Constraining the Particle Nature of Dark Matter: Model-Independent Tests from the Intersection of Theory and Observation

Home , Alex Szalay, Cuspy halo problem, Marc Aaronson

CONSTRAINING THE PARTICLE NATURE OF DARK MATTER: MODEL-INDEPENDENT TESTS FROM THE INTERSECTION OF THEORY AND OBSERVATION

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Gregory Daniel Mack, B.A., M.S.

* * * * *

The Ohio State University

2008

Dissertation Committee: Approved by

Prof. John F. Beacom, Adviser Prof. James Beatty Adviser Prof. Eric Braaten Graduate Program in Prof. Terrence Walker Physics

ABSTRACT

Dark matter is one of the greatest mysteries of modern astrophysics. It comprises about 83% of the matter density in the Universe and approximately 22% of the total

energy density, yet its identity and particle properties are unknown. Gravitational

interactions reveal its presence, but it does not readily interact with light or nor-

mal matter. The purpose of this dissertation is to provide insight into the particle properties of this exotic type of matter in a model-independent fashion. Dark matter is expected to be its own antiparticle, but the strength of its self-annihilation is not known. It is often assumed to be consistent with that which gives the correct

abundance if dark matter were produced as a thermal relic in the early Universe,

but that has not been proven. Constraints on the dark matter self-annihilation cross

section are found over a wide range of masses, both for the separate cases of monoen-

ergetic neutrino and monoenergetic photon production, and the corresponding limits

on the total self-annihilation cross section. This is done by comparing the theoretical

ﬂux from a region of annihilating dark matter to observational data of that region.

While larger than the thermal relic value, the resulting upper bounds are surpris-

ingly stringent and among the ﬁrst model-independent limits of their kind. A speciﬁc

application of residual dark matter annihilations during the time of Big Bang Nu-

cleosynthesis is analyzed, adding a lower limit to the value of the annihilation cross

section for a certain mass range to couple with the calculated upper bounds mentioned

ii above. The interaction strength of dark matter with normal matter is constrained by the case of dark matter capture in Earth and the resulting heat ﬂow from annihilation in the core. When compared to observation, the analysis rules out many possible interaction strengths between dark matter and normal matter, showing that the interaction, as measured by the interaction cross section, must be truly weak for the usually considered mass range. These model-independent investigations help to shrink the range of possible dark matter property values in a generic fashion, aiding in better understanding and more focused analyses.

iii To my parents, family, and friends, who have always shown their support.

To those who ﬁnd the analytic in the artistic, and the aesthetic in the logic.

iv ACKNOWLEDGMENTS

I would like to ﬁrst and foremost acknowledge my adviser, Dr. John Beacom, for mentoring me these last several years. I have learned a great deal from John about

astrophysics, particle physics, doing research, writing papers, giving talks, the politics

of science, ice-cycling in Wisconsin, and even how to grow an impressive goatee. I

would also like to thank the other members of my committee, Drs. Terry Walker, Jim

Beatty, and Eric Braaten, for seeing me through this process, and especially Terry

for encouraging me in joining the Astrophysics group in my initial graduate school

stages (and therefore having me interact with such hooligans as Andrew, Savvas,

and Louie). I also need to thank my undergraduate professors at Ohio Wesleyan

University, Barbara Andereck, Brad Trees, and Bob Harmon, without whom I would

not have made it to graduate school.

I am also grateful for my collaborators, Gianfranco Bertone, Nicole Bell, Hasan

Yuksel,¨ Thomas Jacques, and Gary Steigman, who along with John were essential for

my research and publications. Along those lines, I acknowledge Eric Braaten, Stuart

Raby, Stefano Profumo, Frederick Kuehn, Matt Kistler, Louie Strigari, Jordi Miralda-

Escud´e, Jerry Newsom, Wendy Panero, Ralph von Frese, Laura Baudis, Andy Gould,

Shin’ichiro Ando, Shantanu Desai, Michael Kachelriess, Manoj Kaplinghat, Eiichiro

Komatsu, Robert Scherrer, Shmuel Nussinov, and Casey Watson for discussions and

advice in completing the publications.

v On a more personal note, I would like to thank my parents, Gene and Carlotta, and the rest of my family, especially Ed and Amanda, for always giving me support.

I have to thank my good friends, too, for putting up with me and my irrationalities during this process: Chad Johns, Sara Florkey, Jenn Schmelzer, and Beth Cademar- tori for infinitely many, many, many things including for tolerating me the past ten years; Dave Boley for studying at many different coffee shops with me, sometimes cooking for me, showing me how to cook, scrabble, and the gracious use of his park- ing pass; Marian Homan for innumerable wake-up calls; my “twin sister” Sarah Hixon for having me as a part of Hixon Dance, performing in her works, and sharing ranting sessions; Maggie Page and Noelle Chun, with whom it is always great to dance on- stage; and Ric Rader, Deb Friedes, Andy Vogel, Brian Hauser, and Christina Xydias, for the “Wednesday night grad student gatherings”.

I also like to think that I greatly supported the local business scene in Columbus,

Ohio, by frequenting (rather often) the coffee shops Caffé Apropos and Cup O Joe

(the Short North location, but much more often in German Village). Countless cups of coﬀee contributed to the writing of this dissertation.

vi VITA

5 June 1980 ...... Born – Cleveland, OH

May 2002 ...... B. A. in Physics and Astronomy, Ohio Wesleyan University, Delaware, OH September 2002 – August 2005 ...... Fowler Fellowship, The Ohio State Uni- versity, Department of Physics, Colum- bus, OH August 2005 ...... M.S. in Physics, The Ohio State Uni- versity, Columbus, OH September 2005 – June 2008 ...... Graduate Research Associate, The Ohio State University, Department of Physics, Columbus, OH

vii PUBLICATIONS

“Towards Closing the Window on Strongly Interacting Dark Matter: Far- Reaching Constraints from Earth’s Heat Flow,” G. D. Mack, J. F. Beacom and G. Bertone, Phys. Rev. D 76, 043523 (2007) [arXiv:0705.4298 [astro-ph]].

“General Upper Bound on the Dark Matter Total Annihilation Cross Sec- tion,” J. F. Beacom, N. F. Bell and G. D. Mack, Phys. Rev. Lett. 99, 231301 (2007) [arXiv:astro-ph/0608090].

“Conservative Constraints on Dark Matter Annihilation into Gamma Rays,” G. D. Mack, T. D. Jacques, J. F. Beacom, N. F. Bell and H. Yuksel, (2008) arXiv:0803.0157 [astro-ph].

FIELDS OF STUDY

Major Field: Physics

viii TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vii

List of Tables ...... xii

List of Figures ...... xiii

Chapters:

1. An Introduction to Dark Matter ...... 1

1.1 A General Introduction ...... 1 1.2 A Historical Perspective ...... 4

2. How Much Dark Matter is There? ...... 16

2.1 Necessary Cosmological Formalism ...... 16 2.2 Measuring an Imprint of the Early Universe: the Cosmic Microwave Background ...... 22 2.3 The Universe is “Darker” Than Originally Thought: a Case for Λ . 29 2.4 Correlating Cosmological Measurements ...... 30

3. Dark Matter: Where, What, How? ...... 34

3.1 Where is the Dark Matter? ...... 34 3.2 What Could Dark Matter Be? ...... 40

ix 3.2.1 Supersymmetry ...... 41 3.2.2 Universal Extra Dimensions ...... 43 3.2.3 Other Viable Candidates ...... 44 3.3 How Can We Find Dark Matter? ...... 45 3.3.1 Direct Detection: The Slightest Touch ...... 46 3.3.2 Indirect Detection: Measuring Dark Matter’s Disappearance 50 3.3.3 Unitarity: An Existing Annihilation Constraint ...... 52

4. Towards Closing the Window on Strongly Interacting Dark Matter: Far-Reaching Constraints from Earth’s Heat Flow ...... 57

4.1 Introduction ...... 58 4.2 Review of Prior Constraints ...... 64 4.2.1 Indirect Astrophysical Constraints ...... 64 4.2.2 Direct Detection Constraints ...... 66 4.3 Earth’s Heat Flow ...... 67 4.4 Dark Matter Capture Rate of Earth ...... 69 4.4.1 Maximum Capture Rate ...... 70 4.4.2 Dark Matter Scattering on Nuclei ...... 73 4.4.3 Dark Matter Capture Eﬃciency ...... 75 4.5 Dark Matter Annihilation and Heating Rates in Earth ...... 81 4.5.1 Maximal Annihilation and Heating Rates ...... 81 4.5.2 Equilibrium Requirements ...... 83 4.5.3 Annihilation and Heating Eﬃciencies ...... 86 4.6 Discussion and Conclusions ...... 88 4.6.1 Principal Results ...... 88 4.6.2 Comparison to Other Planets ...... 89 4.6.3 Future Directions ...... 90

5. Upper Bound on the Dark Matter Total Annihilation Cross Section . . . 92

5.1 Introduction ...... 92 5.2 Probing Dark Matter Disappearance ...... 96 5.3 Revealing Neutrino Appearance ...... 97 5.4 Cosmic Diﬀuse Neutrinos: Signal ...... 98 5.5 Cosmic Diﬀuse Neutrinos: Backgrounds ...... 102 5.6 Conclusions ...... 104

6. Conservative Constraints on Dark Matter Annihilation into Gamma Rays 107

6.1 Introduction ...... 108 6.2 Cross Section Constraints ...... 110

x 6.3 Calculation of Dark Matter Signals ...... 112 6.3.1 Dark Matter Halos ...... 112 6.3.2 Milky Way and Andromeda Signals ...... 115 6.3.3 Cosmic Diffuse Signal ...... 116 6.4 Specific Observations and Derived Annihilation Constraints ...... 117 6.4.1 COMPTEL and EGRET ...... 120 6.4.2 H.E.S.S...... 121 6.4.3 INTEGRAL ...... 122 6.4.4 Andromeda Halo Results ...... 123 6.4.5 Cosmic Diffuse Results ...... 124 6.5 Discussion and Conclusions ...... 125 6.5.1 Limits on the Cross Section to Gamma Rays ...... 125 6.5.2 Limits on the Total Cross Section ...... 128 6.5.3 Conclusions and Prospects ...... 131

7. The Eﬀects of Residual Dark Matter Annihilations on Big Bang Nucle- osynthesis ...... 133

7.1 Annihilations in the Early Universe ...... 134 7.1.1 Derivation of Dark Matter’s Relic Abundance ...... 134 7.1.2 Residual Annihilations ...... 139 7.2 The Light Elements from Big Bang Nucleosynthesis ...... 142 7.3 Modiﬁcation of Previous Work ...... 146 7.3.1 Frieman, Kolb, and Turner ...... 147 7.3.2 McDonald, Scherrer, and Walker ...... 148 7.4 Analysis ...... 149 7.5 Conclusions ...... 153

8. Conclusions ...... 154

8.1 A Checklist for Potential Dark Matter Candidates ...... 154 8.1.1 Point 7: Direct Dark Matter Searches ...... 156 8.1.2 Points 8 and 9: Indirect Gamma-ray Constraints and Other Astrophysical Bounds – Neutrinos ...... 157 8.1.3 Point 4: Consistency with Big Bang Nucleosynthesis . . . . 158 8.2 Closing Summary ...... 159

Bibliography ...... 160

xi LIST OF TABLES

Table Page

2 2.1 The latest values of the total matter density in terms of h (Ωmh ), 2 normal matter density in terms of h (Ωbh ), dark matter density in 2 terms of h (Ωχh ), dark energy density (ΩΛ), and the Hubble constant (H0) in units of km/sec/Mpc. [13] ...... 32

3.1 The essential parameters for the four main dark matter density proﬁles in order of decreasing inner slope [8]...... 36

4.1 Relevant heat ﬂow values. The top entries are measured, while the lower entries are the calculated potential eﬀects of dark matter (DM) [45, 67, 68]...... 62

4.2 Comparison of potential dark matter constraints using various planets [76]. A greater diﬀerence between dark matter and internal heating rates give greater certainty. The minimum cross section probed scales roughly with the surface gravity. Earth is the best for setting reliable and strong constraints...... 91

8.1 The 10 qualities that a particle needs to satisfy to be a successful dark matter candidate, as displayed in Ref. [212]...... 155

xii LIST OF FIGURES

Figure Page

1.1 An example of a galactic rotation curve, speciﬁcally for M33. The dashed line shows how the velocities of the stars should behave if they were only governed by the normal matter. The actual observed velocities are shown by the solid line ﬁt to the data points [4]...... 7

1.2 The Bullet Cluster: Comparisons of gravitational force contours with the galaxies in the cluster, and therefore the dark matter (top panel), and the ionized gas that was stalled between the collisions (bottom panel). The gravitational contours are clearly centered around the galaxies, not the gas which is the bulk of the normal matter in the galaxy clusters. If there were no dark matter, the contours would be centered on the gas. The closer the contours, the greater the gravitational force. The cluster on the right has punched through the cluster on the left (like a bullet) and they are continuing to move away from each other. [5] ...... 15

2.1 The power spectrum of the Cosmic Microwave Background Radiation. It is a measurement of the temperature variance of the background as a function of angular scale. A higher multipole moment corresponds to a lower angular scale. The behavior of this power spectrum is indicative of the state and contents of the early Universe. [13] ...... 24

2.2 The relationship between angular scale and the geometry of the Uni- verse. For two geometries beginning at the same angular scale, a ﬂat geometry (green, D) will take less distance to arrive at the same length of the triangle λ than will a positive curvature universe (orange, d). More simply, if λ is equal between the two, D < d. Therefore, a feature in a positive curvature universe would show up at a smaller angular scale in a ﬂat universe. [11] ...... 27

xiii 2.3 A correlation of the supernovae Ia constraints on the acceleration of the Universe with the CMB measurements of the contents of the Universe. The perpendicular nature of the regions allows for particularly few possible solutions. The important region is where the line for Ω = 1 (Flat) coincides with the blue contour lines. [17] ...... 31

3.1 A representation of a dark matter halo enveloping a spiral galaxy. The halo has a much larger extent than the galaxy, and pervades the normal matter therein. [21] ...... 35

3.2 The behavior of three density proﬁles as a function of radius, normalized to the Milky Way. Beginning with the steepest inner proﬁle, they are Moore (dotted), NFW (dashed), and Kravtsov (solid). The position of the Sun (8.5 kpc) is pointed out by the solid vertical line. [23] ...... 39

4.1 Excluded regions in the σχN –mχ plane, not yet including the results of this dissertation. From top to bottom, these come from astrophysical constraints (dark-shaded) [46, 47, 48, 49], re-analyses of high-altitude detectors (medium-shaded) [46, 55, 56, 57], and underground direct dark matter detectors (light-shaded) [33, 50, 51, 52, 53, 54]. The dark matter number density scales as 1/mχ, and the scattering rates as σχN /mχ; for a ﬁxed scattering rate, the required cross section then scales as mχ. We will develop a constraint from Earth heating by dark matter annihilation to more deﬁnitively exclude the window between the astrophysical and underground constraints [45]...... 59

4.2 A graphic illustration of the assumed scattering scenario. The dark matter will scatter oﬀ particles in Earth, become gravitationally captured, and drift down to the center where it will be annihilated by other dark matter captured in the same fashion. The resulting heat will propagate out of the core, which can be compared to the measured internal heat ﬂow of Earth...... 71

4.3 Inside the heavily-shaded region, dark matter annihilations would over- heat Earth. Below the top edge of this region, dark matter can drift to Earth’s core in a satisfactory time. Above the bottom edge, the capture rate in Earth is nearly fully eﬃcient, leading to a heating rate of 3260 TW (above the dashed line, capture is only eﬃcient enough to lead to a heating rate of & 20 TW). The mass ranges are described in the text, and the light-shaded regions are as in Fig. 4.1 [45]...... 80

xiv 5.1 Annihilation of dark matter into SM ﬁnal states. Since all ﬁnal states except neutrinos produce gamma rays (see text), we can bound the total cross section from the neutrino signal limit, i.e., assuming Br(Invis.) 100% [111]...... 95 '

3 5 5.2 Upper: Diﬀuse ν¯ν annihilation signal for mχ = 10, 10 , and 10 GeV, added to the atmospheric background, both as (ν¯µ + νµ) and versus neutrino energy. As noted, the signals are most accurate for Eν & m /3. Lower: Ratio of this sum and background. The σ v values χ h A i at each example mχ are chosen to be detectable by our conservative criteria; the data and assumed uncertainty scales are also indicated [111].101

5.3 Upper bounds on the dark matter total annihilation cross section in galaxy halos as a function of the dark matter mass, calculated as discussed in the text [111]...... 105

6.1 The behavior of the integrated line of sight J(ψ) for the different profiles as a function of angular radius ψ as measured from the Galactic Center. Also shown is the averaged line of sight integral for a cone of radius ψ, J∆Ω, displayed as a thicker line than its J(ψ) counterpart. This is shown for three dark matter density profiles: Moore (dotted), NFW (dashed), and Kravtsov (solid) [23]...... 114

6.2 Example dark matter annihilation signals, shown superimposed on the Galactic and extragalactic gamma-ray spectra measured by COMP- TEL and EGRET. In each case, the cross section is chosen so that the signals are normalized according to our conservative detection criteria, namely, that the signal be 100% of the size of the background when

integrated in the energy range chosen (0.4 in log10 E, shown by horizontal arrows). The narrow signal on the right is the Galactic Center flux due to annihilation into monoenergetic gamma rays, for mχ = 1 GeV; the signal is smeared as appropriate for a detection with finite energy resolution. The broad feature on the left is the cosmic diffuse signal for annihilation into monoenergetic gamma rays at mχ = 0.1 GeV, smeared by redshift [157]...... 119

xv 6.3 The limits on the partial cross section, σ v , derived from the var- h A iγγ ious gamma-ray data. Our overall limit is shown as the dark shaded exclusion region. For comparison, the light-shaded region shows the corresponding limits for the NFW (rather than the Kravtsov) proﬁle [157]...... 126

6.4 The gamma-ray and neutrino limits on the total annihilation cross section, selecting Br(γγ) = 10−4 as a conservative value. The unitarity and KKT bounds are also shown. The overall bound on the total cross section combines the strongest values from these upper limits [157]. . 130

7.1 The number of particles N as a function of x. As the Universe cools, x increases, and the particles begin to freeze out of thermal equilibrium. During freeze-out, annihilations exponentially cease, eventually resulting in an asymptotic value as the remaining abundance. The Z values correspond, from top to bottom, with photons, neutral leptons (neutrinos), charged leptons, and hadrons [201]. The progression from top to bottom also correlates with increasing σ v [24]...... 141 h A i 7.2 The correlation between the predicted and observed amount of the light nuclei abundances. The predicted values are the curves, which are functions of η, the baryon-to-photon ratio or the baryon density. (A baryon is a nucleon, and the term is used to represent all normal matter, since the masses of nucleons are much bigger than the leptons.) The vertical bar for the CMB is the allowed range of the baryon density, while the BBN vertical line shows the concordance region. The (yellow) horizontal rectangular boxes are the observed values with uncertainties. Deuterium, helium-3, and lithium-7 are presented as relative to hydrogen, while helium-4’s abundance is shown as a mass fraction of the total amount of normal matter [208]...... 145

7.3 Constraints on the dark matter annihilation cross section to monoenergetic photons σ v . This combines previous work [157] with the h A iγγ modiﬁed results from FKT (blue) and McSW (green). The shaded regions are excluded; the results of Ref. [157] are an upper limit, while those of this work are a lower limit. For the stronger constraint from FKT, there is no allowed region, while a small wedge remains for that of McSW...... 150

xvi 7.4 Constraints on the total dark matter annihilation cross section. The results of Fig. 7.3 have been aﬀected by the conservative branching ratio Br(γγ) = 10−4. A great amount of parameter space has now been removed for the mass range 20 MeV to 1 GeV...... 152 ∼

xvii CHAPTER 1

AN INTRODUCTION TO DARK MATTER

1.1 A General Introduction

The Universe is a vast expanse, ﬁlled with many beautiful sights. Bejewelled spiral

arms wrap lazily around the X-ray-emitting supermassive black holes at their centers.

Curtains of glowing gas enshroud the remnants of stars whose lives as they know it have ended, transitioning like a caterpillar becoming a butterﬂy. Stars of various sizes and colors shine and twinkle, some with planets swirling about them. However, part

of the beauty in this masterpiece is hidden from view.

This elusive, and in fact necessary, beauty is hidden not by such things as dust,

eclipses by celestial bodies, or something as cosmological as an existence outside the

visible horizon, but by the laws of physics. It is astounding that about eight-three percent (83%) of all that comes under the deﬁnition of “matter” in the Universe is

a type with which humans have no hope of touching. This matter interacts mainly

by gravity, and since most of the processes on cosmological scales are governed by

this fundamental force, it has a large impact. While its presence is revealed by the

attraction between it and normal matter, it also should possess an extremely small

coupling to light and normal matter. However, this means that in the human realm

1 it cannot be seen or touched, and therefore is very diﬃcult to detect. Due to these

qualities, it has been given the moniker “dark matter”.

Some have postulated that the understanding of the gravitational force is ﬂawed

on very large scales, suggesting that modiﬁcations of Newton’s Laws beyond General

Relativity will reproduce the same eﬀects as dark matter. These investigations have

not, however, been able to present a truly convincing argument, and so the quest to

ﬁnd a very isolationist particle continues.

The extent of dark matter’s isolationism (aside from the ever-present pull of grav-

ity) has not been exactly determined. However, because of its lack of interactions

and other inferences from the early Universe, it most likely exists on the order of

the weak scale. Many underground detectors are looking, shielded from most normal

matter, hoping the nuclei of the detector might be given a slight nudge by a dark

matter particle passing through. The idea is that if such a chance event occurs, the astrophysicists could calculate the strength of dark matter’s interactions with normal matter and gain knowledge about its mass based on the amount of energy transferred to the lucky nucleus.

Telescopes are not searching the sky hoping to catch a photon that might have interacted with a dark matter particle. The chance of such an event is inﬁnitesimal, and any signal would undoubtedly be swallowed up by the other numerous photons in the searched area of the sky. Rather, data from telescopes/detectors that are aimed

at regions where dark matter should be clumped can be searched for hints. The dark

matter appears to encompass each spiral galaxy, encasing and pervading the normal

matter in a sphere called a halo. This halo should be denser towards the center. Two

dark matter particles should annihilate each other, meaning that in coming together

2 their masses are converted into energy, reforming as normal matter particles and/or photons. (The original particles do not exist anymore, hence “annihilation”.) There is a possibility these detectors could see a distinctive signal of the resulting particles or photons (or photons generated from those particles), something which could be distinguished from all the other signals generated by normal matter astrophysical

sources. Such an observation would be revolutionary; the energy and intensity of the

signal would give access to the knowledge of the dark matter’s mass, the strength of its

self-annihilation, and the type of normal matter particle to which it could annihilate.

Testing both the extent of dark matter’s interaction with normal matter (as

measured by their interaction cross section) and the strength of dark matter’s self-

annihilation (as measured by its self-annihilation cross section) are important in un-

locking the mystery of dark matter. This dissertation discusses some aspects of these

tests and presents new calculated constraints on the values of both types of cross

sections by comparing theoretical signals and observational data. Simplistically, it

can be thought of in this manner: If dark matter had property “A” with a certain

strength, it would have a theoretical eﬀect “B”. If the observational data does not

display theoretical eﬀect “B”, then dark matter cannot possess property “A”, either

with the assumed strength or at all. This is the main method to study what dark

matter is – by ruling out what it is not. Hopefully, placing constraints on what prop-

erties dark matter can possess could rule out certain models and narrow the search

for the aloof particle.

This dissertation places new limits on both the dark matter – normal matter

interaction cross section and the dark matter self-annihilation cross section. The

remainder of this chapter provides the historical context for dark matter’s existence,

3 while the cosmological formalism necessary to establish the amount of dark matter in the Universe is in Chapter 2. Chapter 3 concludes the background material with discussions of general dark matter properties, including its distribution and possible

candidates, and its primary detection methods. The research projects are detailed in

Chapters 4 through 7. Constraints on the dark matter – normal matter interaction cross section due to Earth’s heat ﬂow are presented in Chapter 4. Chapters 5 through

7 concern the dark matter self-annihilation cross section; Chapters 5 and 6 place

constraints using data from neutrino and photon observations, respectively. The case

of dark matter annihilations around the time of Big Bang Nucleosynthesis is discussed

in Chapter 7, where a limit can be found by the observed abundances of the primordial

elements. Chapter 8 provides a summary and conclusion.

1.2 A Historical Perspective

Even ﬁfty years ago, the Universe was only made of normal matter – or at least,

that is what the scientists who studied it thought. 1933 heralded the beginning of a viewpoint that something was missing from the understanding of the Universe, an idea that then would lie dormant for decades. It was in 1933 when Fritz Zwicky

concerned himself with the Coma Cluster, a nearby group of over a thousand galaxies.

He compared the total gravitational mass of the cluster (the mass needed to keep all

of the galaxies bound to the cluster) with the mass from adding up the individual

galaxies. He based his measure of mass for each galaxy on all the luminous material –

the stuﬀ that shone. He found that the total gravitational mass was greater than the

mass found where the light was ([1] 502-503). Putting forth the idea of “missing mass”

in the Universe did nothing to change people’s attitudes toward his cantankerousness,

4 and the issue was not, in general, taken seriously ([2] 305), although in 1936 Sinclair

Smith conﬁrmed Zwicky’s result for the Virgo Cluster ([1] 503, [3] 101).

The issue was not explored again for quite a while. Even then, the revisitation was not intentional. In the late 1960s, Vera Rubin and Kent Ford built one of the

ﬁrst electronic spectrographs with an original intent to study quasars at Kitt Peak.

At this time, however, a harsh battle over the ways to measure redshifts was ensuing, and they decided to try something less controversial, especially since Rubin’s graduate work on rotational motion in the local supercluster had caused a bit of a stir. Instead, they decided to study plain-old spiral galaxies, speciﬁcally their dynamics ([2] 302,

303).

Little did Rubin and Ford know that it would be one of the most controversial

studies done. In 1970, they published their measured rotation curve of the Andromeda

Galaxy, M31 ([2] 304). A “rotation curve” is a graph of the velocities of stars in a

galaxy, as a function of each star’s distance from the galaxy’s center. According to

Isaac Newton’s Laws of Gravity, a star with speed v and mass m is directly related

to the mass enclosed inside its orbit M and its distance from the center r:

GMm v2 GM 2 = m v = (1.1) r r −→ r r (G is Newton’s gravitational constant). The bulk of the visible mass of the galaxy is enclosed by stars with medium size orbits, so there is not much more additional

visible mass to be enclosed by stars at the further reaches of the galaxy. This means

that stars at the “edge” of the galaxy should move more slowly than stars in the

middle. However, what Rubin and Ford measured was a ﬂat rotation curve – the

velocities of the stars did not drop oﬀ at large distances as Newton said they should.

