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The limiting background in dark matter search at shallow depth

THE LIMITING BACKGROUND IN A DARK MATTER SEARCH AT
SHALLOW DEPTH

by
THUSHARA PERERA

Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy

Thesis Advisor: Daniel S. Akerib

Department of Physics
CASE WESTERN RESERVE UNIVERSITY

January, 2002


To my parents


Table of Contents

Dedication . . . .
Table of Contents
List of Tables . .
List of Figures . .
Acknowledgments
Abstract . . . . .

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1 WIMP Dark Matter
1.1 Introduction . . . . . . . . . . . . . . . . . . . .
1.2 Present-Day Cosmology . . . . . . . . . . . . .
1.2.1 Theoretical Framework . . . . . . . . . .
1.2.2 Constraints on Ωm and ΩΛ . . . . . . . .
1.3 Evidence for Non-Baryonic Cold Dark Matter .
1.3.1 Dark Matter . . . . . . . . . . . . . . . .
1.3.2 Baryonic and Non-Baryonic Dark Matter
1.3.3 Hot and Cold Dark Matter . . . . . . . .
1.4 Weakly Interacting Massive Particles . . . . . .
1.5 WIMP Detection . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
2 The CDMS I Experiment
2.1 Introduction . . . . . . . . . . . .
2.2 Backgrounds and Shielding . . . .
2.2.1 Photon Backgrounds . . .
2.2.2 Neutron Backgrounds . . .
2.3 CDMS Detectors . . . . . . . . .
2.3.1 The Phonon Measurement
2.3.2 Charge Measurement . . .
2.4 Cryogenics and Electronics . . . .
2.4.1 The Icebox . . . . . . . .
2.4.2 Mounting of Detectors and
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Electronics

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. xvi

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2.4.3 Room Temperature Electronics and Data Acquisition . 50
2.5 The Future of CDMS . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Monte Carlo Tools and Their Use in Interpreting Calibration
Data
56
3.1 The Need for Detailed Monte Carlo Simulations . . . . . . . . 57
3.2 Monte Carlo transport code used in CDMS . . . . . . . . . . . 58
3.2.1 Specialized tools for GEANT in CDMS Simulations . . 59
3.3 Geometry Definition for Monte Carlos . . . . . . . . . . . . . 63
3.4 Output of the Monte Carlo . . . . . . . . . . . . . . . . . . . . 65
3.5 Neutron Calibration . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.2 Setup for Simulation . . . . . . . . . . . . . . . . . . . 66
3.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.4 Interesting Features in the Neutron Calibration Data
and Monte Carlo . . . . . . . . . . . . . . . . . . . . . 80
3.6 Veto-Coincident Neutrons . . . . . . . . . . . . . . . . . . . . 93
3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6.2 Monte Carlo Setup . . . . . . . . . . . . . . . . . . . . 93
3.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.7 Photon Calibration . . . . . . . . . . . . . . . . . . . . . . . . 99
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4 Data and Results from CDMS I
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Run 19 Data Set and Analysis . . . . . . . . . . . . . . .
4.2.1 Trigger, Charge Search, and Analysis Thresholds
4.3 Software Cuts and Their Efficiencies . . . . . . . . . . .
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . .
4.3.2 Trace Quality Cuts . . . . . . . . . . . . . . . . .
4.3.3 Physics Cuts . . . . . . . . . . . . . . . . . . . .
4.4 Veto-Coincident Data . . . . . . . . . . . . . . . . . . . .
4.5 Veto-Anticoincident Data . . . . . . . . . . . . . . . . . .
4.6 Dark Matter Analysis . . . . . . . . . . . . . . . . . . . .
4.6.1 Veto-Anticoincident Nuclear Recoils . . . . . . . .
4.6.2 The Neutron Interpretation . . . . . . . . . . . .
4.6.3 Upper Limits on WIMP Dark Matter . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 The Neutron Background in CDMS I
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Possible Sources of External Neutrons . . . . . . . . . . . . .
5.2.1 Neutrons from Cosmic-ray Muons . . . . . . . . . . .
5.2.2 Neutrons from Natural Radioactivity . . . . . . . . .
5.2.3 Rates and Spectra of External Neutrons . . . . . . .
5.3 Studies of Neutron Shielding and Detection in CDMS I . . .
5.3.1 Importance of Neutrons from Hadron Showers . . . .
5.3.2 Spectrum Independence of Results . . . . . . . . . .
5.4 Predictions of External Neutron Monte Carlo and Comparisons with Data . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Additional Shielding of External Neutrons in CDMS I . . . .
5.6 Neutron Background for CDMS II . . . . . . . . . . . . . . .
5.7 Direct Simulation of External Neutrons Through Muon Transport in Rock . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 Monte Carlo Setup . . . . . . . . . . . . . . . . . . .
5.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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183

