# Measuring small-scale anisotropy in the Cosmic Microwave Background with the Sunyaev-Zel'dovich Array

код для вставкиСкачатьTHE UNIVERSITY OF CHICAGO MEASURING SMALL-SCALE ANISOTROPY IN THE COSMIC MICROWAVE BACKGROUND WITH THE SUNYAEV-ZEL'DOVICH ARRAY A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS BY MATTHEW K. SHARP CHICAGO, ILLINOIS AUGUST 2008 UMI Number: 3322639 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 3322639 Copyright 2008 by ProQuest LLC. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 E. Eisenhower Parkway PO Box 1346 Ann Arbor, Ml 48106-1346 Copyright © 2008 by Matthew K. Sharp All rights reserved For Nanny, who will fly over us with presents forever. ABSTRACT Using the Sunyaev Zel'dovich Array (SZA), an eight-element interferometer observing at 30 GHz, we make a measurement of the fluctuation power spectrum of Cosmic Microwave Background (CMB) anisotropy on angular scales corresponding to several arcminutes. Within this range, we expect the primary CMB to be greatly damped and dominated by secondary contributions, especially the Sunyaev Zel'dovich Effect from intermediate-mass galaxy clusters. We measure 93 ± 66yuK2 of power in 44 fields with the SZA. We attribute 30/uK2 of this power to primary CMB, and a further ~ 5 — ICty/K2 to galactic dust grains that are spinning. We believe that we have accounted for all power contributed by radio point sources. The remaining 60±66//K 2 of power we attribute to the Sunyaev Zel'dovich Effect, consistent with simulations of structure with ag—Q.8. While previous measurements have shown some tension with such a low value of eg, we see no such tension. We introduce the measurement with a brief description of erg and its role in the standard cosmological model, and then describe the SZA and our observations. The analysis that we have conducted is described in detail, as well as our interpretation of the results. IV ACKNOWLEDGMENTS My colleagues at the Univerisity of Chicago form the most uniformly intelligent group of people I have met in my life; they are the people who, through their lives, have found easy what other people find difficult, and who have come here looking for a great challenge. This dissertation chronicles work that I found to be immensely challenging, and these acknowledgments recognize the people without whom I would certainly have failed in it. Being John's student is a like being a passenger in a car barreling down a curvy road in the fog; he's comfortable much closer to disaster than most of us are, if only because he sees the road ahead more clearly than we do. The best part of graduate school has been working with John; that such a blisteringly brilliant person can at the same time be so much fun to spend time with gives me faith that there are some people who are truly good. The hardest part of graduate school has been coming to grips with the fact that I will never be as good at anything as John is at this. I hope to consider myself his student for many more years. The original SZA students, Mike, Ryan and Chris in Chicago and Stephen and Tony at Columbia, have been very much a group of brothers, fighting through our shared struggles and becoming closer than many coworkers do. They are the closest friends I've made during these years. Dan Marrone joined the SZA just in time to turn us around, when we needed him most. His broad knowledge, generous patience, and quick wit were immeasurably helpful; he is a great scientist and friend, and I am confident I would not have completed this work in his absence. This does not excuse the fact that he voted for Ralph Nader. Tom Culverhouse made the hard times easier with deck chairs, curry, and a willv vi ingness to think about other people's problems when he had his own. Clem Pryke is better than anyone I know at ferreting out the most important detail and seeing clearly the path forward. His generous advice was uniformly on-target and illuminating; had I talked with him twice as often, I would have finished the project twice as quickly, and with a more refined sense of cursing. James Lamb is the most curious, broadly intelligent, and kind person I have ever met. Cecil Patrick showed me that there's more to the Owens Valley than the inside of our equipment trailer. Erik Leitch's wide-ranging conversation would make him a great person to spend time with even if he wasn't so terribly clever and even if his fish tacos weren't so tasty. Amber Miller, Marshall Joy, Val Smith, David Woody, Dave Hawkins and Ben Reddall completed the SZA team; they were all wonderful to work with, and the only word to describe the feeling at leaving this group is sadness. I gratefully acknowledge the National Science Foundation's Graduate Research Fellowship Program and the Grainger Fellowship, which shared the burden of supporting me while I indulged the filthy habit of physics. Finally, I cannot fail to thank my parents and Emily, who went so far as to bring Christmas to OVRO, and my friends. Christine was at the center of my life for much of my time here. Thanks to Raf and Margo for the camping, to Robert, Keith, Dan and Jenny for the parties, to Reid, Sam and Chris for exploring the underbelly of Baltimore with me, and to Abby for introducing me to swimming in the lake when I was most desperate to find the good parts of Hyde Park. You made my years in Chicago very happy ones. Thank you all. TABLE OF CONTENTS ABSTRACT iv ACKNOWLEDGMENTS v LIST OF FIGURES x LIST OF TABLES xii 1 1 1 4 7 9 11 13 14 16 17 21 22 24 28 INTRODUCTION 1.1 Cosmology: Confronting Data at Last 1.1.1 Measuring the Universe: Cosmological Parameters 1.1.2 The Collapsing Universe: Structure Formation 1.2 Normalizing the Power Spectrum -<jg 1.2.1 Defining a$ 1.2.2 Structure Formation and ag: The Press-Schechter Approach . 1.2.3 Previous Measurements of ag 1.3 Measuring ag with the SZ Effect 1.3.1 The CMB Power Spectrum 1.3.2 Secondary Anisotropy 1.3.3 SZ Effect 1.3.4 SZ Power Spectrum 1.3.5 Power Spectrum Measurements 2 THE SUNYAEV ZEL'DOVICH ARRAY 2.1 The Antennas 2.1.1 Optical Path 2.1.2 Beam Patterns 2.1.3 Array Layout 2.2 Receivers 2.2.1 Amplification and Mixing 2.2.2 Phase Lock System 2.3 Downconverter and Correlator 2.4 System Temperature 2.5 Calibration 29 29 29 31 34 36 36 40 42 44 48 3 52 52 53 55 57 63 OBSERVATIONS 3.1 Field Selection 3.2 Observation Strategy 3.3 Data Calibration and Quality Checking 3.4 Filtering for Antenna Crosstalk 3.5 VLA 8 GHz data vii viii 3.5.1 3.5.2 VLA Data VLA Reduction 63 66 4 LIKELIHOOD FORMALISM 4.1 Visibilities and the Power Spectrum 4.2 The Covariance Matrix 4.2.1 Noise Estimation 4.2.2 Constraint Matrices 4.3 Putting it into Practice 69 69 73 74 78 79 5 FOREGROUNDS 5.1 Removing Sources 5.1.1 Subtracting Known Sources 5.1.2 Marginalizing over Source Fluxes 5.1.3 Projecting Sources with Constraint Matrices 5.1.4 Spectral Indices 5.2 Known Synchrotron Sources 5.2.1 SZA Sources 5.2.2 NVSS/FIRST Sources 5.2.3 8 GHz VLA Sources 5.3 Point Source Simulations 5.3.1 Source Distributions 5.3.2 Simulation Results 5.4 Other Foregrounds 5.4.1 Dusty Protogalaxies 5.4.2 Thermal Galactic Dust 5.4.3 Galactic Spinning Dust 5.5 Primary CMB 82 83 84 85 86 88 88 88 89 91 92 92 95 98 98 99 100 103 6 POWER SPECTRUM RESULTS 6.1 Jackknife Tests 6.2 Data Splits 105 105 107 7 COSMOLOGICAL INTERPRETATION 7.1 Simulation Strategy 7.2 Simulated Maps 7.2.1 Full Gas Simulation 7.2.2 Semianalytic Gas 7.2.3 Pinocchio 7.3 Non-Gaussian Sample Variance 7.4 Power and ag 7.5 Correlations between Point Sources and Clusters Ill Ill 112 113 113 114 115 118 118 ix 8 CONCLUSIONS 123 A RADIO SOURCES DETECTED IN CMB FIELDS 125 B NUMBERS OF SOURCES CONSTRAINED 128 REFERENCES 130 LIST OF FIGURES 1.1 1.2 1.3 1.4 1.5 1.6 1.7 The Milky Way Simulated and Observed Structure Constraints on ag The CMB Sky The CMB Power Spectrum The Spectrum of the SZ Effect The SZ Power Spectrum 2 10 17 19 20 25 27 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 SZA optics path An SZA antenna SZA beam Beam and Illumination Patterns Measured SZA Beams Holographic phase measurements SZA Array Layout SZA Synthesized Beams The receiver chain at 30 GHz Geometrical phase delay The system noise temperature vs time for a typical track 31 32 34 35 36 37 38 39 40 42 48 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 The SZA observation fields Calibration of the Data Bad x2 data Ring Sidelobes An example of extreme crosstalk The second level of crosstalk filtering The final stage of crosstalk filtering The 19-pointing VLA mosaicking scheme 53 56 58 59 60 61 62 65 4.1 4.2 Examples of the Covariance Matrix An SZA scan 74 80 5.1 5.2 5.3 5.4 An example of an SZA field with a bright radio source in it 30 GHz radio source flux distributions Comparison of source models and spectral indices Measured power after 30 GHz point source constraints in simulation and data Measured Power vs VLA cutoff Thermal dust power spectra SZA window function Spinning dust power spectra 90 93 95 5.5 5.6 5.7 5.8 x 96 97 100 103 104 XI 6.1 6.2 6.3 Autocorrelation Function and Jackknife u — v Power measured by SZA Power in different splits of the data 106 108 109 7.1 7.2 7.3 Power Distribution for Gaussian and non-Gaussian skies Non-Gaussian Sample Variance Power vs cr8 116 117 119 LIST OF TABLES 3.1 Flagged Data Volume 63 3.2 The SZA field locations, integration times, and phase calibrators. 5.1 Measurements of spinning dust emissitivities, scaled to IRAS 100 /im 6.1 Jackknife Data Results for 4a 30 GHz and 8a 8 GHz source removal. . 107 6.2 Contributions to total power measured . . 67 102 110 A.l Sources detected at 30 GHz with the SZA 125 B.l Number of Sources Constrained 128 xii CHAPTER 1 INTRODUCTION In this thesis, I will describe a measurement of small-scale power in the Cosmic Microwave Background (CMB) made with the Sunyaev Zel'dovich Array (SZA). In the following chapters, I will discuss the SZA instrument (Chapter 2) and the data that I've taken with it and also the Very Large Array (Chapter 3). In Chapter 4, I will discuss the likelihood techniques with which I have made the measurement. Chapter 5 will deal with the challenges of dealing with our foregrounds, and results will be given in Chapter 6. A brief discussion of the cosmological implications of these measurements will be given in Chapter 7. In this introduction, I attempt to sketch the background and motivation of this work. I describe the recent advances in observational cosmology, emphasizing the parameter ag and its role in models of structure formation as a motivation for measuring the power spectrum of the CMB on small scales. 1.1 Cosmology: Confronting Data at Last Figure 1.1 shows our galaxy, the Milky Way. Although Chicago residents could be forgiven for not realizing it, the Milky Way is visible with the naked eye from many parts of the world. Indeed, it is difficult to stare up at the sky without experiencing, at risk of violating the distant formality of a dissertation, a sense of wonder at the majestic vastness of the universe around us. As far back as traces of human culture exist, we see evidence that people have wondered at the night sky, constructing mythologies to explain its existence and structures to study its workings from Machu Pichu to Giza to Salisbury. Cosmology, the study of the broadest reaches of the universe, is in this sense part of the common heritage that binds humankind together. 1 2 Figure 1.1: The Milky Way 3 For all of its universal interest, the enormous scale of cosmology has long dwarfed our modest ability to study it, and as a result our progress has been slow. By 500 BC, Pythagoras had attempted to describe the motion of the earth around a "central fire" in terms of mathematics; he tried in particular to understand the movement of the heavenly bodies in terms of harmonics, like the standing waves on a string. It took nearly two millennia before Brahe's measurements enabled Kepler to replace Pythagoras's circles with ellipses. Within another century, Newton's gravitational theory assigned mathematical rules to the movement of objects in the universe, a model replaced only in the twentieth century by Einstein's theory of General Relativity. The elegance of scientific progress is encapsulated in the periodic table of the elements: diverse phenomena in chemistry can be simply understood through the table, which introduces simplicity and order to a vast range of observations. Cosmology has long headed in the opposite direction, introducing greater and greater complexity; as observations pushed beyond what Pythagoras and Kepler had to describe, it seemed that the universe made less sense, and not more. Einstein's theory, and the work that followed, provided an elegant framework for describing the universe's behavior in the presence of various kinds of material. But the range of possible universes, all of which fell under the rules of general relativity, was vast, and our ability to discern which of these possible universes we inhabit was frustratingly inadequate. As recently as several decades ago, it could be debated whether cosmology could be an empirical science at all; although models were plentiful, observations to confirm or refute them were very difficult to come by, and it seemed that understanding the universe on its largest scales might remain beyond the limits of human knowledge. There were two barriers to moving cosmology forward: the difficulty of the measurements, and the complexity of the theory. Both of these barriers have been breached in recent years. 4 1.1.1 Measuring the Universe: Cosmological Parameters The universe is a big place, and the objects and events that a cosmologist would like to study are far away in time and space. Laboratory experiments are nearly impossible, and so the inquisitive cosmologist is faced with a bleak landscape; the questions are large and difficult, but there is frustratingly little information available upon which to form answers. The past decades have seen the birth of "observational cosmology," in which scientists manage to evade this difficult constraint. In this era, experiments can measure cosmological parameters that the standard model leaves free, thereby constraining which of all possible universes we live in. It is a testament to the enormous recent advances in observations that these parameters have gone from being essentially unconstrained to being known at high precision in several decades. It is a further testament to the strength of the standard model that the universe, complex as it is, can be described with so few of these parameters. These numbers summarize the holes in our understanding: they are the things that the theory cannot tell us, but that we must instead tell the theory. A perfectly predictive theory would have zero parameters- the fewer the better. All observations to date are consistent with the so-called Lambda Cold Dark Matter cosmology, which in its simplest implementation has only six parameters. These include the amount of baryons, dark matter, and dark energy that the universe contains (given by f^, f2^m, a n a - ^A> respectively, with the requirement that Q-Tot ~ 1)' tne amplitude and spectrum of fluctuations in the initial density field (erg and ns), and the Hubble constant, which tells us how fast the universe is expanding today (Ho)- Given these six numbers, the theory describes the rest, and the entirety of the large scale universe is apparently comprehensible. By comparison, I had to input nearly 30 unique numbers to figure out how much federal income tax to pay in 2007, perhaps revealing that the universe is simple, and life (or 5 at least the United States Tax Code) is not. There are several recent experimental breakthroughs that have measured these parameters and pushed cosmology into the regime where the data are up to the task of confronting the theory: • Dark Matter- The existence of dark matter was suggested by Zwicky in 1933 to explain the fact that the light from galaxies did not suggest enough mass to explain the rotation of galaxies within clusters. However, the details of the dark matter distribution in the universe took fifty years to understand. Estimates of the amount of dark matter in the universe came from clusters of galaxies; Big Bang Nucleosynthesis (BBN) constrains the amount of baryons in the universe based on the abundance of many light elements today. Clusters, which are large enough to be representative of the global abundances of various matter components, allow one to measure the mass in baryons (through the optical and X-ray emission of stars and hot gas) and also the total mass (through the dynamics and distribution of galaxies). Measuring the ratio between visible and invisible matter in clusters, and knowing the total amount of baryons from BBN, allows us to constrain the total amount of unseen, dark matter in the universe (Q,M ~ 0.3) (White et al., 1993). Additional constraints on the amount and the clustering of the dark matter follow from studies of the absorption of radiation from very distant quasars by the intervening matter (Cen et al., 1994), and from the deflection of background light from the gravitational influence of dark matter haloes (Kaiser, 1992). • The Cosmic Microwave Background (CMB)-Shortly after the Big Bang, the universe was hot enough to ionize Hydrogen, and the free ions interacted closely with the photons, locking them in a thick soup. When the temperature 6 dropped low enough for the hydrogen to recombine, the photons were released and streamed freely to us, 14 billion years later. With them, they took a detailed picture of the temperature at this early stage, and their temperature reflects the density pattern in this early photon-baryon soup. The dynamics of this early, relatively simple environment depend strongly on the various components that are present, and so anisotropy in the surface of last scattering contains a wealth of information about the age, geometry, expansion, and composition of the universe (for a complete review of the cosmological import of the CMB, see Hu & Dodelson (2002)). Mapping the temperature of the last scattering surface is difficult, as the anisotropy is at the level of 10 . The CMB monopole was detected by Penzias and Wilson in 1965, but it wasn't until thirty years later that COBE detected the anisotropy (Smoot et al., 1992). In the years that followed, a suite of experiments mined the details of this anisotropy and delivered unprecedented constraints on cosmological models, culminating in the recent WMAP satellite mission, the data from which have become a cornerstone of modern cosmology (Komatsu et a l , 2008). • Type l a Supernovae- In 1998, two groups found surprising results in an experiment to measure the brightness of a certain type of supernova (Perlmutter et al., 1999; Riess et al, 1998). Properly calibrated, the actual brightness of these objects is known, and so their apparent brightness, together with their redshift, can be used to understand how far away they are. Both groups found that distant supernovae were dimmer than expected for a matter-dominated universe, indicating that the universe had expanded at a faster rate recently than it had early in the light's journey. Only by including a cosmological con- 7 stant term, QA> could the observations be explained. This was in remarkable harmony with measurements of the CMB, which dictated that the universe was flat {Q-Tot — I)? an d with observations of the distribution of galaxies that re- quired the amount of matter to be much less than this (fi_^f=0.3); with the new component (VL^ — 0.7), all of the observations were consistent. 1.1.2 The Collapsing Universe: Structure Formation The six cosmological parameters, only measured relatively recently, define the canvas upon which the standard cosmological model paints. The fis define the dynamics of spacetime itself, the expansion and stretching of the fabric in which the matter is embedded. This universe, however, could be a very uniform and boring place. What gives our cosmology its particular flavor (and the option of harboring life) is the presence of parts that are very non-smooth-the extremely overdense regions that form galaxies and stars, separated by vast emptiness. The standard model presupposes that at some point very early in its history the universe underwent a period of "inflation," during which some form of a scalar field finds itself in a false vacuum energy state. Quantum fluctuations find their way to the true vacuum in some spots, and these bubbles of true vacuum have high energy on their boundaries (in contact with the false vacuum state), and low energy within; it is therefore highly energetically favorable for them to expand, which they do rapidly, blowing the universe up in size exponentially in a very short amount of time. When the process has finished, the entirety of the observable universe has been stretched from a very small initial point, and so it is very uniform, very flat, and tiny quantum energy fluctuations in the initial field have been expanded to macroscopic density perturbations. It is these perturbations that will collapse to form the large scale 8 structure visible today in a process called structure formation. In structure formation, the tiny density ripples that inflation leaves in its wake begin to coalesce under the influence of gravity. These fluctuations in the otherwise smooth universe are just slightly more dense than their surroundings. They therefore have a slightly stronger gravitational attraction, and so nearby material tends to move into the overdense regions. Of course, with more material, the spot is now even more overdense, with an even stronger gravitational pull, and so things continue. Eventually, the points that were initially only slightly overdense become very overdense; it is in these collections of matter that galaxies and clusters of galaxies are found, surrounded by the empty space that they have swept up. These initial fluctuations are Gaussian, and can be specified by their power spectrum, which has a shape (P(k) ~ kUs~l) and a normalization (erg), both among our parameters. The initial collapse of inhomogeneities into more pronounced structure is straightforward to calculate (see section 1.2.2) because it is linear, which is to say that the density field can be Fourier transformed, and the various Fourier components evolve independently of one another. However, at some scale between the smooth initial conditions and the tangled web of structure we see today the process becomes nonlinear, and thus much more difficult to calculate. In addition to the inaccessibility of experimental results for many years, the complexity of this nonlinear structure formation prevented the forward movement of cosmology for some time. In recent years, as the computer age has dawned and our ability to do complex calculations has grown dramatically, this roadblock has also been removed; realistic computer simulations of structure formation are now possible, and enable us to understand at a much deeper level how the process of structure formation occurs. This advance forms the second pillar upon which the current age of cosmological sophistication is built. Because most of the matter in the universe is dark matter that interacts only 9 gravitationally, we can simulate the evolution of structure by keeping track only of gravitational interactions among dark matter particles. If the standard model actually describes the way in which structure forms, then we expect that with a big enough computer and enough time we can reproduce the web of structures that we observe. The results of the Millenium Simulation, for example, can be directly compared to observations of the universe that we live in (Figure 1.2); the degree to which the predictions of the model match the observations is remarkable (Springel et al., 2006). This agreement attests to the power of our standard model to turn our list of six parameters into the beautiful web of cosmic structures that have mystified humans since they first looked into the sky, and our ability to describe this complex process enables us to unravel basic cosmology from observations of structures that are not simple at all. Together, the big steps forward in observations and the computer-driven advances in our understanding of how structure forms have pushed the state of cosmological physics forward tremendously within recent decades. The remarkable growth in our understanding of the kind of universe we inhabit has led some to refer to the present as a "golden age" of cosmology. 