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Angular power spectrum of the cosmic microwave anisotropy at declination 40

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ANGULAR SPECTRUM OF THE COSMIC
MICROWAVE ANISOTROPY AT DECLINATION 40
by
Shafinaz Ali
A dissertation submitted to the Faculty of the University of Delaware
partial fulfillment of the requirements for the degree of Doctor of Philosophy
Physics
Summer 2000
(c) 2000 Shafinaz Ali
All Rights Reserved
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UMI Number 9982670
Copyright 2000 by
Ali, Shafinaz
All rights reserved.
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ANGULAR SPECTRUM OF THE COSMIC
MICROWAVE ANISOTROPY AT DECLINATION 40
by
Shafinaz Ali
Approved:
■ ^
7
* 7 ^
_______________________
Henry R. Glyde, Ph.D.
Chair of the Department of Physics and Astronomy
Approved:
Conrado M. Gempesaw II, Ph.D.
Vice Provost for Academic Programs and Planning
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I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
___________________________________
Signed:
Lucio Piccirillo, Ph.D.
Professor in charge of dissertation
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
Qaisar Shafi, Ph.D.
Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
Edward H. Kerner, Ph.D.
Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
William B. Daniels, Ph.D
Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
Peter T. Timbie, Ph.D.
Member of dissertation committee
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ACKNOWLEDGMENTS
There are many people whom I wish to thank for making the years at grad­
uate school a valuable and an enjoyable experience. I thank my advisor Dr. Lucio
Piccirillo for providing me with the opportunity to get involved with this experiment,
also for his advice and suggestions during these years. I wish to thank my disser­
tation committee, Dr. Qaisar Shafi, Dr. Edward Kerner, Dr. William Daniels and
Dr. Peter Timbie for consenting to participate in the dissertation defense. The 1996
campaign would not have been possible without the efforts of many people. I wish
to thank Dr. Rafa Rebolo for his collaboration and help with the 1996 campaign. I
also thank the members of scientific staff at the Obseravatorio del Teide (Tenerife,
Spain)who helped us tremendously during our stay in Tenerife. Especially, I thank
Bruno Femenia for helping me with running the experiment.
Michele Limon helped me getting started in the laboratory. He also helped me
with my numerous programming and other computational and technical questions.
Bob Schaefer helped me to decipher COBE and BATSE data and advised and guided
me through my first project. He has been a constant source of support through out
my graduate student years. The friends at graduate school helped me get through
the years with a good attitude and a positive outlook. Giuseppe Romeo has been
good friend to work with. Vivek Agrawal, Yuqiao Yang, Dave Huber, Judi Provencal
and many others provided good company and laughs. Friends like Jim Hill and
Elhamul Hai encouraged me. My family has been the greatest support throughout
my life. Their love and patience have seen me through the years far away from
iv
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home. My mother’s love and inspiration has been my greatest gift. Finally, my
husband Jonathan’s support, love and friendship have been invaluable.
v
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TABLE OF CONTENTS
LIST OF F IG U R E S .............................................................................................. viii
LIST OF T A B L E S .................................................................................................. x ii
A B S T R A C T .......................................................................................................... xiv
Chapter
1 W H Y M EASU RE TH E A N ISO TR O PY OF COSMIC
BAC K G RO UN D R A D IA T IO N ? ..............................................................
1
1.1 Introduction................................................................................................
1
1.2 Big Bang and In flatio n .............................................................................
2
1.3 Cosmic Microwave Background Radiation ............................................ 7
1.4 Origin of CMB A nisotropy........................................................................... 10
1.4.1
1.4.2
1.4.3
Dipole C o m p o n en t.......................................................................... 11
Sources of Primary anisotropy .........................................................13
Secondary anisotropy....................................................................... 14
1.5 Angular Power Spectrum of CMB A n iso tro p y ......................................... 15
1.5.1
1.6
Cosmological parameters from CMB stu d ie s.................................16
Experimental detection of CMB A nisotropy............................................ 17
2 IN STR U M E N T
................................................................................................ 19
2.1 T elescope.......................................................................................................19
2.2 Cold O p tic s ................................................................................................... 21
2.2.1
2.2.2
Refrigerator......................................................................................... 22
Cold optics b lo c k ............................................................................. 24
vi
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2.2.3
2.3
3
Electronics S y s te m ..................................................................................... 33
OBSERVATIONS A N D CALIBRATION
3.1
3.2
3.3
3.4
Moon Crossing Sim ulation............................................................. 40
Calibration P a ra m e te rs................................................................. 42
Estimation of Instrumentalnoise
.............................................................. 49
DATA R E D U C T I O N .....................................................................................52
4.1
4.2
4.3
4.4
Effect of Atmosphere on d a t a .................................................................... 53
Atmospheric Subtraction............................................................................ 55
Implementation of Atmospheric Subtraction T e c h n iq u e ........................58
Success of the Technique............................................................................. 59
4.4.1
4.4.2
4.4.3
4.4.4
5
............................................. 36
Observing S tra te g y ..................................................................................... 36
Demodulation of the D a t a ......................................................................... 37
Calibration ..................................................................................................38
3.3.1
3.3.2
4
26
Bolometers
Comparison of distribution of tem p eratu re ..................................59
Simulation of Atmospheric su b tra c tio n ........................................ 59
Power Spectra of D a t a ....................................................................60
Final data s e t ...................................................................................60
ANALYSIS A N D R E S U L T .........................................................................79
5.1
Likelihood A nalysis..................................................................................... 79
5.1.1
5.1.2
5.2
Mapping function and Window f u n c tio n ..................................... 80
Likelihood analysis resu lt.................................................................84
Conclusion..................................................................................................... 86
A ppendix
....................................................................................................................................... 90
B IB L IO G R A P H Y .................................................................................................. 94
vii
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LIST OF FIGURES
1.1
Angular power spectrum of CMB anisotropy according to the
Standard CDM model with
= 0.05,Q c d m = 0.95,Qx = 0,ns = 1,
and h = 0.5. The plot was generated using CMBFAST [50]................ 11
1.2
Current Observational results in CMB Anisotropy [23]. The black
curve: AC D M Qm = 0.35, h = 0.65, Qft/i2 = 0.02, n = l; red curve:
ACDM Qm = 0.4, h = 0.8,
= 0.03, n = 1 (Courtesy Wayne.Hu,
2000)
18
2.1
The telescope with the cryostat and the data acquisition box . . . .
20
2.2
Schematic of the telescope with the inner conical shield and the
outer hexagonal shield................................................................................ 22
2.3
C ry o stat....................................................................................................... 23
2.4
Miniature 3He R e frig e ra to r......................................................................24
2.5
Resistance of the thermometer on the baseplate during a cooling
cycle of the refrigerator. The resistance of the thermometer is
proportional to the temperature................................................................25
2.6
The 4.2K and 0.33K filters. A 10 mil gap separates the 4.2K optics
from the 0.33K optics ............................................................................. 26
2.7
Normalized transmission of the Bandpass f ilte r s ....................................27
2.8
Schematic of a composite bolometer. a)the radiation absorber, b)the
substrate, c)the thermometer, d)the thermal link and e)the heat
sink............................................................................................................... 28
2.9
Load curves of the four bolometers looking at a black bodies at
20.43K.........................................................................................................31
viii
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3.1
The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 1........................ 45
3.2
The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 2........................ 46
3.3
The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 3........................ 47
3.4
The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 4........................ 48
3.5
Power spectra for the quadrature component of the first harmonic
for all four channels. At higher frequency the power spectra
gradually flatten out and approach white noise behavior. The upper
limit of Instrumental noise is estimated from the fiat part of the
spectra.......................................................................................................... 51
4.1
Power spectra for the first harmonic for all four channels. The power
spectra falls sharply over 0.01 Hz to 0.2 Hz. and then gradually
flattens out. The sharply falling region is dominated by atmospheric
noise.............................................................................................................. 62
4.2
The logarithm of power spectra 0(k) versus the logarithm of k is
plotted. The slope of the linear fit is ~ - 8 /3 ......................................63
4.3
Atmospheric transmission as a function of frequency (solid line).
The normalized filter transmissions for each of the detectors are also
shown..................................................................................................... . 64
4.4
Data collected during May 11 1996 has been reduced using the 1994
method (blue dotted line) and the present method (red solid line).
The difference between these two methods is almost negligible. . . 65
4.5
Normalized auto-correlation of data from two different days versus
time lag (each point is 0.5 sec). The correlation decreases with
harmonic number. Also the day with less precipitable water vapor
(less atmospheric fluctuation) has less correlation..................
ix
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66
4.6
Normalized cross-correlation of data from two different days versus
time lag (each point is 0.5 sec). This clearly shows the data from
two different days are not correlated.........................................................67
4.7
Sum versus difference of the temperature of observing channel and
channel 4. The top panel is for channel 1 and the bottom panel is
for channel 2. The correlations between the sum and the difference
are given on the figure.................................................................................68
4.8
Effect of using 32,64,128 and 256 points in doing atmospheric
subtraction....................................................................................................69
4.9
Histogram of the rms value of the first harmonic of the demodulated
data................................................................................................................70
4.10
Histogram of the temperature values of the first harmonic of June 9
1996 datafile after averaging (red thin lines) and after atmospheric
subtraction (blue line). 256 points were usedfor both methods. . . 71
4.11
Noise file for atmospheric subtraction simulation.....................................72
4.12
Simulated data with signal added (S/N 0.1,0.5 and 1.) are shown on
the left panels. The right panels shows the data after atmospheric
subtraction ( solid red line) and averaging (blue dotted) lines. The
number of data points used for atmospheric subtraction and
averaging is 256........................................................................................ 73
4.13
Power spectra of Channel 1. The blue dotted lines are data reduced
by atmospheric subtraction with a bin size of 128, and the solid red
lines are data averaged over the same number of data points
74
Power spectra of Channel 2. The dotted blue lines are data reduced
by atmospheric subtraction with a bin size of 128, and the red solid
lines are data averaged over the same number of data points
75
Power spectra of Channel 3. The blue dotted lines are data reduced
by atmospheric subtraction with a bin size of 128, and the solid red
lines are data averaged over the same number of data points
76
4.14
4.15
x
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4.16
D ata from June 8 1996. The open circles on the lower panel are the
data from channel 2 ( second harmonic) before removing the base
line. The solid black line is the sinusoidal fit function. The residual
data is plotted on the top panel with open red circles........................... 77
4.17 Data from June 8 1996 second harmonic. The top panel shows raw
data before atmospheric subtraction, the middle panel after
subtraction, and the last panel after removing the long time drift. . 78
5.1
Mapping functions for ail six harmonics at declination 0 = 0 ° . . . .
5.2
1996 Window functions.............................................................................. 85
5.3
1994 and 1996 Window functions for the first two harmonics
5.4
Results of Maximum Likelihood analysis for channel 2. The solid
line is the Standard CDM model with fia = 0.05, Qcdm =
0.95, QA =
= 0, n = 1, HQ= 5Qkms~lMpc~l , and the
dashed line is the AC D M model with the following
parameters: Da = 0.05, Dcdm = 0.25, DA = 0.7 D„ =
0, n = 1, and H q = 65km s~l M ps~x. At the bottom of the
graph are the window functions(scaled up by the same factor) for
each modulation. The theoretical angular spectrums were generated
by using CMBFAST [50].......................................................................... 87
5.5
Results of Maximum Likelihood analysis for channel 2 plotted
against the same theoretical models along with recent results from
BOOMERang [12] and MAXIMA-1 [21]. Within the error bars our
result is consistent with these two experiments....................................... 88
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82
86
LIST OF TABLES
2.1
Characteristics of the F ilte rs..................................................................... 26
2.2
Structure of data frame
3.1
Phase angles for each channel and harmonics. The values given are
in radians..................................................................................................... 39
3.2
Brightness temperature of the Moon at our w av elen g th s.................... 39
3.3
Rayleigh-Jeans factors and the atmospheric attenuation factors for
each of the four Channels........................................................................... 40
3.4
Summary of the Moon Calibration F i l e s ................................................ 42
3.5
Chopping angle amplitude for each Channel and Harm onic................. 43
3.6
FWHM for each Channel and H arm onic................................................ 43
3.7
Calibration factors for each channel and harmonics from the May 2nd
Moon Crossing. The values are in iiV /K for Antenna temperature 44
3.8
Calibration factors for each channel and harmonics from the May
2nd Moon Crossing. The values are in /iV /K for thermodynamic
temperature .............................................................................................44
3.9
Upper Limits of Instrumental Noise (m K — s 1/2) Estimation . . . .
5.1
The mapping functions for six harmonics of “Dec40” observation as
a function of R.A.........................................................................................81
5.2
Window functions for all six harmonics for 1996 data............................. 84
5.3
Results of likelihood analysis bandpower in n K for Channel 1 . . .
........................................................................... 35
xii
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50
89
5.4
Results of likelihood analysis bandpower in fi K for channel 2
xiii
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...
89
ABSTRACT
In this thesis I present the results of the Cosmic Microwave Background
(CMB) anisotropy experiment carried out at the Obseravatorio del Teide (Tenerife,
Spain) between May 1996 and July 1996. This dissertation is based on the data
collected in the daily drift mode from May 6 1996 to May 12 1996, and June 8 to
June 15 1996. The telescope consists of an off-axis parabolic mirror with a focal
length of 1.33 m and diameter of 45 cm coupled to an off-axis hyperbolic mirror of
diameter 28-cm. The primary mirror chops in a sinusoidal manner with a reference
frequency of 2 Hz, and peak-to-peak amplitude of 5°.8 on the sky. The telescope is
fixed in elevation and azimuth, and the beam is centered on declination 40°. The
detector system consists of 3He cooled bolometers with spectral wavelength centered
at 3.3,2.1,1.3 and 1.1 mm. The instrument is characterized by a Full Width Half
Maximum (FWHM) of ~ 1°.35. Application of a robust atmospheric contamination
removal technique allowed us to subtract most of the atmospheric noise from the
data. We successfully show that using the shortest wavelength (1.1mm) channel as
the atmospheric monitor we can greatly reduce the atmospheric contamination in the
other three. Data from each channel were demodulated into six different harmonics
corresponding to the six higher multiples of the reference frequency. The improved
instrumental setup allows us to measure anisotropy in six different multipole ranges
£ = 38 to £ = 136. In 2.1 mm channel we detect significant signals in £ = 38,61,81
and 100 and upper limits in £ = 119 and 136. In channel 1 (3.1 mm) we detect only
upper limits. The detected signal is consistent with our 1994 result and the results
of other CMB experiments probing the same £ range.
xiv
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Chapter 1
WHY MEASURE THE ANISOTROPY OF COSMIC
BACKGROUND RADIATION?
