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Cosmological parameter and parity violation constraints using Cosmic Microwave Background polarization spectra measured by the QUaD instrument

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COSMOLOGICAL PARAMETER AND PARITY VIOLATION
CONSTRAINTS USING COSMIC MICROWAVE BACKGROUND
POLARIZATION SPECTRA MEASURED BY THE QUAD
INSTRUMENT
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF PHYSICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Edward Yousen McCloskey Wu
August 2009
UMI Number: 3382915
INFORMATION TO USERS
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I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
J>J<J"zJ- t_ LWvrtA(Sarah E. Church) Principal Adviser
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(Robert Wagoner)
a
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
V
<4fyfc
(Steven Allen)
Approved for the University Committee on Graduate Studies.
fati.f,.A~i*~
m
Abstract
The QUaD instrument observed the Cosmic Microwave Background (CMB) in
temperature and polarization for three austral winters from 2005 to 2007 on a
roughly 120 square degree field. QUaD's 143 days of data from the last two seasons are processed, calibrated and analyzed to create high-resolution maps of
the CMB and derive CMB spatial anisotropy spectra. QUaD is the first experiment to observe a number of harmonic peaks and troughs in the E-mode polarization spectrum to small scales, affirming our understanding of the plasma
physics of the early universe and the ACDM model. Cosmological parameters
are derived using Markov Chain Monte Carlo methods from QUaD data solely
with the Hubble Key distance scale to quantify concordance with ACDM. QUaD
data combined with other measurements of the CMB and large scale structure
are also used to quantify QUaD's contribution to precision measurements of cosmological parameters, including a limit on the tensor-to-scalar ratio of r < 0.20.
QUaD establishes an upper limit on the conversion of polarization power by weak
gravitational lensing of < 0.77 ^K2 at 95% confidence, compared to the ACDM
expectation value of 0.054 \iK2. Finally, QUaD constrains the total possible rotation of the polarization directions of photons due to cosmological-scale electrodynamic parity violation to Aa = 0.82° ± 0.49° (random) ±0.50 (systematic).
This is equivalent to constraining isotropic Lorentz-violating interactions to <
10~43 GeV at 95% confidence. QUaD produces the strongest constraint to date on
cosmological-scale parity violating electromagnetic interactions.
IV
Preface
"Plenum ingenni pudoris fateri per quos
profeceris."
Natural History
PLINY THE ELDER
The science results derived from observations made with the QUaD instrument are the collective efforts of dozens of people, many of whom invested years
of their lives in the success of our collaboration. It is particularly cruel fact that
in our field of study, the lion's share of publications and citations goes to those of
us, like myself, who are so blessed to have the chance to analyze data of incredible quality gathered from an instrument largely designed and built by others. In
addition, I have had the privilege to work among first-rate analysts of astrophysical data who have been extraordinarily open with their methods and code, from
whom I have learned much and whose influence is visible throughout the whole
of this work.
The university rightly states that "An important aspect of modern scholarship is the proper attribution of authorship for joint or group research. If the
manuscript includes joint or group research, the student must clearly identify
his/her contribution to the enterprise in an introduction."
As such I would like to clarify my participation in QUaD. In Chapter 2,1 provide a summary of the salient points of receiver and telescope design in order to
provide relevant background for the analysis to come -1 make no claims on any
part of the initial design or construction of the experiment. I did, however, deploy to the South Pole three times after each of our three observing seasons to
V
vi
help replace broken detectors, prepare and test the receiver for calibration measurements, and ultimately disassemble and return the instrument. I also made
minor modifications to the software code and electronics rack to accommodate
the data acquisition of external electrical inputs and helped install the correcting
secondary mirror used in the 2006 and 2007 seasons.
I inherited responsibility for the low-level data processing pipeline from Ben
Rusholme when he left Stanford in 2006. My initial contribution to this pipeline
were the deconvolution and decimation routines. Since then, I have made major modifications to virtually all parts of the pipeline, and the processed data
from this pipeline form the basis for one of two sets of results in every spatial
anisotropy paper QUaD has published (1; 2; 3; 4).
I owe an enormous debt to both Michael Brown and Clem Pryke for their assistance and example in constructing a Monte Carlo analysis to convert raw data
into CMB spectra. Dr. Brown's code formed the initial reference for my own analysis efforts, while many of Professor Pryke's analysis algorithms were incorporated into my own code in an attempt to maintain two separate analyses that
were nonetheless comparable to verify our results and catch errors. Dr. Brown
should be rightly credited with the ground template subtraction algorithm that
ultimately allows us to recover about 40% more data.
Nonetheless I have endeavored to write the code for every piece of analysis
for the results presented in this work, and although many algorithms have been
freely learned from others, I have contributed to the science results of the collaboration by constructing and demonstrating algorithms of my own for analysis, inserting and testing for systematic contamination effects in our simulation
pipeline, and catching major errors in implementation when comparing my results to those derived from the other pipelines in the collaboration.
This is reinforced by my choice of C++ as a working language separate from
others in the collaboration. I also built the small computer cluster on which this
code runs. In the Pryke et al. QUaD paper presenting the field-difference spatial anisotropy spectra (2), the totally independent results from my pipeline were
published alongside those from the pipeline at the University of Chicago.
VI
vii
For science results derived from these CMB spectra, I was the only member of
the collaboration to work extensively on the parity violation results presented in
chapter 6, culminating in a first author paper in Physical Review Letters. Patricia
Castro was instrumental in adapting the MCMC methods used by other experiments to the particular difficulties of QUaD. However, the cosmological parameter estimation results in this work come from C++ code I have written independently. Sujata Gupta was also of great assistance in understanding the science
behind the parameter fits and demonstrating constraints similar to those derived
in this work, with a wider number number of model scenarios.
Throughout this work, anyfigurewithout attribution is my own. Where others
have produced plots or diagrams, or provided photos, I will provide attribution
at the end of the caption.
Vii
Acknowledgments
The personal support of my friends, family and colleagues has been as important
to the successful completion of this work as the professional attributions detailed
in the previous section. My adviser, Sarah Church, has not been only everything
one could hope for in a mentor, teacher and boss, but a close and terrific friend
who I will dearly miss after leaving Stanford.
My lab mates have also taught me most of what I know about astrophysics
and also provided me with great company at home, on top of Mauna Kea and at
the South Pole for the past five years. I cannot thank Jamie, Ben and Mel enough
not only for building QUaD and teaching me most of what I know about it, but
also demonstrating to me admirable examples of good humor, hard work and
unfailing politeness. Jamie has to be particularly thanked for marrying me and
Simone with the powers vested in him by the Great State of California. Thank you
to Keith for laying my basic foundations for microwave astronomy and virtually
everything I know about optical astronomy and telescopes. Thank you to Dana
for making my life so much easier when dealing with Stanford bureaucracy.
I will dearly miss Judy, Matt and Patricia, and I apologize that I did not hold
more promised Rock Band and hot tub parties before I left town. I will also miss
sharing an office with Stephen and bouncing research ideas off of each other in
between discussing Top Gear.
I've been incredibly fortunate to have so many friends in the physics department. Catherine, Chad, Derek, Mike, Brian, Susan, Yvonne, Fen, Wells, Dan, Ian,
Melissa, Anika, Dave and Vince have pulled me through problem sets, necessary
indulgence before heading to the South Pole, and what would have been lonely
Viii
ix
weekends when Simone was still in San Diego.
I'd also like to thank Chao-Lin Kuo for motivating much of my research towards the end of my time at Stanford. Within the QUaD collaboration, Clem
Pryke and Michael Brown have been instrumental in teaching me most of what I
know about data analysis and have entertained every question of mine from the
bothersome to the trivial.
Although they are all several thousand miles away, my mother, father, grandmother and little sister Mindy have only grown closer to me over my time at Stanford and I am incredibly grateful for that. I'm thankful that Judy, Jim, Leah and
Chloe have welcomed so warmly into their family, especially when I can't visit
the Philippines or Taiwan. And, of course, I still falling more and more in love
with my wonderful wife Simone every day. Thank you for fixing many parts of
this thesis and getting me through graduate school, and for the future we have
together.
Contents
Abstract
iv
Preface
v
Acknowledgments
viii
1 Introduction and background
1.1
1.2
1.3
1.4
1.5
Overview of the CMB and history of observation
Qualitative modern understanding of the universe
1.2.1 Properties of the universe
1.2.2 Constituents of the universe
Cosmological history and evolution
1.3.1 Robertson-Walker metric and Friedmann equation
1.3.2 Inflation
1.3.3 Consequences of inflation
Perturbation evolution
1.4.1 Perturbation modes from inflation to recombination
1.4.2 Generation of polarization at recombination
1.4.3 Reionization
1
....
1
4
4
6
7
7
10
12
16
16
18
20
CMB Observables
21
1.5.1
Characterization of CMB temperature anisotropies
21
1.5.2
Characterization of CMB polarization anisotropies
22
1.6
Weak gravitational lensing
26
1.7
The QUaD instrument
29
X
Contents
xi
1.7.1
Overview
29
1.7.2
Location
30
1.7.3
Observing
field
2 Data characterization and history of QUaD
34
2.1 Telescope and beams
2.2
34
2.1.1
Optical design and characteristics
2.1.2
Mainlobe and sidelobe beam predictions and measurements 38
2.1.3
MCMC beam parameter analysis
34
41
Receiver and focal plane
46
2.2.1
Polarization Sensitive Bolometers
46
2.2.2
Focal plane
48
2.2.3
Cryostat and cooling system
52
2.3 Data acquisition and time ordered data
2.4
31
54
2.3.1
JFETs and warm electronics
54
2.3.2
Time constant modeling
56
2.3.3
Time constant measurements
57
2.3.4
Deconvolution
60
2.3.5
Despiking and identification of contaminated data
63
2.3.6
Relative gain calibration
64
Observation History
65
2.4.1
Observation Strategy
65
2.4.2
Data organization
67
2.4.3
Observation efficiency
69
3 Mapmaking and Monte Carlo power spectrum estimation
70
3.1
Overview
70
3.2
Mapmaking
74
3.2.1
Mapmaking formalism
75
3.2.2
Temperature and polarization timestreams
77
3.2.3
Polarization mapmaking
80
3.2.4
Timestream
filtering
82
xii
3.3
Ground contamination removal
83
3.2.6
Maps
85
3.2.7
Variance maps and apodization
91
Spatial anisotropy spectra
92
3.3.1
2-dimensional FFT Temperature Maps
94
3.3.2
2-dimensional FFT Polarization Maps
97
3.3.3
Annular ID spectra
98
3.4
MASTER formalism
3.5
Noise simulations
103
3.5.1
Timestream noise properties
103
3.5.2
Noise generation
106
3.5.3
Noise simulation results
108
3.6
99
Signal-only simulations
109
3.6.1
Bandpower Window Functions
113
3.6.2
Beam and filter transfer function
116
3.7 Absolute Calibration
117
3.8
Systematics tests
121
3.8.1
Jackknife subtraction maps and spectra
121
3.8.2
Quantifying null-signal difference tests
125
3.9
4
3.2.5
Final spatial anisotropy spectra
126
3.9.1
129
Frequency combined spectra
3.10 Limits on gravrtationally lensed polarization
130
Optimal mapmaking and filter functions
4.1 Maximum likelihood bandpower analysis motivation
131
132
4.2 Approximately optimal mapmaking
134
4.3 Mapmaking results
4.4 Pixel filter function
137
138
5 Cosmological parameter estimates
142
5.1
Cosmological Parameters and C/s
142
5.2
Method
147
Contents
5.3
5.4
5.5
6
xiii
5.2.1
Posterior and likelihood functions
147
5.2.2
Basic Metropolis used by WMAP
150
5.2.3
Modified Metropolis using optimized step size
151
5.2.4
Priors
154
5.2.5
Convergence Criterion
155
External Data Sets
157
5.3.1
Hubble Key Measurement
157
5.3.2
WMAP Satellite
158
5.3.3
SDSS Luminous Red Galaxy Survey
158
Results
159
5.4.1
6 parameter ACDM
159
5.4.2
ACDM with tensors
161
Conclusions
162
Electrodynamic Parity Violation
165
6.1
Background
6.1.1 CPT violation induced by a Cherns-Simons term
165
166
6.1.2
168
Ni's Lagrangian, P asymmetry
6.2 Analysis
169
6.3
Current Limits and QUaD Results
172
6.4
Systematic effects and checks
174
6.4.1
Systematic bias caused by beam offsets
174
6.4.2
Systematic rotation
177
6.4.3
6.4.4
Overall rotation measured by near-field polarization source 178
Overall rotation measured through beam offsets
182
6.4.5
Overall rotation and systematic errors
7 Conclusion
183
185
7.1
QUaD in context
185
7.2
Limits on new physics
190
A Robustness of near-field polarization source measurements
191
Contents
xiv
Bibliography
195
ist of Tables
2.1 Measured beam parameters for all operational 2006-2007 channels
2.2 Lab and Gunn source time constant values
2.3 List of observing days used in results and mean declination for day
47
59
68
3.1 Table of jackknife probability to exceed values computed from x2
statistics
124
5.1 ACDM model parameters and 68% confidence intervals from MCMC161
5.2 ACDM with tensor modes model parameters and 68% confidence
intervals from MCMC
163
6.1 Aa non-bias corrected measurement
6.2 Aa systematic bias corrected measurement
6.3 Cross-polarization and deviations from design angle as fit to measurements from near-field polarization source for all detectors . .
6.4 Absolute rotation angle fit from beam offset measurements
xv
172
173
. 181
183
List of Figures
1.1
1.2
WMAP full sky map
Polarization generation due to Thomson scattering of quadrupolar
anisotropics due to scalar and tensor perturbations
1.3 WMAP CMB temperature anisotropy spectrum
3
18
21
1.4
ACDM predicted anisotropy spectra
23
1.5
1.6
Lensing power spectrum Cf^
Toy model of CMB temperature field distortion by lensing
27
28
1.7
Aerial photograph of South Pole Station and MAPO
31
1.8
QUaD observation field
32
2.1
QUaD primary and secondary diagram
35
2.2
Residual warp in primary mirror
36
2.3 Beam ellipticity illustrated by RCW38 maps
37
2.4 Example of moon contaminated data from foam cone reflection . . 38
2.5
2.6
Central feedhorn broadband physical optics model
CoaddedmapsofquasarPKS03537-441atl00andl50GHz
39
40
2.7
ID marginalized posterior distribution from MCMC beam parameter fits for central feedhorn
43
2.8
Comparison of measured beam data to model in annular rings . . .
45
2.9
QUaD focal plane assembly and cryogenic isolation system
49
2.10 2006-2007 operating PSB offsets and polarization sensitivity directions
51
2.11 2006-2007 interstage and ultracold temperatures
53
xvi
xvii
List of Figures
2.12 Response function of a two time constant channel
56
2.13 Comparison of time constants from lab and Gunn source measurements
58
2.14 Cosmic ray incident and post deconvolution timestream
60
2.15 Scan direction subtraction test of deconvolution in primary CMB
field near PKS0537-441
62
2.16 Raw time-ordered sum and difference data and power spectra verification plot
63
2.17 Timestream from elevation nod used for relative gain calibration . . 65
2.18 Eight scan sets comprising one hour of QUaD observing strategy . . 66
3.1
Schematic diagram of Monte Carlo APS estimation pipeline
73
3.2
Effect of cross-polarization efficiency and detector alignment errors on angular power spectra
79
3.3
Demonstration of time ordered data
84
3.4
Field differenced 150 GHz temperature map
86
3.5
3.6
Non-field differenced 150 GHz temperature map
Non-field differenced 100 GHz temperature map
87
88
3.7
Non-field differenced 150 GHz Q Stokes polarization map
89
3.8
Non-field differenced 150 GHz U Stokes polarization map
90
3.9
Non-field differenced 150 GHz inverse variance pixel weight mask . 91
filtering
3.10 2D FFT of 150 GHz temperature map
95
3.11 Noise-dominated Fourier plane and signal-to-noise mask at 150 GHz 96
3.12 150 GHz E and B mode 2D Fourier Maps
97
3.13 Unprocessed 150 GHz spatial anisotropy spectra
3.14 100 and 150 GHz atmospheric and bolometer noise spectra
99
104
3.15 100 and 150 GHz atmospheric and bolometer simulated noise spectra
106
3.16 150 GHz simulated noise-only map
109
3.17 150 GHz simulated noise-only spatial anisotropy spectra
110
3.18 150 GHz simulated signal-only map
Ill
xviii
List of Figures
3.19 2D Fourier space representation of bandpower window function
calculation
113
3.20 Bandpower window functions
114
3.21 Beam/filter transfer functions
116
3.22 Repixelized BOOMERanG calibration and reference maps
117
3.23 100 and 150 GHz computed absolute calibration as a function of
multipole£
118
3.24 150 GHz deck jackknife map
122
3.25 150 GHz deck jackknife spectra
123
2
3.26 Scan direction jackknife x statistics from data simulated distributions
125
3.27 Final 100 GHz temperature and polarization spatial anisotropy spectra
127
3.28 Final 150 GHz (top) and cross-frequency (bottom) temperature and
polarization spatial anisotropy spectra
128
3.29 Final combined spatial anisotropy spectra utilizing information from
100 and 150 GHz auto-spectra as well as cross-frequency spectrum. 129
4.1
150 GHz optimal map eigenvalues
136
4.2
150 GHz optimal map with 25000 eigenvalues corrected
138
4.3
Matrix representation of single row filter functions
140
5.1
Dependence of Q on ACDM 6 parameter set
143
5.2
Convergence of ACDM 6 parameter fit for QUaD and HST data as
a function of chain step
155
5.3
One dimensional marginalized parameter estimates for QUaD + HST160
5.4
Two dimensional marginalized parameter estimates for QUaD + HST164
6.1
Parity violation measurements comparing data to suite of QUaD
signal and noise simulations
6.2
174
Parity violation measurements by bandpower derived from QUaD
EB spectra
175
List of Figures
xix
6.3 Near-field polarization source mounted near QUaD telescope . . . 178
6.4 Near-field polarization source fit for March 2006, central feedhorn . 179
6.5 Absolute rotation angle measured by each detector
184
7.1 Final QUaD temperature spatial anisotropy spectrum compared to
other experiments
187
7.2 Final QUaD polarization spatial anisotropy spectra compared to
other experiments
188
7.3 Final QUaD polarization spatial anisotropy spectra compared to
other experiments, log I scale
189
Chapter 1
Introduction and background
When I behold, upon the night's starr'd face,
Huge cloudy symbols of a high romance,
And think that I may never live to trace
Their shadows, with the magic hand of chance;
When I have Fears that I may Cease to Be
JOHN KEATS
1.1
Overview of the CMB and history of observation
The Cosmic Microwave Background (CMB) was discovered in 1965 by Penzias
and Wilson (5) when they noticed a source of extra signal entering their microwave
telescope that could not be accounted for by the measured noise of their instrument, or atmospheric contributions. It came from all directions of the sky uniformly, and was not polarized as far as they could tell. The significance of their
discovery was recognized immediately (6) as evidence that the universe had once
existed as a hot fireball of disassociated protons and electrons, and had at some
point cooled down to a temperature at which hydrogen could be formed, called
recombination. The standard model for the history of the universe now recognizes this excess microwave radiation at a temperature of about 2.725 degrees K
Overview of the CMB and history of observation
2
as the elongated and cooled light waves emitted by the hot plasma that was the
early universe at the epoch of recombination, providing evidence that not only
has the universe expanded in order to cool down, but also that the expansion
history of the universe points clearly to a single point, the Big Bang.
In 1992, structure in the CMB was discovered in what was previously only seen
as a homogeneous field. Instead of the entire sky glowing at a single temperature of 2.725K, the COBE satellite discovered anisotropies in the microwave background at about 1000 times less power than the the background itself (7). These
anisotropies are theorized to originate from the forces governing the constiuents
of the early universe, a plasma of radiation and matter. Underdense perturbations of of the early universe experienced gravitational collapsing forces due to
the matter component while overdense regions experienced outward radiative
pressure from the photon component. The combination of these opposing forces
led to an oscillating behavior in the early universe plasma with certain resonant
scale sizes. Like most harmonic oscillators with a characteristic driving force, oscillations of a certain frequency are preferred as are their harmonics.
As the early universe grew and further and further reaches of the early universe plasma could transmit information via these acoustic waves, perturbations
of greater and greater scale began to collapse. Once they reached a scale of maximum overdensity, they rebounded and began expanding again. The 1998 long
duration balloon flight of the microwave observing experiment BOOMERANG
over Antarctica published data in 2000 that conclusively observed a peak in the
scale size of the anisotropies on the sky of about 1° (8). This characteristic size of
the anisotropies observed on the sky corresponds to the final collapsed size of the
last, largest harmonic mode that had time to enter the horizon and collapse before recombination occurred, and can be easily seen on the full sky as measured
by the WMAP satellite in figure 1.1.
Soon afterward, numerous other experiments, notably the WMAP satellite (9),
observed anisotropies at higher resolution, tracing out to high signal-to-noise a
Overview of the CMB and history of observation
3
Figure 1.1: Map of CMB temperature anisotropics over the full sky as measured by the
WMAP satellite (WMAP Collaboration).
full spectrum of temperature anisotropies with multiple peaks and troughs corresponding to modes that entered the horizon and oscillated to maximum compression or rarefaction respectively.
The spatial oscillations in density at the time of recombination also provide
the conditions required to generate polarization. A predictable fraction of the
perturbations in the CMB that generate the observable temperature anisotropies
will be arranged in a quadrupole fashion around electrons at the "surface of last
scattering" during recombination, generating polarization through Compton scattering. For example, an electron seeing slightly colder photons arriving from the
left and right, and slightly hotter photons arriving from above and below at the
time of recombination will scatter linearly polarized radiation.
The polarization also has a spatial anisotropy spectrum that is closely coupled to the physics that generate the temperature spatial anisotropy spectrum,
and therefore its detection and characterization is an important test of our understanding of the physics of the early universe. In addition, the particular pattern of
polarization orientations observed on the sky can be decomposed into two parts,
Qualitative modern understanding of the universe
4
the "E-modes" that couple to the temperature anisotropies as described above,
and "B-modes" that arise from gravitational waves generated by tensor perturbations in the early universe plasma and conversion of E-mode power through
gravitational lensing by large scale structure between the Earth and the surface
of last scattering. The DASI experiment, an microwave interferometer that observed from near the South Pole, first observed E-mode polarization in the CMB
in 2002 (10).
This introduction will provide an overview of the relevant physics to understand how the CMB and its temperature and polarization observables were generated and subsequently modified by cosmic events on their way from the recombination to observation.
1.2
Qualitative modern understanding of the universe
Although we will delve into more quantitative details of the universe's formation
later in the introduction, it is important to begin with some qualitative understanding of the properties of the universe and its constituents. This will motivate
the physics presented later.
1.2.1
Properties of the universe
• Isotropy - we can observe that the universe appears to have the same largescale properties in all directions from our vantage point on Earth. Assuming
that the Earth is not in a privileged position in the universe (the Copernican
principle), we can deduce that the universe has the same large-scale statistical properties throughout no matter from which point one observes it and
that the large-scale properties measured from Earth are typical.
• Super-horizon homogeneity - However, the observed homogeneity leads to
another question. Distant parts of the sky as observed from Earth have
very similar properties as observed in the CMB, but from our understanding of the universe's current constituents and evolution, there is no way
Qualitative modern understanding of the universe
5
that those regions could have ever been in causal contact. This is known
as the "horizon problem" and is solved by hypothesizing that some as yet
undetermined force expanded the universe exponentially quickly early in
its lifetime, allowing distant regions to equilbriate to similar properties before being driven out of causal contact. This hypothetical process is known
as "inflation."
• Finite size and age - The dark night sky is sufficient proof that an infinite and
static universe consisting of an infinite number of stars like those we can
observe is impossible. Popularly known as Olbers paradox, such a universe
would have an average sky brightness at the temperature of the surface of a
typical star, as the diminished amount of light due to the increased distance,
r from each successive "shell" of stars would be exactly balanced out by the
additional number stars enclosed in that shell, with both effects scaling as
r2.
• Expansion over time - Hubble's observation that galaxies are receding at a
velocity proportional to their distance is consistent with a universe that is
expanding. For a universal scale a this has been quantified at the current
time as H0 = a/a = 72 ± 8 (km/s) / Mpc by the Hubble Key project (11).
• Spatial flatness - Einstein's equations for General Relativity can be distilled
to the Friedmann equations that describe how the expansion of the universe is governed by its matter and energy constituents. The curvature of
spacetime is determined by the overall density of matter and energy in the
universe, p, and at a special density pc the curvature is flat. When observations of the CMB are combined with measurements of the Hubble Constant
and observations of Type 1A supernovae, we have convincing evidence that
the universe's curvature is indeed flat. As p = pc is a priori a special condition that requires fine tuning, one attractive aspect of the inflationary hypothesis is its implementation often requires that the resulting universe is
flat.
Qualitative modern understanding of the universe
1.2.2
6
Constituents of the universe
The expansion history and future of the universe is driven by the pressure-density
relationships of its constituents, which in turn determines how their densities
evolve as the universe expands. We can define the universal scale factor as a,
and the pressure density relationship w = Vjp which characterizes each of the
constituents.
• Baryonic matter - Normal, unremarkable matter that we interact with only
forms about 4.4% of the total matter-energy content of today's universe. As
free matter in space typically has little or no pressure, it has w = 0 and
scales as p& ~ a~3, like a volume element as the universe expands, as one
might expect.
• Non-baryonic matter - Observations of the rotation rate of stars around the
core of spiral galaxies imply that there is an order of magnitude more matter present than can be observed by the emitted light. Such matter interacts gravitationally with normal baryonic matter but to our knowledge does
not interact electromagnetically. A somewhat direct observation of this behavior was observed recently in the Bullet Cluster by observing the noninteracting "cold dark matter" through weak lensing (12). Insofar as the only
observed differences between the two types of matter are photon interactions, cold dark matter also has w = 0 and scales as pm ~ a - 3
• Photons - Well before matter decouples from the photons, the universe's
mass-energy density was dominated by radiation. However, as the universe
expanded, their contribution to the total energy density dropped. This is
beacuse the expansion of the universe also stretches the wavelength of each
photon linearly with a, so that the energy density drops as pr ~ a~4. The
pressure-density relationship for radiation is w = 1/3.
Cosmological history and evolution
7
• Dark energy - Our current understanding of the other three constituents
indicates that they can make up no more than 30% of the current matterenergy density needed for a flat universe. The observation that the expansion of the universe is accelerating from Type 1A supernovae strongly implies a negative pressure, w < —1/3 "dark energy" constituent. In models in
which dark energy is described as a cosmological constant in the equations
of General Relativity (defined explicitly later), w = — 1 and the density of
dark energy remains constant as the universe expands, pA ~ a0. As a result,
the later history and future expansion of the universe is dominated by dark
energy.
A flat universe consisting largely of the above ingredients is consistent with
many current cosmological probes (13), and is known as the ACDM model.
1.3
Cosmological history and evolution
Dispensing with the details of general relativity, it is instructive to examine how
straightforward applications of the qualitative observed properties of the universe derived above to the equations of general relativity combine with the constituents of the universe and delineate our modern understanding of cosmology.
The results of this review then can be shown to motivate the inflationary hypothesis. This section will the further discuss how the anisotropies in the CMB are
influenced by the cosmological dynamics derived from general relativity and the
initial conditions implied by inflation. Note that much of the following section is
well-known and understood physics, but the following explanation owes much
to (14)
1.3.1
Robertson-Walker metric and Friedmann equation
From the observed isotropy and implied homogeneity of the universe, but apparent evolution in time, the line element describing the Robertson-Walker metric
Cosmological history and evolution
8
is a natural one to choose. It requires homogeneity in space, but allows for an
evolution of the metric over time:
2
2
ds = -dt
2
+ a (t)
dr2
ar
2
1 — KT2 +r
dn2
(1.1)
where a(t) is the universal scale factor as a function of time, K is the curvature of
the universe, with +1 for open geometries, 0 for flat geometries, and -1 for closed
geometries and dD,2 is the standard solid angle line element on a sphere.
The metric can then be used to derive the Christoffel symbols, which in turn
can be used to derive the Ricci tensor. The corresponding Ricci tensor to this line
element is diagonal with elements:
Roo =
R n
-3a
aa + 2d2 + 2K
=
l-kr2
2
(1.2)
(L3)
2
R22 = r (a'd + 2a + 2K)
#33 = r2{ad + 2d2 + 2K) sin2 9
(1.4)
(1.5)
where d = da/dt and d = d2a/dt2. For a stress-energy tensor T^u with elements
defined by the pressure and density, p and p, of the universal constituents in question, we can use the Einstein equation defining General Relativity:
R„v = 8vrG(T^ - i ^ T ) + 9llvk
(1.6)
where G is the gravitational constant and g^u is the metric. The cosmological
constant term containing A can then be folded into the the stress-energy tensor
T^v, the Ricci tensor elements plugged in, and due to spatial isotropy we only
derive two equations for the two space and time degrees of freedom, known as
the Friedmann equations:
Cosmological history and evolution
9
d\2
aJ
a
a
8TTG
_
p--2 2
3 '
aa1
AnG
(p + Sp)
3
(1.7)
(1.8)
The Hubble parameter at any given moment of the universe's history is given
by H = a/a, so we can define the density parameter as a function of time as:
The first Friedmann equation then clearly shows that the curvature of the universe depends on its matter-energy density:
The observation that the universe is flat then requires that Q = 1, or p = | ^
at any given time, and in fact if the universe is flat then Q = 1 remains for all
time. There is no clear reason for why the current density must be this particular
"critical density."
Finally, we can use the Friedmann equations along with the scaling relationships described in the qualitative section above to reduce the dynamics of the
universe at any given time to the densities of its constituents at the current time.
If we define the current time critical density as pcrit = ^ , then the currenttime density of baryons as a fraction of the critical density is Qb = p&/pcrit, and
likewise, Qm = pm/pCnt, ^r = pr/pcrit a n d ttA = pA/pCnt. For a flat universe,
s2{, + i Zm + i lr + i l\ = 1.
Defining the current scale size a0 = 1, we can then use this refinement of the
energy density to show how the universe evolves as a function of the current day
energy and matter density parameters:
10
Cosmological history and evolution
Alternatively, since the redshift z is an observable, it is useful to recast this
equation using a0/a — 1 + z:
- 2 = Qb{l + zf + Q c (l + zf + fir(l + zf + nA
(1.12)
Admittedly this formulation omitting curvature is a bit of a deception - flatness was deduced from observations by incorporating the contribution of the
equivalent energy density from curvature QK (1 + z)2 into the above equations and
showing that the data was consistent with flK = 0.
1.3.2
Inflation
There is a natural solution to the flatness fine tuning problem in equation 1.10. If
we impose the condition that
~
<0
at an
(1.13)
then over time any curvature term becomes negligible and Q is driven very close
to 1. If aH = d is a very large number at some point in the early universe's history,
the geometry of space-time could be nearly, although not perfectly flat. Thus,
any perceptible deviation would not show up until much later in the universe's
history.
This condition also solves the horizon problem. (aH)'1 turns out to be the
"comoving Hubble radius", or roughly the distance over which light can travel
while the scale factor doubles (15), or a rough measure of the distance of causal
contact in the universe at any given time. We can also define the comoving horizon rj, which measures the limiting distance at which particles were ever in causal
contact anytime during the history of the universe:
rda'
i
v=L^vm
M i^
(L14)
The above condition, ^(aH)'1 < 0, implies that the Hubble radius is shrinking
and forcing areas previously in causal contact unable to exchange information,
Cosmological history and evolution
11
leading to r\ much larger than (ai/) _ 1 . This is precisely the solution to the horizon
problem we are seeking - particles once in causal contact are causally separated
due to inflation.
How are physical models of inflation achieved? If the universe expands exponentially with nearly constant positive H, then we have:
a(t) = aeeH{t~te)
(1.15)
where te is the time at the end of inflation. In such a case we have ^§ > 0, implying accelerating expansion of the universe. What kind of matter or energy
could produce such an expansion? We have already seen that the scale size of
our current universe appears to be accelerating due to the influence of a negative
pressure dark energy. Many cosmologists hypothesize that inflation arises from
an as yet undetected homogeneous scalar field that evolves with time, <p = (p(t),
with negative pressure-density w < 1/3, associated with a hypothetical particle
called the "inflaton." Following (16) we can define the density and pressure for
such a field:
P<t> = \4>2 + V(<t>)
(1.16)
v*
(1.17)
=
\tf-v{4>)
where V(4>) is a potential for the scalar field. Substituting this into the Friedmann
equations for a flat universe yields (with h = c = 1):
H2 = ^{\tf
+ Vm
4> + 3H<j> = -%d(p
(1.18)
(1.19)
The "slow-roll" inflation model is the simplest and most commonly used. In
this limit, two parameters, e{4>) and rjfy) are defined and the conditions
12
Cosmological history and evolution
,
W
=
8
,
G
^ * ! « l
,1.21)
are derived by eliminating terms in the Friedmann equation and restricting 4>2 <C
V{4>) and |0| <c \3H<fi\. This leads to a simplification of the Friedmann equations
for the inflaton field:
H> « ^ M
3Hcp « - d F / #
„. 2 2 )
(1.23)
Note that in this formulation e = — ^ = ^ , and for any potential such that
e > 0, our condition for inflation ^ ^ < 0 is satisfied. At some point inflation
must end as the scalar field rolls into an appropriately steep point of its potential,
and "reheating" occurs after dozens of e-foldings (multiplications by a factor of
e), allowing the inflaton field to decompose into the products of the universe we
see today.
1.3.3
Consequences of inflation
Above, we have postulated that inflation stems from a homogeneous scalar field
that evolves identically and simultaneously throughout the universe during the
inflationary period. However, the existence of a particle horizon creates a situation analogous to the quantum harmonic oscillator - the ground energy level for
each mode has a degree of quantum fluctuation related to the the size and shape
of the potential. A nonzero "Gibbons-Hawking" temperature of the vacuum state
can be associated with the horizon (17; 14):
r-I-^
Cosmological history and evolution
13
We can characterize the quantum fluctuations induced by this temperature in
isotropic three-dimensional Fourier space by wavenumber k. The temperature
induces an equal amount of fluctuation in the scalar field at each wave number,
and this leads to fluctuations in the energy density p:
|A0|fc = TGH
Sp
= ^-5<t>
(1.25)
(1.26)
dcp
The fluctuations in the scalar field 54> thus have equal power at all scales, and
follow a "scale-free" spectrum for constant H:
PS4> ~ (A0fc)2 ~ U
(1.27)
The spectrum for what we term "scalar" density perturbations to the metric
incorporates a factor of (fr) 2 into the denominator, coupling the slow-roll parameter e. Unfortunately, conventions for incorporating the factor of k~3 vary
widely, as do prefactors of 2TT and the units of the h, c and G. What is important to
remember is that as a result of ^ converting the scalar field perturbations S<f> into
density perturbations Sp, as modes exit the horizon during inflation they end up
with a nearly scale-invariant "primordial scalar spectrum" P$(k) of
p
* « ~ "ST ~ S?