1.2 Normalizing the Power Spectrum -as, The project presented below is related to understanding one of the cosmological parameters related to structure formation, the normalization of the spectrum of matter density fluctuations <7g. In this section, I will define and explore the role of ag, in structure formation models and briefly review previous measurements of the parameter. 10 Figure 1.2: Simulated and Observed Structure A comparison of maps of the simulated (red) and actual universe (blue). The similarities between the two reflects the power of the standard cosmological model to describe the universe in which we live (Springel et al., 2006). 11 1.2.1 Defining ag We will consider the perturbations in the density field p(x) of the early universe, when structure was very linear, and the perturbations were homogeneous, isotropic, and Gaussian. I will closely follow Zentner (2007) in defining the terms. We can consider the departures from the mean density through the density contrast, defined as 8(x) = [p(x) — p]/p, where p is the mean matter density of the universe. Gaussianity implies that the distribution of S(x) is defined by only its two-point correlation function £,(x\,X2) = (^(^1)^(^2))) or> equivalently, its power spectrum P{k), the Fourier transform of this quantity. Homogeneity implies that the correlation function depends only on the separation of x\ and X2, and isotropy further implies that the dependence is only on the magnitude of the separation vector \x\ — x*2\Preferring to work in Fourier space (like real space, but better), we can write the density contrast as: 5(k) = fd3x5(x)eits. (1.1) Because of the dependence only on the magnitude of the separation, all of the information in the S(k) is contained in its power spectrum P(k): P(A;)=y- 1 (|5(fc)| 2 ), (1.2) where we've had to assume that there is a cutoff scale beyond which there is no power (so that the Fourier integral of power converges), and V is the volume associated with this scale, which is large enough to be uninteresting to us. Because P(k) has dimensions of volume, it will turn out to be handy to define the dimensionless version: A2(k) = k3P(k)/2ir2. (1.3) 12 The variance of mass (8 (x)) can be expressed in Fourier coordinates, such that A (k) is the contribution to that variance from a given scale. Often, it is more useful to consider the contrast smoothed on some scale R: S(x,R) = IdSx'W(\x'-x\,R)S(x') (1.4) where W is the window function with which we're smoothing the field. It is common to use as a window function a sphere in real space of radius R, but in Fourier space this window function has the unfortunate quality of severe ringing, leading to power on all scales. A Gaussian function is more pleasant in having no such feature, remaining Gaussian in both spaces. Whichever window function we have chosen, we can form the variance of the contrast field with this window: a2(R) = (82(x,R)} = IdlnkA2(k)\W{k,R)\2. (1.5) In the early, smoothly Gaussian universe, this implies that the probability of finding a smoothed value of 5(x) between 5 and 6 + dS is given by: P(5, R)d5 = l exp[-52/2a2 (R)]d5, (1.6) 2 ^2na (R) a simple Gaussian. Using the spherical window function, with R = 8h Mpc, is only one possible case of this, and is notated as <rg. While there is nothing special about this particular smoothing window, it is commonly used as a normalization of the power spectrum P(k), which has a known shape. On the face of it, ag specifies the RMS of the density contrast smoothed on 8h~~'-Mpc spheres and has value ~ 1, implying substantial probability for p < 0. This does not pose a problem, however, because the density field is only linear in the distant past, when the amplitude was much less and 13 far from the physical limit. Although we can linearly extrapolate to current times on these large scales, and the quantity defined in Equation 1.5 is still well-defined, the structures will be highly nonlinear at this point, and so the distribution will be far from Gaussian. The parameter remains a useful way of normalizing the model, but this does not imply that linearity applies at late times, or that Equation 1.6 remains valid in this nonlinear regime. As Patsy might say, "it's only a model." 1.2.2 Structure Formation and erg.' The Press-Schechter Approach The role of erg in structure formation can be seen by considering the Press-Schechter description of the process of halo formation (Press h Schechter, 1974). There are problems with Press-Schechter that are resolved in more advanced models, but its simplicity makes it useful for understanding the role of the parameters in the process of structure formation. The model predicts the number of dark matter haloes at a given mass. We start with the assumption that the probability of a given overdensity is given by Equation 1.6, which is to say that the perturbations are Gaussian. Now we assume that when the density within a certain scale exceeds a critical density, 5C, the region will collapse, eventually becoming a cluster. The probability of reaching this overdensity is just the integral of Equation 1.6 from 8C to oo: />oo J5C = i _ s2 V27T0- 2 2 x erfc ( —=L= ) (1.7) (1.8) (1.9) Where erfc is the complimentary error function. The factor of two was added 14 by Press and Schechter in a rather ad hoc way, but has since been explained in the context of Excursion Set Theory. Equation 1.7 gives the probability of forming a region of sufficient density to collapse into a halo. To convert to a number density distribution, we differentiate by M and divide by the volume of the structure, which is M/pb, with pfr the background mass density of the universe. *L = ^L£L = -±e-^J^^. dM dMM 2 ^ ^/2a dM (i io) V ; We can define cr(R) as: / ( * * )= R \-i(R) ^8FI^) (L11) where R and M are two equivalent ways of labeling the same halo. 7 and 8C can be fit from simulations. 0-3 has a redshift dependence that is determined by the cosmology (Carroll et al., 1992), but the overall amplitude remains a free parameter. At any redshift, we can therefore use Equation 1.10 to calculate the number of clusters of a given mass that will form; we can get everything we need from the theory or from simulations except for the value of <7g, which in this sense normalizes the spectrum of the matter. 1.2.3 Previous Measurements of as Because it plays such an important role in distinguishing models of structure formation, many measurements of the value of <jg have been made in a wide variety of ways. Many of the methods used to measure the parameter depend on cluster counts; as we saw in Section 1.2.2, the distribution of clusters with mass depends strongly on the value of <rg, and the number of high mass, high redshift clusters expected with 15 <7g = 0.6 is orders of magnitude less than if <jg — 1. Many of these measurements are performed by counting clusters and comparing their mass distribution to models. The issue in such measurements is in how to measure the mass of a detected cluster. What follows is an incomplete list of recent measurements. Voevodkin & Vikhlinin (2004) accumulated a catalog of 52 clusters detected with the ROSAT X-ray satellite. Each was bright enough to fit a profile to the density of gas, yielding a total gas mass for the cluster. Assuming that ^as = jy^-, which is to say that the portion of the cluster's mass in the gas is equal to the universal average, they derive total masses for their clusters. They find erg = 0.72 ± 0.04. Mantz et al. (2008) also use X-ray clusters from ROSAT to place a constraint. They use the luminosity function of nearly 250 clusters to do so, which requires a model for the relationship between flux and mass. They choose the relationship to match theory and simulation, and carefully treat the many systematics involved in this, finding a8 = 0.78+JJ;^. Gladders et al. (2007) measure <rg in a very similar way, but with optically-selected clusters in place of the X-ray catalogs. In this case, the observable that correlates with mass is the cluster's richness, essentially a measure of how correlated galaxies are with cluster centers. By allowing the parameters of the relationship between richness and mass to be fit simultaneously with the cosmological parameters, they arrive at a8 = 0.67_ 0 13 . Pu et al. (2008) measure the mass of clusters directly, through weak lensing. In a large and deep optical survey, they statistically compute the magnitude of lensing that occurs as light from background sources is distorted by encountering the gravitational influence of a cluster. Because the distortion depends directly on the mass of the cluster, and not only on the gas, weak lensing avoids some of the pitfalls of other cluster-based methods. This analysis finds erg = 0.78 ± 0.04. 16 The WMAP 5 year results (Komatsu et al., 2008) also constrain ag. In this case, the constraint comes not from the distribution of clusters, but rather from the amplitude of small-scale power in the primary CMB. Of course, the perturbations on small scales in the CMB are where the clusters probed by other measurements will come from. A value of erg can be constrained by WMAP data alone, but this constraint is quite weak and highly degenerate with other parameters. However, by combining WMAP 5 year data with Baryon Acoustic Oscillations and supernovae data, Komatsu et al. (2008) find that a 8 = 0.80 ± 0.04. Although there was a time in which different measurements of erg produced different values (see, eg, Tegmark et al. (2004)), as constraints have improved it seems that there is ct growing consGnsus on o"§ f*^ 0.8. The constraints described above are shown graphically in Figure 1.3, along with one glaring exception to this consensus: previous attempts at measuring ag through the power spectrum of Sunyaev Zel'dovich effects imply high values (~ 1) for the parameter that are increasingly at odds with the other methods. The work that this thesis chronicles is aimed at better understanding this discrepancy. 1.3 Measuring a8 with the SZ Effect The Sunyaev Zel'dovich (SZ) Effect, observable in the power spectrum of the Cosmic Microwave Background (CMB), offers another way of measuring the value of ag. In this section, I will define notation for the CMB power spectrum, discuss the role of secondary sources of CMB anisotropy, and discuss in more detail the SZ Effect and its dependence on the value of erg. 17 1.4| ! ! ! ! T 1.2 00 •i i 0.8 0.6 0.4L _i C>^ i_ (N% <3P _i f(\X <6 f$) .<5P f?J .CSP i V i_ ^ QT .CJ^ ) f *V A° Measurement Figure 1.3: Constraints on erg A compilation of measurements of erg. Komatsu's measurement is the WMAP 5 year result, the red area shows the somewhat controversial measurements from CMB power spectrum measurements, and the rest use cluster counts to reach a constraint. 1.3.1 The CMB Power Spectrum One would be hard-pressed to overstate the importance of observations of the Cosmic Microwave Background in the rapid growth of our ability to describe the universe's dynamics and components in recent years. The CMB is so powerful a tool because it offers us an exquisitely detailed glimpse of what the universe was like in its very early stages (about 400,000 years after the Big Bang). At this point in its evolution, the universe was very dense and hot enough to keep hydrogen ionized, such that the resulting soup of free electrons scattered thermal photons with a very high crosssection, effectively locking the electrons and photons together in a fluid. 18 As a fluid, pressure from the photon component of this plasma countered the gravitational desire of the baryons to collapse. Places where the gravitational forces had collapsed a region, increasing the pressure, became hot, while places in which the pressure had pushed the baryons outwards would cool. As the entire universe expanded, the temperature eventually dropped to the point at which the hydrogen recombined, soaking up the free electrons. The probability of scattering the photons correspondingly dropped dramatically, and so the plasma separated; the baryons began their task of forming structure, and the photons streamed into the sunset, carrying with them the temperature distribution from their equilibration. It is the pattern of these hot and cool spots, encoded in the photons, which we can measure when their journey ends 14 billion years later by intersecting a cosmologist. The surface of last scattering, the point at which the decoupling between photons and electrons occurs, is a shell centered on the Earth of radius 12 billion light years, and so the hot and cool photons arrive here from every direction. The pattern of these photons, showing the temperature variations described, has been mapped across the entire sky by the WMAP satellite, and can be seen in Figure 1.4. This pattern is on a sphere, and so it is pleasant to decompose it into the spherical harmonics Y\m: A ^ ) = ^alro7k(x) l,m (1.12) where x is the position on the sky, and AT is the departure from the mean temperature (2.73 K). We believe the universe to be rotationally and translationally invariant, and so all of the information is contained in the power spectrum of this sky, given by C t = i\alm\ >• (1.13) 19 Figure 1.4: The CMB Sky The fluctuations of the CMB temperature as measured by the WMAP satellite. The mean and dipole have been subtracted, as has the galaxy, and the image is of the entire sky. Here, £ describes an angular scale, with £ ~ 9 . Here in Fourier space, the fluid equations that govern the dynamics of the photon-electron plasma become wave equations that do not entangle the different modes described by different £; each mode oscillates independently from the other modes, involving a set of acoustic oscillations of the plasma on a certain scale. There is a single mode (£ ~ 200, or one degree, as it turns out), which has exactly enough time to collapse to its maximal compression (and highest temperature) by the time the recombination cuts the oscillations short. Since inflation ensures that all of the modes begin their oscillations at the same place, there are harmonics of this mode which also are at extrema of their oscillation when the music stops, freezing their temperature maxima and minima in place. It is this series of harmonics which give rise to the famous harmonic peaks seen in the power spectrum (Figure 1.5). 20 6000 5000 £ 4000 + *~~ 3000 u u £ 2000 1000 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 multipole / Figure 1.5: The CMB Power Spectrum The power spectrum of the CMB sky shows the acoustic oscillations of the initial plasma and the damping towards small scales. While the relative heights and positions of these peaks encodes much of our knowledge of cosmology, the part of the spectrum we are interested in here is at scales of several arcminutes, £ ~ 2000. The damping in the CMB power on these scales, obvious in Figure 1.5, is due to two effects. First, even within the tightly coupled baryon-photon plasma there is a finite mean free pathlength for the photons, allowing them to leak from hot to cold regions and wipe out power. Second, the last scattering surface has a thickness to it, and modes on scales smaller than this thickness can interfere with one another across it. While the primary CMB has little power left on these scales, it is here that we begin to find secondary contributions to the anisotropy. 21 1.3.2 Secondary Anisotropy The standard cosmology predicts with great accuracy the power spectrum that we expect of the CMB, but this power spectrum is not exactly what we measure. Instead, the CMB photons can interact with their surroundings during their passage from the last scattering surface to the observer, and these interactions can distort the power spectrum. These distortions are referred to as secondary anisotropics, and result when primary CMB photons interact with either a gravitational potential or free electrons. A nice summary of the variety of these effects is given in Aghanim et al. (2008), which I follow here. Sources of secondary anisotropy include: • Integrated Sachs-Wolfe Effect: As the photons cross a potential well, they are blue-shifted dropping in and red-shifted coming out. The potential can change during the photon's crossing, as the balance between dark energy or radiation and the gravity of dark matter shifts and the potential decays; such a decay leaves a net blue-shift in the photons involved. Such an effect is relevant on scales of about 10°. • Rees-Sciama Effect: This is identical to the Sachs-Wolfe Effect, but happens in the recent universe in large, nonlinear structures like galaxy clusters. These objects are at the scale of several arcminutes. • Gravitational Lensing: As the photons bend in response to the curvature of space, power can be redistributed, mostly towards smaller scales. • thermal Sunyaev-Zel'dovich Effect: The SZ effect is the brightest at lower frequencies. Photons get a boost towards higher energy as they scatter off of hot electrons. Dominated by the electron gas within galaxy clusters, it also appears at the arcminute scale. We will go into this in much greater detail below. 22 • kinetic Sunyaev-Zel'dovich Effect: In the same scattering of CMB photons and intracluster gas, motions in the gas can be passed along to the photons via the Doppler effect. • Ostriker-Vishniac Effect: This is the same as the kinetic SZ Effect, but in the linear case of the early universe. Also active on arcminute scales. • Patchy Reionization: As the universe reionizes, changing its optical depth to the CMB photons, the ionization occurs in spheres around early stars, gradually filling in the spots in between. The patchwork of the ionized and neutral hydrogen also imprints itself as anisotropy in the CMB. 1.3.3 SZ Effect The thermal Sunyaev Zel'dovich (SZ) Effect is expected to be the brightest of these secondary anisotropics, exceeding the brightness of the damped primary CMB above £ of about 2000 (Sunyaev k Zeldovich (1972), Carlstrom et al. (2002)). The effect is the result of simple physics, described in detail in Birkinshaw (1999). Fundamentally, the effect results when CMB photons encounter hot gas within clusters of galaxies. The deep potential well of the cluster heats the gas to ionization, and so there are free electrons available to Thomson scatter off of. Because the electrons are much hotter than the CMB photons are, the net energy transferred is just the amount of thermal energy that the electrons have to give: ~ —f. As the photons traverse the cluster, the probability of an interaction rises with the number density of electrons ne and the length of the photon's path L. Including the Thomson scattering cross section <7y, we find that the change in the CMB temperature due to the SZ effect is proportional to the Compton y-parameter, given by: 23 AT (JT t —-<xy = - ^ / kTenedl (1.14) which is a simple measure of the integrated pressure of the gas. The magnitude of the effect is set by the typical parameters of clusters. Galaxy clusters, the largest gravitationally bound objects in the universe, can have masses of 3 x 1O 15 M 0 and radii on the order of a Megaparsec (Mpc). Ionized gas, spread through the cluster and making up ~ 10 — 15% of the cluster's mass, will reach hydrostatic equilibrium in this deep potential, reaching temperatures of several keV. This hot, ionized gas can Thomson scatter the traversing CMB photons, with probability P ~ nea^R, where ne is the number density of the electrons (about 10 - 2 ), ay is the Thomson cross section (6.7 x 10 era ), and R is the cluster radius (~ IMpc). The probability of a scattering is therefore ~ 1 0 - 2 . When a photon does scatter, the electron will, on average, give it a fractional energy change equal to the ratio of the electron's thermal to rest energy, about 1CP2 for the ~ 5keV temperatures found in large clusters. The total fractional intensity change in the CMB brightness from its interaction in the cluster is therefore ~ 10 ~ IOO/IK for such a large cluster. Were the electron scatterers at rest, the photon would be as likely to give energy as to get it, and there would be no net distortion. However, properly accounting for the angular distributions of the scattering and the boost from the rest frame of the electron to the observer's frame, one finds that as the velocity of the electron increases, the distribution of energy gained by the photon becomes less and less symmetric, and it becomes more likely that the photon's energy will increase. Because the number of photons cannot be changed by scattering processes, the increase in energy of photons that follow a thermal spectrum results in a distortion to that spectrum (Figure 1.6), such that there is a deficit of low-energy photons and a corresponding surplus of higher 24 energy photons. The frequency dependence of the effect in terms of temperature distortions is given by: f(x) = x*L±±-4: (1.15) x e —1 where x = ,y —. The crossover from the regime in which the effect shows up as a deficit of photons (low frequencies) and the regime in which photon numbers are increased (high frequency) occurs at ~ 217 GHz. We now write the SZ effect as: ^ = f(x)y (1.16) The redshift-independence of this quantity is paramount in establishing the SZ Effect's role as a cosmological probe. Alternatively, we can write the analog of Equation 1.15 for intensity units: 9(x) = J^^fi*) (1-17) such that MSZE = g(x)I0y (1.18) 1.3.4 SZ Power Spectrum Counting the distribution of clusters in a survey is a well-known method of constraining cosmology, and especially erg, with equally well-known systematic problems along the lines of those discussed above in the context of X-ray and optical surveys. One of these is the necessity of building an unbiased sample of clusters; understanding the selection effects in most methods of cluster detection is difficult. The near redshift-independence of the SZ Effect helps to avoid this problem, as an SZ survey 25 8 6 4 _ 2 3 0 -2 -4 -6 0 50 100 150 200 250 300 350 400 450 500 Frequency [GHz] Figure 1.6: The Spectrum of the SZ Effect We plot g(x)-Equation 1.17, in intensity units. This shows the relative sign and strength of the SZ effect as a function of frequency. will be essentially mass-limited to very high redshift, in marked contrast to surveys via X-Rays or optical signatures of clusters. A second pitfall of using cluster surveys to measure cosmological parameters is the necessity for converting an observable to mass. The cosmological dependence of the cluster distribution depends on cluster mass (section 1.2.2), but mass is quite difficult to measure directly; the conversion from SZ flux or optical richness to mass presents another opportunity for error. In this sense, measuring the angular power spectrum of SZ fluctuations is a cleaner measurement of the mass distribution of clusters. Instead of detecting individual clusters, we measure the angular power spectrum of many, less massive clusters. Now we need not understand the mass of individual objects, but only the statistical relationship between cluster mass and SZ signal. Analytic calculations of the angular power spectrum from the SZ effect and its dependence on cosmology have been performed 26 by many groups (eg Komatsu & Seljak (2002),Komatsu & Kitayama (1999),Holder k Carlstrom (2001)). The SZ power spectrum depends on the mass distribution of haloes, — ^ - ' , the volume element as a function of redshift, 4^-, and the profile of SZ brightness of the haloes, gvy{M^ z). We write the power spectrum Ci as: 2 fZmax , dV fMmax „,dn(M,z).„ , „ , ,l2 with <jv as the spectral behavior of the SZ Effect, Equation 1.15, and yg as the transform of a projected Compton y-profile. Although the Press-Schechter approach discussed above is a powerful tool for understanding the mass distribution of dark matter haloes 4JJJ, fits to cosmological simulations can provide more accurate results (e.g., Jenkins et al. (2001)). For the y-profile of the clusters, isothermal and spherical P models have been used in the past, yielding small scale, transformed profiles that scale like f , which flatten the shape of the power spectrum. The behavior of these models at large radii is unphysical, though, and there are efforts to replace them with more realistic profiles. Komatsu & Seljak (2002) discuss a "universal pressure profile" for cluster gas, based on hydrostatic equilibrium and a close tracking of gas density and the dark matter arranged in a universal NFW profile. Using these pieces, Komatsu & Seljak (2002) find that at £ ~ 3000, the SZ angular power spectrum is dominated by redshift ~ 1 haloes of mass scale ~ 1014M©. They find the level of power on these scales to be ~ 30 — 200(iK , with a strong dependence on the value of erg (C^ oc erg); the dependence on other cosmological parameters is weak. Cosmological simulations exist within which we can calculate the power spectrum of SZ fluctuations from hydrodynamical simulations of cluster gas (Springel et al. 27 (2001),da Silva et al. (2000), Refregier et al. (2000)). Komatsu & Seljak (2002) find that, taking into account the different cosmologies and simulation details, simulated power spectra agree with one another (and with their analytic calculation) to within a factor of two. This represents a 10% systematic error in determinations of erg from power spectrum measurements. Because of the incredibly steep dependence of SZ power on the value of erg, power spectrum measurements are a powerful probe of this cosmological parameter. Example SZ power spectra with ag of 0.8 and 1.0 are shown in Figure 1.7, along with current measurements. 