Multi-Year Search for a Diffuse Flux of Muon Neutrinos
with AMANDA-II
by
Jessica Louise Hodges
A dissertation submitted in partial ful?llment of the
requirements for the degree of
Doctor of Philosophy
(Physics)
at the
University of Wisconsin { Madison
2007
?
c
Copyright by Jessica Louise Hodges 2007
All Rights Reserved
Multi-Year Search for a Diffuse Flux of Muon
Neutrinos with AMANDA-II
Jessica Louise Hodges
Under the supervision of Professor Albrecht Karle
At the University of Wisconsin | Madison
Neutrinos are valuable messengers that are expected to help answer fundamental ques-
tions about our Universe, including the origin of cosmic rays and the nature of cosmic
accelerators. Neutrino astrophysics may even open a window to processes or objects
we have never yet imagined. The AMANDA-II detector was constructed to search for
and identify distant neutrino sources with non-thermal components. This analysis of
AMANDA-II data collected between February 2000 and November 2003 searches for a
di?use ?ux of TeV - PeV muon neutrinos from unresolved astrophysical sources across
the entire northern sky. Since astrophysical neutrinos are expected to have a harder
energy spectrum than the atmospheric muon and neutrino backgrounds, an energy-
dependent parameter was used to separate the signal and background event classes.
No excess of events was seen in the data over the expected background, therefore up-
per limits were placed on the di?use ?ux of muon neutrinos based on several di?erent
astrophysical neutrino models. Because of their harder spectra, prompt atmospheric
neutrino predictions were also tested and constrained.
Albrecht Karle (Adviser)
i
Acknowledgments
This work would not have been possible without the support of many people. You
encouraged and guided me. You corrected me when I was wrong and supported me
when I was right. You listened. You comforted. You laughed with me. For this, I am
very grateful. My thanks especially go to:
Albrecht Karle, for advising me.
Gary Hill, for always having an open door and willingly explaining it one more time.
Teresa Montaruli, Francis Halzen, Bob Morse, Paolo Desiati and Chris Wendt,
for helpful discussions and support.
John Kelley and Jim Braun, my o?ce mates, for putting up with me every day.
And for listening, answering and advising. I will miss Team J!
Markus Ackermann and the Zeuthen ?ltering crew, for providing this data.
Tom Gaisser, Todor Stanev, Spencer Klein, Kurt Woschnagg, Marek Kowalski
and Philippe Herquet, for serving as reviewers of my publication.
Amy, for listening to me talk about cuts and unblindings at the dinner table.
My parents, who told me they would always be my #1 fans.
Dan, for loving me every day.
Thank you all for believing in me!
ii
Contents
Acknowledgments
i
1 Introduction
1
1.1 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Neutrino Astronomy
6
2.1 What is a neutrino? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Neutrinos as Cosmic Messengers . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Producing Neutrinos in Astronomical Sources . . . . . . . . . . 8
2.2.2 Fermi acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Finding Astrophysical Neutrinos . . . . . . . . . . . . . . . . . . . . . . 12
3 Astrophysical Neutrino Models and Limits
14
3.1 ? / E
? 2
Neutrino Models . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 Waxman-Bahcall Upper Bound . . . . . . . . . . . . . . . . . . 16
3.1.2 Nellen, Mannheim and Biermann Model . . . . . . . . . . . . . 17
3.1.3 Becker, Biermann and Rhode Model . . . . . . . . . . . . . . . 17
3.1.4 Mannheim, Protheroe and Rachen Upper Bound for Thick Sources 18
iii
3.2 Neutrino Spectra Di?erent than ? / E
? 2
. . . . . . . . . . . . . . . . . 18
3.2.1 Stecker, Done, Salamon and Sommers AGN Core Model . . . . 18
3.2.2 Mannheim, Protheroe and Rachen Upper Bounds for Thin Sources 19
3.2.3 Mannheim, Protheroe and Rachen Upper Bound for Neutrinos
from AGN Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.4 Loeb and Waxman Starburst Model . . . . . . . . . . . . . . . . 20
3.3 Existing Astrophysical Neutrino Upper Limits . . . . . . . . . . . . . . 21
4 Atmospheric Neutrinos
24
4.1 Conventional vs. Prompt Atmospheric Neutrinos . . . . . . . . . . . . 24
4.2 Prompt Atmospheric Neutrino Models . . . . . . . . . . . . . . . . . . 28
4.3 Existing Prompt Neutrino Upper Limit . . . . . . . . . . . . . . . . . . 33
5 Neutrino Detection with AMANDA
34
5.1 Search Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 AMANDA Detection Principle . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 AMANDA Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.1 AMANDA Coordinate System . . . . . . . . . . . . . . . . . . . 41
5.4 Optical Properties of South Pole Glacial Ice . . . . . . . . . . . . . . . 43
6 Preparation of the 2000 - 2003 Sample
47
6.1 Livetime and Triggers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 Rejection of Atmospheric Muon Background . . . . . . . . . . . . . . . 49
6.3 Reconstruction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4 Techniques to Further Improve Background Rejection . . . . . . . . . . 53
iv
6.4.1 Zenith-weighted (Bayesian) Reconstruction . . . . . . . . . . . . 53
6.4.2 Removal of Electronic Crosstalk . . . . . . . . . . . . . . . . . . 54
6.4.3 Removal of Non-Photon Events . . . . . . . . . . . . . . . . . . 56
6.4.4 Paraboloid Reconstruction . . . . . . . . . . . . . . . . . . . . . 57
6.4.5 Velocity of Line Fit . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.5 Event Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.5.1 Preparation of Simulated Events . . . . . . . . . . . . . . . . . . 60
7 Obtaining an Upgoing Neutrino Sample
63
8 Separating Atmospheric Neutrinos from Astrophysical Neutrinos 70
9 E?ective Area
73
10 Systematic Uncertainty
77
10.1 Theoretical Uncertainties in the Background . . . . . . . . . . . . . . . 78
10.1.1 Conventional Atmospheric Neutrino Flux based on the Barr et
al. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
10.1.2 Conventional Atmospheric Neutrino Flux based on the Honda
et al. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
10.1.3 Prompt Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . 79
10.1.4 Additional Neutrino Flux Uncertainty . . . . . . . . . . . . . . 79
10.2 Normalizing the Simulation to the Data . . . . . . . . . . . . . . . . . . 81
10.3 Simulation Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10.3.1 Inverted Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.3.2 Uncertainty in Detector Response . . . . . . . . . . . . . . . . . 85
v
10.3.3 Relationship between Up and Downgoing Events . . . . . . . . . 86
11 Results
89
11.1 Results for ? / E
? 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
11.2 Results for Other Energy Spectra . . . . . . . . . . . . . . . . . . . . . 91
11.2.1 Astrophysical Neutrino Upper Limits . . . . . . . . . . . . . . . 94
11.2.2 Prompt Atmospheric Neutrino Upper Limits . . . . . . . . . . . 95
12 Future Techniques for Di?use Analyses
101
13 Conclusions
104
A Q&A for the Non-Physicist
112
A.1 What are neutrinos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.2 Why do we care about neutrinos? . . . . . . . . . . . . . . . . . . . . . 113
A.3 What objects in space are we studying? . . . . . . . . . . . . . . . . . . 116
A.4 How we detect neutrinos? . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.5 How does the AMANDA detector work? . . . . . . . . . . . . . . . . . 117
A.6 How does my analysis work? . . . . . . . . . . . . . . . . . . . . . . . . 118
A.7 Summing it up, plain and simple . . . . . . . . . . . . . . . . . . . . . 120
1
Chapter 1
Introduction
For thousands of years, people have looked to the heavens and wondered what lies in
the deepest regions of space. We have developed from a culture that once thought the
Earth was the center of everything. Now, our imaginations cannot even begin to grasp
how insigni?cant the Earth is in the grand scheme of the Universe. Unless the wildest
science ?ction stories come true, we will never be able to visit or send scienti?c probes
to objects in distant space. In order to learn about the most distant astronomical
objects, we must study the clues they send us. Cosmic particles, accelerated far from
Earth, bombard our atmosphere constantly. These particles carry information that
allows us to better understand the Universe.
The earliest studies of the heavens focused on easily observable objects: the Sun,
Moon and stars. After centuries of questions about the Sun, the particle interactions
that control its burning mechanism are now fairly precisely understood. Now our
focus stretches beyond the stars. We hope to understand the interactions occuring
between particles in objects in distant galaxies. Neutrinos were a key component in
understanding how the Sun burns and they promise to be equally informative about
processes occuring in other galaxies.
2
This analysis focuses on the search for astrophysical neutrinos from distant (non-
solar) sources. First, I will discuss cosmic rays and why neutrino studies are needed
to answer the most fundamental questions about our Universe. Then I will address
the neutrino production models in astrophysical sources. The last several chapters
will describe the search for astrophysical neutrinos using the AMANDA-II detector
located at the South Pole.
1.1 Cosmic Rays
Cosmic rays are charged particles traveling through space at very high energies.
Most of the cosmic rays are protons (˘ 85%) [1]. About 12% of the cosmic rays are
helium nuclei, while other heavier nuclei make up about 1%. Electrons make up the
?nal 2%. Man-made particle accelerators are limited in size and hence limited in the
particle energies that they can achieve. The world's biggest accelerator, the Large
Hadron Collider (LHC), is currently being constructed at CERN and is expected to
accelerate protons to collision energies of about 14 TeV [2]. Cosmic rays, on the other
hand, are some of nature's most energetic particles. They have been observed with
energies as high as 10
20
eV. This is seven orders of magnitude larger than the LHC! This
gives rise to interesting questions: what astrophysical sources are producing the ?ux
of cosmic rays and how are the particles accelerated to such large energies? Cosmic
ray and neutrino astrophysics are closely related since accelerated cosmic rays will
sooner (at the acceleration source) or later (at another target) interact and produce
neutrinos.
The cosmic ray energy spectrum follows an inverse power law over many orders
of magnitude. The di?erential ?ux is described by the following equation [3]:
3
Figure 1.1: The cosmic ray spectrum. (Image credit: Swordy, University
of Chicago).
4
dN
dE
/ E
? (?+1)
:
(1.1)
Several features in the cosmic ray spectrum are worth noting. At energies less
than about 10
9
eV, the ?ux is much lower than predicted by the power law. This
e?ect is known as solar modulation. Before arriving at Earth, cosmic rays must travel
through the solar wind. Low energy particles have a more di?cult time traveling
through this wind due to disturbances in the magnetic ?eld [1]. Hence, the low energy
?ux of cosmic rays at Earth is attenuated.
The knee is the region around 4 ?10
15
eV where a slope change occurs [3, 4]. At
lower energies than the knee, ? ˘ 1.7. However, at energies above the knee, ? ˘ 2.2
[3]. At 5 ?10
18
eV, the slope changes again at the ankle and returns to ? ˘ 1.7. In
the highest observed regions of the spectrum, a sudden cuto? in the cosmic ray ?ux
is expected at 5 ? 10
19
eV. At the Greisen-Zatsepin-Kuzmin (GZK) limit, cosmic rays
are at the threshold energy to interact with cosmic microwave background (CMB)
photons. These interactions produce pions. Current measurements of the cosmic ray
?ux in the energy region above 10
19
eV are con?icting, so the existence of the GZK
cuto? will need to be veri?ed by the next generation of ultra high energy cosmic ray
experiments.
For energies below 10
15
eV (the knee), supernova remnants (SNR) in our galaxy
are considered the most likely source to accelerate cosmic rays [5]. These non-thermal
sources are suspected because they follow power law spectra. They are also powerful
enough to accelerate the particles and have chemical abundances similar to the cosmic
rays [5]. Above 10
15
eV, it is likely the cosmic rays come from outside of the galaxy.
5
Many current theories predict mechanisms for particle acceleration in di?erent astro-
physical objects. Because the models include both charged particle acceleration and
neutrino production, the search for astrophysical neutrinos is linked to the question
of how cosmic rays of such high energies can be formed. This analysis focuses on the
search for neutrinos from distant sources.
6
Chapter 2
Neutrino Astronomy
2.1 What is a neutrino?
Neutrinos are very tiny, chargeless particles from the lepton family. They interact
via the weak force. There is a corresponding neutrino for each of the three lepton
?avors: electron, muon and tau. From their inception, neutrinos were thought to be
massless particles, but recent evidence from neutrino oscillations experiments such
as SNO (Sudbury Neutrino Observatory) and Super-Kamiokande suggests otherwise.
Neutrino oscillations (where neutrinos transform from one ?avor to another) have been
observed, which can only happen if neutrinos have non-zero mass and they are not
degenerate.
2.2 Neutrinos as Cosmic Messengers
One of the ultimate goals of astrophysics is to piece together information from
cosmic ray and neutrino telescopes and telescopes measuring electromagnetic radia-
tion (gamma-ray, optical, infrared, radio, X-ray) into a coherent picture of the inner
workings of distant astrophysical objects. Questions about cosmic rays, gamma rays
7
and neutrinos all seem to be connected. Each of these three types of particles can
provide valuable clues and there are advantages and disadvantages to each type of
study.
Photons are the traditional way of studying the sky. The ?rst astronomers
studied the heavens based on the light that they could see with their eyes. It is, of
course, possible to study photons in other ranges of the electromagnetic spectrum, for
instance, the radio, infrared, ultraviolet, X-ray and gamma-ray regions. Despite the
fact that photons are abundant and easy to observe, photons do not lend themselves
to high energy studies because of their limited distance range. High energy photons
tend to be absorbed by matter before they can reach the Earth.
Cosmic rays are easily detected. However, at most energies (including the TeV
{ PeV energy range of this analysis), cosmic rays do not carry directional information
because charged particles are de?ected by magnetic ?elds. At ultra high energies
(& 10
19
eV), galactic cosmic rays experience only a small de?ection from their original
direction and hence point backward to their source [6].
Neutrinos are chargeless and hence are not de?ected by magnetic ?elds. Since
they travel in straight lines, they carry directional information about their point of
origin. Unfortunately, detecting a neutrino event is rather challenging. Neutrinos are
weakly-interacting particles with very small cross-sections.
Neutrinos are a source of information about the physical processes occurring in
cosmic accelerators. First, we would like to identify what types of sources are capable
of producing the highest energy cosmic rays. Theories suggest that these ultra high
energy particles are being accelerated in processes that should also include neutrino
8
production. If neutrinos originate from the direction of a speci?c source, we know that
hadronic interactions (mainly pp and p?) are occuring in that source. In that case, it
is possible that protons are being accelerated up to the ultra high energies observed.
Gamma ray telescopes, such as HESS (High Energy Stereoscopic System) [7]
and MAGIC (Major Atmospheric Gamma Ray Imaging Telescope) [8], have identi?ed
many gamma-ray sources across the sky. It is hoped that neutrino production can
also be linked to these same sources.
2.2.1 Producing Neutrinos in Astronomical Sources
Active galactic nuclei (AGN), gamma ray bursts (GRBs), supernova remnants
(SNR) and starburst galaxies are among the astronomical objects that could be pro-
ducing neutrinos. They are all considered possible sources in which particles are
accelerated to high energies through shock acceleration processes. The most widely
accepted model of shock acceleration is ?rst order Fermi acceleration, which will be
described later in this chapter. First, it is important to discuss what particles must
be present and what interactions must occur in order to produce neutrinos.
Hadronic proton-proton (pp) and proton-photon (p?) interactions are expected
to occur in astrophysical sources. These interactions create charged and neutral pions
and kaons. As can be seen in the following equations, neutrinos result from charged
pion and kaon decay. If neutrino and ?-ray production can be identi?ed in the same
sources, that would be strong evidence that pp and p? interactions like those described
below are occuring in those astrophysical sources. Note that pions and kaons follow
similar decay chains, so only the pion chain is shown. Charged pions (kaons) create
neutrinos, but neutral pions (kaons) decay into two gamma rays.
9
ˇ
+
! ?
+
+ ?
?
! (e
+
+ ?
e
+ ??
?
)+ ?
?
ˇ
?
! ?
?
+ ??
?
! (e
?
+ ??
e
+ ?
?
) + ??
?
(2.1)
ˇ
?
! ??
The ?avor ratio at the time of the neutrino production in the source is expected
to be ?
e
:?
?
:?
˝
= 1:2:0. (? and ?? are both counted in the ? ?ux as it is labelled
here.) Since neutrinos can change ?avors, or oscillate, the expected ?avor ratio of the
astrophysical neutrinos at the detector is not 1:2:0; ?
?
? ?
˝
mixing leads to a 1:1:1
?avor ratio at Earth [9]. However, Kashti and Waxman [10] have pointed out that at
high energies (E & 100 TeV) ? decay in the source region becomes suppressed. This
results in a ?avor ratio at Earth of 1:1.8:1.8.
2.2.2 Fermi acceleration
First order Fermi acceleration describes the process in which particles are accel-
erated in strong shocks [1, 11]. A shock wave occurs as gas or particles are ?owing (or
exploding) out of an object and travel faster than the speed of sound in the medium.
As particles pass through a shock wave, a number of conservation laws must be up-
held. The conservation of mass, energy ?ux, and momentum ?ux across the shock all
allow a series of relations to be derived about the particles. In the case of a strong
shock (Mach number >> 1),
ˆ
2
ˆ
1
=
(?
h
+ 1)
(?
h
? 1)
(2.2)
where ˆ is the gas density and ?
h
is the ratio of speci?c heats. For a monatomic
10
Figure 2.1: Fermi accleration from di?erent points of view. In the frame of
reference of the upstream and downstream gas, the particles gain energy
as they cross the shock.
gas, ?
h
=
5
3
.
This leads to the relation
ˆ
2
ˆ
1
=
8=3
2=3
= 4
(2.3)
which will be used later in further calculations.