1.1
Introduction
Since the beginning of time human beings have attempted to define their place
in the universe by trying to understand the origin of life, the universe and everything.
Almost every civilization has developed its own description of the cosmos. From the
early ideas of Hindu mythology of a universe created from the nose of Brahma,
to the Abrahamic theology of creation in seven days, explanations of the cosmos
have engaged our spiritual and philosophical interest. Even with the advent of
“modern science” in the nineteenth century our knowledge of cosmology was at
best nebulous. Only in the last century have we seen the emergence of theories
that can describe our Universe in a quantitative manner. The hot Big Bang model
is the theory of choice for many cosmologists. The hot Big Bang theory along
with Inflation can in fact describe the evolution of the Universe in an “acceptable”
manner. This theory describes the creation of the Universe from a hot dense mass.
Over a period of ten to twenty billion years the Universe expanded and cooled
down to its present state. In 1965 the discovery of the isotropic Cosmic Microwave
Background Radiation (CMBR) provided strong support for the hot Big Bang model
since the model predicts the presence of an isotropic black body radiation of a few
Kelvin - a remnant of the fiery past of the Universe.
1
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I must point out that the Big Bang model is certainly not the only theory
that ensued after the discovery of the expanding Universe by Hubble in 1920. The
alternative competing theory is called the steady state model [6],[22] which pro­
poses that, in general, the Universe does not change with time. In contrast to the
explosive creation at t = 0 of the Big Bang model, this model postulates slow but
continuous creation of matter to reconcile with observed expansion. But this model
cannot satisfactorily explain the cosmic helium abundance nor the cosmic microwave
radiation.
The isotropic black body spectrum of the CMB is remarkable, but no less
important is its 10-5 spatial temperature anisotropy. The various theories concerned
with the formation of structure in the Universe predict that the density fluctuations
in the early Universe are imprinted in the anisotropy of the CMB. The study of the
anisotropy in the CMB spectrum is the topic of this dissertation.
1.2
B ig Bang and Inflation
In this section I will give a very brief description of the Big Bang model
and inflation. This review is in no way complete or comprehensive. A detailed
treatment of the topic can be found in books by Kolb and Turner [26], Peebles [37],
and Padmanabhan [35]. First let us review the observations that are the basis for
our present cosmological theories.
• Isotropy and Homogenity : Even though the Universe is inhomogeneous
and anisotropic in a small scale (evident from presence of clusters of galaxies
and nebula), on larger scale d ~ 300Afpc the distribution of observable masses
is approximately isotropic and homogeneous. This is the first observational
basis of cosmology.
• Expansion: In 1920 Edwin Hubble made one of the most important discov­
eries of the twentieth century. By observing that the atomic lines detected in
2
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the spectra of distant galaxies always appear at wavelengths slightly greater
than the laboratory wavelengths of the same atomic spectra (red shifted) and
the magnitude of the redshift is proportional to the distance d of the galaxy
from us, he interpreted this redshift as instances of the Doppler effect due to
the recession velocities v < c (c is the speed of light). The redshift z is defined
as:
*+ l = T ^
Mab
(1-1)
where X0bs is the observed wavelength, and the A/a&is the laboratory wave­
length. Now assuming the redshift being proportional to the distance:
z = - oc d.
c
(1.2)
The constant of proportionality between v and d is the Hubble constant H0,
which is the measure of rate of expansion of the Universe.
v = H0d
(1.3)
In the light of General Relativity, one interprets this redshift as not due to the
recession velocities of the galaxy and the observer but due to the expansion
of the spacebetween the two. The dimensionless timedependent scale factor
R(t) denotes thisexpansion.
The distancesbetween
any twoobjects were
shorter in the past as R > 0. If we denote the scale factor at present to be
R (t0) = 1 then
H
q
= R(t0).
• A ge o f Universe: If the Universe has been expanding for all time at the
same rate, then at some finite time in the past the scale factor must have
gone to zero or the density to infinity at the very moment of the origin of
the Universe in the Big Bang scenario. If there were no forces to slow down
the expansion, the time elapsed since the Big Bang is
H
q
1.
But since there
is m atter in the Universe, gravity works to slow down the expansion, and
3
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the age of the Universe is less than
H
q
1.
The current value of
Ho
is 100
h km s-1 per megaparsec, with 0.4 < h < 1.0. This makes the age of the
Universe about 10-20 billion years. The age of the Universe is independently
measured by geophysical and astronomical measurements of various long lived
radio-isotopes found in meteoritic materials (11-12 billion years) in reasonable
agreement with
H ~ l.
The starting point for modern cosmology is definitely Einstein’s equation:
K - \ 6 t R = ^GTZ
( 1.4 )
where R£ is the Ricci tensor containing the metric and metric derivatives, R =
g^Rfiu is the scalar curvature, T£ is the energy momentum tensor for the m atter
field, and G is the gravitational constant.
The cosmological principle postulates the idea that the Universe is isotropic
and homogeneous on a large scale. Accordingly, the underlying geometry of the
Universe must also be isotropic. The space-time geometry of the Universe may be
completely defined by giving its metric tensor g
In the presence of gravity and
mass, the appropriate metric for an expanding isotropic Universe is the RobertsonWalker metric [32]. In spherical coordinates the metric is
j
ds2 = c2dt2 — R2(t)
+ r2^ 2 + r2sin26d<j)2 .
(1.5)
The quantities r, 6 and <j>are coordinates fixed in the expanding geometry. The spa­
tial part of the Robertson- Walker metric can have three global curvatures, depending
on the value of k. For k = 0 the spatial geometry of the Universe is Euclidean or flat.
For k = +1 the spatial geometry is positively curved and is closed, and for k = - 1
the geometry is negatively curved and open. Recent CMB anisotropy experiments
[12],[21] seem to indicate that the Universe is flat. I will return to this point again
in the section 1.6.
4
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To understand the time evolution of the Universe we need to examine the
evolution of the scale factor R(t). The specific time dependence of the scale factor
is dependent on the type of matter in the Universe. For the isotropic homogeneous
Universe the energy-momentum tensor takes the simple diagonal form
T? = diag[—p, p, p, p]
(1.6 )
where p is the energy density and p is the pressure of the matter filling the Universe.
Local energy conservation requires that the density and pressure satisfy the following
f ? « - s f f r + P).
(1.7)
Using the formforenergy-momentum tensor 1.7 and
ric 1.5Einstein’sequation
the Robertson-Walker met­
can be reduced to a much simpler form knownas the
Friedmann equations:
SirGn
k
— ' “ jp
(?'(R\
( f ij
47rG .
=
. .
>+3p)
(i st
(1-8)
_.
(1-9)
These equations can be used to find the time dependence of R for different kinds of
matter.
If we assume th at we live in a “flat” Universe, i.e k=0, and the Universe is
filled with non-relativistic pressure-less dust, then 1.7 reduces to
Ii =
(1.10)
which is solved with p oc f?“3. The first Friedmann equation yields the solution
R(t) oc t2/3. On the other hand, if the Universe is dominated by relativistic matter
such
that p = p/3 then R{t) oc£1^2. If we consider the Universe to be filled with
a mixture of m atter and
written as
vacuumenergy, then the first Friedmann equation can be
„
(a-
87rG
3
A A:
D9
3
R2
Pm + _
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(1A1)
where pm is the matter density, pyac is the vacuum energy density, and A = 8ttGpvacA is referred to as the “cosmological constant,” which Einstein introduced initially to
counteract gravity in 1917 in order to obtain a static solution, but later he regarded
it as a big mistake after E. Hubble discovered that the Universe is in fact expanding.
Equation 1.11 illustrates the fact that while the vacuum energy is constant
the m atter density decreases as R~3, so for any A > 0 vacuum energy will quickly
dominate the matter energy density . This simplifies to R2 = (A/3)R2, and leads to
R = em ,
(1.12)
an exponential expansion. In the inflationary cosmology scenario [20], in the first
fraction of a second of the history of the Universe, vacuum energy dominates, A > 0.
This changes the relation between space curvature and density. We can define a
density parameter ft as
n s
= T f r U ioZM^/QTTU
r
Pcritical
<113)
where Pcritiad is the value of matter density of the Universe with no curvature or
vacuum energy. It follows from 1.11
k<? - £ = (Cl - 1) H l
(1.14)
This illustrates the fact that a nonzero value of A allows k = 0, a flat Universe for
matter density less than 1. In recent years in the light of observations of distant
exploding stars, supemovae(SNe), the existence of vacuum energy or “dark energy”
is gaining broader consideration. Observers have found evidence that the Universe is
accelerating by using type la SNe as standard candles. A more detailed description
of the supemovae results can be found in e.g. [43], [44],[15], and discussions on the
“dark energy” can be foundin e.g. [8],[4], [14]. Recent CMB anisotropy experiments
[12] seem to lend support to such models indicating that the value of Q \ could be
0.7.
6
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1.3
C osm ic Microwave Background R adiation
According to the “Big Bang” theory the Universe must have started as some­
thing extremely dense at some finite time in the past. During this epoch the Universe
was also extremely hot. The high temperature created an opaque “primordial” soup
of m atter and radiation at thermal equilibrium. But as the universe expanded, the
temperature started falling. Around 4000 K, m atter and radiation started decou­
pling as free electrons joined the protons to form neutral Hydrogen, and the opacity
dropped sharply. The radiation that suddenly became free as the “re-combination”
occurred can still reach us after being scattered from the Last Scattering Surface
(LSS) at z « 1100. Detection of this radiation will allow us to peer deep in time
and space into the dawn of cosmic history. Since the opacity dropped sharply as re­
combination occurred, the present radiation background should have a black body
spectrum, with temperature T0.
The first prediction of the microwave background radiation was made in a
paper by Alpher and Herman [2] in 1948. The idea was later taken up by Gamow
that the hot Big Bang would leave the present Universe at a non zero temperature.
In 1949, Alpher and Herman predicted the “background temperature” ~ 5 K. The
first observational breakthrough came in 1965 with Penzias and Wilson [39], who
observed the isotropic black body spectrum of the CMB with their radio antenna
working at 7.35cm. The explanation of the “excess noise" as the relic of the hot
Big Bang theory was proposed by Dicke and his colleagues [13], who were at the
time preparing to observe the background radiation. The near perfect Planckian
spectrum observed by the FIRAS instrument on board the COBE (Cosmic Back­
ground Observer) satellite confirmed the cosmological origin of the radiation. The
black body temperature is T = 2.726 ± 0.010A' (95% confidence) [30]. The slight
imperfection in the isotropy of the CMB was first discovered by Smoot et ai in
1992 [51].
7
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The standard Big Bang cosmology is very successful in providing a physical
description of the Universe from about 10-2 sec onward [37]. But it fails to provide
answers for some fundamental questions or can answer them only by invoking some
line tuned initial conditions. The key problems are the “flatness problem,” the
“horizon problem” and the u5p/p problem.”
• flatness problem : The Friedmann equation 1.11 with A set to zero can be
written as
(1.15)
Since the m atter density pm scales as R~3 and the curvature term k scales as
R~2, unless the curvature term k is tuned initially to be very small, it will
dominate over the m atter density. The current limits on pm ranges from
0* lP critico / — P — 2 p critical
Thus the question is, how did the density of the Universe get tuned to be so
close to the critical density?
• horizo n problem : One of the features of the Robertson-Walker-Friedmann
Universes is that light can only have travelled a finite distance since the Big
Bang. This puts a physical limit on the distance out to which we can see
objects at a given time. The distance r^ travelled by a photon from the Big
Bang to a time t0 is
(1.16)
For a m atter dominated Universe (like today’s) we can use R = (t/t0)2^3,
which gives
(1.17)
The above equation defines the horizon size today. So in a matter or radiation
dominated Universe the horizon size grows faster than the size of the Universe.
8
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The relic from the thermal radiation of the Big Bang, the CMB radiation has a
Planck distribution with a well defined single temperature accurate to within
a part in 104 over the entire sky. If the photons from different directions of
the sky were never in causal contact, how did these photons conspire to share
the same temperature so accurately?
• 6p/p problem : Here the question is how small inhomogeneities in the density
5p could have developed into such diverse scales of structures today. If these
structures are the results of gravitational amplifications of small density fluc­
tuations, then such fluctuations must have been present in all length scales
since today we see structures in all possible sizes. Where did these density
fluctuations originate?
This is where the Inflationary Cosmology came in about two decades ago
[20]. W ith the inclusion of cold dark matter(CDM), it can extend the standard
cosmology as early as 10-32 sec [53]. The key elements of the Inflationary Cosmology
are a flat Universe, nonbaryonic dark matter and nearly scale invariant adiabatic
density perturbations. The aforementioned pressing questions in cosmology can
be answered within the framework of inflation. I will not go into the details of the
Inflationary Cosmology but briefly focus on how Inflationary Cosmology solves some
of the puzzles of the standard Big Bang theory. Detailed treatment of the topic can
be found in [19], [1] and [26].
According to the model during the period of inflation (~ 10-32 second or
longer) the diameter of the Universe increased by a factor of ~ 105°. After the
expansion the Universe went through a phase transition, and the phase of broken
symmetry took place. The energy density of the false vacuum went into produc­
tion of tremendous amount of particles. The Universe reheated to a temperature
of almost 1027 degrees, and after this point the region continued to expand and
cool at the rate described by the standard Big Bang model. The flatness problem
9
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can be solved by arguing that the rapid expansion of the Universe causes the space
to become flatter and drive the curvature to a very small value. From Friedmann
equation we can see that the ratio Q is driven rapidly towards 1. Without getting
into the details of particle horizons in the exponentially expanding Universe, the
large-curvature scale homogeneity can be explained in terms of the expansion of
the causally connected region before and after inflation. Before inflation the the
scale of the particle horizon can be approximated as r = ct, with t ~ 10-34s. For
scales smaller than r we could expect physical processes to produce homogeneity.