(L28)
where $ is an alternative formulation of the density perturbation Sp to the metric
element #0o- Scalar fluctuations also have curvature perturbations on the diagonal spatial metric elements g^Sij for i = 1,2,3 and are parametrized by *. For
adiabatic perturbations, P^(k) = Py(k).
The nearly scale-invariant spectrum is conventionally parametrized in the literature using a "scalar spectral index" ns that represents the deviation from scaleinvariance:
14
Cosmological history and evolution
n.,-1
k
1
z
~T5\-ET
k
Ho
P*W
(1.29)
where again factors have been dropped to emphasize the primordial power spectrum's shape rather than its amplitude. A scale-invariant spectrum corresponds
tons = 1, and data from CMB experiments can be analyzed for deviations from
1. Such a discovery is of interest because the slow-roll parameters from inflation
couple into ns as:
1 - 4e - 2r]
n.
(1.30)
Inflation also produces "tensor" perturbations to the metric in the form of
gravitational waves. Solutions with gravitational waves have perturbations to the
metric characterized by two functions, h+ and hx and have a metric g^:
/
-1
0
0
\ o
0
0
a hx
(1 + M
2
2
a (l + h+)
a hx
0
0
2
2
0 \
0
0
(1.31)
a2 )
which is a simple extension of the Robertson-Walker metric. The solution to the
Einstein equations for this metric turn out to be oscillatory waves with h+/x oc
e%kn where k is the wave number and r\ is the conformal time (not the slow roll
parameter).
Tensor perturbations caused by inflationary models are also generated in the
nearly scale-free manner as they exit the horizon. However, they are coupled
directly to the energy level of inflation and do not require a factor of ^ like the
scalar modes. Again omitting factors of the Planck mass, the tensor mode power
spectrum is nearly scale-invariant:
PT{k)
v{4>)
k
3
k3
(1.32)
15
Cosmological history and evolution
This is parametrized with the "tensor spectral index" nT and "tensor amplitude" AT as:
pT(k) = ATknT~3
(1.33)
The slow-roll inflation conditions impose the constraints that nT = —2e and
ATIAs
= -8nT, where As is equal to the non-fc dependent terms of P$(fc) and
defines the amplitude of the scalar modes. A measurement of tensor modes in
some fashion would be a direct measurement of the potential V during inflation,
but unfortunately for observationalists there is no lower constraint for the energy
at which inflation must take place. If it takes place at levels sufficiently smaller
than the Planck mass mp = J^
then looking for gravitational waves as a means
to probe inflation is a fool's errand. Nonetheless, a detection of tensor modes
would be the first sign of physics at energy scales of 1015 GeV or greater, far beyond any possible terrestial probe of high-energy physics (18).
Furthermore, during the radiation- and matter-dominated phase of the universe after the end of inflation but before recombination, the horizon expands
and tensor modes that fall within the horizon decay. A super-horizon tensor
mode is not affected by the physics of the early universe plasma and emerges
from recombination as a pristine remnant of inflation. Unfortunately, this means
that the only tensor modes that remain are well before the first peak in the temperature spatial anisotropy spectrum which signifies the acoustic horizon at recombination, so we expect that if inflation-generated tensor modes are observationally feasible they will most likely be found at t < 100. This poses a major
problem for a ground-based instrument with limited sky coverage, like QUaD.
Perturbation evolution
1.4
1.4.1
16
Perturbation evolution
Perturbation modes from inflation to recombination
The Boltzmann equations for radiative transport allow us to relate the initial conditions of scalar perturbations in the metric to the behavior of modes within the
horizon. The full description of these equations is somewhat complicated and
tedious process which will be omitted here. Although analytic solutions can be
found that describe how superhorizon modes and modes that cross into the horizon in either the radiation-dominated epoch or the matter-dominated epoch,
describing modes that cross into the horizon around the time that the matter
and radiation energy densities are roughly equal requires numerical methods. In
practice, observational cosmologists seeking to model the evolution of the nearly
scale-invariant primordial spectrum for a given set of universal constituents use
a high-accuracy numerical code like CAMB (19).
However, we can sketch the outlines of how these equations work. In a collisionless system of particles the total phase space density f(x, q, t) is conserved
due to Liouville's theorem, such that df/dt = 0, where x and q are the generalized
particle positions and momenta. A photon in the early universe plasma, however, is not collisionless. From its perspective phase space density is changing
due to baryon velocities and photon density perturbations, coupled by Thomson
scattering off of charged baryonic particles in the fluid (20). Cold dark matter
interacts indirectly in the matter dominated epoch before recombination by altering the gravitational potentials in the plasma, influencing both the photons
and baryons in the area. Schematically, the condition is df/dt = C[f] where C[f]
are the accumulated "collision" terms.
Perturbations to photon density A can be written as
A = l*3jp.
(1.34)
where / is Planck's blackbody function. The evolution of the perturbations then
follow the Boltzmann equation (20)
Perturbation evolution
<9A
A + c7i o— - Zliljhij = aTcnea(A0 - A + k)iVlB/c)
17
(1.35)
where dots specify differentiation with respect to the conformal time rj = J dt/a,
7i are the direction cosines defined by the direction of the momentum q, ne is the
free electron density, a is the scale factor, vB is the baryon peculiar velocity, A0 is
the isotropic part of A, hij is the metric perturbation and aT is the Thomson cross
section of the electron.
In the literature, this equation is then Fourier transformed and linearized to
allow analysis of modes of different sizes to be examined independently. Qualitatively speaking, there are three regimes to the evolution of scalar perturbations:
• Super-horizon modes that are driven out of causal contact by inflation and
are too large at the time of recombination to fall back into the horizon essentially do not evolve througout their time in the early universe. The anisotropics are only affected by the Sachs-Wolfe effect, whereby photons emerging from hotter, denser regions of space at recombination actually appear
colder than photons emerging from underdense areas. This surprising result comes from the fact that the photons are redshifted by emerging from
the gravitational potential wells. The transfer function of these modes from
inflation to recombination is roughly 1, and when we measure the corresponding modes on the sky in the CMB their spectrum looks flat in equal
power units.
• Modes that enter the horizon from the end of inflation to recombination experience gravitational collapse, and subsequent radiation pressure from the
resulting overdensity leads to oscillatory expansion and contraction whose
behavior is dictated by the baryon density. The baryons, interacting both
gravitationally with the underlying dark matter density and electromagnetically with the photons via the electrons, essentially are a damping inertial
force on the driven harmonic oscillator created by the two competing radiation and gravitational forces. The transfer function of these sub-horizon
modes depends on the amplitude of the perturbation with respect to the
Perturbation evolution
18
Figure 1.2: Polarization generation due to Thomson scattering of quadrupolar anisotropics
(left), scalar density perturbations (middle) and tensor density perturbations (right). Perturbation waves travel either upwards or downwards with respect to the center and right
figures. Scalar modes shown in the central figure propagate like longitudinal waves and
produce patterns of polarization called "E-modes" on the sky that are parity symmetric.
Tensor modes shown in the right-most figure can propagate with perturbations hx and h+
in equation 1.31, yielding both parity asymmetric and symmetric patterns of polarization
along the sky respectively. Figures are taken from (21).
plasma background when the oscillatory process is cut off due to recombination.
• Small-scale subhorizon modes are increasingly dominated by photon diffusion processes. For small enough modes at the surface of last scattering,
photons from hotter regions diffused to nearby cooled regions undergoing
recombination. This process is sometimes called "Silk damping" and it suppresses the small-scale spatial anisotropy power of the CMB.
1.4.2
Generation of polarization at recombination
Quadrupolar anisotropies around an electron during recombination generate polarization patterns on the sky that can be traced to either scalar or tensor perturbations of the metric. The physical method for generating polarization is Thomson scattering of photons off of electrons during recombination - recall that the
cross section for Thomson scattering prefers matching of the input and output
polarization directions, e and e respectively (21):
Perturbation evolution
19
(lax
—— oc e-e
(1.36)
ail
To visualize how polarization is generated from this process, consider the Thomson scattering process from the perspective of the electron for various moments
of unpolarized radiation temperature anisotropies around it. In a uniform background, with unpolarized radiation coming in from all sides, one might reasonably expect from the symmetry of the problem that no net polarization is generated.
Although a dipole of radiation temperature around an electron sets up a preferred direction in the problem, a dipole results in no net polarization, because
the "hot" and "cold" linear polarization states match and cancel at the electron
in every direction. A quadrupole can clearly generate linear polarization aligned
with the "hot" axis of the quadrupole in a outgoing wave normal to the quadrupole,
as shown in figure 1.2.
Both scalar and tensor perturbations to the metric produce polarization, albeit of different types. Scalar density perturbations can travel like longitudinal
pressure waves, and the alternating series of hot and cold moments along the
direction of propagation can generate polarization maximally emitted orthogonal to the direction of propagation. The polarization pattern observed on the sky
propagating orthogonal to the observer's line of sight appears like:
Note that the above pattern of polarization is invariant under parity switch.
Tensor perturbations of type h+ from equation 1.31 also produce a parity invariant pattern of polarization on the sky. As can be seen in figure 1.2, each halfphase of the tensor perturbation generates polarization maximally emitted along
Perturbation evolution
20
the direction of propagation (upwards in this picture). However, tensor perturbations can also propagate in a "cross-polarization" configuration rotated 45°,
where the "cross-polarization" in question refers to the polarization of the gravitational wave. When viewed obliquely from the direction of propagation of the
wave, the electromagnetic polarization pattern on the sky generated from such a
gravitational is a parity asymmetric pattern that appears as:
/
1.4.3
\
/
\
/
\
/
\
Reionization
Although the baryons and photons decouple during recombination at redshift
z ~ 1100, once matter overdensities left over from the scalar perturbations collapsed into the first generation of population III stars the ultraviolet radiation
emitted by these stars' fusion processes reionized the neutral hydrogen in the
universe and made the universe partially opaque. The presence of the GunnPeterson Trough in the Lyman-Alpha forest spectrum of quasars of z > 6 indicates that at these redshifts the universe's neutral hydrogen content was not substantially ionized (22). This is somewhat in conflict with CMB data returned from
WMAP which indicates that the reionization started around z ~ 11.
Regardless, reionization changes modes propagating from the CMB to observation in important ways. In temperature, except for the longest modes larger
than the horizon during reionization, the anisotropy spectrum is dampened in
a way consistent with lowering the overall amplitude. In polarization a new signal is created by Thomson scattering of the temperature anisotropics quadrupole
and higher moments off of electrons in the newly reionized universe. This results
in an overall increase of super-horizon modes in polarization.
CMB Observables
1.5
1.5.1
21
CMB Observables
Characterization of CMB temperature anisotropics
6000
100
500
Multipole moment I
Figure 1.3: CMB temperature spatial anisotropy spectrum Ct as a function of multipole
moment / as measured by the WMAP satellite from 5 years of observations. The locations
of the peaks, their relative size, and the overall amplitude and shape of these spectra
encode a wealth of information about the nature of our universe and its constituent parts.
The red line is the best fit 6 parameter ACDM cosmological model to the data in the first
graph. Figure from WMAP collaboration (23).
Density and tensor perturbations in the early universe evolved until the epoch
of recombination when the baryons decoupled from the photons, and with the
exception of photon diffusion the photons began to move freely through the universe until the reionization. When the modes reach us, we see them on a twodimensional spherical surface - how can we quantify the scale and power of spatial anisotropy that we observe on the sky?
22
CMB Observables
Current research on the CMB focuses on measuring maps of anisotropies in
the background on the microwave sky and deriving the constituents of our universe from the statistical properties of the spatial information encoded in those
maps by assuming that the fluctuations at a given scale are randomly and isotropically distributed across the sky.
Following Dodelson's methodology (15) we identify each point on the maps
above as a point on the surface of a sphere, with coordinates 9 and <f>. If the temperature field is 0(0,0) across the full sky we can decompose this field into the
orthogonal spherical harmonic basis:
oo
—I
0(0, <j>) = J2 E
e=i
a
emYem(e, <P)
(1.37)
m=-i
where a^m are the power of a particular spherical harmonic mode Yem(9,0) on the
full sky. Assuming that the universe does not have some preferred orientation or
direction, we expect (agm) = 0 for a given £ since our coordinate system should
not matter. However, the variance of the aem's is the power of all the modes corresponding to a specific multipole moment I on the sky, which in turn corresponds
to characteristic sizes of oscillating plasma in the early universe. The power for
on a characteristic length scale, the multipole moment £, is designated Ce- The
spectrum of Ce is the primary science result of any modern CMB experiment:
Ci = ^2aema*im
(1.38)
Although the individual aem coefficients depend on our coordinate definitions
the value Ce is invariant upon rotation.
1.5.2
Characterization of CMB polarization anisotropies
Linear polarization directions are invariant under 180° rotation, and therefore
transform as a spin-2 quantity. It is convenient to represent the linear polarization magnitude and direction as the two Stokes parameters Q and U. First, we
CMB Observables
23
100
multipole, I
10
10 4
1000
|
I
!
I
I
TT
10^ --
CM
»^;
EE
10°
/
BB lensing
o
[•x
/
"/I
10" 4
'
'^-"V
^^-^-_^BB 0.2
/^-NBBO.01^
500
"
-
1000
1500
multipole, i
-
2000
2500
Figure 1.4: Predicted CMB temperature spatial anisotropy spectrum Ci (black), E-mode
polarization auto-spectrum CfE and various B-mode polarization auto-spectra CfB as
a function of multipole moment I derived from best fit six parameter ACDM model to
WMAP satellite data (13). Separate B-mode contributions are shown for initial tensor
modes for r = 0.2 and r = 0.01 (blue), as well as conversion of E-modes to B-modes
from weak gravitational lensing (red). The same spectra are plotted in both log (top) and
linear (bottom) scales for multipole moment £.
24
CMB Observables
define the Stokes intensity parameter / which is consistent with the temperature
fluctuations for the electric fields Ex and Ey for a wave incident on the observer:
/ = \EX\2 + \Ey\2
(1.39)
We can then define a second basis by rotating the Cartesian basis for Ex and
SJ/by45°:
E
° = ^ - TI
(1 40)
Ex + ^L
Ej
Eb = -^
(1.41)
V2
-
V2
We then define the two Stokes parameters:
Q = \EX\2 - \Ey\2
2
U = \Ea\ - \Eb\
2
(1.42)
(1.43)
A Q-only positive signal is therefore a totally horizontally polarized wave (in
the electric field) when incident on the observer, and a Q-only negative signal
is vertical, whereas the U signals are diagonals, with bottom-left to top-right for
positive U and top-left to bottom-right for negative U.
Similarly to the temperature field, we can decompose the polarization signals into a series of spherical harmonics using the s = —2,2 generalized spinweighted spherical harmonic functions sYim. Full-sky maps of Q and U parameters can be decomposed into harmonic coefficients Eim and B^m analogous to
the aem for temperature with these functions (24):
CMB Observables
25
oo
—I
P(9,<j>) = Q{0,<f>) + iU{9,<l>) = Y,
l ] ( ^ m + ^m)2y<m(e,0)
£=1 m = - «
oo
—Z
P*(0,0) = Q ( 0 , 0 ) - i t f ( M ) = £ X ; ( ^ m - ^ m ) - 2 * * „ ( 0 , 0 )
d-45)
The £^m coefficients correspond to modes with (—1)* parity upon coordinate
reflection on the full sky, while the Bim coefficients correspond to modes with
(—l)e+1 parity. Armed with these coefficients, we can then define the auto- and
cross-spatial anisotropy spectra, corresponding to polarization power and temperature -polarization cross-correlations:
CfE = Y,E^Elm
d-46)
m
CfB = 5 > * A
(1-47)
m
CJE = J > m £ ; m
(1.48)
m
B
CJ
= Y,aemB*em
(1.49)
m
CfB = J2E^BL
d-50)
m
Again, the spatial anisotropy spectra are invariant under coordinate rotation and
are the fundamental result. Scalar perturbations and certain tensor perturbations can result in non-zero CfE power, whereas only tensor perturbations can
produce CBB power at recombination. These two spectra are often referred to
as "E-mode" and "B-mode" power. Nonzero B-modes can be produced after recombination, however, by weak lensing of the E-mode signal and foregrounds.
Since the scalar perturbations that produce temperature anisotropics also create
the quadrupoles at the epoch of recombination that produce E-modes, we also
(1.44)
26
Weak gravitational lensing
expect to see non-zero temperature-polarization cross-correlation in CjE spectrum.
The level of non-lensed cosmological B-modes is usually characterized by the
tensor-to-scalar ratio of the primordial power spectra:
where k0 is an arbitrary reference scale of the primordial power spectrum. Current limits on r < 0.2 (95 % CL) are generated by closely examining the temperature spectra for residual power from tensor modes while using the scalar spectral
tilt index ns to constraint the inflation slow-roll parameter e. The most ambitious
proposed experiments in the future will attempt to measure r < 0.01.
In a parity conserving universe the CfB and CjB spectra are predicted to be
zero. These spectra can be used to probe for cosmological scale parity violation,
and these tests are described further in chapter 6.
1.6
Weak gravitational lensing
Dark matter gravity potentials between the surface of last scattering at recombination and our location in the universe distort the Gaussian fluctuations of the
CMB on the sky. Recovering the strength and spatial anisotropy spectrum and
perhaps even mapping these potentials by examining their subtle effects on both
the CMB temperature and polarization anisotropics offers a window into highredshift structure formation possibly unreachable by methods in other frequencies.
First, define the gravitational deflection potential <p (unrelated to the inflaton
scalar field) based on the gravitational potential distribution $(£, D) (26):
(f>(h) = - 2 /dD Ds ~ C] <$>{Dh,D)
J
(1.52)
uus
where D is the comoving distance along the line of sight, h is the pointing direction of observation such that x = Dn and Ds is the distance to the surface of last
Weak gravitational lensing
27
io-«
*,
u 10-7
10-*
I I lllll
M
M Mil
1 I I I Mil
1 I I Mill
10-8
flat
all
10 9
§" ~
u"
en
10-10
i
i ) t i i n
1
10
100
/
Figure 1.5: Lensing power spectrum Cf for ACDM model originating from large scale
structure between Earth and the surface of last scattering. This spectrum is convolved
with the E-mode power spectrum to generate non-tensor B-modes. Figure taken from
(25).
scattering during recombination. The temperature fluctuations on the sky 0(n)
and polarization anisotropics Q(h) and U(h) are distorted by the deflection angle
V $ and remapped from their original fields 0(n), Q(h) and U(n) (27; 26):
9(n) = 0(n + V$)
Q{h)±iU{n)
= Q(h + V$) ± iU(n + V$)
(1.53)
(1.54)
The Gaussian random field <fi(h) is the series of gravitational potential wells
in three dimensional space projected onto the surface of a sphere. Its spatial
anisotropy spectrum can be derived in a similar manner to the temperature and
polarization anisotropies above. This results in the gravitational potential power
Weak gravitational lensing
28
spectrum Cf^, which by unitless conventions has a typical power £4Cg ~ 10~7 to
10"6 in ACDM models, peaking at £ ~ 100.
B-modes are generated from E-modes as a convolution of the CfE and Cf4,
spectra in the weak lensing limit. As can be seen in figure 1.4, this results in a
B-mode lensing signal that begins to dominate the tensor mode signal at £ > 200.
Figure 1.6: Toy model of a 32° by 32° CMB temperature field distorted by an enormous
lens, illustrating the use of small-scale CMB power deflected in a non-Gaussian manner
to detect gravitational weak lensing on larger scales. Figure adopted from (27).
The weak lensing distortions have a minute effect on the CMB temperature
spectrum, essentially smoothing out the peaks and troughs by a small amount.
However, examining the power spectrum, which is constructed under an assumption of isotropy and Gaussianity, is a suboptimal method of searching for distortions of the temperature anisotropies from gravitational potentials. The induced
signal in the temperature anisotropy maps is decidedly non-Gaussian - it creates
bulges and swirls that correlate modes on scales of about a degree. An estimator that looked for characteristic non-Gaussian deflections of the temperature
field could either obtain a statistical detection of the lensing power spectrum or
perhaps even recover a map of the lensing potential itself (27). The small-scale
The QUaD instrument
29
anisotropies in a CMB temperature map arising from the I > 2000 damping tail
of the CMB would be seen to be distorted on arcminute scales on the outer circumference of ~ l degree lensing potentials.
1.7
1.7.1
The QUaD instrument
Overview
Careful characterization of microwave background polarization on small angular scales is a crucial test of the current theories of the content and governing
physics of the early universe plasma. In addition, such measurements are an important adjunct to wide-field surveys of the microwave background with large
beams that attempt to maximize their sensitivity to inflationary B-modes (28).
A microwave polarimeter with sensitivity to smaller angular scales of multipoles
£ > 100 has higher sensitivity to gravitationally lensed B-modes, and can resolve
point sources and galactic foreground emission that will be smeared out by a survey with a larger beam.
QUaD's design and observation strategy was optimized to make sample variance limited measurements of the cosmic microwave background's (CMB) polarization on scales of 100 < £ < 3000. It observed a roughly 15° x 9° patch of
the sky from a site near the South Pole during the austral winters of 2005, 2006
and 2007. The main science results presented in this work are derived exclusively from 143 days of observations in the 2006 and 2007 seasons, as a warp in
the primary mirror led to highly elliptical, temperature-dependent beams during
the 2005 season. The experiment itself consisted of a cryogenic receiver with 18
operating pairs of polarization-sensitive bolometeric detectors at 150 GHz and 9
operating pairs at 100 GHz mounted on a 2.6 m Cassegrain telescope.
This work will provide only a brief overview of the siting and design of QUaD,
as these topics are covered in depth in other documents (29; 30). Characterization and calibration activities pertinent to the low-level processing of the data
The QUaD instrument
30
and necessary for the more advanced stages of the analysis pipeline are emphasized and covered in greater detail, as well as those aspects of characterization
that reflect the whole history of the instrument and are not present in other documents.
1.7.2
Location
QUaD was located at the Martin A. Pomerantz Observatory (MAPO), located about
1 km away from the Amundsen-Scott South Pole Station and the Geographic
South Pole. The telescope was mounted on the mount that was built for and
formerly housed the DASI experiment that first detected polarization in the CMB
(10).
The South Pole is a unique observing site with many desirable qualities for
microwave observations. The high altitude, extremely cold weather averaging
-60°C and six-month-long night provide both low precipitable water vapor and
relatively high atmospheric stability (31). In addition, the substantial logistical support of the National Science Foundation's Office of Polar Programs at the
South Pole was instrumental in QUaD's success. In particular, as a bolometeric
experiment in which the detectors were cooled to roughly 250 mK, QUaD consumed between 20 and 25 liters of liquid helium every day for several months,
necessitating the presence of a dedicated facility and staff for cryogens. Few sites
in the world offer this combination of infrastructure and observational opportunities.
The MAPO building surrounds the base of QUaD's mount, and the telescope
was essentially on the roof of the building located within a large, reflective ground
shield extending several meters in height. The ground shield and building were
mechanically connected to the outer of two concentric towers extending from
the ground, while the telescope is mechanically isolated on the inner of the two
pillars. The receiver could be lowered into a heated room in MAPO under the
telescope and wheeled into a neighboring room in the same building to perform
repairs and laboratory calibration tests.
The QUaD instrument
31
Figure 1.7: Aerial photograph of Amundsen-Scott South Pole Station, support facilities
and science buildings. The Martin A. Pomerantz Observatory (MAPO) is visible as the
second building from the left on the lower half of the photograph. QUaD observed from
within the ground shield visible as the wooden "salad bowl" perched on top of MAPO.
The Geographic South Pole is located about 100 m from the main elevated station, seen
in the center of the photograph as the largest wooden-covered building. Photo taken on
October 25, 2007 by Robert Schwarz, QUaD's winterover scientist.
1.7.3
Observing field
From a data analysis standpoint, observing from the South Pole is somewhat
pathological. Regardless of the time of year, the same portion of sky is always
visible, and QUaD's chosen 15° wide field circles the sky at constant elevation
endlessly. We are able to continuously observe the same field during the entirety
of the austral winter, and in theory much of the summer as well. Since QUaD is
located only about 1 km from the Geographic South Pole, mount azimuth and elevation are trivially convertible into right ascension and declination coordinates.
32
The QUaD instrument
0.00024 B I ^ ^ M M M I
mm 0.10 mK
Figure 1.8: QUaD's 15x9 degree observation field (in white) superimposd on top of 94
GHz projection of Finkbeiner-Davis-Schlegel Galactic dust emission model (32). The
orthographic projection shown here corresponds to the half of the sky visible from the
South Pole year round.
However, due to the large signal injected into the detectors while scanning the
telescope in elevation from differences in airmass we are restricted to azimuthal
scans on the sky and are therefore only able to observe any given point in the
sky from two directions - back and forth, parallel to the ground. It is impossible
to construct maps of the CMB polarization and temperature from "cross-linked"
scans that could simultaneously solve for pixel values approaching a point in the
sky in multiple directions. Atmospheric noise will therefore create a level of correlation between adjacent map pixels in a row of constant declination during
the mapmaking process due to their constant neighboring positions in the raw
timestream. This correlation is visible as residual "striping" in the naively coadded maps. This has been demonstrated not to be a problem in the Monte Carlo
The QUaD instrument
33
analysis detailed in later chapters because the effects are easily identified and filtered in the two-dimensional Fourier transforms of the coadded maps. Furthermore, with proper noise modeling and a full solution of the pixel-pixel noise covariance matrix eigenproblem, we can eliminate residual striping from the coadded maps. This will also be demonstrated in later chapters.
3^
Chapter 2
Data characterization and observation
history of the QUaD instrument
"Why do we have to send scientists? Why can't we
send somebody... normal?"
Contact
CARL SAGAN
2.1
2.1.1
Telescope and beams
Optical design and characteristics
The QUaD instrument uses a Cassegrain telescope with a 2.64 m primary mirror molded from a single piece of aluminum and based on the design for the
COMPASS experiment (33; 34). The 0.47 m secondary mirror was made out of a
thin sheet of aluminum supported by a carbon fiber backing. A 48 mm hole was
designed into the center of the secondary in order to prevent power originating
inside the receiver from reflecting back into the detectors.
The initial secondary mirror for the 2005 season was azimuthally symmetric. However, after deployment of the receiver it quickly became evident that a
Telescope and beams
Q
I I I
secondar
35
y mirrw
|j
on-axis b»am
Figure 2.1: Schematic of QUaD telescope and optical paths. Figure courtesy of Gary
Cahill and Creidhe O'Sullivan, University of Maynooth.
measured saddle-shaped "Pringles" residual warp in the primary mirror deviating from the intended parabolic shape created significant ellipticity in the beams
(see figure 2.2).
The initial major-minor axis ratio of a Gaussian fit to the beams was roughly
1.7. After correcting the secondary-primary distance on July 10,2005, the ellipticity of the beams was roughly 30%. Between the 2005 and 2006 observing seasons,
the initial secondary mirror was replaced with one which has a shape specifically
calculated to correct the beam shape from the non-ideal primary shape. When
installed, this resulted in nearly circular, Gaussian beams with an ellipticity well
under 5%. The residual ellipticity and non-Gaussianity of the 2006 and 2007 season beams using the new secondary and relevant to the science results presented
are discussed further in section 2.1.2.
The secondary mirror is supported by using an nearly azimuthally symmetric, microwave-transparent Zotefoam (PPA-30) cone. When mounted over the
primary mirror, the foam cone traps air over the window of the receiver, allowing
warm air drifting up from the heated space underneath the telescope to stabilize
Telescope and beams
36
Chicago fit residual
Figure 2.2: Measurements of residual deviation in mm from parabola of QUaD 2.64 m
primary mirror taken before deployment at University of Chicago. Measurements were
taken using a ROMER CimCore 3000i CMM arm. Plot and measurements courtesy of
Clem Pryke, Univ. of Chicago.
the temperature within the cone at about 15°C. As the minimum temperature
of the South Pole winter can reach —75°, this is a significant benefit to protecting the instrument as well as electronics in the secondary mirror assembly used
for calibration. However, the thermal differential between the interior and exterior environments induces a small amount of unpredictable flexure. After the
installation of the correcting secondary, the mainlobe beam widths are nearly invariant to this day-to-day variation, and for the purposes of analysis we utilize a
consistent set of beams for the whole of both the 2006 and 2007 seasons.
The cone was constructed from 12 pieces of 1.1" thick Zotefoam in two layers
with 6 pieces in each layer, with each piece overlapping with half of two pieces in
the opposite layer. Scotch 924 adhesive was used to bond the layers. Measurements of the foam and adhesive performed at Caltech yielded a transmission loss
of 2% at 150 GHz, attributed to reflections. Without the adhesive, the reflection is
bounded at 0.5%.
Telescope and beams
37
Corrected Secondary (July 7, 2007)
76.7% ellipticity
33.5% elliptioly
5.3% elliptlcity
Figure 2.3: Successive maps of dusty Galactic star-forming region RCW 38 as observed by
central feedhorn detector at 150 GHz, illustrating the effect of secondary mirror movement
and correction on ellipticity of beams. The first two images show RCW 38 observed with
the original isoazimuthal secondary mirror before and after adjusting secondary distance
from primary mirror. The third image shows a similar observation after a new secondary
mirror designed to correct for a warp in the primary mirror was installed between the 2005
and 2006 observing seasons. Note that the data used for the science results presented are
exclusively from after the correcting secondary was installed. Diffuse emission surrounding
the unresolved core of the star-forming region is actually observed and does not represent
beam sidelobe power.
The microwave reflective proprieties of the adhesive have the effect of coupling about 1% of the total main beam power into a roughly annular sidelobe
100° from the boresight of the telescope. Due to the slight flexure of the foam
cone with ambient temperature, the location of this sidelobe is not precisely predictable and fluctuates on the order of 5°. As this power is distributed roughly
equally along the whole of the ring, the only time that substantial contamination
of data occurs is when the moon is high enough over the horizon that some part
of the sidelobe sees it over the ground shield surrounding the entire telescope assembly. Residual power from the galactic plane is expected to couple in through
the sidelobe as well, but the total power is far below QUaD's sensitivity level.
For the set of data used in the science results, an extremely conservative data
cut is applied - all data from when the moon was above the horizon is cut from
Telescope and beams
38
RA offset
Figure 2.4: Single-day polarization raster map of a 5° x 0.5° patch of sky in the main
QUaD observing field by the central 150 GHz pixel on July 10, 2006. A roughly annular
100° sidelobe induced by reflections off of the Zotefoam cone supporting the secondary
mirror couples power from the moon into the beam of the central 150 GHz pixel.
the final data set. Although it was shown to be possible to algorithmically identify
moon contamination by applying Hough transforms to single-channel, singleday maps similar to figure 2.4 to find stripes induced by moon contamination,
ultimately the extra sensitivity gained from the small number of additional days
where no clear moon contamination is seen but the moon is known to be above
the horizon was too small to justify the risk of contaminating the science results.
2.1.2
Mainlobe and sidelobe beam predictions and measurements
The mainlobe beams at 100 and 150 GHz are both well fit by elliptical Gaussians
with a ellipticity of about 5% arising from the residual uncorrected effect of the
warp in the primary mirror. Broadband physical optics models of QUaD's beams
using Zemax software were performed at the NUI Maynooth after the primary
warp was discovered and the correcting secondary installed. These models predict a roughly -25 dB sidelobe arising at 6 arcminutes from boresight at 150 GHz
(34). Neglecting to include contributions from such a sidelobe in our analysis
39
Telescope and beams
-0.2
-0.1
0.0
0.1
0.2
-0.2
-0.1
0.0
0.1
offset on sky {in degrees)
0.2
-0.2
-0.1
0.0
0.1
0.2
Figure 2.5: Broadband physical optics model of central feedhorn at 150 GHz shown in
linear scale (left), log scale (middle), and with an elliptical Gaussian fit and subtracted out
to 2.75 a (right). Models of the sidelobes for each feedhorn derived from a physical optics
model similar to the right plot are used as templates in the Markov Chain Monte Carlo
(MCMC) beam parameter fit routines to estimate total sidelobe power in conjunction with
elliptical Gaussian fits to data from QSO PKS03537-441.
pipeline would result in a major systematic error in our measured temperature
anisotropy spectra from 1000 < £ < 1500.
Despite its relative brightness at 145 ± 7 Jy at 150 GHz (35), the unknown extent of the dust emission at microwave frequencies around the bright core of
star-forming region RCW 38 prevents precision measurements of QUaD's beams.
While RCW 38 was used for gross beam measurements and pointing centering
during QUaD's operation, the final beam measurements were taken from several
day-long observations on the quasar PKS03537-441 that is present in QUaD's science field at RA, Dec. 05:38:50.362,-44:05:08.94 (J2000).
PKS03537-441 is a BL Lac type active galaxy with a light curve that follows a
roughly two year cycle, peaking first in the austral winter of 2005 during QUaD's
first year of observation, and peaking again during June and July of 2007 (3), ranging from about 2 Jy to 9.2 Jy at 100 GHz. At redshift z=0.893, although there appear
40
Telescope and beams
to be four companion galaxies near the AGN (36), the BL Lac object itself is unresolved out to 8", making it a good point-source-like beam calibrator for QUaD
despite its dimness.
QSO P K S 0 5 3 7 - 4 4 1 at 150 GHz
QSO P K S 0 5 3 7 - 4 4 1 at 100 GHz
0.03 1
0.03 1
0.02
0.02
0.01
0.01
-0.00
-0.00
-0.01
-0.01
-0.02
-0.02
-0.03
-0.03
-0.02 -0.01 -0.00
0.01
0.02
0.03
-0.03 1
-0.03
-0.02 -0.01 -0.00
0.03
150 GHz sidelobe f r o m
0.03 JJ^jfyj
0.02
0.02
0.01
0.01
100 GHz sidelobe f r o m
PKS0537-441
0.01
0.02
0.03
PKS0537-441
-0.00 3'i
-0.01
-0.02
- 0 . 0 3 Ktnn • vw.Tt,
-0.03
fX7\jti. m
' • • • . i > * J > . > i . • . • i > . a t i ii_i. j. _ K 1 L !