Confused by the puzzling observation, they stopped researching it, not understanding

5 the ramiﬁcations. An example of a rotation curve, displaying both the expected and actual velocities for a particular galaxy, is shown in Fig. 1.1 [4].

6 Figure 1.1: An example of a galactic rotation curve, speciﬁcally for M33. The dashed line shows how the velocities of the stars should behave if they were only governed by the normal matter. The actual observed velocities are shown by the solid line ﬁt to the data points [4].

7 However, Kenneth Freeman did realize the implications when he studied two other

galaxies, NGC 300 and M33 (the galaxy coincidentally featured in Fig. 1.1), indepen-

dently of Rubin and Ford ([1] 506). In Australia, Freeman analyzed the neutral

hydrogen gas at the edge of each galaxy. This neutral gas, which shows itself by

light in the form of radio waves, extends further than the gas that Rubin and Ford observed, which emits a red visible light due to the recombination of electrons with

the ionized hydrogen ([3] 94). The rotation curves, measured to greater radii than

Rubin and Ford’s, were ﬂat. The paper in which he published this result garnered a great deal of attention, but not for those observations. In fact, he relegated his

discussion of the rotation curves to an appendix ([3] 93).

These data have relatively low spatial resolution; if they are correct, then there must be in these galaxies additional matter which is undetected, either optically or at 21 cm. Its mass must be at least as large as the mass of the detected galaxy, and its distribution must be quite diﬀerent from the exponential distribution which holds of the optical galaxy – Kenneth Freeman ([3] 95).

This statement holds both his idea of missing mass and the reason it was not well received: radio observations were just not that certain. The results were held as inconclusive, since some scientists could conﬁrm them while others could not ([3] 93).

Thankfully, Rubin and Ford decided to look at galactic rotation again and studied numerous spiral galaxies with their spectrograph. The resulting rotation curves, just like M31, were ﬂat. “ ‘This time,’ said Rubin, ‘we knew immediately we had something phenomenal ’ ” ([2] 305). They could think of only two explanations: (i) Newton’s

Laws were incorrect, or (ii) there was more mass than could be seen through the

electromagnetic radiation it emitted (luminous mass), and that it was not distributed in a galaxy the same way as the stars. If the latter were correct, there was on the

8 order of ten times more missing mass than luminous material. However, important astronomers told them it must be a selection eﬀect, something they found because of a biased sample of galaxies which were all bright, and suggested Rubin study dimmer galaxies. Though she did indeed do this, she was already convinced that she had found what Zwicky had put forth decades before.

This occurred around the same time as James Peebles and Jeremiah Ostriker’s investigations of spiral galaxy formation using computer simulations. In 1973, they realized that the disks of spiral galaxies were unstable – unless each was immersed within a sphere of other matter, “like a hamburger patty sandwiched between two halves of a bulky roll ([2] 306).” The next year, with the help of Amos Yahil, they agreed with Zwicky. The missing mass was not just there, but necessary. Rubin’s observations made great sense with this sphere of matter, called a spherical halo.

She began preaching her model of galaxies, the basic view of which exists today, combining her observed rotation curves with the spherical halos. “ ‘Nobody ever told us all matter radiated,’ she [would blurt] in her blunt style, ‘we just assumed it did.’ ”

([2] 307). She again met with resistance, but that gradually gave way to acceptance.

With this hesitant acquiescence came the question of what the missing mass, or

“dark matter” as it came to be known, was. First proposed were of course nonluminous versions of normal matter, such as brown dwarfs (celestial bodies that almost became stars, but were not massive enough to start fusion) and black holes, but there was also speculation of some other kind of new elementary particle ([2] 308). On another front were the abundant neutrinos, particles of normal matter that were known to interact very little and not radiate light, both essential qualities of dark matter.

When these weakly-interacting relativistic particles were found to have at least a tiny

9 bit of mass in 1980, they became the frontrunner candidate for the dark matter. This was especially true since Yakov Zeldovich and Alex Szalay had already done a great

deal of research on that very issue ([2] 321).

However, there were a number of problems with neutrino dark matter, the fore-

most of which was its relativistic nature. Since neutrinos possess very little mass,

their velocities must be close to the speed of light to account for their total energy.

This meant that it would be very hard for them to form dense, intricate galactic

structures since their high energies would actually have a tendency to smooth out

details, hindering gravitational concentration. The killing blow for this “hot dark

matter” came in 1982, when Marc Aaronson observed that Draco, a dwarf galaxy

orbiting the Milky Way, also had a ﬂat rotation curve. This deﬁed convention, since

dwarf galaxies were assumed to be empty of dark matter. Being around one hundred

to one thousand times less massive than regular galaxies, they were not thought to be

massive enough to house dark matter. Neutrinos were extremely light and energetic

and therefore moved too quickly to be gravitationally bound to a dwarf galaxy the

size of Draco ([2] 335, [3] 142). Cold dark matter – particles possessing more mass and

moving more slowly – became the favorite, championed by Peebles, but it, too, had

its limitations. In about 1983, simulations showed that too much galactic structure

was created if the dark matter were too cold. Overly-detailed objects were created

and there were no true voids (areas lacking galaxies) which had been observed in the

actual Universe ([2] 348). However, a cold version still is the leading candidate today.

Dark matter ﬁrst gained real acceptance and respect as an actual phenomenon “in

the spring of 1985 when an International Astronomical Union symposium on the sub-

ject was convened at Princeton ([2] 356).” The cold dark matter candidates abounded,

10 drawn from models based in particle physics such as supersymmetry, string theory,

and supergravity. Each model had a new type of light, stable elementary particle

that ﬁt into the paradigm. A few of these candidates will be discussed in Chapter

3. Most of them fell under the moniker of WIMPs, Weakly-Interacting Massive Par- ticles, but in the 1990s MACHOs were also highly investigated as a last chance for the normal matter to be a candidate. MACHO stands for MAssive Compact Halo

Object, essentially a chunk of normal matter in the galactic halo not heavy enough to emit light, such as asteroids, failed stars, and even white dwarfs ([3] 162). It was a term coined by Kim Griest in response to the name WIMP ([3] 165), and the search for MACHOs was headed by Bodhan Paczynski. This solution was favored by the astronomers over the physicists’ introduction of a new particle, since it did not require some exotic material. The way to ﬁnd these dim celestial bodies involved small-scale gravitational lensing, or microlensing. As noted by Newton and Einstein, gravity bends light, and something that has a larger gravitational force will bend the light

more. A massive galaxy will bend the light of bright objects behind it, acting like a

lens to refract the light into arcs centered on the foreground object, an eﬀect called

gravitational lensing. In microlensing, the foreground object is not massive enough to

refract the light very far from the background star, but rather it would “[bend] what

would otherwise be diverging light rays together so that, as observed from Earth, the

background star would appear to brighten, then fade back to its normal brightness”

([3] 162).

The 1990s endured a great number of debates regarding WIMPs and MACHOs.

The eﬀect was indeed observed, but the claims of total events kept changing, oscil-

lating up and down as results were presented and then invalidated. MACHO surveys

11 eventually were able to rule out brown dwarfs, loose planets, and neutron stars. Black holes and white dwarfs, while still somewhat possible within the limits of the surveys, became unlikely for other scientiﬁc reasons. In the late 1990s, however, it became clear that the recorded events might actually have been located in the Small or Large

Magellanic Clouds orbiting the Milky Way, instead of the Galactic halo. This meant that the percentage of the dark matter halo that could actually be normal matter must be very small, meaning that the physicists’ preferred solution, the WIMPs, must dominate the dark matter halo ([3] 169-171).

Recently, even more convincing evidence of dark matter has been found in the details of an astrophysical skirmish. The Bullet Cluster (1E0657-558) is a system of two galaxy clusters in the process of merging [5]. The smaller of the two has a much higher velocity, and essentially has punched through the other. However, an astrophysical collision is much diﬀerent than an automobile accident. The stars and galaxies pass by each other unnoticed, perhaps somewhat disturbed by the other’s gravitational pull (in this case, because of the high velocity of the smaller cluster, there should not have been much gravitational distortion). It would be like trying to throw loose confetti up while snow was falling down – it would be hard to get a piece of confetti and a snowﬂake to hit. However, most of the normal matter in a galaxy cluster is in the form of gas. The gas in both clusters will indeed collide and heat up, emitting some kind of radiation. The collision stalls the separation, and so the gas is essentially stuck in between. The dark matter in the clusters should also pass unnoticed, because, just like dark matter and normal matter, dark matter should not interact very much with itself.

12 In such a system as the Bullet Cluster, a very particular comparison can be made using gravitational lensing and the radiation of the heated gas. The stars and galaxies have passed through each other a great distance, but the gas is still stuck between. As stated above, the amount of gas in a cluster greatly outweighs the stars and galaxies.

If there were no dark matter, then the gravitational lensing, the bending of the light behind the cluster due to mass, would be centered on the gas. If there were dark matter and it comprised most of the total matter of the clusters, the gravitational lensing would be strongest where the stars and galaxies are, since the dark matter would be with them, having also passed through. At the end of 2006, Douglas Clowe,

Scott Randall, and Maxim Markevitch presented results from X-ray observations of the gas in the Bullet Cluster and a mapping of the gravitational lensing [5]. They showed that the lensing was centered on the galaxies – not the gas, meaning of course that the dark matter hypothesis was seemingly validated. See Figure 1.2 for the graphical depiction of this separation of the gas from the gravitational eﬀects.

13 Even with this convincing evidence, there is still argument as to whether the implications of the Bullet Cluster completely establish dark matter’s existence. The

scientists who study modiﬁed theories of gravity believe that the Bullet Cluster does

not completely rule out their ideas. However, it is unclear if the theories are able to reproduce the necessary eﬀects, and so, for most, the belief swings highly in the

direction of dark matter’s existence. Supported by this and much other evidence, the

quest to determine the nature of this exotic elementary particle that somehow exists

in greater amounts than normal matter and yet maintains its aloofness continues

today.

14 Figure 1.2: The Bullet Cluster: Comparisons of gravitational force contours with the galaxies in the cluster, and therefore the dark matter (top panel), and the ionized gas that was stalled between the collisions (bottom panel). The gravitational contours are clearly centered around the galaxies, not the gas which is the bulk of the normal matter in the galaxy clusters. If there were no dark matter, the contours would be centered on the gas. The closer the contours, the greater the gravitational force. The cluster on the right has punched through the cluster on the left (like a bullet) and they are continuing to move away from each other. [5]

15 CHAPTER 2

HOW MUCH DARK MATTER IS THERE?

About eighty-three percent of matter in the Universe is dark, and there is much more dark matter in galaxies than normal matter. From the previous chapter, it is obvious that observations of galaxies have provided the latter conclusion. How is it, though, that the amount of dark matter in the Universe is known? In order to answer that question, some information must be understood regarding the way in which the proportions of the contents of the Universe are represented. Cosmology provides those answers.

2.1 Necessary Cosmological Formalism

This primer in cosmological formalism and terminology begins with the fact that the Universe is expanding. This phenomenon was first noticed by Edwin Hubble in the 1920s. He was able to use specific astrophysical objects as measuring devices. For example, there is a type of star called a Cepheid variable. These stars have a uniform property: each varies its brightness in a predictable way. This variation occurs with a specific period, and the length of one of these periods was found by Henrietta Leavitt to be a function of the star’s luminosity ([6] 271). There is a standard equation to relate a star’s apparent luminosity b (as viewed from Earth) and its actual luminosity

16 (found by its period of variation) to its distance from Earth d ([6] 270):

L d = . (2.1) r4πb

Using this basic property that made a Cepheid unique, Hubble was able to determine

the distance to the stars. Objects with regular properties used to ﬁnd distances are called “standard candles”. This actually produced a breakthrough, because until this time the Milky Way was thought to be the only galaxy in the Universe, containing

all of the observed phenomena in the sky. However, some of the distances he found

were too large for this to be true any longer, meaning that those Cepheids must be in their own separate galaxies ([6] 271).

By measuring the stellar spectra, he was also able to tell how much the light

was redshifted. When the spectrum of a star is measured, emission and absorption

lines appear as bright and dark, respectively. These lines are at speciﬁc wavelengths

corresponding to either an elemental signature in a star’s emitted radiation or the

absorption of a star’s light by material between the star and Earth. If a spectrum

is shifted uniformly toward longer wavelengths (as indicated by the spacing between

spectral lines), the light has uniformly lost energy on the way to the spectrograph.

Since the lines shift toward the red end of the spectrum in this fashion, this is called

redshift. Hubble noticed that there was a direct relation between the amount of

redshift and the distance to a star. In fact, he could translate the amount of redshift

(corresponding to energy lost) into a velocity. From this it was determined that

the galaxies with a larger distance were moving away with a faster velocity – the

Universe itself was expanding, stretching out the wavelengths of the light and taking

the galaxies along for the ride ([6] 273-275). The factor that speciﬁes just how quickly

17 the Universe is expanding (the direct relation of the velocity to the distance) is called the Hubble parameter, H; the cosmological redshift is quantiﬁed by z.

Theorists were able to determine a generalized way to describe the expanding universe in terms of a metric, deﬁned by the distance between points. The coordinates

of these points were three dimensions of space and one of time. This metric was based on three assertions: the Universe was expanding, homogeneous (made of the same material throughout) and isotropic (looks the same, generally, in all directions). H. P.

Robertson and A. G. Walker found this metric independently, most easily expressed in polar coordinates, to be ([6] 284, [7] 29)

∆r2 ∆s2 = (c∆t)2 R2(t) + r2∆θ2 + r2 sin2 θ∆φ2 . (2.2) − 1 kr2 − The scale factor R(t) is essentially a unit of length for the Universe. The geometry of the Universe, or its curvature, is expressed as k. In everyday life, humans deal with

a Euclidean, or ﬂat, geometry. The distance between two points in this form of the

metric is simple, since it is the same as the distance between two points on a ﬂat piece

of paper. For a ﬂat universe, k = 0. However, there are other simple geometries for

something of such a large scale as the Universe which would result in humans (living

on a much, much smaller scale) interacting in a ﬂat geometry. They are a sphere

(positive curvature, k = +1) and a saddle (negative curvature, k = 1). − Analyzing a triangle on the surfaces of these three geometries leads to a greater

realization of their diﬀerences. The three interior angles of the triangle when drawn

on a spherical surface sum to a value greater than 180◦; the sum of the three angles

do not reach 180◦ for a triangle on a saddle.

While this metric does explain distances in a universe of a given curvature, the dynamics of the Universe are held in the scale factor’s time dependence ([7] 47). In

18 order to understand the scale factor’s evolution, its time dependence needs to be explicitly solved. This is done with the help of the Einstein equations of General

Relativity, a set of tensor equations:

1 Rµν R gµν Gµν (2.3) − 2 ≡

Gµν = 8πG Tµν . (2.4)

Newton’s gravitational constant G is not to be confused with the tensor Gµν, and similarly, the Ricci tensor Rµν and Ricci scalar R are not the scale factor R. This

Gµν is called the Einstein tensor, while the Tµν is the stress-energy tensor. The

indices µ and ν range from 0 to 3, the convention for describing the 4 dimensions of

space and time. Without going into the details of General Relativity, the important

thing to note about this equation is that it relates the geometry of the Universe

(the information for which is in the Einstein tensor) to the energy content of the

Universe (as described by the stress-energy tensor). Using the Robertson-Walker

metric (Eq. (2.2)) and rewriting the Einstein equations in the form of the scale factor

R, one arrives at the Friedmann equation ([6] 297):

dR 2 8πG = ρR2 kc2, (2.5) dt 3 − This relates the expansion – the time dependence of the scale factor – to the curvature and content of the Universe, expressed by the density of the Universe ρ. As a matter of formalism, dR/dt can be written as R˙ . Dividing by R2 rewrites the equation as

([7] 49) 2 ˙ 2 R 2 8πG kc = H = ρ 2 . (2.6) R! 3 − R This more transparent form shows how the expansion of the Universe depends upon the content of the Universe and its curvature.

19 For the moment, assume the curvature is k = 0, ﬂat. The density needed to satisfy this equation is then 3H2 ρ = . (2.7) c 8πG

This is called the critical density, as it is the special density for the Universe to be

2 ﬂat. Today (at time t0 and redshift z = 0), the critical density is equal to 3H0 /8πG,

which is 1.88 h2 10−29 g/cm3 ([7] 503). H is the Hubble constant today (as opposed × 0 to the varying Hubble parameter), and its value is around 70 km/(sec Mpc) in its

traditional units. Since this value is uncertain, quantities that depend on H0 most

often contain instead the unitless h:

H sec Mpc h = 0 , (2.8) 100 km

which therefore ranges between about 0.4 and 1. Astrophysicists more commonly

use electronvolt (eV) units instead of grams, where 1 GeV/c2 is 1.7827 10−24g and × c = 1 in the scheme of Natural Units. For reference, the masses of the neutron and

the proton (the nucleons) are both approximately 1 GeV. In these units, the critical

density is 1.05 h2 10−5 GeV/cm3. × Using this deﬁnition, the Friedmann equation can be written in the more conve-

nient form ρ 1 3 kc2 1 = 2 or 1 = Ω + Ωk, (2.9) ρc − ρc 8πG R where Ω is now the ratio between a density and the critical density ([6] 301, [7] 50), and the curvature density is deﬁned to be (3/8πG)(kc2/R2). The term Ω or Ωh2 − is used, therefore, when talking about proportions of the Universe’s contents. The density ρ is actually the sum of the matter density ρm and the radiation density

(essentially the photons) ρr, meaning Ω = Ωm + Ωr.

20 Another term can be added to the Friedmann equation for energy of the vacuum

of space, a term most often characterized by Λ, a cosmological constant or dark energy which has a negative energy density. The motivation for this term will be

presented and slightly discussed below, but note that a detailed description of dark energy is beyond the scope of this dissertation. Adding this energy density to the

Friedmann equation, which also means adding a term Λgµν to the Einstein equations

(Eq. (2.4)) [8], leads to

1 = Ωm + Ωr + ΩΛ + Ωk = Ω. (2.10)

However, the more relevant form is simply

1 = Ωm + ΩΛ (2.11) due to two reasons: (i) humans exist in a matter-dominated Universe (Ω 0) and r → (ii) the Universe appears to be flat (Ω 0). k → The matter and radiation densities evolve differently with the changing scale radius, due to each being a different form of energy. The radiation density scales as R−4, but the matter density scales as R−3. Current thinking of the exotic vacuum energy density holds that it is independent of the scale factor’s evolution ([7] 48-49). In the early Universe the radiation dominated the dynamics, but as the scale factor grew with the expansion, matter became the more important factor, as it is today. At some point in the future, the vacuum energy will most likely dominate. The importance of this is that the radiation density can be neglected today (Ω Ω ). The second m r point in the above paragraph can be made clear by a study of the Cosmic Microwave

Background.

21 2.2 Measuring an Imprint of the Early Universe: the Cosmic Microwave Background

The Cosmic Microwave Background (CMB) is a view of the light from the early

Universe. In a paper by Ralph Alpher and Robert Herman [9] and the infamous

April Fool’s Day paper of Alpher, Bethe, and Gamov [10] in the late 1940s and early

1950s, the authors postulated that if the Universe started oﬀ as a hot, extremely dense point and expanded to the way it is today, then the radiation from that initial stage should have lost energy with the expansion, the wavelengths stretching to become microwaves with a temperature between 5 and 10 Kelvin. Even though there

were some fundamental mistakes in these ﬁndings, Arno Penzias and Robert Wilson

accidentally found in 1965 this isotropic background noise of microwave radiation at a

temperature of about 3.5 Kelvin, vindicating the Big Bang Model ([6] 321-325). This

temperature has since been measured more accurately to be about 2.7 Kelvin [8]. The

behavior of the scale factor R motivated the idea of the Big Bang, since the continual

expansion observed from the mathematics meant that at some point, R = 0 ([7] 50).

In the very early Universe, the temperature was so hot that the photons and

matter formed a plasma, facilitated by the exceptionally frequent interactions of both

with the free electrons. The matter tried to condense under gravity while the photon

pressure provided a counteraction. The interactions of these diﬀerent types of energy

therefore set up pressure waves in the plasma: the matter would compress, the photons

would expand (rarefy) the collection until the matter was too diﬀuse for the photons

to have the dominant eﬀect, and the cycle would repeat [11, 12]. Through the nature of this process, the behavior of the waves was directly related to the proportions of

matter and photons.

22 Simultaneously, the Universe was expanding and cooling, stretching the wavelengths of the photons. Nuclei were able to form, but the photons kept the electrons apart from the ions. Around a time of 1013 s after the Big Bang, at a temperature of about 3000 K and a redshift of about 1100, the photons cooled enough that they no longer interacted frequently enough with the matter to maintain the plasma. Unhin- dered by the matter, they “freely streamed” out. Since they all had the same speed, they carried with them the imprint of the plasma – including the regions of compression and rarefaction. This time, also (confusingly) called “recombination”, is when the matter and free electrons are ﬁrst able to form neutral atoms ([7] 77-80) [11, 12].

23 Figure 2.1: The power spectrum of the Cosmic Microwave Background Radiation. It is a measurement of the temperature variance of the background as a function of angular scale. A higher multipole moment corresponds to a lower angular scale. The behavior of this power spectrum is indicative of the state and contents of the early Universe. [13]

24 In analyzing the CMB, the most informative representation is a power spectrum such as the most accurate one to date from the ﬁve-year analysis of the Wilkinson

Microwave Anisotropy Probe (WMAP), shown in Figure 2.1. This power spectrum shows the temperature variance as a function of angular scale, where in the ﬁgure increasing l corresponds to decreasing angular scale. The “temperature variance”

is a measure of the ratio of temperature diﬀerences to the total temperature; the

greatest diﬀerences manifest as the peaks. The compression phases are represented

by the odd-numbered peaks, while the even ones are the rarefactions. The radiation is highly isotropic, with diﬀerences on the order of merely 10−5 [13].

Since there exist such direct relations between the peaks in this representation and the photon-matter plasma, the peaks’ positions and heights can provide a great deal of information about this stage of the early Universe. The angular scale at which features are seen corresponds to the distance the radiation traversed to reach the point of observation or the time which it took to get there. The “free streaming” was not instantaneous; the first electrons would have come from looser photon-matter couplings (such as those in early stages of the first compression cycle) and therefore would be seen at larger angular scales. The position of the first peak is therefore a measure of the Universe’s curvature, since a given angular scale corresponds to different traversed distances for different curvatures.

Begin with a set angle between two lines of the same length (defining a distance), add a third line of length a to make a triangle. A positive curvature geometry requires a greater distance than a flat geometry to reproduce the same value of a; a negative curvature Universe would need a shorter distance. Now, hold a fixed instead of the angular scale. A positive curvature’s angle must be larger than the flat geometry’s

25 in this situation. So, a feature in a flat universe would show up at a larger (smaller) angular scale for a positive (negative) curvature. Therefore, the value of the angular scale for the first peak in the power spectrum is a measure of the geometry, and the data strongly suggest that the Universe is flat, Ω = 1. Figure 2.2, taken from Hu and Dodelson (2002) [11], is a good visualization of this relationship between angular scale and distance.

26 Figure 2.2: The relationship between angular scale and the geometry of the Universe. For two geometries beginning at the same angular scale, a ﬂat geometry (green, D) will take less distance to arrive at the same length of the triangle λ than will a positive curvature universe (orange, d). More simply, if λ is equal between the two, D < d. Therefore, a feature in a positive curvature universe would show up at a smaller angular scale in a ﬂat universe. [11]

27 The heights of the peaks are determined by the temperature variance. As the Uni- verse expands, the matter density decreases (the photon density would, too, though photons are constantly being produced at this point), meaning that there should be less and less compression each time the cycle is revisited, resulting in lower and lower peaks. As seen in the figure, this is not the case. What could cause the observed behavior? An overall increase in the amount of normal matter would lead to enhanced compression (the odd peaks) because of the greater gravitational force, but it is the addition of dark matter that significantly affects this. Since dark matter is not subject to the photon pressure, it provides a stable gravitational well that only expands with the Universe. This does lead to enhanced compression, staving off some of the effects of the density dilution. The third peak is very close in height to the second, but slightly smaller, and so this gives evidence for dark matter while allowing for a calculation of the amount of total matter present in proportion to the photons [11, 12, 13].

While the CMB provides evidence the Universe is ﬂat (Ω = 1), it and other observations such as those from the Sloan Digital Sky Survey [14] show that Ωm < 1.

The introduction of the vacuum energy density Λ following the realization of the acceleration of the Universe’s expansion helps to reconcile this problem, because there must be something else making the Universe ﬂat if the total matter is not enough.

While lending credence to Λ’s existence, the CMB does not really constrain ΩΛ independently. Before the acceleration’s measurement, described below, the discrepancies between ﬂatness and Ωm < 1 was called the “Ω problem” ([7] 390).

28 2.3 The Universe is “Darker” Than Originally Thought: a Case for Λ

Recent measurements of distant type Ia supernovae, another type of standard

candle like Cepheid variables, have shown that the Universe is not just expanding, but accelerating [15, 16, 17]. The supernovae are fainter than they should be, meaning they also are farther away than expected. In order for this to be so, the Universe must have increased its expansion – accelerated. There must be some component of the

Universe counteracting gravity, something with a negative pressure that only came to fruition after matter domination began. This is the Λ mentioned earlier, whether it be vacuum energy, a cosmological constant, or something else, often referred to as dark energy. Note: the nature and details of the dark energy are beyond the scope of this dissertation, but it is helpful to discuss the ramiﬁcations of studies of dark energy for the amount of dark matter in the Universe. The acceleration measures the diﬀerence between the gravitational force (matter density) and the negative energy pressure force (dark energy density).

Before the acceleration was realized, a positive curvature geometry was referred to as “closed”, meaning it would eventually collapse back on itself, losing the battle with gravity. A universe with negative curvature was “open” and would continue expanding forever. In the special case of a ﬂat universe, the expansion would coast to a stop at inﬁnity. The term “critical density” referred to the special density needed for this to happen, as an island between two extremes. The dark energy changes these ideas, however, and the terms no longer apply, except as monikers for universes with Ω = 1. For example, a universe can have positive curvature and expand forever 6

29 with the addition of acceleration. The density for ﬂatness is certainly no longer as

“critical” as it once was [15, 16, 17].

2.4 Correlating Cosmological Measurements

Measuring the acceleration essentially calculates the diﬀerence between the matter density and the dark energy density (Ω Ω ) while the CMB constrains the m − Λ combination of the two densities (Ω = Ωm + ΩΛ) [17]. This opposing nature results in strong limits from the correlation, leading to a very good determination of the individual values. A plot showing these overlapping contraints is in Figure 2.3. The intersection of the line “Flat” with the contour ellipses provides for a very narrow range of allowed values.