Because of conservation of mass across the shock wave, ˆ
1
v
1
=ˆ
2
v
2
, the velocities
of the gases on either side of the shock are related by the following:
v
2
=
1
4
v
1
:
(2.4)
Consider a shock wave travelling outward from an astrophysical object (Figure
11
2.1 upper left). Its speed of travel is U. To move into the reference frame of the shock
(Figure 2.1 upper right), add -U to either side of the shock. The upstream material
comes at the shock (at rest) at a speed v
1
= ? U. Using equation 2.4, the downstream
material moves away from the shock at v
2
= ? 0:25v
1
= ? 0:25U.
To move into the reference frame of the upstream gas (Figure 2.1 lower left),
add U to the either side of the shock rest frame. The downstream material went from
moving at .25U to the left to 0.75U to the right.
Similarly, to move into the reference frame of the downstream gas (Figure 2.1
lower right), add 0.25U to both sides in the shock's rest frame. The upstream material
now has a speed -U+0.25U = -.75U. Remarkably, no matter which direction it comes
from, a particle always approaches the shock at 0.75U.
When a particle crosses the shock, it always gains energy, independent of the
direction it is crossing. Furthermore, particles may cross the shock multiple times.
They gain an amount of energy, ?, every time they cross. Assuming that a particle
crosses a shock k times, its energy, E, will be:
E = ?
k
E
o
(2.5)
Assuming P is the probability of crossing the shock, the number of particles with
energy E (E > E
o
) is:
N = P
k
N
o
(2.6)
where N
o
is the number of particles with the original energy E
o
. Equations 2.5
and 2.6 can be combined as:
12
N
N
o
=(
E
E
o
)
ln P
ln ?
(2.7)
The spectral index, ?, of the charged particles is de?ned as
? =
ln P
ln ?
:
(2.8)
Equation 2.7 suggests that N ˘ E
?
, hence the di?erential energy spectrum is
dN
dE
˘ E
?? 1
:
(2.9)
A few assumptions can be used to determine the value of the spectral index, ?:
the energy gain ? is proportional to
v
1
? v
2
c
, and the probability of crossing the shock is
proprotional to
v
1
c
. The detailed calculation is not shown here, but these assumptions
lead to the conclusion that ? = ? 1. Hence, the predicted energy spectrum for ?rst
order Fermi acceleration is:
dN
dE
˘ E
? 2
:
(2.10)
This generic energy spectrum is used as a common benchmark in the search for
astrophysical neutrinos.
2.3 Finding Astrophysical Neutrinos
In the early 1960s, it was realized that neutrinos should play an important role
in accelerators of cosmic rays and that we could detect neutrinos on Earth. When
neutrinos interact with matter, the resulting Cherenkov radiation can be detected in
13
a transparent medium. K. Greisen, F. Reines and M.A. Markov and I.M. Zheleznykh
pioneered the design of future astrophysical neutrino detectors based on this principle.
In 1960, K. Greisen wrote [12]:
Let us now consider the feasibility of detecting the neutrino ?ux. As a de-
tector, we propose a large Cherenkov counter, about 15 m in diameter, located
in a mine far underground. The counter should be surrounded with photomul-
tipliers to detect the events, and enclosed in a shell of scintillating material to
distinguish neutrino events from those caused by ? mesons. Such a detector
would be rather expensive.... About 500 reactions per year would be expected
from neutrinos produced in the atmosphere, but these would have a steeper
energy spectrum and a di?erent angular distribution from that of the primary
neutrinos, with a maximum in the horizontal direction, where the longer path
length in the atmosphere permits more of the mesons to decay. The atmo-
spheric neutrinos would serve to verify the neutrino cross section and calibrate
the apparatus.
Also in 1960, F. Reines predicted that, at least half the time, neutrinos would
continue in the same direction as the pion to within 10% [13]. As a result, he predicted
studying the astrophysical neutrino ?ux by building a detector that consisted of \a
water target in a white container at the end of which are located a few hundred 5-in.
photomultiplier tubes."
M.A. Markov and I.M. Zheleznykh stated in 1961 [14], \All known particles with
the exception of neutrinos are absorbed by scores of kilometres of the substance and
thus are entirely screened by the planet..."
These ideas became the foundation of a new ?eld: neutrino astrophysics.
14
Chapter 3
Astrophysical Neutrino Models and Limits
With the ?eld of neutrino astronomy wide open to speculation, many models have been
developed to predict the neutrino ?ux from extraterrestrial sources over a wide energy
range. Astrophysical neutrino models come with various shapes and normalizations.
Some models have been normalized based on the extragalactic gamma ray background,
while others try to pinpoint the neutrino ?ux based on ultra high energy cosmic ray,
X-ray, or radio measurements
Even if no extraterrestrial signal is detected, upper limits on the neutrino ?ux can
be determined that rule out or constrain existing neutrino production theories. Some
models predict an energy spectral shape and normalization, while other predictions
are only upper bounds.
3.1 ? / E
? 2
Neutrino Models
As mentioned previously, ? / E
? 2
is a generic spectrum predicted by ?rst order
Fermi acceleration. A number of theories have been developed based on this spectral
shape. A common benchmark in neutrino astronomy is the Waxman-Bahcall (WB)
upper bound [15, 16, 17], which assumes the ? / E
? 2
spectrum. Current detectors
15
log
10
[E
n
(GeV)]
3456789
]
-1
sr
-1
s
-2
dN/dE [GeV cm
2
E
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
Atmospheric n [44,45]
SDSS [21,22]
MPR AGN jets [20]
MPR t
ng
<1 model [20]
Starburst [23]
Nellen et al. [18]
MPR optically thick [20]
Becker et al. steep sources [19]
W&B limit/2 (transparent sources) [15]
Becker et al. flat sources [19]
Figure 3.1: Astrophysical neutrino models span many orders of magnitude
and have varying energy spectra.
16
are striving to attain the sensitivity of the WB bound. Astrophysical neutrino models
and bounds developed by Nellen et al., Becker et al. and MPR all have a ? / E
? 2
spectral shape.
3.1.1 Waxman-Bahcall Upper Bound
The Waxman-Bahcall upper bound [15, 16, 17] assumes that cosmological sources
of protons have a ? / E
? 2
injection spectrum. When protons interact with the
radiation ?eld of a source, charged and neutral pions are produced in equal amounts.
Neutral pions do not decay into neutrinos. When a charged pion decays, half of the
energy of the pion is carried by the muon neutrinos and antineutrinos. The neutrino
spectrum is predicted to remain the same as that for protons, ? / E
? 2
.
The upper bound was determined assuming that the energy production rate of
protons is
E
2
CR
dN
_
CR
dE
CR
ˇ 10
44
ergMpc
? 3
yr
? 1
:
(3.1)
To derive the maximum value of the neutrino ?ux, it was assumed that protons
do not lose any energy before escaping the source. The present day neutrino ?ux at
the source was described by WB with the following equation.
E
2
?
dN
?
dE
?
ˇ 0:25t
h
E
2
CR
dN
_
CR
dE
CR
(3.2)
The Hubble time, t
h
ˇ 10
10
years, is the inverse of the Hubble constant, here
assumed to be about 65 km s
1
Mpc
? 1
. Using the relation ?
?
=
dN
dE
?
c
4ˇsr
and substi-
tuting the energy production rate from equation 3.1, the maximum neutrino (?
?
+ ??
?
)
?ux is proposed to be ˇ 1:5 ? 10
? 8
GeV cm
? 2
s
? 1
sr
? 1
.
17
After a correction for redshift evolution, the predicted ?ux is a factor of 3 higher.
However, due to neutrino oscillations, the predicted muon neutrino ?ux is not the same
at the source and the Earth. Approximately half of the muon neutrinos are expected
to oscillate into other ?avors. The upper bound for the ?ux of muon neutrinos and
anti-neutrinos is 2:25 ? 10
? 8
GeV cm
? 2
s
? 1
sr
? 1
at Earth. This upper bound only
applies to optically thin sources, meaning that the optical depth ˝ is small.
3.1.2 Nellen, Mannheim and Biermann Model
In 1993, Nellen, Mannheim and Biermann [18] predicted an astrophysical neu-
trino ?ux of E
2
? = 1:7 ? 10
? 6
GeV cm
? 2
s
? 1
sr
? 1
. They assumed that pp interactions
in the source would lead to electromagnetic interactions that result in the produc-
tion of neutrinos, X-rays and ?-rays. Their model for neutrino production in AGN is
normalized based on X-ray background measurements by ROSAT (R?ontgen Satelit).
Their prediction is valid for the GeV energy region up to 4 ? 10
5
GeV.
3.1.3 Becker, Biermann and Rhode Model
Becker, Biermann and Rhode (2005) [19] predicted the neutrino ?ux from FR-II
radio galaxies (steep spectrum sources) and blazars (?at spectrum sources) since AGN
jets are expected to be the site of p? interactions that lead to neutrino production.
These models were normalized by deriving the relationship between the neutrino lu-
minosity, the disk luminosity and the radio luminosity. Each model (steep and ?at) is
highly dependent on what proton spectral index is used. Hence, results were presented
for numerous spectra, although only the ? / E
? 2
model is compared here.
For steep sources with an assumed ? / E
? 2
spectrum, the predicted ?ux is
18
˘ 1:0 ? 10
? 7
GeV cm
? 2
s
? 1
sr
? 1
for energies up to 10
9
GeV. For ?at sources and a
? / E
? 2
spectrum, the model predicts a ?ux of ˘ 6:3 ? 10
? 10
GeV cm
? 2
s
? 1
sr
? 1
for
the same energy region.
3.1.4 Mannheim, Protheroe and Rachen Upper Bound for Thick Sources
Mannheim, Protheroe and Rachen [20] proposed several models for the astro-
physical neutrino ?ux that will be described in more detail in the next section. They
placed an upper bound on the neutrino ?ux for optically thick sources in which neu-
trons cannot escape (˝
n?
˛ 1). The prediction is normalized to measurements of
the extragalactic gamma ray background. The MPR upper bound for optically thick
sources follows a ? / E
? 2
spectrum and lies just above 10
? 6
GeV cm
? 2
s
? 1
sr
? 1
.
3.2 Neutrino Spectra Di?erent than ? / E
? 2
The following models for astrophysical neutrino production do not follow a
? / E
? 2
spectrum.
3.2.1 Stecker, Done, Salamon and Sommers AGN Core Model
Stecker, Done, Salamon and Sommers (SDSS) [21, 22] developed a neutrino ?ux
prediction based on their studies of AGN cores. The model only makes a prediction for
radio-quiet AGN and was normalized to X-ray data for the AGN luminosity function
and redshift information collected by ROSAT. In 2005, Stecker published a revision to
the original prediction, citing new observational evidence. The original model assumed
that 100% of the X-ray background was nonthermal radiation from AGN. However,
it is now shown that AGN emit mainly thermal radiation. For the galactic black hole
19
source Cyg X-1, the energy spectrum can be explained if 90% of the power contributes
to a thermal electron distribution while 10% of the power goes to nonthermal ?-rays.
Instead of assuming 100% of the AGN output was nonthermal, now the SDSS model
assumes only 10%. This leads to a reduction in the SDSS model by a factor of 10. In
addition, neutrino oscillations were con?rmed after the SDSS model was issued. This
means that the muon neutrino ?ux should decrease by a factor of two by the time it
is detected on Earth. After the Stecker revision in 2005, the predicted ?ux is a factor
of 20 smaller.
3.2.2 Mannheim, Protheroe and Rachen Upper Bounds for Thin Sources
MPR [20] based their models for astrophysical neutrino production on the ob-
served cosmic ray spectrum. Instead of using the ? / E
? 2
spectrum assumed by
Waxman and Bahcall, they assumed a cosmic ray injection spectrum that is consis-
tent with the observable cosmic ray ?ux. Based on results from Havarah Park, Akeno
Giant Air Shower Array (AGASA), Fly's Eye, Yakutsk and the KASCADE air shower
experiment, they assumed an extragalactic cosmic ray spectrum of the form
N
p;obs
(E) = 0:8 ? (E/GeV)
? 2:75
cm
? 2
s
? 1
sr
? 1
GeV
? 1
(3.3)
for energies between 3 ? 10
6
GeV and 10
12
GeV. To form the upper bound for
optically thin sources, the cosmic ray spectrum was calculated in small energy ranges
characterized by an energy E
max
between 10
6
and 10
12
GeV. Each of these cosmic
ray proton spectra was normalized such that the peak of the distribution for each
E
max
reached the extragalactic cosmic ray spectrum in equation 3.3. Since the cosmic
ray and gamma ray outputs are correlated, the cosmic ray normalizations also had
20
to agree with the observed gamma ray background. For that reason, the peak of the
normalized curve for E
max
= 10
6
falls below the observed cosmic ray spectrum so as
not to overproduce gamma rays. The neutrino spectrum was then derived from each
normalized (and maximized) cosmic ray spectrum for the particular values of E
max
.
An envelope was drawn that connects the peak of each neutrino spectrum and this
represents the MPR upper bound for neutrinos from sources that are optically thin to
neutrons.
3.2.3 Mannheim, Protheroe and Rachen Upper Bound for Neutrinos from
AGN Jets
MPR also projected an upper bound on the ?ux of neutrinos from AGN jets.
They followed the same procedure just described to normalize the neutrino spectrum
to the observed cosmic ray ?ux. For this calculation, they used the same generic
cosmic ray and neutrino production spectra as were used in the determination of the
thin sources upper bound. In this case, they ?xed the value of E
max
= 10
11
GeV. The
break energy (E
b
) was allowed to vary between 10
7
and 10
11
GeV. The upper bound
is the maximized superposition of the spectra with di?erent input parameters, E
b
.
3.2.4 Loeb and Waxman Starburst Model
Loeb and Waxman [23] proposed a neutrino production model for starburst
galaxies. Since evidence suggests that the magnetic ?eld in starburst galaxies is 100
times larger than in the intergalactic medium, the protons lose all of their energy to
pion production before they escape from the source. The ?ux prediction was nor-
malized based on the observed synchrotron radio ?ux of electrons. The predicted
21
energy spectrum was derived by considering both the cosmic ray proton spectrum on
Earth (? / E
? 2:75
) and the con?nement time in the source (t / E
? 0:6
). The predicted
?ux is E
2
?
?
SB
?
ˇ 10
? 7
(E
?
/1 GeV)
? 0:15?0:1
GeV cm
? 2
s
? 1
sr
? 1
. However, the authors
felt that large uncertainties should be taken into account and hence the model en-
compasses a large region on a neutrino ?ux plot due to uncertainties in the proton
spectrum, the con?nement time spectral index, and the location of the knee in the
cosmic ray spectrum.
3.3 Existing Astrophysical Neutrino Upper Limits
The most widely tested spectrum by neutrino experiments is the generic ? / E
? 2
spectrum. Fr?ejus, MACRO, Baikal and AMANDA have all published upper limits on
the di?use ?ux of neutrinos with this energy spectrum.
A precursor to this muon neutrino analysis was conducted with data collected in
1997 by the AMANDA-B10 detector [24]. (In 1997, the AMANDA detector consisted
of 10 sensor strings, a subset of the 19 strings in the ?nal AMANDA-II con?guration.)
This analysis focused on the ? / E
? 2
spectrum, but also set upper limits on several
other spectral shapes.
Other AMANDA analyses have focused on the search for a di?use ?ux of neu-
trinos using particle showers or cascades [25] instead of muon neutrinos. Cascades
are caused by ?
e
and ?
˝
charged current interactions and all-?avor neutral current
interactions in the ice near the detector. Upper limits from these analyses constrain
the ?ux of neutrinos of all ?avors, not just muon neutrinos.
The Fr?ejus [26], MACRO [27], and Baikal [28]) experiments have set upper limits
on the ?ux of astrophysical neutrinos in the same energy region as this analysis (TeV
22
Experiment
Upper Limit Energy Range
[GeV cm
? 2
s
? 1
sr
? 1
] log
10
[E
?
(GeV)]
Muon neutrinos only
Fr?ejus [26]
5:0 ? 10
? 6
˘3.4
MACRO [27]
4:1 ? 0:4 ? 10
? 6
4.0 { 6.0
AMANDA-B10 [24]
8:4 ? 10
? 7
3.8 { 6.0
All neutrino ?avors
Baikal [28]
8:1 ? 10
? 7
4.3 { 7.7
AMANDA-B10 [29]
0:99 ? 10
? 6
6.0 { 9.5
AMANDA-II [25]
8:6 ? 10
? 7
4.7 { 6.7
Table 3.1: Upper limits for the di?use ?ux of extraterrestrial neutrinos
as reported by a number of experiments. The ?rst four analyses only
constrain the ?ux of ?
?
+ ??
?
, while the last three constrain the total
neutrino ?ux, (?
e
+ ??
e
+ ?
?
+ ??
?
+ ?
˝
+ ??
˝
).
- PeV). Published upper limits from these experiments assuming a ? / E
? 2
spectrum
are summarized in Table 3.1. Depending on the detector and the speci?c analysis, the
reported upper limit constrains either the muon neutrino ?ux or the all-?avor neutrino
?ux. Upper limits obtained from all-?avor analyses are not directly comparable to ?
?
upper limits. However, for a wide range of neutrino production models and oscillation
parameters, the ?avor ?ux ratio at Earth can be approximated as 1:1:1 [9]. In that case,
either a single-?avor limit can be multiplied by three and compared to an all-?avor
result, or an all-?avor limit can be divided by three and compared to a single-?avor
result.
The Baikal experiment has placed limits on models with spectra other than
? / E
? 2
[28], which are compared to the results from the 1997 AMANDA-B10 analysis
in Table 3.2.
In this analysis, nine di?erent spectral shapes are tested, including the search
23
Upper Limit
Experiment
SDSS
?
MPR AGN Jets
Baikal [28]
2.5 ??
