Data Analysis of Cosmic Microwave Background Experiments A dissertation Submitted to the Faculty of the Graduate School Of the University of Minnesota By Matthew Edmund Ahroe In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy Shaul Hanany, Advisor July 2004 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3137151 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 3137151 Copyright 2004 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. @ Matthew Edmund Abroe 2004 ALL RIGHTS RESERVED Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF MINNESOTA This is to certify that I have examined this bound copy of a doctoral thesis by Matthew Edmund Abroe and have found th at it is complete and satisfactory in all respects and th at any and all revisions required by the final examining committee have been made. Professor Shaul Hanany (Faculty Adviser) GRADUATE SCHOOL Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ata A nalysis of Cosm ic Microwave Background E xperim ents by Matthew Edmund Abroe Under the supervision of Professor Shaul Hanany ABSTRACT The cosmic microwave background (CMB) is a powerful tool for determining and constraining the fundamental properties of our universe. CMB photons last scattered off of free electrons when the universe was roughly one part in 1 0 ®of its present age, and have traveled through the universe practically unhindered until they reach us today. Because these photons last interacted with m atter when the inverse was a small fraction of its present age, the CMB provides us with a picture of what the universe was like in its infancy. Modern experiments that measure spacial fluctuations in the CMB intensity are producing data sets of ever increasing enormity. In this thesis we present various computational and statistical techniques used to analyze datasets from CMB experiments, and apply them to both simulated and actual datasets. Also, when necessary, the algorithms are generalized to run on multi-processor distributed memory supercomputers. The algorithms presented in this thesis perform a variety of tasks in relation to the goal of extracting scientific information from CMB data sets. The CMB anisotropy power spectrum is sensitive to numerous parameters that determine the evolutionary and large scale properties of our universe. Using the power spectrum measurements from a combined m a x i m a - i and c o b e d m r data set we place tight constraints on ~ 10 cosmological parameters using a Bayesian statistical technique. Data from Supernovae type IA are included in this analysis to break parameter degeneracy and place tighter parameter constraints. It is found that the MAXIMA-I data set is consistent with a flat universe, provides evidence for a third peak in the anisotropy spectrum, and tightly constrains the m atter and dark energy content of the universe when combined with a Supernovae lA data. The results of a Bayesian parameter analysis can have a strong dependence on the assumed prior probability distribution of the theoretical parameters. Therefore, we use a frequentist statis tical approach to set confidence intervals on the values of cosmological parameters again using the MAXIMA-I and C O B E DM R data. We define a statistic, simulate the measurements of m a x i m a - i and COBE DMR, determine the probability distribution of the statistic, and use it and the data to set confidence intervals on several cosmological parameters. We compare the frequentist confidence intervals to Bayesian credible regions. The frequentist and Bayesian approaches give best estimates for the parameters that agree within 15%, and confidence interval-widths that agree within 30%. The results also suggest that a frequentist analysis gives slightly broader confidence intervals than a Bayesian analysis. The frequentist analysis gives values of U = 0 . 8 9 = 0 . 0 2 6 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n = 1 . 0 2 and the Bayesian analysis gives values of n = 1.181*^° 2 3 , all at the 95% confidence level. = 0 . 9 8 ^ 3 /1 ^ = 0 . 0 2 9 and Now that numerous experiments have mapped the CMB intensity fluctuations on overlapping regions of the sky it is important to ensure that the various experiments are indeed observing the same signal. We cross-correlate the cosmic microwave background tem perature anisotropy maps from the W M A P , MAXIMA-I, and MAXIMA-II experiments. We use the cross-spectrum, which is the spherical harmonic transform of the angular two-point correlation function, to quantify the correlation as a function of angular scale. It is found that the probability that there is no correlation between the maps is smaller than 1 x 10“ *. We also calculate power spectra for maps made of differences between pairs of maps, and show that they are consistent with no signal. The results conclusively show that the three experiments not only display the same statistical properties of the CMB anisotropy, but also detect the same features wherever the observed sky areas overlap. We conclude that the contribution of systematic errors to these maps is negligible and that m a x i m a and W M A P have accurately mapped the cosmic microwave background anisotropy. Due to a quadrapole anisotropy at last scattering it is predicted that the CMB photons should be linearly polarized, and that the polarization intensity will be roughly an order of magnitude lower than the intensity fluctuations. Experiments that attem pt to measure fluctuations in the CMB po larization are now coming to fruition. With this in mind, we develop a number of simulation and data analysis tools which can be applied to CMB polarization data sets. Two computationally in tensive methods for simulating the CMB polarization signal on the sky are presented. The statistical properties of these maps are confirmed on small patches using the flat sky approximation. These simulated maps are used to assess the amount of mixing in the polarization components due to various amounts of instrumental polarization. A computationally intensive technique for estimating the maximum likelihood temperature and polarization auto and cross spectra is developed. This technique uses a Newton-Raphson iterative technique, also known as the quadratic estimator, for finding the peak of the multivariate Gaussian likelihood function, which is dependent on the CMB temperature and polarization maps. The quadratic estimator takes as input the tem perature and polarization anisotropy maps, their associated pixel-pixel noise correlation matrix, and the beam profile of the detectors. Now that CMB polarization experiments are currently producing data sets new algorithms for analyzing polarization time stream data must be developed and tested. We demonstrate how to gen erate simulations of a polarization experiment in the temporal domain and apply these simulations to the M A XIPO L case. We develop a maximum likelihood map making algorithm and apply it to the temporal simulations to generate simulated M A XIPOL temperature and polarization maximum likelihood maps, and the pixel noise correlation matrix. We show that the output from the map m aking cod e is in agreem ent w ith th e exp ected results. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A cknow ledgem ents T h e d a ta analyzed in this thesis is th e p roduct of countless hours of h a rd work by the M A X I M a /m a x ip o l experim ental team , and I extended th anks to anyone and everyone who has contributed to th a t project. I th an k my advisor, Shaul Hanany, for being very dem and ing (a good quality, especially in a thesis advisor), and for providing m e th e o p p o rtu n ity to do th e work contained herein. I learned a great deal from Shaul, and not ju s t ab o u t physics. N ext I m ust give special thanks to Em ory “Ted” B unn, who was trem endous help th roughout my years of research. Ted wa.s very encouraging during my early “form ative” years, and was always quick to answer any random questions on sta tistic s or CM B d a ta analysis. R adek Stom por was invaluable in b o th his work and ideas, an d a great deal of this work is a result of collaborating w ith Radek. A ndrew Jaffe, P edro Ferreira, and P ro ty W u also provided great assistance throughout m uch of th is work, a n d graciously answered any theoretical questions I had, now m a tte r how trivial. M ost of th e software was developed a t th e U niversity of M innesota Supercom puting In stitu te (M SI), and I m ust extended a well deserved th anks to th e user su p p o rt staff at MSI, especially B irali R unesha and Shuxia Zhang. B oth R unesha a n d Shuxia herocially answered all my supercom puting question and provided critical help in im plem enting my software onto th e MSI machines. Two of th e projects in this thesis are based on extensions of th e M A D C A P software package, w ritte n by Ju lian Borrill. Ju lian was of great assistance early on w hen I was atte m p tin g to learn (and actually compile) M ADCAP, an d also a great supercom puting reference throughout. I also th an k Ju lian for providing me w ith a n account on th e Seaborg m achine a t N ERSC , which I m ade great (and som etim es a bit to much) use of. A shout out to P ete, Jam es, Paul, Roy, and all my o ther friends w ho’ve m ade music an excellent and necessary escape from g raduate school. C onstantly being forced to actually get away from th e com puter and go out and play live music was so great, an d probably the only th ing th a t kept me sane during all these years. You guys rock!! I would like to th an k Jyotirm oy Saha for being a great friend, and actually m aking it 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. out to see me play music. Also thanks to B rad Johnson for being a kick ass lab m ate, a n d always being quick on th e draw to answer any UNIX system or CM B experim ental questions. Also, I extend th anks to Shaul, Liliya W illiam s and K eith Olive for reading this thesis an d providing great feedback. Finally I ’d like to th an k th e people who deserve it m ost. T his thesis has been a long tim e in th e m aking, a n d would not have been possible w ithout the constant love and su p p o rt of my family. Mom, D ad, Betsy and H annah, I love you and do not exaggerate w hen I say I could not have done th is w ithout you. Matthew E. Abroe June 2004 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ed ication For m y loving parents, Michael S. and Dr. Mary J. Abroe Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table o f C ontents A bstract i A cknow ledgem ents iii D edication v List of Tables ix List of Figures x 1 Introduction 1 1.1 A G uide to th is T h e s is ................................................................................................... 2 1.2 Big Bang C o sm o lo g y ...................................................................................................... 5 1.2.1 B a s i c s ..................................................................................................................... 5 1.2.2 Friedm ann E q u a t i o n .......................................................................................... 6 1.2.3 Cosmological P aram eters ............................................................................... 7 1.3 I n f l a ti o n .............................................................................................................................. 9 1.4 T he Cosmic Microwave Background ........................................................................ 10 1.4.1 T he A ngular Power S p e c tr u m ......................................................................... 11 1.4.2 Cosmological P aram eter D e p e n d e n c e ........................................................... 12 1.4.3 CM B D a ta Analysis B a s i c s ............................................................................ 14 2 Bayesian Cosm ological Param eter E stim ation ............................................................................... 19 2.1 Bayesian Credible Regions 2.2 T he Offset-lognorm al A p p ro x im a tio n ........................................................................ 21 2.3 T he MAXIMA-I A n a l y s i s ................................................................................................ 23 2.3.1 P aram eter S p a c e ................................................................................................ 23 2.3.2 R esults 25 ................................................................................................................. vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 3 4 5 6 2.4 D is c u ss io n ........................................................................................................................... 28 Frequentist Cosm ological Param eter E stim ation 30 3.1 In tr o d u c tio n ........................................................................................................................ 30 3.2 D a ta and D atabase of cosmological m odels ........................................................... 32 3.3 The S ta tis tic ................................................................................................................... 32 3.4 D eterm ining Confidence L e v e l s ................................................................................... 34 3.4.1 Frequentist Confidence Intervals .................................................................. 34 3.4.2 Bayesian C redible R e g io n s ................................................................................ 39 3.5 R e s u lts .................................................................................................................................. 40 3.6 D is c u ss io n ........................................................................................................................... 43 Correlating CM B M aps 45 4.1 In tr o d u c tio n ........................................................................................................................ 45 4.2 T he Cross S p e c t r u m ..................................................................................... 47 4.2.1 Full Sky A p p ro x im a tio n ................................................................................... 49 4.2.2 R ealistic E x p e r im e n t s ...................................................................................... 54 4.3 T h e M a p s ........................................................................................................................... 59 4.4 R e s u lts .................................................................................................................................. 63 4.5 D is c u ss io n ........................................................................................................................... 65 4.5.1 A uto- and C ro ss-S p e c tra ................................................................................... 65 4.5.2 C om putational Issues 6 6 4.5.3 Foregrounds and System atic E r r o r s ............... ...................................................................................... 6 6 4.6 C o n c lu s io n s ...................................... 6 8 4.7 Single Value C orrelation S t a t i s t i c s ............................................................................ 6 8 CM B M ap Sim ulations 74 5.1 P olarization B a s i c s ...................................................... 75 5.2 G enerating M a p s ..................... 77 5.2.1 Spherical H arm onic E x p a n s i o n ..................... 78 5.2.2 Sm all Sky A p p ro x im a tio n ................................................................................ 82 5.2.3 Two Point C orrelation M e t h o d ..................................................................... 85 5.3 S tatistical P r o p e r tie s ....................................................................................................... 8 6 5.4 Cross P o l a r i z a t i o n .......................................................................................................... 91 Power Spectrum E stim ation 94 vn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1 Full Sky A p p r o x im a tio n .............................................................................................. 95 6.2 R eal Space Likelihood F u n c tio n ................................................................................. 97 6.2.1 T he Signal Covariance M a tr ix ........................................................................ 98 Q u ad ratic E s t i m a t o r ..................................................................................................... 100 6.3 7 MAXIPOL 7.1 Sim ulations 105 T im e S tream S i m u l a ti o n s ........................................................................................... 105 7.1.1 S i g n a l .................................................................................................................... 106 7.1.2 Noise .................................................................................................................... 107 7.2 M ap M aking ................................................................................................................... 108 7.3 T e s t s ......................... 7.4 Ill 7.3.1 B45 MAXIPOL sim ulation ............................................................................... Ill 7.3.2 Sm all M ap t e s t ................................................................................................... 112 D is c u ss io n ......................................................................................................................... 115 A ppendix A. Probability D istribution o f th e Experim ental D ata 117 A ppendix B. Spin Spherical Harm onics 119 A ppendix C. C Source Code 123 C .l Pixel Noise C orrelation M a t r i x ................................................................................. 123 C.2 Pixel Signal Covariance M a t r i x ................................................................................. 125 R eferences 132 Vlll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Tables 2.1 Parameters Estimates from the 3.1 A comparison of Bayesian, frequentist and maximization 95% confidence intervals. The table also gives the 95% m ax im a-i and cobe dm r d a t a ................................ 26 thresholds from the simulations. Maximization confidence intervals are taken from [3]; they do not give confidence intervals for all the parameters. 41 4.1 Correlations between Bin Spectra E s tim a te s .................................................................. 58 4.2 Cross Spectrum 64 4.3 Difference Spectrum V a lu e s......................................................................................... 64 4.4 Correlations between Bin Spectra E s tim a te s .................................................................. 67 4.5 Correlation Coefficients V a lu e s......................................................................................... 73 5.1 Values for Simulated M a p s ......................................................................................... 91 V a lu e s ................................................................................................ 5.2 RMS fluctuations from cross polarization ..................................................................... 93 6.1 Correlations between Temp/Polarization Bin Spectra Estimates .................. 104 7.1 Map Making computational re q u ire m e n ts ..................................................................... Ill 7.2 MAXIPO L Map Making software t e s t ............................................................................... 116 A .l The average values with sample standard deviations of the marginalized Ziweight matrix entries for both large and small map simulations. The large map simulations were based on pointing from the m a x im a -i 8' map, and the small map pointing was based on a center patch of the map. Units are dimensionless MADCAP units ..................................................... IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 List o f Figures 1.1 The CMB angular power spectrum, Ci, as measured by the (blue squares), m ax im a-ii (red diamonds), and The solid line is the best fit model to the w m ap WMAP ARCHEOPS (black circles), m ax im a-i (green bow ties) experiments. data points. The horizontal axis is spherical harmonic multipole number, i, which is inversely proportional to angular separation on the sky and the vertical scale is //K^. The horizontal scale is log/linear to illustrate the relative coverage of the various experiments........................................................................................................ 1.2 12 Examples of how the Ci amplitude changes by Vcirying the parameters 0 , n, and Q,bh?■ The solid line all four panels represents an inflationary cold dark matter model with cos mological parameters; (//o, fib, flcm6, Ha, « s,'tc )= (5 0 , .05, .9 5 ,0 ,1 ,0 ), while the remaining dashed lines represent models whose parameters vary as shown...................................................... 2.1 The M A X I M A -I 15 high resolution angular power spectrum plotted with two models. The solid blue line is the best fit model to the entire database, while the dashed red line is the best fit with the BEN prior, 2 .2 = 0.02 25 Marginalized one dimensional likelihoods for cosmological parameters as determined by the combined m a x im a-i and COBE DMR data. The circles represent normalized likelihood values in the grid points and the solid blue lines are obtained by cubic spline interpolating between points............................................................................................................................................... 2 .3 Constraints in the ftm- f^A plane from the combined m ax im a-i and cobe dm r 27 data. The shown contours correspond to 68, 95, and 99% likelihood ration confidence levels. The bounds obtained form high redshift supernovae data are also overlaid as well as the joint confidence levels............................................................................................................................................. 2 .4 A spectral index - optical depth contour plot obtained for the DMR data. Shaded contours show 68, m ax im a-i combined 29 cobe , 95 &: 99% likelihood-ratio confidence levels. Lines show an approximate relation between n and t fitted to this result. Central values are shown by the solid line and 95% bounds bythe dashed lines........................................................... X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 3.1 Distribution of values computed for the 10,000 simulations by solving for and u analytically using the m ethod described in the text (dashed line), and numerically using Brent’s root-finding algorithm (solid line). Both histograms have a bin size of one and are normalized to integrate to one....................................................................................................... 3 .2 33 The histogram gives the Ax^ distribution for the entire six-dimensional parameter space from 10,000 simulations of the cobe dm r and m ax im a-i band powers. The dashed curve is the standard x^ distribution for six degrees of freedom. The vertical solid line (vertical dashed line) is Ax^ = 16.5 (12.8), which corresponds to the 95% Ax^ threshold for the histogram (standard x^ distribution). The histogram has a bin size of 0.5, and is normalized to integrate to one............................................................................................................................. 3 .3 34 Simulated one-dimensional Ax^ distributions for all the parameters in the database. The vertical dotted lines correspond to the 95% Ax^ threshold level; numerical values are given in Table 1. Each histogram has a bin size of 0.05 and is normalized to integrate to one. The 95% threshold for a standard x^distribution with one degree of freedom 3 .4 is x^ = 3.8. 35 Simulated two-dimensional Ax^ distributions in the ( H o, ^ b ) and (f2m,DA) planes. The vertical dotted lines correspond to the 95% Ax^ cutoff level, which are 6.25 and 7.70 for (Ho, f ls ) and ($2m, ^ a ) respectively. Each histogram has a bin size of 0.05 and is normalized to integrate to one. The 95% threshold for a standard x^ distribution with two degrees of freedom is x^ = 6............................................................................................................................... 3 .5 Ax^ calculated with the m ax im a-i and cobe dm r 36 data as a function of parameter value for each of the parameters in the database. Solid circles show grid points in parameter space, and the solid lines were obtained by interpolating between grid points. The parameter values where the solid line intercepts the dashed (dotted) line corresponds to the 68% (95%) frequentist confidence region........................................................................................................ 