30 Figure 2.3: A correlation of the supernovae Ia constraints on the acceleration of the Universe with the CMB measurements of the contents of the Universe. The perpendicular nature of the regions allows for particularly few possible solutions. The important region is where the line for Ω = 1 (Flat) coincides with the blue contour lines. [17]

31 Parameter Value 2 Ωmh 0.1326 0.0063 2 Ωbh 0.02273 0.00062 Ω h2 0.1099 0.0062 χ ΩΛ 0.742 0.030 +2.6 H0 71.9−2.7

2 Table 2.1: The latest values of the total matter density in terms of h (Ωmh ), normal 2 2 matter density in terms of h (Ωbh ), dark matter density in terms of h (Ωχh ), dark energy density (ΩΛ), and the Hubble constant (H0) in units of km/sec/Mpc. [13]

The values are listed [13] in Table 2.1. For the relevant parameters in Fig. 2.3, the values in Table 2.1 are more detailed and constraining due to the WMAP 5 analysis.

2 Note: as ρc is often left in terms of h due to the uncertainty in the Hubble parameter,

so are the densities. Since the total matter is the sum of the normal matter and the

dark matter, the CMB also measures the dark matter density.

The CMB’s value of the normal matter density is in accord with that from Big

Bang Nucleosynthesis (BBN), the theory of the formation of the light elements from

the Big Bang ([7] 95) [18]. The elements and light isotopes which could form from

the Big Bang after the Universe cooled to a low enough temperature are hydrogen,

deuterium, helium-3, helium-4, lithium-7, and beryllium-7 which decayed to lithium-

7. Atomic mass numbers of 5 and 8 are unstable, and there was not enough energy

to go further. All the heavier elements were created in stars or through spallation

processes. Analysis of present-day values of these elements can help to determine

their primordial abundances. The known processes which create or destroy the light

elements can be applied over the lifetime of the Universe to determine how much

of them should be left today. Deuterium, for example, is not created in stars in

32 any sustainable way due to its very low binding energy, but it is destroyed through

processes in stars, and so concentrations of deuterium today are thought to be less than the original primordial abundance. BBN will be discussed more in Chapter 7,

2 but its implied value Ωbh (the normal matter density) is between 0.018 and 0.023 [8].

When calculations are done to determine the amount of dark matter that should be produced in the early Universe if it were in thermal equilibrium with the other matter (called a “thermal relic”), they are consistent with the observed abundance

if dark matter interactions are of the weak scale. Speciﬁcally, this concerns the

strength of dark matter self-annihilations, setting a “natural scale” if the dark matter

is indeed a thermal relic [8, 19]. More on this will be discussed in Chapter 7, which

also includes a detailed derivation of the dark matter relic abundance. This quality of

dark matter is commonly considered, though the actual dark matter self-annihilation

cross section has not been constrained by observations to be of this natural scale.

However, substantial progress has been made by this dissertation, and is detailed in

Chapters 5 and 6.

33 CHAPTER 3

DARK MATTER: WHERE, WHAT, HOW?

3.1 Where is the Dark Matter?

The dark matter comprising about 22% of the content of Universe (about 83% of

all matter) is not uniformly distributed. Rather, it is highly clustered with the normal

matter in galaxies, an idea established by the ﬁrst realizations of its existence (see

Chapter 1). In fact, the clustered dark matter is about 105 times denser in galactic

halos than if it were truly homogeneous. The standard picture is a halo within which the normal matter is immersed [20]. Since dark matter cannot radiate heat, it cannot lose enough energy to form a disk or even a star, leaving a spherical distribution after equilibrium (See Figure 3.1). The form of this distribution is suggested by the diﬀuse gray sphere encompassing the spiral disk in Fig. 3.1 [21].

34 Figure 3.1: A representation of a dark matter halo enveloping a spiral galaxy. The halo has a much larger extent than the galaxy, and pervades the normal matter therein. [21]

35 Proﬁle α β γ rs (kpc) Moore 1.5 3.0 1.5 28.0 NFW 1.0 3.0 1.0 20.0 Kravtsov 2.0 3.0 0.4 10.0 Isothermal 2.0 2.0 0 3.5

Table 3.1: The essential parameters for the four main dark matter density proﬁles in order of decreasing inner slope [8].

The simplest form of the dark matter halo is a sphere, although structures such

as ellipsoids with axes of three diﬀerent lengths are possible as well. Regardless, the

dark matter distribution needs to reproduce the ﬂat rotation curves of the numerous

galaxies that have been measured. The generic form of a spherical structure that

satisﬁes the rotation curves (as determined by numerical simulations) is [4, 8]:

ρ0 ρ(r) = γ α (β−γ)/α . (3.1) r 1 + r rs rs h i The ρ0 is picked to be a normalization which can reproduce a speciﬁc value of ρ

at a given rs, sometimes referred to as the scale radius. While simulations using

these density proﬁles do reproduce to a good level the middles and outer edges of

galactic rotation curves, there is diﬃculty determining the speciﬁc radial behavior of

the innermost regions. For large radii, almost all of the most common proﬁle choices

agree that the behavior scales as 1/r3. The most commonly used proﬁles (and their

values of α, β, and γ) are Moore (1.5, 3.0, 1.5), Navarro-Frenk-White or NFW (1.0,

3.0, 1.0), Kravtsov (2.0, 3.0, 0.4), and Isothermal (2.0, 2.0, 0). These are listed here

in order of decreasing slope in the inner region, and the distinctive parameters are

also shown in Table 3.1.

36 A more intense slope leads to an inﬁnite spike in the density at the center, called

a cusp; a uniform density beginning at a certain inner radius is called a core. There formerly was cusp-core controversy, with the theoretical simulations preferring cusps and the observations suggesting cores. This is no longer so severe as to constitute a controversy [8], but the inner region is still debated. The density behavior has speciﬁc ramiﬁcations for studies of dark matter annihilations, which will be discussed in more

detail in Chapters 5 and 6. As a result of this, studies which focus on large angular

regions encompassing the center are more generic than those which consider just the innermost radii. The prominent belief is that there is an enhanced density at the center.

The overall profile of the Milky Way is difficult to determine, as the observers are inside of it, but some details from measured rotations can be found. This includes the local dark matter density. While there is some disagreement (although within one order of magnitude), it is generally accepted that the ρ0 for the local dark matter density is between 0.2 and 0.8 GeV/cm3 [4, 8]. The Sun is located 8.5 kpc from the center of the Milky Way and the scale radius for the NFW profile (the most commonly assumed profile) is 20 kpc. If instead the density were to be expressed in terms of galactic scales (the mass of the Sun (M ) and the length unit parsec (pc)), a useful relation to note is that [22] M GeV = 30 . (3.2) pc3 cm3

The NFW, Moore, and Kravtsov proﬁles are shown in Figure 3.2, normalized to

reproduce the rotation curves at large galactic radii [23].

37 It is important to stress that the measured or inferred quantity is the mass density, meaning that the number density and the dark matter particle mass are un-

determined. Correspondingly, there could be a large amount of light dark matter,

or a little amount of heavy dark matter. This unknown number density is often

needed for studies involving rates (interaction, annihilation, etc.), and so an inherent

mass dependence accompanies those calculations from the necessary substitution of

n = ρ/m.

38 Figure 3.2: The behavior of three density proﬁles as a function of radius, normalized to the Milky Way. Beginning with the steepest inner proﬁle, they are Moore (dotted), NFW (dashed), and Kravtsov (solid). The position of the Sun (8.5 kpc) is pointed out by the solid vertical line. [23]

39 The velocities of the dark matter particles in the halo appear to follow a Maxwell-

Boltzman velocity distribution,

1 2 2 3 −v /v0 3 f(v) d v = 3 e d v, (3.3) (√πv0)

where v0 is the rotation speed at the limit of large radii. The average velocity of the

distribution v2 1/2, the most important value with which calculations are concerned, h i is most often taken to be 270 km/s. For reference, the Sun’s movement around the

center of the Milky Way is about 220 km/s [19].

3.2 What Could Dark Matter Be?

Dark matter is something exotic, set apart from the normal matter of the Standard

Model (SM) of physics. For review (and completeness) the SM consists of six quarks

(up, down, strange, charm, top, bottom), three charged leptons (electron, muon, tau), each with its corresponding neutrino, along with the gluon, photon, the Z0 and W gauge bosons, and a Higgs boson. The corresponding antiparticles are also part of the Standard Model. The four fundamental forces of physics, in order of decreasing strength, are Strong, Electromagnetic, Weak, and Gravitational [8].

Dark matter’s new type of matter must interact with the SM on the weak scale or below. The assumption, or hope, is that dark matter interacts with normal matter at some level beyond gravitation. However, this must be a very weak interaction, or it would have been noticed. As determined from observations and simulations, the general requirements for a particle to be the dark matter are that it: (i) is some new kind of matter that interacts at best on the weak scale, but deﬁnitely gravitationally;

(ii) is neutral (if it were charged it would interact with the electromagnetic force); (iii) is cold (non-relativistic); (iv) makes up 22% of the Universe, produced in the correct

40 amount in the beginning if it is a thermal relic; and (v) is stable on timescales greater than the lifetime of the Universe (or there would be no dark matter today) [8]. It is

important to note that the total dark matter may be made up of more than one type of particle. Usually, the assumption of just one type is made for simplicity.

While there are many options, there are two candidates that arise from attempts

to extend the Standard Model that satisfy these qualities. They have the beneﬁt of being theoretically motivated not just for the sake of dark matter, but instead are part

of a theoretical structure with calculable properties. These two theories are Super-

symmetry and Universal Extra Dimensions. They both propose many new particles,

and the lightest of each theory is a candidate for dark matter; the candidates are

collectively grouped under the term WIMP, or Weakly Interacting Massive Particle.

3.2.1 Supersymmetry

As the name implies, Supersymmetry (SUSY) is a way to introduce a sort of

extreme symmetry to the particles that exist in the Standard Model. One of the

motivations for SUSY is to aid in unifying the four forces and in understanding why

the strengths of the forces have their speciﬁc heirarchical ranking. The following

discussion comes from the extensive review of Supersymmetry as dark matter by

Gerard Jungman, Marc Kamionkowski, and Kim Griest [19].

There are two types of SM particles, namely fermions and bosons. For the quan-

tum mechanical quality of spin, a fermionic particle has a half-integer value (1/2, 3/2,

...) and obey the Pauli exclusion principle. Bosons, on the other hand, have integer

values for their spins. In the SM, they are represented in diﬀerent proportions. In a

simplistic view, the particles that experience the forces are fermions, while those that

41 mediate the forces, including photons, are bosons. In SUSY, every SM fermion has

a bosonic superpartner, and a superpartner fermion exists for every standard boson.

The convention is to add the preﬁx “s-” to achieve the name of a bosonic super-

partner, and to add the suﬃx “-ino” to create the fermionic superpartner’s moniker.

For example, the bosonic superpartner of a quark and the electron are “squark” and

“selectron”, while the fermionic counterparts of photons and gluons are “photinos”

and “gluinos”.

To this end, many new particles are needed. The masses of the superparticles

cannot be equal to that of the SM particles, or they would have been detected in particle colliders; SUSY is a broken symmetry. By nature of the theory, the superparticles are also weakly interacting. There is a quantum property called R-parity

which is required to be conserved in this framework. Superparticles have R = 1, − while SM particles have R = +1. As such, a superparticle cannot decay into a SM particle alone. In order to conserve R-parity, a superparticle must decay into an odd

number of superparticles, which must outnumber the SM products. An example is

a ﬁnal state of three superparticles and two Standard ones. This means that the

superparticle with the lightest mass must be stable. The lightest, neutral superpar-

ticle is therefore a candidate for dark matter, and is abbreviated the LSP (Lightest

Supersymmetric Particle).

The neutral superparticles are the fermionic counterparts of the photon, Z0 boson,

and the two Higgs bosons required in SUSY. The LSP, symbolized as χ, is therefore

represented as a linear combination of these four superpartners:

˜ ˜ 3 ˜ 0 ˜ 0 χ = a1B + a2W + a3H1 + a4H2 . (3.4)

42 This has been expressed in terms of the superpartners B˜ and W˜ 3 of B and W 3 which

are the U(1) and SU(2) gauge ﬁelds that mix to generate the photon and Z0 boson.

This form of the LSP is referred to as the “neutralino” since it is neutral and a fermion. There are more speciﬁc cases where sneutrinos, sleptons, or gluinos can be the LSP, but the most generic and simplistic LSP is the neutralino. One of the drawbacks of SUSY is the large amount of parameters the theory has, which number more than sixty beyond those of the actual SM. The Minimal Supersymmetric

Standard Model (MSSM) has fewer parameters, while the Constrained Minimal Su- persymmetric Standard Model (CMSSM) has only six. Complex computer codes have

been created to deal with this large playing ﬁeld of numbers. However, it is known that the mass of the neutralino is most generally and roughly between 100 GeV and

10 TeV.

3.2.2 Universal Extra Dimensions

Rather than introducing new particles to explain the diﬀerent fundamental force strengths, extra dimensions can be added, and the forces allowed to propagate into them. In the original Kaluza-Klein extra-dimensional scenario, only gravity could reach the other dimensions, diluting its strength and explaining why it would be much weaker than the weak force. In the Universal Extra Dimension (UED) scenario, all the forces propagate. Stefano Profumo and Dan Hooper have recently compiled a comprehensive review for UED dark matter [24].

These additional dimensions are hidden, wrapped upon themselves. When they are compactiﬁed in this manner, they are accompanied by a quantization of the

momenta of all the ﬁelds of the forces, creating a number of energy states. This is

43 analogous to the quantum mechanical particle-in-a-box, and the energy states can be interpreted as particles. The UED analogue of R-parity, called KK-parity, also allows for a lightest stable particle through each odd-level KK state having odd parity. This

Lightest Kaluza-Klein Particle (LKP) is the ﬁrst excitation of the KK photon, or, again, more correctly that of hypercharge gauge boson B, and so it is symbolized as B(1). There are considerably fewer parameters for UED than SUSY, and another important diﬀerence is that the LKP is a boson, unlike the fermionic LSP. However, like the LSP, the LKP possesses a mass around 1 TeV.

3.2.3 Other Viable Candidates

While SUSY and UED dark matter can be thermal relic candidates, that preferred quality is not a requirement. An example of a non-thermal relic candidate is the axion, ﬁrst put forth by Peccei and Quinn [25]. The axion is motivated to solve the “strong CP” problem, the main kink in the otherwise robust theory of Quantum

Chromodynamics (QCD). It is very interesting as a dark matter candidate because it undergoes the Primakoff process, and that participation might allow for its discovery [26, 27, 28]. Through this mechanism, the axion can be converted into a photon in the presence of a magnetic field, and vice versa. A main search for the axion involves the conversion of photons to axions inside the Sun due to its strong inner magnetic fields, and the subsequent reconversion to X-ray photons in magnetic fields outside the Sun. The axions would be very light, about 10 µeV, but yet they would be nonrelativistic and therefore cold [4, 8].

Light dark matter has been proposed recently to explain the 511 keV excess of gamma rays near the Galactic Center (e.g. [29, 30]). While the Lee-Weinberg bound

44 prohibits weakly interacting fermionic dark matter less massive than about 1 GeV, scalar particles (bosons) might exist instead [31]. Even more exotic particles have been proposed, some with no motivation except as a potential dark matter solution. These other options include Q-balls (balls of quarks and therefore not point particles), self- interacting dark matter (possessing a non-negligible interaction between dark matter particles aside from annihilation)), WIMPzillas (superheavy WIMPs), superweakly- interacting massive particles (extremely weak WIMPs), sterile neutrinos (another neutrino that does not interact with the known three, and is very massive), and more [4, 8].

The purpose of mentioning the various candidates in this dissertation is to alert the reader to the existence of particle-types other than the most-often-studied SUSY and UED models. There is quite a zoo of possible particles. As the actual nature of dark matter is uncertain, one must keep an open mind as to the true particle identity, including whether it be a thermal relic or non-thermally produced. Therefore, it makes sense to analyze possible general properties of this exotic matter, and use the results to provide support for or against various candidates. For this main reason, and others, the dark matter investigations presented in the later chapters of this dissertation do not assume a speciﬁc theoretical model’s dark matter candidate.

3.3 How Can We Find Dark Matter?

Dark matter is elusive, but the hope is that it does interact in some other capacity than just the gravitational force. If so, there are ways that this connection to normal matter could reveal dark matter’s existence. These are: (i) direct detection, (ii) indirect detection, and (iii) collider production. The third method will not be addressed

45 in this dissertation, but it is possible that dark matter could be created in particles

accelerators of great enough energy and appear as “missing energy” after a reaction.

The soon-to-be-operating Large Hadron Collider (LHC) holds this possibility, as it

will be the highest energy accelerator constructed to date [8, 32].

3.3.1 Direct Detection: The Slightest Touch

Dark matter may have a ﬁnite probability to hit (scatter with) and transfer energy to a normal matter nucleus. The interaction rate is:

Γ n σ v , (3.5) int ∝ χ χtarget χ where nχ and vχ are the number density and velocity of the dark matter respectively, and σχtarget is the interaction cross section of the dark matter-target interaction. A

larger cross section yields a greater annihilation strength, and a higher probability for interaction.

In the current thinking of a particle that is weakly interacting, the appropriate

experiment to detect such an infrequent event transferring very little energy is a detector located underground consisting of nuclei that have very little energy themselves. The placement underground reduces a great deal of noise produced by the cosmic ray background, and cryogenically cooling the nuclei allows a smaller transfer of energy to be noticed. The region of parameter space in the dark matter-normal matter interaction cross section (σχN ) and dark matter mass (mχ) plane covered by these detectors is a ﬁnite region. The bottom is limited by the weakest interaction

(smallest transfer of energy) that could result in a detection [8]. Most publications of the experimentalists only show the left and bottom edges; the neglect of the others is due to the lack of belief of a dark matter candidate with values corresponding to

46 the top and right edges. Personal communication with experimentalists associated

with diﬀerent detectors revealed that some had not even considered the idea of an

upper limit to their exclusion region. The right edge is governed by the size of the

detector and the dark matter number density [33]. An increase in the dark matter

mass means a decrease in the dark matter number density, since the mass density is

ﬁxed. At a certain high value of the mass, there is not enough dark matter around

to be observable in the lifetime of the experiment. The top edge is set by too much

interaction [34]. For a large enough interaction strength, the dark matter will lose too

much energy before it reaches the detector nuclei, due to scattering in the atmophere

or ﬁrst few layers of rock. Such a dark matter particle will not register in the detec-

tor, and so the region is truncated at those interaction strengths. This basic region is

shown graphically in Chapter 4 in Fig. 4.1. Below, a few of the most prominent dark matter direct detectors are described.

CDMS

The second incarnation of the Cryogenic Dark Matter Search (CDMS II) is located

in the Soudan mine in Minnesota; the prior conception was at Stanford University, in

a tunnel underneath the campus [35]. It consists of bolometric detectors monitoring

cryogenically-cooled germanium and silicon. A dark matter particle that hits these

crystal lattices will create vibrations. These vibrations propagate as “phonons”, and

when they encounter the aluminum in the detector, the energy transfers to electrons,

which will then transfer their electrical energy to a piece of tungsten needing only a

little more energy to transition out of a superconducting state. The resulting change

in the resistance of the tungsten can then be recorded, allowing for the calculation of

the amount of energy it encountered. Since both the phonons and the charge can be

47 detected, this helps in ruling out backgrounds; outside charged particles registering on the detector would not have corresponding phonons. The sensitive nature of the experiment allows the probing of dark matter on the weak scale, with null results

establishing a constraint.

EDELWEISS

EDELWEISS (Experience pour DEtecter Les Wimps En Site Souterrain) is also in its second generation, and is located in the Laboratoire Souterrain de Modane in France’s Frejus mountain, meaning that it is eﬀectively covered by 1.6 km of rock [36]. EDELWEISS is very similar to CDMS in its conception, consisting of bolometric detectors with cryogenic monocrystals of germanium. EDELWEISS-II is more sensitive than the initial installment, and the experimental group is also looking at the use of scintillation to improve the sensitivity for the next generation.

CRESST

An experiment located 1.4 km underground in a laboratory at Gran Sasso in

Italy takes advantage of the physics of scintillation to aid in even greater reduction of backgrounds [37]. CRESST (Cryogenic Rare Event Search with Superconducting

Thermometers) has detectors made of calcium tungstate (CaWO4), which is a scin-

tillator. In a scintillating material, an incoming particle can ionize an atom, and the

resulting electron will emit a photon when it returns to its lower energy state. This

photon then shows itself as luminesence of the material. Simply put, particles that are

not dark matter will make the material glow, while dark matter will not. Detecting

this light means that the event was a background event, allowing for a much more

sensitive analysis.

48 DAMA

The only detector claiming to have registered and measured a dark matter event is the DArk MAtter experiment (DAMA), which also utilizes scintillation and the location of Gran Sasso [38]. One version of the detector employs sodium iodide (NaI), and

another uses liquid xenon (LXe), also. The NaI version claimed to have a positive de-

tection of dark matter in 2003, and claim recently to have veriﬁed their ﬁndings [39].

They appear to see a seasonal modulation signal in the dark matter interaction rate.

The Sun moves around the center of the Milky Way and Earth moves around the

Sun. Consequently, a trough in the interaction signal should be when Earth’s orbit

has it traveling in the same direction as the Sun around the Galactic Center, with a

corresponding peak when Earth’s orbit is against the Sun’s. Other experiments have

not found this, and in fact rule out the parameter space where the positive detec-

tion exists. A drawback of this experiment is that it can only measure counts, and

therefore it is possible there is some other background that is not understood. More

analysis is needed.

Other detection experiments such as XENON [40] and ZEPLIN [41] exist, and

all are trying to access weaker and weaker interaction cross sections. They hope to

either have a positive detection or continue to use the null results to exclude parameter

space. As has been stressed earlier, it is important to explore all possible values of

the parameter space, not just the ones present in preferred theories. To this end, the

exploration in this dissertation involves cross sections of somewhat stronger values

than those relevant for underground detectors. In this case, it is indeed shown in

Chapter 4 that the dark matter interaction strength should be weak and therefore

49 the detectors are searching the correct region of parameter space. The interaction cross section of main concern in this dissertation is the spin-independent variety. This assumes that the dark matter interacts with the whole nucleus, and not the valence spin of a target, which has its own spin-dependent limit. Most nuclei in Earth have a net spin of zero. More details of the interaction cross section are discussed in Chapter

4 (see references therein).

3.3.2 Indirect Detection: Measuring Dark Matter’s Disap- pearance

If dark matter can interact with normal matter, then there also must be the possibility that two dark matter particles can annihilate into normal matter particles.

Dark matter is most often assumed to be its own antiparticle, and here that preference is taken. In SUSY and UED, the antiparticle of the LSP and LKP are by deﬁnition the same as the particle. Such a case deserves the term “Majorana” [8, 19]. If this were not assumed, then the dark matter density must be comparable to the anti-dark matter density, or at the very least there must not be a large discrepancy between the two. The possibility of annihilation into normal matter particles can be visualized as a simple leg-crossing of the Feynman diagram that describes interaction. The annihilation rate per volume is:

Γ n2 σ v . (3.6) A ∝ χh A i

Since the annihilation depends on two dark matter particles coming together, two powers of the dark matter density are needed. If the dark matter were not its own

2 antiparticle, then the nχ would be replaced by 2 nχnχ¯ in more precise equations. Most often, the annihilation cross section is paired with the root-mean-square velocity of

50 the dark matter (the total units of which are cm3/s), and constraints are placed on this combination, not just the pure cross section. This is because, in general, the annihilation cross section is not energy independent. The factor σ v can be h A i expanded in powers of the velocity, and interpreted as a partial-wave expansion [8, 19]:

σ v a + bv2 + ... (3.7) h A i ≈

The a term corresponds to s-wave annihilation, while b includes both s-wave and p-wave annihilations. Low energy scenarios typically involve s-wave annihilations.

Dark matter indirect detection is concerned with the observation of signals from dark matter annihilation. Since the annihilation depends on the dark matter density, areas with enhanced concentrations of dark matter are good regions to observe. Areas of interest include the Galactic Center and external galaxies such as Andromeda [4,

8, 19]. Dark matter will annihilate into diﬀerent channels with diﬀerent probabilities, all of which must sum to unity. These are called branching ratios. In fact,

σ v = Br σ v ; (3.8) h A ii ih A itotal

ΣiBri = 1. (3.9)

By looking for the signal of one of the possible annihilation channels, one can constrain not just the annihilation rate into that speciﬁc channel, but also the total annihilation cross section as a function of the branching ratio to that channel. This translates into a limit on just the total annihilation cross section if the branching ratio were known.

The basic principle involves comparing the theoretical annihilation signal from a given region of dark matter concentration to the observed ﬂux of the relevant annihilation product as measured by detectors. In Chapters 5 and 6, the considered annihilation products are neutrinos and gamma rays, respectively, and methods to

51 calculate theoretical signals are presented. Details about individual gamma-ray detectors (INTEGRAL, COMPTEL, EGRET, CELESTE, HEGRA and H.E.S.S.) are found in Chapter 6. While often the annihilation cross section is assumed to be

3 10−26 cm3/s, since that strength is what reproduces the measured Ω h2 of 0.11 × χ if dark matter is produced in thermal equilibrium in the early universe (See Chapter

7), the actual value has not been tightly constrained. Chapters 5 and 6 of this disser-

tation deal with this issue, however, placing new constraints using information about

neutrinos and photons (see references therein for more detailed citations).

3.3.3 Unitarity: An Existing Annihilation Constraint

The annihilation cross section does have a pre-existing mathematical contraint,

one that is derived from quantum mechanics. This mathematical limit of the interac-

tion between two particles comes from the fact that the S-matrix is unitary, meaning

SS† = 1. The S-matrix (Scattering matrix) is a mathematical object that helps to

relate the initial and ﬁnal states of particle interaction. Its relation to an elastic or

inelastic cross section is through the T-matrix (Transfer matrix). While the S-matrix

describes the scattering of, for example, two particles in general, the details of the

reaction are in the T-matrix. For two dark matter particles, inelastic scattering is

annihilation, as it would be for any particle-antiparticle pair. The unitarity of the

S-matrix means that the total probabilities for elastic and inelastic scattering sum to

1 – the process must be one or the other. The limit does not depend on the exact

nature of the ﬁnal state, so the bound is generic [42, 43].

In general, the S-matrix for two particles a and b in initial state i scattering to

ﬁnal state f (which we will assume is also made of two particles, c and d) is related

52 to the T-matrix by [42]:

f S i = f i + i(2π)4δ4(P P ) f T i . (3.10) h | | i h | i f − i h | | i

The 4-dimensional delta function enforces the conservation of energy and momentum for this process, as represented by the 4-vector P . In elastic scattering, the final state equals the initial state, whereas in annihilation/inelastic scattering, the final state is different from the initial state.

