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Effects of buckyballs and cosmic strings on the cosmic microwave background

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U n iv e r s it y
of
C a l if o r n ia
Los Angeles
E ffects o f B uckyballs and C osm ic S trin gs on th e
C osm ic M icrow ave B ackground
A dissertation submitted in partial satisfaction
of the requirements for the degree
Doctor of Philosophy in Astronomy
by
A m y S h iu -M ei Lo
2005
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© Copyright by
Amy Shiu-Mei Lo
2005
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The dissertation of Amy Shiu-Mei Lo is approved.
/? w ' O
Rene Ong
Veerayalli Varadarajan
Kastushi Arisaka
Edward L. Wright, Committee Chair
University of California, Los Angeles
2005
ii
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T able
of
C ontents
1 In tr o d u c tio n ....................................................................................................
1
2
Buckyball R a d ia tio n ....................................................................................
6
2.1
Anomalous Free-Free E m is sio n s ..........................................................
6
2.2
Small dust g r a i n s ....................................................................................
11
2.3
The Monte Carlo S im ulation............................................................... . 14
2.4
3
2.3.1
Buckyball Initial Conditions . . ; ..........................................
15
2.3.2
Hydrogen Collision Process
....................................................
18
2.3.3
Photon Collision P ro c e s s ..........................................................
21
2.3.4
A Short Justification for Classical M echanics......................
24
Simulation A nalysis.................................................................................
27
2.4.1
Typical Simulation N u m b ers....................................................
29
2.4.2
Emission C h a racteristics..........................................................
32
2.4.3
Anomalous Emission?
.............................................................
39
2.4.4
Possible O bservations.................................................................
47
Signatures o f C osm ic Strings in th e C osm ic M icrowave Back­
ground ....................................................................................................................
3.1
Topological Defects
50
.............................................................................
50
3.1.1
Symmetries and G r o u p s ..........................................................
55
3.1.2
Fundamental Homotopy G r o u p .............................................
59
3.1.3
Cosmic Strings and G U T s .......................................................
64
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3.2 WMAP
.....................................................................................................
69
3.3 String Search: The E dgefinder..............................................................
72
3.3.1
Simulated M a p s ...........................................................................
81
3.3.2
Edgefinder L im its ........................................................................
83
3.3.3
Edgefinder Limits fromEdgeworth Coefficients......................
86
3.3.4
Multiple String M a p s .....................................................................100
3.4 WMAP R e s u lts ........................................................................................... 102
3.4.1
WMAP 2nd Year Sim ulation........................................................103
3.5 A Cosmic String C an d id ate?.....................................................................104
3.5.1
String S e a r c h ..................................................................................105
3.6 Simulated PLANCK R e s u lts .....................................................................108
4
C onclusions and Future W o r k ................................................................... 113
5 B ibliography
................................................................................................... 117
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L is t
2.1
of
F
ig u r e s
The correlated intensity fluctuations of the microwave and farIR emission of various experiments, The y-axis is the emissivity measured in units of optical depth at 100 microns.
From
Finkbeiner, 2003. The data points are from: Wilkinson Microwave
Anisotropy Probe (WMAP) (Bennett et al, 2003), Green Bank
(GB 140) (Finkbeiner et ah, 2002), Tenerife (de Oliveira-Costa
et ah, 1999), Cottingham 19.2 GHz (de Oliveira-Costa, Tegmark,
Page, & Boughn, 1998), OVRO (Leitch et ah, 1997) Saskatoon
(de Oliveira-Costa, Tegmark, Page, & Boughn, 1998), and, Dif­
ferential Microwave Radiometer (DMR) (Kogut et ah, 1996). An
explanation of all the lines are in the te x t.........................
2.2
8
A graph of the emissivity (ju) per Hydrogen atom (n# is hydro­
gen density). Graph taken from Draine & Lazarian (1999a). The
data points with error bars are data from five experiments listed
in the legend. Also plotted are dashed, dotted, and solid lines
representing vibrational dust emissivity of four different dust tem­
perature and spectral index, /3T . The grey band is the dust corre­
lated free-free emission. A schematic of our buckyball radiation in
the anomalous free-free emission frequencies plotted in the thick
dotted line labeled Buckyball....................................................................
2.3
A picture of a Buckminsterfullerene molecule, C 2 4 #
22-
10
The light
grey spheres are the carbon orbitals, and dark grey spheres are the
hydrogen orbitals. The two dehydrogenation sites are indicated by
arrows.............................................................................................................
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16
2.4
A plot of the photon probability distribution used to generate the
photon density in each simulation.........................................................
2.5
Plot of the emission frequency vs.
22
simulation time of the C2o
buckyball in the CNM region. Solid lines indicate times during
which the buckyball is radiating. The dashed lines correspond to
a collision event..........................................................................................
2.6
25
The Energy-frequency plot for C2o buckyball in the WNM region.
The peak is taken as the peak emission frequency, and most graphs
show a clear drop off in the energy emitted vs. frequency after the
peak..............................................................................................................
2.7
28
Spectral index vs. simulation Hydrogen density for each of the 4
tem peratures simulated. Crosses indicate hydrogen tem perature
T = 25 K, Squares are T = 6000 K, Diamonds are T = 8000 K,
Triangle are T = 100 K ............................................................................
2.8
35
Spectral index vs. simulation tem perature for each of the 4 densi­
ties simulated. Crosses indicate hydrogen number density, n H, =
100000 cm-3 , Squares are nn = 0.4 cm-3, Diamonds are n # = 0.1
cm-3 , and triangle are nn = 30 cm ” 3 ...................................................
2.9
36
Peak emission frequency vs. simulation tem perature for each of the
4 densities simulated. Crosses indicate hydrogen number density,
n Hl = 100000 cm” 1, Squares are n H = 0.4 cm” 1, Diamonds are
Tin = 0.1 cm” 1, and triangle are n H = 30 cm ” 1 .................................
37
2.10 Peak emission frequency vs. simulation Hydrogen density for each
of the 4 temperatures simulated. Crosses indicate simulation tem­
perature, T = 25 K, Squares are T = 6000 K, Diamonds are T =
8000 K, Triangles are T = 100 K ...........................................................
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38
2.11 Power vs.
Frequency of the composite emission from all three
buckyballs in equal proportions for the WIM simulation. The thin
line is a nearest
8
neighbor sum to show a smooth spectrum. The
thick dot dashed line is a flat (P oc zT) spectrum for comparison. .
42
2.12 Power vs. Frequency and a smoothed version of the composite
emission from all three buckyballs in equal proportions for the
CNM simulation.........................................................................................
43
2.13 Power vs. Frequency and a smoothed version of the composite
emission from all three buckyballs in equal proportions for the
CNM simulation........................................................................................
44
2.14 Power vs. Frequency and a smoothed version of the composite
emission from all three buckyballs in equal proportions for the
CNM simulation....................................................
3.1
45
Graph a): the double welled potential which leads to a p 4 -kink,
with A = 1 and
77
= 1.7. Graph b): the solution to the potential
depicted in a). The kink through the origin which interpolates
between the two vacuum solutions is the defect called the </>4 -kink.
3.2
54
Figure A: / (the solid lines) and g (the dashed lines) are homotopic
to each other, but not to c (the dotted lines). The grey filled circle
is a hole in the manifold. Figure B: / and g are freely homotopic,
because a path c can be constructed to wind around the defect
separating them .........................................................................................
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60
3.3
A 3-D representation of the Edgefinder. The z-axis represent filter
values, and the x and y axis are the pixel numbers. The right
bottom inset is a plot of a slice along the y = 0 plane of the filter,
with the axis in units of pixels................................................................
3.4
Edgefinder values around a string horizon. The Edgefinder is ori­
ented North (up) to South (down).
The light blue circles are
masked pixels in the WMAP data due to foreground sources.
3.5
75
. .
77
The response function of the Edgefinder Filter. Note the peak
at I ~ 200, around where the first Doppler peak of the CMB
anisotropy angular power spectrum occurs.
The different lines
correspond to different a angles of the orientation of the filter
with respect to the north-south Galactic axis.....................................
3.6
Plot of the input string tem perature Tf vs. the maximum EV of
the set..........................................................................................................
3.7
80
84
Detail of the plot of the input string temperature T f vs. the max­
imum EV. Also plotted are the multi-string set data as well as
limits from the No-String sets. The stars are the Max EV for each
simulated map containing a string of strength shown on the x-axis;
the square indicate simulated maps containing multiple strings; the
triangles indicate maps containing no inserted strings......................
3.8
Plot of the input string temperature Tf vs. the 3rd Edgeworth
coefficient....................................................................................................
3.9
86
95
Plot of the input string temperature Tf vs. the 5th Edgeworth
coefficient................................................................................................. .
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96
3.10 Plot of the input string tem perature T f vs. the 7th Edgeworth
coefficient.....................................................................................................
98
3.11 Detail of the plot of the input string tem perature Tf vs. the 7th
Edgeworth coefficient. Also plotted are the multi-string set data
as well as limits from the No-String sets..............................................
3.12 Transfer function for the PLANCK simulation; the filter had R A D
99
=
0.5.....................................................................................................................109
3.13 Input string tem perature vs. the maximum EV of the EV set for
simulated PLANCK d ata............................................................................ I l l
3.14 Blow up of input string tem perature vs.
the 7th Edgeworth coef­
ficient of the EV set for simulated PLANCK d ata.................................112
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L is t
of
T ables
2.1
Simulation Parameters of the 3 Buckyballs
...................................
2 .2
Environmental Properties of Simulation E n v iro n m e n t...........
2.3
Buckyball Power Law Spectral Indices, /?, and their oy fitting co­
efficients
2.4
13
19
30
Peak Frequency, u p of buckyball emission and power at peak fre­
quency, P ( u p)
31
2.5
Photon vs. Hydrogen Impact
2.6
Simulation Rotational Energy of B u c k y b a lls ...........................
33
2.7
Thermal Rotational Frequency of Buckyballsa ........................
40
3.1
Relevant WMAP Characteristics
3.2
Partially Exposed Strings
3.3
Edgefinder Gain C a lib ra tio n ........................................................
3.4
Number and Strength of Input Strings for Multi-String Simulated
.............................................................
32
.......................................................
71
....................................................................
78
79
Maps......................................................................................................101
3.5
Select Properties of the WMAP composite QVW D ata.............103
3.6
Edgefinder Values for CSL-1
................................................................. 106
x
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A cknow ledgm ents
Getting a PhD. is hard work, and the following people deserve mention: Ned
Wright, for content, reading N drafts (N —>■oo as t —>■graduation), and always
having funding.
Steve Warwick, for support, a proposal, and the occasional
smack.
XI
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V
1975
Born, Taipei, Taiwan
1997
B.S.Honors (physics),
it a
Brown University, RI
1999
M.S. (Astronomy and Astrophysics),
UCLA, Los Angeles, California.
1998--2000
Cota Robles Fellowship
1999--2001
Teaching Assistant
Physics and Astronomy Department, UCLA
1999--present
Research Assistant
Physics and Astronomy Dept, UCLA
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A bstra ct
of the
D i s s e r t a t io n
E ffects o f B u ck yb alls and C osm ic S trin gs on th e
C osm ic M icrow ave B ackground
by
A m y S h iu -M ei Lo
Doctor of Philosophy in Astronomy
University of California, Los Angeles, 2005
Professor Edward L. Wright, Chair
This thesis consists of two Cosmic Microwave Background related projects: a
simulation of an anomalous foreground component, and a search for a distinct
background signature. The Cosmic Microwave Background forms one of the three
major pillars of support for the Big Bang theory of the origin of the universe,
and is an important source of information about the early universe. The first of
two components of this thesis proposes a possible explanation for an anomalous
component of our Galaxy’s foreground contribution to the Cosmic Microwave
Background. The second component of this thesis searches for signatures of early
universe phase transition products called Cosmic strings.
We propose that Fullerene molecules, or, buckyballs, may compose part of the
interstellar medium. Their thermal rotational angular velocity is on the order of
GHz, which lies within the range of the “anomalous free-free emission” that is
correlated with interstellar dust found in Galactic foreground maps of the Cos­
mic Microwave Background. We have written a Monte Carlo code to simulate the
radiation from spinning partially hydrogenated fullerene molecules. We quantify
the emission, compare it to the Galactic foreground, and find that if C2o com-
xiii
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prises 0.5% of the Galactic carbon budget, then there are sufficient buckyballs to
make it possible for fullerene molecules be responsible for the anomalous free-free
emission.
We also performed a search for signatures of cosmic strings in the the Cosmic
Microwave Background data from the Wilkinson Microwave Anisotropy Probe.
We used a digital filter designed to search for individual cosmic strings and found
no evidence for them in the WMAP CMB anisotropies to a level of A T / T ^ 0.29
mK. This corresponds to an absence of cosmic strings with Gfi ^ 1.07 x 10" 5 for
strings moving with velocity v = c / \ / 2. We have searched the WMAP data for
evidence of a cosmic string recently reported as the CSL-1 object. We found that
if the signatures at CSL- 1 were produced by cosmic strings, these strings would
have to move with a velocity ^ 0.94c. We also present preliminary limits on the
CMB data that will be returned by the PLANCK satellite for comparison. With
the available information on the PLANCK satellite, we calculated that it would
be twice as sensitive to cosmic strings as WMAP.
xiv
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CHAPTER 1
In tro d u ctio n
The Cosmic Microwave Background (CMB) is a rich source of information about
the early universe. Studies of the CMB carried out by satellite surveys can also
give us information about the contents of cur own Galaxy. This thesis deals with
both aspects of the CMB: it contains an investigation into a possible source for an
anomalous component of the CMB foreground, as well as a search for signatures
of topological defects in the CMB signal itself.
The very early universe was an extremely hot place; electron and positron
pairs coexisted in abundance. As the universe expanded, it cooled, and these
“
pairs annihilated to create many photons. The photons interacted with charged
particles in the early universe and remained in equilibrium, so th at the photons
we see from the early universe traces the distribution of m atter at that epoch. As
the universe cooled, the energy dropped below the electron-proton binding energy,
and the universe became neutral when hydrogen atoms were formed. Photons
could no longer interact in a significant way with the surrounding m atter, and
they were able to free-stream with the expansion of the universe. This epoch
is known as Last Scattering (LS), and it occurred approximately 300,000 years
after the Big Bang.
Structures of the universe were frozen into the CMB at LS. These photons
have been redshifted since they decoupled from m atter, and they now fall into
the microwave region of the electromagnetic spectrum. For this reason, these
1
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photons are called the Cosmic Microwave Background.
Due to the fact th at the CMB traces m atter distribution at LS, it has been the
focus of numerous studies ranging from the extraction of the primordial power
spectrum (which gives us information on the seeds of the current distribution of
clusters and galaxies), to searches for the formation of the first stars (whose UV
radiation would trigger an episode of reionization of the hydrogen in the universe).
The second portion of this thesis contributes to our knowledge of particle physics
by searching in the CMB for signatures of by-products of certain particle physics
theories known as topological defects. In particular, we search the CMB signals
returned by NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) for a form
of topological defect known as cosmic strings.
Launched in 2001, the satellite WMAP continues to conduct an all-sky survey
of the CMB by observing the microwave universe. Any measurement it makes of
the CMB is actually the cosmic CMB signal combined with non-cosmological
noise.
The strongest source of noise is microwave processes within our own
Galaxy. An accurate measurements of the CMB must carefully remove the Galac­
tic foreground. Many maps of the Galactic foreground exist to serve this purpose.
However, because the processes within our Galaxy are very complex, not every
identified component can be adequately explained.
It is, in fact, well known th at there is an anomalous component in the Galactic
foreground in the frequencies observed by WMAP. The first portion of this thesis
is a simulation of radiation from Buckminsterfullerene molecules motivated by the
presence of this unidentified component of Galactic emission, frequently referred
to as the anomalous free-free emission (Leitch et ah, 1997). This emission was
first discovered in studies of the Galactic foreground (e.g. Leitch et al. 1997 and
de Oliveira-Costa et al. 1999) as a component with a spectral index resembling
2
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free free emission which could not be satisfactorily fitted by thermal dust emission
or dust-correlated gas emission. The emission starts to deviate from fits to known
sources below 60 GHz, and the mechanism responsible for this emission has been
difficult to identify. One of the most popular explanations for this emission came
from Draine & Lazarian (1998), who argued that its source is ultra small spinning
dust grains.
Recent debate over the origin of this emission was kindled by the WMAP
mission itself. Bennett et al (2003) used a Maximum Entropy Method (MEM)
analysis to fit known components of the Galactic foreground to the WMAP data,
and they found th at their model showed that less than 5% of the anomalous
emission came from spinning dust grains.
Finkbeiner (2003), however, performed an alternate analysis on the same
WMAP foreground data and concluded that spinning dust grains can still con­
stitute a major portion of the anomalous free-free emission. This claim does not
challenge the authenticity of the CMB data produced by WMAP, as the chief
concern of Bennett etal. was an accurate representation of the flux from the
Galactic foreground, regardless of the underlying mechanism. W ith renewed de­
bate about the source of the anomalous emission, I followed Draine & Lazarian
(1998) and put forward a specific candidate for this emission in the form of par­
tially hydrogenated Buckminsterfullerene molecules, which are called buckyballs
for short. This portion of the thesis is detailed in Chapter 2.
With the Galactic foreground removed, the WMAP science team has produced
first year maps of the CMB. Chapter 3 of this thesis contains a search for cosmic
strings in the CMB. Topological defects (TD) such as cosmic strings have been
proposed as large scale structure (LSS) candidates but have fallen out of favor due
to the lack of evidence for their existence. TDs form as results of phase transitions,
3
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many of which occurred in the first year after the Big Bang. They are ideal LSS
candidates for three reasons. First, defects reflect the energy scale of the phase
transition, and, therefore, the earlier the phase transition, the higher the energy
density of the topological defect, and the larger the gravitational potential well
for structure formation. Second, these high energy phase transitions occurred a
long time before observational evidence of the first stars and galaxies. This would
give gravity enough time to coalesce m aterial to form structures. Last, cosmic
strings, the most popular class of topological defects, are geometrically similar to
the filamentous LSS first observed in deep redshift surveys (see early papers on
LSS especially Vachaspati 1986).
However, despite their convenience, results of the COsmic Background Ex­
plorer (COBE), an all-sky CMB satellite, showed th at cosmic strings were not
responsible for large scale structure formation, because the CMB anisotropies
were consistent with a Gaussian signature, while numerical simulations of all
topological defects show th at they would leave distinct non-Gaussian signatures
in the CMB (see, e.g., a review by Allen et ah 1997). The COBE results were
interpreted to mean th at the observed LSS’s are products of perturbations with
Gaussian seeds, and hence cosmic strings fell out of favor as a possible mechanism
for structure formation.
To date, there does not exist any conclusive observational evidence of the
existence of cosmic strings or any other TDs such as magnetic monopoles. On the
other hand, COBE data do not preclude their existence in small enough numbers
such th at they do not appreciably affect the CMB angular power spectrum; there
could be on the order of a few cosmic strings in the visible universe. Furthermore,
TDs are necessary products of certain types of phase transitions in the early
universe, and they are useful as possible explanations for variety of phenomenon
4
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such as Gamma Ray Bursts (e.g. Berezinsky et al. 2001), and ultra high energy
cosmic rays (see e.g. review article Torres & Anchordoqui 2004 and references
therein). This persistent interest in topological defects motivated our production
of their observational limits.
The cosmic strings th at do exist need to be few in number. W ith th at in
mind, I performed a search for individual cosmic strings based on all-sky CMB
survey data. The data returned from WMAP have an angular resolution of 13
arc-minutes which enabled searches for individual cosmic strings, because I expect
to start to be able to detect cosmic strings on angular scales of approximately
2
degree scale. While maps with greater resolution have been produced by balioonhorne experiments (e.g. Bouchet, Peter, Riazuelo, & Sakellariadou 2002), these
experiments only observe a very small area of the sky. WMAP covers the entire
4n steradian of sky and is therefore uniquely suited for searches of individual
cosmic strings. Our search involves a pixel by pixel filtering method, as opposed
to the wavelet analysis th at some others have performed on the small scale CMB
experiments (e.g. Barreiro & Hobson 2001), although the underlying principles
of searching for non-Gaussian signatures are similar.