700 600 !ft 500 f l CBI 400 • =? 300 + 200 BIMA / , " 100 / / 0 ^0***' •^-""~'""~~ 1 **• 1 **•* 1 1000 2000 3000 4000 5000 6000 7000 8000 / Figure 1.7: The SZ Power Spectrum Shown are model power spectra for ag = 0.8 (red) and ag = 1.0 (black). Primary CMB anisotropy power is shown as dashed. The CBI and BIMA measurements are also shown. 28 1.3.5 Power Spectrum Measurements Several attempts at measuring this power spectrum have been made. The Cosmic Background Interferometer (CBI) is composed of 13, 0.9 m antennas, observing from Chile at 30 GHz. Although principally aimed at measuring larger scale fluctuation power from the primary CMB, the CBI measured Sdb^-^fiK2 of power in a bin of I > 2000, an excess over expected primary CMB signal at 98% confidence (Readhead et al, 2004). Separately, measurements with the Berkeley Illinois Maryland Association (BIMA) interferometer, also observing at 30 GHz, constrained the power at £ = 5237 to be 22Qt\f^K'1 (Dawson et al., 2006). A third measurement at 150 GHz with ACBAR was consistent with non-thermal signal at small scales (Reichardt et al., 2008). These measurements have been used to constrain the value of erg, yielding erg = 0.96±g;g§ for CBI and a 8 = 1.03±g;|g for BIMA. The apparently high value of <r8 preferred by these measurements is in contrast with the emerging consensus on a lower value for the parameter from other measurements (section 1.2.3). It is our aim to explore this tension with a higher-confidence measurement of SZ power on small scales. CHAPTER 2 THE SUNYAEV ZEL'DOVICH ARRAY The Sunyaev Zel'dovich Array (SZA) is an eight-element interferometer in the Owens Valley, California. A collaboration among The University of Chicago, Columbia University, the California Institute of Technology, and NASA's Marshall Space Flight Center, the SZA was constructed in 2004-2005 at the Owens Valley Radio Observatory (OVRO), operated by Caltech near Bishop, CA. The SZA sits at the bottom of the valley, at an elevation of ~ 4000 ft. The array spent 2005-2008 collecting data for this CMB anisotropy analysis, conducting a 6 deg2 cluster survey, and studying known clusters via their SZ signature. In the summer of 2008, the SZA will be moved to the 7,000 ft Cedar Flat site, where it will continue to observe autonomously and also as a part of the Combined Array for Research in Millimeter-wave Astronomy (CARMA). As an interferometer, the SZA is composed of three major components: a set of antennas, which collect an astronomical signal, receivers, which detect and amplify the signal, converting it into an electrical radiation, and a correlator, which computes the cross correlations among all pairings of the eight antennas. In this chapter, I will describe each of these components, discuss measurements that have been made of their performance, and then discuss the calibration of the system as a whole. 2.1 The Antennas 2.1.1 Optical Path Each of the SZA antennas is composed of an assembly of three mirrors and a feed horn (Figure 2.1). The 3.5 meter primary dish is composed of 27 aluminum panels of 29 30 roughly one-third of a square meter. This main reflector focuses astronomical radiation onto a 35 centimeter diameter secondary mirror, which is suspended above the primary mirror by three feedlegs. The secondary mirror, in turn, reflects radiation onto a tertiary mirror, located behind the primary's surface at the Cassegrain focus. Finally, the third mirror reflects radiation into a cooled receiver dewar, where a corrugated feedhorn provides a smooth electrical transition between the mirrors and the waveguide within the receiver. Without this feed, the impedance mismatch at the receiver would reflect much of the astronomical signal back into the sky. This chain of components is shown in Figure 2.1, and a photograph of an SZA antenna is shown in Figure 2.2. The mirrors within the SZA's optical path are not perfectly smooth; deviations can be characterized as having a Gaussian distribution about the nominal mirror shape, with RMS a. These imperfections scatter radiation away from the receiver, and so the effective collecting area of the telescope is reduced. The magnitude of this reduction depends on how large the deviations are with respect to the radiation's wavelength, and so the problem becomes more pronounced at higher frequencies. It can be shown that the effective area of the antenna is altered like A = Aoe~^7rcri*> , with A as the observing wavelength, known as the Ruze formula (Thompson et al., 1991). The current surfaces of the SZA antennas are set to an RMS surface accuracy of 30//m. At 30 GHz, this corresponds to a loss of less than 1%. At this level, the RMS is dominated by panel alignment, and not by actual imperfections in the surfaces of the panels themselves, which are much smoother; in principle, the SZA antennas can be effectively used up to 350 GHz. 31 <> A f > Source Secondary Mirror (20 inches) Primary Mirror (3.5 meters) Tertiary mirror To Receiver Figure 2.1: SZA optics path The optics path of an SZA antenna, with a dashed ray showing the path of radiation within the system. The source emits radiation which is reflected by the primary, secondary, and tertiary mirrors, ultimately striking the cold feedhorn, which puts ths signal onto waveguide in the receiver. 2.1.2 Beam Patterns The antenna has a primary beam pattern, which describes its response to a point source at various positions in the field. The SZA beam is very nearly Gaussian, with a 12 arcminute full-width half-max at 30 GHz, which means that a source 6' from the center of a field will appear to have half of the flux that it would were it at the pointing center. The width of the beam is inversely proportional to frequency. Figure 2.3 shows the SZA antenna's primary beam pattern; the sidelobes that are visible on a log scale are problematic, as they indicate a sensitivity of the antenna to objects 32 Figure 2.2: An SZA antenna. very far from the region of the sky being observed. To understand the nature of the sidelobes visible on the log scale, we must consider the antenna's illumination pattern. We can imagine the antenna as a transmitter, instead of a detector. In this framing, the feedhorn is said to illuminate the primary mirror with transmitted radiation, following the optics path in Figure 2.1 in reverse. The pattern of this illumination of the primary mirror is determined by the optics of the system, and it is related to the far-field beam pattern of the antenna through a Fourier Transform. The design of the antenna illumination pattern involves a compromise between sensitivity and sidelobes. An extreme example would be to illuminate the mirror with a Gaussian profile, with very small illumination at the dish's edge (blue in Figure 2.4). The transformed pattern would yield a beam pattern that is also Gaussian, with no sidelobes, 33 but by tapering the illumination pattern to be small at the edge we have sacrificed substantial sensitivity by giving almost zero weight to contributions from the large fraction of the mirror's area that is near the edge. In the other direction, we could illuminate the dish uniformly, using the entire area of our antenna (red). In this case, the illumination pattern would have a sharp drop to zero at the edge of the mirror, and the transform of this shape would produce large sidelobes on the sky, spreading the instrument's sensitivity across a wide area. The SZA compromises between these two extreme cases with a Gaussian illumination pattern that is wide enough that the edge of the dish truncates it at its -11.6 dB power point (green). We have some sidelobe structure with this pattern, but also use most of our dishes' collecting area. Each of these illumination functions and the corresponding beam is shown in Figure 2.4. Another important source of structure in our sidelobes comes from the fact that a 35 cm disk is blocked from the center of our aperture by the secondary mirror. The design illumination pattern may not be realized due to alignment or dimensional errors; we can measure the beam profile by scanning the telescopes across a bright source. This was done for all antennas, and a beam width was fit at each of the SZA's 16 frequency bands. The fits, which match well with the expected values for the predicted edge taper, are shown in Figure 2.5. The size of the aperture in wavelengths is the relevant scale here, and so the beam is smaller at higher frequencies. With somewhat more sophistication, we can repeat this observation, offsetting from the source in two dimensions instead of one. The two-dimensional Fourier transform of these data will describe the illumination of the dish in two dimensions. We can use the phase of this illumination pattern to see errors in the surface of the mirror, in a process called holography. In Figure 2.6, we see the phase of the illumination pattern. Discontinuities in these maps were used to adjust the panels so as to achieve the desired surface. 34 -100 -50 0 50 100 Angle from Pointing Center [arcmin] -100 -50 0 50 100 Angle from Pointing Center [arcmin] Figure 2.3: SZA beam The SZA antenna primary beam, at 31 Ghz. Here we show the amplitude response of the antenna to a point source of flux 1 as a function of offset angle. The left plot clearly shows the Gaussian shape of the beam, which has a full-width half-max of about 12 arcminutes. The right plot, the same curve in log scale, allows you to see the bumps and wiggles of the antenna's sidelobes. 2.1.3 Array Layout The SZA, an interferometer, consists of eight of these antennas. The angular scales on which our interferometer is sensitive are determined by the distances between pairs of antennas, called our "baselines." A single baseline of the SZA illuminates coherently a fringe pattern on the sky with a wavelength inversely proportional to the baseline length. Objects that are larger than this scale are said to be "resolved" by the interferometer, and we will not detect flux from these larger scales. We have positioned six of the eight SZA antennas close together, providing sensitivity on scales of ~l-2 arcminutes, the size of galaxy clusters. The remaining two antennas (participating in half of the baselines) are further away, providing sensitivity on scales closer to ~15 arcseconds. The short baselines, with cluster-scale sensitivity, are what will be used 35 0.04 8 ^ 0.03 1 0.02 "8 S 0.01 . i JT\ i . S -200 0 200 distance on the aperture [cm] -50 0 50 Distance from pointing center [arcmin] Figure 2.4: Beam and Illumination Patterns On the left, we see illumination patterns for the SZA primary: uniform (red) uses the whole dish, Gaussian (blue) tapers so as to have no sidelobes, and the actual SZA pattern (green). On the right, we see the corresponding beams on the sky (power), revealing the different resolution and sidelobe structure that results from each. in this analysis (although see Chapter 7). The longer baselines are principally used to identify point sources (Chapter 5). The positions of the SZA antennas are shown in Figure 2.7, along with the sensitivity pattern on the sky for two baselines of different lengths. These patterns represent the response of the baseline to flux from a given position on the sky. Figure 2.8 shows which Fourier components are measured by the SZA; several features are obvious here. First, the SZA's long baselines (large u — v radii) form an entirely separate array that has sensitivity on small scales. The radial repetition in this plot comes from the 16 frequency bands, the different groupings are individual baselines, and the rotation about the center follows the rotation of the sky during the track. The response of the short and long baselines to a single point source at the center of a field is also shown in Figure 2.8; this pattern is simply the sum of the 36 xlO 0.2 -O.l 0 0.1 pointing offset [deg] 27 28 29 30 31 32 33 34 35 Frequency [GHz] Figure 2.5: Measured SZA Beams The SZA antenna primary beam width, as fit to one-dimensional cuts across a bright source. The left panel shows the data for a single antenna in a single frequency band; at various offsets from a bright source, the measured flux is shown. These data are fit to a Gaussian, and the sigma of the fits to all antennas, all frequencies, are shown in the right-hand plot. The red curve is from theoretical calculations of the beam width, including diffraction effects. sinusoids for each u — v sample. 2.2 2.2.1 Receivers Amplification and Mixing The corrugated feedhorn, which is the final optics component in the SZA antennas, is inside of a receiver cryostat. This cryostat is cooled to a typical temperature of 15 K. There are two feedhorns within the cryostat-one feeds a 30 GHz receiver chain, and the other feeds a parallel 90 GHz chain; the tertiary mirror can be moved to select between these two receivers. All of the CMB anisotropy data has been taken at 30 GHz, and so I will focus on this receiver in the discussion below. 37 Figure 2.6: Holographic phase measurements The phase of the antenna illumination pattern for one telescope, as measured through holography. The phase has been converted here to microns. The 30 GHz receiver chain, all cooled within the cryostat, is shown in Figure 2.9. The feedhorn acts as a smooth transition for the radiation between the open air and the waveguide within the receiver system. Following the horn is a circular polarizer. Standard, rectangular waveguide has a defined linear polarization; because astronomical sources are often linearly polarized at the ~ 10% level, if we connect waveguide directly to the feedhorn (and the antenna), then the flux of a source will depend on the relative alignment of the instrument's polarization with the source's. To avoid the calibration problems that such a situation would entail, we insert a 38 ,-. 60 S I 40 '8 \ o CM | -^•H^^BSBS! -o.o5 m -0.05 B ^ 9 | ^ ^ ^ R B ^ & -a 20 CO i o -o.i | | H O u o H I OBI I^Hl 0.05 I ^ H •HI -8 is^BBB^s °<°5Bs^^BiSBI o-tflgflj^^BBH -20 -20 0 20 40 East-West position (m) -0.1 0 degrees 0.1 -0.1 -0.05 0 0.05 0.1 degrees Figure 2.7: SZA Array Layout The position of the eight SZA antennas (left), the synthesized beam for a short baseline (middle, corresponding to the baseline in red), and a long baseline (right, the green baseline) polarizer behind the horn; the polarizer serves to convert the linear polarization of the instrument to circular polarization on the sky. Since sources rarely have any net circular polarization, we are always sensitive to half of the source's total flux, and the sources will have the same flux when observed by the SZA in any orientation. The polarizer feeds the signal into the heart of the receiver, a low-noise, High Electron Mobility Transistor (HEMT) amplifier (Pospieszalski et al., 1995). The SZA amplifiers were built at the University of Chicago for the earlier DASI experiment, and can provide gain of 30-40 dB across a 26-36 GHz band with four stages of amplification while at the same time introducing minimal noise into the signal. Thus amplified, the signal from an antenna is in principle ready to be cross-correlated with that from another antenna. However, at the 30 GHz frequency of the SZA, transporting the signal across the meters that separate the antennas is not straightforward. Instead, we mix the high radio frequency signal (RF) with a local oscillator (LO), a lower frequency tone that, when multiplied by the signal, will produce an intermediate frequency (IF), which can be more easily transported to the correlator. By multiply- 39 1000 I "3 > S3 •1000 -6000 -5000 0 u (wavelengths) 5000 •1000 0 1000 u (wavelengths) 0.05 0 0.05 0.1 degrees Figure 2.8: SZA Synthesized Beams The top two panels show the u — v coverage of an SZA track, with a zoomed view at the inner array on the right. The response of the array to a point source at the center of the field, its synthesized beam, is shown for short baselines (bottom left) and for long baselines (bottom right). ing the signals together, mixers convert R F —> IF, where R F = L O ± IF (the mixer produces both the sum and the difference frequencies-in a single sideband system like the SZA, we must select the IF that we want by means of filters). If the same LO (identical in phase) is used at each antenna, then the phase information of the highfrequency R F is retained in the IF; the correlation to come demands that this be the case. In the case of the SZA, the LO is 35.938 GHz and comes from a varactor-tuned gunn oscillator located on each antenna. The mixer, which is located within the cold 40 (from telescope) / \ Corrugated Feedhorn I Polarizer HEMT Amplifier (27-35 GHz) (to correlator) Mixer LO (36 GHz) • J (1-9 GHz) Figure 2.9: The receiver chain at 30 GHz Within the receiver, the incoming signal is amplified and mixed to lower frequency. The components here are all within the cold cryostat. cryostat, uses this 36 GHz tone to convert the 27-35 GHz observing frequency of the SZA to a more manageable 1-9 GHz IF; it is this IF that is transported by optical fiber to the correlator. 2.2.2 Phase Lock System We must mix the input signal to lower frequency in order to transport it, and the 36 GHz LO from the varactor must be delivered with the same phase to each antenna to avoid losing the coherence information within our data in this mixing. The phase lock system ensures that the LO has the same phase at each antenna. 41 This system relies on several common reference frequencies, which are generated centrally and then distributed to each antenna. We send one such tone at 0.998 GHz; it is sent to the antennas, multiplied by 9, and mixed with the 8.972 GHz output of a YiG oscillator on each antenna. The 10 MHz difference between the two is phasecompared with a separate 10MHz reference that is also shared among the array. The phase of the YiG output is adjusted until the difference matches the phase of the 10 MHz reference-if the YiG matches the reference and the reference is the same, then all YiGs in the array are "phase-locked" together. Now that the YiGs are all outputting 8.972 GHz with the same phase, we multiply this signal by four and compare to the output of the varactor in a second phase lock loop. This time, the difference between the two is compared to a third common reference, this one at 50 MHz. Now the varactor is locked so that it has frequency uVar = 50MHz + 4 x uYiG, where vyiG = 9 x 0.998 GHz - 10 MHz. The varactor is then forced to be: uVar = 50 MHz + 4 x (9 x 0.998 GHz - 10 MHz) = 35.938 GHz, (2.1) and the phase of this tone on each antenna, referenced to the common 10 and 50 MHz signals, will be constant across the array. Because of this, we can use the varactor's LO to mix the 27-35 GHz SZA band to a 1-9 GHz IF band without losing the phase information in the data. It is this preservation of phase that allows us to do interferometry. A final detail about the phase lock system is its relationship to the fringe-rotation of the array. Interferometry relies on the coherence of radiation from a source; however, the different pathlengths this radiation must cross in order to reach the different antennas in a baseline breaks this coherence. In Figure 2.10, we see this geometrical 42 Figure 2.10: Geometrical phase delay The phase delay S between two antennas when viewing the same source. The delay depends on the baseline and the position of the source in the sky, and must be adjusted through fringe-rotation. phase delay between two antennas, which we must compensate for if we want to coherently detect the source. Because the phase difference between the antennas will change as a function of the source position in the sky, we must constantly adjust the relative phases of the antennas in order to keep them all looking at the same place, the phase center, through a process called fringe rotation. We add this phase through the phase lock chain by adjusting the phase of the 50 MHz signal being sent to each antenna. 2.3 Downconverter and Correlator The signal from each antenna, having been amplified and converted to a 1-9 GHz band within the receiver, is further amplified and put onto optical fiber at the antenna. The 43 fibers then take each antenna's signal underground to the correlator trailer, where the interferometer's heart sits. The actual cross correlations are performed within 500 MHz-wide frequency bands; the downconverter takes the 8 GHz bandwidth from the antenna and chops it up into 16 of these bands, each of which is mixed down to span 1-1.5 GHz before being fed into the correlator. The SZA correlator is a digital device, making its calculations with FPGA chips that require digital inputs. Before the analog signals from the antennas can be crosscorrelated, they must first be digitized. We digitize with two bits, allowing four possible values for each antenna; we reduce the amount of computation needed to process the data by assigning the same value to four of the sixteen possible crosscorrelation states, in what is called a "deleted inner-product" digitization scheme. This strategy represents a tradeoff between increased signal-to-noise that comes with more precision and the reduced memory requirements that less precision allows. Reducing the antenna information to two bits only reduces our efficiency by about 10% compared to the undigitized case. The SZA correlator, while a sophisticated piece of computer engineering, is conceptually a simple device. Given two input time-streams, we would like to output their correlation as a function of frequency. One way to do this would be in the frequency domain; we could take the component of each antenna at a given frequency, multiply the two together, and then integrate over many samples to find the correlation. Equivalently, we could operate in the time-domain, and convolve the two time-streams with one-another, Fourier transforming the result to get the same resulting correlation spectrum. This second option is the one that the SZA follows. The convolution is done by means of "lags." we take the two timestreams, mulitply them and sum over many samples (the computation of our correlator follows a 1 GHz clock cycle, so that an individual 0.5 second SZA integration is composed of 500 44 million samples). One of the two signals is then offset by a lag of one sample, and the process is repeated. In all, the SZA correlator uses 32 lags, fifteen of which are negative and one of which is zero. These 32 lags correspond to 32 evaluations of the convolution of the two signals. When we Fourier transform the results we end up with sixteen independent frequency measurements, each of which occupies a 31.25 MHzwide piece of our original 500 MHz band. The other, negative lags encode information about the "Upper Side Band" (USB), which contains the 8 GHz of frequencies that mix into our IF from above. While our signal is mixed to 1-9 GHz from 27-35 GHz with our 36 GHz LO, frequencies from 37-45 GHz also mix into this IF. Because the amplifiers have no gain at these frequencies, these channels contain no signal, but they are useful as a diagnostic. The complex values of these channels, which we call "visibilities," are formally identical to what we would have gotten had we taken the original time-streams, transformed them and selected the component from each antenna that lies in the spectral channel we are after, multiplied the two antenna's values, and integrated over time. The visibility is, in the end, the cross-correlation of signals from the two antennas at a single frequency. 