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

код для вставкиСкачатьCOSMOLOGICAL 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 The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3382915 Copyright 2009 by ProQuest LLC All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 © Copyright by Edward Yousen McCloskey Wu 2009 All Rights Reserved 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 rcj u J. o -hi 1UIW c an sky u o 3 C M C M C M C N C M C S C < ) C ^ C M C M ^ C M C M C N C ^ C ^ C ^ C M C O t > 0 C M C O C M C M C ^ I T—I i—1 -—i i—I I I M M ^ T f ^ T f M M H H W W ^ ^ ^ ^ M M 1 1 1 1 1 i 1 1 ! I 1 1 ! 1 1 1 1 1 1 H d t d c o d ^ i c ^ o c i i O Q ^ t d o j H H o o t d -H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H ^ O i O H Q C O a N H W C C N O O ' T f ^ ^ C C M l O G O M N O O I N a i n O i C O l i O M O O O t D © c s ^ o ^ o i W H O t - c o t o o ^ i n m i N c o CM O CD O I ' ^ O ^ N H O N O l M H i N M O J H ^ N ^ N l M H t D H i O h H C S t O j ^ O O j j j ^ N N H C f i O O i C 0 O O 0 0 C 0 0 3 N 0 0 N N L 0 t 0 0 ) i f l N > - H > - i O r ^ r H O O O O O O r O O C N O O T - i O O O O O O O O O O O O O O O O O O O d d d d d d d d o o d d d d d d o o -H-H-H-H-H+l-H-H-H-H-H-H-H-H-H-H-H-H i C T f C O H C O i O N N I M i O t O H i N O C O I N C l ^ H O O M N O i O C O i O M C N l N G O r O i O l N ^ C O C V 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - H T - I O O ^ ^ ^ ^ ^ ^ ^ . - H ^ ^ ^ ^ ^ ^ ^ ^ , ^ ^ loioioioioi^wioioioioinioioioioioio -H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H M O l C S C O C S ( O i n ^ © ' * O t O N N ( D Q i , i O (MrHtHt-HCO'-HOrHqCNO'-Hi-ti-IOO'^CN'-l O O O O M M O i N N l M i O a i C D ^ ^ O O O l ^ i O COO!CNCOCO(NM(MCO(NNCOCMtOCSCStDO) O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O -H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H m t o i o N i o m t o c s o o ^ i N H r f c s ^ i o c s N O G O O H O i O O O t O H O M l N i n « 0 ) N H n C O O O O O O O O O O O O O O C M r H ^ O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d o o o d d d d 1 1 1 1 1 H I M i O N i O N N O H i n c O c O ^ O H N O W NCS(M(MiMCS(NC^COM(N(M(MrOCO(MCON O O O O O O O O O O O O O O O O O O q q q q q q q q q q o q q q q q q q 1 l ^ l M t N M i M I N M I N I M ^ C S C S I M C S I M C N l O C S O O q q q q q q q q q q q o o o o o o o o d d d H - H - H -H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H N G ^ O • ^ H H O O N I N ^ N H i f l P O O H Q O O i N t O O O c O H ^ 0 ) O O O O N ^ 1 0 H O W l f l O O ) O H t D H i O i O 0 ) Q H H t o ' t d N N t O ( d i X ) N o i d N N o 6 a 6 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d o d d o d o d d d d d d d d d d d d d d d c5<5d>d>d>d>d>d>d)cid>d>d>d>d>d>d>d> 4 1 - H ^ - H ^ - H - H - H - m H - H - H ^ ^ - H - H - H - H - H - H - H - H ^ - H - H - H - H - H - H - H - H - H - H - H - H - H -H-H-H^-H-H-H-H-H-H-H-H-H-H-HHH-H-H O ^ ^ I N ^ C O C O n 0 0 1 N 0 1 0 0 0 0 I N « a i « N M a t D W T t t D C D O i O M N © ^ H N N t O i f l t O C O N O i O i C O t O t O i O N O i O O ) ^ O N N N O O N O O O O ^ ^ O O l i O t O I S O O O l O O O O H i M N N C ^ i O i O O O N N W ^ a N i O C N N C O N W H C S H i O t D I ' t O a i X I ^ C O ( N H O i c s d o i N N H H i O T J d d H H l O i O ^ ^ T f ^ M C O H H H H M M ^ ^ M W H H H H W r t n C 0 « M C O C O H H 1 1 rH r-H CO CO ,-H i—1 1 1 1 1 1 1 1 1 1 1 1 11 1 I 1 1 1 1 1 1 i