This thesis is divided into two sections.
Chapter
2
details our buckyball
simulation to model an anomalous component of the CMB foreground. Chapter 3
is a search for signatures of cosmic strings in the CMB data. I present conclusions
to both projects in Chapter 4.
o
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CHAPTER 2
B u ck y b a ll R a d ia tio n
2.1
A n o m a lo u s F ree-F ree E m issio n s
The CMB foreground contains more power at mid GHz frequencies than can be
explained by conventional theories. The term “anomalous emission” was first
used by Leitch et al. (1997) to describe a component of Galactic foreground that
could not be fit by thermal dust emission. In fits to the 14.5 GHz Galactic
foreground data, Leitch et al. (1997) discovered a component with a temperature
index of 3t ~ 2 (where T oc v~ 0T, v is the frequency) th at was in excess of what
was expected. This tem perature index corresponds to the signatures of free-free
emission, which has a power spectral index of {3 =
0
, where f3 is defined as,
P(v) oc iA
(2.1)
where P is the power. This component has now acquired the name “anomalous
free-free emission” , although “free-free” may be misleading as we now know that
free-free emission constitutes only a fraction of the total anomalous emission.
Due to the fact that its signature was discovered via experiments designed to
quantify the Galactic foreground, this anomalous free-free emission is sometimes
called “Foreground X” (de Oliveira-Costa, Tegmark, Page, & Boughn 1998).
In general, the Galactic foreground at GHz frequencies is composed of three
parts: synchrotron, free-free, and dust emission. As the source of the anomalous
6
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free-free emission is unclear, we discuss evidence for and against each mecha­
nism to motivate our work in explaining this anomalous emission with a new
mechanism.
Galactic synchrotron emission comes from the interaction of magnetic fields
and accelerated electrons. Bennett et al (2003) found th at the spectrum of syn­
chrotron emission at GHz frequencies typically has a tem perature index of fix ~
2.5 to 3, while the anomalous emission is best fit with pT ~ 2. In addition, the
data taken by Reich & Reich (1988) indicated th at the anomalous emission does
not correlate with the observed synchrotron emission at the lower frequencies of
408 MHz and 1.42 GHz. We rule out synchrotron processes as the source of the
anomalous emission.
The anomalous emission is spatially associated with Galactic free-free emis­
sion. Finkbeiner (2003) recently mapped the free-free “haze” permeating the
Galaxy; this haze accounts for a portion of the anomalous emission, as shown
in Figure 2.1, taken from Figure 2.1 of Finkbeiner (2004). The free-free haze is
the solid line labeled as “free-free” in the plot. In addition to the free-free haze,
there is a component due to thermal emission from vibrational dust, extrapolated
from higher frequency data, which also constitutes a small part of the anomalous
emission.
W hat is the source of the free-free haze? Galactic free-free radiation arises
from plasma interactions in the interstellar media.
Free electrons come from
ionized gas, usually as a result of UV photodissociation. This is an equilibrium
process, which necessarily means there should be recombination radiation, so a
search for free-free emission involves looking at Galactic H a regions. McCullough
(1997) showed th at the H a gas that is spatial correlated with the anomalous freefree emission did not account for all of the anomalous emission. Typically, the
7
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- - - - - - - - - ,- - - - - - - - ,- - - - - - - J- - - - - - - - - - (— 1- ,— j- - - - - - - - - - - - - - - - - - - - - - - - - - - ,- - - - - - - - - - - - - - - - ,- - - - - - - - - - - !- - - - - - - - - 1- - - - - - - 1- - - - - - 1- - - - - 1- - - - 1
1000.0
j-
_ • WMAP high latitude "spin"+thermal
: X GB 140ft - Lynds 1622 & LPH 201.663+1.643
: .4 Tenerife (b>20)
' □ Cottingham (t» 3 0 )
; ■ O VRO
O Saskatoon
100.0
w
—3
10.0
ooi
1.0
WNM
WIM ■
0.1
10
100
v (GHz)
Figure 2.1 The correlated intensity fluctuations of the microwave and far-IR emis­
sion of various experiments, The y-axis is the emissivity measured in units of
optical depth at 100 microns.
From Finkbeiner, 2003.
The data points are
from: Wilkinson Microwave Anisotropy Probe (WMAP) (Bennett et al, 2003),
Green Bank (GB 140) (Finkbeiner et ah, 2002), Tenerife (de Oliveira-Costa et
ah, 1999), Cottingham 19.2 GHz (de Oliveira-Costa, Tegmark, Page, & Boughn,
1998), OVRO (Leitch et al., 1997) Saskatoon (de Oliveira-Costa, Tegmark, Page,
& Boughn, 1998), and, Differential Microwave Radiometer (DMR) (Kogut et al.,
1996). An explanation of all the lines are in the text.
8
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amount of dust correlated Ho emission was a factor of 3 to 10 times less than
the anomalous emission. This can be seen in Figure 2.2, from Draine & Lazarian
(1999), where the dust-correlated free-free emission is plotted as a band of grey
dots, under-representing the measured emission (the data points) by up to an
order of magnitude. Finally, gas above 104 K may emit free-free radiation without
significant contributions of recombination radiation. These gas molecules emit
mostly in the X-ray band, and searches for spatial correlations between X-ray gas
with the anomalous free-free emission have come up negative (Finkbeiner et al.
1999). Known mechanisms are insufficient to explain all the anomalous emission.
The main interstellar dust grain emission mechanism is thermal emission; for
the majority of the cool dust in the Galaxy, the peak of this emission at frequen­
cies greater than 100 GHz. According to Draine & Lazarian (1999a), at GHz
frequencies there are two other major dust radiation mechanisms: electric dipole
and magnetic dipole radiation. Simulations dene by Draine & Lazarian (1999)
showed th at unless the magnetic dust grains are specifically tailored, they con­
tribute too little rotational emission to fully account for the anomalous emission.
This leaves electric dipole radiation, which is radiation coming from rotating dust
grains with electric dipoles. Only very small dust grains can rotate fast enough
to emit in the frequency range of the anomalous emission. Small dust grains is
therefore a possible explanation for the anomalous emission.
Finkbeiner (2004) investigated Draine & Lazarian (1999) dust emission for
various interstellar regions. These regions include Cold Neutral Medium (CNM),
Warm Neutral Medium (WNM), and Warm Ionized Medium (WIM). These dust
models span a large range in emissivity, which reflects the large uncertainty in
the dust grain models. This work hopes to limit some of these uncertainties by
modeling radiation from a specific dust candidate. We hope to shed some light
9
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— I—
10
-3 9
I— I— I— I--------------------------- 1-------------------1--------
A de O l i v e i r a - C o s t a e t a l 1998
♦ de O l i v e i r a - C o s t a e t a l 1997
■ L eitch e t al 1997
▼ Lim e t al 1996
.♦*
O K o g u t e t al 1996b
1 0 -40
\
I
N
X
S-,
m
w1 0
-41
iao
Sh
<D
K
U
d u st-c o rre la te d
free-free
/
/cP/qr /<o
10 - 4 2
A'A A
/
' //
/
10 -4 3
10
A »V V/
/V
A
m
20
30
50
F r e q u e n c y (GHz)
100
200
Figure 2.2 A graph of the emissivity ( j / per Hydrogen atom (n# is hydrogen
density). Graph taken from Draine Sz Lazarian (1999a). The data points with
error bars are data from five experiments listed in the legend.
Also plotted
are dashed, dotted, and solid lines representing vibrational dust emissivity of
four different dust temperature and spectral index, 0t - The grey band is the
dust correlated free-free emission. A schematic of our buckyball radiation in the
anomalous free-free emission frequencies plotted in the thick dotted line labeled
Buckyball.
10
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on the process of spinning dust grains as the source of the anomalous free-free
emission.
2.2
S m all d u st grain s
In order to chose a candidate, we narrow the list by examining what we already
know about the anomalous free-free emission. Since the grains radiate by rota­
tion, we can do a first estimate on the grain size based on the necessary rotational
frequency. A “normal” dust grain of radius
1000
A contains roughly
2
x
car­
108
bon atoms. Typical thermal angular speed, cu, of such a dust grain at tem perature
T is given by,
( 2 .2)
where I is the moment of inertia for the dust grain, and k the Boltzmann constant.
The factor 3 reflects the 3 degrees of rotational freedom for a sphere. At T =
50 K, the angular velocity of this
1000
A spherical dust grain is ^ 3.6 x
105
rad s e c '1, or 57 kHz. Compared to the anomalous emission at ~ 30 GHz, this
normal dust grain spins too slowly. For these large dust grains, suprathermal
rotation may be caused by mechanisms such as multiple recombination events
with interstellar hydrogen (Purcell, 1979). However, even these mechanisms only
have a maximum rotational speed of
uj
109 rad s
1
(W hittet, 1992), causing
MHz emission: still too low.
On the other hand, an ultra-small dust grain can achieve a much higher
rotational frequency. At 50 K, a 3
A dust grain containing 30 carbon atoms will
have a thermal rotational velocity of cu ~ 3.1 x 10u rad s e c '1, or, 50 GHz, which
is in the range of the anomalous emission. There is one further advantage to ultra
small dust grains: according to Mathis et al. (1977), the number density, dN, of
11
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interstellar dust obeys the following relationship,
d N oc a ~ 3 Sda
(2-3)
where a is the radius of the dust grain. There are many more small dust grains
than large grains, but most of the mass of dust is in large grains.
The major drawback of small dust grains is that they are fragile. Barely larger
than a molecule, ordinary ultra-small dust grains form either by adhesion of small
molecules, or by fragmentation of larger particles. Many of these dust grains are
easily destroyed by UY photons. The mean Galactic UV field is ~ 102 photons
cm - 2 s - 1 sr-1 ; our ultra-small dust grains candidate must be stable against this
UV field. G uhathakurta & Draine (1989) have calculated the survival time of
dust grains against UV photons as a function of their size. Graphite dust grains
with 20 carbon atoms would have a mean lifetime of 1011 seconds, or about 3000
years. Silicate dust grains with 30 atoms will have a mean lifetime of a little
more than 1010 seconds. Such dust grains are not the ideal candidates because
they would need a constant replenishing source.
These considerations led us to propose that Buckminsterfullerenes, or buckyballs, are better spinning dust grain candidates than “normal” silicate or graphite
dust grains. Aside from C&o, the famous geodesic dome-shaped molecule, there is
a whole family of buckyballs ranging from 20 to 300 carbon atoms in ellipsoidal
configurations composed of various combinations of benzene and penta-carbon
rings. By the virtue of these benzene rings, buckyballs belong to the class of
polycyclic aromatic hydrocarbons (PAHs). Specifically, we consider partially hy­
drogenated buckyballs, e.g. C 6 oHeo_n where n is the number of lost hydrogens
via the dehydrogenation process (where a hydrogen is dissociated from the rest of
the molecule). In interstellar conditions, is usually the result of an impact with
a photon of specified energy. We discuss dehydrogenation later in more detail.
12
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Table 2.1.
B uckyball
Simulation Parameters of the -3 Buckyballs
R ad iu s
T ypical D ipole M om ent
^initial
(p h o to n rate)
(M OI)
(A)
(Debye)
(GHz)
( s ' 1)
g cm2
C 20
3.41
2.0612
38
4.50 x 10- 7
l . l x l 0 “ 37
c24
3.57
4.9937
29
4.71 x 10- 7
1.9x 1 0 ~ 37
C60
4.96
8.1410
13
6.55 x l O -7
1 1 .4 x l0 " 37
Buckyballs make compelling ultra small dust grain candidates. First, they are
small; three species of buckyballs and their characteristics are listed in Table 2.1.
CqqH^q has a radius of ~ 5A, and C 2 0 H 20 has a radius of ~ 3.5
A. The compact
structure of buckyballs means that buckyballs have the smallest moment of inertia
for a given mass of carbon. Larger dust grains are more stable than smaller dust
grains, but their moments of inertia are too large to spin in the GHz range. At the
same time, buckyballs are not large enough so that their collision cross section to
UV photons and hydrogen atoms cannot be neglected. This means that they will
encounter enough photons to significantly alter their emission characteristics. In
this way, their emission spectra will be very non-thermal, and the spectra will
probably (as we indeed will show) obey a power law.
The second reason to consider buckyballs is that they are very strong. Unlike
most dust grains whose formation relies on adhesion, buckyballs are chemically
bonded molecules. Their hollow spherical structure allows deformations under
collision to absorb impact energy which prevents fragmentation. Buckyballs can
also withstand high temperatures.
Experiments have shown that buckyballs,
specifically, Cgo, remain stable up to 1000 K. The smaller PAHs can be excited
to temperatures as high as 3000 K without breaking apart (Omont, 1986). The
general class of PAHs has been proposed to be a component of interstellar dust
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by Dorm as early as 1968.
Aside from physical reasons to consider buckyballs, they are appealing for
simulations. These molecules are nearly spherical which enables much simpli­
fication when considering their angular momenta and inertia. Their particular
dehydrogenation process meant that their dipoles moments can be easily calcu­
lated.
We note th at there are at the present, no unambiguous interstellar detections
of signatures of buckyballs. There have been extensive observations of general
PAH spectral features (starting with, e.g. Duley 1973, to e.g. Rapacioli et al.
2005), and benzene has been discovered in many circumstellar regions (Cernicharo
et ah, 2001). More and more PAH emission lines are being discovered, (see e.g.
Peeters et al. (2004), and a discussion of their relation to the anomalous free-free
emission (Iglesias-Groth, 2004)). To date, there are only two unambiguous extra­
terrestrial signatures of buckyballs. The first is in the vaporized product of the
Allende meteorite (Becker et al., 1994). The second was found in craters formed
on the Long Duration Exposure Facility (di Brozzolo et ah, 1994). Terrestrial
buckyballs are formed by vaporizing graphite. We speculate that similar circum­
stances around carbon stars allow the formation of buckyballs as condensates.
2.3
T h e M o n te C arlo S im u la tio n
This portion of the thesis describes a Monte Carlo simulation of the electric
dipole radiation coming from buckyballs experiencing collisions with interstellar
hydrogen atoms and Galactic photons. My code tracks the angular momentum
of an individual buckyball as it suffers collisions that alternately spin up or spin
down the molecule. From the changes in the angular momentum, we deduce
14
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the energy output and obtain a spectrum of the buckyball’s radiation from the
energy-time analysis. The code was written in C + + .
2.3.1
B uckyball Initial C onditions
Buckyballs are composed of a core shell of carbon atoms, with the hydrogens
radially extended from the carbons.
We simulate three different buckyballs:
C2 0 H 2 0 -T1 , C 2 i H 2 i ~n, and CeoHeo^n, where n is the number of hydrogens lost
during the dehydrogenation process. Throughout the simulation, I trace the an­
gular momentum of the buckyball and generated from th at the energy, rotational
frequency, and radiated power. This requires a knowledge of the moment of in­
ertia of the buckyball. As part of the initial conditions, I explicitly calculate the
moment of inertia tensor, I, for each buckyball, in the buckyball center of mass
frame,
= E m *(c2 - 4 )
(2-4)
i
Ikj
=
- ' 5 2 m irijrik,
(2.5)
i
where j and k runs from 1 to 3, representing the three Cartesian coordinates, + is
the distance from the center of the buckyball to the ith particle in the system, r^
is the j th coordinate distance from the center of the buckyball to the ith particle,
and rrii is the mass of the ith particle. The number and position of the carbon
atoms is fixed and their positions are static relative to each other.
With the interstellar medium (ISM) hydrogen to carbon ratio of 103'5, bucky­
balls are likely to have formed fully or nearly fully hydrogenated. A fully hydro­
genated buckyball is neutral and symmetric with no dipole moment. However,
the bond between the hydrogen and the carbon is relatively weak and can be
dissociated. The leftover electron is easily lost so the whole molecule stays neu-
15
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Dehydrogenation
Figure 2.3 A picture of a Buckminsterfullerene molecule, CzaH-2 2 - The light grey
spheres are the carbon orbitals, and dark grey spheres are the hydrogen orbitals.
The two dehydrogenation sites are indicated by arrows.
tral. However, the carbon atoms surrounding the missing hydrogen site will incur
an excess local charge, so a resonance structure forms in the surrounding carbon
bonds, creating an electric dipole. Once one hydrogen is detached, the next hy­
drogen detachment will likely be from a carbon adjacent to the first detachment,
since this allows the two carbons which have lost their hydrogens to form a car­
bon double bond, which is a more stable structure than two single bonds (Rubin,
2000). A picture of a partially dehydrogenated buckyball is shown in Figure 2.3.
The light grey spheres indicate carbon orbitals and the dark grey spheres are the
hydrogen orbitals. This is a C 2 4 H 2 2 molecule, with the two dehydrogenation sites
indicated by the arrows.
16
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The value of the dipole moment can be calculated by standard chemistry
programs such as Spartan Pro, which semi-empirically solves the Schrodinger
equation for a given molecule. These calculations show th at one missing hydrogen
generates a dipole moment of 0.8 debye. W ith two missing hydrogens, the carbon
double bond creates an electric dipole moment of 1.4 debye, a little less than two
times th at for the single dehydrogenation case.
The number of dehydrogenations that takes place in interstellar space is un­
certain. We have therefore followed the suggestion of Omont, (1986): for our
C 2 0 H 2 0 -T1 buckyball, the median number of dehydrogenations is set at n = 3.
I scale the number of dehydrogenations to the number of carbon atoms for the
larger molecules. There is an upper limit to the number of dehydrogenations,
because, with the loss of two hydrogen atoms, the excess local negative charge of
the surrounding carbons maintains tighter bonds with the remaining hydrogen
atoms.
In my code, I set the number of dehydrogenations as part of the initial con­
ditions of the simulation. Once n is determined for an individual buckyball, it
is static for the rest of the simulation. I acknowledge that in real situations, the
buckyball may lose a few hydrogen atoms to UV photons, and this will affect the
strength of the dipole moment. However, the magnitude of the dipole moment
only affects the overall scale of the emitted power, but it does not change the
shape of the power spectrum. The random orientation of the dipoles results in a
dilution of the strength of individual dipoles, as two dipoles pointing in opposite
directions cancels each other out. Unless the dipoles are aligned serendipitously
on top of each other, different reasonable distributions give ~ ± 5% change from
the mean value of the dipole moment. I then generate the locations of the de­
hydrogenation in pairs. For example, with n — 9, we generate four sites in the
17
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buckyball for double bonded carbons and one site for a resonance structure. The
value of the total electric dipole moment is a superposition of the five individual
electric dipoles moment.
Once I determine the position of the dehydrogenations, I can calculate the
moment of inertia tensor I and the strength and orientation of the electric dipole
of the molecule. I obtain the rotation axis by diagonalizing the moment of inertia
tensor. I then switch to the principle axis coordinates of the buckyball, denoted
by primes, where, I z>zi > Ix>xi , I yryi. In these coordinates, calculations of the
properties of the buckyball becomes simpler. The angular momentum vector L,
becomes,
hi/
y \ iji j i u j j .
y
( 2 .6 )
The oniy non-zero component of rotational velocity is cjz/ by definition, so that
the angular momentum is simply, L\ = Jj/z/wz/. The rotational kinetic energy of
the molecule is therefore
(2.7)
The buckyball and impacting particles are allowed full three dimensional degrees
of freedom. I now let photons and interstellar hydrogen atoms collide with the
buckyball while monitoring the buckyball angular momentum.
2.3.2
H ydrogen C ollision P rocess
In my simulation, I consider buckyball collisions with interstellar hydrogen atoms
and photons. We first describe the simpler hydrogen atom collisions.
Upon collision with a hydrogen atom, I make the approximation that all
of the angular momentum of the hydrogen atom is transferred to the buckyball.
18
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Table 2.2.
Region
Environmental Properties of Simulation Environment
H D ensity
H T em p eratu re
T im e Betw een Collision
H Velocity
(# cm - 1 )
(K)
(s)
(cm s _ 1 )
W IM
0.1
8000
2.10938 x 109
813395
W NM
0.4
6000
6.08926 x 10s
704421
CNM
30
100
6.28897 x 107
90940
VCM
100000
25
1.88669 x 104
78742
The hydrogen atom was immediately ejected from the buckyball with a negligible
percentage of its incoming kinetic energy. This approximation is justified because
the buckyball is at least
200
times more massive than the hydrogen atom so the
re-emission process does not appreciably affect the buckyball’s internal energy.