6 Tests of a Z-sensitive Ionization and Phonon mediated Detector
186
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.2 The ZIP Phonon Technology . . . . . . . . . . . . . . . . . . . 187
6.2.1 Transition Edge Sensors . . . . . . . . . . . . . . . . . 187
6.2.2 Voltage Bias and Electrothermal Feedback . . . . . . . 189
6.2.3 Production and Trapping of Quasiparticles . . . . . . . 191
6.2.4 Biasing and Readout Scheme . . . . . . . . . . . . . . 193
6.2.5 Design Considerations for ZIP Detectors . . . . . . . . 195
6.2.6 Advantages of Using ZIP Detectors . . . . . . . . . . . 199
6.3 Tests of a CDMS II ZIP Detector . . . . . . . . . . . . . . . . 200
6.3.1 Detector Characterization at C.W.R.U. . . . . . . . . . 200
6.3.2 Diagnostics and Testing of a ZIP Detector . . . . . . . 202
6.3.3 SQUID and QET biasing . . . . . . . . . . . . . . . . . 210
6.3.4 Description of Data . . . . . . . . . . . . . . . . . . . . 213
6.3.5 Position Dependent Phonon Energy Calibration . . . . 217
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7 Conclusion
224
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

vi


A Output of the GEANT Based Monte Carlo
228
A.1 Event-by-event Quantities . . . . . . . . . . . . . . . . . . . . 229
A.2 Hit-by-hit Quantities . . . . . . . . . . . . . . . . . . . . . . . 231
Bibliography

233

vii


List of Tables
3.1
3.2
3.3
3.4
5.1
5.2
6.1

Results of fiducial volume calculation. . . . . . . . . . . . . . .
Comparison of data and Monte Carlo for neutron calibrations.
Information on the 73 Ge nuclear excitations. . . . . . . . . . .
Comparison of data and Monte Carlo for veto-coincident neutrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72
81
86
98

Production rates, fluxes and detection rates for the three possible sources of external neutrons. . . . . . . . . . . . . . . . . 158
Comparison of rates and ratios between the external neutron
Monte Carlo and the veto-anticoincident nuclear recoils. . . . 168
Transition temperature, normal resistance, and critical current
at base temperature for the four phonon sensors. . . . . . . . . 203

viii


List of Figures
1.1
1.2

Rotation curves of spiral galaxies. . . . . . . . . . . . . . . . .
Measurements of mass-to-light ratio as a function of dynamical
scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1
2.2
2.3
2.4
2.5
2.6

The Stanford Underground Facility (SUF). . . . . . . . . . .
The CDMS I shield. . . . . . . . . . . . . . . . . . . . . . .
Nuclear- vs. electron-recoil discrimination used in CDMS. . .
BLIP detector. . . . . . . . . . . . . . . . . . . . . . . . . .
The NTD thermistor based phonon readout circuit. . . . . .
Approximate band structure of intrinsic Ge crystals used in
CDMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Simplified version of the ionization readout circuit. . . . . .
2.8 The readout circuitry of a BLIP detector. . . . . . . . . . . .
2.9 Cartoon of blocking electrodes. . . . . . . . . . . . . . . . .
2.10 The CDMS I cryostat. . . . . . . . . . . . . . . . . . . . . .
2.11 Detector mounts and tower . . . . . . . . . . . . . . . . . . .
2.12 Block diagram of the CDMS data acquisition system . . . .
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