-0.02 -0.01 -0.00
0.01
0.02
0.03
- 0 . 0 3 ' | - - ' > • • •-'• 1' • • • .".MMi. . L . . i j . l K i . . i ; n . l l l
-0.03
-0.02 -0.01 -0.00
0.01
0.02
0.03
Figure 2.6: Coadded maps of quasar PKS03537-441 at 100 and 150 GHz. Offset in degrees
in both RA and Declination shown on x and y axes in degrees respectively. Color scale
ranges from 0 to 10 mK in equivalent blackbody fluctuations at 2.7 K. After subtraction
of an elliptical Gaussian mainlobe a residual sidelobe can clearly be seen slightly above
the noise level in the maps, seen in the bottom two plots of the figure.
Telescope and beams
41
Three full days of observations of PKS03537-441 were obtained by QUaD in
good weather on Sept. 7-9, 2007, near the peak of its light curve, for use as
beam measurements. Estimates of the pointing centers for each are derived as
described in section 2.1.3 and used to subtract the detector offsets on the sky and
day-to-day residual pointing wander before coadding the time-ordered data into
the images shown in figure 2.6. The maps are comprised of 404 raster scans slewing 3.5° in azimuth (right ascension) across the quasar and traversing 2° in elevation (declination). A mask is created that is 5.1' wide at 150GHz and 9' wide at 100
GHz prior to timestream filtering so that the source itself does not constrain the
filter parameters, and then an eight-order polynomial is fit to the non-masked
region and removed from the whole of each of the 404 scans in azimuth. Since
on any given scan the signal to noise of the quasar is relatively small and no two
feedhorns are located on quasar at the same time, the mean of all the channels
in a frequency is subtracted from the timestreams in order to reduce the contribution of atmospheric noise. More details of the naive mapmaking process used
to create these maps will be provided in chapter 3.
2.1.3
MCMC beam parameter analysis
Although naive maps of an unresolved quasar are useful in illustrating the presence of sidelobes, unfortunately the quasar is not bright enough to directly map
and measure a — 20 dB sidelobe, nor did QUaD have access to any other far-field
calibration sources that could provide such a measure. The maps shown in figure
2.6 are added across all channels at 100 and 150 GHz; in any single channel the
sidelobe is barely visible above the noise level after an elliptical Gaussian is fit to
the map.
In addition, a robust estimate of the errors on the beam measurement are necessary for proper cosmological parameter estimation. As beam errors typically
scale quadratically with increased multipole moment £, an inaccurate estimate
of beam errors can artificially improve the significance of the high-£ portion of
Telescope and beams
42
the spatial anisotropy spectrum, leading to overly constrained estimates of parameters like ns.
Markov Chain Monte Carlo (MCMC) parameter estimation is a standard technique using Bayesian statistical principles used in cosmological parameter estimation, and will be discussed in further detail in chapter 5. It offers a natural way
not only to derive estimates of mainlobe and sidelobe beam parameters in the
presence of low signal-to-noise measurements of the beam, but also to derive uncertainties on those estimates by examining the width of the posterior distributions. The approach to estimating the sidelobe power from single channel measurements of the quasar is to sample the sidelobe maps from templates created
by subtracting a fit elliptical Gaussian from the beams predicted by the physical optics model for each channel and fit for the amplitude of the sidelobe from
the templates, while simultaneously fitting the mainlobe to an elliptical Gaussian. The beam is parametrized over the 3 days with 11 parameters (flat prior
constraints in parentheses):
1) 0 - mainlobe ellipse orientation (0° to 180°)
2) r - ellipticity ratio of the mainlobe (1 to 1.3)
3) a - mainlobe width (1' to 5').
4) A - amplitude of the mainlobe (20% to 100% of range of input data)
5) w - ratio of peak of sidelobe power to peak of mainlobe power (—10% to 10%
of range of input data)
6-11) ra 1; deci, r"a2, dec2, ra3, dec3 - pointing centers for detector on sky for each of
the 3 days of measurements (within 20' of the presumed feed offset centers
as determined by measurements of RCW38)
The 2,763,360 time-ordered data points for each channel over the three days
were reduced to consider only the roughly 50,000 points of data for each channel
when the pointing and prior knowledge of the feed offsets from measurements
43
Telescope and beams
phi :
2.0x10*
h
( \
1.5x10*
1.0x10*
/
5.0x10s
/
/
145
-0.005
0.005
\
155
V.
0.015
1.10
0.025
1.14
2.0X10
1.5x10*
1.5x10*
• A
J A
0.0245
0.0250
0.0255
0.0260
0.0265
2.75x10 2.85x10 2.95x10 3.05X10 3.15X10
2.5x10* •
2.5x10
2.0x10* :
2.0x10*
i.5xio*:
1.5x10*
1.0x10*:
1.0x10*
5.0x10a:
5.0X103
04.720
0.035
1.12
2.0X10
84.722
84.724
84.726
04.71
84.719
B4.721
84.723
84.725
-44.087
-44.086
-44.085
-44.089
-44.088
-44.067
1
"
84.728
i \
i
84.730
84.732
84.734
84.736
2.5x10
2.0X10*
1.5x10*
1.0X10*
5.0x10*
.089
-44.088
-44.087
-44.086
84.717
.091
-44.090
Figure 2.7: I D marginalized posterior distribution from MCMC beam parameter fits for
central feedhorn from 3 days of observations of unresolved quasar PKS0537-441, Sept.
7-9, 2007. Beam parameters and uncertainties used in spatial anisotropy analysis are
derived from expectation values and widths of the marginalized posterior distributions.
Note positive detection of sidelobe amplitude parameter w at roughly —30 dB level.
of RCW 38 indicated that the feed was pointed within 20' of the quasar to greatly
reduce the time necessary to compute the likelihood for a given set of the 11 parameters. The likelihood for a single data point in a channel, di on the j-th day
of observation given a set of beam parameters with nominal telescope pointing
location raj, decj was therefore:
£(di\(f), r, a, A, w, ra^, dec.,e x p ^ - A
/J
(2.1)
exp
((^s0(rat
- ra,) - sin0(dec, - dec,))2
^
(J
(sin</>(ra; — ra,-) + cos0(decj — dee,-))"
) (2.3)
c2/r
(2.4)
AwS{j&i — raj, decj — dec,-))2}
Telescope and beams
44
S(x, y) is a value sampled from a sidelobe template created by fitting and subtracting an elliptical Gaussian to the physical optics model described in section
2.1.2 and shown for the central feedhorn at 150 GHz in figure 2.5. All the r ^
pointing values and fhj pointing centers are multiplied by a factor of cos(decj)
and cos(decj) respectively to compensate for the flat sky projection. The noise
between neighboring points of time-ordered data is approximated to be white
and uncorrelated, and therefore the total likelihood for a vector of data across 3
days is:
C(d\cj),r,a,A,w,mi,deci,fh2,
dec 2 ,ra 3 ,dec 3 ) = TT£(dj|0, r, a, A,w,fhj, decj) (2.5)
The general algorithm for MCMC is presented in more detail in chapter 5. For
this analysis, flat priors were used in the parameter regions described above, and
a Metropolis stepping algorithm was used with a uniform proposal distribution
within 15% of the prior range from the current point in parameter space for the
first 60000 steps. This is switched to 0.05% of the prior range until the chain has
taken 600,000 steps. The posterior distributions are smooth and display excellent
convergence, as seen in the one dimensional marginalizations of the posterior
distribution for a detector in the central feedhorn shown in figure 2.7. The full
list of derived parameters for all detectors used in the primary science results is
shown in table 2.1.
The sidelobe template amplitude is detected at high significance in the central
rings of detectors (150-01-A through 150-07-B and 100-01-A through 100-12-B),
but is more consistent with zero on the outer ring of 150 GHz channels (150-08-A
through 150- 19-B). A reasonable explanation for this observation is that the exact
spatial dependence of the sidelobe predictions from the physical optics model
break down as one considers detectors towards the edge of the focal plane array due to deformations from temperature fluctuations in the foam cone, mount
and telescope. The overall sidelobe template amplitude used in the final analysis
takes the mean of the sidelobe amplitudes using only the central pixels; at 150
Telescope and beams
45
Observed beam from PKS and model fit from MCMC (red), 100 GHz
|
0.0100
E
flj
m
0.0010
10
Radius (arcmin)
0.015
:
—
0.010
0.005
r
0.000
r
0 005
-
f\
'
<
I"
=
—
T
W-*—
T
M 1
1
k
ri-LliT
T I
T
1-
/iyi i r "frrri'
:
10
Radius (arcmin)
15
20
Observed beam from QSO PKS0537-441 and MCMC model fit (red), 150 GHz
1.0000
0.1000
0.0100
!—*==^,
r
r
I
n
^ \ t .
^
^
t
^
T
T
^
1
==;
0.0010
T
"S
A
0.0001
6
8
10
12
-1
14
Radius (arcmin)
Radius (arcmin)
Figure 2.8: Comparison of beam fits from MCMC method using data from QSO PKS0537441 at 100 and 150 GHz and predictions of sidelobes using physical optics model to
measured beam. Measured data and errors from variance within annular rings of increasing
radius from the unresolved source is displayed in black, and the model sidelobe template
fit directly from the time-ordered data is shown as the red line.
Receiver and focal plane
46
GHz this leads to a sidelobe template amplitude of 80% of that predicted by the
physical optics model, and 30% at 100 GHz. When these templates are used to
construct simulated model beams and then compared against the coadded data
from all channels in a frequency observing the quasar binned in annular rings,
the sidelobe template amplitude model appears to be a good fit as shown in figure 2.8. These represent the highest signal-to-noise measurements of the beams
we have available, as they incorporate all data from all channels, binned in annuli around the quasar, and the model is clearly within the uncertainty of the
measurement.
A slight day-to-day pointing wander is clearly present across the three days
of quasar measurements at the level of less than 0.5', most likely attributable to
long-term temperature fluctuations that flex the mount and foam cone. As the
MCMC fitting algorithm fits the pointing offsets for each detector on a day-byday basis independently for each day, the final beam widths do not take the slight
enlarging of the beams that such an effect would produce into account; therefore in the Monte Carlo simulations described in 3 a day-to-day pointing wander
drawn from a Gaussian distribution of width 0.5' is inserted into each day's overall pointing.
2.2
2.2.1
Receiver and focal plane
Polarization Sensitive Bolometers
QUaD utilizes a set of 18 polarization sensitive bolometers (PSBs) sensitive to
100 GHz radiation and 36 PSBs sensitive to 150 GHz radiation to make polarized
measurements of the CMB. The PBSs utilized in QUaD are identical to those used
in the BICEP experiment (37), and similar to those used on the BOOMERanG
balloon experiment and the Planck satellite (38; 39).
Bolometers detect incident optical power at high sensitivity by measuring the
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Table 2.1
vations o
Receiver and focal plane
48
temperature increase of an absorber weakly connected to a thermal bath. Typically, a bolometer comprises of a thermistor that changes its resistance in response to changes in temperature and an antenna or mesh that responds to incident microwave radiation thermally coupled to the thermistor. In QUaD's detectors, the microwave detection mechanism is a silicon nitride grid metallized
with 120 A of gold over 20 A of titanium. The grid is spaced at 150 //m and metallized in only one of the two directions of the grid, efficiently coupling the PSB
with only one direction of linear polarization (29). A microwave polarization detector is made by stacking two PSBs with orthogonal directions of metallization,
and placing both PSBs into the same feed horn. Assuming that the optical properties of both PSBs in the pair are identical, each detector couples to unpolarized
radiation and one of two orthogonal linear polarizations. Subtracting the derived
signals leads to an instantaneous measurement of polarization.
Differences in responsitivity sensitivity due to loading changes between different PSBs in the array result in time-varying relative gains that need to be calibrated out in order to derive accurate science results. Instead of using detailed
bolometer parameters and physical models, we empirically measure the relative
gains by injecting a small ramp into every bolometer channel by moving the telescope up and down in elevation by about 1°. This process is detailed in section
2.3.
2.2.2
Focal plane
QUaD's bolometers depend on being weakly connected to a cold bath, typically
at < 300 mK. The focal plane assembly serves as both a mechanical support
and thermal conductor to the cryogenic components necessary to couple the
bolometers to these low temperatures. QUaD's focal plane base, seen in figure 2.9,
is a 350 mm diameter bowl milled from a single piece of aluminum 6061 and goldplated to improve thermal conductivity.
The bowl and feed horns coupling microwave radiation to the PSBs are cooled
Receiver and focal plane
49
Figure 2.9: QUaD focal plane assembly and cryogenic isolation system. 3 concentric
rings of horns are seen in a spherical arrangement on the top, with the larger horns
corresponding to 100 GHz and the smaller ones to 150 GHz channels. PSBs are attached
directly under the bowl and within the light-tight shield, as well as thermometer and
resistor heaters for thermal feedback control. The bowl, horns and detectors are coupled
to the 250 mK cold stage of a 3 stage sorption refrigerator. The 250 mK components are
supported by 6 short Vespel legs connected to the middle ring, which is thermally coupled
to the intermediate stage of the refrigerator at 430 mK. A Vespel truss extends from the
baseplate, which is connected to the 4 K liquid helium tank. The box seen on the bottom
right of the assembly contains the JFET amplifiers which stabilize the electrical signals
from the bolometers for routing out to warm electronics.
Receiver and focal plane
50
to ~ 250 mK by thermal link to the coldest of 3 stages of a sorption fridge. An external Stanford Research Systems PID controller powers a heating resistor taped
down to the back edge of the bowl and stabilizes the temperature in a feedback
loop with a thermistor on the focal plane.
Cryogenic optics consisting of two 18 cm diameter anti-reflection coated highdensity polyethylene (HDPE) lenses couple microwave radiation from the Cassegrain
focus of the telescope to the horns on the focal plane. The lenses are thermally
coupled to the 4K bath of the toroidal liquid helium tank that surrounds the entire focal plane assembly.
Corrugated feed horns are arranged on the bowl so that their phase centers
lie along the spherical focal surface created by the optics (29). These feedhorns
propagate a linear combination of the TEU and TMU modes with low sidelobes
and low cross-polar response. The narrow waveguide throat acts as a high-pass
filter, while a stack of metal-mesh optical filters at the aperture of the horns establishes the low-pass cutoff. For the 100 GHz channels the band established
by the combination of the horns and filters is 78-106 GHz, while for 150 GHz the
band is at 126-170 GHz, near the maximum of the 2.7 Kblackbody CMB emission
but avoiding the water vapor absorption lines of the South Pole atmosphere.
Pairs of PSBs with orthogonally aligned metallization, corresponding to sensitivity of orthogonal linear polarization, are mounted on the backside of the focal
plane bowl, coupling to the telescope and optics through the feedhorns. A lighttight shield is mounted around the backside of the focal plane. Each pair corresponds to an instantaneous measurement of either Q or U on the sky through
identical optics, although when empirically measured there is a slight offset between the beams of the pairs of PSBs in some channels. This effect is attributed to
residual birefringence caused by stress on the lenses from their mounting. This
"A-B offset" is inserted into the simulation pipeline and contributes to the errors
of the polarization results. The offsets of all the PSBs on the focal plane and their
polarization sensitivities as measured in situ from observations of PKS0537-441
and a near field polarized source are shown in 2.10.
Receiver and focal plane
51
1.0
150-11-A
150-11-B
150-10-A
150-10-B
0.5
150-09-A
150-09-B
150-08-A
150-08-B
100-04-A
100-04-B
100-03-A
100-03-B
150-03-A
150-03-B
100-02-A
100-02-B
150-12-A
150-12-B
100-05-A
100-05-B
150-04-A
150-04-B
100-06-A
100-06-B
150-13-A
150-13-B
03
c
6
0.0
100-01-A
100-01-B
Q
150-19-A
150-19-B
150-02-A
150-02-B
150-01-A
150-01-B
100-12-A
100-12-B
150-07-A
150-07-B
100-07-A
100-07-B
150-05-A
150-05-B
150-14-A
150-14-B
150-06-A
150-06-B
-0.5
150-18-A
150-18-B
m
150-17-A
150-17-B
150-16-A
150-16-B
-1.0
-1.0
-0.5
0.0
RA offset
0.5
1.0
Figure 2.10: Measured offsets and polarization sensitivity angles for all detectors used in
2006-2007 science results rotated to deck=57.8. Beam offsets are derived from 3 days
observations of quasar PKS0537-441 described in 2.1.3, and individual sensitivity directions are derived from near-field polarization source measurements described in chapter
6. The 150 GHz PSBs are shown in blue, and 100 GHz PSBs are shown in red. Slight
offsets between pairs of PSBs within the same horn can be seen in some channels, for
example 100-11-A and 100-01-B. Although the beams of pairs of bolometers pass through
identical optics when projected on the sky, the slight offset in a pair is postulated to be
attributable to birefringence in the lenses.
Receiver and focal plane
2.2.3
52
Cryostat and cooling system
The focal plane assembly is housed within a custom-made cryostat manufactured by AS Scientific within two surrounding toroidal tanks of liquid helium
and liquid nitrogen with capacities of 21 L and 35 L respectively. The tanks are
both thermally connected to internal radiation shields to isolate the focal plane
assembly from warmer black-body radiation. The aperture of the cryostat consists of a window made out of anti-reflection coated ultra-high-molecular-weight
polyethylene and a flat cylindrical ecosorb baffle around the window to prevent
reflections from the top of the cryostat that extends through the primary mirror
from creating a sidelobe.
As radiation comes into the cryostat through the window, it first encounters
a set of IR radiation blockers and a 360 GHz edge filter cooled by the 77 K liquid
nitrogen tank, followed by a 270 GHz edge filter, the field and camera lenses and a
"cold stop" all cooled to 4 K by the liquid helium tank. The cold stop consists of a
cylindrical knife-edge aluminum aperture coated with microwave-black ecosorb
on the bottom. The cold stop intercepts the sidelobes of the beams emerging
from the feed horns and ensures that there are no unpredictable sidelobes created from reflections of the sidelobes within the cryostat.
The 3-stage He4-He3-He3 sorption fridge is located next to the focal plane
assembly on the 4K baseplate. Each successive stage consists of a sealed assembly with two chambers, a still and a pump linked by thin-walled, stainless steel
tube called the condensation point. During normal operation, liquid helium in
the still boils off slowly at a reduced temperature due to activated charcoal in the
pump which reduces the vapor pressure above the liquid. After liquid helium in
the still is expended, the still can be refilled by heating the charcoal and driving
the helium gas out of its pores, and thermally coupling the condensation point
with a temperature at or below the boiling point of the He4 or He3 gas.
In normal operation, the He4 stage is condensed first by thermal linkage to
the 4K liquid helium bath, and the enthalpy of its liquid is then used to cool the
interstage He3 and ultracold He3 stages. QUaD's fridge cycle took about 5 hours
Receiver and focal plane
53
422.5
258.5
422.0
2"
E
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E
0)
<D
TO
to
2
421.0
420.5
256.0
420.0
255.5
0
50
100
150
Observation day
Figure 2.11: Mean interstage (black) and ultracold temperatures (blue) during each of
143 days of observations in 2006 and 2007 seasons used for primary science result.
and was run daily, achieving a hold time of 30 hours on the ultracold stage and 24
hours on the interstage. In practice, after allowing the focal plane's temperature
to stabilize, our primary CMB science observation schedule took about 19 hours,
and in order to observe our field from the same azimuth on every day we cycle
the fridge earlier than its hold time would allow.
Over the two seasons of QUaD's 2006-2007 science run the ultracold and interstage temperatures were very stable, implying that no cryogenic touches or
internal loosening of thermal couples developed within the cryostat during this
time. The stability of the temperatures can be seen in figure 2.11, although it
should be noted that in normal operation the ultracold stage is thermally coupled to the focal plane whose temperature is stabilized in a PID feedback loop.
Data acquisition and time ordered data
2.3
2.3.1
54
Data acquisition and time ordered data
JFETs and warm electronics
The fundamental physical quantity measured from the detectors is the voltage
off a nearly constant-current measurement through the thermistor coupled to
the membrane of the PSB. Schematically, the bolometer is wired in a series circuit with two load resistors, RL — Rboi0 — RL, forming a balanced bridge circuit.
Electrical leads from each bolometer/load resistor junction are buffered through
cold JFET followers located on the 4K stage after passing through low thermal
conductance manganin wire (29). The high-impedance signals emerging from
the bolometers needs to be routed carefully through tied-down wiring before
the JFET followers to prevent microphonic pickup from influencing the detector
measurements.
The JFETs themselves are mounted on a silicon nitride membrane with thin
wires formed by lithography in order to minimize thermal conductance, since
the JFETs need to be above 50 K to operate and close 100 K to minimize noise. Although heaters are located on the JFET membrane, in practice the ~ 6.5 mW dissipated by the amplifiers in normal operation is sufficient to keep the JFETs working nearly optimally. The signal that emerges from the JFETs is a low impedance
measurement of the bolometer voltage with a gain of 0.99, which is then routed
out of the cryostat.
During science observations the bolometers are biased with an AC current of
1.25 nA at 110 Hz (29). The differential signal read out of the JFETs is then fed
into warm electronics on the exterior of the cryostat, starting with a preamplifier
with a gain of 100, followed by a low-pass 4-pole Butterworth filter with a cutoff
frequency of 475 Hz and a 2-pole high-pass Butterworth filter with a cutoff frequency of 2.8 Hz to suppress noise away from the bias frequency. The two latter
filters have a gain of 5.
The AC signal is then demodulated with a reference signal from the same electronics that generate the original AC bias current. The demodulator can be tuned
Data acquisition and time ordered data
55
with an adjustable phase to maximize the output signal. After demodulation the
signal is low-pass filtered at 20 Hz with a 6-pole Butterworth filter.
At this point the signals typically consist of pN fluctuations on top of a 1 to
10 V DC offset. In order to utilize more readily available 16-bit analog to digital
converters (ADC) with lower resolution, the DC offset is subtracted via another
software tunable parameter, and the remaining signal is amplified with another
gain of 100 before being digitized. Including an extra factor of 2 due to the differential measurement, the total gain of the amplification chain after offset removal
is 100,000.
The actual ADC conversion takes place in the data acquisition system (DAC)
derived from the original hardware used for the DASI experiment. It consists of
a VME controller running a real-time VXWorks system, and two VMIVME-3122
64-channel 16-bit ADC cards. The data is then transmitted via TCP/IP to a PC
running Linux that archives the data in preparation for return to the United States
via satellite link. This control computer also allows the user to observe the data
in real-time and control the telescope and tune the offset and phase parameters
as needed, either manually or via automated observation scripts.
In practice, due to atmospheric drifts in loading on the bolometers, unless
the offsets and phases are retuned at about once an hour, the bolometer voltages
drift off-scale after amplification and information is lost. The proper offset level
is obtained by sampling the signals before the offset removal and final gain of 100
amplification. QUaD's observation strategy is detailed in section 2.4.1, but before
discussing deconvolution it is important to note that during our primary science
run the offsets were sampled and reset about once every half-hour. Due to the
change to low-gain mode that this necessitates, this means that every half-hour
the timestream from all channels encounters a discontinuity, limiting our ability
to perform manipulations on the transfer function of the bolometers or measure
their noise properties to a minimum of about 1 mHz.
Data acquisition and time ordered data
56
150-05-A lab measured response
10
15
Frequency, Hz
20
25
Figure 2.12: Response function for 150-05-A, showing strong two time constant behavior.
Data gathered during lab calibration run using ecosorb chopper and 77 K cold load,
December 4th, 2005. Blue line shows single time constant model with r = 8.99 ms.
Red line shows fit best fit to two time constant model T\ = 6.88 ms, r2 = 372 ms and
a = .209.
2.3.2
Time constant modeling
An ideal bolometer weakly connected to a thermal bath has an impulse response
that decays exponentially with time as heat is transferred out of the absorber into
the focal plane and ultrastage cooler due to the temporary temperature differential. The temporal response is thus:
hit)
-t/T
(2.6)
where r is a characteristic bolometer time constant that varies from device to
device and is typically in the 10 to 50 ms range. This corresponds to a frequency
space response:
#iM
I
1 + IUJT
(2.7)
57
Data acquisition and time ordered data
However, about half of the QUaD PSBs were found to better fit a long two time
constant model:
ff2M = - i ^ - + — ^ —
1 + lUTi
(2.8)
1 + ILUT2
where a < 0.5. r2 can be very long, up to 1 s, although more typically it is in the
100 ms range. This behavior is attributed to debris on the PSB membrane that
have substantial heat capacity but weak thermal linkage. Debris on the PSBs was
actually observed when the focal plane was taken apart and rearranged between
the 2005 and 2006 seasons. Channels that were observed to have a pathological
extremely long time constant were correlated to observations of large pieces of
blue "crud" present on the bolometer webs, which were shaken out before the
bolometers were replaced on the focal plane.
The color and size of the granules implied that the debris were leftover copper
salts dislodged from the feed horns. The QUaD feedhorns were electroformed on
top of a mandrel because the interior corrugations of the waveguide could not
be machined. After the mandrel is dissolved some small amount of copper salt
granules can remain lodged within the sub-millimeter feed horn corrugations,
despite repeated attempts at washing the horns in a sonicator bath.
2.3.3
Time constant measurements
The injection of a large and regular square wave signal into the bolometers is a
suitable means to probe the time constants. Using a reference from the square
wave generator, the convolved rising and falling edges in the bolometer signals
can be isolated, individually Fourier-transformed, divided by the frequency response of a step function, and then averaged together to obtain a stable measurement of the bolometer response function.
The "lab" time constant measurement was taken after reassembly of the focal plane in December 2005, and utilized a chopper wheel placed between the
cryostat window and a microwave transparent Styrofoam box containing a cold
load of 77 K liquid nitrogen. The circular chopper alternated between a 300 K
Data acquisition and time ordered data
58
Lab time constants(blue) vs. Gunn source (black)
0.12
0.10
~
0.08
r
0.06
CO
"~ 0.04
0.02
0.00
10
20
30
Bolometer
40
50
60
Figure 2.13: Comparison of first (fast) time constant from lab time constant measurements in December 2005 and measurements from near-field Gunn source with receiver
mounted on telescope during February 2007. Systematic difference is attributed to loading
differences on bolometers in internal environment compared to sky, but overall coincidence
of channel-to-channel time constants over two years reinforces validity of measurement.
blackbody ecosorb and a transparent foam support that allowed signal from the
77 K load to be seen. The slices of ecosorb were arranged such that a full rotation of the chopper would correspond to 8 rising and 8 falling transitions. Since
the loading on the bolometers while viewing the sky is closer to 27 K, the time
constants measured in this setup are not expected to be accurate for CMB observations but nonetheless provide a test of our modeling and the relative variation
of the time constants on the focal plane. Since the injected signal was large, the
lab tests were taken by biasing the bolometers with a DC signal and bypassing
the Butterworth low-pass filters further down the electronics chain. The chopper
wheel was rotated at about 1 Hz, corresponding to an injected square wave of 8
Hz.
The Gunn oscillator measurements were taken in February of 2007 by mounting the oscillator on top of the secondary mirror with the telescope pointing at
zenith. A narrowband 100 GHz or 150 GHz signal was modulated with a 0.01
Hz square wave and several hours of observations were taken. Again, the reference signal from the modulator was used to isolate the step transitions, Fourier
59
Data acquisition and time ordered data
Table 2.2: Derived time constants from lab measurements using a 77K/300K chopper
wheel and Gunn oscillator source using sky loading. Blank spaces for r2 and a indicate
channels where single time constant model is a better fit to observed response functions.
Gunn measurements and derived values courtesy Clem Pryke, University of Chicago.
Detector
150-01-A
150-01-B
150-02-A
150-02-B
150-03-A
150-03-B
150-04-A
150-04-B
150-05-A
150-05-B
150-06-A
150-06-B
150-07-A
150-07-B
150-11-A
150-11-B
150-12-A
150-12-B
150-13-A
150-13-B
150-14-A
150-14-B
150-15-A
150-15-B
150-16-A
150-16-B
150-17-A
150-17-B
150-18-A
150-18-B
150-19-A
150-19-B
100-01-A
100-01-B
100-02-A
100-02-B
100-03-A
100-03-B
100-04-A
100-04-B
100-05-A
100-05-B
100-06-A
100-06-B
100-07-A
100-07-B
100-09-A
100-09-B
100-10-A
100-10-B
100-11-A
100-11-B
100-12-A
100-12-B
lab
0.009
0.010
0.007
0.008
0.011
0.013
0.009
0.007
0.007
0.006
0.028
0.017
0.062
0.016
0.027
0.006
0.006
0.007
0.019
0.011
0.028
0.010
0.017
0.006
0.034
0.006
0.017
0.006
0.037
0.021
0.016
0.071
0.007
0.010
0.008
0.014
0.006
0.009
0.010
0.007
0.009
0.007
0.008
0.012
0.006
0.007
0.131
0.080
0.008
0.010
0.031
0.017
0.014
0.010
TI
T2 lab
a lab
0.030
0.066
0.139
0.298
0.117
0.018
0.017
0.283
0.067
0.372
0.103
0.055
0.038
0.014
0.697
0.016
0.035
0.030
0.038
0.127
0.080
0.058
0.024
0.010
0.011
0.014
0.008
0.209
0.295
0.280
0.384
0.273
0.101
0.466
0.128
0.207
0.035
0.384
0.214
0.449
0.331
0.455
0.322
0.447
0.087
0.249
0.012
0.099
0.048
0.097
0.332
0.210
0.307
0.141
11.556
0.251
0.591
0.403
0.130
0.150
0.028
0.032
0.017
0.038
0.035
0.027
0.062
0.033
0.018
0.025
0.031
0.016
0.449
0.040
0.053
0.072
0.043
0.195
0.281
0.176
0.200
0.295
0.011
0.100
0.351
0.437
0.464
0.487
0.030
0.115
0.072
0.089
0.150
0.079
0.076
0.135
r\ Gunn
0.012
0.018
0.009
0.009
0.015
0.022
0.013
0.009
0.009
0.012
0.027
0.028
0.014
0.026
0.029
0.009
0.010
0.010
0.033
0.017
0.062
0.020
0.016
0.013
0.022
0.011
0.027
0.010
0.026
0.033
0.026
0.102
0.010
0.014
0.012
0.020
0.010
0.012
0.019
0.012
0.014
0.008
0.012
0.016
0.010
0.009
0.065
0.046
0.013
0.014
0.042
0.023
0.021
0.016
T2 Gunn
a Gunn
0.055
0.212
0.654
0.231
0.086
0.171
0.071
1.153
0.140
0.058
0.050
0.105
0.292
0.097
0.278
0.151
0.166
0.194
0.420
0.022
0.559
0.112
0.017
0.103
0.173
0.022
0.274
0.067
0.043
0.027
0.089
0.234
0.871
0.426
0.061
0.248
4.001
4.909
1.892
0.032
0.300
0.430
0.180
0.015
0.192
0.118
0.993
0.898
0.080
0.058
4.038
2.458
4.160
0.180
0.055
0.050
0.069
0.027
0.016
0.227
0.160
0.098
0.065
14.749
0.316
0.021
0.288
0.336
0.094
0.091
0.288
0.148
0.194
0.024
0.122
0.100
Data acquisition and time ordered data
60
transform and average them to produce a measurement of the response function \H(u>)\ to which the time constant model was then fit. The signal emitted
by the Gunn diode was low enough that the data acquisition and lockin system
had to be run in high-gain mode, including the low-pass Butterworth filters in
the measured response function. However, since the receiver was mounted on
the telescope and pointed at the sky, the loading in this setup was very close to
that of normal observations and the time constants derived from this measurement are those used in the eventual deconvolution algorithm and primary CMB
analysis. A comparison of the lab and Gunn source measurements can be seen
in figure 2.12, and the values themselves can be seen in table 2.2.
2.3.4
Deconvolution
Cosmic ray on 150-01-B, 060329
4
3
.
2
o
>
1
0
-1
0.0
0.5
1.0
1.5
Time (s)
Figure 2.14: Cosmic ray incident on detector 150-01-B, March 29, 2006 (black) generating
near impulse response. Ringing is due 6-pole low-pass Butterworth filter, while exponential
decay is attributed to detector time constant. Post-deconvolution and 5 Hz digital lowpass filter timestream in blue.
The purpose of the time constant measurements is to remove the effects of the
electronics and bolometer response functions from the time-ordered data and
Data acquisition and time ordered data
61
accurately reproduce the original fluctuations on the sky observed by the PSBs.
In addition to the bolometers, in high-gain mode there are two Butterworth lowpass filters. The first filter is a 2-pole filter with a 30 Hz cutoff:
2
#2o(s) = 1/(^2 + 1.4142— + 1)
(2.9)
where s = icu and ojb = (27r)30 Hz. This is followed by a more aggressive 20 Hz
6-pole Butterworth filter:
#30 (s) = —-5
W
3
5
(2.10)
( ^ + A ^ + l ) ( ^ + 5 ^ + l ) ( 4 + C ^ + l)
where A = 0.5176387, £ = \/2, C = 1/A and wc = (2vr)20 Hz. As the frequencies of interest and cutoff bands are far from the Nyquist frequency of 50 Hz, no
conformal mapping techniques to account for the continuous to discrete time
sampling, like a bilinear transform, are used to compute the digital equivalent of
these filters. The total response function is therefore:
H(s) = H20(s)H30(s)Hbolo(s)
(2.11)
where Hboi0{s) is the appropriate one or two time constant bolometer response
function for this particular channel. For a roughly 30 minute timestream sampled
from a single bolometer at 100 Hz, d, the deconvolution procedure is:
ddeconv = F-1{^JT\HLP{S)}
(2.12)
H(s)
where HLP(s) is a cosine taper filter from 5 to 10 Hz that is used to eliminate
the high-frequency noise that is generated by dividing the high-frequency components of the white noise in the time stream by a nearly zero deconvolution
response function.
Cosmic ray events occur several times per day, and deposit energy on the PSBs
nearly instantaneously. The resulting signal is a near impulse response, and successful deconvolution of the exponential bolometer time constant should show
62
Data acquisition and time ordered data
a symmetric timestream around the cosmic ray event. Figure 2.14 demonstrates
the timestream around a cosmic ray event before and after the deconvolution
and digital low pass filter process. After deconvolution and low-pass filtering,
the data is safely decimated to 20 Hz without any possibility of aliased highfrequency power contaminating the timestream.