2.4 System Temperature The noise in our system, which we will focus on in Chapter 4, has two distinct components, both of which we have gone to some trouble to minimize. The first component is the contribution from the instrument itself, which we will describe as the receiver temperature (TRX), and the second is the atmosphere through which we look at the sky. The total system noise, Tsys, is the combination of these two, and determines the sensitivity of the SZA; the system temperature tells us the noise level 45 (in temperature units) of a measurement with a single baseline integrating for one second with one Hertz of bandwidth; our total system temperatures at 30 GHz are ~40K. Part of the noise in our measurements comes from the instrument itself. As mentioned above, the optics of the SZA are such that the primary mirror is overilluminated by the beam, and there is some sensitivity of the instrument to the hot ground behind the mirror. Measurements of this spillover contribution put it in the range of ~ 5K. The rest of the instrumental contribution to the noise comes from the receiver itself, and is labeled as TRX\ SZA receiver temperatures are ~ 2§K. Inside of the receiver, the polarizer, amplifier, and mixer have taken the signal detected by the antenna, amplified it, and converted its frequency to facilitate transportation. At each of these steps there is noise added to the system, and each component of the receiver can be assigned a noise temperature T{. A component's contribution to the total system noise temperature will go like rf, where G{ is the total amplification gain of the system up to that point (this is because the noise from previous components has been amplified along with the signal). The first gain stage in our system is the HEMT amplifier, which has a gain factor of more than 1000 x, and a noise temperature of ~ 15K. A component that comes after the HEMT would therefore need a noise temperature of at least 15,000K to contribute the same amount to the total system temperature. By placing the low-noise, high-gain HEMT first in our signal chain, we minimize the noise contribution of all later components. We measure the receiver noise contribution by looking at two sources of known temperature, TJJ and T@, and assuming that the power P coming out of the antenna is proportional to the temperature of the system plus TJJ or TQ. Then we have: PH = k{TH + TRx) (2.2) 46 Pc = k(Tc + TRx) (2.3) and TRx = ^ Pc FC (2.4) i Using a load soaked in liquid nitrogen for PQ and an ambient temperature load for P/j, we can thus directly measure TRX. We find 25 K receiver temperatures are normal for the SZA receivers as installed on the antennas, much of which comes from the amplifier. Extensive work was done by the DASI collaboration to reduce this contribution through the design of the amplifiers, and further work was done by the SZA to bias the amplifiers in such a way as to maximize gain with minimal noise. The second major contribution to our system noise is that from the earth's atmosphere. Water and Oxygen at various altitudes can absorb and radiate at 30 GHz; since much of the atmosphere is at temperatures ~ 300K, this radiation forms another large contribution to our instrument's noise. We measure the total system temperature, including the spillover, receiver temperatures, and atmospheric contributions, every twenty minutes or so during observations. This is done similarly to the method by which we have measured the receiver temperature, through power measurements of two different loads. For the hot load, we use an ambient-temperature calibrator that can be positioned in front of the receiver window automatically. The power measured on the load is given by: PH = HTioad + Tspill + TRx) (2.5) where Tioaci is the temperature of the load, measured constantly with a thermometer embedded in the load itself. We assume that ground spillover has the same temperature as the load (T;oa^ = Tspni), and TRX is the noise temperature discussed above. 47 Now we remove the hot load, and observe the power from the sky: Psky = HTspill + TRx + TCMBe-^c(z) + T ^ m ( 1 _ e -r o S ec(,) ) (26) where rg is the zenith opacity, z is the zenith angle at which we are observing, Tatm is the atmospheric temperature, and T^MB Y% the cosmic microwave background temperature. We form the combination TSys = P p -*H *_% (Tload - TcMBe-W"**) - Tatm(l - e -t>"*(*)) (2.7) *sky Tspiii + TRx + T C M J 3 e-^ g e c (^) + TgtmQ. - e~ W * ) ) Tload - TCMBe-^<*) x(Tioad - TCMBe-Tosec{z) = Tspiu + TRx + TCMBe-T^c(z) - Tatm(l - e " W ( * ) ) - Tatm(l - e - ^ e c W ) + Tatm{1 _ e-rasec{z)) This is exactly the set of contributions that we want to know in order to understand our system noise, Tsys. We measure PJJ, Tioad and Psky, we know TQMB already, and so we need only calculate TQ and Tatm. We do both with a model of the atmosphere that contains information about its spectral behavior, and which builds a model of the atmosphere's pressure and temperature based on weather station measurements on the ground. With this information, our hot-cold measurement can give us the total system temperature from Equation 2.7. By locating the telescope in a location with little water vapor (despite its role in supplying water to Los Angeles, the Owens Valley is in fact a desert), we reduce this atmospheric contribution to the system noise. A final step is to multiply the system temperature value by T sec z e ° (\ effectively scaling the noise level to a point above the atmosphere, where the signal is. The total 48 system temperature is in the range of 35-45 K. Tsys as a function of time is shown for a track in Figure 2.11; over the track, the source being tracked rises, transits, and sets, and the thickening atmosphere away from transit shows itself as a rise in noise temperature. T for a single Antenna, Band sys Time [hrs] Figure 2.11: The system noise temperature vs time for a typical track. 2.5 Calibration There are certain properties of the array that must be monitored or adjusted in order to keep it well-calibrated. These include the pointing of the antennas, their absolute flux calibration, and the antenna positions. The SZA control system sends desired encoder positions to the antennas' motion control computers, and this system engages the azimuth and elevation motor on the 49 antennas until the encoders report the desired values. It is the job of the pointing model, part of the control system, to translate a position on the sky that we want to observe into the encoder values that will bring us there. To first order, this model has only two parameters: encoder offsets for the two axes that keep track of where zero angle is for the encoder. Our ability to slew the antennas to a precise position on the sky is called the antennas' pointing. In order to point the antennas at a given part of the sky to within 0.5 arcminute accuracy, we need a somewhat more complex pointing model. The parameters in this widely-used model are all physically motivated, describing actual mechanisms of creating pointing errors. In addition to the two encoder offsets, our model uses seven more parameters to describe the physical imperfection of the antenna mount. The first two parameters describe the flexure of the antenna under its own weight; when the antenna is pointed at the horizon, all of this weight is hanging from the mount, and there is some sagging. When the antenna is pointed straight up, the weight is supported from below, and there is no such sagging. The flexure terms in the pointing model account for this sagging in an elevation-dependent way. There are two more parameters to describe the fact that the antenna's azimuth axis is not perpendicular to the ground; although the antennas have been leveled with some care, there remains at a small level a wobble if you rotate the antenna around this axis. Another parameter accounts for the angle between the azimuth and elevation axes; ideally this angle would be 90°, but when it is not we must account for it. Two final parameters allow for a collimation between the antenna itself and the receiver within it; these terms are important when switching between two receivers that are aligned differently. Altogether, these nine parameters are sufficient to tell the antenna exactly where it needs to slew so that the encoder offsets, sagging, tilts and collimation all conspire to leave it pointing at the right spot to within our desired tolerance. 50 On a monthly basis, we fit for the parameters in the pointing model by taking observations of many bright sources of known position. We point at each source and offset in the X and Y directions, forming a ten point cross, and fit the measured flux vs offset to a Gaussian beam; the center of these fit Gaussians tells us the offset between where we thought we were pointing and where we actually were. Because each term in the model has a different dependence on azimuth and elevation, by measuring many offset positions at many points in the sky we can fit for all of the parameters in the pointing model. These fit parameters are then fed back into the control system, and used in determining the azimuth and elevation that correspond to a requested position in the sky. In general, with ~ 100 pointing offsets scattered across the sky we can fit the entire model and achieve pointing with less than 0.5 arcminute RMS offsets. Note that the pointing of the antennas affects where on the primary beam the field center is located, but does not imply that we will measure sources to be at the wrong position on the sky. The reconstruction of source positions instead depends on the accuracy of our fringe-rotation, which is good enough that SZA measurements of known sources fall within a few arcseconds of their correct positions. Once we determine that the antennas are pointed where we think they are, we can ask how to convert the measured flux to an actual flux on the sky. The difference between the two is accounted for by a factor that we call the aperture efficiency. This term generally refers to the fact that through diffraction and surface errors, among other causes, an antenna has an effective area which is less than its geometrical area. However, we include in our aperture efficiency all factors which reduce the observed response to a source of a given flux. The SZA aperture efficiencies are in the range of 60%, within about 10% of the aperture calculated from the illumination of the mirror. We measure the aperture efficiency of each antenna across its sixteen bands 51 via regular observations of Mars. With a model describing the temperature of Mars on a day by day basis that is accurate to ~ 5% (Rudy, 1987), we can compare our measured flux with what the model predicts for the planet and derive the aperture efficiencies. These efficiencies are then used in data reduction to rescale all of our measured fluxes to accurately reflect the brightness of the objects that we observe. Finally, understanding which Fourier components of the sky that we are measuring at a given time depends on knowing accurately the positions of the antennas. Having these positions well-determined is also critical for transferring the phase of a calibrator (at one position in the sky), to a target (at a different position). Similarly to the antenna pointing, we periodically measure the phase of bright point sources at various positions on the sky. All sources should have zero phase when they are at the center of the field, and so the observed phase is a contribution from the instrument, which should be constant or slowly varying with time, and a contribution from an error in the phase delays due to errors in our assumed antenna positions, which will depend on the position of the source in the sky. We can fit the phase offsets observed in sources of different positions on the sky, minimizing them by adjusting the antenna positions. With the calibration in place, the antenna, receiver, and correlator can accurately measure the Fourier components of the sky. CHAPTER 3 OBSERVATIONS In this chapter, I describe the data that we have taken with the SZA instrument for the purpose of this power spectrum analysis. I will describe the fields that we have observed, as well as the observation strategy employed. I will discuss the calibration of these data, and also the extensive filtering that was invoked in order to produce a clean dataset. Finally, I will discuss the Very Large Array (VLA) observations undertaken to account for the contamination of our data by compact radio sources. 3.1 Field Selection The SZA was intended to simultaneously execute two experiments: the measurement of the CMB anisotropy power spectrum described here, and a six deg^ survey of the sky to detect massive clusters directly. The observations from 2005-2007 were largely split between these two projects, and the location of target patches of sky for the two projects were selected to accommodate both within a 24-hour observing day. The blind cluster survey requires deep optical data to produce redshifts for any detected clusters, and so these fields were placed so as to overlap existing optical surveys at 9 and 14 hours of Right Ascension. CMB observations were then performed from 0-6 hours Local Sidereal Time (LST). When the survey was completed in 2007, CMB observations filled available observing time, spreading across the sky. Within the range of RA observable during these hours, fields were selected to lie at a declination that passes close to zenith at the SZA's latitude of 37°. This block of observable sky is labeled as "SZA4" in Figure 3.1. The CMB observations cover 44 fields in total; the position and integration time for each field is listed in Table 3.2. Within this block, some effort was made to minimize the potential contamination 52 53 Figure 3.1: The SZA observation fields the bulk of the 44 fields with CMB observations fall within the oval marked "SZA4". The remaining fields were targeted for the SZA's blind cluster survey. from galactic synchrotron emission, based on higher frequency measurements with IRAS. No effort was made to avoid radio sources in our field selection, as there is evidence that sources and clusters are correlated (Coble et al., 2007), and we wanted to avoid biasing ourselves against having large-scale structure in our fields. Each field was observed for approximately 25 hours; with calibration, this project took nearly 1700 hours of data with the SZA. 3.2 Observation Strategy The SZA antennas are packed as closely as possible to increase sensitivity on large angular scales. Diffraction around the mirror edges produces sidelobes at large angles from the pointing direction; when these sidelobes on two antennas overlap on the ground or horizon, we expect "false fringes" to contaminate the data. This contamination should largely have been wiped out by the fringe rotation, but we designed the 54 observations in such a way as to allow the removal of any residual effect. To enable the subtraction of this ground contamination from the ground (in fact, anything that is fixed with respect to the antennas), we observed fields in groups of four. The fields in a group are separated by 4 minutes on the sky, and were observed for 4 minutes each. We thus observe each field for exactly as long as it takes for the next field to move into the same position, thereby observing all four fields over the exact same range of azimuth and elevation, and so with the same contamination from the ground. Since all four sources are observed over the same patch of ground, subtracting the mean of the four fields in a group, visibility by visibility, allows us to remove this potential source of contamination. The visibilities that we aim to measure have an amplitude and a phase, and the instrument makes a contribution to both that can vary on timescales of minutes. We included frequent calibration observations to try and remove this instrumental contribution. Because of the way in which the correlator normalizes the cross-correlations, the physical scale of the correlation coefficients depends on the system noise temperature Tsys. As described in Chapter 2, we used a hot/cold measurement to determine TSyS; because the value of Tsys changes with observing elevation and weather, we measured it every few minutes. The instrumental phase can also change on short timescales, as temperatures change and cable pathlengths are altered. These changes were monitored through periodic observations of a phase calibrator, which is simply a bright, unresolved quasar near to the observed fields. Similarly, drifts in amplitude gain between antennas or frequency bands, and even within a 500 MHz frequency band must be tracked. While monitoring the phase calibrator, we also removed any amplitude differences among antenna or frequency, ensuring that a source has the same flux across our array. The calibrators used for each field are listed in Table 3.2. A noise source within our 55 correlator, which provides correlated signal to each correlator channel, can be used for further phase monitoring, but we found that the noise source and astronomical phase calibrator followed each other very closely, and this step of calibration was deemed unnecessary. Observations of the phase calibrator bracket observations of the four fields in a group: PhCal - F l - F2 - F3 - F4 - Phcal, with system temperature measurements in between each step. The cycle lasts ~ 20 minutes. We repeated this cycle of observations for ~ six hours each day until we had acquired ~ 25 hours of good data on each field, and then proceeded to the next set of four fields. 3.3 Data Calibration and Quality Checking Data is averaged and recorded to disk every 20 seconds. These data are loaded into MATLAB for calibration with custom-built and automatic calibration software. The first step is to flag data with any indication of hardware problems. These most commonly include failures in phase lock or tracking of an antenna, and also the partial obscuring of one antenna by another, or "shadowing." Next, the hot and cold load measurements are used to compute the system temperature, which is interpolated in time and used to convert the data into physical brightness units from correlation coefficients. System temperature values for a track are shown in Figure 3.2. Next, we use the calibration observations to find the spectral response of each band and antenna. One example is shown in Figure 3.2. Because the antenna components have gains that vary with frequency, these spectra, or "bandpasses," are non-flat. We remove this shape from the data, so that a flat spectrum source appears that way. The amplitude and phase of the calibrator are then computed as a function of time, interpolated to cover the entire observation, and removed from the data so that 56 the calibrating source has constant amplitude and zero phase, as it should once the drifting instrumental phase is removed. Pre and post calibration measurements of these steps are all shown in Figure 3.2. System Temperature for Antl, Band 1 Antenna Passband Phase of Calibrator 0.2 o -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 ""o 2 4 time 6 """o 5 10 15 Frequency Channel 20 0 2 4 Time [hrs] 6 Figure 3.2: Calibration of the Data For a single band and antenna/baseline, we show the system temperature (left), the amplitude of the bandpass (center), and the phase of the calibrator (right). In the latter two, blue shows the raw measurements on the calibrator, and the red shows the same thing after calibration. At each step of calibration, data that cannot be interpolated or that appear to indicate equipment problems are nagged. Such data include very large or variable values of Tsys, evidence of bad weather. Discontinuities in calibrator phase or amplitude often reflect correlator problems, as does decoherence on the calibrator. Aberrantly high noise is also flagged. Nearly a quarter of data is removed with conservative flagging parameters. Flagged and calibrated data are then loaded for each day of observation, and two further tests are performed to check for bad weather and equipment failures. We plot, day by day and for each baseline-band combination, the variance of the system temperature over the day, and also the chi-square of target data (the blank sky target data is well-modeled by Gaussian noise of zero mean). In some cases, equipment 57 problems or clouds revealed themselves in these plots as clusters of aberrant points (Figure 3.3). In many cases, hardware problems could be discovered through this filter; even if the data problems could not be explained, this data was excluded. 3.4 Filtering for Antenna Crosstalk In addition to the sidelobes from our illumination pattern, discussed in Chapter 2, the three feedlegs supporting the secondary mirror scatter radiation onto very broad circles on the sky, creating a sidelobe that can extend up to 180 degrees from the pointing direction (Figure 3.4). We found that in certain configurations, one of the two antennas in a baseline can coherently detect radiation from the HEMT of its partner, as the smooth-edged feedlegs of the two antennae are perfectly aligned so that these rings cross one another and radiation from one HEMT is reflected into the other receiver. This radiation has a short wavelength, such that it wraps many times over the 500 MHz of one of our bands. Consequently, a band average over this 500 MHz wipes out all sign of coherent signal, but an RMS measured across one of the bands shows large scatter. The RMS and band average for an extreme case of crosstalk is shown in Figure 3.5. We found that applying crinkled Aluminum foil to the feedlegs broke up the edge for coherent scattering and reduced the magnitude of crosstalk, but data issues seem to remain at a lower level, and so we use further filtering to remove contaminated data. In the reduction process, periods of dramatically elevated RMS, as shown in Figure 3.5, are simply discarded. For the next stage of data filtering, the ratio r between the observed and theoretical variance is computed for each visibility: 2 r(time, baseline, band) = i^e^ured expected 58 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728 baseline 2000 1500 1000 1.5 X 2 /v 2.5 Figure 3.3: Bad %2 data An example of problematic data. Plotted is the x °f the d a t a for each day (y axis), shown for each baseline and band combination. For each baseline, all 16 frequency bands are shown sequentially. On day 31, e.g., obviously exotic data is seen in all bands of baselines 2 and 14. The histogram is of the x values, with the high tail showing the problematic points. 59 Figure 3.4: Ring Sidelobes The sidelobes created by scattering from the three feedlegs. The antenna is sensitive to objects on the red rings (not inside of them). The pointing center is at the center of this plot, and the rings show points on the sky at 50,100,150 degrees from the center of the beam. time [hre] time [hrs] Figure 3.5: An example of extreme crosstalk. At hour 1.5 and again at 3, this baseline show huge jumps in the RMS across the channels within a 500 MHz band (left). However, this rapid oscillation averages out, and is not visible in the plot of visibilities averaged over the same band (right). The signature is present in all bands of this baseline. Now we compute the band-to-band covariance of these ratios within a 2 hour block of time. The covariance of bands i and j is given by: C(baseline)ij = (r(time, baseline, i) x r(time, baseline, j)) where the average is taken over time. The total covariance of a baseline, C, is just the mean of the C{j over all i and j . The distribution of C among different baselines is assumed to be Gaussian. The baselines to antennas 6 and 7, which should have no crosstalk, are used to estimate the width of this distribution, and baselines that are outliers at the 5 a level are discarded. In Figure 3.6, the same extreme data is shown, and it is obvious that baseline 3-5, shown also in Figure 3.5, is one of these far outliers. Adding this filter removes < 1 % of the data, but substantially improves the results of jackknife tests in the process (jackknife tests are discussed in Chapter 6). 61 band-band covariance on Jan25s 2006 baseline 1-0 5 10 baseline 2-0 5 15 baseline 5-0 10 baseline 3-0 5 15 baseline 6-0 10 15 baseline 7-0 baseline 4-0 5 10 15 baseline 2-1 5 10 15 5 10 5 15 baseline 3-1 5 10 5 15 10 10 5 15 10 5 10 M 15 baseline 6-3 5 10 15 baseline 7-4 15 baseline 7-2 - _ wm 15 baseline 3-2 baseline 6-2 "mm 15 baseline 4-1 baseline 7-1 5 10 5 10 15 baseline 5-1 5 10 15 baseline 4-2 5 10 15 baseline 4-3 5 10 15 baseline 6-1 5 10 15 baseline 5-2 5 10 15 baseline 5-3 ^2»a " E 10 15 baseline 7-3 5 10 15 baseline 6-5 5 10 15 baseline 5-4 5 10 15 baseline 7-5 10 15 5 10 15 baseline 6-4 5 10 15 baseline 7-6 5 10 15 Figure 3.6: The second level of crosstalk filtering. Here, the ratio of observed and theoretical RMS is computed, and covariance of different bands is computed for each baseline and averaged over time. Vastly different levels of covariance, as seen here in baseline 3-5, are flagged. All panels use the same color scale, with white at the maximum. 