Cosmological Dark Matter and the Isotropic Gamma-Ray ...317422/FULLTEXT01.pdfConstraints on Cosmological Dark Matter Annihilation from the Fermi-LAT Isotropic Diffuse Gamma-Ray Measurement

Cosmological Dark Matter and theIsotropic Gamma-Ray Background

Measurements and Upper Limits

Alexander Sellerholm

Cover image: A 15 h−1 Mpc thick slice of the Millennium simulation at redshift z=0. The simulated box has asize of 500 h−1 Mpc and contains 1010 dark matter objects. Credit to V. Springel et al. (2005).

c© Alexander Sellerholm, Stockholm 2010

ISBN 978-91-7447-082-6 (pp. i-xiii, 1-98)

Printed in Sweden by US-AB, Stockholm 2010

Distributor: Department of Physics, Stockholm University

Abstract

This thesis addresses the isotropic diffuse gamma-ray background, as mea-sured by the Fermi gamma ray space telescope, and its implications for indi-rect detection of dark matter. We describe the measurement of the isotropicbackground, including also an alternative analysis method besides the onepublished by the Fermi-LAT collaboration. The measured isotropic diffusebackground is compatible with a power law differential energy spectrum witha spectral index of −2.41±0.05 and −2.39±0.08, for the two analysis meth-ods respectively. This is a softer spectrum than previously reported by theEGRET experiment. This rules out any dominant contribution with a signifi-cantly different shape, e.g. from dark matter, in the energy range 20 MeV to102.4 GeV. Instead we present upper limits on a signal originating from anni-hilating dark matter of extragalactic origin. The uncertainty in the dark mattersignal is primarily dependent on the cosmological evolution of the dark mat-ter distribution. We use recent N-body simulations of structure formation, aswell as a semi-analytical calculation, to assess this uncertainty. We investigatethree main annihilation channels and find that in some, but not in all, of ourscenarios we can start to probe, and sometimes rule out, interesting parameterspaces of particle physics models beyond the standard model.

We also investigate the possibility to use the angular anisotropies of theannihilation signal to separate it from a background originating from conven-tional sources, e.g. from active galactic nuclei. By carefully modelling the per-formance of the Fermi gamma-ray space telescope and galactic foregroundswe find that this method could be as sensitive as using information from theenergy spectrum only.

iii

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I E. A. Baltz, B. Berenji, G. Bertone, L. Bergrström, E. Bloom, T.Bringmann, J. Chiang, J. Cohen-Tanugi, J. Conrad, Y. Edmonds,J. Edsjö, G. Godfrey, R. E. Hughes, R. P. Johnson, A. Linonetto,A. A. Moiseev, A. Morselli, I. V. Moskalenko, E. Nuss, J. F.Ormes, R. Rando, A. J. Sander, A. Sellerholm, P. D. Smith, A.W. Strong, L. Wai and B. L. Winter.Pre-launch estimates for GLAST sensitivity to Dark Matter. JCAP,0807 013 (2008) .

II A. A. Abdo et al. [Fermi-LAT collaboration]Constraints on Cosmological Dark Matter Annihilation from theFermi-LAT Isotropic Diffuse Gamma-Ray Measurement. JCAP04014 (2010) .

III A. A. Abdo et al. [Fermi-LAT collaboration]Spectrum of the Isotropic Diffuse Gamma-Ray Emission Derivedfrom First-Year Fermi Large Area Telescope Data. Physical Re-view Letters, 104 (10):101101 (2010).

IV A. Cuoco, A. Sellerholm,. J. Conrad, S. HannestadAnisotropies in the Diffuse Gamma-Ray Background from DarkMatter with Fermi LAT: a closer look. To be submitted (2010).

Reprints were made with permission from the publishers.

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Gamma-ray Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Detection techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Sources of gamma-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 The Fermi Gamma-Ray Space Telescope . . . . . . . . . . . . . . . . . . . . 93.1 Instrumental overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Instrument Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Fast detector simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Galactic diffuse gamma-ray emission . . . . . . . . . . . . . . . . . . . . . . . . 174.1 The Milky Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Radiative processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Pre-Fermi measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 The first year Fermi measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Measurements of The Isotropic Gamma-Ray Background . . . . . . . . 315.1 Pre-Fermi measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Origin of Diffuse Extragalactic Gamma-Rays . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Fermi analysis of the IGRB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.2 The residual background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.3 Additional cuts - the dataclean class . . . . . . . . . . . . . . . . . . . . . . . 345.2.4 Estimating the error in the Monte Carlo simulations of the esidual back-

ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.5 Estimating the residual background using cosmic ray trigger rate . . . . . 365.2.6 Comparing the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3 Linear Extrapolation analysis of the Fermi data . . . . . . . . . . . . . . . . . . . . . 395.3.1 Photon data treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.2 Data and model format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.3 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.4 Analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.5 Model Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.3.6 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

viii CONTENTS

5.3.7 First year Fermi results and discussion . . . . . . . . . . . . . . . . . . . . . . . 456 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.1 The evolving Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Nature of Dark Matter - Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 Nature of Dark Matter - Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4 WIMP Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.5 Looking for Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.5.1 Direct Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.5.2 Indirect searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.5.3 At accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.5.4 Dark Matter Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Cosmological Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.1 Particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Dark matter distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.3 High energy γ -ray environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.4 Other isotropic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8 Upper limits by Fermi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.1 Fermi sensitivity to Cosmological WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . 778.2 Upper limits on Cosmological Dark Matter by Fermi . . . . . . . . . . . . . . . . . . 80

8.2.1 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

9.2.1 Angular power spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889.2.2 Fermi Exposure and PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909.2.3 Sensitivity estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9410 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9911 Sammanfattning på svenska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Preface

This thesis investigates the capabilities of the Fermi gamma-ray space tele-scope to detect a signal from dark matter. There are many potential sourcesfor such signals and here we focus on a particular one of extragalactic ori-gin, and compare that to the measurement of the isotropic gamma-ray back-ground. The extragalactic dark matter signal, the extraction of the isotropicbackground and how the latter can constrain the former have been my maintopics of research. These areas of research bring together many interesting anddiverse fields of physics and astronomy. Covering them all to the extent thatthey deserve is beyond the scope of this thesis. I have tried to touch brieflyon some of the aspects relevant for the work presented in the accompanyingpapers and put them into context.

My contributionsMy initial contribution to the Fermi-LAT collaboration was the part on cos-mological WIMPs in the pre-launch sensitivity estimate to dark matter annihi-lation signals. I wrote the section, did theoretical calculations and performedthe analysis. This is published in [1] (PAPER I).

A natural follow up on this work was to search for this signal in the actualdata once the Fermi-LAT was launched. Since a dominating signal from darkmatter was lacking in the data, this resulted in upper limits on a possible con-tribution. This work is published in [2] (PAPER II). For this paper I wrote theinitial draft and did the statistical analysis.

I was also involved in the measurement of the Isotropic gamma-ray back-ground during a visit to SLAC National Laboratories. My main contributionwas to the initial investigation of the charged particle contamination of thesignal, summarised in section 5.2.5. I also developed an alternative analysismethod of the isotropic signal, described in section 5.3. This analysis con-firmed the result of the official measurement, published in [3] (PAPER III).

A detailed investigation of the sensitivity of Fermi to small scaleanisotropies of a dark matter signal is presented in PAPER IV, which is in thefinal stages of publication. I provided the background models and did all theFermi related simulations for this paper.

ix

x PREFACE

AcknowledgmentsDuring my graduate studies I have been depending on help and aid from manypeople, in particular my supervisor Jan Conrad. Thank you! Not only forthe science discussions and general guidance, but also for the endless moralsupport, for always believing in your students and for spreading those goodvibes wherever you are. I would also like to thank my secondary supervisorsLars Bergström and Joakim Edjsö for help and support. I would like to thankMichael Gustafsson and Gabrijela Zaharijas for bringing the Cosmo WIMPproject together, Alsessandro Cuoco for being the master of anisotropies andMarkus Ackermann for the support with the linear analysis of the IGRB, An-tje Putze for helping me with cosmic rays, Barbro Åsman for always beinga supportive mentor, Erik Lundström for all the years we shared office, TomiYlinen for all the travels we endured, Elisabet Oppenheimer, Mona Holger-strand and Marieanne Holmberg for taking care of me, and Joakim Lundborgfor not charging me for all those hours of computer support.

I would like to thank the hard working people of the Fermi-LAT collabora-tion, especially members of the Dark Matter and New Physics working group.My work has in particular been influenced by people from the Bay Area:Markus Ackermann, Marco Ajello, Elliot Bloom, Jim Chang, Seth Diegel,Yvonne Edmonds, Gudlagur Johannesson, Stuart Marshal, Simona Murgia,Troy Porter, Stefe Ritz and Ping Wang. A special thank to Marco Ajello, theStanford Italians and the Stanford Alpine Club for making my time at SLACsimply fantastic.

I would like to express my sincere gratitude to my colleagues in the CoPSgroup, for interesting discussions, all those fika-times and late dinners.Especially my physics family that means so much to me: Marianne Johansen,Rachel Rosen, Sara Rydbeck, Cecilia Marini Bettolo, Rickard Ström andKatarina Bendtz.

I would also like to thank my collaborators in the work conducted outsidemy main field of research. Sören Holst, Stefan Sjörs and Sofia Sivertsson forall the effort we spend on Hollywood Physics. A special thanks to Pelle Beck-man, Andreas Bergfeldt, Joakim Edsjö and Rickard Ström for helping with theawesome movie posters. The equal opportunities group for always offering anenvironment for stimulating discussions. The Phd-council and the PCPC, fortrying to improve the work and party situation at Fysikum and the membersof CoWS for all the precious drops of whisky we shared.

Thanks to my family, without you nothing would have been possible: mymother Britta, my uncle Pelle and his family; Helena, Majken and Linnéa, mygrandfather Sven, my pack of sysslingar and their mothers. And the best offriends: Niklas, Pelle, Joakim, Klara, Stina, Oscar, Apostolos and the climbersand skiers from Bergsidan with whom I have shared so many thrilling mo-ments.

PREFACE xi

Finally, thanks to Marianne for having the courage to form a family withme.

I acknowledge generous financial support from the Swedish research council(through HEAC) and Carl-Erik Levins stiftelse, John Söderbergs donnation-sstipendium and STINT for travelling support.

Overview of thesisThe thesis is organised as follows: chapters 1 - 4 review aspects of gamma-rayastronomy, the Fermi Gamma-Ray Telescope and a short introduction to thegalactic diffuse emission and cosmic ray propagation. Chapter 5 describesthe measurement of the isotropic diffuse gamma-ray background. Chapter 6contains a brief summary of dark matter, in particular WIMP dark matter,and its structure in the Universe. Chapter 7 describes in some detail thecosmological dark matter signal as it will be used in chapter 8. The resultsof chapter 5 and 7 are combined in chapter 8 to set upper limits on selfannihilating cosmological dark matter. Chapter 9 describes the prospects forFermi to use the anisotropies in the angular distribution of the annihilationsignal to detect dark matter. Finally chapter 10 contains a summary andoutlook.

Alexander SellerholmMay 2010

1

1. Introduction

The emergence of precision cosmology has given rise to a concordance model.In this model the Universe is flat and described by forces and constituents thatto the most extent are of unknown nature. The main part is in the form ofdark energy, a mysterious force that accelerates the expansion of the Universeat present time. The second largest contribution to the energy density of theuniverse is dark matter (DM). Although the precise physics behind the darkmatter is not known, there are a large number of independent observations thatpoint towards the existence of non-baryonic, collisionless and dissipation-freematter; cold dark matter (CDM).

A large group of theoretically well motivated candidates are weaklyinteracting massive particles (WIMPs). These are particles that appear invarious extensions of the standard model of particle physics (SM), withmasses of typically 100 GeV to a few TeV and couplings at the electroweakscale. Usually the WIMP is the lightest of an entire family of new particlesthat are prevented to decay into standard model particles by symmetries ofthe theory.

Even if the present indications of dark matter are of gravitational nature, thereare prospects to detect it also by other means. These are usually grouped intotwo kind of methods; direct and indirect detection. The basic principle behinddirect detection is to measure the recoil energy as WIMPs scatter off atomicnuclei in detectors.

Indirect detection experiments search for astronomical signals that wouldappear from pair annihilation or decay of dark matter particles into stan-dard model particles. These kinds of interactions are believed to occur if thedark matter was once in equilibrium with the early Universe through pair-production and pair-annihilation. The annihilation products could either bemeasured directly or they could decay to detectable particles. Thus, one ex-pects to see signatures of dark matter in gamma-rays, neutrinos and cosmicrays from regions with enhanced dark matter density.

With the launch of the Fermi Gamma-Ray Telescope [4, 5], a new windowhas been opened to the gamma-ray sky. The prospects for Fermi to detectsignals from annihilating dark matter of various kinds have been a subject ofintense research up until it launched in July 2008. By now there are already

2 INTRODUCTION CHAPTER 1

numerous constraints on signals from dark matter using the data from theFermi Gamma-Ray Telescope.

3

2. Gamma-ray Astronomy

Unlike many other fields of astronomy, in particular optical which has beenapplied for centuries, gamma-ray astronomy is a rather new field pioneeredin the late 1950’s [6]. There are many reasons for this. In particular gamma-rays where discovered as late as 1900, by French chemist and physicist PaulVillard, and gamma-rays does not easily penetrate the Earth atmosphere. Thenature of gamma-rays also presents astronomers with the challenge that theycan not be detected in the way electromagnetic waves usually are. Observa-tional astronomy is based on the fact that electromagnetic radiation can bereflected on surfaces of different geometries and therefore focused on a smalldetector area. This is not possible with gamma-rays since if they (at all) in-teract with material, the large amount of energy deposited will disrupt it indifferent ways. This is why gamma ray telescopes either are designed as par-ticle physics detectors or aimed to see the traces of the gamma-rays as theycollide with the Earth atmosphere, rather than the photons themselves. Sincegamma-rays are more like particles than waves they are typically describedby their energy, given by Eγ = hν , where h is Planck’s constant and ν is thefrequency of the electromagnetic wave. Typically energies are in units of 106

or 109 electron volts (MeV or GeV).Due to the high attenuation of gamma-rays in the Earth atmosphere, the

first successful gamma-ray astronomy experiments were spaceborne. The ne-cessity to have experiments in space is of course the main limiting factor onapparatus design and size.

2.1 Detection techniquesThe principle gamma-ray - matter interactions are via the photoelectric effect,Compton scattering, and pair production into pairs of electrons and positrons.The photoelectric effect is dominant below ∼ 1 MeV (and therefore of nogreat interest for our purpose) pair production above∼ 10 MeV and Comptonscattering in between. The energy dependent cross sections for these processesdetermine what kind of detector which will be suitable for detecting photonsof a particular energy. In chapter 3 we will review in more detail the princi-ples behind pair production detectors, when discussing the Fermi Gamma-RaySpace Telescope.

4 GAMMA-RAY ASTRONOMY CHAPTER 2

To accurately measure the energy and direction of high energy gamma-raysvia pair production, and accumulate high enough statistics, the detector has tobe built larger and larger (or more massive) as the energy of the gamma-raysincrease. Around ∼ 1 TeV this makes spaceborne experiments impractical.Instead the cherenkov light from the particles produced when the gamma-ray interact with matter in the atmosphere, is used to reconstruct its directionand energy. This kind of telescopes are called cherenkov telescopes and aretypically one or several mirror arrays built on high ground.

The era of spaceborne gamma-ray astronomy started with the Explorer-11satellite in 1962. During its seven months mission it detected 22 spatially ran-dom distributed gamma-rays and 22000 cosmic rays [7]. The OSO-3 satellite,launched in 1968 [8], found a concentration of gamma-rays in the galacticplane. In 1972 the SAS-2 satellite [9] could correlate the gamma-ray emis-sion with the galactic structures and gave the first all-sky image of the sky ingamma-rays. A more detailed observation of the gamma-ray sky and detectionof 25 individual sources was done by the COS-B satellite in 1975 [10].

In 1991 the Compton Gamma-Ray Observatory (CGRO) was launched[11]. The CGRO carried onboard four different instruments, of which theEnergetic Gamma Ray Experiment Telescope (EGRET) was the biggest one.The observatory covered in total an energy range from 20 keV to 10 GeV.EGRET had high enough resolution to correlate gamma-ray sources withprevious detected objects and established that active galactic nuclei (AGN)where the primary source of extragalactic high energy gamma-rays. EGRETalso offered the first detailed study of the diffuse galactic and extragalacticgamma-ray emission, which will be discussed in chapter 4 and 5. The thirdEGRET catalog of sources is shown in figure 2.1.

2.2 Sources of gamma-raysBecause of their high energy, gamma-rays are typically not produced inthermal processes. The most important radiation processes giving riseto gamma-rays are inverse Compton-scattering of light on high energyelectrons, bremsstrahlung from charged particles deflected by atomic ormolecular nucleus and the decay of π0-mesons produced by cosmic rayinteractions with the interstellar medium. These processes will be discussedin more detail in chapter 4 when discussing the galactic diffuse emission.

The astrophysical objects that emit gamma-rays all have in common thatthe environment in which the radiative processes take place are extreme insome sense. For comprehensive reviews on the subject see e.g. [14] or [15].

AGNs are believed to host supermassive black holes (with masses in therange 105 to 109 solar masses). The large amount of matter accreting around

SECTION 2.2. SOURCES OF GAMMA-RAYS 5

+90

-90

-180+180

Third EGRET CatalogE > 100 MeV

Active Galactic NucleiUnidentified EGRET Sources

PulsarsLMCSolar FLare

Figure 2.1: Above: The third catalog of high energy EGRET soures (3EG) based on[12]. The size of the markers are proportional to the highest measured intensity (>100MeV). Below: First Fermi-LAT catalog [13]. Comparing the figures illustrates thesuperior sensitivity of the Fermi-LAT.

6 GAMMA-RAY ASTRONOMY CHAPTER 2

the black hole produces huge collimated jets. When the relativistic jets aredirected towards the Earth we observe strong emission in gamma-rays. Thisclass of AGNs are referred to as blazars. When viewed over a wide rangeof wavelengths the blazars energy spectrum seems to feature two prominentbumps. One is located somewhere between IR and UV wavelengths and be-lieved to originate from synchrotron radiation of the high energy electrons inthe jet. The second peak is located at gamma-ray energies and believed tooriginate from inverse Compton-scattering.

41

42

43

44

45

46

47

48

49

-5 0 5 10

8 13 18 23

Log

10 ν

Lν[

erg

s-1]

Log10 Eγ [eV]

Log10 ν [Hz]

Figure 2.2: An empirical spectral energy density sequence of blazars from [16]. Lu-minosity decreases counting the spectra from top to bottom and at the same time thepeaks are shifted to higher energies The right most data points are average fits toEGRET data [17].

The exact physics of blazars is still a matter of debate [18]. The locationof the peaks have been found to dependent on the total luminosity of theblazar, with high luminosity sources having their peaks shifted towards lowerenergies and vice versa. A number of spectra where the trend is visible isshown in figure 2.2. Typically one therefore divides blazars into two classes,flat spectrum radio quasars (FSRQs) with high luminosity and emissionlines in the optical band, and BL Lacertae objects (BL Lacs) with lowerluminosity, no emission lines and the peaks at higher energies. Whetherthese two classes are intrinsically different or part of the same population atdifferent stages of evolution is still unknown. Blazars are also highly variablesources, where the emission at certain wavelengths can vary significantlywithin an hour or less. From the point of view of gamma-ray astronomy in

SECTION 2.2. SOURCES OF GAMMA-RAYS 7

the GeV range, blazars appear to have power-law energy spectra. The FSRQsare typically softer than the BL Lacs with average spectral indices ∼ −2.5and ∼ −2.2 respectively as observed by Fermi [19] and ∼ −2.2 and −2 byEGRET [20].

The remnants of supernovae, both the pulsar (or neutron stars) and the ejectedshell from the explosion, are strong sources of gamma-rays. Gamma-raybursts are another kind cosmic explosions that are some of the most luminousevents observed in the Universe [21]. Their origin are not completelyunderstood but a common theory is that they are the collimated jet from corecollapsing supernovae, viewed head on. The gamma-ray explosion typicallyoccur during a very short period of time, from a few milliseconds to less thanan hour. Shorter bursts have also been observed. Their origin are even moreuncertain but the merging of binary neutron stars has been suggested as apossible explaination.

From the point of view of Earth, the Milky Way is the most dominant sourceof gamma-rays (c.f. chapter 4). Also other galaxies have been observed ingamma-rays. EGRET detected the Large Magellanic Cloud [22] which hasbeen confirmed by the Fermi-LAT [23]. The Fermi-LAT has also discoveredgamma-ray emission from sources that were not detected by EGRET, for ex-ample starburst galaxies [24] and radio galaxies (i.e. non blazar AGNs) [25].The full first Fermi-LAT source catalog can be seen in figure 2.1.

9

3. The Fermi Gamma-Ray SpaceTelescope

This chapter briefly describes the Fermi-LAT. For a full description of pre-launch and in-orbit performance see [4, 5].

3.1 Instrumental overviewThe Fermi gamma-ray telescope consists of two independent instruments: theLarge Area Telescope (LAT) and the Burst Monitor (GBM). The LAT is theprimary instrument, consisting of a tracker, a calorimeter and a surroundinganti-coincident detector (ACD). It has a modular design, consisting of 16 iden-tical towers with a base area of 40 × 40 cm2 . The towers are arranged in afour by four matrix, where each tower has its own setup of tracker, calorimeterand data acquisition device. A schematic overview of the LAT is shown in fig-ure 3.1. The LAT is, as EGRET was before, a pair conversion telescope. Thismeans that it detects gamma-rays by inducing them to pair produce, and thenmeasures the deposited energy of the following cascade in the tracker andcalorimeter. By accurately reconstructing the track of the electron- positronpair in the tracker, the direction of the incident photon can be calculated. TheGBM is sensitive to gamma-rays in the energy range 10 keV-30 MeV and isdesigned to provide a large field-of-view for localising gamma-ray bursts. TheGBM will not be used in any search for dark matter and will therefore not bediscussed any further here.