3 .6 37 Two-dimensional frequentist confidence regions in the (Dm, Da) plane. The dark to light regions correspond to the 68%, 95%, and 99% confidence regions respectively. The dashed line corresponds to a flat universe, Q = Dm -t- Da = 1...................................................................... 3 .7 39 Two-dimensional frequentist confidence regions in the {Ho,Hb) plane. The dark to light regions correspond to the 68%, 95%, and 99% confidence regions respectively. Standard cal culations from big bang nucleosynthesis and observations of D j H predict a 95% confidence region of Dnh^ = 0.021 3 .8 indicated by the shaded region.............................................. 40 Bayesian likelihood functions for each of the parameters in the database. Solid circles show grid points in parameter space, while the solid lines were obtained by interpolating between grid points. The parameter values where the solid line intercepts the dashed (dotted) line corresponds to the 68% (95%) Bayesian credible regions................................................................. XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 3 .9 A comparison of the 95% frequentist confidence intervals (FCI, solid line), Bayesian cred ible regions (BCR, drished line), and maximization regions (M, dashed dot line) for the parameters in the database, and for Q and 17b 4.1 42 A comparison of the CMB angular power spectrum as measured by I. The black circles are the WMAP and MAXIMA- binned power spectrum, which were produced from a w m ap weighted combination of cross spectra from 28 detectors. The blue squares are the I angular power spectrum, which is a composite spectrum from the m ax im a-i m ax im a- maps with a 5' and 3' square pixelization [56]. The red diamonds are the CMB angular power spectrum computed from the m ax im a-ii experiment, which is described in detail in Section 4.3. Note that no adjustments have been made to the calibration for any of the data sets.................... 4 .2 46 Results from various tests of the auto- and cross-spectrum estimation code using simulated M AXIM A-I maps. In each case the red and green points represent the auto-spectra for the first and second maps, respectively, and the blue points represent the cross-spectrum. The top panel shows the three spectra estimated from maps with near perfect correlation. The middle panel shows the spectra estimated from maps with 50% correlation (the dotted line is the input cross-spectrum model), and the bottom panel shows the spectra estimated from maps with no correlation.................................................................................................................. 4 .3 A comparison of the shows the w m ap m ax im a-i map in the and w m ap m ax im a-i 57 CMB anisotropy maps. The upper left panel region, smoothed with a Gaussian window function with FWHM of 13.2' as described in the text. The m ax im a-i map is shown in the upper right hand panel, and the sum and difference of the two maps are shown in the lower left and lower right panels, respectively. All the maps are pixelized in HEALpix pixelization with resolution parameter nside=512. Note that we use the raw data in all analyses. 4 .4 A comparison of the m ax im a-i and upper right panels show the raw m ax im a-ii M A X I M A -I 60 CMB anisotropy maps. The upper left and and a vertical gradient removed from each. The . . m a x im a -ii m ax im a-i maps, respectively, with only map is shown in the upper right hand panel, and the sum and difference of the two maps are shown in the lower left and lower right panels, respectively. All the maps are pixelized in HEALpix pixelization with resolution parameter nside=512. Note that we use the raw data in all analyses..................... 4 .5 The beam filter functions for m a x im a -i , m ax im a-ii , and w m ap 61 considered in this analysis. They are the dotted line, dashed line, and solid line, respectively. N ote that due to higher angular resolution of the m a x im a beams the observations reach to higher £s and this is accounted for in the comparisons. The horizontal axis is spherical harmonic multipole. XU Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . 63 4 .6 The auto-spectra and cross-spectrum estimated for the maps described in the text. From top to bottom; the results for Ii/W M A P , m ax im a -i/m a x im a -ii , and m ax im a- . The horizontal axis is spherical harmonic multipole number, and the vertical axis is 4 .7 m a x im a -i/w m a p ............................................................................................................................... A visual comparison of the power spectrum made by taking the difference maps from all three experiments. The diamonds (red) are the power spectrum of the difference map, the squares (blue) are the power spectrum of the II m a x im a -i /w m a p M A X I M A -I / m a x i m a - difference map, and the triangles (green) are the power spectrum of the /w m ap M A X I M A -I I difference map. We find all three spectra to be statistically consistent with the null spectrum.............................................................................................................................................. 4 .8 70 Simulated probability distributions for the single value correlation statistics. In each panel the horizontal axis in statistic value, and the vertical axis is frequency per bin normalized such that the maximum probability is equal to one. The black distributions were computed under the null hypothesis, i.e. that both are pure noise. The blue distributions were computed under the hypothesis that each map contains signal and noise as described in the text. The vertical dotted red lines are the actual values computed for the M AXIM A-II 5.1 m a x im a-i m aps................................................................................................................................ - 72 Simulated CMB anisotropy maps using using the spherical harmonic expansion method, with only TT , T E , and E E fluctuations. Each maps contains 65, 536 3' square pixels. No beam smoothing was included in these simulations............................................................... 5 .2 Various real space intensity maps for the Stokes vectors Q and U for either E or B type fluctuations. This illustrates the fact that ss E B, then Q U and U —Q. The vertical and horizontal axis are pixel number.......................................................................... 5.3 82 Simulated CMB anisotropy maps using the m ax im a-i 84 pointing. These maps were generated using the two point correlation, and contain only T T , E E type fluctuations, and T E correlation. The each pixel contains 1/xK^ noise.................................................................... 5 .4 87 The T T , T E , E E , and B B power spectra estimated from simulated maps generated using the spherical harmonic expansion method. The solid lines are the theoretical models used to generate the maps, and the solid circles with error bars are computed using the FFT m ethod described in the text. There is a residual B B component at small scales comparable 5 .5 with the pixel size of the maps..................................................................................................... 89 The real space E and B maps computed from the maps shown in Figure 5.1.............. 90 X lll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 5 .6 The E E and B B power spectra estimated from simulated Q and U maps with various amounts of cross polarization. For each set of power spectra the corresponding amount of cross polarization is given in the upper right hand corner of the figure. The solid line in the E E figure is the model used to generate the pure signal maps with no cross polarization. There are no B B fluctuations in the simulated maps....................................................................... 6.1 92 Estimated power spectra using the quadratic estimator. The circles with error bars are the estim ated power spectra, and the solid lines are the underlying models of the maps. . . 103 7.1 The number of hits per 3' pixel for the m axip ol-i pointing solution for the B45 channel. This pointing was used to generate the simulations shown in Figure 7.2................................... 112 7.2 An example of the map-making code using the pointing information from the B45 channel of the MAXIPOL-I flight. The left panel contains the pure signal input maps to the simulation. On the right panel are the recovered maps. Only a small amount of noise was input in the time domain................................................................................................................... 113 7.3 The pixel noise covariance matrix for the small map making test described in the text. The B .l test is for a simulation with 1,000 total time stream samples spread evenly over 5pixels. 114 Two coordinate systems which differ by a rotation of the angle 6 ........................................ 120 B.2 Examples of the spin-0 spherical harmonics Y(m for various multipole coefficients £ and m. These figures show only the real component (spherical harmonics are complex functions). N ote that the m = 0 harmonics have no azimuthal dependence, whereas the m ^ 0 (and especially the m = £) harmonics have strong azimuthal dependence........................................... XIV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 Chapter 1 Introduction W hen you can m easure w hat you are speaking ab o u t a n d express it in num bers, you know som ething ab o u t it; b u t w hen you cannot m easure it, w hen you cannot express it in num bers, your knowledge is of a m eager an d unsatisfactory kind. -Lord K elvin To a great extent Lord K elvin’s statem ent reflected th e sta te of cosmology as a science until the la tte r p a rt of the 20th century. However, w ith th e inform ation gleaned from m odern m i crowave background experim ents cosmology was transform ed from a d a ta starved to a d a ta rich science. Prom these experim ents th e therm al spectrum of th e microwave background was found to be consistent w ith th a t of a black body [61], confirm ing its cosmological ori gin, and for th e first tim e slight spatial fluctuations were found in th e nearly hom ogeneous tem p e ra tu re of th e microwave background [85, 8 6 , 102]. U nder the big bang paradigm of cosmological evolution fluctuations in th e tem p eratu re of th e microwave background, comm only referred to as th e cosmic microwave background (CM B), correspond to tiny density fluctuations in th e early universe. These initial fluctu ations eventually evolved into galaxies and clusters of galaxies via grav itatio n al instability [52, 6 8 , 69, 70]. T hrough an accurate characterization of the CM B angular fluctuations we can discrim inate betw een com peting cosmological theories, an d determ ine num erous cosmological param eters to a few percent accuracy. T he CM B provides us w ith a picture of w hat th e universe looked like in its earliest stages, and it is th e goal of some experim ents to m ap th e sp atial anisotropy of its intensity. These experim ents have now reached m aturity. Balloon borne experim ents such as MAXIMA [36] and BOOMERANG [7], and later ARCHEOPS [6 ], and ground based interferom etric experi m ents, CBi [60], DASI [35], and VS A [29] have accurately m apped th e angular tem p eratu re 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fluctuations in th e CM B to high spatial resolution. Recently WMAP [4] produced a full sky high resolution m easurem ent of th e CM B tem p eratu re anisotropy. T here are num erous publications on the experim ental aspects of CM B detection. For a discussion of th e MAXIMA experim ent see [36, 55, 74, 100], and for a focus on polarization m easurem ents see [46, 47]. Cosmological m odels do not predict the tem p eratu re of the CM B a t any point on th e sky. R ather, they predict th e statistical properties of the sp a tia l intensity fluctuations. In order to ex tract cosmological inform ation from m easurem ents of the CM B we m ust first distill th e d a ta from tim e stream sam ples into the form of a pixelized m ap a t high resolution. T hen, from th e pixelized m ap com pute th e anisotropy spectrum as a function of angular scale. In m any cases b o th of these tasks proves to be com putationally daunting. It is the angular anisotropy spectrum which encodes th e sta tistic a l properties of th e fluctuations on th e sky, a n d allows us to discrim inate betw een com peting cosmological m odels. W ith in the last several years d a ta provided by experim ents tightly constrained th e m ultiple acoustic peak stru c tu re in th e anisotropy spectrum , an d num erous cosmological p aram eters have been estim ated to a few percent accuracy. 1.1 A G uide to this Thesis T he goal of this thesis is to present b o th sta tistic a l and com putational algorithm s for an a lyzing a n d extracting cosmological inform ation from experim ents th a t mea.sure anisotropy in th e CM B. In m any cases the d a ta sets considered are large, requiring th e aid of high perform ance d istrib u ted m em ory supercom puters. In these cases we give specific exam ples of how th e actu al com puter software scales w ith the num ber of processors. Also, w hen pos sible we use idealistic and simplified sta tistic a l argum ents to analytically derive expected estim ators w ith error bars from the CMB d ata. These idealistic estim ators can th en be com pared to th e actu al results com puted from the real data. T he algorithm s presented in this thesis perform a variety of tasks in relation to the goal of extracting scientific inform ation from CMB d a ta sets. F irst, in th is in tro d u cto ry chapter we present a basic description of global cosmology, including a discussion cosmological param eters, inflation, and the CMB. We also give an in tro d u ctio n to CM B d a ta analysis, which will provide a foundation for subsequent chapters. In C h a p te r 2 we place tight constraints on ~ 10 cosmological param eters using a Bayesian sta tistic a l technique from a com bined m a x im a - i and c o b e d m r d a ta set. D a ta from Supernovae type lA are included in this analysis to break p aram eter degeneracy and place tighter p a ram eter constraints. Bayesian param eter estim ation is th e m ost com m on technique for determ ining cosmological p aram eters from CM B d a ta sets, and we review its basic assum ptions and m ethodology in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th is C hapter. Here we also introduce th e offset lognorm al approxim ation. A t th e tim e of its release, this sim ultaneous analysis of th e MAXIMA-I a n d COBE DMR d a ta placed th e tightest constraints on cosmological param eters using CM B d ata. T he results of a Bayesian param eter analysis can have considerable dependence on the as sum ed prior probability d istrib u tio n of the theoretical param eters. Therefore, a frequentist statistical m ethod for cosmological param eter estim ation from CM B d a ta sets is developed in C hapter 3, and th is m ethod is again applied to th e MAXIMA-l an d COBE DMR data. At th e tim e, this was th e m ost fully frequentist analysis of CMB d a ta . A frequentist analysis asks a com pletely different question of th e d a ta th a n the Bayesian analysis, and there is no guarantee th a t confidence intervals determ ined by the diflferent m ethods will coincide. It is im p o rtan t to note th a t th e MAXIMA-I d a ta used in this analysis is slightly different th an th e version used in Bayesian analysis discussed in C h ap ter 2. T he Bayesian analysis uses th e high resolution 3' pixelized d a ta discussed in [56], whereas th e frequentist analysis uses th e 5' pixelized d a ta of [36]. It is shown th a t th e confidence intervals estim ated using our frequentist m ethod is consistent w ith confidence intervals estim ated using b o th a Bayesian a n d m axim ization [3] techniques. C hapter 4 presents m ethods for com paring and quantifying th e am ount of correlation betw een two anisotropy m aps. T his chapter specifically focuses a great deal on th e cross spectrum , which is th e spherical harm onic transform of th e two point correlation function betw een two m aps. T he cross-spectrum provides a sim ple way for accounting for different experim ental properties betw een m aps such as differing pixelizations or beam profiles. A d ditionally, unlike other m ethods of correlating CMB m aps, the cross-spectrum provides the am ount of correlation on m ultiple angular scales. In this C h ap ter we correlate d a ta from th e WMAP, MAXIMA-I, and MAXIMA-II d a ta and rule out the no correlation hypothesis to greater th a n 1-10~* confidence using the cross spectrum . We also com pute th e difference sp ectra for these m aps and show th a t it is consistent w ith zero. A t th e end of th e chapter we present a num ber of single value statistics th a t can be used to correlate CM B m aps, and com pute th eir probability distrib u tio n s using M onte Carlo sim ulations. We also com pute th e values of these statistics for the MAXIMA-I and MAXIMA-II m aps, an d d em o n strate they are consistent w ith th e distrib u tio n s of m aps which share a com m on cosmological signal. By correlating CM B anisotropy m aps from different experim ents we can gain confidence in the spatial reconstruction of the m aps, and confirm th a t m odern experim ents have accurately m apped th e CM B anisotropy In C hapter 5 we discuss for th e first tim e polarization of th e CM B. Due to a local ized quadrapole anisotropy photons th a t Thom son scattered off electrons a t last scattering should be linearly polarized. A n a rb itra ry polarization field can be decom posed into curl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. free and divergence free com ponents. T his is th e so called E - B decom position, in analogy w ith electrom agnetism , and different cosmological m odels predict various am ounts of fluctu ations in th e various com ponents. T his decom position also removes th e coordinate system dependence found in th e usual Stokes p aram eter representation. M apping th e polarization anisotropy in these com ponents as a function of angular scale is th e current goal of CMB experim ents. W ith th is in m ind we develop three m ethods for sim ulating th e tem p e ra tu re an d polarization anisotropy, two of which are com putationally intensive. These sim ulations can be used to test d a ta analysis software developed in o ther chapters, a n d provide some insight into th e n a tu re of CM B polarization. T he statistical properties of th e sim ulated m aps are testing using F F T ’s, and it is shown th a t they are indeed consistent w ith the expected results. We also use these sim ulations to constrain th e am ount oi E B m ixing due to various am ounts of experim ental cross polarization. A m ethod for extracting th e E and B power spectrum from m easurem ents of th e Stokes p aram eter anisotropy is presented in C hapter 6 . T his m ethod also provides for sim ultaneous estim ation of th e various tem p eratu re and polarization cross-spectra, which is expected to be non-zero for certain com ponents (nam ely th e T E cross-spectrum ). T his technique uses a N ew ton-R aphson iterative technique, also known as th e q u ad ratic estim ator, for finding th e peak of th e m ultivariate G aussian likelihood function, which is dependent on th e CMB tem p e ra tu re and polarization m aps. T he q u ad ratic estim ator takes as in p u t the tem p era tu re and polarization anisotropy m aps, their associated pixel-pixel noise correlation m atrix, a n d th e beam profile of th e detectors. Because m axim izing th e likelihood in this natu re involves m atrix inversion and m atrix -m atrix m ultiplication w ith very large dense m atrices, we require th e aid of high perform ance supercom puters. M axim um likelihood tem peraturepolarization power sp ectra estim ators for th e full sky idealized case are analytically derived, as well as expressions for th e correlation betw een power sp ectra estim ates using th e full sky approxim ation. We th en show th a t the correlation results from th e q u ad ratic estim ato r are consistent w ith th e analytical results. We finish w ith a dem onstration of how to generate sim ulated tem p e ra tu re and polar ization CM B d a ta in the tem poral dom ain, and apply m axim um likelihood m ap m aking techniques using these sim ulations in C h ap ter 7. T he m ap m aking software is tested using two different sim ulations, and it is shown th a t th e o u tp u ts are in agreem ent w ith expected results. T his software is used to generate tem p eratu re and polarization anisotropy m aps and the pixel noise correlation m atrix from sim ulated tim e ordered d a ta , and is applied to sim ulated d a ta using th e actual pointing from the m a x i p o l - i experim ent. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 B ig B ang Cosm ology In this section we review basic elem ents of th e big bang and m odern cosmology, which is by no m eans intended to be com plete. For a m ore thorough tre a tm e n t of m odern cosmology see the excellent tex ts [52, 6 8 , 69, 70]. Here we give a sim ple derivation of th e Friedm ann equation a n d a basic description of cosmological param eters. 1.2.1 B a sic s In the classical big bang fram ework the Universe is considered to be hom ogeneous and isotropic w hen viewed on sufficiently large scales. T his is the cosmological principle. T he m ost general m etric satisfying these assum ptions is th e R obertson-W alker m etric ds^ — —d t ^ + a ^ { t ) d l ‘^ = —dt^ + a^{t) 1 ( 1. 1) ( 1.2 ) —k r “ ^ w here {r,6,(f)) are th e sta n d a rd spherical coordinates. A: is a constant related to th e curvature of space, an d a{t) is th e expansion factor. T he tim e separation and sp a tia l separation betw een two points in our “space-tim e” are given by dt an d a{t)dl, respectively. N ote th a t the cosmological principle is different from the the perfect cosmological principle, which states th a t th e universe should look the sam e to all observers a t all tim es. U nder the big bang paradigm th e early universe would look very different from th e present universe. Therefore, th e big bang theory is inconsistent w ith th e perfect cosmological principle. T he m ain observational evidence which su pports the big bang paradigm are • T he observed light elem ent abundance in th e universe, which is correctly predicted by big bang nucleosynthesis (BBN). • T he existence and therm al spectrum of the CMB. T h e existence of th e CM B and its near perfect isotropy across th e sky is a rem arkable confirm ation th a t th e universe is b o th homogeneous and isotropic, a n d evolved from a hot dense p ast [70]. T he high degree of isotropy in th e CM B im plies th a t th e universe was near isotropic a t th e epoch of last scattering to roughly one p a rt in 10® on scales from 90° down to a few arc-m inutes. Also, th e observed expansion of the universe is consistent w ith (but not unique to) th e big bang model. Edw in H ubble found th a t th e red shift of galaxies is proportional to th eir ap p aren t m agnitudes [42]. Assum ing th a t they are equally lum inous th en their redshifts are proportional to th eir distances, or v (x d. T his is know n as H ubble’s Law, and can be directly obtained if one assum es a hom ogeneous expansion of th e universe. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Given a length Zq? th en after some tim e t th e length becomes I — a{t)lo- D ifferentiating gives H ubble’s Law: V — = -I (1.3) H{t% (1.4) which states th a t th e velocity w ith which two objects move away from each o ther is pro p ortional to th eir distance. T he value of proportionality is th e H ubble p aram eter H {t) , and th e current value of the H ubble p aram eter is H q. 1.2.2 Friedmann Equation Given th e assum ption th a t our universe is b o th homogeneous an d isotropic on large scales, th e history and evolution of th e universe can be determ ined by specifying several funda m ental param eters. Cosmological evolution equations can be derived w ith either E in stein ’s field equations, or w ith N ew tonian gravity. T his is possible because from th e cosmological principle we can take a volume elem ent sm all enough for N ew tonian gravity to apply as being representative of th e entire universe. T he basic equation of cosmological evolution in relation to several fundam ental param eters is given by th e Friedm ann equation. Here we give a brief derivation. T he continuity equation for a non-relativistic fluid is |^ + V-(pu) = 0 (1.5) T h e assum ption of hom ogeneity implies th a t V p = 0, an d from isotropy we have V • v — 3a /a = 3H{t). P u ttin g th is all together yields p = -3 -(p -t-p ). a (1.6) E quation 1.6 states th a t th e density will be changing in tim e, a n d it will be decreasing (assum ing a fluid positive pressure), which is consistent w ith an expanding universe. New to n ’s law of m otion for a homogeneous gravitating sphere w ith density p is d = —AnGap(3. To include relativistic com ponents we sim ply replace th e density w ith a pressure term to include for “gravitating m ass” , or p —> p-|- 3p. T his, com bined w ith E qu atio n 1.6, implies th a t d a 47tG M ultiplying by da leads to d f I .2 - a dt 47tGp 2 —a ) = 0 , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 1 .8 ) w here G is th e g ravitational constant. Integrating E q uation 1.8 gives a? 8nGp k a? 3 a? (1.9) w here k is th e constant of integration. T he N ew tonian in te rp re ta tio n of k is th e to ta l energy, whereas th e relativistic in terp retatio n is th a t k is the curvature of space. T he Friedm ann E quation is th u s w ritten as SttGo k a where th e A term was introduced by E instein to o b tain a static solution to th e equation^. T he universe expands adiabatically because the sym m etry im posed by th e cosmological principle im plies th a t there is no net heat flow th rough any surface. T herefore as the universe expands it cools, and will thus be dom inated by different species of particles a t different epochs. Each species of particles can be param eterized uniquely w ith an equation of state p = wp. ( 1 -1 1 ) Some exam ples of different equation of states are: 1) tc = 0 (m atter), 2) ro = 1/3 (radia tion), and 3) u; = —1 (cosmological constant). Plugging E quation 1.11 into th e continuity E quation (1.6) yields p = poa-3(i+“'), (1,12) where po is th e present day density. Now, w ith E quation 1.12 a n d the F riedm ann equation we can determ ine th e tim e evolution of the scale param eter a{t) for different com ponents of particle species, i.e. for different cosmological models. 1.2.3 C o sm o lo g ica l P a ra m eters In this section we derive and discuss several cosmological param eters which determ ine the type of universe we live in. D ensities Based on th e Friedm ann equation we define th e following param eters. T h e critical density Pcrit is defined as the density necessary to o b tain a flat universe {k = 0 ) in a m a tte r only ^This was before Hubble’s discovery of the recession of galaxies in ail directions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. universe. It is P crit = = 1.88 X 10“ ^® gm cm “ ^. (1-13) ottG where th e dim ensionless param eter h = Ho/{10Qkm/sec/Mpc). We can now define the dim ensionless norm alized m a tte r density param eter Om = — , (1.14) Per w here th e density p is from ordinary m a tte r only. We can re-w rite E q u atio n 1.10 as 1 w here Qk = —k/{HQOQ), and = (1.13) = A / { 3 H q), which is th e effective density from th e cosmo logical constant. A flat universe is defined as one in which th e curv atu re isequal to zero, fifc = 0. We define th e to ta l density p aram eter = 1 —fife = O a + w hich is th e to ta l density from b o th ordinary m a tte r and cosmological constant. T he F riedm ann equation can now be w ritten as H {z ) = = Ho{nK + nm{i + z f + + ( i.ie ) HoE{z). (1.17) N ote th a t we’ve w ritten the H ubble param eter as a function of redshift z, instead of scale factor a. T he relationship betw een redshift and the scale factor a t some earlier tim e t is 1 + z = (1.18) w here oq is typically set to unity. We call H{ z ) th e H ubble function. It is interesting to investigate how difi'erent values for these cosmological param eters (i.e. different cosmologies) effect th e dynam ics of th e universe. A fiat m a tte r dom inated universe is the well known Einstein-D e S itter universe [70], which has flm = fl = 1. Using th e Friedm ann equation wefind th a t for an Einstein-D e S itter universe implies th a t this type of universe is expanding (a > 0 a{t) oc T his ),b u t th a t th e ra te of expansion is decreasing (a < 0). A flat radiation dom inated universe {w = 1/3) has solution a{t) oc which is also expanding a t a decreasing rate. A nother interesting case is a A dom inated {w = - 1 ) fiat universe, which has solution a{t) oc In this case th e universe undergoes exponential expansion. Indeed, for any w < —1/3 dom inated universe th e expansion will have positive acceleration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Age of the universe T he tim e separation betw een some point a t redshift z' (w ith scale factor a') an d the present tim e is given by /'• = i's f 1 r ' dz H o i i^ + z ) E { z y where we have set th e present day scale factor ^ to unity. T hus, in a m a tte r dom inated the age of th e universe is to — — Ho [ T— —— r ( Q a + O m ( l + .2;)^ + ^ 0 (1 ( 1 — f l ) ( l + 2; ) ^ ) ^ ■ (1-21) + ^) Specifically, for an Einstein-D e S itter universe the expression reduces to an d for A > 0 m odels A fu rth er discussion of cosmological param eters is given in Section 1.4.2. T here we define more param eters, and give a discussion of how these param eters afllect th e angular power spectrum of the CMB. 1.3 Inflation W hile th e big bang m odel of th e universe is very successful, passing some very crucial observational tests (see Section 1.2.1), it is not w ithout its problem s. Recent observations im ply th a t th e universe is indeed close to flat, or 1. Since th e universe has long been m atte r dom inated [70] the to ta l density drops as Therefore, th ere is no a priori reason to expect th a t th e present day curvature should be close to zero. T his is th e so called flatness problem . T he second problem w ith sta n d a rd big bang cosmology is th a t it provides no explanation for th e uniform ity of th e CM B (see Section 1.4), which is known as th e horizon problem . CM B photons observed from opposite sides of th e sky ap p ear to be in alm ost perfect therm al equilibrium (fluctuations do exist on the order of 1 p a rt per m illion). T he m ost n a tu ra l explanation for this is th a t a t some earlier tim e these regions were in causal contact, b u t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 under th e sta n d a rd big bang this is not possible. In fact, any regions sep arated on th e sky by m ore th a n ~ 2 ° were not in causal contact a t the epoch of decoupling. A dditionally, th e unw anted relics of th e early universe such as m agnetic m onopoles would dom inate the universe to d ay according to sta n d a rd big bang. Inflation provides a solution to th e flatness and horizon problem s, a n d dilutes any un w anted particle relics such as m agnetic m onopoles. For thorough reviews of inflationary cosmology see [57, 58]. We saw in Section 1.2.3 th a t th e universe undergoes rap id expan sion for any w < —1/3 m aterial. In the very early universe very sm all patches on th e scale of 1 / H are in causal contact. According to inflation the universe th e n undergoes a period of rapid expansion driven by a rc < - 1 / 3 m aterial. T his causes all observable m odes to be m uch larger th a n the H ubble length, causing hom ogeneity an d driving th e to ta l density to = 1. W hile inflation does a tte m p t to insure perfect hom ogeneity it cannot defeat the uncertainty principle, which ensures th a t there are always some irregularities left behind. Inflation sim ultaneously solves a num ber of problem s w ith th e sta n d a rd big bang m odel, and provides an explanation for th e slight density p e rtu rb atio n s which should be present in th e early universe. A nother im p o rtan t consequence of inflation is th e p roduction of prim ordial gravitational waves [106, 48, 41]. A lternative term inology for th e density p er tu rb a tio n s an d gravity waves are scalar and tensor p e rtu rb atio n s, respectively, due to their transform ation properties. 1.4 T he Cosm ic M icrowave Background D uring th e early universe electrons act as a scattering m edium for prim ordial photons, m eaning th e universe is opaque. As th e universe expands it cools, an d electrons bind w ith protons to form n eu tral hydrogen [70]. Using the Saha ionization equation one can find the tem p eratu re a t which th e fraction of ionized nuclei drops su b stan tially is T ~ 3000K. T here fore once th e tem p e ra tu re of the universe reaches this tem p e ra tu re th e scattering m edium is no longer present, an d th e photos freely escape w ithout being scattered or absorbed. At this point th e universe becomes tran sp aren t, an d th is is known as th e epoch of decoupling. T he existence of this prim ordial rad iatio n was first predicted by Gam ow in th e 1948 [25, 26]. It was discovered serendipitously by Penzias and W ilson in 1965 [71], and the entire spectrum was determ ined to high accuracy by th e C O B E satellite [61]. T he CODE satellite found th a t th e CM B has a near black-body spectrum , w ith a m ean tem p eratu re of 2.725 ± 0.002 [23, 24, 61]. T he tem p eratu re is extrem ely uniform except for a dipole, which is due to our peculiar m otion through the CMB. However, because CM B photons were tightly coupled to m atte r a t the epoch of decoupling we expect some anisotropy in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 CM B. T h e D M R in strum ent aboard the CO BE satellite first m easured th e CM B anisotropy out to an angular scale of 7° on th e sky [85]. 1.4.1 T h e A n g u la r P ow er S p ec tr u m To quantify th e CM B anisotropy we expand th e CM B tem p e ra tu re function on th e sphere, T ( f) , as a function of spherical harm onics [43] oo = E vn—i <^imYemir). E (1.24) £—0 m = —£ T his is analogous to taking th e two dim ensional Fourier transform of a field in ordinary Euclidean space. Here, the m ultipole num bers i and m correspond to sp ectral indexes, and th e m ultipole coefficients, correspond to am plitudes of wave vectors. T he m onopole is sim ply th e average tem p eratu re, and th e dipole (£ = 1) term is on th e order a ~ 3mK [23]. All higher order fluctuations are on th e order of a few hundred of [4, 7, 36]. T he spherical harm onics are discussed in A ppendix B, and exam ples are given in Figure B.2. According to the cosmological principle the anisotropy should be hom ogeneous and isotropic, i.e. th e m ean tem p eratu re should be constant over th e sphere, an d there are no special directions in th e m ean. T his implies th a t th e covariance betw een two points is a function only of the angular separation, and th e variance of th e m ultipole coefficients is independent of m: ~ CiSi^iSirnm' • (1.25) T he Ci values are com m only referred to as the angular power spectrum . For theories w ith isotropic G aussian initial conditions Ci contains th e entire sta tistic a l content of th e theory in th e sense th a t it com pletely determ ines the d istrib u tio n of th e coefficients. Exam ples of the CM B angular power spectrum , Q , are given in Figure 1.1^. Shown are the estim ates of Q from NA SA’s WMAP satellite [4], and from th e MAXIMA-I [56], MAXIMA-II [2], and ARCHEOPS [6 ] balloon-borne experim ents, an d th e b est fit m odel to th e WMAP d a ta [87]. T he shape and am plitude of the angular power spectrum depends on a num ber of cosmological param eters (i.e. th e to ta l density U). Before recom bination the photons and baryons were tightly coupled, oscillating acoustically due to gravity on sub-horizon scales. These oscillations m anifest them selves as a series of peaks and troughs a t different characteristic scales. T he characteristic scales of these fluctuations are frozen ^Note that it is common practice to plot Ct£{£+1)/2TT instead of Ct- This is because the power spectrum is relatively flat, especially at low ^’s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 8000 • WMAP mMAXIMA I ♦ MAMMA 11 6000 2000 2000 10 100 200 400 600 800 1000 1200 Figure 1.1; The CMB angular power spectrum, Ci, as measured by the w m a p (black circles), M A X I M A -I (blue squares), M A X I M A -I I (red diamonds), and A R C H E O P S (green bow ties) experiments. The solid line is the best fit model to the w m a p data points. The horizontal axis is spherical harmonic multipole number, I, which is inversely proportional to angular separation on the sky and the vertical scale is fjK^. The horizontal scale is log/linear to illustrate the relative coverage of the various experiments. in a t decoupling, and w hen projected on to the sky they produce a peak stru c tu re in the angular power spectrum (see Figure 1.1). T he anisotropy on large scales is produced by the Sachs-Wolfe effect [101], which produces a som ew hat flat power spectrum . T his is because these regions were not in causal contact a t the epoch of decoupling. 1.4.2 C o sm o lo g ic a l P a r a m e te r D e p e n d e n c e We identify three m ajor characteristics of the CMB anisotropy sp ectru m which in tu rn depend on various cosmological param eters. These are th e location of th e peaks, th e relative am plitude of fluctuations betw een peaks, and th e overall am plitude of fluctuations. T he fundam ental scale on which the location of th e peaks depends is th e angular size of th e sound horizon a t recom bination. Before decoupling over-densities in th e photon-baryon fluid are in th e linear regim e {6p/p <C 1). T he density field can be decom posed into Fourier m odes which evolve independently. If we freeze th e oscillations a t th e tim e of decoupling rideci each m ode will be in a different stage of oscillation. T he to ta l power will have the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 largest contributions coming from m odes having [40] ^- = where Cg is th e velocity of sound and ^ ‘S (1.26) 3 is th e sound horizon a t decoupling. To determ ine th e size of an object w hen projected on th e sky we use th e angular diam eter distance, Da, defined as: physical size = Da x angular size. For an O = 1 universe it is defined as f!z 1 = im T )L w ith sim ilar expressions existing for open and closed m odels [70]. Therefore, each k m ode corresponds to a n angular size on the sky such th a t k — 1/ {ODf^‘^), where D*® is th e angular diam eter distance to decoupling. T he spherical harm onic m ultipole is approxim ately related to angular separation on th e sky by ^ « n /9 , so p u ttin g all this together gives a series of peaks in th e angular power spectrum w ith m axim a a t mvrDf®® dec T 's T he quantities constant, and (1.28) are b o th functions of the net m atte r density, fim) cosmological and th e to ta l density fl. T his dependence leads to a nearly exact degeneracy am ong these param eters, which determ ine the geom etry of the universe. A specific exam ple of th e Ci dependence on is shown in th e u p p er right panel of Figure 1.2. To und erstan d th is dependence consider E quation 1.28. A n increase in th e to ta l density would cause a decrease in th e angular diam eter distance^ (i.e. objects would ap p ear larger on th e sky). T his in tu rn shift th e acoustic peaks of the power spectrum to larger angular scales. A decrease in would have th e exact opposite effect, i.e. the fluctuations would shift to sm aller angular scales. T he tem p e ra tu re fluctuations for sub-horizon m odes are actually a superposition of two waves. T he first is from a com bination of gravitational and intrinsic tem p e ra tu re variation of th e photon. T he second is from a D oppler shift resulting from th e line of sight velocity of th e photon-baryon fluid. W hen the density of baryons is negligible com pared w ith the density of cold dark m atter, the tem p eratu re fluctuations from b o th gravity and velocity are roughly th e same. However, an increase in th e baryon density causes th e effective m ass of th e oscillator to increase, creating a larger contribution to tem p e ra tu re variation from gravity. For adiab atic fluctuations the result is an increase in th e relative am plitude of the odd num bered peaks [40] (Figure 1.2). ^This is true for fiat, open, and closed models, even though we only show £>« for a fiat $1 = 1 model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 M ost cosmological theories assum e an initial prim ordial power sp ectru m of th e form P{k) oc A;” . C hanging th e spectral tilt of prim ordial fluctuations, n, alters th e overall am plitude of fluctuations and tilts th e angular power spectrum (Figure 1.2). T he intergalactic m edium is ionized out to some redshift. If this redshift is relatively low th en reionization will have a negligible effect on CM B anisotropy. However, if reionization occurred a t sufficiently early tim es th en the num ber of CMB photons observed which did not last sc atter a t decoupling is no longer negligible. Define r as th e optical d e p th to reionization. R eionization sm ooths out tem p e ra tu re fluctuations by a factor of 6 “ ’’ [40]. T he reason is simple: if we have early enough reionization, then a ph o to n th a t comes tow ard us from a p a rticu la r direction need not have originated from th a t direction. In m odels w ith very early reionization (high r ) , the peaks are com pletely washed away (Figure 1.2). T he degeneracy in param eters which determ ine th e geom etry of th e universe and am plitude of fluctuations can be sum m arized as follows. M odels will have alm ost th e same angular power spectrum if they satisfy th e following criteria [2 0 ] ^ • Identical values for th e physical densities and Q.cdm.h'^- • A specific com bination of n and r which results in th e sam e am p litu d e of fluctuations. • Identical values for th e ratio of the angular diam eter distance to th e speed of sound horizon size a t decoupling. 1.4.3 C M B D a ta A n a ly sis B a sics T he prim ary goal of any tem p e ra tu re CMB experim ent is to accurately m easure th e angular power spectrum of tem p e ra tu re fluctuations^. By assum ing an idealized m easurem ent of the CMB tem p e ra tu re anisotropy we can derive some useful relations. According to sta n d a rd inflationary scenarios th e CM B is a G aussian random field. T his m eans th a t specifying the m ean for every point on th e sky f, and th e covariance for every pair of points on th e sky com pletely determ ines th e d istribution. A dditionally, because of hom ogeneity and isotropy th e spectral dom ain th e spherical harm onic coefficients are uncorrelated [98], b u t th e correlation stru c tu re for th e real space tem p e ra tu re fluctuations are more complex. Given a pixelized m ap of the tem p e ra tu re fluctuations, T ( f) , th en from E quations 1.24 and 1.25 the covariance m atrix for T (f) is M rr' = (T (f)T (fO ) (1.29) '‘The criteria presented here axe the same as in [20], except that they did not include r in their analysis. ®We will see later that modern experiments are now attem pting to measure polarization as well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 CDM Varying fl 6000 5000 i 4000 ^ 3000 6000 5000 n=i 4000 3000 2000 1000 500 1000 1500 0 500 1000 1500 I Varying n Varying Qih^ 8000 6000 i 6000 5000 ^ 3000 ^ 2000 n=1.0 4000 n = .9 4000 2000 1000 500 1000 500 1500 I 1000 1500 I Figure 1.2: Examples of how the Ci amplitude changes by varying the parameters fl, n, and The solid line all four panels represents an inflationary cold dark matter model with cosmological parameters: {Ho, fib, flcmb, fiAjUs, r c )= (5 0 , .05, .95, 0,1, 0), while the remaining dashed lines represent models whose pa rameters vary as shown. (1.30) tm i'm ' = Y ,C iY {f)Y {r') (1.31) tm (1.32) where 6 is th e angular separation on th e sky betw een pixels r and r', and in th e last step we used th e spherical harm onic addition theorem (see A ppendix B). If we assum e th a t our m easurem ent is over the entire sky w ith negligible noise, then each spherical harm onic coefficient can be com puted from T (f) by perform ing th e inverse spherical harm onic transform ation atm = J T {f)Y em {r)dn . (1.33) We can w rite down a likelihood in either real space ■^real ~ l^ e x p { 4 r ^ M - ‘r } , 27t|M | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1.34) 16 w here N^ix is th e num ber of pixels, or in spherical harm onic space To estim ate to m axim um likelihood power spectrum Q , we could use either £real or £gp. Since £reai involves a dense m atrix of size iVpix x iVpix, where iVpix ~ 10^ for m odern day experim ents such as m ax im a-i and A^pix ~ 1 0 ® for w m ap, com putationally speaking £sp is m uch sim pler to deal w ith. However, for experim ents w ith p a rtia l sky coverage such as MAXIMA-I and MAXIMA-II we are forced to deal w ith £reai because we cannot com pute the individual spherical harm onic coefficients from E quation 1.33. Proceeding w ith th e likelihood in th e spherical harm onic dom ain, we can w rite down a log likelihood for each spherical harm onic i up to an uninteresting constant as -21n£, = (2«+l)lnC ,+ Y ' (1.36) To o b tain th e m axim um likelihood estim ate for Ci we com pute th e following ^d\nC( __ 2 £ 4-1 ^ /I Q-7^ ra= -i ^ Setting th is equal to zero we obtain ^ 1 m ——i This is our m axim um likelihood estim ato r for the angular power sp ectru m a t a given i. To find the variance in our estim ator we com pute th e following {C i-{C e)f (1-39) = {C f)-{C if (1.40) = (1.41) Var(C^) = 2 ^ -h l ^ So the “error b a rs” on our estim ator for the power spectrum C(, th e square root of the variance, has th e interesting property th a t its proportional to Ci- It is also inversely pro portional to th e square root of £, which m eans we can m ore accurately constrain the power spectrum a t sm aller scales (higher £). T he expression in E q uation 1.41 represents an upper lim it on how well an experim ent can constrain th e power spectrum , and comes from the fact th a t we only have one universe to observe. T his is known as “cosmic variance” . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 Let us relax some of our assum ptions. A realistic experim ent would have some noise in each m easurem ent and sm oothing of th e sky signal due to a finite beam coverage of the detectors. T hen th e d a ta in pixel i can be w ritten as d* = Si + rii, w here Si is th e beam sm oothed CM B tem p eratu re, and rxj is the in stru m en tal noise contribution. For m any cases th e real space beam function can be assum ed to be a spherically sym m etric G aussian, in which case the beam profile, can be w ritten as [13] Bi = e x p |- ^ c r ^ £ ( £ + 1 ) |, (1.42) w here a is F W H M x 0.4247. Again assum ing a full sky m easurem ent th e n th e noise could be expanded in term s of spherical harm onic, rum- Now our d a ta can be w ritte n as d im ~ ^ if^ tm T f^im - To determ ine th e m axim um likelihood estim ator for th e to ta l d a ta power sp ectru m we again calculate th e following _ d h ^ Bj ^ dCi where Ci + § l 2 Q ..x {B]Ci + N i f ^ _ , = ^ i - S etting this equal to zero gives Ci = (1-45) ^i where m = —i If we assum e th a t th e pixel noise is uncorrelated and has th e same RM S value in each pixel, (niTij} = th en th e second term in E quation 1.45, th e noise variance in th e spherical harm onic dom ain, can be w ritten as Ni = (jniJ^} = (^ ^ y 'E E M Y e m im im ir j)* P i (L47) j 47T 2 M . P>^’ 4’piX Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (1.49) 18 Again com puting th e variance in our m axim um likelihood estim ate for th e power spectrum we find V a r(Q ) = 2/ + 1 I + B ] N,p ix . (1.50) T his is equivalent to w hat was found in [49]. It is com m on to express th e pixel noise in term s of detector sensitivity, s. In this case th e pixel noise is cTpix = w here tpix is th e tim e spent observing each pixel. For exam ple, th e sensitivity of th e MAXIMA detectors are roughly s = S O ^ K V ^ [36]. From E quation 1.50 we can see th a t th e expected error in our power spectrum estim ator has two term s, a “cosmic variance” term , and a term th a t contains contributions from instrum ental noise and beam sm oothing. A t large angular scales th e cosmic variance will dom inate, whereas for sm aller scales th e sm all causing the in stru m en tal term to dom inate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. becomes very C hapter 2 B ayesian C osm ological P aram eter E stim ation In C hapter 1 we saw th a t the CM B provides us w ith a unique perspective of th e very early universe, a n d th a t th e statistical properties of th e spatial anisotropy depends on a num ber of fundam ental param eters. universe 1 2 These param eters include th e to ta l density in the , th e physical density of baryons and cold dark m atte r, f2 *h^ a n d the H ubble expansion param eter H q, and the spectral index of initial p e rtu rb a tio n s to nam e a few. Several experim ents have m apped th e tem p eratu re spacial fluctuations a n d used these m aps to place tight constraints on cosmological param eters. In this C h ap ter we discuss th e m ost comm on m ethod for ex tractin g cosmological param eters from CM B d a ta , Bayesian inference, and apply th e technique of param eter extraction to a com bined MAXIMA-I [56] and COBE dmr [32] d a ta set. Some o th er exam ples of Bayesian p aram eter estim ation applied to other CM B d a ta sets are given in [63] ( b o o m e r a n g ), [44] ( b o o m e r a n g & M A X iM A-l), [35] ( d a s i ), and [6 ] ( ARCHEOPS). T he tig h test constraints yet on cosmological param eters are given by [87], which com bined th e WMAP, ACBAR, and CBI d a ta and estim ated ~ 20 param eters to a few percent accuracy. T hey used a Bayesian p aram eter estim ation technique and M onte Carlo M arkov C hains (M CM C) to explore the m ultidim ensional likelihood space. T he rest of this C hapter is organized as follows: In Section 2.1 we discuss in general how to place Bayesian credible region on a set or subset of p aram eters, or on an individ ual p aram eter via th e likelihood function. In Section 2.2 we discuss th e oflFset lognorm al approxim ation, a m ethod of radical com pression of CM B d ata. T his will be a very useful approxim ation, allowing us to compress a CM B d a ta set from th e m ap dom ain w ith Np d a ta points and an Np x Np covariance m atrix, where Np ~ 1 0 ,0 0 0 to th e power sp ectru m dom ain 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 w ith N c i d a ta points and an Nc^ x Ncg covariance m atrix, w here Nc^ ~ 30. T he offset lognorm al approxim ation will also be useful in the frequentist estim ation of cosmological param eters, discussed in C hapter 3 We apply the Bayesian m ethod of param eter estim ation to a com bined th e m a x im a - i high resolution d a ta and COBE DMR d a ta in Section 2.3. In th is section we also discuss th e database of inflationary cosmological m odels a n d likelihood software used to com pute the probability distrib u tio n s of th e individual param eters, and present th e results of the MAXIMA-I analysis. A brief discussion is given in Section 2.4. 