The unitarity condition S†S = SS† = 1 allows the following derivation:

(1 S)†(1 S) = 1 S S† + S†S − − − − = (1 S†) + (1 S) (3.11) − −

As seen from Eq. (3.10), the T-matrix is (with multiplicative factors) equal to 1 - S, and so Eq. (3.11) is important to the behavior of the T-matrix.

The T-matrix can be expanded as a series of partial waves, employing Legendre

polynomials and the quantization of total angular momentum J [44], with a T-matrix

between states i and f with angular momentum J represented as (TJ )if . This requires

the explicit use of p˜, a diagonal matrix of the 3-dimensional momentum of the initial

and ﬁnal states, and means the S-matrix can be expanded in terms of partial waves

as well. In this formalism, T T † = 2i T p˜ T †, which follows from the deﬁnition J − J J J 1/2 1/2 SJ = 1 + 2i p˜ TJ p˜ [42]. Here is how they are related via Eq. (3.11):

1 S = 2i p˜1/2 T p˜1/2 ; 1 S† = 2i p˜1/2 T † p˜1/2 (3.12) − J − J − J J (1 S ) + (1 S)† = (1 S)(1 S)† − J − − − 2i p˜1/2 T T † p˜1/2 = 4 p˜1/2 T p˜1/2p˜1/2 T †p˜1/2 − J − J J J T T † = 2i T p˜ T †. (3.13) J − J J J 53 Note: since p˜ is a diagonal real matrix, p˜† = p˜. For the case of inelastic scattering, this implies the relation 1 Tf=6 i = Sf=6 i 1/2 1/2 . (3.14) 2i pi pf † Due to the unitarity of the partial wave S-matrix (SJ SJ = 1), the amplitudes of the scattering matrices can also be written, as alluded to earlier, as [42]

S 2 + S 2 = 1, (3.15) | f=i, J | | f=6 i, J | Xf or

1 S 2 = S 2 = 1 A2 , (3.16) − | f=i, J | | f=6 i, J | − J Xf 2 representing the square of the elastic scattering amplitude as AJ , a positive real number between 0 and 1.

Since both the S-matrix and the T-matrix can be expanded as partial waves in

J, so can the cross section: σ = ΣJ σJ . In the standard quantum mechanical/particle physics fashion, the partial wave cross section is related to the T-matrix by

4π(2J + 1) pf 2 σJ = Tif, J , (3.17) (2sa + 1)(2sb + 1) pi | | Xλ Xf where sa and sb are the spins of the incoming particles, and the sum over λ is a sum

over the helicities of the outgoing states. A particle’s helicity state refers to whether its spin momentum is parallel or antiparallel to its 3-momentum p. Employing Eqs. (3.14) and (3.16) for annihilations, where i = f, the cross section expression simpliﬁes to 6 2 4π(2J + 1) p S σ = f inel, J inel, J (2s + 1)(2s + 1) p 1/2 1/2 a b i 2i pi p Xλ Xf f 2 π(2J + 1) 1 AJ = 2 − . (3.18) pi (2sa + 1)(2sb + 1)

54 2 An upper bound can be obtained by setting AJ = 0, yielding π(2J + 1) σJ 2 . (3.19) ≤ pi (2sa + 1)(2sb + 1)

The upper bound is largest for the case sa = sb = 0 (possible values of spins are

2 2 2 0, whole-integers, and half-integers). The relevant pi is approximately mχv , or

2 2 mχ vrel/4, meaning 4π(2J + 1) σinel, J 2 2 (3.20) ≤ mχ vrel 4π(2J + 1) or σinel, J vrel 2 . (3.21) ≤ mχ vrel In terms of relevant quantities, in unitless form, this constraint on the annihilation cross section is [43]:

GeV 2 300 km/s σ v 1.5 10−13 (2J + 1) cm3/s . (3.22) J rel ≤ × m v χ rel As mentioned earlier, the average velocity of dark matter in a galactic halo is around

270 km/s. Which J’s contribute signiﬁcantly to the cross section? Above, it is stated

that the partial waves are expanded in terms of Legendre polynomials, and therefore

the angular dependence of the inital system. The J = 0 partial wave corresponds

to no angular dependence; any angular dependence includes an additional velocity

2 J dependence of the form (vrel/4) . This is the same situation displayed in Eq. (3.7). In the low velocity limit, such as in galactic halos, the J = 0 term has the greatest

contribution [42, 43]. The J = 0 term refers to s-wave annihilation. The resulting

upper limit is

GeV 2 300 km/s σ v 1.5 10−13 cm3/s . (3.23) J rel ≤ × m v χ rel This constraint is from quantum mechanics, concerns the annihilation of two particles to an undetermined ﬁnal state in the low velocity limit, does not depend on the nature

55 of the products, and therefore is a very general constraint for dark matter annihilation [42, 43]. As will be seen in the following chapters, general methods to explore dark matter’s properties can be very constraining and well worth investigating.

56 CHAPTER 4

TOWARDS CLOSING THE WINDOW ON STRONGLY INTERACTING DARK MATTER: FAR-REACHING CONSTRAINTS FROM EARTH’S HEAT FLOW

This investigation into the dark matter – normal matter interaction cross section was done with John Beacom and Gianfranco Bertone, and has been published in the journal Physics Review D [45]. I present a slightly modiﬁed version here, to be consistent with this dissertation. We point out a new and largely model-independent constraint on the dark matter scattering cross section with nucleons, applying when this quantity is larger than for typical weakly interacting dark matter candidates.

When the dark matter capture rate in Earth is eﬃcient, the rate of energy deposition by dark matter self-annihilation products would grossly exceed the measured heat

ﬂow of Earth. This improves the spin-independent cross section constraints by many

orders of magnitude, and closes the window between astrophysical constraints (at very

large cross sections) and underground detector constraints (at small cross sections).

In the applicable mass range, from 1 to 1010 GeV, the scattering cross section ∼ ∼ of dark matter with nucleons is then bounded from above by the latter constraints,

and hence must be truly weak, as usually assumed.

57 4.1 Introduction

While weakly interacting dark matter is the preferred idea, it remains unproven, and so it is important to systematically test the properties of dark matter particles using only late-universe constraints. In 1990, Starkman, Gould, Esmailzadeh, and

Dimopoulos [46] examined the possibility of strongly interacting dark matter, noting that it indeed had not been ruled out. Many authors since have explored further constraints and candidates. In this literature, “strongly interacting” denotes cross sections signiﬁcantly larger than those of the weak interactions; it does not necessarily mean via the usual strong interactions between hadrons. We generally consider the constraints in the plane of dark matter mass mχ and spin-independent scattering cross section with nucleons σχN .

Figure 4.1 summarizes astrophysical, high-altitude balloon/rocket/satellite detector, and underground detector constraints in the σχN –mχ plane. Astrophysical limits such as the stability of the Milky Way disk constrain very large cross sections [46, 47].

Accompanying and comparable limits include those from cosmic rays and the cosmic microwave background [48, 49]. Small cross sections are probed by CDMS and other underground detectors [33, 50, 51, 52, 53, 54]. A dark matter particle can be directly detected if σχN is strong enough to cause a measurable nuclear recoil in the detector, but only if it is weak enough to allow the dark matter to pass through Earth to the detector.

58 -10

-15 Cosmic Rays MW redone SKYLAB -20 ] MW Disk Disruption 2 IMP 7/8 IMAX

-25 XQC [cm N χ

σ -30 RRS

log Underground Detectors -35

-40

-45 0 5 10 15 20 log mχ [GeV]

Figure 4.1: Excluded regions in the σχN –mχ plane, not yet including the results of this dissertation. From top to bottom, these come from astrophysical constraints (dark-shaded) [46, 47, 48, 49], re-analyses of high-altitude detectors (medium-shaded) [46, 55, 56, 57], and underground direct dark matter detectors (light-shaded) [33, 50, 51, 52, 53, 54]. The dark matter number density scales as 1/mχ, and the scattering rates as σχN /mχ; for a ﬁxed scattering rate, the required cross section then scales as mχ. We will develop a constraint from Earth heating by dark matter annihilation to more deﬁnitively exclude the window between the astrophysical and underground constraints [45].

59 In between the astrophysical and underground limits is the window in which σχN

can be relatively large [46]. High-altitude detectors in and above the atmosphere have been used to exclude moderate-to-strong values of the cross section in this region

[46, 55, 56, 57]. However, there are still large gaps not excluded. There also is some doubt associated with these exclusions, as some of the experiments were not speciﬁcally designed to look for dark matter, nor were they always analyzed for this

purpose by people associated with the projects. In fact, the exclusion from the X-

ray Quantum Calorimetry experiment was recently reanalyzed [57] and it changed

substantially from earlier estimates [56]. If this intermediate region can be closed,

then underground detectors would set the upper limit on σχN . That would mean that

these detectors are generally looking in the right cross section range and that dark

matter-nucleon scattering interactions are indeed totally irrelevant in astrophysics.

We investigate cross sections between the astrophysical and underground limits,

and show that σχN is large enough for Earth to eﬃciently capture dark matter.

Incoming dark matter will scatter oﬀ nucleons, lose energy, and become gravitationally

captured once below Earth’s escape velocity (Section 4.4). If this capture is maximally

eﬃcient, the rate is 2 1025 (GeV/m ) s−1. The gravitationally-captured dark matter × χ will drift to the bottom of the potential well, Earth’s core. Self-annihilation results

if the dark matter is its own antiparticle, and we assume Standard Model ﬁnal state

particles so that these products will deposit nearly all their energy in the core.

Inside a region in the σχN –mχ plane that will be deﬁned, too much heat would be

produced relative to the actual measured value of Earth’s heat ﬂow. The maximal

heating rate obtained via macroscopic considerations is 3330 TeraWatts (TW), ' which follows from the maximal capture rate and the assumption that Earth is opaque

60 with a geometric cross section. Note that the ﬂux of dark matter scales as 1/mχ,

while the heat energy from annihilations scales as mχ, yielding a heat ﬂow that is independent of dark matter mass. The eﬃcient capture we consider leads to a very

similar heating rate, though it is based on a realistic calculation of microscopic dark

matter-nucleon scattering, as discussed below. Dark matter interactions with Earth

have been previously studied in great detail, e.g., Refs [46, 58, 59, 60, 61, 62, 63,

64, 65, 66], but those investigations generally considered only weak cross sections for

which capture is ineﬃcient.

In our analysis, the σχN exclusion region arises from the captured dark matter’s

self-annihilation energy exceeding Earth’s internal heat ﬂow. This region is limited

below by the eﬃcient capture of dark matter (Section 4.4), and above by being weak

enough to allow suﬃcient time for the dark matter to drift to the core (Section 4.5).

These two limits deﬁne the region in which dark matter heating occurs. Why is it

important? Earth’s received solar energy is large, about 170, 000 TW [67], but it is all

reﬂected or re-radiated. The internal heat ﬂow is much less, about 44 TW (Section

4.3) [68]. Inside this bounded region for σχN , dark matter heating would exceed the

measured rate by about two orders of magnitude, and therefore is not allowed. We

will show that this appears to close the window noted above in Fig. 4.1, up to about

m 1010 GeV. In order to be certain of this, however, we call for new analyses of χ ' the aforementioned constraints, especially the exact region excluded by CDMS and

other underground detectors. Our emphasis is not on further debate of the details

of speciﬁc open gaps, but rather on providing a new and independent constraint. In

Table 4.1, we summarize the heat values relevant to this paper. While the origin of

61 Heat Source Heating Rate Solar (received and returned) 170,000 TW Internal (measured) 44.2 1 TW DM annihilation (opaque Earth) 3330 TW DM annihilation (our assumptions) 3260 TW DM kinetic heating 3000 10−6 TW ∼ ×

Table 4.1: Relevant heat ﬂow values. The top entries are measured, while the lower entries are the calculated potential eﬀects of dark matter (DM) [45, 67, 68].

Earth’s heat ﬂow is not completely understood, we emphasize that we are not trying to account for any portion of it with heating from dark matter.

There has been some previous work on the heating of planets by dark matter annihilation [66, 69, 70, 71, 72, 73, 74, 75]. These papers have mostly focused on the

Jovian planets, for which the internal heat flow values are deduced from their infrared radiation [76]. In some cases [66, 69, 70, 71, 72], dark matter annihilation was invoked to explain the anomalously large heat flow values of Jupiter and Saturn, while in other cases [73, 74, 75], the low heat flow value of Uranus was used to constrain dark matter annihilation. An additional reason for the focus on these large planets is that they will be able to stop dark matter particles of smaller cross section than Earth can

(Ref. [72] considered Earth, but invoked an extreme dark matter clumping factor to overcome the weakly interacting cross section). However, as we argue below in

Section 4.6, the more relevant criterion is how signiﬁcant of an excess heat ﬂow could be produced by dark matter annihilation, and this is much more favorable for Earth.

(If this criterion is met, then the ranges of excluded cross sections will simply shift for diﬀerent planets.) Furthermore, the detailed knowledge of Earth’s properties gives

62 much more robust results. In this paper, we are presenting the ﬁrst detailed and systematic study of the broad exclusion region in the σχN –mχ plane that is based on not overheating Earth.

Our constraints depend on dark matter being its own antiparticle, so that annihilation may occur (or, if it is not, that the dark matter-antidark matter asymmetry not be too large). This is a mild and common assumption. The heating due to kinetic energy transfer is negligible. Since the dark matter velocity is 10−3c, kinetic ' heating is 10−6 that from annihilation, and would provide no constraint (Section ∼ 4.4). The model-independent nature of our annihilation constraints arises from the nearly complete insensitivity to which Standard Model particles are produced in the dark matter annihilations, and at what energies. All ﬁnal states except neutrinos will deposit all of their energy in Earth’s core. (Above about 100 TeV, neutrinos will, too.) Since the possible heating rate (> 3000 TW) is so large compared to the measured rate ( 40 TW), in eﬀect we only require that not more than 99% of the ∼ ∼ energy goes into low-energy neutrinos, which is an extremely modest assumption.

Some of the annihilation products will likely be neutrinos, and these may initiate signals in neutrino detectors, e.g., as upward-going muons [46, 77, 78, 79, 80, 81,

82, 83, 84]. While the derived cross section limits can be constraining, they strongly depend on the branching ratio to neutrinos and the neutrino energies. Comprehensive constraints based on neutrino ﬂuxes for the full range of dark matter masses appear to be unavailable; most papers have concentrated on the 1–1000 GeV range, and a few have considered masses above 108 GeV. We note that the constraints for dark matter masses above about 1010 GeV may require annihilation cross sections above

63 the unitarity bound, as discussed below. As this paper is meant to be a model-

independent, direct approach to dark matter properties based on the dark matter

density alone, we do not include these neutrino constraints.

We review the current dark matter constraints in Section 4.2, discuss Earth’s heat

ﬂow in Section 4.3, calculate the dark matter capture, annihilation, and heating rates in Sections 4.4 and 4.5, and close with discussions and conclusions in Section 4.6.

4.2 Review of Prior Constraints

Figure 4.1 shows the current constraints in the σχN –mχ plane. As we will show, the derived exclusion region found by the requirement of not overheating Earth using dark matter annihilation lies in the uncertain intermediate area between the astrophysical and underground constraints.

4.2.1 Indirect Astrophysical Constraints

If σχN were too large, dark matter particles in a galactic halo would scatter too frequently with the normal matter disk of a spiral galaxy, and would signiﬁcantly disrupt it. Using the integrity of the Milky Way disk, Starkman et al. [46] restrict the cross section to σ < 5 10−24(m /GeV) cm2. A more detailed study by Natarajan χN × χ et al. [47] requires σ < 5 10−25(m /GeV) cm2. Both of these limits consider χN × χ dark matter scattering only with hydrogen. As shown below in Eq. (4.12), the spin-

4 independent dark matter-nucleon cross section scales as A for large mχ, and though the number density of helium (A = 4) is about 10 times less than that of hydrogen

(A = 1), taking it into account could improve these constraints by 256/10 25. ' ' Chivukula et al. [85] showed that charged dark matter could be limited through its

64 ionizing eﬀects on interstellar clouds; this technique could be adapted for strongly

interacting dark matter.

Strong scattering of dark matter and normal matter would also affect the cosmic microwave background radiation. Adding stronger dark matter-normal matter interactions increases the viscosity of the normal matter-photon fluid [48]. A strong coupling of normal matter and dark matter would generate denser clumps of gravitationally-interacting matter, and the photons would not be able to push them as far apart. The peaks in the cosmic microwave background power spectrum would be damped, with the exception of the first one. The resulting constraint is

σ < 3 10−24(m /GeV) cm2 [48], and is not shown in Fig. 4.1. These results χN × χ do take helium into account, but do so only using A2 instead of A4. This possible

change, along with the much more precise cosmic microwave background radiation

data available currently, calls for a detailed re-analysis of this limit, which should

strengthen it.

Cosmic ray protons interact inelastically with interstellar protons, breaking the protons and creating neutral pions that decay to high-energy gamma rays. A similar situation could occur with a cosmic ray beam on dark matter targets instead [49]. The fundamental interaction is between the quarks in the nucleon and the dark matter; it

is very unlikely that all quarks will be struck equally, and the subsequent destruction

of the nucleon creates pions. If the dark matter-nucleon cross section were high

enough, the resulting gamma rays would be readily detectable. From this, Cyburt et

al. [49] place an upper limit of σ < 7.6 10−27(m /GeV) cm2. Improvements could χN × χ probably be made easily with a more realistic treatment of the gamma-ray data.

65 4.2.2 Direct Detection Constraints

Underground detector experiments have played a large role in limiting dark matter that can elastically scatter nuclei, giving the nuclei small but measurable kinetic energies. Due to the cosmic ray background, this type of detector is located underground. The usual weakly interacting dark matter candidates easily pass through the atmosphere and Earth en route to the detector. However, for large σχN the dark matter would lose energy through scattering before reaching the detector, decreasing detection rates.

Albuquerque and Baudis [33] have explored constraints at relatively large cross sections and large masses using results from CDMS and EDELWEISS. In Fig. 4.1, we present a crude estimate of the current underground detector exclusion region. The top line is deﬁned by the ability of a dark matter particle to make it through the atmosphere [34] and Earth to the detector without losing too much energy [33]. The

lower left corner and nearby points are taken from the oﬃcial CDMS papers [50, 51, 52]

with the aid of their website [86]. The right edge is taken from DAMA [54]. As the

mass of the dark matter increases, the number density (and hence the ﬂux through

Earth) decreases. At the largest mχ values, the scattering rate within a ﬁnite time

vanishes. Finally, we have extrapolated each of these constraints to meet each other, connecting them consistently. We call for a complete and oﬃcial analysis of the exact region that CDMS and other direct detectors exclude. Our focus is on the cross sections in between the underground detectors and astrophysical limits.

To investigate cross sections in this middle range, direct detectors must be situated above Earth’s atmosphere, in high-altitude balloons, rockets, or satellites. Several

such detectors have been analyzed for this purpose, though they were not all originally

66 intended to study dark matter. Since these large σχN limits have in some cases been calculated by people not connected with the original experiments, some caution is required. Nevertheless, in Fig. 4.1 we show the claimed exclusion regions, following

Starkman et al. [46] and Rich et al. [55], along with Wandelt et al. [56] and Erickcek et al. [57] (including the primary references [87, 88, 89, 90]). We are primarily in accordance with Erickcek et al. These regions span masses of almost 0.1 GeV to 1016

GeV, and cross sections between roughly 10−33 cm2 and 10−11 cm2. These include the

Pioneer 11 spacecraft and Skylab, the IMP 7/8 cosmic ray silicon detector satellite, the X-ray Quantum Calorimetry experiment (XQC), and the balloon-borne IMAX.

These regions are likely ruled out, but not in absolute certainty, and there are gaps between them. The Pioneer 11 region is completely covered by the IMP 7/8 and

XQC regions, and is therefore not shown in Fig. 4.1. The region labeled RRS is Rich et al.’s analysis of a silicon semiconductor detector near the top of the atmosphere, truncated according to Starkman et al., and adjusted with the appropriate A-scaling as in Eq. (4.12).

4.3 Earth’s Heat Flow

Heat from the Sun warms Earth, but it is not retained. If all the incident sunlight were absorbed by Earth, the heating rate would be about 170,000 TW [67]. Some of it is reﬂected by the atmosphere, clouds, and surface, and the rest is absorbed at depths very close to the surface and then re-radiated [76]. Earth’s blackbody temperature would be about 280 K, and it is observed to be between 250 and 300 K, supporting the idea of Earth-Sun heat equilibrium. Internal heating therefore has minimal eﬀects on the overall heat of Earth [76].

67 Our focus is on this internal heat ﬂow of Earth, as measured underground. Ge-

ologists have extensively studied Earth’s internal heat for decades [91]. To make a

measurement, a borehole is drilled kilometers deep into the ground. The temperature

gradient in that borehole is recorded, and that quantity multiplied by the thermal

conductivity of the relevant material yields a heat ﬂux [91, 92].

The deepest borehole is about 12 kilometers, which is still rather close to Earth’s

surface. Typical temperature gradients are between 10 and 50 K/km, but these cannot

hold for lower depths. If they did, all rock in the deeper parts of Earth would be molten, in contradiction to seismic measurements, which show that shear waves can

propagate through the mantle [92]. Current estimates place temperature gradients deep inside Earth between 0.6 and 0.8 K/km [92].

More than 20,000 borehole measurements have been made over Earth’s surface.

Averaging over the continents and oceans, there is a heat flux of 0.087 0.002 W/m2 [68, 91]. Integrating this flux over the surface of Earth gives a heat flow of

44.2 1 TW [68, 91]. Again, the heat ﬂux is directly measured underground, all over Earth, and is independent of the solar ﬂux, Earth’s atmosphere, and anything else

above Earth’s surface. Obviously, the possibility to make direct heat ﬂow measure-

ments under the surface is unique to Earth.

While the heat ﬂow value is known well, the origin of the heat is not, and in fact

is undergoing much theoretical debate [93, 94]. Some speciﬁc contributors are known,

however. The decay of radioactive elements produces a signiﬁcant amount; uranium

and thorium decay in the crust generates about forty percent of the total [68]. Potas-

sium adds to this, though there is much less of it in the crust. However, there is

68 potentially a large amount in the mantle and perhaps even the outer core [68]. Kam-

LAND has a hint of detected neutrinos coming from uranium and thorium decays [95], and it (along with other detectors) could potentially help to make the heat contribu-

tion from them more accurate [96, 97, 98, 99, 100, 101, 102, 103, 104, 105]. Larger concentrations of uranium and thorium are excluded by KamLAND, and theoretical

predictions from the Bulk Silicate Earth model are consistent with the forty percent

value [96, 97, 98]. The remaining heat is due to processes in the core and perhaps

even the mantle, although speciﬁc knowledge of Earth’s interior is limited [106].

The residual heat ﬂow, which we assume to be 20 TW [93, 94], we use as the

target limit for the heat ﬂow from dark matter annihilation. Models give values

of the core’s heat output between 2.3 TW and 21 TW, supporting the conservative choice of 20 TW [106]. Annihilation scenarios creating heat ﬂows greater than 20 TW

are therefore excluded. In fact, if heating by dark matter annihilation is important at all, we show that it typically would exceed this value by more than two orders of magnitude. It is important to note that we are not trying to solve geological heat problems with dark matter, and in fact our analysis implies it is very unlikely that dark matter is contributing to Earth’s internal heat ﬂow, which is interesting in itself.

4.4 Dark Matter Capture Rate of Earth

If a dark matter particle scatters a suﬃcient number of times while passing through

Earth, its speed will fall below the surface escape speed, 11.2 km/s. Having therefore been gravitationally captured, it will orbit the center of Earth, losing energy with each subsequent scattering until it settles into a thermal distribution in equilibrium

with the nuclei in the core. For the usual weak cross sections Earth is eﬀectively

69 transparent, and scattering and capture are very inefficient. In contrast, we will consider only large cross sections for which capture is almost fully efficient. Note that for our purposes, the scattering history is irrelevant as long as capture occurs; in particular, the depth in the atmosphere or Earth of the first scattering has no bearing on the results. The energies of the individual struck nuclei are also irrelevant, unlike in direct detection experiments. We just require that the dark matter is captured and ultimately annihilated. We remind the readers of the relevant dark matter mass density ρ = n m 0.3 GeV/cm3 and the average dark matter particle speed of χ χ χ ' 270 km/s. A cartoon of the situation is shown in Figure 4.2.

4.4.1 Maximum Capture Rate

We begin by considering the maximum possible capture rate of dark matter in

Earth, which corresponds to Earth being totally opaque. Although our ﬁnal calculations will involve the microscopic scattering cross section of dark matter on nuclei, this initial example deals with just the macroscopic geometric cross section of Earth.

The ﬂux per solid angle of dark matter near Earth is nχvχ/4π, where nχ is the dark matter number density, and vχ is the average dark matter velocity. Since Earth is taken to be opaque, the solid angle acceptance at each point on the surface is 2π sr. Thus the ﬂux at Earth’s surface is nχvχ/2. The capture rate is then found by multiplying by Earth’s geometric cross section, σ = 4πR2 5.1 1018 cm2. Since ⊕ ⊕ ' × nχ is not known, this is (ρχ/mχ)σ⊕vχ. For vχ = 270 km/s, this maximal capture rate is GeV Γmax = 2 1025 s−1. (4.1) C × m χ

70 Figure 4.2: A graphic illustration of the assumed scattering scenario. The dark matter will scatter oﬀ particles in Earth, become gravitationally captured, and drift down to the center where it will be annihilated by other dark matter captured in the same fashion. The resulting heat will propagate out of the core, which can be compared to the measured internal heat ﬂow of Earth.

71 We will show that our results depend only logarithmically on the dark matter velocity, and hence are insensitive to the details of the velocity distribution.

This maximal capture rate estimate is too simplistic, as it assumes that merely coming into contact with Earth, interacting with any thickness, will result in dark matter capture. Instead, we deﬁne opaqueness to be limited to path lengths greater than 0.2 R⊕, a value that incorporates the largest 90% of path lengths through

Earth. This reduces the capture rate, but only by about 2%. We therefore adopt

0.2 R⊕ as our minimum thickness to determine efficient scattering. This length, translated into a chord going through the spherical Earth, defines the new effective area for Earth. The midpoint of the chord lies at a distance of 0.99R⊕ from Earth’s center. Thus, practically speaking, nearly all dark matter passing through Earth will encounter sufficient material. The above requirements exclude glancing trajectories from consideration, for which there would be some probability of reflection from the atmosphere [73, 75]; note also that the exclusion region in Section 4.4 would be unaffected by taking this into account, since the dark matter heating of Earth would still be excessive.