150GHz T signal
150GHz T scan jack
Figure 2.15: Scan direction subtraction test of deconvolution in primary CMB field near
PKS0537-441.
The ultimate test of the success of the deconvolution procedure can be seen in
the scan direction subtraction maps of the primary CMB field. PKS0537-441 lies
within the primary CMB field and is repeatedly observed over the course of the
2006-2007 observations. As the telescope rasters across the sky, it always observes
a point on the sky twice moving from opposite directions roughly 30 seconds
apart. One map of the primary CMB field can be made from the left-traversing
timestream, while another can be made from the right-traversing timestream.
When subtracted, if the deconvolution procedure is successful the resulting map
should be consistent with noise. As seen in 2.15 the quasar is successfully removed after deconvolution and scan direction jackknife, indicating successful
deconvolution.
Data acquisition and time ordered data
2.3.5
63
Despiking and identification of contaminated data
The cosmic ray events shown in 2.14 happen multiple times in each observation day, and need to be excluded from the data time stream. In addition, other
"glitch" events occur, including the occasional sudden temperature drift on the
focal plane that produces a spike in the timestream of many channels, and very
infrequent "bit tearing" events when the VME crate incorrectly clocks the bits
read from the ADC channels causing a momentary spike in the digitized timestream.
S^^.^..
^**&to***Jf****m+f*^'M*^»**^
<V\ r
i W t y l . * I l-yi'.^ Iljj
i i ^ i iir-n
_, i •> „ ir-iiin p j i .J
JUI
mt»tM»r^itfiff)mmmimmm»vmhi^mmmi»
•H««*«>»*iWM»if^^
^^H
•M«|l<llMI4allM«*«M*mi*<««^^
MIVlMMWIWVkMaaWWMH
==
nmii<H«i ti<i •
mi
mtfI*WI»NHI>I|I"IIII
"•MlMMW<wHMflMM>MMMll#M«in*iMI*MM>^^
sc u,t
E^ i
pi
i
_i.
L_J
,
[
i 11
u
i... I...
... i . .
LtEj
_
w i _ j __u
„_,
i
. i.
j j j .
.... ..... L .
£X-L^
LT
i
.
1
L... L.
1. .
1
L .
. L_. L.
.
...
L
.
1 .
I
_L
.
1
1
,
, . i
-i.
...Li Li.L,
L,.
•
.-
1 1 .
L i
L,
L.i
L_
(
-L- L i . L L
,
=
1[ ==
.
L .,
_j
.
L_i_j—1_=
lilu
_J—
,
,
L
=
_ 1 .. 1 L
1_=
.. 1 1 l . l . I . I . I I 1 1 -
Figure 2.16: 20 minutes of raw time-ordered data sum and difference (top) and power
spectra (bottom) plot used for human examination of timestream. Timestream marked in
red has been flagged by despiking algorithm as either a suspect thermal event or a cosmic
ray event.
The despiking algorithm locates these events by examining the timestream on
the same 30 minute basis in which the data is deconvolved, finding the standard
deviation of the data within that time interval, and looking for 8 a events. Since
the peak of the event typically occurs well after its start, due to the response time
Data acquisition and time ordered data
64
of the bolometers, the algorithm continues to search for the full extent of the
event by expanding the presumed contaminated area until the timestream drops
below one a. Although the events are typically only only a second long, 78 seconds of timestream around the event are flagged and tossed from consideration.
The despiking algorithm looks for 8 a events in both the individual detector
timestreams and the difference timestream of two PSBs within the same feed
horn. The reason for this is that over the 30 minute basis considered long-term
atmospheric drifts can be of a higher magnitude than the spikes caused by cosmic rays; the differenced data tends to be much more stable and spikes in any
individual detector show up quite clearly there.
Finally, every bit of timestream data has been manually examined to search
for thermal events that do not provoke the despiking algorithm in all channels,
and this data has been manually flagged and tossed. An example of a humanflagged event indicates in the channels not marked in red - the correlated bump
in all timestreams occurs that a fault of some kind has occurred but is not of sufficient magnitude to be flagged by the despiking algorithms. There are a total
of 13,278 of these plots from the final set of 143 days of 2006-2007 observations.
Each was examined, and 9 events were found and manually flagged.
2.3.6
Relative gain calibration
The PSBs within a given frequency have differences in gain of about ± 20 %.
These differences are calibrated out on a half-hourly basis after the high-gain
offsets and phases are reset by performing an "el nod" - the telescope is slewed
upwards in elevation over 20 seconds by 1°, and then back down. The entails
a change in airmass, which injects a nearly identical signal into every channel
on the focal plane, and using the known offsets of the detectors on the sky each
detector is fit to a volts per airmass number. The mean of the volts per airmass
number for a fixed subset of channels in each frequency is then taken to be the
reference for that 30 minute period and the relative gains of all channels in the
frequency are defined based on that number.
Observation History
65
Elevation nod, 060329
0
10
20
30
40
50
time (s)
Figure 2.17: Deconvolved timestream for all active detectors during an elevation nod.
The telescope is slowly elevated about 1°, injecting a large signal into all channels due to
the changing amounts of airmass each PSB observes. Each channel is fit to a volts per
airmass number, and the relative gains of the detectors are defined by the mean of a fixed
subset of the detectors within a frequency, either 100 or 150 GHz.
2.4
2.4.1
Observation History
Observation Strategy
QUaD observes two adjacent fields in right ascension, the first from 5h to 5h30'
and the second from 5h30' to 6h, at declinations ranging from —53.2° to -46.8°.
These coordinates are for the telescope pointing and central feedhorn - the field
is somewhat expanded by the roughly 1° radial extent of the full focal plane on
the sky.
The observing strategy is fairly straightforward and the pattern seen in 2.18
is exactly repeated 16 times in a day throughout the observing seasons, except
that the declination offset is different in each set. The first field is rastered back
and forth 5 times in right ascension at each declination for 4 declinations spaced
0.02° apart. In later processing the telescope turnarounds as well as data from
Observation History
66
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
-46.78
-_
-46.82
-46.84
z
-46.88
()
500
1000
1500
2000
2500
3000
time (s)
Figure 2.18: Eight scan sets comprising one hour of QUaD observing strategy showing
two fields in RA at 4 declinations. Excluded data from telescope turnaround periods and
outer 80 % of azimuth range shown in red. Pointing data shown is not entirely contiguous
- telescope turnaround times between the 8 scan sets shown are not included, and time
scales within the 8 scans have been slightly stretched to show proper time axis for the
whole of the one hour set.
the outer 80% of the azimuth range are discarded, resulting in 40 "half-scans" of
30 seconds or less from the constant velocity portion of the scans. In the half
hour that has elapsed since the beginning of observations on the first field, the
second has rotated into the same azimuth range and a duplicate scan strategy is
performed on the second field. This has the advantage of creating a redundant
Observation History
67
observation of every azimuth and elevation on a half-hour time scale with different parts of the sky observed.
The most naive technique for subtracting an azimuth-synchronous ground
signal would therefore be to subtract data from the first field of observations from
the second. For an analysis that assumes Gaussianity of fluctuations in the microwave background, like the typical derivation of the spatial anisotropy spectrum, this yields no change in the underlying statistical properties of the differenced signal field and effectively removes ground contamination from the data.
2.4.2
Data organization
Although there is contiguous, Fourier transformable data on the half hour time
scale of observations on a single field, analysis of noise properties of the data
on these scales is impossible after ground template filtering schemes described
in detail in chapter 3, because after filtering the uneven azimuth coverage results in differing noise properties for each point in the half hour block. The
timescale on which analyses of the noise properties in the timestream uncorrupted by ground pickup are possible are therefore limited by field differencing
time scales. Since the data is most reasonably field differenced on a single declination basis, the fundamental unit of timestream organization in the processed
data is a 6.5 minute single declination scan 7800 points at 20 Hz, with 10 halfscans and 9 telescope turnarounds contained within.
A data analysis pipeline programmed in IDL (an analysis language maintained
by ITT Visual Information Solutions), takes in raw digitized detector voltages and
telescope pointing information produced by the VME crate and archived by the
control computers, performs deconvolution, despiking and relative gain calibration for all channels, then divides the data up into 128 7800 data point chunks
for each detector. This is written to a FITS file and then distributed to the rest of
the QUaD collaboration for further analysis. Although considerations for filtering and noise measurement of the timestream are taken into account in the data
organization design, the IDL pipeline does not perform any of these operations
Observation History
68
Table 2.3: List of all observing days used in result and mean declination for each day.
Days are in Y Y M M D D format.
date
060329
060330
060331
060401
060402
060403
060404
060407
060408
060409
060424
060425
060426
060427
060430
060501
060502
060503
060504
060505
060506
060507
060521
060522
060523
060524
060525
060526
060529
060531
060601
060602
060603
060604
060617
060618
dec.
-47.11
-47.75
-48.39
-49.03
-49.67
-50.31
-50.95
-51.59
-52.23
-52.87
-51.75
-52.39
-47.43
-48.07
-48.71
-49.35
-49.99
-50.63
-51.27
-51.91
-52.55
-47.59
-49.03
-49.67
-50.31
-50.95
-51.59
-52.23
-47.91
-48.55
-49.19
-49.83
-50.47
-51.11
-52.55
-47.59
date
dec.
060619 -48.23
060620 -48.87
060621 -49.51
060622 -50.15
060623 -50.79
060624 -51.43
060625 -52.07
060626 -52.71
060627 -47.11
060628 -47.75
060629 -48.39
060630 -49.03
060701 -49.67
060714 -50.47
060715 -51.11
060716 -51.75
060717 -52.39
060719 -47.43
060720 -48.07
060721 -48.71
060722 -49.35
060725 -51.27
060726 -51.91
060727 -52.55
060728 -47.59
060729 -48.23
060805 -47.11
060810 -49.67
060811 -50.31
060813 -50.95
060814 -51.59
060816 -52.23
060820 -47.91
060821 -48.55
060823 -49.19
060824 -49.83
date
dec.
060907 -52.55
060908 -47.59
060910 -48.23
060911 -48.87
060913 -49.51
060914 -50.15
060915 -50.79
060916 -51.43
060917 -52.07
060918 -47.11
060919 -52.71
060920 -47.75
060921 -48.39
061004 -49.83
061006 -50.47
061008 -51.11
061011 -52.39
061012 -47.43
061013 -48.07
070329 -51.59
070330 -52.23
070331 -52.87
070415 -49.35
070416 -49.99
070417 -50.63
070419 -51.27
070420 -51.91
070421 -52.55
070422 -47.59
070423 -48.23
070424 -48.87
070425 -49.51
070427 -50.15
070511 -51.59
070512 -52.23
070513 -52.87
date
070514
070515
070516
070518
070519
070520
070521
070522
070523
070608
070609
070610
070612
070613
070614
070615
070616
070706
070709
070710
070711
070712
070713
070714
070716
070717
070718
070803
070804
070805
070806
070807
070809
070811
070814
dec.
-47.27
-47.91
-48.55
-49.19
-49.83
-50.47
-51.11
-51.75
-52.39
-51.43
-52.07
-52.71
-47.75
-48.39
-49.03
-49.67
-50.31
-48.71
-49.99
-50.63
-51.27
-51.91
-52.55
-47.59
-48.23
-48.87
-49.51
-47.75
-48.39
-49.03
-49.67
-50.31
-50.95
-52.23
-52.87
Observation History
69
except a mean removal of every scan.
2.4.3
Observation efficiency
190 days of observations were taken on the primary science field during the 2006
austral winter, and 133 days of observations were taken during the 2007 winter.
In 2006, 40 days are excluded for bad weather or other observing problems. Another 59 days are excluded from periods when the moon is above the horizon,
leaving 91 good days used from 2006. In 2007 another 40 days are excluded for
bad weather, and another 41 days are excluded for moon contamination, leaving 52 good days of observations. The total observing efficiency after data cuts is
therefore 143/323 = 44.27%. The full list of days used can be seen in table 2.3.
During the timestream filtering process used to remove azimuthal synchronous
ground pickup described in 3, the ground pickup templates at the outer azimuthal
edges of a scan set are overfit because there are only 3 or fewer samples from
that azimuth since the telescope is rastering the sky, rather than the ground, and
moves in azimuth over the 30 minute period of a single field observation. To
compensate for this, the data from outer 80% of the azimuth range is excluded
after the filtering process from consideration, resulting in the loss of 2.394 % of
the data that would normally be included.
In the final set of 143 observation days, 105982 cosmic ray events were recorded,
and 78 seconds of timestream surrounding the event were excluded from further
use. This results in 0.67% of the total dataset being discarded.
lo
Chapter 3
Mapmaking and Monte Carlo power
spectrum estimation
"Any one who considers arithmetical methods of
producing random digits is, of course, in a state of
sin."
Monte Carlo Method
JOHN VON NEUMANN
3.1
Overview
The main science results from QUaD are derived from a Monte Carlo analysis
software pipeline. Monte Carlo estimation of the CMB spatial anisotropy spectrum was first proposed to analyze the BOOMERANG balloon experiment (40)
and has favorable characteristics for the analysis of a high-resolution experiment
with large amounts of data in which modes from a wide variety of spatial anisotropy
scales are retained and analyzed. The originators of the method termed it MASTER (Monte Carlo Apodised Spherical Transform EstimatoR) and subsequent authors affiliated with the QUaD experiment have made the necessary mathematical adaptations to the method to extend it to polarization results (24).
Overview
71
This chapter will delve into each step of the MASTER-derived process that derives both temperature and polarization spatial anisotropy spectra in detail, documenting the data products that result from each step in turn, but it is important
to have a feel for the whole of the analysis process before continuing. In broad
detail, what we desire to measure is our universe's realization of the cosmological
power spectrum, Ce, which manifests itself under an assumption of Gaussianity
as the average power in the coefficients of a spherical harmonic transform of a
noiseless full sky map, m(n):
aim = / m(n)Ye*m(n)dn
1
(3.1)
m=(.
Ce =
2£ + 1 ^ 'a<?m'2
^3'2')
m=-e
Note that the Ce calculated in equation 3.2 will be from a single realization,
i.e. the universe we live in. Under an assumption of Gaussianity, the measured
Ce can be assumed to be drawn from a x2 distribution with an average value of
^theoretical a n d 2£ + 1 d e g r e e s of freedom (40).
A real, ground-based experiment like QUaD can only sample a small fraction
of the sky (about 1 %), and of course is not noiseless and has a response function
that is convolved with the true sky. The measured C/s are therefore distorted. If
we were to repeat QUaD's observations with identical scan strategy, noise and
contamination properties in thousands of "different universes" with the same
underlying cosmology then the alm will in each case be drawn from a Gaussian
distribution with variance Cjh, then the assembled average of the distorted, measured "Psuedo-C/' Ce would look like:
< Ce >= J2 Mu,Ft,Bi <Ce> + <Ne>
(3.3)
v
The finite area of sky creates a coupling between the previously statistically
independent Ce in the form of the matrix Ma<. The filtering used to clean the
Overview
72
data of contamination and the instrumental beam functions suppresses the true
realized Cg by F^Bf. Finally the instrumental noise adds an effective bias of Ne.
These are all thankfully linear functions that have no dependnce on the underlying true Cg. The Monte Carlo approach advanced by the MASTER algorithm
addresses these effects by creating hundreds of realistic simulations of the underlying data set including both noise and signal properties. The ensemble averages
of the noise simulations are used to measure the noise bias < Ne > and the ensemble averages of the signal-only simulations are used to measure the filter and
beam signal suppression factor Fg>Bj. The matrix Mu> depends only on the sky
cut used and can be semi-analytically calculated without resorting to full end-toend simulations. Finally, the scatter of the signal and noise simulations and the
covariances between the measured Ce from the ensemble of simulations gives a
robust estimate of our errors, and can be propagated to a curvature matrix (the
inverse of the covariance matrix) that can be used for parameter estimation.
While the theory is linear and not dependent on the "theoretical" signal used,
we can use the results of previous surveys of the large scale temperature anisotropics
of the CMB like WMAP (41) for the simulation process and, as a bonus, can compare our results to that of previously measured a previously measured cosmology.
This emphasizes a strength of the Monte Carlo approach - because we create a full end-to-end simulation of the noise and signal distortion properties of
our instrument we can validate our analysis by examining whether the simulations and consistency checks look "close enough" to similar consistency checks
on the data itself ("close enough" will be rigorously quantified later on in this
chapter). Furthermore, the existing simulation pipeline allows us to propagate
systematic errors from hypotheses simply by adjusting their low-level effects on
the timestream and then examining the result - for example, the effects of misestimating the fraction of unintended polarization power picked up by a detector
aligned in an orthogonal direction can be tested by adjusting a parameter in the
software pipeline and propagating through to errors in the Q.
All software, other than that used for the pre-processing of the data, for the
reults presented here was written in C++. This includes the entirety of the data
Overview
Figure 3.1: Schematic diagram of Monte Carlo APS estimation pipeline
73
Mapmaking
74
analysis and simulation software pipeline (hereafter referred to as "pipeline").
Other similar pipelines exist within the QUaD collaboration built in FORTRAN
and Matlab, and although some of the details in each pipeline differ they are
nearly algorithmically identical. The presence of multiple pipelines as an error
checking method within the collaboration was justified by discovery of mistakes
due to inconsistencies that became apparent in pipeline-to-pipeline comparisons of intermediate data products and final results.
Each set of 4 end-to-end signal and noise simulations of the full 143 days of
QUaD data takes about 12 hours on a single PC using an Intel quad-core Core
2 Q6600 processor running at 2.4 GHz. The simulations were multi-threaded
up to the number of cores per PC to minimize the number of disk I/O operations needed to produce a set of simulations. The combination of the Monte
Carlo analysis algorithm, the usage of very fast, openly available numerical analysis libraries like FFTW and LAPACK, and optimized, multi-threaded C++ code
resulted in a dramatic lowering in the computation costs necessary to analyze
CMB data with QUaD. While many other CMB experiments like ACBAR (35) and
BOOMERANG (42) have depended on supercomputing time at the National Energy Research Scientific Computing Center (NERSC), the results below were computed from 416 simulations requiring 10 days of computation on 5 desktop-class
PCs costing $4000 total. This made the analysis process much more flexible and
allowed many small effects in the data to be traced quickly.
3.2
Mapmaking
The core of any modern CMB data analysis process for data from an experiment
utilizing direction detection is the conversion of timestream from the detectors
into spatial maps of the microwave background. Ideal mapmakers are a lossless
compression scheme (43) of the underlying time ordered data, although as we
will see below QUaD's particular noise and contamination properties make such
an ideal compression computationally infeasible and we resort to other methods.
Nonetheless, all of these methods begin with the mapmaker.
75
Mapmaking
QUaD's full field size of 18° x 10° is small enough that a flat-sky approximation
with nearly square pixels can be used. With a resolution of dpix = 0.02° per pixel,
we define the number of rows and columns in a rectangular map as:
_ decmax - decmin
""row
j
_ (ramax - ramin) cos(decref)
'"col
7
V*3.T:J
where decref = deCmax^deCmin. The flat-sky analysis also yields benefits because
its harmonic transform, a two-dimensional Fast Fourier Transform, is far more
readily interpreted than the aem produced by a spherical harmonic transform and
QUaD's particular patterns of noise contamination are easily identifiable in this
basis.
3.2.1
Mapmaking formalism
First, let us parameterize a map of nrow x ncoi pixels as a npix = nrow x ncorlong
vector m, the full 143 days of used timestream from a single QUaD detector as an
n^-long vector d and a nd x npix pointing matrix connecting data points to locations on the map consisting entirely of Is and Os as A, where each row consists of
all Os except for a 1 corresponding to the column indexed to the appropriate map
pixel. The most naive, unbiased mapmaker would simply derive an estimate for
each point on the sky by averaging every data point that hit that map pixel according to the pointing matrix. The number of times a pixel is hit is simply the
diagonal matrix A T A and therefore the averaged map vector is:
m = (A T A)" 1 A T d
(3.5)
However, this is not an optimal combination of the data unless the timestream's
noise properties are stationary and totally white. In the presence of non-white
noise and covarying errors, we want to minimize the x2 of the covarying quantities d — A T m, as a function of the npix parameters in m. This results in the optimal map-making equation cited often in CMB literature (15):
76
Mapmaking
A T N ^ A r a = ATN^1d
(3.6)
where N _ 1 is the inverse of the timestream covariance matrix. In full generality,
this is a very difficult problem to solve - 0{n2d) data points must be binned into an
nPiX x npix inverse pixel noise covariance matrix, and then this npix x npix must be
inverted. At the full resolution of 0.02°, half of the size of QUaD's 150 GHz beams,
this results in the necessary inversion of a 300,000 x 300,000 element matrix.
The processed can be simplified in two ways. First, the noise matrix inversion
process can be simplified by by either degrading the resolution of the map to
reduce the number of pixels or imposing simplifying assumptions on the matrix
A T N _ 1 A to make it more invertible. Second, a "noise correlation length" can
be introduced such that we only have to consider a band matrix version of N"1,
resulting in 0(nd x ncorr) operations.
The natural limit of both of these simplifications is a weighted mapmaker,
where we assume that the noise properties are not stationary but that the data
is uncorrelated. This results in a diagonal inverse noise matrix with diagonal elements wl,w%, wj, etc. This results in a diagonal pixel noise covariance matrix.
Since the inversion is trivial we can individually estimate each map pixel from
the data and a set of weights:
rrii = - = - ^
2^=1
(3.7)
w
j
In the QUaD Monte Carlo pipeline the weights used are the inverse of the variance of the surrounding points in the data. This is the logical extrapolation of the
full inverse timestream covariance matrix to a correlation length of zero. In particular for every half-scan of 600 points at 20 Hz for a single pair of detectors, we
assume that the timestream is noise dominated and calculate the variance a?.
The weights for all 600 points in the scan are then Wj = er|.
77
Mapmaking
3.2.2
Temperature and polarization timestreams
So far we have avoided discussion of the nature of the signal being integrated
into the maps. QUaD is a polarimeter that can measure both total intensity and
polarization, and with minor modification the inverse variance weighted formalism can be applied to both. First, let's look at the voltage a single PSB observes as
a function of the beam-smoothed temperature and Stokes polarization fields on
the sky, T(rajjfe, decjtk),Q(r&jtk, deci;fc) and U(va,jik, deciifc) (44):
Vj,k = - ^ ( ( 1 + efc)T(rai,fc,deciifc)
+(1 - ek)[Q(mjtk, decjjfc) cos(2ajifc) + C/(raj)fc, deciifc) sin(2cr/, k)])
where we retain the convention of timestream indexing by j and introduce an index over detectors k. The cross-polarization efficiency ek is an intrinsic property
of the PSB that denotes the contribution to the signal from the linear polarization
component orthogonal to the metallization direction of the PSB grid that the PSB
detects. Its net effect is to introduce an error into the temperature-polarization
power ratio at the level of e4 if not properly accounted for, and in QUaD the detectors have e ~ 7%. The gain conversion factor gjtk converts from the astrophysical
intensity units of interest (either equivalent blackbody temperature or Janskys) to
detector voltage. Finally, a^k is the detector's polarization sensitivity orientation,
which is a combination of fixed orientation of the detector on the focal plane and
the total rotation of the instrument and the current time that we denote the "deck
angle", dkj.
When reconstructing the maps from the timestream signal, we assume that
the two orthogonally aligned detectors within a horn point at the same ra^, dec^
and are aligned 7r/2 + Sak from each other, where 8ak is the deviation from perfect
orthogonality, which has been measured in QUaD to be about 1%. Folding the
gain calibration factor into the voltages, our reconstructed astrophysical signal is
Sjtk = Vjtk/(gjtk(l + ejt)). assuming that the relative gains gjyk have been calibrated
out and are identical, the sum of two detectors with indices kA and kB within a
78
Mapmaking
horn is therefore is:
Sj,kA + SjtkB
—
^(T(ra.j^,deCjtk)
(<3(rai)fc, deci)fc) cos(2ajiA:yl + 5ak)
(1 + ekB)
+U(iSi^k, deci>fe) s\n(2ajtkA + 5a)))
+
-(T(rajjfe,deci;fc)
+ 7——A-(Q(rajjfc, decjjfc) cos(2a i M + TT - 5afe)
(J- + efcj
+[/(ra,-ife, deci>fe) sm(2ttj>fcA + n - 5a)))
During reconstruction we assume 5a = 0, which is consistent with the population means from our measurements using a near-field polarization source, with
c<5a ~ 1°> and we assume that all of the ek for the same observing frequency are
the same. The measurements of the detector properties ek and pair properties
5ak are described in more detail in conjunction with overall polarization angle
calibration in chapter 6. The sum of the signals can then be reduced to:
SjtkA + Sj>kB = T(r&j^k,deCjtk)
+
2—(1 + ^^( r a ^ f c ' d e c ^)( c o s ( 2 Q J> f c A ) ~~ cos(2ajM))
+U(r&j,k,decjrk)(cos(2aj,kA)
=
-
cos(2ajikA)))
T(rajjfc,deci)fe)
If a non-zero 5ak is left in, the net result is to prevent perfect cancellation of
the polarization components of the signal and create "polarization to temperature" signal leakage. Likewise, we can compute the difference between the signals
in the two PSBs in a single horn:
Mapmaking
Sj,kA — Sj^B
79
=
-(T(ra i ; f c , deci>fe) - T(ra ijfc , decj;fc))
+
o—(] + \(Q(raj,k^ecj,k)(cos(2ajM)
+C/(rajjfc, dec.,-,k)(cos(2a^kA)
=
(1
+
+
cos(2ajjkA))
cos(2ajtkA))
JQ(T&j,k, decj,fc) cos(2a i)fcA ) + C/(raiiA;, deci;fc) sin(2aj>fcA)))
TT ratio by I to unmodified simulation for epsilon = 0.07 +/- 0.03, delta alpha = +/- 3 deg.
1.0015
1.0010
1500
EE ratio by I to unmodified simulation for epsilon = 0.07 +/- 0.03, delta alpha = +/- 3
500
BB ratio by I to unmodified simulation for epsilon = 0.07 +/- 0.03, delta alpha = +/- 3
Figure 3.2: Ratios of cut-sky psuedo-Q angular power spectra from simulations of ACDM skies with detector cross-polarization efficiency errors and detector misalignments
errors turned on at the level of 7 ± 3 % and 3° respectively to ideal detectors in green.
Mean of all simulations shown in red. Note that on any given realization of the bolometer
characteristics this results in a roughly 5% calibration error on the polarization power
but over many simulations there is virtually no bias. The temperature power spectra are
largely unmodified in comparison to the errors induced by noise.
Mapmaking
80
So, if we assume during reconstruction that the detectors are perfectly aligned
and their relative gain has been properly adjusted, the sum of the detector signals
within a horn is sensitive only to the temperature field on the sky and the difference is sensitive only to the polarization. The effect of relaxing these assumptions
to the level of uncertainty that we observe in calibration tests can be determined
by inserting the appropriate effect into the signal-only component of the simulation pipeline on a detector by detector basis.
Using these tools it is found that the inclusion of the uncertainties in detector characteristics induces an error in the reconstructed polarization power of at
most 5% and nearly no change in the observed temperature field, seen in figure
3.2, due to the sufficient number of bolometers on the focal plane that average
out these effects in the total reconstructed angular power spectra. After averaging over many possible realizations of the focal plane these effects are nearly unbiased and negligible due to the error bars induced by noise. For our final science
results these simulated random errors in e and 8a are included in the signal-only
simulation pipeline and contribute marginally to the total error of the results.
3.2.3
Polarization mapmaking
The formalism for making temperature maps is now straightforward - just substitute in the sum of the detectors within a pair for all the data points dj to make
an estimate of each map pixel using an inverse variance weight scheme:
^Aiij}k=l(SJ,kA
m i
=
SjtkB, Wj
+
"*•'•*_;
Z^Ai>j>k=i
(3.8)
w
i
Polarization is somewhat more complicated. At an single point in time a detector difference pair can either measure a purely Q Stokes parameter signal, a
purely U signal, or a linear combination of the two depending on its current rotation with respect to the sky. As there is only one differenced signal, this means
that multiple observations of a single point on the sky are necessary to constrain
Mapmaking
81
both Q and U, either from the same detector at a different orientation or a different detector with a different orientation.
The recipe for decoding Q and U from multiple observations of a single point
can be derived by defining a \2 for a single map pixel with two parameters, Qt
and Ui (45) and assuming that the noise is uncorrelated both between adjacent
detectors and adjacent timestream points:
X2 = Y1 wi,k((Sj,kA ~ SJ!kB) - Qi cos(2ajjk) - Ut sin(2ajifc))2
(3.9)
j,k
Differentiating and maximizing the \2 with respect to Qt and Ui yields a 2 x 2
system:
fY,j,kWJ,kC0S(2aj,k){Sj,kA
- Sj,kB)\
;
\Ej,fc« j,fcSin(2aj)fc)(SJ-,feA - SjikB) J
/£^W2^)cos(2a,
f c
w
\l2j,k j,kSm(2aj^k)cos(2ajjk)
)
=
sm(2a,fc)cos(2^)\
/Qi\
sin(2aiJfe)sin(2ai)fc)/
\Ui)
This system can be inverted to yield:
_ (Hj,k wJ,k cos(2ai]fc) cos(2aj]fe)
w
\12j,k i,k sm(2ajtk) cos(2ajtk)
sm(2ajtk)
cos(2ajtk)\
sin(2ajjk)
sm(2ajtk)J
\J2j,kWj,ksm(2aJ!k)(SjM
- Sj>kB)J
In practice we therefore accumulate 5 values for every map pixel as we proceed through the pointing strategy and identify which differenced timestreams
are applicable to which point on the sky. The inverse variance wjik of the differenced timestream and as the atmosphere is unpolarized the differenced timestream
it has nearly white noise properties when properly filtered of residual polarized
contaminating ground emission. The assumption of uncorrelated noise that goes
Mapmaking
82
into this derivation is therefore quite good and a lack of residual striping from atmospheric contamination can be seen in the Q and U maps seen in figures 3.7
and 3.8.
3.2.4
Timestream filtering
The unfiltered timestreams are dominated by slow-varying atmospheric fluctuations that are well fit by third to fifth-order polynomials on 30-second scale. As a
result, filtering the data results in maps that are visually less contaminated; however it is important to understand the filtering operation in detail as the assumption that we can recover the spatial anisotropy signal after timestream filtering
through Monte Carlo signal-only simulations lies at the heart of the MASTERderived technique.
A polynomial fit simply solves the least squares problem (46):
y = Xa
(3.10)
where y is a n-long vector of timestream data points like Sjyk, and a are the solved
polynomial coefficients. X is a Vandermonde matrix for a k-th order polynomial
that looks like:
'1
xi
x\
... x'p
1
x2 x\
... x\
yl
xn xn
...
( 3 n )
xn)
Typically x\, x2, etc. are regularly spaced timestream points at 20 Hz and are
therefore just indexed 1,2,.... Solving the least squares problem gives us the polynomial coefficients:
a = (X T X)" 1 X T y
The "pure polynomial" resulting from this fit is:
(3.12)
83
Mapmaking
X(X T X)" 1 X T y
(3.13)
And therefore, it is clear that the polynomial filtering can be described as a
linear operation with an x n matrix operating on our original data vector:
y = (I-X(X T X)- x X T )y
(3.14)
The filtering process therefore not only subtracts power from the timestream,
but potentially induces systematic correlations. These effects are mitigated further on in the analysis process by examining signal-only simulations. In QUaD,
we fit and remove a third order polynomial to the data on 30 second "half-scan"
length timescales. As the Vandermonde matrix-derived filter disproportionately
distorts the data at the end of these 600-point halfscans, when integrating the
data into a map the last 70 points on each end are deweighted by aggressively
tapering weights wjik associated with those points to low values.
3.2.5
Ground contamination removal
As can be seen in figure 2.18 QUaD's scan strategy has a high degree of redundancy in azimuth. Since QUaD observes two adjacent fields separated by 0h30'
in right ascension separated at half hour intervals, the most direct means of filtering out an azimuthally synchronous ground signal is to difference data taken
at half hour intervals from the two fields.
Because we implicitly assume that the CMB fluctuations are a Gaussian random field, differencing the timestream from two adjacent fields does nothing to
final spatial anisotropy power spectrum except scale it. In addition, the field differencing operation leaves the data with uniform noise properties since every
data point has a one-to-one correspondence with a point in the opposing field.
Field differenced data therefore allows us to perform Fourier-space operations on
10 half-scans and turnarounds totaling 390 seconds. However, this operation effectively costs us a factor of \/2 in sensitivity because half of our data is discarded
Mapmaking
84
10
5
o
0
>
|i*^yv>^^
-5
-10
200
time (seconds)
Figure 3.3: 390 seconds of original and filtered time ordered data from central detector
150-01A on April 25th, 2006 at 20 Hz. From top to bottom, plots are unaltered data,
data after field differencing, data after ground template subtraction, and after third-order
polynomial fit subtraction on 30 second halfscan timescales.
Mapmaking
85
to remove the azimuthally symmetric signal.
An alternative ground contamination removal scheme utilizes the overlap in
azimuth across multiple half-scans. However, for a single elevation any particular point of azimuth is scanned anywhere from 2 to 20 times, yielding estimates
of the ground synchronous signal with differing variances across the whole of
the 390 second "full scan". Therefore, any ground subtraction scheme using azimuth templating rather than field differencing will inevitably degrade the maximum time scales on which atmospheric noise properties can be measured - this
will become relevant when we consider noise-only simulations of the timestream
(cite Michael when his paper comes out).
In practice, the ground template subtraction is implemented by dividing the
azimuth range of a full scan and its field partner's 7800 data points into 120 azimuth bins. Since long-term atmospheric wander will obscure the ground synchronous signal, the 20 half-scans (10 in each field's full scan) have a mean component removed from them, and then the data from the central 80% of the azimuth range of the scans is averaged within the 120 azimuth bins to obtain an
estimate of the ground signal in those regions with high degrees of azimuth redundancy. The outer 20% of the scans' total azimuth range is tossed from consideration because those portions are traversed very infrequently.
Regardless if the data is field-differenced or ground-template subtracted, what
we ultimately consider are the 10 600 data point, 30 second-long half-scans from
each field. After the ground template removal process these are polynomial filtered under the assumption that the residual long-term variations are dominated
by atmospheric drifts.