After inflation, the size of such horizons inflated by ~ 1043. At present the size of
such horizon would be much larger still. Since the observable Universe was con­
tained within a causally connected region at a very early epoch, we expect it to be
approximately homogeneous today.
The quantum fluctuations in the scalar field associated with inflation natu­
rally invoke the density fluctuations responsible for structure in the Universe.
1.4
Origin o f C M B Anisotropy
The CMB photons that reach us today have last scattered at the LSS (last
scattering surface) at z cs 1000. At the early epoch of the Universe the radiation
and highly ionized plasma were coupled together. As the Universe expanded and
became cool enough for protons to capture electrons (the recombination epoch),
the photons decoupled from the matter. The mean free path of the photons became
extremely large. The recombination took place over a width of Az ^ 100. The epoch
and the thickness of the last scattering surface is not dependent on any particular
cosmological model [56], but the amount of scattering is dependent on the baryon
density parameter, and the angular scales depend on the total density parameter.
The total isotropy of the CMB spectrum is expected as long as the Universe is
perfectly isotropic and homogeneous in all angular scales. But we already know that
the Universe is very inhomogeneous in all scales up to many megaparsecs, so this
10
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90
80
Standard CDM
70
60
SO
40
30
20
100
10
1000
I
Figure 1.1: Angular power spectrum of CMB anisotropy according to the Standard
CDM model with
= 0.05,SIcdm = 0.95,Da = 0,n4 = 1, and h = 0.5.
The plot was generated using CMBFAST [50]
must produce anisotropy in the CMB temperature at some level. In this section I will
outline few of the main processes responsible for producing temperature fluctuations
in the CMB.
1.4.1
D ipole Com ponent
The dipole moment of the CMB anisotropy has an amplitude of
ATdipo,e/T0 = 1.2 x 10-3.
The most widely accepted explanation for this anisotropy is the “Doppler effect.”
If an observer is in motion with respect to a frame in which the CMB is isotropic,
11
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she/he will see the photons blueshifted to higher energy in the direction of her/his
motion. On the other hand, the photons will be redshifted to lower energy in
the direction opposite to her/his motion. In the non-relativistic case, assuming a
blackbody spectrum for the CMB, the dipole moment can be written as [38]
T ^u
= v£
T0
c
(1 lg )
v
’
where v is the velocity of the observer. Observations have established that the
Earth is moving at 360 ± 20km s -1 in the direction of right ascension a = 11.2 h
and declination S = -7°.
On the basis of chronological order, the sources that produced anisotropy in
the CMB can be listed as follows:
• Prim ary Anisotropy: caused at the time of recombination
- (i) Gravitational potential
- (ii) Doppler effect due to peculiar velocity
- (iii) Density fluctuations
- (iv) Damping
• Secondary Anisotropy: caused after recombination from processes that
affected the photons as they traveled from the last scattering surface (LSS) to
the observer.
- (i) Gravity
- (ii) Local reionization
- (iii) Global reionization
12
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1.4.2
Sources of Prim ary anisotropy
The gravitational potential 0, the peculiar velocity vp, and the density fluctu­
ation S of the region from which the CMB photons last scattered left their imprints
on these photons. The resulting fluctuation in the temperature can be written as :
Y ~ (r) =
r) - r.v(r) + icf(r).
<j>(
(1.19)
The speed of light c = 1 in the above expression. The first term is contributed
from the photons that last scattered in a gravitational potential well 0 < 0 and
were redshifted as they climbed out of it. The second term is due to Doppler
redshift of the photons that scatter from m atter moving away from us with peculiar
velocity vp, and the last term arises due to the photons emanating from over-dense
regions (£ > 0 ), which are hotter because denser regions are at higher temperature.
The vector r is the comoving distance to the surface of the last scattering. The
gravitational potential, peculiar velocity and density 6 are calculated at the time of
re-combination z ~ 103. On very large scales, 6 > few degrees, S « -20, so the right
side of equation 1.19 is 0/3. This is called the Sachs-Wolfe effect. The flat part of
the CMB spectrum is dominated by this effect.
Before the period of recombination, when the temperature of the Universe
was T > 3000 K, the photon-baryon plasma could be considered as a single fluid.
This plasma underwent acoustic oscillations under gravity until it recombined [24].
Primordial perturbations gravitationally attracted this fluid and compressed it into
high density regions. The fluid collapsed under its own gravity and became more
and more over-dense until photon pressure halted the collapse and it rebounded
up setting up acoustic waves. Regions of compression and rarefaction at the last
scattering surface represent hot and cold spots respectively in the CMB. After recom­
bination, as the photons climbed out of the potential wells, they suffered additional
13
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gravitational redshift. The combination of these effects produced the temperature
anisotropy that we observe today.
The fourth term listed under primary anisotropy is a negative source of
anisotropy in the sense that it smears out the fluctuations. The finite duration
of the recombination period leads to the damping of the anisotropies at the smallest
scales [49]. The finite thickness of the last scattering surface A z ~ 100 produces two
damping effects. One of them is due to random walk of photons which decreases
fluctuations by diffusion, and the other one is due to the destructive interference
of acoustic waves for wavelengths A less than the thickness of the last scattering
surface. Both of these effects lead to the suppression of peaks at large is.
1.4.3
Secondary anisotropy
After recombination, the photon temperature can only be altered through
gravitational effects. If the gravitational potential is time dependent then the tem­
perature fluctuation can be obtained by integrating the derivative of the potential
along the photon trajectory. This is called the integrated Sachs-WoIfe(ISW) effect.
The ISW effect on a single photon can be represented as
AT
—
r •
=
J
(f> [r(t),t
]dt;
(1.20)
where 0 is the conformal time derivative of the gravitaional potential 0 at a given
position. The nonzero value of 0 gives rise to early ISW, late ISW and the ReesSciama effects, respectively.
Right after recombination, the gravitational potential 0 decays slightly due
to the non-negligible contributions from photons. This causes the early ISW effect.
The effect of early ISW shows up on large angular scales t < 200. In models with
^matter + A / 1 the 0 is also non zero at low z. This gives rise to late ISW effect. It
also leaves its imprint on large scales. The Rees-Sciama effect is the fluctuations in
14
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the CMB caused by the nonlinear structure formation. In standard CDM models
the contributions from this effect are much smaller than the primary anisotropies.
If the baryons get reionized after the photons free-stream to us after recombi­
nation, the other two fields v and 5 can influence the CMB photons. The reionization
can happen either locally or globally. The kinematic Sunyaev-Zel’dovich (SZ) effect
is due to the Thomson scattering of the CMB photons off of free electrons in the
hot intra-cluster gas. This causes a blueshift in the direction of the cluster.
The thermal Sunyaev-Zel’dovich (SZ) effect is independent of cluster velocity.
The high temperature of the cluster gas distorts the Planckian spectrum of the CMB
by depleting the Rayleigh Jeans region and overpopulating the Wein region. The
frequencies above 218 GHz are blueshifted while the frequencies below are redshifted.
Both the kinematic and the thermal SZ effect are examples of the local reion­
ization processes. The effect of these two processes on the overall CMB spectrum
is not appreciable. On the other hand, a global reionization can have a dramatic
effect on the CMB power spectrum. The fluctuations in smaller angular scale will
be drastically suppressed, but the power on large scales will remain unaffected.
1.5
A ngular Power Spectrum o f CM B Anisotropy
The temperature fluctuations A T / T in CMB can be expressed in terms of
spherical harmonics
—
(M ) =
If we assume that the anisotropy of the CMB is a Gaussian random field then
the mean and the two point correlation function will completely define it. Since a
priori the mean of the fluctuations are known to be zero, we need to evaluate only
the correlation function on the sky. The sky correlation function can be defined as
[56]
AT(n t ) AT(nj)
15
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( 1 .21 )
where the average is taken over the sky, and
&21
is the separation angle of two points
on the sky. Using the properties of spherical harmonics the correlation function is
expressed as
C(#2 i) =
£ (2 « + l)C,P,(cos«)
(1.22)
where the Cts are the coefficients of the Legendre decomposition of the correlation
function, and they are related to the ae<ms by
^m,m''
The statistical properties of the AT / T are completely specified by the angular
power spectrum Ct. The results of anisotropy experiments are usually expressed in
terms of [£(£ + l)CV/27r]lyf2.
1.5.1
Cosmological parameters from CM B studies
The total anisotropy spectrum contains a wealth of information about pa­
rameters of the cosmological model. The amplitude and the position of the power
spectrum can be used to distinguish between different cosmological models.
The heights of the peaks are dependent on the baryon to photon ratio. In­
creasing the baryon fraction fit will make the peaks higher. Since baryon number
density is a function of flb/i2, the peak heights are ultimately dependent on the
quantity Qf,h2.
The physical scale associated with the first peak is the horizon size at last
scattering. The angle subtended by the horizon, at redshift 1000, depends on the
geometry of the Universe. For a fiat Universe the photon geodesics are straight
lines. However, for a closed or an open Universe the geodesics converge and diverge
respectively. Thus a given physical scale at high redshift would subtend a larger
or a smaller angle today. For an open Universe the angle subtended today by the
horizon would be smaller by a factor of Q ^2. The angular position of the first peak
is thus sensitive to the spatial curvature of the Universe.
16
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If the Universe is flat
(fim + fiA
=
1)
then increasing the cosmological constant
A will increase the height of the lowest multipoles via late ISW.
1.6
Experim ental detection o f CM B A nisotropy
Since the successful detection of CMB anisotropy at large angular scale (9 ~
7°) by the COBE satellite experiment [51] almost a decade ago a plethora of ground
based and balloon experiments have detected anisotropy in different angular scales.
With the advent of newer and better technology, we can get more reliable results
from experiments. Advances in data analysis have made it possible to compare
results from different experiments. We will discuss this in detail in Chapter 5.
At large angular scale, the detection of the CMB quadrupole by COBE pro­
vided the normalization for the power spectra. The CMB anisotropy results have
favored a low baryon density, fifth2 ~ 0.03. The combined result of the anisotropy
experiments so far strongly indicates the presence of a peak at I as 250. The available
data suggest a flat Universes with fim -f-
«s 1. The new data from the two most
recently published experiments BOOMERANG [12] and MAXIMA [21] apparently
confirm this and the data also indicate the presence of a second and even possibly
a third peak. In figure 1.2 the current observational data are depicted against the
theoretical model of the CMB power spectra. CMB anisotropy results along with
Big Bang nucleosynthesis and large scale structure results can ultimately measure
the cosmological parameters.
17
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I— I
I I 11111------------1— I I I 11111
BOOMERanG
MAXIMA
• Previous r
W.Hu 3/09/00
Figure 1.2: Current Observational results in CMB Anisotropy [23]. The black
curve: \ C D M Qm = 0.35, h = 0.65, fl^/i2 = 0 .02 , n = l; red curve:
K C D M n m = 0.4, h = 0.8, fift/i2 = 0.03, n = 1 (Courtesy Wayne.Hu,
2000 )
18
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Chapter 2
INSTRUMENT
Even though Penzias and Wilson measured the CMB spectra as early as 1965
the intrinsic anisotropy of the CMB (apart from its dipole anisotropy) was not de­
tected before 1992. The main reason for this delay was the inadequate sensitivity of
detectors to the CMB fluctuation of A T /T 10-5. Advancement in detector tech­
nology during the last couple of decades has made it possible to detect fluctuations
of this magnitude with reasonable accuracy.
Our instrument consists of three major components: telescope, cold optics
and data acquisition system. The same telescope coupled with a different cryostat
system was used in 1994 to measure anisotropy from Tenerife [40], [17]. The cold
optics consists of a cryostat that houses the four bolometric detectors with wave­
lengths centered at 3.3,2.1,1.3 and 1.1mm, and a miniature 3He refrigerator. The
telescope is surrounded by two layers of shields to minimize ground radiation. The
smaller inner shield is attached to the antenna and moves with it. A much larger 45°
hexagonal shield, which is attached to the ground, surrounds the whole telescopedetector assembly. Figure 2 shows the schematic of the telescope along with the
cold optics and the data acquisition box.
2.1
Telescope
The telescope is an off-axis Gregorian telescope that was first used to measure
CMB anisotropy in Antarctica [41]. The primary mirror is an off-axis parabolic
mirror with a focal length of 1.33 m and a diameter of 45 cm. The secondary mirror
19
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Figure 2.1: The telescope with the cryostat and the data acquisition box
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
consists of an off-axis hyperbolic mirror of diameter 28 cm. The mirrors are made
with silver-coated carbon fiber reinforced plastic(CFRP) [40]. The moving metallic
parts connected with the mirror are made of aluminum and titanium, and the fixed
structures of the antenna are made out of stainless steel. The beam is switched on
the sky by sinusoidally wobbling the primary mirror at 2 Hz, and the magnitude of
the wobbling angle peak to peak is 5°.8. The advantage of wobbling the primary
mirror instead of the secondary mirror is that this method reduces spillover of the
ground radiation synchronous with the modulation frequency, since the receiver horn
is coupled only to the stationary secondary mirror. Moreover, this technique also
produces less aberration and smaller offset in the data.
The primary and the secondary mirror are connected by a fixed arm. The
primary mirror can be moved along a semicircular guide rail which allowed us to set
the elevation of the antenna. The optical axis of the receiver and the elevation axis
of the antenna coincide. The antenna can be used for performing an elevation scan
without moving the detectors. Sky scans were performed in two different modes: a)
drift-scan: the antenna was fixed towards the North at an elevation of ll°.5 degree
and the sky rotation was used to explore a strip of the sky at fixed declination. The
beam oscillated, along the declination 40 strip of the sky. b) Zenith Observation :
The primary mirror pointed at the local zenith and the sky rotated perpendicular
to the wobbling mirror.