i—t i—1 — i 1 i—1 1 1 i—i — i 1— i 1 T—1 1 I 1 1 1 1 i—I T—I I I I CMCMCMC<lC^CMCMCMCMCMe^CNCMC<ICOeOCM<N^^CMCMCMCNCMC^ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d ^^^^^^^-H-H-H-H-H-^-H-H-H-H^-H-H-H-H-H-H-H-H-H-H-H-H-H+l-HO M ^ t D N ( M O ( N © H t D W ^ 0 1 i O O O ) C O c O W 0 0 0 0 0 1 N 0 1 0 ) C O M Q O N N O O O O N ^ W C O C O ^ ^ C O ! X ) C N n O ) O O Q 0 1 ^ t D ( X ) O a i O O H ^ M H H « ^ ^ | m © G O G O C O W C O O ) O N l C C O N H O O O i N O O O O C O l O l 0 ^ l O © t C l D C J t O N M M 0 1 0 ) 3 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o q o q q q q q q q q q q o o o o o o q q q q q q q q q q d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d -H^-H-H-H-H-H-H-H-H-H^-H-H-H-H-H-H-H-H^-U-H-W^-H-H-H-H-H-H-H-H-H-H-H H N C O O N ( D © I N C O H N n O ( M ^ ( O N « N N H r O O i t D H i O G O i O W H O H T j 0 1 0 0 • ^ W Q O ^ O N o O ^ m c O i M C O O O O t D i O N M N O i O C O O O H O T ^ O l M O t O i O O O O C O O ^H^HCNCNCN'-H>-H(M'-H'-i'-ir-ICSlrHOOOOO(M--iOO'-iOOOO<-l<-HOCOTHOrHCO o o o o o o o o o q q q q q q q q q q q o o o o o o o q q o q q q q q q d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d I I I I I I b O l M C N i O Q O I M N ^ H ^ r O H H C O H O C O N H i n O J O i f l m ^ C J O i a i C O O J r o C N ^ C D O C D in CO 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ci o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o cocococococoeoeocococ^cocococococococ^ s N©- HWC- HOt --OH- ^O- H! C-NHOO-CHl iS- DHI -WNH©a- Hit H-DHOi -OHOH-lH^O-OHN O-CHt S-OHi H-OHO^- lHt HO40IN^2-MHC i-OHOH-CHOl-COH O-OH(i-MOH m-^HcO-oHD O-OH^O-aHNi- ^HC h-OHCN-OH^ O-aH^i -(OHM-^ HC^ -NOH 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 H N N r O O O O O O O O O O o O O O c o ^ ^ m m t O O O O O O O O O O O O O O O O D O O O t O N N O ' - H ' O O O O O O H H i N - H ^ ' O O O O O O N H O O <pa<ffl<M^w<M<m<m<m<Pp 0 <^D3<fpq<^cQ<m<m<m<pp<pp<ffl<Pp<ffl 0 H O O O 0 wiOLninLnLnLnLnini/5iowinLniniOLnininininLnLDLninLni^inLnioinininininin 0 HFHC\JMCncO^^LOintOCONNCOO:O^C7300rHfHCS|tNfO«^^tDCDNNCOCOO,3Cr! 4H-H-H-H-H-H-H-H-H-H-H-H-H^-H-H-H-H-H~H4I-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H W N Q O r O « « i M N ^ a i N i O i O ( D O C D N 0 5 0 c O « H C O N N ( M ^ « H ( O a i C O H I X ) ^ H Q O O r o c O O ^ N H O I N C O O H D G M O O W a K O ^ O l C O O ^ W I M C D H C O N O J C O N O O N C O " * M t O O H O ) « 0 © © C l C ) O H H H r o H H O O ^ l O N C O H C O C O O ) T f L O C O ( N 1 0 C 0 0 5 ID m co ^ - ^ to | to io N | t o ^ t o i O H N ^ ^ ^ G O H o i • ^ c o ^ i C ' — i | H Tf io to <M | OS O ^ ( M ^ c s c O Q i O ^ O N f O M O i O O O C O O i n H N O N M N O l M N C O r - f O ^ N C O O i m H M C ^ M M H i X ) G ^ m ( » O O O i O ^ C O H i O © H i O ( r O ' ^ I N C O a ! i O C O C , O N ^ H ^ O ) O O H ^ O i ^ O r H O O r H i - f O i H O i - H f - H i - l ' - l i - l i - I O q ^ - I ^ H ^ H ^ H C S O r H i - l i - l i H i H i - l O q O i - H ' - l ' - l J- o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d -H-H-H-H^^-H^-H-H-H-H-H-H-H-H^^-H-H-H^41-H-H-H-H-H-H-H-H-H-H-H-H-H ^ ^ © Q O W H n t D i O O ^ O ^ ^ H O ^ N O C O N H O O l N N N O T f O O O O N i O O i O N O N N i O M O l O O O I N O i O C O m a t D ^ C O O c O O H i O H C O O Q O t O i O i O T f K O i O H H ^ O O O ' - H i - l O O ^ r H T - H O r H r H r H O O f - I C N ' - l ' - l i - l i - H O O O O r H O O O O ' - l i - H f - i O O ^ ^ ^ ^ ^ ^ ^ ^ r H ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ r - H ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ , o 00 g. • • Decijffset i iffset cin sky sidelobe Ratio, Rotation, <f> Det OJ > '(U u <u DC earn para meters for all operat:ionail 2006-2007 channels derived fr asar PKS 0537-441. All angul;ar si;zes and offsets on sky are in arc 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 ~a. 421.5 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 « « •£ c E .y o <u "a a: N o .E u Q. N (TJ 1/1 tJ * > o E o ro +-> °--o l/> OJ -* 0) > OJ o IE <" S > a c o E E- ™ ^ 8 "8-g faO O 4-> o bp O '+-> N "i_ J3 o a. LU o 0) N I/) LO o o ID T3 U C 4) E o u— c 3 CO O O o •4-> T3 0) o in 01 i_ _o o u N UOI^DUI|03Q 3 O LO E^ a. a . o .i-s Si i/l O Mapmaking 90 (L) ro O •j3 ro (sj - <u +-> 03 U 1— -C T3 *-< E O .5 ra bO _c a; ^ a E ° in 2 JE o g_- b i/i <u •J* O "O <D > ra O <[) •;= OJ <" % Q ro 3 a ra <u > +-> o o E £ c O c 1ro C +3 .^P ro 'i/i o E o ra c E o u "8 -a ^ =3 hO O +J JZ V o Q. LU id c 3.. Q . .2 o £ ra > V £ O ~ O £"? E •E o o o 0) 2 3? <L> _o •= — _ a. a> c o J5 y .O Ura 3 o 2 UOipUI|08Q <» x N d - o o OO ^ LD tu o bO ra '+-> (J „. <U ra "o -Q E 1/1 LU Mapmaking 3.2.7 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 (/) CD D 00 bO 3 *r "O <"> 2 - l/> OJ :?. ci' i/i .X DJ O o <— 3 cz cr Q. in' Q- cr fD fD fD E_ 3 CD _= l/l N n (/) 3 o' r+ fD 3 O to O in in X 3 ET CD Cn O to h-1 0 0 0 0 bo 00 0 t—' 1—I 0 0 O 00 b 1—• 4^ CO 0 0 CD CD CO cn 0 O ^ 0 on en 0 b0 0 CD 4=> 0 CO CO Ul CO 0 0.0 0.0 0.3 O O O In 4^ b CD O 0 0 CD 0 0 0 c? CO C/5 0 n N X N O O h—' n fD N X 0 P 0 0 0 O 0 0 CD • O 0 O CO 0 O M O 0 ^j 00 0 CO 1—I 1—I 1—' h-1 p 1—' O O £> CO 1—1 co cn 0 O O 0 O H^ CD CD b CD 4^ cn CD CO to CD CD O CD CO CO 4^ - J CD ^J to In b0 •<l O cn 0 O O 1—' CD 00 4^ CD CO 0 b0 to ^ 4^ O O O O to to CO 4^ O O *4=> CD O 0 1—' ^ 3 n O Cn O 1—> N O O O 1—1 Cfi C/3 O n N HI N X to kl cn t o O0 CD cn 0 CO CO 1—I - J CO CD CO ^11 CO C/3 1 — ' CO to to 4=> CO to I— H cn O kl ^ O 0 0 0 CO b b t—' to CO O CD CD O CO 00 4* to ^ CD cn 0 O O CD »—' 0 0 p 0 0 ^ CO CD 1 — > 0 CO CD cn CO p 0.6 — h CD © 0.5 0.0 h-' O Cn 4^ CO O O CD b CO t—' CO CD N deck 0 O 0 1-1 £> CD b 0 CO 1—• CO C/5 00 00 CD IZ> CD CO h-1 CD 0.0 Q- O 0 00 en •vj CO 00 CD en 4^ *> 0 0 to to CD to O 0 ^ 3 H H dd Cd W M H td W H W H H dd dd dd m M & CD CD O 3 0 spl it seas CD o 3 ^ O 0 ID 4^ t o £> CO CO 00 O to to O p p CO J — • 1—1 ^J >—* to <! On cn 0 0 O 4^ b b CD CD 0 to CD 0 w H H a m H 0.0 (/) cr 3 7 3 n o ^ r+ ^ o q cr> - L R3 fD o o h- ° O O/) "0 r*> < ' —I — 3 Q- n X° fD t—' ^ 0 0 0 to OI ^ CO CD 0 CO 4^ 0 0 Dd dd H Cd l-( Td scan tion 150 G CD S X N W w a W H H a td td I-! Td alpl 150 G <D 52. r+ o 3 .£ £ § =: >L =T c r -1- 1/1 ( 7 N O ) T o o fD o CT o i/> 3 " CD -"t - \ 3 ^P CD O00 T3 fD O r+ O00 Typ' eque OOST Typ^ eque to 4^ r+ fD r+_ n" i/i 3El) U) r+ fD ^< 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 / eo 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* * * _ * { {- 0|L- •.nM- i ^ijl ^ ^ T | BB K i 40- i „ ; _: : 20- l „t^ ^ - i !l h T{ l I i*t**.^}I-*^}H„i , H ; - *!