The time scale between hydrogen collisions and the energy of the incom­
ing hydrogen atoms depends on the characteristics of the type of interstellar
media I model. I follow the Draine and Lazarian division of the interstellar re­
gion into three types: Warm Neutral Medium (WNM), Cold Neutral Medium
(CNM), Warm Ionized Medium (WIM), and we added a new region, the Very
Cold Medium (VCM) to expand the range of density and temperatures. The
temperature and hydrogen density characteristics of the four regions are summa­
rized in Table 2.2. The temperature range simulated went from 25 K to 8000
K.
The hydrogen-buckyball collision has a characteristic time scale r = (n Ha v ),
where nn is the number density of the hydrogen atoms, o is the cross-sectional
area (collision cross section) of the buckyball, which I take to be its physical size,
7rr 2
(ignoring Coulomb interactions since the buckyball is neutral), and v is the
velocity of the hydrogen atoms. The probability density function for the arrival
19
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time,f, or the time between hydrogen collisions, is proportional to exp (—t / r ) .
The hydrogen atoms are assumed to be in thermal equilibrium and therefore
their speeds obeyed a Maxwellian distribution
p W = 4Tr( ^ r r
^
’”’!/2tT
(2.8)
where m is the mass of a hydrogen atom, k the Boltzmann constant, and v is
the velocity of the hydrogen atom. I obtained average cross section, < av >, by
computing,
A A g A
J p(v)dt!
(2.9)
in our case the cross section, a, does not depend on velocity.
The hydrogen atom collision site on the buckyball is chosen by randomly
generating the x and y coordinates, [px,Px\, of a point, p, in the unit circle per­
pendicular to the incoming velocity, v, of the hydrogen, where x and y are unit
vectors in the plane of the unit circle, perpendicular to the unit vector, v, of
the incoming hydrogen atom. The buckyball has a rotational velocity of vrot.
At the start of the simulation, this is set to the thermal rotational velocity. As
the simulation progresses, it changes according to the angular momentum
of the
buckyball. The angular momentum imparted to the buckyball, AL,from the
hydrogen atom is given by,
AL =
where A v = \ rot + v, and
b
m A v
x
b
(2-10)
is the impact parameter, given by
b = r ■[px ■i + py ■y - ^/l - p2
x - p 2yw] .
(2 . 1 1 )
Since the buckyball is a rigid rotator, its rotational velocity is vTOt = no. This
alters the rotational kinetic energy, which is simply L 2/21 and produces a new
angular velocity, given by uo' = y 2 E / I Z
20
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Between collisions, the buckyball radiates away its energy according to the
classical Larmor formula
2 u 4d2
P = sin
3 c3
( 2 . 12 )
0
where 9 was the angle between the electric dipole moment, d, and the rotation
axis, z', and c is the speed of light. Since to is proportional to y/E, then d E /d t oc
—E 2, so th at the final energy, E f , at some time t seconds after the collision, is,
1
e
,
1 8 d 2 sin 2 9
e,
3 c > /y ,
(2.13)
where jE* is the energy of the buckyball right after the collision.
2.3.3
P h o to n C ollision P rocess
The buckyball is also bombarded by Galactic background photons. I calculate
the distribution and energy of the photons according to the M athis et al. (1983)
estimates of the energy density of ambient radiation in the solar neighborhood.
The energy density of this radiation is given by,
zl 7r
= u ux v +
E
4-7T
W i — B x (Ti)
c
+
— B X(2.725K)
c
where u\ is in units of erg cm 2 pni-1 , (W2, W 3, IT4 ) =
(1
(2.14)
x 10~1 4 ,10~13, 4 x 10~13),
with (T2, T3, T4) = (7500,4000, 3000)K. 4wJ\ has units of erg cm 2 pm -1. B \ is the
blackbody function. The UV components of the Galactic background photons are
given, in units of flux, F\ = 4:irJ\ , and relates to energy density by, i n J x / c = u\.
The UV components are,
47Tj \
0
0 —>0.0912 fim
38.57A3-4172
0.0912 —>• 0.110 n m
2.04510-2
0.110 —> 0.134 pm
7.115 x 10- 4 A^
6678
0.134 —>■0.246 pm
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(2.15)
_Q
_ Q
~0
0.05
0.00
10
6
10
5
10
-4
Log Wavelength of Incident Photon ( c m )
Figure 2.4 A plot of the photon probability distribution used to generate the
photon density in each simulation.
The probability density function of ambient solar neighborhood photons is plotted
in Figure 2.4. I use this photon distribution in my simulation.
Bombardment by a photon raises the internal energy and therefore the tem­
perature of the buckyball. The buckyball then emits photons to radiate away
the energy. The actual number of photons emitted by the buckyball is a stochas­
tic process, but the average number is dependent on the tem perature to which
the buckyball was excited. The tem perature is calculated from the enthalpy,
Un (T) (given in ergs) of the buckyball; I use the G uhathakurta & Draine (1989)
approximation,
4 11 x 10_ 22 T 3 -3
Un (T) = (1 - 2 / N ) N 1 + g 5l x l 0 __3 T + L 5 x l Q _ 6 T 2 + g 3 x
1 0 _ 7 r 2 .3
(2-16)
where N is the number of carbon atoms in the buckyball, and T is the temperature
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of the buckyball in Kelvin.
If we assume th at all the energy of the absorbed photon went into heating
the buckyball, then equating U^(T) to hv, the energy of the photon, yields the
temperature increase of the buckyball after photon absorption. The number of
photons emitted by a buckyball with energy E suffering a collision with a photon
of energy hv is,
E+hv
r N
A N = / -rdE
E
E
(2.17)
^
where,
N = I
E = I QuB v(T)dv
(2.18)
Qv is the absorption cross section for micron-sized graphite dust grains calculated
by Draine and Lazarian (1999), scaled for the buckyballs, and B u is the Planck
function. The dotted variables means the time derivative, for example, E is the
rate of change of the energy. The total number of photons emitted is therefore,
» iV
S hQ ,B „(T,)dv
SQ.B „ (T ,)
S
S Q M T i)
’
’
where T; is the tem perature of the buckyball with energy E, T f is the temperature
of the buckyball with energy E + hi/, both obtained by solving Equation 2.16.
I constrained the number of photons emitted to be a positive integer. The
minimum number of photons emitted is
0
and the maximum, given by a collision
with the highest energy Galactic photon, is 27. The emitted photon has both
orbital angular momentum and spin, but the orbital angular momentum of a
photon about the buckyball is much less than h = h /2 tt. Therefore, I only
need to consider the spin of the photon. Each photon emitted will change the
angular momentum of the buckyball by ±h. The sign of the change in angular
23
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momentum due to photon emission is determined randomly. I set up two random
numbers to separately generate a) the time between successive photon impacts,
and b) the number of photons emitted by the buckyball according to Equation
2.19. Between photon impacts, the buckyball radiates classically according to
the Larmor formula as it did between hydrogen atom impacts. The photon and
hydrogen collision processes drive up the variance in the angular momentum, as
the buckyball angular momentum executes a random walk due to the random
directions of the collision particles. The competing radiation process reduces the
angular momentum of the buckyball. An equilibrium between these two processes
produces the spectrum for my buckyball radiation.
Each simulation was set for 10 years, with as little as 1000 events in the
rarefied region, and as many as 100,000 events in the densest region. The ini­
tial conditions of the simulation is wiped out in as little as five events, so th at
running the simulation for
10
years produces a good representation of the dis­
tribution of buckyball radiation. Plotted in Figure 2.5 is the emission frequency
vs. simulation time for a C 20 buckyball in the CNM region. The distribution in
the Figure is representative of the direct output of the simulation from which I
derive parameters such as spectral index and emissivity.
2.3.4
A Short Ju stification for C lassical M echanics
The fact th at buckyballs are very small may raise concerns about using classical
mechanics in our simulation. I show that the rotational quantum number of
C 2 0 H 2 0 is large enough for a classical analysis (for references, see, e.g. Thompson
(1994)).
Consider a rigid body rotator. The Hamiltonian of this system is exactly the
24
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cr 20
520
54 0
560
58 0
600
S im u la tio n T im e (M s )
Figure 2.5 Plot of the emission frequency vs. simulation time of the C 20 buckyball
in the CNM region. Solid lines indicate times during which the buckyball is
radiating. The dashed lines correspond to a collision event.
25
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kinetic energy of the system in the absence of outside forces (e.g. gravity)
H
h2
t 2
p
°xi
_|_ y V _|_
2
I'Xl
lyl
j
2
zi
( 2 .2 0 )
Izl
where J is the angular momentum operator, and I x,T y, andlz are the principle
moment of inertia for each axis. The system is characterized by three angular
momentum quantum numbers: J for the rotation of the rotator, M for its projec­
tion onto the z axis, and K the projection on the z’ axis defined previously. The
energy eigenstates are given by (J M K \ H | J M K ) .
Buckyballs are very symmetric molecules (dehydrogenations introduce small
deviations which
1
took pains to include in the simulation, but contribute very
little moments of inertia), so we can easily find a coordinate system with I x —
Iy which makes all off-diagonal elements of I approximately zero. In fact, this is
exactly the Principal Axis Coordinates (the primed coordinate in our simulation).
The energy eigenvalues are,
E t = h2
j(j + 1)
/
1
\Iz
2/ x
1
\ k2^
IxJ
( 2 .2 1 )
2 j
where j is the quantum number of state J, and k is the quantum number for state
K. Compare this to the energy level of the diatomic molecular vibrational bands,
Ej = —
( 2. 22)
Different J ’s represent different bandheads, with K ’s being the bands within
a bandhead.
For simplicity, consider K = 0, the case where the rotation is
perpendicular to the z' axis; the energy difference between adjacent states is,
A E j = E J+i —E j ~ J + 1. The rotational kinetic energy of a C 2 0 H 2 0 molecule
at 50 K is 3.5 xlO - 1 5 ergs, and the moment of inertia is 2 x l 0 ~ 37 g cm2. This
gives J = 35. The energy difference between the two states is about
6
%, or,
about 10- 1 6 ergs. The fact that the energy bands are so close together leads us
to believe treating the rotational energy level as a continuum was justified.
26
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2.4
S im u la tio n A n a ly sis
At each time-step in the simulation, we recorded the energy and rotational fre­
quency of the buckyball. Our main goal was to extract the spectral index (f3)
of the three buckyball emissions as well as the emissivity. Using the energy-time
data, we made a series of energy bins and tabulated the amount of time the
buckyball spent in a particular energy bin. Specifically, we divided the energy
range into 500 uniform bins so that later, when we wanted a composite spectrum
of the three buckyballs, we could just sum the power in each bin. From this in­
formation, we obtained an e n e rg y -fre q u e n c y relationship: how much energy the
buckyball radiated at what frequency. The peak of this graph we defined to be
the peak emission frequency of the buckyball. We have plotted an example of
this in Figure 2.6. The peak emission,
lup
and the power at peak emission, P ( ojp),
are listed in Table 2.4, for the three buckyballs and four regions. The units of the
lop
are given in GHz, and the units of P(cop) are given in 1CT31 erg s_ 1 Hz-1. For
simplicity, in this section, we use C 2 o, C'24, C 6 o, to denote
6 2 0 /^ 20 - 71,
CftoHfto—n, where n is the number of dehydrogenation.
We divided the total amount of energy in each energy bm by the total time
a given buckyball spent in the bin to get the power radiated at each frequency.
Dividing this by the central frequency of the bin, gave us the power per Hz
emitted by the buckyball in each bin. The slope of this graph is the spectral
index of the radiation, /?, given by Equation
2 .1
where P( uj) is the power of the
buckyball, in units of erg Hz- 1 s-1, and to is the frequency of the buckyball, given
in units of Hz. As a goodness-of-fit parameter, we cite the residual, ar, of the
least squares fit to the spectral index,
ar =
m
(togio-PmM - f i t ) 2 /m ,
27
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(2.23)
peak emission
107
1C8
109
1 0 10
1 0 11
LOG Emission Frequency (Hz)
Figure 2.6 The Energy-frequency plot for C2o buckyball in the WNM region. The
peak is taken as the peak emission frequency, and most graphs show a clear drop
off in the energy emitted vs. frequency after the peak.
28
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where Pm(a>) is the data, f i t is the best fit to the logarithm of the data, and m
is the number of data points. In the cases where the spectral index was better
fit by splitting into two power laws, we cite the spectral index for the higher
frequency power law, labeled t for top, at ar ~ 0.07, so th at the assessment was
the same for each simulation. The lower frequency power law is labeled b for
bottom. Exception to the oy ~ 0.07 rule was made for some C qq cases because
the scatter in these data was much larger than that for the other two buckyballs.
In order to have oy ~ 0.07 for these cases, I could only use less than 10% of the
data points for fitting the top power law which was not an accurate representation
of the distribution. Therefore, for the Ceo data, we typically cited the slope that
fitted a reasonable number of data points while maintaining a small oy. Where
this actually occurred was judged on a case by case basis, but typically, oy was
0.3 for data with larger scatter. The spectral indices of all buckyball emission
and their oy fitting coefficients are in Table 2.3.
For a measure of the relative effects of photon vs.
hydrogen impacts, we
have included, for each simulation environment, the percentage of the angular
momentum transferred from photons and hydrogen atoms in Table 2.5. The
numbers cited are the number of hydrogen or photon impacts in the given region
for each of these buckyballs; the numbers in the parenthesis are the percentage
of the total angular momentum transferred to the buckyball by those collisions.
In the more rarefied regions (WIM), there were no hydrogen impacts, and in the
denser regions (VCM), hydrogen impacts dominated.
2.4.1
Typical Sim ulation N um bers
As a first check, we can compare the buckyball rotational velocity, vrot, to the
mean impacting hydrogen velocity,
v h , to
verify our hydrogen absorption assump-
29
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Table 2.3.
Buckyball Power Law Spectral Indices, ft, and their oy fitting
coefficients. . .
Region
W IM
W NM
CNM
VCM
P ro p e rty
C 24
C-20
C60
ta
bb
t
b
t
b
3
2.691
0.871
2.436
0.973
1.165
0.934
Tr C
0.0755
0.419
0.0666
0.324
0.172
0.297
0
2.360
0.935
2.498,
0.912
1.470
0.917
(7x
0.105
0.293
0.0772
0.352
0.193
0.393
3
2.603
0.883
2.580
0.940
1.867
0.962
CTr
0.0761
0.460
0.0721
0.278
0.171.
0.267
3
0.981
0.885
1.016
0.862
1.009
0.894
(Tx
0.0527
0.299
0.0540
0.298
0.0419
0.287
aT op (high frequency) p o rtio n of the em ission
b B o tto m (low frequency) p o rtio n of th e em ission
cuy for linear fits to th e sp ectral index
30
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Table 2.4.
Peak Frequency, u p of buckyball emission and power at peak
frequency, P ( u p).
Region
W IM
W NM
CNM
VCM
C haracteristic
C 20
C 24
Ceo
LOpa
37.25
23.03
9.13
P ( u p )b
1.688
1.760
1.324
UJp
22.91
13.11
7.63
P(udp)
0.6333
0.6573
1.639
UJp
16.84
20.75
5.41
P(uip)
0.9861
1.639
1.553
UJp
31.23
18.29
12.24
P(UJp)
19.82
16.32
145.8
ain units of GHz.
bin u nits of 10" 31 erg s _1 H z- 1 .
tions. Taking the largest value of the C 2 o peak frequency, u p from Table 2.4, and
the C*2 o radius, we obtain the C*2 u rotational velocity to be vrot = ujpeak x radius,
1260 cm s - 1. Comparing this to the mean vh from Table
2 .2
, for the coolest
region (VCM), showed that on average the impacting hydrogen atom velocity
was almost 100 times faster than the rotational velocity of the buckyball
In
the buckyball reference frame, the much larger speed of the impacting hydrogen
atoms meant that our assumption of the buckyball completely absorbing the hy­
drogen atom ’s angular momentum is feasible. For the other buckyballs, vrot is
similarly small compared to the hydrogen atom speed: C 2 a- 821 cm s” 1, Ceo: 607
cm s_1.
I present some characteristic results of the simulation. For a given simulation,
the rms amount of angular momentum from a single photon absorption and sub­
sequent emission process is ~ 2.5h, or about
10%
of the total angular momentum
31
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Table 2.5.
Photon vs. Hydrogen Impact
Region
P ro p e rty
W IM
# P h o to n Im p act (%}
729 (100%)
792 (100%)
731 (100%)
# H ydrogen Im pact(% )
0 (0%)
0 (0%)
0 (0%)
W NM
# P h o to n Im pact (%)
731 (100%)
746 (89.48%)
1036 (65.01%)
# H ydrogen Im p act (%)
0 (0%)
2 (10.52%)
2 (34.99%)
CNM
# P h o to n Im p act (%)
667 (95.38% )
735 (97.65%)
1018 (94.3%)
# H ydrogen Im p act (%)
20 (4.62% )
17 (2.35%)
49 (5.7%)
# P h o to n Im p act (%)
668 (6.02%)
760 (5.78%)
1062 ( 0.1% )
# H ydrogen Im p act (%)
42150 (93.98%)
46154 (94.22%)
88621 (99.9%)
VCM
C 20
C 24
C60
aPercentage of to ta l angular m o m e n tu m tran sferre d
of the buckyball. Approximately 30% of all photon collisions im parts an angular
momentum of one or two units of h to the buckyball. The average change in
the rotational energy of a VCM region t72o buckyball, given in units of kT, is
8
E / k T = 0.0055. A typical hydrogen atom collision imparts ~ l / 1 0 th of the
angular momentum of a photon impact, which means that hydrogen impacts are
irrelevant when the hydrogen atom density is low compared to the photons. I
list in Table 2.6 the mean rotational energy of the buckyball from the simulation,
in temperature units kT, as well as the 97th percentile value for the rotational
energy in temperature units. The mean rotational energy of the buckyball is subthermal in all cases, a fact reflected in the peak rotational frequency discussed in
the next section.
2.4.2
Em ission C haracteristics
Here I describe the simulation results, and I discuss our analysis of the processes
which led to these results. In general, the b (lower frequency) power law for
32
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Table 2.6.
Simulation Rotational Energy of Buckyballs
C 2 0 #20-71
C 60 # 60—n
C 2 4 #24 —71
Region
< E >a
E (97t/l %-tile)
W IM
0.00314
0.00560
0.00191
0.00301
0.00188
0.00378
W NM
0.00188
0.00396
0.00104
0.00241
0.00225
0.00386
CNM
0.0719
0.131
0.125
0.2.34
0.0943
0.196
VCM
0.658
1.17
0.356
0.949
0.766
1.14
<E>
E (9 7 fA % -tile)
<E>
E (97t '1 % -tile)
ain te m p e ra tu re u n its of k T
all buckyballs in all regions has a power spectral index of /3 ~ 1, with average
(3 = 0.91. The t spectra (at higher frequencies), are steeper and obeys a f3 ~ 2
power law, with the average spectral index (3 = 1.9. The breaking of the power
law into t and b is most pronounced in the CNM region, and it is least pronounced
in the VCM region, where breaking the fit into two power laws produced a less
than 10% improvement in oy. For the other regions, breaking the power law
produces a 25% to 50% improvement in oy of the fits. The break in the power
law is also less pronounced in the C qo buckyball. It is noteworthy that the t
spectra contains significantly less scatter than the b spectra.
Table 2.4 gives the peak frequency, cjp, and the power emitted by the buckyball
at the peak frequency, P ( ujp).
I define peak frequency as the frequency bin
which contain the highest total emitted energy.
I plotted a sample graph of
the total energy emitted vs. the frequency in Figure 2.6. In every case, the
buckyball emission showed a clear drop off past the peak emission frequency, so
I am confident th at the buckyball emission range was limited. The value of u p
from all buckyballs and simulation regions spans 5.4 to 37.3 GHz. The peak
frequency of emission by the C$o buckyball is lower in every region than the
33
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smaller buckyballs.