TrackerThe tracker is made out of 18 trays, each consisting of two Silicon Strip De-tectors (SSD) and with intermediate layers of tungsten conversion foils, in thetop 16 trays. The trays are all rotated 90 with respect to each other and allhave separate read-out. In this way each lower SSD forms a x-y grid with theupper SSD on the tray below, except for the top and bottom trays. This gives17, stacked, x-y coordinates which accurately can pin-point the position of theincident photon. The strip-silicon technique is one of the mayor advantages ofthe LAT as compared to EGRET, which used a spark chamber.

10 THE FERMI GAMMA-RAY SPACE TELESCOPE CHAPTER 3

CalorimeterThe LAT calorimeter is, like the tracker, a layered device made out of 1536scintillating CsI(Tl) crystals. The entire device weighs about 1500 kg andmakes up 60 % of the entire Fermi weight of 2500 kg. The main purposeof the calorimeter is to accurately measure the energy of the incoming pho-tons. When high energy photons interact in the calorimeter or tracker con-version material they produces e+e− pairs which are energetic enough toproduce bremsstrahlung photons which again pair produce. Alternation be-tween bremsstralung and pair production leads to a multiplication of photons,positrons and electrons which is called an electromagnetic (EM) shower.

Each calorimeter module consists of 8 layers, with 12 CsI(Tl) crystals ineach, that are rotated 90 with respect to each other to form a x-y grid. Eachcrystal has four diods that read out the scintillation light. The segmented, gridstructure of the calorimeter gives both longitudinal and transverse informationof the energy deposited. The ability to follow the showers in all directions alsoenables to discriminate between EM-showers and hadronic showers and cantherefore be used to reject hadronic background.

The anti-coincidence detector and data acquisition setupThe tracker is covered by an ACD, which is sensitive to charged particles andwhose primary purpose is to reject the background originating from chargedcosmic rays. The ACD is made out of 1 cm thick scintillator tiles which areread out by miniature photomultiplier tubes. The ACD generates a veto when-ever a charged particle traverses it. This means that even if the trigger andcalorimeter picks up a signal it is rejected since it most likely was causedby a charged cosmic-ray that entered the detector. The charged, cosmic-raybackground (mainly protons) outnumber the high-energy gamma-rays by afactor of 105, requiring a similar efficiency of the background rejection capa-bility. Cosmic rays that pass the background rejection analysis, and are mis-interpreted as gamma-rays are called the residual (cosmic ray) background.The tiling of the ACD is made to prevent self-veto events from "backsplash".Backsplash occurs when particles, created by the incident photon, are scat-tered backwards. On their way out of the detector, the charged particles gener-ate a veto in the ACD and they are mistaken for background events. The tilingof the ACD enables to crosscheck the veto trigger location with the recon-structed path of the incident photon, thus reducing self-veto events. The tilesare made smaller towards the calorimeter to make this procedure more accu-rate for particles that enter the detector from the side. The EGRET instrumentonly had a single ACD shield and suffered almost a 50% loss in detectionefficiency, above 10 GeV, due to backsplash.

Because of the large amount of data that are collected in the LAT and thelimited data transfer rate down to Earth, the instrument must process the data

SECTION 3.1. INSTRUMENTAL OVERVIEW 11

Figure 3.1: A pictorial sketch of the LAT with one tower module highlighted. See textfor detail on the different parts of the LAT. Courtesy NASA/ Fermi-LAT collaboration.


Figure 3.2: Parametrisations of the Fermi-LAT IRF: PSF (upper row), effective area(middle row) and energy dispersion (lower row). Left column shows on-axis (zerophoton inclination angle) energy dependance. Right column shows the IRF at 10 GeVand dependance of inclination angle. From [26]

SECTION 3.2. INSTRUMENT RESPONSE FUNCTIONS 13

onboard and only send the most useful to the ground. This is achieved by anumber of algorithms and filters integrated in the system. The primary task isto distinguish between true gamma-ray events and cosmic-ray events, but alsoto separate interstellar gamma-rays from those produced in the Earth atmo-sphere due to interaction with cosmic rays. An accepted event is characterisedby more than one track from the same point in the tracker (at least three SSDsmust be activated and aligned) and an EM-shower in the calorimeter, or nosignal in the tracker but a sufficiently high energy deposition from the EM-shower in the calorimeter. If there is a signal in the ACD it must match thepreliminary reconstructed track of the incident photon otherwise it is vetoed.

3.2 Instrument Response FunctionsThe Instrument Response Function (IRF) is a mapping between the parame-ters that characterise an incoming photon to the same parameters as measuredby the detector. In the following sections we will focus on how the IRF is de-fined for Fermi and how it is incorporated in the Fermi collaboration softwarepackage Science tools 1. The LAT IRF is modeled as an effective area timesthe probability that a photon with a given set of parameters is detected with aset of observables. The photon parameters for the LAT are the energy E andinclination angle φ between the true source location and the LAT normal. Theevent parameters are the apparent energy E ′ and the apparent source positionφ ′.

Prior to launch there were three analysis classes, each with a different back-ground rejection level. These are called transient, source and diffuse, and havean expected averaged on-orbit residual background rate of 2, 0.4 and 0.1 Hzrespectively [4] (c.f. section 5.2.3 for an additional, post-launch analysis classwith an even lower residual background level). Each of the analysis classes hasits own set of IRFs. Also, starting from the front of the instrument, the LATtracker has 12 layers of Tungsten conversion foils of 3% radiation lengths(referred to as the front section), followed by 4 layers of Tungsten with 18%radiation lengths (referred to as the back section). These sections have intrin-sically different IRF due to multiple scattering. Therefore there are three IRFs,each with a back and front version. The full IRF is written as:

Rb/ fi (E ′, φ

′; E, φ) = Ab/ fi (E, φ)Db/ f (E ′; E, φ)Pb/ f

i (φ ′; E, φ), (3.1)

where A is the effective area, D is the energy redistribution function (or energydispersion), P is the Point Spread Function (PSF), b/ f represents front or backand i runs overt the analysis classes transient, source and diffuse. Note that A

1http://glast.gsfc.nasa.gov/ssc/data/


has units of area while D and P are probability distributions over the measuredparameters.

The IRF is a parametrization of the full detector response which is modeledby the detailed Monte Carlo code GLEAM (GLAST LAT Event Analysis Ma-chine) [4] based on GEANT42.

The Point Spread FunctionThe PSF is a distribution of measured angles φ ′ from which it is possible toreconstruct the most probable inclination angle φ . Effectively it determines theangular resolution of the instrument. Note that the angle φ ′ is really a vectorsince there are two independent directions on the sky and each has its ownprobability distribution. However we assume that the PSF is circular aroundthe true source position and that P is really P(θ ; E, φ) where θ is the anglebetween the true and apparent source position.

The bulk of the energy dependence of the PSF is contained in a scalingfunction:

θ(E) =[(p1(E/100)−0.8)2 + p2

2]1/2

, (3.2)

where p1 and p2 are constants that depend on the event class. In each logE−cos i bin the scaled angle deviation, δ = θ/θ , is fitted to the the distribution:

d lnNdδ

= 2δ

σ

(1− 1

γ

)[1+

12γ

(δ

σ

)2]−γ

, (3.3)

where σ is a function of both energy E and inclination angle φ . The index γ

characterises the PSF tail at large angular separations. For γ = 2 we have thatθ95%/θ68% = 3. For large γ equation 3.3 approaches a Gaussian. Parameteri-zations of the PSF can be seen in the upper plot of figure 3.2.

Effective AreaThe LAT effective area is a function of the incident photon energy E andand inclination angle φ . There is no simple parametrisation of the effectivearea available. Instead interpolations, such as the one seen in the middle plotof figure 3.2, of Gleam simulations are used. For all practical purposes theeffective area is typically integrated over the instrument lifetime, which givesthe exposure in units of cm2 s.

Energy dispersionThe energy redistribution reflects the energy resolution of the LAT instrument.It is typically of order 10% which can be compared with the huge energy rangeof four orders of magnitude (30 MeV to 300 GeV) where Fermi is sensitive.Therefore, in many applications, the energy redistribution can be neglected.

2http://geant4.web.cern.ch/geant4/

SECTION 3.3. FAST DETECTOR SIMULATIONS 15

This is not the case for spectral lines, where the spectral shape is determinedprecisely by the energy dispersion [27].

The same empirically determined energy redistribution function is used forall event categories. The redistribution from actual photon energy E to appar-ent photon energy E ′ is parameterised as:

dNdx

= A(1+ x)p1

1+ exp(x/p2), where x = (E ′−E)/E, (3.4)

A is a normalisation factor and p1, p2 depends on E and φ . Parameterizationsof the energy dispersion can be seen in the lower plot of figure 3.2.

3.3 Fast detector simulationsFor many applications it is necessary to make relatively fast simulations ofhow a source would be seen by the Fermi-LAT. This can be done using thescience tool gtobssim.

An event is generated as follows :1. A candidate event, with energy E and inclination angle φ , is selected from

the source flux using a cross section area A0 = max(A).2. This event is accepted if ξ < A(E, φ)/A0, where ξ is uniformly distributed

in the open interval [0, 1).3. Apparent energy and inclination is then drawn according to the energy dis-

persion and PSF:

E ′ 7→ D(E ′; E, φ),φ′ 7→ P(φ ′; E, φ).

The fast detector simulation uses the parameterised IRFs as described above.

17

4. Galactic diffuse gamma-rayemission

The diffuse emission from the Milky Way completely dominates the gamma-ray sky, even at high latitudes. The main part of the galactic diffuse emissionoriginates from cosmic ray, primarily proton and electron, interactions withgas in the interstellar medium and the interstellar radiation field. Any calcu-lation of the galactic diffuse emission is therefore primarily dependent on theunderstanding of the cosmic ray spectra, the gas and interstellar radiation fielddistribution throughout the galaxy.

This chapter reviews in brief the structure of the galactic gas, the cosmicray composition, the interstellar radiation field and how these componentscombine with radiative processes to give rise to the galactic diffuse emission.

4.1 The Milky WayThe Milky Way is a spiral-bar galaxy with a radius of about 30 kpc, where oursun is located at a distance of approximately 8 kpc from the centre. Typicallydifferent regions of the galaxy are labelled by the spiral arms, as viewed fromthe Earth. Figure 4.1 shows a recent estimate of the spiral arms structure ofthe Milky Way. The stellar is believed to consist of about 1011 stars, whichare confined to the galactic plane and increase in density towards the galacticcentre.

Gas distributionThe interstellar gas is mainly composed of hydrogen in one of the threeforms: atomic (HI), molecular (H2) and ionised (HII).

The HI distribution has been measured accurately by radio observations of thehyperfine transition, giving rise to an emission line at a wavelength of 21 cm.Atomic hydrogen is found all over the galaxy, but concentrated in the galacticplane. Typical densities of HI are about 1 atom cm−3 with a total mass ofabout 109M [28].

There is no direct way of tracing molecular hydrogen. However, it is as-sumed to correlate with carbon monoxide 12CO which has a distinct emissionline at 2.6 mm. The ratio of H2 column density (N(H2)) and the integrated

18 GALACTIC DIFFUSE GAMMA-RAY EMISSION CHAPTER 4

intensity of the CO line (WCO), XCO = N(H2)/WCO ∼ 1020 mol cm−2 (K kms−1)−1, is typically taken as a free parameter when fitting the galactic diffuseemission model to data. The H2 gas appears to form huge molecular clouds inand around the galactic plane. The densities can be very high, up to 104 atomscm−3 but with a similar total mass as the HI gas [29] .

The ionised hydrogen is on average only a few percent of that of neutral gas.It is typically found in regions around massive stars. However, a large halo ofionised gas has also been found to extend up to 1 kpc away from the galacticdisc [30]. Due to its large extension, the HII contribution to the gamma-rayemission is non negligible.

Illustrations of how the different distributions of hydrogen give rise to dif-fuse gamma-rays can bee seen in figure 4.2.

The second most abundant atomic gas is helium, with a ratio in numbersto hydrogen of about n(H)/n(He) ∼ 0.1. Helium therefore has to be takeninto consideration when evaluating the gas contribution to the galactic diffuseemission.

Instead of using CO and HI as the tracers of gas in the Galaxy, dust has beenshowed to correlate to the total hydrogen column density [31]. By observingthe emission of dust in the infrared, correcting for temperature variations andassuming a gas to dust ratio one can convert the dust-reddening maps to hydro-gen tracers. This is applied in part for the analysis of the isotropic gamma-raybackground in chapter 5.

The interstellar radiation and magnetic fieldsThe interstellar radiation field is a combination of all the light produced in thegalaxy. Knowing its distribution is important when estimating radiation lossesfrom propagating electrons and diffuse gamma-rays from inverse Compton-scattering. The main contributions originate from the cosmic microwave back-ground (dominates at wave lengths > 1000µm∼ 0.1 meV), emission from mi-croscopic dust grains (peaks between 10 and 30 µm) and starlight (dominates< 10µm) [32]. The interstellar radiation field extends several kpc away fromthe galactic disk and radially it follows the progenitor distribution. The stellarfield is concentrated in the galactic centre and decrease with increasing radius,which is imprinted in the starlight distribution. The dust emission correlateswith the neutral gas (HI and H2) and peaks around 5 kpc from the galacticcentre. The cosmic microwave background is uniform.

A weak magnetic field is believed to exist throughout the galaxy. Its shapeis not very well understood but is believed to consist of a regular compo-nent, tracing the galactic arms, decreasing towards the outskirts and awayfrom the galactic plane, and a random component with a roughly constantaverage value throughout the galaxy. The field strength is hard to estimate and

SECTION 4.1. THE MILKY WAY 19

Figure 4.1: An image of the Milky Way, reconstructed using Spitzer data. Accordingto this the Milky Way has only two spiral arms that winds around the bar shapedcentre. Courtesy NASA/JPL R. Hurt -Caltech.


Figure 4.2: Spatial distribution of the differental energy spectrum of gamma-rays,at 1.2 GeV, originating from the HI (top), H2 (middle) and HII (bottom), throughbremsstrahlung and π0-decay. The maps are in galactic co-ordinates and the colourscale is logarithmic. They were generated using GALPROP and the differential energyspectra can be seen in figure 4.5.

SECTION 4.2. COSMIC RAYS 21

depends on the size of the scale under consideration, measurements point to avalue around ∼ 2− 6µG [33]. Knowledge of the magnetic field is importantwhen calculating energy losses of cosmic ray electrons that emit synchrotronradiation

4.2 Cosmic RaysCosmic rays are not just mediators carrying information from some distantastrophysical phenomena, but are viewed as one of the components of the in-terstellar medium. The energy spectrum of galactic and extragalactic cosmicrays have been measured from 106 eV up to ∼ 1020 eV, and can be seen inthe left panel of figure 5.1. In the energy range 109 < E < 1015 eV the cos-mic rays are primarily protons and the energy spectrum is well described by apower law distribution ∝ E−2.7. This is the relevant energy range in the con-text of galactic diffuse gamma-ray emission and the cosmic rays are believedto be galactic of origin. Even though the origin of cosmic rays is not com-pletely understood, the primary galactic sources are supposed to be supernovaremnants. The acceleration mechanism can be described by first order Fermiacceleration, where charged particles gain energy through repeated crossingof a shock front expelled by the supernova (see e.g. [34]). This accelerationwould naturally result in a power law energy distribution of the acceleratedparticles with spectral index ∼ −2. However, the shock acceleration in su-pernova remnants can not accelerate particles above ∼ 1015 eV. Around thisenergy there is a softening of the observed cosmic ray spectrum. This featureis called the ’knee’ and could be due to the cut-off in the super nova acceler-ation mechanism. A complementary (or alternative) explanation of the ’knee’could also be that cosmic ray with such high energies are able to escape theconfinement of the galactic magnetic field.

At energies ∼ 1018 GeV there is a hardening of the observed cosmic rayspectrum. This is called the ’ankle’. Its origin is not well understood but sincethe transition energy between the galactic and a possible extragalactic compo-nent of the spectra is not well determined, the ’ankle’ could be a sign of that.Indications of a correlation between high energy cosmic rays and AGNs hasbeen found [35], which would support their extragalactic origin. At these highenergies cosmic rays can no longer be detected directly but has to be inferredindirectly from showers created as the high energy particle collide with theEarth atmosphere. This introduces large uncertainties when determining thenuclear composition of cosmic rays above 1015 eV, and therefore it is not wellknown.

The most energetic cosmic rays detected have energies ∼ 1020 eV, with anoccurrence rate of less than one per km and year. At this energy the cosmic


rays start to interact with the CMB photons through the ∆ resonance. Theexcited state decays into a proton or neutron and a π-meson with the appro-priate charge. The process continues until the energy of the cosmic ray fallsbelow the π production threshold. This presents a limiting horizon for theultra high energy cosmic rays and if they are of extragalactic origin, the spec-trum should exhibit a cut-off. This cut-off is known as the Greisen-Zatsepin-Kuzmin (GZK) cut-off, after its discoverers in 1966. A cut-off at energies and∼ 1020 eV has been observed recently [36, 37], which could be due to theGZK effect.

PropagationCharged cosmic rays do not propagate on straight lines from the source ofproduction to their arrival at Earth, like gamma-rays and neutrinos do. Insteadthey experience multiple scattering on magnetic field homogeneities (i.e. dif-fusion), they loose energy due to ionisation and radiative processes and theyinteract with the interstellar medium. To model these different processes thedensity of a given cosmic ray species, φi, as a function of position x, time tand energy in the the interval E +dE evolves in time according to (simplifiedequation adopted from [38]):

∂φi

∂ t= ∇ · (D(x, E)∇φi)− (∇ ·V)φi−

∂

∂E

(dEdt

φi

)+Qi(x, E, t)

−(

1τcol

+1

τdecay

)φi +

βcρ

m ∑k≥i

σk→iφk. (4.1)

The terms on the first line on the right hand side of equation 4.1 represent(from left to right) diffusion, where D is the diffusion coefficient, convectionwith a galactic wind of coherently moving cosmic rays of velocity V, energylosses and the source term of cosmic rays, Qi. The second line contains theinteraction terms with losses of nuclei by collision and decay, represented bythe mean time scale of these processes, τcol and τdecay respectively. The lastterm of equation 4.1 represents fragmentation of heavy nuclei that travel atspeed βc, interact with the interstellar medium, represented with mass m andmean density ρ and produce lighter nuclei with the total cross section σk→i.

Solving the partial differential equation 4.1, with realistic boundary con-ditions, a proper treatment of the interstellar medium and taking all cosmicray species into account, with their individual cross sections, in a self consis-tent way is a difficult task. There exist a wide span of models for cosmic raypropagation that are to a different extent simplifications of equation 4.1.

Different cosmic ray propagation models are typically evaluated by compar-ing predictions of secondary φS and primary φP cosmic rays to observations.Pure primary cosmic rays only have the production term QP and not the frag-mentation term, and vice versa for secondary cosmic rays. The best studied

SECTION 4.2. COSMIC RAYS 23

almost pure secondary and primary cosmic ray species are boron and carbonrespectively. Their measured ratio, B/C, can be seen in figure 5.1, which athigh energies it is observed to be compatible with a power law ∝ E−δ .One of the simplest propagation models is the Leaky Box model. It assumesthat cosmic rays are homogeneously distributed in the galaxy and allowed topropagate freely, with a constant probability per unit time to escape τesc. Thisreplaces the diffusion term in equation 4.1 with −φi/τesc. The escape time isusually expressed in terms of the escape length X = ρβcτesc g cm−2, whichis interpreted as the mean amount of matter traversed by a cosmic ray prop-agating through a homogenous interstellar medium. If convection and energylosses are ignored the steady state solution (i.e. ∂φi/∂ t = 0) of equation 4.1 inthe Leaky Box model gives (for i = S) φS/φP ∝ X ∝ E−δ .

This can be used to estimate the injection spectra of cosmic rays. Oncethe spectrum of φP ∝ E−γ is measured, equation 4.1 (for i = P) givesQ ∝ φP/X ∝ Eδ−γ . This shows that even in this very simple model of cosmicray diffusion, the propagated cosmic ray spectrum should be softer than theinjection spectrum.

A more refined propagation model is the Diffuse Halo model, where the dif-fusion coefficient D is related to the mean free path λD between scattering onthe magnetic field homogenietes, D ∝ βcλD. If particle interactions, energylosses and convections are ignored equation 4.1 turns into:

∂φi

∂ t= ∇ · (D∇φi)+Qi. (4.2)

The solution to this differential equation, for a delta function source term atthe origin, is a Gaussian distribution. This means that φ can be interpretedas the probability to find a random walking particle at a distance x from theorigin at time t.In the simplest case of the Diffuse Halo model it is assumed that the gas isconfined to a galactic disc of height h (from observations h ≈ 100 pc) withdensity ρg, which is imbedded in a diffusion halo of height H >> h (couldbe at least 1 kpc given the extend of the HII region). A sketch of the diffusehalo model can be seen in the right panel of figure 4.4. Solving equation 4.2,the average amount of matter traversed by a cosmic ray particle diffusing thedistance H away from the galactic plane is given by [38]

Xdi f f = βcρghH/D g cm−2. (4.3)

If this is taken to be equal to the corresponding escape length of the Leaky Boxmodel, the diffusion constant can be evaluated using the secondary to primaryratio D ∝ X−1 ∝ Eδ which gives constraints on the diffusion halo geometry.This is valid for stable nuclei and when the energy per nucleon is high. At


Ek/n (GeV/n)1 10 210

B/C

rat

io

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.33 E∝

IS

IMP7-8 (Garcia-Munoz et al. 1987)

Voyager1&2 (Lukasiak 1999)

ACE 97-98 (De nolfo et al. 2006)

ACE 97-98 (George et al. 2009)

ACE 01-03 (George et al. 2009)

AMS-01 (Tomassetti et al. 2009)

HEAO-3 (Engelmann et al. 1990)

Spacelab-2 (Swordy et al. 1990)

CREAM 04 (Ahn et al. 2008)

Figure 4.3: Left: The cosmic ray energy spectrum from [39]. Right: The secondary toprimary ratio B/C, as a function of energy per nucleon. The data points above 3 GeVper nucleon are well fitted by a power-law ∝ E−3.3. The data is taken from [40].