*M 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. Bibliography [1] QUaD Collaboration: P. Ade, Bock, J, Bowden, M, Brown, M. L, Cahill, G, Carlstrom, J. E, Castro, P. G, Church, S, Culverhouse, T, Friedman, R, Ganga, K, Gear, W. K, Hinderks, J, Kovac, J, Lange, A. E, Leitch, E, Melhuish, S. J, Murphy, J. A, Orlando, A, Schwarz, R, O'Sullivan, C, Piccirillo, L, Pryke, C, Rajguru, N, Rusholme, B, Taylor, A. N, Thompson, K. L, Wu, E. Y. S, & Zemcov, M. (2007) ArXiv e-prints 705. [2] Pryke, C, Ade, P, Bock, J, Bowden, M, Brown, M. L, Cahill, G, Castro, P. G, Church, S, Culverhouse, T, Friedman, R, Ganga, K, Gear, W. K, Gupta, S, Hinderks, J, Kovac, J, Lange, A. E, Leitch, E, Melhuish, S. J, Memari, Y, Murphy, J. A, Orlando, A, Schwarz, R, O'Sullivan, C, Piccirillo, L, Rajguru, N, Rusholme, B, Taylor, A. N, Thompson, K. L, Turner, A. H, Wu, E. Y. S, & Zemcov,M. (2008) ArXiv e-prints 805. [3] QUaD collaboration: R. B. Friedman, Ade, P, Bock, J, Bowden, M, Brown, M. L, Cahill, G, Castro, P. G, Church, S, Culverhouse, T, Ganga, K, Gear, W. K, Gupta, S, Hinderks, J, Kovac, J, Lange, A. E, Leitch, E, Melhuish, S. J, Memari, Y, Murphy, J. A, Orlando, A, O' Sullivan, C, Piccirillo, L, Pryke, C, Rajguru, N, Rusholme, B, Schwarz, R, Taylor, A. N, Thompson, K. L, Turner, A. H, Wu, E. Y. S, & Zemcov, M. (2009) ArXiv e-prints. [4] QUaD collaboration: M. L. Brown, Ade, P, Bock, J, Bowden, M, Cahill, G, Castro, P. G, Church, S, Culverhouse, T, Friedman, R. B, Ganga, K, Gear, W K, Gupta, S, Hinderks, J, Kovac, J, Lange, A. E, Leitch, E, Melhuish, S. J, Memari, Y, Murphy, J. A, Orlando, A, O'Sullivan, C, Piccirillo, L, Pryke, C, Rajguru, N, Bibliography 196 Rusholme, B, Schwarz, R, Taylor, A. N, Thompson, K. L, Turner, A. H, Wu, E. Y. S, & Zemcov, M. (2009) ArXiv e-prints. [5] Penzias, A. A & Wilson, R. W. (1965) Astrophysical Journal 142, 419-421. [6] Dicke, R. H, Peebles, P. J. E, Roll, P. G, & Wilkinson, D. T. (1965) Astrophysical Journal 142, 414-419. [7] Smoot, G. F, Bennett, C. L, Kogut, A, Wright, E. L, Aymon, J, Boggess, N. W, Cheng, E. S, de Amici, G, Gulkis, S, Hauser, M. G, Hinshaw, G, Jackson, P. D, Janssen, M, Kaita, E, Kelsall, T, Keegstra, P, Lineweaver, C, Loewenstein, K, Lubin, P, Mather, J, Meyer, S. S, Moseley, S. H, Murdock, T, Rokke, L, Silverberg, R. F, Tenorio, L, Weiss, R, & Wilkinson, D. T. (1992) Astrophysical Journal Letters 396, L1-L5. [8] de Bernardis, P, Ade, P. A. R, Bock, J. J, Bond, J. R, Borrill, J, Boscaleri, A, Coble, K, Crill, B. P, De Gasperis, G, Farese, P. C, Ferreira, P. G, Ganga, K, Giacometti, M, Hivon, E, Hristov, V. V, Iacoangeli, A, Jaffe, A. H, Lange, A. E, Martinis, L, Masi, S, Mason, P. V, Mauskopf, P. D, Melchiorri, A, Miglio, L, Montroy, T, Netterfield, C. B, Pascale, E, Piacentini, F, Pogosyan, D, Prunet, S, Rao, S, Romeo, G, Ruhl, J. E, Scaramuzzi, F, Sforna, D, & Vittorio, N. (2000) Nature 404, 955-959. [9] Hinshaw, G, Barnes, C, Bennett, C. L, Greason, M. R, Halpern, M, Hill, R. S, Jarosik, N, Kogut, A, Limon, M, Meyer, S. S, Odegard, N, Page, L, Spergel, D. N, Tucker, G. S, Weiland, J. L, Wollack, E, & Wright, E. L. (2003) Astrophysical Journal Supplement 148, 63-95. [10] Kovac, J. M, Leitch, E. M, Pryke, C, Carlstrom, J. E, Halverson, N. W, & Holzapfel, W. L. (2002) Nature 420, 772-787. [11] Freedman, W L, Madore, B. F, Gibson, B. K, Ferrarese, L, Kelson, D. D, Sakai, S, Mould, J. R, Kennicutt, Jr., R. C, Ford, H. C, Graham, J. A, Huchra, J. P, Hughes, S. M. G, Illingworth, G. D, Macri, L. M, & Stetson, P. B. (2001) Astrophysical Journal 553, 47-72. Bibliography 197 [12] Clowe, D, Bradac, M, Gonzalez, A. H, Markevitch, M, Randall, S. W, lones, C, & Zaritsky, D. (2006) Astrophysical Journal Letters 