Figures 2.7 to Figure 2.10 are plots comparing simulation region quantities
against each other. Figure 2.7 compares the effect of hydrogen density on the
spectral index; Figure 2.8 plots the effect of hydrogen tem perature on spectral
index. Figure 2.9 plots the effect of hydrogen temperature on peak emission fre­
quency, and Figure 2.10 plots the effect of hydrogen density on peak emission
frequency. In Figures 2.7 and 2.8, the spectral indices of the different tem per­
atures simulated are offset by the labeled amount for clarity. For example, the
dot-dashed diamond lines (indicating the 8000K simulation) is offset by +1.0,
which means 1.0 was added to all spectral index values. This is done so the spec­
tral behavior of each simulation is distinguishable from the other. The trends
shown in these Figures are discussed together in the rest of the subsection.
Two major effects governed buckyball dipole emission.
1)
Buckyball size dic­
tates frequency response. The smaller buckyballs responded faster to changes
in energy, largely because they could spin faster, and therefore emitted their
absorbed energy quicker. This resulted in the presence of the hyper and quies­
cent states. The hyper states were emissions right after an angular momentum
increasing impact which spun up the buckyball. The quiescent states were lulls
between collisions. The large buckyball had more inertia and therefore took more
energy to be spun up to a hyper state. This explained the fact th at it generally
had a lower spectral index, as well as a lower peak frequency, as shown in Table
2.4.
2) The hydrogen density of the emission region drove the variance in the
buckyball angular momentum. Even with the higher temperatures and therefore
harder impacts, the rarefied regions has substantially subthermal emission. In
Figure 2.10, the high temperature and high hydrogen density produces drastically
34
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4
: offset = +1
C20
offset = +0.5 Q.
"A ,
x
0
\
"O
_c
"cC
o
offset = -0.5
-t—'
0
Q.
+
A
□
O
CD
1-4
25K
1 00 K
600 0K
8 00 0 K
10
-2
"
Hydrogen Density (g/cm
0
10
ur
f
Figure 2.7 Spectral index vs. simulation Hydrogen density for each of the 4 tem­
peratures simulated. Crosses indicate hydrogen tem perature T = 25 K, Squares
are T = 6000 K, Diamonds are T = 8000 K, Triangle are T = 100 K.
35
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4
3
offset = +0.5
x
■a
0
_c
Io 2x>ffset = -0.5
0
Q.
co
1
0
10°
101
102
10 3
10 4
IQ5
Hydrogen Temperature (K)
Figure 2.8 Spectral index vs. simulation temperature for each of the 4 densi­
ties simulated. Crosses indicate hydrogen number density, n Hl = 100000 cm-3,
Squares are
uh
= 0.4 cm-3 , Diamonds are n # = 0.1 cm-3 , and triangle are nn
= 30 cm~3.
36
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140
+
A
□
O
N
I
o
>
120
o
c
a;
100
cr
80
3
<D
C
o
w
'E
C20
10 0 0 0 0 / c m 3
30/cm 3
0.4/cm 3
0.1 / c m 3
/
60
'c/5
40
LU
CC
a>
20
CL
0
10
100
10 00
1 0 000
Hydrogen Temperature (K)
Figure 2.9 Peak emission frequency vs. simulation tem perature for each of the
4 densities simulated. Crosses indicate hydrogen number density, n H, = 100000
cm-1 , Squares are n H = 0.4 cm-1, Diamonds are n H = 0.1 cm-1, and triangle
are n # = 30 cm-1 .
37
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120
N
X
0
o
|
80
+
A
□
O
25K
1 00K
60 0 0K
80 0 0 K
C20
C7
£
^
60
c .
o
'w
•|
40
LU
®
o
20
CL
4
2
Hydrogen Density (g/cm
6
f
Figure 2.10 Peak emission frequency vs. simulation Hydrogen density for each of
the 4 temperatures simulated. Crosses indicate simulation temperature, T = 25
K, Squares are T = 6000 K, Diamonds are T — 8000 K, Triangles are T = 100
K.
38
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higher peak frequencies, compared to the lower temperatures. However, this is
not a tem perature effect, because at high temperature but low density, only the
high density simulation shows a jum p in peak emission frequency, as can be seen
in Figure 2.9. The three other simulations have low peak densities even with
higher temperatures. This behavior is due to the fact th at im pact rate is more
im portant than impact strength. The ratio of the RMS hydrogen atom velocity
between the WIM region and the VCM region is ~ 18, but the ratio of the
mean free time between hydrogen atom arrival is
1/50000. The random walk
executed by the buckyball’s angular momentum has larger variance with more
steps, as opposed to larger steps. As a result, the highest density region has the
highest emission frequency.
Larger variance is also responsible for a flatter spectrum, as the averaged
effect is to distribute the energy more evenly among frequency bins. Figures
2 7 and 2.8 illustrates this. Figure 2.8 shows that the spectral index does not
vary as a function of the temperature, whereas Figure 2.7 shows that at all
temperatures, the spectral index drops with respect to hydrogen density.
In
effect, the frequent and random collisions provides a cooling mechanism to the
buckyballs. The rarefied regions are dominated by rare but powerful impacts
which imparts large jumps (up or down) in the buckyball angular momentum,
which in turn contributes to a higher spectral index.
2.4.3
A nom alous Em ission?
Are buckyballs candidates for anomalous free-free emission? The anomalous freefree emission was measured to have a temperature index of fix ~
2
, or a power
spectral index of (5 ~ 0. This is different than the buckyball power spectral index
that we measure, typically around 2 for t spectra and 1 for b spectra. However,
39
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Table 2.7.
Thermal Rotational Frequency of Buckyballs3,
Region
C 20#20—n
C 2 4 R 24 —n
C qqH qq—n
W IM
681.1
547.3
223.4
W NM
589.9
474.0
193.5
CNM
76.15
61.19
24.98
VCM
38.07
30.60
12.49
ain u n its of GHz.
we are encouraged by two facts:
of the anomalous emission, and
1)
2)
the peak emission frequencies fits in the range
buckyball dipole radiation drops off sharply
at frequencies higher than the peak emission frequency. These two facts suggest
that individual buckyballs do not produce the flat free-free power spectrum, but
a composite of spectra from a population of buckyballs having many peaks in
this region may contribute to the observed spectrum.
In general, we expect the spectrum of the composite buckyball emission to
resemble the C qq simulation result, because terrestrial graphite condensates pro­
duce more C§q than any other buckyball. Figures 2.11 to 2.14 are plots showing
a composite spectrum of the three species of buckyballs for the WIM region.
The power from each species is given equal weight, so the spectra contain equal
proportion of C2o to C2± to Cgo- The scatter plot in each figure is the sum of
the power for each frequency bin of all three buckyballs, fitted with a best fit
straight line. I include a nearest
8
neighbor sum of the above scatter plot, offset
by a factor of 103 below the scatter plot in each Figure for clarity. The equal
proportion probably overestimates the expected number density of C2o and C 2 4 ,
however, it illustrates the point that, as expected, the composite spectra of all
four regions strongly resemble the C e0 spectrum, with minor modifications at the
40
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turn-off due to the higher peak frequencies of the smaller buckyball.
The best fit line of the composite spectra gives the composite spectral index
for each region: for VCM, (3 = 0.873; WNM, (3 = 1.12; WIM, /? = 1.01; CNM,
/T = 1.15. In all cases, the composite spectral index was very similar to the
spectral index for the region listed in Table 2.3. The smaller buckyballs also
emitted with less power than
6 *
60,
as shown in Table 2.4. We conclude based on
the composite spectra that, in general, buckyball emission is dominated by C6o
emission. However, in the higher frequency regime (above ~20GHz), the emission
from most regions was dominated by the C2 o buckyball because the C 6o emission
was restricted to lower frequencies due to its slower rotation rate. Therefore,
we use the emission characteristic of C2o to fit the observed anomalous freefree emission between 35-45 GHz in order to estimate the number of buckyballs
required to produce the anomalous free-free emission. Both the WIM and VCM
regions contain emissions in this range.
Draine & Lazarian (1999a) calculated the emissivity per H nucleon of the
observed microwave emissions by cross-correlating the sky brightness with a FarIR sky map. Their data are shown in Figure 2.2. Also plotted on their graph is
the emissivity of various populations of dust with at different temperatures and
a spread of tem perature indices as references. We have plotted a schematic of
our WIM composite data on top of their graph for a comparison. Our buckyball
simulations produced more than enough power to account for the excess emission.
The schematic for the emission from our VCM region would be another order of
magnitude higher. In the WIM region, the buckyballs spent 3.6 % of simulation
time in the 35-45 GHz range; the number for VCM is 3.1%.
We calculate the power emitted per Hertz from each buckyball in this range
by multiplying Equation 2.12, the Larmor formula, by the percentage of time the
41
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5
o
a.
P = 0 reference line
-3 6
10
,-38
best fit
nea rest 8 neighbor s m o ot hi n g
0
-40
i
i
I
I i I
10
100
Frequency (GHz)
Figure 2.11 Power vs. Frequency of the composite emission from all three buck­
yballs in equal proportions for the WIM simulation. The thin line is a nearest
8
neighbor sum to show a smooth spectrum. The thick dot dashed line is a flat
(.P oc i/°) spectrum for comparison.
42
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I
7/5
03
10
34
03
o
Q_
10
36
offset=10
10
38
best fit
nadrest 8 neighbor s m o o t h in g
10
40
10
5
10
6
10
7
10
9
10
10
11
10 1
LOG Emission Frequency (Hz)
Figure 2.12 Power vs. Frequency and a smoothed version of the composite emis­
sion from all three buckyballs in equal proportions for the CNM simulation.
43
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o
offset=10
Q_
0 10
-36
o
10'
-38
best fit
nearest 8 n e ig hb or sm o o t h in g
10'
-40
10
'
10"
.
10
10
10
10
10
1
?
LOG Emission Frequency (Hz)
Figure 2.13 Power vs. Frequency and a smoothed version of the composite emis­
sion from all three buckyballs in equal proportions for the CNM simulation.
44
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10
1 0
-28
'
I I I I I ] --------------1------- 1------1— I I T T T l-------------- 1--------- 1---- 1— I I I I I
-30
^ i °-3 2
IsT
x
o5
0
oHO
$
-34
<^o
0
5
o
^
0
o_ J
n'~ 36
^o
offset=10v
1
10
' 38
best fit
nearest 8 nei ghbor sm oo th in g
10
-40
I 1.1__________i
1 0
'
-.9
10"
10w
10
i
i
i
10
i i m
0
I
11
LOG Emission Frequency (Hz)
Figure 2.14 Power vs. Frequency and a smoothed version of the composite emis­
sion from all three buckyballs in equal proportions for the CNM simulation.
45
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buckyball spends in the frequency band, assuming an isotropic distribution in
the emitted radiation:
P (C 20,35 - 45Gbfz)
=
3 % ~ l ~ ~ s m 2d
4-/r 3 cs
(2.24)
=
3 % i | (247r 3
(2.25,
40x3 la9),d2.
Given a typical value of the dipole moment of 1.3 debye, the power per molecule
per Hz is,
P(G 2 o, 35 - 45GHz) = 4.54 x
1 0 “ 3 4 er^
(2.26)
Comparing this to the data, the free-free emissivity at 40 GHz from Figure 2.2
is,
eo//(4 0 GHz) = 4 x 10" 4 1 er^ s~l s r ' 1 H z ~ l H ~ ' .
(2.27)
We obtain the number of buckyballs needed to radiate with this emissivity by
equating the following,
taff = P( C 2 o, 35 - 45GHz) x 109H z x 1^ 1 ^ —
[Oj n H
(2.28)
where 109 Hz is the width of the band, [G2 o]/[G] is the concentration of C2o to the
amount of carbon in the Galaxy, n c is the carbon density of the Galaxy, and n # is
the hydrogen density of the Galaxy. The necessary C 20 to carbon ratio is ~0.5%
for buckyballs to constitute all of the anomalous free-free emission. This is a
large but not unreasonable amount of carbon to be in these ultra small molecules
(compare to 0.9% obtained by Foing & Ehrenfreund 1997). If we assume spherical
dust grains, the grain mass, M, scales with grain radius, a, as, M oc a05. The
mass contained in 3 A grains (size of the smallest buckyball) is
of the mass in
1000
^3/1000
0.06
A grains, which is comparable to what we need to explain
the galactic emission.
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2.4.4
P ossib le O bservations
We expect buckyballs, like many other PAHs, to form in a variety of interstel­
lar conditions. Buckyball searches have been conducted near carbon rich stars,
specifically around R Coronae Boreal (RCorB) is stars (Clayton et al., 1995).
RCorB itself is thought to have a fossil dust shell (Gillett et ah, 1986), extending
from 1.3’ to 9’. Conservative estimates of carbon dust content of this fossil shell
is around 0.013 M sun. Given our predicted buckyball abundance, we expect to
see radio emission of approximately 0.75 mJy at 40GHz from RCorB, a value th at
is very small and difficult to disentangle from other emissions processes at this
wavelength. In our opinion, it would be virtually impossible to directly observe
buckyball dipole emission around stars.
High latitude (\b\ > 15) cirrus clouds (HLC), on the other hand, present ideal
conditions for the detection of buckyball dipole emission. These clouds contain
large numbers of carbon atoms, but have relatively simple chemistry so that the
observation of a single process is possible. HLCs are by definition found at high
Galactic latitudes where there are less objects interfering in the line of sight.
They span a range of hydrogen densities from 100 cm - 3 in rarefied clouds to 105
cm - 3 in dense knots and filaments (Ingalls et ah, 1997). HLCs also have low
kinetic temperatures, around 10 K. These factors make our VCM region very
representative of conditions expected in HLCs. We present buckyball emission
estimates for two HLCs, one in the Northern hemisphere and one in the Southern
hemisphere.
Meyerdierks et al. (1990) presented a study of LVC (Low Velocity Cloud)
127, whose center is located at (l,b) = (127°, 20°), part of the North Celestial
Loop. They found this cloud to be a clumpy distribution of knots; the most
prominent knot is centered at (l,b) = (128.2,20.8), 7’ (0.2 pc) in diameter. From
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CO measurements, they have derived a carbon nuclei abundance of N(C) = 5.4 x
1016 cm-2 . This corresponds to a C /H ratio of 3 x 10-4, similar to the mean
Galactic ratio. If we assume a uniform distribution, the total number of carbon
atoms in the clump is therefore
Ctot = N{C) - 7r(r ) 2 = 5.4 x 1016 ■tt(0.1 • 3 x 1018) 2
=
4.86 x 1051atoms
(2.29)
(2.30)
Again, assuming 0.5% in C 2 0 , we get,
N c 20
=-■
=
x 4.86 x. 1051
(2.31)
1.2 x 1048molecules
(2.32)
The emission is therefore 0.61 mJy from this knot. An instrument like the Very
Small Array (VSA) may be able to detect this level of emission. The VSA (see
website: http://w w w .m rao.cam .ac.uk/telescopes/vsa/) is located at Tenerife and
can observe the sky between DEC of -7 to 63. It is designed to obser ve between
26 and 36 GHz, with a tem perature sensitivity of 7 fiK. Our buckyball emission
translates to a radio telescope main beam tem perature of 14 /j,K at 40 GHz. The
intensity of a source given in terms of the main beam temperature of a radio
telescope is given by,
= 2 6 5 T MB{ K ) e H a r c , m n )
Xl {cm)
where 9 is the source radius, and A is the wavelength.
A sample of HLCs was measured in the Southern hemisphere by Ingalls et al.
(1997). Of these, the cloud G225.3-66.3 (Keto & Myers, 1986) is a good southern
hemisphere candidate. Images of this cloud show that it is a thick arc, spanning
25’ in length, and the thickest region is 5’-10’ in width. The cloud is located
at (RA, DEC) = (02h 36m 45.6s, -19h 49m 01s), and we assume this cloud to
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have an approximately triangular structure with a length of 25’ and a width of
5’. Ingalls et al. (1997) derived a carbon column density of 2.5 x 1016 cm - 2 for
this cloud. The total number of carbon in this clump is,
Nc
= pc x A = 2.5 x 1016 • 0.5(0.7pe x 0.15pc)
=
1.18 x 1052 atoms,
(2.34)
(2.35)
and therefore 2.95 x 1048 buckyball molecules. Given our buckyball luminosity, we
expect this cloud to have a buckyball radiation of 1.5 x 10~2&erg s~l cm~~2 Hz'~l ,
or 1.5 mJy. In the Southern hemisphere, the Cosmic Background Imager (CBI,
see website http://w w w .astro.caltech.edu/ tjp /C B I/), located in the Andes, has a
sensitivity of 13 to 41 p K . Our buckyball emission has a main beam tem perature
of 14 p K across this source, if we assume an equivalent source beam of 10’, and
should be detectable by CBI.
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CHAPTER 3
S ign atu res o f C osm ic S trin g s in th e C osm ic
M icrow ave B ack grou n d
W ith the Galactic foreground removed, the intrinsic CMB signal contains a wealth
of information. Almost any process in the early universe will leave its trace on
the CMB. The photons permeate the universe and is a background to every
process th at has taken place since the era of Last Scattering, which occurred
about 300,000 years after the Big Bang. In this chapter, we describe how we
constructed and calibrated a digital filter to set detection limits for a class of
topological defects known as cosmic strings. Topological defects are predictions
of Grand Unification Theories which seek to unite all the forces we observe as
one. These theories predict physics behaviors at such high energies that it is
currently impossible for us to perform experiments to verify the theories. This
lack of experimental constraints has led to a proliferation of different Grand
Unified Theories, and we hope by producing limits on a necessary product of
these theories, we can start to differentiate between viable theories.
3.1
T o p o lo g ica l D e fe c ts
As we go back in time, the universe was smaller and hotter.
The extremely
hot and dense conditions of the very early universe represent the perfect lab-
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oratory to examine fundamental particles and interactions th at the subject of
particle physics seek to explain. The Standard Model of particle physics gives us
a good description of the physics of the universe up to the electroweak unifica­
tion, ~ 10- 1 0 seconds after the Big Bang. The interplay between particle physics
and cosmology has intensified in the recent years as cosmological observations
gained enough sophistication to produce evidence in support of particle physics
theories. For example, the observed deuterium abundance of the universe reflects
the weak interaction cross section which governs free neutron decay. The fraction
of neutrons which had not decayed became incorporated into atoms when deu­
terium formation became energetically favorable. Similarly, topological defects
are a possible observable consequence of certain types of particle physics theories.
This chapter is a description of theories that produce cosmic strings.
More conventional sources for topological defects can be found in condensed
m atter systems and liquid-crystal systems. Nematic liquid crystals, widely used
in digital displays, provide a good analogue for the formation of cosmological
topological defects. In fact, the correspondence is so good th at laboratory ex­
periments with nematic crystals are used to predict properties of cosmic string
networks (Ray & Srivastava, 2004). Liquid crystals are long organic compounds
which flow like liquids, but their flow contains directional information akin to
crystal lattice orientations in a solid. Nematic liquid crystals consists of rods of
crystals whose flow has the crystals’ long sides oriented parallel to each other.
The solid-liquid transition tem perature for these types of crystals range between
10 to 200 ° C.
As their phase goes from liquid to solid, the crystals lose their ability to flow
and the orientation of the crystal alignment in the original liquid flow is locked
into the solid. Different domains with different orientations may develop in the
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absence of an external aligning electric field. Boundaries of the different domains
form defects, where the parallel orientation of the crystals is disrupted, resulting
in a band of crystals with random orientations. These defects come from crystals
maintaining continuity through the different values of the degenerate ground
state. Hence, they are a consequence of the topology of the ground state. These
topological defects are stable because in order to bring either sides of the defect
into alignment, it is necessary to flip the orientation of all the crystals on one side
of the defect to match the other, which is impossible without an external energy
source.
The formation of cosmic topological defects rests on the same principle of
degenerate ground states for phase transitions. We begin with a simple model;
consider a general Lagrangian density
L = y 2 ( 9 ^ ') ( 9 ^ ) - V(<f).
(3.1)
where the </>represents a scalar field in one time and one spatial dimension, x, and
V{(f)) is the field potential. For now, we are not concerned about the details of
the transformation of the Lagrangian or with choosing any particular gauge. Let
us consider the simplest ground state degeneracy: a discrete, two-valued solution,
where the potential is
V 'M =
- x2)2;
(3-2)
A and r/ are some positive constants. This is the familiar double welled potential.
The Lagrangian with a potential given by Equation 3.2 leads to a field equation
solution th at has the form,
(x) = ?7 tanh
This solution is known as the c/>4 -kink, one of the simplest topological defects.