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51

Schematic depicting the definition of “clumps”. . . . . . . . .
Average clump size vs. energy deposited for Ge and Si. . . . .
Geometry definition used in Run 19 simulations. . . . . . . . .
Charge Yield versus Recoil energy in the first neutron calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radii of BLIP5 inner and outer contained events from Monte
Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detector-by-detector comparison of data and Monte Carlo spectra for the first neutron calibration. . . . . . . . . . . . . . . .
Comparison of summed spectra from the first neutron calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detector-by-detector comparison of data and Monte Carlo spectra for the second neutron calibration. . . . . . . . . . . . . .

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3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13

Comparison of summed spectra from the second neutron calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Charge versus recoil energy from the second neutron calibration. 82
Energies of photon scatters in the neutron calibration. . . . . . 85
Proposed method for measuring the high-energy neutron flux. 90
Detector-by-detector comparison of data and Monte Carlo spectra for veto-coincident neutrons. . . . . . . . . . . . . . . . . . 97
Comparison of summed spectra between data and Monte Carlo
for veto-coincident neutrons. . . . . . . . . . . . . . . . . . . . 98
Charge yield vs. recoil energy for inner and shared events in
the 6 V photon calibration data. . . . . . . . . . . . . . . . . . 100
Charge yield in BLIP1 from Run 18 data. . . . . . . . . . . . 103
Charge yield for BLIP1 in Run 18 as estimated by the Monte
Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Distributions of low charge yield events in the Run 18 photon
calibration Monte Carlo. . . . . . . . . . . . . . . . . . . . . . 106
BLIP4 charge yield vs. BLIP3 charge yield for electron-calibration
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Cumulative raw livetime for low-background data in Run 19. . 115
Phonon trigger efficiencies in BLIPs 3 through 6. . . . . . . . 117
Phonon χ2 vs. phonon energy for typical low-background data
form Run 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Charge-yield distributions for 10-100 keV veto-anticoincident
inner events in BLIPs 3 through 6. . . . . . . . . . . . . . . . 127
Distribution of veto-trigger times relative to charge triggers. . 128
Distribution of veto-trigger times relative to the inferred charge
pulse time for phonon triggers. . . . . . . . . . . . . . . . . . . 129
Recoil-energy spectra for veto-coincident inner events. . . . . . 131
Recoil-energy spectra for veto-coincident shared events. . . . . 132
Ionization yield vs. recoil energy for veto-anticoincident single
scatters in BLIPs 4 through 6. . . . . . . . . . . . . . . . . . . 134
Single-scatter photon and electron spectra for veto-anticoincident
inner events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Single-scatter photon and electron spectra for veto-anticoincident
shared events. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Recoil energy distribution of inner nuclear-recoil candidates. . 137
Scatter plot of ionization yields for veto-anticoincident double
scatters in BLIPs 4 through 6. . . . . . . . . . . . . . . . . . . 139

x


4.14 Ionization yield vs. recoil energy for veto-anticoincident events
in the Run 18 Si ZIP detector. . . . . . . . . . . . . . . . . . . 141
4.15 Schematic comparison of simulated and observed numbers of
nuclear-recoil events. . . . . . . . . . . . . . . . . . . . . . . . 142
4.16 Spin-independent σ vs. M limit plot. . . . . . . . . . . . . . . 147
5.1