SDSS
4.0 ??
MPR
AMANDA-B10 [29] 23.2 ??
SDSS
not tested
? = model lowered by factor of 10 by Stecker in 2005
Table 3.2: Upper limits for the di?use ?ux of extraterrestrial neutrinos
from spectra di?erent than ? / E
? 2
Note that all upper limits on the
SDSS ?ux are adjusted to take into account the 2005 revision by Stecker.
.
for prompt neutrinos from the decay of charmed particles described in Chapter 4.
Since this analysis is optimized on energy-dependent parameters, the optimization
was performed individually for each energy spectrum.
24
Chapter 4
Atmospheric Neutrinos
Atmospheric muons and neutrinos form the primary background for the search for
astrophysical neutrinos. When cosmic rays interact in Earth's atmosphere, muons and
neutrinos are among the particles that are formed. While the so-called atmospheric
muons are the largest background to the search for astrophysical neutrinos, they are
also the easiest background to remove since they are unable to travel all the way
through the Earth. On the other hand, both atmospheric neutrinos and the desired
astrophysical signal can approach the detector from any direction. It is important to
thoroughly understand this important background.
4.1 Conventional vs. Prompt Atmospheric Neutrinos
When cosmic rays hit the atmosphere, interactions lead to the generation of
many types of particles. Two di?erent event classes are used to describe the resulting
atmospheric muons and neutrinos: conventional and prompt.
The conventional atmospheric muon and neutrino ?uxes are the product of pion
and kaon decays in the atmosphere. Pion and kaon decays occur via similar processes,
hence only the pion decay chain is shown below:
25
Figure 4.1: Cosmic ray interactions in the atmosphere. (Image credit:
Milagro)
26
ˇ
?
! ?
?
+ ?
?
(??
?
)
(4.1)
?
?
! e
?
+ ?
e
(??
e
) + ??
?
(?
?
)
(4.2)
The second class of events results from the semileptonic decay of charm particles,
meaning mesons that contain either a charm (c or c?) quark. Since these charm particles
decay quickly before they can lose much energy, the resulting neutrinos are called
prompt neutrinos. The following D meson and ?
c
particle decays are among the most
common semileptonic charm decays [30, 31].:
D
+
! K
?
0
+ ?
+
+ ?
?
(4.3)
D
+
! K
?
0
+ e
+
+ ?
e
(4.4)
?
+
c
! ?
0
+ ?
+
+ ?
?
(4.5)
The total atmospheric neutrino ?ux is dependent on the critical energies of the
particles that can decay into neutrinos, ?
crit
, where ?
crit
is the energy for which the
decay length and interaction length are equal [32]. As the energy increases, it is
more likely that a particle will interact rather than decay. If interactions take place
instead of decays, no neutrinos are immediately produced. Since the critical energies
for pions, kaons and charm particles are all di?erent, each of these neutrino decay
sources is dominant at di?erent energies.
27
Particle ?
crit
(GeV)
?
?
1.0
ˇ
?
115
K
?
850
D
?
3.8 ?10
7
D
0
,D
?0
9.6 ?10
7
D
?
s
8.5 ?10
7
?
+
c
2.4 ?10
8
Table 4.1: Critical energy for di?erent particles.
?
crit
=
mc
2
c˝
h
o
(4.6)
The critical energy, ?
crit
, is calculated from the particle's rest energy mc
2
and
the mean life time ˝. The constant h
o
comes from the assumption of an isothermal
atmosphere [32].
Table 4.1 lists the critical energy of several particles that contribute muons and
neutrinos to the atmospheric ?ux when they decay. Pions and kaons have critical
energies of 115 and 850 GeV, respectively. However, the critical energy for charm
mesons is approximately ?ve orders of magnitude larger.
Below ?
crit
, particles are more likely to decay than interact and the secondary
particles will have the same energy spectrum as the parent. However, for particle de-
cays above the critical energy, the energy spectrum of the secondary particles increases
by one over the spectral index of the parent [33]. For instance, pions less than 115
GeV (and their decay products) will follow a ? /E
? 2:7
spectrum just as the cosmic
rays that created them. However if the energy is above ?
crit
, the resulting neutrinos
have a ? /E
? (2:7+1)
spectrum.
28
In the TeV ? PeV energy range of this analysis, conventional atmospheric neu-
trinos come from particle decays that occur above the critical energy. As a result,
conventional atmospheric neutrinos follow a ? /E
? 3:7
spectrum for the entire energy
region covered by this analysis. Atmospheric neutrinos from the decay of charm parti-
cles are still below ?
crit
and hence follow a ? /E
? 2:7
spectrum. Somewhere above 850
GeV (the kaon critical energy), charm particle decays become the dominant source of
atmospheric muons and neutrinos.
Many parameters go into the calculation of the atmospheric neutrino ?ux and
the models are highly subjective. Every model makes di?erent assumptions about
the input parameters for calculating the prompt ?ux. Since the particle interactions
have branching ratios that indicate that neutrinos are not always formed, Monte Carlo
simulations are the best way to predict the atmospheric neutrino ?ux. The conven-
tional atmospheric neutrino simulation will be discussed in detail in Chapter 6. Since
prompt neutrinos dominate over the conventional atmospheric neutrino ?ux at high
energy because of their harder spectrum, it is possible to treat prompt atmospheric
neutrinos as signal and search for these events in the detector.
4.2 Prompt Atmospheric Neutrino Models
There is a great deal of uncertainty in the prompt atmospheric neutrino ?ux since
particle accelerators are not energetic enough to probe the energy region relevant to
charm production in the atmosphere. As a result, models predict prompt neutrino
?uxes that vary widely over several orders of magnitude. There are many di?erent
parameters that go into each model, for instance the primary spectral index, ?, the
critical energy, and the interaction and decay lengths.
29
The ?ux of secondary charm particles, ?
i
, can be described by an equation with
three terms [33]:
d?
i
(x; E
i
; ?)
dx
= ?
1
?
i
?
i
(x; E
i
; ?) ?
?
crit
E
i
x
?
i
(x; E
i
; ?) +
Z
1
E
i
1
?
N
?
N
(x; E
N
; ?)
dW
iN
dE
i
dE
N
:
(4.7)
In Equation 4.7, i stands for the type of charm particle, for instance, D
?
, D
?
,
?
c
. The ?ux is dependent on three parameters: x is the depth of the atmosphere
that has been penetrated in g/cm
2
, E
i
is the particle energy and ? is the angle of the
particle's approach. This equation can be simpli?ed by considering each of the three
terms separately.
The ?rst term in Equation 4.7 represents the loss of charm particles due to
interactions in the air. The nuclear interaction mean free path is ?
i
.
The second term in Equation 4.7 represents charm particle decay. This term is
directly proportional to the critical energy, ?
crit
.
The last term accounts for the production of new charm particles throughout the
shower development in the atmosphere. The initial nucleon ?ux is ?
N
. The nucleonic
interaction length, ?
N
is the mean free path of nucleons in the atmosphere [30].
?
N
=
Am
p
˙
N? air
in
(4.8)
In equation 4.8, A is the atomic number for an air nucleus (on average, A=14.5).
The proton mass is m
p
. The total inelastic cross section for nucleon (N) and air
interactions is ˙
N? air
in
.
The ?nal parameter in the third term of equation 4.7 is
dW
iN
dE
i
. This is the energy
30
distribution of the secondary charm particles that are produced.
The production of charm particles in the atmosphere is often described by a
parameter called the Z-moment. It is related to the charm production cross section
and varies greatly between the di?erent prompt atmospheric neutrino models. Before
de?ning the Z-moment, one further parameter must be described. x
F
, also called
Feynman x, is the fractional momentum that the charm particle receives from its
parent. The Z-moment is de?ned as [30]:
Z
Ni
(?) =
Z
1
0
x
?
F
dW
Ni
dx
F
dx
F
(4.9)
The large variation in the Z-factor across the di?erent models leads to the wide
spread in prompt atmospheric neutrino predictions. A few of the models that were
used in this analysis are described below.
The Martin GBW (Golec-Biernat and Wustho?
?
) [34] model utilizes perturba-
tive quantum chromodynamics to predict the charm cross section. pQCD is currently
the most widely accepted model of charm production based on accelerator data. This
model for deep inelastic scattering includes gluon saturation e?ects which lower the
predicted charm production cross sections. The Z-moment can be de?ned by the
following equation with the framework of the Martin GBW prediction.
˙Z
Ni
=
Z
d˙
c
dx
x
2:02
dx
(4.10)
where 2.02 is the spectral index of the incoming primary cosmic ray ?ux above
10
6
GeV and x is the gluon distribution of the proton. ˙ is the charm production
cross section.
31
log
10
[E
n
(GeV)]
3456789
]
-1
sr
-1
s
-2
dN/dE [GeV cm
2
E
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
Conventional atmospheric n
Martin GBW prompt atmospheric n
Naumov RQPM prompt atmospheric. n
CharmC prompt atmospheric n
CharmD prompt atmospheric n
Figure 4.2: Prompt atmospheric neutrinos are predicted to follow a harder
spectrum than conventional atmospheric neutrinos. The ?ux of prompt
atmospheric neutrinos is highly uncertain and predictions range over sev-
eral orders of magnitude.
32
The Naumov RQPM (Recombination Quark Parton Model) [35, 36] incorpo-
rates intrinsic charm into the calculation of the prompt atmospheric ?uxes. Intrinsic
charm assumes that there are charm quark/anti-quark pairs in the incoming particle
[37]. For instance, a proton is usually thought of as having three components - two up
quarks and one down quark (uud). Assuming intrinsic charm means that charm quarks
have non-negligible contributions to the proton wave functions. Instead of de?ning
a proton as (uud), it is de?ned as (uuc?cd). The charm quarks and anti-quarks arise
from gluon-gluon fusion. When particle collisions occur, the quarks recombine into
new groupings, hence forming new particles, possibly with charm (D
+
= cd
?
, D
?
= cu?,
?
+
c
= udc, ...).
For the Naumov RQPM model, the Z-moment is energy dependent [37].
Z
Ni
(?) = Z
?
(
E
N
E
?
)
˘
(4.11)
where ˘, E
?
and Z
?
are constants directly dependent on the primary cosmic ray
spectral index, ?.
Zas, Halzen and Vazquez [33] suggested ?ve di?erent models of the prompt
neutrino ?ux. Each one has a di?erent parameterization of the energy dependence of
the charm production cross section. Only two of the models are tested in this analysis,
Charm C and Charm D.
The Charm C model assumes an energy dependence for the charm cross section
that is ?tted to experimental data [38] with a log
2
(s) ?t. (
p
s is the center of mass
energy of the particle) Charm D uses the charm cross section from Volkova [39].
33
4.3 Existing Prompt Neutrino Upper Limit
The predecessor of this analysis was a 1997 AMANDA-B10 analysis that put an
upper limit on the Charm D ?ux from Zas et al.. The upper limit was 4:8 ? ?
CharmD
.
When the published limit is compared to the atmospheric neutrino spectrum, the
Charm D ?ux upper limit crosses over the atmospheric neutrinos between 3 ? 30
TeV. Thus, the true prompt ?ux is expected to begin dominating over the conventional
atmospheric neutrino ?ux at an energy greater than a 3 ? 30 TeV, although the most
currently accepted models suggest a much high region for the crossover (50 TeV - 1
PeV).
34
Chapter 5
Neutrino Detection with AMANDA
AMANDA, or Antarctic Muon and Neutrino Detector Array, is an ice-Cherenkov
detector located at the South Pole. A team of collaborating scientists from around
the world designed and built the detector during the 1990s. Since AMANDA analyses
are searching for a small astrophysical neutrino signal among a large background, the
search methods have been developed to maximize the signal retention and background
rejection.
5.1 Search Method
This analysis uses the Earth as a ?lter to search for upgoing astrophysical
neutrino-induced events. The background for the analysis consists of atmospheric
muons and neutrinos created when cosmic rays interact with Earth's atmosphere.
The majority of the events registered in the detector are atmospheric muons traveling
downward through the ice.
Conventional atmospheric neutrinos arise from the decay of pions and kaons
created in cosmic ray interactions with the atmosphere. Atmospheric neutrinos are
able to travel undisturbed through the Earth. They can be separated from atmo-
35
Figure 5.1: Types of events in the detector.
spheric muons by their direction, namely by demanding that the reconstructed track
is upgoing. The conventional atmospheric neutrino ?ux asymptotically approaches a
? ˘ E
? 3:7
spectrum in the multi-TeV range.
In the initial sample of 5:2 ? 10
9
events, many downgoing events were misre-
constructed as upgoing tracks. Misreconstruction happens because photons scatter
in the ice, causing directional and timing information to be lost. Hence, the selected
upgoing event sample not only contains truly upgoing neutrinos, but a certain fraction
of downgoing atmospheric muons.
An energy-correlated observable was used to separate neutrino-induced events
since the predicted astrophysical neutrino ?ux has a much harder energy spectrum
(? / E
? 2
) than the conventional atmospheric neutrinos from pions and kaons. Any
excess of events at high energy over the expected atmospheric neutrino background
36
indicates the presence of a signal.
The search method can be summarized with the following three steps.
1. Use the zenith angle from the reconstructed track to reject obviously downgoing
events.
2. Select events that have observables more consistent with typical long upgoing
tracks. This separates truly upgoing events from misreconstructed downgoing
events.
3. Use an energy-related observable (number of OMs triggered) to separate upgoing
atmospheric neutrinos from upgoing astrophysical neutrinos.
5.2 AMANDA Detection Principle
Since neutrinos have no charge and are hence hard to detect, they are studied
indirectly by the Cherenkov light that they induce. Cherenkov light occurs when a
charged particle travels faster than the speed of light in a transparent medium. Light
travels slower in media other than a vacuum and the speed of light is adjusted by a
refractive index, n.
c
medium
=
c
vacuum
n
(5.1)
The particle travels faster than the light can propagate away from it and hence
the light lags behind in a wavefront with a ?xed angle [1]. The critical angle formed
by the Cherenkov cone obeys the equation:
37
Figure 5.2: A muon neutrino and the Cherenkov cone.
cos? =
1
n?
(5.2)
where ? = v/c and v is the particle velocity. In ice, n=1.32 and the Cherenkov
angle is ˘ 41
?
for relativistic particles (v ˘ c).
When a neutrino interacts with matter, charged particles are created. If a
neutrino-matter interaction occurs in water or ice, then the resulting particles emit
Cherenkov light that can be detected by photosensors. If many sensors are distributed
throughout a three-dimensional volume of water or ice, then the position and timing
information available for each detected photon can be used to reconstruct the path of
the charged particle. A neutrino-induced muon travels in the same direction as the
neutrino with a median space angle di?erence of [5]
0:7
?
(E
?
=TeV)
0:7
:
(5.3)
As charged particles such as muons travel through matter, they lose energy.
There are two types of energy loss in matter, continuous and stochastic. Ionization of
38
the ice through which a muon travels leads to a continuous loss of ˘ 2MeV=(g=cm
2
)
(roughly independent of energy, although it rises slightly above 1 GeV). Stochastic
losses occur from bremsstrahlung, electromagnetic interactions with nuclei and e
+
e
?
pair production [3] and lead to losses proportional to the energy. The energy loss rate
for muons in matter is:
dE
dX
= ? ? ? E=˘
(5.4)
where ? represents continuous losses and ˘ is a factor that combines the e?ects of
bremsstrahlung, hadronic interactions and pair production. At the critical energy, ?,
continuous and discrete losses are equally important. After traveling through matter
to a depth of X, a particle with an initial energy E
0
has energy
< E(X) >= (E
0
+ ?)e
? X=˘
? ?
(5.5)
Because of muon energy losses, atmospheric muons can not travel to the detector
from the far side of the Earth. However, neutrinos can. The prediction that neutrinos
could be observed with Cherenkov radiation in a large transparent medium led to the
idea of water/ice Cherenkov detectors in the 1960s.
5.3 AMANDA Detector
The AMANDA detector is comprised of 677 photomultiplier tubes installed
upside-down in glass spheres, attached to large cables and lowered into deep holes
in the polar ice cap.
Each AMANDA string was deployed after drilling through the polar ice with a
39
hot water drill. As the main cable was lowered into each hole, optical modules (OMs)
were attached at regular spacings. An OM consists of a glass pressure vessel that
surrounds an 8-in photomultiplier tube (PMT) and the electronics used to operate it.
Each OM is connected to the main cable and information from each OM is transmitted
up to the surface when photons are detected. The OMs are located at depths between
1500 m and 2000 m below the Antarctic surface.
The detector was called AMANDA-B10 when only 10 strings were deployed.
The strings were layed out so that the instrumented ice volume was cylindrical. In
this con?guration, AMANDA took data from 1997-2000. In 2000, an outer ring of
strings was added and AMANDA-II began operation with 19 strings. Each string was
connected on the surface to a computer system in MAPO, a building at the South
Pole. An event is recorded in the detector every time at least 24 OMs report seeing a
photon within a 2.5 ?s window.
AMANDA is operated in conjunction with SPASE, the South Pole Air Shower
Experiment. SPASE consists of 30 surface tanks that are used to measure downgoing
air showers caused by cosmic ray interactions in the atmosphere. As discussed in later
sections, downgoing atmospheric muons are a calibration source for the detector.
The AMANDA collaboration grew to include members from about 20 collab-
orating institutions. After years of successful operation, AMANDA merged with
the IceCube collaboration in order to build a much larger, cubic kilometer detec-
tor. AMANDA is now considered the inner core detector of IceCube. The IceCube
strings surround AMANDA and the digital optical modules (DOMs) lie between 1450
and 2450 meters. IceCube will eventually contain 80 strings and instrument one cubic
40
Figure 5.3: Layout of the AMANDA detector.