648, L109-L113. [13] Komatsu, E, Dunkley, J, Nolta, M. R, Bennett, C. L, Gold, B, Hinshaw, G, Jarosik, N, Larson, D, Limon, M, Page, L, Spergel, D. N, Halpern, M, Hill, R. S, Kogut, A, Meyer, S. S, Tucker, G. S, Weiland, J. L, Wollack, E,, & Wright, E. L. (2009) The Astrophysical Journal Supplement Series 180,330-376. [14] Carroll, S. M. (2004) Spacetime and geometry. An introduction to general relativity. [15] Dodelson, S. (2003) Modern Cosmology. (Academic Press). [16] Liddle, A. R & Lyth, D. H. (2000) Cosmological Inflation and Large-Scale Structure. [17] Gibbons, G. W & Hawking, S. W. (1977) Phys. Rev. D 15, 2738-2751. [18] Bock, J, Aljabri, A, Amblard, A, Baumann, D, Betoule, M, Chui, T, Colombo, L, Cooray, A, Crumb, D, Day, P, Dickinson, C, Dowell, D, Dragovan, M, Golwala, S, Gorski, K, Hanany, S, Holmes, W, Irwin, K, Johnson, B, Keating, B, Kuo, C L, Lee, A, Lange, A, Lawrence, C, Meyer, S, Miller, N, Nguyen, H, Pierpaoli, E, Ponthieu, N, Puget, J.-L, Raab, J, Richards, P, Satter, C, Seiffert, M, Shimon, M, Tran, H, Williams, B, & Zmuidzinas, J. (2009) ArXiv e-prints. [19] Lewis, A, Challinor, A, & Lasenby, A. (2000) Astrophys. J. 538, 473-476. [20] White, M, Scott, D, & Silk, J. (1994) Annual Review of Astronomy and Astrophysics 32, 319-370. [21] Hu, W & White, M. (1997) New Astronomy 2, 323-344. [22] Venkatesan, A. (2000) Astrophysical Journal 537, 55-64. [23] Dunkley, J, Komatsu, E, Nolta, M. R, Spergel, D. N, Larson, D, Hinshaw, G, Page, L, Bennett, C. L, Gold, B, Jarosik, N, Weiland, J. L, Halpern, M, Hill, Bibliography 198 R. S, Kogut, A, Limon, M, Meyer, S. S, Tucker, G. S, Wollack, E, & Wright, E. L. (2009) Astrophysical Journal Supplement 180, 306-329. [24] Brown, M. L, Castro, P. G, & Taylor, A. N. (2005) Monthly Notices of the Royal Astronomical Society 360, 1262-1280. [25] Hu,W. (2000) Phys. Rev. D 62, 043007. [26] Smith, K. M, Hu, W, & Kaplinghat, M. (2004) Physical Review D 70, 043002-+. [27] Hu,W. (2001) Astrophysical Journal Letters 557, L79-L83. [28] Chiang, H. C, Ade, P. A. R, Barkats, D, Battle, J. O, Bierman, E. M, Bock, J. J, Dowell, C. D, Duband, L, Hivon, E. F, Holzapfel, W. L, Hristov, V. V, Jones, W. C, Keating, B. G, Kovac, J. M, Kuo, C. L, Lange, A. E, Leitch, E. M, Mason, P. V, Matsumura, T, Nguyen, H. T, Ponthieu, N, Pryke, C, Richter, S, Rocha, G, Sheehy, C, Takahashi, Y. D, Tolan, J. E, & Yoon, K. W. (2009) ArXiv e-prints. [29] Hinderks, J, Ade, P, Bock, J, Bowden, M, Brown, M. L, Cahill, G, Carlstrom, J. E, Castro, P. G, Church, S, Culverhouse, T, Friedman, R, Ganga, K, Gear, W. K, Gupta, S, Harris, J, Haynes, V, Kovac, J, Kirby, E, Lange, A. E, Leitch, E, Mallie, O. E, Melhuish, S, Murphy, A, Orlando, A, Schwarz, R, O' Sullivan, C, Piccirillo, L, Pryke, C, Rajguru, N, Rusholme, B, Taylor, A. N, Thompson, K. L, Tucker, C, Wu, E. Y. S, & Zemcov, M. (2008) ArXiv e-prints 805. [30] Zemcov, M. (2006) Ph.D. thesis (Cardiff University). [31] Lane, A. P. (1998) Submillimeter Transmission at South Pole, Astronomical Society of the Pacific Conference Series eds. Novak, G & Landsberg, R. Vol. 141, pp. 289-+. [32] Schlegel, D. J, Finkbeiner, D. P, & Davis, M. (1998) Astrophysical Journal 500, 525-+. [33] Farese, P. C, Dall'Oglio, G, Gundersen, J. O, Keating, B. G, Klawikowski, S, Knox, L, Levy, A, Lubin, P. M, O'Dell, C. W, Peel, A, Piccirillo, L, Ruhl, J, & Timbie,P.T. (2004) Astrophysical Journal 610, 625-634. Bibliography 199 [34] O'Sullivan, C, Cahill, G, Murphy, J. A, Gear, W. K, Harris, J, Ade, P. A. R, Church, S. E, Thompson, K. L, Pryke, C, Bock, J, Bowden, M, Brown, M. L, Carlstrom, J. E, Castro, P. G, Culverhouse, T, Friedman, R. B, Ganga, K. M, Haynes, V, Hinderks, J. R, Kovak, J, Lange, A. E, Leitch, E. M, Mallie, O. E, Melhuish, S. J, Orlando, A, Piccirillo, L, Pisano, G, Rajguru, N, Rusholme, B. A, Schwarz, R, Taylor, A. N, Wu, E. Y. S, & Zemcov, M. (2008) Infrared Physics and Technology 51, 277-286. [35] Kuo, C. L, Ade, P. A. R, Bock, J. J, Cantalupo, C, Daub, M. D, Goldstein, J, Holzapfel, W. L, Lange, A. E, Lueker, M, Newcomb, M, Peterson, J. B, Ruhl, J, Runyan, M. C, & Torbet, E. (2004) Astrophysical Journal 600, 32-51. [36] Heidt, J, Jager, K, Nilsson, K, Hopp, U, Fried, J. W, & Sutorius, E. (2003) Astronomy and Astrophysics 406, 565-577. [37] Yoon, K. W, Ade, P. A. R, Barkats, D, Battle, J. O, Bierman, E. M, Bock, J. J, Brevik, J. A, Chiang, H. C, Crites, A, Dowell, C. D, Duband, L, Griffin, G. S, Hivon, E. F, Holzapfel, W. L, Hristov, V. V, Keating, B. G, Kovac, J. M, Kuo, C. L, Lange, A. E, Leitch, E. M, Mason, P. V, Nguyen, H. T, Ponthieu, N, Takahashi, Y. D, Renbarger, T, Weintraub, L. C, & Woolsey, D. (2006) The Robinson Gravitational Wave Background Telescope (BICEP): a bolometric large angular scale CMB polarimeter, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. Vol. 6275. [38] Masi, S, Ade, P. A. R, Bock, J. J, Bond, J. R, Borrill, J, Boscaleri, A, Cabella, P, Contaldi, C. R, Crill, B. P, de Bernardis, P, de Gasperis, G, de Oliveira-Costa, A, de Troia, G, di Stefano, G, Ehlers, P, Hivon, E, Hristov, V, Iacoangeli, A, Jaffe, A. H, Jones, W. C, Kisner, T. S, Lange, A. E, MacTavish, C. J, Marini Bettolo, C, Mason, P, Mauskopf, P. D, Montroy, T. E, Nati, F, Nati, L, Natoli, P, Netterfield, C. B, Pascale, E, Piacentini, F, Pogosyan, D, Polenta, G, Prunet, S, Ricciardi, S, Romeo, G, Ruhl, J. E, Santini, P, Tegmark, M, Torbet, E, Veneziani, M, & Vittorio, N. (2006) Astronomy and Astrophysics 458, 687-716. Bibliography 200 [39] Jones, W. C, Bhatia, R, Bock, J. J, & Lange, A. E. (2003) A Polarization Sensitive Bolometric Receiver for Observations of the Cosmic Microwave Background eds. Phillips, T. G & Zmuidzinas, J. pp. 227-238. [40] Hivon, E, Gorski, K. M, Netterfield, C. B, Crill, B. P, Prunet, S, & Hansen, F. (2002) Astrophysical Journal 567, 2-17. [41] Spergel, D. N, Bean, R, Dore, O, Nolta, M. R, Bennett, C. L, Dunkley, J, Hinshaw, G, Jarosik, N, Komatsu, E, Page, L, Peiris, H. V, Verde, L, Halpern, M, Hill, R. S, Kogut, A, Limon, M, Meyer, S. S, Odegard, N, Tucker, G. S, Weiland, J. L, Wollack, E, & Wright, E. L. (2007) Astrophysical Journal Supplement 170, 377-408. [42] Masi, S, Ade, P, Bock, J, Bond, J, Borrill, J, Boscaleri, A, Cabella, P, Contaldi, C, Crill, B, de Bernardis, P, De Gasperis, G, de Oliveira-Costa, A, De Troia, G, Di Stefano, G, Ehlers, P, Hivon, E, Hristov, V, Iacoangeli, A, Jaffe, A, Jones, W, Kisner, T, Lange, A, MacTavish, C, Marini-Bettolo, C, Mason, P, Mauskopf, P, Montroy, T, Nati, F, Nati, L, Natoli, P, Netterfield, C, Pascale, E, Piacentini, F, Pogosyan, D, Polenta, G, Prunet, S, Ricciardi, S, Romeo, G, Ruhl, J, Santini, P, Tegmark, M, Torbet, E, Veneziani, M, & Vittorio, N. (2005) ArXivAstrophysics e-prints. [43] Tegmark, M. (1997) Phys. Rev. D 55, 5895-5907. [44] Jones, W. (2005) Ph.D. thesis (California Institute of Technology). [45] Ganga, K. (2006) QUaD Analysis Notes. Internal