Figure 4.4: A schematic picture of the Halo Diffuse Model. Particles originate fromthe galactic disc of height h, and can diffuse into the larger halo of height H. Adoptedfrom [41]

SECTION 4.3. RADIATIVE PROCESSES 25

lower energies, other processes come into play; energy losses, convection,solar modulation etc. Each of them influences the secondary to primary ratioin its own way. The diffusion coefficient index δ is typically in the range 0.2-0.8 depending on the propagation model assumed. For a state-of-the-art useof the Halo Diffuse model, for several choices of Halo properties, convectionand energy loss terms see [42], or [40] where a detailed Markov chain MonteCarlo scan have been done to fit parameters of various cosmic ray models todata.

To solve equation 4.1 for a geometry with a gas distribution and cosmicray injection distribution as complex as in the Milky Way, the analyticalapproach becomes unfeasible. Instead a numerical solution can be obtain ona spatial grid where gas and source distributions can be arbitrarily specified.An example of this is the GALPROP 1 code which propagates all cosmic rayspecies on a three dimensional spatial grid until a steady state is reached.GALPROP also calculates the galactic diffuse gamma-ray emission foreach point in space using the obtained cosmic ray spectra together with gassurveys and a model for the interstellar radiation and magnetic fields. In thisway, the spatial distribution of the gamma-ray spectrum is obtained in a selfconsistent way.

There are still many challenges in cosmic ray physics. Some of the biggestare to explain the cosmic ray origin and the shape of the secondary to primaryratio. The propagation halo region is still badly constrained and the individualrole of the propagation terms in equation 4.1 is not accurately known [43].

The cosmic ray spectra have also been interpreted in the light of dark mat-ter. Recent measurements of the electron and positron fluxes by PAMELA [44]and Fermi [45] have revealed excesses relative to what is expected from con-ventional cosmic ray models. This has resulted in dark matter models whichhave their dominant annihilation channel into charged leptons and will be dis-cussed further in chapter 7.

4.3 Radiative processesThe main processes giving rise to the galactic diffuse gamma-ray emissionare believed to originate from interactions of cosmic ray protons (through π0-decay) and electrons (through bremsstrahlung) with galactic gas, and elec-trons with the interstellar radiation field (through inverse-Compton scatter-ing). These processes determine the energy spectral features of the galacticdiffuse emission.

1http://galprop.stanford.edu


Since the individual contributions of the different processes vary in bothshape and absolute contribution over the galaxy, an average spectrum of thediffuse emission contains little information about the underlying processes.Instead spectra of specific regions are very different and have to be studiedseparately. Example spectra of the three processes can be viewed in figure4.5, for two regions of the sky.

Pion productionNeutral pions, that decay to two gamma-rays, are most commonly producedin collisions between protons and hydrogen.The interaction has the form:

p+N→ p′+N′+n1π0 +n2(π+ +π

−), (4.4)

where p is the proton (could also be a helium nucleus α) colliding with nu-cleus N, producing secondary nucleons (p′ and N′) and some number (n1 andn2) of π-mesons. The neutral π0 has a lifetime of ∼ 10−16 s and decays totwo gamma-rays, the charged π+/− has a lifetime of∼ 10−9 s and decays intomuons and neutrinos.

The gamma-ray spectrum is determined by the velocity of the π0 meson. Ifit decays at rest, the photons are emitted in opposite directions with an energyof half the π0 rest mass, mπ0 = 135 GeV. If the decay happens in-flight thephoton energy depends on the angle of emission relative to the flight directionof the π0. The geometric mean of the two photons emitted is however alwaysequal to mπ0/2, which makes the gamma-ray spectrum symmetric around thisenergy in a log-log plot. At higher energies the differential energy spectrumhas a power law behaviour with a spectral index −(4/3)(γ−1/2), if the pri-mary cosmic ray spectrum is ∝ E−γ . The ’pion bump’ is clearly visible infigure 4.5.

BremsstrahlungWhen a charged particle is deflected by the electromagnetic field of a nu-cleus it emits radiation. The radiation energy is proportional to the accelera-tion caused by the deflection. The cross-section of the interaction is highly de-pendent on the energy of the incident electron, and the complete quantum me-chanical treatment is complicated by the screening of the bound electrons andthe presence of the nucleus. The gamma-ray energy spectrum is proportionalto the electron and gas distribution in the galaxy. The shape of the gamma-rayenergy spectrum is the same as for the initial electrons.

Inverse Compton-ScatteringPhotons colliding with charged particles can transfer parts of their energy inthe collision. This is the typical case of Compton-scattering, and the energydependent cross section is given by the Klein-Nishina formula [46]. In astro-physical scenarios, the inverse process is common when high energy charged

SECTION 4.4. PRE-FERMI MEASUREMENTS 27

particles (mostly electrons) interact with the interstellar radiation field. Theresulting gamma ray spectrum depends on the distribution of the electronsand of the interstellar photons. If the electrons have a Lorentz factor of γ andthe soft photons a characteristic energy of hν , the inverse Compton-scatteredphotons will have a characteristic energy of γhν and a differential energy dis-tribution ∝ E−(α+1)/2, where α is the spectral index of the electron differentialenergy distribution.

4.4 Pre-Fermi measurementsThe first detection of the galactic plane in gamma-rays was done by the OSO3 satelite in 1968 [8]. Although the angular resolution of the instrument wasnot good enough to resolve the galactic plane, the detection was consistentwith an elongated shape along the galactic plane, with an enhancementtowards the centre. This was one of the first detections of an extraterrestrialgamma-ray source.

The first detailed study of diffuse gamma-rays from the galactic plane(|b| ≤ 10 in galactic coordinates) was done using EGRET data [47] (referredto as the Hunter model). Three basic assumptions where made regardingcosmic rays when deriving the diffuse gamma-ray energy spectra: that cosmicrays are galactic in origin, that there exists a correlation between cosmic rayand interstellar matter density in the galaxy and that the cosmic ray spectrathroughout the galaxy are the same as measured in the solar vicinity. Theresults confirmed that the observed galactic diffuse emission to a large extentagrees with the expected from cosmic ray interacting with the galactic gasand radiation fields. However, above 1 GeV the measured emission showedan excess over the expected spectrum. This excess is known as the EGRETGeV-excess.

A later re-analysis of the EGRET data, but inferring the cosmic ray distribu-tion from a numerical propagation using GALPROP, confirmed the existenceof the GeV-excess in all directions on the sky [48]. By relaxing the conditionthat the cosmic ray spectra in the galaxy are the same as measured locally,the same authors could produce a galactic diffuse emission model that incor-porated the GeV-excess and therefore was compatible with the EGRET ob-servation [49]. This galactic diffuse emission model was called the optimizedmodel, as opposed to the conventional of [48].


102 103 104 105

E [MeV]

10-6

10-5

10-4

10-3

10-2

E2·dφ/dE

0[M

eVcm

−2

s−1

sr−

1]

10 <|b|<20 ICSPiBremssTotal GDE

102 103 104 105

E [MeV]

10-6

10-5

10-4

10-3

10-2

E2·dφ/dE

0[M

eVcm

−2

s−1

sr−

1]

|b|>60 ICSPiBremssTotal GDE

Figure 4.5: Differental energy spectrum of galactic diffuse gamma-rays originatingfrom π0-decay (dashed), bremsstrahlung (dash-dot) and inverse Compton scattering(dotted). The upper panel shows an average over low latitudes and the lower panel forhigh latitudes. These specific spectra were generate using the GALPROP code. Thespatial distribution can be seen in figure 4.2

SECTION 4.5. THE FIRST YEAR FERMI MEASUREMENT 29

4.5 The first year Fermi measurementOne of the first Fermi results on the galactic diffuse gamma-ray emission wasthat the existence of the GeV-excess could not be confirmed all directions ofthe sky [50]. Instead the gamma-ray spectrum at intermediate latitudes is infair agreement with a model based on the locally observed cosmic ray spec-trum, see figure 4.6. Also the local emission from HI has been measured to bein good agreement with the same cosmic ray spectrum [51].

The new modelling is based on a refined GALPROP code that is planned tobe released at the same time as the first detailed study of the galactic diffuseemission by the Fermi-LAT collaboration is published.

10-4

10-3

10-2

102 103 104

Eγ2

Jγ

(Eγ)

(M

eV c

m-2

s-1

sr-1

)

Eγ (MeV)

0° ≤ l ≤ 360°, 10° ≤ |b| ≤ 20°

LATIsotropicSourcesπ0-decay

ICBremsstrahlungTotal

Figure 4.6: The Fermi-LAT measurement and modelling of the galactic diffuse emis-sion at intermediate latitudes, from [50]. The observation does not confirm the GeV-excess seen by the EGRET satellite [47, 48]

31

5. Measurements of The IsotropicGamma-Ray Background

This chapter describes measurements and models of the isotropic gamma-ray background (IGRB). Some, if not all, of the IGRB is believed to be ofextragalactic origin, referred to as EGRB . In the following we will separatebetween the two when discussing contributions of different kinds. However,the measurements are always of the IGRB, which is the sum of the EGRBand any other isotropic components, whether it is of galactic or instrumentalin origin.

From a cosmologist point of view, the EGRB is one of the most interest-ing signal in gamma-rays. This is because of the high penetration power ofgamma-rays, which enables them to carry information to us from the earlyUniverse. The EGRB can constrain many exotic models of physics that wouldproduce an abundance of gamma-rays throughout the evolution of the Uni-verse. Also, if the origin of EGRB is mostly from unresolved point-sources, itwill contain information of the population as a whole.

5.1 Pre-Fermi measurementsThe first indications of an isotropic, possibly extragalactic, flux came from theSAS-2 satellite [9] in 1972 and the first energy spectrum was obtained by theEGRET experiment [52]. The IGRB is a weak component of the total galacticdiffuse emission and hard to disentangle. Even in the direction of the galacticpoles the IGRB is expected to be comparable to the galactic emission.

The analysis [52] (which we will referee to as the "Sreekumar et al. " anal-ysis) relies heavily on the model used for galactic diffuse emission. The basicidea behind the method is to extract the IGRB from the total measured inten-sity where the galactic contribution is extrapolated to zero. Assuming that thetotal measured intensity consists of a galactic part, that varies as a function ofgalactic coordinates, and a constant IGRB contribution, uniform over the sky,this can (for a fixed energy bin) be written as:

Φ(ΦGal) = k ·ΦGal +ΦIGRB, (5.1)

where Φ is the intensity in number of photons per unit area, time and solidangle. From this the IGRB can be extracted as the value of Φtot where ΦGal =

32 MEASUREMENTS OF THE ISOTROPIC GAMMA-RAY BACKGROUND CHAPTER 5

0. What kind of model and sources that are included in ΦGal will of-courseinfluence the outcome of ΦIGRB.

The Sreekumar analysis used the Hunter et al. galactic diffuse gamma-ray model [47] and obtained a spectrum compatible with a power-law withspectral index −2.10± 0.03, close to that of gamma-ray blazars (on average−2.15± 0.04 [20]), with a total flux of (1.45± 0.05)× 10−5 photons cm−2

s−1 sr−1 above 100 MeV.This analysis was later re-examined [53] where it was concluded that using

the Hunter et al. model for the galactic diffuse gamma-rays led to an overes-timation of the IGRB. They estimated, only by using EGRET data from thegalactic poles where it should be easiest to disentangle the IGRB from thegalactic componnet, that the IGRB should be significantly lower than previ-ously measured, even compatible with zero. They placed an upper limit on thetotal flux above 100 MeV to < 0.5× 10−5 photons cm−2 s−1 sr−1 at a 99%confidence level. However, they did their estimate based only on the latitudi-nal profiles of the EGRET data and galactic model, not taking into accountthe spectral shape of the IGRB.

A subsequent analysis of the EGRET data, using the same method as in theSreekumar analysis but employing the optimized GALPROP model to rep-resent the galactic diffuse gamma-ray emission, was presented in [54] (re-ferred to as the "Strong et al. " analysis ). They found a slightly lower fluxof (1.11± 0.01)× 10−5 photons cm−2 s−1 sr−1 above 100 MeV and that theIGRB spectrum was not compatible with a power-law. Instead it had a clearspectral feature around 3 GeV.

The IGRB spectra from the Sreekumar [52] and Strong et al. analyses [54]can be see in figure 7.1.

5.2 Origin of Diffuse Extragalactic Gamma-RaysThe IGRB has been used to constrain many theories of cosmologicalevents that potentially could result in a large production of gamma-rays.A few of them are decay of primodial heavy leptons [55], models for thebaryon-antibaryon asymmetry [56, 57], evaporation of primordial blackholes [58, 59] and (as investigated in this work) annihilations of dark matterparticles [60, 61, 62, 63, 64, 65, 66, 67].

Unresolved point sources are believed to contribute significantly to the ex-tragalactic background. Constructing a model for the unresolved contributionfrom a specific source class requires knowledge of the population as a wholeand how it evolves with redshift. This information is not always available tothe required accuracy and often properties of the population at lower energieshave to be extrapolated to gamma-ray energies.

SECTION 5.2. ORIGIN OF DIFFUSE EXTRAGALACTIC GAMMA-RAYS 33

Typically the unresolved emission is derived in an empirical approach, start-ing from observed properties of a population (energy spectra, luminosity dis-tributions etc.) and try to estimate how these have evolved in time. The to-tal contribution of the entire population is then integrated in redshift belowthe point source sensitivity of the instrument under consideration. This hasbeen done for blazars [68, 61, 63, 69, 70, 71, 72, 73], star forming galax-ies [69, 74], clusters of galaxies [75], gamma-ray bursts [76] and even un-identified sources [77].

The first estimate of the contribution from blazars [68] seemed to agreequalitatively with the EGRET measurements by Sreekumar et al. [52]. It hadan energy spectrum close to ∝ E−2 with a slight concavity, depending on theassumed distribution of blazar spectral indices.

This model was not consistent with the subsequent analysis of the EGRETIGRB by Strong et al. [54], that had a non power law behaviour and a sharpspectral feature at ∼ 3 GeV. In general it was hard to make any of the conven-tional astrophysical unresolved source models to resemble the spectral fea-tures of the IGRB and it was unlikely that any of the models could account forit on its own, neither in shape or normalisation. However, appropriate com-binations of models could make out the total contribution at certain energies[76, 78]. The spectral feature of the IGRB has been interpreted in terms ofa signal from annihilating dark matter [65, 66]. For the background modelsconsidered it was certainly possible to account for the spectral features of theStrong et al. [54] IGRB energy spectrum with a signal from annihilating darkmatter.

5.2.1 Fermi analysis of the IGRBThe GaDGET tool was developed especially for diffuse analysis within theFermi collaboration [79]. It uses a multi component likelihood fit of predictedcounts versus observed counts. The predicted counts are obtained by convolv-ing the model intensity maps with the exposure of the observation. The IGRBis modelled as an isotropic component with the normalisation as a free param-eter in each energy bin. The final IGRB spectrum was obtained by subtractingthe remaining residual background (described below) from the fitted isotropicintensity.

The final IGRB measurement is dominated by systematic uncertainties re-lated to the subtraction of the residual background and by the uncertainties inthe effective area of the instrument. The uncertainty in the effective area wasestimated from careful comparisons between observations and simulations ofthe Vela pulsar. The effects on observed intensity is estimated to be 10% be-low 10 MeV, 5% at 560 MeV and 20% at 10 GeV. These systematic errorswhere added in quadrature to the formal error from the fitted IGRB spectrum.


5.2.2 The residual backgroundThe number of cosmic ray incedenting on the Fermi LAT outnumbers the highenergy gamma-rays by more than four orders of magnitude. The pre-launchrequirement on the level of residual background was set to be 10% of theSrekumar et al. IGRB [52]. The standard background rejection is based on apre-launch scheme using simulations and beam-tests of the detector. This re-sulted in the three background rejection classes: source, transient and diffuse,each with increased background rejection power (c.f. section 3.2). A moreefficient background rejection class comes with a penalty of reduced effec-tive area. The standard low-background class is the diffuse class, as describedin [4]. However, on-orbit studies indicate that the residual background washigher than predicted prior to launch.

Figure 5.1 shows pre-launch estimate of the contributions of different kindsof cosmic rays as compared to the diffuse gamma-ray emission.

[GeV/nu]kinE10 210 310

]-1

sr s

ec

Ge

V)

2F

lux

[(m

-910

-810

-710

-610

-510

-410

-310

-210

-110

1

10

210

J. Conrad, 2009

EGBdiffuse gammapositronselectronsHeprotons

Figure 5.1: Estimates of the energy spectrum of gamma-ray photons and cosmic raysincidenting on the LAT. The energy is given per nucleon for the charged cosmic rays.The cosmic rays outnumbers the galactic diffuse emission photons by more than fourorders of magnitude and the estimated extragalactic photons by more than five.

5.2.3 Additional cuts - the dataclean classA more stringent event selection was developed for the IGRB analysis, the dat-aclean class, which resulted in a reduced residual background. For this classthe requirement on the probability that an event is a photon was increased, and


made energy dependent. Tighter constraints where imposed on photons inci-denting on areas where the ACD has a lower efficiency. The average chargedeposition in the tracker was required to be smaller and the transverse showershape in the calorimeter was restricted to be more similar to electro-magneticshowers.

Using these additional cuts the residual background was estimated to bereduced by 25 -95% in the energy range 20 MeV to 102.4 GeV, and there-fore subdominant to the IGRB. The reduction in effective area between thedataclean and diffuse classes, as functions of energy, can be seen in figure 5.2.

Energy [MeV]

310 410 510

]2ef

fect

ive

area

[m

0.3

0.4

0.5

0.6

0.7

0.8

0.9

diffuse classthis analysis

Figure 5.2: On-axis effective area of the diffuse and dataclean background rejectionschemes. From PAPER III

5.2.4 Estimating the error in the Monte Carlo simulations of theesidual backgroundAfter applying the dataclean background rejection scheme, the remainingresidual background can be estimated by Monte Carlo simulations. The un-certainty in the simulation is estimated by comparing it with the observedcosmic ray rate. To do this two cosmic ray dominated LAT data sets were cre-ated, with a different level of gamma-ray contamination. The first data set Acontains all events that pass the LAT onboard filtering, but is subject to no fur-ther processing. Basically this set is only cosmic rays. The second sample Bare those events that pass the low-background-rejection source analysis filter,but not the diffuse filter. This sample has a much higher level of cosmic rayrejection than data set A, but is still dominated by cosmic rays compared tothe diffuse class.

To remove the cosmic ray contribution which is not modelled well by thestandard classification analysis scheme, the dataclean filter is applied to bothdata sets and only events from |b| > 45 were included, to further reduce


gamma-ray contamination. The gamma-ray content of data set A is now neg-ligible but there might still be a significant fraction in set B. To estimate theremaining gamma-rays in set B, the intensity model maps used for the IGRBanalysis were convolved with the exposure of data set B, and subtracted. Infigure 5.3 the comparison between the observed cosmic ray rate of the A andB data set and the simulated rates are showed. The agreement is within 20%for the minimal rejection sample A, and within +50%/-30% for data set B. Thelarger errors in the latter sample indicate the uncertainties to accurately modelthe cosmic ray content in the high energy tail of the distribution. When sub-tracting the simulated remaining residual background from the IGRB analysis,it is the larger error on the estimate that is employed.

Measured Energy [MeV]

310 410 510

rat

e [H

z]

−510

−410

−310

−210

−110LAT CR events (dataset A)

LAT CR events (dataset B)

simulation

0.01×rate

Figure 5.3: Comparison between two simulated and measured cosmic ray dominatedorbit-averaged data samples. Dataset B was used to estimate the uncertainty in theremaining residual charged particle contamination. From PAPER III.

5.2.5 Estimating the residual background using cosmic raytrigger rateAn alternative method to subtract the residual background is to look at subsetsof the data with different cosmic ray trigger rate. Subsets with a high cosmicray trigger rate should contain higher contamination of cosmic rays, and viceversa. Measuring the IGRB for datasets of different cosmic ray trigger rateand extrapolating the results to zero cosmic ray trigger rate should yield ameasurement free from contamination. There are however a few things thatcomplicate this method. One thing is that the gamma-ray detection and se-


lection efficiency decreases with increasing cosmic ray trigger rate, due to a"pileup" effect in the detector [80]. Depending on what dominates in a par-ticular energy bin, increment due to more residual background or decrementdue to the pileup, the measured IGRB would be an increasing or decreasingfunction of cosmic ray trigger rate. For different energy bins It is hard to pre-dict which effect that would dominate. However, at the extrapolation point thepure gamma-ray intensity should be regained.