2.2
Cold O ptics
The cold optics consists of the detectors, different band-pass filters, and the
refrigerator all housed in a 4He cryostat. The operating temperature of the detectors
is 0.33 K. To achieve this low temperature the detector system must be very well
isolated from the ambient, and the heat transfer must be minimized. The cryostat is
vacuum pumped to ensure the least possible heat conduction between the different
parts of the dewar. A hollow outer wall separates the liquid nitrogen bath at 77
21
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F igure 2.2: Schematic of the telescope with the inner conical shield and the outer
hexagonal shield.
K from the ambient temperature. The inner shell is regulated at 4.2 K by a liquid
Helium bath. A copper thermal link connects the bath to the 3He refrigerator. The
refrigerator, in turn, is in thermal contact with the cold optics block carrying the
four bolometers. Figure 2.3 depicts the cryostat.
2.2.1
R e frig e rato r
We used a m iniature3He refrigerator working at 0.3 K. A detailed description
of the refrigerator can be found in [10]. Figure 2.4 depicts the different parts of the
refrigerator. The refrigerator consists of a charcoal cryopump and an evaporator
(Helium pot) connected by two steel tubes to the condenser plate. The condenser
plate is in thermal contact with the 4He cryostat cold base plate. The cryopump
includes a heater and a heat link to the 4He bath. It connects to the condenser by
means of a steel tube. The thermometers on the cryopump, the evaporator and the
condenser plate (base-plate) are monitored during the cooling process.
The cooling cycle [52] starts with all the parts of the the refrigerator at the
liquid 4He bath temperature of 4.2 K. To achieve the operating temperature we
22
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Liquid
Liquid
<He
Mylar
window
3He refrigerator
Filter
Optics
block
F igure 2.3: Cryostat
need to pump on the liquid AHe bath until the temperature of the bath reaches 1.7
K. At this point, the heater connected with the cryopump is turned on at 11 V with
a current of 68 mA for 25 minutes. As the temperature of the cryopump rises to 35
K, almost no 3He remains adsorbed in the charcoal. This raises the pressure and
exceeds the saturated vapor pressure o f 3He at the bath temperature causing it to
condense. When all of the 3He is collected in the evaporator, the heater is turned
off and the cryopump is activated as it relaxes back to the bath temperature. This
reduces the vapor pressure above the liquid 3He and causes it to evaporate. The
evaporator temperature quickly drops and stabilizes around 0.33 K. The evaporator
is connected by a copper thermal link to the optics block carrying the bolometers,
which stabilize to the operating temperature in 2-3 hours. The hold time for the
23
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F ig u re 2.4: Miniature 3He Refrigerator
temperature is approximately 24 hours. Figure 2.5 shows the temperature of the
base plate during data collecting.
2.2.2
C old optics block
The photons that are reflected by the primary mirror onto the secondary
mirror pass through a Mylar window on the cryostat. These photons then travel
through two back-to-back Winston cones. A Winston cone is a high-collection non­
imaging optics piece which only allows light in its field of view to enter the system
and reflects back all other photons. Thus the first Winston produces a well defined
beam on the sky. The input aperture of this / / 3 Winston cone is 1.27cm and the
half angle is 9°.5. So the throughput of the system is:
ACl = 0.11cm2str
(2.1)
where A is the area of the input aperture and Q is the solid angle subtended by the
beam.
The second Winston cone terminates in a combination of fluoro-gold (FG),
black polyethylene(BP) and Quartz(QZ) glass filters. These filters block the high
24
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Figure 2.5: Resistance of the thermometer on the baseplate during a cooling cycle
of the refrigerator. The resistance of the thermometer is proportional
to the temperature.
frequency photons from reaching the bolometers. Figure 2.6 shows the 4.2 K and
the 0.33 K filters.
The photons passing through the 4.2K filter can then propagate through the
detector system. A gap between the 0.33 K optics block and the 4.2 K filter system
thermally isolate them. The 0.33 K optics block is thermally connected to the
refrigerator. To reduce heat leak it is suspended by Kevlar wires inside a support
cage. The four bands are defined by using a combination of beam splitters and
band pass filters. Residual out of band high frequency photons are blocked by a
combination of fluoro-gold, black polyethylene and Quartz glass filters. Table 2.1
and figure 2.7 summarize the characteristics of the filters.
25
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Q_ N
CQ O'
e>
=cTO BOLOMETER
WINSTON
CONE
4.2 K
0.3 K
F ig u re 2.6: The 4.2K and 0.33K filters. A 10 mil gap separates the 4.2K optics
from the 0.33K optics
T able 2.1: Characteristics of the Filters
______________________ Flat Spectrum_____ CMB Spectrum
Channel Apeo* (mm) £(GHz) A//(GHz) £(GHz) Ai/(GHz)
1
97
26
3.3
97
25
2
32
2.1
169
146
30
3
45
238
44
1.3
243
4
1.1
270
47
265
46
2.2.3
B olom eters
After passing through the Winston cones, the incoming photons are incident
on the four bolometers. The main components of a bolometer are [45], a radiation
absorber, a supporting substrate, a thermometer, a thermal link, a heat sink and
a mechanical support. To function efficiently, each of the components must meet
certain criteria described below. Figure 2.8 depicts the principal components of a
bolometer.
• Radiation Absorber: It must have appropriate dimension to intercept incident
radiation. Also it requires large absorptivity over the range of frequency of
26
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1.0
0.8
0.6
0.4
z 0.2
taJ
0.0
100
400
Figure 2.7: Normalized transmission of the Bandpass filters
interest, and low heat capacity.
• Supporting Substrate: It must also have a low heat capacity and a large
thermal conductivity to ensure that it remains isothermal during the operation
of the bolometer.
• Thermometer: It is usually connected to the substrate or the absorber itself.
It must have low heat capacity, low electrical noise and adequate thermal
dependence of electrical resistance to measure the incident radiation efficiently.
The thermal resistance dependence of a thermometer is characterized by the
coefficient of resistance given by
°(T) = Z § -
<«>
• Thermal Link: It transfers the incident power from the absorbing material
to the heat sink. For efficient transference it must have appropriate heat
27
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conductance and low heat capacity. The response time of the bolometer is
dependent on good thermal contact.
• Heat Sink: The heat sink absorbs the heat input and keeps the temperature
of the system unchanged. To do so it must maintain stable temperature.
• Mechanical Support: The mechanical support must be stiff enough such that
the resonant mechanical frequencies are much higher than the operating fre­
quencies of the bolometers. To avoid heat leaks the support must have low
heat capacity and low thermal conductance.
.
^ .7
'W'
Figure 2.8: Schematic of a composite bolometer. a)the radiation absorber, b)the
substrate, c)the thermometer, d)the thermal link and e)the heat sink.
Bolometers are square law detectors. As photons strike the radiation absorber
of the bolometer and deposit energy onto it, the temperature of the bolometer
changes, and subsequently the resistance of its thermometer. In our system a voltage
drop across the thermometer proportional to the incident power is measured. A
constant bias current / Wo4 is maintained through the thermometer. To calculate
the responsivity of a bolometer, let us assume power P0 + P i e (here ui refers
to the frequency of incoming radiation) is incident on a bolometer with resistive
28
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thermometer R and thermal conductance G. The resulting temperature of the
bolometer is give by:
TB = T 0 + T leiut.
(2.3)
The resistive thermometer in a bolometer is biased with a constant current I, which
generates time varying heat
P R {T ) = / 2[/?(T.) + — Tie*11}.
(2.4)
The power lostby the bolometer to the heat sink is G(T b - T0).To calculate the
responsivity 5 b of a bolometer we first equate the the inputpower to the output
power and the heat stored in heat capacity C,
P0 + Pie *"4 + I 2R(T0) +
= G(T0 - T,) + G T ^ 1 + i u C T ^ .
dT
(2.5)
Equating the time dependent parts and the time independent parts, respectively,
yields
P„ + I 2R(T,) = 5 (7 ; - T,)
^
(2.6)
= G + i w C - ^ I 2.
(2.7)
The voltage responsivity of a bolometer can be defined as
A voltage drop
A absorbed signal power
Using equation 2.7 we can write S b as:
9
- V'
S b ~ Pi
[ G - P ’( g ) + iu C l
[V/W],
(2.8)
The equation for responsivity 2.8 is derived for an idealized bolometer, but in gen­
eral, the responsivity is influenced by thermal feedback, which in turn changes the
conductance G of the bolometer. A less accurate but simpler method of calculating
29
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the responsivity of a bolometer is to measure its response to changes in the electrical
power dissipated in the thermometer. This responsivity is often called the electrical
responsivity S e [25]. Briefly, one measures the dc I-V curve of the bolometer for a
range of currents and voltages near the anticipated operating point. The load-curves
for our four channels at three different temperatures are shown in figure 2.9. The
values of R = V / I and Z = dV/d.1 are calculated from the I-V curve for a range of
bias points and the dc electrical responsivity is calculated as each point from Jones ’
expression
_
E=
Z
—R
21R '
*
This equation can be recovered from equation 2.8 in the special case of no optical
power. S e in that case can be written as
aK
S° = G ^ P Let us now discuss the noise in bolometeric detectors. Instrumental noise is
typically expressed in terms of Noise Equivalent Power, or NEP. It is defined as the
required signal power such that the Signal to Noise ratio is equal to 1. The sources of
noise in a bolometer detector system can be divided into two groups, intrinsic noise
and extrinsic noise [29]. Intrinsic noises are photon noise, Johnson noise and phonon
noise resulting from conductive heat flow out of the bolometer. Photon noise in the
bolometer is caused by the random fluctuations of the incoming photons. The mean
square power fluctuation P% per bandwidth B is given by
f i = 2 / P„ / u ^ + /
^
\W*/Hz]
(2.10)
where P„ is the spectral power absorbed in the bolometer, v is the frequency of
the incident radiation, h is the Planck’s constant, and AQ is the throughput of the
detector system. The absorbed power in a bolometer in the frequency range
P =
T)
m .
30
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to
( 2. 1D
30
25
l(nA)
20
- e - Channel 1
Channel 2
—
&r~ Channel 3
- e - Channel 4
15
10
@ T = 20.43K
5
20
40
60
80
100
V(mV)
Figure 2.9: Load curves of the four bolometers looking at a black bodies at 20.43K
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where k is Boltzmann’s constant and x = h u /k T . The expression for NEP for
photon fluctuations in bolometers is then given by:
photon
( 4fc5 \ T M fier / r
[<*&)
7]
[ I
(2.12)
The Johnson noise is generated by the thermal motion of the electrons in the ther­
mometer resistor R at temperature T. If T is the bolometer temperature, Pe is the
electrical power dissipated in the bolometer, ui is the angular frequency of incoming
radiation, and r is the time constant of the bolometer response then Johnson noise
is given by [36],
Johnson
The phonon noise is caused by the passage of quantized carriers of energy through
the thermal conductance G. The resulting NEP is
{NEP)phonm = 4kT2G.
(2.14)
This noise, similar to photon noise, is independent of the measurement frequency.
Since all three of these noise sources are uncorrelated, the total NEP is the
square root of the sum of the equations 2.12 2.13 and 2.14. Both Johnson and
phonon noises are dependent on temperature, so to reduce these noises bolometers
are cooled to very low temperature.
The temperature fluctuation in the heat sink (N E P )2HS, the amplifier noise
(N E P )lmp and the low frequency noise also contribute to the total noise spectra
of a bolometer. Bolometers are also plagued by excess low frequency noise, often
called the 1 / f noise. In most experiments the background photon noise from the
atmosphere, thermal radiation etc., dominates other sources of noise.
32
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2.3
Electronics System
The electronics system is comprised of detector electronics, housekeeping
electronics and the computer interface. Circuit diagrams of the electronics can be
found in [34]. The detector electronics system involves the power supply and relays
for the bolometers and the amplifiers. The bolometers are powered by 12V' batteries
and the relays are powered by 18V voltage regulators.
Each of the bolometers along with its 20MQ resistor are maintained at 0.33K
operating temperature while the rest of the electronics are maintained at ambient
temperature. The signal from each of the bolometers is amplified in two stages before
being sent to the micro-controller electronics box. Adjustable gain factors allow us
to detect both a strong and a faint signal with good resolution without saturating
the ADC converters. The electronics box is well shielded against external RF noise.
In addition, all external connections pass through a n filter which prevents external
RF noise from entering the system.
The housekeeping electronics maintain and regulate the cryogenic system,
keep time and temperature, monitor power supply voltage, regulate mirror oscil­
lation and the orientation of the antenna. Stable low temperature of the cryostat
is paramount for normal operation of the detectors. The housekeeping electronics
keep track of the detector temperature by means of two Germanium thermometers,
one attached to the 4He bath and the other attached to the *He refrigerator. The
frequency of the mirror oscillation is controlled and monitored by measuring the
phase differential between a known oscillator and the optical encoder. The actual
wobbling frequency is sent to the computer interface, and the frequency is changed
accordingly if it is off the desired value. The pointing of the antenna is controlled
by a programmable servo controller which drives two motors, one in azimuth and
the other in elevation. It monitors the position of the antenna via two encoders.
The controller receives and sends back information through a RS232 serial line to
33
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the computer.
A 68//C705C8 micro-controller interfaces with the computer via an R S 232
bus. For every cycle of the mirror (every half second) one long frame and 63 short
frames are sent to the computer. A long frame is 30 bytes long and a short frame
is 12 bytes long. Table 2.2 summarizes the structures of the frames. If any error
occurs during the cycle, it is reported on the long frame and the controller then
waits for the next reference signal to begin converting again. A calibration is done
at power up and every 60 minutes.
Data are collected and stored in a computer for analysis. The data collecting
software is also capable of performing lock-in on the data in real time.