3.2.6
Maps
-52
5.4
5.6
Right Ascension
Figure 3.4: Field differenced inverse variance weighted map of the QUaD's temperature field at 150 GHz. Color scale
runs from -150 to 150 fiK. Ground contamination removal achieved through field differencing.
Q
-46
en
00
(W
CD
7T
ED
5.2
5.4
5.6
Right Ascension
5.8
6.0
1 -
Figure 3.5: Non-field differenced inverse variance weighted map of the QUaD's temperature field at 150 GHz. Color
scale runs from -150 to 150 fj,K. Ground contamination removal achieved through azimuth template subtraction.
5.0
I
QJ
CO
-J
<ra
I-a
Figure 3.6: Non-field differenced inverse variance weighted map of the QUaD's temperature field at 100 GHz. Color
scale runs from -150 to 150 /iK. Ground contamination removal achieved through azimuth template subtraction.
5.4
5.6
Right Ascension
Mapmaking
89
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Mapmaking
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91
Variance maps and apodization
Figure 3.9: Non-field differenced inverse variance weight pixel mask used for apodization
and computation of spatial anisotropy spectra. Accumulated variances for each pixel have
been inverted, then convolved by a a = 0.05° Gaussian kernel in map space, and known
locations of detected point sources have been downweighted.
Since the variances of the half-scans are used to compute the weights going
into the mapper, we can also choose to map the variances themselves, but accumulating the map pixel variance values of:
-, = S^-.°W =
1
,3.15)
The inverse of the variance maps are a useful way of apodizing the data before
taking two-dimensional Fourier transforms. To diminish the row to row variation
observed as a result of non-uniform observation times, the entire inverse variance map is convolved with a two-dimensional Gaussian kernel with a width of
a = 0.05°.
Spatial anisotropy spectra
92
There are 8 point sources are easily identifiable in QUaD 100 and 150 GHz at
5 a or above when the data maps are convolved with a matched filter (3). These
sources match to corresponding sources in the PMN catalog (47). After smoothing, Gaussian "divots" of beam-sized widths are inserted into the inverse variance
weight map so that the immediate area around known point sources are heavily
deweighted in the computation of spatial anisotropy spectra. The final inverse
variance weight mask used for 150 GHz can be seen infigure3.9.
3.3
Spatial anisotropy spectra
For sufficiently high multipoles £ > 60 the spatial anisotropy spectra Ce can be approximated by visibilities in the Fourier space equivalent of the temperature field
at increasing accuracy with increasing I (48). Due to filtering andfieldsize constraints QUaD is sensitive only to t > 100 and therefore a flat sky approximation
where Fourier visibilities are used to compute the spatial anisotropy spectrum is
appropriate. These approximations are typically used in interferometers where
the fundamental observables are visibilities in the Fourier plane, and indeed the
first experiment to detect polarization in the CMB was an interferometer, DASI,
that mosaiced many smallfields(10).
Following White et al, let us explicitly state the Fourier space observables of an
interferometer, the convolution of the temperature field with the primary beam
of the instrument:
VWxf*B{*)W*>*~*
(3.16)
where B{x) is a map of the primary beam and AT(x) is the map of the CMB fluctuations. Although we only use the temperature field here we will see later that
using Fourier transforming the Q and U fields leads to a simple transform to the
E and B mode Fourier fields.
The fundamental science observable regardless of the type of instrument or
whether we use the flat-sky approximation that we are trying to discern is the
93
Spatial anisotropy spectra
correlation function of two points on the sky that depends solely on the cosmological spatial anisotropy spectrum, Cf.
C(xl-xl)
=
/AT
AT
^ — (*!) — (x2) )
=
— Y(2£+l)CePe(x[-x2)
(3.17)
oo
1
(3.18)
where Pe(x) is a Legendre polynomial function. The double Fourier transform of
the correlation function can be taken to convert x[ and x2 to Fourier visibilities
u[ and u2:
I dx\dx2C(xi • £2) exp[27ritTi • (x[ — £2)] exp[2iri(ul — u2) • x[]
(3.19)
White et al expand the exponential as a Bessel series:
00
e2niu-x =
j0(27ru)
+ 2 ^2 Jm(2yra) cos[m arccos(u • x)]
(3.20)
m=l
Since orientation should not matter for a Gaussian random field, there is a
rotational invariance that allows us to eliminate the higher order Bessel terms,
leaving us with a sky spectrum S(u) that depends only on the magnitude of the
visibility in the Fourier plane and the underlying correlation function:
f2
S(u) ex / udujC(io)Jo(27TULo)
(3.21)
Jo
with ui = \x~i — x2\. Inserting the appropriate correlation function we obtain that
the sky spectrum as a function of d is generally:
S(u) = ^ - Y(2£
Z7TU z — /
+ l)C £ J 2m (47ra)
(3.22)
For sufficiently large £ White et al point out that J2e+i is sharply peaked, and
therefore on small angular scales the flat-sky approximation yields
Spatial anisotropy spectra
94
u s u
< 3 - 23 )
" ( )«-izrw-Ct
with £ = 2KU due to the peak of the associated Bessel function.
Although packages like HEALPix (49) allow for pixelization and harmonic analysis of CMB temperature and polarization maps on spherical surfaces to obtain
spatial anisotropy spectra without invoking the flat sky approximation, the analysis of the QUaD data presented here uses the flat sky approximation because it
allows the data maps to be interpreted and filtered in two dimensional Fourier
space. Interpreting and filtering the a^m coefficients of the spherical harmonic
Ygm functions is far less straightforward, especially in the QUaD maps where the
dominant atmospheric noise is scan and azimuthally synchronous. Parallel analysis pipelines of QUaD have used the HEALPix package and curved sky formalism
and obtain nearly identical spatial anisotropy spectra (1).
3.3.1
2-dimensional FFT Temperature Maps
Although interferometers measure the visibilities u and can therefore directly
measure the sky spectrum S(u), we can arrive at an equivalent representation
by performing a 2D discrete Fourier transform on our pixelized flat sky map:
T x
( > y)e-2ni{ux/M+vv/M)
f(u, v) = Y,J2
x
= F{T(x, y)}
(3.24)
y
where u and v are the components of the visibility u.
In practice we zero-pad the 579 x 420 pixel maps up to 1024 x 1024 before
performing a two-dimensional Fast Fourier Transform using the FFTW package
(50) in order to optimize the speed of the transform and simplify the interpretation of the baseline spacings in Fourier space. We can downweight the noisy
portions of the map and enforce an apodization using the smoothed inverse variance mask, 4z described above and account for the normalization factor due to
the mask and padding:
Spatial anisotropy spectra
95
Figure 3.10: Two dimensional Fast Fourier Transform of 150 GHz temperature data map.
Concentric red rings are spaced at I = 500,1000,1500,2000,2500... demonstrating correspondence of Fourier plane baselines to spatial anisotropy spectra. Trench shown down
the middle of image is the effect of scan synchronous third-order polynomial subtraction
anisotropically filtering atmospheric noise. Color scale from 0 to 10~3[iK2.
where the pixel number i and its coordinates x, y are used interchangeably and
nx and ny are the original number of pixels in the non-padded map. We can determine the effective t value of every point on the Fourier plane by converting the
Spatial anisotropy spectra
96
Figure 3.11: Scan direction subtracted "jackknife" map of temperature 150 GHz field
showing noise-dominated Fourier plane (left), and signal-to-noise weight mask formed from
100 noise-only realizations of the QUaD instrument and scan strategy. Note locations of
high noise in scan direction subtracted Fourier map are excised from signal-to-noise mask
and correspond to correlated noise in the temperature maps due to characteristic angular
spacing of detectors on the focal plane when projected onto the sky. Color scale from 0
to 10~3fiK2.
pixelization width, 8 = 0.02° per pixel to the Fourier plane spacings:
5u v
' = 1024(^/180.0°
5£ = 2n8Uu
(3-26)
(3.27)
which specifies the spacing in equivalent spherical harmonic multipole H for each
pixel from the center of the 2D Fourier plane and follows from the flat-sky approximation. To plot the power in the plane we then examine the auto-spectrum
f'(u, v)f'*(u, v), shown in figure 3.10. Signal is readily apparent when compared
to the scan direction subtracted temperature map shown in the left half of 3.11,
where the residual bright stripe along the y-axis of the 2D Fourier plane reflects
the remaining scan-synchronous power due to atmospheric noise. Regular "spots"
Spatial anisotropy spectra
97
can also be seen in the plane representing the contribution of correlated noise
across the whole of the map due to the fixed spacing of the beams of the detectors when projected out onto the sky.
3.3.2
2-dimensional FFT Polarization Maps
Figure 3.12: 2D Fourier plane representation of QUaD 150 GHz E (left) and B (right)
auto-spectra as derived from linear combinations of Q and U Stokes parameter Fourier
maps. Note visible detection of E mode signal compared to non-detection of B mode
signal, and lack of noise leakage around vertical trench signifying unpolarized nature of
atmospheric noise. Red concentric circles spaced at I = 500,1000,..., and color scale
from 0 to 10" 3 ^K 2 .
The Q and U Stokes parameter data maps returned by the polarization mapmaking formalism above can be apodized and converted to the Fourier plane in
the exact same manner as the temperature maps. There is then a trivial conversion to convert the Q and U Fourier space maps into E and B Fourier space maps
in the flat-sky approximation (51):
Spatial anisotropy spectra
98
arctan
'V
(3.28)
-U.
E'(u,v)
= Q'{u,v)cos(2(/>) + U'(u,v)8m(2<t))
B'(u,v)
= -Q'{u,v)sm(2(t>) + U\u,v)cos(2(f))
(3.29)
(3.30)
Polarization auto-spectra can then be obtained by examining the combination E'(u, v)E'*(u, v) and B'(u, v)B'*(u, v), while temperature-polarization crosscorrelations can be obtained from maps of f'(u, v)E'*(u, v) and f'(u, v)B'*(u, v).
Finally, polarization curl-gradient cross-correlation, particularly relevant for tests
of electrodynamic parity violation, can be obtained from E'(u, v)B'*(u, v). The
150 GHz two dimensional auto-spectra for E and B clearly show evidence of signal in E and only noise in B, and are clearly far less contaminated by atmospheric
noise in figure 3.12.
3.3.3
Annular I D spectra
After obtaining the two-dimensional Fourier representations of the auto-spectra
TT, BB, EE and cross-spectra, TB, TE and EB, conversion to spatial anisotropy
spectra is obtained by averaging the values £2X'(u, v)Y'* (u, v) within annular, concentric bins spaced we = 81 apart, with I = 2ny/u2 + v2 and X, Y refer to either T,
EorB.
Note that the one-dimensional spatial anisotropy spectra derived from this
method are distorted by noise bias as well as filter and beam suppression and
need to be corrected using the MASTER method described below. As can be seen
in 3.13 the noise bias is considerable when compared to the signal in the autospectra, but the temperature and polarization noise is sufficiently uncorrelated
that the temperature-polarization cross-spectra largely have no noise bias.
MASTER formalism
99
1000
1500
2000
2500
1000
1500
2000
2500
1000
1500
2000
2500
iiV
,* ******
:„ o • • 8 ?:
500
1000
1500
2000
2500
i*«*t»t***{i*;t
. * » • » « * * !
500
1000
500
1500
2000
2500
500
1000
1500
r
i*}iii
i'-
2000
Figure 3.13: Unprocessed 150 GHz spatial anisotropy auto- and cross-spectra bandpowers
from data (red points) and simulation ensemble (gray lines) created by averaging points
in annular bins with widths of we = 81 from Fourier planes like those shown in figures
3.10 and 3.12. Signal shows evidence of beam and filter suppression, and auto-spectra
show evidence of significant noise bias increasing with £. Blue lines are equivalent WMAP
five year best-fit A-CDM model and blue points are equivalent bandpowers.
3.4
MASTER formalism
Although we are concerned with finding the cosmological Cg, any actual experiment only is able to measure a fraction of the full sky and can only obtain an
estimate of the cut-sky "psuedo-C/, which we denote as Ce in accordance with
the literature (24). Even a full-sky satellite experiment like WMAP or Planck cannot actually obtain a direct unbiased measurement of Ce due to the presence of
foregrounds that necessitate the elimination of certain areas of the sky like the
MASTER formalism
100
galactic plane from consideration. The cosmological spectrum Ct encodes all
the necessary information as a Gaussian random field under the assumption that
there are no phase correlations among moments in the field, that is (20):
(a\mai>mi) = Ci5u>8mm>
(3.31)
However, the presence of a fractional sky cut destroys the independence of
the measured multipoles from one another through the "mode-coupling matrix",
M'e'er.
ct = YsMu'Ci'
(3 32)
-
e
Mee> is solely a function of the sky cut and can be computed semi-analytically
using numerical integration of the spherical harmonic moments of the sky cut
mask (24). In practice we actually measure a more limited set of "bandpowers"
that we will denote as Ce:
Y, pbeCi
(3.33)
1
Pu
1-K
e(b+i)_jb)
•
low
low
= { ^hov'-hov
2 < f(b) <f<
Z
—
l
low —
t
£{b+1)
—
l
low
(3.34)
0 : : otherwise
where an implicit assumption is made that £(£ + \)Ci is constant within a bandpower and the I associated with a bandpower index b for bandpower Q is the
midpoint |(4oi +4ow )• Although this notation is somewhat confusing, throughout this text we will often index bandpowers by their associated £ value rather
than the bandpower index b even though there are matrices like the binning operator Pu that convert between the two.
This bandpower binning operation of spherical harmonic coefficients is equivalent to the averaging of power within an annular bin in Fourier space and the derived data products are directly comparable in the flat-sky approximation limit.
MASTER formalism
101
For sufficiently narrow bins in I space this is a valid assumption, and can be compared to theoretical spectra if the mode coupling matrix MK can be discerned.
Any real instrument also measures a temperature map that is convolved with
its beam function. This is equivalent to a multiplication in u, v space and therefore just acts as a filter on the individual £ moments:
f(u) oc I dxB{x)AT(x)e2™-£ oc BtCt
(3.35)
?=2TTO
A core assumption of the MASTER method is that the timestream filtering
acts in an identical way to the beam, although this is demonstrably not the case.
Much of the analysis above assumes that the fluctuations are isotropic, but in
the case of azimuthally synchronous filtering this is clearly not the case and in
fact on average the filter introduces some amount of non-isotropic correlation
~
~
~ 2
< a*emai'mi >y£ 0 and bandpower covariance < C^C^ > ^ A Q 6W. Nonetheless
treating the filter function F£ solely as a suppression that is a function of t, rather
than a full mixing matrix Fu> over an ensemble average of isotropic temperature
fields has been shown to be a valid unbiased approximation on a variety of experiments (40; 52; 1) for a high-pass filter like our polynomial fit.
For a noiseless experiment we then arrive at the ensemble average measured
spatial anisotropy spectrum bandpowers for a given sky cut, beam and filter suppression and bandpower binning:
< Q > = Y,PuYJM^F^B^<c'^>
i
(3 36)
-
a
< Ce > = F*B| Y, pw Y, Ma' <C't>
t
(3 37)
-
i>
where we now define the beam and filter suppression functions, Fe and B£ on a
per bandpower basis rather than in the unbinned spectrum. Note for any given
single sky realization due to the anisotropic nature of the filter, Fe will not act
MASTER formalism
102
identically and |f ^ ^ > > but on average this is an unbiased estimate. In a noiseless experiment, we can therefore obtain an unbiased estimate of the cosmological bandpowers by computing the quantity:
v
(.estimate
—
-'(.measured
^TT}2
<-q QQ"\
I.O.OOJ
beBe
F^Bf is obtained from a suite of signal-only simulations detailed below. This
estimate is nominally comparable to theoretical spectra when converted to bandpowers using a combination of the sky cut Mw and the binning Pbe. We denote
this "bandpower window function" as Wa and describe a numerical method for
computing it below.
Finally, the auto-spectra of a real experiment with noise display evidence of
a noise bias that increases with I, as seen in figure 3.13. In general, the filtered
timestream consists of signal, Sj and noise rij components, which when binned
into a two-dimensional map induces cross-correlations consistent with the noise
spectra of the underlying data. Uncorrelated timestreams, such as temperature
and polarization, result in cross-spectra with no noise bias. Since every step of
our process, including the Fourier transforms of the maps, is linear, we can propagate the noise term alongside the signal and derive an equivalent
dj --=
mi
(3.39)
Sj+Hj
(3.40)
=
m ==
1
=1
—
i,fc=l
F{m}
== f{u) + N{u)
< (f{u) + N{u))*{f{u) + N(u)) > --= Ce + Nt
Kjwj
(3.41)
3
(3.42)
(3.43)
where the last step assumes that the filtered, beam-smoothed signal has no correlation with the instrumental and atmospheric noise. We then arrive at our full
103
Noise simulations
expression for the ensemble average signal and noise bandpowers of a series of
experiments with identical noise properties and different sky realizations with
the same underlying spatial anisotropy spectrum:
<Ce> + <Ne>=Y,PbtJ2
(Mu'Fi'B% < Cv >) + < Nv >
e
e
(3.44)
Subtracting off the ensemble average of the noise spectra over many instrumental realizations, < Ne > is an unbiased method to recover the filtered, beamsmoothed and sky cut signal. We obtain an estimate of < Ne > through a Monte
Carlo method of instrumental noise simulations detailed below.
3.5
Noise simulations
< Ng> > is derived in a MASTER-type analysis by creating many realistic simulations of the noise component of the timestream, creating noise-only maps of
those timestreams, and propagating them to a set of noise-only spatial anisotropy
spectra. The mean of the set of noise only spectra is the estimated noise bias
< Ne> > and the variances correspond roughly to the noise-induced portion of
the error bars.
3.5.1
Timestream noise properties
We characterize the noise properties of the time ordered data by averaging many
measurements of the noise-dominated power spectrum density in Volts per root
Hertz, defined as:
d{u) = F{d(t)}
pM _
sg^y
y
-'" Jsamp
(3.45)
(346)
Noise simulations
104
100 GHz
150 GHz
10000.0 f
10000.0
1000.0 k
1000.0
100.0 k
100.0
10.0 k
10.0
1.0 k
1.0
0.1
0.1
0.01
0.10
1.00
Frequency (Hz)
10.00
;
;
K
:
[ V^
i
No^
n
:
'
,
0.01
,
,
0.10
1.00
Frequency (Hz)
1
]
10.00
Figure 3.14: 100 and 150 GHz atmospheric and bolometer noise spectra measured from
March 29, 2006 observations of CMB field. Y-axis is power spectrum density (PSD)
in units of nanovolts per root Hz. PSDs derived from temperature sensitive pair sum
timestreams in black, PSDs derived from polarization sensitive pair difference timestreams
in red. PSDs integrated over 64 6.5 minute field-differenced full-scans and all good
detectors in each frequency. Observed noise is dominated by atmospheric fluctuations
at low frequencies tending as 4, detector and electronics white noise at intermediate
frequencies, and microphonic coupling spikes at high frequencies. Roll-off at 6 Hz due to
digital low-pass filter. Note that polarization atmospheric noise (in red) is far smaller than
temperature (in black). The nominal science band is between 0.1 and 1.0 Hz. Individual
power spectra like these are used to construct the model for simulation noise in the QUaD
MASTER-type analysis.
where T represents the Fast Fourier Transform operation, N is the number of
samples in the timestream (typically 7800 points, or 6.5 minutes worth of data),
fsamp is the sampling frequency, 20 Hz and d(t) is the windowed data and zeropadded timestream. The factor of N is necessary due to the conventions of the
FFTW package (50).
Noise simulations
105
We need to obtain realistic noise spectra on scales where the individual frequency space modes are Gaussian distributed in order to reach our goal of creating simulated timestreams that are indistinguishable from those obtained by the
instrument itself. We can largely ignore the signal component of the CMB when
measuring the original timestream for its noise properties, since on any given
single scan the signal-to-noise of our measurement is very low - it requires the
integration of dozens of days over many detectors to obtain a reliable measurement of the primary anisotropy of the temperature field.
However, we know that there is a strong azimuthally synchronous ground
pickup signal present in the data that does not have these properties, so we cannot naively Fourier transform our data timestream and determine its noise spectrum without some degree of preprocessing to account for the non-Gaussian,
systematic ground contamination. The ground template subtraction technique
is not suitable for producing long timestreams with coherent noise properties because any given section of azimuth is sampled anywhere from 2 to 20 times in the
creation of the template. The ground template estimates themselves therefore
have quickly varying quality, and when subtracted from the unprocessed time
ordered data create data with incoherent noise properties that are unsuitable
for long timescale measurements of noise properties. This is particularly problematic because atmospheric temperature fluctuations in the microwave regime
have a j-like spectrum that is dominated by long time scale drifts, as can be seen
in figure 3.14.
Field differencing the data allows us to construct noise spectra that are free
from ground contamination while retaining constant noise properties over the
whole of a 6.5 minute set of 10 half-scans. We therefore construct noise power
spectra on a 6.5 minute scale for a single full scan and its field difference partner
using the power spectrum of their difference. Although we could obtain a higher
signal to noise measurement of the noise power spectrum by integrating over
several scans, using a single scan and its field difference partner limits the time
scale on which we believe the noise properties to be stationary and Gaussian to
36.5 minutes.
Noise simulations
3.5.2
106
Noise generation
100 GHz
,—r
10000.01
150 GHz
"
10000.0
1000.0 k
1000.0
100.0 k
100.0
10.0 k
10.0
1.0 k
0.1
0.01
0.10
1.00
Frequency (Hz)
10.00
0.01
0.10
1.00
Frequency (Hz)
10.00
Figure 3.15: 100 and 150 GHz atmospheric and bolometer noise spectra from simulations
based on March 29, 2006 observations and averaged over all operating channels within
a frequency over an entire simulated observation day. This plot is directly comparable
to the spectra shown in figure 3.14. Y-axis is power spectrum density (PSD) in units of
nanovolts per root Hz. PSDs derived from temperature sensitive pair sum timestreams
in black, PSDs derived from polarization sensitive pair difference timestreams in red.
Note that 60 log-space frequency bins are used to estimate the noise spectra, except at
the lowest frequencies where each individual mode has its own bin. Effects of binning
can be seen in the broadening of microphonic spikes and low-pass drop-off tail at higher
frequencies. Note that relevant science band is between 0.1 and 1 Hz.
In practice the noise from bolometers across the focal plane is somewhat correlated because although the beams are separated on the sky they are nonetheless viewing similar parts of the atmosphere. In addition, small thermal fluctuations on the focal plane couple directly into signal drifts in all channels. As a
result, we measure the unnormalized "one sided cross-spectrum":
107
Noise simulations
di(u) = T{d(t)}
P«M - Jl^f^l
y
(3.47)
(3.48)
1 * J samp
where the indices i, j refer to individual channels within a frequency of the focal
plane. The noise spectra are then binned into 60 log-spaced bins:
Bijk = J2b"kP^)
(3-49)
uik
where bwk is a binning matrix averaging the noise spectra that is zero except in
those rows and columns where bin k and frequency UJ match and the corresponding elements is 1/Nk, the number of frequency elements averaged into that bin.
To produce noise simulations, we assume that the measured cross power spectrum is an un-biased measurement of the variance of the magnitude of a Gaussian distributed, complex Fourier mode. For example, for a single detector, we
would generate an FFT mode from our measurement of its auto-spectrum Pu{ui)\
buk = yBuk
(3.50)
r
d'i(u) = bUk(N(0,v L/2) + iN(0,^/lj2))
(3.51)
where N(0, y/T/2) are random draws from a normal distribution, andrf-(w)is an
individual simulated Fourier mode that be run through an inverse FFT to yield a
simulated timestream.
In the context of a full covariance matrix, Bfc for each frequency bin across
the focal plane the equivalent operation to the square root is the Cholesky decomposition, Lfc such that LfeLf = B^. We therefore generate a set of simulated
noise timestreams by drawing a iVCftan„e/s-long vector of random complex numbers with an average magnitude of 1 and multiplying this against the Cholesky
decomposition:
Noise simulations
108
r = N(0, y/lj2) + iN{0, y/lfi)
d'(u) = Lfcf
d'iit) = T-\d'M)
(3.52)
(3.53)
(3.54)
where in the last step the simulated timestream for detector i, d-(i) is recovered
through an inverse FFT transform of the full set of a single detector's simulated
Fourier modes d'i(u).
Although in principle a physical covariance matrix should be positive definite
with only non-zero, positive and real eigenvalues, due to the limited number of
samples of each element in the covariance matrix at low frequencies where the
log-spaced bins often allow only one sample per bin, our estimates of the frequency bin noise covariance matrix can often be non-positive definite. Since the
Cholesky decomposition requires that the input matrix Bk be positive definite
this can pose a problem.
To correct noisy estimates that result in non-positive definite covariance matrices, we use LAPACK to eigendecompose the matrix, find any negative or complex eigenvalues, set them to 1% of the minimum positive eigenvalue, and reconstruct the positive definite corrected matrix Bk = QAQT where Q is a matrix of the eigenvectors aligned in columns and A is a diagonal matrix of corrected positive eigenvalues. This results in sub-percent level changes in the reconstructed covariance matrix because the original matrix is dominated by the
unaltered modes with large positive eigenvalues to begin with.
3.5.3
Noise simulation results
The simulated noise timestream d[(t) can be propagated through the same mapping and spatial anisotropy analysis as the normal data detailed above. We can
therefore create noise-only maps, which are converted to noise-only spectra, which
are then averaged over 416 simulations to produce the estimate of < Ne > and
Signal-only simulations
109
Figure 3.16: 150 GHz simulated noise-only temperature map integrated from 143 days of
simulated data. Noise is dominated by azimuthal striping due to constant elevation scan
strategy.
subtracted from the measured data spatial anisotropy spectrum. The scatter of
this set of simulations corresponds directly to the contribution of instrumental
and atmospheric noise to the covariance matrix of the bandpowers and the plotted errors.
3.6
Signal-only simulations
We recover the beam and filter correction factor F^Bj as a function of multipole t
through many signal-only simulations in which we convolve realizations of temperature and polarization fields constructed from a fiducial spectrum with our
Signal-only simulations
0
500
110
1000
1500
2000
2600
0
500
1000
I
BB
eo r
0
'
•
'
•
•
•
500
•
>
500
•
•
1500
•
•
2000
'
2
2SO0
2°°^
0
'
'
500
'
'
'
2500
'
'
1000
1500
2000
2500
1500
2000
2500
_JB
1000
2000
TE
•
1000
____________^_
0
•
1500
I
'
'
'
' 3
EB
1500
2000
2500
0
500
1000
Figure 3.17: Computed noise bias of 150 GHz spatial anisotropy spectra (red) and set of
416 simulated noise-only spatial anisotropy spectra (gray). Noise bias is computed from
mean of simulations in each bandpower, and random noise contribution to final error bars
and covariance matrix is computed from scatter of simulations.
instrumental beam and resample them into flat sky maps using QUaD's scan
strategy and timestream filtering scheme.
Although in principle any non-zero spatial anisotropy input spectrum would
suffice because the beam smoothing and filtering processes are linear in both
map and spatial anisotropy spectrum space, in practice we use the 5-year WMAP
temperature, EE-mode and TE-mode spectra (53) as inputs into the signal-only
simulation process. These spectra are used in conjunction with a modified version of the "synfast" utility of the HEALPIX software package to generate partialsky map realizations (54). If we use an input spectrum Ce, then HEALPIX generates a random realization of normally distributed aem such that:
111
Signal-only simulations
5.0
5.2
5.4
5.6
5.8
6.0
Right Ascension
Figure 3.18: 150 GHz simulated signal-only temperature map integrated from 143 days
of simulated data.
< aem > = 0
<ajm> = Ct
(3.55)
(3.56)
The "synfast" utility then converts these aim into sky realizations on constant
elevation rings tessellated on a spherical surface. The HEALPIX resolution scheme
is parametrized by an Nside value that is typically a power of 2, with Npix = l2N^ide
pixels. The areas of the pixels are equal and are arranged in rings of constant
latitude. In order to avoid pixelization effects we generate CMB realizations in
HEALPIX at the very high resolution of Nside = 8192, corresponding to pixels
112
Signal-only simulations
roughly 0.006° wide.
Only the subset of rings needed for QUaD's declination range of —45° to -55 c irc
is computed in order to save time and disk space, as Nside = 8192 corresponds to a
roughly 6.4 GB full-sky map in memory. Simulating QUaD's full observation field
requires only about 1% of this total amount.
The tessellated spherical HEALPIX maps are then interpolated to a flat-sky
map with 0.012°-wide pixels. The beam parameters derived from the Monte Carlo
routines described in Chapter 2 are then used to generate flat-sky beam maps for
each detector:
B(ra, dec) = exp
x
+
((ra — rao) cos (p — (dec — dec0) sin 0)5
((ra — rao) sin <j) + (dec — dec0) cos 0)2
2^f
wS(ra — ra0, dec — dec0)
where ra0, dec0 are the centers of the QUaD observing field and aa,ab,cj) and w are
the detector elliptical Gaussian and sidelobe template strength parameters, and
S(x, y) is the sidelobe template computed from the physical optics model. This
centered, simulated beam is then convolved with the simulated map, M(ra, dec)
in both temperature and polarization by using FFTW to compute:
M'(ra, dec) = T~x {T[B(ra, dec)}T[M(ra, dec)]}
(3.57)
The simulation process creates timestream samples from these simulated maps
for each detector, then performs ground template and polynomial fit subtraction
on the timestreams equivalent to the normal data. The simulated, filtered signalonly timestream is then accumulated into maps in the same manner as the data.
A sample map from one of the hundreds of signal-only simulations used to compute the beam and filter suppression functions can be seen in 3.18.
Signal-only simulations
3.6.1
113
Band power Window Functions
Figure 3.19: 2D Fourier space representation of numerical bandpower window function
calculation. Annular rings of complex numbers with unity magnitude and random phase
comprising | of a bandpower in t space are injected into an otherwise empty Fourier space
representation of the QUaD field (left image). This Fourier map is inverse transformed to
normal map space, where it is windowed and apodized by the QUaD inverse variance weight
map. The Fourier transform of this windowed map is then taken, and the distribution of
power due to the weight mask is measured in Fourier space (right image in logarithmic
scale).
Computing the filter/beam transfer functions from the signal-only simulations on a bandpower-by-bandpower basis requires that we have some means to
convert the fiducial input spectrum into a set of bandpowers. We cannot simply
use the binning operator Pu from equation 3.37 due to our non-full sky coverage
which couples power into the bandpowers from adjacent multipole modes. We
therefore need to numerically compute the "bandpower window functions" Wu
that convert a theoretical spectrum into theoretical bandpowers and depends
solely on the sky weight mask used in map space:
^(theory
/
J
e
WbeCetheory
(3.58)
Signal-only simulations
114
TT150
II1"'
iM!
" nil'I
..JtiJJ
1^
Iblu
:
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
500
1000
1500
2500
3000
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
1
2000
0.4 |
500
1000
1500
2000
2500
3000
500
1000
1500
2000
2500
3000
Figure 3.20: Bandpower window functions for 30 QUaD bandpower spaced at Se = 81.
where we again use the midpoint £ to index the bandpower C(theory instead of
its associated bandpower index b. Although it is possible to derive the bandpower window functions analytically from the spherical harmonic transform of
the weight mask (24), we instead use a numerical approach that takes advantage
of the existing analysis software. Under the assumption that the spectrum-tobandpower conversion is linear we can compute the window functions by computing the window function individually for each multipole. Since we are multiplying the mask and map in real space, in multipole space this is a convolution:
M(u,v)
Wu
= T
oc
a2(ra, dec)
p
Yl
£> =
w I du'dv'M{u -u',v-
^u2+v2
(3.59)
v')8[2iry/u2 + v2 - £} (3.60)
Signal-only simulations
115
Due to the size of the two dimensional Fourier transform and the number of
modes that need to be computed, performing this computation for every point
in u, v space in both temperature and polarization is not feasible. Furthermore,
each point in u, v space requires that we choose a phase associated with that
mode. In order to simplify the computations, we set all pixels in u, v space within
an £ range | the size of a full bandpower to unity magnitude in power with random phase, as shown in figure 3.19. This Fourier space map is then inverse transformed back to normal space, multiplied with the inverse variance weight mask,
and then forward transformed back into 2D Fourier space and multiplied against
the Fourier space signal-to-noise weight mask. The contribution of this fraction
of a bandpower to each of the final 30 bandpowers is then measured. We therefore measure the unnormalized Wbe over N realizations as:
1
N
T
2=1
e=Vu2+v2
T-x{Ru{u,v)}
l
a- (ra,dec)
K(u,v)
(3.61)
where R^t{u, v) is a ring of unit magnitude and random phase in Fourier space
with radius I and K(u, v) is the Fourier signal-to-noise weight mask.
Due to the random nature of the phases, a robust measurement requires that
we average over many different realizations of each ring. In practice the bandpower window functions converge within N = 10 iterations, but for the final set
we use iV = 40 iterations. The samples taken at 8 times the resolution of the
bandpower spacing are then interpolated to yield the final bandpower window
functions seen in figure 3.20.
Polarization adds a small degree of complication to this measurement. TE
cross-spectra are trivially calculated by this technique by injecting equivalent
rings of power into both the temperature and E-mode Fourier planes and measuring the power of the cross spectrum:
Signal-only simulations
TE
W,
1
116
N
PiU
i=1
2
T
ax2 (r a, dec)
2
e=Vu +v
T
T'1
{RJ^E{U,V)}
2
(TE {ra,dec)
K(u,v)
K(u,v]
(3.62)
The flat sky approximation and sky cut convert a small amount of E-mode
power into B-mode power on large scales. To account for this effect, rings of
power are injected into the E-mode Fourier plane, and then measured in both
the E and B-mode Fourier planes at the end of the process. After measurement,
the total contribution to each bandpower is normalized to unity power, and the
EE and BB bandpower window functions are normalized together.
3.6.2
Beam and filter transfer function
500
1000
1500
multipole moment I
2000
2500
Figure 3.21: Beam/filter transfer functions computed from signal-only simulations for 100
GHz (red), 150 GHz (blue) and cross-frequency temperature spectra (black).
The bandpower window functions allow us to convert the five-year WMAP
best-fit model spatial anisotropy spectra that we have used as inputs into the
Absolute Calibration
117
"synfast" facility to bandpowers that are directly comparable to the spectra obtained from signal-only simulations. We can compute the beam and filter transfer function:
F{B2
=
< Clsignal-only
>
(3_g3)
^^theory
In practice there is a noise in the observed transfer function at the 5% level
and we smooth the transfer function by fitting an eight-order polynomial to the
observed points. The derived transfer functions are shown in figure 3.21. At low
multipoles before the peak the transfer function is dominated by the effect of the
ground subtraction, and at 500 < £ < 1000 the transfer function is suppressed by
the effect of the beam sidelobes and the roll-off at high I is due to the Gaussian
mainlobe width of the beam.