The type of nucleus with which dark matter scatters depends on its initial trajectory through Earth. For a minimum path length of 0.2 R⊕, this trajectory runs through the crust, where the density is 3.6 g/cm3 [107], and the most abundant element is oxygen [60]. Choosing this path length and density are conservative steps.

Any larger path length would result in more eﬃcient capture, and a higher density and heavier composition (corresponding to a larger chord and therefore a diﬀerent target nucleus, such as iron, which is the most abundant element in the core) would as well. A more complex crust or mantle composition, such as 30% oxygen, 15% silicon,

72 14% magnesium and smaller contributions from other elements [60], would stop dark

matter 2 times more eﬀectively. ∼ 4.4.2 Dark Matter Scattering on Nuclei

When a dark matter particle (at v 10−3c) elastically scatters with a nu- χ ' cleus/nucleon (at rest) in Earth, it decreases in energy and velocity. After one scat-

tering with a nucleus of mass mA, dark matter with mass mχ and initial velocity vi

will have a new velocity of

v m m f = 1 2 A χ (1 cos θ ) , (4.2) v − (m + m )2 − cm i r χ A mA 1 2 (1 cos θcm). (4.3) mχ−→mA − m − r χ

All quantities are in the lab frame, except the recoil angle, θcm, which is most usefully

deﬁned in the center of mass frame (see Landau and Lifschitz [108]). Here and below we give the large mχ limit for demonstration purposes, but we use the full forms of

the equations for our results. After scattering, the dark matter has a new kinetic

energy,

χ KEf = 1 m m m v2 1 2 A χ (1 cos θ ) (4.4) 2 χ i − (m + m )2 − cm χ A χ mA KEi 1 2 (1 cos θcm) . (4.5) mχ−→mA − m − χ

73 The nucleus then obtains a kinetic energy of

KEA = KE χ KE χ i − f 1 m m = m v2 1 1 + 2 A χ (1 cos θ ) (4.6) 2 χ i − (m + m )2 − cm χ A χ mA KEi 2 (1 cos θcm) (4.7) mχ−→mA mχ − = m v2(1 cos θ ) (4.8) A i − cm

From the kinetic energy, the momentum transfer in the large mχ limit is:

2 ~q 2 KE = | | = mAvi (1 cos θcm) (4.9) 2mA − ~q 2 = 2(m v )2(1 cos θ ). (4.10) | | A i − cm

In order to maintain consistency with others, we work with n and σ in nucleon units even though the target we choose (oxygen) is a nucleus. This means that nA (where A represents the mass number of the target) is

n ρ nA = = . (4.11) A mN A

In turn, the cross section for spin-independent s-wave elastic scattering is represented as

µ(A) 2 σ = A2 σ (4.12) χA µ(N) χN 4 A σχN . mχ−→mA

Here A is the mass number of the target nucleus, which equals mA/mN , and

µ(A or N) is the reduced mass of the dark matter particle and the target.

The A2 factor arises because at these low momentum transfers, the nucleus is not resolved and the dark matter is assumed to couple coherently to the net “charge” –

74 the number of nucleons. (If this coherence is somehow lost, a factor A would still remain for incoherent scattering.) The momentum transfer q = √2m KEA m v A ' A i corresponds to a length scale of 10 fm for oxygen, much larger than the nucleus. ' We ﬁnd that the corresponding nuclear form factor when the dark matter mass is

comparable to the target mass is 0.99. The square of the reduced mass ratio ' arises from the Born approximation for scattering, which is based on the two-particle

Schr¨odinger equation cast as a single particle with relative coordinates and reduced

mass [109]. The spin-dependent scattering cross section does not have the A2 factor

in Eq. (4.12) [46]. Our constraints could be scaled to represent this case by also taking

into account the relative abundance of target nuclei with nonzero spin in Earth, which

is of order 1%.

Note that if mχ = mA, and θcm = π, the dark matter can transfer all of its

momentum to the struck nucleus, losing all of its energy in a single scattering [60].

Taking this into account would make our constraints stronger over a small range of

1 2 masses, but we neglect it. The nuclear recoil energy from this resonance is then 2 mχvi .

Since vi is on average 270 km/s, this means that the maximum energy transferred

from a collision is 10−6 that of the annihilation energy, m c2. ∼ χ 4.4.3 Dark Matter Capture Eﬃciency

From the full or approximate form of Eq. (4.4), we see that the dark matter

kinetic energy is decreased by a multiplicative factor that is linear in cos θcm. If, in

each independent scattering, we average over cos θcm, the average factor by which the

kinetic energy is reduced in one or many scatterings will simply be that obtained

75 by setting cos θcm = 0 throughout. (For s-wave scattering, the cos θcm distribution is uniform.)

We will deﬁne eﬃcient capture so that the heating is maximized. To be gravitationally trapped, a dark matter particle must be below the escape speed of Earth (vesc

1 2 = 11.2 km/s), or equivalently, its kinetic energy must be less than 2 mχvesc. After one scattering event, the dark matter kinetic energy is reduced:

χ χ KEf = KEi f(mχ). (4.13)

In successive collisions, this is compounded until

1 1 m v 2 = m v 2 [f(m )]Nscat . (4.14) 2 χ esc 2 χ i χ

Note that for collinear scatterings, the velocity loss in Eq. (4.2) is also speed loss, leading to the same deﬁnition of Nscat.

Therefore, on average, the number of scatterings required to gravitationally cap-

ture the dark matter is

2 ln (vi/vesc) Nscat = − (4.15) mAmχ ln 1 2 2 − (mχ+mA) mhχ i ln (vi/vesc), (4.16) mχ−→mA mA where we have set cos θcm = 0, since this corresponds to the average fractional change

in the kinetic energy. Again, for simplicity the same element is taken to be the target each time. Note that since the initial dark matter velocity is inside the logarithm,

Nscat is insensitive to even large changes in the assumed initial velocity.

The number of scatterings for a given mass is large. A dark matter particle that

has the same mass as the target nucleus will scatter about 10 times before it is

76 captured. Note that the required Nscat scales as mχ in the large mass limit, becoming

3 very large: for mχ above 16 TeV (10 times the target mass), Nscat is already larger

than 3000. The actual energy losses in individual collisions are irrelevant for our analysis, as we require only that the dark matter is captured after many collisions.

For large values of Nscat, all scattering histories will be well-characterized by the average case.

So far, these equations have just been kinematics; the required Nscat for stopping

has not yet been made speciﬁc to Earth. It becomes Earth-speciﬁc by relating Nscat

to the path length in Earth L and the mean free path λ:

L N = = Ln σ . (4.17) scat λ A χA

The column density of Earth then deﬁnes the required cross section to generate Nscat

scatterings. The shortest path the particle could travel in is a straight line, so we use that as the minimum. Any other path would be longer, and hence more eﬀective at

capture. This therefore deﬁnes the most conservative limit on σχA. Since we have

ﬁxed cos θcm to be 0 on average, in fact the path will not be completely straight.

However, the lab frame scattering angles are small.

For elastic collisions between two particles, the range of scattering angles in the

lab frame depends on the two masses, m1 and m2. There is a maximum scattering

angle when one mass is initially at rest in the lab frame (in this case, m2) [108]. If

m1 < m2, there is no restriction on the scattering angle, which is deﬁned in relation

to m1’s initial direction (m2 is at rest). However, if m1 > m2, then

max sin θlab = m2/m1. (4.18)

77 Our main focus is m1 = mχ > m2 = mA. For mχ somewhat greater than mA, note that the dark matter scattering angle in the lab frame is always very forward.

Combining Eqns. (4.12), (4.15), and (4.17), the minimum required cross section to capture a dark matter particle is

2 min mN σχN = 2 Nscat(mχ), (4.19) µ(A) mA µ(N) ρL 2 ln (v /v ) m2 = − i esc N , 2 mAmχ µ(A) ln 1 2 2 (mχ+m ) m ρL − A A µ(N) h 4 i mN 1 mχ ln (vi/vesc). mχ−→mA m ρL A

Again, we choose a path length of 0.2 R⊕, to select about 90% of the possible lengths through Earth. Taking this length as a chord through Earth, the location corresponds to the crust, with an average density of 3.6 g/cm3, where the most common element is oxygen. We also choose an incoming dark matter velocity of 500 km/s, which eﬀectively selects the entire thermal distribution. A slower dark matter particle is more easily captured. These parameters give a required cross section of 2 1.8 10−33 cm2 µ(1) min − × µ(16) σχN = (4.20) 16 GeVmχ ln 1 2 2 − (mχ+16 GeV) h i mχ 2.2 10−37 cm2 . (4.21) mχ−→mA × GeV Note that we use the unapproximated version, Eq. (4.20), for our ﬁgure, and give the

large mχ limit in the equations for demonstrative purposes. When mχ is comparable

min to mA, the functional form of σχN is signiﬁcantly, and importantly, diﬀerent from the

approximated, large mχ case.

min The resulting curve for σχN is shown in Fig. 4.3, as the lower boundary of the heavily-shaded exclusion region. The straight section of this constraint is easily seen

78 from Eq. (4.21), as the required cross section for our eﬃcient capture scenario scales as mχ, due to the large number of collisions required for stopping, as in Eq. (4.16).

At lower masses, the curved portion has its minimum at the mass of the target. Dark matter masses close to that of the target can be captured with smaller cross sections because a greater kinetic energy transfer can occur for each collision. At very low masses, much less than the mass of the target, the dark matter mass dependence in the logarithm is approximated diﬀerently. In this limit, σ is 10−32 (GeV/m ). χN ' χ As the dark matter mass decreases, it becomes increasingly more diﬃcult for the

dark matter to lose energy when it strikes a nucleus. As noted above, the cross section constraints in the spin-dependent case could be developed, and would shift the results up by 3 or 4 orders of magnitude. All of the other limits that depend on

this A2 coherence factor would also shift accordingly.

79 -10

-15

-20 ] 2 -25 [cm N χ

σ -30 log -35

-40

-45 0 5 10 15 20 log mχ [GeV]

Figure 4.3: Inside the heavily-shaded region, dark matter annihilations would over- heat Earth. Below the top edge of this region, dark matter can drift to Earth’s core in a satisfactory time. Above the bottom edge, the capture rate in Earth is nearly fully eﬃcient, leading to a heating rate of 3260 TW (above the dashed line, capture is only eﬃcient enough to lead to a heating rate of & 20 TW). The mass ranges are described in the text, and the light-shaded regions are as in Fig. 4.1 [45].

80 The lower edge of the exclusion region is generally rather sharp, because of these parameters. For example, consider the case of large mχ, where Nscat is also large.

If the corresponding cross section is decreased by a factor δ, so is the number of scatterings, and by Eq. (4.14), the compounded fractional kinetic energy loss would only be the 1/δ root of that required for capture. For small cross sections, as usually

considered, the capture eﬃciency is very low. To eﬃciently produce heat, the mini-

mum cross section must result in 90% dark matter capture. We stress again that ∼ we are not concerned with where the dark matter is captured in Earth, so long as it

is. The probability for capture can, however, be decreased using Poisson statistics

(shown in Fig. 4.3 as the dashed line with the accentuated dip at low masses) to yield

just 20 TW of heat ﬂow. This extension and the upper edge of the exclusion region

are described below.

4.5 Dark Matter Annihilation and Heating Rates in Earth

4.5.1 Maximal Annihilation and Heating Rates

Once it is gravitationally captured, dark matter will continue to scatter with nuclei

in Earth, losing energy until drifting to the core. Once there, because of the large

cross section, the dark matter will thermalize with the nuclei in the core. The number

of dark matter particles N is governed by the relation between the capture (ΓC) and

annihilation (ΓA ) rates [110]:

1 2 1 2 Γ = AN = ΓC tanh (t ΓCA). (4.22) A 2 2 p We neglect the possibility of evaporation [59] for the moment, which will aﬀect our

results for low mχ and σχN , as we will explain further below. The variable t is the age of the system. A is related to the dark matter self-annihilation cross section σχχ

81 by

< σχχv > A = , (4.23) Veff where Veff is the effective volume of the system [110]. For the relevant cross sections considered, equilibrium between capture and annihilation is generally reached (see below), so the annihilation rate is

1 Γ = Γ (4.24) A 2 C

The eﬀective volume is determined by the method of Griest and Seckel (1988) [110], which is essentially the volume of the dark matter distribution in the core. The

number density of dark matter is assumed to be an exponentially decaying function,

exp( m φ/kT ), like the Boltzmann distribution of molecules in the atmosphere. The − χ temperature of the dark matter in thermal equilibrium is T . The variable φ is the

gravitational potential, integrated out to a radius r, written as

r GM(r˜) φ(r) = 2 dr˜; (4.25) 0 r˜ Z r˜ M(r˜) = 4π r02ρ(r0)dr0. (4.26) Z0 The resulting eﬀective volume using the radius of Earth’s outer core, approximately

3 0.4R⊕, a temperature of 5000 K, and a density of 9 g/cm [107], is

Rcore m φ 2 − χ Veﬀ = 4π r e kT dr. (4.27) 0 Z 3 mχ 2 0.5√ 100 GeV GeV 2 = 1.2 1025 cm3 u2e−u du × mχ 0 Z 3 100 GeV 2 5.3 1024 cm3 . (4.28) mχ−→mA × m χ

For increasing mχ, the integral (without the prefactor) in the second line of Eq. (4.27)

quickly reaches an asymptotic value of about 0.44.

82 At very large masses, the eﬀective volume for annihilation becomes very small. For

10 instance, at mχ & 10 GeV, the radius of the eﬀective volume is . 0.1 km. With such

a large rate of energy injected in such a small volume, the core temperature would

be increased, requiring a more careful treatment. However, in Section 4.5.2, we state

how a limit on the annihilation cross section from the unitarity condition [42, 43, 111]

10 makes us truncate our bound at mχ = 10 GeV, as reﬂected in Fig. 4.3. Therefore,

this small eﬀective volume is not a large concern for our exclusion region.

We assume that the dark matter annihilates into primarily Standard Model par-

ticles, which will deposit nearly all of their energy into Earth’s core (with small cor-

rections due to particle rest masses and the escape of low-energy neutrinos). When

all of the dark matter captured is eﬃciently annihilated, as speciﬁed, the heating rate

of Earth is in equilibrium with the capture rate:

Γ = Γ m = n σ v m heat C × χ χ eﬀ χ χ ρχ 2 = 2π(0.99R⊕) (270 km/s) mχ (4.29) mχ = 3260 TW.

This heat ﬂow is independent of dark matter mass, since the ﬂux (and capture rate,

when capture is eﬃcient) scales as 1/mχ, while each dark matter particle gives up mχ in heat when it annihilates. The value is much larger than the measured rate of

44 TW we discussed in Section 4.3.

4.5.2 Equilibrium Requirements

Does the timescale of Earth allow for equilibrium between capture and annihilation

2 in our scenario? In order for Eq. (4.22) to be in the equilibrium limit, tanh (t√ΓCA)

must be of order unity. This is true if t√ΓC A has a value of a few or greater. Since

83 Earth is about 4.5 Gyr old, we conservatively require that the time taken to reach equilibrium should be less than about 1 Gyr. From this, a realistic annihilation cross section is found. The condition

2 tanh (t ΓCA) = 1; t ΓCA few (4.30) ' p p allows the relation

< σ v > (few)2 χχ & = A 2 (4.31) Veﬀ ΓC t V & eﬀ < σχχv > 10 2 . (4.32) ΓCt

For an eﬃcient capture rate (Eq. 4.1), the time of 1 Gyr, and the limit of large mχ,

this requires an annihilation cross section for equilibrium of

GeV 1/2 < σ v > & 10−30 cm3/s. (4.33) χχ m χ Since this required lower bound is much smaller than that of typical weakly interacting

dark matter particles that are thermal relics (< σ v > 10−26 cm3/s [8]), it should χχ ' be easily met. One expects large scattering cross sections to be accompanied by large

annihilation cross sections, so that even the possibility of p-wave-only suppression of

the annihilation rate should not be a problem.

For very large masses, the required annihilation cross section, while small, ap-

10 proaches a quantum mechanical limit (See Section 3.3.3). For example, for mχ & 10

GeV the s-wave cross section exceeds the unitarity bound [42, 43, 111]. We note that

this may also aﬀect constraints on supermassive dark matter based on neutrinos from

annihilations [77, 78, 79]. To be conservative, we therefore do not extend our con-

straints beyond this point, though they may still be valid.

84 The timescale also has to be long enough for dark matter to drift down to the core.

If σχN is too large, the dark matter will experience too many scatterings and will not

settle into the core, and thus may not annihilate eﬃciently. Following Starkman et

al. [46], we deﬁne the upper edge of our exclusion region to require a drift time of .

1 Gyr. This places a restriction of

(m /GeV) σ . 7.7 10−20cm2 χ χN × A2(µ(A)/µ(N))2

−20 2 mχ/GeV . 7.7 10 cm 2 × A4 mχ+mN mχ+mA −23 2 mχ σχN . 2.5 10 cm , (4.34) −→ × GeV mχmA for a target of iron. However, a more detailed calculation might relax this requirement.

For example, Starkman et al. [46] show that for large values of the capture cross section

and certain other conditions, annihilation may be eﬃcient enough to occur in a shell,

before the dark matter reaches the core. This would generally still be subject to our

constraint on heat from dark matter annihilation, and hence our exclusion region

might extend to larger cross sections than shown. The two features of this drift line

at low mass occur around the mass of the target and the mass of a nucleon, due to

the various dominances of the mass-dependent term in the denominator. The details

of the shape of this drift line at low masses are irrelevant, because the astrophysical

constraints already exclude the corresponding regions.

Aside from drifting to the core, the question of whether heavy dark matter can

actually get to Earth has been asked [112, 113]. The low-velocity tail of the high- mass dark matter thermal distribution in the Solar System may be driven into the Sun by gravitational capture processes [112, 113], especially because this dark matter’s

velocity is on the order of the orbital speed of Earth in the Solar System, which is

85 about 30 km/s. However, this would aﬀect only a tiny fraction of the full thermal

distribution that we require to be eﬃciently captured.

4.5.3 Annihilation and Heating Eﬃciencies

We are not picking a speciﬁc model for the annihilation products, aside from

considering only Standard Model particles, which will deposit their energy in Earth, with the exception of neutrinos. Our constraint thus has a very broad applicability.

As noted the calculated heat ﬂow if dark matter annihilation is important is 3260

TW, which is very large compared to our adopted limit on an unconventional source

of 20 TW (or even the whole measured rate of 44 TW). Typically then, either dark

matter annihilation heating is overwhelming or it is negligible, inside or outside of the

excluded region, respectively. As shown in Section 4.4, the kinetic energy transferred

from dark matter scattering on nuclei is about 6 orders of magnitude less than the

energy from dark matter annihilations. This contribution to Earth’s heat is too low

to be relevant for global considerations. However, it would be interesting to consider

the more localized eﬀect of the kinetic energy deposition in the atmosphere for very

large cross sections.

There are circumstances in which the heating from dark matter annihilations can

take a more intermediate value, including down to the chosen 20 TW number. As ex-

plained above, typically the number of scatterings required to gravitationally capture

the dark matter is very large. Therefore, a small decrease in σχN and the propor-

tionate change in the expected number of scatterings means that the compounded

kinetic energy loss is nearly always insuﬃcient. However, at low mχ, the number is small enough that upward ﬂuctuations relative to the expected number can lead to

86 min capture. If Nscat collisions typically lead to efficient capture for a cross section σχN , as defined above, a new and smaller N may be defined by the condition that the

Poisson probability Prob(N N ) = 20 TW / 3260 TW = 1/163. With this N, ≥ scat

and its proportionately smaller σχN , upward Poisson ﬂuctuations in the number of

scatterings lead to eﬃcient capture for a fraction 1/163 of the incoming ﬂux. Note

that this small capture fraction is not just the low-velocity tail of the dark matter

thermal distribution, since we have deﬁned these conditions for the highest incoming

velocities, vi = 500 km/s.

The resulting constraint on σχN is shown by the dashed line that dips below the

main excluded region in Fig. 4.3. The enhanced valley around 16 GeV again arises

from the ease of capture when the dark matter mass is near the target mass. Note that

for each mass the required < σχχv > is increased by the same factor that decreased

min the original required σχN . Since most of this exclusion region is already covered by

underground detectors, its details may not be so important.

For low dark matter masses, evaporative losses of dark matter from the core

due to upscattering by energetic iron nuclei may be relevant [59]. Simple kinematic

estimates show that dark matter masses below 5 GeV might be aﬀected. However, '

this is only potentially important if σχN is small enough that scattering is very rare

– since otherwise any upscattered dark matter will immediately downscatter. From

the considerations above about Poisson ﬂuctuations in the number of scatterings,

we expect that this should only be relevant between the dark-shaded region and the

dashed line.

87 4.6 Discussion and Conclusions

4.6.1 Principal Results

As summarized in Fig. 4.1, while very large dark matter-nucleon scattering cross

sections are excluded by astrophysical considerations, and small cross sections are

excluded by underground direct dark matter detection experiments, there is a sub-

stantial window in between that has proven very diﬃcult to test, despite much ef-

fort [46, 55, 56, 57, 59, 60, 61, 62, 65, 66, 71, 73, 77, 80, 81, 82, 87, 88, 89, 90, 114, 115].

High-altitude experiments have excluded only parts of this window. In this window,

dark matter will be eﬃciently captured by Earth. We point out that the subsequent self-annihilations of dark matter in Earth’s core would lead to an enormous heating rate of 3260 TW, compared to the geologically measured value of 44 TW.

We show that the conditions for eﬃcient capture, annihilation, and heating are all quite generally met, leading to an exclusion of σχN over about ten orders of

magnitude, which closes the window on strongly interacting dark matter between

the astrophysical and direct detection constraints. These new constraints apply over

a very large mass range, as shown in Fig. 4.3. We have been quite conservative, and so

very likely an even larger region is excluded. These results establish that dark matter

interactions with nucleons are bounded from above by the underground experiments,

and therefore that these interactions must be truly weak, as commonly assumed.

This means that direct detection experiments are looking in the correct σχN range

when sited underground and motivates further theoretical study of weakly interacting dark matter [116, 117, 118, 119]. Furthermore, it means that dark matter-nucleon

scattering cannot have any measurable eﬀects in astrophysics and cosmology, and

this has many implications for models with strongly or moderately interacting dark

88 matter [75, 114, 115, 120, 121] and other astrophysical constraints on the dark matter- nucleon interaction cross section [122, 123]. This exclusion region also completely

covers the cross section range in which strongly interacting dark matter might bind

to nuclei [124, 125, 126, 127].

To evade our constraints, extreme assumptions would be required: that dark mat-

ter is not its own antiparticle, or that there is a large ( & 163) particle-antiparticle

asymmetry of dark matter, or that dark matter self-annihilations proceed only to

purely sterile non-Standard Model particles, at the level of & 100:1. (Although Chap-

ter 5 features such a large branching ratio to neutrinos, it is emphasized that this

is used only to set the most conservative bound on the dark matter annihilation

cross section, and not to be representative of a realistic model, as will be shown.)

While our constraints are based on Earth’s measured heat ﬂow, it is important to

emphasize that dark matter capture and annihilation generally cannot contribute

measurably without being overwhelming, and hence are excluded. Thus, in the ongo-

ing debate over the unknown sources of Earth’s heat ﬂow, it seems that dark matter

can play no role. The most important next step in reﬁning our understanding of

the known generators of the measured 44 TW will come from isolating the contri-

bution from uranium and thorium decays by measuring the corresponding neutrino

ﬂuxes [95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105].

4.6.2 Comparison to Other Planets

Starkman et al. [46] calculated the eﬃcient Earth capture line for dark matter,

but only had model-dependent results. We have now considered the consequences of

annihilation in Earth, and have shown that it gives a model-independent constraint.

89 Other planets have been discussed, such as Jupiter and Uranus [66, 69, 70, 71, 72,

73, 74, 75], but Earth is the best laboratory. It is the best understood planet, with internal heat ﬂow data measured directly underground from many locations, and

Earth’s composition and density proﬁle are well known [68, 91, 92, 93, 94, 106, 107].

Importantly, the relative excess heat due to dark matter annihilation would be much greater for Earth than the Jovian planets.

What about other planets? The maximal heating rate due to dark matter scales with surface area, and can be compared with the internal heating rates estimated from

min infrared data [76]. If a constraint can be set, the minimum cross section σχN that can be probed scales with the planet’s column density as (nL)−1 (see Eq. (4.17)) up to nontrivial corrections for composition (see Eq. (4.12)). Note that the column density nL is proportional to the surface gravity GM/R2, which varies little between the ∼ planets, as noted in Table 4.2 [76]. Due to its known (and heavy) composition and well-measured (and low) internal heat, the strongest and most reliable constraints will be obtained considering Earth. As an interesting aside, it may then be unlikely that heating by dark matter could play a signiﬁcant role in explaining the apparent overheating of some extra-solar planets (“hot Jupiters”) [128, 129, 130, 131, 132].

4.6.3 Future Directions

While it appears that the window on strongly interacting dark matter is now closed over a huge mass range, more detailed analyses are needed in order to be absolutely certain. Our calculations are conservative, and the true excluded region is likely to be larger. It would be especially valuable to have new analyses of the astrophysical limits and the underground detector constraints. This would give greater certainty that no

90 Planet DM Max. Heating Internal Heat Surface Gravity (TW) (TW) (units of ) ⊕ Earth 3.3 103 0.044 103 1.00 × × Jupiter 420 103 400 103 2.74 × × Saturn 290 103 200 103 1.17 × × Uranus 53 103 103 0.94 × ≤ Neptune 50 103 3 103 1.15 × ×

Table 4.2: Comparison of potential dark matter constraints using various planets [76]. A greater diﬀerence between dark matter and internal heating rates give greater certainty. The minimum cross section probed scales roughly with the surface gravity. Earth is the best for setting reliable and strong constraints.

sliver of the window is still open. For dark matter masses in the range 1 GeV to 1010

GeV, the upper limits on the dark matter scattering cross section with nucleons from

CDMS and other underground experiments have been shown to be true upper limits.

Thus the dark matter does indeed appear to be very weakly interacting, and it will be challenging to detect it.

91 CHAPTER 5

UPPER BOUND ON THE DARK MATTER TOTAL ANNIHILATION CROSS SECTION

This first installment in a two-part series of investigations into the dark matter self- annihilation cross section was done with John Beacom and Nicole Bell, and published in the journal Physical Review Letters [111]. As with the research in Chapter 4, it has been modified slightly to be coherent with this dissertation. We consider dark matter annihilation into Standard Model particles and show that the least detectable final states, namely neutrinos, define an upper bound on the total cross section. Calculat- ing the cosmic diffuse neutrino signal, and comparing it to the measured terrestrial atmospheric neutrino background, we derive a strong and general bound. This can be evaded if the annihilation products are dominantly new and truly invisible particles.