Also, the extrapolation function is a-priori unknown and differences be-tween reasonable choices has to be considered as a systematic uncertainty ofthe method. If the cosmic ray contamination is too high, the resulting uncer-tainty would make this method inapplicable.

The photon event variable that is closest related to the cosmic ray triggerrate is the livetime fraction of the instrument. A high background trigger ratecorresponds to low livetime fraction and vice-versa. The 10 months datasetwas divided into five livetime fraction bins, as showed in figure 5.4. The firsttwo bins corresponds to to about 12.5% of the total instrument live time eachand the last three bins to about 25% each. Exposures maps corresponding tothe individual livetime fraction subsets where generated and the IGRB analy-sis was performed on all of them individually. The "contamination free" IGRBwas obtained by extrapolating to live time fraction = 1. The extrapolations foreach energy bin is shown in figure 5.5.

Figure 5.4: Distribution of livetime fraction for the 10 month data sample. The binswhere used for the cosmic ray trigger rate extrapolation method.


Figure 5.5: Extrapolations to zero cosmic ray trigger rate (livetime fraction of one)for each energy bin. The trend is either decreasing or increasing depending on whicheffect that dominates for each energy: increment due to cosmic rays or decrement dueto lower photon detection efficiency.

SECTION 5.3. LINEAR EXTRAPOLATION ANALYSIS OF THE FERMI DATA 39

5.2.6 Comparing the MethodsFigure 5.6 shows the IGRB, determined by the linear analysis (see next sec-tion), for the diffuse and dataclean analysis classes. Also shown are the tworesidual background subtraction methods applied to the dataclean spectrum.It is clear that the extended cut of the dataclean analysis class reduces almostall residual background at high energies while leaving a considerable amountat low energies.

The residual background subtracted spectra agree with each other within er-rors. However, the differences illustrates a systematic uncertainty introducedby the method of subtraction. The two methods are attached to different sys-tematic uncertainties. The Monte Carlo method depends heavily on an accu-rate modelling of the instrument and of the in-orbit cosmic ray environment,which are hard to model to the precision required. The Monte Carlo methodis also very computational demanding, with roughly 40000 CPU hours for ev-ery day of background simulation. The cosmic ray trigger rate extrapolationmethod on the other hand inherits all of the systematic uncertainties of theanalysis method used; uncertainties of the diffuse galactic background, the ef-fective area etc. and suffers statistically from the fact that the dataset has to bedivided in several parts. Due to the many uncertainties of the livetime fractionextrapolation method, and since it is not really clear from the results (i.e. fig-ure 5.5) that it is applicable, the Monte Carlo method was used to estimate theresidual background of this work and PAPER III.

5.3 Linear Extrapolation analysis of the Fermi dataThis section describes an alternative measurement of the IGRB, as comparedto the one published in PAPER III. The method is inspired by the originalanalysis by Sreekumar et al. [52] to obtain the energy spectrum.

The purpose of having two methods doing the same measurement is to in-dependently verify the result. Also, using two methods give an estimate ofthe systematic uncertainty involved with the analysis method. The principledifferences between the methods are summarised in table 5.1.

5.3.1 Photon data treatmentTo minimise the contamination of Earth albedo gamma-rays a cut in the anglebetween the LAT normal and the Earth zenith axis (< 100) was made. To re-move photons for which the space craft effective area is most uncertain, a cuton the inclination angle of incoming photons to the LAT normal (i.e. field ofview < 65) was also made. These are standard cuts and used in more or lessall analysis of Fermi data. The data used are from the first ten months of nom-


102 103 104 105

E[MeV]

10-4

10-3E∗φ

[MeVcm

−2s−

1sr−

1]

Diffuse-class fitDataclean-class fitMC subtracted CRLivetime frac. extrap.

Figure 5.6: Comparison between the IGRB obtained with the diffuse (green dash-doted) and dataclean (black solid) analyses classes. The dataclean IGRB after sub-tracting the residual background from the Monte Carlo simulation (red dash-dot withcrosses) and using the cosmic ray trigger rate extrapolation method (blue dotted) arealso shown. The errors are purely statistical for the non-subtracted spectra and in-cluding the errors from the subtraction methods in the subtracted spectra respectively.As can be seen, adopting the dataclean analysis class removes most of the residualbackground at high energies.

Table 5.1: Summarising the main differences between the GaDGET analysis PAPERIII and the linear extrapolation analysis (this work).

Method aspect Linear analysis GaDGET

Fitted quantity Observed intensity Observed counts

Fit method Linear χ2 fit Multi component Likelihood

IGRB Extrapolation to zero foregroundintensity in each energy bin.

Isotropic component with freenormalisation in each energybin.

Point sources Power law template for allsources.

Template for weak source,strong sources fitted individu-ally.


inal operation, between mid-August 2008 to mid June-2009. The backgroundrejection scheme used was dataclean as described above.

5.3.2 Data and model formatThe counts data was binned in Nside = 64 Healpix [81] maps . This divides thesky in 49152 pixels of equal area, corresponding to a resolution of about 0.9.The equal area bining is suitable for this kind of all-sky analysis, since alsothe galactic poles (|b|> 85) can be included without introducing numericalinconveniences as the solid angle gets smaller and smaller. This is of particularimportance for the IGRB analysis since the relative contribution to the galacticdiffuse emission is highest at the poles. The galactic poles where not includedin the pre-Fermi analysis of the energy spectrum of the IGRB. To limit theuncertainty from the galactic model, only photons from galactic latitudes |b|>10 where included, where the galactic diffuse emission is more than an orderof magnitude weaker than in the galactic plane.

5.3.3 DeconvolutionIn order to performe the linear analysis, observed photon counts have to beconverted to observed intensity. This process will be referred to as deconvolu-tion of the data. In principle, to obtain the intensity, one has simply to dividethe observed counts by the total exposure of the measurement. The problemis that the effective area of the LAT is energy dependent. Therefore, one hasto know the energy spectrum of the observed photons in order to select theenergy at which it is appropriate to calculate the effective area. This primar-ily becomes an issue at low energies (E < 1 GeV ), where the effective areachanges rapidly with energy.

The energy at which the effective area is calculated is the median energycalculated from the model intensity maps. For each energy bin (Emin, Emax) ofthe model spectrum, the median energy inside that bin is calculated as:

E =

(E1−α(φ)

min +E1−α(φ)max

2

) 11−α(φ)

, (5.2)

where α(φ) is the spectral index of a power law interpolation of the differen-tial energy spectrum as a function of sky coordinates.

Skymaps of measured flux and deconvolved flux can be seen in figures 5.7.

5.3.4 Analysis methodThe basic idea of the linear analysis, as used in [52], [54] is to obtain theIGRB, for each energy bin, as the offset when the observed intensity is extrap-


Figure 5.7: Deconvolved intensity (upper skymap) and model predicted intensity(lower skymap), at 300 MeV. Units are in cm−2 s−1 sr−1 and the color scale is loga-rithmic.


olated to zero predicted foreground intensity. In practice, a linear regressionis made between the observed and predicted intensities for all pixels in theregion of interest in the sky:

Φiobs =

N

∑j=1

k j ·Φi, jf ore +ΦIGRB, (5.3)

where i runs over all pixels on the sky and j over the N foreground componentsof the fit. If the predicted foreground intensities are in good agreement withthe observed ones one would expect the scaling factors k j to be close to unity.

All the fits are done using the MIGRAD routine in the MINUIT [82] fittingpackage, as implemented in the python package pyminuit1.

The errors attached with the observed intensity are calculated using the in-tensity and counts Healpix maps as δ I = I δC

C , where C are the counts andδC is an approximation to asymmetrical Poisson errors [83] valid also for lowstatistics. To ensure that the statistics is high enough, the maps are downgradedin resolution if any of the pixels contains less than 10 photons. Downgradinga Healpix map corresponds to summing (or averaging) the neighbouring fourpixels, which reduces the resolution by a factor of two.

5.3.5 Model ComponentsThe obtained IGRB spectrum depends heavily on the accuracy of the fore-ground model assumed. For this work the foregrounds are the galactic diffuseemission, the Fermi-LAT detected point sources and the gamma-ray emissionfrom the sun. These components are similar to the ones used in the GaDGETanalysis of PAPER III.

Galactic Diffuse EmissionThe principle components of the galactic diffuse emission is the inverseCompton emission and emission arising from cosmic ray interacting withatomic hydrogen (HI and HII) in the local Galaxy (at galactic radii 7.5 kpc< R < 9.5 kpc ). In PAPER III these components are individually fitted to theLAT data. The other galactic diffuse components are cosmic ray interactingwith H2 and non-local HI and HII, and not varied in the fit. For the linearanalysis all galactic diffuse components where fitted together. The local HIand HII are traced by the dust extinction maps instead of the usual HI andHII tracers (c.f. section 4.1).

The cosmic ray propagation is done using GALPROP with a rigidity de-pendent diffusion coefficient consistent with available secondary to primaryratio data, cf. chapter 4. The halo size is taken to be 4 kpc and the injection

1Pyminuit is available from http://code.google.com/p/pyminuit/


spectra for cosmic ray protons and primary electrons are chosen to repro-duce the locally observed spectra, including the electron spectrum measuredby the Fermi LAT [45]. When the cosmic ray spectra and distributions havebeen propagated to a steady state, they are folded with the gas distributions(HI,HII and H2), to calculate the gamma-ray emissivities from π0-decay andbremsstrahlung radiation, and a recent calculation of the interstellar radiationfield [32] for the inverse Compton-scattering. The gamma-ray intensity mapsare obtained by integrating the obtained emissivities over the line-of-sight.The intensity maps are further convolved with the Fermi-LAT PSF in order tomake an appropriate comparison with the deconvolved data intensity maps.

The IGRB measurement is attached with systematic uncertainties related touncertainties in the galactic diffuse model. The effect of these where investi-gated in PAPER III and divided into the categories: inverse Compton-scattermodel and cosmic ray halo size, cosmic ray source distribution and gradientsin the XCO factor, differences in derived HI column densities using radio ordust reddening data. Individually these errors are typically smaller than theerror of the measured IGRB and none of them dominates significantly overthe other. Also, the individual errors are not independent of energy so there isno simple way of combining them.

Another systematical uncertainty is introduced by the choice of region ofinterest. When studying sub-region of the |b| > 10 sky, the measured IGRBchanges slightly but never significantly.

In figure 5.11 the systematic errors related to the galactic diffuse gamma-ray emission is added linearly and displayed by the grey area.

Point SourcesAll point sources are modelled using power-law spectra with spectral indexand total flux from the 11-months Fermi catalog [13]. A total of 1451 sourceswere added to Healpix maps as "apparent intensity", meaning that their indi-vidual extension on the sky is set by the Fermi LAT-PSF. The normalisationof the entire point-source map was fitted for each energy bin.

Solar EmissionThe sun is visible in gamma-rays through inverse Compton-scattering emis-sion from solar radiation on cosmic ray electrons [84]. The intensity is smallcompared to other sources, but including it does improve the fit.

The residual backgroundThe residual background intensity was obtained by convolving an isotropicspectrum with the exposure of the dataset, and normalising the counts in eachenergy bin by the simulated number of cosmic ray events in the correspond-


ing bin. The counts map was then deconvolved using the same method asdescribed above.

5.3.6 TestsTo verify that the software works to satisfaction and to investigate the accuracyof the analysis method, the analysis software was tested using mock data.

At a first stage the analysis chain was applied to mock data directly derivedfrom the foreground models together with an isotropic component. These wereconvolved with a simulated exposure map (similar to the real exposure mapof the data) to obtain a mock counts map. This was deconvolved, as outlinedabove, to obtain a mock intensity map to which the linear analysis could beapplied. The energy spectrum of the isotropic component was a power lawdΦ(E)/dE = 9.5 ·10−3E−2.41 MeV−1cm−2 s−1 sr−1 . The model componentsused were the total galactic diffuse emission and the 11 months source catalog.The result can bee seen in figure 5.8 and the extracted isotropic energy spec-trum is in good agreement with the input. The errors are the formal statisticalerrors originating from the 10 month exposure map used. The deviations at the1% level can be traced to numerical artefacts of the deconvolution method.

At a second stage, the procedure above was repeated but by adding realisticstatistical fluctuations to the data. This was done by creating a Poission countmap using the mock count map, from the previous test, as the mean. The resultcan be seen in figure 5.9. The isotropic component and model are the same asin the previous test. As can be seen the input spectra is again retrieved and theand all the individual linear fits have reasonable χ2.

As a final check we use a fast detector simulation to simulate theforeground components and the isotropic background. The modelcomponents where the galactic diffuse emission, split into two components,one of the inverse Compton-scattering and one of the bremsstrahlung and π0

decay. The isotropic component was generated using a power-law energyspectrum dΦ/dE = 9 · 10−4E−2 MeV−1 cm−2 s−1 sr−1 . The obtained datafiles were run through the analysis pipeline and the input isotropic energyspectrum was obtained with the same precision as in the previous test. Theresult can be seen in 5.10.

All of the above tests regain the input isotropic energy spectrum and thestatistical properties of the fits are under control.

5.3.7 First year Fermi results and discussionThe GaDGET analysis of the energy spectrum of the IGRB derived from thefirst ten months of Fermi data is at high confidence compatible with a power-law of spectral index −2.41±0.05 and itegrated flux above 100 MeV (based


102 103 104 105

E[MeV]

10-5

10-4

10-3

E·φ

[MeVcm

−2s−

1sr−

1]

Fit (this analysis):A ·E γ

γ=−2.41±0.0078

A=0.0095±0.00047

χ2 =0.0778/7

Linear Analysis−//− Power law fit

Figure 5.8: Upper plot: Results of the linear analysis (data points) using mock datagenerated by convolving the model components with a simulated exposure of 10months. The isotropic component was modeled by an energy spectrum dΦ/dE =9.5 ·10−4E−2.41 MeV−1 cm−2 s−1 sr−1 (dashed line) and regained with a power lawfit (dotted line) perfectly. Lower plots: The individual linear fits.


102 103 104 105

E[MeV]

10-5

10-4

10-3

E·φ

[MeVcm

−2s−

1sr−

1]


γ=−2.41±0.0076

A=0.0096±0.00044

χ2 =6.48/7


Figure 5.9: Upper plot: Results of the linear analysis (data points) using mock datagenerated by convolving the model components with a simulated exposure of 10months. Poission statistics was added to the mock counts. The isotropic componentwas modeled by an energy spectrum dΦ/dE = 9.5 · 10−4E−2.41 MeV−1 cm−2 s−1

sr−1 (dashed line) and regained with a power law fit (dotted line) within errors. Lowerplots: The individual linear fits.


102 103 104 105

E[MeV]

10-4

10-3

E·φ

[MeVcm

−2s−

1sr−

1]


γ=−1.997±0.004

A=0.00087±2.3e−05

χ2 =6.07/7


Figure 5.10: Upper plot: Results of the linear analysis (data points) using mock datagenerated by 10 months fast detector simulation of the model and isotropic com-ponents. The isotropic component was modeled by an energy spectrum dΦ/dE =9 · 10−4E−2 MeV−1 cm−2 s−1 sr−1 (dashed line) and regained with a power law fit(dotted line) within errors. Lower plots: The individual linear fits.


on the power-law fit) of (1.03± 0.17)× 10−5 cm−2 s−1 sr−1 , in the energyrange 20 MeV to 102.4 GeV. This is a softer spectrum than reported by thefirst analysis of the EGRET data by Sreekumar et al. [52]. The flux is alsosignificantly lower when taking into account the improved point-source sen-sitivity of the LAT. However, the flux is compatible with the re-analysis ofthe EGRET data by Strong et al. [54], but does not show any of the harderspectral features above 1 GeV. However, it is intriguing to note that the mea-sured spectral index is again in good agreement with the Fermi LAT measuredaveraged spectrum of Blazars, 2.40±0.02 [19]. However, using the first popu-lation study of blazars from the Fermi collaboration it does not seem like theycan make up more about 15% of the total measured IGRB flux [19].Other studies, however using the EGRET blazar catalogue, shows that unre-solved blazars [16] and (or) starforming galaxies [85] can account for most ofthe Fermi measured IGRB.

The measurement of the IGRB using the linear analysis is shown in table5.2. The formal error from the linear fits, the error in the effective area (cf.section 5.2.1) and the error from subtracting the residual background (cf sec-tion 5.2.2) were added in quadrature to obtain the final error of the measure-ment. Table 5.2 also show foreground model predicted and fitted intensities.The χ2 of the individual fits are not perfect from a statistical point of view.However, the error of the effective error has not been taken into account inthese fits. This is because this error is not independent within each energy binand can therefore not be added to the statistical error of each data point in astraight forward way. Keeping this in mind and considering the uncertainty inthe galactic foreground emission and the crude way in which the sources havebeen modelled, the χ2’s does not seem unreasonable.

In figure 5.11 the linear fits for each energy bin are shown together with theobtained energy spectrum. Also shown is a compression with the GaDGET,simulated residual background spectrum and the systematic uncertainty fromthe foreground modelling.

A minimum χ2 fit was done between the measured integrated intensityspectrum a power law differential energy spectrum. The best fit value for thespectral index is −2.39± 0.08 with an integrated intensity (above 100 MeV)of (0.99±0.76)×10−5 cm−2 s−1 sr−1 . Both are in agreement with the GaD-GET result within errors.

This analysis confirms the result of the GaDGET analysis PAPER III andshows that the obtained IGRB spectra is not significantly dependent on theanalysis method used. However, it seems the linear analysis results are at-tached with larger errors.


Table5.2:Integrated

intensitiesand

fitresultsforthe

IGR

Band

theothercom

ponentsin

cm2

s −1

sr −1.

Energy

inG

eV0.2-0.4

0.4-0.80.8-1.6

1.6-3.23.2-6.4

6.4-12.812.8-25.6

25.6-51.251.2-102.4

Fluxfactor

×10 −

6×

10 −7

×10 −

7×

10 −8

×10 −

8×

10 −9

×10 −

9×

10 −9

×10 −

10

EG

B2.3±

0.69.1±

2.03.6±

0.613.8

±2.4

5.0±

0.916.2

±4.3

7.8±

1.93.2±

1.010.7

±4.6

GalD

iff(model)

4.825.0

11.141.7

14.146.3

15.25.0

16.5G

alDiff(fit)

4.6±

0.524.3

±2.8

11.6±

1.546.2

±6.9

16.0±

2.746.5

±8.7

15.5±

3.23.6±

0.915.1

±4.7

Sources(m

odel)1.0

3.71.5

6.52.8

12.55.7

2.713.4

Sources(fit)

0.9±

0.13.9±

0.51.8±

0.27.5±

1.12.8±

0.57.2±

1.42.4±

0.62.4±

0.76.8±

3.1Sol(m

odel)0.1

0.70.3

1.60.7

2.61.0

0.31.1

Sol(fit)0.1±

0.020.5±

0.10.2±

0.040.9±

0.20.4±

0.11.5±

0.50.6±

0.30.3±

0.23.0±

1.8C

Rbackground

1.3±

0.63.8±

1.70.9±

0.42.5±

1.20.7±

0.35.7±

2.91.3±

0.80.5±

0.40.8±

1.2

LA

T9.4±

0.942.3

±4.9

18.4±

2.571.9

±10.8

25.0±

4.278.6

±14.5

28.0±

5.710.7

±2.4

37.3±

8.9

χ2

offit(perd.o.f.)14695.7

(1.5)12618.1

(1.3)4543.6

(1.9)3401.4

(1.4)946.4

(1.8)756.8

(1.4)265.5

(1.9)22.0

(1.1)21.8

(1.1)


102 103 104 105

E[MeV]

10-5

10-4

10-3

E·φ

[MeVcm

−2s−

1sr−

1]


γ=−2.387±0.08

A=0.0083±0.0053

χ2 =0.846/7

Linear Analysis−//− Power law fitGaDGET analysis−//− Power law fitSystematic errorsCharged particles

Figure 5.11: Upper plot: The measured energy spectrum for the linear analysis (bluesquares) together with the best fit power law (blue dashed line). This is compared withthe results of the GaDGET analysis from PAPER III (black dots) together with thecorresponding best fit power law (back dotted lines). The charged particle backgroundis represented by the green band, and the systematic uncertainties related to the galac-tic diffuse foreground is represented by the grey band. Lower plots: The individuallinear fits.

53

6. Dark Matter

Most of the matter in our universe is believed to be made out of invisible, atmost weakly interactive, yet-to-be discovered particles known as dark matter.The gravitational influence of dark matter on ordinary visible matter can beeseen on almost all cosmological scales. However, as of today it is only thegravitational influence that has been observed.

This chapter first gives a short introduction to how the evolution of the Uni-verse is modelled. Then follows a brief description of some of the compellingindications for the existence of dark matter, and how that has led to the currentcold dark matter (CDM) paradigm of modern cosmology.

6.1 The evolving UniverseThe evolution of the Universe if often described by the Hubble parameter H,named after Edwin Hubble, who already in 1929 observed the first indicationof an expanding Universe [86]. Using Einstein’s equations of general relativ-ity, and assuming an isotopic and homogeneous Universe (the cosmologicalprinciple), H obeys the so-called Friedmann equation:

H2 ≡ aa

=8πGρ +Λ

3− k

a2 . (6.1)

In the above equation G is Newton’s constant of gravity, ρ(t) the energy den-sity of the Universe, Λ a constant, a(t) an unconstrained function of timeand k = −1, 0, +1 depending on if a negative, flat or positive curvature hasbeen adopted for the metric of the Universe1. Given an equation for ρ(a) andboundary conditions, equation 6.1 can be solved for a(t) which completelyspecifies the dynamical evolution of the Universe.