2.1 B ayesian Credible R egions Assum ing th e d ata, d, has some probability density function th en given a set of m easure m ents a likelihood function, JC,{ap), can be w ritte n down. T his is sim ply th e p ro d u ct of th e individual probability densities when the d a ta are independent variables [8 , 72]. T he set of p aram eters th a t m aximizes th e likelihood function are ap tly nam ed th e m axim um likelihood param eters, dp. We can use Bayes’ theorem to place credible regions around th e m axim um likelihood param eters. Bayes’ theorem states th a t the probability d istrib u tio n of th e theoretical p a ram eters given a d a ta set is P{ap\d) (X C{ap)P{ap) w here P{ap) is th e prior probability d istrib u tio n for th e param eters. (2.1) Unless otherw ise noted, we take P{ap) to be flat over th e range of param eters considered. A dditionally, E q uation 2.1 allows us to “u p d a te ” P{ap\d) once new inform ation is acquired. T his will be especially useful for p u ttin g jo int constraints on param eters from different astrophysical m easurem ents. A Bayesian credible region is defined as th e region which contains a fraction c of the probability volume C{ap)dap = ^ J P{ap)dap ( 2.2) T he boundaries of the volume V define the credible region in th e entire m ultidim ensional param eter space. In m any cases we are interested in a credible region for one, or a t m ost two param eters. To o b tain a credible region for a specific p aram eter, a*,, we m arginalize over the uninteresting param eters J^ J T he range to C{ap)ddp = c J C{ap)dap defines the credible region a t confidence c for th e p a ram eter a^. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (2.3) 21 2.2 T he Offset-lognorm al A pproxim ation S tan d ard inflationary theories unam biguously predict th a t th e CM B fluctuations obey G aus sian statistics. Therefore, given a m easurem ent which consists of a pixelized signal plus noise m ap, d = s + n, w ith Np pixels, and a noise correlation m atrix, J\f = (n n ^ ), th e likelihood is C{ap) = ------ — -—} = exp I — ^ d ^ (5 (a „ ) + A/')~^d, 1 (27r)^p/V I*5(«p)+A T| 7 (2.4) w here th e signal covariance m atrix, S{ap) — {ss^), is a function of cosmological param eters an d given by E quation 1.32. We have explicitly assum ed th a t th e signal and noise are uncorrelated. T he m ost straightforw ard m ethod for ex tractin g cosmological p aram eters dp from a pixelized CM B m ap and associated noise covariance m atrix would be to m axim ize E quation 2.4. T he d a ta provides d and W , and sta n d a rd software tools such as CM BFast^ [80] could be used to form S{ap). However, this m ethod is extrem ely com putationally expensive. C om puting the likelihood for one m odel involves inverting a m atrix th a t has 15,000 X 15,000 elem ents for th e MAXIMA-I 5' d a ta , and 23,000 x 23,000 for th e MAXIMA-I 3' d ata. We wish to m axim ize the likelihood as a function of 7 cosmological param eters, or ~ 6 X 10® m odels. C learly com puting E quation 2.4 6 x 10® tim es is a challenging task. A less com putationally expensive m ethod for param eter estim ation lies in th e notion of d a ta compression. If we could reduce the num ber of d a ta points from 10^ m easurem ents down to 10^ we would drastically reduce the com putational complexity. T his is done via th e angular power spectrum , or Cf, defined in E quation 1.25. From th e m aps th e m axim um likelihood Ci are estim ated, and these Q are th en used to com pare w ith theoretical models. For exam ple, estim ating Q for a full sky m easurem ent of the CM B anisotropy is given in E quation 1.38, and for b o th tem p eratu re and polarization power sp e ctra is discnssed at length in C h ap ter 6 . T he question th en lies in how to com pare Q w ith theoretical predictions. T h e sim plest th ing to do would be to w rite down a G aussian likelihood as a function of Q . However, Ci is clearly non-G aussian d istributed. To be specific, from E q u atio n 1.50 we see th a t Ci has the p roperty th a t its error is proportional to its value, which is uncharacteristic of a G aussian d istrib u ted variable. T he solution is to find a variable which is a function of Ci whose d istrib u tio n is more G aussian th a n th a t of C/. T he following argum ents follow th a t of [10]. Suppose we had a full sky m easurem ent of the CM B anisotropy which is a com bination of signal and noise. i www.cmbfast.org Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 22 and we neglect beam sm oothing in th e following argum ents for sim plicity. spherical harm onic could be com puted individually: dim = T hen, each w here aim riim are the contributions from CM B and noise, respectively. We can assum e these variables are G aussian d istrib u ted w ith {aimat'm') = Cidui and {nimni'm') — N i S w ^ . O ur log likelihood (L = - 2 l n £ ) would be: L = \dn ( 2 £ + l ) l n ( C + iV )+ (2.5) w here we have suppressed th e I index because each Q can be estim ated individually (i.e. we can w rite down th e above equation for each Cr). O f course we can w rite down th e ML estim ator, C = ^ • A nd th e curvature ab o u t th e ML estim ate is dC^ c = c = (2 -6 ) {c + AT)2 which im plies th a t the “G aussian error” on th e power (the inverse square root of th e curva ture) is proportional th e power itself. If we change bases to a new variable, Z — ln((7 -f iV), th en Z = ln (C + N ) , and th e curvature in our new variable is 9 2 - In £ dC^ 2i + \ z=z /(c +n ) (C+ iV)2 V = ) 2i+ l. ( 2 .8 ) (2.9) Thus, our new variable Zi has th e desirable p roperty th a t th e curvature (and th u s th e error) does not depend on its own value, as was tru e for Q . It was also shown [10] th a t assum ing Z i is G aussian d istrib u ted is a very good approxim ation to the entire likelihood. In practice th e q uantity is a com bination of pixel noise and beam properties, a n d an explicit recipe for its calculation is given in [1 0 ]. By assum ing th a t Zi is G aussian d istrib u ted , we can compress all of th e inform ation of th e d a ta into a log likelihood (up to an irrelevant additive factor) of th e form: L{ap) = '^ ^ { Z g - Z g ) M g g ,{Z g l ~ Zgi) (2.10) BB' where Z g = is th e offset lognorm al curvature m atrix, and the sum is over band powers^. \n { C g + N g ) is th e d a ta , and Z g = \n{Cg{ap) + N , is the theoretical offset lognorm al ^It was shown in Chapter 1 that the CMB is expected to be Gaussian and isotropic, and Gaussian white noise would also be isotropic on the sphere ^Experiments cannot determine each Ci separately due to incomplete sky coverage. Rather, they deter mine the average power in a given band whose width is typically A l ~ 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 quantity, which is dependent on cosmological param eters Op. We know have an explicit recipe for estim ating the m axim um likelihood cosmological p aram eters given a CM B m ea and experim ental factor N b com pute surem ent. F irst, from th e m easured ban d powers Z g . T hen, find th e set of cosmological param eters which m axim izes E q u atio n 2.10. 2.3 T he M A X IM A -I A nalysis In our analysis we use the MAXIMA-I high resolution d a ta described in [56], and th e cobe DMR d a ta of [32]. At th e tim e of this analysis, th e MAXIMA-I power sp ectru m gave th e largest angular scale coverage (35 < i < 1235), and highest sensitivity of any CM B experim ent [36, 56]. Because th e m a x im a - i m ap only covers ~ 120 deg^ on th e sky, we include the COBE DMR d a ta to o b tain large scale inform ation. Given th e com bined m a x im a - i and cobe dm r d a ta set, we w ish to find th e set of cosmo logical p aram eters th a t m axim izes our offset lognorm al likelihood, an d generate confidence regions for these param eters using E quation 2.2. T he problem is essentially minimizing^ E quation 2.10 as a function seven param eters. Since we have no functional dependence in th e theory we cannot com pute the first derivative of the function w ith respect to the cosmological param eters, which im m ediately rules out a num ber of sta n d a rd m inim ization techniques. Also, grid search m ethods which only require function evaluations are not ap plicable due to num erous local m inim a, and by the fact th a t we w ish to com pletely m ap the likelihood space. For this reason we a d a p t the technique of sim ply com puting a very large d atab ase of theoretical m odels, and calculating the likelihood for every p aram eter grid point in our seven dim ensional volume. A flat prior w ithin th e p aram eter ranges of th e database is assum ed. T hen, to ob tain probability distrib u tio n s on individual p aram eters, or subset of param eters, we sim ply m arginalize over th e rem aining “uninteresting” p aram eters. T he results of th e Bayesian cosmological param eter analysis from th e MAXIMA-I high resolution d a ta are published in [89], which are now discussed in the subsequent Sections. 2.3.1 Param eter Space To com pute th e d atabase we used th e publicly available software CMBFast® [80, 96]. T he set of cosmological param eters we chose to vary are {fl, O a, n, r , Cio}. These are the param eters m ost sensitive to changes in th e angular power spectrum . We let the ^The log likelihood h is L = —2 In C, where C is the full likelihood. Thus, minimizing the log likelihood is equivalent to maximizing the full likelihood. ®http://www.cm bfast.org Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 param eters take th e following values: • CiQ is continuous. • T - 0, .025, .05, .075, .1, .15, .2, .3, .5 • ^cdmh^ = 0.03,0.06,0.12,0.17,0.22,0.27,0.33,0.40,0.55,0.8 • flA = 0 ,.1 ,.2 ,.3 ,.4 ,.5 ,.6 ,.7 ,.8 ,.9 ,1 • = 0 .3 ,0 .5 ,0 .6 ,0 .7 ,0 .7 5 ,..., 1.2,1.3,1.5 = 0.00325,0.00625,0.01,0.015,0.02,0.0225,0.025,0.0275,0.03,0.0325,0.035, • 0.0375,0.04,0.045,0.05,0.075,0.1 • n = 0.6,0.7,0.75,0.075,0.8,0.085,0.0875,..., 1.2,1.25,..., 1.5 T he range and values chosen for th e param eters was sim ilar to those of [54]. We increased th e resolution of th e grid in areas of expected high likelihood for th e param eters fl, n, and and kept th e griding th e same for param eters which we were less sensitive to. These were t, O a , and flcdmh^- T he num ber of m odels in each param eter dim ension is {r, = {9, 10, 11, 16,17, 24} = 6,462,720 cosmological inflationary m odels. T h e m ost recent version of CM B Fast im plem ents a k-splitting m ethod which decreases com pute tim e by an order of m agnitude [96]. This, along w ith the use of m ultiple processors on an IBM SP m achine, kept th e com pute tim e of the entire d atabase w ithin reason (a couple of days). Also, only ~ 49 ^ points were stored per model. T his is because C M B Fast only com putes th e actu al Q power for a specified num ber of points, and cubic spline interpolates to obtain th e power in th e range 2 < (. < 1500. A software w rapper was w ritte n th a t th e com putes C M B Fast for each grid point in the database, and stores only those Q values th a t were actually com puted by the B oltzm ann code. T he kept th e to ta l size of th e d atab ase under 1.5 GB. Software for com puting the entire seven dim ensional likelihood volum e of th e entire database was w ritten in C. T his was parallelized using M PI to d istrib u te th e likelihood volume up am ongst a num ber of processors. It was designed to ru n in parallel for either two, five, nine or ten processors. If two, five or ten processors were requested th en the ^cdmh^ values would be d istrib u ted to th e various processors, otherw ise th e t values were distrib u ted . T he likelihood code would read in a m odel from th e d atabase, cubic spline interpolate to generate th e values for the entire range 2 < i < 1235, and would si m ultaneous m inim ize the log likelihood as a function of the norm alization (Cio) and the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 M A X IM A - I DMR best fit 6000 \ 4000 2000 0 200 400 600 80 0 1000 1200 Figure 2.1: The m a x i m a - i high resolution angular power spectrum plotted with two models. The solid blue line is the best fit model to the entire database, while the dashed red line is the best fit with the BBN prior, ilbh^ = 0.02 calibration uncertainty.® 2.3.2 Results We com bined the d a ta from th e MAXIMA-I and COBE DMR experim ents, an d com puted likelihoods for th e m odels using th e database and software described in th e previous section. C alibration uncertainty and uncertainty in th e beam shape of th e detectors were included in th e calculation of th e likelihood. We also restricted our analysis to th e cosmologically interesting regim e of 0.4 < h < 0.9, 0.m > 0 .1 , and age of th e universe > 10 Gyr. T he best fit cosmological param eters to th e d a ta are (Sib, S lcd m , SIa, t , n, h)= (0.07, 0.68, 0.1, 0.0, 1.025, 0.63). These are characterized by the high m atte r density an d low vacuum energy content. T his best fit m odel is plo tted w ith th e d a ta in Figure 2.1. A dditionally, one dim ensional distrib u tio n s were determ ined for each p aram eter in th e d atabase, an d nu m erous o ther cosmological param eters which are com binations or functions of th e database param eters. T he results are given in Figure 2.2, where we show th e entire one dim ensional probability distributions, and Table 2.1, where the 6 8 % and 95% confidence regions are ®Minimizing the log likelihood as a function of the normalization and calibration is discussed at length in Chapter 3 in the context of frequentist parameter estimation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Table 2.1: Parameters Estimates from the MAXIMA-I parameter symbol ML estimate total density physical baryonic density physical CDM density® scalar spectral index optical depth to reionization®*’ power spectrum normalization [/xK^] cosmological constant® m atter density® age of the universe [Gyr] ratio of 3'‘‘*peak to 2"®*peak n 0.90 0.033 0.17 1.025 0.05 690 0.65 0.32 8.70 Dcdmh n T Cio Da Dm to n 0 .6 8 and 6 8 CO BE DM R data % region [0.80,1.00] [0.024,0.036] [0.16,0.28] [0.960,1.120] < 0.13 [610,820] [7.12,12.50] [0.60,0.77] 95% region [0.74,1.08] [0.020,0.046] [0.10,0.33] [0.910,1.250] < 0.28 [565,890] [0.49,0.80] [0.21,0.46] [ 6.120 , 15.50] [0.55,0.95] The maximum likelihood estimates with 6 8 % and 95% Bayesian credible regions for cosmological parameters estimated using the high resolution m a x i m a - I data [56] and c o b e d m r data [32]. Units axe given when the parameter is not dimensionless. Note that the top seven parameters are the parameters actually varied over in the database, while the bottom three are either combinations of the other parameters (Qm and to) or a function of the anisotropy spectrum (7^). a These estim ates are dominated by the priors. b Estimate with an n = 1 prior. c Constraint from combining CMB and supernovae lA data. Only 95% confidence regions were computed. given for a num ber of param eters. Note th a t th e m axim um likelihood param eter values are not necessarily equal to those given by the b est fit model. T his would be the case if th e likelihood function were G aussian w ith respect to th e fitted p aram eters, which is not th e case. Also note th e considerable Because of degeneracies th a t exist betw een various param eters, we use stronger priors in order to place constraints which are sm all com pared to th e d atabase prior. For exam ple, th e d istrib u tio n for r is assum ing a n = 1 prior. Also, th e d istrib u tio n was found to be highly dependent on th e h, to priors. D ue to a degeneracy in th e Dm — Da plane, there were m odels which fit b o th th e MAXIMA-I, COBE DMR, and supernovae constraints to high likelihood (see Figure 2.3). T he ML estim ates for Dm and D a given Table 2.1 use a Supernovae prior, and are therefore considerably different from the values given by th e best fit m odel. Those values are characterized by a high m a tte r content and low Da= 0.0325 ± We calculated th e 95% credible regions for the following param eters: 0.0125, Dcdm^^ = 0.17])]Q gy, D = 0.9])]g’Jg. T he likelihood functions for these param eters are shown in figure 5. T h e d a ta was consistent w ith a flat universe (D = 1) a t 6 8 % confidence, and consistent w ith th e physical baryonic density determ ined from light elem ent abundance (D*h^ = 0.02) a t 95% c.r. We found th e d a ta was consistent w ith BBN in th a t there were Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 27 0.8 I 0,6 V 0.2 0 .2 0 .4 0 .6 0 .8 1 . 0 1 . 2 1 . 4 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .0 0 .2 0 .4 0 .6 0 .6 20 30 ^0 [G y r] 0 .0 0 .8 1 .0 1.2 1 .4 1.6 j 0.6 0.2 0.0 0 .0 0.1 0 .2 0 .3 0 .4 0 200 400 600 8001000 0 2 C,o[mK] 10 0 .5 1.0 1.5 Figure 2.2: Marginalized one dimensional likelihoods for cosmological parameters as determined by the combined m a x i m a - i and c o b e d m r data. The circles represent normalized likelihood values in the grid points and the solid blue lines axe obtained by cubic spline interpolating between points. m odels which fit th e com bined of m a x im a - i, c o b e d m r datasets well assum ing a BBN prior = 0.02 (see Figure 2.1). We also found a strong degeneracy betw een th e optical d e p th to th e last scattering surface, t and th e prim ordial power spectrum index, n (Figure 2.4). T he degeneracy allowed us to derive a com bined constraint: n ~ 0 . 46 t -I- (0.99 ± 0.14) (95% c.r.), for t < 0.5. Independent of th e value of the optical depth, we found a 95% credible region for the spectral index of n = 1 . 0 7 5 • A ssum ing no reionization (r = 0), we got n = 0.99 ± 0 .1 4 for th e 95% c.r. A lternatively, fixing n a t unity, gives us an upper lim it on th e optical d ep th of T < 0.26, a t 95% c.r. T he MAXIMA-I d a ta was characterized by an increase in power from th e region 410 < I < 785 to 786 < I < 925, and subsequent decrease in power in th e range 926 < I < 1235. T his was suggestive of th e existence of a th ird peak in th e angular power spectrum , which was previously undetected. We defined the quan tity TZ = Ci I C j , w here C f and C} are the power in I bands 410 — 785 and 786 — 925 respectively. A likelihood and 95% c.r. of 77. = 0.68 ^q ' ^ 3 were obtained by m arginalizing over the rem aining param eters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 2.4 D iscussion We have developed an entire Bayesian param eter estim ation m ethod for ex tractin g p ro b a bility distrib u tio n s of cosmological param eters given a m easurem ent of th e CM B anisotropy spectrum . T his m ethod also provides a sim ple m ethod for inclusion of ad d itio n al infor m ation, such as confidence regions from SNIa m easurem ents. T h e p aram eter estim ation software consists of a large m ulti-dim ensional d atabase of theoretical inflationary models, and parallel likelihood code th a t com putes th e entire likelihood volum e for a given d a ta set. T his code will also m arginalize over uninteresting param eters to o b tain probability distrib u tio n s for single param eters or probability contours for two param eters. W ithin th e chosen fam ily of inflationary ad iabatic m odels we placed stringent constraints on various cosmological param eters. T he results confirm ed the near flatness of th e universe and, w hen com bined w ith d a ta from supernova, indicated a non-zero vacuum energy density. T h e d a ta also indicated a rise in power in th e region 786 < I < 925, which is consistent w ith th e location of a th ird peak for m odels w ith high likelihood. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 r a M AXIM A-1 + COBE Figure 2.3: Constraints in the ilmplane from the combined m a x i m a - i and c o b e d m r data. The shown contours correspond to 68, 95, and 99% likelihood ration confidence levels. The bounds obtained form high redshift supernovae data are also overlaid as well as the joint confidence levels. 03 0 . 2 0.8 1.0 1.2 s p e c t r a l in d e x [n^] 1.4 Figure 2.4: A spectral index - optical depth contour plot obtained for the m a x i m a - i combined c o b e d m r data. Shaded contours show 68, , 95 & 99% likelihood-ratio confidence levels. Lines show an approximate relation between n and r fitted to this result. Central values are shown by the solid line and 95% bounds by the dashed lines. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 3 Frequentist C osm ological Param eter E stim ation 3.1 Introduction M ost CM B analyses to date have used a Bayesian statistical approach to estim ate the values of th e cosmological param eters. T he results of these analyses can have considerable dependence on th e priors assum ed, (e.g., [12, 44, 54]), and it is therefore instructive to a tte m p t an estim ate of th e cosmological param eters th a t is independent of priors. Bayesian a n d frequentist m ethods for setting lim its on param eters involve quite different fundam ental assum ptions. In th e Bayesian approach one a tte m p ts to determ ine th e proba bility d istrib u tio n of th e param eters given th e observed d ata. A Bayesian credible region for a p aram eter is a range of p aram eter values th a t encloses a fixed am ount of th is probability. In th e frequentist approach, on th e other hand, one com putes the probability d istrib u tio n of th e d a ta as a function of th e param eters. A p aram eter value is ruled out if th e probability of g etting th e observed d a ta given th is param eter is low. Because th e questions asked in the two approaches are quite different, there is no guarantee th a t uncertainty intervals obtained by th e two m ethods will coincide. Frequentist analyses quantify th e probability d istrib u tio n of th e d a ta in term s of a sta tis tic th a t quantifies th e goodness-of-fit of a m odel to th e d ata. estim ator is probably th e m ost widely used statistic. T he m axim um -likelihood W hen th e d a ta are G aussian- d istrib u ted and th e m odel depends linearly on th e p aram eters, th e sta tistic is d istrib u ted and sta n d a rd x^ tables are used to determ ine confidence intervals [72]. It has becom e com m on to com pare CMB d a ta to theoretical predictions via th e angular power spectrum , which depends on a num ber of cosmological p aram eters. T he d a ta points 30 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 31 are usually th e m ost likely levels of tem p e ra tu re fluctuation power w ithin certain bands of spherical harm onic m ultipoles. However, the b an d powers Q are not G aussian dis trib u te d [1 0 ], and th e theoretical angular power spectrum does not depend linearly on the cosmological p aram eters. T hus a sta tistic m ay not be x^-d istrib u ted . F urtherm ore, the com plicated probability d istrib u tio n of th e d a ta points and th e dependence of th e theoretical predictions on th e p aram eter values m ake th e analytic calculation of th e probability d istri b u tio n of x^ impossible. T hus, there is no guidance on how to set frequentist confidence intervals. A frequentist analysis was previously used to assess th e probability of a sta n d a rd CDM cosmological m odel given th e d a ta from th e UCSB S outh Pole and [32, 8 8 cobe dm r experim ents ]. M ore recently a frequentist approach was used to determ ine confidence intervals on several cosmological param eters [34, 67]. These recent analyses (im plicitly) assum e th a t th e band power Ci is G aussian d istrib u ted and th a t th e cosmological m odel is linear in th e cosmological param eters, and thus th a t sta n d a rd x^ values can be used to set confi dence intervals on various cosmological param eters. These analyses also do n ot account for correlations betw een betw een band powers. It was argued in [28] th a t a frequentist analy sis is b e tte r suited th a n Bayesian for answering th e question of how consistent p aram eter estim ates from CM B d a ta are w ith estim ates from other astrophysical m easurem ents. A m ethod for estim ating th e angular power spectrum which uses frequentist considerations was presented in [38]. In this C h ap ter we present a m ore rigorous approach to frequentist p aram eter estim a tion from CM B d a ta th a n previous analyses. We use th e d a ta from cobe dm r [32] and MAXIMA-I experim ents [36] and sim ulations to determ ine th e probability d istrib u tio n of an app ro p riate A x^ statistic, and use th is d istrib u tio n to set frequentist confidence intervals on several cosmological param eters. We com pare th e frequentist confidence intervals to Bayesian credible regions obtained using th e sam e d a ta and to th e likelihood-m axim ization results of [3]. T he stru c tu re of th is C hapter is as follows: in Section 3.2 we discuss th e MAXIMA-I and COBE DMR d a ta and th e d atab ase of cosmological m odels used in our analysis. In Section 3.3 we present th e x^ sta tistic used in our analysis. Section 3.4 describes th e process of settin g frequentist and Bayesian confidence regions on cosmological param eters. T h e results an d a discussion are given in Sections 3.5 and 3.6. T his C h ap ter is based on work published in [!]• Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 32 3.2 D ata and D atabase of cosm ological m odels We use th e angular power spectrum com puted from th e 5' m a x im a - i CM B tem p e ra tu re anisotropy m ap [36] an d th e 4-year COBE DMR angular power spectrum [32]. T he MAXIMA-I a n d COBE DMR power sp ectra have 10 and 28 d a ta points in th e range 36 < £ < 785 and 2 < £ < 35, respectively. More inform ation ab o u t th e m a x im a experim ent and d a ta is provided in [36, 55]. It was shown in [79, 104] th a t th e tem p e ra tu re fluctuations in the MAXIMA-I m ap. are consistent w ith a G aussian d istribution. T h e MAXIMA-I analysis was extended to sm aller angular scales in [56], b u t these d a ta are not used in th is analysis. To perform our analysis we constructed a database of 330,000 inflationary cosmological m odels [80] th a t has th e following cosmological p aram eter ranges an d resolutions: • T = 0 .0 ,0 .1 ,0 .2 ,0 .3 ,0 .4 ,0 .5 • Q b = 0.005,0.01,0.02,0.03,0.04,0.05,0.075,0.10,0.15 • Qm = 0 .0 5 ,0 .1 ,0 .1 5 ,0 .2 ,0 .2 5 ,0 .3 ,0 .3 5 ,0 .4 ,0 .5 ,0 .6 ,0 .7 , 0.8,0.9,1.0 • Oa = 0 .0 ,0 .1 ,0 .2 ,0 .3 ,0 .4 ,0 .5 ,0 .6 ,0 .7 ,0 .8 ,0 .9 ,1 .0 • Ho ^ 4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 0 • n = 0 .6 ,0 .7 ,0 .8 ,0 .9 ,1 .0 ,1 .1 ,1 .2 ,1 .3 ,1 .4 ,1 .5 , and Cio, th e m odel norm alization, is continuous. N ote th e difference in choice of param eters and num ber of m odels in this database com pared w ith th e datab ase used in th e Bayesian analysis from C h ap ter 2. T his is because th e analyses were interested in investigating the p aram eter dependence of different cosmological p aram eters, and th e k -splitting m eth od of C M B Fast, which provides an order of m agnitude increase in the form ing of a d atabase, was not yet available a t th e tim e of this analysis 3.3 T he S tatistic To set frequentist confidence intervals we choose th e m axim nm -likelihood estim ato r goodness-of-fit statistic. We use th e 3 3,8 a S defined in equation (39) of [10] (hereinafter B JK ) (3.1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 0 .050 0 .040 0 .030 oT 0.020 0.010 0.000 0 50 150 1 0 0 Figure 3.1: Distribution of values computed for the 1 0 , 0 0 0 simulations by solving for Af and u ana lytically using the method described in the text (dashed line), and numerically using Brent’s root-finding algorithm (solid line). Both histograms have a bin size of one and are normalized to integrate to one. = ln{AfCf + Z f = \n{u Cf + Xi) (3.2) Xi). (3.3) T h e sum in equation (3.1) is over the COBE DMR and MAXIMA-I bands. T he d a ta and theory b a n d powers are denoted as C f and C\, respectively, Mjj is th e inverse covariance m atrix for th e Z f quantities, and N is th e norm alization of th e m odels to th e d ata. T he Xi param eters are th e so called “offset-lognorm al variables” are depended on properties of th e experim ental d a ta (i.e. beam size, pixel noise), and are defined in [10]. T he variable u accounts for th e calibration uncertainty of th e MAXIMA-I d a ta , which is 8 % in th e power spectrum [36], i.e. (t„ = 0.08. For th e COBE DMR bands u is defined to be one. Each tim e a is calculated we solve for the norm alization Af and calibration factor u th a t sim ultaneously m inim ize x^- Because we did not find a closed-form analytical solution to th e m inim ization of equation ( 3 . 1 ) w ith respect to Af and u, and a num erical m inim ization would have been com putationally prohibitive, we used th e following approxim ation. We assum e th a t equation (3.1) is well approxim ated by Y ^ i u C f - AfCt)F^,{uCf - AfCj) + {u - W here th e Fisher m atrix Fj,- for th e C f quantities, is related to 1 )^ 2 (3.4) by Mij = F i j i C f + X i ) i C f + x j ) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.5) 34 0.14 0.12 0.10 Qh 0.06 0.04 0.02 0.00 0 5 10 20 15 25 30 35 Ax' Figure 3.2: The histogram gives the Ax^ distribution for the entire six-dimensional parameter space from 10,000 simulations of the C O B E D M R and M A X i M A - i band powers. The dashed curve is the standard distribution for six degrees of freedom. The vertical solid line (vertical dashed line) is Ax^ = 16.5 (12.8), which corresponds to the 95% Ax^ threshold for the histogram (standard x^ distribution). The histogram has a bin size of 0.5, and is normalized to integrate to one. M inim ization of equation (3.4) w ith respect to J\f and u gives two coupled equations which we solve for u by assum ing th a t M — 1. We th en use th a t value of u to solve for M . We com pared this approxim ate solution to a rigorous num erical m inim ization of equation (3.1) for 10,000 cases a n d found an RMS fractional error of less th a n 1.5% (see Figure 3.1). Once th e factors N and u have been determ ined using equation (3.4), th e exact equation (3.1) is used to find th e value of • 3.4 D eterm ining Confidence Levels 3.4.1 Prequentlst Confidence Intervals Let a denote a vector of param eters in our six-dim ensional param eter space, a n d atrue be th e unknow n tru e values of the cosmological param eters th a t we are try in g to estim ate. By m inim izing we find th a t th e best-fitting m odel to th e M AXiM A-i a n d cobe dmr d a ta h a s th e fo llo w in g p a r a m e te r s ag: (Hq, O b , Om, O a , n, r ) = (60, .075, .7, .2 , 1 , 0). T his m odel gives a, — 36, which is an excellent fit to 38 d a ta points. We define A x ^(a) = X^(a) - X^(ao), (3.6) where th e first term on th e right hand side is a x^ of th e d a ta w ith a m odel in th e database, an d the second is a x^ of th e d a ta w ith th e best-fitting model. To quantify th e probability Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 n 0.05 0.04 0.03 <I 0.02 0.01 0.00 0 1 2 3 4 5 0 1 2 3 4 2 5 3 4 5 6 4 5 6 AX^ A/ A/ T 0.05 0.04 X 0.03 <1 0.02 0.01 0.00 2 3 AX^ 4 5 2 3 A/ 4 5 0 1 2 3 Ax' Figure 3.3: Simulated one-dimensional Ax^ distributions for all the parameters in the database. The vertical dotted lines correspond to the 95% Ax^ threshold level; numerical values are given in Table 1. Each histogram has a bin size of 0.05 and is normalized to integrate to one. The 95% threshold for a standard x^ distribution with one degree of freedom is x° = 3.8. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 (H ojH e ) 0.025 0.020 cv X 0.015 <3 0 .0 1 0 0.005 0.000 0 2 4 6 8 0 2 4 6 8 10 Ax Figure 3.4: Simulated two-dimensional distributions in the {H o,CIb ) and planes. The vertical dotted lines correspond to the 95% Ax^ cutoff level, which are 6.25 and 7.70 for { H o , Q b ) and (ffm, Ha) respectively. Each histogram has a bin size of 0.05 and is normalized to integrate to one. The 95% threshold for a standard x^ distribution with two degrees of freedom is x^ = 6. d istrib u tio n of th e d a ta as a function of th e param eters we choose a threshold 0 , and define TZ to be th e region in p aram eter space such th a t < ©. 7^ is a confidence region a t level a if th ere is a probability a th a t TZ contains th e tru e cosmological param eters atme- In other words, if m any vectors agy) and regions TZj are generated by rep eatin g th e experim ent m any tim es, a fraction a of th e ensem ble of TZj would contain atrue- Since th e m ay not be statistic d istrib u ted , we use sim ulations to determ ine its probability d istrib u tio n as a function of th e cosmological param eters. T he sim ulations m im ic 10,000 independent observations of th e CM B by th e MAXIMAI and C O B E experim ents. T he CM B is assum ed to be characterized by the MAXIMA-I an d C O BE b est estim ate for the cosmological param eters, Uq. A pplying th e equivalent of equation (3.6) [see equations (3.7) and (3.8)] for each of th e sim ulations gives a set of 10,000 values A x j (i = 1, .■•, 10^), and by histogram m ing these values we associate threshold levels 0 w ith probabilities a. T his relation betw een 0 and a is applied to th e d istrib u tio n of A x^ th a t is calculated using equation (3.6) to determ ine a frequentist confidence interval TZ on the cosmological param eters. Note th a t our procedure assum es th a t th e probability d istrib u tio n of A x^ around Uq closely mimics th e probability d istrib u tio n of A x^ around atrue- T his is a sta n d a rd assum ption in frequentist analyses [72]. T he alternative approach Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 6 4 'X < 2 0 40 50 60 70 80 0 .6 O.B 1.0 1.2 1.4 0 .0 0 0 .0 0.1 0 .2 0.3 0,4 0 .0 0 .2 0.4 0.6 0 .8 0 .0 0 .0 5 0 .2 0.4 0 .1 5 0.10 0 .6 0 .8 1 Figure 3.5: Ax^ calculated with the m a x i m a - i and c o b e d m r data as a function of parameter value for each of the parameters in the database. Solid circles show grid points in parameter space, and the solid lines were obtained by interpolating between grid points. The parameter values where the solid line intercepts the dashed (dotted) line corresponds to the 68% (95%) frequentist confidence region. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 o f determ ining th e probability d istrib u tio n around each grid point in p aram eter space is com putationally prohibitive. Because finding th e best-fitting ban d powers from a tim e stream or even a sky m ap is com putationally expensive [11], we perform 10,000 sim ulations of th e quan tities Zf^ which are related to th e ban d powers Q as defined in equation (3.3). We assum e th a t th e Z f are G aussian-distributed [10] and we discuss and ju stify th is assum ption in A ppendix A. T he q u antities w here j denotes one of th e 10,000 sim ulations, are draw n from two m ultivariate G aussian distrib u tio n s th a t represent th e MAXIMA-l (10 d a ta points) and GO BE (28 d a ta points) b an d powers. T he m eans of th e distrib u tio n s are th e Z j quantities as determ ined by ao, and the covariances are taken from th e data. Each of th e is thus a vector w ith 38 elem ents representing an independent observation of a universe w ith a set of cosmological p aram eters ao . We include uncertainty in th e calibration an d beam -size of the MAXIMA-I experim ent by m ultiplying th e MAXIMA band-pow ers by two G aussian random variables. T he calibration random variable has a m ean of one and sta n d a rd deviation of 0.08, and the beam -size random variable has a m ean of one an d a variance th a t is ^-dependent [36, 104]. For each sim ulation j the entire database of cosmological m odels is searched for th e vector of p aram eters ao(y) which m inim izes And we calculate ^ X j - = Xj(ao) - Xj(ao{j))- (3-7) T he first a n d second term s on th e right hand side are th e x^ of sim ulation j w ith th e m odel ao, and of sim ulation j w ith its best-fitting m odel, respectively. A norm alized histogram of A x j for all 10,000 sim ulations is shown in Figure 3.2 an d gives th e p robability d istrib u tio n of A x^ over th e six-dim ensional param eter space. T he 95% threshold is A x^ == 16.5, th a t is, 95% of th e probability is contained in th e range 0 < A x^ < 16.5. Figure 3.2 also shows a sta n d a rd x ^ ~ d istrib u tio n w ith six degrees of freedom and its associated 95% threshold level. T he difference betw een th e results of the sim ulations and th e sta n d a rd x^ d istrib u tio n is a ttrib u te d to th e non-linear dependence of th e m odels on the p aram eters a n d m inim izing X^ over J\f an d u. G ontour levels in th e six-dim ensional p aram eter space th a t are provided by different thresholds of th e d istrib u tio n of A x | cannot be used to set confidence intervals on any individual param eter. To find a confidence interval for a single p aram eter p we com pute the probability d istrib u tio n ^X^p) the following way. We search th e d ata b a se for th e m odel th a t m inim izes th e x^ w ith sim ulation j under th e condition th a t p is fixed a t its value in ao, and for th e m odel th a t m inim izes the x^ w ith sim ulation j w ith no restrictions on the param eters. We com pute Ax(p) = x|(a(p)) - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3-8) 39 c 0.8 1.0 Figure 3.6; Two-dimensional frequentist confidence regions in the (flnuflA) plane. The dark to light regions correspond to the 68%, 95%, and 99% confidence regions respectively. The dashed line corresponds to a flat universe, O = flm -f = 1where a(p) is th e vector of param eters th a t m inim ize subject to th e co n strain t th a t p is fixed. A histogram of ^ x f p ) provides th e necessary distrib u tio n . T h e one-dim ensional d istributions for all six param eters in the database are shown in Figure 3.3. G eneraliza tion of th is process for finding the probability d istrib u tio n for any subset of param eters is straightforw ard. T he two-dim ensional distrib u tio n s in th e {H o, Q b ) and (Qud^^a) planes are shown in Figure 3.4 and th e corresponding 95% thresholds are = 6.25 and A x^ = 7.70, respectively. Using th e sim ulated one- and two-dim ensional probability d istrib u tio n s of A x^ we set 68% and 95% threshold levels on the d istrib u tio n of A x^ th a t are calculated using th e d a ta and th e d ata b a se of m odels, i.e. the one calculated from equation (3.6), an d we determ ine corresponding confidence intervals on th e cosmological param eters. Figures 3.5, 3.6, and 3.7 give th e association betw een A x^ and cosmological p aram eter values for each of the p aram eters in the d atab ase and in the (H q, Ob) and (Gm, Ga) planes. 3 .4 .2 B a y esia n C red ib le R eg io n s A detailed description of placing Bayesian confidence intervals in th e context of CM B d a ta analysis is given in C h ap ter 2. Here we give a brief review. According to Bayes’s theorem th e probability of a m odel given the d ata, the posterior probability, is pro p o rtio n al to the p ro d u ct of th e likelihood C (a) = e xp( — a prior probability d istrib u tio n of the param eters. If th e prior is constant, as we shall assum e, th en th e posterior probability is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 0.00 0.10 0.05 0.15 Qp Figure 3.7: Two-dimensional frequentist confidence regions in the {Ho, H b ) plane. The dark to light regions correspond to the 68%, 95%, and 99% confidence regions respectively. Standard calculations from 0 0 2 1 + 0 °® ® big bang nucleosynthesis and observations of D / H predict a 95% confidence region of - 0 .0 0 3 [17], indicated by the shaded region. directly proportional to the likelihood function. To set a Bayesian credible region for any param eter or subset of param eters of interest we calculate th e likelihood £(a) for all models in th e database, assum e a flat prior probability d istrib u tio n for all p aram eters, and integrate th e likelihood over th e rem aining param eters. T he 95% credible region is th e region th a t encloses 95% of th e probability. T h e likelihood functions for each p aram eter in th e database are shown in Figure 3.8. 3.5 R esults T h e 95% frequentist confidence intervals and Bayesian credible regions for each p aram eter in th e datab ase are given in Table 1 and Figure 3.9. We found th a t th e optical d e p th to last scattering r was degenerate w ith other param eters in th e d atabase, m ostly w ith the spectral index of th e prim ordial power spectrum n. Because of th is degeneracy th e 95% confidence interval of r covers nearly th e entire range of values considered, and th e 95% credible region is disjoint. T he 95% confidence intervals and credible regions for Hq, Ob, Om a n d Oa include m ost of th e p aram eter values because th e angular power spectrum of th e CM B is not very sensitive to any one of them alone. It is m ore sensitive to the com bination O eh^, an d to th e to ta l energy density p aram eter 0 = Ojh + O a . T o set confidence intervals and credible regions for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 6 s 0 40 50 60 70 80 0.6 0.8 0.1 0.2 0.3 0.4 0.0 0 1.0 1.2 1.4 0.00 0.05 0 . 10 0.15 6 0.0 Figure 3.8: Bayesian likelihood functions for each of the parameters in the database. Solid circles show grid points in parameter space, while the solid lines were obtained by interpolating between grid points. The parameter values where the solid line intercepts the dashed (dotted) line corresponds to the 68% (95%) Bayesian credible regions. P aram eter Ho n fls r Ojji Oa '^X(95%) 2.80 4.50 2.10 2.45 3.40 3.40 95% Confidence Regions Bayesian frequentist [42,79] [0.92,1.33] [0.046,0.135] < 0.48"^ > 0.32^ < 0.67^ [0.70,1.15] 5.1 [0.015,0.046] 5.6 a: Sets only u p p er or ower lim its on p aram eter B albi et al. [41,79] [0.95,1.28] [0.052,0.146] — [0.37,0.95] < 0.64®^ — [1.00,1.32] — — — — [0.79,1.12] [0.019,0.044] [0.70,1.25] [0.020,0.048] Table 3.1: A comparison of Bayesian, frequentist and maximization 95% confidence intervals. The table also gives the 95% Ax^ thresholds from the simulations. Maximization confidence intervals are taken from [3]; they do not give confidence intervals for all the parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 J 1 ~ BCR FC l 11 11 4T 1 1 40 50 60 BCR - h\I FCl 0 .6 0 70 0 .7 0 0 .8 0 0 .9 0 - - - I nJ 1 1 .0 0 1 ,1 0 1 .2 0 1 .3 0 1 .4 0 1 .6 0 - BCR L r - rL 1 1 FC l 1 1 0 .0 1 0 .0 3 0 .0 3 0 .0 4 0 .0 5 0 .0 7 5 1 1 BCR U1 1 FCl 0 ,0 0 0 ,1 0 0 .1 0 0 .2 0 0 .3 0 0 .4 0 0 .5 0 n* M BCR BCR FC l FCl 0 .1 0 0 .3 0 0 .3 0 0 .4 0 0 .5 0 0 .6 0 0 .7 0 0 .6 0 0.90 1.00 0.00 0.10 0.20 0.30 0 .4 0 0 .6 0 0 .6 0 0 .7 0 O.BO 0 .9 0 1 ,0 0 n M M BCR BCR FCl FC l 0 .8 5 0 .3 5 0 .4 5 0 .5 5 0 .8 5 0 .7 5 0 .6 5 0 .9 5 1 .0 5 1 .1 5 1 ,2 5 1 .3 5 1 .4 5 0 ,0 0 5 0 .0 1 3 0 .0 2 1 0 .0 2 9 0 .0 3 6 0 ,0 4 4 0 ,0 5 2 0 .0 6 0 Figure 3,9: A comparison of the 95% frequentist confidence intervals (FCl, solid line), Bayesian credible regions (BCR, dashed line), and maximization regions (M, dashed dot line) for the parameters in the database, and for SI and flsh^- Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 43 these new p aram eters we form ed all possible com binations of and binned th em in th e following bins for an d in our database {0.00500, 0.0129, 0.0207, 0.0286, 0.0364, 0.0442, 0.0521,0.0600}, and for 0 : {0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95, 1.05, 1.15, 1.25, 1.35, 1.45}. T he center value in each bin is considered a new d ata b a se grid point. We repeat th e process used to find th e 95% threshold for and tre a tin g them as one-dim ensional param eters. We also calculate th e ap p ro p riate in teg rated likelihood functions. Table 1 lists the 95% threshold, confidence interval, an d credible regions for ft and a n d ft, the We determ ined th e frequentist and Bayesian central values for n, param eters to which th e CM B power spectrum is m ost sensitive. In th e frequentist approach th e central value of a p aram eter is the value given by th e best-fitting m odel, a n d th e Bayesian central value is th e m axim um m arginalized likelihood p aram eter value. and Bayesian analyses give respectively a value of T he frequentist = 0.89}|q^9 an d 0 . 9 8 flBh? = 0 . 0 2 6 a nd 0.029+° °}^, and n = 1.02+“;^J and 1.18+g;23, ^^e 95% confidence level. 3.6 D iscussion A com parison of th e frequentist confidence intervals an d the Bayesian credible regions is shown in Figure 3.9. We have also included the results of [3], who set p a ram eter confidence intervals using th e sam e d a ta set considered in this p ap er b u t use m axim ization ra th e r th a n m arginalization of th e likelihood function. In th is m ethod th e likelihood function for a p aram eter is determ ined by finding th e m axim um of th e likelihood £ ( a ) as a function of the rem aining param eters. W hen £ ( a ) is G aussian m axim ization a n d m arginalization are equivalent. T he central values and the w idths of confidence intervals derived from all three m eth ods give consistent results w ithin ab o u t 15% and 30% respectively. A closer exam ination suggests, however, th a t frequentist confidence intervals are som ew hat b roader th a n the Bayesian ones. For five out of eight param eters (n, Om, IIa, fi, a n d th e frequentist confidence intervals are som ew hat larger th a n th e Bayesian credible regions, a n d for Ho the intervals are nearly identical. For the three param eters for which we have results from all three m ethods th e Bayesian intervals are either the narrow est an d ft) or identical to m axim ization (n). Also, we find th a t th e 99% frequentist confidence intervals are som ew hat w ider th a n th e 99% Bayesian credible regions for every p aram eter considered except O b . D espite th is p a tte rn which suggests th a t a frequentist analysis gives broader confidence intervals, it is difficult to claim such a p a tte rn conclusively. F urtherm ore, it is not useful to Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 44 quantify th e p a tte rn exactly because the difference in confidence interval w idths is usually w ithin one p aram eter grid point. M uch finer gridding and hence a m uch larger database would be necessary to claim such a p a tte rn w ith high confidence. A larger databa.se would also provide a m ore accurate determ ination of th e A x^ functions in Figure 3.5, and th e like lihood functions in Figure 3.8. However, a larger d atabase would have been com putationally prohibitive. T he difference betw een th e confidence interval and credible region for th e baryon density is of some interest. M axim ization [3] and Bayesian [44, 89] analyses of th e an d C O BE d a ta gave consistency betw een m a x im a - i from CM B m easurem ents and a value of 0.021 from some determ inations of D / H from quasar absorption regions [65] only a t the edge of th e 95% intervals. T his was in terpreted as a tension betw een CM B m easurem ents and either deuterium abundance m easurem ents or calculations of BBN [18, 34, 95]. A value of 0.021 for is consistent w ith th e frequentist confidence interval a t a level of 75%, an agreem ent a t a confidence of ju st over l a . Recent analysis of new CM B d a ta is consistent w ith a value of = 0.021 w ithin a l a level [63, 73]. T h e com parison betw een th e Bayesian- a n d frequentist-based analyses raises th e ques tion of w hether agreem ent a t the level observed was in fact expected. T he Bayesian and frequentist approaches to p aram eter estim ation are conceptually quite different. A Bayesian asks how likely a param eter is to take on any p a rticu la r value, given th e observed d ata. A frequentist, on th e o ther hand, asks how likely th e given d a ta set is to have occurred, given a p a rticu la r set of param eters. Since the two questions are com pletely different, there is no guarantee th a t they will yield identical answers in general. In certain specific situations Bayesian and frequentist approaches can be shown to yield th e sam e results. For exam ple, in the p a rticu la r case of G aussian-distributed d a ta w ith uniform priors an d linear depen dence of th e predictions on th e param eters, th e two approaches coincide. However, these hypotheses (particularly the last one) do not apply to th e case we are considering. Bayesian an d frequentist m ethods also coincide asym ptotically, i.e. in th e lim it as the num ber of independent d a ta points tends to infinity [21]. In th a t lim it, all confidence regions would be sm all in com parison to the prior ranges of th e p aram eters, and th