Flux-normalized neutron spectra at the SUF from simulations
of neutron production mechanisms external to the shield. . . .
5.2 Penetration and detection probability of neutrons as a function
of neutron energy outside the shield. . . . . . . . . . . . . . .
5.3 Spectra of neutrons incident on detectors for a range of initial
neutron energies. . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 The dependence of mean recoil energy and multiples fraction
on initial neutron energy. . . . . . . . . . . . . . . . . . . . . .
5.5 Production (dark) and ambient (light) spectra of external neutrons at the SUF. . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 The neutron spectra incident on detectors due to external and
internal neutrons. . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Recoil energy spectra from the veto-anticoincident germanium
data set and the corresponding external neutron Monte Carlo.
5.8 Comparison of observed and predicted cumulative spectra for
veto-anticoincident neutrons. . . . . . . . . . . . . . . . . . . .
5.9 Geometry setup for the FLUKA Monte Carlo. . . . . . . . . .
5.10 Muon spectra at ground level and at the SUF tunnel from
FLUKA simulations. . . . . . . . . . . . . . . . . . . . . . . .
5.11 Ambient neutron spectra inside the SUF tunnel from GEANT
and FLUKA simulations. . . . . . . . . . . . . . . . . . . . . .
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9

Resistance versus temperature for a Transition Edge Sensor
(TES). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pictorial representation of quasiparticle trapping and diffusion
in ZIP detectors. . . . . . . . . . . . . . . . . . . . . . . . .
The biasing and readout scheme for phonon sensors in ZIP
detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Present aluminium fin and TES design in ZIP detectors. . .
Phonon side of a ZIP detector. . . . . . . . . . . . . . . . . .
IbIs data from sensor A. . . . . . . . . . . . . . . . . . . . .
IbIs data from sensor B. . . . . . . . . . . . . . . . . . . . .
IbIs data from sensor C. . . . . . . . . . . . . . . . . . . . .
IbIs data from sensor D. . . . . . . . . . . . . . . . . . . . .

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157
159
162
164
166
167
170
171
177
179
180

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197
199
205
206
207
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6.10 The noise power spectral density for the four phonon channels
and two charge channels of a ZIP detector. . . . . . . . . . . . 212
6.11 Ionization energy versus phonon energy from detector G6 before applying the position dependent phonon energy calibration.215
6.12 Histogram of ionization energy in G6 . . . . . . . . . . . . . . . 215
6.13 Y delay vs. xdelay for 241Am and 137Cs photons in G6 . . . . . 216
6.14 Position dependence of phonon pulse height. . . . . . . . . . . 218
6.15

× 100 for a grid in xdelay and ydelay . . . . . . . . . 219
6.16 Ionization energy vs. calibrated phonon energy. . . . . . . . . 220
7.1

Projected sensitivity of CDMS II and WIMP upper limits from
recent experiments. . . . . . . . . . . . . . . . . . . . . . . . . 225

A.1 Raw output of GEANT based Monte Carlo. . . . . . . . . . . 229

xii


Acknowledgments

The time I have spent working on CDMS has been a very enjoyable and exciting one. In addition to the wide variety of physics problems and challenges
that have come my way, my life has been enriched by the people I have associated with during this time. Of these people, I mention first Dan Akerib,
my thesis advisor. I have benefitted immensely from Dan’s broad knowledge
of physics, perspective on issues, his ability to quickly grasp and explain new
concepts, and his talent for separating the essentials from the details. Before
long, I realized how fortunate I was to have an advisor like Dan. I am proud
to be his first graduate advisee. I hope that many more young physicists will
have the opportunity to experience the leadership, humility, encouragement,
and friendship that he brings as a thesis advisor.
I thank the two research associates I have worked with in the CWRUCDMS group, Alex Bolozdynya and Richard Schnee, for their guidance and
friendship. I thank Alex for giving me a feel for the practical, hardwareoriented side of experimental physics which I now relish. I also treasure the
many conversations we have had on more esoteric topics. I have learned much
from Richard on subjects ranging from poker to data analysis. I admire his
overall knowledge of the experiment, and had a good opportunity to make use
of it during the last few months of thesis writing. Special thanks to him for
helping me out in many ways during the last few months. Don Driscoll was
my first office mate and colleague at CWRU. I have greatly enjoyed talking
physics and solving problems with him. I have happily become addicted to his
xiii