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The potential, V(4>), and the solution, </>(x), are both plotted in Figure 3.1,
where A = 1 and
77
= 1.7.
The <J)4-kink interpolates between the two ground state values, ±r). The field,
(f)(x), can be thought of as a mapping which maps x, the space dimension, to the
ground state, or, vacuum values ±rj. The kink around the origin is the result of
the field rising to the top of the central hump of the potential in order to preserve
continuity between the two vacuum values. The </>4-kink is stable because, in order
to remove the kink, we need to lift one of the sides of the solution (e.g. —oo to 0 )
over the potential hump. This process would cost an infinite amount of energy.
In 4 dimensions, </>4-kinks are 2-dimensional sheet-like defects known as domain
walls, which would occur at boundaries of regions with different vacuum values
of a discrete symmetry. However, the existence of domain walls is ruled out by
the observed homogeneity of the universe, because the large energy density of
domain walls cannot be dissipated - they would create large discontinuities in
the energy density of the universe.
This simple model illustrates two important ideas critical to understanding
topological defects. First, the mapping 4>(x), which maps the real spatial values
(—oo < x < oo) to the vacuum values (±?7 ), is what determines the characteristics
of the defect. Second, the stability of the defect is a consequence of the topology
of the mapping, and while transient defects exist (a cosmic string formed with
one side bounded to a monopole is possible, though unstable), topological defects
are generally stable objects with measurable energy density and size.
At energies much higher than the central hump, the potential of Equation 3.2
has only one effective solution at zero, as the field is energetic enough to traverse
the potential barrier of the central hump. However, as the energy in the field
drops, for example, due to an adiabatic expansion, the field falls into either of the
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3.0
5
2.0
5
0
5
0.0
3
2
0
2
3
Figure 3.1 Graph a): the double welled potential which leads to a </>4 -kink, with
A = 1 and
77
= 1.7. Graph b): the solution to the potential depicted in a). The
kink through the origin which interpolates between the two vacuum solutions is
the defect called the
0 4 -kink.
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two solutions, and it is less energetic than the central hump. The symmetry of the
held has been spontaneously broken. Even though the Lagrangian in Equation
3.1 is symmetric, the physical states of the vacuum given by the potential in
Equation 3.2 are not.
3.1.1
Sym m etries and Groups
The symmetries of a system can be characterized by the set of operations which
leaves the system invariant. Noether’s theorem states that this symmetries is the
result of the conservation of some quantity. For example, a Lagrangian which
has rotational symmetry is invariant under rotational operators; this symmetry
implies the conservation of angular momentum. We can perform rotations on
a vector x through an angle 0, in 2-dimensional Cartesian coordinates where
x —>■x ' , and x' =- Rgx. The matrix R.q is
cos 9
Rd
sin 9
(3.4)
- sin 9 cos 9
In 3 dimensions, rotations in any arbitrary direction can be written as a composite
of rotations about the three Euler angles: a , j3, 7 , corresponding to rotation about
the Euclidean x-, y-, and z-axes, with
/
RJa) =
1
0
0
0
cos a
sin a
0
\
^ cos (5 0 —sin f3 ^
Ry(P) =
—sin a cos a
0
^ sin (3 0
I
R z{j)
0
=
V
cos 7
sin 7
—sin 7
cos 7
0
0
0
1
0
(3.5)
cos /? ^
\
These rotations have the following properties: two successive rotations is a rota­
tion (closure); with a = /3 =
7
= 0, the operation produces no rotation (identity);
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three rotations, R a)
R c, produce the same result, whether they are grouped
(RaRb)Rc, or R a(Rt)Rc) (associativity); for any rotation operation R a, there ex­
ists an operation O ” 1 such that R ^ R a brings the system back to the original
orientation (inverse). These properties makes the set of rotational operations
a mathematical group. The matrices in Equation 3.5 are one representation of
the rotation group. Therefore, the symmetries of a system or a theory can be
described using groups.
Mathematically, the group of rotations in 3-D space is given by 5 0 (3 ), which
are represented by the group of 3x3 matrices which have determinant 1 and are
orthogonal (an orthogonal matrix A and its transpose, A T, produce the identity,
A TA = I). The 5 0 (3 ) group is closely related to the 5 0 (2 ) group, which governs
spin and associated spinor transforms. One representation of 5 0 (2 ) is given by
2x2 unitary (a unitary m atrix A and its transpose conjugate of the matrix, A \
produces the identity: A^A = I) matrices whth determinant 1 . The 5 0 (2 ) group
was first used to describe spin associated with conservation of angular momentum
and is now used to describe a variety of phenomenon. Heisenberg discovered that
5 0 (2 ) can be used to describe isospin: the interchange of protons and neutrons
in strong interactions as a result flavor independence. Glashow, Weinberg and
Salam used 5 0 (2 ) to describe their theory of weak interactions mediated by the
W and Z bosons.
Every group can be described by group generators, successive applications of
which generate all members of the group. For 5 0 (2 ), the group generators are
the Pauli matrices, given by,
To =
T =
v
1
o
( 0 -i
1
.
T3 =
I.
0
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(3.6)
These generators can also be given as commutation relations,
(3.7)
[T„,Tt] = 2itawTc
where eai,c is the permutation tensor, given by,
+ 1 if (afcc) is (1,2, 3), (2, 3,1), (3,1,2)
£abc={ - 1
0
if (ate) is (3, 2,1), (1,3, 2), (2,1,3)
(3.8)
>
otherwise
In this way, complex operations on the Lagrangian can be simplified and catego­
rized. Although it is not necessary to use group representation to discuss particle
physics, it is particularly useful because we can exploit the mathematical prop­
erties of groups. I will return to the S U (2 ) when discussing specific examples of
symmetry breaking related to cosmic strings.
Symmetry breaking reduces the number of invariant quantum mechanical op­
erators of the system. For example, the rotation matrix given above is the matrix
representation of the operation of rotating a vector through an angle
6
. If the
angle 9 is infinitesimally small such th at the state of the system is effectively
unchanged under this small rotation, then the Lagrangian of this system is said
to be rotational invariant. In other words, the mathematical group that describes
rotations is an invariant group of the system. Any process th at may happen to
the system can be described using an operator. Some familiar operators include
the aforementioned rotation, spatial translation, time translation, and charge
conjugation.
Before symmetry breaking, the set of invariant operations, g, on a Lagrangian
is defined to be the group G. The vacuum state, labeled |0), gives,
(OHO) = rjo
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(3.9)
where r]0 is a possible value for the vacuum state (e.g.
+77
for the double-welled
potential). After symmetry breaking, the now smaller set of invariant operations,
h, is defined by the group if, a subgroup of G,
(3.10)
H={ g(EG\D (g) <t >0 = <l>o}
where D(g) denote the matrix representation of the operations g, and
</>0
is the
vacuum state. All of the information about G and H is encoded in the quotient
space M,
M = G/H.
(3.11)
M is called the vacuum manifold. In general, we can define a topological space
for any group. The mathematical definition of a topology is the following: let A
be a set and T a family of subsets of X satisfying the following three conditions,
a) the set X and {0} (the singleton containing the element 0) belong to T
b) the union of any members of T is a member of T
c) the intersection of any finite family members of T is a member of T.
T is then a topology for A, and the ordered pair (A, T) is called a topological
space. Since both G and H are groups and have topologies, M is a topological
space, and I will take the liberty of calling it a manifold without proving that it ’s
locally Euclidean (locally flat, like the surface of the earth on human scales). The
topology of the vacuum manifold, M , determines the properties of the defect.
It is complicated to determine the characteristics of defects from vacuum man­
ifolds of real phase transitions. Fortunately, our task is simpler if we only want
to know whether a given theory will produce cosmic strings, without worrying
about the exact structure of the strings. We only need to know the homotopy
class of the vacuum manifold.
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3.1.2
Fundam ental H om otop y Group
Homotopy is a concept th at can be applied to general m athem atical objects: two
objects are homotopic if one can be continuously deformed into another. Thus
a circle and a square drawn on a flat sheet are homotopic: the circle can be
continuously deformed into the square without any of the points ever leaving the
flat sheet, such th at a small change in the input coordinates (of the flat sheet)
produces a small change in the path traced by the circle. The homotopy between
these two objects is a function of both the objects themselves and the manifold
on which the objects are defined. Mathematical objects such as a circle may also
be functions which define a path on its manifold. In appropriate cases, this path
can be considered a mapping, a function that takes values in the manifold and
assigns it different values, representing a transformation between spaces,
f i x ) . S ■> l\
(3.12)
The mapping, f(x ), takes values given in a space S, the source, and maps them
to values in a space T, the destination. The circular and the square paths may
both be mappings, and if the topology of their destination is appropriate, these
mappings may also be homotopic.
As a more rigorous example, consider two functions, / and g. On a manifold
M, let / be a mapping of the interval 0 < x < 1 onto M, where the mapping f ( x )
defines a closed loop, / ( 0) = / ( l ) = P , where P is a point on the manifold M.
The mapping f ( x ) defines a path on M. If g is also a mapping on M of the same
interval, 0 < x < 1, and a loop, g(0) = (/(l) = P , and if the path traced by f ( x )
can be continuously deformed to the path traced by g(x), f and g are homotopic.
This is illustrated in Figure 3.2A, where the loops / and g wind around a hole in
the manifold. The two loops / and g are homotopic. In contrast, loop c cannot
be deformed in any way to form / or g.
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g
g
Figure 3.2 Figure A: / (the solid lines) and g (the dashed lines) are homotopic
to each other, but not to c (the dotted lines). The grey filled circle is a hole in
the manifold. Figure B: / and g are freely homotopic, because a path c can be
constructed to wind around the defect separating them.
We define the product, / ■g (sometimes just written as f g ), as,
/ ( 2 x),
o < x < y2
/ ' 9(x) = .
g( 2 x -
1 ),
y2 < x <
1
(3.13)
.
In general, two mappings such as / and g do not need to share a base point
(P) in order to be homotopic. Mappings are called “freely homotcpic” if they
are homotopic but do not share a common base point. The two loops / and g
in Figure 3.2B wind around the same hole in the manifold; however, with the
addition of a second hole in the manifold near the point P , there is no way to
continuously deform / to form g. Yet we know th a t / and g trace the same hole
in the manifold. In this case, the two mappings / and g can be said to be freely
homotopic, because we can construct a third mapping, c, such that,
cfc~l ~ g
(3.14)
where the loop composition c/cT 1 takes / around the first hole in the manifold,
making it conjugate (equivalent) to the loop g.
From here on, we make no
distinctions between homotopic and freely homotopic. In Figure 3.2B, we have
shown / and g to originate from the same point for clarity. In principle, / and g
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are freely homotopic as long as g starts on any point of c. A homotopic group,
therefore, is the group of all functions which are the same up to homotopy; / and
all loops which can be continuously deformed to / form a homotopy group.
Figure 3.2 showed th at homotopy between two functions is dependent on the
topology in which they reside. If we disregard the mappings, the homotopy of a
topology is a characteristic of the topology itself. Mathematically, the homotopic
classification of a topology is based on mappings of the hypersphere, S n: the first
homotopic class is based on homotopic mappings to the circle, S'1; the second
homotopic class is defined based on homotopies to the sphere, S 2; and so on.
They are given designations ivn(T) where n denotes the order of the class, S n,
and T denotes the space to which this classification belongs.
To clarify exactly how this classification works, we start with the simple case
of a continuous, Euclidean space. Pick a point P in this space and consider all
loops starting and ending at P. There is essentially only 1 loop that can be
drawn, up to homotopy. This loop, in fact, is contractible to P. We therefore say
that this space has a trivial first homotopy group. Every simply connected space,
Tsc has a trivial vri(Tsc) group; 7ti(M ) is called the fundamental group. This is
because on all simply connected spaces, loops are contractible to a point. Note,
however, that Tsc may have non trivial 7r2 (Tsc), as in the case of the surface of a
hollow sphere. The usual group identities apply to homotopy groups: members
of this group include the composite function / • g; the inverse, / - 1 (:z), equivalent
to traversing f ( x ) in the reverse direction; and the identity, / , is defined as all
loops contractible to the point P.
Consider the loop group and the manifold given in Figure 3.2B; there are two
holes in the manifold, and since the rest of the manifold is continuous, drawing
loops on this manifold is equivalent to drawing circles around the holes. To start
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with, there are three choices, a loop can wind around no holes, one hole, or two
holes. A given path may wind around the first hole a number of times before
winding around the second hole, and so on. There is an infinite number of possible
configurations, and, clearly, this manifold has a non-trivial fundamental group.
Similarly, if a manifold contains non-contractible points, they possess a non­
trivial 7r2(T) homotopy group because all non-contractible points are equivalent
to traversing the surface of a sphere, and so on for the higher dimensions.
We now look at a more realistic potential for cosmic strings: the Mexican hat
potential. It is the 3-dimensional analog of the double-welled potential given in
Equation 3.2. The functional form remains the same, except now <p is <p(x, y, z, t),
a function in four-space.
The vacuum state of this potential is continuously
degenerate, defined by the circle of minima at \<f)\ =
77.
The topology of the
ground state of the Mexican hat potential is isomorphic to the unit circle, S'1. as
the rest of the space is energetically forbidden. If we traverse this minima circle,
the loops of our path are defined only by the number of times we go around
the circle, since partial loops are contractible to the loop origin. Therefore, the
only unique characterization of paths in this space is the winding number, given
by integers, of how many times around the circle we pass the loop origin. For
this vacuum manifold, 7r1 (S’1) is isomorphic to the group of integers, Z. Note
that while the vacuum state degeneracy is continuous, the homotopy group of
the vacuum manifold isn’t. Each possible value of the vacuum state corresponds
to an integer. There may be an infinite number of integers corresponding to an
infinite number of states, each state is separate from each other, as the group
of integers is discontinuous. This is the reason why the fundamental group is
im portant for cosmic strings: this class of defects arises from the non-trivial
winding around holes in the vacuum manifold, a result of the topology of G /H .
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Likewise, different potentials yield different topological spaces for the vacuum
state, and we give a summary of the isomorphisms for the fundamental group
as found in Vilenkin k, Shellard (1994). In general, for compact connected Lie
groups G (Lie group operations are differentiable), the fundamental group 7rx(G)
gives,
Z
tti(G)^<
G = U(n)(n > 1), 5 0 (2 )
Z2 G = SO(n)(n > 3)
I
(3.15)
others
where Z 2 denotes the discrete group consisting of two elements from Z, and I
denotes the identity group, sometimes denoted as {1}. The differences between
the double-welled potential and the Mexican hat potential is a good illustration
of the differences between two homotopy groups. In the first case, the manifold
M consists of two distinct and disconnected components, ± 77. This type of topol­
ogy belongs to the homotopy group 7r0(M). Manifolds with non-trivial 7t0(M)
produced domain walls. The Mexican hat potential, on the other hand, has a
non-simply connected manifold with holes around which winding numbers for a
defect may develop. This corresponds to a 7Ti(M) homotopy group and there­
fore produced cosmic strings. Higher dimensions of homotopy groups produce
monopoles (non-contractible spheres for 7r2) and textures (non-trivial mapping of
5 3 where 7t3(M) / I).
A mathematical theorem allows us to further simplify the condition for a
cosmic string topology: Let G be a connected Lie group, with subgroup H ,
and H may be composed of disconnected parts. We designate H 0 to be the
part containing the identity, I, in which case G / H 0 is then simply connected.
The quotient group tto(H) = H / H 0 is isomorphic to the fundamental group of
7ti(G/H )
(James 1984, Proposition 5.22 and 5.23). By this theorem, if H is
disconnected, then there exists a non-trivial 7ti(G /H ), and one would expect
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cosmic strings to form. We can see the validity of this theorem by considering
the reverse: if H is simply connected, and since H is a group, which by definition
contains the identity element, all loops within H can be contracted to the identity.
However, if H is disconnected, then there exists some loops g(t) in H th at cannot
be contracted to the identity. To be more precise,
M = 7ii(G/H) = t v0( H ) .
(3.16)
The existence of cosmic strings boils down to whether the unbroken symmetry
group H is connected. If H is unconnected, we expect the formation of cosmic
strings. In fact, the homotopic group to which. H belongs delineates the type
of topological defects th at are produced in a theory with the given phase transi­
tion. It is really the topology of the group which describes the vacuum state th at
determines the defect characteristic, the most obvious of which is the defect’s
dimension. For this reason, objects such as cosmic strings are called topologi­
cal defects. For the Mexican hat potential, the vacuum manifold is n o n - s i m p l y
connected because Z is discrete. We expect cosmic strings to form under phase
transitions described by the Mexican hat potential.
3.1.3
C osm ic Strings and G U T s
I am particularly interested in using cosmic strings as a possible probe of Grand
Unification Theories (GUT). GUTs are complete theories describing the funda­
mental forces (except gravity) as the result of breaking down one unifying force.
The motivation for GUTs has its roots in the successes of the electroweak union.
The Glashow-Weinberg-Salam theory (Weinberg, 1967) showed that electromag­
netism and the weak force are part the same force, known as the electroweak
force, at energies slightly larger than the masses of their force mediators. The
mass of the W boson is 81.4 GeV, the mass of the Z boson is 91.1 GeV and the
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photon is massless (Hagiwara et ah, 2002). The mass difference of the media­
tors for the two forces is responsible for the spontaneous symmetry breaking th at
occurred when the tem perature of the universe dropped below ~100 GeV. The
electroweak force belong to the group S U ( 2 ) x U(l), a union of the two groups
formed by the weak force and electromagnetism.
Efforts to unite the electroweak with the strong force have had more limited
success due to the higher energies involved and therefore the lack of experimental
constraints. The strong force, which proceed via Quantum Chromodynamics, is
mediated by eight gluons of three colors: red, green and blue, and has a 517(3)
symmetry. The current Standard Model is therefore a 517(3) x SU(2\ x U( 1)
theory. GUTs (which are not part of the Standard Model) seek to explain this
517(3) x 517(2) x 1/(1) theory as the result of spontaneous symmetry breaking
of some fundamental group G. Candidates for G include: the simplest minimal
group, 517(5), which has 24 mediators representing the
8
(QCD)x 3 (QED)x 1
(E-M) mediators. For many reasons, including esthetics, 50(10) is more popular.
The overall scheme of GUT symmetry breaking is the following,
G
H
... ->• 517(3) x 517(2) x 17(1)
517(3) x 17(1)
(3.17)
where the arrows indicate a phase transition, and H indicates a possible inter­
mediate state between the GUT symmetry group G and the Standard Model.
If we believe that a unifying GUT can be constructed, at each point where
a phase transition occurs, there is the possibility of forming topological defects.
In the absence of other evidence, the existence of topological defects can give us
limits on the types of phase transitions that are allowed in the early universe.
As an example given in Vilenkin & Shellard (1994), consider the breakdown of
an 517(2) symmetry to Z 2 (the subscript
2
denotes two elements of Z), via the
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following mechanism,
SU(2)
{/(l)
Z2,
(3.18)
where the superscript 3 indicates the phase transition is mediated by a triplet. We
break the 517(2) symmetry with a potential similar to the Mexican hat potential,
given by
V((p,Tp) = ^ ( d ) 2 - r / l ) 2 +
■4>)2
(3.19)
where <f> and ijj are in C 3, the three dimensional complex space.
The extra
7J)2
+
quadratic term is motivated by the fact th at we are using two triplets to break
the symmetries of the potential. W ith respect to the first triplet, </>, the ground
state occurs when
</>2
= rj^. The orientation of this triplet can be in any direction,
and for simplicity, we take it to lie along the z-axis,
(3.20)
0
V^
Rotations in the z-direction, given by the operator R = exp(—iOT:i) leaves 4>q
invariant. Generators of rotations in the z-direction, T3, is given by,
T, =
i
0
0
0
0
0
(3.21)
There remains an unbroken subgroup, 17(1). The vacuum manifold is therefore,
G / H = SU(2)/U(l) ~ S 2
(3.22)
We see that no cosmic strings form here, because the leftover manifold, 5 2 is a
simply-connected sphere.