e Bayesian priordependence would becom e negligible. CMB d a ta are clearly not yet in th is lim it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 4 C orrelating C M B M aps 4.1 Introduction N A SA ’s WMAP satellite has produced a full sky high resolution m easurem ent of th e CMB tem p e ra tu re anisotropy [4]. T his m ap has been used in conjunction w ith o ther CM B and cosmological d a ta to constrain num erous cosmological param eters to unprecedented ac curacy [87]. Previous to WMAP a num ber of experim ents produced high quality m aps CM B anisotropy m aps. These included b o th balloon borne bolom etric experim ents such as MAXIMA-I [36, 56], boom erang [7], and ARCHEOPS [6], and gronnd based interferom et- ric experim ents, CBi [60, 66], DASi [35], and VS A [29]. T ight constraints were placed on cosmological param eters from these experim ents as well. For exam ple see [1, 44, 63, 73, 89]. It is im p o rta n t to ensure th a t th e signal m easured by these experim ents is indeed cos mological in origin. A n excellent m ethod for detecting any system atic errors in a p articu lar dataset is to cross correlate it w ith another experim ent th a t has overlapping sky coverage and sim ilar angular resolution. T he point is to determ ine if the signal is tru ly cosmological. T h a t m eans th a t it is not instrum ental, procedural, or foregrounds. C om paring observa tions m ade by very different instrum ents w ith different observational an d d a ta analysis procedures as well as different frequency bands so th a t one m ight expect different residual foreground signals is a very powerful check. Some difficulties arise if th e two experim ents under consideration have different pixel resolutions and beam profiles, which is usually the case. In th a t case a straig h t forw ard pixel to pixel com parison is no longer accurate because th e CM B signal contained in corresponding pixels is not th e same. M any analysis com paring CM B anisotropy m aps have been perform ed. For a com parison of th e FIRS an d COBE DMR d a ta see [27], Tenerife and COBE DMR [59], an in tern al com parison of P y th o n d a ta [78], an internal com parison of Saskatoon d a ta [92], MSAM an d Saskatoon 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 • WMAP 6000 • M A X I M A -I • M/IXIAIA n 4000 2000 / M) 200 400 600 800 1000 1200 I Figure 4.1: A comparison of the CMB angular power spectrum eis measured by w m a p and m a x i m a - i . The black circles are the w m a p binned power spectrum, which were produced from a weighted combination of cross spectra from 28 detectors. The blue squares are the m a x i m a - i angular power spectrum, which is a composite spectrum from the m a x i m a - i maps with a 5' and 3' square pixelization [56]. The red diamonds are the CMB angular power spectrum computed from the m a x i m a - I I experiment, which is described in detail in Section 4.3. Note that no adjustments have been made to the calibration for any of the data sets. [51], and qm ap and Saskatoon [105]. Because the WMAP m aps are b o th full sky and high resolution a cross com parison betw een and extrem ely useful. w map and o ther CM B experim ents is b o th possible Such a com parison would provide a consistency check th a t b o th experim ents are indeed detecting the same signal on th e sky. Here we present a first cross correlation betw een WMAP and d a ta from another CM B experim ent. T his C h a p te r is organized as follows: in Section 4.2 we develop a m ethod for quantifying th e am ount of correlations betw een two CMB anisotropy m aps a t various angular scales, nam ely th e cross-spectrum . T his m ethod has th e benefit th a t it accounts for different pixel resolutions an d beam window functions betw een experim ents in a straig h t forw ard m anner. F irst, using th e full sky approxim ation we derive estim ators for th e auto- a n d cross-spectra, and derive estim ates for the am ount of correlation betw een th e various sp ectra estim ators. We th en develop a m ethod to com pute the power sp ectra in the m ore realistic case w ith noise a n d p a rtial sky coverage. T his m ethod is tested w ith sim ulations w ith various am ounts of correlation, and it is shown th a t th e results are consistent w ith th e in p u t power spectra. We also com pute th e correlations betw een sp ectra estim ates, a n d show th a t they are consistent w ith the analytically derived results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 In Section 4.3 we discuss the CM B anisotropy m aps used in our analysis, which are the WMAP 93 GHz m ap, th e m a x im a - i and m a x i m a - ii m aps. We com pute th e power spectra and cross-spectra for th e m aps, and discuss our results in Section 4.4. T he cross-spectrum for two m aps which are uncorrelated would be consistent w ith th e null spectrum . Thus, using the cross-spectrum we can rule out th e hypothesis th a t two m aps are uncorrelated a t some specified sta tistic a l significance. We reject the null hypothesis for th e m aps considered using a sta tistic in Section 4.4, and finish w ith a discussion of our results in Section 4.5 and present a conclusion of th e entire analysis in Section 4.G. T he results of this entire analysis follow w hat was published in [2 ]. We finish th is C h ap ter by presenting a num ber of single value sta tistic s th a t can be used to correlate CM B m aps. T he d istributions of these statistics are calculated w ith M onte Carlo sim ulations under th e assum ption of a shared signal in th e m aps, and assum ing the m aps are pure noise. T h e values of these statistics are com puted w ith th e m a x im a - i and MAXIMA-II m aps, an d it is shown th a t these values are consistent w ith th e assum ption of a shared signal. W hile these single value statics do not carry th e sam e am ount of power in determ ining th e am ount of correlation betw een d a ta sets th a t th e cross spectrum does, they are com putationally triv ial to com pute. As we will see com puting th e cross spectrum for large d a ta sets is a com putationally expensive procedure th a t requires th e use of high perform ance supercom puters. 4.2 T he Cross Spectrum T here are a num ber of m ethods for com paring and quantifying th e am ount of correlation betw een experim ents which produce CM B anisotropy m aps. T he sim plest th in g to do is a “correlation-by-eye” , or sim ply p lo ttin g th e two m aps next to each o th er and see if they contain sim ilar stru ctu re. W hile this says nothing statistically a b o u t how consistent the two m aps are, it does give an im m ediate im pression if th e m aps are com patible. One could also calculate for th e difference m aps, which should be consistent w ith noise. However, a x^ has th e lim iting feature th a t only overlapping sections of th e m aps can be included, and the m aps m ust have identical pixelizations. A dditionally, if th e m aps have diflTerent beam s then, as previously stated , the signal in corresponding pixels is different a n d x^ would not be x^ d istrib u ted . Also, there are a num ber of correlation coefficients which can be calculated betw een two CMB m aps to quantify th eir consistency. See C h a p te r 14 in [72]. U nfortunately, these correlation coefficients also suffer from the sam e difficulties as th e x^ statistic. Previously statistics have been derived which do take into account p a rtia l overlapping Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 m aps, different pixelizations and beam profiles. [51] derive b o th Bayesian and frequentist techniques for correlating CMB m aps, and apply these statistics to d a ta from th e MSAM92, MSAM94, and Saskatoon experim ents. [94] defines a “null-buster” s ta tistic which is also useful for quantifying th e correlation betw een CM B d a ta sets w ith different experim ental properties. These statistics can be used b o th as an internal consistency check betw een detectors for th e sam e experim ent, and to cross correlate m aps from different experim ents. In our analysis we com pute the cross-spectrum to m easure th e am ount of correlation betw een two CM B m aps from different experim ents. T he prim ary difference betw een the cross-spectrum and the techniques previously discussed is th a t th e cross-spectrum gives th e am ount of correlation present in th e m aps a t various angular scales, w hereas th e other techniques distill th e inform ation into a single value. T he cross-spectrum is essentially the spherical harm onic transform of th e real space correlation function for two m aps. T he power sp ectra estim ated from th e WMAP m aps are actually the cross-spectra from different detectors from th e sam e frequency band [37]. T his WMAP power sp ectru m along w ith the b est estim ate power spectrum from MAXIMA-I [56] is shown in Figure 4.1. T he two d a ta sets agree very nicely despite th e fact th a t no calibration ad ju stm en ts axe m ade to either power spectrum . O f the th irte e n m a x im a - i binned powers, all b u t two agree to b e tte r th a n Icr w ith th e WMAP d ata. Shown in the figure is also th e CM B angular power spectrum as m easured by th e MAXIMA-II experim ent. T he MAXIMA-ii spectrum m easures fluctuations in th e CMB on angular scales ranging from 10° to 10' in 10 bins of = 75. T he experim ent and m ap used to calculate this spectrum is described in detail in Section 4.3. We find th a t th e MAXIMA-II spectrum is also in close agreem ent w ith b o th th e WMAP an d MAXIMA-I spectra. All 9 of th e 10 m a x i m a - ii bin powers agree w ith m a x im a - i an d all 10 agree w ith th e WMAP bin powers to w ithin la. However, close agreem ent of th e power spectrum m easured by different experim ents does not necessarily im ply th a t the experim ents are actually m easuring th e sam e signal. T he CM B is only one realization of the actu al angular power spectrum , an d th ere are infinite m any realizations (or different CMBs) of th a t power spectrum . T hus, it is possible to have two m aps th a t have th e sam e exact underlying power spectrum , b u t are com pletely uncorrelated. In th a t case th e au to-spectra of th e m aps would agree, b u t th e cross-spectrum would be consistent w ith zero. It is therefore instructive to com pute th e cross-spectrum for CMB m aps from different experim ents. Here we give a brief description of th e crossspectrum , and how we estim ate it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 4.2.1 Full Sky Approximation In th is section we consider two idealized hypothetic m easurem ents of th e tem p e ra tu re CMB anisotropy in order to derive some basic properties of th e cross-spectrum , a n d how it affects estim ates of th e auto-spectra. In our hypothetical m easurem ents we neglect any beam sm oothing by th e experim ental ap p aratu s, and assum e negligible noise in th e detectors. We first derive estim ators for th e all three spectra, an au to-spectrum for each m ap and the cross-spectrum . We th en com pute th e expected variances in th e estim ato rs for all three spectra. We finish by com puting the covariance and correlation betw een th e au to -sp ectra and cross spectrum estim ators. Estim ators Let Ti{r) and T2 (f) be th e tem p e ra tu re anisotropy m aps of th e CM B m aps from two different hypothetical experim ents, which are b o th full-sky, have negligible noise, and there is no beam sm oothing from th e experim ental ap p aratu s. T hen (4.1) = (4.2) I T2{f)Yem{r)dn, w here as usual th e Yimir) are spherical harm onics. Assum ing G aussianity and statistical isotropy im plies th a t and are b o th G aussian random variables w ith m ean zero, and ) = (2 )^( 2 )* V _ CY' Su'Sjnm' (4.3) Cf'^SuiSmm' (4.4) (4.5) where a n d C^''^ are w hat we com m only refer to as th e power sp ectru m for th e first and second m ap, respectively, and is w hat we define as th e cross power spectrum . Prom th e full sky tem p e ra tu re m aps from b o th experim ents each an d can be estim ated for every I and m using E quations 4.1 and 4.2. T h en for each i a n d m our d a ta vector is (4.6) dem which has th e p roperty a {dJmdem) = ^e = ( 1) c:(C) c ;(2 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.7) 50 where (..) designates and ensemble average over m any hypothetical universes. T hen, given and a '^ , for each i we can w rite down a likelihood our collection o f (4.8) m i e ‘^ ^ V W e As usual it is easier to work w ith th e log likelihood, which is -2 1 n £ = + k (2f+ l)ln|M ,|+ (4.9) m=~t where k is an uninteresting constant. To find th e m axim um likelihood estim ates of th e cross spectrum and au to sp ectra we take th e derivative of th e log likelihood w ith respect to our param eters and find th a t _ a i^ , gln£ ^d c f ^ _ _ (2 £ + l ) c f ^ \Me\ - 2 (2 f + 1 ) 0 ^ ^ W i\ ^ 1 i |M^|2 2 ^ ^ W t? J z _ r r ( ‘^)'l2 |„( 2 ) | 2 _ .p { 2 )^2 |„ (l ) | 2 > \^em\ ) I £m\ t ( c f o f >|oSP - Cj‘> o f V S P (4.10) (\Me\ + 2 ( c f V ) (4.11) where \Me\ = -{ c P f. From E quations 4.9 and 4.12, we get th e condition th a t (4.12) S etting the first derivatives of the log likelihood w ith respect to our p aram eters equal to zero, and sim ultaneously solving all three equations yields th e result m = —£ ^ (2 ) ^ E i» S P m = —i c,{C) 2f + m = —t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ('i-w ) 51 Variances We now com pute th e variances in th e estim ators of th e two a u to -sp ectra an d th e cross spectrum . T h e variance in th e estim ator for is Var(Cf)) = ( ( ( ? (" )-( c f ) ) ) ') (4.16) = (4-17) If w, X, y, and z are G aussian random variables w ith m ean zero, th en th e following relation is useful: (wxyz) — { wx ) { y z ) -t- { wy ) { x z ) -H (4.18) { wz ) { xy ) . Prom this a n d E q uation 4.13 it follows th a t ' ' mm' ' ' mm' — tm^ M ! i m ^ im ’ > + (4™“S '> (4 ™ '“tm')l = Ef ^ = +(of*)' ^ mm' 2^ 4-20) +(<^!'T'*’””•'1 ) where we have used th e fact th a t a^m = (—l)"*a^_,^. By com bining E quations 4.17 and 4.22 we see th a t V a r(c f> ) = ^ ( C f ’) ' ’ from which it follows th a t V a r ( C f ') These are th e wellknown = (4.24) cosmic variance term s in the individual m aps. T he variance in th e cross-spectrum estim ator is V a r (C f^ ) ={ { c f ^ y ) - { C P ) \ (4.25) T he first term is (1 ) (2 >x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M26) 52 {21 + 1)^ ^ im ^ tm 'f tm ^em' / mm' + ( 4 m 4 m * ) ( 4 m * 4 m ' >] ( 4 '2 7 ) , 2 ^ E (^fO'+ ....... +(^rO {2i + i y 1 We now see th a t th e variance is >{C)n V a r (C r ) — I— f = -4-ny^r{^'> r c r ) + C . (4 .3 0 ) Thus, in th e ideal case th e error bars (square root of th e variance) of th e cross spectrum is proportional to th e cross spectrum plus the root of th e au to -sp ectra m ultiplied together. A nd in th e no correlation case (i.e. zero cross-spectrum ) th e error b ars in th e cross-spectrum is p ro p ortional to th e root of th e au to -sp ectra m ultiplied together. Covariances and Correlations We would like to com pute the covariance and correlation betw een our estim ators for the au to -sp ectra and cross-spectrum . covariance o f and We begin by considering th e two auto-spectra. T he is defined as C o v (C '('\C 'f^ ) = = ( C 'f^ -(c f^ ))) (4-31) (C ^ ^ ^ cf^ )-(cf^ )(C f^ ). (4 .3 2 ) T h e first term is (2 ) (2 )* J .4 3 3 ^ (2i + 1 ) ^ (2 ^ + 1 )^ C f c f > E + + ( c f > ) % - w + ( c f >)'«,'mm' ^ ( c f ' ) f (4 .3 4 ) (4 .3 5 ) from which we get C ov(C < ‘ > ,c f > ) = 27: f ^ ( c f ' ) . (4 .3 6 ) Therefore correlations in the m ap dom ain (positive cross spectrum ) induce correlations betw een th e estim ators of th e auto-spectra. T he correlation betw een C o „ (C f.C f> ) = and , ^ V a r(c f is ( « ^ ) ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 (C) \ 2 [c, J we expect a correlation of So in the perfectly correlated case where unity betw een th e au to -sp ectra estim ates. E verything we’ve com puted th u s far is for a noiseless experim ent. If we introduce noise th en (4.39) where c f> ^ c f> + v f> (4.40) c f> ^ c f> , (4.41) are th e noise estim ated in a given i for th e first an d second experim ents, and respectively. For a noisy experim ent In th e m ore realistic case we expect a high correlation betw een th e estim ators a t low i because th e signal to noise is m uch b e tte r in these regions. As we go to higher I th e noise increases causing less correlation betw een au to -sp ectra estim ates. We now investigate th e correlation betw een the estim ators for th e a u to -sp ectra and cross-spectrum . T he covariance betw een the auto-spectrum a n d cross-spectrum estim ators is C o v C c f^ c f^ ) = (4 .4 3 ) T he first term is ^ = '' mm! 7 ^ '■ 7 ^ ' 1 + E [cf’c f ’+ + c f ' c f (4 .4 5 ) mm! <Jf> c f . (4 .4 6 ) an d the covariance is C o v (c f> ,(7 f> ) = 27^ c f > c f > . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.47) 54 We can now com pute the correlation betw een th e au to-spectrum and cross-spectrum esti m ators. Using E quations 4.23, 4.30, and 4.47 we get =.(1 ) C o rr(c j^ c f)) = ^ y/2C f^ (4.48) (4.49) It is easy to see th a t Covi{C^P,cf^) (1 ) MC)s C o rr(q "^ = (4.50) A gain in th e idealized case w hen all sp ectra are equal, th e correlation betw een au to -sp ectra and cross-spectrum estim ators is unity, and th e noise is only present in th e denom inator causing less correlation a t high i values. 4 .2 .2 R e a listic E x p e rim en ts Up to now we have been considering full sky m aps in our toy m odels of CM B m easurem ents. Since in practice this is near im possible we now consider th e case of p a rtia l sky coverage. W ith o u t com plete sky coverage th e individual spherical harm onic com ponents cannot be determ ined, so in practice the auto and cross-spectra are estim ated from d a ta in the real, i.e. pixel dom ain. We can w rite th e first and second m aps as rpW (4.51) ! (2 ) w here * is a pixel index, J2) (2 ) (4.52) and s„(Y2 )’ are th e CM B signal in pixel i, an d n (Y1 ’) and n. (Y2') are th e pixel noise for th e first and second m ap, respectively. Let th e num ber of pixels in the and N p ^ \ respectively. Now, we w rite th e d a ta vector in first and second m aps be pixel space as W rp{2) t where d is now a colum n vector of length (4.53) . Assum ing th e signal and noise w ithin each experim ent is uncorrelated and th a t th e noise betw een experim ents is uncorrelated, using E quations 4.51 and 4.52 we find th a t {dd^) = M = g[C)T g{2) ^ jy { 2 ) Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (4.54) 55 where and are th e pixel noise covariance m atrices for th e first and second experim ent, respectively. and are th e CM B signal covariance m atrices, which can be w ritten as I (4.56) t t and are th e beam profiles for the first and second experim ent, Pn are Legendre polynom ials, an d is th e angle betw een pixels i and j . Again up to an uninteresting constant, we can w rite down our likelihood for d as -2 1 n £ = ( i V W (4. 58) We th en m axim ize th e likelihood as a function of two au to -sp ectra a n d th e cross-spectrum . To m axim ize C we adopt a N ew ton-R aphson technique for finding th e zero of the first derivative of th e log likelihood functions, described in detail in [9]. However, there is a fundam ental difference. In [9] they advocate using a q u ad ratic estim ato r technique, which involves replacing th e curvature m atrix of th e likelihood function w ith its ensem ble average, th e Fisher m atrix. T his b o th simplifies calculations, an d speeds up convergence of the algorithm . However, th e likelihoods we are considering in this analysis are m ore complex in n a tu re th a n those in [9], and using the Fisher m atrix approxim ation results in steps in power which yield unphysical results (non-positive definite pixel correlation m atrix an d non positive definite curvature m atrices in power). We therefore use th e full curvature m atrix when calculating N ew ton-R aphson steps in power. T his slows down th e algorithm , b u t it does lead to a definite convergence in our param eters. Due to incom plete sky coverage and beam sm oothing it is not possible to estim ate the power sp ectra a t each spherical harm onic index for some specified w idth. W hen and and we estim ate th e power in an i bin are la r g e \ which is case for all recent CMB experim ents including th e d a ta we consider in th is docum ent, th e m axim ization technique is com putationally intensive. For a description of th e com putational requirem ents see [11]. Note th a t th is m ethod involves estim ating all three power sp e ctra sim ultaneously. Be cause one or b o th of th e m aps under consideration are only p a rtia l sky, th e n there will be a ^Any dataset with roughly > 2000 would fall under this category. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 56 dependence betw een our estim ates for th e bin powers for th e various power spectra. T his is in contrast to th e full-sky estim ators derived in E quations 4.13-4.15, an d to th e m ethod used to estim ate cross-spectrum for th e WMAP m aps, outlined in [37]. T hey com pute th e cross spectra betw een various channels of the sam e frequency band, a n d these cross-spectra are w hat form th e basis of th eir final com bined CM B power spectrum . T hey use a frequentist approach w hen estim ating the cross-spectrum [38], which enables th em to w rite down an estim ator from th e cross-spectrum alone, i.e. one which is independent of th e auto-spectra. We find the cross-spectrum a powerful probe for com paring two CM B anisotropy maps. It provides an estim ate of the am ount of correlation on m ultiple angular scales between two m aps w ith error bars th a t are relatively uncorrelated betw een bins. O ur m ethod for com puting th e cross-spectrum is very general, and accounts for different beam shapes, pixel resolutions an d sky coverage in a sim ple and straightforw ard way. It is also possible to com pute th e cross-spectrum for two m aps th a t do not overlap a t all, except th a t in th a t case you would not be sensitive to any sm all scale correlation, although th is could allow for th e recovery of large-scale inform ation from separate experim ents. If two CM B m aps have a high degree of correlation th en we expect th e cross-spectrum to resem ble th e au to-spectra for the m aps, w ith a well defined first peak a t I ~ 2 1 0 , and higher order peaks if b o th m aps are sensitive to those scales. Two m aps which have no correlation would produce a cross spectrum consistent w ith zero a t all angular scales. Sim ulations To test th e auto- and cross- spectrum estim ation code we used sim ulated CM B m aps where th e statistical properties were already known. T he sim ulated m aps were generated using the spherical harm onic expansion m ethod described in C h ap ter 5. T he three cases considered were: 1) th e m aps had no correlation, 2) the m aps were correlated a t the 50% level, and 3) the m aps contained alm ost perfect correlation. In th e no correlation case, we sim ply considered two m aps w ith different realizations of th e sam e underlying power spectrum . In th e 50% correlation case th e spherical harm onic coefficients of th e m aps were correlated using th e m ethods described in C hapter 5. T he m aps th a t are alm ost perfectly correlated were form ed by taking th e same m ap, and th en adding a sm all am ount of noise (10 jj,K w hite noise p er pixel). We estim ated all three sp ectra from th e m aps for all three cases. In th e no correlation and 50% correlation case th e power sp ectra were binned w ith an in itial bin o f i — 2,35, and bins of A £ = 75 up to ^ = 1500. For th e near perfect correlation case an initial bin of £ = 2,110 was used, and again using bins of A £ == 75 up to £ = 1500. T h e large in itial bin for the perfect correlation case was necessary to get th e code to converge because we ran into Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 6000 ft s + 2000 0 200 400 600 800 0 200 400 600 800 6000 + 2000 c r o s s s p c c t 'o 6000 0 200 400 600 BOO Figure 4.2: Results from various tests of the auto- and cross-spectrum estim ation code using simulated maps. In each case the red and green points represent the auto-spectra for the first and second maps, respectively, and the blue points represent the cross-spectrum. The top panel shows the three spectra estimated from maps with near perfect correlation. The middle panel shows the spectra estimated from maps with 50% correlation (the dotted line is the input cross-spectrum model), and the bottom panel shows the spectra estim ated from maps with no correlation. M A X I M A -I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 Table 4.1: Correlations between Bin Spectra Estimates Simulation 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% No No No No No No No No No No No Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation Correlation N o C orrelation bin C o v r { C ^ i\C ^ i^ ) Corr( 4 ^ \ 4 ^)} Corr(C^^\C<f)} [ 2, 110] [ 111, 185] [ 186, 260] [ 261, 