humor and optimism as well as the wonderful company of his family, Diana
and Astra. Gensheng Wang, the next graduate student to join the group
has been a pleasure to work with. In addition to the general comradery we
have developed, I have enjoyed and learned much from our interesting and
often lively discussions. I also thank him and his family for treating me on
many occasions to the best Chinese food I have ever tasted. In the last year,
I have had the pleasure of working with Sharmila Kamat. Her hard work,
and questions have been very helpful to me. I hope I have done a decent
job of answering some of those questions. I would also like to thank the
undergrads, Matt, Mo, Tim P, Peter, Aaron, and Tim J for their tireless
work and for making the lab a fun and lively workplace. Thanks also to
Cheshana Marshal, and Mike Stamatikos for their excellent company.
I would like to thank all the secretaries in the physics department for
making my life much easier than it could have been. I also thank my thesis
committee for their interest and patience. I am grateful to Lawrence Krauss
for pointing me in the direction of Dan and CDMS when I started looking
for research work a while back.
I thank Bernard Sadoulet and Blas Cabrera for welcoming me to the
CDMS collaboration. My exposure to the rest of the CDMS collaboration
occurred mainly over the month-long visits I made to Stanford. During these
stays and later, I have had the opportunity of working closely with Rick
Gaitskell whose guidance and encouragement have been valuable to me. I
am also grateful to Tom Shutt for his help and patient explanations during my
first days in CDMS. I also thank Angela Da Silva whose work was the starting
point for many of the studies described here. I was fortunate to have been
paired with Steve Eichblatt on most of these studies. I have greatly enjoyed
his company and friendship. I have enjoyed many interesting and informative
xiv


discussions with Dan Bauer, Sae Woo Nam, Andrew Sonnenschein, Roland
Clarke, and Sunil Golwala during my visits to Stanford. I am especially
thankful to Sunil for his perseverance and hard work in making Run 19 the
success that it was. I thank Steve Yellin, Dennis Seitz, Laura Baudis, Maria
Isaac, Paul Brink, Patrizia Meunier, Tarek Saab, and Vuk Mandic for helping
me with much of the work described in this dissertation. Although I haven’t
worked on specific projects with many of the other members of CDMS, I owe
all of them a big thank you because I have benefitted from all of their hard
work.
I thank all my friends and relatives all over the world! Special thanks to
Chaminda, Kathy and the kids, and my friend Bala for their encouragement
and support during the last few months of thesis writing. I am grateful to
Christine for her company, and friendship. I also thank my brother Gehan
for his support and words of encouragement during trying times. Finally but
most importantly, I thank my parents. I would not be who I am or where I
am if not for the love, freedom, and opportunities they have given me. Thank
you.

xv


The Limiting Background in a Dark Matter Search at Shallow Depth

Abstract
by

Thushara Perera

A convincing body of evidence from observational and theoretical astrophysics suggests that matter in the universe is dominated by a non-luminous,
non-baryonic, non-relativistic component. Weakly Interacting Massive Particles (WIMPs) are a proposed particle candidate that satisfy all of the above
criteria. They are a front-runner among dark matter candidates because their
predicted contribution to matter in the universe is cosmologically significant
and because they may arise naturally from supersymmetric (SUSY) models
of particle physics. The Cryogenic Dark Matter Search (CDMS) employs
advanced detectors sensitive to nuclear recoils caused by WIMP scatters and
capable of rejecting ionizing backgrounds.
The first phase of the experiment, conducted at a shallow site, is limited
by a background of neutrons which are indistinguishable from WIMPs in
terms of the acquired data. By accounting for and statistically subtracting
these neutrons, CDMS I provides the best dark matter limits to date over a
wide range of WIMP masses above 10 GeV/c2. These results also exclude
the signal region claimed by the DAMA annual modulation search at a >71%
confidence level.
xvi