unpublished QUaD collaboration document. [46] Weisstein, Eric (MathWorld-A W. Wolfram (2009) Web Least Squares Resource), Fitting-Polynomial, Technical report. http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html. [47] Gregory, P. C, Vavasour, J. D, Scott, W. K, & Condon, J. J. Astrophysical Journal Supplement 90,173-177. (1994) Bibliography 201 [48] White, M, Carlstrom, J. E, Dragovan, M, , & Holzapfel, W. L. (1999) The Astrophysical Journal 514,12-24. [49] Gorski, K. M, Hivon, E, Banday, A. J, Wandelt, B. D, Hansen, F. K, Reinecke, M, & Bartelmann, M. (2005) Astrophysical Journal 622, 759-771. [50] Frigo,M& Johnson, S.G. (2005) Proceedings of the IEEE 93,216-231. Special issue on "Program Generation, Optimization, and Platform Adaptation". [51] Seljak, U. (1997) The Astrophysical Journal 482, 6-16. [52] Piacentini, F, Ade, P. A. R, Bock, J. J, Bond, J. R, Borrill, J, Boscaleri, A, Cabella, P, Contaldi, C. R, Crill, B. P, de Bernardis, P, De Gasperis, G, de Oliveira-Costa, A, De Troia, G, di Stefano, G, Hivon, E, Jaffe, A. H, Kisner, T. S, Jones, W. C, Lange, A. E, Masi, S, Mauskopf, P. D, MacTavish, C. J, Melchiorri, A, Montroy, T. E, Natoli, P, Netterfield, C. B, Pascale, E, Pogosyan, D, Polenta, G, Prunet, S, Ricciardi, S, Romeo, G, Ruhl, J. E, Santini, P, Tegmark, M, Veneziani, M, & Vittorio, N. (2006) Astrophysical Journal 647, 833-839. [53] Nolta, M. R, Dunkley, J, Hill, R. S, Hinshaw, G, Komatsu, E, Larson, D, Page, L, Spergel, D. N, Bennett, C. L, Gold, B, Jarosik, N, Odegard, N, Weiland, J. L, Wollack, E, Halpern, M, Kogut, A, Limon, M, Meyer, S. S, Tucker, G. S, & Wright, E. L. (2009) Astrophysical Journal Supplement 180, 296-305. [54] Gorski, K. M, Hivon, E, Banday, A. J, Wandelt, B. D, Hansen, F. K, Reinecke, M, & Bartelmann, M. (2005) Astrophysical Journal 622, 759-771. [55] Hinshaw, G, Nolta, M. R, Bennett, C. L, Bean, R, Dore, O, Greason, M. R, Halpern, M, Hill, R. S, Jarosik, N, Kogut, A, Komatsu, E, Limon, M, Odegard, N, Meyer, S. S, Page, L, Peiris, H. V, Spergel, D. N, Tucker, G. S, Verde, L, Weiland, J. L, Wollack, E, & Wright, E. L. (2006) ArXiv Astrophysics e-prints. [56] Bond, J. R, Jaffe, A. H, & Knox, L. (1998) Phys. Rev. D 57, 2117-2137. [57] Reichardt, C. L, Ade, P. A. R, Bock, J. J, Bond, J. R, Brevik, J. A, Contaldi, C. R, Daub, M. D, Dempsey, J. T, Goldstein, J. H, Holzapfel, W L, Kuo, C. L, Bibliography 202 Lange, A. E, Lueker, M, Newcomb, M, Peterson, J. B, Ruhl, J, Runyan, M. C, & Staniszewski, Z. (2009) Astrophysical Journal 694,1200-1219. [58] Stompor, R, Balbi, A, Borrill, J. D, Ferreira, P. G, Hanany, S, Jaffe, A. H, Lee, A. T, Oh, S, Rabii, B, Richards, P. L, Smoot, G. F, Winant, C. D, & Wu, J.-H. P. (2002) Physical Review D 65, 022003-+. [59] Seljak, U & Zaldarriaga, M. (1996) Astrophysical Journal 469, 437-+. [60] Harrison, E. R. (1970) Phys. Rev. D 1, 2726-2730. [61] Zeldovich, Y. B. (1972) Monthly Notices of the Royal Astronomical Society 160, 1P-+. [62] Kosowsky, A, Milosavljevic, M, & Jimenez, R. (2002) Phys. Rev. D 66, 063007. [63] Pryke, C, Halverson, N. W, Leitch, E. M, Kovac, J, Carlstrom, J. E, Holzapfel, W. L, & Dragovan, M. (2002) Astrophysical Journal 568, 46-51. [64] Perlmutter, S, Aldering, G, Goldhaber, G, Knop, R. A, Nugent, P, Castro, P. G, Deustua, S, Fabbro, S, Goobar, A, Groom, D. E, Hook, I. M, Kim, A. G, Kim, M. Y, Lee, J. C, Nunes, N. J, Pain, R, Pennypacker, C. R, Quimby, R, Lidman, C, Ellis, R. S, Irwin, M, McMahon, R. G, Ruiz-Lapuente, P, Walton, N, Schaefer, B, Boyle, B. J, Filippenko, A. V, Matheson, T, Fruchter, A. S, Panagia, N, Newberg, H. J. M, Couch, W. J, & The