Our bound is much stronger than the unitarity bound at the most interesting masses, shows that dark matter halos cannot be signiﬁcantly modiﬁed by annihilations, and can be improved by a factor of 10–100 with existing neutrino experiments.

5.1 Introduction

The self-annihilation cross section is a fundamental property of dark matter, as discussed in Chapter 3. For thermal relics, it sets the dark matter mass density,

92 Ω 0.3, and in these and more general non-thermal scenarios, also the annihi- DM ∼ lation rate in gravitationally-collapsed dark matter halos today [8]. How large can the dark matter annihilation cross section be? There are two general constraints that bound the rate of dark matter disappearance. (Throughout, we mean the cross section averaged over the halo velocity distribution, i.e., σ v , where v 10−3c.) h A i rms ∼ The ﬁrst is the unitarity bound, developed for the early universe case by Griest and Kamionkowski [42], and for the late-universe halo case by Hui [43] (see Section

3.3.3). In the plane of σ v and dark matter mass m , this allows only the region h A i χ below a line σ v 1/m2 (this will be made more precise below). The second h A i ∼ χ is provided by the model of Kaplinghat, Knox, and Turner (KKT) [133], in which

signiﬁcant dark matter annihilation is invoked to resolve a conﬂict between predicted

(sharp cusps) and observed (ﬂat cores) halo proﬁles. Since this tension may have

been relaxed [8], we reinterpret this type of model as an upper bound, allowing only

the region below a line σ v m . That the KKT model requires σ v values h A i ∼ χ h A i & 107 times larger than the natural scale for a thermal relic highlights the weakness

of the unitarity bound in the interesting GeV range. However, there have been no

other strong and general bounds to improve upon these.

While these bound the disappearance rate of dark matter, they say nothing about

the appearance rate of annihilation products, instead assuming that they can be made

undetectable. To evade astrophysical limits, the branching ratios to speciﬁc ﬁnal

states can be adjusted in model-dependent ways. However, a model-independent fact

is that the branching ratios for all ﬁnal states must sum to 100%. A reasonable

assumption is that these ﬁnal states are Standard Model (SM) particles. We show

that the most diﬃcult SM ﬁnal state to detect is neutrinos; but that surprisingly

93 strong ﬂux limits can be simply derived from recent high-statistics data; and that we

may interpret these as bounding all SM ﬁnal states, and hence the dark matter total

annihilation cross section. See Fig. 5.1.

If dark matter is not its own antiparticle and if there is a large particle-antiparticle

asymmetry, then annihilation could be prohibited, making all bounds inapplicable or

irrelevant. Our bound can be evaded if the ﬁnal states are dominantly new and truly invisible non-SM particles, in which case all dark matter annihilation searches will

be more challenging; we quantify an upper bound on the branching ratio to SM ﬁnal

states below.

94 χ “Visible” states:  + −  All Standard γγ, qq, e e , ...    Model = +  ﬁnal states  “Invisible” states:   χ   νν

Figure 5.1: Annihilation of dark matter into SM ﬁnal states. Since all ﬁnal states except neutrinos produce gamma rays (see text), we can bound the total cross section from the neutrino signal limit, i.e., assuming Br(Invis.) 100% [111]. '

95 5.2 Probing Dark Matter Disappearance

For dark matter that is a thermal relic, the cross section required to ensure

Ω 0.3 is σ v 3 10−26 cm3 s−1 [8]. KKT discussed several models in DM ∼ h A i ∼ × which the dark matter is not a thermal relic, e.g., it might have acquired mass only

in the late universe, or have been produced through the late decays of heavier par-

ticles [133]. As emphasized in Refs. [43, 133], it is interesting to directly ask how

large the annihilation cross section could be in halos today, irrespective of possible

early-universe constraints.

Unitarity sets a general upper bound on σ v , and can only be evaded in certain h A i unusual cases [42, 43, 134]. In the low-velocity limit where the cross section is assumed

to be s-wave dominated, σ v 4π/m2 v, or h A i ≤ χ cm3 GeV 2 300 km/s σ v 1.5 10−13 . (5.1) h A i ≤ × s m v χ rms In the KKT model, the required cross section to suﬃciently distort the dark matter

proﬁles of galaxies is

cm3 m σ v 3 10−19 χ . (5.2) h A iKKT ' × s GeV h i (Similar eﬀects are attained via elastic interactions [135]. A large self-annihilation cross section implies a large elastic self-scattering cross section, but not vice-versa [43].) Hui [43] showed that unitarity restricts the KKT model to relatively small masses; for vrms = 300 km/s, mχ . 80 GeV. There have been no other model-independent methods to constrain the KKT model. We argue next that dark matter disappearance must be accompanied by the appearance of something, and the bound on the weakest ﬁnal state bounds all of them. The appearance rate bounds

σ v directly, independent of which partial waves dominate σ , i.e., its v-dependence. h A i A 96 5.3 Revealing Neutrino Appearance

We assume that annihilation proceeds to SM particles, and express the cross section in terms of branching ratios to “visible” and “invisible” ﬁnal states, such as

gamma rays and neutrinos, respectively, as in Fig. 5.1. If the branching ratio to a

speciﬁc ﬁnal state were known, then a bound on that appearance rate would yield

a bound on the total cross section, inversely proportional to this branching ratio.

However, the branching ratios are model-dependent, and any speciﬁc one can be

made very small, making that bound on σ v very weak, e.g., for m = 1 GeV, KKT h A i χ require Br(γ) . 10−10 to allow their total cross section [133]. Note that gamma-ray

data constrain only the product σ v Br(γγ), and the bounds vary with m . This h A i χ will be discussed in greater detail in Chapter 6.

KKT [133] and Hui [43] assume invisible but unspeciﬁed ﬁnal states. It is clear that

most SM ﬁnal states produce gamma rays. Quarks and gluons hadronize, producing

pions, where π0 γγ; the decays of weak bosons and tau leptons also produce → π0. The stable ﬁnal state e+e− is not invisible, since it produces gamma rays either

through electromagnetic radiative corrections [136] or energy loss processes [137]; the

ﬁnal state µ+µ− produces e+e− by its decays. Thus the only possible “invisible” SM

ﬁnal states are neutrinos. This will discussed again and in greater detail in Chapter

6, but it is important to realize that the ﬁnal products of monoenergetic photons and

monoenergetic neutrinos can be thought of as opposite extremes for the detection of

dark matter annihilations, as most sensitive and least sensitive, respectively.

Of ﬁnal states with neutrinos, we focus on ν¯ν. Similar bounds could be derived

for ν¯ν¯νν, but we assume that these are suppressed and/or that the Rube Goldberg-

ish Feynman diagrams required would contain charged particles, and hence gamma

97 rays through (model-dependent) radiative corrections. Even though we essentially treat the appearance of monoenergetic neutrinos as devoid of gamma-rays, it must be said that due to electroweak bremsstrahlung, ﬁnal-state neutrinos are inevitably accompanied by weak bosons and hence gamma rays, primarily with E m /2; γ ' π however, these gamma-ray constraints on σ v are weaker than or comparable to h A i what we obtain directly with neutrinos [138, 139, 140]. Therefore, we are still valid in treating the limit from neutrinos as a case apart from gamma-rays.

To derive our bound on the total annihilation cross section, we assume Br(ν¯ν) ' 100%. This is not an assumption about realistic outcomes, but it is the right way to derive the most conservative upper bound for SM final states. Why is this a bound on the total cross section, and not just on the partial cross section to neutrinos? Suppose that Br(ν¯ν) were reduced enough that the 1/Br(ν¯ν) correction for an impure final state were necessary; at our factor-two precision, this occurs when another SM final state has a comparable branching ratio. For the total cross section set by the neutrino bound, any other pure final state would be more strongly constrained, thus making this cross section disallowed for all final states in the SM. Therefore, while setting this bound using neutrinos can be too conservative, it can never overreach.

5.4 Cosmic Diﬀuse Neutrinos: Signal

The most direct approach to bound the χχ ν¯ν cross section is to use the cosmic → diffuse neutrino flux from dark matter annihilations in all halos in the universe as the signal. Since this is isotropic and time-independent, it is challenging to detect above the background caused by the atmospheric neutrino flux. A complementary approach uses the Milky Way signals, which have somewhat different uncertainties

98 on the predictions and data [23]. The data to test the diﬀuse signal are available

in the full energy range now, but this is not yet true for all the directional signals.

While the latter will likely be stronger eventually, going beyond our rough estimates

will require proper experimental analyses.

The cosmic diﬀuse signal from χχ ν¯ν annihilations depends on the radial → density proﬁle of each dark matter halo, the halo mass function (the relative weighting

of halos of diﬀerent masses), and how those halos evolve with redshift. We follow the

calculations of Ullio et al. [141, 142]; see also [143, 144, 145, 146]. The signal spectrum

is dΦ σ v c Ω2 ρ2 zup ∆2(z) dN (E0) ν = h A i DM crit dz ν , (5.3) dE 2 4πH m2 h(z) dE0 0 χ Z0 −1 −1 where the 1/2 is for assuming χ is its own antiparticle, H0 = 100 h km s Mpc is the Hubble parameter, and ΩDM is the dark matter density in units of the critical density ρcrit. We assume a ﬂat universe, with ΩDM = 0.3, ΩΛ = 0.7, h = 0.7, and

3 1/2 h(z) = [(1 + z) ΩDM + ΩΛ] .

The factor ∆2(z) accounts for the increase in density due to the clustering of dark matter in halos, defined so that ∆2 = 1 corresponds to all dark matter being at its average density in the universe today. The concentration of halos and thus the value of ∆2 evolves with redshift. (Note that we have absorbed a factor of (1 + z)3 into the definition of ∆2, as in Ref. [142].) However, to collect most of the signal, we only need neutrino energies near mχ, and hence will be sensitive only to modest redshifts where it is accurate to take ∆2(z) ∆2(0) [141]. Note that only ∆2 matters, and ' not its individual factors. The value of ∆2 does depend on the halo profile chosen.

We adopt ∆2 = 2 105 for Navarro, Frenk, and White (NFW) halos with moderate × assumptions about the halo mass distribution. For cuspier Moore proﬁles, ∆2 could

99 be 10 times larger, while for ﬂatter Kravtsov proﬁles, it could be 2 times smaller; ' ' see the discussions in Ref. [23].

For χχ ν¯ν, the source spectrum dN /dE0 is a delta function; neutrinos pro- → ν duced with energy E0 are redshifted to the observed energy E = E0/(1 + z), i.e.,

dN (E0) 2 2 m ν = δ(m E0) = δ z χ 1 , (5.4) dE0 3 χ − 3E − E − h i where we have accounted for 2 neutrinos per annihilation, equally divided among 3

ﬂavors. (Note that νµ has a large fraction in every neutrino mass eigenstate, so any initial mix of mass or ﬂavor eigenstates would be close to this.) In Fig. 5.2, we show example dark matter signals compared to the atmospheric neutrino background.

100 ]

-1 0 sr

-1 -2 s

-2 -4 -6 -8 /dE) [cm

Φ -10 -12 log (E d 10 assumed data

5 Ratio

0 -1 0 1 2 3 4 5 6 log E [GeV]

3 5 Figure 5.2: Upper: Diﬀuse ν¯ν annihilation signal for mχ = 10, 10 , and 10 GeV, added to the atmospheric background, both as (ν¯µ + νµ) and versus neutrino energy. As noted, the signals are most accurate for Eν & mχ/3. Lower: Ratio of this sum and background. The σAv values at each example mχ are chosen to be detectable by our conservative criteria;h thei data and assumed uncertainty scales are also indicated [111].

101 Using the neutrino signal, we can also derive a constraint from the relativistic energy density. Requiring Ω < 0.2 at low redshift [147] leads to σ v < 10−17 rad h A i × 3 (mχ/GeV) cm /s. While this bound applies to any light ﬁnal state including non-SM

particles such as purely sterile neutrinos, it is weak, and would require even greater halo modiﬁcations than the KKT model.

5.5 Cosmic Diﬀuse Neutrinos: Backgrounds

How large of a neutrino signal is allowed by present data? As shown in Fig. 5.2,

the signal spectrum is sharply peaked. To be insensitive to the spectrum shape,

i.e., the redshift evolution, we deﬁne the signal as integrated over a bin of width

∆ log10 Eν = 0.5, just below Eν = mχ, (i.e., we consider z . 2, so we can ignore the

tails at low energy.) To be detectable, we require that the signal be 100% as large

as the angle-averaged atmospheric neutrino (νµ + ν¯µ) background, integrated in the

same way. Signal and background are both somewhat smeared from received neutrino

energy to detected energy, but well within this bin. This conservative approach allows

us to simply derive the ﬂux and annihilation cross section constraints over the very

5 wide mass range 0.1–10 GeV. Our model predicts equal ﬂuxes of (νe + ν¯e), (νµ + ν¯µ),

and (ντ + ν¯τ ), any of which can be used to derive bounds.

The atmospheric neutrino (νµ + ν¯µ) spectra as a function of neutrino energy have

been derived from data from the Fr´ejus (0.25–104 GeV, in 9 bins) [148] and AMANDA

(1.3 103–3.0 105 GeV, in 10 bins) [149] detectors. Neutrino attenuation in Earth × × will only be signiﬁcant above 105 GeV. The agreement with theoretical predictions

against upward ﬂuctuations in the data is very good, well below the 100% uncertainty

102 that we adopted. These spectra were derived from neutrino-induced muon data by a regularized unfolding technique, which might miss a narrow signal.

4 We thus considered the data in more detail, ﬁnding that for Eν = 0.1–10 GeV, such a signal is deﬁnitely excluded, especially using both the (νµ + ν¯µ) and (νe + ν¯e) signals. The most useful data are from the Super-Kamiokande detector. In Ref. [150], visible-energy spectra for each of e-like and µ-like events from 0.1–100 GeV are given in 4 log-spaced bins per decade. The agreement with predictions including neutrino oscillations is excellent; the moderate exceptions in some of the highest-energy bins are explainable [150]. Neutrinos with E 10–103 GeV are probed by the count ν ∼ rates (no spectra) of upward throughgoing muons [150], and similarly for E 102– ν ∼ 104 GeV and upward showering muons [151]; both are also in excellent agreement with predictions.

Dedicated analyses of the measured data could improve the signal sensitivity by a factor 10–100, depending on the energy range. First, using the sharp feature in the spectrum at mχ; Fig. 5.2 shows that while the signal is comparable to the background when integrated over an energy bin, in the endpoint region it is much larger. Second, the uncertainties below 10 GeV are actually below 10%, and apply to narrower bins in energy than we assumed [150]. Third, by 10 GeV, the (νe + ν¯e) to (νµ + ν¯µ) background ﬂux ratio is 1/3 and rapidly falling [152, 153]; in addition, the (νe + ν¯e)

ﬂux is strongly peaked at the horizon [152, 153], while the dark matter signal is isotropic. Fourth, detailed analyses of Super-Kamiokande and AMANDA upward throughgoing and upward showering muon data should be more sensitive than the simple count rates we used.

103 5.6 Conclusions

We have shown that the dark matter total annihilation cross section in the late universe, i.e., the dark matter disappearance rate, can be directly and generally bounded by the least detectable SM states, i.e., the neutrino appearance rate. This can be simply and robustly constrained by comparing the diﬀuse signal from all dark matter halos to the terrestrial atmospheric neutrino background. Our bound on σ v is h A i shown in Fig. 5.3. Over a large range in mχ, it is much stronger than the unitarity bound of Hui [43]. It strongly rules out the proposal of Kaplinghat, Knox, and

Turner [133] to modify dark matter halos by annihilation. Our bound can be evaded with truly invisible non-SM ﬁnal states. For a cross section above our bound, its ratio to our bound yields an upper limit on the branching ratio to SM ﬁnal states required to invoke that large of a cross section.

Annihilation flattens halo cusps to cores of density ρ m /( σ v H−1) [133]. A ∼ χ h A i 0 Our bound implies that for all m & 0.1 GeV, this density is ρ & 5 103 GeV/cm3, χ A × which only occurs at radii . 1 pc in the Milky Way for an NFW profile. Annihilation should thus have minimal effects on galactic halos.

104 -15 -16 Bound -17 Ω rad Unitarity Bound -18 KKT Model /s] 3 -19 -20 Neutrino Bound -21 v> [cm A

σ -22 -23

log < -24

-25 Natural Scale -26 -27 -1 0 1 2 3 4 5 6 log m χ [GeV]

Figure 5.3: Upper bounds on the dark matter total annihilation cross section in galaxy halos as a function of the dark matter mass, calculated as discussed in the text [111].

105 Detailed analyses by the Super-Kamiokande and AMANDA Collaborations should

be able to improve our bound by a factor 10–100 over the whole mass range. Halo

substructure or mini-spikes around intermediate-mass black holes could increase the signal by orders of magnitude [143, 144, 145, 146, 154, 155, 156]. The sensitivity

could thus become close to the natural scale for thermal relics, making it a new tool

for testing even standard scenarios.

106 CHAPTER 6

CONSERVATIVE CONSTRAINTS ON DARK MATTER ANNIHILATION INTO GAMMA RAYS

Hasan Yuksel,¨ Shunsaku Horiuchi, John Beacom, and Shin’ichiro Ando built upon the work presented in the previous chapter in order to improve our constraints on the dark matter self-annihilation cross section due to the production of neutrinos.

They did this in their paper “Neutrino Constraints on the Dark Matter Total An- nihilation Cross Section” [23]. They were able to modify the original method to analyze neutrino signals from regions inside the Galactic halo. Thomas Jacques,

Hasan Yuksel,¨ John Beacom, Nicole Bell, and I applied these same methods, including the original and the speciﬁc addition of the Andromeda galaxy, to the case of photons (gamma-rays). This research has been submitted to the journal Physical

Review D [157] and has been altered here to maintain consistency with this dissertation. Using gamma-ray data from the Milky Way, Andromeda (M31), and the cosmic background, we calculate conservative upper limits on the dark matter self- annihilation cross section to monoenergetic gamma rays, σ v , over a wide range h A iγγ of dark matter masses. If the ﬁnal-state branching ratio to gamma rays, Br(γγ), were known, then σ v /Br(γγ) would deﬁne an upper limit on the total cross h A iγγ section; we conservatively assume Br(γγ) & 10−4. For intermediate dark matter

107 masses, gamma-ray-based and neutrino-based upper limits on the total cross section are comparable, with the former dominating for small masses and the latter for large masses. We comment on how these results depend on the assumptions about astrophysical inputs and annihilation ﬁnal states, and how GLAST and other gamma-ray experiments can improve upon them.

6.1 Introduction

If the dark matter is a thermal relic of the early universe, the annihilation cross section must be σ v 3 10−26 cm3 s−1 in order to obtain the observed relic h A i ∼ × abundance, Ω 0.3. It is possible that dark matter is not a thermal relic, e.g., DM ' Ref. [133, 158], which makes it even more interesting to consider direct late-universe constraints on the annihilation cross section, e.g., Refs. [4, 8, 19, 23, 111, 140, 141,

159].

Even if the total annihilation cross section is set by the relic abundance, the branching ratios to specific final states are model-dependent. The dark matter disappearance rate due to annihilation can be constrained by the appearance rates of various SM particles. If the dark matter is the lightest stable particle in some new physics sector, then it can be natural to have the final-state branching ratio to SM particles, Br(SM), be 100%, and we assume this. If the final states include new particles that do not interact with normal matter, then all appearance-based results are weakened proportionally to Br(SM). Again, we assume that dark matter is its own antiparticle, or that there is not a large particle-antiparticle asymmetry.

Here we calculate the constraints that can be placed on the annihilation cross section using gamma rays, the most detectable ﬁnal states, over a wide range of dark

108 matter masses. We ﬁrst focus on the monoenergetic gamma-ray ﬁnal state (γγ), as it would be a very clean signature of dark matter annihilation, with Eγ = mχ, e.g.,

Refs. [160, 161, 162, 163]. Unfortunately, in typical models, this is small, Br(γγ) ∼ 10−4 10−3; in some models, it can be larger [164, 165], but one cannot be certain − that these predictions match with nature. Since gamma rays will be ubiquitously

produced, directly in SM ﬁnal states, or through radiative corrections and energy-loss

processes, we also discuss more general outcomes, in which the gamma-ray energies

are in a broader range below mχ.

We consider constraints on the dark matter annihilation cross section over a large

mass range of 10−5 – 105 GeV. At all but the highest energies, gamma-ray data is

available to test the annihilation cross section, provided that we combine constraints

deﬁned using the Milky Way halo, the Andromeda halo, and all the halos in the

universe. We hope that our results will be useful in challenging experiments to report

stronger limits using new data and focused analyses. With the launch of GLAST

(Gamma-ray Large Area Space Telescope) this year, and with new studies by TeV-

range experiments, these prospects are good. Using our upper limits on the dark

matter annihilation cross section to gamma rays, and a conservative assumption about

the branching ratio to monoenergetic gamma rays, we deﬁne upper limits on the total

cross section and compare to other constraints.

Since the dark matter annihilation rate scales with the dark matter density squared

and the density proﬁles are uncertain, we are mindful of how our constraints on the

cross section are aﬀected by astrophysical uncertainties. We are conservative in our

input choices and analysis methods, and we show how our results depend on these.

In light of these considerations, we do not consider corrections below a precision of

109 a factor of 2, which also allows some simpliﬁcations. Our upper bounds on the ∼ annihilation cross section to gamma rays would only be improved by more optimistic

assumptions.

In Section 6.2, we discuss important general bounds on the total annihilation cross

section. In Section 6.3, we review the analysis methods used for the case of gamma-

ray lines from various dark matter concentrations. The experiments and observations we use are discussed in Section 6.4. In Section 6.5, we summarize and interpret our results.

6.2 Cross Section Constraints

The annihilation cross section sets the dark matter disappearance rate, for which there are two important constraints. The reader is briefly reminded here of these constraints, which are mentioned in Chapters 3 and 5 and repeated here for consistency. The first is unitarity [42, 43], which sets a general upper bound that can only be evaded in unusual cases [134]. In the low-velocity limit where s-wave annihilation dominates, the unitarity bound is σ v < 4π/m2 v, or h A i χ 300 km/s GeV 2 σ v 1.5 10−13 cm3 s−1 . (6.1) h A i ≤ × v m rms χ For m & 106 GeV, this would require that σ v be smaller than that for a thermal χ h A i relic. However, for smaller mχ the unitarity limit is much less constraining. The second comes from the requirement that annihilation does not drastically alter the density profiles of dark matter halos in the Universe today. In the KKT model of

Ref. [133], a large self-annihilation cross section was invoked in order to reconcile predicted cuspy density proﬁles with the ﬂatter ones inferred from observation, requiring m σ v 3 10−19 cm3 s−1 χ . (6.2) h A iKKT ' × GeV h i 110 We re-interpret this result as an approximate upper bound, beyond which halo density

proﬁles would be signiﬁcantly distorted by dark matter annihilation. Note that this

limit is very weak for all but the lightest masses.

We now discuss limits which arise from the appearance rate of dark matter an-

nihilation products, assuming Br(SM) = 100%. As discussed in Chapter 5, all ﬁnal

states except neutrinos obviously produce gamma rays, either directly or as sec-

ondary particles. Quarks and gluons hadronize, producing pions and thus photons

via π0 γγ, while τ , W and Z0 also produce π0 via their decays. Charged particles → produce photons via electromagnetic radiative corrections [136, 166], and electrons

and positrons also produce photons via energy loss processes [137, 167]. Therefore we

expect a broad spectrum of final state photons, even though the branching ratio to the monoenergetic γγ final state may be small. We use the gamma-ray data to place upper limits on σ v , and these are of general interest for their own sake. With an h A iγγ assumption about Br(γγ), these results also define an upper limit on the total cross

section of σ v /Br(γγ). h A iγγ The research presented in Chapter 5 showed an important, general, cosmological

limit on the total annihilation cross section obtained by considering annihilation into

the least detectable ﬁnal state, namely neutrinos. This will be referred to as BBM

[111], while the analysis by Yuksel¨ et al., which extended these results to include the

Galactic Halo, is deemed YHBA [23]. Given that stronger constraints will exist on all ﬁnal states other than neutrinos, we can set a conservative upper bound on the total dark matter annihilation rate by assuming the branching ratio to neutrinos is

100%. (Unlike all other constraints, the neutrino constraint, being the weakest, is not to be divided by a realistic branching ratio; this follows from the fact that the sum of

111 all branching ratios must be 100%. See Chapter 5.) By requiring that the neutrino

ﬂux produced by annihilation be smaller than the measured atmospheric neutrino

background, robust bounds on the total annihilation cross section were obtained

over a wide mass range. For all masses considered, these limits are much stronger

than the KKT limit; they are also stronger than the unitarity limit except at high

masses. While neutrinos are the least detectable annihilation products, even they are

accompanied by gamma rays via electroweak radiative corrections; these results lead

to constraints on σ v that are comparable to those obtained directly with neutrinos, h A i placed by M. Kachelriess and P. Serpico in response to BBM [140].

6.3 Calculation of Dark Matter Signals

Where the dark matter density is largest, at the centers of halos, the uncertainties are the largest; these regions contribute relatively little to the gravitationally-

measured mass of a halo. To cover as large an energy range as possible, we have to

consider gamma-ray data for the Milky Way, Andromeda, and all of the dark matter halos in the universe. In all cases, though the astrophysical and analysis uncertainties vary in their severity, we make conservative choices for the dark matter density and hence the cross section limits (smaller choices of densities mean larger upper limits on the cross section).

6.3.1 Dark Matter Halos

A standard parameterization of the dark matter density proﬁle in a halo is

ρ0 ρ(r) = γ α (β−γ)/α . (6.3) (r/rs) [1 + (r/rs) ]

112 Of the proﬁles mentioned in Chapter 3, consider speciﬁcally the Navarro-Frenk-White

(NFW) [168] and Kravtsov proﬁles [169], which are deﬁned by (α, β, γ) = (1, 3, 1) and

(α, β, γ) = (2, 3, 0.4) respectively. Near the center of the halo, the density of an NFW

proﬁle scales with radius as 1/r, while the Kravtsov proﬁle scales less steeply, as

0.4 1/r . For large radii, r & rs, these two proﬁles coincide more closely. For the Milky

Way, rs is 20 kpc for the NFW profile and 10 kpc for the Kravtsov profile, while the normalization, ρ0, is fixed such that the density at the solar circle distance, Rsc = 8.5

−3 −3 kpc, is ρ(Rsc) = 0.3 GeV cm (0.37 GeV cm ) for the NFW (Kravtsov) proﬁle.

There does not appear to be a consensus on the values of the halo parameters for

Andromeda. For example, compare the NFW profiles in Ref. [170] with Ref. [171], where both ρ0 and rs are quite different. Thus we have chosen to model Andromeda using the Milky Way parameters, as an appropriate compromise between the com- peting profiles.