41
Figure 5.4: Optical Module.
kilometer of ice. Currently, 22 IceCube strings have been deployed and each con-
tains 60 DOMs spaced by 17 meters. The IceCube string layout forms a hexagonal
structure, with each string separated by 125 meters.
5.3.1 AMANDA Coordinate System
AMANDA aims to separate atmospheric muons from neutrino-induced muons
by their directional information. Since the desired neutrino signal can travel all the
way through the Earth, it is natural to design the detector to best identify this type
of event. Hence, the PMTs face downward.
An event which travels from the South Pole surface directly downward into the
ice has a zenith angle of 0
?
. An event which travels from the far side of the Earth is
considered upgoing. Exactly upgoing events have a zenith angle of 180
?
.
42
Figure 5.5: When the IceCube detector is complete, it will contain 80
strings and cover one cubic kilometer of ice. The AMANDA detector lies
within the IceCube array and is represented by the cylinder in the ?gure.
43
Figure 5.6: AMANDA coordinate system
5.4 Optical Properties of South Pole Glacial Ice
Unfortunately, photons emitted in the ice do not necessarily travel in straight
lines. Due to properties of the ice, the photons may be absorbed or may scatter before
being detected by an OM. Scattering and absorption in the ice delay the photon arrival
times. This leads to larger values of the space angle di?erence between the true and
reconstructed track of the particle.
Photons scatter more in dirty (dust-?lled) or bubble-laden ice. At shallow
depths, the ice contains many bubbles. Hence, AMANDA-II was built deeper than
this region in order to alleviate scattering di?culties. Layers of dust in the relatively
pure, deep polar ice reveal di?erent periods of climatological change [40].
44
In order to model the Cherenkov cone correctly, it is necessary to simulate the
clean and dirty layers of the ice. Lasers installed with the OMs are used to measure
the scattering and absorptivity of the ice at di?erent depths.
The geometric scattering length, ?
s
, is the average distance traveled by a par-
ticle between successive scatters [40]. After one scatter, the average change in the
particle's direction is < cos ? >. After being scattered n times, the photon has direc-
tion < cos ? >
n
. In this situation where a high energy photon is scattering o? dust
particles in the ice, the scattering is mostly forward and hence anisotropic. The e?ec-
tive scattering length, ?
e
, is used because of the anisotropy. After each scatter i, the
photon has moved a distance ?
s
< cos ? >
i
. Assuming a large number of scatters, n,
leads to the following relation:
?
e
= ?
s
X
n
i=0
< cos ? >
i
=
?
s
1 ? < cos ? >
:
(5.6)
The e?ective scattering coe?cient is the reciprocal of the e?ective scattering
length:
b
e
=
1
?
e
:
(5.7)
The e?ective scattering is depth and wavelength dependent and is shown in
Figure 5.7. The power law ?t with AMANDA data indicates that the wavelength
dependence of b
e
can be described by b
e
(?) / ?
? ?
, where ? = 0:9 ? 0:03 [40].
The absorption length, ?
a
, characterizes the distance at which the survival prob-
ability is 1/e. The absorption coe?cient, a, is the reciprocal of ?
a
:
45
a =
1
?
a
(5.8)
A two-component absorption model was used [40] to characterize the absorption
coe?cient as a function of depth, z.
a(z) = C
dust
(z)?
? ?
+ A
IR
e
? ?
0
=?
(5.9)
C
dust
is the dust concentration factor which varies as function of depth. The
values of the constants were determined to be A
IR
= (6954?973)m
? 1
and ?
0
= (6618?
71) nm. A ?t to AMANDA data indicates that ? = 1:08 ? 0:01. The absorption
coe?cient as a function of depth and wavelength is shown in Figure 5.7.
The results of the extensive AMANDA analyses of the optical properties of the
ice have been incorporated into the ice models used in this analysis. Three dirty
layers of ice were found at AMANDA depths and were included in the simulation.
This analysis uses ice simulated with the Muon Absorption Model, MAM. Although
the dust layers are simulated, optical properties of the ice remain one of the largest
systematic uncertainties in the detector.
46
Figure 5.7: The scattering (left) and absorption (right) properties of light
vary as a function of depth and wavelength in the polar ice.
47
Chapter 6
Preparation of the 2000 - 2003 Sample
6.1 Livetime and Triggers
An event triggers the detector when 24 or more optical modules are hit within
2.5 ?s. During the four-year span from 2000 to 2003, 7.14 billion events were recorded
by AMANDA-II.
2000
2001
2002
2003
Number of Triggers 1:37 ? 10
9
2:00 ? 10
9
1:91 ? 10
9
1:86 ? 10
9
The detector is not always running or accepting data, hence the livetime is 807
days rather than the four years spanned by the data ?les. Detector deadtime occurs
for a fraction of a second while an event is being recorded. Upgrades and maintenance
during the South Pole summer months also reduce the livetime. The adjusted livetime
is listed below for each year. The ?rst and last days of acceptable data for the year are
listed in the ?nal two columns. Due to di?erent instrument settings and the behavior
of the detector, the year 2000 is best characterized by three separate periods.
48
Day of the Year
0
50
100
150
200
250
300
350
Number of Events
0
200
400
600
800
1000
1200
´
10
6
Data 2000
Data 2001
Data 2002
Data 2003
Figure 6.1: Cumulative number of events.
Year/Period Detector Livetime (days) First Day Last Day
2000p1
58.9
47
125
2000p2
86.5
126
244
2000p3
51.6
245
309
2001
193
44
293
2002
204
43
323
2003
213
43
315
Searching for an astrophysical neutrino signal among 7.14 billion events could be
likened to searching for a needle in a haystack. As already mentioned, the detector is
dominated by events triggered by downgoing muons created from cosmic ray interac-
tions with the atmosphere. The ?rst step in reducing the data to a more manageable
size is to remove all events that appear to have traveled downward through the de-
tector. Unfortunately, ?rst-guess techniques for the event direction are not enough
to reject all downgoing backgrounds, as many events may be misreconstructed. As
the data volume is reduced, more accurate, but computer-intensive reconstruction
49
techniques can be applied to the remaining events.
The ?ltering process includes reconstructing the track direction of each event
and then removing any events that appear to be background. The ?ltering process
occurred in ?ve stages and is summarized in Table 6.2. The following sections explain
the ?ltering techniques applied.
6.2 Rejection of Atmospheric Muon Background
At Filtering Stage 1, a ?rst-guess reconstruction was applied to every data event.
The procedure, known as a direct walk reconstruction, considered a straight line track
between every pair of triggered optical modules [42]. A track between two hits was
only labelled as a candidate if the two hits occurred within 30 ns of the time it would
take for light to travel between the two optical modules. If multiple candidates existed,
the selected track was the one with the most neighboring candidates within a space
angle di?erence of 15
o
. Figure 6.2 shows the zenith angle of the reconstructed tracks
for all events at the beginning of the analysis. Vertically downgoing tracks have a
reconstructed zenith angle ? of 0
?
(cos(?)=1).
After using the direct walk reconstruction to infer the incoming zenith angle of
every event, all events reconstructed with zenith angles less than 70
?
were removed. To
improve the background rejection, another reconstruction method, JAMS, was then
applied.
The JAMS reconstruction uses a pattern matching algorithm when ?tting a track
to the data [43]. JAMS is particularly designed to eliminate coincident downgoing
atmospheric muons in the detector since they can be misreconstructed as a single
upgoing event. When JAMS is applied, 50 di?erent input directions are searched for
50
Cosine of Reconstructed Zenith Angle
-1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1
Back to top
Events
10
6
10
7
10
8
10
9
Back to top
level 0
Data 2000 - 2003
CORSIKA atms. m
Figure 6.2: The cosine of the reconstructed zenith angle is shown for
every event at the beginning of the analysis. The experimental data is
dominated by downgoing atmospheric muons. Events reconstructed as
upgoing appear on the left side of the plot and downgoing events appear
on the right.
51
clusters of events. A cluster has at least 7 close, on-time hits for the suggested track. A
simpli?ed Gaussian likelihood is then maximized to get the track parameters for each of
the identi?ed clusters. A neural net trained to separate good and bad reconstructions
is used to rank each of the clusters. The track from the highest ranking cluster is the
JAMS track. If the JAMS reconstruction indicated that the zenith angle of the track
was less than 80
?
, the event was removed.
6.3 Reconstruction Methods
Track parameters were adjusted to maximize the log likelihood, given the ob-
served light pattern [42]. Many of the Cherenkov photons scatter multiple times as
they travel through the ice and this changes their direction and delays the times at
which they are likely to be detected. For a given photon observation and proposed
track, a probability can be calculated that suggests how likely it is that a particular
hit was caused by that track. We denoted this the probability density function or
p.d.f., p(x
i
j~a), where x
i
describes a particular hit and ~a describes the parameters of
the track. The likelihood of a track is the product of the values of the p.d.f. for the
observed photon arrival times.
L(xj~a) =
Y
i
p(x
i
j~a)
The Pandel function was developed to describe the arrival times of light from the
Cherenkov cone for the Baikal experiment [44]. This function describes propagation
of light in media with both di?fusion and absorption e?ects. As the light travels, it
scatters instead of taking a direct path to the optical modules. Simulation samples
52
2000
2001
2002
2003
Number of Triggers 1:37 ? 10
9
2:00 ? 10
9
1:91 ? 10
9
1:86 ? 10
9
Filtering Stage 1
4:54 ? 10
7
8:18 ? 10
7
6:83 ? 10
7
6:53 ? 10
7
Filtering Stage 2
5:50 ? 10
6
6:87 ? 10
6
7:59 ? 10
6
8:02 ? 10
6
Filtering Stage 3
1:63 ? 10
6
1:90 ? 10
6
2:10 ? 10
6
2:22 ? 10
6
Table 6.1: The total number of data events at each ?ltering stage.
were used to ?t the parameters of the function for the AMANDA detector and its
local ice [42]. The function was then modi?ed to account for PMT time jitter. The
Pandel function is used to parameterize the p.d.f.
Because the initial track hypothesis is not always optimal, an iterative technique
was used in which each event was reconstructed 32 times using the Pandel parame-
terization. Each iteration of the track reconstruction shifts the zenith and azimuth
of the track and moves it to pass through the center of gravity of the hits. The best
track resulting from an iterative reconstruction maximizes the log likelihood of the
observation. Hence, in the Pandel reconstruction, the log likelihood of the Pandel
p.d.f. is maximized.
Because iterative reconstructions require more computing time, they were not
used until the Stage 1 ?ltering removed the obviously downgoing events from the
data set. The 32-iteration Pandel reconstruction was considered the most accurate
reconstruction performed on this data. At Filtering Stage 3, all events with Pandel-
reconstructed zenith angles less than 80
?
were removed.
53
6.4 Techniques to Further Improve Background Rejection
Several other techniques were used to improve background rejection. The ?rst
technique described here is an additional 64-iteration reconstruction. This zenith-
weighted (Bayesian) reconstruction provided valuable information for removing back-
ground later in the analysis. Methods were also employed to remove electronic crosstalk
and other fake events. In the ?nal ?ltering stages, two other reconstructions were per-
formed and their outputs used to characterize the events.
6.4.1 Zenith-weighted (Bayesian) Reconstruction
The zenith-weighted reconstruction uses our prior knowledge about the atmo-
spheric muon ?ux to perform each track reconstruction [42]. Based on previous mea-
surements with AMANDA, it is known that most of the events that trigger the detector
are atmospheric muons. Even without knowing any information about an event, it
is much more likely that the event actually traveled downward through the detector
rather than upward. Hence, a weighting function was derived based on the atmospheric
muon zenith angle distribution. If, for instance, an upgoing and downgoing track are
being considered as possible ?ts for a given event, the downgoing track would be given
a much larger weight and would be the preferred reconstructed track direction.
The zenith-weighted reconstruction is based on Bayes' theorem. Bayes' theorem
de?nes the probability of a situation A occuring given that situation B is true. The
theorem states that
P(AjB) =
P(BjA)P(A)
P(B)
:
(6.1)
In order to use this theorem, one must have prior knowledge of the situation.
54
In other words, one must know P(BjA) (the probability of B occurring if A is true),
P(A) (probability of A occurring) and P(B) (probability of B occurring).
In this scenario, P(BjA) corresponds to the probability that the reconstructed
track (assumed to be true) produces the observed events. This is the likelihood de-
scribed earlier. P(A) is the prior probability of observing a track in the direction
indicated by the reconstruction. Since it is known that atmospheric muons traveling
downward through the detector make up most of the background (downgoing atmo-
spheric muons:upgoing atmospheric neutrinos = 10
5
:1), there is a very strong chance
that the true track of the particle was downgoing. A weighting function was derived
based on the zenith distribution of downgoing atmospheric muons and this represents
P(A). P(B) is a normalization factor for the probability function.
When the zenith-weighted reconstruction is performed, the maximized quantity
is the p.d.f., P(BjA), times the prior function, P(A). The assumed prior weights the
function so that the resulting tracks are downgoing. Hence, truly upgoing events have
low likelihood values for this reconstruction.
6.4.2 Removal of Electronic Crosstalk
Electrical crosstalk between channels causes fake hits to register in the detector
[45]. AMANDA strings 5 - 10 are twisted pair cables which means that several cables
are wound together in the ice. When an electrical signal is generated by an optical
module, the signal travels up the cable to the counting house. Sometimes, the signals
leak o? into neighboring cables that are twisted into the same string bundle. This can
induce hits in optical modules attached to other cables, but located in the same ice
hole. Crosstalk is characterized and removed in two ways during ?ltering.
55
Figure 6.3: Time-over-threshold for the observed light pulse is on the
x-axis while amplitude is on the y-axis. Fake hits caused by crosstalk
appear in the lower left corner, at small values of the amplitude and the
time-over-threshold.
First, crosstalk events can be identi?ed by their pulse shape signatures. Crosstalk
hits are very narrow signals and hence they show unnaturally short time-over-threshold
values. When examining a two-dimensional plot of ADC (amplitude of light detected)
vs. time over threshold, crosstalk hits cluster in the low amplitude, short time-over-
threshold corner.
Crosstalk can also be identi?ed by the pattern of fake hits induced. Some optical
modules are known as talkers because they tend to induce fake hits preferentially in
56
other receiver modules. Measurements were done in which a a di?user ball was ?ashed
near an optical module. All other optical modules were tuned to high threshold
conditions, essentially turning them o?. In an ideal situation, only the optical module
hit by the di?user ball light would register a signal. However, hits on other optical
modules were recorded, hence indicating that crosstalk occurred.
A map was created that indicated which optical modules were most likely to
induce crosstalk in others. During crosstalk cleaning, hits were removed if a correlation
was seen between hits and the talker-receiver map. Hits were also removed if they fell
in the low amplitude, short time-over-threshold region described above.
6.4.3 Removal of Non-Photon Events
As described in the crosstalk section, not all hits that appear in the detector are
real. Flare events represent another class of fake events. Flare events are dominated
by hits that are not induced by photons triggering the optical modules. It is not
known what causes these non-photon hits.
Flare checking is a procedure intended to remove events that have too many
non-photon hits [46]. Nine indicators were de?ned and established to quantify the
number of non-photon hits in an event. Unfortunately, some good events also contain
non-photon hits. The problem is to separate good events from ?are events based on
the number of non-photon hits. Using the nine indicators, events were removed if they
contained too many non-photon hits.
The nine ?are indicators were developed based on strange behavior seen in some
event observables [46]. Non-photon events can be identi?ed because they do not show
typical pulse lengths. They can be either too long or too short. The beginning or end
57
of the pulse may not be observed because it occurred outside of the trigger window.
This is called a missing edge. A number of optical modules are listed as dead and hits
in these channels indicate ?ary events. Also, an event is most likely ?ary if there are
many hits on the twisted pair channels but very few hits along other types of strings.
Events were removed if they had more than 10 hits attributed to the following
three indicators:
1. number of hits that were too short on the twisted pair strings
2. number of hits in strings 1-4 of AMANDA-B10 relative to the number of hits in
string 5-10
3. the relative number of hits on strings 5-10 versus strings 11 - 19.
6.4.4 Paraboloid Reconstruction
A paraboloid ?t was performed on each event. L(r,z,a;P ) is the likelihood for a
track direction with point r along the track, zenith z, azimuth a and pattern of hits,
P . L
best
is the likelihood associated with the best reconstructed track for the event (in
this case, the 32-iteration Pandel track). The parameter space is searched in zenith
and azimuth until:
?(? logL) = (? logL
ellipse
) ? (? logL
best
) = 0:5
(6.2)
A con?dence ellipse is drawn in space about the zenith and azimuth coordinates
of the track. The ellipse covers 1˙ in either coordinate, hence having a probabil-
ity of 39.4% of containing the true track direction. The median of this distribution
58
represents a 50% probability that the ellipse contains the true track direction. The
width of the ellipse corresponds to the uncertainty in the track parameters. When
expanded to three dimensions, the likelihood space is ?tted to a paraboloid and a
minimization is performed to determine the paraboloid track. The range covered by
the con?dence paraboloid in each parameter is known as the paraboloid error. The size
of the paraboloid errors is used later in the analysis to estimate the angular resolution
of each event.
6.4.5 Velocity of Line Fit
A line ?t was added in the ?nal step of ?ltering. The ?t constructs a track
assuming that the muon travels a straight path with some velocity, v. It does not take
the Cherenkov cone or the optical properties of the ice into account [42]. The absolute
speed of the ?t can be used to classify events in the detector. Long muon tracks have
larger speeds than cascades.
6.5 Event Simulation
This analysis relied on simulated data sets of background and signal events.