34
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T a b le 2.2: Structure of d ata frame
BYTE
Bit
1
8
2 -4
5,6
7,8
9,10
24
16
16
16
16
11,12
1 3 -1 6
17
18
19
20
21
22
23
24
25
26
2 7 -2 8
2 9 -3 0
32
8
8
8
8
8
8
8
8
8
8
16
16
Content
Flag identifier of short and long frame
Time counter 3.25521 n second resolution
ADC1 signal from channel 1
ADC2 signal from channel 2
ADC3 signal from channel 3
ADC4 signal from channel 4
End of short frame
Independent real-time second counter
Local temperature
-15 Volts power supply voltage
+15 Volts power supply voltage
+5 Volts power supply voltage
Temperature input 1 He 4 bath temperature
Temperature input 2 Cryopump temperature
Amplifier 1 gain
Amplifier 2 gain
Amplifier 3 gain
Amplifier 4 gain
X position
Y position
End of long frame
35
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Chapter 3
OBSERVATIONS AND CALIBRATION
The anisotropy in the Cosmic Microwave background radiation is on the order
of 10-5 or less of the background temperature. To measure such a small signal we
need a very sensitive, efficient and well calibrated experimental setup. The field of
view of the telescope together with the observing strategy determine the angular
scale at which we can make measurements, so it is important to use a system with
well defined beam and judicious choice of an observing technique.
3.1
O bserving Strategy
The observing strategy for the “dec40” mode is similar to the observing
strategy used for the 1994 campaign. This kind of observing strategy involved
chopping the beam in a sinusoidal manner on the sky by chopping the primary
according to : 9(t) = 60, <j>(t) = <j>0 + asin(u)t + ip), where 6{t) and (j>{t) denote
elevation and azimuth, respectively; 90=0° and <j>0=78°.5 denote the initial position
of the antenna and a is the zero to peak wobbling amplitude at a reference frequency
of 2Hz. We chose to observe using the ’dec40’ mode along with the ’zenith’ mode
in 1996 [47] so we could confirm our previous results from 1994.
Data were always collected during the night. The high altitude of the ob­
serving site usually provided us with cloudless nights. When the weather conditions
were too bad, the data collected during that period were not used for final data
36
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analysis. Also since we collected data from the same site using two different ob­
serving techniques, each of them can be used as a check to determine if the signal
detected is truly in the sky.
3.2
Dem odulation o f the D ata
A lock-in is applied to the time series data to reduce low frequency noise. The
lock-in process involves convolving the time series data with a periodic reference sig­
nal. This process attenuates any noise frequency that is different from the reference
frequency. If s(t) is the sky signal at time t, and Ln(t) is the lock-in function, then
the convolved signal is given by
T
1
s» = f l
(3.i)
where ui is the angular reference frequency and T is the time period. We demodulate
the time series data by applying a lock-in function Ln{u)t) which is a combination
of sine and cosine function with frequency nu/, where n is an integer and u; is the
angular frequency of the mirror oscillation:
Ln(t) = cos(nut) + isin(nujt).
(3.2)
Since the time series data is really a discrete function of time, the integral
in equation 3.1 must be replaced by a summation over the number of data points
collected within the complete oscillation of the mirror. The nth component of the
deconvolved data can be expressed as
1
N = 64
O vi
5” = ^
(3.3)
where d(i) is the ith sample of the data collected during the cycle of the mirror,
and N is the total number of data samples during the cycle. Equation 3.3 is very
similar to the discrete Fourier Transformation of the data. In fact applying a lock-in
function 3.2 to the signal is equivalent to taking the Fourier Transformation (FT) of
37
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the signal with the cosine and sine components as real and imaginary parts of the
FT, respectively. The high frequency noise is removed from the time series data by
means of a band-pass filter. This filter attenuates any frequencies higher than 12Hz,
and also removes the dc component from the data. Due to the presence of this filter
we can only demodulate the signal up to the 6th multiple of the reference frequency.
Also the amplitude of the chopping angle along with the beam size constrain us
from demodulating the data to higher multiples of the reference frequency.
If there is no phase difference between the sky signal and the reference lock-in
function, then for odd n the sine component (in-phase component) of the demodu­
lation contains the sky-signal coherent with the reference motion along with instru­
mental noise, and the cosine (quadrature) component contains no sky-signal but
mostly instrumental and other noises. Similarly, for even n the cosine component
should contain all the signal. In practice, the data collecting electronics introduce
a phase delay in the detector output signal, so both the in-phase and the out-phase
components of the demodulated data contain signal. We can write the sine and the
cosine components as
(3.5)
where <j>n is the phase difference. Since, On and 6„ are equivalent to the real and
imaginary part of the FT of the real data, the phase angle <j>n is given by
(3.6)
The phase angles for each channel and each harmonic are tabulated in 3.1.
3.3
C alibration
The detectors used in this experiment were tested and calibrated in the lab­
oratory [41]. We decided to use the Moon as our calibrator in the field because
38
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Table 3.1: Phase angles for each channel and harmonics. The values given are in
radians.
Harmonic
1J‘
2nd
3rd
4 th
5th
6th
Channel 1 Channel 2
-0.97
-0.99
-1.18
- 1.11
-0.87
0.80
-0.58
-0.55
- 0.68
- 0.68
-0.70
0.78
Channel 3
Channel 4
-0.60
-0.55
- 0.68
- 0.68
-0.70
0.78
- 0.88
-0.95
-1.15
-1.15
-0.95
0.78
this would allow us to include the efficiency of the telescope and the atmospheric
efficiencies in our calibration parameters. The Moon has an angular diameter of
0°.5 which makes it a perfect point-like source for our beam. We observed the Moon
near the full phase 4 times during the experiment. It gave us a good way to redefine
our beam size and other telescope parameters. We performed raster scans on the
Moon which provided us with an accurate method for making a 2-D map of the
beam. The brightness temperatures of the Moon [18] at our wavelengths are given
in the following table.
T able 3.2: Brightness temperature of the Moon at our wavelengths
Brightness Temp K
Channel 1 Channel 2 Channel 3 Channel 4
288.74
258.75
275.96
286.15
We used the temperatures quoted in the table 3.2 for doing a simulation of
the Moon crossing. The Moon temperature at each frequency had to be corrected
by the Rayleigh-Jeans factors and the atmospheric attenuation factors. The rele­
vant Rayleigh-Jeans factor and the atmospheric attenuation factors for the May 2nd
39
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Moon crossing are listed in table 3.3. The Rayleigh-Jeans factors convert the moon
antenna temperature to thermodynamic temperature and the atmospheric attenu­
ation factors (a function of the Moon elevation) take into account the reduction of
the Moon signal due to varying amounts of air-mass.
Table 3.3: Rayleigh-Jeans factors and the atmospheric attenuation factors for each
of the four Channels.
Channel
1
2
3
4
R-J Factors
1.288
1.660
3.660
4.820
Atmospheric Attenuation
0.714
0.645
0.485
0.424
The calibration process involves three steps:
• Simulating the Moon crossing through the detector beam
• Finding the correct phase angles
• Normalizing the simulation to the data.
3.3.1
M oon Crossing Sim ulation
We simulated the Moon crossing by assuming the Moon to be a disc of radius
of 0°.5 and the telescope beam a Gaussian of Full Width Half Maximum (FWHM)
wobbled sinusoidally in azimuth with amplitude a . The motion of the moon in both
x and y directions were incorporated in the beam on the sky. The 2 dimensional
Gaussian beam is denoted by B(x(t),y{t))
B(x(t),y(t)) = exp
{x - P)2 + ( y - 7)21
2a2
(3.7)
with P = asin(ut) + vt, and 7 = vyt + C, v the speed of the moon in x direction,
vy the speed in y direction of the moon and C the offset caused by the beam not
40
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crossing the moon exactly horizontally. We find that our beam does not cross the
Moon exactly through its center but rather along a smaller chord, this produces
an asymmetry in the lobes of the harmonics. Temperature T \f0on and the Gaussian
beam integrated over the area of the moon with appropriate normalization expresses
the Moon temperature T* in the ith channel.
B(x(t),y(t))dxdy
(3.8)
where T mooti is the moon temperature in the i th channel, N is the normalization
and B(x(t), y(t)) is the Gaussian beam in 2-D.
The double integral was reduced to a single integral by first integrating with
respect to y
(3.9)
which can then be numerically solved by using Gauss-Legendre quadrature with
A=
and B = e r / ( ^ “ )
We also accounted for the phase of the moon and the atmospheric attenuation
in our fitting. The moon temperatures used in the simulation have 3% uncertainty
[18]. The overall calibration error is 10%.
Moon observations were performed using two observing strategies, drift scan
and raster scan. In the drift scan mode the Moon observations consisted of fixing
the telescope in azimuth, az=0°, and the elevation of the Moon as it transited the
Meridian. The instrument collected data as the Moon drifted through the instru­
mental beam. In the raster scan mode the antenna was initially oriented in the
same manner, but as the Moon drifted through the beam the elevation of the beam
changed by preset increments to scan the Moon. Two horizontal drift scans and two
raster scans of the Moon were collected during the campaign. The elevation, and
percent illumination of the Moon as well as the method of scans for these calibration
files are summarized in table 3.4.
41
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T ab le 3.4: Summary of the Moon Calibration Files
Date
scan mode
May 2 1996 horizontal
May 12 1996 horizontal
raster
May 25 1996
raster
May 28 1996
Percent Illumination
100
27
50
81
Elevation
52° .66
59° .40
56°
54° .82
Due to inclement weather condition, the calibration data collected on May
12 and May 28 were not used. The calibration parameters were mainly estimated
using the May 2nd calibration file, and tested against the parameters estimated from
the May 25th raster scan file.
Fitting of the Moon signal with our Moon simulation allowed us to determine
the wobbling angle a, the beam width F W H M , as well as the phase angles and the
calibration constants r . Since we observed the Moon in both horizontal and raster
scans the degeneracy of the offset in elevation and the Moon temperature in data
could be eliminated by using the raster scan data.
3.3.2
C a lib ra tio n P a ra m e te rs
The chopping angle amplitude and the Full Width Half Maximum (F W H M )
for all four channels and six harmonics are tabulated in tables 3.5 and 3.6. The
parameters are estimated by minimizing the x 2 between the simulation and the
data. Figures 3.1, 3.2, 3.3 and 3.4 shows the real moon crossing data from May 2,
1996 and the simulated moon crossing data for channel 1, 2, 3 and 4, respectively.
The values of a and F W H M seem to vary slightly among different harmonics
for a given channel. For estimating an average value of the chopping angle amplitude
and the beam width, more weight was given to the lower three harmonics, since these
42
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T a b le 3.5: Chopping angle am plitude for each Channel and Harmonic
5th
6th
2.°85
2.°87
2.°85
2.°85
2.°86
00
cO
*o
2.°85
2.°85
2.°85
2.°82
2.°80
2.°76
00
2nd
3rd
Channel 3 Channel 4
o00
c4
vt
Channel 1 Channel 2
2.°86
2.°90
2.°76
2.°83
2.°76
2.°77
to
'00o
(O
Harmonic
2.°87
2.°90
2.°80
2.°85
T able 3.6: FWHM for each Channel and Harmonic
Harmonic
3rd
4 th
5th
Qth
ID
oCO
iH
1st
2nd
Channel 1 Channel 2
l.°35
l.°35
l.°26
l.°26
l.°26
Channel 3 Channel 4
l.°35
l.°35
l.°35
l.°26
l.°26
l.°35
l.°35
l.°35
l.°35
l.°26
l.°26
l.°35
l.°35
l.°35
l.°34
l.°26
l.°26
l.°34
carried the bulk of the signal. The average values chosen are:
a
= 2°.85
(3.10)
FWHM
= 1°.35
(3.11)
= 0°.573.
(3.12)
a
The offset in elevation
which is responsible for slight asymmetry in the
harmonics, is estimated to be
f = 0°.3.
43
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(3.13)
The conversion factors for converting the data in y V to K are given in the
following tables. The factors were estimated by fitting the simulation to the real
data and minimizing the x2-
Table 3.7: Calibration factors for each channel and harmonics from the May 2nd
Moon Crossing. The values are in y V / K for Antenna temperature
Harmonic Channel 1 Channel 2 Channel 3 Channel 4
1st
2nd
3rd
4 th
5th
Qth
2.28
2.25
2.23
1.68
1.96
2.22
23.07
17.39
14.32
9.35
9.95
11.59
13.90
33.38
26.37
16.33
15.97
18.71
114.28
112.67
102.16
70.01
72.44
75.30
Table 3.8: Calibration factors for each channel and harmonics from the May 2nd
Moon Crossing. The values are in y .V /K for thermodynamic tempera­
ture
Harmonic Channel 1 Channel 2 Channel 3 Channel 4
l*t
2nd
3rd
4h
t
5th
6th
1.77
1.75
1.73
1.30
1.52
1.72
13.90
10.48
8.63
5.63
5.99
6.98
11.44
9.12
7.20
4.46
4.32
5.11
23.71
23.37
21.19
14.53
15.03
15.62
44
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3
2
1
0
•1
data
sim ulation
2
0
-2
•2
larmonic
-3
13.1
13.2
13.3
13.4 13.5
13.6 13.7
13.1
13.3
13.4
13.5
13.6 13.7
13.3
13.4
13.5 13.6 13.7
2
2
1
1
0
0
-1
1
-2
13.2
3
4 Hamu
Harmonic
-2
13.1
13.2
13.3
13.4
13.5 13.6
13.7
13.1
2
13.2
0.6
0.4
1
0.2
0
J 0.0
-1
-0.4
0.6
-2
-
13.1
13.2
13.3
13.1
13.4 13.5 13.6 13.7
13.2 13.3 13.4 13.5 13.6 13.7
RAtfiourt)
RA(houra)
Figure 3.1: The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 1 .
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
4
data
simulation
2
0
-2
I Harmonic
-4
-10
13.0
13.4
13.6
13.0
13.6
3
4
2
1
I 0
1
-2
2
0
-2
-4
13.4
13.2
Harmonic
4 Harmonii
*3
13.1 13.2 13.3 13.4 13.5 13.6 13.7
13.1 13.2 13.3 13.4 13.5 13.6 13.7
2
1
0
-1
2
“
13.1 13.2 13.3 13.4 13.5 13.6 13.7
6^(vmonic
13.1 13.2 13.3 13.4 13.5 13.6 13.7
RA(houra)
RA(houre)
F ig u re 3.2: The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 2 .