The second phase of the experiment, located at a deep site, is scheduled
to begin data acquisition in 2002. Due to longer exposures, larger detector
mass, and low background rates at this site, data from CDMS II are expected
to improve on present WIMP sensitivity by about two orders of magnitude.
Emphasized in this work are the research topics in which I have been
directly involved. These include the work described in Chapters 3 and 5
with regard to the development and use of simulation tools, detailed studies
into the limiting neutron background, and the present understanding of this
background in relation to CDMS I and CDMS II. I was also involved in
several detector development projects in preparation for CDMS II. Analysis
of test data from a ZIP detector, planned for use in CDMS II, is presented
in Chapter 6.

xvii


Chapter 1
WIMP Dark Matter
1.1

Introduction

Much of the theoretical and observational work in present-day astrophysics
revolves around the dark matter problem. Particle physics also plays an
integral role in research into this subject. The Cryogenic Dark Matter
Search (CDMS) is designed for the direct detection of Weakly Interacting
Massive Particles (WIMPs), a strongly motivated dark matter candidate.
In this chapter, I will briefly outline the reasoning and evidence behind the
dark matter problem, the need for non-baryonic cold dark matter, the
motivation for WIMPs, and some specifics regarding their detection.
The dark matter problem refers to the lack of luminous matter for
explaining certain astronomical observations under the framework of
conventional gravity. This discrepancy between theory and observation may
be explained by a “dark” component of matter in the universe. The
discussion in this chapter is phrased in terms of this assumption. Another
solution to the dark matter problem may be the discovery of a new theory
of gravity and inertia that does not require additional matter to reconcile
theory and observation. Several such theories have been proposed.
However, none of them have gained widespread acceptance due mainly to
1


2
aesthetic reasons1 . In either case, experimental searches for dark matter,
like CDMS, serve an important purpose. They are useful for detecting or
setting limits on several proposed dark matter candidates.

1.2
1.2.1

Present-Day Cosmology
Theoretical Framework

The development below follows several standard text books on
cosmology [3, 4, 5]. The details may be found in these references.
The Robertson-Walker metric, which is the outcome of assuming
that the universe is homogeneous and isotropic on large scales, is given by
dr2
+ r2 dθ2 + r2 sin2 θdφ2
1 − kr2

ds2 = dt2 − R2 (t)

(1.1)

where R(t) is a scale factor with units of length. It is scaled to give k the
values +1, −1, or 0, for positively curved, negatively curved, and flat
universes. The Einstein equations for this metric simplify to the Friedmann
equations given below.

R

2

+

8πG
Λ
k
=
ρ+
2
R
3
3

¨
R
4πG
Λ
=−
(ρ + p) +
R
3
3

(1.2)

(1.3)

Here, units with c = 1 have been used. The matter density and pressure of
the “fluid” filling the universe are given by ρ and p respectively. Newton’s
constant is represented by G while k is the curvature index as given above.
Terms involving the cosmological constant Λ can be absorbed into the first
term of each r.h.s. above, if it is viewed as a “fluid” component having
energy density ρΛ = Λ/8πG and pressure pΛ = −ρΛ . For matter and
radiation the pressure p is given by 0 and ρ/3, respectively.
1

Several authors have recently claimed that the predictions of one such theory, MOdified
Newtonian Dynamics (MOND) compare unfavourably with existing data. [1, 2]


3
The Hubble parameter is given by
H=


.
R

(1.4)

According to equation 1.2, a flat universe (k = 0) implies that
ρtotal = ρ + ρΛ = 3H 2 /8πG. This value of ρtotal is referred to as ρc , the
critical density. Using this definition, equation 1.2 may be rewritten as
1+

k
= Ωm + ΩΛ = Ω
H 2 R2

(1.5)

where the Ω’s are obtained by dividing the respective densities (ρ and ρΛ )
by ρc . Note that both Ω’s are functions of R. Also note that a Ω geater
than, less than, or equal to unity correspond to positively-curved,
negatively-curved, and flat universes, respectively. The Hubble constant H0
is the present value of H. It is usually quoted to be [6]
H0 = 71 ± 7 km/sec/Mpc = 100h km/sec/Mpc