3.7
Absolute Calibration
Figure 3.22: Repixelized flat sky BOOMERanG calibration (left) and reference (right)
maps made using scan strategy and coverage of QUaD 150 GHz detectors. Higher signalto-noise upper half of the field is cross-correlated against QUaD to derive absolute gain.
One important caveat to the relative gains technique for QUaD detailed in the
previous chapter is that we have no absolute measurement of the power incident
on the bolometers. Unfortunately due to QUaD's location at the South Pole and
Absolute Calibration
118
Absolute calibration from B03, QUaD 100 GHz
200
400
600
800
multipole moment I
1000
1200
Absolute calibration from B03, QUaD 150 GHz
8x10 s
6x10 s
-
:
_.
A
:
4x10 s
•/-
2x10 s
-
-
.
.
.
.
.
.
.
.
x
^
457580. ±4944.03 pK 1 volt
0
!
200
|
400
!
600
800
multipole moment I
!
1000
1200
Figure 3.23: 100 GHz (top) and 150 GHz (bottom) compute absolute calibration factors
as a function of multipole moment £. Each bandpower yields a nearly independent measurement of the overall absolute calibration factor, which is averaged from 200 < £ < 800
in the region of QUaD and BOOMERanG's scale overlap in sensitivity to temperature
anisotropies. Computed overall calibration factors are shown, as well as errors from random noise as computed by the scatter of measured points. Unnormalized QUaD beam
suppression (blue) and BOOMERanG 150 GHz beam suppression (green) shown.
Absolute Calibration
119
the height of the ground shield, planets like Jupiter or Neptune along the ecliptic are too low in elevation for QUaD to map and use as external calibrators like
other microwave experiments. We therefore perform absolute calibration using
the overlap of our field with the BOOMERanG balloon experiment's measured
temperature CMB field (42), which is in turn calibrated off the WMAP satellite
(55). This chain of calibration is necessary because WMAP is most sensitive to
fluctuations at £ < 100 due to its large beam and full sky survey, while QUaD's
small field, long integration time and relatively small beams make it most sensitive to £ > 300. BOOMERanG's sensitivity bridges this gap.
The cross-frequency measurements 100 and 150 GHz correlation and crossspectrum temperature-polarization correlation spatial anisotropy spectra show
little evidence of bias because of the uncorrelated nature of their noise. Similarly,
the BOOMERanG experiment provides maps of their primary CMB field made
from two sets of detectors on their focal plane with almost totally uncorrelated
noise and the cross-correlation of these maps has insignificant noise bias. With a
QUaD temperature map in volts and two calibrated BOOMERanG maps in Kelvin,
we can therefore compute an unbiased gain factor in spatial anisotropy space as:
„. _
^ uim a£m
^
QUaD „ B-2 >
rg g^l
where af^1 and af^2 correspond to the multipole moment coefficients from the
first and second BOOMERanG maps respectively and af^aD is from the measured
QUaD temperature field at either 100 or 150 GHz.
Since we work in the flat-sky Fourier space approximation, we instead compute the equivalent factor from the two dimensional Fast Fourier Transforms of
the respective repixelized flat sky maps:
_ _ <fB-^u,v)f*B-\u,v)>
<fQUaD(u,v)f*B
(3 _ 65)
2
{u,v) >
where the usual conversion of £ = 2TT\U\ is made between the equivalent Fourier
modes and multipole moments.
Absolute Calibration
120
In order to get the BOOMERanG maps comprising a much larger field to be
directly comparable to QUaD, we repixelize the provided maps using the same
simulation pipeline that is used for signal-only simulations detailed above. A
simulated signal-only timestream is created by sampling off the BOOMERanG
maps, filtered in the same manner as normal QUaD data, and then accumulated
into maps with the same pixelization and field size as those used by QUaD.
However, these equations assume that both instruments have identical beam
suppression and suppression at higher multipoles £ due to pixelization effectively
smoothing the map at small scales. As the QUaD 0.02° pixels are far smaller than
the nside = 1024 HEALPIX pixels used by the BOOMERanG maps, we only need
to correct for the BOOMERanG pixel window function we- Both the pixel window
function and the beam suppression are provided to use by the HEALPIX software
and BOOMERanG collaboration respectively; QUaD's beam suppression bfUaD
can be easily measured in the absence of filtering by simply injecting a single
"hot" pixel into a map of QUaD's full field, convolving it with our beam model,
and measuring the spatial anisotropy spectrum. The corrected absolute calibration gain factor taking these effects into account is:
<fB-1(u,v)f*B-\u,v)>b?UaD
9e = —z UaD
r-oZo
< TQ (u,v)T*
\u,v) > bfOOMERanGwe
.„3 _„
( - 66 )
Once we have the corrected calibration factor by multipole ge, if we assume
that the absolute gain factor should be constant as a function of £ we can average
ge bandpowers in the multipole range where we can reasonably believe that there
is sensitivity overlap between QUaD and BOOMERanG, 200 < £ < 800:
800
9 = Y, 9e
(3-67)
£=200
We can also determine the errors on our gain measurement by looking at the
scatter of these values:
Systematics tests
121
800
A^=,
Y.^-~9f
(3-68)
\ £=200
ge for 100 and 150 GHz is shown in figure 3.23, and the computed value at 100
GHz is 548579 ± 13487.4 //K/volt, and at 150 GHz is 457580 ± 4944.03 //K/volt.
These calibration factors can then be used to scale the noise-only simulated maps
and data maps appropriately and derive the final spatial anisotropy spectra.
3.8
3.8.1
Systematics tests
Jackknife subtraction maps and spectra
Splitting our data into two nominally equivalent halves, processing them separately and equivalently to our normal process should result in null "jackknife"
spectra composed only of noise content. This process tests for possible systematic errors in our analysis that can result in false measured signals. In addition,
since the jackknife auto-spectra also have a noise bias that is roughly twice that
of the normal data, we can test our noise simulation model by verifying that simulated jackknife auto-spectra resemble that measured from data.
The four jackknife tests that QUaD uses are:
• Scan direction jackknife - QUaD's scan strategy consists of constant elevation raster scans. We can therefore split the data into one half for left
traversing scans and one half for right traversing scans. This is a fairly sensitive test of proper deconvolution of the timestreams.
• Split season jackknife - QUaD's 143 days of observation can be split into two
nearly equal halves of the first 72 and the last 71 days. Differencing these
two halves is a test of absolute gain drifts in the focal plane array or other
long term changes in detectors.
• Focal plane jackknife - QUaD's focal plane can be rotated on the sky such
Systematics tests
122
Right Ascension
Figure 3.24: 150 GHz temperature deck jackknife map created by splitting QUaD data
into two halves measured with the receiver rotated 60 degrees with respect to one another.
The QUaD observation strategy observes the first and second "deck" rotation angle for 8
hours each with one transition daily. Near perfect cancellation of the CMB temperature
field signal seen figure 3.5 is observed.
that half of the designed detector pairs are sensitive to the Q Stokes parameter exclusively and the other half are sensitive only to the U parameter. However, in the 100 GHz channels uneven attrition of working detector
pairs results in more detectors in one half than the other.
• "Deck" (dk) rotation jackknife - the DASI mount that the QUaD telescope
occupies allows for rotation of the receiver along the boresight axis. The
receiver orientation is restricted by the angle required for the cryogenic refrigerator to operate properly. QUaD observes at two dk angles every day for
Systematics tests
J
QEO
123
;
«
6
s
. • S M
* « * « .
500
1500
2000
•
i5
tt
i
o[e.«ii»»«»«tti|S|i|^n}»'»*>i»i
} } { ,
}
{
^
1500
2000
1500
2000
A
.J
1500
2000
II
U
1500
/n/'i^i'u
2000
Figure 3.25: 150 GHz deck jackknife spectra (red). WMAP 5 year best-fit model and
equivalent QUaD bandpowers in blue. Note near perfect cancellation of signal consistent
with simulations (grey).
equal amounts of time separated by 60 degrees. Due to the thermal fluctuations that occur when the receiver is rotated, only one change between
the dk angles occurs each day. This test is particularly important for testing
polarization systematics because the polarization sensitivity orientation of
every bolometer changes on the sky. It is also likely the most difficult jackknife test to pass due to the long time scale (8 hours) between the different
dk angles and the systematically different portions of azimuth range that
with respect to the South Pole geography for each dk observation. The 150
GHz deck jackknife map is shown in figure 3.24
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Systematics tests
125
100GHzTTDOF;28
100GHzEEDOF:28
100QHzBBDOF:2B
100GHzTEDOF:28
100GHzTBDOF:28
PTE dist: 0.010
PTEdist: 0.264
PTE dlst 0.674
PTE dist: 0.761
PTE dist 0.306
PTE dist: 0.703
PTE sims: 0.106
PTE aims: 0.260
PTE sims: 0.6S3
PTE sims: 0.750
PTE aims: 0.315
PTE sims: 0.709
10QQHzEBDOF:28
Cross ET DOF; 28
150
100
ft
A
HI
..,
20
40
60
80
100
A
40
/I
/
eo
do
100
SO
«
60
SO
100
/
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PTE sims: 0.089
100
„
,1s..,
40 eo
PTE dist: 0.090
too
150GHzTTDOF:2B
/
\
.
V,
A
/ '
40
80
BO
100
Cross BT DOF: 28
Figure 3.26: Scan direction jackknife \2 statistics from data (red line), suite of 416
simulations (histogram) and theoretical \2 distribution for 28 degrees of freedom for all
spectra. Null test for scan direction systematics like deconvolution failure pass when data
falls within histogram of simulations. Probability to exceed (PTE) is shown for the data
based on empirical distribution of simulations and theoretical \2 distribution.
3.8.2
Quantifying null-signal difference tests
To formally test whether our observed jackknife spectra are consistent with zero
and also those spectra derived from signal and noise simulations, we can define a
X2 statistic for the jackknife bandpowers from the bandpower covariance matrix,
MUi as observed from the scatter of the signal and noise simulations:
X — 2^{^n ~ Ce2)Mu>(Ci>1 — CV2)
(3.69)
LV
where Q a and Q 2 are the spectra derived from each half of the jackknife data.
Ideally the difference of these two spectra are zero on a per-bandpower basis and
Final spatial anisotropy spectra
126
therefore the resulting x2 values should follow a x2 distribution with degrees of
freedom equal to the number of bandpowers used in the statistic. In practice,
the simulations "know" about some of the non-idealities that result in non-null
statistics and instead of comparing the x2 statistic from the data to a theoretical
X2 distribution, we compare it to the ensemble of x2 statistics from our simulations to verify whether our data could reasonably be drawn from the simulations.
Figure 3.26 shows schematically how this process works for all the spectra using the scan direction jackknife. A x2 statistic is computed for each simulation
based on the bandpower covariance matrix of the rest of 416 simulations based
on the upper 28 bandpowers. The first 2 bandpowers are removed from consideration due to the large suppression induced by timestream filtering. Almost
all of the spectra pass, although the low probability to exceed (PTE) values of
the 150 GHz and cross-frequency temperature spectra indicate that there is extra power in these spectra beyond what could be predicted by signal and noise
simulations. This is likely because the signal-to-noise of the temperature spectra
is so high that even very minor analysis problems can create spurious jackknife
power from the relatively bright temperature signal. For example, a small error
in the measurement of the time constant of the detectors used to determine the
bolometer response function used for deconvolution will have a much greater
effect in the high signal-to-noise regime of the temperature spectrum than the
polarization spectra. Figure 3.25 shows that although the temperature spectra
sometimes fail the formal x2 tests their absolute power is visibly nearly consistent with zero when compared to the sky signal. Table 3.1 details the data PTE
from x2 statistics for all spectra and jackknives - note that there are no failures in
any of the polarization spectra.
3.9
Final spatial anisotropy spectra
Once we have verified that our simulation process produces a reasonable set of
realizations that the bandpowers we derive from data are consistent with in jackknives, we can derive the spatial anisotropy spectra from the data using our suite
Final spatial anisotropy spectra
127
of simulations to correct for the noise bias and filter/beam suppression. In addition, the empirically measured scatter of the addition of the signal and noise
simulations provides us with the error bars for each bandpower, as well as the
full covariance matrix between all the bandpowers. This covariance matrix can
then be used to constrain cosmological parameters.
i
H#
o[8 « < » » * j i i
*Tt
^ T i
500
1000
Figure 3.27: Final 100 GHz temperature and polarization spatial anisotropy spectra
Final spatial anisotropy spectra
128
I H
a
:
i i 1
i
L__.il .111J
1000
n,.
°|
-_ j , , ? . i-j-i-il .
i'l"!)'!!-"
{
|
} ii i* * * _ * { {-
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|
BB
K
i
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l I
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;
-
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200
SI
100
L
r\
k flf
1 • SJ
^
-100
'-•
vA A
V\\\y
\ 1/
j
,
U,
XT fk^yM^r^
- . 11__ i
j-r-j-f-^
i
• ^ . M ^ M ^ L :
Figure 3.28: Final 150 GHz (top) and cross-frequency (bottom) temperature and polarization spatial anisotropy spectra
Final spatial anisotropy spectra
3.9.1
129
Frequency combined spectra
..•••«
4*
§
i ' t M l
»»fitiif-ijijj lll'-H
)
j
H H " ,
! ) )
t
)
M u t
) (
j
'••i"}i}ilil
TT
t+
Figure 3.29: Final combined spatial anisotropy spectra utilizing information from 100 and
150 GHz auto-spectra as well as cross-frequency spectrum.
Each bandpower has three separate frequency measurements for the 100 and
150 GHz auto-spectra as well as the cross-frequency spectrum. When normalized
for the differences in blackbody emission at the various frequencies the anisotropies
should have an equivalent spectrum shape, although the signal to noise will be
higher at 150 GHz due to the greater number of detectors and closer proximity to
the blackbody emission peak.
The covariances between the spectra offer a natural means with which to optimally weight the three measurements of each bandpower in order to create a
combined spectrum. For each bandpower we can define a 3 x 3 covariance matrix across frequencies, Qtj. The optimal weighted bandpowers is then:
Limits on gravitationally lensed polarization
130
C / = ^VQV1Cei
(3.70)
where i, j run over the frequencies 100 GHz, 150 GHz and cross. We can perform a
similar operation for every bandpower of every simulation, yielding a covariance
matrix and estimates for the errors on each bandpower shown in the combined
spectrum in figure 3.29.
3.10
Limits on gravitationally lensed polarization
We can use the bandpower covariance matrix specified by the distribution of the
simulated combined spectra to combine all of the BB auto-spectra bandpowers
with median multipole I > 200 to form a limit on B-mode power arising from
the gravitational lensing of E-modes. Essentially, we are averaging the value £(£ +
l)CfB for 200 < t < 2500, for the bandpower covariance matrix Qif
C,BBlens- =
E
^
1 C
f
B
(3.71)
Using CAMB, the ACDM expectation of this "single-bandpower" estimate of
B-mode lensing power is 0.05 fiK2. After correcting for E to B mode leakage from
sky cut, the limit we obtain for the frequency combined BB spectrum is < .77 \iK2
(95 % confidence).
Chapter 4
Optimal mapmaking and filter functions
Although the Monte Carlo methods for deriving CMB spatial anisotropy spectra are computationally efficient, the presence of significant anisotropic atmospheric noise in the temperature maps makes a non-Gaussianity analysis very
difficult. For example, looking for weak gravitational lenses as described in section 1.6 by examining the small-scale distortions of the CMB around a lens relies
on examining the map for deviations from a Gaussian field.
Creating an optimal mapper that properly accounted for the noise characteristics of the time ordered data should in principle produce a temperature map
with far less residual noise. The formalism for solving such a map also gives
us the pixel-pixel noise covariance matrix "for free" which can be used to derive maximum likelihood bandpowers. A natural start for a pixel-based nonGaussianity analysis would therefore be a maximum likelihood bandpower analysis that could be directly compared to the temperature anisotropy spectrum derived using Monte Carlo methods.
The purpose of this chapter is to demonstrate an optimal mapping technique
and the creation of data products necessary for a maximum likelihood bandpower analysis from the QUaD data.
Maximum likelihood bandpower analysis motivation
4.1
132
Maximum likelihood bandpower analysis motivation
Maximum-likelihood pixel-based estimators are an alternative method to the
Monte Carlo analysis for deriving the measured bandpowers and errors. Provided we can fully describe the noise properties of the time-ordered data, such
an analysis can produce an optimal estimate of the bandpowers, eschewing the
approximations made in the Monte Carlo analysis of the previous chapter. In
particular, the ansatz used in the MASTER-style analysis that the timestream filtering function operates on a per-bandpower basis in a noiseless is somewhat
worrying:
y-^
^-'(•estimate
^ (measured
-p T-)2
(A~\\
V*-*-J
In reality the filter function can concievably induce leakages in power from
one multipole to another. A trivial way to visualize this effect is to imagine a
timestream that measures exactly one cycle of a perfectly sinusoidal mode on the
sky. When estimated by a fourth-order polynomial, much but not all of the signal is fit, and after subtraction the residual is a dispersed mode of roughly twice
the frequency of the original sine wave. As the most aggressive filter we employ
is a third-order polynomial across 7.5° of one half of the field, this power blending effect is most worrying only at the largest scales in the first bandpower. The
signal-to-noise in the first two bandpowers is typically low due to the aggressive
filtering on these scales and these bandpowers are not reported or used in the
cosmological parameter or parity violation analyses.
The MASTER analysis depends in part on the fact that despite the anisotropic
filtering schemes of ground-based CMB experiments (particularly at the South
Pole where exclusively right ascension synchronous scans are used), the isotropic,
Gaussian nature of the primordial fluctuations are on average suppressed in an
unbiased manner on a per-multipole basis when the timestream is filtered. This
assumption becomes increasingly worrisome when considering hypothetical nonGaussian signals in the QUaD field. For example, if we were to consider weak
gravitational lensing of the CMB by closely examining the QUaD temperature
133
Maximum likelihood bandpower analysis motivation
maps for characteristic lensing distortions, an analysis that did not include a
pixel-based estimate of the temperature fluctuation shears caused by timestream
filtering from those caused by gravitational lenses.
A waypoint on an eventual weak lensing or non-Gaussianity analysis for QUaD's
temperature field would therefore be greatly advantaged by first obtaining a pixelbased maximum-likelihood estimate of the QUaD bandpowers that could demonstrate the usage of a pixel-based methods for correcting for QUaD's filtering scheme.
In broad outlines, let us define the temperature map as a vector of pixels, fh. For
a given pixel covariance matrix C we can define a Gaussian likelihood (56; 35):
^ocicr 1 / 2 exp
1
--m
2
„
^
Cm
(4.2)
where C = C s 4- C N is the addition of the pixel signal covariance matrix C s
and the pixel noise covariance matrix C N . The pixel signal covariance matrix
is derived from a hypothesized spatial anisotropy spectrum (57). To see how to
get from the spectrum to the covariance matrix, first for any real experiment the
measured signal timestream Sj is a convolution of the true temperature map T(f)
with the beam B{r):
I' d2rT{r)Bi{r)
(4.3)
Following Reichardt et al, after timestream filtering we can combine the beam
and filter transfer functions into Fi(r) and define the signal component of a single
map pixel as:
mi=
f d2rT(r)Fi(r)
(4.4)
The elements of the theory covariance matrix derived from the spatial anisotropy
spectrum C? are then (57):
Approximately optimal mapmaking
CT,ij =
(mimjfeaal=
134
I I dPr<Pr'Fi(r)Fj{r')(T(r)T(P))
(4.5)
Or*, = J J dWr'F^F^') J ^Qe^-?)
CT,H = J ^C^-^nH^YHFAr')}
(4.6)
(4.7)
How do we determine the bandpowers? First, parametrize the spatial anisotropy
spectrum Cg by a set of bandpowers, qb. The partial derivative dCT,ij/dqb can be
analytically derived and the maximum likelihood bandpower, ql can be determined through iterative numerical methods by finding dCT,ij/dqb\qb=q* = 0 (35).
What derived data products are therefore required for such an analysis?
• m0, an unbiased estimate of the our map distorted only by the filtering
scheme, optimally weighted to remove noise.
• C N , the pixel space noise covariance matrix derived from the noise properties of the filtered data.
• Fi(r), the beam and filter function for each pixel in the map m 0 .
Although a full maximum likelihood bandpower analysis is not presented here
due to computational and time limitations, the derived data products have been
computed and such a bandpower analysis, as well as future non-Gaussianity studies, are future work for the QUaD collaboration.
4.2
Approximately optimal mapmaking
Recall that the core mapping equation is:
A T N " 1 A m = A T N" 1 (f
(4.8)
Approximately optimal mapmaking
135
where N " 1 is the inverse time space covariance matrix, d is the vector of timeordered data, A is the pointing matrix and fh is the ultimate map. Following the
first ACBAR analysis (35), for experiments that do not require filtering, the optimal minimum variance map is formed as:
fh = Ld
L =
(A T N- 1 A)"" 1 A T N- 1 cT
(4.9)
(4.10)
How do we derive these results in practice? In the Monte Carlo analysis pipeline,
we estimate the noise properties of ground-template subtracted data using the
noise spectra of field differenced data. Assuming that the noise between channels is uncorrelated and that our nearly-circulant time-ordered data noise covariance matrix is totally circulant, then the elements of the inverse noise covariance
matrix just depends on the noise auto-power spectrum, Pu, and the spacing between the points in the timestream (58):
where n is the number of points in the noise power spectrum Pw. N^} is derived
from binning up the auto-spectra of each channel individually over the whole of
half an observing day, or one deck angle, from field-differenced data, and then
taking the inverse Fourier transform of its reciprocal. An appropriate extension
of the spectra white noise level needs to be made to compensate for the low pass
filter and decimation process, so PW(LU > a>Nyquist) = ^L(^Nyquist)- Note that any
symmetric form for N _ 1 is an unbiased estimator for the binned map pixels fh.
However, only the true timestream inverse covariance matrix accounting for all
of the correlations between points at different time slices within the same channel and the cross-correlations between channels produces map pixels with minimum variance and is optimal. For example, in the Monte Carlo analysis for purposes of computational speed we use a diagonal approximation for N _ 1 where
Approximately optimal mapmaking
136
the diagonal elements are the inverse variance of the half-scans. This results in
a perfectly valid weighting of the map pixels but leaves excess correlated noise
between pixels seen in the characteristic horizontal striping in the temperature
maps of figure 3.5.
1x10
2x10
3x10
4x10
Eigenvalue number
Figure 4.1: Ordered eigenvalues of the 150 GHz optimal map pixelized at 3 arcminutes.
After deriving a "semi-optimal" estimate of N _ 1 we can therefore begin to
solve the map equation by using the pointing data to bin the inverse pixel-space
noise covariance matrix C N _ 1 = A T N _ 1 A.
Mapmaking results
137
However in practice this matrix is not sufficiently well-defined and has negative eigenmodes, making it not positive definite and indicating we have insufficient amounts of data to properly estimate the true noise covariance matrix. We
can correct the matrix in an unbiased fashion by setting all eigenvalues under a
cutoff to the same value as the cutoff, which must be positive. Schematically, the
optimal mapping process is therefore:
• For each channel, take the auto-spectra of the field-differenced data. Average the auto-spectra over the 32 field-differenced, single deck rotation
scans of half a day, correct for the low-pass filter rolloff in the white noise
level, and take the inverse Fourier transform of the reciprocal.
• Bin C N _ 1 = A T N _ 1 A, the pixel space inverse noise covariance matrix. Also
bin ATN"1(C the "unnormalized map".
• Eigendecompose CN x —* QAQ T . For a cutoff at a positive eigenvalue,
set all eigenvalues below it to this value, producing A —> A. Invert and
reconstruct the matrix using C N = QA _ 1 Q T .
• Solve for the filtered optimal map m = C N A T N _1 ci
4.3
Mapmaking results
At 3' resolution there are 36298 non-zero pixels at 150 GHz and the same number
of modes in the inverse pixel covariance matrix. Cutting off a successively higher
number of modes clearly improves the noise qualities of the map, although figuring out where to stop seems somewhat arbitrary. There are probably no more
than 10000 good modes in the map, and figure 4.2 shows the resulting map when
25000 of the 36298 eigenmodes are corrected.
The eigenvalue problem currently takes about 7 to 8 hours on an 8-core, 64
GB Linux machine using the multi-threaded version of the Intel MKL.
Pixel filter function
138
Figure 4.2: 150 GHz optimal map with 25000 of 36298 eigenvalues corrected. Map is
binned using 3 arcminute pixels.
4.4
Pixel filter function
Before we can derive bandpowers using the noise covariance matrix and map, we
need to address an important complication - we apply a non-trivial set of timespace linear filters to our data:
d
Ud
(4.12)
(4.13)
Pixel filter function
139
As described in 3.2.4, on a halfscan-by-halfscan basis the slowly varying atmospheric noise component in our data can be well-described by a third-order
polynomial, so with the exception of the ground-template subtraction our filters
can be described as:
d' = (I - X(X T X)- 1 X T )d
(4.14)
= I-X(XTX)-1XT
(4.15)
n
where X is a Vandermonde matrix of the appropriate order. Since the timestream
filtering operations are demonstrably linear we can estimate the filter functions
for each pixel F^r) by simply examining the resulting power from the filter operation for power in each pixel in isolation. Schematically the process is:
• For each pixel in the QUaD field, create an empty map with the exception
of that single pixel with an unnormalized value of 1 in arbitrary units.
• Using the pointing strategy, extrapolate the resulting signal-only timestream
from this one "hot" pixel for the entire of the QUaD 2006-2007 observation
seasons in all detectors.
• Filter the resulting signal-only timestream using QUaD's normal groundtemplate subtraction and polynomial fit subtraction.
• Use the pointing data to rebin the resulting filtered timestream. The result
is the filter function Fi(r).
However, implementing this algorithm in turn for each of the 36298 5' pixels would take roughly 14 years of computational time on a modern single-core
machine. We can exploit significant redundancy in the QUaD scan strategy to
drastically cut the amount of time needed to compute each filter function. In
particular, QUaD's scan strategy simply repeats the same set of 8 full scans (80
half-scans) at four declinations in the two different right ascension fields at different declinations. Each ground template subtraction and filtering process is
Pixel filter function
140
restricted to two of these full scans in each of the two RA fields separated by 30
minutes in time. As a result, in the absence of cross-detector filtering the scan
strategy restricts our filtering to single declinations and therefore single rows of
the map.
Figure 4.3: 100 GHz (top) and 150 GHz (bottom) single row filter function matrix representation for pixels in the central row of map at deck=57. Filter matrix demonstrates
coupling of power due to timestream filtering operations from a high-resolution map corresponding to the long, horizontal axis, to 5 arcminute lower resolution pixels equivalent
to a row of pixels in the map shown in figure 4.2.
We can therefore build up our filter functions by simulating only two full scans
worth of data over 13 minutes at 20 Hz and constructing the filter function for a
single set of scans for each detector along a single row. A small additional complication is that we have to do this twice for each detector based on its position during each of the two instrumental (deck) rotations that QUaD observes
Pixel filter function
141
at throughout the 2006-2007 seasons. The single-row, single-detector filter functions can then be binned using the pointing strategy for each of the declinations
throughout the 2006-2007 seasons. The total process requires less than 12 hours
using non-multi threaded code, and the resulting filter matrix showing the mixing of power from an oversampled, high-resolution map to the 5 arcminute pixel
optimal map used in the previous sections is shown in figure 4.3.
The filter functions are defined using pixels 3 times smaller than the final 5
arcminute pixels in order to avoid aliasing effects. The total beam/filter functions can be derived by convolving the filter-only functions operating on individual rows of the map with the appropriate coadded instrumental beam derived in
chapter 2.
/V z.
Chapter 5
Cosmological parameter estimates
"It is said there is no such thing as a free lunch. But
the universe is the ultimate free lunch."
A Brief History of Time
ALAN GUTH, AS QUOTED BY STEPHEN HAWKING.
5.1
Cosmological Parameters and C / s
The emerging consensus among both observational astrophysicists and cosmological theorists is that the simplest possible model for the universe explained
by known astrophysical data can be expressed in six parameters. This is known
as the A-CDM model of cosmology, which predicts that the universe is flat, its
expansion history is dominated by dark energy expressed as a cosmological constant A in the present day, and that there is a substantial component of "cold dark
matter" in additional to our normal known baryonic matter.
The spatial anisotropy spectrum in both temperature and polarization, as well
as the TE cross-correlation, can be predicted exactly by solving the differential
Boltzmann equation to propagate perturbations in the temperature field from
primordial fluctuations to the epoch of recombination, for a given set of cosmological parameters that describe the nature of the early universe plasma. This
Cosmological Parameters and C / s
omegabaryon h
143
omegacdm h
omegabaryon h
omega_cdm h
500 10001500200025003000
10
10.000
1.000
0.100
0.010
0.001
10
100
1000
10
100
1000
7000[~
6000 [
5000|
4000 [
3000 k
2000 k
1000 P
500 10001500200025003000
tau
\
amplitude
7000
6000
5000
4000
3000
2000
1000
tau
100
1000
amplitude
10.000
1.000
0.100
0.010
0.010
0.001
0.001 L
Figure 5.1: Dependence of temperature and polarization spatial anisotropy spectra on
each of ACDM parameters for flat, non-running spectrum 6 parameter model. Black line
is best fit model for WMAP 5 year data. Red and blue lines signify spatial anisotropy
spectrum when a given parameter is increased and decreased respectively. Physically
relevant linear combinations of parameters shown from top are baryon density, cold dark
matter density, Hubble constant, primary fluctuations spectral index, optical depth to
reionization, and primordial fluctuation amplitude. Y axes are *• ~^J e in \iK2, X axes are
multipole moment t. All spectra were generated using CAMB (19).
computation can be drastically sped up by numerically integrating two terms
along the photon past light cone from the epoch of recombination back to the initial primordial scalar fluctuations (59), and was originally embodied in a program
called CMBFAST that allowed relatively rapid computation of predicted spatial
anisotropy spectra to arbitrary precision for non-closed universes.
The ideas in CMBFAST were later improved and computations sped up by an
order of magnitude by the authors of CAMB, the software of choice used by most
experiments to derive anisotropy spectra from cosmological parameters today
(19). Although there are a dizzying array of parameter options available in CAMB
Cosmological Parameters and C / s
144
reflecting the varied nature of cosmological theories today, for this analysis we
focus on a minimal set. The combinations of parameters have been chosen to be
• Qb - the "baryonic" or normal matter density at the current day. Increasing this value tends to boost the odd numbered peaks in the temperature
spectrum.
• Vtcdm - cold dark matter density at the current day that acts gravitationally like normal matter but does not interact through the other forces like
electromagnetism. Increasing this value tends to boost the even numbered
peaks in the Ct temperature spectrum.
0
JJ0 = da^t _ ^ e Hubble constant, describes the current expansion rate of
the universe. Equivalently used in this analysis with h = H0/100. HQ1 is
essentially a distance scale at the surface of last scattering in the context
of a flat universe and changing its value shifts the peaks left and right by
changing the angular size of fluctuations on the sky.
• ns - the spectral index of primordial scalar fluctuations, such that the primordial scalar power spectrum scales as A2s(k) = (^r)n"_1. A spectrum with
ns = 1 means that the initial scalar perturbation power does not depend
on scale, or is "scale-invariant." A near-scale invariant spectrum with ns
slightly less than 1 is a natural prediction of slow-roll inflation (16). However, a scale-invariant spectrum is also known as a Harrison-Zel'dovich spectrum since the idea's origination in the early 1970s (60; 61). Increasing this
value adds an overall upward tilt to the spectrum.
• T - optical depth to reionization. A consensus has begun to emerge that
light from the first metal-poor population III stars reionized the intergalactic medium around redshift z ~ 6. This dampens the temperature anisotropy
on scales that are causally connected at the time of reionization, roughly
I > 100, and also induces a small amount of polarization signal via Thomson scattering of the primary quadrupolar temperature anisotropy off the
newly freed electrons at the epoch of reionization (22).
Cosmological Parameters and Cg's
145
• A = log(1010As) - the amplitude of the initial scalar perturbations at the
reference scale k* - acts as an overall normalization constant on the entire
spectrum.
However, we would like to use a set of parameters that is roughly orthogonal
and linear with respect to small deviations in the spatial anisotropy spectrum. We
therefore use the "physical parameters" Qbh2 and SlCdmh2 for the matter densities,
which decouples the effect of changing these parameters from the distance scale
expressed as h (62). Some degeneracies cannot be avoided - as can be seen in 5.1,
A and r are nearly totally degenerate at I > 100, which is also the scales on which
QUaD is most sensitive.
While earlier experiments in the field resorted to "grid-based" searches of the
parameter space using weeks of time on supercomputing clusters to find the best
fit of cosmological parameters to their data (63), modern experiments have recognized that Markov Chain Monte Carlo (MCMC) techniques are ideal for divining cosmological parameters while running a minimal number of instances of
relatively slow spectra generating programs like CAMB. The main difficulty is that
the calculation the spatial anisotropy spectrum at an appropriate level of accuracy is relatively computationally slow when considering the millions of points
that would need to be investigated for a grid-based parameter search approach.
This problem is of course exacerbated if ones consider more parameters than the
standard six, such as the scalar to tensor ratio or running of the spectral index.
MCMC techniques for searching the parameter space can reduce the number of
points needed to explore the posterior likelihood distribution of the parameters
from millions of CAMB instances to tens of thousands.
The goal of the following analysis is not to investigate the wide range of alternative cosmological theories that can be constrained with CMB data - exhaustive
analyses by the WMAP team of their five year data show that the six parameter
model can sufficiently describe their data and that there is no significant detections of other parameters. Some alternative parameters include:
• r - tensor to scalar perturbation spectrum amplitude ratio. Tensor metric
Cosmological Parameters and C / s
146
perturbations are couples to gravity waves generated during a hypothesized
inflationary epoch of the universe, and the amplitude of the tensor perturbations is directly related to the speed of expansion H during inflation. The
tensor spectrum is directly coupled to the low-£ BB spectrum, and QUaD's
limited sensitivity at £ < 100 means that the QUaD's polarization contribution to a limit on r is negligible. In fact, the WMAP team has found that
the primary constraint on the tensor/scalar ratio comes from constraining
the tensor contribution to their t < 32 temperature spectrum (13) to which
QUaD has very limited sensitivity. Later in this analysis we will show that
QUaD's high-£ temperature spectrum helps to break a nslr degeneracy and
contributes a small amount of additional constraining power on r above
WMAP.