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This remaining symmetry can be broken again by a second triplet. We set
the second scalar field, ip, orthogonal to <fi in the y-direction,
i ’o -
w
(3.23)
The remaining 17(1) symmetry is now broken. However, if we look closely, there
are two elements, 1 and -1, th at are mapped to the identity I. This means that
there is a little group, H = Z 2 (where Z indicates integers) left over. Z 2 contains
only two elements, 1,-1, and is disconnected. Cosmic strings will form during
this phase transition.
Cosmic strings are created as either infinite filaments which span the horizon,
or as closed loops. Infinite cosmic strings leave tell-tale gravitational signatures.
The metric of a long string in cylindrical coordinates, to linear order in mass per
unit length of the string, yu, is given by,
ds2 = dt2 — dzA — (1 —h){dr2 + r2d02)
(3.24)
where h = 87rGpTn(r/r0), r 0 is a constant of integration, and r = (x2 + y2)
where x and y are the Cartesian coordinates for a string lying along the 2 -axis.
In general, yu is expressed as a dimensionless quantity, G/i, and,
(3.25)
where m pi is the Planck mass, and
is the string symmetry breaking scale
Numerical simulations of cosmic string networks have shown that string evolution
is self-similar and approaches a scaling solution (see, e.g. Bennett & Bouchet
1990). The correlation length of the scaling solution, £, is set at string symmetry
breaking. Strings have length £ in a volume defined by £3. The constant £ grows
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proportional to time, £ =
factor of the universe.
7
where 7 ^ is a constant dependent on the expansion
Therefore the energy density of the string is roughly
2The electroweak symmetry breaking scale is around 100 GeV, the energy scale
defined by the electroweak mediators, the W and Z bosons whose masses were
given earlier. An electroweak string would have G/i
would have
77
~ 1016GeV, and G/j,
10 34; a GUT scale string
10-6 . Therefore, knowing the masses of the
strings can give us the energy scale of the phase transition th at created them.
Alternatively, the absence of strings would give an upper limit to the energy
of possible phase transitions. Strings are created with relativistic velocities. In
order for strings to not dominate the energy density of the universe, numerical
simulations have shown th at strings could intersect and break off loops in a
process called intercommutation. The amount of intercommutation can be tuned
so that infinite strings have many intersections. Closed loops can wiggle and
emit gravitational radiation, which causes the loop radii to decrease. Eventually,
within the lifetime of the universe, string loop radii can reach zero. This is their
primary energy dissipation mechanism (see e.g. recent numerical simulations,
Moore, Shellard, & Martins 2002) th at prevents cosmic strings from dominating
the energy-density of the universe.
Cosmic strings grow along with the horizon size, and because they are rela­
tivistic, string wakes will span the horizon. At last scattering (LS), the horizon
has angular size 6h given by,
(3.26)
where
zls
is the redshift of LS. Since the CMB freezes gravitational signatures at
LS, we expect to find most of the signatures of cosmic strings in the CMB at the
horizon scale of LS. However, different numerical simulations give different string
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densities and velocities and there is little agreement on the size and number of
strings we should see in the sky. The one fact that all simulations agree on is
th at string wakes leave edge-like signatures in the CMB.
We emphasize th a t cosmic strings are necessary products of phase transitions
whose unbroken symm etry group, H, is disconnected. The detection of a string
would certainly be very interesting, but the absence of these defects is equally
telling. Were a Z 2 kind of topology to arise from the previous discussion, one
defect would be created per causal horizon This leads to a possibility of ~ 13000
cosmic strings in the present day sky. Even if only 0.1 % of these strings survived,
there should still be 13 strings in the sky. Given our 2/3 sky coverage (see Section
3), we have the possibility of seeing 8 strings in the WMAP sky. Therefore, a non­
detection on our part is a genuine limit. The complete absence of the detection of
strings would be difficult to reconcile for theories containing disconnected vacuum
topologies.
3.2
W M AP
Photons produced in the early universe, when the ambient tem perature was high
enough to keep hydrogen ionized, remained in equilibrium with the baryons and
traced baryon density.
At a red-shift of z ~ 1100, electrons and protons re­
combined and the universe became transparent to photons. These photons have
redshifted as the universe expanded and we now observe them as the 3K Cosmic
Microwave Background. The photons retained the gravitational signatures of the
early universe density distribution.
The primary effect of a cosmic string on the CMB is to create temperature
anisotropies due to the wake of the moving string. Along a particular line of
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sight, the tem perature anisotropy, 5 T / T induced across a string moving in the
plane of the sky is given by,
A = SxG vff j.
where
7
(3.27)
is the Lorentz factor and (3 = v/ c is the velocity of the string, G is the
gravitational constant, and n is the string linear mass density..
The Wilkinson Microwave Anisotropy Probe (WMAP) is the latest NASA
satellite to measure the tem perature anisotropies of the CMB (see the LAMBDA
website for a complete list of publications and public release data products:
http://lam bda.gsfc.nasa.gov/product/m ap). WMAP has two back-to-back Gre­
gorian telescopes which observe two patches of the sky separated by 141°. A
set of differencing assemblies obtains the tem perature difference between the two
patches. The final data product of WMAP is the CMB temperature anisotropy
of the each pixel in the sky, except where there are microwave sources and the
pixels are masked. These masked pixels include most of the Galactic plane, the
Galactic bulge, and some scattered sources off the Galactic plane.
WMAP was launched on June 30, 2001, and it arrived at its L2 orbit on Oct.
1, 2001. The L2 is the second Lagrange point, a quasi-stable point where the
gravitational pull of the earth balances the gravitational pull of the sun to. The
first year data release in February, 2003, contained data taken by WMAP from
Aug. 10, 2001 to Aug. 9th,
2002
. WMAP is still taking data and is expected
to last at least until 2005. The following work is based on the first year data.
As more data are released in the future, the sensitivity of this project will be
improved as the pixel noise is reduced by repeated observations.
WMAP observes the sky at five frequencies, from 23 to 94 GHz. Relevant
WMAP characteristics are tabulated in Table 3.1. The data products released
by the WMAP science team include the thermodynamic temperatures of each
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Table 3.1.
Relevant WMAP Characteristics
Frequency
Resolution
Wavelength
Sensitivity
GHz
FWHM, arcmin
mm
/rK, 0.3° x 0.3°
23
56
13.6
35
33
41
10.0
35
41
32
7.5
35
61
21
5.0
35
94
13
3.3
35
WMAP pixel at all five frequencies. I use the data from the three highest fre­
quency bands (Q, V, W) in our analysis. The reason for using the highest three
frequencies was to eliminate as much Galactic contamination as possible, which
the lower two frequencies are more sensitive to.
I first present the theoretical limits on the mass of cosmic strings based on
WMAP sensitivity, following Vilenkin & Shellard (1994). An ensemble of strings
will have strings moving in different directions with different velocities, so it is
more useful to consider the root mean squared (rms) tem perature fluctuation
from a group of strings. For this, I need to consider the fact th at cosmic strings
grew with the horizon size, so that one may have different values for different
redshifts. W ith the results of numerical simulations, we can average over string
velocities between redshift z and 2z. The rms fluctuation induced in the CMB
by string wake, between z and 2z is shown to be,
5oT
- j r = 6Gii.
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(3.28)
and is independent of z. Therefore, I can use this result generally. In real obser­
vations, one cannot see CMB signatures from before the LS cut-off. I therefore
correct for the fraction of strings observable since LS,
(3.29)
RMS
For
zls
= 103, this gives an rms tem perature fluctuation from an ensemble of
strings of 19G/j.
WMAP has a maximum beam resolution of full width at half maximum of uj =
13 arc-minutes, so th at the maximum redshift observable for WMAP is zmax ~
(4uj)~2. I replace zLS with zmax in Equation 3.29. WMAP RMS tem perature
sensitivity to cosmic strings is 20Gn. As a comparison, the COBE satellite had
a field of view of 7 degrees, which resulted in an expected A T /T of 8.6Gji. The
observed fluctuations for COBE was,
(3.30)
RMS
which resulted in a limit for cosmic strings with Gji < 1.3 x 10 6. For a similar
observed RMS tem perature fluctuation, WMAP improves the limit for the mass
per unit length of cosmic strings to Gji < 5.3 x 10“ 7. This is the limit for an
ensemble of strings; if we allow the existence of very few cosmic strings, these
rare objects can be much heavier than this limit while not having an appreciable
effect on the CMB. This paper presents a method of searching for the limits of
individual cosmic strings.
3.3
S tr in g Search: T h e E d g efin d er
For my string search, I used the WMAP map of tem perature differences, A T /T ,
of the full sky. At 13 arc-minute resolution, the WMAP sky is divided into 12 x 49
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pixels. WMAP uses the Hierarchical Equal Area isoLatitude Pixelisation of the
sphere (HEALPix: http://www.eso.org/science/healpix, Gorski et al 1999) to
pixelize the sky. We have designed a digital filter for the WMAP data in search
of signatures of filamentous string wakes, and we named this filter the Edgefinder.
The Edgefinder took an input pixel and defined a window with radius RA D . Each
pixel within this circular window was assigned an [x, y\ pair, with the input pixel
at [0,0]. Pixel displacements from the origin, [0,0], are calculated and each pixel
within R A D is assigned a normalized (to R A D ) x displacement and a normalized
y displacement. The y-axis was defined to be parallel to the line connecting the
North and South Galactic poles. Our filter window was small enough that we
could consider the sky inside to be flat. For each pixel within the window, the
Edgefinder multiplied the filter value, F ( x , y ), by the pixel tem perature at [x, y\
and stored the sum of this product for all pixels in the window at the position
of the input pixel. Therefore, the output of this digital filter was a map where
the value at each pixel was a sum of the effects of the filter on the surrounding
pixels. I called this output the Edgefinder value map, EV for short.
For the WMAP data and associated simulations, the Edgefinder had an
R A D = 1° window in the sky, which corresponded to the size of the horizon
at LS. This window had a radius of 18 pixels and usually contained 260 pixels
in total. The filter value, F ( x , y ), was designed to pick out edges in the sky
aligned with the y-axis of the filter. The specific values of each of the filter were
generated separately, because pixel centers were slightly offset from each other
depending on the pixel latitude, and the edges of the filter were ragged due to
the diamond grid of the HEALPix scheme. This, coupled with the fact th at I
only have 260 pixels in the window, meant that the sum of the filter values didn’t
always sum to a perfect zero. We therefore generated the filter values twice at
each pixel: the first time to collect the excess, a, for each window, and the second
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time to distribute a evenly among the pixels. This forced filter values in the R A D
window to sum to zero exactly.
The filter values, F ( x , y ), were given by,
F(x, y) = N ( x =f A) x exp
)
(3‘31)
the minus sign applied to pixels with x < 0 and plus for x > 0, A was the
normalized height of the filter, and N was the total normalization factor for the
filter, discussed later. The exponential smoothing function ensured th at the filter
is smoothed and compact.
The normalized radius, r n., is rn = yfx2+ y 2. The
filterfunction is designedto be insensitive to a constant value, a gradient,
and to
be spherically symmetric except for the edge detection. Insensitivity to a constant
demanded
J
F(x, y)dxdy = 0.
(3.32)
Insensitivity to a gradient demanded
J
xF (x , y)dxdy = 0,
(3.33)
and the value of A was adjusted to ensure this. The Edgefinder is depicted in
Figure 3.3 as a shaded surface plot with the z axis representing the filter value,
F(x, y) . In addition, a plot of the cross section of the filter, is also shown.
The filter could be rotated in the sky to detect strings of different position
angles with respect to the North-South Galactic alignment. This was achieved
by rotating through an angle a by altering the [x,y] values of a pixel to [x',y']
by the following transformation,
x, \
y
X
y
\ ( cos a —sin a
y s in a
(3.34)
cos a
The filter values are then generated with x' in place of x and y' in place of y. In
most simulations, I ran the filter at 20 different a values equally spaced between
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side view
- 0 .0 2
0 .0 1
0 .0 0
0 .0 1
0 .0 2
Figure 3.3 A 3-D representation of the Edgefinder. The z-axis represent filter
values, and the x and y axis are the pixel numbers. The right bottom inset is a
plot of a slice along the y=0 plane of the filter, with the axis in units of pixels.
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0 and 180 degrees. This represented a shift through one pixel at the edge of the
filter window, and should pick out edges aligned in all directions.
The characteristic Edgefinder behavior around a string horizon is depicted in
Figure 3.4. In this figure, the top picture is a simulated input temperature map,
where the dark disk is the simulated string horizon. The disk edges are fuzzy
due to the noise added to the map (as a real string in the sky would be seen
by WMAP). The bottom picture is the Edgefinder response to the input map.
The Edgefinder was oriented North to South, and produced a “hot” signal when
encountering a rising edge, and a “cool” signal when encountering a dropping
edge. If I rotated the filter window by 180°, I would get the opposite signal,
meaning that the sign of the EVs was not im portant, so I used absolute values
in my statistics. Running through 20 a angles from 0 to 180° really represented
running through 40 a angles from 0 to 360°.
Visible on the same figure are pale circular disks which indicate masked pix­
els in the WMAP data, usually where there were strong foreground microwave
sources. These regions were not used in my simulations. I used the combined KpO
and Kp2 masks (see Bennett et al 2003 for an explanation of the masks) which
masked 1109593 out of 3145728 pixels (35.7%). If strings were partially blocked
by masked pixels, the Edgefinder can still detect them, but at a reduced sensitiv­
ity. I ran tests where strings were partially blocked by a mask, with 5 different
exposure levels from 100% to 18%, where 100% indicated an unblocked string.
These results are tabulated in Table 3.2. For strings which are more than 50% ex­
posed, there were no differences in the peak EV compared to an unblocked string.
For strings th at are less than 50% exposed, the peak EV dropped off rapidly as
the string centers moved behind the mask, but were still detectable, their EVs
dropping to around 15% of an unblocked string. This test indicated that strings
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with centers located outside of a masked area will be picked up normally by the
Edgefinder, and therefore the masks are not blocking more sky than their actual
area.
Masked
Figure 3.4 Edgefinder values around a string horizon. The Edgefinder is oriented
North (up) to South (down). The light blue circles are masked pixels in the
WMAP data due to foreground sources.
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Table 3.2.
Partially Exposed Strings
Exposure %
Normalized Max EV
Non-zero Pixels
100
0.961
575
73.6
0.956
339
48.7
0.569
210
32.6
0.160
135
18.0
0.143
71
The Edgefinder filter values, F (x,y), were calibrated to give a response of
Edgefinder value, EV, of 1 for a 1 mK input signal. For example, if the input CMB
map had all its southern hemisphere pixels at 1 mK, and northern hemisphere
pixels at 2 mK, the Edgefinder would return EV = 1 for equatorial pixels, and
EV — 0 at other pixels. The normalization constant for the R A D = 1° WMAP
Edgefinder was N = 0.33. Due to fluctuations in the normalization and the excess
value collection of the filter, the actual response varied between 0.97 and 1.08,
but averaged to 1.0 across the equator. Strictly speaking, the EV is unit-less, but
for the sake of clarity, we will sometimes give it units of mK. Unless specifically
mentioned, all tem perature units in this paper are in milliKelvin.
I created a set of calibrators to verify the gain of the Edgefinder. The cali­
brating set contained edges, or string horizons, of various tem peratures ranging
from A T /T = 1.1 mK to A T /T = 1 nK. The results of the calibration are in
Table 3.3. In most cases, the EV was the same magnitude as the input edge to
within 2%; we are therefore confident that the Edgefinder had a linear response
to the input string horizon over the range of pertinent input temperatures.
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Table 3.3.
Edgefinder Gain Calibration
n -i a
1s
EV G a i n | Q=0b
EV G a i n | Q=7r/ 2
1.000000
0.9567
0.9956
0.100000
0.9985
0.9732
0.010000
1.0800
0.9085
0.005000
0.8456
0.9630
0.001000
0.7639
1.0150
0.000500
0.8046
0.9842
0.000100
0.3296
0.8974
0.000050
0.8974
0.9794
0.000010
0.9559
1.0200
0.000001
1.0440
1.0910
aT, is the input cosmic string horizon tem ­
perature
bEV Gain is —EV /TS— , here quoted for 2
different a angles.
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o
CD
o
>
a —0
CD
a —
7v/1 0
O
a =ix/b
a = 3-n / 1 0
a = 2 7 r/6
Z3
a = 7T/ 2
on
-2
0
100
1000
Figure 3.5 The response function of the Edgefinder Filter. Note the peak at
I ~ 200, around where the first Doppler peak of the CMB anisotropy angular
power spectrum occurs. The different lines correspond to different a angles of
the orientation of the filter with respect to the north-south Galactic axis.
The size of the Edgefinder was chosen to match the most likely string size that
we may be able to find. For WMAP, this was a problem since this scale was also
where the CMB Gaussian anisotropy signals were the strongest. The response of
the filter to I values from 1 to 1000 is plotted in Figure 3.5 for representative a
angles. The largest filter response was around I = 200; the WMAP data indicated
th at the first Doppler peak of the CMB anisotropy angular power spectrum is
located at I ~ 220. This was undesirable as we wanted to insensitive to as much
much of the Gaussian signal as possible so we could focus on the non-Gaussian
signals.
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One easy solution would be to make the filter smaller. However, this is not
feasible with WMAP data. At R A D = 1°, there were 273 pixels in the filter; at
R A D = 0.5°, there would be only 69 pixels in the filter, spanning less than 10
pixels. This resolution was too coarse to run the Edgefinder. For now, I report
results of our good enough filter while we work on ways to eliminate the response
at i ~ 200. One certain method is to wait until PLANCK data is available, and
with the improved resolution, we will get 270 pixels in the filter at R A D = 0.5.
This is discussed in Section 6.
3.3.1
Sim ulated M aps
I knew from the calibrator set described in the previous section th at the Edgefinder
had a linear response to the input edge.
1 next needed to find out how the
Edgefinder responded to noisy maps like the ones from WMAP. 1 anticipated
th at the Edgefinder would not be able to detect strings when the string hori­
zon signals became swamped with noise. In order to find this limit, I produced
simulated maps to quantify the behavior of the Edgefinder. Once I understood
how the Edgefinder responded to maps containing strings of known magnitudes,
I could then set detection limits of the Edgefinder for the WMAP data. Because
they are so vital, I describe in detail how we generated the simulated maps:
1.
Coefficients, Ci, of the CMB angular power spectrum were generated
by CMBFAST (see webpage http://www.cm bfast.org) using the cosmological
constants derived from the WMAP experiment, a ACDM cosmology, given by:
Zmax = 1500, Kmax = 3000, n b = 0.044, Qc = 0.218, Qx = 0.738, H 0 = 71.6,
T cmb = 2.725, Yue = 0.24, N^^massiess) = 3.04,
tlss
= 0.099, and n = 0.955
(Bennett et al, 2003).
2. I used HEALPix associated software SYNFAST to generate a CMB map
81
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which matched the Ct generated in Step 1. SYNFAST generates random Gaussian
fields on a sphere based on the input power spectrum (the CVs).
3. I added string horizons at various temperatures, Ts, into the map by adding
the value Ts to circular regions within a string radius R s.
4.
I used HEALPix associated software SMOOTHING to convolve three
beams with the map made in step 3. A W-Band map was made by convolving
a 13 arc-minute Gaussian beam with the cpm’s of the tem perature map created
at Step 3, and the new <pm’s were used in SYNFAST to generate a new map.
A V-Band map was made from the convolution of a 21 arc-minute beam, and
a Q-Band map from a 32 arc-minute beam. For each map generated in Step 3,
three maps convolved with beams appropriate for the three WMAP bands were,
made.
5. The 3, W-, V-, and Q-Band, maps were averaged to get the final SMOOTHed
map. Noise was then added to the map with the following prescription: the an
for the noise of each pixel was generated by combining the noise characteristics
of the 3 WMAP bands,
\JvQ + °V + a W
3
where
oq
(3.35)
is the pixel noise in the Q band,
where N q ^
s
”Q
=
av
=
-p^L=
(3.37)
aw
=
- ,
\j Nw,obs
(3.38)
'obs
(3 36)
r)Q\
is the number of times the pixel had been observed by the WMAP
satellite in the Q band; ctq^ = 2.211, aVt0 = 3.112, ovyo = 6.498 are the noise
weights given by the WMAP team. The noise values were generated by a Gaus-
82
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sian random number generator with a = an. The reason for this noise prescrip­
tion, and for the convolution with three beams in Step 4, was th at the data I fed
to the Edgefinder are a composite of data from the three WMAP bands Q, V and
W. Due to the SMOOTHING and the added noise, the final string horizon tem ­
perature was at a slightly different tem perature than the initial Ts. I sometimes
also report, for comparison, the average tem perature of the final input horizon,
•which I designated Tf.
