a

(1)plusmn G

(1)∓

2M (1)

1 minus m2h

gh(0)G

(1)plusmn G

(1)∓

= minusg2mW

2 (A56)

gA(0)ξ(0)ξ(0) = Qe (A57)

gA(0)ξ

(1)sdξ(1)sd

= Qe (A58)

gZ(0)ξ

(1)dξ(1)d

gZ(0)ξ

(1)s ξ

gZ(0)ξ

(1)s ξ

(1)3 ξ(0)ξ

gB(1)ξ(0)ξ

(0)+ U

(0)i D

=gradic2VijPL (A67)

(0)+ U

(1)diD

=gradic2Vij cosα

cosα(1)Dj

(0)+ U

(1)siD

=gradic2Vij sinα

sinα(1)Dj

(0)+ U

(1)diD

=gradic2Vij cosα

sinα(1)Djγ5 (A70)

(0)+ U

(1)siD

=gradic2Vij sinα

cosα(1)Djγ5 (A71)

(1)+ U

(0)i D

=gradic2Vij cosα

(1)DjPL (A72)

(1)+ U

(0)i D

(1)DjPL (A73)

(1)+ U

(1)diD

=gradic2Vij cosα

(1)UiPL (A74)

(1)+ U

(1)siD

(1)UiPL (A75)

S igSξ1ξ2

gh(0)ξ

(1)dξ(1)d

= minusg mξ

gh(0)ξ

(1)s ξ

= minusg mξ

gh(0)ξ

(1)dξ(1)s

= minusg mξ

γ5 (A79)

gh(1)ξ(0)ξ

= minusg mξ

gh(1)ξ(0)ξ

= gmξ

ga(1)0 ξ(0)ξ

minusigT3mξ

ga(1)0 ξ(0)ξ

+igT3mξ

ga(1)+ U

(0)i D

= minusi gradic2

cosα(1)DjPL

+igradic2

ga(1)+ U

(0)i D

= igradic2

sinα(1)DjPL

+igradic2

ga(1)+ U

(1)diD

= minusi gradic2

sinα(1)UiPL

minusi gradic2

ga(1)+ U

(1)siD

= igradic2

cosα(1)UiPL

minusi gradic2

dagger =

A(1) micro

c∓(1)

c∓(1) q

= plusmnie qmicro

c∓(1)

= minusi g2mWξ

V micro1

igV1V2V3V4 middot(

(0)minus W

(0)+ W

= g2 (A88)

(1)minus W

(1)+ W

2g2 (A89)

(10)minus W

(01)+ W

= g2 (A90)

(10)minus W

(01)minus W

(10)+ W

= g2 (A91)

gZ(0)Z(0)W

(0)minus W

= minusg2c2w (A92)

gZ(0)Z(0)W

(1)minus W

= minusg2c2w (A93)

gA(0)A(0)W

(0)+ W

(0)minus

= minuse2 (A94)

gA(0)A(0)W

(1)+ W

(1)minus

= minuse2 (A95)

gZ(0)A(0)W

(0)+ W

(0)minus

= minusegcw (A96)

gZ(0)A(0)W

(1)+ W

(1)minus

= minusegcw (A97)

(1)3 A

(1)3 W

(1)+ W

(1)minus

= minus3

2g2 (A98)

(1)3 A

(1)3 W

(0)+ W

(0)minus

= minusg2 (A99)

(1)3 Z(0)W

(01)+ W

(10)minus

= minusg2cw (A100)

V micro1

iηmicroνgV1V2S1S2

gZ(0)Z(0)h(0)h(0) =g2

2c2w(A101)

(0)plusmn W

(0)∓ h(0)h(0) =

2(A102)

gB(1)B(1)h(1)h(1) =3g2

4(A103)

gB(1)B(1)a

(1)0 a

M (1)2

2 (A104)

(1)3 A

(1)3 h(1)h(1) =

4(A105)

(1)3 A

(1)3 a

(1)0 a

M (1)2

2 (A106)

(1)plusmn W

(1)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A107)

gB(1)B(1)a

(1)+ a

(1)minus

M (1)2

2 (A108)

(1)3 A

(1)3 a

(1)plusmn a

(1)∓

2+ 13m2

2 (A109)

(1)plusmn W

(1)∓ h(1)h(1) =

4(A110)

(1)plusmn W

(1)∓ a

(1)0 a

2+ 13m2

2 (A111)

(1)∓ W

(1)plusmn a

(1)∓

2minus 13m2

2 (A112)

gB(1)A

(1)3 h(1)h(1) = minus3gY g

4(A113)

gB(1)A

(1)3 a

(1)0 a

= minus3gY g

M (1)2

2 (A114)

gB(1)A

(1)3 a

(1)plusmn a

(1)∓

=3gY g

M (1)2

2 (A115)

gB(1)W

(1)plusmn h(1)a

(1)∓

= ∓i3gY g

(A116)

gB(1)W

(1)plusmn a

(1)0 a

(1)∓

= minus3gY g

M (1)2

M(1)W M

(A117)