Often the energy density ρ is expressed in fractions of the critical densityρc, introducing the parameter Ω = ρ/ρc with

ρc =3H2

8πG. (6.2)

In the same way the constant Λ and the curvature k, can be associated withenergy densities, ΩΛ ≡ Λ

8πGρcand Ωk ≡ −k

a2 (we will simply refer to the Ωs

1Natural units have been adopted and will be so for most of the thesis., i.e. speed of light c andthe reduced Planck constant h equals to one.

54 DARK MATTER CHAPTER 6

as densities). The constant term will not be diluted by the expansion of theUniverse. Therefore, at some point it will start to dominate the energy contentof the Universe and start to accelerate the expansion. Observations of the ap-parent magnitude of Type-1a supernovae [87] indicate that this has started tohappen in the recent stage of the evolution of the Universe. Whether or not theacceleration is caused by a cosmological constant or a mysterious dark energyis another outstanding question in modern cosmology and particle physics.

The density Ω is usually separated into a baryonic component Ωb (i.e. or-dinary matter), cold dark matter ΩCDM and a radiation/relativistic particlescomponent Ωr. The densities evolve in different ways with the expansion ofthe Universe. Therefore they have dominated the energy content at differenttimes.

Since the Universe is expanding it is not trivial to calculate distancesto far away sources. As Hubble noted, the light from distant galaxies wasDoppler shifted proportional to their distance. This defines the redshift z:1 + z = λobs/λemit, where λemit and λobs is the wavelength of the light atemission and observation respectively. In terms of the expanding Universethis is related to the scale factor a(t) as 1+ z = a(tobs)/a(temit).

The distance to a far away source, at redshift z, is for a flat Universegiven by the luminosity distance dL = (1 + z)

∫ z0 dz′/H(z′), where

H(z) = H0√

(Ω0M(1 + z)3 + Ω0

Λ) and H0,Ω

0M,Ω0

Λare the present day Hubble

constant and densities respectively.

Today the Universe is estimated to be 13.7 billion years old, and its energybudget is: ΩΛ ∼ 74%, ΩCDM ∼ 22%, Ωb ∼ 4% and Ωr ∼ 0.005% [88].

6.2 Nature of Dark Matter - ObservationsThe need of invisible dark matter to explain the dynamics of distant galaxieswas first pointed out by Zwicky in 1933 [89], when studying the Coma cluster.At present the evidence for the existence of cold dark matter is overwhelming.The observational evidence comes from studies of rotation curves of galax-ies [90], dynamics of clusters of galaxies [91], the Sunyaev-Zel’dovich effect[92], big bang nucleosynthesis [93], gravitational lensing [94] and the cosmicmicrowave background (CMB) [88]. Detailed reviews of motivations for darkmatter can be found in for instance [95, 96, 97]. Below follows two of themost frequent discussed observational indications.

Rotation CurvesObservations of the rotational properties of our Galaxy indicate that some-thing other than the stellar population must contribute significantly to the

SECTION 6.2. NATURE OF DARK MATTER - OBSERVATIONS 55

mass of the Galaxy [98]. Assuming that it is the gravitational force that keepsthe starts is circular orbit in the galaxy, the velocity at a galactic radius r isgiven by v(r) =

√GM(r)/r, where G is Newtons’s constant of gravity and

M(r) ∝∫

r2ρ(r)dr the mass within r, with a density given by ρ(r). The stel-lar population density is observed to decrease exponentially with radius whichwould imply a constant mass distribution at large radii and hence a velocityrotation curve vstar(r) ∝

√r at large radii. On the other hand, the velocity curve

is observed to be constant at large distances from the galactic centre. This canbe achieved by adding a mass component with MDM(r) ∝ r, or equivalentlyρDM(r) ∝ r−2. This is called an isothermal profile and was the first estimateof the dark matter halo introduced to explain the rotation curves of galax-ies. To fit the rotation curves at small galactic radii a pseudo-isotermal profileρ ∝ (1 + 1/r)−2 was introduced, which approached a constant value at thecentre of the galaxy.

An example of an rotation curve, that illustrates the discussion above, canbe seen in figure 6.1, for the galaxy NGC 3198.

Figure 6.1: Measured rotation curve of NGC 3198 from [99], with an example ofthe ordinary contribution ’disk’ and the dark matter contribution ’halo’. Note that thestars of this survey could only be detected out to a radius of 15 kpc. Beyond that 21cm measurement of hydrogen was used to estimate the disk mass distribution.

The Bullet ClusterA recent, and maybe one of the most striking evidences of dark matter, is theBullet cluster [100]. The cluster is observed to undergo a high velocity merger


and two clusters can bee seen emerging from the collision (one of them has acharacteristic ’bullet’ shape). The hot gas in the merger emits X-rays and tem-perature estimates give a good indication of the total mass of the gas, which isthe dominant component of baryonic matter in the cluster. However, the totalmass of the cluster is estimated by weak gravitational lensing. This measuresmany small distortions of objects behind the cluster due to the gravitationalbending of light as it is influenced by the gravitational force of the cluster.

Not only do the two mass estimates not agree quantitatively, i.e. the totalmass is much larger than the observed mass of the gas, implying existence ofdark matter, but they do not agree qualitatively either. The hot gas, deformedby its radiative interaction, does not coincide with the bulk of the mass ofthe cluster, but lags behind. Not only is a clear separation seen between theradiating matter and the main gravitational source, but the latter also showsno deterioration in shape by the collision, as the former do. This can easilybe understood if the bulk of the gravitational source is made out of non in-teracting dark matter. The dark matter components of the colliding clusterswould pass trough each other with no interaction other than gravitational. Thegas components on the other hand would interact and radiate, causing a shockfront which would lag behind the dark matter. The separation and shapes, ofthe (intact) dark matter halos and (deformed) shock fronts of the two clusters,can be seen in figure 6.2.

Previously it had been known that the hot gas of clusters could not be con-tained by the mass of the visible matter, but should evaporate away. Also itwas known that mass estimates of clusters by gravitational lensing, did notagree with the mass of the visible matter in the cluster. The Bullet cluster addsanother piece of evidence for the non interactive nature of dark matter.

6.3 Nature of Dark Matter - PredictionsThe matter density field of the early Universe can be described by small fluc-tuations around the mean density. The over-density field can be modelled asa superposition of plane waves, where the perturbation amplitudes for eachscale (wavelength) are assumed to have a random Gaussian distribution. Afterinflation, matter density fluctuations are assumed to be scale invariant. Duringradiation domination (i.e. when Ωr is larger than the other densities), densityperturbations can only grow until they enter the horizon of the causally con-nected Universe, ∼ 1/H. Perturbations within the horizon will oscillate dueto radiation pressure and not grow significantly. Enhancements of perturba-tions at scales smaller than the horizon will therefore be suppressed. As thehorizon expands larger and larger scales of perturbation will cease to grow. Atthe time of matter-radiation equality (when the Universe is about ∼ 7× 104

SECTION 6.3. NATURE OF DARK MATTER - PREDICTIONS 57

Figure 6.2: The merging Bullet cluster (named after the shape to the right). The X-rayemitting gas (red) and the main mass (blue), from weak lensing, is showed on topof an optical image of the region. Courtesy X-ray: NASA/CXC/CfA/M.Markevitchet al.; Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al.; Lensing Map:NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.

years old), the radiation pressure will no longer prevent perturbations to growat any scale, and they will all increase at the same rate. Therefore there is asuppression of structures smaller than the size of the horizon of the Universeat matter-radiation equality. Today this corresponds to a suppression of struc-tures smaller than ∼ 1 Gpc.When matter dominates the energy density of the Universe, collision-lessmatter (as for example CDM) perturbations can grow on all length scales.Baryonic matter on the other hand is coupled (electromagnetically) to radia-tion and experiences the pressure as the baryonic-radiation perturbations growtoo dense. This will prevent baryon perturbations to amplify below a certainlength scale.

After decoupling (i.e. when the CMB is emitted, when the Universe is about∼ 3×105 years old) the baryonic matter, now free from the radiation pressure,rapidly fell into the previously formed gravitational potential wells of the darkmatter. Without the early growth of dark matter structure, baryons would neverhave had time to form the structures observed today. The average distance themost extended pressure waves in the coupled baryonic-photon plasma couldhave travelled until decoupling leaves an imprint, corresponding to the firstacoustic peak of the CMB power spectrum. Today this imprint corresponds to


an increased probability of finding galaxies separated by a distance of ∼ 150Mpc [101].

The linear growth is only applicable for small enough perturbations. Atsome point over-densites will start to collapse and form gravitationally boundobjects, halos, which will start to interact with each other. The collapse of ob-jects can be described analytically in the linear regime by e.g. Press-Schechtertheory [102] and its extension. However, a proper treatment of interactionbetween halos of different masses becomes to complicated to handle analyti-cally. The linear regime breaks down when the density perturbations no longercan be approximated by a random Gaussian field. Today, only scales largerthan ∼ 8 h−1 Mpc can be treated using the linear formalsm. The non-linearregime is often modelled by large simulation (so called N-body simulations).They start from an initial condition (e.g. from the primodial power spectrumof fluctuations) and evolve a large number of gravitationally interacting parti-cles in an expanding universe. They show that structure forms in a hierarchi-cal manner, with small halos forming first and subsequently merging to formlarger halos.

Halo profilesLarge N-body simulations were pioneered in the 90’s [103] and it was earlyestablished that the halo profiles were well fitted by a power law ρ ∼ r−1 inthe central part.Subsequent simulations, such as the Millennium [104] and Via Lactea [105],confirmed this result and concludes that dark matter halos in a large massrange were well fitted by a universal density profile of the form:

ρ(r) =ρs

(r/rs)γ[1+(r/rs)

α](β−γ)/α

. (6.3)

This is a broken power law that scales as r−γ close to the centre of the halo,r(β−γ)/α at an intermediate distance ∼ rs and r−β in the outskirts of the halo.Halos of different masses and sizes are then determined by the values of ρsand rs.

A profile that fits N-simulation is the Navarro, Frenk and White (NFW)profile [106] with (α, β , γ) = (1, 3, 1) in equation 6.3. Different values of thepower-law indices have been reported from different simulations and the innerprofile could be as steep as r−1.5, as obtained in [107]. The kind of profiles thatare divergent at the origin are referred to as cuspy.

Sometimes it is convenient to parametrize a halo by its virial mass M andconcentration parameter c instead of the parameters ρs and rs in equation 6.3.The concentration parameter is defined as c = R/r−2, where r−2 is the dis-tance at which the profile falls as r−2 (r−2 = rs for the NFW profile). R isthe virial radius within which the halo has the mass M and a mean density of


10−2 10−1 100 101 102

r/rs

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

ρ/ρs

Moore (1999)NFWEinasto, α = 0.17Einasto, α = 0.2Einasto, α = 0.23Isothermal

Figure 6.3: Examples of dark matter halo profiles. In terms of equation 6.3 they cor-respond to: pseudo-Isothermal (α, β , γ) = (2, 2, 0), NFW (α, β , γ) = (1, 3, 1) [106],Moore (1999) [107] (α, β , γ) = (1.5, 3, 1.5). The Einasto profile is showed for dif-ferent values of α (in equation 6.4), and rs and ρs are r−2 and ρ−2 correspondingly.


δ (z)ρ(z) where ρ(z) is the mean matter density of the Universe (δ (z = 0) is tosome extent a matter of convention, common values are ∼ 100 or 200). Thesedefinitions will be used when calculating the flux from cosmological WIMPannihilations in chapter 7. The term virial refers to the virial radius, withinwhich the virial theorem holds. It states that the average gravitational poten-tial energy is twice the average kinetic energy. Practically it defines a gravi-tationally bound system where the dark matter particles has a fairly isotropicdistribution of velocities.

When adopting a NFW profile, the mass of the Milky Way halo has been es-timated to ∼ 1012 M [108], with R = 255 kpc and c = 18 (which correspondsto rs = 14 kpc). From N-body simulations the concentration parameter hasbeen found to depend on the halo mass and change with redshift. The relationc(M, z) is an important prediction of ΛCDM N-body simulations. For a MilkyWay sized halo (i.e. M ∼ 1012 M) the concentration parameter at z = 0 isfound in a range between 10 and 20. Even if the empirical models proposed todescribe c(M,z) agree fairly well within the mass interval resolved in N-bodysimulations, they can differ substantially when extrapolated to lower masses[109, 110, 111, 112]. This has a large effect on the predicted extragalacticgamma-ray signal from dark matter, as will be discussed in chapter 7

It has been suggested that another kind of profile, the Einasto profile [113],can cure systematic differences that appear over large mass scales betweenNFW profile fits to N-body simulations. The Einasto profile introduces a thirdparameter, α that regulates the transition of the power-law slopes:

ρ(r) = ρ−2 exp(− 2

α

[(r

r−2

)α

−1])

, (6.4)

where ρ−2 and r−2 are the density and radius where the logarithim slop is -2, respectively. The parameter α has been found to depend systematically onthe formation history of halos [110]. A value that fits N-body simulations isα ∼ 0.2. Examples of profiles can be seen in figure 6.3.

X-ray observations of galaxy clusters have revealed mass distributions com-patible with the NFW profile. The c(M,z) relation on the other hand seem toevolve less with redshift and decrease faster with mass, compared to N-bodysimulations [114].

When studying halos of smaller mass, i.e. galaxy size and below, observa-tions and simulations are harder to reconcile. The cuspy profiles do not pro-duce the observed rotation curves of galaxies (cf. section 6.2), that requiresa profile approaching a constant value at the centre ρ ∼ r0, refereed to ascored. This discrepancy between observations of rotation curves of galaxiesand predictions of cold dark matter simulations has been a long standing issuefor the cold dark matter paradigm and is referred to as the core-cusp problem[107, 115, 116].


Core versus CuspIt was first believed that the method that was used to measure rotation curvesdid not offer high enough resolution and therefore could mistake the innercusp for a core. However, as observations techniques have improved it seemsthat systematic effects of the measurements are not large enough to explainthe discrepancy, rotation curves still preferred cored profiles. Moreover, whencusp profiles can be fitted to rotation curves, the structural properties of thedark matter halo can not be made compatible with ΛCDM predictions, .i.e.they do not follow the c(M,z) relation discussed above.

Alternative suggestions have been proposed to resolve the problem, manyof them have to do with the fact that typically baryons are not included inN-body simulations. This is due to computational limitations or that the bary-onic effects would come into play below the time and mass resolution of thesimulations. Feedback from starformation, relativistic jet outflows or the ef-fect of merging galaxies could potentially erase the cusp that is predicted fromdark matter only simulations. However, the effect of including baryons doesnot always shallow out the profile, it can also make it more cuspy since largeamount of baryons in the center of halo can accumulate dark matter, makingthe problem more severe [117].

Also it is not clear that the formation history scenarios that have been sug-gested to levitate the problem can be applied to the special kind of galaxiesfor which the rotation curves are obtained. The galaxies of interest are lowsurface brightness (LSB) galaxies. These are faint, gas-rich, late-type galax-ies that are dark matter dominated. They seem to have have had only spo-radic star-formation and show little evidence of interactions or mergers withother galaxies or any other events that might have disrupted the dark matter orbaryonic components. In general LSB galaxies seem to have had a quiescentevolution.

Also it has been suggested that if the full triaxility of dark matter halos aretaken into consideration, they can appear as cored, even though they are cuspy[118]. However, this solutions turns out to depend on the observations angleof the halo and the rotation curves should sometimes be observed to be evencuspier than simulations predict. However, this is very rarely observe.

The only way to resolve the issue, whether it is a real dead end for CDM ornot, seems to be to self consistently include baryons in high resolution cosmo-logical N-body simulations. A recent attempt to resolve the in-homogeneousinterstellar medium in a galaxy, assuming a ΛCDM universe was presented in[119]. They found that strong outflows from supernovae reduced the centraldensities of dark matter and made the profile as shallow as ρ ∼ r−0.6.


Missing SatellitesAnother long standing problem facing the ΛCDM scenario is that the numberof small scale halos should be many more than the observed galaxies canaccount for. According to N-body simulations, a Milky way sized halo shouldbe surrounded by a few hundred dark matter satellites big enough to host agalaxy. This is sometimes referred to as "the missing satellite problem" [120,121].

Originally the discrepancy between observed and predicted satellites wasmore than one order of magnitude; about 10 observed and about 500 predicted(for instance by the Via Lactea simulation [122]).

Proposed solutions to this problem are usually of two kinds, change theproperties of ΛCDM or change the astrophysics of galaxy formation. Onecould change the primordial power spectrum of gravitational fluctuations atsmall scales or make the dark matter particles slightly "warm", i.e. give themhigher velocities, which would suppress structure formation at small scales.

From the astrophysical point of view, starformation could be prevented indwarf galaxies by suppressing allocation of enough gas due to reionisation.Alternatively tidal stripping could have reduced once very bright galaxies totheir present form.

Using data from the Sloan Digital Sky (SDSS) Survey [123] several more,very faint and dark matter dominated galaxies have been observed. Takingtheses galaxies into account, and modelling the two astrophysical scenariosmentioned above, the number of predicted and observed satellite galaxies con-verge more and more [124, 125], .

6.4 WIMP Dark MatterProbably the most studied class of dark matter candidates are the WeaklyInteractive Massive Particles (WIMPs). They were in chemical and thermalequilibrium with the primordial plasma, through the electroweak interactionand the processes:

χ +X ↔ χ +Y (6.5)

and

χ + χ ↔ X +Y , (6.6)

where χ represent the WIMP and X and Y are particles of the standard model.The number density of WIMPs nχ will develop in time around the thermalequilibrium value nχ,eq, according to:

dndt

=−3Hnχ −〈σv〉[n2

χ −n2χ,eq

]. (6.7)

SECTION 6.5. LOOKING FOR DARK MATTER 63

Here 〈σv〉 is the thermal average of the total annihilation cross section timesthe relative velocity of the WIMPs (hereafter cross section). The dilution ofparticles due to the expansion of the Universe is given by the term −3Hnχ . Ifnχ becomes smaller than nχ,eq the process 6.6 will happen more often in the← direction and nχ will approach the equilibrium value again and vice versa.

At some point the interaction rate Γ = nχ〈σv〉 fell below the expansionrate of the Universe H and the processes 6.6 ceased. When this happened theWIMPs decoupled chemically from the rest of the plasma, the number densityof WIMPs was "frozen out". A complete calculation of the relic density afterfreeze out depends on the specific model of WIMPs, but a good estimate isgiven by [95]:

Ωχh2 ≈ 3 ·10−27 cm−3 s−1

〈σv〉, (6.8)

where h ∼ 0.7 is the Hubble constant in units of 100 km s−1 Mpc−1. Toobtain the observed value of the relic abundance Ωχh2 ∼ 0.1, the crosssection has to be 〈σv〉∼ 10−26 cm−3 s−1. The ’WIMP miracle’ is that ifthe WIMP has a mass at the electroweak scale mχ ∼ 100 GeV, the crosssection comes out about right automatically σv ∼ α2/m2

χ ∼ 10−25 cm−3 s−1,where α is the fine structure constant. The electroweak scale is also wherewe expect to find extensions of the standard model of particle physics, e.g.supersymmetry. This is what makes the WIMP class of models so appealing.

There are also other particle physics models that contain suitable dark mattercandidates. The most studied are probably the axion and gravitino. They areusually not in equilibrium with the primordial plasma but are produced withthe correct relic abundance in some other way. See for instance [126] for areview of non-WIMP dark matter.

6.5 Looking for Dark MatterEven though dark matter has only been seen trough its gravitational influenceof baryonic matter, there are prospects for other means of detection.

6.5.1 Direct SearchesTypically set in remote places, like deep underground, the direct detectionexperiments search for dark matter scattering of nuclear material. The recoilenergy spectrum is generally given by [127]

dNdEr

=σρ

2µ2mχ

F2∫ vesc

vmin

f (v)v

dv, (6.9)


where ρ ∼ 0.3−0.4 GeV cm−3 is the local dark matter density, mχ the WIMPmass, µ = mχmN/(mχmN) is the WIMP-nucleus reduced mass (assuming nu-cleus mass mN), σ the interaction cross section and F2 the nuclear form fac-tor. The integral is over the velocity distribution f (v) of dark matter particles,where vmin is the minimum velocity able to generate a recoil energy Er andvesc is the maximum escape velocity set by the halo model. Typically a spher-ically symmetric Maxwell-Boltzmann velocity distribution is chosen with aroot mean square velocity in the range 170-270 km/s. The interaction crosssection has both a spin-independent and spin-dependent part. Depending onthe interaction properties of the dark matter particle one or the other domi-nates. These kind of scattering processes are the same that once kept the darkmatter in thermal equlibrium with the primordial plasma of the early Universe.

There have been claims that excess event has been detected and could beinterpreted as dark matter. In particular the DAMA collaboration has reportedan excess correlated with the annual modulation when the earth is movingtowards or with the mean dark matter wind in the Galaxy [128, 129]. However,this result has been hard to reconcile with the upper limits from other directdetection experiments when assuming a standard halo model.

Recently an excess of 2 event, over an expected background of 0.9± 0.2was reported by the CDMS-II experiment [130] located in the Soudan minein Minnesota.

6.5.2 Indirect searchesInstead of looking for dark matter interacting on Earth, one can look for sig-nals that would appear from annihilation of dark matter particles into standardmodel particles: indirect detection. This kind of interactions are believed tooccur since the WIMPs where once in chemical equilibrium with the primor-dial plasma through pair-production and pair-annihilation. The annihilationproducts could either be measured directly or they could decay to detectableparticles. Thus, one expects to see signatures of WIMPs in gamma-rays, neu-trinos and cosmic rays from regions with enhanced dark matter density.