Sixty-three days of downgoing atmospheric muons were simulated with CORSIKA
[41] version 6.030 and the QGSJET01 hadronic interaction model. The events were
simulated with a ? / E
? 2:7
primary energy spectrum. These downgoing events are
so frequent (˘80 Hz at trigger level) that two atmospheric muon events produced
by unrelated primaries often occur in the detector during the same detector trigger
window of 2.5 ?s. These coincident muon events may be caused by two muons which
are each individually incapable of triggering the detector with at least 24 OM hits.
59
Cut Applied
Fit
Number
Filtering Stage 1
Hit and OM Cleaning
Low ToT / amplitude ?lter
Direct Walk ?t
0
Zenith (direct walk) > 70
o
Direct WIMP ?t
1
CFirst (cascade) ?t
2
Pandel (16-iter.)
3
Cascade SPE
4
Cascade MPE
5
JAMS
6
Filtering Stage 2
Zenith(JAMS) > 80
o
Crosstalk Cleaning
Pandel (32-iter., JAMS seed)
7
MPE (16-iter., Pandel seed)
8
Bayesian (16-iter, Pandel seed)
9
Filtering Stage 3
Zenith(Pandel 32-iter.)> 80
o
Smoothness calculation
Cascade likelihood (16-iter.)
10
Paraboloid
11
Filtering Stage 4
Bayesian (64-iter., Pandel seed)
12
FlareChecker
Flrshrtm+Flrndcb1+Flrndc11 < 10
Filtering Stage 5
Line Fit
13
Table 6.2: The atmospheric muon background was reduced by three orders
of magnitude by a series of ?ltering requirements.
60
However, their combined hits trigger an event and the timing patterns of the light
from the two tracks may be such that the reconstruction results in a single upgoing
track. These coincident muon events were simulated for 826 days of livetime and have
a frequency of about ˘2-3 Hz at trigger level.
Muon neutrinos with a ? / E
? 1
spectrum were simulated with nusim [48] and
reweighted to atmospheric neutrino ?ux predictions [49, 50, 51, 52, 53], as well as an
astrophysical muon neutrino ?ux of E
2
? = 1?10
? 6
GeV cm
? 2
s
? 1
sr
? 1
. The normal-
ization of the test signal spectrum, which is irrelevant when setting a limit, was taken
to be approximately equal the previous upper limit from the AMANDA-B10 di?use
analysis [24].
6.5.1 Preparation of Simulated Events
Every operating season, the conditions and electronic settings for the detector are
slightly di?erent. For this reason, simulated events were generated to mimic the exact
conditions of the detector during a given period or year. For instance, a particular
optical module may have been acceptable for use in a 2001 analysis, but not in 2003.
Simulation was generated for atmospheric muons and neutrinos, coincident at-
mospheric muons, and astrophysical neutrinos. The amount of each type of simulation
that was generated varied based on available computer resources. It would be too com-
puter intensive to generate one day of simulated livetime for every day of livetime that
the detector was actually running during 2000 to 2003. Hence, the simulation repre-
sents smaller livetimes than in the actual data. The simulation events were scaled to
match the livetime of the data during a particular period or year.
61
Year / Period Coincident Muon Atmospheric Muon Detector
Simulation
Simulation
Livetime
Livetime
Livetime
(days)
(days)
(days)
2000 period 1
139.1
9.9
58.9
2000 period 2
136.6
9.9
86.5
2000 period 3
139.4
9.9
51.6
2001
138.3
10.1
193
2002
133.6
12.1
204
2003
139.1
10.9
213
Atmospheric muons and coincident muons are simulated with their supposed
energy spectrum, ? / E
? 3:7
. The generation of events with the true energy spectrum
makes it very time consuming to accumulate acceptable statistics at high energy. How-
ever, nusim is generated with a ? / E
? 1
spectrum. By generating a ?at spectrum in
cosine of the zenith angle, the nusim events can be reweighted to represent atmo-
spheric or astrophysical neutrinos. Hence, the generated nusim [48] livetime cannot
be listed in the table. Instead, each event receives a weight based on equation 6.3. Not
all of these events trigger the detector. Only a fraction survive to the higher ?ltering
levels.
nusim weight = I ??? E ? A ? L ??? ln(E
high
/E
low
) / ( N
files
? N
events
) (6.3)
where
I = interaction probability
? = ?ux from model [GeV
? 1
cm
? 2
s
? 1
sr
? 1
]
E = Energy [GeV]
A = Area [cm
2
]
L = Livetime [s]
62
? = Solid angle [sr]
ln(E
high
/E
low
) = Energy range over which the events were generated
N
files
= number of ?les generated
N
events
= number of events generated per ?le
Number of Events
2000
2000
2000
2001 2002 2003
Generated
period 1 period 2 period 3
?10
6
?10
6
?10
6
?10
6
?10
6
?10
6
?
1.00
1.00
0.60
2.39 2.52 2.60
??
1.00
1.00
0.60
2.39 2.55 2.59
The simulated events underwent the same reconstruction procedures as the data
and had to satisfy the same zenith angle requirements.
After using the ?ltering levels to require that events must have entered with
zenith angles greater than 80
o
, there would ideally have been no remaining simulated
atmospheric muons that truly came from 0
o
to 80
o
. However, after Filtering Stage
5, there were 3:6 ? 10
6
simulated downgoing muons remaining. These muons recon-
structed with angles between 80
o
and 180
o
, despite the fact that they were generated
in the opposite hemisphere. These are known as misreconstructed events. The next
chapter will describe how these events were removed.
63
Chapter 7
Obtaining an Upgoing Neutrino Sample
In order to prevent any inadvertent tuning of the event selection criteria that would
bias the result, a blindness procedure was followed which required that further event
selections were developed only on low energy data and simulation, where the signal is
negligible compared to the background. The number of OMs triggered (from now on
indicated by N
ch
, or number of channels hit) is the energy-correlated observable used to
separate atmospheric neutrinos from astrophysical ones (Figure 7.1). Only low energy
data events (low N
ch
values) were compared to simulation. High energy data events
(high N
ch
values) were only revealed once the ?nal event selection was established.
Energy and N
ch
are correlated since high energy events release more energy in the
detector causing more hits than low energy ones. However, the correlation is not
perfect since high energy events occurring far from the detector may trigger only a
few OMs.
Event selection was based on observables associated with the reconstructed
tracks [42]. In order to separate misreconstructed downgoing events and coincident
muons from the atmospheric and astrophysical neutrinos, events were required to have
observables consistent with long tracks and many photons with arrival times close to
64
N
ch
0
20
40
60
80
100 120 140 160 180 200
Normalized Counts
0
0.05
0.1
0.15
0.2
0.25
0.3
1.9 < log
10
(E/GeV) < 2.1
2.9 < log
10
(E/GeV) < 3.1
3.9 < log
10
(E/GeV) < 4.1
4.9 < log
10
(E/GeV) < 5.1
5.9 < log
10
(E/GeV) < 6.1
Figure 7.1: The number of OMs hit during an event (N
ch
) was used as an
energy-correlated observable. Each line on this N
ch
distribution represents
events with approximately the same simulated energy. High energy events
may not be contained within the detector and hence can trigger a wide
N
ch
span.
65
those predicted for un-scattered propagation. The number of photons arriving be-
tween ? 15 and +75 ns of their expected un-scattered photon arrival time is referred
to as the number of direct hits (N
dir
). The direct length (L
dir
) is the maximum sep-
aration of two direct hits along the reconstructed track. The smoothness (S) is a
measurement of how uniformly all hits are distributed along the track and it varies
between ? 1.0 and 1.0. Positive values of the smoothness indicate more hits at the
beginning of a track and negative values indicate more hits occur toward the end.
Evenly distributed hits will have smoothness values near 0. The median resolution
(MR) is calculated from a paraboloid ?t to the likelihood minimum for the track [47].
This method analyzes the angular resolution on an event-by-event basis. Lastly, high
quality events have higher values of the logarithm of the up-to-down likelihood ratio,
?L = (? logL
down
) ? (? logL
up
). The likelihoods L
up
and L
down
are the product of the
values of the probability density function for the observed photon arrival times, for the
best upgoing and zenith-weighted downgoing track reconstruction [42], respectively.
The likelihood ratio requirement was more strict for vertical events than for events
near the horizon. Horizontal events tend to have smaller likelihood ratios since the
zenith angle di?erence between the best upgoing and zenith-weighted downgoing track
hypothesis is often small.
As seen in Figure 7.2, requiring a minimum value of the track length, for instance,
can be a powerful method of rejecting misreconstructed downgoing backgrounds. The
event selection requirements for L
dir
, N
dir
, smoothness, median resolution and like-
lihood ratio were established to remove many orders of magnitude more misrecon-
structed background than upgoing atmospheric neutrinos or signal neutrinos. Events
66
Track length [m]
0
50
100 150 200 250 300 350
Back to top
Events
1
10
10
2
10
3
10
4
10
5
10
6
Data
Atms. n
Signal n
Coincident m
Atms. m
Back to top
remove
keep
Figure 7.2: The reconstructed track length within the detector is shown.
In order to identify muon neutrino tracks, events were required to have
long tracks of at least 170 meters. This removed a large fraction of the
atmospheric muon simulation, but had a smaller e?ect on the atmospheric
neutrino and signal simulations.
67
L0
L1
L2
L3
L4
L5*
Zenith Angle [
o
]
>80
>80
>80
>80
>100
Number of
Direct Hits
>5
>8
>8
>13
Track Length [m]
>100
>130
>130
>170
jSmoothnessj
<j0:30j
<j0:30j
<j0:25j
Median
Resolution [
o
]
<4.0
<4.0
Likelihood
Ratio (?L)
vs. Zenith
?L > ? 38:2cos(Zen:)
+27:506
Number of
Remaining Events 5:2 ? 10
9
7:8 ? 10
6
1:2 ? 10
6
3:5 ? 10
5
1:8 ? 10
5
465
* = level of the ?nal analysis
Table 7.1: The table summarizes the event quality requirements as a func-
tion of quality level. Events only remained in the sample if they ful?lled
all of the parameter requirements for a given level. The removal of all hor-
izontal events (zenith < 100) contributed to the large decrease in events
from L4 to L5.
which did not meet an optimized minimum or maximum value of each parameter were
removed.
The event selection requirements were successively tightened, based on the recon-
structed track parameters, establishing ?ve quality levels. The requirement is de?ned
for each parameter in Table 7.1. The plots in Figure 7.3 show the zenith angle distri-
bution of all events ful?lling the zenith angle >80
?
and event observable requirements
at the chosen level. Although the entire zenith angle region is being studied, the event
selection requirements preferentially retain vertically upgoing events. Horizontal and
vertical events must pass the same requirements for track length and number of direct
hits, however this is more di?cult for horizontal events since the detector is not as
wide as it is tall.
After the zenith angle criteria was ful?lled at Level 1, the data mostly con-
68
Data 2000 - 2003
Uncertainty in atmospheric n flux
Barr et al. atms. n
Honda et al. atms. n
CORSIKA atms. m
Signal n: E
2
f = 10
-6
GeV cm
-2
s
-1
sr
-1
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
level 1
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
level 2
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
level 3
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
level 4
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
10
2
10
3
10
4
10
5
10
6
10
7
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
-2
10
-1
1
10
10
2
10
3
10
4
horizontal
events
removed
level of final analysis
level 5
Cosine of Reconstructed Zenith Angle
-1
-0.8
-0.6
-0.4
-0.2
-0
Events
10
-2
10
-1
1
10
10
2
10
3
10
4
Figure 7.3: The cosine of the zenith angle is plotted for all events surviving
the event quality criteria at a given level. Events at cos(zenith) = ? 1 are
traveling straight up through the detector from the Northern Hemisphere.
The initial zenith angle requirement removed events from 0
o
to 80
o
(level
1 - top right). Events reconstructed just above the horizon appear at the
right side of each plot. Each level represents an increasingly tighter set
of quality requirements. As the quality level increased, misreconstructed
downgoing muons were eliminated. To ensure a clean upgoing sample,
events coming from the horizon were discarded by requiring reconstruction
angles greater than 100
o
. The ?nal analysis was performed at level 5
(bottom right) with horizontal events removed.
69
tains misreconstructed atmospheric muons (top right, Figure 7.3). As the quality
parameters become more restrictive, the data begins to follow the atmospheric neu-
trino simulation in the upgoing direction and the atmospheric muon simulation in the
downgoing direction. At Level 5, the event quality requirements were strong enough
to have removed all of the misreconstructed downgoing atmospheric muon events that
were simulated. However, just to be sure that the ?nal data set only included at-
mospheric and astrophysical neutrinos and no misreconstructed downgoing events, an
additional zenith angle requirement was imposed. All events were kept if they were
reconstructed between 100
?
and 180
?
. The analysis continued with the data sample
shown at Level 5.
70
Chapter 8
Separating Atmospheric Neutrinos from
Astrophysical Neutrinos
Figure 8.1 shows the N
ch
distribution for events at Level 5. The optimal place for the
energy-correlated event observable requirement was established with the simulation by
minimizing the expected Model Rejection Factor (MRF) [54]. The Feldman-Cousins
method was used to calculate the median upper limit [55]. The MRF is the median
upper limit divided by the number of predicted signal events for the ?
?
signal being
tested. The MRF was calculated for every possible N
ch
value and was at its mini-
mum at N
ch
? 100. Hence, the optimal separation of astrophysical and atmospheric
neutrinos is achieved with this N
ch
requirement.
The ?nal event sample was composed of events which pass all event selection
requirements (Level 5) and have N
ch
? 100. After the high N
ch
requirement, the
atmospheric neutrino simulation peaked at 10 TeV, while the signal simulation peaked
around 100 TeV (Figure 8.2). The energy range de?ned by the central 90% of the signal
with N
ch
? 100 is the energy range for the sensitivity or limit. For this search, the
central 90% signal region extends from 16 TeV to 2.5 PeV.
71
N
ch
0
20
40
60
80 100 120 140 160
Events
10
-2
10
-1
1
10
10
2
Data 2000 - 2003
Uncertainty in atmospheric n
Barr et al. atms. n + prompt n
Honda et al. atms. n + prompt n
E
2
f
signal
= 1.0 x 10
-6
GeV cm
-2
s
-1
sr
-1
E
2
f
signal
= 7.4 x 10
-8
GeV cm
-2
s
-1
sr
-1
final event sample
atms.
n
rescaled
N
ch
0
20
40
60
80 100 120 140 160
Events
10
-2
10
-1
1
10
10
2
Figure 8.1: N
ch
, or number of OMs hit. The full data set is revealed. Both
conventional and prompt atmospheric neutrinos are shown and their un-
certainties are represented by the gray band. The central prompt neutrino
?ux used here is the average of the Martin GBW [34] and Naumov RQPM
[35, 36] models. The signal is rescaled to re?ect the new limit.
72
log
10
(True
n
Energy [GeV] )
234567
Back to top
Events
10
-1
1
10
10
2
10
3
Barr & Honda avg. atms n, no N
ch
cut
Barr & Honda avg. atms n, N
ch
>=100
Signal n, no N
ch
cut
Signal n, N
ch
>=100
90% signal region
Figure 8.2: The true energy of the simulation is shown for atmospheric
neutrino and signal events. The thin dashed (atms. ?) and solid (signal ?)
curves represent the number of events before the N
ch
? 100 requirement.
The thick dashed and solid lines represent only the events in the high
energy sample.
73
Chapter 9
E?ective Area
The e?ciency of the detector for neutrinos is quanti?ed by the e?ective area. In the
energy range relevant to this analysis, it increases with energy and is further enhanced
by including uncontained events. The e?ective area is described by the following
equation where N represents the number of observed events and T is the detector
livetime:
N
T
=
R
A
?
e?
(E
?
; ?)?
?
d?dE:
(9.1)
The e?ective area as a function of energy is shown for di?erent zenith angle
regions in Figure 9.1 (and is tabulated in Tables 9.1 and 9.2). At energies greater
than 10
5
GeV, the Earth begins to be opaque to neutrinos depending on direction
and the highest energy events are most likely to come from the region around the
horizon [56]. In Figure 9.1, the e?ective area decreases at high energy because tracks
with zenith angles between 80
?
and 100
?
were discarded. Most of the events that were
removed were high energy events from the horizon.
74
Log
10
(E
n
/GeV)
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
)
2
area (cm
n
Back to top
Effective
10
-2
10
-1
1
10
10
2
10
3
10
4
10
5
10
6
-1.0 < cos(True Zenith) < -0.8
-0.8 < cos(True Zenith) < -0.6
-0.6 < cos(True Zenith) < -0.4
-0.4 < cos(True Zenith) < -0.17
all angle
Figure 9.1: E?ective area for ?
?
as a function of the true simulated energy
in intervals of cosine of the true neutrino zenith angle. The e?ective area
is the equivalent area over which the detector would be 100% e?cient for
detecting neutrinos. The absorption of neutrinos in the Earth is taken
into account. The angle-averaged e?ective area is represented by the solid
black line.
75
Energy
-1 <cos(Zenith) <-.8 -.8 <cos(Zenith) <-.6
log
10
(E/GeV)
?
?
??
?
?
?
??
?
[10
3
cm
2
]
[10
3
cm
2
]
[10
3
cm
2
]
[10
3
cm
2
]
3.6
0.487
0.166
0.279
0.0673
3.8
1.04
1.1
0.652
0.646
4
3.36
2.85
1.82
1.89
4.2
8.74
7.54
4.97
5.56
4.4
18.8
16.2
15.3
12.4
4.6
29.3
30.4
34
26.9
4.8
44.9
46.4
52.7
58.8
5
59.6
65.5
92.6
88
5.2
75.7
69.7
128
121
5.4
72.6
84.4
153
163
5.6
63.5
77.8
180
179
5.8
63.3
66.9
183
188
6
51.9
49.3
170
177
6.2
36.6
39.1
145
151
6.4
27.8
22.6
110
113
6.6
9.97
14.7
72.3
77
6.8
7.8
8.73
54.2
48.2
7
3.39
3.08
29.6
29.5
7.2
3.12
1.44
16.5
15.2
7.4
0.939
0.718
7.97
9.64
7.6
0.864
0.791
5.12
4.15
7.8
0.492
0.521
2.59
2.08
Energy
-.6 <cos(Zenith) <-.4 -.4 <cos(Zenith) <-.17
log
10
(E/GeV)
?