62 observed variance/ expected significance of outlying point Figure 3.7: The final stage of crosstalk filtering. The ratio r of observed and expected variances is computed and averaged over band. The histogram of these averaged ratios for a track is formed (left, blue) and the inner region is fit to a Gaussian (red). The percentage of points beyond Na within a baseline is computed (right), and baselines that account for more than 30% of the outlying points are excluded for this track. Baseline 3-5, which for this day passed all other filters, is shown in red. A final stage of filtering also uses the ratio of observed and expected RMS, given by r. For each 6 hour track, r is averaged over band (the crosstalk tends to be similar across the bands of a baseline) yielding a statistic for each scan and baseline. A histogram of these numbers is generated and fit to a Gaussian (Figure 3.7). We then compute the percentage of significant outliers that comes from a given baseline; if a baseline makes up 30% or more of the 4-7<r outliers, then the baseline is flagged for that track. We have experimented further with flagging data taken when the sun or moon is within 25 degrees of our pointing, but ultimately it seems that this cut is ineffective. We discard scans that have fewer than 10 days of observations in them, as well as all scans with fewer than seven degrees of freedom from which to estimate the variance. 63 Cut No cuts [hrs] Pipeline Crosstalk Tsys/x2 sun> 25° moon> 25° sun AND moon ground matching < 10 days <7dof Field set 2 131 15% 7% 0% 0% 7% 7% 4% 7% 4% 3 117 21 5 0 25 7 28 11 4 9 4 126 13 6 0 4 12 16 13 1 11 5 171 12 5 0 6 8 14 8 0 11 6 127 12 7 0 12 15 27 7 1 10 7 114 21 4 0 0 0 0 4 1 7 8 103 24 2 0 0 0 0 3 1 7 9 102 21 3 0 0 0 0 3 6 8 10 114 29 1 0 0 0 0 2 0 7 11 123 23 3 1 3 3 3 5 3 7 total: 12 108 24 3 0 0 0 0 7 8 6 total time flagged [hrs] 1336 254 hrs 58 hrs 1 hrs 74 hrs 70 hrs 125 hrs 85 hrs 37 hrs 108 hrs 541 hrs (40%) Table 3.1: Flagged Data Volume These cuts appear as "10 days" and "7 dof" in Table 3.1. The parameters for these various filtering techniques have been tuned in order to pass several stringent jackknife tests of data quality, described in Chapter 6. The total amount of data nagged by the various steps is listed in Table 3.1. We see here that ~ 30% of the 1300 hours of data taken are flagged if we neglect the sun cut. The final three rows of cuts are related to data that is poorly matched among a group of fields (ground matching) or that don't have enough samples to accurately calculate the variance of the data (10 days, 7 dof). With the filtered data in hand, we are ready to begin the analysis described in Chapter 4. 3.5 VLA 8 GHz data 3.5.1 VLA Data As we will discuss at length in Chapter 5, the dominant foreground in our measurement is the population of compact, synchrotron-emitting radio sources; this contaminant can be dealt with, provided that we know the positions of the brightest sources in our field, and residual contamination from sources that are undetected can be accounted for if we have an understanding of the characteristics of the population of these sources. We acquired the information we require with observations at the Very 64 Large Array (VLA), using 36 hours in observation program AS882 in November 2006 to cover half of our SZA fields, and covering the remaining fields in summer 2008 through observation program AM961. The Very Large Array, located in Socorro, New Mexico, is one of the flagship instruments of the National Radio Astronomy Observatory. The VLA consists of 27 antennas, each of which is 25 meters across. The antennas are movable, and cycle through configurations ranging from the close packed D array, with baselines no longer than 600 meters, to the extended A array, which spans the three, 21 kilometer-long arms of a Y. The VLA can observe at 74 MHz, 300 MHz, 1.4, 5, 8, 14, 22, and 45 GHz. Our 2006 observations were in the C configuration at 8 GHz, while the 2008 observations were in the D configuration, also at 8 GHz. At 8 GHz, the VLA primary beam is about 5 arcminutes across. To cover the field of view of the SZA, which has a 12 arcminute beam, we required multiple pointings with the VLA, which could then be mosaicked together into a single image. The noise across an SZA pointing is not uniform, growing at large radius from the center as the beam drops to low amplitude. We distributed our integration time across the mosaic pointings with the VLA so that the final, mosaicked noise map at 8 GHz has the same profile as the noise in our SZA 30 GHz maps; this profile is simply determined by the primary beam of the SZA antennas. In this way, our detection threshold in the VLA maps is uniform across the field not in the flux of the sources detected, but rather in the detection significance of these sources in the SZA maps. Equivalently, we have a flat sensitivity in the VLA map to the flux of sources when weighted by the SZA beam at their position. This is, in fact, the relevant quantity for our purposes, and it can be achieved with dramatically less VLA observing time than would be required to acquire an 8 GHz flux catalog of uniform flux sensitivity. The VLA 8 GHz sensitivity at the center of the fields is approximately 30 fiJy RMS, and allows us to detect at 65 - SZA 0.1 power poini 0 arcmin o 5.6 arcmin A 9.7 arcmin 11.2 arcmin + 15 10 -%*^ 5 0 -5 / o A o 5* \ + O O F A / A 10 - -15 A \ O o A - A -10 -5 10 15 Figure 3.8: The 19-pointing VLA mosaicking scheme The circle marks the SZA's 10 i/l power point, and the axes are shown in arcminutes. 5cr all sources with flat or steeper spectra that will show up at 30 GHz at the la level in the SZA maps. Mosaicking to cover the SZA's primary beam to the tenth power point at l l ' requires 19 VLA pointings at 8 GHz (Figure 3.8). At the SZA beam center, we reached 0.2 mJy sensitivity at 30 GHz. We achieved ~ 30/j,Jy sensitivity in the center pointing of the VLA mosaic. For pointings away from the SZA beam center we aimed for 6 pointings at a radius of 5.6', with a noise sensitivity of 0.062 mJy, 6 pointings at 9.7' with sensitivity of 0.2 mJy, and 6 pointings at 11.2' with sensitivity of 0.375 mJy. For total integration on all 19 pointings, phase calibration and slewing, we used a total time of ~ 1.5 hours of VLA time for each SZA pointing. 66 3.5.2 VLA Reduction The data taken during our VLA observations were calibrated, and radio sources were detected, using the NRAO's Astronomical Image Processing System (AIPS), a standard set of custom tools created by NRAO for the processing of data from the VLA. Deciphering the intricacies of AIPS was far from straightforward, and I am grateful for the help of Chicago postdoc Dan Marrone and NRAO employees Eric Greisen and Dave Meier. The AIPS cookbook, also of great help in understanding the program, hysterically gives a wide variety of recipes involving bananas. At no point in the frustration of trying to understand the perverse syntax of AIPS was the presence of these comical recipes irritating. I list the tasks used in AIPS to calibrate and mosaic our data here, not because I imagine them to be interesting or illuminating to my committee members, but rather because I wish that such a list had been available to me when I was trying to understand how to use AIPS. I attach my AIPS scripts as an appendix. The data was loaded with task FILLM. VLANT updates baseline corrections, LISTR was used to have a quick look at the data quality, and UVFLG nagged data that seemed suspect. VPLOT was used to further inspect the data, SETJY sets the flux scale, CALIB produces a calibration table, GETJY and CLCAL do some calibration steps. I used VPLOT and SNPLT to look at the calibrator data, and nagged data until the calibrator looked like a point source. With the data flagged and calibrated, I used SPLIT to select the pointings from a given field and IMAGR to make a map of each one. FLATN does the mosiacking, RMSD calculates a noise map, and COMB creates the signal-to-noise ratio in a map. SAD then hunts through the map and finds all 5<r sources, which I output to a catalog of 8 GHz-detected sources. This 8 GHz catalog is in no sense complete; we have not detected all 150 /^Jy sources in our fields. Instead, we have found at 5<r all sources that have a beamweighted flux of 150 /iJy, where the weighting is with the SZA's primary beam, centered at the mosaic center. Such sources, if they had perfectly flat spectra, would contribute below the 1 a level in our 30 GHz maps, and so we presume that, for flat or slightly rising spectra, we will have detected most sources that have any possibility of contaminating our higher frequency data. Table 3.2: The SZA field locations, integration times, and phase calibrators. Field Name CMB07 CMB08 CMB09 CMB10 cmbAl cmbA9 cmbAl7 cmbA25 cmbll cmbI9 cmbll 7 cmbI25 cmbRl cmbR9 cmbRl 7 cmbR25 cmbYl cmbY9 cmbY17 cmbY25 cmbEEl cmbEE3 cmbEE4 cmbEE2 cmbXXb cmbXXa RA 02:07:37.0 02:15:14.0 02:22:51.0 02:30:28.0 02:12:00.0 02:16:11.8 02:20:23.7 02:24:35.5 02:12:00.0 02:16:10.8 02:20:21.5 02:24:32.3 02:12:15.6 02:16:25.2 02:20:34.7 02:24:44.3 02:12:00.0 02:16:08.7 02:20:17.3 02:24:26.0 14:18:39.2 14:27:03.6 14:31:15.6 14:22:48.0 21:28:50.6 21:24:38.7 Dec +34:00:00 +34:00:00 +34:00:00 +34:00:00 +33:00:00 +32:59:44 +32:58:56 +32:57:37 +32:37:08 +32:36:52 +32:36:05 +32:34:46 +32:11:24 +32:11:07 +32:10:18 +32:08:57 +31:51:24 +31:51:08 +31:50:21 +31:49:03 +35:31:52 +35:31:49 +35:31:48 +35:31:50 +24:59:35 +24:59:37 unflagged hours 20.23 20.23 20.23 20.23 20.47 20.47 20.47 20.47 21.42 21.42 21.42 21.42 35.00 35.00 35.00 35.00 21.56 21.56 21.56 21.56 20.90 20.90 20.90 20.90 19.86 19.86 phase calibrator J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J0237+288 J1331+305 J1331+305 J1331+305 J1331+305 J2139+143 J2139+143 Table 3.2: SZA field locations, continued Field Name cmbXXl cmbXX2 cmbDDl cmbDD2 cmbDD3 cmbDD4 cmbAAl cmbAA2 cmbAA3 cmbAA4 cmbCCl cmbCC2 cmbCC3 cmbCC4 cmbBBl cmbBB2 cmbBB3 cmbBB4 RA 21:33:02.6 21:37:14.5 14:18:40.1 14:22:52.2 14:27:04.2 14:31:16.3 21:24:38.7 21:28:50.7 21:33:02.6 21:37:14.6 02:11:31.3 02:15:43.2 02:19:55.1 02:24:07.1 21:24:38.1 21:28:50.0 21:33:01.9 21:37:13.9 unflagged hours Dec +24:59:33 19.86 +24:59:32 19.86 +35:01:42 18.06 18.06 +35:01:40 18.06 +35:01:39 +35:01:37 18.06 21.31 +25:29:37 21.31 +25:29:35 21.31 +25:29:33 +25:29:32 21.31 21.51 +33:27:43 21.51 +33:27:45 +33:27:47 21.51 +33:27:48 21.51 +25:59:24 21.07 21.07 +25:59:23 +25:59:21 21.07 21.07 +25:59:19 phase calibrator J2139+143 J2139+143 J1331+305 J1331+305 J1331+305 J1331+305 J2139+143 J2139+143 J2139+143 J2139+143 J0237+288 J0237+288 J0237+288 J0237+288 J2025+337 J2025+337 J2025+337 J2025+337 CHAPTER 4 LIKELIHOOD FORMALISM In this chapter I outline the likelihood techniques used to turn our visibilities into a measurement of the power spectrum. 4.1 Visibilities and the Power Spectrum The visibilities that are the data output of an interferometer are the correlation coefficients between a pair of antennas, called a baseline. The SZA, with eight antennas, has 28 baselines. The SZA antennas convert radiation from the sky into a voltage; this voltage, Vi, has a component which is signal and one that is noise. vi = Si + Ni (4.1) In many cases, the noise term is much larger than the signal term; interferometers are able to detect weak signals by combining the voltages from the two antennas in a baseline. The interferometer's correlator mulitplies the voltages; the product, called a visibility (V{j) is then given by: Vij = (viVj) = {(Si + Ni)(Sj + Nj)) = (SiSj) + {NiNj) + (8^) + {SjNi) (4.2) where i, j label antennas. If the noise in one antenna is uncorrelated from the noise in the other, as we expect it to be, then the last three terms integrate to zero, and the visibility contains only the signal (and the small part of the noise which is correlated). The signal part of the antenna output contains information about the sky. The antenna's primary beam, 69 70 A(x), describes its relative sensitivity to radiation coming from different places x on the sky. The SZA antennas have primary beams which are nearly Gaussian, with a full width at half power point of about 12 arcminutes. Thus the antenna is maximally sensitive to things at x = 0, which is the place on the sky where the antenna is pointing; it is less sensitive as a source of radiation moves further from the pointing center. Neglecting the noise term, which we have seen will soon disappear, the power that antenna i outputs can be written as an integral of the sky brightness, I(x), weighted by the antenna beam: Pi = Sf = (dxAi(x)I(x) (4.3) If two antennas are pointing at the same place in the sky, we expect the signal power from each to be the same. However, we combine the voltages from the two antennas, not the powers, and so we are also able to keep track of any phase difference between them. The visibility is thus complex: Vij = SiSj = ! dxA(x)I{x)ei(t>ii, (4.4) where A(x) is the common beam, and 0^ is the phase difference between the two antennas. We can force the antennas to have the same instrumental phase (see the discussion of fringe-tracking in Section 2.2.2), so that radiation from the pointing center will arrive at the two antennas with the same phase and <^j = 0. However, radiation from a spot away from the center of the pointing will reach the two antennas with a differential phase that will depend on the distance of the source from the pointing center and the separation of the antennas in wavelengths, which we will call u. This phase is 2irux, and will weight the emission from every spot x on the sky, so that we write the visibility as: 71 f dxA(x)I(x)e-i2m V(u)= This is simply a Fourier transform. (4.5) Each visibility that the SZA measures is simply a Fourier component of the sky brightness times the primary beam of the antennas. Equivalently, the visibility can be written as the convolution of the Fourier Transform of the sky brightness with the transform of the antenna's beam: V>) = ^ i ( u , A ) * ^ ( u ) (4.6) Just as the beam A(x) describes the area in the image plane over which we are integrating signals, its transform A(u) describes the area within the plane of Fourier components of the sky which contribute to an individual measurement at position u. It is clear that such a function must exist; although we label each measurement with a Fourier component that depends on the distance between the centers of two antennas, slightly longer or shorter baselines are also being measured by these antennas, so long as the ends of these baselines land somewhere on the two antennas' 3.5 meter surface. Since the beam is the far field pattern squared, and the far field pattern is the transform of the aperture field, we can see from the convolution theorem that A(u) is the autocorrelation of the aperture field. The beam is normalized so the A(0) = 1, so we equivalently require J A(u) — 1. In the flat-sky approximation, appropriate in the case of the SZA's 12' primary ,-4 beam, the power spectrum of the sky is defined in terms of these same Fourier components: 2 Ct AT (4.7) 72 The C^s are what we aim to measure, and Equation 4.6 shows that our visibilities contain all of the information that we need to make this measurement, without ever needing to turn our data into an actual map of the sky. Specifically, a measurement of the variance of the visibilities is a direct measurement of the amplitude of the power spectrum. Computing the covariance of two visibilities is a bit algebra-intensive, but you can see that there will be a term like ( I ^ T - I ) = Q> and a term like A(u)2. For the variance of the real (or imaginary) part of the visibility measured at Uj we get: Cfj = \ (jf) J duCe(u)A(Ui - u)[A(Uj -u) + A(Uj + u)} (4.8) All we need to do to measure the power spectrum is to collect our visibilities, measure their covariance, and read off the CMB power. In practice, we use a maximum likelihood technique that follows Bond et al. (1998), whereby we maximize a likelihood function £(C\A). Here C is the covariance matrix, which we relate to the CMB power through Equation 4.8, A is the vector of our visibilities, and C tells us how likely a given variance is, given our data. We assume that the data are Gaussian distributed. If we were to ask what the probability of making measurement x is, given that it came from a Gaussian distribution of variance a , we would answer: p( 2) * - -jhr*-^ (49) We want to know the converse: given a measurement x, what is the likelihood that it came from a distribution of variance a , or P(<r2|x)? It turns out to also be Gaussian, because of Bayes's Theorem, which says that: 73 P{a2\x) = P(x\a2)P{a2) (4.10) In our case, P ( a 2 ) , the prior on what values of the variance are possible, is flat, and so: P{ailx) = P{x^) = _L_exp(_^ (4.u) Our experiment adds the complication that our visibilities can be covariant, and so we use matrix multiplication. To the question of how likely a covariance matrix is, given our data, we answer: £ C A < I > = (2x)W/2|C|l/2^4ArCrlA)' <412> where N is the length of the data vector A. All we have to do is take the data, evaluate this function for various values of CMB power, and read off the most likely value. 4.2 T h e Covariance M a t r i x The reality is only slightly more complicated than this. While Equation 4.8 tells us how the CMB power enters into the covariance matrix, there are other sources of covariance that must also be considered. We can write the total covariance matrix, which we must use to evaluate Equation 4.12, as the sum of three pieces: C = CcMB + Cnoise + Cconst CQMB (4-13) i s the contribution from the CMB sky that we've just described, given by 74 Equation 4.8. Cnoise is a diagonal piece describing the noise variance of our many independent measurements. We will attempt to measure this noise term from the data themselves in section 4.2.1. Finally, the CconS£ term, called a constraint matrix, is our means of removing the contamination from non-CMB sources of emission in the sky, principally radio point sources and the ground; the constraint matrices through which we do this are discussed in section 4.2.2. Examples of these various components are shown in Figure 4.1, with CQMB in the left panel. Each component is described in detail below. CMB covariance Instrument Noise covariance Constraint covariance 5 10 15 20 visibilities (4 fields) 5 10 15 20 visibilities (4 fields) 5 10 15 20 visibilities (4 fields) Figure 4.1: Examples of the Covariance Matrix Shown are the components of the covariance matrix for a small number of visibilities in a set of four fields. The left panel shows the C(JMB component, given by Equation 4.8, which shows how the overlapping aperture functions in the u — v plane introduce correlations among the visibilties in one field (the eight blocks are real and imaginary parts of four fields. The middle panel shows the instrumental component, Cnoise, which is diagonal. The right panel shows constraint matrices; the large blocks are from point sources within each field, the off-block-diagonal components are the ground constraint. 4-2.1 Noise Estimation In the absence of any signal at all, our visibilities would still fluctuate with some instrumental noise. In temperature units, the noise component is described by the system temperature, Tsys. The measurement of the system temperature is described 75 in Section 2.4, and its contribution to the variance of the data drops with the number of samples, 2(3T, where (3 is the bandwidth (500 MHz in the SZA bands), r is the integration time (typically 20 seconds), and the factor of two comes from the fact that we have two antennas in the baseline. Meanwhile, the temperature contribution of the signal can be written in terms of an antenna temperature, T4, where \AS = kTA. Here S is the energy flux of the source, A is the effective area of the antenna, and k is Boltzmann's constant. The effective area is given by the product of the geometrical area of the SZA's 3.5 m diameter dishes and an aperture efficiency factor, described in Section 2.5 and typically around 60%. Thus, the signal to noise of a measurement of a source of flux S is given by: _ TA KSN = Tf^ 1 sys ASVWF = -57™ (4-14) Unisys There is an additional noise contribution from the digitization of signals from the antennas before the cross-correlation is computed. This digitizer noise is accounted for with another efficiency factor, r)c, about 87% for the SZA's 2-bit digital correlator. In flux units, the RMS contribution of the antennas is thus given by: '** = w§? (4 15) ' Putting these pieces together, the typical noise level in a single SZA visibility is about 0.1 Jy (the signal we are seeking is ~ 1000 times dimmer). Our estimation of the CMB power relies critically on an accurate estimate of the noise of our data; in Equation 4.13, you can see that an underestimate of our instrumental noise will translate directly into an overestimate of the CMB power, or vice versa. We can estimate the noise in our measurements in several ways, either by measuring the scatter directly from the data or by attempting to directly measure or 76 calculate the factors in Equation 4.15. M e a s u r i n g t h e S c a t t e r of t h e D a t a w i t h a Small N u m b e r of P o i n t s While knowing the various quantities in Equation 4.15 with enough precision may not be possible, we can instead directly measure the scatter of the data and use this scatter as an estimate of the true variance of the distribution. If we have many time samples, the sample variance of the data gives a very robust estimate of the instrument variance. Unfortunately, we do not have many time samples on the time scales on which the system noise is changing; in our ~ 4 minute scans, we normally only get 10 time samples on a given source, and there are statistical complications that come from using estimates with so little information in them. While the estimate of the variance from ten samples (C 2 Q) is an unbiased estimate of the true variance (a ), which is to say that {(T\Q) = <r2, this is not enough for our purposes. Our analysis, based on the likelihood function given in Equation 4.12, depends on the x given in the zero covariance case by \ statistic of our data, — (^l)- With perfectly Gaussian x and good knowledge of the a2, we would find that x 2 = 1Although it is true that our variance estimates are unbiased, the distribution that the variance estimates from v + 1 samples follow is a chi-squared distribution, given by: f(vo2\v) = ^ U = e —n . (4.16) For v < ~ 40, this distribution is highly skewed, favoring values smaller than the mean. The nonsymmetric distribution of the a2 becomes a problem for us because although (a2,) = a2, (-^-) ^ 4 j . The distribution of our %2 statistic is skewed away 77 from one: A.V = <s> = <«•}< (4.17) / 1 \ /•oo ^ 1 /-I J—oo A , /2TT o /•«> ( a 2 r / 2 - l e - a V 2 1 2 x ax x v I = TT xda -oo r(il/)2"/ 2 *2 V (4.18) where the second line follows from the first because estimates of the variance are independent from the samples themselves and we have normalized the problem such that the true value of a — 1. We take the approach that our goal is to find the variance that forces x = 1> and so to compute the variance of our data we take the sample variance of the 10 visibilities within each 4 minute scan and rescale them by the number of degrees of freedom they were estimated from according to Equation 4.18, thereby forcing the x to one. In order to evaluate our likelihood function, we are forced to invert the visibilityvisibility covariance matrix; some compression of our ~ 1 million visibility data vector is therefore necessary to make this inversion possible. Such compression is possible because many of our visibilities are redundant, in particular sequential time steps are highly correlated, and each of the ~30 days of observations sample the same u — v components. When we combine these correlated measurements, we are forced to do it with uniform weights for reasons that will be discussed below. While not the optimal way of combining measurements, this does have the benefit of producing variances for the compressed visibility vector which are sums of variances of the 78 uncompressed numbers. Because the sum of chi-square distribution with z^ degrees of freedom is given by another chi-square distribution with J2 V{ degrees of freedom, it is easy to keep track of how many samples are contained in each final variance estimate. We combine the variances of the real and imaginary parts of each of our original 10 samples in this way (the real and imaginary components of the visibilities should follow the same statistics), and then combine the variances from many days of observations. In this way, the total number of degrees of freedom contained in the variance estimates for the final, compressed visibilities is quite large, and the rescaling necessary from Equation 4.18 is quite small. The variance estimates that we have now accumulated are put on the diagonal of a matrix of size Nvis; this matrix forms the detector noise component of our total covariance matrix, Cnoise) given in Equation 4.13. Cnoise is the middle panel of Figure 4.1. 