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dam
simulation
-10
I Harmonic
-20
-10
13.0
13.2
13.4
13.6
13.0
8
6
4
4
2
2
0
•2
13.4
13.6
I 0
-2
-4
•6
-6
13.2
'}
13.0
Harmonic
13.2
4 Harmonic
-4
13.4
13.6
13.0
13.2
13.4
13.6
0.8
2
1
0
-1
2
0.6
0.4
I
0.2
0.0
- 0.2
-0.4
-
13.1
13.2 13.3 13.4
13.5 13.6 13.7
0.6
13.1
RA(houn)
13.2 13.3 13.4 13.5 13.6 13.7
RA(hoor»)
F ig u re 3.3: The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 3.
47
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30
data
simulation
-20
I
-10 -
Harmonic
\
P* Harmonii
•40
13.0
13.2
13.4
13.6
13.0
13.2
13.4
13.6
13.4
13.6
I
-10
-10
-20
13.0
13.2
13.4
13.6
4 Harmonic
13.0
8
2
1
4
0
0
1
•4
-2
13.1
13.2 13.3 13.4 13.9 13.6 13.7
13.1
RA(hourt)
13.2
13.3 13.4 13.9 13.6 13.7
RA(houra)
F igure 3.4: The red solid line is the Moon crossing data from May 2, 1996, and
the blue dotted line is the simulated data for channel 4.
48
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3.4
E stim ation o f Instrum ental noise
In this section we estimate the instrumental noise present in the data. Since
the other sources of noise, for instance atmosphere, introduce additional noise over
all spatial frequency ranges, the high frequency noise is in effect a combination of
instrumental and atmospheric noise. The power spectra of the data can be used for
estimating the contribution from instrumental noise.
According to the Wiener Khinchin Theorem for a stationary random function
x(t) the power spectral density (PSD) S(v) is the Fourier transformation pair of the
Auto-correlation Function(ACF) R{r).
s„ = f°° R(r)e~2*iuTdr.
J -00
(3.14)
For a discrete uncorrelated random time series x n(N A t) with variance a 2, the power
spectral density at positive frequency can be expressed as [54]
S„ = So = AtRo = a2A t
(3.15)
where A t is the sampling time and N is an integer. Since we are interested in
estimating variances for each channel and each harmonic, we plot the square root
of the quantity S(v) versus spatial frequency in figure 3.5. These figures clearly
illustrate that at the low frequency region the noise does not have a fiat spectrum;
it only exhibits flat behavior at the high frequency range. At the low frequency
range, the spectrum is dominated by the characteristic \ / v bolometer noise and the
1 /t/ atmospheric noise. At high frequency both of these spectra flatten out and
become indistinguishable from each other. We can apply equation 3.15 in this fiat
portion to estimate the instrumental noise. But as already pointed out, this would
only give us a gross upper limit of the instrumental noise, since part of the signal is
coming from atmospheric noise. For the best estimation of the noise we have used
only the quadrature component for each harmonic and each channel to estimate the
instrumental noise. As the noise level for a given harmonic changes with each night
49
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of observation, we use the minimum values observed during the whole campaign
table 3.9.
T able 3.9: Upper Limits of Instrumental Noise (m K - s lf2) Estimation
Harmonic
1
2nd
3rd
Ath
5th
Qth.
Channel 1 Channel 2 Channel 3 Channel 4
5.5
5.8
5.9
7.7
6.0
2.7
1.0
1.3
1.5
2.2
1.9
1.1
2.4
2.9
3.1
4.3
4.4
2.5
2.8
2.6
2.6
2.7
2.2
1.7
50
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!4
mod t (3J mm)
of ptawccwnponwil offlmtanncnfc
0.01
to
T
T
T
i
» « rTf
▼
▼
T
i i i
Channel 2 (2.1 mm)
out of pliamcompoocBtofflm hanmmic
aoi
T
▼r n
ai
i i
»■)
▼
I
T
I I I Ij
Channel 3 (IJmm)
out of pbmr mpaMm offlm bmmooic
% to
at
aoi
T
T
T
I I I I I|
T
T
I II
dmaod4 <t.l mm)
out of pfcmecomponent of flm harangue
o.oi
0.1
Frequency (Hz)
Figure 3.5: Power spectra for the quadrature component of the first harmonic for
all four channels. At higher frequency the power spectra gradually
flatten out and approach white noise behavior. The upper limit of
Instrumental noise is estimated from the flat part of the spectra.
51
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Chapter 4
DATA REDUCTION
Data collected from a ground based telescope at millimeter wavelengths suf­
fers mainly from atmospheric contamination, as well as other sources of contamina­
tion e.g. instrumental and Galactic foreground. From the experiment conducted in
1994 at the same site using the same basic instrument, we know that the Galactic
foreground is not very relevant for our purpose. The high altitude (~ 2400m.) of
Izana (Tenerife) is very advantageous for ground based telescopes. Almost 70% of
the days, the inversion layer of clouds is well below the observatory, and typical
precipitable water vapor is < 3mm. The smooth laminar air-flow around the island
reduces spatial irregularities in water vapor content. During our observing cam­
paign, 1996 May till 1996 June, we collected data while the skies were reasonably
clear, but that doesn’t necessarily mean that all the data collected are usable. The
data reduction procedure consisted of the following:
• Removing instrumental glitches from raw time series data
• Removing outlying points from the data by applying three sigma cuts and also
removing offsets
• Applying the atmospheric subtraction technique to data to remove atmo­
spheric noise, or where appropriate, averaging deconvolved data to reduce
noise
52
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• Removing a long time (low frequency) residual offset from the data by fitting
a combination of sine and cosine functions
• Stacking data from each night’s observation into bins of 0.066 to 0.135 hours
in Right Ascension (RA) to form the final data set
4.1
Effect o f Atm osphere on data
The earth’s atmosphere absorbs and emits in the CMB spectra rather effi­
ciently. All ground based CMB experiments suffer from the noise contributed from
the variable emission from the atmosphere. Even when experiments are carried out
in high and dry sites, the largest unwanted signal usually comes from the atmo­
sphere. Excessive atmospheric fluctuations can cause uncertainty and poor data
efficiency. Experimental techniques like beam switching at constant elevation can­
cels out the constant atmospheric signal but leaves atmospheric fluctuations in the
signal. One way to get rid of the atmosphere is, of course, to go above it, but since
satellite experiments are virtually outside the scope of any small research group, we
need to figure out how to cope with this problem.
Several models of the atmosphere are discussed in the literature [55] [28] [11] [5].
I will briefly review some of the basic characteristics and feature of these models
here. The input variables required by these models are mainly the distributions
of temperature, water vapor content and pressure with altitude, and a parametric
description of the absorption lines and continuum spectra of O 2 and H 2 O. The
antenna temperature due to a signal from outside the atmosphere can be written
as:
T(h0) = Tooexpl-TuiO, s0)] + f Tphys(s )exp[-T„(s - s0)kl/(s')ds
J0
(4.1)
where Too is the temperature of the astronomical source (emission from galaxy,
extra-galactic sources and CMB), h0 is the height of the observing site, TPhys is the
53
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temperature, and r„ and ku are, respectively, optical depth and volume attenuation
coefficient at frequency v. The second term in the above equation is the temperature
of the atmosphere at a given frequency v. The atmospheric temperature has at least
two components, one from 0 2 and the other from H20 . In other words
= Ta ,02(v ) + U>TA,Hto(v)
(4.2)
where w is the precipitable water vapor content of the air given in mm. As stated
above, the fluctuations of atmospheric emission are more relevant for cosmic back­
ground anisotropy experiments than its absolute flux. The macroscopic changes in
the temperature, pressure, and the content of oxygen and water vapor cause fluctu­
ations in the emission that can have time-scales of tens of second to several hours or
days. When turbulence occurs in a layer with a temperature gradient different from
the adiabatic one, it mixes air of different temperature at the same altitude produc­
ing temperature variations. The equations that describe the velocity fluctuations
can also be employed to describe the temperature fluctuations. The turbulence in
the atmosphere is very difficult to model given the large number of parameters that
can influence it.In one dimension following a simple turbulence model [27) we can
write the spatial spectral density of the temperature fluctuations as
$»(«) oc * r 5/3
where
k
is the wave number of the fluctuations and
<&*(*) = 4tm2 | I I I G{r)e2inT Kdr \2
and 0 (r) is the temperature fluctuation about the mean < T (r) >
0 (r) = T (r)— < T(r) > .
Equation 4.3 is valid for
L0 < K < l a
54
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(4.3)
where la and L0 are the internal and external scale of the turbulence respectively.
Near the ground level, the value of external scale L0 varies between several meters
to hundreds of meters. Inside this inertial range the turbulence is assumed to be
homogeneous. The three dimensional spectrum is given by
$0(k) a /c-ll/3.
(4.4)
This power law dependence of the spectral density is called the Kolmogorov spectrum.
This type of model does not take into account the variation of wind speed with height
or any convective motions. In practice the power spectrum is dependent on wind
speed and the height of the emission, which smears out the power law dependence.
Taking into account these different effects current models [9] predict a power index
between -1 1 /3 and —8/3.
We see the evidence of a power law effect resulting from atmospheric turbu­
lence in our data. In figure 4.1 the “power spectra” of the first harmonic of all four
channels are plotted from the file T9605111916. At the middle frequency range the
“power spectrum” falls steeply, and at high frequency range it flattens out. In figure
4.2 the power law fit is shown. It is clear from these figures that the power index
for this particular day is very near -8 /3 .
4.2
Atmospheric Subtraction
Many (e.g [5],[11],[31]) have suggested using multi-frequency photometers to
take measurements of the CMB anisotropy to correct for the atmospheric noise. In
this type of atmospheric subtraction technique one channel is used as the atmo­
spheric monitor channel and then the data from the other channels are corrected
by using this monitor channel. The method described below has been proposed by
Boynton [7], [16] and was used by Andreani [3] without much success.
Data is taken with photometers operating at different frequencies. The high­
est frequency channel, which is most dominated by atmospheric noise, can be used
55
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to get rid of the atmospheric contribution from the data of other channels and to
recover the signal due to the astronomical fluctuations. The signal detected at any
channel i is the sum of astronomical signal (ATAstro) attenuated by the atmospheric
emission coefficient and the contribution from atmospheric emission (ATAtm) in the
frequency band of the channel. There is also a constant offset O* present in the
data. So the total signal detected in any channel is given by
ATi = ATAstro e~°' + OliATAtm + 0{.
(4.5)
For small values of a, e-Qi can be replaced by (1 —a ).
ATi = ATJ44tro(l —a,) + atiATAtm + Oi-
(4.6)
We make the simplifying assumption that the atmospheric absorption in different
channels are linearly correlated:
ai = kam
(4.7)
where a m is the atmospheric absorption coefficient of the monitor channel and k is
a constant. In our experiment, the wavelengths of the four bolometric detectors are
matched with maximum atmospheric transparencies as shown in figure 4.3. We used
our highest frequency channel, channel 4 (1.1mm) as the monitor channel. By using
equations 4.6 and 4.7 we can write the temperature in any channel 4 and channel i
as
AT4 = A T ^ ro(l —ct4) + a 4ATAtm + 0 4
(4.8)
ATi — A T Astral ~ kot4) + OliATAtm + 0{.
(4.9)
Before proceeding further with the analysis, we subtract the offset from each har­
monic of each channel.
56
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By summing and subtracting the equations 4.8 and 4.9
Si, 4 =
=
A ,4 =
AT4 + ATi
2ATAstro + ^ 4(1 + k)(ATMm ~ AT^atro)
(4 . 10)
AT4 —ATi
— Q;4(l —k)(ATAtm
ATAstro)
(4.11)
By manipulating equations 4.10 and 4.11, we rewrite the term 5 ,-,4 as
S iA — 2A T Astro +
(4.12)
Equation 4.12 shows that a linear fit between ^,4 and A ,4 determines the value for
A T Astro •
A very similar procedure was used for analyzing the data from our second
campaign in 1994 [17]. T hat technique required prior knowledge of the atmospheric
transparencies. The advantage of the current technique is that it does not require
the absolute value of the coefficients of the atmospheric transparencies, but only
demands that the coefficients for different frequencies are linearly correlated. Figure
4.4 shows the same night’s data reduced by both methods. As evident from figure
4.4 the difference between these two techniques is not appreciable.
The high correlation between the raw data collected from the same night in
all four channels and the lack of such correlation between d ata from different nights
clearly demonstrates that the correlation is due to local atmospheric fluctuations
rather than due to some astronomical source, e.g.CMB or galactic signal. The
signal amplitude of the raw data is too high to be attributed to astronomical sources.
Figure ?? shows the normalized auto-correlation between data from the same day
for the first three harmonics, and figure 4.6 shows the cross-correlation of data from
first harmonic for two different days.
57
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4.3
Im plem entation o f Atm ospheric Subtraction Technique
The key to the success of this subtraction technique is the high linear cor­
relation between the monitor channel and the observing channel. A line is fitted
to the sum and the difference of the channels to determine the astronomical signal.
To achieve a statistically robust value for the value of AT,iatro we should use as
many data points as possible. But we must take into account that the atmospheric
conditions often are not stable for a very long period of time, so we must choose
our binning size for applying the fit judiciously. After doing a study of the effect
of binning size on the result of the subtraction, we decided to use a binning size of
256 points or 128 seconds. Figure 4.8 shows the effect of different binning size on
the reduced data. The binning size of 128 sec. is on one hand short enough that we
can assume a stable atmospheric condition, and also on the other hand, 256 points
is large enough for the fit to be statistically significant. In figure 4.7 the sum vs
difference of each of the channels with channel 4 are plotted for all six harmonics.
We also studied the auto-correlation of data to choose the time-period for binning.
There are a few things to notice in these figures. First, the correlation be­
tween the sum and the difference for any channel i and channel 4 worsens as the
harmonic number increases. Second the correlation increases as the frequency of
the channel i increases for the lower harmonics. Both of these effects are to be
expected if the dominant noise present in the data is atmospheric in nature. Since
atmospheric fluctuations are inversely proportional to angular scale [33] for the same
throw angle and the same beam size the lower harmonics are more sensitive to the
atmospheric noise. The successful implementation of this technique demands at
least a 75% correlation between the monitor channel and the observing channel for
each of the harmonics. The atmospheric subtraction process is successful for the
lower harmonics one,two and three but not as successful for the higher harmonics.