?
??
?
?
?
??
?
[10
3
cm
2
]
[10
3
cm
2
]
[10
3
cm
2
]
[10
3
cm
2
]
3.6
0.108
0.0562
0.0752
0.0451
3.8
0.282
0.163
0.178
0.0818
4
0.845
0.93
1.13
0.543
4.2
3.73
3.39
1.98
1.66
4.4
9.74
8.22
7.23
6.02
4.6
21.1
19.9
17.9
18.2
4.8
49.7
43.3
33.2
36.9
5
86.2
77.5
74.2
68.3
5.2
118
119
119
113
5.4
179
165
163
167
5.6
232
217
264
230
5.8
243
232
306
310
6
271
286
377
373
6.2
269
258
418
389
6.4
251
229
441
452
6.6
212
197
437
391
6.8
154
149
417
437
7
105
114
413
380
7.2
79.8
61.4
328
327
7.4
46.3
32.9
285
274
7.6
31.8
19.4
209
212
7.8
17.7
10.3
142
146
Table 9.1: E?ective area as a function of the energy and zenith angle of
the simulation.
76
E?ective Area in cm
2
Energy
All angle
log
10
(E/GeV)
?
?
??
?
[10
3
cm
2
] [10
3
cm
2
]
3.6
0.164
0.0572
3.8
0.381
0.343
4
1.24
1.07
4.2
3.33
3.15
4.4
8.9
7.51
4.6
17.9
16.7
4.8
31.8
32.6
5
55.6
52.7
5.2
78.9
75.8
5.4
102
103
5.6
136
127
5.8
144
145
6
161
162
6.2
164
155
6.4
157
155
6.6
139
130
6.8
121
126
7
112
102
7.2
86
83.4
7.4
67.8
63.6
7.6
51.2
49
7.8
35.5
34.6
Table 9.2: The angle-averaged neutrino e?ective area as a function of
energy.
77
Chapter 10
Systematic Uncertainty
A discovery is made if an excess of events over the predicted background is observed
in the data. However, due to uncertainties in the simulation, the number of sig-
nal and background events predicted may not accurately re?ect the true signal and
background. Theoretical uncertainties exist in the atmospheric neutrino ?ux mod-
els for several reasons. The cosmic ray spectrum is very uncertain at high energy
and hadronic interactions for this energy range are not well understood. There are
also detector-related uncertainties. Photons scatter more in dirty or bubble-laden ice.
Hence, our incomplete understanding of the dust layers in the ice and the bubbles in
the hole ice (formed from water that refroze after deployment of the OMs) add un-
certainty to our models [40]. There are also uncertainties in the simulation associated
with the modeling of light propagation in the ice and with the optical module sensitiv-
ity. These contributions are considered individually to see how they a?ect the number
of simulated events in the ?nal sample. The number of experimental data events re-
maining after the ?nal energy requirement (N
ch
? 100) is then compared to the range
of predicted background and signal events when uncertainties are considered.
78
10.1 Theoretical Uncertainties in the Background
For this analysis, two models based on the work of Barr et al. [50, 52, 53] and
Honda et al. [51] were considered equally likely options for the conventional atmo-
spheric neutrino simulation. These two models are recent calculations that cover the
highest and lowest portion of the atmospheric neutrino ?ux band created by uncertain-
ties in the primary cosmic ray ?ux and the high energy hadronic interaction models.
Since these models do not extend to the high energies needed for this analysis, the
models were extrapolated to higher energies.
10.1.1 Conventional Atmospheric Neutrino Flux based on the Barr et al.
Model
For this analysis, the Barr et al. ?ux below 10 GeV was taken from [50]. From
10 GeV to 10 TeV, the ?ux tables from [52], based on the primary spectrum of [53],
were used. Above 10 TeV, the weight was derived by performing a 2-dimensional ?t
with a ?fth degree polynomial to the log
10
E vs. cos(zenith) tables of the atmospheric
neutrino ?ux values from lower energies just mentioned. The TARGET version 2.1
[57] hadronic interaction model was used [50].
10.1.2 Conventional Atmospheric Neutrino Flux based on the Honda et
al. Model
In an attempt to better ?t the AMS [58] and BESS [59, 60] data, Honda et al.
changed the power law ?t to the proton cosmic ray spectrum from -2.74 to -2.71 above
100 GeV [51]. Other parameters in the cosmic ray ?t remained similar to the Barr et
al. ?ux mentioned above [61], although the DPMJET-III [62] interaction model was
79
used. The atmospheric neutrino weights from [51] were used up to 10 TeV. Above
that energy, a 2-dimensional ?t of the lower energy values was again used as described
above. The result was a lower atmospheric neutrino ?ux prediction than the Barr et
al. ?ux.
10.1.3 Prompt Atmospheric Neutrinos
Conventional atmospheric neutrinos from the decay of pions and kaons are not
the only source of atmospheric background. Above 50 TeV - 1 PeV, the source of
atmospheric neutrinos is expected to change [33, 35, 36, 34, 30]. Semileptonic decays of
short-lived charmed particles become the main contributor to the atmospheric neutrino
?ux.
10.1.4 Additional Neutrino Flux Uncertainty
Uncertainties were included for both conventional atmospheric neutrino models.
The uncertainty in the cosmic ray spectrum was estimated as a function of energy
based on the spread of values measured by many cosmic ray experiments [63]. These
uncertainties were added in quadrature with the estimated uncertainty due to choosing
di?erent hadronic interaction models [50, 51, 61]. Uncertainties were also estimated
based on the spread of predictions surrounding the unknown prompt neutrino ?ux.
Unless mentioned otherwise, when prompt neutrinos were included in this work, the
average of the Martin GBW (Golec-Biernat and Wustho?
?
) [34] and Naumov RQPM
(Recombination Quark Parton Model) [35, 36] models is shown. This is henceforth
called the central prompt neutrino model.
All of the uncertainty factors for the total (conventional + prompt) atmospheric
80
log
10
[E
n
(GeV)]
12345
Back to top
Percent Uncertainty
20
Back to top
30
40
50
60
70
80
90
100
Figure 10.1: The estimated uncertainty in the atmospheric neutrino ?ux as
a function of energy. Due to the large uncertainty in the prompt neutrino
?ux at greater than 10
4
GeV, the total uncertainty rises sharply.
neutrino ?ux were combined and are shown as a function of energy in Figure 10.1. Since
the true energy of every simulated event is known, each event was given a weight based
on the maximum uncertainty estimated for that neutrino energy. As a result, three
predictions for the number of atmospheric neutrinos in the ?nal high energy sample
were made (the model, the model plus maximum energy-dependent uncertainty, the
model minus maximum energy-dependent uncertainty). Since both the Barr et al.
and Honda et al. ?uxes were considered equally likely, the central prompt neutrino
?ux was added to both predictions. Then uncertainties were added and subtracted
to both of these total atmospheric neutrino ?uxes, creating six di?erent background
possibilities.
81
10.2 Normalizing the Simulation to the Data
After all but the N
ch
event selection requirements were ful?lled, the N
ch
dis-
tribution for the observed low energy events was inconsistent with that for the total
atmospheric neutrino simulation in normalization. Each of the six atmospheric neu-
trino background predictions was renormalized to match the number of data events
observed in the low N
ch
region, where the signal was insigni?cant compared to the
background. By rescaling the simulation to the number of observed data events, the
uncertainty in the atmospheric neutrino ?ux was reduced to the uncertainty in the
spectral shape.
The number of low energy conventional atmospheric neutrinos (second column
of Table 10.1) is added to the 4.0 prompt neutrinos predicted with the central prompt
neutrino model. The total atmospheric background prediction before renormalization
is shown in the third column of Table 10.1. Instead of renormalizing the simulation
based on all events with N
ch
< 100, the renormalization was only based on the region
50 < N
ch
< 100. Because of the di?culty of simulating events near the threshold of the
detector, the atmospheric neutrino simulation did not faithfully reproduce the shape of
the N
ch
distribution for the data at N
ch
below 50. Atmospheric neutrino models were
scaled to match the 146 events seen in the experimental data for 50 < N
ch
< 100. The
total number of high energy events predicted to survive the ?nal energy requirement
is shown before renormalization in the sixth column and after renormalization in the
last column.
Since some of the atmospheric neutrino models predicted more events than the
data while others predicted less, renormalization of the models to the data brought the
82
Atms. ? Model
Conv.
Conv. ? +
Scale
Conv.
Conv. ? + Background
Atms. ?
prompt ?
Factor
Atms. ?
prompt ?
Predicted in
50 <N
ch
<100 50 <N
ch
<100
to 146
N
ch
? 100
N
ch
? 100
N
ch
? 100
Low
Sample
Energy
after
Data
Scaling
Events
Barr et al. Max
249.4
253.4
0.58
13.3
14.5
8.3
Barr et al.
193.5
197.5
0.74
9.1
10.3
7.6
Barr et al. Min
137.6
141.6
1.03
4.9
6.1
6.3
Honda et al. Max
191.0
195.0
0.75
9.3
10.5
7.9
Honda et al.
148.7
152.7
0.96
6.4
7.6
7.3
Honda et al. Min
106.5
110.5
1.32
3.4
4.6
6.1
Table 10.1: Number of atmospheric neutrino events predicted by the sim-
ulation. Uncertainty in the high energy cosmic ray ?ux was incorporated
into the maximum and minimum predictions.
simulated models into closer agreement. Because this renormalization aimed to correct
for theoretical uncertainties in the atmospheric neutrino background prediction, it was
not applied to the simulated neutrino signal.
10.3 Simulation Uncertainties
To assure that the detector response to high energy events (N
ch
? 100) is under-
stood, it is important to study high energy events while simultaneously keeping the
high energy upgoing events blind to the analyzer. To this end, an inverted analysis
was performed in which high quality downgoing tracks were selected from the initial
data set. The advantage of studying high quality downgoing tracks is that large data
sets are available to study both the high and low energy events. When the data and
simulation observable distributions are not perfectly matched, imposing event quality
requirements may result in removing di?erent fractions of the simulation in comparison
with the data. The inverted analysis was used to study this systematic e?ect.
83
10.3.1 Inverted Analysis
For the inverted analysis, all event quality requirements described previously
(Table 7.1) were applied, but events were selected based on a high probability of being
downgoing rather than upgoing tracks.
When compared to the downgoing experimental data, small shifts were observed
in the peaks of the simulated distributions for the number of direct hits (N
dir
), the
smooth distribution of hits along the track (S), the event-by-event track resolution
(MR) and likelihood of being downgoing muon tracks rather than upgoing (Inverted
Likelihood Ratio, ILR). These discrepancies are most likely due to inaccurate mod-
eling of optical ice properties in the simulation, since it is technically challenging to
implement a detailed description of photon propagation through layered ice.
If multiple parameters are correlated, it is possible that mismatches in one pa-
rameter may a?ect the agreement between data and simulation in another. In order
to study these e?ects, the di?erences in the data and simulation were analyzed at the
level where no quality criteria had been applied. The simulated distributions needed
to be shifted to larger values by approximately 10% for N
dir
, 8% for S, 5% for MR
and 1% for ILR. When simultaneous corrections to the simulation for all of these
e?ects were applied, the downgoing data and simulation were in better agreement for
all parameter distributions. Later in the analysis, these shifts were applied to the
upgoing analysis. The number of background and signal events appearing in the ?nal
upgoing sample was recalculated based on these simulation shifts.
84
N
ch
0
50
100
150
200
250
Back to top
Events
10
5
10
6
10
7
Data
CORSIKA atms. m
Figure 10.2: In the inverted analysis, the highest quality downgoing events
were studied. The N
ch
distribution is shown for all events which survive
the inverted quality requirements.
85
10.3.2 Uncertainty in Detector Response
The downgoing sample from the inverted analysis was also used to study how
well the detector response was simulated in the high energy (N
ch
? 100) regime. Using
downgoing data and atmospheric muon simulation, a ratio of the number of events was
taken as a function of N
ch
from the histograms shown in Figure 10.2. If the simulation
perfectly described the data, the shapes of the N
ch
distributions would match and this
ratio would be ?at. The downgoing ratio was mostly ?at, but slightly increased at large
N
ch
where low statistics introduced large uncertainties. The statistical uncertainty
aside, a scenario was considered in which the downgoing data to simulation ratio truly
increased as N
ch
increased. Under this scenario, the simulation is renormalized by a
larger factor at high N
ch
to replicate the data. This N
ch
-dependent renormalization
was then applied to the upgoing simulation used for the main part of the analysis. This
non-linear normalization factor had a negligible e?ect in the number of atmospheric
neutrinos predicted in the ?nal sample of events with N
ch
? 100. However, the high
energy signal simulation event rate increased by 25% when this non-linear N
ch
e?ect
was included. This uncertainty was incorporated in the ?nal limit calculation that
will be described in the next section.
Detection e?ciency also depends on the OM sensitivity. This parameter of the
simulation was modi?ed and new simulated events were generated. After comparing
the data and simulation with di?erent OM sensitivities, a 10% uncertainty in the
total number of events due to inaccurate modelling of the OM detection sensitivity
was incorporated into the ?nal upper limit calculation.
The systematic errors due to the neutrino interaction cross-section, rock density
86
Number of Channels (Nch)
0
50
100
150
200
250
300
350
400
ratio = data / dCorsika
0
0.5
1
1.5
2
2.5
3
ratio = data / dCorsika atmospheric muons (inverted analysis)
Figure 10.3: Ratio of the number of data events to the number of simulated
events as a function of N
c
from the inverted analysis.
(below the detector), and muon energy loss do not contribute signi?cantly to this
analysis [64].
10.3.3 Relationship between Up and Downgoing Events
In addition to using the inverted analysis to study high energy events and the
bias introduced by inaccurate simulation, the downgoing events can be used as a
calibration beam for the upgoing atmospheric neutrino ?ux. To do this, the same
model (CORSIKA) was used to describe the downgoing atmospheric muons and the
upgoing atmospheric neutrinos [65].
As shown in Table 10.2, the ratio of experimental data to CORSIKA downgoing
muon simulation was relatively constant as the event selection became more discrim-
inating. The simulation does not match the data normalization and this may be a
87
consequence of the theoretical errors in the CORSIKA simulation (mainly due to the
hadronic interaction model (QGSJET01) and uncertainty in the primary spectrum
(? / E
? 2:7
)). Another contributing factor to the normalization di?erence may be
that light propagation in the layered ice is modeled inaccurately. When the upgoing
CORSIKA atmospheric neutrinos are rescaled based on the downgoing muons, then
the upgoing experimental data and CORSIKA atmospheric neutrino simulation are
in good agreement for the number of low energy events in the ?nal sample. This can
only be seen when the tightest criteria are applied because misreconstructed muons
and coincident muons contaminate the data sample when the quality requirements are
loose. For instance, at level 5 in the inverted analysis, the ratio of downgoing data
to simulation was 1.22. For the upgoing analysis at level 5, 146 events were observed
and 124.9 CORSIKA atmospheric neutrinos were predicted. When adjusted based on
the inverted analysis, 152.4 (= 124:9 ? 1:22) CORSIKA atmospheric neutrinos were
predicted, which is in good agreement with the observed value. This shows that it is
possible to adjust the normalization of the upgoing events based on the downgoing
observations (when the up and downgoing simulation use the same input assumptions
about the spectrum and interaction model).
88
L1
L2
L3
L4
L5
?
Downgoing
data (?10
8
)
7.88
6.70
6.05 5.89 2.59
CORSIKA
atms. ?(?10
8
)
6.63
5.75
5.12 5.01 2.12
ratio
1.19
1.17
1.18 1.18 1.22
Upgoing
signal
325.4
241.0 190.8 184.8 103.2
coinc ?
2572.8 267.6 45.8 29.4 0
misreconstructed
CORSIKA
atms. ?
37801.7 2574.8 147.7 34.2 0
Barr et al.
atms. ?
680.6
525.9 392.9 379.9 193.5
Honda et al.
atms. ?
513.2
399.6 299.9 290.0 148.7
Martin GBW prompt ?
1.9
1.9
1.6
1.5
0.7
Naumov RQPM prompt ? 18.9
18.9
16.0 15.5 7.5
CORSIKA
atms. ?
439.9
335.2 251.0 242.5 124.9
Adjusted
CORSIKA
atms. ?
523.5
392.2 296.2 286.2 152.4
data
276894 24422 1269 531
146
?
L5 = level of ?nal analysis
Table 10.2: The number of low energy events (50 <N
ch
<100) at a given
level for the di?erent types of simulation and experimental data. The top
portion of the table presents results from the inverted analysis. The main
upgoing analysis is summarized in the lower portion of the table. Note
that the upgoing data and adjusted CORSIKA atmospheric neutrino ?ux
are in good agreement when the CORSIKA neutrino events are adjusted
by the scale factor determined in the downgoing analysis. This agreement
can be seen at the tightest quality levels because all misreconstructed
backgrounds have been removed.
89
Chapter 11
Results
We calculated a con?dence interval based on the number of events in the ?nal N
ch
?