The flux energy spectrum of gamma-rays2 from annihilating dark matter, asdetected on Earth, in a direction ψ on the sky is of the form.

dΦ(ψ)dE

=〈σv〉8π

dNγ

dE

∫l.o.s.

(ρ(ψ)mχ

)2

dl, (6.10)

where we have used previous notations and the integral over the dark matterdensity is along the line of sight in the direction ψ . The gamma-ray annihila-

2This expression is also valid for neutrinos, since they propagate freely to us from the site ofannihilation. For charged particles the situation is different and the line of sight integration hasto be replaced by a propagated emission spectra.


tion yield dNγ/dE is given by

dNγ

dE= ∑

fbf

dNf

dE+2bγγδ (E−Mχ)+bXγδ (E−M2

X/4Mχ), (6.11)

where bx is the branching ratio into the x final state. The first term in equation6.11 is the contribution from WIMP annihilation into the full set of tree-levelfinal states, containing fermions, gauge or Higgs bosons, whose fragmenta-tion/decay chain generates photons. These processes give rise to a continuousenergy spectrum. The second and third terms correspond to direct annihilationinto final states of two photons or one photon and some other neutral particle(such as a Z or Higgs boson), respectively. Although of second order (loopprocesses), these terms can give rise to significant amounts of monochromaticphotons.

If assuming a non standard dark matter profile, the EGRET "GeV excess"could be interpreted as an indirect signal from annihilating dark matter [131].However, that particular model was showed to overproduce anti-protons [132].

Another signal that has been interpreted as an indirect signal from darkmatter is the 0.511 MeV line observed by the Integral instrument [133]. Thedark matter candidates suggested are light mχ ∼ 50 MeV [134] and do not fallinto the standard WIMP class of models.

Recently measurements of cosmic charged particles has also been inter-preted as dark matter, as will be described in section 6.5.4 below.

6.5.3 At acceleratorsEven if a signal from dark matter is discovered from direct or indirectsearches, this will be far from establishing its full nature. A clear line signalin gamma-rays would for instance give a good estimation of the mass ofthe particle and an estimate of the degenerate quantities density and crosssection. However, it would say little about the underlying particle physicsscenario. So far we have mainly learned about the physics of heavy particlesat large accelerators.

The large hadron collider (LHC) at CERN, an accelerator with a circum-ference of almost 30 kilometres, built 100 meters underground at the borderbetween France and Switzerland, is about to start operation. It will collide pro-tons with a centre of mass energy of 14 TeV and therefore probe new physicsbeyond the standard model [135].

If dark matter is WIMP-like it might be produced at the LHC and its prop-erties can be studied. However, to confirm that a particle produced in a lab onEarth is really the same as the one that makes up most of the dark matter inthe Universe, it has to be confirmed by astronomical observations.


6.5.4 Dark Matter CandidatesThere exist many extensions of the standard model of particle physics thatcontain suitable dark matter candidates. However, there is no reason to believethat there is a single species of particles that constitute all of the observeddark matter. When identifying regions of parameter spaces that produce theobserved relic abundance, which constrains masses and cross sections of themodel and therefore also specifies the indirect signal, one typically assumesonly one dominant dark matter component. If this condition is relaxed, sig-nals could be much stronger without violating relic abundance measurements(even though the signal might be in conflict with other observations). Thiscan be worth keeping in mind when looking at the various parameter spacespresented in chapter 8.

The WIMP candidate is the lightest of what can be an entire family of newparticles predicted in the specific particle physics scenario. They are usuallyprevented from decaying further into SM particles by an imposed discrete Z2symmetry. In this section we only touch briefly on the three scenarios thatare used in the context of upper limits on the cosmological dark matter signalin chapter 8. There are of course many more scenarios existing, but many ofthem have the same observable consequences, in terms of gamma-rays, as theones described below.

SupersymmetryProbably the most studied WIMP candidate is the neutralino, the lightest neu-tral particle that arises in supersymmetric extensions of the SM (see e.g. [95]).The neutralino is often used as the archetype for majorana fermionic dark mat-ter. Supersymmetry (SUSY) introduces a symmetry between every fermionicand bosonic degree of freedom. From a theoretical point of view SUSY offersone solution to the so called hierarchy problem, i.e. it removes fine tuning ofhigh energy parameters as they contribute, through radiative corrections, tothe mass of scalar fields responsible for electroweak symmetry breaking. TheZ2 symmetry that prevents the lightest SUSY particle to decay into standardmodel particles is called R-parity. Effectively it is imposed to be conserved inevery interaction vertex, where the superpartners have R = −1 and ordinaryparticles have R = 1.

The smallest supersymmetric extension of the standard model is called theminimal supersymmetric model (MSSM) in which, in addition to one morescalar doublet field, every particle has corresponding super-partners, fermionsfor the bosons and bosons for the fermions. So far no super-partners havebeen observed which means, if SUSY exists, that it is not an exact symmetryof the theory at lower energies, i.e. that SUSY is broken. However, there aremany ways to break SUSY and the breaking scheme introduces many new


parameters in the theory. The MSSM contains more than one hundred newparameter that describe couplings, masses and mixings of the new fields.

In order for SUSY to solve the hierarchy problem, and offer viable WIMPcandidates, it has to be broken at TeV energies. The MSSM also offers a wayto unite the couplings of the standard model gauge groups U(1)SU(2)SU(3),induce a spontaneous electroweak symmetry breaking and SUSY also seemsto be a natural ingredient in super string theories.

It is very hard to handle a parameter space as big as that of MSSM. There-fore many of the terms in the full theory are removed due to phenomenologicalconsiderations (i.e. charge-parity violating terms and flavour changing neutralcurrents at tree level) and many mass terms are set equal. MSSM-7 [136] is areduction of the full MSSM parameter space with only seven free parameters,all defined at the electroweak scale.

Leptophilic dark matterRecent measurements of the electron and positron fluxes by PAMELA [44]and Fermi [45] has revealed excesses over what is expected from conventionalcosmic ray models . This has led to suggestions of dark matter models witha dominant annihilation into charged leptons, which could give a signal ingamma-rays via inverse Compton scattering on background radiation and viafinal state radiation. In order to fit the observation the dark matter particlehas to be heavy > 10 TeV, or the annihilation has to be into leptons only[137]. The level of signal required is a few orders of magnitude higher thanwhat is expected from the standard freeze-out scenario. However, such highsignals can be obtained by increasing the cross section through the so-calledSommerfeld enhancement effect [138], and (or) by increasing the dark matterdensity, e.g. with substructure. From a particle physics point of view, modelswith such high cross sections, mass and particular annihilations channels arenot what you would expect from the "standard" WIMP scenario. However,there are certainly models that allow for a dark matter interpretation of thePAMELA/Fermi cosmic ray data [139].

The Inert Doublet ModelThe inert doublet model (IDM) [140] is an extension of the standard modelwith only one extra Higgs doublet H2. It has no couplings to fermions due toan imposed unbroken discrete Z2 symmetry (i.e. H2 is inert), under which H2is odd and all remaining fields are left invariant. One feature of IDM is thatthe mass of the particle that plays the role of the standard model Higgs can beas high as about 500 GeV and still fulfill present experimental precision tests[140].

Since conservation of the Z2 symmetry implies that the lightest of the twoinert Higgs particles is stable, it is also a viable dark matter candidate. One


of the interesting features of the IDM is that it offers very high annihilationbranching ratios into γγ and Zγ final states, compared to the branching ratiosinto quarks which yield the continuum spectra [141]. The range of dark mattermasses is just in the range where the sensitivity of the Fermi-LAT is optimal.

69

7. Cosmological Dark Matter

This chapter is devoted to the properties of the diffuse gamma-ray signal orig-inating from cosmological dark matter. The signal was early estimated in [60]and found not to give rise to a significant gamma-ray flux. However, it waslater realised that this signal is highly dependent on the structure of dark mat-ter [62]. Taking this properly into account, the signal is enhanced by severalorders of magnitude. However, since the distribution of dark matter is not fullyunderstood, this also gives the largest uncertainty when estimating the signal,as will be discussed in chapter 8.

The signal from dark matter annihilating throughout the Universe can becalculated in several ways. It differs from that of equation 6.10 since halos atall redshift have to be accounted for. Here we follow the procedure of [63],where the number of photons per unit effective area, time and solid angle inthe redshifted energy range E0 to E0 +dE0, is given by:

dΦ

dE0=〈σv〉8π

ρ2

m2χ

∫dz(1+ z)3 ∆2(z)

H(z)dNγ(E0(1+ z))

dEe−τ(z,E0), (7.1)

where we use notations defined in the previous chapter. The structure of darkmatter is encoded in the quantity ∆2(z), which is the average enhancementof the signal due to clustering. If dark matter was uniformly distributed∆(z) would be equal to one. The attenuation of gamma-rays is described byτ(z, E0), the optical depth of the Universe for photons.

In the following we will discuss the various quantities contributing to thesignal given by equation 7.1.

7.1 Particle physicsThe preferred particle physics model enters the differential gamma-ray fluxvia the cross section 〈σv〉, the dark matter particle mass mχ and the differentialgamma-ray yield per annihilation dNγ/dE, equation 6.11.

In contrast to the integral in equation 6.10, the emission spectra will bedistorted by the integration over redshift. Since the spectrum of the continuumand line signal are very different in shape, the result of the integration overredshift, in equation 7.1, is quite different. The continuum spectra becomeslightly broadened and the peak is red-shifted to lower energies.

70 COSMOLOGICAL DARK MATTER CHAPTER 7

101 102 103 104 105

E0 [MeV]

10−5

10−4

10−3

10−2

E2 0·dφ/dE

0[M

eVcm−2

s−1

sr−1

]

µ+µ−

bb γγ

1.2 Tev µ+µ−

200 GeV bb

180 GeV γγ

–//– , with energy disp.–//– , τ - Stecker et al.

EGRET (Sreekumar et al. 1997)EGRET (Strong et al. 2004)Fermi (Abdo et al. 2009)

Figure 7.1: Measurements of the IGRB by Fermi-LAT [3] and EGRET [52, 54], to-gether with three types of gamma-ray spectra from annihilating dark matter. The over-all normalization of the dark matter spectra are given by assuming the MSII-Sub1∆2 model, and for this visualization we have chosen the following cross sections〈σv〉 = 5× 10−25 cm3 s−1 (for bb), 1.2× 10−23 cm3 s−1 (µµ) and 2.5× 10−26 cm3

s−1 (γγ). The solid lines are with the Gilmore et al. [142] absorption model applied,and the dotted lines with the Stecker et al. [143] absorption. Also shown is the linespectra convolved with the energy resolution of the Fermi-LAT experiment (dashedline). The dotted line passing through the Fermi data points is a power law with thespectral index of -2.41. From PAPER II.

The line signal is different since all photons are emitted at the sameenergy, E = mχ (in the case of a 2γ-final state) and are observed at theenergy E0 = E(1 + z)−1. At high redshifts the universe becomes opaque tohigh energy photons and the signal decreases dramatically at lower E0. Thisresults in the characteristic spectral feature of a sharp cut-off at mχ , witha tail to lower energies as seen in figure 7.1. The line signal is particularlysensitive to the absorption model adopted, as illustrated in figure 7.1.

In chapter 8 we will use three kind of emission spectra when calculating upperlimits on the cosmological dark matter signal. These are final states of bottomquarks, final states of leptons and of monochromatic photons, i.e. from thegamma-ray line.

SECTION 7.1. PARTICLE PHYSICS 71

Model 1: bbMany dark matter candidates (e.g., within supersymmetry, c.f. section 6.5.4)have their dominating annihilation channel into quarks and heavy gagebosons. These give rise to gamma-rays via the hadronization and decay ofπ0 (cf. section 4.3). For simplicity we will assume annihilation only into bb.Also annihilations into other quarks as well as into W/Z gauge and Higgsbosons would all give fairly similar spectra. We use the DarkSUSY package[144] to obtain the gamma-ray yield dNbb/dE, when calculating the darkmatter signal.

Model 2: µ+µ−

We model a ’leptopilic’ (cf. section 6.5.4) dark matter candidate by assumingcomplete annihilation into µ-leptons. The yield of gamma-rays has two contri-bution: one from inverse Compton-scattering of background light on electronsand positrons originating from the decaying µ+µ− pairs, and one from finalstate radiation. The inverse Compton part is modelled following [145, 146]:the differential Klein-Nishina cross section is convolved with the number den-sity of the CMB and this is folded with the spectral distribution of electronsand positrons given by the DARKSUSY package [144], that utilize tabulatedPYTHIA 6.154 results [147].

When the final state of annihilation is into charged leptons, a photon canbe emitted from one of the final legs (final state radiation (FSR)) or from theinternal virtual process (internal bremsstrahlung (IB)). This gives an enhance-ment of photons at high energies with a cut-off at the mass of the dark matterparticle. The flux of final state radiation photons from the prompt µ+µ−γ

process was calculated by using the approximate formula given in [148, 149].

Model 3: γγ

If a multi-GeV line would be detected, this would be a very strong indica-tion of dark matter, since it is hard to imagine any conventional astrophysicalsource producing a line at these energies. The location of the line would alsobe directly correlated to the dark matter particle mass.

Even though branching ratios into two body final states containing at leastone photon are typically less than 1%, there are dark matter candidates, likethe inert doublet model (cf. section 6.5.4), that can have this as their mainannihilation channel. In chapter 8 we consider only the two photon channel.

There is also a dedicated line search by the Fermi collaboration [27]. In thatcase the shape of the line is set by the energy resolution of the Fermi-LATand therefore different in shape compared to the cosmological line signal. Inprinciple also our line signal has to be convolved with the energy dispersionbefore compared to data, since the IGRB measurement was not corrected forthat. However, the effect can be anticipated to be very small since the energy


bin width of the measurement is much larger than the energy resolution. Infigure 7.1 we convolve the signal from a 180 GeV dark matter particle with anenergy dispersion that follows closely the one used in [27], which is also validfor the IGRB measurement. The effect on the upper limits is, as expected,negligible.

7.2 Dark matter distributionSince the annihilation signal is proportional to the dark matter densitysquared, varying the steepness of the density profile can radically change thegamma-ray signal. Comparing a cored profile to a steep, cuspy profile givesa normalization difference of six orders of magnitude if the source is thegalactic center [63]. This difference tends to be much smaller if one integratesthe dark matter density over larger volumes. Therefore the cosmologicalsignal is not as sensitive to the exact choice of profile. Also, averaging overthe mass function of halos makes the cosmological signal more robust thanpoint-source signals.

The evolution of the dark matter structure is encoded in the quantity ∆2(z) inequation 7.1, which describes the averaged squared over-density in halos, as afunction of redshift.

In PAPER II we use two ways to calculate this quantity. The main approachis based on the recent N-body simulation ’Millenium-II’ [150] as used byZavala et al. [151] to determine the cosmological signal from dark matterannihilations. Using two basic structural properties of dark matter halos, themaximum rotational velocity of the halo and the radius at which this velocityoccurs, and by approximating the internal mass distribution of the halo by aNFW profile, Zavala et al. are able to calculate the luminosity from each halodown to masses of about 7 ·108 h−1 M, which is the resolution of resolvedhalos in the simulation.

However, from a theoretical point of view we expect to find halos withmasses down to 10−9 - 10−4 M [152]. How the extrapolation down to lowermasses is done is currently a big factor of uncertainty. This is because a sig-nificant part of the luminosity is expected to originate from small halos.

Also within larger halos, N-body simulations show that there should ex-ist smaller, bound halos that have survived tidal stripping. Although not asmassive as the primary halos the substructure halos arise in higher density en-vironments which makes them denser than their parent halo. Also the subhalosare tidally stripped from the outside in, which further increases their concen-tration. The phenomena of halos within halos seems to be a generic featuresince detailed simulations reveal substructures even within subhalos [111]

SECTION 7.2. DARK MATTER DISTRIBUTION 73

To take into account the contribution from unresolved halos (and subhalos)Zavala et al. fit a power law to the differential luminosity contribution, perlogarithmic halo mass bin, versus halo mass, at different redshifts. These fitsare then exrapolated down to a a minimum mass of 10−6h−1 M. How theseextrapolations are done defines a range of uncertainty on the normalisation ofthe cosmological dark matter signal. We define a conservative case when onlyresolved halos are taken into account (MSII-Res) and a very optimistic case(MSII-Sub2) where the extrapolations where done in a way as to maximisethe signal from small halos. We define a moderate case (MSII-Sub1) whereextrapolations to the smallest masses where done in a conservative way, whichwill be used as the reference scenario.

Each individual ∆2(z) is also affected by other parameters, such as cosmo-logical parameters (in particular σ8, the rms mass fluctuations on the 8 Mpch−1 scale) and the dark matter profile. However, these uncertainties are smallerthan the differences between the individual ∆2(z) models, so we consider themas bracketed by those.

An alternative approach is to calculate ∆2(z) in a semi-analytic way, whichwas done in [63] and PAPER I. Then one integrates the contribution from ha-los of all masses, weighted by the halo mass function, and for each mass takeinto account the spread of halo shapes as a function of redshift. Doing thisit is convenient to parametrize a halo by its virial mass M and concentrationparameter c. The concentration parameter depends both on mass and redshift,as defined in section 6.3, and we model this according to the model by Bul-lock et al. (2001) [109]. Here the concentration parameter is also treated as astochastic variable with a log-normal distribution for a fixed mass. Simplifiedwe can write the quantity ∆2(z) as:

∆2(z) =

∫dM

dndM

∫dcP(c)

⟨ρ2(M, c)

⟩〈ρ(M, c)〉2

(7.2)

where dn/dM is the halo mass function, P(c) the log-normal distribution〈ρ(M, c)〉 is the spatial average of the dark matter halo profile. The effectof subhalos is taken into account by assigning to them a fraction of the totalhalo mass, a different mass functions and higher concentration parameters (asin [63]). We set 10% of the halo mass in subhalos and associate with them a 4times higher concentration parameter, as in PAPER I. Using an NFW profilewe refer to this scenario as the semi-analytical NFW Bullock et al. substruc-ture model (BulSub).

In the upper panel of figure 7.2 the quantity ∆2(z)(1+z)3/h(z) for the differ-ent halo evolution scenarios (MSII-Res, MSII-Sub1, MSII-Sub2 and BulSub)can be seen. The extreme scenarios MSII-Res and MSII-Sub2 differs by threeorders of magnitude in signal strength and both the MSII-Sub1 and BulSub


give a value in between, with the BulSub signal being slightly stronger thanMSII-Sub1.

7.3 High energy γ-ray environmentAny extragalactic gamma-ray signal is affected by absorption in theintergalactic medium, especially at high energies. The absorption isparameterized by τ , the optical depth. The dominant contribution tothe absorption in the GeV-TeV energy range is pair production on theextragalactic background light emitted in the optical and infrared. For theoptical depth, as function of both redshift and observed energy, we use theresults by Gilmore et al. (2009) [142] and compare with a slightly oldercalculation by Stecker, Malcan and Scully (2007) [143]. One aspect of thedifferent absorption models can bee seen in the lower panel of figure 7.2,where the quantity ∆2(z)(1 + z)3/h(z) · exp−τ(z, E0) is plotted for differentenergies E0.

The absorption model by Gilmore et al. predicts s lower optical depth than theone by Stecker et al. In the latter, the universe is optically thick (τ > 1) for 10GeV photons at about z∼ 3 and for 25 GeV photons at about z∼ 1, whereas τ

is about 0.03 and 0.5 for the corresponding values in the Gilmore et al. model,and gets optical thick for 25 GeV photons only at about z ≈ 9.

One of the reasons for the differences is that the modelling is very dif-ferent in the two cases. The Gilmore et al. model is based on a calculationand measurements of the extra galactic background light (EBL) in differentwavebands, originating from different populations of sources, for differentredshifts. Whereas the Stecker et al. model is based on measurement of cur-rent brightness in one band and then assumed to brighten with redshift as apower law in (1+ z).

There also exists other models for the optical depth of gamma-rays, forinstance [61, 153, 154, 155, 156, 157, 158, 159]. However, the Gilmore et al.model predict a weaker absorption than all of these (except maybe for [157]at z > 2) and the Stecker et al. model a stronger absorption, which to someextent bracket the uncertainty in the gamma-ray absorption.

A high absorption has observational implications for measurements ingamma-rays for an experiment such as Fermi. High redshift sources wouldexhibit a cut-off in their spectra at high energies. Observing (or lack ofobserving) these features offers another way to measure the optical depth tophotons, in contrast to the other approach of calculating it starting form theEBL.

SECTION 7.3. HIGH ENERGY γ -RAY ENVIRONMENT 75

0 1 2 3 4 5 6 7 8 9z

104

105

106

107

108

(1+

z)3 ∆

2 /h

(z)

MSII-Sub2

BullSub

MSII-Sub1

MSII-Res

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5z

104

105

(1+

z)3 ∆

2 /h

(z)·e−τ

(z,E

0)

MSII-Sub1, τ = 0

10 GeV20 GeV

80G

eV

400G

eV 10GeV

20G

eV

80G

eV

400G

eV

τ = Gilmore et al. τ = Stecker et al.

Figure 7.2: Upper: Comparison of the different models used to calculate the enhance-ment of annihilation signal due to structure formation; ∆2(z) based on the MillenniumII simulation (MSII-models) [151] and the semi-analytic model (BulSub) [63]. Allthe enhancement factors ∆2(z) are multiplied by the factor (1 + z)3/h(z) in order toreflect the relevant part of the integrand in equation 7.1. Lower: Comparison of thegamma-ray absorption models of Gilmore et al. [142] (solid) and Stecker et al. [143](dashed), and their affect on the signal in the MSII-Sub1 structure formation scenariofor the energies E0 = 10,20,80 and 400 GeV. The upper most (black solid) line is ifno absorption is present. From PAPER II


7.4 Other isotropic signalsIt is not only the cosmological dark matter signal presented here that cangive rise to isotropic gamma-ray emission. Potentially decaying gravitino darkmatter, of galactic and extragalactic origin [67, 160], and unresolved galacticsubhalos [161] can give rise to an approximately isotropic signal. Also, a pop-ulation of intermediate mass back holes could accrete dark matter and enhancethe cosmological signal [162]. All of these sources would add to our signal andmake the upper limits calculated in chapter 8 more conservative.