3.3.2
Edgefinder Lim its
In all, 82 input simulated maps were made according to the prescription given
in Section 4.1. Out of these, 15 of the input maps were baseline maps, where no
string horizons were inserted. I called the EV output of these maps the No-String
sets. The maximum EV of the No-String sets indicated the maximum EV due to
background signals (i.e., not from the strings). In other words, the No-String sets
produced the noise limit; I considered anything above this limit to be signal. Both
the CMBFAST generated CMB signal and noise are basically Gaussian random
variables, so I needed to have a range of No-String sets in order to ensure I have
proper coverage of the possible noise values. To this end, I created 15 No-String
input maps.
I then created the 67 “Stringy” maps, which contained different number of
string horizons with various sizes and temperatures. The input string horizon
temperatures ranged from Ts = 1.0 mK to 0.0001 mK, the inserted string radius,
R s, ranged from 1.0 to 4.0 degrees. Most maps were made with one inserted
string, but one map contained as many as 60. The Stringy maps are further
divided into single and Multi maps, where Multi maps had more than 1 inserted
string horizons. There are 16 Multi maps.
83
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0.8
0.6
£
0
No —String Largest
Average No —String
No —Strina Smallest
.4
X
o
0.2
A No —String Values
□ Multi String Values
.0
0” 6
10
-5
10
°
-4
.-3
LOG Input String Te m pe r a t u r e (K)
Figure 3.6 Plot of the input string tem perature Ty vs. the maximum EV of the
set.
84
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Comparison between the maximum EV (Max EV) of the Stringy and NoString sets will yield the input string value th at gives a Max EV above the
No-String limit. The No-String limit is given by the largest value of the Max
EVs of the No-String sets. The average of the maxima of the 15 No-String sets
was EV = 0.242 mK; the largest of the 15 sets was EV = 0.269 mK. I plotted
the Max EV of all single Stringy sets against the input string temperature, Ts in
Figure 3.6. The No-String limits are plotted as the dashed and dotted horizontal
lines. A blow up of the region around the No-String limit is in Figure 3.7. The
data indicates th at an input string horizon of 0.345 mK and cooler yielded similar
Max EVs as the No-String sets. I can consider T =-- 0.354 mK as one of the limits
to the sensitivity of the Edgefinder. If I took this limit to be wakes formed by
strings traveling at c/y/2 (the mean absolute velocity of strings from numerical
simulations), according to Equation 3.27,1 would have limits for the cosmic string
at G\x £ 1.37 x 10“ 5.
This is the crudest but most robust method of obtaining the sensitivity limit
of the Edgefinder. The problem posed by the Edgefinder is th at I am looking for
very small non-Gaussian features in a largely Gaussian data set. For an input
string of R s = 2°, the number of pixels whose filter window contained the edge
constituted 0.0087% of the total pixels. The binning of the EV set data resulted in
the non-Gaussian signatures occurring in the edges of the histogram. To improve
upon the Max EV limits, I needed a set of descriptive statistics that could pick
out small non-Gaussian signals at the outer edges of a Gaussian function. I found
the Edgeworth Series to be suitable for my purpose.
85
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0.3 5
N o - S t r i n g Largest
Average N o - S t r i n g
0.30
0.20
A No —String Values
□ Multi String Value
10
6
10
5
10
-4
10
LOG Input String T e m pe ra tu re (K)
3
Figure 3.7 Detail of the plot of the input string tem perature Tf vs. the maximum
EV. Also plotted are the multi-string set data as well as limits from the No-String
sets. The stars are the Max EV for each simulated map containing a string of
strength shown on the x-axis; the square indicate simulated maps containing
multiple strings; the triangles indicate maps containing no inserted strings.
3.3.3
Edgefinder Lim its from E dgew orth C oefficients
The Edgeworth Series uses the Method of Moments to obtain coefficients of the
terms in the series. It has been proposed as a tool for picking out non-Gaussian
signatures in the CMB by various authors (e.g. Cayon et al. 2003). Here, we will
only illustrate the derivation of the Edgeworth Series to introduce the components
necessary to calculate the coefficient. A full derivation can be found in Kendall
(1987), discussions relevant to astrophysics can be found in e.g. Juszkiewicz et
al. (1995) and Blinnikov & Moessner (1998).
86
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The Edgeworth series is an asymptotic expansion of a distribution using the
normalized Gaussian function and its derivatives as the basis set. Coefficients
of series terms give how well the given distribution approximates a Gaussian.
Collection of coefficients of like terms in the Gram-Charlier series results in the
Edgeworth series. Specifically, the Edgeworth series is a linear combination of
the Chebyshev-Hermite polynomials using cumulants of the distribution as coef­
ficients of the terms in the series. In this section, we use the abbreviation D r to
mean drj d x r.
Before beginning the derivation, there are a few things th at need to be intro­
duced. Let a(x) be the Gaussian distribution,
a(x)
= j— e~^x2.
V2 tt
(3.39)
The Chebyshev-Hermite polynomials (referred to hereafter as Hermite polynomi­
als) are generated by successive derivatives of a(x),
Da(x)
= —xa(x)
D 2a(x)
= (x 2 — l)a(a:)
D 3a(x)
= (3x —x^)a(x)
(3.40)
and we obtain the nth Hermite polynomials, H n(x), as the polynomial resulting
from the n th derivation of a(x),
(—D)ra(x) = H n(x)a(x).
(3-41)
Hn{x) = ( - 1 )ne 2 x2D n (e“ ^ 2) .
(3.42)
ft has the general form,
The first 5 Hermite Polynomials are,
H,
=
1
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#1
x
h2
x2 —1
H3
x 3 —3x
Ha
x 4 — 6x2 + 3.
The Hermite polynomials are orthonormal and have the following property which
is easily verified,
/
0, ?n
OO
H m(x) Hn(x)a(x)dx
n
(3.43)
nl, m = n
-OO
The characteristic function of a distribution is the P'ourier transform of the
probability density of the distribution. It gives statistical information about the
distribution in a compact form and, if it exists, uniquely defines the distribution.
The characteristic function, <f>(t), is defined as,
m
ei,xdF
=
where dF is the distribution over x.
(3.44)
For clarity, we may think of dF as the
distribution of pixel tem peratures and x as the range of possible temperature
values. Taylor expanding <f>(t) in Equation 3.44, we get at t = 0,
« (i) ~ i + Y , ^ ( a y
(3.45)
r —1
where /ir is the rth moment of the distribution $(£). The moments, fi, are more
familiarly expressed as,
/
OO
(x — hi )rdF
(3.46)
-OO
where n i is the mean value of the distribution. Taking the natural log of <h(t),
we obtain the following Taylor expansion,
In <F(f)
(3.47)
j=i J-
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where Kj is the j th cumulant of the distribution d>(f). The j th order cumulant,
Kj, is defined by,
(3.48)
The cumulants,
k,
are related to moments, p, by,
(3.49)
where the bracket is the binomial bracket.
Expanding the terms, the first 4
cumulants expressed in terms of central moments (moments about the mean) are
given by (note th at Ki = pi = 0 for central moments),
.2
«3
=
«5
=
A*3
-
1 0 p 3h 2 -
Cumulants provide a measure of how the distribution dF differs from Gaussian.
For example,
k2
is the dispersion (a2) of the distribution;
is known as the
skewness of dF, which measures the asymmetry of the distribution relative to a
Gaussian; k4 is the kurtosis, which measure how quickly the tail of the distribution
flattens compared to a Gaussian.
The Edgeworth series is essentially a recollection of terms of the GramCharlier series. We introduce here the Gram-Charlier series and motivate the
re-collection of terms later. The Gram-Charlier series expands a distribution in
terms of the derivatives of the Gaussian function, a(x). Such a series would look
like the following,
i =o
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where the cjs are coefficients to the Hermite polynomial, H f s. The coefficients,
Cj, of this series is obtained by multiplying both sides of the above equation by
Hr(x) and integrating from —oo to oo, using the orthonormality condition of the
Hermite polynomials (Equation 3.43) we get,
(3.51)
Substituting in the Hermite polynomials and re-expressing the coefficients in
terms of central moments, the series becomes the Gram-Charlier series. The first
five coefficients of the Gram-Charlier series expressed in terms of cumulants,
are,
1
Cl
=0
C2
= (M2 ~ l)./2
C3
= 113 / 6
c4
= (p4 —6 / j ,2 + 3) / 24.
(3.52)
In essence, this is also the Edgeworth series. The difference is th at the GramCharlier series is not truly asymptotic, because no m atter where the series is
truncated, the remainder terms are of similar order due to the mixing of linearorder /i terms in the coefficient cjs. For example, c2 is (fi2 — 1)/2 and c4 is (/i4 —
6 /i 2
+ 3)/24, containing both ji2 and /z3 to linear order. We need to re-order the
series in such a way th at successive terms are negligible compared to the previous
term. Instead of using moments, /r, in the coefficients, the Edgeworth series uses
cumulants,
k.
By transforming the Gram-Charlier series using the relationship
between moments and cumulants (Equation 3.49), one obtains coefficients that
have the desired properties.
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To show th at such a thing is reasonable involves more work. Consider ob­
taining the characteristic function of a distribution based, again, on derivatives
of the Gaussian function, a(x). This time, however, we will build in the fact that
the distribution is expressed as exponents of derivatives of a(x) and consider our
starting point the expression,
{exp(ftr.Dr )}o;(x).
where we have suggestively labeled the coefficients
(3.53)
k,
as we hope to show th at in
fact, these are cumulants. Referring to Equation 3.44, the characteristic function
of this expression is given by,
r°o
r°°
/ e*<x exp(KrD r)a(x)dx = / eltx ^
J —oo
J —oo
( kH DrV \
I -J—r.— ) a(x)dx
\
J■
(3-54)
/
Using the definition of Hermite polynomials, and moving terms without x depen­
dence outside the integral, the characteristic function becomes,
5Z
■] roo
I
eltx( —i y rH ^ ( x ) a ( x ) d x
(3.55)
This now resembles a Fourier transform of the expression H r ( x ) a ( x ) . We can
derive this starting from the Fourier transform of a(x),
V 2 ^ a { t ) = e- ^ 2 = [ elte- L e ~ k2x2dx.
J
y 27r
(3.56)
Each derivative of a ( t ) generates an (ix) term in the Fourier transform. So we
can write the following,
V 2 ^ D ra{t) = ( - l ) r v /2* H r {t)oi{t) = /° ° e ^ ^ L e ^ 2/2) dt.
3—oo
\ / 2 tt
(3.57)
Writing the second equality again, with a little re-arrangement,
fOO
1
ir H r (t)a(t) =
.— /
e~ltx x r a ( x ) d x ,
V27T 3—oo
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(3.58)
which tells us that the Fourier transform of x ra ( x ) is,
foo
1
xra(x) = — /
.
__
e~UxirV27rHr(t)a(t.)dt.
(3.59)
Z7T 7 - o o
Interchange x and t and moving some constants,
poo
.__________
( ~ i ) rV2TTtra (t) = /
(3.60)
e~lxtH r ( x ) a ( x ) d x .
J — OO
This gives the Fourier transform of H r (x)a{x) to be \ / 2 n i Tt ra{t). Putting this
into Equation 3.55, the characteristic function of Equation 3.53 becomes.
T
7'!
\/2 n { - i ) rn rja(t i =
'
=
v ¥ r a ( t) ¥
¥ (
j!
v¥ro;(f) exp[«;r (—it)7"].
(3.61)
We artificially construct another expression to be considered, based on Equa­
tion 3.53, and motivated by the form of the Gram-Charlier series,
exp
1!
2!
3!
4!
(3.62)
a{x).
The characteristic function of this expression can be obtained by analogy to the
characteristic function the Expression 3.53. Again, using the fact that derivatives
of a(t) results in another (i x ) term in the Fourier transform, the characteristic
function of the previous expression is given by,
4>(t) = y/2na(t) exp
(3.63)
+ ~ { i t f - ^ ( i t f + ^ ( ^ ) 4-2!
3!
Looking back to Equation 3.47, the constant that we have labeled
k
is indeed
the cumulant. This is shown by taking the logarithm of Equation 3.63. This
means that if
k
were the cumulants of the distribution $ , then the expression
in 3.62 is the proper expansion in powers of derivatives of a(x). We are now
justified to take the coefficients of the Gram-Charlier series and turn the moments
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into cumulants. The first 7 Edgeworth series coefficients, E n are given below,
coefficients,
Ei = 1
(3.64)
E2 =
^3 / 6
E3
K 4/24
=
Ea =
K5 / I 2 0
E5 =
(k &4 10K.g)/720
E
=
(k
=
(k.s T 56/?5/?3 4 35k^)/40320.
q
£7
j
4 35/v4«;3)/5040
In its usual form, the Edgeworth series is given by,
ff({x)\ = exp 1
1)3
Di
a(x)
(3.65)
or, alternately, using Hermite polynomials,
f ( x) = a{x) ( l + — H 3 + — H a + — H 5 -f
W V
6 3 24 4 120
—
720
4
J
.
((3.66)'
It is clear that each successive term contains cumulants of higher order.
We tested the Edgeworth Series on the histogram of a perfect Gaussian dis­
tribution, for which we knew all E n coefficients should be zero. We found th at
the biggest effect on the coefficients was how quickly the edges of the Gaussian
function went to zero. For example, in two trial runs we used 1 million random
numbers to represent a normal Gaussian distribution, with zero mean and unit
variance. The one run where we allowed the x variable to go to x = 16, generated
zeros at a level of 10-17. The other run where we only allowed x to go to x = 3,
had zeros on the level of 10~8.
This meant that we had to choose the same set of bins for every EV set
instead of binning the data with bin-size set by the maximum and minimum of
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each distribution. We experimented with the number of bins to see what our binsize should be. W ith too few bins, the non-Gaussian aspects of the distribution
were hidden. W ith too many bins, the residuals (^(d ata —fi t ) ) increased from
errors at the wings of the distribution, and since this was exactly the location
of the non-Gaussian signals we wished to detect, we had to be cautious. W ith
the number of bins from 500 and 1000, the variation in the coefficients generated
were within 10%. The optimal number of bins turned out to be 574, which struck
a balance between generating a good zero value for the perfect Gaussian case,
as well as having enough room at the wings for the non-Gaussian signals. We
binned the absolute value of the EV set for each of the simulated maps into
574 bins, with the uppermost bin at 0.3 mK. The upper bin was chosen because
maps containing EV greater than 0.3 mK were well above the Max EV of the NoString set and therefore were already known to contain strings. Since we used the
absolute value of the EVs, we mirrored the histogram across the y-axis to create
a negative half for the distribution; the whole distribution was then normalized
to one.
Due to the fact th at wre reflected the distribution across the y-axis, the his­
tograms were completely symmetrical. This meant th at the odd moments (even
cumulants) were zero. Thus, we only needed to look at the odd Edgeworth coef­
ficients; E n where n is 3, 5, or 7. For our purposes, the 7th Edgeworth coefficient,
E -(, was a good discriminator of whether a map contained a cosmic string hori­
zon. We have included plots of the other coefficients, E 3 and E 5, in Figure 3.8
and Figure 3.9 respectively; note that the y-axis of Figure 3.8 is on a log scale to
include all data points. Compared to the E-j coefficient, E 3 did not pick out as
many points, so at best, it can used as corroborating evidence. The E 5 coefficient
was similar to E 3, but it included more points.
94
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X
No —String Largest
Average No —String
No —String Sm al les t
O
O
O
□
□
CD
O 3
o
*
-3L-
o
AT
JL XC
s
CD
cn
ID
A No —String Values
□ M u l t i - S t r i n g Values
X)
K)
□ q
□
At
^ * . - _ XG
n ix T f ^ j g
*
0
10
—6
-4
^ —5
1 0 °
10
LOG Input string t e m p e r a t u r e (K}
10'
Figure 3.8 Plot of the input string tem perature Tf vs. the 3rd Edgeworth coeffi­
cient.
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.0
X
*
No String Largest
------------- Average ■No —String
.............. - N o String s m a ll e s t
o
o
o
0.5
0)
o
o
_c
□
□
-
-
*
□
-
*
□
H—'
o
5
o
cn
TD
_
*
A
A N o - S t r i n g Values
□ M u lt i - S t r i n g . V a l u e s
x:
in
-
j _g__
0.0
Lu
-
-
-0.5
10 - 6
10~5
10"4
LOG Input String Te m per at ur e, (K)
10
-3
Figure 3.9 Plot of the input string temperature Tf vs. the 5th Edgeworth coeffi­
cient.
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Out of the 15 sets of No-String maps, the E 7 coefficients ranged from 6.51 x
10- 6 to —3.92 x 1CT6. Inspection of E 7 and other coefficients showed that there
was a clear trend: the Stringy sets which contained hot string horizons produced
bigger non-Gaussian tails that were picked up by E 7. The fact that the E 7
coefficients of the No-String set were distributed around zero tells us th at a small
E 7 indicated a lack of non-Gaussian features like those produced by a string
horizon.
It also indicated the magnitude at which the E 7 coefficient can be
considered noise. This was corroborated by the fact that the higher the Tf, the
larger and more positive the E 7 coefficient. Therefore, I could use the No-String
E 7 values as a discriminator between maps with strings and maps without strings.
I took the second largest No-String (14th out of 15) E 7 value as our limit of the
Edgefinder’s single string sensitivity, at EV = 5.06 x 10~ 3 mK.
The type of map th at produced the smallest E 7 signals were maps with only
one string horizon inserted. I made
66
maps of single string with input tem pera­
tures ranging from 1 mK to 1 fiK. The E 7 coefficients and input string horizons
for all sets are graphed in Figure 3.10. It shows clearly that a large and positive
value of E 7 is indicative of the presence of hot string horizons. Figure 3.11 is a
blow up of the region around the No-String limit. String horizons with Ts > 0.27
mK were clearly above threshold E 7 value. At Ts = 0.260 mK I encountered the
first string horizon with E 7 below the threshold. I therefore set T f = 0.27 mK
as our 100% confidence level of string detection. Examining Figure 3.11, I can
see th at the E 7 values mostly fall within the boundary defined by the maximum
No-String and minimum No-String E 7, with a few peeking above the threshold.
The scatter in the data is very constant. Therefore the limit of our detection
is firm at 0.27 mK; all input strings cooler this threshold looks like noise to the
Edgefinder. For a cosmic string moving at mean simulation velocity, (3 = l / \ / 2
this corresponds to a Gji = 1.07 x 10~ 5 string.
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0.6
No —String Largest
Average No —String
No —String Smallest
X
O
o
o
0.4
*
0)
o
o
n
0.2
o
3
0
cn
TD
0.0
A No —String Values
□ Multi —String Values
_c:
-o.:
.-6
10
J
10
r
LOG Input String Tem pe ra tu re , (K)
Figure 3.10 Plot of the input string temperature Tf vs.
the 7th Edgeworth
coefficient.
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0.02
c
0)
o
Q)
o
0.0
o
0.00
sz
0.0
A N o - S t r i n g Values
0.02 L _
- 0
10
°
10
-4
10
3
Input String Tem per at ur e, (K)
Figure 3.11 Detail of the plot of the input string tem perature Tf vs. the 7th
Edgeworth coefficient. Also plotted are the multi-string set data as well as limits
from the No-String sets.
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3.3.4
M ultiple S tring M aps
17 simulated maps were made with multiple string horizons inserted in the map,
ranging from 1 to 60 horizons a,nd R s ranging from 1 to 4°. The number of
string horizons and their sizes are summarized in Table 3.4. The purpose of
the multiple string sets were twofold, first to determine if multiple strings had
a similar effect on the Edgeworth coefficients as the single strings, and if so,
determine the number of weak strings needed to generate an E 7 signal above the
No-String discriminant. The Max EV and the E 7 coefficients for the multiple
sets are plotted as letters A to P, according to their set name in Table 3.4, in
Figures 3.7 and Figure 3.11, respectively. The bold items in the table represent
sets with E 7 coefficients larger than the No-String limit.