In the innermost regions of halos, the uncertainties in the dark matter annihilation rate are largest. For larger central regions, or whole halos, these uncertainties are much less. For the Milky Way, these effects can easily be seen in Fig. 2 of YHBA; for larger angular regions centered on the Galactic Center, the dark matter annihilation signals for different profiles become much more similar to each other. While the uncertainties at small angular scales can be orders of magnitude, those at large angular scales are not more than a factor of about two. To be conservative, we neglect the possibility of halo substructure, e.g., Refs. [172, 173], or mini-spikes around intermediate-mass black holes [154, 156], which would lead to enhanced annihilation

signals. For completeness, the relevant ﬁgure from YHBA is displayed here as Fig. 6.1.

113 Figure 6.1: The behavior of the integrated line of sight J(ψ) for the different profiles as a function of angular radius ψ as measured from the Galactic Center. Also shown is the averaged line of sight integral for a cone of radius ψ, J∆Ω, displayed as a thicker line than its J(ψ) counterpart. This is shown for three dark matter density profiles: Moore (dotted), NFW (dashed), and Kravtsov (solid) [23].

114 6.3.2 Milky Way and Andromeda Signals

We ﬁrst consider annihilations in our Galaxy, following the conventions of YHBA,

and generalize this to the nearby galaxy Andromeda (M31). The intensity (ﬂux per

solid angle) of the annihilation signal at an angle ψ with respect to the Galactic

Center (GC) is proportional to the square of the dark matter density integrated over

the line of sight,

`max 2 2 2 (ψ) = J0 ρ Rsc 2`Rsc cos ψ + ` d` , (6.4) J 0 − Z p where J = 1/[8.5 kpc (0.3 GeV cm−3)2] is an arbitrary normalization we use to 0 × make a dimensionless quantity, and which cancels in the ﬁnal results. The upper J limit of the integration is given by ` = (R2 sin2 ψR2 )1/2 +R cos ψ. We deﬁne max MW − sc sc as the average of over a cone of half-angle ψ centered on the GC, J∆Ω J 2π ψ = (ψ) sin ψdψ , (6.5) J∆Ω ∆Ω J Z0 where ∆Ω = 2π(1 cos ψ) is the angular size of the cone in steradians. The values of − (ψ) and can be read directly from Fig. 2 of YHBA (below, we do not explicitly J J∆Ω show the sr−1 units of (ψ) and ). J J∆Ω Eq. (6.4) can easily be generalized to external halos [174, 175] (such as the An-

dromeda galaxy at a distance of D 700 Mpc) using M31 '

`max 2 2 2 (ψ) = J0 ρ DM31 2`DM31 cos ψ + ` d` , (6.6) J ` − Z min q where the result is independent of the upper and lower limits of integration (`min,max) as long as they cover most of the halo under consideration.

For extragalactic dark matter sources, the annihilation signals will include a contribution from the dark matter in our own galaxy along the line of sight. However, in

115 the case of an external galaxy like Andromeda, this can be eliminated by subtracting

the background intensity from a region close to the source.

With these deﬁnitions, the intensity of the dark matter annihilation gamma-ray

signal is

dΦγ σAv ∆Ω 1 dNγ = h i J 2 , (6.7) dE 2 J0 4πmχ dE

where dNγ/dE is the gamma-ray spectrum per annihilation. In the case of annihilation into two monoenergetic gamma rays, we simply have dN /dE = 2δ(m E); γ χ − we generalize this below. Similarly, the total ﬂux (per unit energy) from a region of solid angle ∆Ω is

dφγ dΦγ σAv ∆Ω∆Ω 1 dNγ = ∆Ω = h i J 2 . (6.8) dE dE 2 J0 4πmχ dE

6.3.3 Cosmic Diﬀuse Signal

The calculation of the cosmic diﬀuse annihilation signal is detailed in Chapter 5 for

the case of neutrinos and in YHBA, and can be easily modiﬁed for the case of photons.

For example, Refs. [141] and [142] analyze a cosmological ﬂux of photon annihilation

products from external galaxies including dark matter clustering to establish a signal.

The cosmic diﬀuse ﬂux, arising from dark matter annihilation in halos throughout

the Universe is

2 2 dΦγ σAv c ΩDM ρcrit = h i 2 dE 2 4πH0 mχ zup f(z)(1 + z)3 dN (E0) γ e−z/zmax dz , (6.9) × h(z) dE0 Z0 −1 −1 where H0 = 70 km s Mpc is the Hubble parameter and ΩDM is the dark matter density in units of the critical density. We assume a ﬂat universe, with ΩDM = 0.3,

3 1/2 −z/zmax ΩΛ = 0.7, h(z) = [(1 + z) ΩDM + ΩΛ] . The factor e , taken from Ref. [141],

116 accounts for the attenuation of gamma rays, a modest eﬀect for the energies considered here. The factor f(z) in Eq. (6.9) accounts for the average increase in density

squared due to the fact that dark matter is clustered into halos, rather than uniformly distributed, and the evolution with redshift of the halo number density. (The ∆2 factor in BBM is equal to f(z)(1 + z)3.) Following YHBA, we use the parameterization log (f(z)/f ) = 0.9 [exp( 0.9z) 1] 0.16z, where f depends on the halo profile. 10 0 − − − 0 Choosing the Kravtsov (NFW) profile, f 2 (5) 104. 0 ' × Gamma rays that are produced with energy E0 are observed at Earth with redshifted energy E = E0/(1 + z). For annihilation into monoenergetic gamma rays, the delta function source spectrum is modified by redshift as

dN 2 m = 2 δ(m E0) = δ z ( χ 1) , (6.10) dE0 χ − E − E − which shows that the observed ﬂux at an energy E is contributed by sources at redshift

mχ 1. E − 6.4 Speciﬁc Observations and Derived Annihilation Constraints

We have collected gamma-ray ﬂux measurements and limits from a wide variety of

experiments, spanning an extensive energy range from 20 keV to 10 TeV. In most of

the observations, the energy spectra are given in log-spaced energy intervals. We cal-

culate annihilation gamma-ray ﬂuxes for the Galactic, Andromeda, and cosmic dark

matter sources, using the methods outlined in Section 6.3 above. These are compared

−0.4 with observational data over an energy range, conveniently chosen as 10 mχ –mχ,

117 that is comparable to or larger than the energy resolution and bin size of the experiments. If only upper limits on the ﬂux are given, we instead compare our predictions directly with these upper limits.

Our constraints on the dark matter annihilation rate are determined by demand- ing that the annihilation ﬂux be smaller than 100% of the observed (presumably not produced by dark matter) gamma-ray background ﬂux at the corresponding energy

range. In Fig. 6.2, we show the GC and cosmic diﬀuse signals from dark matter anni-

hilations which fulﬁll this criterion, superimposed upon the Galactic and extragalactic

spectra, respectively, as measured by COMPTEL and EGRET.

118 -2 10

-3 10 ] -1 sr -1

s -4

-2 10 [ cm -5 /dE 10 Φ E d -6 10

-7 10 -3 -2 -1 0 1 2 10 10 10 10 10 10 E [ GeV ]

Figure 6.2: Example dark matter annihilation signals, shown superimposed on the Galactic and extragalactic gamma-ray spectra measured by COMPTEL and EGRET. In each case, the cross section is chosen so that the signals are normalized according to our conservative detection criteria, namely, that the signal be 100% of the size of

the background when integrated in the energy range chosen (0.4 in log10 E, shown by horizontal arrows). The narrow signal on the right is the Galactic Center flux due to annihilation into monoenergetic gamma rays, for mχ = 1 GeV; the signal is smeared as appropriate for a detection with finite energy resolution. The broad feature on the left is the cosmic diffuse signal for annihilation into monoenergetic gamma rays at mχ = 0.1 GeV, smeared by redshift [157].

119 The experiments report their results as either intensity (as in Eq. 6.7) which requires that we calculate , or ﬂux from a given angular region (as in Eq. 6.8) for J∆Ω which we need ∆Ω. We present the values of these parameters which correspond J∆Ω to the Kravtsov proﬁle, as this results in the most conservative upper limits on the annihilation cross section. Our limits on the dark matter annihilation cross section

are reported in Fig. 6.3, where we also show how our results would change if the

NFW proﬁle were adopted instead. The details of the experiments and our analyses are summarized below for each observation.

6.4.1 COMPTEL and EGRET

COMPTEL [176], the imaging COMPton TELescope aboard the Compton

Gamma Ray Observatory (CGRO) satellite, measured gamma rays in the energy

range 1–30 MeV. EGRET [177], the Energetic Gamma Ray Experiment Telescope,

also aboard the CGRO, measured gamma rays in the energy range 30 MeV to nearly

100 GeV. For both COMPTEL and EGRET, the full sky was studied with an angu-

lar resolution of at worst a few degrees (for the large regions we consider, this makes

no diﬀerence). The energy resolution was modest, and the data were given in a few

logarithmically-spaced bins per decade in energy.

Both COMPTEL and EGRET observed the Galactic Center region, and the mea-

sured gamma-ray intensity energy spectra are reported in Refs. [178] and [179] for the

region 30◦ < l < 30◦ and 5◦ < b < 5◦ (Galactic longitude and latitude, respec- − − tively). The disk-like morphology of the emission region makes it clear that nearly all of this emission is due to ordinary astrophysical sources; to be conservative, we do not attempt to deﬁne a limit on the small component of this that could be due

120 to centrally-concentrated dark matter, and simply use the total observed intensity to bound any dark matter contribution. Also, we evaluate the dark matter signal as if

from a circular region of ψ = 30◦; accounting for the rectangular shape of the region

would lead to a higher value than the 13 that we adopt. Using a less conserv- J∆Ω ' ative set of assumptions than we employ, stronger limits on σv were derived from h iγγ the EGRET data in Ref. [162].

6.4.2 H.E.S.S.

H.E.S.S. (High Energy Stereoscopic System), a system of multiple atmospheric

Cerenkˇ ov telescopes, is presently in operation in Namibia [180]. H.E.S.S. has observed

the Galactic Center region in the energy range 0.3–15 TeV. An apparent point source

at the Galactic Center was observed, as was an extended source ( 1◦) known as the ∼ Galactic Center Ridge [181]. While the origin of the point source is unknown, the

Ridge emission is almost certainly astrophysical, and is consistent with being caused

by cosmic rays colliding with a gas cloud (again, we do not attempt to account for this,

and will simply bound any dark matter contribution by the total observed intensity).

Since the uncertainties in the dark matter proﬁle increase for smaller angular

regions around the Galactic Center, it is more robust to deﬁne our results using the

extended region instead of the point source. The Ridge emission was observed in

an angular region 0.8◦ < l < 0.8◦ and 0.3◦ < b < 0.3◦ in Galactic coordinates, − − and the resulting ﬂux reported by H.E.S.S. reﬂected a background subtraction from a

nearby region ( 0.8◦ < l < 0.8◦ and 0.8◦ < b < 1.5◦) to help account for cosmic rays. − Thus, we have to consider not the whole dark matter signal, but just its contrast

121 between the central and adjacent regions by accounting for this subtraction in our

analysis.

We approximate the intensity from the rectangular region of the Galactic Center

Ridge with a circle of radius 0.8◦. We also estimate the adverse eﬀect of the back-

ground subtraction on our limits by choosing to be subtracted at its maximum, J i.e., ψ = 0.8◦. This means

◦ 2π 0.8 = ( (ψ) (0.8◦)) sin ψdψ 3 . (6.11) J∆Ω ∆Ω J − J ' Z0 Had we not made this subtraction correction, our limits on the cross section would

be stronger by about an order of magnitude.

6.4.3 INTEGRAL

The space-borne INTEGRAL (INTErnational Gamma-Ray Astrophysics Labora- tory) observatory [182] has searched for gamma-ray emission in the Milky Way over the energy range 20–8000 keV, using the SPectrometer on INTEGRAL (SPI). Teegar- den and Watanabe [183] presented results of an INTEGRAL search for gamma-ray line emission from the Galactic Center region (we use their zero-intrinsic-width results, as appropriate for the low dark matter velocities of the halo). Other than the

expected positron annihilation [184] and 26Al decay signatures, no evidence of other

line emission was found.

To reduce backgrounds and improve the sensitivity of the line search, the measured

intensity from large angular radii ( > 30◦) was subtracted from that in the Galactic

Center region ( < 13◦), resulting in a 3.5-σ constraint on the ﬂux of very roughly

. 10−4 photons cm−2 s−1 in the energy range 20–8000 keV. Our calculations must

reﬂect this subtraction, which will somewhat weaken the sensitivity to the dark matter

122 signal. A similar correction was used in Ref. [185]. We implement this as

13◦ ∆Ω = 2π ( (ψ) (> 30◦)) sin ψdψ 2 . (6.12) J∆Ω J − J ' Z0 Due to the decreasing trend of the dark matter proﬁle, the intensity outside the

Galactic Center region will be largest at 30◦, and accordingly we choose this value to be as conservative as possible (a larger subtraction leads to a weaker upper limit on

σ v ). Had we not made this correction, our limits on the cross section would be h A i stronger by up to a factor of 2.

6.4.4 Andromeda Halo Results

The Andromeda galaxy has been observed by several gamma-ray experiments, all of which placed upper limits on the flux. EGRET, CELESTE, and HEGRA all observed Andromeda, each encompassing a respectively smaller angular region of that extended object. As the results were reported as flux limits from specified angular regions, we compare to these using ∆Ω, which is an input to Eq. (6.8). J∆Ω EGRET viewed Andromeda with an angular radius of 0.5◦ and set a 2σ upper limit on the gamma-ray flux of 1.6 10−8 photons cm−2 s−1 from 0.1 GeV to 2 GeV, × since no signal was seen [186]. For the angular region of this observation, the flux will be proportional to

0.5◦ ∆Ω = 2π 0(ψ) sin ψdψ 2 10−3 . (6.13) J∆Ω J ' × Z0 CELESTE (CErenkˇ ov Low Energy Sampling and Timing Experiment) is an atmospheric Cerenkˇ ov telescope in the French Pyrenees which studies gamma rays with energies greater than 50 GeV [187]. It viewed Andromeda in the energy range of

123 50–700 GeV, and again no signal was seen [188]. A 2σ upper limit on the energy- integrated ﬂux from Andromeda was reported as . 10−10 photons cm−2 s−1, employing an angular radius of θ = 0.29◦ and yielding ∆Ω 1 10−3. obs J∆Ω ' × HEGRA (High Energy Gamma Ray Astronomy experiment) was an atmospheric

Cerenkˇ ov telescope, located in La Palma in the Canary Islands [189]. It took data in the range 0.5–10 TeV, with better energy resolution than that of CELESTE [190]. It

has an even smaller angular radius of θ = 0.105◦, which yields ∆Ω 2 10−4. obs J∆Ω ' × HEGRA reported 99% C.L. upper limits for the gamma-ray line ﬂux, and these can

be used directly.

6.4.5 Cosmic Diﬀuse Results

INTEGRAL [191], COMPTEL [192] and EGRET [193] have all made measure-

ments of the gamma-ray ﬂux at high latitudes, and these can be used to set a limit on

the cosmic dark matter annihilation signal. The INTEGRAL data were collected in

broad energy bins, much like those of COMPTEL and EGRET. The cosmic gamma-

ray background was also measured by the Gamma-Ray Spectrometer aboard the Solar

Maximum Mission (SMM) [194] over the energy range 0.3 – 8 MeV, for a ﬁeld of view

of 135◦ in the direction of the Sun [195], and we include this.

For the cosmic diﬀuse analysis, the framework detailed in Section 6.3.3 can be

applied. Note that for simplicity we calculate only the true cosmic diﬀuse dark

matter signal, neglecting any Galactic contribution along the lines of sight. This

contribution from the Galactic halo (which would add to the signal and thus make

our limits stronger) is signiﬁcant for NFW or steeper proﬁles and can even dominate

over the true cosmic dark matter signal; see YHBA and Ref. [196].

124 6.5 Discussion and Conclusions

6.5.1 Limits on the Cross Section to Gamma Rays

In Fig. 6.3, we combine all of the upper limits on the partial cross section to monoenergetic gamma rays, choosing the strongest limit for each value of the dark matter mass. The shaded exclusion region shows our combined bound. These searches for dark matter signals are limited by astrophysical backgrounds, and the general trend of how the limits vary with mass follows from how these backgrounds vary with energy.

We now consider how the cross section limit should scale with mass and the assumed spectrum of final-state gamma rays. Since typical astrophysical signals (and backgrounds) have ever-diminishing fluxes with increasing energy, the sensitivity of gamma-ray detectors must likewise improve with energy, at a minimum being able to detect the backgrounds well. The gamma-ray number flux of the signal integrated in a logarithmic energy bin is EdΦ/dE σ v /m2 . If the gamma-ray number flux ∼ ∼ h A i χ of the background integrated in a logarithmic energy bin scales as EdΦ/dE E/E α, ∼ then we expect σ v m3−α. For example, for the EGRET diffuse data, α is h A ilimit ∼ χ slightly greater than 2, and so the cross section limits scale slightly less rapidly than the mass.

125 -20 10

-22 10 HEGRA M31

CELESTE M31 -24 HESS GC RIDGE ] 10 -1 s 3 -26 EGRET M31 10 [cm γγ

-28 DIFFUSE PHOTON BACKGROUND

v > 10 CGRO MW A σ

< -30 10

-32 INTEGRAL 10

-34 10 -5 -3 -1 1 3 5 10 10 10 10 10 10 mχ [GeV]

Figure 6.3: The limits on the partial cross section, σ v , derived from the various h A iγγ gamma-ray data. Our overall limit is shown as the dark shaded exclusion region. For comparison, the light-shaded region shows the corresponding limits for the NFW (rather than the Kravtsov) proﬁle [157].

126 Almost all of these experiments had fairly poor energy resolution. To be con-

servative, we assumed an analysis window with a logarithmic energy width of 0.4 in

log10 E for the Galactic and cosmic diﬀuse analyses; this is at least as wide as the

energy bins reported by the experiments. That is, even though we nominally assumed

two monoenergetic gamma rays at Eγ = mχ, our results have not taken advantage of

this fact. The exception is the INTEGRAL line search, where the excellent energy

resolution is what leads to this limit being stronger than expected from the general

trend in Fig. 6.3. This is similar to the analysis in Chapter 5, which assumed widths

of 0.5 in log10E.

For a narrow signal at mχ, the cross section limit is deﬁned by requiring that

the signal be comparable to the background at an energy E m . Now consider a γ ' χ

broader energy distribution, or rather an extreme case where a signal at Eγ is pro-

duced by dark matter of mass mχ = aEγ, with a > 1. Compared to the original

case, that annihilation rate will be down by a factor a2 due to the reduced num-

ber density, and, accordingly, the required cross section for the signal to match the

2 background at Eγ would have to be a times larger. That is, for the upper limits,

σ(m ) a2σ(m /a) . In eﬀect, we have already assumed a 3, given the χ continuum ' χ line ∼ wide analysis bins we have used. Thus our main results would be only moderately

degraded if we assumed that the gamma rays were within some larger range of energy

below the dark matter mass: for a spectrum of two gamma rays distributed within

one order of magnitude below mχ, we would expect that our cross section limits would

be weakened by roughly only one order of magnitude. It is trivial to adjust the limits if fewer gamma rays are assumed. Given the large range on the axes in Fig. 6.3,

127 and our intention to deﬁne approximate and conservative limits, this shows that our results are much more general than they ﬁrst appear.

How sensitive are our limits to the choice of density profile? As noted, we chose the rather shallow Kravtsov profile to be conservative. If we adopt an NFW profile, which increases much more rapidly toward the Galactic Center (scaling with radius as r−1 rather than r−0.4) the annihilation rates will be larger and the cross section limits correspondingly stronger. In Fig. 6.3, we show how our results would change if we had used an NFW profile instead of the Kravtsov profile. At most energies, the changes are modest, and illustrative of the potential uncertainties. The only significant change

to the combined gamma-ray limit is for the H.E.S.S. Galactic Center Ridge case. In

the NFW case, the steeper proﬁle gives an overall larger intensity and a smaller signal

cancelation when the background is subtracted (See Fig. 6.1.

6.5.2 Limits on the Total Cross Section

Unsurprisingly, the cross section bounds derived under the assumption of monoen-

ergetic gamma rays are substantially stronger than those deﬁned similarly for ﬁnal-

state neutrinos in BBM and YHBA. Indeed, this was an assumption in those two

works that we have now justiﬁed in more detail than before.

It is unrealistic to have Br(γγ) = 100%, of course, if one is trying to set a limit

on the total cross section. If Br(γγ) is known, then a limit on the total cross section

can be determined by dividing the limit on the partial cross section to that ﬁnal state

by the branching ratio: σ v σ v = h A iγγ . (6.14) h A itotal Br(γγ)

128 In typical models, this branching ratio is typically 10−3 or smaller [19, 8, 4]. To be conservative, we simply choose a value such that it is implausible that the true branching ratio could be smaller. We therefore assume Br(γγ) = 10−4, but this choice could still be debated. As noted, our analysis uses wide logarithmic energy bins, and so, at the very least, would capture the gamma rays near the endpoint due to internal bremsstrahlung from charged particles [136, 166]. (Similarly, limits on the total cross section deﬁned by assuming only W +W − ﬁnal states [159, 197] would have to be corrected by dividing by Br(W +W −).)

Figure 6.4 summarizes various limits on the total cross section, including the one just described, the unitarity bound mentioned earlier, and the neutrino bound from

YHBA (based on the Milky Way signal and the Kravtsov proﬁle). The standard cross section for a thermal relic is also shown. Note that our limits bound σ v directly, h A i independent of whether σ is s-wave or p-wave dominated.

When shown in this way, it becomes clear how surprisingly strong the neutrino bound on the total cross section is, as it is comparable to the gamma-ray bound. It is very important to emphasize that while the gamma-ray bound on the partial cross section has to be divided by a realistic Br(γγ), this is not the case for the neutrino bound, as explained above. If we assume only Standard Model ﬁnal states, then all

ﬁnal states besides neutrinos lead to appreciable ﬂuxes of gamma rays, and hence are more strongly excluded. Of course, the gamma-ray and neutrino cross section limits can both be weakened by assuming an appreciable branching ratio to new and truly sterile particles.

129 -16 10 Unitarity Bound KKT -18 10

] -20

-1 10 s Neutrinos

3 Gamma Rays -4 Br(νν)=1 Br(γγ)=10 -22 10 [cm

total -24 10 v > A σ -26 Natural Scale < 10

-28 10

-30 10 -5 -3 -1 1 3 5 10 10 10 10 10 10 mχ [GeV]

Figure 6.4: The gamma-ray and neutrino limits on the total annihilation cross section, selecting Br(γγ) = 10−4 as a conservative value. The unitarity and KKT bounds are also shown. The overall bound on the total cross section combines the strongest values from these upper limits [157].

130 6.5.3 Conclusions and Prospects

Using gamma-ray data from a variety of experiments, we have calculated upper

limits on the dark matter annihilation cross section to gamma rays over a wide range

of masses. These limits are conservatively defined in terms of our analysis criteria, our assumptions about the uncertain dark matter density profiles, and the gamma-ray spectrum. While our results were nominally defined for monoenergetic gamma rays with Eγ = mχ, we have shown that all of our results except the INTEGRAL line flux

limit are only weakly dependent on this assumption.

There are good prospects for improved sensitivity with present and upcoming

gamma-ray experiments, particularly GLAST [198, 199] and the TeV ACT detectors.

More detailed searches and analyses by the experimental collaborations themselves

should also lead to improvements, which we encourage. These searches for dark matter

signals are already background-limited, which will limit the possible improvements.

GLAST and other experiments should be able to make reductions in the backgrounds

by taking advantage of better energy and angular resolution, and by reducing the

residual diﬀuse emission by subtracting astrophysical components and resolving in-

dividual sources. The high statistics expected for GLAST and other experiments

should also make it possible to deﬁne detection criteria in terms of the uncertainty

on the background, instead of the whole measured background.

Using a conservative choice on the branching ratio to gamma rays, namely

Br(γγ) 10−4, we deﬁned an upper limit on the total dark matter annihilation ' cross section by dividing our limits on the partial cross section to gamma rays by

this branching ratio. At intermediate energies, the upper limit on the total cross

131 section defined this way is comparable to previous upper limits defined using neutrinos [111, 23]. The combined limit is considerably stronger than the unitarity bound or the cross section of Ref. [133], which would lead to substantial modifications of dark matter halos. Additional work is needed to push the sensitivity of these and

other techniques down to the expected cross section scale for thermal relics.

132 CHAPTER 7

THE EFFECTS OF RESIDUAL DARK MATTER ANNIHILATIONS ON BIG BANG NUCLEOSYNTHESIS

This research with Gary Steigman deals with a more specialized situation con- cerning the dark matter annihilation cross section. We examine the case of residual annihilations of dark matter during Big Bang Nucleosynthesis. Annihilations do not cease when a particle freezes out of thermal equilibrium, but rather follow an exponential tail. If dark matter freezes out of thermal equilibrium during Big Bang Nucle- osynthesis and produces electromagnetic products such as photons, the abundances of the light elements formed during BBN would be diﬀerent. The photodissociation of helium-4 would result in an overproduction of the light elements. The analysis results in a lower limit on the dark matter annihilation cross section over a small mass range of 20 MeV to about 1 GeV. This interesting limit behavior is combined with the monoenergetic photon upper limits from dark matter annihilations in the

Milky Way, Andromeda, and cosmic halos, producing a wedge of available parameter space. Limits are found for the speciﬁc annihilation cross section to photons and the total annihilation cross section.

133 7.1 Annihilations in the Early Universe

The early Universe was a seething mess of energy. The photons and particles

existed together in thermal equilibrium and interacted very frequently, especially

through the reaction particle + antiparticle γ + γ and its reverse. However, the −→ Universe expanded, spreading out the distances between the particles, thereby slowing

their rate of interaction. As the particles and antiparticles spread out, it became

increasing more diﬃcult to “ﬁnd each other” to annihilate. The Universe also cooled,

dropping the energy of the photons and limiting the reaction γ + γ particle −→ + antiparticle. If the photons did not have the necessary energy, they could not

produce the massive products. Particles decoupled from the thermal equilibrium in

this manner. The change in the number of a particle was then only subject to its own annihilation. When the temperature of the Universe (approximately) dropped below the mass of a particle, and its interaction rate was less than the expansion rate, that

particle-type “froze out” of equilibrium, setting the particle’s relic abundance [19].

7.1.1 Derivation of Dark Matter’s Relic Abundance

This derivation of dark matter’s relic abundance follows the methods of Refs. [7]

and [200] and their formalisms, unless otherwise stated. Details are explicitly shown.