77

8. Upper limits by Fermi

This chapter is divided into two parts. First we review the pre-Fermi sensitiv-ity estimates to the cosmological dark matter signal. The result was publishedin PAPER1, together with sensitivity estimates for many other potential darkmatter sources in gamma-rays. The second part reviews the actual upper limitsplaced by the first Fermi-LAT measurement of the isotopic gamma-ray back-ground. This result was published in PAPER2.

There are several different uncertainties associated with the upper limits.Not only is there a large uncertainty in the shape and normalisation of the darkmatter signal, but the background is also to a large extent hard to quantify. Be-cause of this we derive several different limits. For the sensitivity estimate weused only one kind of signal, with two different normalisations (correspondingto including subhalos or not) and two limiting cases for the background. Forthe upper limit analysis we use three different shapes of the signal (cf. section7.1), and two analysis approaches motivated by the difficulty to quantify thebackground.

In a given dark matter scenario, it is not at all clear from which region onthe sky the signal would dominate. Each region that potentially could giverise to a dark matter signal is attached with different observational difficultiesthat could make it more or less useful (e.g. different level of background andunderstanding of it) and different theoretical uncertainties. It has been arguedthat a signal from the galactic centre would always be stronger than the ex-tragalactic and therefore be discovered earlier [163]. However, local sources,like the galactic centre are more sensitive to the exact properties of the centraldensities of the dark matter profile, while the extragalactic signal, which isan average of many sources, is not as sensitive [63]. Since there is no uniquerelation between different sources of dark matter signals, it is important toestablish independent limits for each corresponding measurement.

8.1 Fermi sensitivity to Cosmological WIMPsThe Fermi sensitivity was estimated using fast detector simulations of a cos-mological dark matter signal and of the background.

78 UPPER LIMITS BY FERMI CHAPTER 8

A generic WIMPThe emission spectrum is treated in a very generic way, assuming annihilationinto bb quarks and a branching of 0.5 ·10−3 into the monochromatic gamma-ray line. In general a specific model, defined as a point in a possibly very largeparameter space, does not correspond to this generic case since annihilationchannels into other fermions, leptons, gauge or Higgs bosons could be sub-stantial. However, the bb emission spectrum is quite similar to the other casesand basically the only other annihilation channel that results in a significantdifferent emission spectrum is the τ+τ− channel (see [164] for a discussion).Therefore the generic WIMP is a good approximation to many particle physicsscenarios.

The normalisation of the signal was computed following [63], and is similarto the BulSub scenario employed for the upper limit calculation.

Background sourcesThe background sources used are of two kinds, one solely consisting of unre-solved blazars from [63] and one which is the best fit power-law to the EGRETdata according to the first analysis by Sreekumar et al. [52]. Both backgroundswere simulated for ten years to minimize the presence of statistical fluctua-tions.

The purpose of having two background models is to investigate how dif-ferent amounts of astrophysical background would change the sensitivity ofFermi. The blazar background represents an optimistic choice of backgroundand the Sreekumar measurement a conservative one. Note that the blazar back-ground used is now ruled out by the Fermi measured IGRB [3].

Before launch it was hard to estimate the residual background in the data.Since the pre-launch requirement was 1/10 of the EGRET measured IGRB(by [52]) we model the charged particle background by adding it as 10% ofthe astrophysical background.

EGRET exclusionThe regions marked as Excluded by EGRET in figure 8.1 correspond to theWIMP signal that is as large as the EGRET observed IGRB spectrum in anyenergy bin. We use the IGRB spectrum obtained by Strong et al. [54].

Sensitivity calculationsThe Fermi sensitivity was calculated using a χ2-test on the histograms of thesignal and background events from the simulation:

χ2 =

N

∑i=1

(bi−ni)2

σ2b,i

, (8.1)

SECTION 8.1. FERMI SENSITIVITY TO COSMOLOGICAL WIMPS 79

[GeV]!m40 50 60 70 80 90 100 200

]-1 s3

v>[c

m"<

-2710

-2610

-2510

-2410

-2310

"GLAST 1yr,5 NFW

Excluded by EGRET

"GLAST 1yr, < 5 NFW

[GeV]!m40 50 60 70 80 90 100 200

]-1 s3

v>[c

m"

<

-2710

-2610

-2510

-2410

-2310

"GLAST 1yr,5 NFW+subhalo

Excluded by EGRET

"GLAST 1yr, < 5 NFW+subhalo

Figure 8.1: Fermi 1-year, 5σ sensitivity for generic, thermal WIMPs annihilation intobb and a branching of 0.5 ·10−3 into two-photon line. The lower sensitivity region isextended to illustrate the uncertainty in the background (see text for details). In theupper panel the WIMP signal was calculated using a NFW profile only and in thelower panel using a NFW profile with 10% of its mass in substructures, with concen-tration parameters four times that of the parent halo. By introducing substructures thesensitivity increases by roughly a factor of ten.


where bi in the number of events in each bin with only background, ni isthe number events in each bin containing both signal and background andσ2

b,i is the error squared of bi. The number of entries in each bin is assumedto be a Poisson variable yielding σb,i =

√bi. To transform the χ2 obtained

from eq. 8.1 into a probability (or P-value) that the WIMP signal wouldbe a fluctuation in the background one integrates the χ2-distribution, of Ndegrees of freedom, from the obtained χ2-value to infinity. The sensitivity isusually expressed in number of σ , obtained from a corresponding gaussiandistribution. In this way a 5σ effect corresponds to a probability of ≈ 4 ·10−7

and 3σ to 10−3.

ResultsIn figure 8.1 the Fermi sensitivity to a generic WIMP is presented as limits inthe Mχ -σv parameter space. The limits shown in figure 8.1 are for the genericWIMP without any specific underlying particle physics scenario. However, arate typical for WIMP annihilation that would result in the observed presentday relic density of Ωh2 ≈ 0.1 is σv ≈ 3 ·10−26 cm3 s−1. For Fermi to probethis rate for a large range of WIMP masses, a boost of about a factor of tenis required. This could be achieved by adding substructures, as shown in thelower panel of figure 8.1.

8.2 Upper limits on Cosmological Dark Matter byFermiIn this section we calculate upper limits on a signal from cosmological darkmatter set by the Fermi IGRB measurement (table I in PAPER III). We do thisfor three different signals, originating from annihilations into final states ofbb, µ+µ− and γγ (c.f. section 7.1).

As described in chapter 5 the first Fermi measurement of the isotropicgamma-ray background, in the energy range 0.2 to 102.4 GeV, is to high con-fidence compatible with a single power law with spectral index -2.41. A singledominant source with a significantly different spectral shape can therefore beexcluded in this energy range. In the light of this we derive upper limits ondark matter annihilation cross sections, 〈σv〉, for our set of generic dark mat-ter scenarios. Somewhat different limits can be derived depending on what isassumed for the astrophysical background. To bracket this we derive limitsin two ways. The conservative analysis assumes that as much as possible ofthe IGRB originates from dark matter. We simply scale a dark matter spec-trum until it overshoots the IGRB in any energy bin with a value n timesthe error of the IGRB in that bin. This allows for a strong signal and there-

SECTION 8.2. UPPER LIMITS ON COSMOLOGICAL DARK MATTER BY FERMI 81

fore the limits are weaker. This method assumes that an arbitrary adjustablebackground can be superimposed on the dark matter signal to account for theIGRB in the rest of the energy bins, while not contributing to the bin wherethe limit was set. Since this is clearly a non-realistic scenario (but still indi-cating the maximum allowed dark matter signal) we also derive limits usingan approach that almost all the measured IGRB originates from astrophysicalbackgrounds, on top of which a small dark matter signal can be superimposed.This we call the stringent method. In practice, a χ2 test is performed by com-paring the IGRB spectrum to the best fit background including a dark mattersignal for a given 〈σv〉 ≥ 0. The background is chosen as the sum of twopower laws N1E−2.4 + N2E−2.7, where the normalisations N1 and N2 are freeparameters and fitted for each value of 〈σv〉. The power law spectral indicesare motivated from studies of blazars [165, 19] and starforming galaxies [74]respectively. Note that the astrophysical extragalactic signal is not necessar-ily expected to be a single power law, or a sum of two power laws. Differentsources are expected to have different spectral indices, and a sum of differentpower laws is not a single power law (cf. section 5.2). Another considerationis that gamma-ray absorption against the extragalactic background light wouldaffect the spectral shape at the high end of the LAT energy range [165]. Thestringent limit is defined as when increasing 〈σv〉 forces the χ2 value of thefit to deviate from its best fit value by n2.

The upper limits, for both methods, corresponding to 90, 95 and 99.999%confidence level, are when n = 1.28,1.64 and 4.3. All our upper limits arederived under the approximation that the probability distributions for the in-tensity in each energy bin are independent and take Gaussian functional forms.The values of n simply correspond to what is needed to retrieve stated confi-dence levels for one-sided (upper) limits on the 〈σv〉 parameter.

8.2.1 Results and DiscussionsFigure 8.2 and 8.3 show the upper cross section limits for our, in section 7.1,considered annihilation channels. We show the 90, 95 and 99.999% confi-dence limits for the reference structure formation scenario MSII-Sub1. Forthe other ∆2(z) models we show only the 95% limit for direct comparison. Alllimits are presented using the Gilmore et al. [142] absorption model. How theStecker et al. [143] model would change the limits is for clarity only showedfor the MSII-Res ∆2(z) scenario.

The outcome from the conservative and the stringent limit analysisprocedures are shown in the left and right panel of each figure, respectively.In the case of the conservative limits, the 90 and 95 % confidence level linesturns out to be close to overlapping, and only the 90% level is then shown.


The upper plot of figure 8.2 shows the limits on the bb channel over which ascan in MSSM-7 (cf. section 6.5.4) parameter space, performed using Dark-SUSY [144], is superimposed. A model is selected if it is consistent withpresent accelerator constraints and have a neutralino thermal relic abundancecorresponding to the inferred cosmological dark matter density by WMAP[88]. The dark grey dots correspond to points where the branching to bb isdominating (>80%). Light grey points also include models with branching(>80%) into states with gamma-ray emission spectra similar to that of bb, i.e.from other quarks or heavy gauge bosons. The change in the limits caused bya different absorption model is small and only matters when the limit is set bythe high energy part of the dark matter spectrum. In the bb case this happensfor a dark matter mass around ∼ 600 GeV.

The lower plot of figure 8.2 shows the limits on the ’leptophilic’, µ+µ−

channel. Also shown is the best fit regions to the PAMELA and Fermi e+e−

spectra from [139]. For dark matter masses below ∼ 400 GeV the limits areset by the final state radiation bump at high energies. Therefore the differentabsorption models affect the limits below these masses.

Figure 8.3 shows the upper limits on the gamma-ray line signal. Also shownis the IDM parameter space that produces the observed relic abundance ofdark matter, from [141]. They wiggly behaviour of the limits are caused bythe energy binning of the Fermi IGRB energy spectrum. When the narrowline is split between two energy bins the limit gets weaker. The two gamma-ray final state limits can be converted to limits on final states containing onegamma-ray and an additional particle, X, with mass mX. The correspondinglimits are given by:

〈σv〉mDMγX,limit = 2

m2DM

m′2DM×〈σv〉m

′DM

γγ,limit ,wherem′DM =mDM

2

(1+

√1+

m2X

m2DM

).

(8.2)The annihilation limit 〈σv〉mDM

γγ,limit is directly read from figure 8.3 at thecorresponding dark matter mass m′DM. For this expression to be exactly valid,the X particle has to be stable. Otherwise the finite lifetime of the X particlewill introduce a broadening of the monochromatic lines, which is the case forthe Z0 boson.

The limits set by the MSII-Sub2 structure formation scenario are probablysome of the strongest indirect limits to date. They cut deep into the MSSM-7parameter space and exclude the best fit regions to PAMELA and Fermi e+e−

data completely. However, it is likely that the contribution from small halosand substructures is overestimated in this scenario or that such as scenarioalready would be constrained by other dark matter sources, e.g. the sub-halosin the Galaxy. The MSII-Sub1 and BulSub scenarios resembles more what has


been considered in other studies of galaxy clusters [166], dwarf galaxies [167,168] and diffuse emission [169, 170] using Fermi data. The cluster [166] anddwarf galaxy study [167] resembles our stringent analysis using the BullSubscenario. The cosmological limits are similar to the dwarf limits and somewhatstronger than the cluster limits. The analysis in [169, 170] is more similar tothe our conservative analysis and when they use a smooth NFW or Einsatoprofile their limits are similar to our MSII-Sub1 limit.

Dark matter models adjusted to fit PAMELA and Fermi cosmic ray datawere already in tension with other observations. The results presented in thelower plot of figure 8.2 confirms the results that these models tend to overpro-duce the isotropic gamma-ray background [145, 146].

The monochromatic line constraints are, even in the optimistic MSII-Sub2halo scenario, not probing models with relative strong line signals, such asthe inert doublet model [141]. The more reliable result using the MSII-Sub1scenario is significantly weaker than the dedicated galactic line search by theFermi-collaboration [27]. This seems to indicate that the cosmological linesignal is far from the sensitivity of the Fermi-LAT.


102 103

WIMP mass [GeV]

10−27

10−26

10−25

10−24

10−23

10−22

〈σv〉bb

cm3

s−1

MSSM-7

90%95%

99.999%

95%

95%

95%

Stringent limitsMSII-ResMSII-Sub1

BulSubMSII-Sub2

bb like > 80%bb > 80%

102 103

WIMP mass [GeV]

10−27

10−26

10−25

10−24

10−23

10−22

〈σv〉bb

cm3

s−1

MSSM-7

90%99.999%

95%

95%

95%

Conservative limits

bb like > 80%bb > 80%

102 103

WIMP mass [GeV]

10−26

10−25

10−24

10−23

10−22

〈σv〉µ

+µ−

cm3

s−1

PAMELA fit

Fermi fit

90%95%99.999%

95%

95%

95%

102 103

WIMP mass [GeV]

10−26

10−25

10−24

10−23

10−22

〈σv〉µ

+µ−

cm3

s−1

PAMELA fit

Fermi fit

90%99.999%

95%

95%

95%

Figure 8.2: Cross section, 〈σv〉, limits on dark matter annihilation into bb (upper plot)and µ+µ− (lower plot) final states. The coloured regions mark the (90, 95, 99.999)%exclusion regions in the MSII-Sub1 ∆2(z) DM structure scenario, other scenarios only95% upper limit lines. The effect of using the Stecker et al. [143] absorption modelcan be seen by the branching of the dash-dotted line. Conservative limits are shownon the left column and stringent limits in the right column. See text for details.


101 102

WIMP mass [GeV]

10−29

10−28

10−27

10−26

10−25

10−24

〈σv〉γγ

cm3

s−1

IDM2γ

90% 95%99.999%

95%

95%

95%

Stringent limitsMSII-ResMSII-Sub1

BulSubMSII-Sub2

101 102

WIMP mass [GeV]

10−29

10−28

10−27

10−26

10−25

10−24

〈σv〉γγ

cm3

s−1

IDM2γ

90%99.999%

95%

95%

95%

Conservative limits

Figure 8.3: Cross section, 〈σv〉, limits on dark matter annihilation into γγ final states.The coloured regions mark the (90, 95, 99.999)% exclusion regions in the MSII-Sub1∆2(z) DM structure scenario, other scenarios only 95% upper limit lines. The effectof using the Stecker et al. [143] absorption model can be seen by the branching ofthe dash-dotted line. Conservative limits are shown on the left column and stringentlimits in the right column. See text for details.

87

9. Anisotropies

This chapter summarises the possibilities for the Fermi-LAT to detect a sig-nal from dark matter using the angular power spectrum of the isotropic signal.This would be an independent and complementary way to detect a signal orig-inating from dark matter in gamma-rays. The results are published in PAPER4.

9.1 IntroductionEven though the IGRB is assumed to be isotropic, small scale fluctuationscould be imprinted depending on the origin of the emission. We know fromlarge scale structure studies of galaxies that they tend to cluster together infilaments, with enormous voids in between [171]. This is also confirmed byN-body simulations where dark matter clusters in the same way. If the originof the IGRB is extragalactic and consists of unresolved point sources (blazarsor dark matter halos) that traces these large-scale filaments, we expect to seeanisotropies in the IGRB.

It was first proposed by Ando and Komatsu (2006) [172] that the anisotropyproperties of the Cosmological WIMP signal could be used as a detectionsignature, complementary to the energy spectrum. Even if the backgroundsources trace the same large scale structure, the gamma-ray signal would bedifferent since the flux from annihilating dark matter scales as ∝ ρ2 whilethe background IGRB scales as ∝ ρ . This could potentially make the angu-lar power-spectrum of dark matter sources significantly different from that ofconventional astrophysical sources [173, 174, 175].

Also, the signal from unresolved galactic dark matter subhalos would giverise to an approximately isotropic signal [161] for which the angular power-spectrum could be measurable [176].

In PAPER 4 we perform a detailed study of anisotropies related to the cos-mological dark matter signal and galactic subhalos. For the first time we takeinto account the galactic foreground and resolved point sources in a consis-tent way using the Fermi instrument response functions. We use 3 dimen-sional power spectra for the density field and the square of the density field(originating from simulations of large scale structure [177]), and use randomrealisations of these as templates of the extragalactic background originating

88 ANISOTROPIES CHAPTER 9

from conventional and dark matter sources. From the constructed maps wecalculate the corresponding angular power spectra and the cross correlationangular power spectra between different energy bins.

9.2 AnalysisModel maps of point sources and the galactic diffuse foreground are preparedin energy dependent Healpix skymaps (c.f. section 5.3.2). We use the GAL-PROP conventional model [48] for the galactic diffuse emission and the firstyear Fermi catalog to model the point-sources [13] . The extragalactic back-ground from astrophysical background sources are spatially modelled by thelarge scale matter distribution from [177]. Each energy bin is normalised tothe best fit power-law from the Fermi IGRB measurement (c.f. chapter 5).

9.2.1 Angular power spectraAny map on a sphere, described in a direction Ω by f (Ω), can be decom-posed in spherical harmonics Y l

m as f (Ω) = ∑l,m almY lm. The angular (auto-

correlation) power spectrum is defined as the average of the square of thecoefficients Cl = 〈|alm|2〉= ∑m

|alm|22l+1 . Since the maps corresponding to the dis-

tribution of the extragalactic background are not independent between differ-ent energy bands (by construction in this case), further information is alsocontained in the cross-correlation power spectra between map i and j: Ci j

l =∑m(ai

lma j∗lm +ai∗

lma jlm). Typically when showing the power spectra, one plots the

quantity l(l +1)Cl/2π . Approximately, a multipole l corresponds to an angleof about l/200 degrees. The angular power spectra presented are all calculatedusing Healpix [81].

The intrinsic angular power spectra of the individual background and signalcomponents are very different. Figure 9.1 shows full sky angular spectra ofthe various components after they have been convolved with a (simulated)five year Fermi exposure and PSF. Only energies > 10 GeV were consideredand for visual purposes all fluxes where normalise to be the same. Alwayswhen dealing with a finite number of counts there will be a noise from thePoisson statistics in the angular power spectrum. This so called ’shot noise’has a constant value 4π/N where N is the number of counts. In the upper andmiddle panel of figure 9.1 the angular power spectra can bee seen before andafter subtracting the shot noise.

The galactic diffuse emission has mostly large scale features and its angularspectrum dominates over the extragalactic ones below l ∼ 200−300 where itdecreases. On all angular scales the point sources are the dominant sources ofanisotropy. To make the foreground subdominant to the extragalactic diffuse

SECTION 9.2. ANALYSIS 89

Figure 9.1: All sky angular power spectra at 10 GeV. All signals are convolved withthe Fermi-LAT Point Spread Function. In the upper panel the spectra before shot noise(represented by the dashed lines) removal are shown. The middle panel shows shotnoise removed spectra. The last panel show the angular power spectra after applyingthe mask. From PAPER IV.


sources we build a mask starting from the foreground sources themselves. Inthis way the dark matter anisotropies dominate the angular spectrum at smallenough scales, as can bee seen in the lower panel of figure 9.1. The mask ofthe galactic diffuse foreground is defined by removing every pixel that exceedsthe astrophysical extragalactic background energy spectrum by more than afactor of 2 in each energy bin. The point source mask is produced in a similarway, but by removing every pixel that exceeds the astrophysical extragalacticbackground by more than 20%. Applying the mask we remove about 30 - 50%of the sky depending on energy.

Applying a mask to the data distorts the intrinsic angular power spectrumof the components. However, the effect of the mask can be deconvolved usingappropriate algorithms [178]. For the purpose of this work it is sufficient tocalculate when the dark matter anisotropies dominate over the astrophysicalextragalactic background. Therefore we are satisfied with studying themasked maps. Skymaps of model predicted counts and correspondingPoisson counts, multiplied by the mask, can be seen in figure 9.2.

The difference between the galactic dark matter signal from unresolved subha-los and the cosmological signal is best illustrated by the angular power spec-trum as a function of energy, for a fixed scale l. Such an energy angular powerspectrum can be seen in figure 9.3. The galactic signal is constant in energywhile the extragalactic signals are increasing with energy. The change in theconventional extragalactic background is not dramatic, and is only due to theeffect of absorption at high energies. The effect on the dark matter signal is5 orders of magnitude between 100 MeV an 100 GeV. This is because of thespecial shape of the dark matter energy spectrum and how that combines withabsorption in the integration over redshift.