This is especially pronounced in the case of Channel 1 which is not very sensitive to
58
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atmospheric noise. Applying the atmospheric subtraction technique can introduce
noise rather than reduce it when the correlation is bad. In the cases where the
atmospheric subtraction process is inapplicable we averaged the data over the same
binning size to reduce noise. The higher harmonics {four,five, and six) of all three
observing channels have been reduced by averaging whenever the correlation was
below threshold.
4.4
Success o f the Technique
In this section the effect of the atmospheric subtraction on the data is pre­
sented. The next sections present the power spectra and the rms values of the data
before and after the atmospheric subtraction. These tests indicate that though we
have achieved significant rejection of atmospheric noise, the noise spectra is still
above the instrumental noise level.
4.4.1
Comparison of distribution o f tem perature
Figure 4.9 shows the distribution of the rms values of the demodulated data
without any atmospheric subtraction and 4.10 shows the same data after applying
the atmospheric subtraction technique and averaging over same number of points
(256 points). The narrow distribution of the temperature values clearly indicates
that the subtraction technique has considerably reduced atmospheric noise.
4.4.2
Sim ulation of Atmospheric subtraction
To ensure that the atmospheric reduction method is not subtracting signal
from the data but is only eliminating atmospheric noise, we performed a series of
simulations. A simulated signal was co-added with a typical night’s time series
data. The signal to noise ratio S / N of the simulation was varied from 0.1 to 1.0.
Figure 4.11 shows the raw data to which the simulated signal was added. The root
mean square (rms) fluctuation of this data file is 160 mK. Simulated signals with
59
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rms values 14, 70, and 140 mK were added to this data hie and the atmospheric
subtraction technique was applied on these simulated data with 256 point binning.
The reduced data is plotted in figure 4.12. The same data was also averaged over the
same number of data points (256 points) and is plotted in the same graph with dotted
lines. It is evident from the graph that as the S / N gets smaller the atmospheric
subtraction technique is far superior in extracting signal than averaging. If the S / N
ratio is decreased further, then the S / N does not improve appreciably. The S / N
ratio of the reduced d ata is increased by at least an order of magnitude.
4.4.3
Power Spectra o f Data
The reduction of noise in the data is evident in the power spectra. When
we compare the power spectra of the atmospheric reduced data with averaged data
binned over the same number of points in figures 4.13, 4.14, and 4.15, we see that for
channel 2 the atmospheric subtraction seems to have worked the best. For channel 3,
even though the overall noise has significantly reduced, the noise level is still high,
indicating that this channel still retains some residual atmospheric contamination.
The power spectra also clearly show that for all three channels the atmospheric
subtraction technique is most applicable for the first two harmonics.
It should be mentioned that this kind of atmospheric subtraction technique
is limited by the ratio of the atmospheric noise to CMB signal. In [31] the authors
have shown that for the detector pair 1.1mm and 1.3 mm this sort of subtraction
technique can be applicable only if the atmospheric noise is 5 —6 times the CMB
signal. On the other hand, under similar conditions for the 1.1mm and 2.1mm pair
this technique is viable even if the noise is 20 times the signal.
4.4.4
Final data set
The atmospheric reduced data from each night’s observation contain a resid­
ual baseline of long wavelength (/time). The frequency of such drift is much lower
60
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than the intrinsic fluctuation in the data. Residual atmospheric noise or thermal
fluctuations of the bolometers could be the sources of such excess oscillation present
in the data. Such baselines are removed from each night’s of observation by fitting
a linear combination of sinusoidal functions. The minimum period of the sinusoidal
function were chosen to be larger than the angular sensitivity of the instrument,
such that the fit would not remove signal from data. The minimum periods for
each of the harmonics were chosen on the basis of the width of their corresponding
window functions (see section 5.1.1). Figure 4.16 shows a typical data set before
and after removing the baseline.
The data sets with the baseline removed are then stacked in 3 - 4 minute
bins to form the final data set for each channel and harmonics. The final data set is
produced by evaluating the weighted averages according to the following equations:
_
• ”
E " i. 1 / 4 _
1
(«. - 1) s ? .. i / 4
'
'
where AT* is the final temperature of the ith R.A. bin composed from A7V,
in ith bin and j th night, and cr» and
are the corresponding standard deviations.
Only data between R.A. 12 - 19 were used for the statistical analysis. Data
collected at lower R.A. were discarded because the bolometers were usually not
stabilized during this period. At R.A. higher than Hour 19 we expect the data to
be contaminated with signal from galactic plane crossing.
61
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‘ ‘ ‘1,1
I 9I 9I 7I 9III
Frequency (Hi)
0.1
F ig u re 4.1: Power spectra for the first harmonic for all four channels. The power
spectra falls sharply over 0.01 Hz to 0.2 Hz. and then gradually flattens
out. The sharply falling region is dominated by atmospheric noise.
62
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10
g
a
7
6
5
-4.5
-4.0
-3.5
-4.5
•3.0
-4.0
•3.5
•3.0
m(>o
13
13
12
12
11
11
10
10
9
9
a
-4.5
-4.0
-3.5
-4.5
-30
-4.0
-30
F ig u re 4.2: The logarithm of power spectra 0(*c) versus the logarithm of
plotted. The slope of the linear fit is ~ —8/3
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
k
is
1.0
1 0.8
Atmospheric
0.6
0.4
0.2
I.......
0.0
50
100
150
200
250
300
350
400
450
500
Frequency (GHz)
Figure 4.3: Atmospheric transmission as a function of frequency (solid line). The
normalized filter transmissions for each of the detectors are also shown.
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
present method
1994 method
2.0
1.0
0.0
u
E
-
1.0
-
2.0
Channel 2 2
harmonic
-3.0
10
11
12
13
14
15
16
17
18
19
R-A.(Hours)
F ig u re 4.4: Data collected during May 11 1996 has been reduced using the 1994
method (blue dotted line) and the present method (red solid line).
The difference between these two methods is almost negligible.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
0.8
jun 9 (pwv-3.l2mm) l“ hnrmooic
jun I l(pwv-1.74mm) l“ harmoaic
I 06
I
0.4
1 .
0.0
•500
-400
-300
-200
-100
0
1*4
100
0.8
S
0.4
300
400
SCO
jun 9 2 harmonic
jun 112“ harmonic
0.8
I
200
0.2
0.0
•500
-400
-300
-200
-100
0
I*
100
200
300
400
500
-80
■80
-40
•20
0
H
20
40
60
80
100
0.8
I 0.6 I
0.4 -
0.2
0.0
-100
F ig u re 4.5: Normalized auto-correlation of data from two different days versus
time lag (each point is 0.5 sec). The correlation decreases with har­
monic number. Also the day with less precipitable water vapor (less
atmospheric fluctuation) has less correlation.
66
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1.0
0.8
.1
T
T
T=1
autocorrelation ch2 notel jn e 9
crosscorrelation june 9 an< juni 1
0.6
1
i
-
0.2
•6000
-4000
±
±
±
_L
±d
-2000
0
2000
4000
6000
lag
Figure 4.6: Normalized cross-correlation of data from two different days versus
time lag (each point is 0.5 sec). This clearly shows the data from two
different days are not correlated.
67
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1000
800
f
?
®
600
400
■400
-300
•200
0
•100
Cbl-CMonK)
1000
800
I
I
3
600
400
p—0.93
200
•600
-400
-900
ch2-cM(mK)
-200
-too
Figure 4.7: Sum versus difference of the temperature of observing channel and
channel 4. The top panel is for channel 1 and the bottom panel is for
channel 2. The correlations between the sum and the difference are
given on the figure.
68
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— 32 pomt bu ling cb2J
—— 64 point binning ch3. l
i
0-10
-15
-20
10
12
14 16
HA.(Houn)
IS
10
12
14
16
18
20
ILA.(Houn)
121 poim binning ch2_l
254 point binning ch2_l
i
1
-10
-10
-15
-20
-20
10
12
14 16
R-MKotn)
18
20
10
12
14
16
fLA.(Houn)
18
20
Figure 4.8: Effect of using 32,64,128 and 256 points in doing atmospheric subtrac­
tion.
69
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600 r » i H r - r T - r - T - i - r
i
300m — i— i— i— i—i
550 -
-
500 -
-
450 -
-
400 -
-
350 -
-
□ Channal 1
Flret harmonic
300
-
200 -
-
ISO -
-
100 -
-
0
0
200
*
250 -
50 -
250 -
-
150 -
□ Channal 2
FIret hormone
100
SO
T - w
■i
20 30 40 SO 60
10
mK
»r
1--- '--- T
I
□ ChannaM _
Flret harmonic
40
00 I
*
40 -
□ Channal 3
Flret harmonic
30
i
■6
20
20 -
10
■
7
Ijinnflisa ■
I
100
200
300
mK
Figure 4.9: Histogram of the rms value of the first harmonic of the demodulated
data.
70
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□ awraQt
—
•100
. -r-n-IT
-80
0
in
MTIMpMltC
wbndHn
80
100
mK
a nm gi
— amxpMrtc
•uBOMIon
•100
•SO
0
SO
100
mK
F ig u re 4.10: Histogram of the temperature values of the first harmonic of June 9
1996 datafile after averaging (red thin lines) and after atmospheric
subtraction (blue line). 256 points were used for both methods.
71
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*
E
15.0 15.2 15.4 15.6 15.8 16.0 16.2 16.4 16.6 16.8 17.0
R.A.(Hours)
Figure 4.11: Noise file for atmospheric subtraction simulation.
72
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400
200
150
100
200
% 0
•SO
-200
-400 r
-100
1 ■ 1 ‘ 11 ■ 1 ■ 1 "
15.2 15.6 16.0 16.4 16.5
15.2
RA(lfoun)
15.6
16.0
16.4
16.5
16.4
16.5
16.4
16.5
HA.(Houn)
400
200
i
% 0
-so
•200
-100
•400
-ISO
15.2
15.6
16.0
16.4
16.5
15.2
&A.(1loun)
166
16.0
R.A.(Houn)
200
100
0
-100
-200
15.2
15.6
16.0
16.4
16.5
15.2
HA.UIoun)
15.6
16.0
R-A.(Koun)
F igure 4.12: Simulated data with signal added (S/N 0.1,0.5 and 1.) are shown on
the left panels. The right panels shows the data after atmospheric
subtraction ( solid red line) and averaging (blue dotted) lines. The
number of data points used for atmospheric subtraction and averag­
ing is 256.
73
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100
I I 11111
100
b21av
b21at
tfl
I
10
a11av
a11at
100
100
a31av
a31at
e
V
10
i
1
100
100 P T
b61av
b61at
aS1av
aS1at
<I/>
10
%
Hz
Hz
F ig u re 4.13: Power spectra of Channel 1. The blue dotted lines are data reduced
by atmospheric subtraction with a bin size of 128, and the solid red
lines are data averaged over the same number of data points.
74
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1000
rrrry
b12av
b12at
100
nd
1 harmonic
2
harmonic
if
i (
i n
1
6
‘ ■•■■■■»>
56
Iff3
2 3 4
Iff4
■
2 3 4 56
b32av
b32at
3 harmonic
1
;
■ ■ ■“ ■■■!
3 4 5 6
10-4
10-3
2
4 harmonic
■
2
OM
3 4 5 6
100 prry
b52av
b52at
6 harmonic t/v'H.i
5 harmonic
4U
Mr
Figure 4.14: Power spectra of Channel 2. The dotted blue lines are data reduced
by atmospheric subtraction with a bin size of 128, and the red solid
lines are data averaged over the same number of data points.
75
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1000
a23«v
«23at
1000
100
100
*
I
b13av
b13at
A lii
1000
b33av
b33at
100
a43av
a43at
100
/«
100
a63av
a63at
b53av
b53«t
100
**•1
*
I
Hz
Hz
F ig u re 4.15: Power spectra of Channel 3. The blue dotted lines are data reduced
by atmospheric subtraction with a bin size of 128, and the solid red
lines are data averaged over the same number of data points.
76
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4
*
2
E 0
-2
-4
12
13
14
15
16
17
18
RJV. (Hours)
F ig u re 4.16: Data from June 8 1996. The open circles on the lower panel are the
data from channel 2 ( second harmonic) before removing the base
line, The solid black line is the sinusoidal fit function. The residual
data is plotted on the top panel with open red circles.
77
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12
13
14
16
16
17
16
fLA.(Houn)
F ig u re 4.17: Data from June 8 1996 second harmonic. The top panel shows raw
data before atmospheric subtraction, the middle panel after subtrac­
tion, and the last panel after removing the long time drift.
78
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Chapter 5
ANALYSIS AND RESULT
5.1
Likelihood Analysis
The final data for a given channel and harmonics consists of a set of tem­
perature differences AT* collected at a specific R.A. bin. The experimental noise in
every observation is characterized by the variance a* associated with it. From this
given data set we want to determine the intrinsic amplitude of the CMB anisotropy
by using the Maximum Likelihood Analysis. The Likelihood is defined by
_ eg p (-^ D TM - 1D)
(2tt)"/2 |M |l/2
(5.1)
where D is the data matrix made out of the temperature observations ATi, and M
is the corresponding covariance matrix. The covariance matrix is defined as the sum
of the sky signal covariance (G r)and the noise matrix (CN)of the data.
M = C n -f- C x
(5.2)
The noise matrix C n is calculated from the instrumental noise tr* associated with
the data and the autocorrelation between different data points. C x is the theoretical
point-to-point covariance m atrix. The theoretical sky signal covariance is dependent
on the model chosen for the correlation function for the sky signal. The theoretical
parameters are adjusted to maximize the likelihood function L.