(1)3 W

(1)plusmn a

(1)0 a

(1)∓

= minusg2

M(1)W M

(A118)

gB(1)B(1)h(0)h(0) =g2

2(A119)

gZ(0)Z(0)h(1)h(1) =g2

2c2w(A120)

(1)3 A

(1)3 h(0)h(0) =

2(A121)

gZ(0)A

(1)3 h(0)h(1) =

2cw(A122)

gZ(0)Z(0)a

(1)0 a

M (1)2

2 (A123)

(0)plusmn W

(0)plusmn a

(1)∓ a

(1)∓

= minusg2 m2W

2 (A124)

gA(0)A(0)a

(1)plusmn a

(1)∓

= 2e2 (A125)

gZ(0)Z(0)a

(1)plusmn a

(1)∓

4c4wm2W

2M(1)W

2 (A126)

(01)+ W

(01)minus h(10)h(10) =

2(A127)

(01)+ W

(10)minus h(01)h(10) =

2(A128)

(0)plusmn W

(0)∓ a

(1)0 a

2+ 3m2

2 (A129)

(0)plusmn W

(0)∓ a

(1)plusmn a

(1)∓

2 (A130)

gB(1)A

2(A131)

gB(1)Z(0)h(1)h(0) =egY

2cw(A132)

gA(0)Z(0)a

(1)plusmn a

(1)∓

2c2wm2W

2 (A133)

gA(0)W

(01)plusmn h(10)a

(1)∓

= ∓i e2

(A134)

gB(1)W

(0)plusmn h(0)a

(1)∓

= ∓i e2

(A135)

gA(0)W

(0)plusmn a

(1)0 a

(1)∓

= minus e2

mW +M(1)W

M(1)W M

(A136)

gZ(0)W

(01)plusmn h(10)a

(1)∓

= plusmni e2

(A137)

gZ(0)W

(0)plusmn a

(1)0 a

(1)∓

2s2wcw

M(1)W M

(A138)

gA(0)A(0)G

(1)+ G

(1)minus

= 2e2 (A139)

gB(1)B(1)χ

(0)+ χ

(0)minus

2(A140)

gB(1)B(1)G

(1)+ G

(1)minus

2 (A141)

gB(1)B(1)G

(1)∓ a

(1)plusmn

mWM(1)

2 (A142)

gB(1)B(1)χ3(0)χ3(0) =g2

2(A143)

gB(1)B(1)G

(1)0 G

2 (A144)

gB(1)B(1)G

(1)0 a

mZM(1)

2 (A145)

gB(1)A(0)G

(1)plusmn χ

(0)∓

= plusmniegY

(A146)

gB(1)A(0)a

(1)plusmn χ

(0)∓

= plusmniegY

(A147)

(1)plusmn A(0)h(0)G

(1)∓

= ∓i eg2

(A148)

igS1S2S3S4

gh(0)h(0)h(0)h(0) = minus6λ (A149)

gh(1)h(1)h(1)h(1) = minus9λ (A150)

ga(1)0 a

(1)0 a

= minus3g2c2wm2ZM (1)2 + 9λM (1)4

4 (A151)

gh(1)h(1)a

(1)0 a

= minusg2m2

Zc2w + 6λM (1)2

2M(1)Z

2 (A152)

gh(1)h(1)a

(1)+ a

(1)minus

= minusg2m2

W+ 6λM (1)2

2M(1)W

2 (A153)

ga(1)0 a

(1)0 a

(1)+ a

(1)minus

= minusg2m2

WM (1)2(1 + c4w)c4w + 6λM (1)4

2M(1)W

2 (A154)

ga(1)+ a

(1)+ a

(1)minus a

(1)minus

= minus2g2m2WM (1)2 + 6λM (1)4

2 (A155)

gh(1)h(1)h(0)h(0) = minusλ (A156)

gh(0)h(0)a

(1)0 a

= minusg2m2

Zc2w + 4λM (1)2

2M(1)Z

2 (A157)

gh(0)h(0)a

(1)+ a

(1)minus

= minusg2m2

W+ 4λM (1)2

2M(1)W

2 (A158)

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P a r t I I

Scientific Papers

P a p e r I

P a p e r I I

P a p e r I I I

P a p e r I V

P a p e r V

P a p e r V I

P a p e r V I I

Abstract

Preface



Special Relativity

General Relativity


Evolving Universe

Initial Conditions


Dark Energy

Dark Matter








Simulation Setups



Nonsphericity

Axis Ratios

Alignments




Halo Substructure















Compactification

Kaluza-Klein Parity

The Lagrangian



Mass Spectrum


Relic Density



Direct Detection

Indirect Detection


Gamma-Ray Continuum

Gamma Line Signal



Supersymmetry

Some Motivations

The Neutralino








More Constraints


Gamma Rays

Continuum

Gamma-Ray Lines


Dark Matter Signals

The Data

The Claim

The Inconsistency



The Status to Date

Summary and Outlook



Vertex Rules

Bibliography


Documents

Light from Dark Matter