• K - curvature. The standard six parameter model assumes a flat universe.
Allowing a non-flat model with a seventh parameter for curvature changes
the scale size of the fluctuations on the sky in a manner nearly degenerate
with modifying H0. This parameter is dropped from our analysis as surveys
of type la supernovae (64) and careful measurements of H0 in conjunction
with previous CMB data strongly support a flat or very nearly-flat universe.
• ^ ^ - scalar spectral index running. Various inflationary models predict
deviations from a scale-invariant scalar perturbation spectrum. Any firstorder deviation can be parametrized by ns, and a small second-order deviation can be parametrized by the change of ns as one scales along the
spectrum. The WMAP 5 year team found no evidence for such a running
spectral index, nor have other analyses of QUaD's cosmological parameters
that have searched for beyond 6-parameter model effects.
• Isocurvature mode fraction - most CMB analyses begin with the presumption that the primordial perturbations were entirely adiabatic - that is they
preserved the relative densities of all constituents parts of matter and radiation in the perturbation and thus were equivalent to a fluctuation of curvature. Isocurvature modes are generated through perturbations that affect
Method
147
only a subset of the constituents of the universe and thus lead to changes in
the relative densities at the perturbation. Previous analyses of QUaD have
constrained such modes using polarization data but find no evidence for
their existence (65).
Instead what this analysis endeavors to provide is a demonstration of QUaD's
fit and constraining power to the standard six parameter model with a minimal
set of other astrophysical data. The addition of QUaD's data to the WMAP 5 year
data release is also investigated to determine how much additional constraining
power QUaD brings to the field at large.
5.2
5.2.1
Method
Posterior and likelihood functions
This analysis is derived from the WMAP team's MCMC methodology paper (66).
First, define the posterior distribution for a set of cosmological parameters, a,
given some measured Ce across the whole sky that they denote as Ce and V(d0) is
the prior distribution, using Bayes theorem as:
V(a\Ce) = £(Q|Cf(5))P(« 0 )
(5.1)
CJh(<5) is the spatial anisotropy spectrum as generated by a cosmological simulation program like CAMB from a set of cosmological parameters a. For an ideal
experiment that can measure the whole sky with no noise, we assume that individual aim are Gaussian distributed, and that there is no covariance between
any of the modes, and so the likelihood of a measured aim for a given theoretical
spatial anisotropy spectrum Ce is:
£(f|^)ocn e X P ( H '"^ ( 2 C '"' ) ) .
em
vCe
(5.2)
Method
148
Since the aim depend on the direction we choose for zenith as a frame of reference, and we assume that the universe is isotropic, the a\m can be collapsed over
£to:
21n£ = J ^ ( 2 £ + l )
e
r
/rth\
^
ln(4-)+Ce/Ci
(5.3)
L V Ce J
However, in practice even a full-sky survey like WMAP has to upweight and
downweight portions of the measured sky because of contaminating emission
from nearby galaxies and quasars, as well as different integration times on different parts of the sky. A ground-based experiment like QUaD operates only surveys
about Y^O °f the total sky. The result is that for a real experiment on a limited sky,
the C/s begin to be covarying. One possible way to account for this is to approximate the likelihood as a Gaussian combination of covariates:
ln£ Gauss ex - \ Yj£t
ee'
~ Ct)Mw{Ct - Ce) ,
(5.4)
where Mee> is the inverse of the covariance matrix of the C/s. Different experiments have chosen various methods to derive this covariance matrix. WMAP
computes this semi-analytically by measuring their instrumental effects at various scales and calculating the effective noise this adds to each mode with some
covariance (9). For QUaD's derived MASTER method, we can determine the covariance matrix by using the addition of the suite of 416 signal and noise simulations. If cf1 is the bandpower indexed by its average multipole £ for a single
signal and noise added simulation indexed by i, then our covariance matrix is:
„-._ g (cf-< Q >)«*>-<c,>)
(55)
where N is the number of simulations.
However, the true likelihood function is not exactly Gaussian, and the assumption introduces a slight "downward" bias to the Q (67). Bond, Jaffe and
Knox propose to instead approximate the likelihood with an offset log-normal
Method
149
distribution:
N,
- •t
1
xe
Ze =
rvth
M{z)
(5.6)
BjFj
(5.7)
ln(Ce + xe)
= \n(Clh + xe)
(5.8)
=
(5.9)
{Ce + xe)Mu>(Ce, + xe)
-z?)
•Mog—norm
(5.10)
1
a
where Ne is the noise bias per bandpower and FeBj is the filter/beam suppression
factor, xe, is meant to represent the noise variance contribution to the error rather
than that induced by the underlying signal; an alternative formulation to derive
xe is:
xe = Ce—
(5.11)
where aN is square root of the variance due to noise and as is from the variance
due to signal. In practice we use the mean of the signal only simulations for the
value Ce in this computation rather than the actual estimated bandpower from
data. Note that as usual the index t runs over bandpowers, not true multipoles,
and that Cf1, the theoretical bandpowers against which the data spectrum is compared is computed using the bandpower window functions described in chapter 3. The zero-crossings in the TE spectrum make the estimation of xe on a per
bandpower basis problematic, and we therefore revert to the Gaussian likelihood
when computing the likelihood of a theoretical TE spectrum.
The offset-log normal likelihood resembles the Gaussian likelihood closely
enough that we can define an "effective" x2 value for both of —2 In £.
Method
5.2.2
150
Basic Metropolis used by W M A P
To determine the posterior distribution of a, modern cosmological parameter
analysis relies heavily on MCMC. Essentially, MCMC technique allow us to sample a limited number of points from the posterior distribution in the region of
highest likelihood based on a guided "wander" through this area of parameter
space.
Although the signal to noise at low I for WMAP is sufficient to run a normal metropolis algorithm and converge within 50,000 steps using 4 chains, for
QUaD's lower signal to noise we must modify the algorithm to take into account
the empirical covariance of the cosmological parameters in the high likelihood
region of parameter space. The basic Metropolis algorithm is described in the
WMAP methodology paper (66) - note that I quote directly from the paper here:
1) "Start with a set of cosmological parameters ai, compute the C\th and the
likelihood/:! = C{Cjth\Ce).
2) Take a random step in parameter space to obtain a new set of cosmological
parameters d2. The probability distribution of the step is taken to be Gaussian in each direction i with r.m.s given by a,. We will refer below to at as the
"step size".
3) Compute the C| th for the new set of cosmological parameters and their likelihood £ 2 .
4.a) If£ 2 /£i > 1, "take the step" i.e. save the new set of cosmological parameters
a2 as part of the chain, then go to step 2 after the substitution «i —>• <52.
4.b) If £ 2 / £ i < 1, draw a random number x from a uniform distribution from 0
to 1. If x > £ 2 / £ i "do not take the step", i.e. save the parameter set dii as
part of the chain and return to step 2. If x < £ 2 / £ i , " take the step", i.e. do
as in 4.a).
Method
151
5) For each cosmological model run multiple chains starting at randomly chosen, well-separated points in parameter space. When the convergence criterion is satisfied and the chains have enough points to provide reasonable
samples from the a posteri distributions (i.e. enough points to be able to
reconstruct the 1- and 2-a levels of the marginalized likelihood for all the
parameters) stop the chains."
When marginalizing the parameters the first 20,000 points are typically discarded because during this time the chain is wandering from its random starting
place in the parameter space to the region of highest likelihood. This is typically
called the "burn-in" period.
5.2.3
Modified Metropolis using optimized step size
It step 2 above, the definition of "random step" is deliberately unclear, because
virtually any random proposal can be used as long as the whole of the valid parameter space has a finite, if very small chance, of eventually being proposed
without going through a cyclical period of proposals (this constraint is also known
asergodicity).
Perhaps the most unbiased method is to define a range throughout which
each of our parameters is physical viable and then choose our step size to be
some small fraction of that total range in each parameter, drawing our new parameters from a uniform distribution in an N-dimensional sphere bounded by
that range fraction around our current parameters d?i.
For every step, the proposed new set of parameters is therefore an uncorrected random step with a distance away set by this range. Obtaining "convergence", or a sufficient number of samples from the posterior distribution that
the addition of more steps does not change our sampled distribution, in an optimal amount of time using the Metropolis algorithm usually requires that about
40 - 60% of the proposals in step 4 above are accepted.
However this method's weakness is that in general the constraining power of
our data set and the valid physical range of each parameter have very little to do
Method
152
with one another. Choosing a fixed fraction of the parameter space may result in
steps much larger than the width of the posterior distribution in some variables,
contributing to a low acceptance rate, or a step too small with respect to the uncertainty in another variable, contributing to a high acceptance rate. One way to
get around this would be to tune our proposals to the observed variance of the
posterior distribution during the chain's sampling, but this is in principal a very
dangerous step to make as it may destroy the chain's ergodicity.
For example, if the chain wanders into an area of the likelihood space where
there is a local but not global optimum of the likelihood, and the likilihood surface is steep in one parameter, a "self-tuning" algorithm may inadvertently adjust the step size of the seemingly tightly constrained parameter down so far that
the chain has very little probability of emerging from the local minima and thus
poorly samples the true posterior distribution.
Although using a step size tuned to a some fraction of the parameter range
can accomplish acceptance rates in the proper ranges, in practice this results in
uneven convergence in each of the parameters. One way to combat this is to use
an exploratory chain get a rough idea of the variance of each parameter.
1) Start a single chain at the current best known cosmological parameter values, as published by the WMAP team in their 3 year data. Use the above
algorithm with a step size 8 of about 1% of the flat prior range, as defined
by the physical limitations of each of the cosmological parameters. Run for
20,000 steps.
2) Derive the covariance matrix of the samples of the parameter posterior distribution from the test run of 20,000 steps. Save the Cholesky decomposition of the covariance matrix.
3) Start several new chains at randomly chosen places in the parameter space
as defined by the flat priors. For the first 1,000 steps, derive the proposal
distribution by generating a ra-long vector of normally distributed numbers
with a a of .01 (the variables are all approximately normalized to have prior
ranges of about 0 to 1, except for the amplitude, which ranges from 0 to 5).
Method
153
Then perform the transformation p = EiV„(0, a) where p is the n-long proposal vector, and E is the eigenvector matrix derived from the parameter
covariance matrix and Nn(0, a) is a vector of normally distributed numbers
with mean zero and standard deviation a. This produces a proposal distribution that has roughly equal variance in all parameters since the eigenvectors are orthonormal, but with the proper covariances for each step. Apply
p to the Metropolis stepping algorithm above.
4) After 1,000 steps, the chain is likely in the general neighborhood of the highest likelihood area. To speed convergence, we need to account for the fact
that CMB data heavily constrains some parameters and loosely constrains
others - for example, f2& is constrained to roughly ±.001 while r is contrained
to about < 0.5 within the same prior range numerically. Incorporating the
previously measured variances as well as the covariances greatly speeds
convergence, but doing this too early makes the chain greatly susceptible
to falling into local minima, because fib is not allowed to range very far on
any given step. Therefore, after 1,000 steps, the proposal distribution becomes p = CNn(0,0.3) where C is the Cholesky decomposition of the parameter covariance matrix. This creates a proposal distribution that retains
both the proper covariances and variances of the previously run chain.
4) Run all chains until they satisfy some convergence criteria, described below.
The sampling of the covariance matrix of the parameters is by no means an
accurate representation of the actual width and covariances of the posterior distribution, but the approximate "order of magnitude" estimate allows us to do two
things: 1. approximate the level of covariance among the parameters such that
we can form a orthogonal basis to step in and 2. get an order of magnitude estimate of the uncertainties of the estimated parameters so that we can tune the
step size to speed convergence.
While the WMAP data is well constrained enough that it can converge without incorporating the parameter covariances into the proposal distribution, tuning the step size o^ for QUaD in order to produce a proposal distribution that has
Method
154
an acceptable acceptance rate without falling into local minima in the likelihood
space is extremely difficult. By measuring the covariances of the parameters directly in a "pre-run" we can consistently run converging MCMC chains without
having to fine-tune the proposal distribution.
5.2.4
Priors
It is typical within cosmological parameter analysis to use flat priors, in part due
to a bias in the field against ruling out any portion of the parameter space that
is explicitly physically disallowed. If a parameter outside the flat prior range is
proposed the proposal is discarded and a new set of parameters drawn without
adding a point to the chain. Although in the Metropolis algorithm the current
parameter set would be added to the chain again, empirically this behavior prevents the chain from spending sampling excessive numbers of points at the edge
of the prior in those cases where the QUaD data cannot provide more than an
upper limit. For example, QUaD has no detection of r, just an upper limit, and as
a result the sampling distribution does not tail off to zero at the end of the prior
range.
The priors used for this analysis are similar to those used by the WMAP team,
but are expanded to accommodate the weaker constraints on r and amplitude
from QUaD:
• 0.0 < Qbh2 < 0.25
• 0.0 < ncdmh2 < 0.5
• 0.48 < h< 1.0
• 0.5 < ns < 1.5
• 0.015 < r < 0.75
• 2.0 < log{1010As) < 4.5
Method
155
There is also a hidden prior imposed by CAMB as a result of the "physical
parameters transformation" transformation, Qb —>• Q,bh2,Q,cdm —> Qcdmh2. CAMB
rejects as unphysical values of Qb > 1.0 or ilcdm > 3.0, so if h ranges to too small a
value this additional prior can be triggered.
5.2.5
Convergence Criterion
c
!Q
3
DC
•
cCO
E
©
2X10 4
4x10
6x10
Chain step
8x10
Figure 5.2: Gelman Rubin convergence statistic f of ACDM 6 parameter fit for QUaD
data with Hubble Key prior of H0 = 72 ± 8 as a function of post burn-in step for 5
chains. Lower values indicate better convergence. Note that the two slowly converging
parameters are As and r, which are largely degenerate on the scales at which QUaD is
most sensitive. These two parameters achieve f < 1.1, while all other parameters achieve
f < 1.01.
The Gelman convergence criteria (68) can be used test multiple MCMC chains
run under the same conditions for convergence and mixing. Quoting from the
156
Method
WMAP methodology paper, for M chains with posterior samples y{ indexed by
chain j with N post-burn-in points indexed by i (66):
"We define the mean of the chain
r,33
I
,
N
y = lNj 2 v i >
=
(5.i2)
and the mean of the distribution
NM
y = T^Y,yiNM
<5-13)
We then define the variance between chains as
B. n
M
1
M
—T,^'vf-
<5 14)
-
and the variance within a chain as
The quantity
w
is the ratio of two estimates of the variance in the target distribution: the numerator is an estimate of the variance that is unbiased if the distribution is stationary,
but is otherwise an overestimate. The denominator is an underestimate of the
variance of the target distribution if the individual sequences did not have time
to converge.
The convergence of the Markov chain is then monitored by recording the
quantity R for all the parameters and running the simulations until the values for
R are always < 1.1. Gelman (Kaas et al. 1997) suggest to use values for R < 1.2.
External Data Sets
157
Here, we conservatively adopt the criterion R < 1.1 as our definition of convergence."
We also adopt the more aggressive criterion of R < 1.01 for all parameters
except T and As as the definition for convergence, r and As are held to the WMAP
criteria of R < 1.1 in the QUaD data only case because they converge far more
slowly due to the degeneracy of these parameters at £ > 100 at which QUaD has
limited sensitivity.
5.3
External Data Sets
While cosmic microwave background data is very good at estimating the angular
size of the horizon of acoustic fluctuations at recombination, interpreting this
angular size as a physical size requires an estimate of how the distance scale
has changed over the history of the universe. Without an assumption of the the
curvature of the universe that constrains the dark energy component, CMB data
alone has reduced constraining power without some tracer of the history of the
universe's expansion in the recent dark-energy dominated period.
Under the assumptions of a flat universe with a dark energy component, since
the curvature is known the expansion history can be constrained by extrapolating the baryon and cold matter densities at recombination by the relative size of
the peaks of the temperature spatial anisotropy spectrum. However the essential
problem remains that we need a late-time distance scale in order to constrain
CMB data well.
5.3.1
Hubble Key Measurement
Adding a reasonable estimate of the current time Hubble parameter H0 greatly
increases the constraining power of the QUaD dataset in the absence of any other
external datasets. As the Hubble Key project value and constraint of 72 ± 8 km /
sec / Mpc from multiple techniques is used throughout the field and consistent
with many other measurements of the Hubble constant (11), we use this as our
External Data Sets
158
primary constraint of the distance scale and determine the constraining power
of QUaD in the absence of any other CMB data with only the Hubble constraint
invoked. To add this constraint to our chains we simply add x\ = (72 ~~ «^)/82 to
—2 In £, where ah is the proposed value of the h = HQ/10Q in the current MCMC
chain.
5.3.2
WMAP Satellite
Measurements of microwave background by the WMAP satellite integrated over
five years have unparalleled sensitivity on the lowest multipole range £ < 10 in
both temperature and polarization, and on the first and second acoustic peaks
of the temperature spectrum. The low-£ data from WMAP effectively break the
r — As degeneracy and the temperature spectrum tightly constrains the baryon
and dark matter physical densities (13).
As WMAP is currently the most powerful single microwave dataset in the field
when constraining the 6 parameter ACDM model, an appropriate question to ask
is how much additional constraining power QUaD brings. WMAP's likelihood
computation is considerably more complicated than the multipole-based likelihoods described above. On large scales WMAP uses a pixel-based likelihood
computed using Gibbs sampling for temperature multipoles £ < 32 and an exact
likelihood for polarization at £ < 23 (23). During MCMC computations used in
this analysis the likelihood code provided by the WMAP team is used to compare
theory spectra output by CAMB.
5.3.3
SDSS Luminous Red Galaxy Survey
In order to provide an estimate of what additional constraining power QUaD
brings to the state of the art in cosmological parameter constraints, in this analysis we choose to use the Sloan Digital Sky Survey (SDSS) matter power spectrum
at z ~ 0.35 as determined by their Luminous Red Galaxy (LRG) sample. By establishing a near-time scale of the matter power oscillations, analyses that integrate
only the SDSS LRG information in conjunction with the WMAP data break the
Results
159
degeneracy between fim and h that pushes us to use the physical parameter Qmh2
in our CMB only analyses.
The combination of the SDSS data alone with WMAP is sufficient to improve
the constraints by a factor of two in the six parameter ACDM models and by an
order of magnitude in models allowing for curvature or tensors (69).
The SDSS collaboration has provided public decorrellated bandpowers of the
matter spectrum P&(fc) that are very easy to integrate into this analysis. An effective xl DSS statistic can be defined assuming a Gaussian likelihood for the matter
bandpowers:
-,nHk)?
Xioss - E "*'*'
b
b
where Plh{k) is the predicted matter power spectrum at z = 0 returned for a given
set of cosmological parameters a, oh is the random error on each bandpower and
c is a bias correction factor that scales the matter power spectrum from z = 0 to
z = 0.35. In practice since there is considerable uncertainty on the proper bias
correction factor due to its sensitivity on models of structure formation, we only
allow the shape of the matter power spectrum to constrain the result by adding a
nuisance parameter to the MCMC chains, allowing the bias to range freely as an
extra fit parameter that is marginalized over for the end result.
5.4
5.4.1
Results
6 parameter ACDM
Five chains were run for three combinations of data: 1. QUaD and the Hubble
Key constraint 2. WMAP 5 year, SDSS LRG and Hubble Key and 3. QUaD, WMAP
5 year, SDSS and Hubble Key. Exploratory chains of 20,000 points were used to
determine parameter covariance matrix for proposal step size optimization, and
then chains of 100,000 steps were run with the first 20,000 points discarded for
burn-in. Again, all parameters reached convergence using the Gelman-Rubin
160
Results
0.020
0.015
0.015
0.010 :
0.005
0.005
A
0.015
0.025
0.035
HyiOO
h omegacdm
h omegab
0.020;
0.010
0.01b
•
A
0.010
•
\ 1\
0.005 ,.
0.005
/
\
0.10
0.15
/
0.000
0.05
0.0257814 +0.00255409-0.00313457
J
\
0.000
0.045
/
0.20
0.25
0.3
0.118986 +0.0143230 -0.0207684
0.5
/
/
\
•
\
\
V
0.7
0.9
1.1
0.712710 +0.0697763 -0.0723606
tau
0.010
0.008
0.010
^'~\
0.008
"A
0.006
0.004
0.006
A. H
\
0.000
0.4
0.6
0.8
1.0
1.2
0.825343 +0.0772543 -0.0839720
0.004
A^N
0.002
0.0
0.2
0.4
0.6
0.002
0.000
0.8
0.268333 +0.0807723 -0.253331
2.5
3.0
3.5
4.0
4.5
3.69857 +0.331581 -0.405265
Figure 5.3: One dimensional marginalized posterior cosmological parameter distributions
using QUaD T T / E E / T E spectra and Hubble Key project H0 constraint. Note that due
to a lack of sensitivity at £ < 100 QUaD can only constrain an upper limit on the optical
depth to reionization, r.
criterion above, with R < 1.01, except for r and As for the QUaD/HST chain.
The sampled posterior distributions were then marginalized in one and two dimensions to produce the plots shown. Parameter estimates are taken from the
one dimensional peaks of the marginalized distributions and uncertainties are
derived from the empirical 68% widths.
The pivot scale for denning As and ns was k* = 0.002Mpc~\ identical to that
used by the WMAP collaboration in order to allow for direct comparison. In those
runs where tensor modes were included the tensor pivot scale was identical.
The derived best-fit parameters and their posterior distributions are summarized for the standard six parameter ACDM model in Table 5.1, and it is clear that
they are all consistent well within the 68% confidence intervals. The rederived
161
Results
Data
QUaD/HST
WMAP5/
SDSS LRG/
HST
QUaD/WMAP5/
SDSS LRG/
HST
nbtf
* ^cdm'l
Ho
ns
T
As
0 119+0014
71 ± 7
OO+0.077
nU O
Z
<0.53
o 7 +0.33
°-'-0.40
0.0227 ± 0.0006
n 114«+00048
70.1 ± 1 . 9
0.96812:211
0.084 ± 0.016
3.08 ± 0.04
0.0224 ± 0.0005
0.1143 ±0.0044
69.8 ± 2 . 0
0.959*0.012
0.084 ±0.016
3.08 ± 0.04
U.U/08_0
0Q31
'
-0.084
Table 5.1: ACDM model parameters and 68% confidence intervals from MCMC for three
combinations of data: QUaD T T / T E / E E spectra and Hubble Key project constraint
H0 = 72 ± 8, W M A P 5 year likelihoods combined with Hubble Key and SDSS Luminous
Red Galaxy survey, and QUaD and WMAP combined with Hubble Key and SDSS LRG.
All confidence intervals are 68% except for upper limits on r for QUaD/HST data, which
is a 95% limit.
WMAP/SDSS parameters and constraints are virtually identical to the values reported by the WMAP team themselves for this combination (13). The addition of
QUaD to the WMAP/SDSS dataset only slightly improves the constraints on the
parameters, mostly by improving the measurement on ns with the addition of
small-scale data.
One item of note is that the QUaD only analysis displays an anomalously low
value of the scalar spectral index ns = 0.82. This result has been confirmed in
other analyses of the QUaD data alone using the standard MCMC software package in the field, CosmoMC.
5.4.2
ACDM with tensors
A natural extension of the ACDM model is to include a seventh parameter, the
tensor-to-scalar ratio r. QUaD's B-mode polarization constraints essentially offer no constraining power on tensor-induced polarization, due to a lack of sensitivity at the scales I < 200 owing to the experiment's small beam and field size.
Conclusions
162
However, results from the WMAP satellite team showed that they had competitive sensitivity to polarization-only experiments by measuring constraining additional tensor mode power in the temperature and temperature/E-mode crosscorrelation spectra. The 5 year WMAP data constraints r < 0.43 (95 % CL), and
we have recovered that result using the MCMC code for the QUaD analysis when
analyzing the WMAP likelihood in isolation (13).
Because the tensor mode power primarily affects the low multipole moments
before the first temperature peak there is a degeneracy between ns and r. Adding
in SDSS LRG data significantly improves the measurements of Qb and Qh> constraining the first peak and allowing for careful limits on the residual power allowed by tensor modes given WMAP's uncertanties (69).
QUaD can add constraining power to WMAP measurements by breaking the
ns and r degeneracies with high-£ small-scale data. When the SDSS LRG sample
is added to the WMAP 5 year constraints with the Hubble prior, we obtain r <
0.23 at 95% confidence. When QUaD is then added to those measurements, a
tensor-scalar ratio constraint of r < 0.20 is obtained. The full set of parameter
constraints can be seen in table 5.2.
When the chains are run, the slow-roll inflation constraint of nT = —r/8 is
enforced.
5.5
Conclusions
The derived posterior distributions for the combination of SDSS and WMAP 5
are consistent with published results using standard MCMC tools in the field,
verifying that the method used to obtain these constraints. QUaD is shown to
be consistent with the reigning ACDM 6 parameter model with tight constraints
on cosmology used only in conjunction with a distance scale prior provided by
the Hubble Key project. Finally, when added to a state of the art analysis QUaD's
high-£ temperature spectra and polarization data slightly improve the constraints
on one of the most competitive cosmological parameter constraint analyses to
date from a limited set of data, the combination of the SDSS LRG sample and
Conclusions
Data
WMAP5/
SDSS LRG/
HST
QUaD/WMAP5/
SDSS LRG/
HST
163
nbh2
^^cdm'^
46
h
ns
T
As
0.0229 ± 0.0006
0 11147+0O0
u- 1^'-0.0054
0u.(u<t_
7 0 4 +0 0 00 22 12
0u.»i
9 7J--0.017
1+0015
0.080 ±0.015
3.08 ± 0.03
0.0224 ± 0.0006
U.il^O_o .0046
n 7n8+0022
U U8
-' -0.020
0U .09=07+0.014
/_0 Q16
0.080 ±0.015
3.06 ± 0.03
Data
WMAP5/
SDSS LRG/
HST
QUaD/WMAP5/
SDSS LRG/
HST
r
< 0.23 (95% CL)
< 0.20 (95 % CL)
Table 5.2: ACDM with tensor modes model parameters and 68% confidence intervals
from MCMC for two combinations of data: WMAP 5 year likelihoods combined with
Hubble Key and SDSS Luminous Red Galaxy survey, and QUaD and WMAP combined
with Hubble Key and SDSS LRG. All confidence intervals are 68% except for upper limits
on tensor to scalar ratio r, expressed as 95% confidence limit.
WMAP 5 year data. The tensor-to-scalar ratio constraint is reduced from r < 0.23
to r < 0.20 at 95% confidence.
Conclusions
*
164
0.01
0.02 0.03
h* omega_b
0.04
0.01
0.02 0.03
h z omegab
0.04
0.05
0.10 0.15 0.20
h2 omegacdm
1.31
1.31
1.04
1.04
0.78
0.78
0 51
0 51
i.
0.01
0.02 0.03
h2 omegab
0.04
0.05
0.10 0.15 0.20
hs omegacdm
0.4
0.6
0.8
H/IOO
1.0
0.01
0.02 0.03
h2 omega_b
0.04
0.05
0.10 0.15 0.20
h2 omegacdm
0.4
0.6
0.8
H/I00
0.51
0.78
1.04
1.31
0.02 0.03
h2 omega_b
0.04
0.05
0.10 0.15 0.20
h2 omega_cdm
0.4
0.6
0.8
H/IOO
0.51
0.78
1.04
1.31
4
0.01
-0.25
0.25
0.75
1.25
Figure 5.4: Two dimensional marginalized posterior cosmological parameter distributions
using QUaD T T / E E / T E spectra and Hubble Key project H0 constraint. 2D contours are
for 68% and 95% confidence level regions.
)(o5~
Chapter 6
Electrodynamic Parity Violation
"Just about as much right," said the Duchess, "as
pigs have to fly...."
Alice's Adventures in Wonderland
LEWIS CARROLL
6.1
Background
Cosmic Microwave Background (CMB) polarization measurements at multipoles
of I > 20 are unaffected by reionization and are an effective means to probe for
cosmological scale electrodynamic parity violation to the surface of last scattering. Using the CMB is particularly attractive because of the long path length to
the surface of last scattering, the well-understood physics of the primordial universe that generated the CMB photons, and two cross-spectra, the temperaturecurl (TB) and gradient-curl (EB) cross-correlations, that should be null in a parityconserving universe (70; 71; 72; 73; 74). As the effect is frequency independent in
many theories that generate parity violation, measurements of the CMB at multiple frequencies can distinguish it from other EB correlation inducing effects like
Faraday rotation from magnetic fields in the intergalactic medium (75; 76; 77).
Background
166
The known parity violation in the weak force is sufficient motivation for investigating electro dynamic parity violation, but it has been shown that parityviolating interactions are a potential solution to the problem of baryon number
asymmetry because they can be a signature of CPT (charge-parity-time) violation
in an expanding universe (78). The CPT theorem implies that Lorentz violation
can also be tested with these models (79; 80).
6.1.1
CPT violation induced by a Cherns-Simons term
One method to generate a CPT violating effect arises by adding a Cherns-Simons
term to the normal electrodynamic Lagrangian, violating Lorentz, P and CPT
symmetries (79; 81):
C = ~Fta/F^+pllAvF^
(6.1)
Here F^v denotes the field tensor, p^ is an fixed, external vector, Av the 4vector potential a n d , F^v is the electromagnetic field tensor dual such that:
F^v = \e»valiFap
(6.2)
This dual tensor is therefore F^v where the components of the electric and
magnetic field have been switched, Ei/c <-*• 5 j . Non-zero time or space components of Pn induce a rotation of the polarization direction of each photon as it
propagates from the surface of last scattering. Note that we only consider in this
model a non-dynamic <9Mp^ = 0. We can then define an equivalent mass of the
external vector and the strength of the interaction (79):
V»P
= rn2
(6.3)
777
£cs
=
-jB-A
(6.4)
Carroll then demonstrates that for a electromagnetic plane wave in free space
the resulting dispersion relation becomes:
167
Background
UJ — k = ±(pok — upcosO)
l - ^2 ^ l2
u> — k
(6.5)
where the ± denotes the relations for right- and left-handed circular polarization
states respectively, p = \p\ and 9 is the angle between k and p. To first order in p^
this becomes:
k = UJ T y (Po ~ P cos 6)
(6.6)
If the change in phase of a circularly polarized plane wave traveling a distance
Lis a = kL, then the resulting frequency-independent relation between the difference between the two polarization states is:
Aa = - (p0 — p cos Q)L
(6.7)
The net effect is to induce an overall rotation of the polarization direction of
each photon as it propagates from the surface of last scattering to observation.
This is equivalent to a local rotation of the Stokes parameters, Q and U, in the
polarization maps made by CMB experiments, inducing gradient (E) to curl (B)
mode mixing and therefore EB correlation.
We can relate the strength of the proposed interaction with an equivalent energy scale if we have a measurement for Aa in degrees. Assuming that L ~ 13.7
gigalightyears is the distance to the CMB (13):
Po — P cos 6 ~ m
2ivhAa
m ~
180 x 13.7 gigayears
m « Aa(5.3 x 1CT44 GeV)
(6.8)
(b.9)
(6.10)
For constraints of Aa < 1.0° that are achievable by QUaD this is the most
competitive means of testing CPT violation from any astrophysical or laboratory
method (82).
168
Background
A generalization of Lorentz violating effects to operators of arbitrary dimension d has been proposed as the "Standard Model Extension" (80). Their generalization only has one CPT/Lorentz violating parameter for d = 3 that results in
equal, frequency-independent rotations of polarization vectors across the whole
sky,fc/y\00.The constraints on m in the Cherns-Simons term model are equivalent
to constraints on fc(y)00.
6.1.2
Ni's Lagrangian, P asymmetry
In 1973, Ni proposed a Lagrangian that refutes Schiff's conjecture, in that it obeys
the Weak Equivalence Principle while violating the Einstein Equivalence Principle while remaining Lorentz-invariant (83):
£N = ^Vd^F^
(6.11)
where g = -det(gfiV) is minus the determinant of the matrix and 0 is a dimensionless scalar function of the gravitational fields in question (like g^u). Observationally, Carroll and Field showed that under this theory two test bodies, such as
photons, with opposite helicities would follow the same trajectory in a laboratory
experiment, but the presence of a gravitational field could be detected by the differing phases of the two particles (84). For a monotonically varying scalarfield4>
we would see a rotation in the apparent polarization directions of photons from
an astronomical source at redshift z of:
A a = cf){z) - <f>(0)
(6.12)
Assuming that 4>{z) varies smoothly on order of the Hubble time we obtain
energy constraints on the possible size of the effect using the same dimensional
analysis as the Cherns-Simons case:
|0| = Aa(5.3 x 10"44 GeV)
(6.13)
Note that this Lagrangian is Lorentz-invariant so CPT symmetry is preserved.
Analysis
169
However, parity (P) symmetry is not conserved, as reflected in the deviant behavior of photons with opposite helicities.
Although far beyond the scope of any possible sensitivity QUaD has, the CMB
also has the potential to distinguish whether the inflaton field is parity violating
through the temperature-curl (TB) cross-spectrum (70). Lagrangians of the form
Cow = m)R\^kr
(6.14)
preferentially amplify inflationary gravity waves with one helicity and attenuate the other while the tensor perturbations were still within the horizon, where
/ ( $ ) is some smoothly varying function of the inflaton field and RxafJiV is the Riemann curvature tensor. This results in non-zero TB power at scales larger than
the acoustic horizon at recombination, and a polarization sensitive CMB experiment would need substantial sensitivity to large scale modes I < 300 to observe
this effect which QUaD lacks.
6.2
Analysis
Assuming that the CMB is a Gaussian random field, the entirety of its statistical
properties can be described by the auto- and cross-correlation power spectra:
Q
=
a
2f I l /
J
t-matm
(6.15)
m
where the aem are the coefficients of the spherical harmonic decomposition of the
temperature or polarization maps. X and Y here denote T,EorB
for the respec-
tive maps of temperature, gradient-polarization and curl-polarization modes.
Normally the CjB and CfB are expected to be null because the spherical harmonic eigenfunctions Yjm and Y ^ have parity (-1) £ and Y^ has parity (—l) m .