(1.6)

where the dimensionless parameter h is useful for expressing the uncertainty
in H0 when quoting cosmological parameters. For example, using this
value, the current critical density is estimated at 1.1 × 10−6 h2 GeV/cm3.
Light emitted in the past is redshifted by a factor 1 + z given by
1+z =

R0
R

(1.7)

where R0 is the present value of the scale factor while R is the value of the
scale factor at the time that light was emitted. According to equation 1.5,
the r.h.s. of equation 1.7 is a function of the curvature k, Ωm , and ΩΛ . The
redshift (z) in the l.h.s. is a measurable.
When a disk and a point are separated by a large distance, the solid
angle subtended by the disk at the point depends on the curvature of space.
Light rays from the point to the edges of the disk will be convergent,
divergent, and straight for positive-curvature, negative-curvature and flat


4
universes respectively. Curvature-measurement experiments where the
point is an observer and the disk is the a far-away object of known size are
called standard ruler tests. Standard candle tests, where the redshift of an
object is recorded against its luminosity distance, also yield information on
curvature.

1.2.2

Constraints on Ωm and ΩΛ

Distance versus redshift measurements on high-redshift supernovae Ia [7, 8]
and measurements of the first-Doppler-peak angular size in the Cosmic
Background Radiation (CBR) [9, 10], are highly successful instances of the
two methods outlined above. Since z is a measurable and the r.h.s of
equation 1.7 depends on curvature and the Ω’s, measuring curvature using
these methods puts constraints on specific functions of Ωm and ΩΛ .
Experimental constraints on Ωm and ΩΛ are obtained using measurements
at several redshifts. While to first order, the angular size of the first CBR
peak is only sensitive to Ω = Ωm + ΩΛ , the shape of the angular power
spectrum of temperature anisotropy and positions of other peaks in that
spectrum can be used to obtain possible ranges in Ωm and ΩΛ [11, 12].
These experiments, together with other observations [13] have in recent
years provided accurate evidence in favor of a particular cosmological
model. This model is described by


1, Ωm

0.3, ΩΛ

0.7.

(1.8)

This model is contrary to previous expectations that Ω = 1 and Ωm ≥ 0.9.
In recent years, the above model has lead to a concentrated effort to find a
good particle physics motivation for a non-zero Λ.
A theoretical prejudice for Ω = 1 exists for two reasons.
Equations 1.2 and 1.3 can be used to show that Ω = 1 is the only static
solution for Ω. Furthermore, values of Ω different from unity will lead to


5
large deviations from unity in a very short time. Therefore, in the early
universe, Ω must have been extremely close to unity in order to be
consistent with the present observation of a nearly flat universe. Therefore,
if Ω is not exactly unity, a mechanism for extreme fine tuning is required to
explain its minute deviation from unity in the early universe. This is the
first theoretical reason for expecting that Ω = 1. The other is due to a
popular class of theories known as inflation. These theories were first
developed to explain the absence of magnetic monopoles. Soon thereafter,
it was realized that they also explain the startling homogeneity of the
cosmic microwave background radiation which is incident on the earth
today from a large number of causally disconnected parts of the universe.
In inflation theories, an exponential expansion of a previously-small
causally-connected patch of space is used to explain this homogeneity. The
exponential expansion drives the scale factor R to a large value, thus
making the second term on the l.h.s. of equation 1.5 negligibly small.
Therefore, even if the curvature k is non-zero, Ω is driven to unity.

1.3
1.3.1

Evidence for Non-Baryonic Cold Dark
Matter
Dark Matter

The galaxy luminosity density in the nearby universe is measured to be [14]
Lg = 3.3 × 108 hL /Mpc3 .