4-2.2 Constraint Matrices The third piece of the covariance matrix given by Equation 4.13 is a constraint matrix that removes from the analysis sources of known contamination. The covariance contains N independent modes, or separate pieces of information, from the N visibilities that we have measured. We can select only those modes, or combinations of visibilities, that we believe to be contaminated and de-weight them by making their variance very large (Bond et al., 1998; Halverson et al., 2002). In practice, this is done by taking a visibility template that matches our contaminant, V, forming the outer product of this length-N vector with itself, Cconst = V V, multiplying the resulting rp N x N matrix by a large prefactor A, and adding XV V to our covariance matrix. The simplest constraint that we use is to remove modes that correspond to the 79 common ground. Because of the observing strategy discussed in Chapter 3, the ground is the same in each of the four fields observed together; removing it is a matter of removing the common-mode of these fields. In this case, we form a visibility data vector that contains all four fields observed at once: A = (VW^n, ^2[l:An> ^3fl:Afl> ^4[l:Afl)> where each field has taken N measurements. We form a ground constraint template for each of the N visibilities, such that VQn(n has value one for all four copies of visibility Vi, and zeros elsewhere. We form the constraint matrix from this template, multiply by our prefactor, and sum over the constraints from all N visibilities. The final result, which removes JV/4 modes from our data, deweights any component which is the same across the four fields, including the ground. The other principle source of constraints is the population of radio point sources that we know to be in our fields. We discuss these constraints at length in the next chapter. An example of a constraint matrix is in the right panel of Figure 4.1. Having now assembled our covariance matrix, we can evaluate our likelihood function and determine the most likely amount of power given our data. We compute one such likelihood curve for each of our eleven sets of four fields; the total likelihood of our experiment is given by the product of these curves. 4.3 Putting it into Practice Over the course of the 1,300 hours of observations taken for this project, we accumulated 100 million visibilities. To put the tools outlined above into use, we must first massage the data into a more manageable form. We break each day's observation into ~ 20, 4-minute integrations, which we will term a "scan." The visibilities that are within each scan are > 99% correlated. The repeatability of the individual tracks is good enough from day to day that each scan contains visibilities from many days. 80 The u — v coverage of a scan can be seen in Figure 4.2. Now we can construct an array of visibilities which is [Integrations] x [Scans] x [Days] x [Baselines] x [Bands]. As seen in Figure 4.2, there are about ten, twentysecond integrations in each scan. There are fewer than twenty scans in a track, and around 30 tracks were taken for each field. Thus, the visibilities are formed into array which is 10x20x30x28xl6=~ 2.5 million entries for each field. If jackknife tests are to be applied (see Chapter 6), we do it at this stage. i i 0 100 11 i 200 300 u (wavelengths) i i 400 500 Figure 4.2: An SZA scan We show the u — v coverage for a single baseline for a single day. The coordinates for a single band are highlighted in red; we show the visibilities in scan number three in green. Each scan has ~ 10 visibilities per baseline per band. Now, we follow the steps above to calculate a variance from the 10 samples within a scan. Since the reals and imaginaries should be drawn from the same distribution, we average the variances computed for each, for a total of 18 degrees of freedom in 81 the 4-minute scan. The visibilities are also averaged, and now we have 20x30x28x16 visibilities, each with a variance. At this point, we combine the different days. The previous sorting ensures that the scans for each day match the scans for the other days. We average over the days, again with uniform weighting. We are left with 20x28x16 visibilities and variances, where each one has been averaged over 2 (Real/Imaginary) x 10 (samples in a scan) x 30 (days)=600 samples. With our data in a form that is considerably more manageable, we divide the data into bins of I and calculate the covariance matrix for each by doing the overlap integrals in Equation 4.8. Elements that are more than 95% correlated are averaged together (still with uniform weighting). Constraints are calculated and added to the covariance matrix, along with the diagonal noise matrix. Because we are constraining the ground, we must build this matrix and visibility vector for an entire group of four fields-the process must then be repeated ten more times, for the other fields. At this point, we have a manageably small list of visibilities and a complete covariance matrix to go with them. We can evaluate Equation 4.12 across a range of CMB power values, and generate a likelihood curve for each group of fields. We multiply the eleven curves together, as the different field groups are independent, to generate the total likelihood curve of our measurement. We simply find the maximum and the 68% confidence region of this curve to describe our result. CHAPTER 5 FOREGROUNDS In Chapter 4, we discussed the tools that we need to extract a power spectrum measurement from our data. In Chapter 6, we will outline powerful Jackknife tests to ensure that this detected power is coming from the sky, and not from some instrumental contamination. In this Chapter, we will try to understand the role that foreground sources play in our measurement. In fact, the power in our fields is dominated by synchrotron-emitting Active Galactic Nuclei (AGN) or star-forming galaxies, which are point-like when seen by the SZA. An example of such a radio point source is obvious in Figure 5.1; most SZA fields contain at least one such source, and they must therefore be removed to correctly estimate the power from the CMB. The constraint matrix formalism outlined in section 4.2.2, whereby modes corresponding to a source with known position and shape are heavily down-weighted in the data, allows us to remove these sources. Because the factor by which the mode is projected out is essentially infinite, the actual flux of the object is irrelevant. This allows us to remove objects that were detected at other frequencies, the 30 GHz flux of which are often highly uncertain. By contrast, if we wanted to simply subtract the contaminants from our data, we would need to measure their fluxes at 30 GHz with an uncertainty less than the tiny flux level of the CMB signal; such an observation program would be prohibitively time-intensive. In this chapter, we will begin with a description of the constraint matrix formalism, discuss at length the various catalogs of synchrotron sources that can be projected from our data, estimate the effectiveness of this removal, and finish with a discussion of several other classes of objects that might influence our results. 82 83 5.1 Removing Sources Ideally, we would like to have a list of sources that are in our data and simply subtract them from the visibilities one by one. While we are lucky enough to have a list of sources that other experiments have observed within our fields, we often do not know their fluxes at 30 GHz at all, much less to an accuracy that matches the low flux level of our signal. The next best thing would be to marginalize over the unknown flux of the sources in our data; this admits our ignorance of the source flux, but renders our result immune to this contaminant, whatever its flux. While conceptually simple, this procedure turns out to be quite computationally demanding. Instead, we will remove the contribution of sources of known position and shape, but unknown flux, by forming a constraint matrix that corresponds to the modes matching the source, multiplying the matrix by a large number, and adding it to our overall covariance matrix. Effectively, this procedure de-weights the modes in our data that a source has contaminated. Here we will provide a prescription for this process, and justify the claim that it is equivalent to marginalizing over the source amplitude. The visibility measured from a point-like source is given by: VPS(u, v) = A0 f^j " ei2^ul+vm) (5.1) where AQ is the unknown amplitude at frequency VQ, a is the spectral index, (u,v) labels the Fourier component being measured by a given visibility, and (l,m) gives the offset of the source from the phase center. For simplicity, we will assume for the moment that our data contains only 2 visibilities, such that our covariance matrix is 2 x 2 and is given by C = f Cn C12 ^ , where the covariance due to noise and to i C\2 C22 J the Gaussian CMB signal are included. We will further assume that a — 0. 84 5.1.1 Subtracting Known Sources The best case scenario would be that in which we know the source position (I, m) and flux A. We can then simply subtract Equation 5.1 from our measurements to remove the influence of this source. If our initial data vector was given by A = [Vi, V2], our subtracted vector will be A' = [Vi — Vps(l), V2 — Vps(2)]. Now the likelihood function becomes: \Tn-lt (5.2) In the simple 2 sample case, we can evaluate this explicitly, using: ( l C~ = \C\ \ C22 -C12 -C12 Cn (5.3) ) and (5.4) \C\ = C11C22 - C12C12 We write: X2 - (A-PSfc^iA-PS) V\ - Api V2 - Ap2 = x |C| V C22 -C12 -C12 Cn f Vi-An^ x ^ V2 - Ap2 j T7T{(C22V{ - Ap1C22V1 - C22ApiV1 + C 2 2 ^, p2 f2 - C ^ V i + C12V2APl C12Ap2Vi CuVf + - C12Ap\p\ - Ci2ViV2 + C12V1AP2 + AC12P1V2 - A2CupiP2 - ClxV2Ap2 - CnAp2V2 + CnA2p22) + (5.5) 85 where we have written the components of Equation 5.1 as Ap^ separating the spatial and flux information. Ugly as it may appear, we can evaluate this function exactly, calculating the likelihood curve without any contamination from the known source. 5.1.2 Marginalizing over Source Fluxes Now let us assume that we know the source's position (/, m), but not the flux. We will not be able to evaluate Equation 5.5 explicitly, not knowing the value of A, and so we cannot compute the likelihood of power, neglecting the source. Instead, we add the flux of the source as an additional parameter, S, calculating a two-dimensional surface of the likelihood of a given amount of CMB power (hidden within the covariance matrix), and a point source at (l,m) of flux S. Having computed the likelihood of CMB power for every possible source flux, we can marginalize over that flux by simply integrating the likelihood surface over the point source dimension. The resulting onedimensional likelihood curve tells us the probability of given levels of CMB power in the presence of a source of any flux. Our final power spectrum constraint is now insensitive to the point source, whatever its flux was. We rewrite Equation 5.5 as: C{C,S) = ^ ^ e x p ^ ^ C ^ - 2 C ^ V ^ +C ^ ))exp{-^{AS2 + BS)) (5.6) with A = (C22Pi-2Ci 2 piP2 + C i i P 2 ) a n d B = (-2C22P1V1+2C12P2V1 + 2C12P1V22C11P2V2). Now we marginalize (integrate) over S. The likelihood function given a source of any flux at a certain position is given by: 86 oo / -00 00 dSC(C, S) 1 e x p ( - ^ i ( C 2 2 ^ 2 - 2C 12 V 2 ^ + CnVi)) 25F|C|V2^V 2|C|" oo X / dSe (5.7) (1(AS*+BS)) -00 completing the square we are left with: £(C) M o r f l = - = c ( =»|0| 2TT e W h / V A (5.8) which simplifies to: / V ;iw a r 5 (4v?-2PlP2VlV2+I%V$) y 27r(C22pf - 2Ci2PiP2 + C i i ^ ) ' To review, the pi depend only on the source position; we've altered our likelihood function to exclude any contribution from a source at this position, regardless as to the source flux. In principle, this will work fine for our purposes; if you'd like to go through the algebra for our ~ 4000 element visibility vector, please go ahead! 5.1.3 Projecting Sources with Constraint Matrices Now we take another tack; instead of altering the visibilities to remove a source, we add a constraint to the covariance matrix to account for its contribution. The rp constraint is given in this case by \pp , where A is the large prefactor and the p{ are given by Equation 5.1 with A = 1, and the new covariance matrix is: 87 Cn + \p\ C'C = C + XppT = Cu + \p\P2 y C\2 + ApiP2 (5.10) C22 + AP2 j such that C. c C22 + Xp\ -C12 - \p\P2 \CC\ ^ -C12 - \p\P2 (5.11) C n + Apf and |C C | = C11C22 + XCnpl + AC22P1 - Cf2 - 2ACi2Pip2- (5.12) If we make A quite large, we can skip the terms that don't grow with A, and write: \cc\ = A(CHP! + C22P? - 2C12P1P2) (5.13) Reevaluating the LH function with this covariance, we have: ( P 2y2 -2p 1 p 2 V2y 1 +p2y|) £c(C) = x 2<K\{CllPl + C22P? e 2(C 1 1 p|+C 2 2P^-2C 1 2 PiP2) (5.14) 2C12P1P2) The overall scale of the Likelihood is irrelevant, so the last remaining factor of A doesn't matter and Equations 5.9 and 5.14 are identical. Marginalizing over the flux that we don't know is the right thing to do, although a bit messy algebraically. The constraint matrix is mathematically equivalent, but substantially simpler to implement. We use it in our analysis. 88 5.1.4 Spectral Indices The sources we detect have unknown spectral indices; the value of this index changes the shape of the constraint matrix, and so we need a way to include the spectrum of our source into the corresponding constraint. In the limit that the spectral behavior of a source is linear across our band, we can write an arbitrary spectral slope as a sum of two fixed-spectrum sources, each with a coefficient. The inifinite prefactor means that by constraining multiple sources at the same position, but with different spectral indices, we are constraining any linear combination of the two. Constraining two spectral indices should therefore be sufficient to remove the contribution of a single source of arbitrary spectral index. Just to be safe, we constrain three sources at every position: one with a flat spectrum, one with spectrum i/ - 0 - , and one with v . Every possible spectral index can be written as a sum of these three fixed terms. 5.2 Known Synchrotron Sources Given a list of source positions, we now know how to remove the contamination they introduce; all we need is to find this list of sources to constrain. Here we discuss three sets of sources that we know about: those that are detected by the SZA at 30 GHz, those that are detected by the NVSS at 1.4 GHz, and those that we have observed with the VLA at 8 GHz. We will later consider how to best combine these three sets of sources to effectively remove the contribution of radio sources to our power measurement. 5.2.1 SZA Sources The most obvious radio sources to constrain from our data are those that are bright enough at 30 GHz to be detected directly by the SZA (Figure 5.1). To search for 89 such sources, we make a map of each CMB field with data from both long and short baselines; this map has nearly twice as much data in it than a map from only the short baselines, which is the data from which the CMB power spectrum measurement is made, and therefore our all-baseline map has a factor of ~ y/2 less noise. Within this map, we search for a maximum pixel, fit for a point source near this point and subtract the fit point source. We repeat this procedure until there are no more high pixels at the 4cr level remaining in the map. Through this procedure, we find 70 sources in our 44 fields; in Appendix A, we show the measured position, flux, detection significance and offset from the pointing center for each of these sources. We constrain each of the sources in Table A.l from our data, and will refer to them below as "SZA Sources." In a single field (CMBI17), we found a source which has a flux of 700 mJy. This source, which is ~ 1 million times brighter than the signal we are searching for, cannot be removed from the data with enough precision to leave no residual trace of itself behind; we are forced to discard this field from the analysis. 5.2.2 NVSS/FIRST Sources As it turns out (see below), constraining the sources that we detect with the SZA is insufficient to remove the power contribution of radio sources-there are additional sources that we do not see that are also contaminating our measurement. Using the constraint matrix formalism, we need not know the flux of these sources at 30 GHz, requiring only the source positions to remove them from the data. Two facts will help us get these positions: first, most radio sources have spectra that fall with frequency, and are easier to detect at lower frequencies. Second, there exist several surveys at low frequencies that cover substantial areas of the sky. While the sensitivity of these surveys does not approach that of the SZA's within our fields, the falling spectral 90 -15 -10 -5 0 5 10 arc minutes Figure 5.1: An example of an SZA field with a bright radio source in it. Clearly, the source dominates any Gaussian CMB present in this field, and so must be removed. indices mean that we can still hope to find sources that contaminate our measurement at 30 GHz within the low-frequency catalogs. We find ~ 300 NVSS sources that have fluxes greater than 0.1 mJy when weighted by the SZA beam at their position. To find the locations of these subliminal sources, we use low frequency data from the VLA. The NRAO VLA Sky Survey (NVSS) has surveyed a large fraction of the sky at 1.4 GHz, producing a catalog of radio sources that is 50% complete at 2.5 mJy and 99% complete at 3.4 mJy (Condon et al., 1998). In addition to the SZA-detected sources, we will project from our data all NVSS sources whose flux, weighted by the 91 SZA beam, exceeds some threshold. Combining the NVSS and SZA source catalogs, we account for sources with a 30 GHz flux above ~ 0.6 mJy (the SZA source detection limit), or a 1.4 GHz flux above 3.4 mJy (the NVSS cutoff). Sources that are at the NVSS detection limit, and have a spectral index a > — 1, will contribute to our 30 GHz measurement at the 0.2 mJy noise level. We expect there to by many such sources. A small number of our fields fall within the area surveyed by the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST) project (Gregg et al., 1996). The resolution of FIRST is better than NVSS and is 95% complete at lmJy. Where possible, we use FIRST data in place of the NVSS. 5.2.3 8 GHz VLA Sources In addition to the sources seen in our data and in the NVSS, we have a list of 108 sources detected in 8 GHz observations with the VLA of our fields, discussed in Chapter 3. The noise level in these maps was 30 /^Jy at the center of the fields, with sensitivity rolling off just as the SZA primary beam rolls off our sensitivity at 30 GHz. In this way, we detect at ~ 5a with the VLA all sources that would contribute to the SZA maps at the 1 a level, assuming that the source has a flat spectrum (a conservative assumption). Because of the tremendous sensitivity reached with these VLA data, we will detect many sources that the NVSS survey was not sensitive enough to see. Additionally, because 8 GHz is closer to our 30 GHz observations in frequency, the extrapolation in frequency is less, ie sources bright at 30 GHz are likely to be bright also at 8 GHz, whileas with a rising spectrum they could conceivably be quite dim at 1.4 GHz. We therefore expect that the bulk of our subliminal sources will be detected by these VLA observations. We thus have three sets of sources, observed 92 at different frequencies and with different sensitivities, with which to constrain the foreground power in our measurement. As discussed in Chapter 3, the 8 GHz coverage of our fields is as yet incomplete. Where 8 GHz data is available, we assume that it supersedes the NVSS/FIRST catalogs and use the 8 GHz catalog alone. NVSS/FIRST sources are constrained only when no 8 GHz catalog is available for a field; when our second set of VLA observations are completed in August 2008, we will not use the NVSS catalog any longer. The list of sources constrained is given in Appendix B. The optimal way of combining these datasets, and also of accounting for the sources that we may have missed, will be addressed through simulation. 5.3 Point Source Simulations We know that our measurement of CMB power is heavily contaminated by radio sources at 30 GHz, and also that we can remove the contribution of a source if we know its position. Here we will use simulations to understand how effectively we can remove sources contributions to the power. The simulations have two components: generating a realistic population of sources and choosing which ones we will constrain away. 5.3.1 Source Distributions To understand the point source contribution to our measurement, we aim to simulate a realistic set of radio sources, detect the ones that we think we've seen in our data, and see how much power the remaining sources contribute. We estimate the source population in two ways: using source counts measured with the SZA and following existing models. 93 The SZA's blind cluster survey has detected ~ 200 sources to a flux limit of ~ lmJy (Muchovej et al, in prep). The 6 deg2 over which these sources were detected were later followed up with VLA observations at 4.8 GHz, allowing us to measure both the 30 GHz number distribution and the spectral indices. The 30 GHz number counts can be fit to a power law of 0.0055 x f J^JZ ) -1.97 arcmin . This result appears to be in agreement with other 30 GHz measurements at higher flux levels (Figure 5.2), including Coble et al. (2007), Kovac et al. (2002) and Mason et al. (2003). o SZA points SZA Best Fit line * Coble points deZotti CBI ffl ^DASI •|VSA(20a3) ]VSA(2005) 1.5 2 2.5 teglO(S[mJyJ) 3.5 Figure 5.2: 30 GHz radio source flux distributions Measured by the SZA (blue points), best fit power law (red), and with previous measurements at 30 GHz. While these measurements of the source distribution at 30 GHz represent the lowest flux characterization of the source population, the sources that we expect to cause us the most trouble are those that fall in the 0.01-1 mJy range, which the SZA 94 has not directly measured. One approach to simulating these sources is to assume that the best-fit SZA power law continues at lower flux levels; this is the approach we take when we generate sources according to the SZA distribution. A second way to estimate the source contribution is to follow existing models. While 30 GHz information about source populations is unavailable at the sub-mJy flux level, extensive work has been done to understand this population based on lower-frequency measurements, de Zotti et al. (2005) have fit a range of low-frequency source measurements to a model with several source populations, and the use this model to predict 30 GHz source properties. This model predicts a turning over of the flux distribution below 1 mJy, such that it agrees with the best-fit SZA distribution above 1 mJy, but predicts fewer sources below this level (Figure 5.3). We will use both the SZA-fit power law and the de Zotti model to estimate source contamination. The other property of sources that we need to know for our simulations is the distribution of spectral indices to lower frequencies, so that we can accurately predict which of the sources would have been detected in our 8 GHz catalog from the VLA. We take this distribution from the 4.8 and 30 GHz measurements of the SZA-detected sources. The distribution of spectral indices is shown in Figure 5.3. In simulating sources, we pick spectral indices according to this histogram and assume that the spectral behavior between 30 and 8 GHz is the same as that between 30 and 4.8 GHz. Our simulations will generate 30 GHz sources according to either the de Zotti model or the SZA power law. We will then assign an 8 and 1.4 GHz flux to each source according to its spectral index. We will add these sources to skies of Gaussian CMB signal. These skies will be Fourier-sampled according to the u — v coverage of our observations, and appropriate noise will be added. This fake data will then be processed just as the actual data is, adding constraints for the point sources we wish to account for and computing likelihood curves, thereby estimating the excess power 95 - 2 - 1 0 1 loglO(S[mJy]) 2 -1.5 -1 -0.5 0 0.5 spectral index 4.8-30 GHz 1 Figure 5.3: Comparison of source models and spectral indices comparison at low flux between the de Zotti model (blue) and the best-fit SZA power law (black) (left) and the distribution of measured spectral indices between 30 and 4.8 GHz (right). in our measurement from sources in various flux ranges. 