Supernova Cosmology Project. (1999) Astrophysical Journal 517, 565-586. [65] QUaD collaboration: P. G. Castro, Ade, P, Bock, J, Bowden, M, Brown, M. L, Cahill, G, Church, S, Culverhouse, T, Friedman, R. B, Ganga, K, Gear, W. K, Gupta, S, Hinderks, J, Kovac, J, Lange, A. E, Leitch, E, Melhuish, S. J, Memari, Y, Murphy, J. A, Orlando, A, Pryke, C, Schwarz, R, O'Sullivan, C, Piccirillo, L, Rajguru, N, Rusholme, B, Taylor, A. N, Thompson, K. L, Turner, A. H, Wu, E. Y. S, & Zemcov, M. (2009) ArXiv e-prints. [66] Verde, L, Peiris, H. V, Spergel, D. N, Nolta, M. R, Bennett, C. L, Halpern, M, Hinshaw, G, Jarosik, N, Kogut, A, Limon, M, Meyer, S. S, Page, L, Tucker, Bibliography 203 G. S, Wollack, E, & Wright, E. L. (2003) Astrophysical Journal Supplement 148, 195-211. [67] Bond, J. R, Jaffe, A. H, , & Knox, L. (2000) The Astrophysical Journal 533, 19-37. [68] Gelman, A, Leenen, I, Mechelen, I. V, & Boeck, P. D. (1992) Bridges between deterministic and probabilistic classification models. [69] Tegmark, M, Eisenstein, D, Strauss, M, Weinberg, D, Blanton, M, Frieman, J, Fukugita, M, Gunn, J, Hamilton, A, Knapp, G, Nichol, R, Ostriker, J, Padmanabhan, N, Percival, W, Schlegel, D, Schneider, D, Scoccimarro, R, Seljak, U, Seo, H, Swanson, M, Szalay, A, Vogeley, M, Yoo, J, Zehavi, I, Abazajian, K, Anderson, S, Annis, J, Bahcall, N, Bassett, B, Berlind, A, Brinkmann, J, Budavari, T, Castander, F, Connolly, A, Csabai, I, Doi, M, Finkbeiner, D, Gillespie, B, Glazebrook, K, Hennessy, G, Hogg, D, Ivezic, Z, Jain, B, Johnston, D, Kent, S, Lamb, D, Lee, B, Lin, H, Loveday, J, Lupton, R, Munn, J, Pan, K, Park, C, Peoples, J, Pier, J, Pope, A, Richmond, M, Rockosi, C, Scranton, R, Sheth, R, Stebbins, A, Stoughton, C, Szapudi, I, Tucker, D, Berk, D. V, Yanny, B, & York, D. (2006) Physical Review D 74, 123507. [70] Lue, A, Wang, L, & Kamionkowski, M. (1999) Physical Review Letters 83, 1506-1509. [71] Lepora, N. E (1998) ArXiv General Relativity and Quantum Cosmology e-prints. [72] Kamionkowski, M, Kosowsky, A, & Stebbins, A. (1997) Physical Review D 55, 7368-7388. [73] Kamionkowski, M, Kosowsky, A, & Stebbins, A. Letters 78, 2058-2061. (1997) Physical Review [74] Zaldarriaga, M & Seljak, U. (1997) Physical Review D 55,1830-1840. Bibliography 204 [75] Scoccola, C, Harari, D, & Mollerach, S. (2004) Physical Review D 70, 063003+. [76] Gardner, S. (2008) Phys. Rev. Lett. 100, 041303. [77] Scannapieco, E. S & Ferreira, P. G. (1997) Physical Review D 56, 7493-+. [78] Feng, B, Li, H, Li, M, & Zhang, X. (2005) Physics Letters B 620, 27-32. [79] Carroll, S. M, Field, G. B, & Jackiw, R. (1990) Phys. Rev. D 41,1231-1240. [80] Kostelecky, A & Mewes, M. (2008) ArXiv e-prints 809. [81] Xia, J.-Q, Li, H, Zhao, G.-B, & Zhang, X. (2008) Astrophysical Journal Letters 679, L61-L63. [82] Kostelecky, A & Russell, N. (2008) ArXiv e-prints. [83] Ni,W.-T. (1977) Physical Review Letters 38, 301-304. [84] Carroll, S. M & Field, G. B. (1991) Phys. Rev. D 43, 3789-3793. [85] Komatsu, E, Dunkley, J, Nolta, M. R, Bennett, C. L, Gold, B, Hinshaw, G, Jarosik, N, Larson, D, Limon, M, Page, L, Spergel, D. N, Halpern, M, Hill, R. S, Kogut, A, Meyer, S. S, Tucker, G. S, Weiland, J. L, Wollack, E, & Wright, E.L. (2008) ArXiv e-prints 803.

1/--страниц