(1.9)

where L refers to the sun’s luminosity. Given this luminosity density, the
matter density will equal ρc when the mass to luminosity ratio is
M
L

1400h

M
L

(1.10)

where M refers to the solar mass. Thus, the matter density in the universe
cannot be equal to the critical density or a significant fraction of it, if most


6
of the matter in the universe is in objects like like stars
(M/L < 10Modot /Lodot ).
It addition to the above argument, there are several direct
measurements that support the existence of a dark matter problem. The
most familiar of these are from the rotation curves of spiral galaxies and
observations of galaxy clusters.
Galactic Rotation Curves
The tangential velocity of stars in the plane of a spiral galaxy can be
measured using redshifts of stellar absorption lines. About 83% of the
luminous matter in a typical spiral galaxy is contained within a radius Ropt
of about 10 kpc. The radio emission line of neutral hydrogen can be used to
trace velocities beyond this point to about 2Ropt [16]. A set of velocity
curves obtained in this way for several typical spiral galaxies is shown in
figure 1.1. The crucial feature of these plots is that the velocity curve
flattens beyond r > Ropt. According to Newtonian gravity, which is
applicable in this case, the tangential velocities are governed by
V2
GM (r)
=
r
r2

(1.11)

If most matter in the galaxy were luminous, the velocity curve beyond Ropt

is therefore expected to vary as 1/ r. The observed rotation curves clearly
indicate the presence of a non-luminous component. The expected rotation
curve due to the luminous matter is also displayed in figure 1.1. The
dashed curve indicates the rotation curve due to a particular “dark halo”
model used by the authors of [15]. Surveys of spiral-galaxy rotation curves
typically yield mass-to-light ratios greater than 10 hM /L .


7

Figure 1.1: Rotation curves of spiral galaxies. The radius is given in units
of Ropt. The velocities are normalized to the velocity at Ropt. Data are
shown with error bars. The dashed curve is the velocity contribution due
to an assumed dark halo, which is modeled to have a density distribution
proportional to r3 /(r2 + a2) where a is a constant [15]. The dotted and solid
curves represent the expected rotation curves due to luminous matter and
the combination of dark and luminous components. Figure taken from [15]


8
Clusters of Galaxies
Galaxy clusters are gravitationally bound systems of up to several thousand
galaxies. Because of their large size, galaxy clusters are expected be a fair
sampling of the universe. Mass-to-light measurements on galaxy clusters
can therefore be used to estimate Ωm for the universe.
Mass-to-Light Ratios
Several methods are used to extract mass-to-light ratios from galaxy
clusters. The first method uses the peculiar velocities of galaxies to
estimate gravitational potential energy in a cluster [17]. The gravitational
potential is obtained from peculiar velocities through the virial theorem
which states that
1
= −
2

(1.12)

where and are the average values of kinetic and potential
energy respectively. This method is valid as long as the cluster is in a state
of dynamic equilibrium. The “first” discovery of the dark matter problem is
attributed to Zwicky, who in 1933 used this method on the Coma cluster.
X-ray emission from hot intracluster gas can also be used for
mass-to-light estimates. Hydrostatic equilibrium is assumed for gas in the
central part of the cluster. The observed x-ray maps are then fit to models
of temperature and density distributions of the gas [18, 19]. Gravitational
lensing of background galaxies by clusters has also been used to measure
the dark matter content of clusters [20]. Most mass-to-light estimates
obtained from galaxy clusters lie in the range (250 − 450)hM /L . This
implies that Ωm is in the range 0.18 to 0.32.
Cluster Baryon Fraction
As with mass-to-light ratios, the ratio of baryon density to matter
density in galaxy clusters is expected to be a fair sampling of the baryon


9

Figure 1.2: Measurements of mass-to-light ratio as a function of dynamical
scale. Figure taken from [22]
fraction of the universe. Once the baryon fraction of a cluster is measured,
it can be combined with constraints on baryon density provided by Big
Bang nucleosynthesis, discussed below, to obtain Ωm . These estimates are
in good agreement with mass-to-light estimates from galaxy clusters, with
typical quoted Ωm s of about 0.4±0.1 [21].
Consistency of Evidence
The above methods and other studies (eg. large scale flows, Virgo infall)
yield a coherent picture of matter density in the universe. All estimates of
Ωm from large scale structures, with fair sampling of the universal matter
density, give values consistent with the picture described by equation 1.8.
Figure 1.2 shows the light to mass ratios from a large number of
measurements carried out at different length scales [22].


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