Figure 9.5 shows the angular power spectrum ’matrix’. This shows the an-gular power spectra on the diagonal plots and the cross-correlation spectra inthe off-diagonal plots. As can be seen, the mask makes out the bulk of theanisotropies on its own. The WIMP in this example is chosen with a massof 200 GeV and normalised to the flux of the extragalactic background. Thebump above the statistical error in the diagonal plot at 12.8 GeV < E < 102GeV, is the signal that would make this model detectable.

9.2.2 Fermi Exposure and PSFThe Fermi exposure is roughly isotropic when averaged over long enoughtime (> 55 days). The anisotropies that exist are on large scales and do notaffect our analysis (see the midle panel of figure 9.1).

The PSF on the other hand limits the accuracy of determining the angu-lar power spectra. The Fermi PSF is energy dependent and improves at high


Figure 9.2: (Left) 5yr average counts map (in Healpix nside=256 format) of our refer-ence no-dark matter model, i.e. the model contains astrophysical extragalactic back-ground, galactic foreground and resolved point sources. An energy dependent maskis applied to suppress galactic foregrounds and point sources. The energy ranges areindicated in the titles and only the region outside the masks are shown. The regionoutside the mask increases with energy since the extragalactic background spectrumis harder than the galactic diffuse one. (Right) A random realization of the expectedcounts. From PAPER IV.


Figure 9.3: Energy dependence of the anisotropies for our models of cosmologicaldark matter, astrophysical extragalactic background and dark matter in galactic sub-structures. The dark matter mass assumed was 1 TeV. From PAPER IV.

energies. The angular resolution of Fermi is about 3 at 100 MeV, 0.6 at 1GeV and 0.1 at 10 GeV. This suppress anisotropies at medium-high mul-tipoles below 1 GeV. However, if the PSF is approximated with a Gaussianwith variance σ , the error on the angular power spectrum due to the PSF isδCl ∝ exp(l2σ2/2). Using this one can still recover information about theintrinsic power spectrum.

In figure 9.4 the energy dependent effect of the PSF on the power-spectracan bee seen for a cosmological dark matter signal. As expected theanisotropies are suppressed at l > 100 for E > 1 GeV. At high energiesE < 100 GeV, given the PSF stated above we do not expect any suppressionin the power spectrum below l ∼ 2000. However, in figure 9.4 there is asuppression already at 100 < l < 200 for E > 3 GeV. This is likely due thelong tails of the PSF.

9.2.3 Sensitivity estimatesAssuming that the maps of different energy bins are statistically independent,a χ2-test can be done between the power-spectra of the simulated maps Ci

l andthe model maps Ci

l . The simulated counts are binnend in 5 bins between 0.4≤E ≤ 820 GeV and the power spectra are calculated for 18 logarithmically


Figure 9.4: Power spectra at energies E = 300,30,3,0.3 GeV from top to bottom,for a cosmological dark matter signal (with mχ =1 TeV annihilating into bb) and theircorresponding PSF convolved spectra. The PSF suppresses anisotropies at l > 100 forE <1GeV. The raise of the power spectrum at high multipoles in the E = 300 MeVcase is a numerical Healpix artefact. From PAPER IV.


spaced multipole bins between 2≤ l ≤ 512. The χ2 is given by:

χ2 = ∑

i,l

(Ci

l−Cil

δCil

)2

, (9.1)

where the denominator is given by

δCil =

√2

fsky(2l +1)∆l

(Ci

l +δΩ

Ncounts

), (9.2)

where ∆l is the number of multipole bins, fsky is the fraction of the sky not af-fected by the mask and δΩ the solid angle of fsky. However, since by construc-tion the random realisations of the anisotropic extragalactic background mapsare not independent between energy bins, equation 9.1 is expected to overes-timate the χ2. To properly take into account the correlations of the anisotropymaps, it is the full cross-correlation power spectra Ci j

l that has to be used inequation 9.1, instead of Ci

l . In that case also the term δCil has to be replaced

by the full covariance of Ci jl , which is given in Appendix B of PAPER IV.

Figure 9.6 shows two examples of sensitivity plots from PAPER IV forannihilations into τ+τ−. The results for the other channels bb and µ+µ− arequite similar.

The results from the full χ2, including cross correlations, is only showedfor the galactic subhalos (lower panel of figure 9.6) because the result did notchange significantly for the cosmological dark matter. This can probably beunderstood by looking at the energy anisotropy spectra in figure 9.3. If thedark matter anisotropies are not significantly higher than the background forenough energy bins, there is probably not much information to gain from thecross correlations. The anisotropies of the galactic subhalo signal are alwayslarger than the astrophysical extragalactic background (at l = 100), which re-sults in a significant improvement of the sensitivity.

9.3 Summary and discussionSimulating five years of data we find that Fermi-LAT is sensitive to a darkmatter contribution at the level of about 1%-10% of the measured IGRB, de-pending on the dark matter particle mass and annihilation channel. In termsof the the cross section, 〈σv〉, this corresponds roughly to 3 · 10−26 cm3 s−1,the expected the value from thermal freeze out, for low masses in the chan-nels that produce the most high energy photons when a boost from a reason-able amount of substructure is added. The sensitivity is similar when look-ing at the anisotropies from the galactic subhalos. We find that including thecross-correlation between power spectra of different energy bins is primar-ily of importance when the dark matter anisotropies are dominating over the

SECTION 9.3. SUMMARY AND DISCUSSION 95

Figure 9.5: Power Spectra Matrix for the 5 average counts maps (including themasks) used in the analysis. Auto-correlations Ci

l = Ciil are in the diagonal and cross-

correlations Ci jl in the off-diagonal elements. The power spectra are binned in 18

logarithmically spaced multipole bins. The dashed lines in the auto-correlation plotsindicate the level of shot noise. The two sigma statistical errors are represented by thegray shaded area. For illustration mχ ≈ 200 GeV is chosen and the dark matter flux isnormalised to the same level of the astrophysical extragalactic background at 10 GeV.The bump above the statistical error in the 2.8 GeV < E < 102 GeV autocorrelationspectrum is the signature that would make this mode detectable. From PAPER IV.


Figure 9.6: Sensitivity estimates to anisotropies from cosmological dark matter (up-per panel) and the the equivalents of anisotropies from galactic unresolved subhalos(lower panel). The normalisation of the annihilation energy spectrum corresponds tothe generic WIMPs of section 8.1 including the enhancement effect of substructure(lower solid lines) and the model without substructures (upper coloured fields). Thedotted lines in the lower panel correspond to calculating the χ2 using the full cross-correlation power spectra. Also indicated is the expected cross section from thermalfreeze-out (dashed line). From PAPER IV.

SECTION 9.3. SUMMARY AND DISCUSSION 97

background over a wide energy range. This is particular important when con-sidering the emission from unresolved galactic subhalos.

Previous work on the subject has assumed a perfect extraction of the fullexragalactic background. We use a realistic foreground modelling and incor-porates the full instrumental response of the Fermi-LAT. From the the fore-ground sources themselves we create an energy dependent mask that cutsabout 30-50% of the sky but renders the remaining emission close to isotropic.

For this analysis we were satisfied with studying the masked gamma-raymaps only. Even if the emission outside the mask is close to isotropic thereis still a considerable contamination from the galactic diffuse foreground. Forthe full analysis, some kind of additional foreground cleaning probably has tobe employed. Our analysis shows that the 68% containment of the PSF (c.f.figure 3.2) does not incorporate the full effect of the PSF on the angular powerspectrum. Therefore the systematic effects of the PSF have to be studied inmore detail.

99

10. Summary and outlook

The isotropic gamma-ray backgroundThe Fermi-LAT first year measurement of the gamma-ray background is com-patible with a featureless power law of spectral index −2.41 in the energyrange 200 MeV to 100 GeV and dominated by systematic errors. This wasobtained using an extended background rejection specially developed for thisanalysis. The remaining residual background was estimated using a MonteCarlo simulation and subtracted. In this thesis we also present a way to sub-tract the residual background by measuring the isotopic background in data-sets with different expected level of residual background. Extrapolating theresults to zero expected residual background would yield a contamination freemeasurement. The two methods agree with each other within errors, but dueto the additional systematic errors present in the latter method, this was notexploited in the final analysis.

We also present an alternative analysis method to determine the isotropicbackground. This method is inspired by the original method used to determinethe isotropic energy spectrum from EGRET data. This analysis method agreeswithin errors with the multi component likelihood analysis finally adopted forthe Fermi measured isotropic background, and shows that the result is notsignificantly dependent on the analysis method.

Upper limits on an extragalactic dark matter signalThe featureless power law spectrum of the isotropic gamma-ray backgroundexcludes any additional component with a significant different energy spec-trum. Instead we derive upper limits on a signal from annihilating dark mat-ter of extragalactic origin. The uncertainty in shape of this signal is mainlydetermined by the underlying particle physics scenario. We investigate threedifferent scenarios with significantly different spectral shapes that are repre-sentative for a wide range of models: annihilation into final states of b-quarks,µ-leptons and monochromatic photons.

The strength of the signal is mainly affected by the dark matter distributionand evolution. We investigate four scenarios where the contribution to thesignal from dark matter halos, especially below the mass resolution of currentN-body simulation, have been taken into account in different ways. The mostconservative and most optimistic scenario differ in signal strength by threeorders of magnitude.

100 SUMMARY AND OUTLOOK CHAPTER 10

The signal is also affected by the intergalactic absorption of gamma-rayphotons and how it evolves with redshift. We consider two models for theoptical depth and find that the upper limits are affected mainly when set bythe high energy part of the energy spectrum.

There is presently no well established model for the bulk of the isotropicemission. The popular alternatives are unresolved gamma-ray pointssources, such as blazars and star-forming galaxies. The lack of well definedbackground introduces an uncertainty when calculating the upper limits.We bracket this by having two approaches to the analysis, one conservativewhere we allow for as much dark matter signal as possible, and one stringentwhere we assume that the bulk of the isotropic emission originate frombackground with power law energy spectra.

We find that in the most optimistic halo scenario the upper limits can ex-clude large parts of the underlying particle physics scenarios under consid-eration. However, our moderate halo formation models can just barely probeinteresting parts of the considered supersymmetric parameter space. In partic-ular the thermal WIMP canonical cross section 〈σv〉= 3× 10−26 cm3 s−1 isout of reach. The situation is similar for the leptonic final state, where regionsproposed to explain the PAMELA and Fermi electron-positron excess can notbe entirely excluded. In the case of monochromatic gamma-ray final stateseven the most optimistic halo model can barely reach regions with interestingdark matter candidates.

Anisotropic extragalactic signalsIf the isotropic gamma-ray background is extragalactic in origin and corre-lates with the large scale structures in the Universe, a dark matter signal isexpected to have different properties at small scales than a signal from conven-tional sources. We investigate the possibility for the Fermi-LAT to be sensi-tive to this using a realistic simulation of the spatial distribution of foregroundsources and the instrument response. Decomposing the gamma-ray sky intospherical harmonics and investigating the angular power spectra of the indi-vidual components, we find that the extragalactic signals are subdominant atall scales. By constructing an energy dependent mask from the foregroundmodels we are able to screen them and thereby making the dark matter signaldominant at small scales. Using this mask we investigate when the dark mat-ter signal can be detected over the conventional extragalactic background toa certain significance. We find that this method offers a similar sensitivity asusing the energy spectrum only. Therefore it can be used as a valuable com-plement when looking for an indirect signal from dark matter in the isotropicgamma-ray background.

101

Outlook and discussionAt the present, the errors in the measured isotropic spectrum are dominatedby systematic uncertainties. These originate mainly from the instrumentaluncertainty of the effective area, the uncertainty when subtracting the esti-mated residual background and the uncertainty in the galactic diffuse emis-sion model. Knowledge of all these areas is continuously increasing within theFermi-LAT collaboration. The Monte Carlo simulations of the residual back-ground are continuously being developed and with time more statistics willbe accumulated. Since not even the dataclean analysis scheme can removeall residual background, there is also room for improving the background re-jection algorithms. This will minimise the uncertainty introduced when sub-tracting the residual background. The diffuse emission model will be iteratedwith the isotropic background and the catalog of point sources will becomemore accurate. Therefore the measurement of the isotropic background is ex-pected to be more and more refined as the systematic uncertainties are betterunderstood. Both the residual background estimates and the galactic diffusegamma-ray emission will benefit from ongoing cosmic ray measurements andin particular from the upcoming AMS experiment [179].

The Fermi-LAT will also increase our knowledge of the source populationsthat potentially account for the bulk of the isotropic emission. Detailed sur-veys of for instance blazars will be important for this. Also, if the origin of theisotropic emission is extragalactic and mainly consisting of unresolved pointsources, it might change over time as more as more sources are resolved. Withsmaller errors and better understanding of the conventional sources contribut-ing to the isotropic background, we will be able to put tighter constraints on asignal from dark matter.

With time the statistics of high energy photons will increase and this willallow us to extend the energy range of the measurements. Presently the al-gorithms and background rejection schemes are optimised for energies below300 GeV. Work is ongoing to increase this energy range to ensure the qualityof measurements at high energies. Increasing the energy range of the IGRBmeasurement is especially interesting since if the soft power law continues, itmay imply that the background is very low for a potential dark matter signalluring at higher energies.

One way to go beyond merely the spectral properties of the isotropicemission is to look at the spatial distribution. This will be challenging, andin particular require a detailed understanding of the various foregroundcomponents. Combining the spectral and spatial information enables us tofurther constrain different models of emission.

At the moment, lacking any other but gravitational indications for the exis-tence of dark matter, its nature is highly unconstrained. Indirect signals are

102 SUMMARY AND OUTLOOK CHAPTER 10

very sensitive to the distribution of dark matter, which presently is hard to inferfrom observations without being obscured by the complicated baryonic pro-cess present. N-body simulations have recently started to model the combinedevolution of dark matter and ordinary matter in detail. However, to accuratelytake these processes into account, including the smallest scale structures, in arealistic simulation of our Universe, is still far from being achieved.

The large uncertainties associated with the distribution of dark matterstrongly affect predictions of indirect signals. Therefore it is not possibleto uniquely specify the signal from different sources given a specific darkmatter scenario. Because of this it is important to constrain the signalfrom all possible sources using the corresponding measurement. As theseuncertainties are better constrained the various diffuse components arisingfrom dark matter has to be considered together with the various componentsof the galactic diffuse emission. This presents a very challenging analysissince these models contain several non linear parameters, for which statisticalerrors are not easily estimated. Presently, due to the large uncertainties in thegalactic diffuse model, it can not be excluded that a small signal from darkmatter has been wrongly accounted for in some of its components.

As simulations and observations improves it will be interesting to see ifthe cold dark matter paradigm can reconcile them. Even if that is the case,the particle nature of the dark matter still has to be verified experimentally.The success of the Big Bang model is that we can apply well understoodphysics to the early evolution of the Universe. Understanding the physics atthe electroweak scale will be the next step to push our understanding of theearly Universe further back in time. If dark matter is WIMP like, it will mostlikely play an important role.

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11. Sammanfattning på svenska

Vårt universum tycks dominerat av en osynlig typ av materia. Denna mörkamateria kan bara påvisas på grund av dess gravitationella inverkan på den syn-liga materian; stjärnor, galaxer och gas. Exakt vad den består av är fortfarandeen obesvarad fråga inom modern fysik och den här avhandlingen söker bidramed en del av lösningen. En populär teori är att den mörka materien var i ter-misk jämvikt med de övriga partiklarna i det tidiga universum. Denna jämviktkunde dock bara upprätthållas då universums täthet var tillräckligt hög. I taktmed att universum utvidgades upphörde jämvikten och den mörka materianhar sedan dess inte nämnvärt kunnat växelverka med den övriga materian,varvid dess antal har bevarats. Hur mycket mörk materia som skapades pådetta sätt beror på styrka i dess växelverkan. Det visar sig att partiklar meden växelverkan på energiskalan för den svaga kraften skapar nästan precis denmängd mörk materia som observeras i universum. Att det dessutom är precisvid den energiskalan vi förväntar oss att finna ny fysik (med bland annat dennya acceleratorn LHC i CERN) har gjort att den här typen av mörk materia-modeller har studerats i stor utsträckning.

Om den mörka materien var i jämvikt med med det tidiga universum finnsdet en möjlighet att den fortfarande växelvärkar, om än väldigt svagt. Dettakan till exempel ske genom par-annihilation mellan två mörk materia-partiklarvilket skulle kunna resultera i partiklar som vi kan mäta på jorden. Det hargenomförts många studier av olika mörk materia-modeller som skulle kunnalämna spår efter sig i form av kosmisk strålning av olika slag; neutriner, lad-dade partilar eller fotoner. Signaler från mörk materia har typiskt en annanenergifördelning är signaler från konventionella astrofysikaliska källor, vilketgör att de skulle kunna särskiljas från dessa.

Sommaren 2008 skickades gammastrålningsteleskopet Fermi upp i om-loppsbana kring jorden. Dess mål är att söka efter fotoner från yttre rym-den med mycket hög energi, cirka en miljard gånger energirikare än synligtljus. Fotoner med sådan hög energi skapas mestadels under väldigt extremaförhållanden. Till de vanligaste källorna hör rester från supernovor, både denkvarvarande stjärnan pulsaren och plasmaskalet som skickas ut från explosio-nen. En annan källa är aktiva galaxkärnor. I centrum av dessa galaxer tror manatt det finns enorma svarta hål vars dragningskraft driver jetstrålar av laddadepartiklar. Energin i jetstrålarna är så stor att partiklarna kan slungas flera ljusårfrån galaxen. När dessa strålar riktas mot jorden syns de tydligt i gammastrål-

104 SAMMANFATTNING PÅ SVENSKA CHAPTER 11

ning. Den typ av aktiva galaxkärnor där strålen är riktad mot jorden kallasblazarer.

En annan tänkbar källa till gammastrålning är partiklar av mörk materiasom annihilera varandra. Signalen från mörk materia förväntas vara somstarkast där den mörka materien är som mest koncentrerad, till exempeli galaxens centrum. Fermi söker efter signaler från mörk materia från enrad olika områden. Mörk materia borde finnas ansamlad kring de flestagalaxer, och därmed borde det komma en svag signal från miljardtals galaxeri universum. Om alla dessa signaler summeras bör de ge upphov till ensignal som ser likadan ut i all riktningar; en isotrop signal av kosmisktursprung. Det är denna signal vi fokuserar på i den här avhandlingen. Det ärdock inte enbart mörk materia som förväntas ge ett bidrag till den isotropagammabakgrundsstrålningen. Till exempel förväntas blazarer, som inte ärstarka nog att detekteras av Fermi, ge ett isotropt bidrag när de summeras.

I PAPPER I uppskattade vi hur stark signalen från mörk materia måstevara för att upptäckas av Fermi. Vi förutsätter att signalen är inbäddad i endominerande bakgrund, från till exempel oupplösta blazarer, och finner att detfinns partikelmodeller som skulle kunna producera signaler starka nog för attdetekteras av Fermi.

En isotrop signal har nu detekterats av Fermi. Analysen har gjort med tvåmetoder, varav den ena publicerats i PAPPER III och den andra presenteras iden här avhandlingen. De båda analyserna finner att antalet fotoner avtar som∼ E−2.4, där E är energin hos fotonerna. Mätningen domineras av system-atiska fel som härrör från osäkerheten i den effektiva ytan hos instrumentetsamt i den förutspådda kvarvarande backgrunden från laddade partiklar.

På grund av gammabakgrundsstrålningens jämna energispektrum kan endominerande komponent med en annan form, till exempel en signal frånmörk materia, uteslutas i mätningens energiintervall. Istället beräknar vi övregränser för hur stark en signal från mörk materia skulle kunna vara för attdölja sig inom mätningens felmarginaler. Dessa är publicerade i PAPPER II.Modeller för mörk materia som resulterar i en starkare signal än den somtillåts av mätningen kan sägas vara uteslutna. Vilken model som svarar motden maximalt tillåtna signalstyrkan kan dock inte bestämmas entydigt dådet beror på hur den mörka materien är fördelad i universum, något somär mycket osäkert. Beroende på hur denna osäkerhet behandlas uteslutermätningen olika modeller för mörk materia.

Ytterligare en aspekt av gammabakgrundstrålningen är att den, beroendepå dess ursprung, inte förväntas vara helt isotrop. En signal från mörk materiaförväntas ha en lite annorlunda fördelning på himlen jämfört med en signalfrån andra källor. Huruvida en sådan skillnad kan upptäckas med Fermi stud-eras i PAPPER IV. Vi modellerar förgrundskällor och instrumentella effekter

105

och visar att signalens fördelning över himlen kan komplettera informationenfrån energifördelning.

Mätningen av den isotropa gammabakgrundstrålningen kommer att förbät-tras i takt med att Fermi samlar mer data, att förståelsen för instrumentet ökarsamt att modeller för den kosmiska strålningen i dess omloppsbana förbät-tras. Detta kommer att leda till att fler modeller för mörk materia kan begrän-sas. Detsamma gäller en rad olika mätningar som Fermi gör, vilka undersökerolika aspekter av en signal från mörk materia.

För att definitivt kunna identifiera den mörka materien skulle en signal igammastrålning bara vara en pusselbit. Kompletterande information från an-dra signalkällor, direkt detektion av mörk materia-partiklarna på jorden samtatt de kan skapas och studeras i acceleratorer är antagligen nödvändigt för enslutgiltig svar på frågan: vad är mörk materia?

107

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