100 sample of the predicted background and signal and the observed data. Statistical
and systematic uncertainties were incorporated into the con?dence interval such that
the true, but unknown, value of the di?use ?ux of astrophysical neutrinos is contained
within the interval in 90% of repeated experiments. A hybrid frequentist-Bayesian
method based on the work of Cousins and Highland [66] was used to construct a con-
?dence belt with systematic uncertainties. The likelihood ratio ordering was based on
the uni?ed con?dence intervals explained by Feldman and Cousins [55]. The uncer-
tainty in the detection e?ciency of the signal was set at 27% (10% for optical module
sensitivity added in quadrature with 25% for non-linearity in the N
ch
spectrum when
data and simulation are compared). Systematic uncertainties on the number of back-
ground events in the ?nal sample were also included in the con?dence belt construction.
Inclusion of the signal and background uncertainties followed the methods described
by Conrad et al. [67] and Hill [68].
In constructing the ?at Bayesian prior for the background, twelve atmospheric
neutrino models were considered equally likely. The twelve predictions were derived
90
as follows. Initially, two background predictions were considered, Barr et al. and
Honda et al., each with the central prompt neutrino ?ux added. To include systematic
uncertainties in the models, maximum uncertainties were added and subtracted from
each model. Hence, the six predictions were named Barr et al. maximum, nominal
and minimum and Honda et al. maximum, nominal and minimum. The number
of events predicted for the background in the ?nal sample is listed in Table 10.1.
To account for systematic uncertainties in the detector response, the simulation was
shifted in four di?erent parameters. This simulation shift was performed on each of the
6 models described above, hence creating a total of 12 di?erent atmospheric neutrino
predictions that were used in the con?dence belt construction. The number of events
predicted by the 6 models with shifted simulation was within 10% of each number
reported in Table 10.1.
11.1 Results for ? / E
? 2
The signal hypothesis consisted of a ?ux E
2
? = 1.0 ? 10
? 6
GeV cm
? 2
s
? 1
sr
? 1
.
At this signal strength, 66.7 signal events were expected in the ?nal N
ch
? 100 data.
(This value assumes half of the correction from the simulation shifts since 68.4 events
were predicted in the ?nal selection, but the number of events decreased to 65.0 when
the simulation shifts were applied.) The sensitivity was obtained from the slice of
the con?dence belt corresponding to zero signal strength. The median observation
assuming no signal was seven events, giving a median event upper limit of 6.36 and
hence a sensitivity of 9.5 ? 10
? 8
GeV cm
? 2
s
? 1
sr
? 1
.
When the data with N
ch
? 100 was revealed, six data events were observed.
This was consistent with the average expected atmospheric neutrino background of
91
Event
1
2
3
4
5
6 required
value
Year 2001 2001 2001 2001 2002 2003
Day of Year
118
186
210
274
226
182
N
ch
102
106
157
116
100
111
?100
Track Length [m]
206.7 221.8 197.7 178.2 180.4 207.6
>170
Number of
Direct Hits
27
32
30
22
29
29
>13
Zenith Angle [
?
] 107.3 121.6 106.1 101.8 123.8
113.3
>100
Median Resolution [
?
]
2.4
1.4
1.8
3.0
1.6
2.8
<4.0
Table 11.1: Observable and reconstructed qualities are shown for the ?nal
six events. In addition, events ful?lled requirements based on the recon-
structed values of their smoothness (S) and their upgoing vs. downgoing
likelihood ratios.
7.0 events (after averaging all models that have been rescaled to the low energy data).
Information about the observable quantities for the ?nal six events can be seen in
Table 11.1. The ?nal N
ch
distribution is shown in Figure 8.1. The total number of
events predicted for the signal and background can be compared to the observed data in
Table 10.2 (50 N
ch
<100) and in Table 11.3 (N
ch
? 100). With uncertainties included,
the upper limit on a di?use ? / E
? 2
?ux of muon neutrinos at Earth (90% con?dence
level) with the AMANDA-II detector for 2000 { 2003 is 7.4 ? 10
? 8
GeV cm
? 2
s
? 1
sr
? 1
for 16 TeV to 2.5 PeV. The results are compared to other neutrino limits in Figure
11.1.
11.2 Results for Other Energy Spectra
Other signal models were also tested with this data set. Due to their di?erent
energy spectra, the N
ch
requirement was reoptimized by minimizing the MRF with
92
log
10
[E
n
(GeV)]
3456789
]
-1
sr
-1
s
-2
dN/dE [GeV cm
2
E
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
this analysis
AMANDA-II 2000 atms. n
m
data (prelim.)
Barr et al. atms. n + prompt atms. n
Honda et al. atms. n + prompt atms. n
Max uncertainty in atms. n
Frejus
MACRO
AMANDA B-10 1997 n
m
diffuse
AMANDA-II 2000 Cascades (all-flavor / 3)*
AMANDA B-10 1997 UHE (all-flavor / 3)*
Baikal 1998 - 2002 ( all-flavor / 3 )*
RICE 1999-2005 ( all-flavor / 3 )*
AMANDA-II 2000 unfolding (prelim.)
AMANDA-II 2000-3 n
m
limit
W&B limit/2 (transparent sources)
Full IceCube 1 yr
* assumes a 1:1:1 flavor ratio at Earth
Figure 11.1: The upper limits on the ?
?
?ux from sources with an E
? 2
en-
ergy spectrum are shown for single and all-?avor analyses. All-?avor upper
limits have been divided by three, assuming that the neutrino ?avor ratio
is 1:1:1 at Earth. The Fr?ejus [26], MACRO [27], and AMANDA-B10 [24]
upper limits on the ?
?
?ux are shown, as well as the unfolded atmospheric
spectrum from 2000 AMANDA-II data [69]. The AMANDA-II all-?avor
limit from 2000 [25], the AMANDA-B10 UHE limit [29], the Baikal ?ve
year limit [28] and the RICE six year limit [70] have all been adjusted for
the single ?avor plot. The ? / E
? 2
limit from this analysis is a factor of
four above the Waxman-Bahcall upper bound. Although not shown, this
analysis excludes the ? / E
? 2
predictions made by Nellen, Mannheim, and
Biermann [18] and Becker, Biermann, and Rhode [19] and constrains the
MPR upper bound for optically thick pion photoproduction sources [20].
The IceCube sensitivity for a full detector was estimated with AMANDA
software [71].
93
each signal model. For signal models with softer spectra than ? / E
? 2
, a lower N
ch
requirement was optimal. Five of these models were optimized with a cut on or near
71, so only one unblinding was used for all of these models, N
ch
? 71. Four prompt
neutrino models [33, 35, 36, 34] and one astrophysical neutrino model [23] were tested
under these conditions. One astrophysical model was optimized at N
ch
? 86 [20]. Two
astrophysical neutrino models with harder spectra than ? / E
? 2
were tested with a
higher energy requirement, N
ch
? 139 [20, 21, 22].
Results of these searches are summarized in Table 11.2. The normalization of
the overall number of low energy atmospheric neutrinos to data was performed over
the region 50 < N
ch
< 100 for the harder spectra (N
ch
? 139), and over 50 < N
ch
< 71
and 50 < N
ch
< 86 for the softer spectra.
When the data from the N
ch
? 139 region were examined, there was good agree-
ment with the expected atmospheric neutrino background (1 event observed on a
backround of 1.55). For N
ch
? 86, 14 events were observed while an average of 12.9
background events were predicted. However, 37 events were observed while only 27.4
events were expected for N
ch
? 71, leading to a two-sided con?dence interval. Since
the chance probability of observing 37 or more events on this background is 4%, we
do not exclude the background-only null hypothesis. The 90% con?dence interval for
? is shown for each model in Table 11.2 and upper limits are calculated based on
the upper bound of each con?dence interval. If the MRF is greater than 1, then the
model is not ruled out based on observations from this four-year data set. Since more
events were observed in the data than were predicted by the background simulation
for N
ch
? 71, the upper limit on those ?ve models is roughly a factor of three worse
94
than the sensitivity.
11.2.1 Astrophysical Neutrino Upper Limits
The ?rst astrophysical neutrino model tested with the N
ch
? 139 requirement
was initially proposed by Stecker, Done, Salamon and Sommers [21]. The ?ux tested
in this analysis includes the revisions by Stecker, Done, Salamon and Sommers in 1992
[21] and the factor of 20 reduction by Stecker in 2005 [22]. It predicts a ?ux (?
SDSS
)
of high energy neutrinos from the cores of AGNs, especially Seyfert galaxies. Based
on the present data, the upper limit on this ?ux is 1.6??
SDSS
. The best previous limit
on this model was established by the Baikal experiment, with an upper limit of 2.5
??
SDSS
[28].
Mannheim, Protheroe and Rachen (MPR) [20] computed an upper bound for
neutrinos from generic optically thin pion photoproduction sources (˝
n?
< 1), as well
as an upper bound for neutrinos from AGN jets. (In addition, they calculated an upper
bound for generic optically thick (˝
n?
˛ 1) pion photoproduction sources assuming
a ? / E
? 2
spectrum, but this is constrained by the results discussed in the previous
section.) The upper bounds do not necessarily represent physical neutrino energy
spectra, but were constructed by taking the envelope of the ensemble of predictions
for smaller energy ranges. Each ?ux prediction within the ensemble was normalized
to the observed cosmic ray proton spectrum.
Nonetheless, the shapes of these two upper bounds were tested as if they were
models. However, one should be careful not to misinterpret the results. A limit on a
model implies a change in the normalization of the entire model. A limit on an upper
bound only implies a change in normalization of the bound in the energy region where
95
the detector energy response to that spectral shape peaks.
The MPR AGN jet upper bound was tested with the N
ch
? 139 requirement.
The upper limit on this spectrum is 2.0??
MPRAGN
. In comparison, the Baikal upper
limit on this spectrum is 4.0??
MPRAGN
.
The MPR upper bound for optically thin sources was tested with a N
ch
? 86
requirement. The limit on this spectrum and normalization is 0.22??
MPR˝<1
.
The remaining neutrino searches were conducted with the lower N
ch
requirement,
N
ch
? 71. A signal hypothesis involving neutrinos from starburst galaxies [23] was
tested. Loeb and Waxman assumed that protons in starburst galaxies with energy
less than 3 PeV convert almost all of their energy into pions. Their work predicts a
range that should encompass the true neutrino spectrum, but the model tested here
uses the most probable spectrum from the paper, ? / E
? 2:15
. This analysis assumed
the ?ux was valid for energies ranging from 10
3
to 10
7
GeV. The upper limit on this
spectral shape and normalization is 21.1??
starburst
.
These astrophysical neutrino models and their observed upper limits based on
this data set are shown in Figure 11.2. Neutrino oscillations are taken into account
for all models where this factor was not already applied.
11.2.2 Prompt Atmospheric Neutrino Upper Limits
Since prompt neutrinos have a harder (less steep) spectrum than the conventional
atmospheric neutrinos, it is possible to search for a prompt neutrino ?ux by separating
the two event classes in energy. The ?nal N
ch
requirement was reoptimized and the
normalization factor was determined based on the interval (50 ? N
ch
< 71).
In the astrophysical neutrino searches described thus far, the range of atmo-
96
log
10
[E
n
(GeV)]
3456789
]
-1
sr
-1
s
-2
dN/dE [GeV cm
2
E
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
Barr et al. atms. n + prompt atms. n
Honda et al. atms. n + prompt atms. n
Max uncertainty in atms n
SDSS model
SDSS 2000-3 AMANDA-II limit
MPR AGN jets model
MPR AGN jets 2000-3 AMANDA-II limit
MPR t
ng
<1 model
MPR t
ng
<1 2000-3 AMANDA-II limit
Starburst model
Starburst 2000-3 AMANDA-II limit
E
-2
AMANDA-II 2000-3 n
m
limit
Figure 11.2: Astrophysical neutrino models and upper limits established
with this analysis. The Barr et al. and Honda et al. atmospheric neutrino
models are shown as thin lines with maximum uncertainties assumed by
this analysis represented by the band. Other models that were tested
included the SDSS AGN core model [21, 22], the MPR upper bounds for
AGN jets and optically thin sources [20], and a starburst galaxy model
[23].
97
spheric neutrinos predicted in the ?nal sample included an uncertainty due to the
unknown prompt neutrino ?ux. For the search for prompt neutrinos, this uncertainty
in the total atmospheric neutrino ?ux was changed so that only conventional atmo-
spheric neutrino uncertainties were included. Since the atmospheric neutrino simula-
tion was still normalized to the low energy data, the overall e?ect in the atmospheric
background prediction for the ?nal sample was small.
Martin et al. predict prompt lepton ?uxes based on the GBW model for deep
inelastic scattering. This model includes gluon saturation e?ects [34] which lower
the predicted charm production cross sections. The predicted ?ux is lower than the
sensitivity of this data set. The upper limit on this model is 60:3 ? ?
MartinGBW
.
The Naumov RQPM [35, 36] model of prompt atmospheric neutrinos incor-
porates data from primary cosmic ray and hadronic interaction experiments. This
non-perturbative model includes intrinsic charm [30]. The upper limit on this model
is 5.2??
NaumovRQPM
.
Prompt neutrinos based on the models of Zas, Halzen and Vazquez were also
simulated [33]. A parameterization was established to describe the energy dependence
of the charm cross section. For the Charm C model, the charm cross section was ?tted
to experimental data. In the Charm D model, the cross section was parameterized
by Volkova [39]. The upper limit for Charm C is 1.5??
CharmC
. However, due to the
upward ?uctuation in the number of events in the N
ch
> 71 region, the upper limit
on the Charm D model is 0.95??
CharmD
. Since the MRF is less than 1.0, the model
is disfavored at the 90% con?dence level. The prompt neutrino models are shown in
Figure 11.3, along with the upper limits based on these data.
98
log
10
[E
n
(GeV)]
3456789
]
-1
sr
-1
s
-2
dN/dE [GeV cm
2
E
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
Barr et al. atms. n
Honda et al. atms. n
Max uncertainty in atms. n
Martin GBW model
Martin GBW 2000-3 AMANDA-II limit
Naumov RQPM model
Naumov RQPM 2000-3 AMANDA-II limit
CharmC model
CharmC 2000-3 AMANDA-II limit
CharmD model
CharmD 2000-3 AMANDA-II limit
Figure 11.3: Prompt neutrino models and upper limits based on this anal-
ysis. The Barr et al. and Honda et al. atmospheric neutrino predictions
are shown for reference. Two charm models [33] were tested, along with
the Naumov RQPM [35, 36] and Martin GBW [34] models.
99
Astrophysical ?
? / E
? 2
SDSS [21, 22] MPR AGN jets [20]
N
ch
100
139
139
n
b
7.0
1.55
1.55
n
s
66.7
1.74
1.42
?
median
(n
b
)
6.36
2.86
2.86
sensitivity
?
median
(n
b
)=n
s
? ? 0.095 ??
E
? 2
1.6 ??
SDSS
2.0 ??
MPRAGN
n
obs
6
1
1
?
90%C:I:
(0,4.95)
(0,2.86)
(0,2.86)
upper limit
?=n
s
? ? 0.074 ??
E
? 2
1.6 ??
SDSS
2.0 ??
MPRAGN
(log
10
Emin; log
10
Emax)
(4.2,6.4)
(5.1,6.8)
(5.0,6.9)
MPR ˝n? < 1 [20]
Starburst [23]
N
ch
86
71
n
b
12.9
29.1
n
s
42.7
1.05
?
median
(n
b
)
8.48
8.24
sensitivity
?
median
(n
b
)=n
s
? ?
0.2 ??
MPR˝<1
7.8 ??
Starburst
n
obs
14
37
?
90%C:I:
(0,9.49)
(0,22.13)
upper limit
?=n
s
? ?
0.22 ??
MPR˝<1
21.1 ??
Starburst
(log
10
Emin; log
10
Emax)
(4.0,5.8)
(3.8,6.1)
Prompt ?
Martin
Naumov
GBW [34] RQPM [35, 36]
CharmC [33]
CharmD [33]
N
ch
71
71
71
71
n
b
27.4
27.4
27.4
27.4
n
s
0.41
4.74
16.05
26.15
?
median
(n
b
)
8.75
8.75
8.75
8.75
sensitivity
?
median
(n
b
)=n
s
? ? 21.3 ??
MGBW
1.8 ??
NRQPM
0.55 ??
CharmC
0.33 ??
CharmD
n
obs
37
37
37
37
?
90%C:I:
(1.29,24.72)
(1.29,24.72)
(1.29,24.72)
(1.29,24.72)
upper limit
?=n
s
? ? 60.3 ??
MGBW
5.2 ??
NRQPM
1.5 ??
CharmC
0.95 ??
CharmD
(log
10
Emin; log
10
Emax)
(3.5,5.5)
(3.6,5.6)
(3.8,5.7)
(3.6,5.6)
Table 11.2: Several ?ux shapes were tested with this data set. N
ch
is the
minimum number of OMs that had to be hit for an event to appear in
the ?nal data set. The predicted number of events for background, n
b
,
and signal, n
s
, were determined by the simulation. The median event
upper limit is ?
median
(n
b
). The sensitivity is the model ?ux multiplied
by the median event upper limit and divided by the number of signal
predicted. The number of events observed in the four year data sample is
n
obs
. The upper limit is calculated from the maximum value of the 90%
con?dence interval for the event upper limit, ?. The upper limit is the
test ?ux multiplied by ?=n
s
. All values quoted here incorporate systematic
uncertainties.
100
Upgoing N
ch
? 100
L0
L1
L2
L3
L4
L5
signal
160.4 123.7 103.7 103.2 68.4
coinc. ?
54.2
4.3
2.8
2.8
0
misreconstructed
CORSIKA
atms. ?
862.1
35.4
0
0
0
Barr et al.
atms. ?
36.0
27.6
19.3
18.9
9.1
Honda et al.
atms. ?
25.2
19.3
13.5
13.2
6.4
Martin GBW
0.42
0.42
0.36
0.36 0.19
prompt atms. ?
Naumov RQPM
4.8
4.8
4.2
4.2
2.2
prompt atms. ?