5.3.2 Simulation Results Now that we can simulate skies with realistic point source populations, we can study the effects of different strategies for removing these sources. The first test that we do is to study the effect of varying the threshold we use to determine which of the 30 GHzdetected sources we constrain. For each simulation, we make maps that include 100 /xK of CMB power and point sources, and we add constraints for any source that exceeds Na in the map, where a is computed from the variances of the data. In Figure 5.4, we can see the power we measure with various levels of removal for both source models. Data is also plotted, and we note that at high significance cutoffs the SZA-fit power law seems to predict more power than is seen in either the de Zotti model or in the data. Because at low flux the two models agree rather well, we 96 interpret this to mean that there is a population of sources at the 4-5cr level (~ 0.6 mJy) that is in the SZA-fit source model, but not in the de Zotti model or the data. This is consistent with the turnover that the de Zotti model predicts below 1 mJy (Figure 5.3). The right panel of this plot shows how much power in our data remains after subtracting the power that simulations predict comes from unconstrained point sources. 400 A ^ 300 A > A <1 A 100 A A • A A t 8 GHz VLA+4 4 5 6 Point Sources Removed at 30 GHz (O) 8 GHz VLA+4 4 5 6 Point Sources Removed at 30 GHz (O) Figure 5.4: Measured power after 30 GHz point source constraints in simulation and data (left) Recovered power from simulations with 100 /iK2 of CMB power and de Zotti model sources (blue) or SZA power law distribution (red), compared to data (green). Sources above Na are constrained, along with 5a detections in our 8 GHz catalog where indicated, (right) Power in data minus the excess over input power in the simulations, where de Zotti simulations are blue and SZA-power law simulations are red. While the SZA-based simulations predict large power from sources at the roughest level of source-cleaning, the different simulations agree that, when 8 GHz data is combined with 4a detections at 30 GHz, there is no excess power from undiscovered sources. In our data, we will try to reduce the magnitude of power from undetected sources by including information from lower-frequency catalogs. We duplicate this in simulation using the spectral indices that we have taken from the distribution in Figure 5.3; sources detected at 5a in our 8 GHz data are considered to be detected and are 97 350 300 8 GHz VLA SZA 250 ^ 200 3- % 1501100 50 0 •••• 5 8 15 20 30 0.1 mJy 4 J _ 5 6 Figure 5.5: Measured Power vs VLA cutoff We show the power in our data as a function of the sources constrained. In the green region, we have constrained the sources that we detect at 4-6<r in the SZA data. In the blue, we constrain 4a SZA sources plus all NVSS sources that have a beam-weighted flux greater than 0.1 mJy. In the pink, we constrain the 4a SZA sources plus those that we detect at 8 GHz at 5, 8, 15, 20 or 30 a. Fields with no 8 GHz coverage have NVSS sources of flux greater than 0.1 mJy constrained within the pink region. constrained. Combining these 8 GHz constraints with our 4a cut, we get the leftmost points in Figure 5.4. The simulations indicate that no power remains with these constraints; our measurement has no residual power from point sources that we have missed. In Figure 5.5, we explore how the measured power responds to constraints from lower frequency catalogs, where we include in all cases the 4a SZA-detected source constraints. Including 5a detections at 8 GHz constrains ~ 100 more sources than 98 the 30cr case, but changes the power by very little; we find that the power detected is very stable as a function of the lower limit of VLA sources being constrained. We take this stability as an indication that we have reached the bottom of the sources that are contributing power, and that there is no more point source contribution in these data. The simulations, with different source populations, indicate that we have found the sources that contribute to our measurement. Further, we find that the faintest sources in our 8 GHz data are not contributing substantial power (leading to the flatness in Figure 5.5), suggesting that still fainter sources will not influence our measurement. We conclude that there is not any residual power in our data from point sources. 5.4 5.4-1 Other Foregrounds Dusty Protogataxies There is evidence of a population of galaxies at high redshift that are enshrouded in dust and which have high rates of star formation going on within them. The dust can absorb ultraviolet radiation from the stars, and re-emit it with a spectrum which can be steeper than thermal (~ ir\ and so they can emit powerfully at sub mm wavelengths (Blain et a l , 2002). These objects, which will not be detected in low frequency radio surveys such as the NVSS, pose a serious problem for CMB experiments at higher frequencies and are often treated in the same detail as the synchrotron sources that we have dealt with above. At 30 GHz, however, it appears that we can safely neglect these objects. Based on observed number counts, even a worst-case spectral index of a = 2 would be far below detectability at 30 GHz, as even at 150 GHz the contribution to power is negligible (Reichardt et al., 2008). There is evidence that some of these sources have a minimum at tens of GHz frequencies, 99 and rise towards lower frequency; we expect these objects are included in our lower frequency catalogs. Further evidence that no problematic population of dusty galaxies exists in our data is the fact that we do not find sources that do not have NVSS counterparts. 5.4.2 Thermal Galactic Dust As mentioned in Chapter 3, many of the observed fields lie close to the galactic plane. Within this region, we positioned the fields so as to minimize dust emission as measured by the IRAS 100/^m maps (Wheelock et al., 1991); large dust grains at ~ 20K cool by thermal radiation, and can therefore contribute small amounts of power to our measurement. Because the SZA has no ability to differentiate SZ and dust (or other foregrounds) spectrally, estimating this contribution requires that we use existing maps of the dust within our fields. This involves several extrapolations: dust maps of the sky are taken from higher frequency observations, such as IRAS (12-100yum). Luckily, the work of estimating 30 GHz emission from these maps has been done by Finkbeiner et al. (1999), who fit the various COBE maps to multiple components of different spectral indices. The power predicted at 30 GHz by the Finkbeiner model within our 44 fields is shown in red in Figure 5.6. Unfortunately, the Finkbeiner maps have a resolution of ~ 5 arcminutes, slightly larger than the scales of interest to us. Because thermal dust emission is dominated by larger grains, the IRAS 100/im maps, which are available at ~2 arcminute resolution, are expected to closely trace the emission. We also show the power spectrum of these IRAS maps of our fields in Figure 5.6, and we see that the power spectrum of thermal dust emission is falling with increasing £, and so the Finkbeiner prediction for our fields is an overestimate of what we expect to see in our data. We therefore take the power 100 2: 1 1 < > S -1 8? -2: 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Multipole moment / Figure 5.6: Thermal dust power spectra Here we measure the power spectrum of thermal radiation from galactic dust in our 44 fields, as predicted at 30 GHz by Finkbeiner et al. (1999) (red), and as traced by the higher-resolution IRAS 100/xra map (green, with arbitrary normalization). We take the normalization of the red curve and the shape of the green curve to estimate the contribution to the SZA; the region of SZA sensitivity is shown by the arrows, and the SZ power from one set of simulations is shown as black. level of the Finkbeiner maps as an upper limit for the contribution to our measurment of thermal dust, < O.OlyuK , insignificant in the face of our ~ 100/iK2 signal. 5.4-3 Galactic Spinning Dust While we see that the contribution from thermal dust is small, as expected, there has been some evidence in the past for an additional mechanism of 30 GHz emission from dust, depending on dipole radiation from smaller, spinning dust grains. Such emission is expected to peak at frequencies of tens of GHz and to be tightly correlated with galactic dust (Draine & Lazarian, 1998). Dust correlated emission within this 101 frequency range was observed at 15 and 30 GHz on scales of ~ 15° (Leitch et al., 1997), and at 30, 50 and 90 GHz on 7° scales with COBE DMR (Kogut et al., 1996), but was initially assumed to be a form of free-free emission. The spectral incompatibility of the signal with free-free emission, and its consistency with spinning dust models, was demonstrated at 10, 15, and 30 GHz with data from Tenerife (de Oliveira-Costa et al., 2002). Finkbeiner et al. (2002) targeted two diffuse clouds at 15 GHz and found more evidence for excess dust emission, and Finkbeiner (2004) reproduced this result in the WMAP data. Finally, Watson et al. (2005) and Dickinson et al. (2007) found more evidence for excess emission in individual objects at similar frequencies with COSMOSOMAS and the CBI, respectively. While it may be too soon to claim a detailed understanding of this source of emission, it seems well-established that there is, in fact, a contribution at SZA-frequencies from spinning dust grains, labeled by some as "foreground X." It seems from the results of COBE and WMAP that the excess emission from this dust is well-correlated with infrared maps of the dust's thermal emission. Assuming that the distribution of dust grain sizes does not vary significantly over the sky, it is sensible to try and measure an emissivity that can relate measured IRAS 100/j,m fluxes to observed emission at lower frequencies; such an emissivity could then be used to estimate the contribution of this foreground to SZA measurements. Dickinson et al. (2007) have tabulated ~30 GHz measurements of this emissivity from various groups, and find that the scaling depends rather strongly on the environment of the observed sky. They find that HII regions, which they targeted, have typical scalings from IRAS of ~ 3.3 ± 1.7iu,K(MJysr~1)~1, whileas higher galactic latitude results favor emissivities in the range 10-25 \xK(M Jysr• ) . We reproduce Dickinson's Table 5 as Table 5.1. To Dickinson's data, we add Leitch et al. (1997)'s emissivity as measured in 36 fields near to the North Galactic Pole. 102 Table 5.1. Measurements of spinning dust emissitivities, scaled to IRAS 100 \im Target H II regions Six HII regions (mean) LPH96 cool dust clouds 15 regions W M A P LDN 1622 G159.618.5 other All-sky W M A P RING5M Dust Emissivity liKiMJysr-1)'1 Reference Estimated SZA Power 3.3 ± 1 . 7 5.8 ± 2 . 3 Dickinson et al. (2007) Dickinson et al. (2006) < lfiK2 < 1/iif2 11.2±1.5 24.1 ± 0 . 7 17.8 ± 0 . 3 Davies et al. (2006) Casassus et al. (2006) Watson et al. (2005) 1 fiK2 5»K2 3fj.K2 10.9 ± 1 . 1 42 Davies et al. (2006) Leitch et al. (1997) lfiK2 16 iiK2 To estimate the contribution of spinning dust to the SZA measurement, we take the power in IRAS lOO^rn maps of our fields, fit it to logioPower = A£ + B, and extend it to the smaller angular scales that we are sensitive to but to which IRAS is not sensitive. Limiting ourselves to the higher emissivities measured in cool dust regions in Table 5.1, we scale the IRAS power spectra by the measured emissivities as an estimate of the dust contribution from each measured value. To calculate the contribution to our measurement of each of these estimates, we must multiply it by our window function and integrate over the range of £ over which we are sensitive. This window function encodes our relative sensitivity on different scales; if we measured only two visibilities, we would have equal sensitivity on the two scales probed by those visibilities. In the £(£+1) units that we use here, this sensitivity falls like £ . Because we have many measured visibilities, we histogram the u — v radii probed by them and scale this histogram by £ to calculate our window function. This function is shown in Figure 5.7. The results range from < 1/j.K to 16/iK^, and are listed in Table 5.1. Note that the emissivity measured by Leitch et al. (1997) is substantially higher than other measurements, and so the corresponding likely a substantial overestimate of this contribution. 16/J,K2 is 103 43.531 2.5 • a — 15 • 1- 0.5" o0 2000 4000 6000 8000 Multipole / 10000 12000 14000 Figure 5.7: SZA window function The window function, which determines our sensitivity on various scales, shows the amount of data on each scale and is then scaled by £ . 5.5 Primary CMB Although the contribution of the primary CMB anisotropy is small on the angular scales we are interested in, we have some sensitivity to very large scales. Although our baselines can not reach below 350 wavelengths at the center of our band (£ = 2200) without the antennas shadowing each other, as described in Chapter 4 there is extended sensitivity in the u — v plane due to the aperture function. Thus, if we reach our smallest baseline of 350 cm, such that two antennas are just touching at the edge, there will be some (small) sensitivity extending all the way to the largest scales. Because the primary CMB is so bright on scales of ~ 1 degree, we therefore expect a small contribution to find its way into our data. We estimate this contribution with simulation. We take SZ skies and add primary CMB to them (we use lambda CDM power spectra to do so). Simulating our 44 fields 104 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Multipole moment I Figure 5.8: Spinning dust power spectra IRAS power spectra for SZA fields, scaled by measured 30 GHz spinning dust emissivities in high galactic latitude dust: Leitch et al. (1997) (Green), Davies et al. (2006) (Blue), Watson et al. (2005) (Cyan), and Casassus et al. (2006) (Red). Simulated SZ signal is in black, and SZA angular sensitivity is shown by arrows. of observations as discussed before, we note the power difference with and without the primary CMB component; we find the offset to be 30/^K2. This contribution will need to be subtracted from our power measurement. CHAPTER 6 P O W E R S P E C T R U M RESULTS With the data filtered in the ways described above, it is possible to compute the likelihood of different levels of CMB power in the data, having constrained the positions of known point sources. To check that the power level extracted from the data in this way is not biased by lingering contamination, we can subject the data to several powerful jackknife tests. We have performed our power spectrum analysis in a blind fashion, in the sense that we did not calculate the final result until we had finalized our data cuts and treatment. Instead, we worked with the data as jackknifed in the ways described below such that the power in the jackknifed data should be zero. We spent considerable effort honing our analysis tools such that discrepancies from zero power in these data were eliminated, and only then repeated the calculation for unjackknifed data. In this way, we believe that we have conducted the analysis without any bias from the power spectrum result. 6.1 Jackknife Tests In these jackknife tests, the data are split into two halves. The visibilities in each half are perfectly matched in the u — v plane, so that each half is measuring exactly the same Fourier components on the sky as the other. Our data are taken in such a way that three of these splits are possible: i) frequency jackknife, wherein we separate even and odd frequency bands, and neighboring bands sample very nearly the same u — v points, ii) time jackknife I, in which we take advantage of the fact that the SZA observes the same fields in the same way for many days, and we split these days into the first half and second half, and iii) time jackknife II, in which the same fact allows us to separate the tracks into even and odd days. In each jackknife splitting, each u-v 105 106 point in one half of the data has an exact match in the other half. Figure 6.1 shows the matching in the u — v plane that enables us to make these splits. 0 100 200 300 400 Separation in the U-V plane (k) 200 300 400 500 U Figure 6.1: Autocorrelation Function and Jackknife u — v In the left panel, we plot the autocorrelation function of the SZA's antenna illumination pattern at a single frequency, which shows us how correlated two u-v points are as a function of how far apart they are in the u — v plane. On the right, we show the u — v points sampled by a single baseline on one day (blue), on the next day in the same band (green), and on the same day in the next band (red). The circle shows the region in which points are 90% correlated with the center of the circle; our jackknife tests are possible because the different colored points are well within this circle. After dividing the data into two halves, we difference the matching pairs of visibilities. Since the pairs of visibilities have the same u — v coordinates, differencing them removes any contribution from the common sky, and anything else that is constant across the two halves. However, any contamination that varies with time or frequency, and is therefore not the same in the two halves, will remain in the differenced (jackknifed) data. We compute variances from the differenced visibilities, and compute the power in each jackknifed dataset just as we do for our unjackknifed 107 Jackknife Test Even Bands - Odd Bands First Half Data- Second Half Even Days - Odd Days Unjackknifed Data Gauss Fit to Likelihood Curve 75 ± 61fiK2 - 4 8 ± 92fiK2 43 ± 88/^K2 93 ± 66fiK2 PTE xl of this data 0.25 0.64 0.33 0.10 Table 6.1: Jackknife Data Results for 4cr 30 GHz and 8<7 8 GHz source removal. data. If the power that we see in our undifferenced data comes from the sky, as we hope that it does, then the jackknifed datasets should ah favor zero power. If there is contamination, which we presume will not be constant across all of the splits, it will show up as a detection of power. Additionally, the jackknife tests pose a strenuous test for our noise model; if we've mis-estimated the noise in our data, then we expect that the jackknifed data, which should have nothing but noise in them, will be poorly described by our covariance matrices. In Table 6.1 , we see that all three jackknife tests pass, and that the data shows a weak detection of power. In Figure 6.2, we plot our result alongside the earlier measurements and with model spectra at the consensus erg = 0.8 and also ag = 1.0, favored by the previous SZ measurements. The point in Figure 6.2 shows the measured power (93/iK2) minus the primary CMB contribution (30/iK2). 6.2 Data Splits The jackknife tests are an extremely powerful means of finding contamination or errors in the noise model within the data. There are many other splits of the data that we can imagine which divide the data into halves with potentially different systematic effects, but which do not allow the perfect u — v matching that enables proper jackknife tests of the kind listed in Table 6.1. With likelihood curves for each set of four fields, we can easily divide the data according to properties of the fields 700 1 • 1 1 I 1 •• — i 1 1 — 600 I - 5 * 500 ¥• 400 - CBI h 300 + • lf— 200 100 - M [ SZA * *\ X_——r 0 , 1000 2000 I BIMA \ **>3000 4000 5000 6000 7000 8000 / Figure 6.2: Power measured by SZA Here we show the SZA measurement alongside those of BIMA and CBI. We include model SZ spectra for ag=0.8 (red) and 1.0 (black). We have subtracted the contribution of spinning dust and primary CMB from our measurement. and compare the power in each half. We show several of these splits in Figure 6.3. We divide according to first season/second season of observations, depending on the date of observation of the field in 2006-2007, the galactic latitude of the fields, by the measured RMS of the IRAS 100 /im maps of the fields, by whether the sun was near or far from the fields at observation (equivalently, night/day), and by the average elevation of the fields during observation, as a proxy for how likely the data are to be contaminated by ground contamination. In many of these splits, the differences among fields are small; for example, the mean solar offset from most of the fields falls in the range 100-155. We split according to the farthest and nearest fields to the sun and check for signs of a discrepancy, despite the fact that the nearest fields are only slightly nearer. 109 Therefore, and because the significance of power detection is so low, these split data tests are considerably weaker than the jackknife tests discussed previously. We find no evidence for systematic effects in these tests; each of the splits shows consistent amounts of power in both halves of the data. Lest one should be tempted to read significance into the sun and elevation splits, we point out that the blue sun offset point contains the data further from the sun, and that the red elevation half is that at higher elevation. We ascribe no statistical meaning to the difference in Figure 6.3. 300 250 200 cs M =L -aCD 150 tec •*•* 100 <D Q 50 ss Co U 0 -50 -100 season galactic lat IRAS RMS sun offset data split elevation Figure 6.3: Power in different splits of the data Red/blue points show different halves of the data across the listed split. This is a very weak test for systematic effects, and we find evidence for none. The blue sun offset point is that for the fields further from the sun, and the red elevation point is the higher elevation fields. Of the ~ 100/^K2 of power we detect, we have argued that the contribution from residual point sources is negligible, that < 10/xK2 of power is contributed by spinning 110 Table 6.2. Contributions to total power measured. Contribution Power Measured [/J.K2] Total Measured Point Source Residual Spinning Dust Primary CMB Attributable to SZ 93 ± 6 6 0 < 10 ~30 60 ± 6 6 dust emission, and that the primary CMB is responsible for an additional 30/iK , leaving us with 60±66//K of power that can be ascribed to SZ signal, consistent with zero. These contributions are summarized in Table 6.2. CHAPTER 7 COSMOLOGICAL INTERPRETATION To understand the meaning of our measurement in the context of as, we compare to several simulations of large scale structure. These simulations have been carried out by different groups and converted into y-maps, two-dimensional projections of the Compton y parameter. These y maps are assembled from lightcones, in which slices of the simulation are taken at various stages of evolution and stacked together to yield a cone that contains realistic contributions from all redshifts. We take these maps as inputs to simulated SZA observations, and attempt to estimate the mean level of power measured by the SZA from these skies, as well as the scatter of these measurements. This scatter includes the non-Gaussian sample variance of the SZ maps, whileas the Gaussian estimate of our measurement error does not. In this chapter, we discuss our simulation strategy, briefly introduce several sets of simulated y-maps, investigate the magnitude of non-Gaussian sample variance, and then compare the power that they imply in the SZA measurement as a function of ag. We also discuss issues related to possible correlations between clusters and radio sources. 7.1 Simulation Strategy Prom various cosmological simulations, we have sets of several hundred independent maps. For each set of maps, we form 50 groups of 44 fields; we pick the maps from the available set randomly with replacement. For each map, we simulate SZA observations. This is done by converting the simulated map into sky brightness units at 30 GHz, transforming the map, and sampling the transformed surface at the u — v coordinates that we have measured in our experiment. To these simulated visibilities, we then add noise of the magnitude we have estimated for each of our visibilities. Ill 112 This fake data therefore has the same u — v coverage and noise properties as our real data. We sort and combine the fake data with the same tools that are used for the actual data, and the likelihood analysis is performed identically, yielding a total likelihood curve for each group of 44 fields, and thus a set of 50 data points equivalent to our measurement. We record the maximum likelihood power for each of our 50 sets of fields. These numbers are used to calculate the scatter about the mean level of measured power; this scatter includes the full, non-Gaussian sample variance of the measurement. We anticipate an underestimation of this variance due to the fact that we have reused the fields multiple times. 