From the results of Sets A to C, I can see that the Ma.x EV of multiple string
sets are generated by individual strings, and therefore from the Max EV alone,
I cannot tell if there is more than one hot string in the data. However, the E 7
coefficient of multiple strings are cumulative. This means that m a situation
where I have a small Max EV, and a large E 7, I know th at the sky contains
multiple strings.
However, looking at the results of Sets D, E, and F, I can see that if the
strings are cooler than the detection threshold, their E 7 signatures are small
enough th at a few strings (less than 10) will not show detection. It takes quite
a few cool strings for the cumulative effect to show up. For very cool strings
(Ts ~ 0.15mK), it takes 20 input strings for the E 7 coefficient to be above the
threshold.
The size of the string has some effect on the coefficients, but it is secondary
to the strength of the edge. Compare Sets D to J and G to K; these sets have
similar number of input strings, and similar Ts, but Sets J and K have twice
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Table 3.4.
Number and Strength of Input Strings for Multi-String Simulated
Maps.
D ata Set
# of Strings
RAD
T
1S
Max. EV
£V(10“6)
M ulti A
2
2 .0
0.30
0.290
84.4
M ulti B
4
2 .0
0.30
0.310
185
Multi C
10
2 .0
0.030
0.231
2 .8 6
Multi D
4
2 .0
0.030
0.236
Multi E
30
2 .0
0.030
0.257
3.59
Multi F
60
2 .0
0.030
0.243
-2.42
Multi G
5
2 .0
0.015
0.231
2.09
Multi H
10
2 .0
0.015
0.245
4.76
M u lti I
20
2 .0
0.15
0.244
8.77
Multi J
2
4.0
0.015
0.249
4.49
Multi K
2
4.0
0.030
0.263
0.609
M u lti L
2
1 .0
0.30
0.265
46.2
Multi M
5
2 .0
0.15
0.234
0.394
Multi N
10
2 .0
0.15
0.230
2.54
M u lti O
5
1 .0
0.30
0.294
120
M u lti P
10
1 .0
0.30
0.294
114
-
2 .2 1
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the R A D , which means 4 times the number of pixels. The E 7 coefficients of
the sets are similar and show little trend reflecting the change in inserted string
size. Furthermore, consider Set L, where the inserted string had R s = 1 degree;
this set had an E 7 coefficient significantly above the detection threshold. Both
these phenomenon point to the fact th at the Edgefinder is most sensitive to the
temperature jum p at the edge of a string horizon, rather than the size of the
string.
The results of the multiple string sets shows th at the Edgefinder limit is
firmly set at Ts ~ 0.27 mK. I can detect cooler strings, but they need to be
numerous: looking at sets M and N shows that I need more than ~10 strings of
0.15 mK for a detection. For very cool strings at Ts < 0.03 mK, the data are
very noisy. The multiple strings sets indicated that the most im portant criterion
for string detection is the temperature of the input string. If the string is above
the detection threshold, within a reasonable range of sizes, it will be picked up
by the Edgefinder
3.4
W M A P R e su lts
The WMAP data I ran through the Edgefinder is an average of the data in
three of the WMAP bands: Q, V and W. I called this the QVW composite
map. Pertinent statistics of the WMAP QVW composite map are in Table 3.5,
including the Max EV and its E 7 coefficient. The resulting EV set was binned in
the same manner as the simulated maps. Comparing the QVW composite map
and the No-String sets, statistics such as mean, median, standard deviations are
similar.
Looking at the results of the WMAP composite QVW data, the maximum EV
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Table 3.5.
Select Properties of the WMAP composite QVW Data.
WMAP Band
Max Map T
Max EV
0 .6 6 6
0.258
QVW
e
7
-3.82 x 10~ 6
is below the threshold set from the max EV of the No-String set, and is in fact,
below the maximum EV of half of the No-String simulated data sets. This means
that there are no hot string horizons in the WMAP data at the level of noise set
by the No-String sets. The
£7
coefficient leaves no doubt that there are no non-
Gaussian signatures in the tail of the distribution of the WMAP data produced
by string horizons similar to those I inserted into the simulated maps. I can
confidently say that the Edgefinder did not find any evidence of string wakes in
the CMB data measured by WMAP, to the single string limit of Gfi < 1.07 x L0“5.
The 20 string limit of less 0.15 mK strings gives G/i < 5.97 x 10-6.
I mention again the fact that in the actual WMAP data, due to Galactic
and foreground contamination, about 1/3 of the pixels were masked.
I have
reproduced this masking in our simulated data so that I have the same number
of pixels per map as the actual WMAP data. I caution that any strings hidden
behind these masked pixels would not be picked up by the Edgefinder.
3.4.1
W M A P 2nd Year Sim ulation
I have performed the analysis necessary in anticipation of the WMAP second
year data release. All of the simulated maps were done in the same manner as
the first year data, with the exception th at the noise is now 1 / y/2 time the noise
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of the first year data. The second year simulated maps produce limits at max
EV = 0.283 mK and E 7 = 0.257 mK. The E 7 threshold yields a single string
limit of Gfx < 1.02 x 10-5 . Most of the noise in our string search is due to the
first acoustic peak in the CMB angular power spectrum; the Edgefinder is cosmic
variance limited, so reduced radiometer noise has little effect. When the second
year WMAP data is release, I will run the filter and if the result is negative, this
is the detection limit.
3.5
A C o sm ic S tr in g C a n d id a te?
Sazhin etal. (2003) reported a discovery of an object which contains two sources
of identical isophotes, color, and fitted 2 -D light profiles in the Osservatorio Astronomico ai Campodimonte Deep Field (OACDF). In addition, spectra of the
sources are identical with a confidence level higher than 99%. Morphological ar­
guments led them to propose th at this object is a background galaxy lensed by acosmic string. They have named this object the Campodimonte-Sternberg-Lens
candidate 1, or CSL-1. The red-shift of both sources in the object is 0.46T0.008;
the separation of the two sources is
2
Precise finder charts for the object were not available. I found this object by
visually inspecting the OACDF deep field and comparing it to the Palomar All
Sky Survey plates. I found CSL- 1 to be located at (J2000) RA 12:23:30.72, Dec
-12:38:57.8. There may be some small uncertainty about the location of CSL-1.
I examine the WMAP data at this location to see if the Edgefinder can detect
a string. As WMAP data has a resolution of 13 arcminutes, by including the
four nearest pixels to th at coordinate, I believe I have covered this object in our
search.
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3.5.1
String Search
From the image separation, I can derive a string mass per unit length, /i. In a flat
universe where
matter = 1, from G ott (1985), I find th at the image separation
Ad of lensing by a cosmic string is related to D, the deficit angle of the conical
space around the string, by,
Ad = D cos a
1 — (1 4 xs) -1/2
l - ( l + ^)-V 2
(3.67)
where zs is the red-shift of the string, zg = 0.48 is the red-shift of the background
object (galaxy), and a is the angle of the straight string with respect to the plane
of the sky. If I assume th at the string is in the plane of the sky, which means
a = 0, there are two limiting results: for zs = 0, I find Ad = D; for zs = 0.4,
Ad =
0 .1 0 2
D.
For a flat universe, D , is related to the string mass per unit length, G/j, by,
D =
87 tG u .
(3.68)
Therefore, with a maximum D of 9.7 x 10~ 6 radians, I get G/j = 3.86 x 10~7.
A string will cause a jum p in the value of the tem perature of the CMB due to
Doppler shifting. The change in tem perature is given by Equation 3.27.
I have processed WMAP data at the position of the CSL- 1 object and com­
pared the Edgefinder values to Edgefinder values of regions with similar sky
coverage and galactic latitude. The CSL- 1 object was small enough that it was
within one WMAP pixel (on pixel 968549). However, I include results from the
surrounding 4 pixels in the event th at I have misjudged the position of CSL-1.
In addition, the alignment of the two images contains some uncertainty, thus I
also ran the Edgefinder along two separate position angles, a = 0 and a = 7t / 2 0 .
The results are in Table 3.6.
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Table 3.6.
Pixel #
Edgefinder Values for CSL-1
WMAP QVW T
EV | Q =0
/% -tile EV |Q=7r/2 o/%-tile
968549
-0.144015
0.07336 / 95.30%
0.08686 / 97.61%
968550
-0.0735366
0.08322 / 97.11%
0.08280 / 97.05%
968548
0.0418802
0.1248 / 99.76%
0.08621 / 97.53%
968527
0.0717491
0.1234 / 99.74%.
0.09720 / 98.65%
Note. — EV of CSL-1 and 3 immediately adjacent pixels. Vahies
listed are for
a
=
0
and
a
=
tv /
20
The Edgefinder values of the 4 pixels are all above the 95th percentile, es­
pecially pixels 968548 and 968527, which have very high E.V.’s above 99th per­
centile. However, these high percentiles can be misleading. First, they are all
significantly under the No-String detection limit. Second, as an upper limit, if I
allow the WMAP tem perature at those pixels to be entirely caused by the pres­
ence of a string, I can say th at the EV is exactly the A T due to the string motion,
due to the Edgefinder gain being
1
. W ith the Gfi given above, this means for
an E.V. = 0.08686 mK, the string needs /Ty ^ 3.3, or v = 0.957c, to account for
the tem perature jum p. The ranges of string velocity for the E.V. in Table 3.6 is
from v = 0.941c to v = 0.979c. These high string velocities makes the case for
the existence of cosmic strings at this location more unlikely, as the rms string
velocity is v ~ 0.7c. I therefore cannot say that I have a significant detection of
a cosmic string at the location of CSL-1 in the WMAP data.
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A recent paper (Shlaer & Tye, 2005) showed th at a correction factor is nec­
essary for measuring the angle deficit of a relativistic string, given by,
A9rei =
+ 0 cos(a)) A 6t
7(1
(3.69)
where a isthe angle between the emitted photons and the perpendicular plane
of the cosmicstring. If I assume an optimal alignment, where cos (a) = —1, the
measured angular separation is the true separation modulated by the correction
factors
7 (1
—0). This significantly modifies our result for the CSL-1 object. The
corrected tem perature difference is given by,
A 9rel =
- P )A 9
(3.70)
A9rei — 7 ( 1 —(3)&7rG[i
(3-71)
7(1
A 9rei
7(1 - P)
=
8 irGfi
(3.72)
and by putting in Equation 5, I obtain,
S T
7
o
n r ,
= 8i W
S T
7 t
A ( ) rel
a
= ^
ST 1
T A9
,0 ^
7
0
1-/3
,3,3)
(3.74)
For our EV = 0.08686 mK, I get v = 0.765. The range of velocities is 0.756 <
v < 0.785. Given a rms string velocity of v ~ 0.7c, these numbers are quite
reasonable. I caution th at given the fact there is only one double image, the
optical evidence for an edge is only suggestive, not conclusive; after all, many
close pairs of galaxies do exist. A better method was suggested by (Shlaer &
Tye, 2005) , who showed that partial images with abrupt edges would be caused
by cosmic string lensing. That is not the case here.
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3.6
S im u la te d P L A N C K R e s u lts
Looking forward, I also made simulated maps of CMB data th at will be gathered
by the European Space Agency’s PLANCK satellite. PLANCK is scheduled to
launch in 2007. Most of PLANCK is still being constructed, so I do not have data
on the noise characteristics of PLANCK. W hat I do know is th at PLANCK will
have a 5 arc-minute resolution, with a tem perature sensitivity of 4 x 10~ 6 K (more
information is available from the PLANCK website: h ttp :/www.rssd.esa.int/ in­
dex.php?project=PLANCK). I expect PLANCK will have about 1/10 the noise
of WMAP. For our simulated PLANCK maps, I followed the same procedure
outlined in Section 4, except in step 5, where for the added noise, I used 1/10 the
average WMAP noise value for the an of each PLANCK pixel. In the HEALPix
scheme, the improved PLANCK resolution means having 12 million pixels in the
sky, a 4 fold increase on WMAP. Running each simulated map at 20 a angles
became computationally infeasible. Therefore, I only ran the PLANCK maps
through
6
different a angles, spaced equally between
0
and 180'L
The increased resolution of the PLANCK satellite will allow us to bypass
the problems of being sensitive to the first Doppler peak at I ~ 200. W ith the
PLANCK Edgefinder at R A D = 0.5°, I still had ~ 270 pixels in the filter window,
which was comparable to the number of pixels in the WMAP Edgefinder window
of R A D = 1.0°. The transfer function for the R A D = 0.5° PLANCK Edgefinder
filter is plotted in Figure 3.12. For the R A D = 0.5° Edgefinder, the peak filter
sensitivity is at i ~ 500, well away from the first Doppler peak. I performed
the bulk of our simulation using the R A D = 0.5° PLANCK Edgefinder. At this
resolution, the PLANCK filter had a similar number of pixels as the WMAP
R A D = 1° filter, so the gain from the filter was comparable.
As with the WMAP set, I created calibrators for the PLANCK Edgefinder and
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0.5
o
0.0
D
CD
D
D
0.5
>
CD
O
5
zs
(/;
2.0
10
100
1000
10000
Figure 3.12 Transfer function for the PLANCK simulation; the filter had R A D
0.5.
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tuned the filter to return EV = 1 for a 1 mK edge. Noise-only, A-CDM-only (a
flat, cold dark m atter universe with the a cosmological constant dominating the
total density of the universe) maps were put through the PLANCK Edgefinder for
calibration purposes. 1 made a total of 30 Stringy PLANCK maps with only one
string horizon per map. In addition,
6
baseline No-String maps were generated.
Since the range of EV for PLANCK is expected to be wider, the optimum number
of bins for the PLANCK data is 704. The total number of data points remain
similar to the WMAP simulation, so the comparison of the two statistics is valid.
The plots of the string tem perature, Ts, of a map vs. the max EV, and vs.
E j are in Figure 3.13 and Figure 3.14, respectively. The limit obtained from the
max EV was very similar to the the limit obtained from the E 7 coefficient, most
probably as a result of the fact th at I have eliminated a lot of the Gaussian signal
by going to a smaller Edgefinder window. This strengthened our confidence in
these limits. Both the Max EV and E7 methods indicated th at the detection
limit for the PLANCK Edgefinder occurred at T = 0.14 mK. This represented
a factor of 2 increase in the single string Edgefinder sensitivity compared to the
WMAP data. This is, however, very similar to the multiple strings limit for the
WMAP data. For a cosmic string moving at mean simulation velocity, (3 = 1 /\/2
this input string corresponds to a string mass of Gji = 5.77 x 10-6 .
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0.8
No —String Largest
Average No —String
No —String Sm al les t
0.6
A
>
LU
X
¥
o 0.4
>K
z
a
z
<
_i
Q_
—
OK
0.2
Z r\_
_ ?K
-
M
O——
M
O—
/IS
X j/_ 7 ix
• M T \-
_ Z \._
................ '
A No —Strinq Values
0.0
0
“
10
—6
10
^
- 4
^
10
-3
'
Input String T e m pe ra tu re , (K)
Figure 3.13 Input string tem perature vs. the maximum EV of the EV set for
simulated PLANCK data.
Ill
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0 .0 2 0
c
CD
o
0 .0 15
(5
0 .0 1 0
°
0 .0 05
--—
No —String Largest
Average No —String
No —String Sm al le s t
*
W
*
4r
CD
_________ tL____________ - 4 . ...
CD
S
o.ooo
ttt
-0.005
-
%------
A No —String Values
0.010
0
-6
, —5
10
10
"
-4
10
-3
Input String T e m p e r a t u r e , (K'
Figure 3.14 Blow up of input string temperature vs. the 7th Edgeworth coefficient
of the EV set for simulated PLANCK data.
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CHAPTER 4
C on clu sion s and F u tu re W ork
I have constructed and calibrated a digital filter for the WMAP CMB data to
search for cosmic strings.
By comparing the WMAP data and a control set
cheated from CMBFAST and SYNFAST, we have concluded th at the CMB data
returned by the WMAP satellite do not contain single strings to the limit of
Gjj,
1.37 x 10~ 5 using the max EV as the threshold and Gn < 1.07 x 10- 5 using
the 7th Edgewort h coefficient as the threshold. This limit may be more stringent
if we allow the sky to have multiple strings, to a limit of G\i < 5.97 x 10
if there
are more than 20 string horizons in the sky. We caution that WMAP effectively
examined 2/3 of the visible sky so it is possible th at we are missing strings in
our analysis. W ith the second year WMAP data, we can improve this limit by
about 5%. This improvement is a result of an expected 30% improvement in the
WMAP radiometer noise. However, because the Edgefinder filter window was 2
degrees, the first CMB Doppler peak generated a large background that cannot
be filtered out, and therefore a large reduction in radiometer noise resulted in a
relatively small improvement in the discrimination threshold.
I have also investigated claims of a possible cosmic string detection of the
object CSL-1, and I found little evidence of a string at this position. For the
proposed string mass of Gfi = 3.86 x 10-7, WMAP CMB temperatures at the
location of CSL- 1 would require the string to have been moving very relativistically, with v ~ 0.96c. We conclude that this possibility is unlikely and that much
113
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more sensitive and higher angular resolution data would be needed for a critical
test of CSL-1.
I ran the Edgefinder method through simulated PLANCK data. As more
information about the PLANCK satellite becomes available, we can do a more
realistic modeling of the noise characteristics and therefore get better limits on
the strength of cosmic strings PLANCK can detect. Currently, I have a projected
limit of Gfj, <; 5.77 x 10~6, a factor of two better than the single string limit from
WMAP. The predicted detector noise of PLANCK is too small to affect the filter
appreciably, and the true limit of the PLANCK data for the Edgefinder method
will be cosmic variance. The factor of two improvement over the first year WMAP
data is mostly a result of the smaller filter window which reduced signals from
the first CMB Doppler peak.
W ith more and more sensitive all sky CMB surveys, we can begin to set firm
experimental limits on the existence of GUT scale cosmic strings, and thereby
iimiting the types of allowed phase transitions. Relic products such as cosmic
strings currently provide the only viable tool for probing conditions of the very
early universe. Limits such as those presented here are useful in the absence of
testable conclusions.
Cosmic strings only exert an influence on CMB photons through their gravi­
tational field. Therefore, if we know the string spatial distribution, their length
scale, and their energy density (the values tabulated in Chapter 3), we should be
able to completely characterize the behavior of CMB photons within a network
of strings. All of the above parameters were explored with numerical simulations
by various groups, e.g. Bennett k, Bouchet (1990), Albrecht & Turok (1985),
Allen et al. (1997). I have laid down the calculations necessary for calculating a
Monte Carlo initial string configuration. A possible future direction would be in
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writing a multi-variable, ray-tracing code to produce the final CMB sky modified
by the gravitational potential of the strings.
I have also demonstrated, in the Chapter 2 , th at buckyballs are viable candi­
dates for an anomalous component of the CMB foreground. Buckyballs have high
frequency (20 to 40 GHz) emissions generally with a power spectral index (j3) of
around 2, and low frequency (1 to 20 GHz) emission of spectral index 1. The peak
emission frequency across the simulated regions spans 5 to 37 GHz, where the
anomalous free-free emission is observed. Buckyball emission from our simulation
was dominated by collision density effects. The composite spectra indicated that
the dominant emission from buckyballs comes from the C$o molecule, while the
highest frequency emissions comes from C2oIf we assume buckyballs are responsible for the anomalous free-free excess
emission in the 35 to 45 GHz range, they would need to comprise 0.5% of the
total: amount of Galactic carbon in order to explain the observed e m iss iv ity . We
believe this is a reasonable number. While the spectral index of the buckyball
emission is not “free-free” , i.e. p ~
0
, the observations can still be explained
by an emission mechanism th at peaks in the range of the anomalous emission
window between bright synchrotron at low frequencies and vibrational dust at
high frequencies.
A composite buckyball spectrum is close enough given the
modest signal-to-noise ratio of the observations to date.
I have identified some high latitude clouds that could have detectable emission
from rapidly rotating buckyballs. If measurements can be made in these high
latitude clouds to confirm the detection of spinning buckyball dipole radiation,
my simulation would yield not only a solution to the anomalous free-free emission
problem, but also another possible identification of interstellar fullerene emission.
Together, these two portions of this thesis explored some issues associated
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with the Cosmic Microwave background. These results represent a unique
of key aspects of the early universe, and of our Galactic content.
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