Data
11456 1347
96
45
6
Downgoing N
ch
? 100
CORSIKA
atms. ? (?10
7
)
7.31
6.53
6.05
6.01 5.09
data (?10
7
) 9.75
8.59
8.07
8.03 6.60
L5 = level of ?nal analysis
Table 11.3: The number of high energy events (N
ch
?100) at a given
quality level for the di?erent types of simulation and experimental data.
101
Chapter 12
Future Techniques for Di?use Analyses
This analysis relied on a cut and count method to set an upper limit on the di?use
?ux of astrophysical and prompt neutrinos. Seven background events were predicted
to exist in the high energy N
ch
? 100 sample. Six events were observed and the
astrophysical neutrino upper limit was established based on the observation of roughly
the same number of data events as were predicted by the background simulation. By
simply cutting and counting, any excess that might be observed could not be identi?ed
certainly as being astrophysical in origin. It could be possible that some or all of the
excess actually comes from prompt atmospheric neutrinos. In order to overcome this
problem in this analysis, assumptions were made about the prompt neutrino ?ux in
order to set an astrophysical neutrino limit. (And vice versa.)
For the astrophysical neutrino search, it was assumed that the prompt atmo-
spheric neutrino ?ux was equal to the average of the Naumov RQPM and Martin
GBW models. Currently, the range in which the prompt neutrino ?ux is expected
to lie is bounded at the high end by the Naumov RQPM model. The Martin GBW
model marks the lower boundary of the predicted ?ux region.
The uncertainty in the prompt neutrino ?ux spans orders of magnitude and
102
assuming a certain background ?ux due to these types of events could potentially be
misleading. Since no excess of events was observed in the astrophysical search, it was
then assumed that no astrophysical neutrino ?ux was present so that an upper limit
on the prompt atmospheric ?ux could be placed. However, it would make more sense
to place two-dimensional limits on the astrophysical and prompt atmospheric ?uxes
at the same time.
Two-dimensional limits show an allowed region in which the astrophysical and
prompt atmospheric neutrino ?uxes may lie and be consistent with the observed data.
By ?nding an allowed region in two-space, assumptions do not have to be made about
either type of ?ux.
Cut and count methods do not consider shape information about the predicted
or observed spectrum of events. The next di?use analysis will compare the shape and
normalization of the observed data to simulations based on di?erent background and
signal ?ux models. N
ch
and the reconstructed zenith angle of the track were power-
ful parameters used in this analysis and they will be used to calculate likelihoods of
the di?erent models based on the observed data. By using shape and normalization
information for N
ch
and the reconstructed zenith angle, it is hoped that the conven-
tional atmospheric neutrino background can be better understood (for instance, we
could determine which conventional atmospheric neutrino model is correct... or most
correct).
Multiple two-dimensional limits will need to be published when this method is
used (for instance, a two-dimensional limit on the di?use ?ux of ? / E
? 2
neutrinos
versus prompt atmospheric neutrinos as predicted by Martin GBW). Another two-
103
Figure 12.1: Two-dimensional upper limits can be established without
making a priori assumptions about the amount of astrophysical or prompt
atmospheric neutrino ?ux in the data. The x-axis represents the upper
limit on the prompt atmospheric neutrino ?ux while the y-axis represents
the upper limit on astrophysical neutrinos.
dimensional region would have to be published in order to constrain the Naumov
RQPM model, for instance.
104
Chapter 13
Conclusions
The experimental data were consistent with the predicted range of atmospheric neu-
trino background. Six high energy events were observed in the ?nal data set, while
the average predicted background was 7.0 events. There is no indication of an astro-
physical signal. At a 90% con?dence level, the di?use ?ux of extraterrestrial muon
neutrinos with an E
? 2
spectrum is not larger than 7.4 ? 10
? 8
GeV cm
? 2
s
? 1
sr
? 1
for
16 TeV { 2.5 PeV.
This analysis also provides upper limits on four astrophysical neutrino models
and four prompt neutrino models. For the hardest signal spectra, the results are con-
sistent with background. The softer spectra were tested with lower N
ch
requirements
and despite the observation leading to a two-sided 90% con?dence interval, the level
of excess is not signi?cant enough to claim a detection.
Before requiring events to ful?ll N
ch
? 100, the observed events were compared
to the atmospheric neutrino simulation with systematic uncertainties included. The
observed low energy data were used to normalize the atmospheric neutrino simulation,
hence narrowing the range of atmospheric neutrinos predicted by the di?erent models
for the ?nal high energy sample. Systematic e?ects of the event selection procedure
105
were studied in the inverted analysis using atmospheric muons. A consistency was
established between the observed downgoing atmospheric muon ?ux and the upgoing
atmospheric neutrino ?ux using the inverted analysis.
This result is the best upper limit on the di?use ?ux of TeV { PeV muon neu-
trinos to date. The upper limit is an order of magnitude lower than the previous
AMANDA result by performing a multi-year analysis [24] and by using a larger de-
tector, AMANDA-II instead of AMANDA-B10. For a ? / E
? 2
spectral shape, this
analysis provides an upper limit that is a factor of three better than the Baikal muon
neutrino upper limit (muon neutrino upper limit = all-?avor limit/3 assuming a 1:1:1
?avor ratio).
This analysis set upper limits on four prompt atmospheric neutrino predictions,
while one of these models is disfavored at a 90% con?dence level. Other spectral shapes
were tested for astrophysical neutrinos. No models were excluded, however constraints
were placed on the existing predictions. The shapes of the MPR upper bounds were
tested in the energy region where the detector response peaks. For the benchmark
? / E
? 2
spectral shape, the current limit is a factor of 4 above the Waxman-Bahcall
upper bound.
AMANDA-II has now been integrated into IceCube. The sensitivity of the Ice-
Cube detector will continue to improve as the detector grows to its ?nal volume, 1
km
3
. Based on estimations with AMANDA software, the full IceCube detector will
have a sensitivity that is a factor of 10 better than this analysis after one year of
operation [71].
106
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112
Appendix A
Q&A for the Non-Physicist
What are neutrinos? How do we detect them? Why is the AMANDA detector located
at the South Pole? What can we learn by studying neutrinos? These are among the
?rst questions asked by people who want to learn about IceCube and neutrino science.
This chapter is intended to explain basic neutrino astrophysics to the non-physicist.
Each section title is a question I've been asked during my time on this project.
A.1 What are neutrinos?
The basic particles that make up atoms are protons (positive charge), neutrons
(no charge) and electrons (negative charge). Protons and neutrons (not neutrinos!)
belong to a family called hadrons. Hadrons are formed from even smaller constituents
known as quarks and gluons. On the other hand, electrons are members of the lepton
family. Leptons cannot be broken down into smaller constituents. Neutrinos are also
in the lepton family and are very tiny, chargeless particles.
There are three types of charged leptons and each one has a neutrino partner.
Electrons (e), muons (?) and taus (˝) are all cousins. Taus are the heaviest, then
muons, then electrons. There is a corresponding neutrino ?avor for each of the charged
113
e
+
, e
?
electron,positron ?
e
, ??
e
electron neutrino, electron anti-neutrino
?
+
, ?
?
muon
?
?
, ??
?
muon neutrino, muon anti-neutrino
˝
+
, ˝
?
tau
?
˝
, ˝?
˝
tau neutrino, tau anti-neutrino
Table A.1: The lepton family.
particles. We call them ?
e
, ?
?
and ?
˝
. Each neutrino also has an anti-neutrino partner
which is denoted by the same symbol with a line or bar placed above it. Neutrinos
were originally thought not to have any mass, although it has since been proven that
they do. The lepton family is summarized in Table A.1.
Every second about one trillion (1,000,000,000,000) neutrinos travel harmlessly
through your body. Because neutrinos are so incredibly small, most of the time they
pass right through solid objects without interacting at all. Neutrinos very rarely
interact with other atoms and they are constantly bombarding everything around us.
They travel at almost the speed of light - that is 300,000,000 meters every second - or
670 million miles per hour.
A.2 Why do we care about neutrinos?
We have much to learn about neutrinos! The neutrino wasn't postulated un-
til 1930 and wasn't discovered until the 1950s. With trillions of neutrinos traveling
through you all the time from all directions, the neutrinos must come from many
sources. For instance, neutrinos are created during radioactive decays, such as the
breakdown of Potassium in the human body. Experiments have detected neutrinos
produced by nuclear reactors, as well as those created by particle interactions in the
upper atmosphere. Further out in space, neutrinos are formed inside stars, including
the Sun.
114
The Sun has been studied for centuries and we now have a good understanding
of the mechanisms that make the Sun burn. As a result, we know how old it is and
how long we expect it to burn. Even though we can never travel there, we have found
ways to explain the particle interactions occuring there. We now understand, for
instance, that the Sun burns because energy is being released every time two protons
fuse together. Neutrinos are created in the Sun and other neutrino experiments have
detected these neutrinos.
On the other hand, the Earth and Sun are only tiny blips on the cosmic radar.
Stars and galaxies abound. They are so far away from us that we will never be able
to send a spaceship or probe to study them. However, just as we want to know how
the particles in the Sun interact, we would like to know how the particles in other
stars and galaxies are interacting. We think that neutrinos are playing active roles in
these distant objects, but we won't know for sure until we detect them. Since stars
and galaxies are so far away from us, we must study them based on the clues that
they send to us.
There are three main types of telescopes or detectors studying distant space.
They each detect a di?erent type of particle.
1. Optical telescopes. These detectors study light from space. The light can ar-
rive anywhere along the electromagnetic spectrum. That means that the light
output could be optical (light that we see with our eyes) or it could be in radio
waves (long) or ultraviolet or x-rays (short), for instance. Gamma-rays have the
shortest wavelengths and carry the most energy.
2. Cosmic ray detectors study charged particles traveling in space. Cosmic rays are
115
Figure A.1: Electromagnetic spectrum. Image credit: Laboratory for
Computational Science and Engineering, University of Minnesota.
mainly protons.
3. Neutrinos telescopes search for neutrinos.
There are advantages and disadvantages to each of these three ways of studying
the Universe. The important point, though, is that all of these methods complement
each other. We should pursue all of the detection methods in order to gain the best
understanding of the Universe. Light is the traditional way in which we have studied
the skies. However, sometimes the light is absorbed by other objects in space and does
not make it all the way to the Earth. Cosmic rays, on the other hand, are also fairly
easy to detect. However, because they have positive or negative charge, the direction
that they are traveling can be changed as they travel through magnetic ?elds in space.
116
That means that we will never know their direction of origin. Neutrinos, on the other
hand, are important messengers because they always travel in straight lines - directly
from the source to the point where they are detected. Because neutrinos do not have
any charge, magnetic ?elds in space cannot change their direction. Neutrinos do not
interact with matter very often. In fact, they often travel through the Earth (and your
body!) without even noticing. Since they rarely interact, large detectors are needed
to observe an interaction.
A.3 What objects in space are we studying?
The explosion of a star is called a supernova and we know that these events pro-
duce neutrinos. They were documented in coincidence with a supernova that occurred
in 1987.
Gamma-ray bursts (GRBs) are yet another type of astronomical object that we
are studying. Gamma rays are at the highest end of the electromagnetic spectrum,
meaning that these photons of light have the highest energy possible. When there is a
quick burst of gamma-rays (light) in the sky (anywhere from about 1 to 100 seconds),
it is called a GRB. Hundreds of GRBs have been identi?ed and clues are indicating
that they may be closely related to to supernovae. We also believe that neutrinos are
involved in these events and we would like to understand better how they happen.
An AGN, or active galactic nuclei, is a galaxy that most likely has a black hole
at its center. These galaxies have very strong light emission. The light can occur
anywhere along the electromagnetic spectrum - in optical, radio waves, ultraviolet
(UV) or x-rays for example. The light output from an AGN can occur in streams
or jets, where most of the particles travel out of the AGN in one of two directions
117
(pointing exactly opposite). We do not yet fully understand what particle interactions
are fueling this behavior, but it seems likely that neutrinos are involved. If we ?nd
neutrinos are originating in AGN, it will be a major step toward understand how AGN
and their associated black holes work.
A.4 How we detect neutrinos?
Because neutrinos rarely interact, we need a very large detector to increase our
chance of an interaction occurring. Here, I will focus on detecting the muon neutrino,
?
?
, since that ?avor was the focus of this analysis. When a muon neutrino hits an
atom (remember that they usually pass right between them), an interaction occurs.
The neutrino ceases to exist and a muon is formed. The muon then gives o? light
as it travels in the same direction as the neutrino was going. The AMANDA and
IceCube detectors are built to detect the light that is given o? as the muon travels.
The reason the detectors are located deep in the ice at the South Pole is that a large,
clear medium is needed. Otherwise, the light couldn't be detected! Water and ice are
ideal for these detectors because they are cheap. Where else would you ?nd a cubic
kilometer of a clear medium? Similar detectors are being built or are operating in
lakes and the Mediterranean Sea.
A.5 How does the AMANDA detector work?
The AMANDA detector contains 677 light sensors. These light sensors are buried
in the ice so that they can detect the light from the muons. The sensors are connected
along 19 cables and form a three dimensional array. To install each string in the ice,
a hot water drill made a very deep, narrow hole. Once the ice was melted and the
118
hole contained water, the cable was lowered into the hole. Every 15 meters, a light
sensor was attached. The 19 AMANDA strings are located such that each string is at
least 25 ? 50 meters from any other. When the muon emits light, the light sensors
closest to the muon record a signal. This continues as the particle travels through the
ice emitting light. The path of the muon is then identi?ed by the sequence in which
the light sensors recorded seeing something. Since the neutrino and muon it produces
always travel in the same direction, we can reconstruct the direction of the neutrino
and point back to the exact direction in space where it originated.
A.6 How does my analysis work?
Some AMANDA analyses perform point source searches of the sky. This means
that we look for neutrinos coming from a particular direction in the sky because we
know that there is an active galaxy or other object there. However, neutrinos have
not yet been found by looking at speci?c objects.
Consider this scenerio (in which none of the numbers are real). Let's say that
if I look in the direction of a speci?c AGN with the AMANDA detector, I expect 50
events in my background. The background events are always there, although maybe
at times there are more or less than 50 (say 46 or 57 background events, for example).
In order to say that the AGN is a neutrino source, I must be able to say that there
are many more neutrinos than I would expect in my background. If I measured 60
neutrinos, I probably would not be able to say with certainty that the neutrinos came
from the AGN. It could have happened that there were 50 background events and
10 signal events. However, it would also be quite possible that the background was
just a little bit higher than normal and all 60 events were background. However, if I
119
measure 100 neutrinos and only expected 50, then it is likely that the AGN is creating
the neutrinos that I detected.
Unfortunately, no point sources of neutrinos have been identi?ed by searching for
neutrinos from a speci?c direction. Another method can be used to make it easier to
detect a neutrino source and the following analysis technique was used in the analysis
described in this thesis. Let's say that we search for a signal from all of the AGN across
the entire sky. For simplicity, let's assume that there are 20 AGN, although this is
obviously a gross underestimate! Across the entire sky, I expect 500 background events.
If I observe 700 events, that is 200 more events than I expected in my background. It
would then seem very likely that a neutrino source is creating these extra events and
they are not background! If there are 20 total AGN and 200 extra events over the
expected background, that means that 10 events had to come from each AGN (that
is 200/20 = 10). Note that 10 neutrinos was not enough to claim an AGN neutrino
detection in the point source search I described earlier. However, if many sources each
contribute 10 events, then together all of the distant AGN are identi?ed as neutrino
sources. This method is called a di?use search for neutrinos.
For this analysis, I began with 5,200,000,000 events that triggered the detector
between February 2000 and November 2003. I then had to throw away all of the events
that were obviously background. In the end, I predicted a background of 7 events.
When I looked at the data, I observed 6 events, a slight ?uctuation from the predicted
value. Since I did not see a large excess of events over the predicted background, I did
not claim to observe any neutrino signal from astronomical objects like AGN.
However, we can still learn from the analysis even though we did not observe
120
neutrinos from other galaxies or objects. We can set an upper limit on how strong
those sources of neutrinos could actually be. Let's return to the scenario from above.
If 500 events were predicted and 520 events were observed, we would not claim to
see any astrophysical neutrino signal. It could have been that each of the 20 AGN
contributed 1 event in addition to the background of 500, however 520 could just be
a background ?uctuation. However, we can say that if the AGN were putting out
10 neutrinos each, then we would have observed 700 events and would have claimed
detection of a neutrino signal. Hence, since that did not happen, we know that each
AGN must emit less than 10 neutrinos. That is the maximum amount of neutrinos
that the source could emit and still be consistent with what we observed. We call this
an upper limit. We have learned something - we have learned that AGN put out less
than 10 neutrinos each. (Don't forget... all of these numbers are fake!)
The longer the detectors are turned on, the more likely it is we will detect a
signal. The IceCube detector will be much larger than the AMANDA detector and
this will increase the probability of identi?ying distant neutrino sources.
A.7 Summing it up, plain and simple
Neutrinos are tiny, chargeless particles. Most of the time, they pass harmlessly
through matter because they do not like to interact.
Neutrinos provide valuable information to physicists and astronomers because
they travel in straight lines. When a neutrino is detected, we can trace backward to
?nd the direction of origin and possibly the source. We hope to identify objects in
space that are producing neutrinos that we can detect here on Earth. These objects
might be supernovae (exploding stars), gamma-ray burts (objects that cause very
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quick, bright ?ashes of light) or active galactic nuclei (galaxies with black holes at the
center).
The AMANDA detector uses a 3-dimensional array of light sensors to search for
the light emitted by particles after a neutrino hits an ice molecule and interacts. Since
we are detecting light, we need a large, clear medium. The detector is buried at the
South Pole because the ice is very pure and the light can travel long distances in the
ice.