If dark matter were indeed a thermal relic, its abundance could be found using the

Boltzmann equation describing the change in the number of particles for a particle

that is its own antiparticle in thermal equilibrium in an expanding Universe: dn + 3Hn = σ v (n2 n2 ). (7.1) dt −h A i − eq (Note: here and in the following equations, n and m are used instead of the understood

nχ and mχ.) The value of the dark matter density when it is in thermal equilibrium

134 is neq, while n is the number density of dark matter actually present. The change in the dark matter density is due to both the expansion of the Universe (the term 3Hn) and loss of particles through annihilation, and is analyzed here by examining the number of dark matter particles in a comoving volume, N = nR3. This “comoving volume” is called such because as the scale length changes, it changes as well, allowing observation in a still environment. It is a similar situation to that of the laboratory and center-of-momentum frames in particle scattering. The left-hand-side is originally

R−3d(nR3)/dt, and this can be seen through the derivative chain rule:

d(nR3) dn dR R−3 = R−3 R3 + 3R2 n (7.2) dt dt dt dn 1 dR = + 3 n (7.3) dt R dt dn = + 3Hn. (7.4) dt

Equation (7.1) is very diﬃcult to solve as written, but greater insight can be gained from using the fact that the entropy in a comoving volume (sR3) is a constant, where s is the entropy density. If sR3 is a constant with respect to time, so is sR3/sR3, and s/s. Placing s/s inside the time derivative on the left-hand-side Eq. (7.2) simpliﬁes the equation:

d( n sR3) d( n ) R−3 s = R−3 sR3 s + 0 (7.5) dt dt d( n ) = s s (7.6) dt dY = s , (7.7) dt

where Y is deﬁned as n/s. Therefore,

dn dY + 3Hn = s , (7.8) dt dt

135 and Eq. (7.2) can now be written independently of the Universe’s expansion through

this new variable Y as dY = s σ v (Y 2 Y 2 ). (7.9) dt − h A i − eq Eventually, n will have to be recovered, which will mean multiplying by the value of the entropy density today s0. Another variable substitution must be made, however,

before continuing. As mentioned earlier, the Universe’s temperature drops as it ex-

pands. The relationship between the time and temperature is understood, although it diﬀers during the various epochs of the Universe (radiation-dominated, matter- dominated, etc.) The time period in question is the radiation-dominated era, where

the Hubble parameter H is related to the time by H = 1/(2t) and

90 M t = g−1/2 pl . (7.10) 32π3 ∗ T 2

The Planck mass M is 1.22 1019 GeV. The eﬀective massless degrees of freedom pl × contributed to the system by the particles in equilibrium is g∗. The particles have diﬀering contributions based on their nature as fermions or bosons. In thermal equilibrium, 7 g = g + g . (7.11) i 8 i i=Xbosons i=fXermions For the time in question, the photons, neutrinos, electrons, and positrons are in

equilibrium, and so g = 10.75. It is also helpful to consider the unitless ratio m/T ∗ ≡ x, since this is the tell-tale ratio for when a particle freezes out of equilibrium (photons with thermal energy T cannot combine to form particles of mass m > T ), and to

express the diﬀerential equation as a function of x instead of t. To this end,

M t = 0.301 g−1/2 pl x2 ; (7.12) ∗ m2 dt M = 2x (0.301) g−1/2 pl . (7.13) dx ∗ m2 136 Recall that H = 1/(2t), and so

2 1/2 m 1 H(x) = 1.67 g∗ 2 ; (7.14) Mpl x 2 1/2 m 2 H(m) = 1.67 g∗ H(m) = H(x)x . (7.15) Mpl −→

These relations are important because they allow the change of variables

d H(m) d = , (7.16) dt x dx

reshaping Eq. (7.9) as

dY x σ v s = h A i Y 2 Y 2 . (7.17) dx − H(m) − eq However, this equation is not fully in terms of x. The entropy in the radiation-

dominated era is

2π2 s = g T 3 (7.18) 45 ∗ 2π2 m3 1 = g = C (7.19) 45 ∗ x3 x3

Regardless, Eq. (7.17) cannot be solved analytically. The form of Yeq is a decreasing

exponential:

g 3/2 −x Yeq = (0.145) x e (7.20) g∗ At times long after freeze-out, Y 2 is much greater than Y 2 would be (Y 2 Y 2 ), and eq eq so the combination of Eq. (7.17) and Eq. (7.19) can be approximated at late-times by dY σ C Y 2 = 0 , (7.21) dx −H(m) x2+n where, for cold dark matter, the partial-wave expansion σ (v/c) σ x−n is used, h A i ≈ 0 adapted from Eq. (3.7), with n being integers beginning at 0. The necessity of the

137 v/c stems from the use of energy units, noting again that x itself is unitless in this formalism.

This new diﬀerential equation can be simply solved, integrating Y from its value at freeze-out Yf to that at late-times Y∞, and x from xf to inﬁnity (at late-times, m/T becomes large due to the decrease in T ). For completeness, the answer to the integral is displayed here:

−(n+1) 1 1 σ C x = 0 f . (7.22) Y∞ − Yf −H(m) n + 1

Due to annihilations after freeze-out, Y∞ < Yf , so this becomes

n (n + 1)H(m) xf Y∞ xf ≈ C σ0 (n + 1) xf 3.8 −n 1/2 . (7.23) ≈ σ0x g∗ Mplm

2 The more relevant quantity is Ωχh , but Y can be easily converted. Multiplying by the current entropy density today recovers the number density:

2900 n = s Y Y (7.24) 0 0 ∞ ≈ cm3 ∞ 4 (n + 1) xf −3 = 1.1 10 −n 1/2 cm . (7.25) × σ0xf g∗ Mplm

The quantity σ x−n is essentially σ (v/c) , and so it will now be written as such, 0 f h A i without the (n + 1), which for s-wave annihilations would be 1, and for p-wave would be 2. The Planck mass is 1.22 1019 GeV, and 1 GeV is equivalent to 5.08 1013 × × cm−1. Using those relations and the fact that ρ is 1.05 10−5 h2 GeV/cm3, this c × yields

ρ cm2 x χ0 h2 = Ω h2 3.33 10−38 f (7.26) ρ χ0 ≈ × σ (v/c) 1/2 c h A i g∗ cm3/s x = 1 10−27 f . (7.27) × σ v 1/2 h A i g∗ 138 Unfortunately, the only way to solve for a value of xf is through iteration of the formula [201]

1 g 15 σAv m xf + ln xf = 45.4 + ln 1/2 10 h 3 i , (7.28) 2 g∗ cm /s GeV! but solving it suggests a value for xf around 20. Putting the relic density equation in the most helpful form results in

6.1 10−27 x 10.75 Ω h2 × f (7.29) χ0 ≈ σ v /(cm3/s) 20 g h A i r ∗ If g is 60, the next highest value it can hold, the coeﬃcient is instead 3 10−27, ∗ × the value quoted most often. This is the relic abundance value, and the reason why

2 the dark matter is usually assumed to be weakly interacting. For Ωχh = 0.11, the

observed value, σ v 10−26 cm3/s, a value that can be held by weakly interacting h A i ∼ particles. The inverse relationship between the annihilation cross section and the relic

abundance is now apparent.

7.1.2 Residual Annihilations

There is a small, but non-negligible, exponential tailing-oﬀ in the freeze-out

process, as seen in Figure 7.1 [201]. The ﬁgure is a plot of the number of particles as a

function of x. As the time progresses, the temperature drops, and x increases. While

not derived here (but mentioned above), the annihilations do not cease entirely, but

rather experience an exponential slowing, so that the asymptotic value is not quite

the abundance at freeze-out. This exponential behavior is alluded to in Eq. (7.28).

These “eternal” [202] or “residual” [203] annihilations may be important for dark

matter, speciﬁcally if its annihilation products include electromagnetic particles and

the residual annihilations occur during Big Bang Nucleosynthesis. If electromagnetic

139 particles, such as photons, are produced in large amounts during BBN, they could

disrupt the formation of the light elements via photodissociation. This would be

possible for a dark matter mass in the approximate range of 20 Mev to 1 GeV, a mass range slightly above one that has garnered recent attention (e.g. the MeV-mass dark matter in Refs. [204], [205], [206], and [207]). The observed light element abundances can place a constraint on dark matter’s annihilation into electromagnetic products during BBN.

140 Figure 7.1: The number of particles N as a function of x. As the Universe cools, x increases, and the particles begin to freeze out of thermal equilibrium. During freeze-out, annihilations exponentially cease, eventually resulting in an asymptotic value as the remaining abundance. The Z values correspond, from top to bottom, with photons, neutral leptons (neutrinos), charged leptons, and hadrons [201]. The progression from top to bottom also correlates with increasing σ v [24]. h A i

141 7.2 The Light Elements from Big Bang Nucleosynthesis

When the Universe was young enough to have a temperature just greater than

about 1 MeV, there were three important reversible reactions between four particles:

ν + n p + e− (7.30) e ←→ e+ + n p + ν¯ (7.31) ←→ e n p + e− + ν¯ (7.32) ←→ e

At around 1 MeV, the left-to-right reactions signiﬁcantly dominated over the reverse, aided by the decoupling of the neutrinos [208]. The neutron density therefore froze

out, since neutron decay was then favored over neutron formation, resulting in a

neutron-to-proton ratio of 1/6. At this point, the reversible reaction n + p D + ←→ γ was occurring [201]. However, deuterium (D) has a very weak binding energy, and

it is very easily photodissociated (as evidenced by the forward and backward nature

of the reaction). Deuterium was a fundamental building block for the formation of

heavier elements, however, and this prevented their formation. Eventually, though,

enough deuterium was produced to get past this obstacle. In this brief lapse of

time, some of the free neutrons decayed, dropping the neutron-to-proton ratio to

1/7. Deuterium is just an isotope of hydrogen; nucleosynthesis truly began when new

elements could form, which was when the Universe cooled to about 0.1 MeV and the

electrons and positrons in thermal equilibrium annihilated each other [7]. Deuterium,

tritium, and helium-3 were produced, enabling the formation of helium-4. In fact,

helium-4 was the most stable element beyond hydrogen formed at this time, and so

142 it was the most abundant. Almost all of the free neutrons were bound into helium-

4 [201, 208]. The essential reactions for producing this abundant element are listed

below [7, 208].

n + p D + γ −→ D + p 3He + γ −→ D + D 3He + n −→ D + D p + 3H −→ D + D γ +4 He −→ 3H + D n + 4He −→ 3He + D p + 4He −→

Since there were no stable nuclei with mass number 5 (or, for that matter, 8), the formation of elements heavier than helium-4 was signiﬁcantly suppressed, due to the lack of necessary energy. However, some lithium-7 was produced, along with beryllium-7, which β-decayed to lithium-7 [7].

3H + 4He γ + 7Li −→ 3He + 4He γ + 7Be −→

In summary, hydrogen, deuterium, helium-3, helium-4, and lithium-7 were produced in signiﬁcant amounts in the early Universe, though helium-4 was formed most abun- dantly beyond hydrogen. Their observed and predicted abundances and their correlations are shown in Fig. 7.2. While deuterium, helium-3, and lithium-7 are shown in relation to hydrogen, helium-4 is, by convention, shown by its mass fraction in the

143 primordial Universe, relative to the total amount of normal matter, most of which at the time was hydrogen. The observed amount of lithium-7 does not quite agree with the predicted value, but the rest ﬁt well and attest to the strength of BBN as a theory.

144 Figure 7.2: The correlation between the predicted and observed amount of the light nuclei abundances. The predicted values are the curves, which are functions of η, the baryon-to-photon ratio or the baryon density. (A baryon is a nucleon, and the term is used to represent all normal matter, since the masses of nucleons are much bigger than the leptons.) The vertical bar for the CMB is the allowed range of the baryon density, while the BBN vertical line shows the concordance region. The (yellow) horizontal rectangular boxes are the observed values with uncertainties. Deuterium, helium-3, and lithium-7 are presented as relative to hydrogen, while helium-4’s abundance is shown as a mass fraction of the total amount of normal matter [208].

145 Based on the state of the early Universe, calculations generally are in agreement with the observed amounts of these elements. The deuterium abundance has a special signiﬁcance. Since it is so easily destroyed and hard to produce, the observed abundance today is a lower limit – it could have been produced in greater numbers in the early Universe [7].

As stated above, if there were a significant amount of dark matter annihilation during this time, and if the annihilation products were electromagnetic particles such as photons, the associated photodissociations would result in different light element abundances. The largest effects would come from the photodissociation of helium-4, creating greater proportions of the lighter nuclei. Equation (7.29) shows how the relic abundance of dark matter is inversely proportional to the annihilation cross section. Assume, for the scenario presented here, that the dark matter indeed froze out with this abundance during the time of BBN, and annihilated into electromagnetic particles. The greater the annihilation cross section, the less dark matter existed at this time. Conversely, the smaller the annihilation cross section, the more dark matter remained, meaning a greater amount of residual annihilation. For dark matter freezing out around the time of BBN, the dark matter self-annihilation cross section cannot be too small, or the resulting electromagnetic particles would result in differences in the light nuclei formation.

7.3 Modiﬁcation of Previous Work

We consider two main publications dealing with this topic. Other earlier work has been done (e.g. [209] and [210]), and a very recent, model-dependent paper has come out on the topic [211]. However, the most useful resources have been utilized here.

146 7.3.1 Frieman, Kolb, and Turner

Frieman, Kolb, and Turner (1990) (FKT) [202] explored this idea in detail. They acknowledged the inverse relationship between the relic abundance and the annihilation cross section, but were interested in the limits on the dark matter mass mχ.

Speciﬁcally, they used the results to explore limits on the mass and possible lifetime of a massive Dirac neutrino as the dark matter candidate.

As an illustration of the importance of these residual annihilations, insert the solution Y∞ (Eq. (7.23)) into the Boltzmann equation (Eq. (7.9)) to show the change in Y with Hubble time [202]:

Y˙ T 1+n | | Y , (7.33) H ' ∞ T f where n is the partial-wave expansion integer. This means that every Hubble (expan-

1+n sion) time, a fraction (T/Tf ) of the relic particles annihilate.

Mentioned earlier, we are concerned with electromagnetic products. Above a certain energy, it is more preferable for an energetic photon to interact with a background photon and make an electron-positron pair than it is to photodissociate a nucleus. Once the temperature drops enough, photodissociation can be considered.

At that point, the competition for photons produced from dark matter annihilations is between Compton scattering on the electrons and the photodissociation of nuclei.

FKT considers the photodissociation of 4He into 3He. If too much photodissociation occurs, there would be an overproduction of 3He. Speciﬁcally, they use the ratios (3He + D)/H and 4He/H, and the idea that 3He < 3He + D [202]. Using the approximate photon distribution appropriate for photodissociation and the conservative constraint that the helium-3-to-helium-4 ratio is 3He/4He < 2.4 10−3, they ×

147 are able to place a limit on the dark matter mass needed for this scenario [202]:

2 −28 3 mχ xf 10.75 0.03 3 10 cm /s 18B 2 × . (7.34) GeV ≥ 10 g∗ Ωbh σAv h i

This assumes s-wave annihilations, and is valid for 20 MeV . mχ . 1 GeV. Above

about 1 GeV, hadronic annihilation products need to be considered. The fraction of annihilation products that are electromagnetic (in this instance, photons) is B, while

2 Ωbh is the abundance of normal matter (see Table 2.1).

Examining the equation, it is trivial to turn it into a lower limit on the dark matter annihilation cross section, using B = 1, xf = 20, and the nominal value for

2 Ωbh . This is GeV σ v & 3.2 10−26cm3/s . (7.35) h A i × m χ Setting B = 1 assumes that the dark matter always annihilates into electromagnetic particles. We have been analyzing the behavior for photons, but the same holds true for showers resulting from electrons and positrions [202].

7.3.2 McDonald, Scherrer, and Walker

A decade later, McDonald, Scherrer, and Walker (2000) (McSW) [203] analyzed this scenario using the deuterium-to-hydrogen ratio instead. They used two conser-

vative measurements of this ratio, a lower value of D/H < 4 10−5 and a higher × value of D/H < 3 10−4 [203]. Since the photodissociation of 4He into D requires × 3 slightly more energy than the product He, this concerns mχ & 26 MeV. However,

2 a main diﬀerence in the calculations is the use of Ωχh – McSW did not write it as

an explicit function of σ v . Consequently, they derive an upper limit on the cross h A i

148 section instead of a lower limit. Their presented limit is [203]

0.02 GeV σ v 10−3B h A i (Ω h2)2 < 1.3 (high D) Ω h2 m 3 10−27cm3/s χ b χ × < 0.2 (low D). (7.36)

2 Inserting the cross section dependence of Ωχh and using the same values as above for the other parameters, this rearranges to

GeV σ v & 1.2 10−29cm3/s . (7.37) h A i × m χ This assumes a value of 1 for the comparison to deuterium, as a compromise between

1.3 and 0.2.

7.4 Analysis

The preliminary results presented here modiﬁed two published results to reﬂect

the dependence of Ω h2 on σ v . While interesting in their own rights as lower χ h A i limits on the annihilation cross section, they are especially signiﬁcant when shown

with the upper limit from Chapter 6. Figure 7.3 shows the limits on dark matter’s

annihilation to monoenergetic photons, for both the wide range of masses from dark

matter annihilations in the Milky Way, Andromeda, and cosmic halos, and the narrow

range of masses considered in this analysis. For the stronger bound, FKT, there is no

allowed parameter space; the weaker McSW bound allows for a small wedge. Again,

this treatment assumes Br(χχ γγ) = 1. −→

149 -15 -16 -17 -18 -19 -20

/s] -21 3 -22 -23 -24 -25

v> [cm -26 relic abundance A -27 σ -28 -29 -30 log < -31 -32 -33 -34 -35 -36 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 log m χ [GeV]

Figure 7.3: Constraints on the dark matter annihilation cross section to monoenergetic photons σAv γγ. This combines previous work [157] with the modiﬁed results from FKT (blue)h andi McSW (green). The shaded regions are excluded; the results of Ref. [157] are an upper limit, while those of this work are a lower limit. For the stronger constraint from FKT, there is no allowed region, while a small wedge remains for that of McSW.

150 Using the same conservative branching ratio of 10−4 mentioned in Chapter 6, the

limits can be expanded to constrain the total annihilation cross section (Figure 7.4),

remembering that Br(χχ γγ) σ v = σ v . Since the new bounds have taken −→ h A i h A iγγ into account the fraction of annihilations resulting in electromagnetic products as B,

they are multiplied by 10−4, instead of being divided by that factor as in the halo

limits. As expected, there is considerably more available parameter space, virtually

centered on the natural scale value if the stronger bound is used. The neutrino and

unitarity bounds could also be shown on this ﬁgure, but their constraints at the

relevant mass range are weaker than the photon limits.

For this narrow range of masses, there are now two important constraints of opposite sign on the total annihilation cross section. The dark matter cannot annihilate too

strongly, or its signal would be observable in halos; it cannot annihilate too weakly, or residual annihilations would cause an overproduction of the light BBN elements.

A more detailed analysis might shrink the remaining parameter space.

151 -15 -16 -17 -18 -19 -20

/s] -21 3 -22 -23 -24 -25

v> [cm -26 relic abundance A -27 σ -28 -29 -30 log < -31 -32 -33 -34 -35 -36 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 log m χ [GeV]

Figure 7.4: Constraints on the total dark matter annihilation cross section. The results of Fig. 7.3 have been aﬀected by the conservative branching ratio Br(γγ) = 10−4. A great amount of parameter space has now been removed for the mass range 20 MeV to 1 GeV. ∼

152 7.5 Conclusions

The preliminary results of the eﬀects of residual annihilations of dark matter during Big Bang Nucleosynthesis seem very promising. Following the theory set forth by Frieman, Kolb, and Turner (1990) [202] and McDonald, Scherrer, and Walker

(2000) [203], constraints have been placed that restrict the dark matter annihilation cross section parameter space between the masses of 20 MeV and 1 GeV. Especially interesting is the behavior of the constraint on the annihilation cross section, since it is a lower limit. The annihilation cross section is limited to be between this lower limit and the upper limit derived in Chaper 6. A deeper analysis will provide greater insight into this situation.

153 CHAPTER 8

CONCLUSIONS

Dark matter is an elusive, but essential, part of the Universe. Gravitational eﬀects strongly suggest its presence, but hopefully more direct evidence will be found shortly.

Its near-invisibility plays on the natural curiosity of scientists, sparking ideas for detecting it, whether directly, indirectly, or in high energy particle colliders. Until it is found, the nature of the dark matter particle is one of the greatest mysteries of current cosmology and astro-particle physics.

8.1 A Checklist for Potential Dark Matter Candidates

As long as the nature of dark matter is uncertain, more and more candidates will be put forth. Marco Taoso, Gianfranco Bertone, and Antonio Masiero (2007) [212] have compiled a very helpful 10-point checklist that any dark matter candidate has to satisfy, which is shown in Table 8.1.

Most of these points have been dealt with at some level in this dissertation, either in the introductory chapters, or in the ones containing new research. For instance,

Points 1, 2, and 3 were addressed in Chapters 1, 2, and 3 (not in that order). A candidate must be able to account for the amount of dark matter observed (Chap- ter 2), along with possessing the generic qualities of “cold” and “neutral”. Hot, or

154 Does it match the appropriate relic density?

Is it cold?

Is it neutral?

Is it consistent with BBN?

Does it leave stellar evolution unchanged?

Is it compatible with constraints on self-interactions?

Is it consistent with direct DM searches?

Is it compatible with gamma-ray constraints?

Is it compatible with other astrophysical bounds?

Can it be probed experimentally?

Table 8.1: The 10 qualities that a particle needs to satisfy to be a successful dark matter candidate, as displayed in Ref. [212].

155 relativistic, dark matter prevents small-scale structure from forming, by not allowing clustering on those scales. Dwarf galaxies are indeed observed, and so the dark matter must not be hot. However, it must not be too cold; simulations currently result in the formation of too many dwarf galaxies, and highly dense cusps (spikes) at the center of dark matter halos instead of more uniform cores, which are suggested by observations. The properties of dark matter must not alter what is known to be true about the Universe, such as Big Bang Nucleosynthesis (Chapters 2 and 7). Points 4,

7, 8, and 9 have been investigated in great detail in this dissertation.

8.1.1 Point 7: Direct Dark Matter Searches

A new bound has been placed on the dark matter–normal matter interaction cross section σχN for a wide range of dark matter masses, as described in Chapter

4. The analysis excludes a large region of cross sections that have larger-than-weak strengths based on the internal heat ﬂow of Earth. Underground detectors can only measure a collision if the dark matter retains a certain amount of energy. Dark matter with higher interaction cross sections will lose too much energy from collisions before reaching the detector, thereby not registering.

However, if the dark matter experiences enough collisions in this manner, it will be gravitationally captured and drift to Earth’s core. Annihilations will then occur due to the concentration of dark matter. Most of the products will deposit their energy in the core, resulting in a heat ﬂow. For eﬃciently-captured dark matter, the heat

flow is about 75 times that of Earth’s total heat flow, which has been measured very well by geologists and geophysicists. The cross sections resulting in efficient capture, therefore, are ruled out. Coupled with the existing constraints, dark matter must

156 have a weak interaction strength if its mass is between 0.1 and 1010 GeV, a strength

that is usually assumed.

8.1.2 Points 8 and 9: Indirect Gamma-ray Constraints and Other Astrophysical Bounds – Neutrinos

Dark matter’s annihilation products can provide indirect evidence for its nature,

whether it be for self-annihilation cross sections for speciﬁc ﬁnal states, or for the

total self-annihilation cross section. The annihilation products can be grouped into

two categories: (i) photons or particles that eventually produce photons and (ii)

neutrinos. Neutrino detectors have become sensitive enough to produce reliable data,

but photons can be detectable over a wider range of energies. The two extremes of

detectability are monoenergetic neutrinos and monoenergetic photons. Seeing a signal

of either would be a “smoking gun” for dark matter. The theoretical signal from dark

matter annihilation must be less than the observed background from astrophysical

sources.

In Chapter 5, neutrinos from dark matter annihilations in halos of galaxies other

than the Milky Way are analyzed. The isotropic atmospheric neutrino background

due to cosmic rays is compared with a theoretical monoenergetic neutrino signal. The

data are broken up into energy bins of half-an-order-of-magnitude, and the signal

equal to the background is looked for in each one. The lack of an observed signal

places a constraint on the dark matter annihilation cross section.

Due to the nature of neutrino and gamma-ray interactions in detectors, neutrino

limits are inherently weaker than those from gamma-rays. Therefore, a conservative

constraint on dark matter annihilation to neutrinos is the same as a limit on the total annihilation cross section. If the branching ratio to monoenergetic neutrinos were

157 known, the limits from the other ﬁnal states would be stronger than the neutrino

limit modiﬁed by Brνν (remember, Bri = 1).

Chapter 6 places a limit on the annihilationP to monoenergetic gamma-rays using

the same principle as Chapter 5. Backgrounds from the Galactic Center region,

the Galactic halo, Andromeda, and the contributions from all external galaxies are

used. The resulting constraint is strong, and it can be transformed to a bound on

the total annihilation cross section by dividing by the appropriate branching ratio.

The conservative value of 10−4 is used here, making it comparable to the limit from

neutrinos. A combination of the gamma-ray, neutrino, and unitarity limits comprise

a model-independent, conservative upper bound on the annihilation cross section

covering a wide range of masses.

8.1.3 Point 4: Consistency with Big Bang Nucleosynthesis

Preliminary results have also been discussed regarding Point 4, speciﬁcally the ef-

fect of residual dark matter annihilations on BBN. The photodissociation of helium-4

in the epoch of light element formation would result in an overabundance of light

nuclei such as deuterium and helium-3. When a particle freezes out of thermal equi-

librium, its annihilations do not cease entirely. If dark matter experienced these

residual annihilations after freezing out, and the annihilation products were photons,

electrons, or positrons, then this photodissociation could occur.

The amount of dark matter left over to experience these residual annihilations is

determined by the annihilation cross section. The smaller the cross section, the more

particles remain to annihilate. The cross section therefore cannot be too weak, or the

overabundance of dark matter would lead to a corresponding overabundance of light

158 nuclei. Chapter 7 details this lower limit, which is combined with the upper limit from monoenergetic photons in Chapter 6 to create a restriction of the total dark

matter annihilation cross section parameter space as a function of mass.

8.2 Closing Summary

The investigations in this dissertation have established new constraints on the

particle nature of dark matter by comparing theory with observation. They have

examined the consequences of dark matter properties, speciﬁcally a strong interaction

cross section with normal matter, a strong self-annihilation cross section, and a weak

annihilation cross section. Eﬀects are not seen; constraints are placed. There are

many more model-independent investigations to explore, involving both direct and

indirect detection. Eventually, striking evidence about dark matter’s particle nature

will be found. That will be a very exciting day.

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