Confidence intervals on the parameters are found from the distribution of
L(T). The value of Tmax that maximizes L is determined as the best value for the
79
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parameter. A nonzero detection can be claimed when L(Q)/L(Tmax) < 0.15. The
upper and the lower error bars are determined by equations 5.3 and equation 5.4:
fo + L(T)dt
c w
w
-
0M 13
( 5 -3 )
J? -L (T )d t
T - m
- °-1587-
*
( 5 -4 )
On the other hand, for I ( 0 ) /I ( T mox) > 0.15, the 95% upper limit is found by finding
Tul such that
j j L(T)dt
&? L (T )d t
U‘y&-
As described in section 1.5, the theoretical sky fluctuations T’(x) can be
expressed in terms of spherical harmonics as
r(*) = EaP7"(*)
(5.5)
t,m
where x is a unit vector, which can be expressed in terms of spherical coordinates
x = (sin6cos<l),sin0sin(t>,cosd). The theoretical sky covariance matrix C x is given
by
c? = i£ (2 £ + l)C ,W f
(5.6)
and Ct = (|a7*|2), and the window function W\* is expressed as
W ? = / dxi j dx2Mi(zi)M j(x 2 )Pi(£{-x2)
(5.7)
where M (x) is the mapping function.
5.1.1
M apping function and W indow function
Any given experiment filters the true sky signal with a mapping function
M which is characteristic of the observation strategy and the beam profile of that
experiment. The temperature Tcxp(x) observed by a given experiment is in general
80
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linearly dependent on the true sky temperature. The observed temperature can be
expressed in the following form [57]:
Imp(x) = J d * V (* ,* '):r(* ').
(5.8)
If we make an approximation of flat sky, with x = <f>and y = (7r/2) —Q (0 denotes
the longitude and 6 denotes the latitude of a point on the sky), then for x = (0,0)
and x = (x, y) the mapping function for our experiment can be written as
M ^ y ) = £ &
f Z . L{-t ) e x v
y2 + [x — asin(uict)]2
dt
2a2
(5.9)
where N is the normalization constant calculated from equation 5.10, u c is the
chopping frequency, a is the wobbling amplitude and a = FWHMjv&Ln2 is the
a of the beam as calculated in Chapter 3. The mapping function is dependent on
the lock-in function L{t). We used sine and cosine lock-in function (see chapter 3
section 3.2) to demodulate the data. The following equation gives the normalization
f
f
J J m >o
M (x, y)dxdy = 1.
(5.10)
The normalizations of the mapping functions are given in table 5.1 and figure
5.1 shows the mapping functions for all six harmonics.
Table 5.1: The mapping functions for six harmonics of “Dec40” observation as a
function of R.A.
Harmonics
1
2
3
4
5
6
Normalization
0.3269519
0.3869875
0.5141101
0.6911262
0.939624
1.3182801
81
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0.6
0.4
0.2
I
|
00
I
-02
-0.4
-
0.6
-6
-4
•2
0
2
4
6
HA.
F igure 5.1: Mapping functions for all six harmonics at declination 8 = 0°
The window function can be calculated from the mapping function A/(x,x )
in the following way [57]. The two point autocorrelation function of the experimental
temperature is given by
= / < “ • / <**,*/,(*!)*/,(*,) ( r t s i m * ; ) )
(5.ii)
which can be rewritten as
<TOT,(xi)TeIp(Sj))
= J d x i f <fx2.M,(*,)Mj(xa) x
j - £ ( M + l ) C , P , (* ,•*»)
(5.12)
4?r (= 1
= j - 5I(2£ + l ) C t W t
47r 1=1
(5.13)
where equation 5.7 defines the window function W*(xi -Xa). For a constant elevation
scan(0 = constant) as in “Dec40” scan one can express W;(xi • Xa) as a function
of the angular separation \<j>i — <f>2| between two points on the sky.The
82
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window
function for our experiment can be written as
WM)
= N 'B l W z g - £
H W M ) |2
L m —- l
xL ^(a)5^(A 0)cos(m 0)
(5.14)
(5.15)
n
^ r
dtL(t)e'mann^
27TJ-ir/uic
- sin(mA(j>/2)
,E
S" i M )
=
(5'16)
Lm(ac)
=
mZ*/2 '
where B /(a) refers to the beam profile function of the experiment. It can written as
Bt(a) = exp[—£{£ + l)cr2].
A0 is the bin size of the data on the sky. For our lock-in functions the Lm(a) can
be written in terms of Bessel functions. In figure 5.2 we have plotted the “zero-lag5’
(A0 = 0, We as a function of I only) window functions for all six harmonics. Since all
channels share the same values of a and chopping angle a, to a good approximation
they also share the same window functions. Table 5.2 lists the effective I for each
of the harmonics calculated by
using equation 5.17 [33], and alsothet ranges over
which the window function has
amplitude larger than el/2 ofits peakvalue.Figure
5.3 shows the 1994 window functions and the 1996 window functions for first two
harmonics (1994 data were analyzed in two i bands only).
The effective i value for each window function is calculated by multiplying
the average
t
value with the flat power spectrum. The
i ef/ective
for each window
function is calculated by using the following expression.
f»
^effective
—
y
LtUl w ?
t+l 1
M - l/2 M /n
2^et[TH)vvt
where W£ denotes the window function for the nth demodulation.
83
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(Z 171
\ ° - L< I
T a b le 5.2: Window functions for all six harmonics for 1996 data.
Harmonic
^effective
I st
2nd
3rd
38if?
6 ii”
81-21
100t£
119^25
136tg
5th
6th
To build the covariance matrix to use in a Maximum likelihood test one needs
to choose a parametric representation for C(. We chose a “flat” radiation spectrum
for Ci given by [48]
where the parameter AT is the quadruple moment which can be varied over a range
of temperature values to maximize the likelihood function.
5.1.2
L ikelihood analysis resu lt
We analyzed data from all six harmonics of channel 1 and channel 2. The
results of the likelihood analysis are tabulated in table 5.3 and table 5.4.
For channel 1 we can only determine upper limits of fluctuations in four of
the six t bands. Due to excessive noise in the first and fifth harmonics, data from
these harmonics were not used for channel 1.
The results of the first four harmonics in channel 2 are determined with a
68% confidence level. The fluctuations in the fifth and the sixth harmonics are upper
limits with 95% confidence level.
84
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2.0
1.5
1.0
0.5
\ \
0.0
0
50
100
150
200
250
300
t
F ig u re 5.2: 1996 Window functions
The channel 2 results from 1996 are plotted in figure 5.4 with the Standard
Cold Dark Matter (Standard CDM) and the A (cosmological constant) CDM models
of the CMB angular power spectrum. The 1994 result is also plotted on this graph.
The result from 1996 is consistent with the result from 1994. Given the error bars
our d ata cannot distinguish between the Standard CDM and the AC D M model,
but our results show a monotonic increase of temperature with £ consistent with the
presence of the first Doppler Peak around £ = 200.
In figure 5.5 we have plotted our result against the same theoretical models
along with the recent results from the two balloon-borne experiments BOOMERang
[12] and MAXIMA-1 [21]. Our data are consistent with the results of these two
experiments.
85
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2.0
1 H arm onic 1996
a^H arm o n ic 1996
1.5
1*1 Harm onic 1994
2nd H arm onic 1994
1.0
0.5 -
0.0
40
80
120
160
200
240
280
(
Figure 5.3: 1994 and 1996 Window functions for the first two harmonics.
5.2
C onclusion
We have detected CMB anisotropy in infective = 38 to ^effective — 100 for
channel 2. Successful implementation of the atmospheric subtraction technique
has made it possible for us to use a ground based telescope to detect such signal.
Improvements in instrumental setup have allowed us to probe a wider range of I
space. We have tested the the cosmological origin of the detected signal by using a
Null test. The 1994 and 1996 results are found to be consistent with each other. Our
result on the low I range is consistent with COBE results, and on the high t range
with the aforementioned balloon-borne experiments BOOMERang and MAXIMA.
The atmospheric subtraction technique developed here can be useful for ef­
ficiently removing atmospheric noise from CMB anisotropy data collected from a
86
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100
5?
Channel 2 (2.1mm) 1996
Channel 2 (2.1 mm)1994
COBE
60 -
Tci
[]
1
10
100
1000
I
F ig u re 5.4: Results of Maximum Likelihood analysis for channel 2. The solid line is
the Standard CDM model with fit = 0.05, Q c d m — 0.95, Q \ =
0 , = 0 , n = 1, Hq = 50km s~ LMpc~l , and the dashed line is the
AC D M model with the following parameters: fit = 0.05, Qcdm =
0.25, Q \ — 0.7 Qu = 0, n = 1, and H0 = 65kms~l Mps~L. At
the bottom of the graph are the window functions (scaled up by the
same factor) for each modulation. The theoretical angular spectrums
were generated by using CMBFAST [50]
87
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100
•
A
O
This Experiment
BOOMERang
MAXIMA 1
60 -
40 -
100
1000
t
F igure 5.5: Results of Maximum Likelihood analysis for channel 2 plotted
against the same theoretical models along with recent results from
BOOMERang [12] and MAXIMA-l [21]. Within the error bars our
result is consistent with these two experiments.
88
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T a b le 5.3: Results of likelihood analysis bandpower in \i K for Channel 1
1.
Channel 1
61±8
81115
lOOtg
136to®
< 92
< 93
54
< 127
T able 5.4: Results of likelihood analysis bandpower in fi K for channel 2
Channel 2
38ti7
61±lo
81115
331 n
4 3 tg
47tH
4 9 t£
< 115
< 120
Mfl
+o 1
o
ie
119tg
136tg
high and dry experimental site, e.g. the South Pole. Also, ground based observa­
tions of CMB polarization is the next step towards determining and constraining
the cosmological parameters.
89
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Appendix
Since there arc quite a few conventions for defining and normalizing power
spectral density, to avoid confusion, in this section I will explain in detail the con­
vention used for calculating the PSD for this work.
The Fourier transformation of any stochastic time series can be defined as
[46):
r-
1 /2 T
„
. 4
xT{y) = /
x{t)e-2vwldt.
J- 1 / 2 T
The mean square expected value is:
E H I - V M I 2} = r j T
j \
-
r i ) R T e - 2' i" d T
where R ( t ) is the ACF of the process x(t)
R t ( t ) = E r{x(t)x(t + r)}
If we define the power spectral density (PSD),as S„, then
S„ =
S„ =
U
m
f° °
J—00
r ^
R
E
r l X
^
f r ^
- ^
d r
}
Su is a even function of frequency and by definition nonnegative. The power spectral
density (PSD) or the spectral density function of a stationary random function is
defined as the Fourier transformation pair of the Autocorrelation Function (ACF)
according to WienerJOiinchin theorem. If we consider a one sided spectral density
function Su defined only for v > 0 then the ACF at r = 0 is just the variance a2 of
90
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the signal. The units of one-sided S„ are X 2 —sec( where X is in units of X (t)).
In practice, d ata are taken at some discrete time interval A t. The function R{ t ) is
zero for all values except r = 0, ±1, ±2...., so the integration goes to a summation
for only positive values of r. For an uncorrelated stationary random series xn(N A t)
where N is an integer, the expected form of the PSD at positive frequency becomes
(34]:
s„ = s 0 = Alfto = <r2At.
Now I will describe the computational method used for estimating the instrumental
noise for our detectors. Calculation of the power spectra was done mainly by fol­
lowing the methods described in Numerical Recipes [42]. The following discussion is
also following the treatment of this subject. As mentioned above the normalization
of PSD is not unique. We follow the following definition of total power for a signal
C(t) sampled at N points, over a total time T = (N —1) A, sampled at time interval
A :
1 -t
^ AC-1
— / \C(t)\2d tfs ~
\Cj\2 = “meansquaredamplitude"
*
N j_0
whereas the “time-integral squared amplitude” is
r m
t v A
Jo
Z
M
j= 0
The discrete Fourier transformation of x(t) is given by:
AC-l
Ck = £ Cje2ltijk/Nk = 0 ,...JV - 1
j= o
then the periodogram estimate of the power spectrum is defined at N /2 -l-1 frequen­
cies as [42]
P( 0 ) = P(/o) = ^ |C
„ |2
P(A) = f ilia l2+ |CW
-*|2]fc = 1,2,
- 1)
P(A) = P U m ) = j^lCwl2
91
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According to Parseval’s theorem, this form of power spectrum is normalized such
that the sum of N /2 + 1 values of P is equal to the mean squared amplitude of the
function Cj. Also by multiplying this powers by A / N we recover the time-integral
squared amplitude. In general the periodogram, a discrete function, is not exactly
equivalent to the continuous power spectrum of ct, but on average we can expect
P(fk) to be some kind of average of P( f ) over a narrow window function centered
on its f k. This produces leakage of power to near and far frequency bin. To reduce
this leakage one needs to apply Data Windowing. Data windowing changes relation
between spectral estimate Pk at a discrete frequency and the actual underlying con­
tinuous spectrum P( f ) at nearby frequencies. By not applying any data windowing
in effect we are applying a window function w = 1., and as this is a square window
function and it turns on and off abruptly, this contributes substantial components
at high frequencies. To remedy this situation one can apply a window function Wj
to the input data Cj that changes more slowly from zero to a maximum and then
back to zero as j changes. The modified equations for the periodogram estimator
are:
N-l
Dk = ^ 2 CjWje2*ijk/Nk = 0, ....N - 1
>=o
P( o) = P(/„) =
P(A) =
+ |£V-*l2]fc = i, 2 , . . . ( f 1
P(fc) = P{ f m ) = w ~ \ Dm
1)
2
S3
k
N
fk = N A k = ° ,1 ,''"'2
where Wss is equal to
W '^ N T T
w 2.
i= o
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An example of such a window function is “Parzen Window” :
Wj = 1 —
~ ;(n - 1)
k (N + 1) 1
Now I would like to establish a relationship between the Periodogram of power
spectra and the noise variance. Since we are interested in extracting the instrumental
noise from the power spectra, and we expect the instrumental noise to be white noise
with constant power spectra amplitude, we are only concerned with the flat part of
the spectra. If the power spectra was flat with constant value Pq over the entire
frequency range then:
N/ 2 + 1
^2 P{fi) = c 2At[mk2 - sec]
j=o
P0(N /2 + 1) = <r2&t
c r^ A t) = yf{P0/{ N /2 + l))[mk - yj{sec)}
93
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