Assuming that there is a parity-violating effect in the electrodynamics equations
that prefers one polarization to another over cosmological scales, let us denote
the average preferred rotation of the polarization direction of a photon from the
Analysis
170
surface of last scattering as it heads towards us as Aa. This corresponds to a rotation of the polarization directions in the maps (70; 78) inducing E to B mixing,
and therefore EB cross correlation. Likewise, since there is already TE cross correlation, TB cross correlation is also induced. The full set of mixing equations is
(85):
nTE,obs
nTB,obs
nEE,obs
nBB,obs
s-iEB,obs
=
CjE cos(2A«)
(6.16)
=
Cj £ sin(2Aa)
(6.17)
=
CfE cos2 (2 Aa) + CfB sin2 (2 Aa)
(6.18)
= CfB cos2 (2 Aa) + CEE sin2 (2Aa)
(6.19)
=
^(Cf*-C?fl)sin(4Aa)
(6.20)
Following (85), we assume that cosmological BB modes are zero to simplify
the equations and maximize the likelihood of a detection:
nTE,obs
nTB,obs
nEE,obs
nBB,obs
nEB,obs
=
CjEcos(2Aa)
(6.21)
=
CjE sin(2Aa)
(6.22)
=
CfE cos2 (2Aa)
(6.23)
£
2
=
Cf sin (2Aa)
(6.24)
=
i(Cf B )sin(4Aa)
(6.25)
For the purposes of plotting and analysis, we can derive a theory-independent
X2 statistic to combine the first two and the last three equations separately to obtain an estimate of Aa, utilizing constraining power from across our 23 reported
bandpowers. First, we assume £(£+l)Cfx'°bs
is constant within a bandpower and
define the quantities below for each bandpower:
Analysis
171
CjB'obscos(2Aa)-CjE'obssm(2Aa)
DTB,e
=
DEBI
'B,obs
= CEB
'obs
•{Ce
(6.26)
(6.27)
+^e
)sm(4Aa)
We can then minimize x2(Aa) for the TB and EB combinations separately to
estimate Aa x:
2
(Aa)
X
2
X
(Aa)
= J2DT^MU'DTB/
w
= J2DEB,eM-}DEB/
ee'
(6.28)
(6.29)
We empirically measure the covariance matrix, Mi(i, of the bandpowers in each
spectrum DEB,e and DTB^ from a set of simulated bandpowers combining realizations of A-CDM cosmology temperature and polarization fields for the signal
component and accurate realizations of QUaD's instrumental noise. Our method
utilizes the set of 416 signal and noise Monte Carlo simulations from the analysis pipeline of QUaD described in previous chapters and thoroughly tested for
evidence of systematic contamination.
Figure 6.1 shows the results of this combination for the data (red line) and
simulations (histogram) for 150GHz in both EB and TB. Overplotted is the total
uncertainty assuming that the simulations reflect a normal distribution and that
the systematic error is 0.5°. It is clear that the observed data can easily be drawn
from the set of simulations in which no parity-violating interactions have been
included; we therefore conclude that there is no detection.
x
It is also possible to estimate Aa by measuring the quantities
EE ohi
bn
o<^ a n d
QTB.obs
,
'-
j(cTE,o„s)2HCTB,obs)2
on a per-bandpower basis, combining them using the covariances as meaV
V
g
5
sured from simulations, and then applying inverse trigonometric functions. However, this is biased in the presence of noise. We thank an anonymous referee for suggesting our current method.
Current Limits and QUaD Results
Spectrum
150 GHz EB
150 GHz TB
100 GHz EB
100 GHz TB
Cross-freq. EB
Cross-freq. TB
150 GHz combined
100 GHz combined
Cross combined
all EB combined
all TB combined
all combined
172
Aa
(random and sys. errors)
1.24°±0.69°±0.5°
-0.09°±2.52°±0.5°
-0.60°±1.51°±0.5°
5.50°±5.32°±0.5°
0.39°±0.75°±0.5°
-1.02 o ±2.60°±0.5°
0.99°±0.69°±0.5°
0.19°±1.44°±0.5°
0.26°±0.75°±0.5°
0.64°±0.55°±0.5°
1.40 o ±2.83°±0.5°
0.81°±0.49°±0.5°
signal-only
simulation scatter
0.08°
0.38°
0.13°
0.44°
0.10°
0.40°
0.09°
0.14°
0.10°
0.08°
0.38°
0.09°
% simulations
exceeding
9.87%
81.3%
57.7%
16.9%
53.0%
56.7%
13.7%
77.8%
61.5%
21.8%
44.0%
7.86%
Table 6.1: Column 1: A a uncorrected measurements from QUaD, including random
and systematic errors. Column 2: Scatter of signal-only simulations, indicating sample
variance. Column 3: Fraction of signal+noise simulations where ||Aa|| exceeds that of
data.
To obtain a visual representation of a "Aa spectrum," We can also estimate
the best fit for Aa on a per-bandpower basis by minimizing:
2
X£ (Aa)
= J2DTB,IMK}DTB/
e
x|(Aa)
=
Y,DBB,tM£DEBj
e
(6.30)
(6.31)
The Aa spectrum using the EB, BB and EE spectra for 100 GHz, 150 GHz, cross
and all frequencies combined is shown in figure 6.2.
6.3
Current Limits and QUaD Results
Komatsu et. al. (85) report their limits from the WMAP 5 year high-£ data as
Aa = —1.2° ± 2.2°. Other authors have found weak evidence for parity violation
by combining the WMAP 5 year data and data from the BOOMERanG balloon
Current Limits and QUaD Results
Spectrum
150 GHz EB
150 GHz TB
100 GHz EB
100 GHz TB
Cross-freq. EB
Cross-freq. TB
150 GHz combined
100 GHz combined
Cross combined
all EB combined
all TB combined
all combined
Aa
(random and sys. errors)
1.24°±0.69°±0.5°
-0.09°±2.52°±0.5°
-0.60°±1.51°±0.5°
5.50°±5.32°±0.5°
0.39°±0.75°±0.5°
-1.02°±2.60°±0.5°
0.99°±0.69°±0.5°
0.19°±1.44°±0.5°
0.26°±0.75°±0.5°
0.64°±0.55°±0.5°
1.40°±2.83°±0.5°
0.81°±0.49° ± 0.5°
173
systematic
bias
0.002°±0.004°
-0.109°±0.019°
-0.022° ±0.006°
0.145°±0.022°
0.007°±0.005°
0.034° ±0.020°
-0.005°±0.004°
-0.012°±0.007°
0.004°±0.005°
0.002°±0.004°
-0.040° ±0.019°
-0.006°±0.004°
bias-corrected
Aa
1.24°±0.69°±0.5°
0.02°±2.52°±0.5°
-0.58°±1.51°±0.5°
5.35°±5.32°±0.5°
0.38°±0.75°±0.5°
-1.05°±2.60°±0.5°
1.00°±0.69°±0.5°
0.20°±1.44°±0.5°
0.26°±0.75° ± 0.5°
0.64°±0.55°±0.5°
1.44°±2.83°±0.5°
0.82°±0.49°±0.5°
Table 6.2: Column 1: A a measurements from QUaD, including random and systematic
errors. Column 2: Bias and standard errors on mean sampled from 496 signal-only simulations. Column 3: Column 2 subtracted from Column 1. Slight bias in results is caused
by uncorrected slight non-alignment of elliptical beams within a single feedhorn causing
spurious temperature to polarization leakage and cross-correlation. This effect is included
in the signal-only simulations and the mean of the signal-only simulations, in column 2
reflects the cumulative effect on our computed A a .
experiment, reporting Aa = -2.6° ± 1.9° (81). (79) derived constraints on Aa 10
high-redshift radio galaxies in 1990, yielding Aa = -0.6° ± 1.5°. The best single
redshift number, for 3C9 at z = 2.012, is Aa = 2° ± 3°.
QUaD's results broken down by individual spectrum and frequency, as well as
combined within and between frequencies, are shown in Table 6.1. Reported errors are 68.2% confidence limits as determined by the distribution of signal and
noise simulations. 150 GHz EB alone is significantly more constraining than any
current result. These results are consistent with a constraint on isotropic Lorentzviolating interactions (80) of k^)0Q < 10~43 GeV (95% confidence limit), and likewise limit the strength of the Cherns-Simons interaction to < 10"43 GeV. At no
frequency, nor in any spectrum, is there a significant detection.
Systematic effects and checks
174
EB parity violation limits
EB parity violation limits
g
40
8 M
20
|
20
101
^
*
10
0
oL
t
hi ^
h
J
i
i
t
i
-20
TB parity violation limits
o
parity violation angle, degrees
TB parity violation limits
10
o
parity violation angle, degrees
EB parity violation limits
EB parity violation limits
TB parity violation limits
TB parity violation limits
I!
\
B
0
parity violation angle, degrees
10
*
10
0
;Y
10
I
\
0
parity violation angle, degrees
10
Figure 6.1: Aa measured from QUaD 100 GHz (top left), 150 GHz (top right), 100-150
cross-frequency (bottom left), and coadded (bottom right) TB and EB spectra; histogram
of simulations and red line for data. Histogram does not account for systematic error.
Dotted line indicates total uncertainty assuming a Gaussian 0.5° systematic error.
6.4
6.4.1
Systematic effects and checks
Systematic bias caused by beam offsets
We know from measurements of the beams of our instrument that the two individual polarization-sensitive detectors within a horn have elliptical beams that
are slightly offset from each other on the sky with slightly different shapes, although our optical design intended for both detectors to have an identical and
co-aligned beam. This is discussed further in Chapter 2. As these detectors are
sensitive to orthogonal polarization directions, at reconstruction we make an instantaneous measurement of microwave polarization at a specific location on
Systematic effects and checks
175
i
1000
multipole moment
iilfM'll
fM
1000
multipole moment
i
l
i i
1000
multipole moment
{U
1000
multipole moment
Figure 6.2: A a per bandpower derived from QUaD 100 GHz (top left), 150 GHz (top
right), 100-150 cross-frequency (bottom left), and coadded (bottom right) EB spectra.
Note that in practice these points are combined before the final transformation to A Q
— the purpose of this plot is to give a visual representation of the relative uncertainties
across the band powers.
the sky by assuming that the detectors are exactly co-aligned at their average location, inducing temperature anisotropies to appear as false polarization signals
and therefore correlating T and B to first order and E and B at second order and
creating spurious power in the TB and EB spectra.
As the effect is far subdominant to our noise we have incorporated the known
beam non-idealities and misalignments directly into our simulation pipeline as
it would be computationally infeasible to account for them while reconstructing
maps from the raw detector signal.
Systematic effects and checks
176
This results in a small amount of TB and EB power that is evident over hundreds of averaged signal-only simulations when we incorporate the timestream
filtering algorithm used on the real data because the effect is quite small in comparison to cosmic variance. This power is responsible for the "signal-only simulation bias" reported in our results and shown in table 6.2.
The effect was isolated by producing 400 signal-only simulations where the
simulated detectors are perfectly aligned with circular, Gaussian beams where
the two orthogonally polarization-sensitive detectors within a feedhorn are coaligned on the sky and sensitive exclusively to their intended polarization direction. These baseline simulations return TB and EB spectra, and parity violation
results, that are consistent with zero averaged over many realizations. After incrementally adding the known detector non-idealities accounted for in our normal
simulation runs, it is clear that the combination of offset, non-ideal beams between two detectors sharing a feedhorn and timestream filtering applied on the
resulting signal produces a slight bias in the TB and EB spectra averaged over
many signal-only simulations, and therefore is the principal cause of the bias in
the reported results.
For example, in 100 GHz TB field differenced results where the effect is most
prominent, the parity violation result averaged over 400 ideal signal-only realizations with perfect beams and no timestream filtering is 0.003 ± 0.018. When
we add in the effects of offset elliptical beams within a feedhorn and timestream
filtering, without adding any of the other known systematic effects, the resultant parity violation signal averaged over 400 realizations is 0.069 ± 0.022. Finally,
when all of our known systematic effects are included, the average over 496 simulations converges to 0.073 ± 0.022. The small difference in these two results is
attributable to the uncertainty in temperature-polarization cross-calibration inserted into our full effects simulation pipeline due to our known errors on the
measured amount of "cross-polarization leakage", or sensitivity of our bolometers to radiation of an orthogonal polarization from the intended design. The full
effects simulation pipeline populates the simulated focal plane with bolometers
drawn from a distribution of cross-polar response of 7.7% ± 1.3%, consistent with
Systematic effects and checks
177
our laboratory calibration measurements and discussed in further detail in Hinderks et al.
We therefore also present values for Aa where the systematic bias induced by
a combination of timestream filtering and the slightly different, non-aligned, and
elliptical nature of the beams of two orthogonally aligned polarization sensitive
detectors within a single feedhorn leading to temperature to polarization leakage has been quantified by signal-only simulations and subtracted off of the data
results in table 6.2. Note that in all frequencies and spectra this bias is an order
of magnitude smaller than our random and systematic errors. After combination
the EB spectra dominate the analysis and there is virtually no bias.
6.4.2
Systematic rotation
The primary systematics concern is that there might be a systematic rotation of
the true detector sensitivity angles, producing a false signal totally degenerate
with that of parity violation; for example, a —3° systematic misalignment and a
Aa = —3° true parity violation signal would produce identical results. We can
artificially induce such an effect in the data analysis by simply transforming the
polarization maps:
Q' = Q cos(2A6) - U sin(2A6)
(6.32)
U' = Q sin(2A9) + U sm(2A9)
(6.33)
The entire dataset has been reanalyzed after inserting an artificial A9 = 2°
local polarization rotation only in the data maps, resulting in a 2 degree shift after
deriving Aa identically to the procedure above, validating the analysis pipeline.
The overall rotation of the instrument was measured using two methods described below. The first measures the polarization sensitivity angle of each bolometer using a near field polarization source. The second constrains the absolute angle of the focal plane by examining the measured offsets of the beams of each
Systematic effects and checks
178
detector from the telescope pointing direction on an astronomical source. These
two methods agree nearly exactly indicating that any systematic rotation of the
bolometers within the focal plane structure is negligible.
6.4.3
Overall rotation measured by near-field polarization source
Figure 6.3: Near-field polarization source mounted on MAPO's roof near the QUaD
telescope and ground shield during mid-March 2006. Photo courtesy of Michael Zemcov,
Caltech/JPL.
The near-field polarization source consisted of a chopped thermal source mounted
on a mast about 50 feet above MAPO, at 40° elevation with respect to the telescope. A wire grid was placed in a 7.5 cm aperture at the front of the box, reliably emitting microwave polarized radiation of a known angle. The wire grid was
crafted and mounted inside a square aluminum holder to tolerances within 0.1°.
The holder could be slotted behind the aperture such that the linear polarization that emerged was either totally vertically or horizontally aligned. A digital
tiltmeter inside the box was then aligned such that the tilt around the axis from
Systematic effects and checks
179
which the radiation emerged could be measured from the ground.
Polarizer grid measurement, 150-01A Run 1
1.2
1.0
0.8
fa
o) 0.6
0.4
0.2
0.0
-150
-100
-50
0
Rotation (degrees)
50
100
150
Figure 6.4: Near-field polarization source fit for detector from central feedhorn. Measurements are taken at multiple orientations of telescope around boresight and fit to a
sinusoidal response indicating detector orientation and cross-polarization efficiency. Measurements taken with polarizing grid mounted in vertical orientation in blue points and
fit is blue line. Red points correspond to measurements with grid aligned horizontally,
and black line is sinusoidal fit. Note that perceived shift in phase between horizontal and
vertical grid alignments is due to sensitivity of horizontal measurements to uncompensated
fore-aft tilt of the box holding the polarization source.
The absolute polarization sensitivity angle of each of the detectors, as well
as their response to the orthogonal polarization, or "cross-polarization leakage"
could then be measured for a given grid orientation by making raster maps of the
polarization source at several telescope deck axis rotation angles. The perceived
signal in each detector integrated over the source at each angle fits a function:
fi(<t>) = A((l - ei) cos(20 - 28ai) + 1 + et)
(6.34)
where A is the overall amplitude, i is an index over detectors, e{ is the crosspolarization leakage of a given detector, 4> is the design orientation of the detector
Systematic effects and checks
180
with respect to the orientation of the grid and 6ai is the deviation from the design
orientation. An example of a single detector being fit to this function can be seen
in figure 6.4.
The bolometers have no net overall systematic rotation with respect to overall
angle of the telescope we use in the analysis pipeline when
< San >= 0
(6.35)
over all the detectors in the focal plane. Two full-day measurements of the nearfield polarization source were performed in March 2006 and May 2006, with the
box being unmounted and remounted on the mast separately for each run. Half
the day was spent measuring the source at 9 telescope rotation angles with the
grid vertical, and another 9 with the grid aligned horizontally. However, after
a discrepancy was discovered in the perceived angle between the vertical and
horizontal measurements for each one, it was realized that despite the presence
of the digital tilt-meter measuring tilt of the box normal to the aperture to high
precision, "left-right" spin of the box on the axis normal to the ground would
contribute an error to first order for measurements when the grid horizontally
aligned, but not when the grid was vertically aligned. The mathematics detailing
this effect are derived in appendix A.
The measurements taken when the grid was aligned horizontally were therefore removed from consideration, and the value at which < 8ai > is zero computed for each run from the vertical measurements only. In the coordinate system used by the telescope encoders and data acquisition computer, the absolute rotation angle at which this occurs is dk = 57.84° ± 0.02° in March 2006 and
dk = 57.84° ± 0.02°. The errors given are standard errors on the mean of the
noise-limited measurements for each individual detector. The 0.19° discrepancy
between the two measurements is attributable to slight wobbling in the alignment of the box observed by the tiltmeter due to wind moving the box over the
course of the several hour measurement runs.
Systematic effects and checks
detector
£i
150-01A
150-01B
150-02A
150-02B
150-03A
150-03B
150-04A
150-04B
150-05A
150-05B
150-06A
150-06B
150-07A
150-07B
150-08A
150-08B
150-09A
150-09B
0.08
0.08
0.07
0.07
0.08
0.07
0.06
0.08
0.10
0.08
0.09
0.09
0.07
0.05
0.07
0.06
0.06
0.06
5ai
-1.94°
-2.60°
1.42°
-0.78°
-0.79°
-0.44°
-1.42°
0.39°
-1.22°
-1.66°
0.80°
1.13°
1.26°
-0.90°
1.24°
0.63°
1.42°
0.28°
181
detector
150-10A
150-10B
150-11A
150-11B
150-12A
150-12B
150-13A
150-13B
150-14A
150-14B
150-16A
150-16B
150-17A
150-17B
150-18A
150-18B
150-19A
150-19B
et
0.08
0.11
0.06
0.07
0.06
0.06
0.11
0.08
0.09
0.07
0.06
0.06
0.22
0.18
0.11
0.10
0.06
0.07
Sat
-2.11°
0.72°
-1.73°
-0.44°
0.10°
-2.09°
-1.12°
-1.99°
-0.38°
1.09°
3.30°
1.69°
0.42°
0.48°
1.80°
-0.01°
2.38°
-0.88°
detector
100-01A
100-01B
100-02A
100-02B
100-03A
100-03B
100-04A
100-04B
100-05A
100-05B
100-06A
100-06B
100-07A
100-07B
100-11A
100-11B
100-12A
100-12B
(-i
0.09
0.08
0.07
0.08
0.06
0.08
0.09
0.09
0.07
0.09
0.09
0.08
0.08
0.07
0.08
0.10
0.07
0.06
5ai
2.01°
-0.22°
0.54°
0.95°
-1.20°
0.28°
-0.23°
-0.16°
-1.95°
0.12°
0.30°
-0.25°
-0.26°
-0.91°
1.43°
0.13°
0.59°
0.58°
Table 6.3: Cross-polarization efficiency (ej) and deviation from design angle (<5«j) measured parameters for all detectors used in 2006-2007 QUaD data analysis. Parameters are
derived from measurements of the near-field polarization source with a wire grid aligned
vertically in the aperture in March and May, 2006
These measurements also provide a robust estimate of eit the cross polarization efficiency for each detector that ultimately determines the temperaturepolarization relative calibration. Although the difference in measured et is constrained to < 0.01 between the measurement runs separated by two months,
in simulation and analysis we use the population mean and sample variance,
ei = 0.07 ± 0.03 at 150 GHz and e, = 0.077 ± 0.03 at 100 GHz, for all detectors
within a frequency during reconstruction of polarization maps and in the signalonly simulation pipeline.
A secondary concern is random scatter in the assumed detector angles, which
can be observed in the sample variance shown in table 6.4.3. This is a different
Systematic effects and checks
182
effect than a systematic rotation of all of the detectors, and has an observed magnitude of about asa = 1°- The Monte Carlo simulation pipeline includes the injection of a degree of uncertainty about the true orientation of each polarization
sensitive bolometer (PSB) into every simulation commensurate with the uncertainty of the measurements. Thus, when constructing a "fake focal plane" for
signal-only simulations of a given CMB realization, we assign every bolometer
a random deviation from its presumed angle at reconstruction, drawn from a
Gaussian distribution with a = 1°. Signal-only simulations with and without the
random orientation scatter and cross-polarization leakage effects included show
that their contribution to the final uncertainty is negligible.
6.4.4
Overall rotation measured through beam offsets
A second method to calibrate the overall rotation of the telescope is to assume
that there is not systematic "twist" to the polarization sensitivity orientations
with respect to the machined offsets of the feeds on the focal plane. By measuring the beam offsets of the feeds on the sky using an astronomical source and
using the known, machined symmetry of the feeds on the focal plane, we can
constrain the absolute orientation of the entire telescope and receiver assembly
by rotating focal plane beam offset maps such as the one shown in figure 2.10
until the offsets on the sky match the design locations.
QUaD's focal plane is designed such that there are three rings of bolometer
pairs - 6 "inner 150 GHz" feeds, 12 100 GHz feeds, and 12 "outer 150 GHz" feeds,
in addition to the central 150 GHz feedhorn. The measured offsets of each of
these feeds on the sky can be centered for each ring and rotated until the bolometers lie in the design orientation of the ring. Again in encoder and data acquisition coordinates, the absolute overall rotation of the system derived from this
method is dk = 57.75° ± 0.13° in 2006 and dk = 57.91° ± 0.16° in 2007. The receiver was unmounted and remounted on the telescope between the 2006 and
2007 observing seasons and the difference is attributable to the "slop" allowed by
the mounting screws.
Systematic effects and checks
183
date (yymmdd)
Source
dk angle
060225
060226
060308
060310
060313
060323
060326
060405
060713
060815
061010
070314
070414
070517
070707
070813
070907
070908
070909
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
RCW38
J0538-440
J0538-440
J0538-440
0°
0°
0°
0°
0°
0°
0°
30°
0°
30°
0°
0°
30°
0°
0°
0°
0°
30°
-30°
fit angle sample variance
57.71°
57.76°
58.01°
57.81°
57.91°
57.92°
57.61°
57.64°
57.64°
57.64°
57.41°
57.67°
57.73°
57.96°
57.86°
58.18°
57.96°
57.94°
57.93°
0.673°
0.457°
0.504°
0.415°
0.333°
0.380°
0.328°
0.393°
0.379°
0.350°
0.419°
0.431°
0.474°
0.477°
0.377°
0.347°
0.535°
0.435°
0.466°
Table 6.4: Absolute rotation angle fit from beam offset measurements, "dk angle" column refers to rotation angle of telescope and receiver as measured by encoders during
measurement, "fit angle" column denotes equivalent rotation angle that matches beam
offsets on sky to design offsets on focal plane indicating absolute orientation calibration.
6.4.5
Overall rotation and systematic errors
The measurements from the near-field polarization source and the beam offset
method are therefore consistent within 0.1°. For the overall rotation method the
calibrated angle is dk = 57.80° ±0.016°, weighting the 2006 season twice as heavily
because it has roughly twice as much observation time, while for the near-field
polarization source dk = 57.75° ± 0.02°.
The almost perfect coincidence of values from these two measurements is
quite informative. Only in conjunction do they suggest that we not only understand the overall rotation of the instrument, but also that the polarization sensitivity orientation of the detectors on the focal plane are not systematically turned
with respect to their design orientation.
Systematic effects and checks
184
150-02 left, row flat dk orientation from beam maps, 2006-2007
10
40
50
30
bolometer
Green: inner 150. Blue: outer 150. red: 100
20
60
Figure 6.5: Absolute rotation angle measured for each detector by comparing beam offsets
on sky to designed offset of feedhorns on focal plane. X-axis runs over detectors, with
the inner 150 GHz detectors in green, the outer 150 GHz in blue, and the middle ring
of 100 GHz detectors in red. Note that each connected light blue line signifies a single
beam offset mapping run on a different day through the 2006-2007 seasons. Although
clearly the sample variance is smallest on the outermost detectors, due to their better
constraining power and farther distances from the central feedhorn, all 3 rings are clearly
measuring roughly the same overall orientation.
However, given that there is a 0.2° difference in the near-field polarization
source measurements between the two runs, we very conservatively assign a systematic error of 0.5° to our measurement, although in principle we have no reason to believe that such a systematic error is much larger than 0.2°.
irr
Chapter 7
Conclusion
then on the shore
Of the wide world I stand alone, and think,
Till Love and Fame to nothingness do sink.
When I have Fears that I may Cease to Be
JOHN KEATS
7.1
QUaD in context
The final QUaD combined polarization spectra trace out the acoustic peaks at
high significance to scale sizes of 0.1°, and the location and series of the peaks
is consistent with a flat universe dominated by dark energy with significant dark
matter content, known as the ACDM model. The measured peaks are also consistent with precision measurements of cosmological parameters obtained from
other CMB experiments and measurements of large scale structure. This is a
strong confirmation that the principles of plasma physics and cosmology used
to analyze the CMB temperature spatial anisotropy spectrum are valid.
It is useful to look closely at how QUaD compares to the most sensitive CMB
experiments currently. As can be seen on figure 7.2 and figure 7.3, although QUaD
is not sensitive to large-scale polarization where inflationary tensor modes would
QUaD in context
186
be discovered, it measures the small-scale, high-£ region of the E-mode polarization spectrum no other experiment has been able to reach and also establishes a
limit on lensed B-mode power in a wide range of the spatial anisotropy spectrum.
Finally, in figure 7.1, it is clear that QUaD has competitive sensitivity at high
multipoles to the temperature spectrum. The cosmological implications of QUaD's
measurement of the high-^ temperature spectrum are discussed in depth in (3),
and after careful calibration and treatment of beam errors determine that there
is no excess measured by QUaD at the higher multipoles I > 2500.
Nonetheless from figure 7.3 we see that there is yet much to be measured in
polarization to test for a tensor-to-scalar ratio down to the level of r = 0.01. The
BICEP experiment's 2009 results have begun to make measurements of the low£ B-mode polarization spectrum, deriving a limit of r < 0.73 at 95% confidence
(28) from B-mode polarization alone. More sensitive successors to BICEP will
need accompanying successor experiments to QUaD, sensitive to the smallerscale polarization signal both for foreground templating and the measurement
and removal of the lensed B-mode spectrum in order to derive a robust result on
inflationary modes.
100
+
500
1000
1500
multipole, /
2000
2500
Figure 7.1: Final QUaD temperature spatial anisotropy spectrum (blue points) shown compared to the data from the
WMAP satellite (black points) and observations from the South Pole by ACBAR 2008 (red) and BICEP 2009 (green).
1000
CN
CN
10000
• of
^
N/
0.1 IT'
1.0
10.0
100.0
1000.0
/
\/
\1/
500
V
\J/
" T
\ / \J/ \k
w
\l/
*•• «
V
*•!••
\l/ y
1
1000
1500
multlpole, /
?.*• • * * . - * • ^ 1
\J/
M/
3*1
i
2500
\y \J/ \l/
* •
2000
M/
•
Figure 7.2: Final QUaD temperature (solid) and E-mode (hollow) spatial anisotropy spectra (blue points) as well as 95
% upper limits on B-modes shown compared to the data from the WMAP satellite (black points) and observations from
the South Pole by ACBAR 2008 (red) and BICEP 2009 (green).
+
O
CM
M
10000.0
10
10
0
-4
-2
10
10 2
5
5
10
i
*
*
*
100
multlpole, /
•vi/
T
.-•?•'
1000
Figure 7.3: Final QUaD temperature (solid) and E-mode (hollow) spatial anisotropy spectra (blue points) as well as 95
% upper limits on B-modes shown compared to the data from the WMAP satellite (black points) and observations by
ACBAR 2008 (red) and BICEP 2009 (green). This is the same plot from the previous page on a log scale for I.
->^
+
O
CN
CN
10'
Limits on new physics
7.2
190
Limits on new physics
As shown in chapters on parameter estimation and parity violation effects, although QUaD's impact on improving estimates of the six parameter ACDM model
are limited, it nonetheless contributes to limiting speculative physics and the as
yet undetected gravitational lensing of E-modes into B-modes.
QUaD establishes an upper limit on the conversion of polarization power by
weak gravitational lensing of < 0.77 fiK2 at 95% confidence, compared to the
ACDM expectation value of 0.054 \xK2. Further limits on, or perhaps even a detection of the gravitational lensing of the CMB are likely possible with more carefully constructed maps of the QUaD's CMB temperature field due to its precise
measurement of small-scale CMB power by looking for particular non-Gaussian
signal induced by weak lensing.
Although QUaD has virtually no sensitivity on the scales in polarization on
which tensor modes are likely to be found, it nonetheless can improve on measurements of the tensor-scalar ratio made by WMAP and reinforced by measurements of large-scale structure by contributing constraining power to the TE spectrum and helping to break the degeneracy of r with the scalar spectral index ns.
When considering QUaD's data with the WMAP 5 year data, the Hubble Key measurement of H0 = 72 ± 8 km s_1Mpc_1 and the SDSS Luminous Red Galaxy sample
a constraint on r < 0.20 is obtained.
As a CMB polarimeter with wide coverage of the spatial anisotropy spectrum
QUaD is perhaps uniquely placed to make strong constraints on the cosmologicalscale parity violation effects. To reiterate, the constraints on the net rotation of
the polarization directions of photons as they propagate from the CMB measured
by QUaD is AQ = 0.82° ± 0.49° (random) ±0.50 (systematic). This is equivalent
to constraining isotropic Lorentz-violating interactions to < 10~43 GeV at 95%
confidence.
Appendix A
Robustness of near-field polarization source
measurements
The projection of the polarization direction from the near-field polarization source
onto the aperture plane of the telescope is sensitive to misalignments of the source
itself. As the source was only utilized twice in order to measure the absolute
orientation of polarization sensitivity for each bolometer on the focal plane, we
need to examine whether an unaccounted rotation can systematically corrupt
our measurements, yielding a false additional rotation angle for each bolometer
and therefore mimicking a cosmological parity violation signal.
The near-field polarization source is detailed in chapter 6. Although we used
a tilt meter that sensitively tracked the rotation of the polarization source box
normal to the grid itself, the fore-aft tilt and left-right spin of the box were "eyeballed" as the source was placed on a mast several dozen feet above the roof of
MAPO.
Since the box was 35 degrees off the ground and parallel to the ground, the net
effect of errors in the "eyeballed" alignment directions is to make the polarization
orientation measurements when the grid was horizontally aligned more prone to
systematic error from left-right spin than the vertical measurements. Here we
describe why this is the case and justify disposing of half of our measured data
when estimating the overall rotation of the telescope.
192
Let 9 be the left-right spin of the box, 4> be the vertical fore-aft tilt of the box,
9T be the azimuthal offset of the telescope pointing from the box and <pT be the
telescope pointing elevation. Using a cartesian frame where i is the right of the
telescope, j is pointing towards the source and k is pointing straight up towards
zenith, we can figure out the vector parallel to grid on the box, where g~fr is for a
horizontally aligned grid and g^ is for a vertically aligned grid:
g~H =
(cos#,sin#cos(/>, sin#sin</>)
g~v =
(sin 9 sin (f), cos 9 sin (/>, cos (f>)
(A.l)
(A.2)
The pointing vector of the telescope is:
tp = (sin 9T
COS 0X, COS 9T COS 4>T,
sin 4>T)
(A.3)
Let i[ be the vertically aligned vector on the aperture plane, and t2 be the horizontally aligned vector:
t\
=
(—sin 9T sin (f)T, —cos 9T sin 4>T, cos
t2
= (cos9T,-sin9T,0)
fix)
(A.4)
(A.5)
For a horizontally aligned grid in the source, the projection onto the aperture
plane is:
t'lH
=
^2i?
=
— sin <pT sin 9T
cos
^c o s
®T —
COS 9
— cos 9T sin <pT sin 9 cos (f> + sin 9 sin <fi cos 0 T (A.6)
sin ^r sin 9 cos 0
(A.7)
To first order the projection of the horizontal grid polarization onto the wrong
vector on the aperture plane is:
193
t'1H tn 6 sin
fa
(A.8)
This means about 57% of any left-right rotation of the box on the pole is turned
into a spurious rotation when projected onto the aperture plane. Assuming that
the vertical measurements are not systematically contaminated, this means that
in the March 2006 run the box was rotated about 2.8 degrees and in May 2006 the
box was rotated 5.1 degrees.
Likewise we can compute the projection of the vertical grid vector:
t'iv
= sin 6T sin cf>T sin 0 sin 9 — cos 9? sin 4>T sin (ft cos 9 + cos (p cos <pT (A.9)
t'2V = sin 0 sin 6 cos 6j< — sin <>
/ cos 6 sin 9T
(A. 10)
The only first order effect here should average to zero across the pol source
map:
t'2V « -(psm9T
(A. 11)
Let us assume then that the vertical measurements are robust to misalignments in the source and free of other systematic rotations. The mean angle rotation angle across the focal plane for March 2006 is .84° and for May 2006 is .65°.
This points to an overall rotation angle where the cryostat is horizontally aligned
atdk = 57.75°.
It seems to be worth getting this overall rotation of the instrument as correct
as possible. Perhaps we can use the fact that the telescope pointing mixes in with
these misalignments of the pol source? From equations 6,7,9 and 10 we can preserve the first-order combinations of telescope pointing and box misalignment
terms:
194
t'1H = —9T sin (ft? — 0 sin 4>T
(A.12)
t'2H = l-96T
(A. 13)
t'iv
(A. 14)
=
— (p sin 4>T + C O S 0 T
t'2V = 4>9T
(A. 15)
(A. 16)
In practice the near-field nature of the source makes such a first-order correction impossible because the raster maps of the source are not smoothly varying they appear to have a "donut" shape of a ring of maximal intensity.
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