7.2 Simulated Maps Advances in computational power over the past decade have enabled large-scale cosmological simulations to come a long way, but there are still limits on how accurately simulators can portray the evolution of structure. For our purposes, there are two components that are relevant to the production of a realistic ?/-map: the clustering of the dark matter into haloes and the heating of gas within these haloes. The SZ effect is dominated by the hot gas in the deep potential wells occupied by clusters of galaxies. The dark matter that creates the potential well can be simulated with relative ease in iV-body simulations, which know only about gravity and dark matter. These simulations can reproduce much of the large-scale structure in the universe, including the cluster haloes that we are interested in; moreover, the computational simplicity of simulating non-interacting dark matter allows for very large scale simulations to be run in a reasonable amount of time. However, clusters from iV-body simulations contain no gas, and therefore produce 113 no SZ effect. Simulators take one of several paths for adding the gas to their simulated halos, and we will use examples of each to compare our result to simulations of known cosmology. 7.2.1 Full Gas Simulation The most model-independent solution is to simply add the gas at the beginning of a simulation, teach the computer about fluid physics so that it treats the gas properly, and see what it comes up with. Hydrodynamical simulations take this approach, which is taxing for even the most impressive computers. Because the task is so demanding, only small areas of the sky can be simulated in this way, and so the sky area available in simulated maps is small. Moreover, there are unknown features in the way that gas behaves within cluster haloes. The means by which gas cools, becoming stars, or heats up, perhaps by interacting with active galactic nuclei, is not well-understood. Hydrodynamical simulations must parameterize these effects in order to match observations. The agreement between the simulated clusters and the observed ones, while becoming better and better, is far from perfect. The simulated y-maps of White et al. (2002) are from a hydrodynamical simulation. This simulation allow the gas to radiatively cool, and injects energy through galactic winds. We have used 60 0.5-degree maps from these simulations, which were derived from a simulation with (78=0.9. They contain haloes of mass 5 x 1013M© — 3 x 1O 14 M 0 . 7.2.2 Semianalytic Gas Less sophisticated in its treatment of gas, but considerably less computationally demanding, are simulations that create cluster haloes with dark matter only, and apply 114 gas after the simulation has finished. This has the virtue of being more manageable computationally, but risks missing the contribution to the power spectrum from gas not associated with haloes. Schulz &: White (2003) have created a set of maps from such a simulation, wherein the gas is added to haloes via a semianalytic prescription. They assume that the gas follows the dark matter distribution, defining the gas density, and also that the clusters are all isothermal. The temperature of the haloes is assumed to scale with mass. These simulations were demonstrated to be in good agreement with the fully hydrodynamic simulations of White et al. (2002). Prom this simulation, we use 10 maps, each 3 degrees on a side, with Q^=0.3 and crg=1.0. These maps contain haloes in the range 4 x 1012M© — 1 x 1015M©. The simulations of Shaw et al. (2007) follow a similar tack; they identify haloes in a large A^-body simulation, and then insert gas with several restrictions. They demand that the gas be in hydrostatic equilibrium with the dark matter profile of the cluster, and additionally parameterize the degree to which dense gas cools into stars and energy is fed from active galactic nuclei back into the gas. These simulations were run with WMAP3 cosmology (<rg = 0.77), and the 100 0.5 deg maps contain haloes of mass greater than 5 x 1013M©. 7.2.3 Pinocchio Holder et al. (2007) go a step further in simplifying the process; they take a shortcut that allows them skip even the full iV-body simulation. Using the Pinocchio algorithm of Monaco et al. (2002), initial Gaussian perturbations are created, and collapse times are calculated at different positions on many scales. Haloes are determined to have formed at the earliest collapse time, and assigned the corresponding mass. Mergers among haloes are then addressed, yielding a catalog of dark matter haloes, complete 115 with merging histories, in a small fraction of the time needed to actually perform a complete iV-body simulation. The results of full simulations and the Pinocchio catalogs has been demonstrated to be in agreement. Holder takes the catalog output from Pinocchio and assigns gas to each halo according to an analytic expression that accounts for many of the heating and accretion phenomena described above. Because of the great speed with which Pinocchio allows these maps to be generated, Holder can compute simulated y-m&ps over a range of cosmologies spanning <rg <G [0.6 — 1]. He provides 900 0.5 deg maps at each value of erg, with halo masses greater than 1 x 1013M©. 7.3 Non-Gaussian Sample Variance The power spectrum is a complete description of the sky only if the sky has Gaussian statistics. The SZ Effect, however, is very non-Gaussian (Figure 7.1). While we measure the power spectrum for consistency with the broader set of CMB observations (and because it is a much easier statistic to measure!), the formalism that we have outlined here is not designed for the problem that we are applying it to. In particular, although the power spectrum of the SZ is still a well-defined quantity, we have computed the error on our measurement assuming that the signal was Gaussian. Since we have surveyed only a small part of the sky, it is very feasible that our measurement could be consistent with a level of power outside of the errorbars we show in Figure 6.2. At issue is the sample variance of our result. The SZ signature will fluctuate from field to field at a higher level than that at which a Gaussian signal would be expected to fluctuate. A single large cluster in a field will give it much more power than fields with no such cluster; such a field would be a many sigma outlier in Gaussian terms. 116 We can see this effect by simply histogramming the power measured in many fields of SZ maps and comparing to that in Gaussian CMB maps with the same level of power (Figure 7.1). Although the two distributions have the same mean power, the scatter of the SZ maps is considerably larger. Thus the sample variance of our measurement has been underestimated. Simulated y-maps (mean= 170) 0 500 1000 Measured Power [|i, K ] Gaussian sky maps (mean=160) 0 100 200 300 Measured Power [|X K2] Minimum power y-map Maximum power y-map -0.2 -0.2 0 0.2 Minimum power map -0.2 -0.1 0 0.1 0.2 0 0.2 Maximum power map -0.2 0 0.2 Figure 7.1: Power Distribution for Gaussian and non-Gaussian skies In the top row, we show the most likely power in 900 signal-only simulated observations of realistic y-map skies. In the bottom row, the same observations are made of Gaussian skies with the same power spectrum. The long tail for the non-Gaussian case reflects the severe non-Gaussianity of the SZ effect. The color scale in all panels is the same We explore the degree of this underestimation through the simulations discussed above. We take 50 sets of 44 maps, ending up with 50 likelihood curves. We use the interval that contains 68% of the most likely powers from these 50 to estimate 117 the width of the actual distribution of the power, including non-Gaussian sampling effects. We compare this to the width of the likelihood curves, which contains our measurement errors, but not the non-Gaussianity. The ratio of the two is shown in Figure 7.2. We see that at low values of <7g, the sample variance is unimportant compared to the noise on the measurement, and so the Gaussian estimate is appropriate. At values of erg above 0.8, however, this no longer is true; the sample variance is being severely underestimated, and so the (Gaussian) error on our measurement underestimates the scatter of the power. This must be born in mind when interpreting our result. Figure 7.2: Non-Gaussian Sample Variance Here we compare the scatter of 50 simulated measurements of t/-maps to the Gaussian width of their likelihood curves. The colors reflect different sets of simulations, including Holder et al. (2007) (blue), Shaw et al. (2007) (green), Schulz & White (2003) (black), White et al. (2002) (red). 118 7.4 Power and erg Using these simulations, we can explore the effect of different values of <7g on our measurement. For each set of simulations, we again form 50 groups of observations of 44 fields. We compute the most likely power for each set, and plot the mean and RMS of these powers in Figure 7.3. These errors therefore include the sample variance effect mentioned previously. We can see the strong dependence of the level of power on the value of erg, but also that there are lingering differences among simulations. If we consider only the simulations of Holder et al. (2007) (blue in Figure 7.3), we see that our measured power is most consistent with values of ag between 0.7 and 0.8, in line with the various other measurements discussed in Chapter 1. However, the range of allowed ag is large, especially in the face of differences among the simulations. Figure 7.3 shows no tension between our measurement and other measurements of as- 7.5 Correlations between Point Sources and Clusters In the simulations discussed in Chapter 5, we distributed the radio sources randomly across our fields, matching observed flux distributions. However, there is some evidence that radio sources are spatially correlated with clusters (e.g., Coble et al. (2007), Lin & Mohr (2007)). This correlation poses two problems to our measurement. First, low-flux, undetected sources in clusters can fill in the SZ decrement, leaving the cluster with less flux and reducing the power from SZ. Second, if we can detect the sources and project them from our data as described in Chapter 5, we will be projecting the cluster's flux along with the point source. The projection will be complete if the cluster is point-like on the scale of our short-baseline synthesized beam, ~ 1-2 arcminutes. 119 4UU • i - T - - i -• - _350 i ^ 300 dL J 250 1 200 S 150 11 < i i_ o p^ i i * I i i i 0.6 0.7 0.8 i 5 loo •5 so £ o i C/3 ^„ -50 0.5 a i 0.9 1.1 8 Figure 7.3: Power vs erg The power measured by the SZA with various sets of simulated y-maps. The colors reflect different sets of simulations, including Holder et al. (2007) (blue), Shaw et al. (2007) (green), Schulz & White (2003) (black) and White et al. (2002) (red). . We demonstrate the magnitude of this problem in simulation. The worst case scenario is that every cluster has at least one point source within it that we detect and project out, removing the unresolved portion of the cluster's flux with it. To estimate the magnitude of the effect in this case, we use the hydrodynamical simulations of White et al. (2002), which provide 60 independent 0.5 degree y-maps with clusters in the mass range 5 x 1O13M0 — 3 x 1014M©. We observe these maps with SZA-like u — v coverage but very little noise, and calculate the mean power for the entire set of fields. Computing the joint likelihood of the fields in our normal way, we find the power to be ~ 100 fiK . Now we insert a constraint at the catalog position of each cluster in the maps. We repeat the process, using the same random number seed so 120 that the noise is identical to the previous simulation. We find that the constraints do wipe out power from the clusters, leaving us with only 27/J.K2. The crux of this problem is that on the arcminute scales that correspond to our short baselines, it is very difficult to tell the difference between a cluster and a point source. However, the longer baselines of the SZA (see Chapter 2) have a much easier time distinguishing the two scales, since for them a cluster is highly resolved. We include the long baselines in the analysis by adding a second bin of CMB power that includes only our long baselines to the covariance matrix. In this way, the magnitude of CMB power in long and short baselines is entirely decoupled, but the constraints for point sources span both long and short baseline data. We fix the CMB power in the higher-resolution bin to be zero, include the full point source constraint matrix, and vary the CMB power level in the lower-resolution bin according to our usual procedure. In simulation, this restores the power to 70fiK2, 70% of the original power. Again, this is a worst case scenario, in which every cluster has a point source at its core. A more realistic treatment follows the measurements of Coble et al. (2007), who found that the density of sources within 0.5 arcminutes of a cluster center could be fit with ^ = 0.174 x ( imJv) arcmin , a factor of 30 greater than the densities the SZA has measured away from clusters. In simulation, we incorporate these data by replacing a 0.5 arcminute disk around each cluster with a point source map generated with the higher source density. We then constrain only those sources that are bright enough to be detected in our data at 30 or 8 GHz, and repeat the test described above. As expected, we find the magnitude of the problem to be less than in the severe case of a source in every cluster. We also replace the simulated u — v coverage with more realistic values, and collect groups of 44 fields at a time, just as in the real data. Now the 135 /xif2 of power we see in the fields is reduced to 121 47 iiK when the sources are projected from the short baselines, and 91 /J,K when the long baselines are used, as well. Note that this remains a pesimistic case, as the OVRO/BIMA sample on which the Coble measurements are based is severely biased towards the most massive clusters, which we expect to contain the most point sources. We consider these numbers to be worst-case values. Now we apply the long baseline constraints to the real data. Because the volume of data becomes quite large (the covariance matrix is ~ 30,000 x 30,000), we must take an additional compression step, combining adjacent time scans across the ~ 20 minutes that separate them in most tracks. Much of the data remains > 90% correlated across these scans-we discard data which does not, about 15%. The power in this set of compressed fields is very close to the power reported in Chapter 6. Adding the long baselines, the power goes from 89±72/j,K^ to 93±63/ii<'2; apparently, we see no sign of this effect in our data. An increase in power that comes with including the long baselines could also have been due to radio sources that are unresolved in the short baselines, but slightly resolved in the long baselines. This would have the effect of leaving unprojected power in the short baselines, which could show up as CMB. None of the ~ 200 sources detected in the SZA cluster survey were deemed to be resolved on the ~ 15" scale relevant for our long baselines. Equivalently, data with decoherence due to weather could smear sources out on the long baselines, leaving residual flux when the point source is removed. Because we see no jump in power by including the long baselines, we conclude that there is no issue in our data with cluster-source correlations, and also that our sources are being successfully projected as point-like objects. Additionally, we note that the brightest sources detected in the fields we have used in this analysis can be fit to our long and short baselines separately, and that the recovered flux is the same for both fits; this further supports our contention that projecting these sources 122 as point sources is reasonable. CHAPTER 8 CONCLUSIONS We have measured the power spectrum on arcminute scales in 44 fields with the SZA. We find that power to be 93 ± 66fj,K2. We have argued that the contribution from undetected radio point sources is negligible, that primary CMB contributes ~ 30/iK , and that spinning dust contributes < 10//K2 to this measurement. We find that including long-baseline data has no significant effect on this result, arguing against any wiping out of power due to correlations between the sources we are constraining and our cluster signal. The long baseline test further suggests that our source removal is robust, and argues against other forms of contamination from the sky. We estimate the allowed SZ power spectrum to be 60±66/iK 2 , consistent with zero. As seen in comparison to a variety of simulated SZ skies, this level of power is very consistent with consensus values of erg of ~ 0.8, suggesting that the moderate tension between previous SZ power spectrum measurements and other measures of erg should not cause us worry. The larger errors on these previous SZ measurements give them some room to be in agreement with the higher-confidence measurement described here. Our measurement does not rule out zero power on these scales. The SZA, when it resumes operating at the Cedar Flat site in Fall 2008, could conceivably continue observations like those presented here to reduce the uncertainties that we have listed. Alternatively, the SZA cluster survey has accumulated ~ 6 times the data volume used in this analysis; these data were also taken in a way that allows the removal of ground contamination, and could in principle be used to substantially improve this measurement. Utilizing this data would demand substantially more computational power, as the pointings in these mosaics are heavily correlated, and must therefore 123 124 be treated in a single covariance matrix. At slightly lower £, the QUaD experiment promises to deliver power spectrum measurements at 100 and 150 GHz in the coming months. Not far behind, the APEX experiment, and then the Atacama Cosmology Telescope and South Pole Telescope, also at higher frequency, will map much larger portions of the sky at comparable depth to what we have achieved here. By the time Planck produces results, multi-frequency experiments will have used spectral signatures to separate the primary CMB, foregrounds, and the thermal SZ effect from one another. This SZA measurement will allow us to refine our cosmological models through these datasets without fear that the elevated CBI power at 30 GHz reflects physics that we have failed to understand in the SZ power spectrum. In this sense, we have steadied this slight wobble in SZ cosmology. APPENDIX A RADIO SOURCES D E T E C T E D IN C M B FIELDS We made maps of each of the fields used in this analysis, using short and long baselines together. The RMS of these maps was in the range 0.12-0.15 mJy. We show here the complete catalog of sources that were detected in these maps at 4cr or greater; all of these sources were projected from our data. Table A.l: Sources detected at 30 GHz with the SZA Field Name CMB09 CMB09 CMB09 CMB09 CMB09 CMB09 CMB09 CMB09 CMB09 CMB10 CMB10 cmbAl cmbA9 cmbAl7 cmbA25 cmbA25 cmbI9 cmbI25 cmbI25 cmbI25 cmbR9 cmbR9 cmbR9 cmbR17 cmbR17 cmbR25 source RA hh:mm:ss 02:22:48.7 02:22:33.8 02:22:22.9 02:22:50.3 02:22:16.6 02:22:46.0 02:22:53.2 02:23:08.0 02:22:13.2 02:30:41.9 02:29:50.3 02:11:40.6 02:16:03.9 02:20:45.1 02:24:36.1 02:24:18.1 02:16:10.4 02:25:00.3 02:24:40.9 02:24:27.3 02:16:45.5 02:16:28.1 02:16:26.8 02:20:24.7 02:20:11.5 02:24:40.6 source Dec dd:mm:ss +33:57:18 +34:05:17 +34:00:11 +33:55:04 +33:53:26 +34:04:32 +34:05:08 +33:55:05 +33:54:44 +33:56:21 +33:51:33 +33:04:17 +33:03:44 +32:57:07 +32:51:48 +32:52:16 +32:38:49 +32:31:37 +32:29:39 +32:34:31 +32:03:57 +32:10:32 +32:08:13 +32:10:43 +32:10:39 +32:02:04 125 source flux mJy 12.8 14.4 5.8 4.3 16.6 1.8 1.6 1.7 6.3 4.0 24.6 2.5 1.2 2.2 4.5 1.6 0.7 7.5 1.2 0.6 3.8 0.7 0.8 1.4 1.2 5.9 detection significance 72.8 37.9 17.9 16.8 10.5 7.5 5.8 4.9 4.5 18.7 4.8 7.3 5.0 8.5 13.7 4.1 4.3 18.9 4.5 4.4 5.3 5.2 5.1 10.3 5.5 15.2 Table A.l: Sources detected with the SZA, continued Field Name cmbR25 cmbR25 cmbY9 cmbY9 cmbY25 cmbEEl cmbEEl cmbEEl cmbEEl cmbEEl cmbEEl cmbEE4 cmbEE4 cmbEE2 cmbEE2 qmbEE2 cmbXXb cmbXXb cmbXXb cmbXXb cmbXXa cmbXXa cmbXXa cmbXXa cmbXXl cmbXXl cmbXXl cmbDDl cmbDD2 cmbDD3 cmbDD3 cmbDD4 cmbAAl cmbAAl cmbAA3 cmbAA3 cmbAA3 cmbCCl source RA hh:mm:ss 02:24:40.6 02:24:59.7 02:16:01.3 02:15:24.7 02:24:07.7 14:18:40.6 14:18:40.1 14:18:40.9 14:18:10.2 14:19:21.9 14:18:13.3 14:31:12.4 14:30:59.9 14:22:59.9 14:22:51.4 14:22:39.4 21:28:19.9 21:28:51.5 21:28:52.9 21:28:41.5 21:24:22.7 21:24:32.2 21:24:13.8 21:24:13.8 21:32:53.7 21:32:33.5 21:33:02.4 14:19:10.1 14:22:38.7 14:27:02.4 14:27:17.2 14:31:10.9 21:25:03.8 21:24:34.2 21:33:14.3 21:33:04.6 21:32:49.2 02:11:43.1 source Dec dd:mm:ss +32:08:43 +32:14:05 +31:48:23 +31:52:51 +31:47:30 +35:30:06 +35:29:50 +35:33:25 +35:31:07 +35:32:49 +35:35:24 +35:35:27 +35:36:27 +35:22:45 +35:32:09 +35:31:45 +25:06:40 +24:58:15 +25:00:59 +25:05:43 +24:55:12 +25:03:18 +25:01:11 +24:55:03 +24:55:14 +25:00:15 +24:53:26 +35:08:29 +34:58:18 +34:59:06 +35:01:27 +35:02:33 +25:25:11 +25:27:49 +25:28:58 +25:20:47 +25:30:11 +33:31:40 source flux mJy 1.3 1.9 2.8 5.8 0.8 5.7 3.2 1.3 1.8 4.4 1.3 4.1 3.1 22.3 0.9 0.8 12.1 1.1 0.8 1.7 9.0 1.9 0.9 2.9 1.5 2.1 1.3 6.5 1.4 3.0 1.3 0.7 4.6 1.0 34.3 9.8 1.4 4.5 detection significance 10.0 6.4 15.8 4.4 4.3 36.7 21.2 8.5 5.3 4.7 3.6 19.2 9.5 16.1 5.6 5.0 6.1 6.6 4.6 4.1 19.5 6.1 1.8 3.8 5.2 4.4 3.1 4.4 5.0 12.8 5.7 3.8 8.1 5.2 128.8 6.6 5.0 7.9 Table A.l: Sources detected with the SZA, continued Field Name cmbCCl cmbCC3 cmbCC3 cmbCC3 cmbCC3 cmbCC3 cmbCC4 cmbCC4 cmbBBl cmbBBl cmbBBl cmbBB2 cmbBB3 cmbBB3 cmbBB3 cmbBB4 cmbBB4 cmbBB4 source RA hh:mm:ss 02:11:31.5 02:19:30.2 02:19:43.8 02:19:11.4 02:19:31.2 02:19:31.1 02:24:23.5 02:23:33.7 21:24:50.2 21:24:26.1 21:24:38.7 21:28:47.4 21:33:02.1 21:33:02.2 21:32:55.8 21:36:42.1 21:36:55.6 21:37:04.6 source Dec dd:mm:ss +33:23:01 +33:28:51 +33:30:26 +33:31:50 +33:21:57 +33:23:55 +33:32:30 +33:24:33 +26:01:55 +26:07:21 +25:51:28 +25:49:20 +26:09:32 +26:04:19 +25:50:13 +25:57:41 +25:56:42 +25:54:48 source flux mJy 4.2 57.4 1.3 9.9 2.7 1.7 3.7 4.1 7.6 4.0 2.8 8.4 14.2 1.3 6.5 15.7 4.7 1.2 detection significance 16.6 179.4 5.7 4.6 4.0 3.9 10.6 6.1 34.9 4.6 3.9 4.0 5.5 4.4 4.5 27.9 17.3 4.2 APPENDIX B NUMBERS OF SOURCES CONSTRAINED Here we list the number of sources constrained in each field. Sources could be used if they were detected with the SZA at 30 GHz, if they were detected in our VLA 8 GHz data, or if they are found in the NVSS or FIRST catalogs. NVSS/FIRST sources are constrained if they exceed a certain 1.4 GHz flux when weighted by the SZA primary beam. 8 GHz sources are constrained if they fall within a certain radius of our field centers. See Chapter 5 for discussion of which of these sources have been constrained in which context. Table B.l: Number of Sources Constrained Field Set 2 3 4 5 6 7 Field Number 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 N 30 GHz 2 0 9 2 1 2 1 1 3 1 0 3 3 3 2 3 1 1 3 1 2 8 GHz 30' 2 2 14 2 7 3 4 7 6 7 9 6 11 7 10 11 7 6 10 7 0 8 GHz 12' 2 2 12 2 3 2 4 3 4 1 0 4 4 5 7 4 3 2 3 3 0 128 NVSS > .1 / 2 mJy 3/1 9/1 13/6 3/1 4/3 9/ 1 5/1 4/3 3/3 7/3 13/5 3/3 6/4 6/3 2/2 6/4 6/3 6/4 6/2 6/3 9/7 FIRST > .1 / 2 mJy 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 0/0 0/0 14/9 Table B.l: Number of Sources Constrained, continued Field Set 8 9 10 11 12 Field Number 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 N 30 GHz 0 2 2 1 4 2 1 0 1 2 0 0 0 3 0 2 0 3 2 2 2 3 2 8 GHz 30' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 3 3 0 0 0 0 8 GHz 12' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 NVSS > .1 / 2 mJy 6/ 1 11/5 9/7 1/1 18/5 12/5 1/1 4/1 8/2 12/7 4/1 2/0 5/0 13/6 2/0 9/4 9/1 7/4 9/4 8/3 8/2 9/6 8/3 FIRST > .1 / 2 mJy 10/2 18/5 14/9 0/0 0/0 0/0 0/0 6/1 14/2 18/7 6/1 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 REFERENCES Aghanim, N., Majumdar, S., k Silk, J. 2008, Reports of Progress in Physics, 71, 066902 Birkinshaw, M. 1999, Phys. 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