335] [ 336, 410] [ 411, 485] [ 486, 560] [ 561, 635] [ 636, 710] [7 1 1 , 785] [ 786, 1500] 1.00 1.00 1.00 1.00 0.99 0.98 0.98 0.97 0.96 0.97 0.8 7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.99 0.99 0.9 7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.99 0.99 0 .9 7 [ 2, 35] [ 36, 110] [ 111, 185] [ 186, 260] [ 261, 335] [ 336, 410] [ 4 1 1 , 485] [ 486, 560] [ 561, 635] [ 636, 710] [7 1 1 , 785] [ 786, 1500] 0.47 0.35 0.31 0.24 0.31 0.19 0.25 0 .27 0.22 0.32 0.18 0.23 0.80 0.72 0.69 0.63 0.69 0.57 0.64 0.66 0.60 0.70 0.56 0.61 0.80 0.72 0.69 0.63 0.69 0.57 0.64 0 .66 0.60 0.70 0.56 0.61 [ 2, 35] [ 36, 110] [ 111, 185] [ 186, 260] [ 261, 335] [ 336, 410] [4 1 1 , 485] [ 486, 560] [ 561, 635] [ 636, 710] [ 7 1 1 , 785] [ 786, 1500] 0.06 0.01 0.00 0.00 0.00 0.01 0.00 0.00 0.01 0.01 0.00 0.00 -0.33 0.12 0.01 0.06 -0.01 0.10 0.09 -0.09 -0.15 0.17 0.02 -0.01 -0.37 0.11 0.01 0.06 -0.01 0.10 0.09 -0.09 -0.15 0.17 0.02 -0.01 I [B] The correlations between auto- and cross-spectra estimates for the simulated M A X I M A - i maps used to test the estimation software, as defined in Equation 4.60. The top 11 rows are for maps with near perfect (100%) correlation, the middle 12 are for maps with 50% correlation, and the bottom 12 are for maps that are not correlated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 num erical problem s in th is situation. These problem s were also encountered w hen running th e real d a ta , and th is is subsequently discussed in Section 4.5. We also w ish to investigate th e correlations betw een estim ated spectra, and see if those agree w ith our results derived in th e Section 4.2.1. T he results of th e auto- and cross- sp ectra estim ation code are shown in Figure 4.2. Prom top to b o tto m th e results are for perfect correlation, 50% correlation, and no correlation. In each cases th e estim ated sp ectra agree w ith the in p u t m odels. All pairs of estim ated au to -sp ectra fall on th e solid curve, which was th e sta n d a rd ACDM m odel used to generate th e m aps. T he estim ated cross-spectra also agree w ith how m uch correlation was in p u t into th e m aps. T he for each estim ated cross-spectra w ith th e in p u t models from the top of Figure 4.2 to b o tto m are 9.3, 9.2, and 9.6 for 9 DoF, respectively. T h e F isher m atrix was com puted for the auto- and cross- sp ectra estim ates, a n d we m ade th e assum ption th a t th e inverse Fisher m atrix is th e covariance m atrix (an assum ption justified in [9]). H enceforth we call th e inverse Fisher m atrix the covariance m atrix, S. T he covariance m atrix will have the form C o v (c W „ c W ,) C o v (C 'W „ C < g .) C o v (c W „ (7 W ,) C o v (c W „ c lS ,) C o v (C -(g „ C < g ,) C o v « 7 < g „ C '® ,) C o v (C « „ C ™ ,) C o v (c S g „ c S ') C o v ((7 ® „ C '< j'l,) (4.59) J Suppose we estim ate th e auto- and cross- sp ectra in 12 bins (as is th e case for th e no cor relation and 50% correlation sim ulations shown in Figure 4.2). T h en E will be a 36 x 36 m atrix, and each block m atrix contained in E will be a 12 x 12. T h e variances for each sp ectra-estim ate (the square of the error bars) are th e diagonal elem ents of th e block diag onal m atrices. We are interested in th e correlation betw een spectra-estim ates, specifically betw een spectrum estim ates for the same i bin. T he correlation betw een bin estim ates is Covv{B,B') = (4.60) , / e (b : W ) ^ e {b >,b >) T he correlation was com puted for all three pairs of sp ectra estim ates, for all three cases. T he results are shown in Table 4.1. Note th a t the near perfect correlation case has only sp ectra estim ates in 4.3 1 1 bins, while th e other two cases have 1 2 bins. T he M aps We cross-correlate CM B tem p eratu re anisotropy m aps from th e WMAP, MAXIMA-I and MAXIMA-II experim ents. T he WMAP m ap used is th e W -band (93 GHz) foreground-cleaned m ap, and only th e p o rtio n th a t overlaps w ith th e MAXIMA-I field. By analyzing WMAP’s derived foreground m aps we find th a t th is p o rtion of th e W -band m ap is free of point Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 MAXIMAI WMAP 3 0 0 [/^K] 300 [^iK] ( 2 3 1 .6 , 5 6 .3 ) E q u a t o r ia l ( 2 3 1 .6 , 5 8 .3 ) E q u a t o r i a l (MAXIMAI+WMAP)/2 (MAXIMAI-WMAP)/3 3 0 0 [/iK] 300 [/liK] -3 0 0 ( 2 3 1 .6 , 5 8 .3 ) E q u a t o r ia l (2 3 1 .6 , 5 8 .3 ) E q u a t o r ia l Figure 4.3: A comparison of the m a x i m a - i and W M A P CMB anisotropy maps. The upper left panel shows the W M A P map in the m a x i m a - i region, smoothed with a Gaussian window function with FWHM of 13.2' as described in the text. The m a x i m a - i map is shown in the upper right hand panel, and the sum and difference of the two maps axe shown in the lower left and lower right panels, respectively. All the maps are pixelized in HEALpix pixelization with resolution parameter nside=512. N ote that we use the raw data in all analyses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 14.8 15.0 (M A X IM A -I - 15,2 15.4 M A X IM A -II)/2 (M A X IM A -I + M A X IM A -II)/2 ..................... I ■' ' ' -133 -4 0 0 14,8 15.0 15,2 15,0 15,2 Figure 4.4: A comparison of the M A X I M A -I and m a x i m a - i i CMB anisotropy maps. The upper left and upper right panels show the raw m a x i m a - i and m a x i m a - i i maps, respectively, with only a vertical gradient removed from each. The m a x i m a - i map is shown in the upper right hand panel, and the sum and difference of the two maps are shown in the lower left and lower right panels, respectively. All the maps are pixelized in HEALpix pixelization with resolution parameter nside=512. Note that we use the raw data in all analyses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 sources, and contains a negligible am ount of d ust, free-free, a n d synchrotron emission^. T h e WMAP d a ta a t 93 GHz has sim ilar angular resolution and is closest in frequency to th e MAXIMA d ata. T his m ap is pixelized using th e HEALPix^ [33] pixelization in Celestial coordinates w ith nside= 512, and contains 7,926 pixels. A n nside=512 corresponds roughly to a pixel w ith a w idth of 7'. Following [4] we assum e th a t there are no noise correlations betw een pixels a n d th a t th e beam p a tte rn is non-G aussian, azim uthally sym m etric and has FW H M of 13.2'. We com puted the beam window function by tak in g th e weighted average of th e individual beam s for each of the four detectors used to form th e final W -band m ap. T he resulting beam profile in i space is shown in Figure 4.5. T h e W iener filtered version of this m ap is shown in [2], and we show a G aussian sm oothed version of th is d a ta in the u p p er left panel in Figure 4.3; we use th e raw d a ta in th e analysis. T h e W iener filtering is perform ed w ith th e corresponding best fit m odels for each m ap. A m ore detailed discussion of th e WMAP m aps is given in [4] and of algorithm s used in their com putations in [37]. T he MAXIMA m ap-m aking procedure is described exhaustively in [90]. T he MAXIMA-I m ap we use in this analysis is th e 8 arcm inute version of th e d a ta published by [36], which covers a larger area of th e sky and has a coarser resolution com pared to th e d a ta published by [56]. T here are a to ta l of 5,972 pixels, and th is m ap covers ~ 120 deg^ on th e sky. It is a com bination of three 150 GHz photom eters and one 240 GHz photom eter. T he beam s for each ph o to m eter have a FW H M of ~ 10' [36] and th e effective window function is calculated using the technique described in [103]. T he MAXIMA-l m ap shown in Figure 4.3 is sm oothed w ith a G aussian beam w ith FW H M of 13.2', which is th e WMAP resolution. T he raw version of th e m ap is used for all quan titativ e analyses. Figure 4.3 also shows th e difference of the MAXIMA-I an d WMAP G aussian sm oothed m aps (lower right panel), a n d sum of th e m aps (lower left panel). T he p a tte rn of tem p e ra tu re fluctuations, which is sim ilar in b o th m aps, disappears in th e difference m ap, whereas the sum m ap retain s th e stru c tu re . T h e MAXIMA-II m ap comes from a flight of th e MAXIMA payload th a t took place on 1999 Ju n e 17. We use th e d a ta from four photom eters a t 150 GHz, an d only th e p o rtio n of the MAXIMA-II m ap th a t overlaps th e m ap of m a x im a - i. M ore details a b o u t th e MAXIMA-II flight, d a ta an d m aps are given in [75, 91]. T he MAXIMA-II m ap is pixelized using an 8' square pixelization in celestial coordinates, contains 2,757 pixels, an d covers ~ 50 deg^ on th e sky. T h e beam profile for th is m ap is ~ 10' FW H M , and again com puted using the te c h n iq u e s d e s c r ib e d in [103]. T h e M A X IM A -II p o w e r s p e c tr u m s h o w n in F ig u r e 4 .1 h a s 10 bins of A i = 75, extending over th e I range 35 < £ < 785. Figure 4.4 shows th e overlap ^http://Iam bda.gsfc.nasa.gov/product/map/m _products.cfm ®http://w w w .eso.org/science/healpix/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 M A X IK iA -I & \ M A X IM A - I I 0 .6 0.4 W M AP 0,2 0.0 0 500 JOOO 1500 2000 Figure 4.5: The beam filter functions for m a x i m a - i , m a x i m a - i i , and w m a p considered in this analysis. They are the dotted line, dashed line, and solid line, respectively. N ote that due to higher angular resolution of the m a x i m a beams the observations reach to higher i s and this is accounted for in the comparisons. The horizontal axis is spherical harmonic multipole. region of th e MAXIMA-I and MAXIMA-II m aps and th e difference an d sum m ap. N ote th a t th e m aps shown in this Figure are th e raw d ata. Identical tem p e ra tu re fluctuations th a t are ap p aren t in each of the m aps disappear in th e difference m ap. 4.4 R esults T he auto- and cross-spectra for all com binations of th e WMAP a n d MAXIMA m aps are shown in Figure 4.6. T he error bars on the sp ectra are th e square root of th e curvature of the likelihood function ab o u t the m axim um likelihood param eter value. In all cases we com pute th e sp ectra in bins of w idth A£ = 75, over th e interval 111 < ^ < 710, and m arginalize over all m odes £ < 110 and I > 711. T he app ro p riate pixel window functions for each m ap were convolved w ith th e beam functions in th e analysis. We found th a t th e cross-spectrum estim ator did not converge when th e initial bin was split in two, and th is is fu rth e r discussed in Section 4.5. In all cases th e cross-spectra are consistent w ith th e a u to -sp ectra giving strong evidence for a correlation betw een th e m aps. We also com pute the power spectrum of th e difference m aps for all th ree pairs of m aps using bins of A£ = 75 over the range 35 < ^ < 785. T h e WMAP window function is used w hen com puting th e MAXIMA-I-WMAP and MAXIMA-Ii-WMAP difference spectra. T he expected residual power resulting from different beam profiles is m axim um a t th e bin centered a t £ ~ 300, and is approxim ately equal to th e Icr error b ar of the m a x im a - I / w m a p Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. difference 64 Table 4.2: Cross Spectrum Values M aps D oF ^cutoff M A X IM A -I / w m a p 8 8 8 53 53 53 M A X IM A -I / m a x i m a - ii M A X IM A -II / w m a p X 191 241 150 Tlie from Equation 4.61 calculated for all three combinations of maps. A y^ greater than 53 implies that the probability that the no correlation hypothesis is true is less than 1 x 10~®. T a b le 4.3: D ifference Spectrum x^ V alues The M aps D oF M A X IM A -I / w m a p 10 M A X IM A -I / m a x i m a - ii 10 M A X IM A -II / w m a p 10 7.5 8.3 17.2 of fhe power spectrum for the difference maps with the null spectrum. power spectrum . T he effect is less th a n Ict for all rem aining bins. We use th e m a x im a - I window function w hen com puting the MAXIMA-l-MAXIMA-II difference spectrum . T he results are shown in Figure 4.7. O f th e 30 band power estim ates for th e difference m aps, 28 are w ithin Ict of zero power. To fu rth er quantify th e level of correlation betw een th e m aps we use a sta tistic to reject the hypothesis th a t th e m aps are uncorrelated. We w rite our s ta tistic as = (4.61) BB' w here the sum is over ban d power estim ates, and F is th e Fisher m atrix for th e cross spectrum . Because th e au to-spectra and cross-spectrum are estim ated sim ultaneously, we m arginalize over th e au to sp ectra when calculating the To test th e null hypothesis we choose a statistical significance a = 1 x 10“ *. If is greater th a n th e critical value 53, th en th e probability th a t the null hypothesis is tru e is less th a n 1 x 10~*. T he results are sum m arized in Table 4.2. In all cases x^ is significantly larger th a n th e critical value giving an essential certainty th a t th e no-correlation hypothesis is false. N ote th a t assum ing th e x^ of E quation 4.61 is x^ d istrib u ted is equivalent to assum ing th a t th e are G aussian d istrib u ted , which is an approxim ation. We also com pute th e x^ of th e difference sp ectra shown in Figure 4.7 w ith th e null sp ectra to determ ine how consistent these are w ith no fluctuations in th e difference m aps. T he results are shown in Table 4.3. T he 10 power spectrum bins com puted from th e MAXIMAi/w M A P and m a x i m a - i / m a x i m a - ii difference m aps have a y^ of 7.5 a n d 8.3, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 w ith the null spectrum . T he m ax im a - 1 1 / w m ap difference m ap gives a is a 7% chance of getting of 17.2. T here > 17.2 for 10 DoF. Overall there is a good fit to th e null spectrum m odel, which im plies th a t differencing th e overlap section of th e m aps removes th e sky signal and is consistent w ith noise. 4.5 D iscussion 4.5 .1 A u to - an d C ro ss-S p ectra T he auto- an d cross-spectra of th e different d a ta sets agree w ith each o th er to w ithin Icr over alm ost all £ bins giving evidence th a t a t each angular scale all experim ents are detecting th e sam e sp atial fluctuations on the sky. All auto- and cross-spectra show th e first acoustic peak in th e power spectrum and th en a level of power th a t is consistent w ith subsequent peaks. These results are consistent w ith sta n d a rd inflationary ACDM m odels. A u to-spectra of th e overlap section of the w m ap d a ta give increased error bars at £ > 486 because of th e lim ited sky coverage of the overlap regions, and because of th e beam profile of th e W -band m ap. We find th a t th e beam p a tte rn alone causes th e WMAP auto-spectrum error bars in th e bins £={{486,560}, {561,635}, {636,710}} to be 2-3 larger th a n those for MAXIMA-I or MAXIMA-II. Negative power was found in th e bin £={486,560} for th e WMAP au to -sp ectra (see th e top and b o tto m panel of Figure 4.6). T here is no requirem ent th a t th e auto-spectrum be positive in our estim ation m ethod. A com parison of th e au to -sp ectra shown in Figure 4.6 reveals th a t th ere is a difference betw een b a n d power estim ates for the sam e d ataset. T his difference arises because th e com p u tatio n of cross-spectra involves estim ating b o th auto- and cross-spectra sim ultaneously, giving rise to correlations betw een the different spectra. T he fractional changes in power averaged over bins are 3%, 6%, and 15% for th e WMAP, MAXIMA-I, an d MAXIM A-ll d a ta sets, respectively. If th e likelihood d istrib u tio n of the ban d powers were strictly G aussian, th en the m axim um likelihood estim ates would be th e sam e regardless of th e correlations be tween spectra. However, th e likelihood as a function of auto- and cross-spectra is som ew hat non-G aussian (see E quation 4.58 in [9], and C h ap ter 2 in this docum ent), so th e correlations do effect th e ban d power estim ates. T he fact th a t the changes betw een estim ates are sm all suggests, however, th a t the d istributions are close to G aussian. We note th a t th e bin-powers are correlated a t the 77% level or higher a t th e lowest £ bin. T he correlation decreases to less th a n 10% a t the highest £ bin for correlations betw een th e au to -sp ectra an d to 20% - 50% for correlations betw een th e auto- a n d cross-spectra. These results are in broad agreem ent w ith expectations given th e high am p litu d e of the cross-spectrum . T h e specific values for the th e correlation betw een power sp ectra estim ates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 are given in Table 4.4. 4.5.2 Com putational Issues T he strong correlation betw een th e different d a ta sets leads to some co m putational dif ficulties w hen a tte m p tin g to find the m axim um likelihood auto- an d cross-power spec tra . As discussed in Section 4.2 the cross-spectrum is lim ited by th e requirem ent th a t We find th a t using a q u ad ratic estim ator (Fisher m atrix) m ethod for (C ) calculating th e N ew ton-R aphson step leads to a m is-estim ate of th e step for 6C\ when sta rtin g w ith a guess significantly far away from the peak in likelihood space. T his conse quently results in a step which leads to a non-positive definite pixel covariance m atrix. For exam ple, an initial guess of a null spectrum leads to an unphysical pixel covariance m atrix in all three cases we are considering. T his is rem edied by using th e curv atu re m atrix to (O com pute SC^ , as discussed in Section 4.2. Once the p aram eter values becom e sufficiently close to th e m axim um likelihood values either th e curvature m atrix or fisher m atrix can be used to find th e m axim um likelihood param eters. B oth techniques converge to th e sam e set of param eters for all three analyses. T he power sp ectra shown in Figure 4.6 have been calculated w ith a broad initial bin at £ = {2,110}. T his is because the N ew ton-R aphson likelihood m axim ization technique did not converge if this low £ bin was split to two. Some binning stru ctu res would cause negative eigenvalues in th e curvature m atrix of th e param eters or steps in p aram eter space th a t would lead to a non-positive definite pixel covariance m atrix, b o th of which are unphysical. We carried out sim ulations and found a sim ilar phenom enon. T h e cross-spectra of un correlated m aps or m aps w ith a sm all value for th e ratio of expected cross-spectrum to auto-spectrum converged to th e expected answer. T h e calculation of th e cross-spectrum also converged w ith sim ulated m aps th a t had perfect correlation (i.e. th e sam e m ap w ith different noise realizations) and a broad first bin w ith £ — {2,110}. However it did not converge w ith sim ulated m aps th a t had perfect correlation and two bins betw een ^ of 2 and 110. Therefore, we a ttrib u te the com putation problem s encountered as a lim itatio n in the m ethod used for com puting the cross-spectrum and not a feature in any o f th e d a ta sets considered in th is analysis. 4.5.3 Foregrounds and System atic Errors T he cross spectrum and difference spectrum analyses conclusively d em o n strate th a t all three experim ents have m apped th e sam e tem p e ra tu re fluctuations on th e sky. However, these analyses are not sensitive to w hether the shared fluctuations are CM B in origin or th e result Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Table 4.4: Correlations between Bin Spectra Estimates Analysis M A X IM A - l/W M A P M A X IM A - l/W M A P M A X IM A -1 /W M A P M A X I M A - l/W M A P M A X IM A - l/W M A P M A X IM A -l/ W M A P M A X I M A -l/W M A P M A X IM A -l/ W M A P M A X IM A -l/ W M A P M A X I M A - l/ W M A P M A X I M A -l/W M A P m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i/ m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m a x im a - ii/ w m a p m a x im a -ii/ w m a p m a x im a - ii/ w m a p m a x im a - ii/ w m a p m a x im a - ii/ w m a p m a x im a - ii/ w m a p m a x im a - ii/ w m a p m a x im a - ii/ w m a p m a x im a - ii/ w m a p m a x im a -ii/ w m a p m a x im a - ii/ w m a p I [B] C o rr(4 ^ \(7 lf)) C orv(C^^\cP) C orr((7i= *\cf)) [ 2, 110] [ 111, 185] [ 186, 260] [ 261, 335] [ 336, 410] [411, 485] [ 486, 560] [ 561, 635] [ 636, 710] [711, 785] [ 786, 1500] 0.88 0.86 0.84 0.45 0.17 0.08 0.16 0.04 0.02 0.02 0.00 0.97 0.96 0.95 0.79 0.54 0.38 0.54 0.27 0.21 0.18 0.06 0.97 0.96 0.95 0.79 0.54 0.37 0.52 0.27 0.20 0.17 0.05 [ 2, 110] [ 111, 185] [ 186, 260] [ 261, 335] [ 336, 410] [411, 485] [ 486, 560] [ 561, 635] [ 636, 710] [711, 785] [ 786, 1500] 0.48 0.95 0.82 0.66 0.37 0.34 0.32 0.15 0.09 0.03 0.01 0.86 0.99 0.95 0.92 0.80 0.79 0.78 0.59 0.47 0.28 0.22 0.79 0.99 0.95 0.88 0.70 0.68 0.67 0.47 0.36 0.20 0.16 [ 2, 110] [ 111, 185] [ 186, 260] [ 261, 335] [ 336, 410] [411, 485] [ 486, 560] [ 561, 635] [ 636, 710] [711, 785] [ 786, 1500] 0.59 0.77 0.70 0.33 0.12 0.03 0.04 0.04 0.07 -0.00 0.00 0.90 0.94 0.93 0.77 0.54 0.30 0.34 0.33 0.45 -0.05 -0.07 0.85 0.93 0.90 0.68 0.42 0.20 0.24 0.23 0.32 -0.03 -0.04 bin The correlations between auto- and cross-spectra estim ates for the w m a p , m a x i m a - i , and m a x i m a - i i maps discussed in this analysis. The top 11 rows are for the m a x i m a - i / w m a p analysis, the middle 11 are for the m a x i m a - i / m a x i m a - i i analysis, and the bottom 1 1 are for m a x i m a - i i / w m a p analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 o f foreground contam ination or a shared system atic effect. A careful foreground analysis was carried out by b o th the w map and m a x im a team s. It was shown th a t th e MAXIMA-I region of the sky a t 150-240 GHz contains a negligible am ount galactic contam ination and th a t it has no detectable point sources [36, 45]. A detailed analysis of the foreground sources in th e WMAP d a ta is presented in [5]. A lthough we use W M AP’s foreground-cleaned m ap, even W MAP’s foreground m aps have negligible am ount of contam ination in th e MAXIMA-I region. T he RM S fluctuations in th e WMAP 93 GHz d ust, synchrotron, and free-free m aps are lower th a n the corresponding WMAP CMB m ap by a factor of 17, 42, and 560, respectively. Also, th e WMAP team find no point sources in th e MAXIMA-I region of the sky. It is unlikely th a t w map and m a x im a share system atic errors. We therefore conclude th a t the com m on signal in th e WMAP and MAXIMA d a ta is th e cosmic microwave background radiation. Since th e cross-spectra agree w ith the a u to -sp ectra we conclude th a t w ithin the signal-to-noise ratio of the tests system atic errors in th e d a ta are sm aller com pared to statistical errors. 4.6 C onclusions We have presented a Bayesian m ethod for estim ating th e cross-spectrum betw een two CMB tem p e ra tu re anisotropy m aps. T he m ethod is advantageous for correlating m aps because it does not require th e m aps to have perfect overlap, identical beam shapes or pixelizations. Using this form alism we found a high degree of correlation betw een th e m aps from m a x im a - I, WMAP, a n d MAXIMA-II ; in all cases the null hypothesis is rejected w ith a probability higher th a n 1 - 10“ ®. A dditionally, we com puted the power spectrum of th e difference m aps for all com binations of th e three d a ta sets considered, an d found th a t in each case th e spectra were consistent w ith the null spectrum . T he results show conclusively th a t the tem p eratu re fluctuations d etected by each of the MAXIMA-I, WMAP, and MAXIMA-II experim ents are reproduced by these experim ents, in overlapping regions of th e sky. T he close agreem ent of th e fluctuations d etected by these experim ents shows th a t current CMB experim ents are now beginning to provide us w ith high precision images of the tru e microwave sky. 4.7 Single Value Correlation S tatistics T hus far in th is we have only considered m ethods for correlating CM B m aps th a t are com putatio n ally expensive, nam ely th e cross-spectrum and th e power sp ectru m of th e difference Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 69 ♦ ^ m \{A.\IMA H' Kf Al ’ • C R O S S SPEC TRA I 4000 400 I ♦ MAXIMA ! ■ MAXIMA • ^ II C R O S S SPEC TRA 4000 ■ 200 400 L 8000 • C R O S S SPEC TRA 6000 4000 I+ G“ 2000 400 L Figure 4.6: The auto-spectra and cross-spectrum estimated for the maps described in the text. Prom top to bottom: the results for M A X i M A - i / w M A P , m a x i m a - i / m a x i m a - i i , and m a x i m a - i i / w m a p . The horizontal axis is spherical harmonic multipole number, and the vertical axis is fiK^. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 70 1 .5 x l0 * [ • MAMMA l/K M A I’ □ MAXI MAI-I/MAXJMA-II ft M A M M M ll/W M M ' 5.0x10^ - \ -5 .0 x 1 0 ^ - - I.O x lQ t - -i.5xio<; 400 I Figure 4.7: A visual comparison of the power spectrum made by taking the difference maps from all three experiments. The diamonds (red) are the power spectrum of the m a x i m a - i / w m a p difference map, the squares (blue) are the power spectrum of the m a x i m a - i / m a x i m a - i i difference map, and the triangles (green) are the power spectrum of the m a x i m a - i i / w m a p difference map. We find all three spectra to be statistically consistent with the null spectrum. m ap. T he cross-spectrum is an optim al m ethod for correlating CM B d a ta sets because it determ ines th e am ount of correlation betw een two m aps a t m ultiple angular scales while providing a sim ple way of accounting for different beam profiles an d pixelizations betw een experim ents. We finish this chapter w ith a discussion of single valued correlation sta tistic s an d their applicability to CM B d a ta sets. T hough single valued correlation sta tistic s do not carry as m uch inform ation as th e cross-spectrum , they are com putationally triv ial to com pute, and still carry useful statistical inform ation. In this chapter we present a num ber correlation statistics, determ ine th eir d istrib u tio n using M onte Carlo sim ulations, a n d apply these statistics to th e M A X iM A -l/M A X iM A -ll d a ta sets. C om puting th e Statistics Given two tem p e ra tu re anisotropy m aps w ritten as colum n vectors, and th e linear correlation coefficient r is defined as (4.62) w here and are w ritten as colum n vectors and defined in E quations 4.51 and 4.52. T he coefficient r gives a m easure of how well th e d a ta fits to a straig h t line. In th e case of perfect correlation (i.e. p lo ttin g verses gives a perfect straig h t line) th en r = 1, Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 71 a n d perfect anti-correlation gives r = —1. In the lim it of large sam ple size and under the null hypothesis r is norm ally d istrib u ted w ith m ean zero and a variance of \ / \ / N [72]. If we have a m easure of th e variance in each d a ta point i, called an d for the first and second m aps respectively, th en we can w rite down a noise weighted correlation coefficient, r^,. T his is calculated by replacing the m aps in E quation 4.62 w ith th e noise weighted m aps. Since th e variance of r is only dependent on th e num ber of d a ta points IV, th en Vyj will have th e sam e variance. W eighting th e d a ta points in th is m ethod allows use to account for different uncertainties am ong m easurem ents. In practice we usually have access to th e full pixel-pixel noise correlation m atrix. In th is case and are sim ply th e diagonal elem ents of th e noise correlation m atrix. Also, If the m aps contain signal, then th e statistic (4.63) should give an acceptable fit, where N is th e sam ple m ean of th e two noise correlation m atrices from b o th m aps. T he rank correlation coefficient is an exam ple of a non-param etric sta tistic , which is a sta tistic whose in terp retatio n is independent of th e pop u latio n distrib u tio n . Replace th e value of each d a ta point w ith its rank am ong all other values in th e sam ple. T hen the resulting vector will contain only integers of value I, 2, 3,..., N . Let rank vectors of th e A and M represent the m aps, respectively. T h e ran k correlation coefficient Vg is and defined as th e linear correlation of th e ranks [72]. T h a t is, r.s w here M = Ei { M i - M ) { A - A ) and A represent th e m eans of th e ranks (which should be (4.64) N / 2 o f course). Note th a t the m eans of th e sam ples in E quation 4.62 were not included because it was explicitly assum ed th a t they were zero. M onte Carlo Sim ulations To test th e correlation statistics given in Section 4.7 we M onte C arlo each of th e statistics by generating m any random d a ta sets, and com puting th e d istrib u tio n of each statistic. We generate sim ulations of th e overlapping regions of th e MAXIMA-i a n d MAXIMA-II m aps. Each m ap contained 2757 8' pixels and beam properties described in Section 4.3. We sim ulate th e MAXIMA-I and MAXIMA-II m aps, an d as follows. T he tem p e ra tu re in pixel i is m odeled as (4.65) Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 72 0.6 0 .6 0.4 0.2 0.2 0.0 0 .0 •0.20 -0 .1 0 0.00 0.10 0.20 0.30 0.40 - 0.0 0 .2 0.2 0 .4 0 .6 0 .6 0.4 0.4 0.2 0.2 o .o L . 0.0 - 0 .2 0.0 0.2 2400 0.6 0.4 2600 2800 3000 3200 F ig u r e 4.8: Simulated probability distributions for the single value correlation statistics. In each panel the horizontal axis in statistic value, and the vertical axis is frequency per bin normalized such that the maximum probability is equal to one. The black distributions were computed under the null hypothesis, i.e. that both are pure noise. The blue distributions were computed under the hypothesis that each map contains signal and noise as described in the text. The vertical dotted red lines are the actual values computed for the M A X I M A -I - M A X I M A -I I maps. n(2) w here and = 4 (2 ) , (2) (4.66) > are th e beam sm oothed signal generated w ith m ethods discussed in C h ap ter 5, a n d and are G aussian random num bers w ith m ean zero and and = We use th e MAXIMA-I best fit m odel to generate th e sim ulated signal in th e m aps. In o ther words we neglect off diagonal noise correlations in th e m aps to keep the com putations simple. We could generate a sim ulated noise stream for b o th m aps th a t uses th e full noise correlation m atrices, b u t th a t would involve Cholesky decom posing a 2,757 X 2,757 m atrix. Using E q uation 4.65 and 4.66 two sets of 2,000 sim ulations were generated. F irst as sum ing th e pure noise hypothesis = 0 for every pixel i), an d second assum ing a c o m m o n c o s m o lo g ic a l s ig n a l. T h e r e s u lts a re s h o w n in F ig u r e 4 .8 , w h ic h p lo ts th e prob ability d istrib u tio n s of th e statistics r , r ^ , r^, and under b o th assum ptions. T he black histogram in each panel are th e d istributions of th e statistics assum ing b o th m aps are pure noise, and the blue histogram s assum e a shared signal. T he vertical d o tte d red lines rep resent th e values of th e statistics com puted for th e correlation coefficients th e m a x i m a - i / m a x i m a - ii m a x i m a - i / m a x i m a - ii m aps. For all values are consistent w ith th e assum ption Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 73 Table 4.5: Correlation Coefficients Values Statistic {X) ax m a x i m a - i/ m a x i m a - ii Nax r Vs 0.00 0.00 0.00 0.02 0.05 0.04 0.19 0.51 0.43 9.50 10.20 10.75 r w Ts 0.20 0.53 0.45 0.04 0.03 0.03 0.19 0.51 0.43 0.25 0.67 0.67 r’to Tlie values for all tliree correlation coefRcients under botli tiypotfieses. Tlie top tfiree rows represent tlie mean and standard deviations for the distributions under the pure noise hypothesis. The bottom three rows represent the mean and standard deviations under the assumption of a shared signal. The first column are the mean of the distributions, the second column is the standard deviations crx = — (X)^. The third column is the value of the statistic for the actual m a x i m a - i / m a x i m a - i i data, and the last column represents the number of a x the m a x i m a - i / m a x i m a - i i value is away from the mean. of a shared signal, and inconsistent w ith th e pure noise hypothesis. T he m a x i m a - i / m a x i m a - ii com puted for is consistent w ith b o th assum ptions. T his is because th e of E quation 4.63 should have a sta n d a rd sta tistic d istrib u tio n under either assum ption, pure noise or shared signal. T he results are fu rth er quantified in Table 4.5. T h e values of th e correlation coefficients r, and for th e m a x i m a - i / m a x i m a - ii d a ta are 9.5, 10.2, a n d 10.75 a away from the no correlation hypothesis, in which the the coefficients would be zero. Also, as th e table indicates, th e values for th e correlation coefficients are all w ithin l a from th e m ean of th e distrib u tio n s assum ing a shared sky signal. These results indicate th a t th e MAXIMA-I an d MAXIMA-II m aps do indeed share a comm on signal, a n d th a t signal is consistent w ith sta n d a rd cosmological models. Reproduced with permission of the copyrighf owner. Further reproduction prohibited without permission. C hapter 5 C M B M ap Sim ulations We now tu rn our a tte n tio n for the first tim e to th e polarization of th e CM B. It is pre dicted th a t th e CM B photons will contain some am ount of linear polarization because they Thom son scattered off of electrons a t th e last scattering surface. M easurem ent of th e CM B polarization would provide a robust test of th e g ravitational instab ility paradigm of sta n d ard cosmological theory [53, 76] because it is unam biguously predicted th a t th e photons contain some am ount of polarization. A dditionally, polarization m easurem ents will help in breaking cosmological param eter degeneracies th a t exist when using only tem p e ra tu re d a ta , and provide inform ation ab o u t th e reionization history of th e universe [19, 107]. T he polarization of the CM B also provides for a direct test of th e inflationary paradigm . P rim o r dial gravitational waves induced during an inflationary epoch would leave a characteristic im print in th e polarization stru c tu re in the CM B [41, 48, 106]. In this chapter we present a com plete form alism for perform ing num erical sim ulations of b o th CM B tem p e ra tu re and polarization m aps based on a given inflationary cosmolog ical m odel. In all cases it is assum ed th a t the fluctuations in th e CM B obey a G aussian probability distrib u tio n . T his m eans th a t to com pletely specify th e d istrib u tio n we only need a m ean and variance, or in th e m ultivariate case a vector of m eans an d a covariance m atrix. Sim ulating tem p e ra tu re and polarization m aps are useful for a variety of reasons. T hey provide us w ith an excellent m ethod for testing d a ta analysis software, which will be subsequently discussed in C hapters 6 and 7. Sim ulating m aps also provides us w ith some insight into th e decom position of polarized m aps into curl free and divergence free parts, or th e so called “E and B decom position” . As discussed in C h ap ter 3 we use sim ulated m aps to test a key assum ption in th e frequentist estim ation of cosmological p aram eters. Also, [103] uses m any thousands of realizations to test th eir m ethod for analyzing CM B m aps w ith asym m etric beam s. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 In this C h ap ter we present three m ethods for generating sim ulated CM B anisotropy m aps. T he first is th e spherical harm onic expansion m ethod, a n d is described in Sec tion 5.2.1. T he CM B anisotropy is a function on th e sphere, and can therefore be expanded in term s of spherical harm onics. Because we are assum ing G aussian isotropic fluctuations th e correlation stru c tu re is m uch sim pler in spherical harm onic space. To be specific, the spherical harm onic coefficients are uncorrelated [98]. T he next m ethod uses a Fast Fourier Transform to generate m aps. It is com putationally m uch faster th a n th e previous m ethod, b u t has th e lim itation th a t is can only be used for sm all m aps. Also, can edge effects can bias the sta tistic a l properties of th e m aps based on th e m ap s tru c tu re (see [14] for a dis cussion of th is ). T he last m ethod we call th e two point correlation m ethod. It is by far the m ost com putationally expensive m ethod for generating a m ap, b u t is useful for generating m any (~ 10®)realizations [103]. We th en present a m ethod for com puting th e polarization a n d tem p e ra tu re power spectra w ith error bars using F F T ’s,and it is applied to sim ulated m aps to test th eir statistical properties. It is shown th a t th e power sp e ctra estim ated from the m aps do indeed agree w ith the sp ectra used to generate the m aps. T his ch ap ter finishes by using th e sim ulated m aps are to determ ine th e am ount of E B m ixing due to cross polarization. It is shown th a t for MAXIPOL th e net effect of cross p olarization on th e E E power spectrum is of a few percent. 5.1 Polarization Basics T h e electric field com ponents for a m onochrom atic electrom agnetic wave of frequency cuq propagating in th e z direction can be w ritten as follows: Ex = ax{t) cos {uot - 9x) (5.1) Ey = ay{t) cos {u>ot - 9 y ) , (5.2) where Ex and Ey sta n d for the E field com ponent in th e x and y direction, respectively. If these two com ponents have some correlation th en the wave is said to be polarized. T he Stokes’ param eters provide a sim ple form alism for quantifying th e p olarization of an elec trom agnetic wave. These are defined in term s of tim e averages: I = (®x) + (®y) Q =(4)-{ap (5.4) U = {2axay cos{9x - 9y)) (5.5) V = {2axaysin{9x - 9y)). (5.6) T he rad iatio n intensity I of th e wave is positive definite. T h e o th er th ree param eters can take either sign, and characterize th e polarization state. For com pletely unpolarized Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 rad iatio n we have th e condition Q = U = V = 0. It is also comm on to define th e polarization intensity, Ip, as Ip = a n d the degree of polarization is Ip/1. 0 is com pletely unpolarized light. + U^ + V"^, Perfectly polarized light has Ip — I, an d Ip = T he V p aram eter represents th e am ount of circular polarization, and for pure linear polarization F = 0. CM B photons T hom son scattered off of free electrons a t last scattering, which produces pure linear polarization. We therefore neglect th e Stokes p aram eter V in our discussion. T he observables in a CM B experim ent are th e tem p e ra tu re T{9,4>), which is equivalent to th e first Stokes param eter, and the linear stokes param eters Q { 9, ^ ) an d U{9,</)), all three of which are functions on the sky. At each point on th e sphere Q{9, </>) an d U {9,4>) are defined w ith respect to some orthonorm al u nit vectors x an d y th a t are orthogonal to the sphere (see A ppendix B). Note th a t an experim ent could in principle a tte m p t to determ ine V, b u t since it is expected to be zero m ost experim ents sensitive to polarization will neglect th a t term . In A ppendix B we show th a t th e two quantities Q + iU and Q — i l l are spin + 2 an d —2, respectively. We can therefore make th e following expansion using spin weighted spherical harm onics: T{9,(l>) = Y,4^Yem{9,4>) (5-7) im E 2YU^,<I>) (5.8) im Q{9,^)-iU{9,cl>) = (5.9) tm I t is useful to introduce th e following quantities (®2,tm — 2 ( “ 2 ,tm + 2,tm) 0,-2,tm) ■ (5.10) (5 -H ) We can now define th e real space quantities E{9,cl>) = Y,af^Ye^{9,ct>) (5.12) im B{9,<t>) = Y.^frnYtm{9A), (5-13) im and it can be shown [106] th a t E{9,(j)) and B{9, (j)) are ro tationally invariant under coor din ate ro tatio n . T his is in direct contrast to th e observables Q{9,4>) and U{9,(j)), which are coordinate system dependent. T he real space E{9,(j)) and B{9,4>) q u antities have the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 im p o rtan t characteristic properties th a t th e E{6^ cj)) field is curl free, V x E{9, (f)) — 0, and B{6,4>) is divergence free, ■B{d,(f)) ==0 [106]. It is interesting to investigate how the real space polarization quan tities behave under th e p arity transform ation x' — —x and y' = y, which is a reflection a b o u t th e y —axis. Prom th e spin-2 p ro p erty of polarization (see A ppendix B) Q'{9,(j)) = Q{9,4>), and U'{9,(j)) = —U{9,(j)) under this p arity transform ation. T his th en im plies th a t E'{9,(f>) = E{9,(j)), and B'{9,(f)) = -B{9,<p) under the sam e transform ations. 5.2 G enerating M aps According to th e cosmological principle th e CM B is statistically isotropic, m eaning there are no preferred sky directions, and the coefRcients are uncorrelated in th e spherical harm onic dom ain [98]. T his implies th a t (5.14) (5.15) m ') — BB) = C rSu'S: ) (5.16) For inflationary theories, th e CMB tem p eratu re and polarization anisotropy is a realization of a G aussian random field. Typically the T B and E B cross sp ectra are zero because B has th e opposite p a rity of T or E. Therefore, in m ost analyses th e T B an d E B cross sp ectra are neglected, b u t we include it here for com pleteness. T his m eans th a t th e spherical harm onic coefficients of th e real space observables {T{ 9 , 0), Q { 9 , 0), and U {9,0)) are G aussian random num bers. Also note th a t all these variables obey the reality condition th a t { - i r a l mi “ to (5.17) w here X — T , E , or B. Thus, given a set of tem p eratu re and polarization power sp ectra we w ant to produce a set of spherical harm onic coefRcients w ith the desired sta tistic a l properties. For each m we define a three elem ent vector a such th a t (5.18) and we w ant a to satisfy th e following statistical properties ■0 (®^m) ~ 0 0 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (5.19) 78 ■^ cr cr 1 {aimaim)—Mi — cr cf cr .cr cr cF . In other words, we take each (5.20) to be G aussian random variable w ith m ean zero, and covariance m atrix given by E quation 5.20. T his is done by choosing six G aussian random num bers w ith m ean zero and u n it variance, and form ing a three elem ent colum n vector x of complex num bers. T hen th e vector ^ (5.21) will have th e desired statistical properties, where L i is th e Cholesky decom position of the covariance m atrix Mu. T he Cholesky decom position is sim ilar to th e square root of a m atrix, an d has th e p roperty = M^. T he ^ 2 factor is in E q uation 5.21 because th e vector x is com prised of com plex num bers. By defining aim as in E q uation 5.21 its easy to see th a t th e p roperty in E q uation 5.19 holds, and {aimaj;,) = \i,L ix{L ix*f) = \ l i {x x *'^)l I (5.22) (5.23) = Ml. (5.24) Once the spherical harm onic coefficients w ith th e desired sta tistic a l properties have been generated th e real space quantities T (0, (^), Q{6,4>), and U{0,(l)) can be calculated. 5 .2.1 S p h erica l H arm on ic E x p a n sio n Tem perature M aps T he tem p e ra tu re m ap can be form ed using E quation 5.7. T h e spherical harm onics Yim{0, (p) and a j ^ coefficients are complex num bers. T he analytical expressions for m ultipole values a n d recursion relations for Yim{0-i<f>) are given in [72]. T he spherical harm onics can be explicitly w ritte n as YimieA) = MimiO)e^^‘^ where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.25) 79 an d [72] give efficient recursion relations for generating th e Legendre polynom ials P ^ { c o s 9 ) . These functions obey the orthogonality condition ^ I Yim{6,4>)Yiim>{0A)d^ = du'dmm', (5.27) th e conjugation relation YimiOA) = { - lT Y l_ ^ { e , c l > ) (5.28) and the spherical harm onic addition theorem states th a t 2/ 4-1 ^ Y , Y U e ,m m {e 'A ') = ^ ^ p r(c o s^ ) (5.29) m—-£ w here ^ is th e angle betw een (0,0) and {6',(j)'). Given a pixelization on th e sky (0,0), and a tem p e ra tu re power sp ectru m C j out to some m axim um m ultipole spherical harm onic /m a x , a sim ulated CMB te m p e ra tu re anisotropy m ap is form ed by choosing (/max + 1)^ random according to E q u atio n 5.21. For each / there are 2 / + 1 degrees of freedom. T he m = 0 coefficient is real (one degree of freedom ), a n d two degrees of freedom for every m > 1 coefficient because is complex. T he coefficients w ith m < —1 are th en determ ined by E quation 5.17. and Yim as We can rew rite th e complex num bers aim = a\^ + ia\^ (5.30) Yim = Yl^ + iY L - (5.31) where th e superscript T has been suppressed for th e m om ent. Now using E q uation 5.7 the tem p e ra tu re a t a given location on the sky is T (0 ,0 ) Y(^^m + iY L M m + ia \J = (5.32) Im Y - + \ I ~ Y L 4 m ) + K Y L 4 m + Y L a U \ ■ (5.33) y \ m ——l / A fter using th e conjugation relations and bit of algebra we get T {0,0) - + I (5.34) lm = l J w h e r e n o w it is e x p lic it ly c lea r t h a t T ( 0 , 0 ) is re a l, a s e x p e c t e d . Sim ulated tem p e ra tu re m aps generated using E q uation 5.34 w ith a ACDM cosmology and MAXIMA-I pixelization w ith various am ounts of correlation were used to generate the results shown in Figure 4.2. Also, num erous exam ples were shown in C h a p te r 3, where sim ulated m aps were used to test basic assum ptions of the frequentist cosmological p aram eter estim ation m ethod. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 Polarization M aps C om puting G aussian random Q{6, (f) and U{0, (j>) m aps is only slightly m ore involved th an th e tem p e ra tu re case. F irst, from E quations 5.10 and 5.11 we have a2,em = - a f m - icifm (5-35) a-2,im = ~(^fm + (5.36) Com bing these expressions w ith E quations 5.8 and 5.9 yields <3(^5 ~ -2Ylm) + iafm ( ^ ~ -2Yem) (5.37) tm u{6,(j)) = + -2Ytm) ~ io-fm ( ^Ytm ~ -2^fm ) > (5.38) tm which we rew rite as Q{OA) = + (5-39) tm U{dA) -^a g ,W em -ia g ,^tm - - (5.40) tm T he quantities Wem = ( 2Ytm + - 2Yt m ) / ‘^ and Xi m = ( 2^£m - ~ 2Y t m ) l ‘^ are given in [48]. T hey are given explicitly by WUOA) - - 2 i v J - f ^ ^ + J ^ (£ -l)')M ,^ (cos0) + (^ + m ) ^ M ( ,_ i ) ^ ( c o s 0 ) |e * - ^ X tm {Q A) = - 2 i V f ^ ^ { ( ^ - l ) c o s 0 M f ^ ( c o s 0 ) - ( £ + m )M (^_i)^(cos0)}e*"^-^, sin 6 f \ sm i) ^ / sm a ) (5.41) (5.42) where ^' = V I t I Prom E quation B.5 comes th e conjugation relations Wim iO A) = (5.44) XUGA) = (5.45) T his implies th a t th e m = 0 com ponent is real for WirniG, 4>) an d im aginary for X(,m{G, <p)Following th e argum ents in Section 5.2.1 we now show th a t th e p olarization m aps Q { 6 A ) and U { 6 A ) are real. T he spin ± 2 spherical harm onics and corresponding coefficients are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 complex num bers, so the polarization m aps as w ritten in E quations 5.39 a n d 5.40 in term s of its real and im aginary p arts. We only show explicitly th a t Q{0, (j)) is real; th e following argum ents can be easily applied to U{9,(f)). E xpanding E quation 5.39 in term s of its real and im aginary p a rts we get Q{d,<P) = - Y ^ { a f : ; , + i a f i ) { W l ^ + i W L ) + i { a Z + i a f i ) { X l ^ + iXi^){bA&) im im + i {^ ilW L + . (5.47) A fter a b it of work, an d m aking use of th e conjugation relations we get { m=i E ag,W i„ - a ^ X } „ - m =l (5.48) where all th e im aginary term s have canceled. A sim ilar relation can be derived for the U{9^ (f>) m ap: f m=i u(e,4.) = - E + 2 E Km=l 1 - “Z » ' L + ) . (5.49) A n exam ple of sim ulated T { 9 ,^ ) , Q{9,4>)-, and U{9,(j)) m aps using a square p atch of sky w ith 256 X 256 3' pixels is shown in Figure 5.1. These m aps were generated using a sta n d a rd ACDM cosmological m odel, using only T T , T E , and E E power spectra. T he power sp ectra for these m aps are shown in Figure 5.4. It is also im p o rta n t to note th a t this m ethod is com putationally expensive because of for each pixel we m ust com pute the double sum s in E quations 5.34, 5.48, and 5.49, w here (. typically goes up to 1500. However, ru n tim e can be cut significantly w ith the aid of a supercom puter. Since no com m unication is necessary betw een pixel calculations th e algorithm is em barrassingly parallel. A ssum ing we are sim ulating a m ap w ith N_pix pixels w ith N_procs processors th e vector of pixels can be d istrib u ted am ongst them w ith th e following pseudo code: MY_pix = N_pix/N_procs if(MY_pe EQ N _procs-l) then MY_pix += N_pix mod N_procs MY_s = MY_pe * (N_pix/N_procs), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 1 0 t 16 0 50 100 150 20 0 250 0 50 (a)T 100 150 20 0 250 (b)Q 1 2 0 50 100 150 20 0 250 (c) U F ig u r e 5.1; Simulated CMB anisotropy maps using nsing the spherical harmonic expansion method, with only T T, T E , and E E fluctuations. Each maps contains 65, 536 3' square pixels. No beam smoothing was included in these simulations. w here MY_pix is th e num ber of pixels assigned to a specific processor, MY_s is th e sta rtin g point in th e pixel array for a specific processor, and th e ’la s t’ processor is given any left over pixels if N_pix is not divisible by N .procs. T he m aps in Figure 5.1 w ith Np = 65536 and iVproc = 224 took ~ 21 m inutes to ru n on th e IBM SP2 m achine a t NERSC^. 5 .2 .2 S m all Sk y A p p ro x im a tio n G enerating sim ulated CM B tem p e ra tu re and polarization anisotropy m aps can be done very efficiently using fast Fourier transform s if we only consider a sm all p a tc h of sky, as opposed to th e full spherical. T his is because for a sm all enough p atch we can assum e th e sky is flat. ^http://www.nersc.gov Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 F irst assum ing we have a,n N x N m ap w ith square pixels of angular size A on a side, th en the i vector takes on discrete values given by „ n N " = “ T N y T he relationship betw een spherical harm onic m ultipole £ and angular separation 9 is ap proxim ately £ = 27t/0, or £ = 2nk. Therefore, a m ap which has 3' pixels, and 256 x 256 to ta l pixels would have a N yquist frequency of £Nyquist = 3600, an d th e sm allest m ultipole com ponent probed is £i = 28.12. Given some tem p e ra tu re m ap T{x), this could be expanded into Fourier com ponents f(k) J ( f x T{ x) = (5.51) w here b o th x = [x, y) and k = ( k i , k 2) are two dim ensional vectors. T h e Fourier com ponents obey the reality condition f{ki,k2) = f { N - k u N - k 2 ). (5.52) O f course th e polarization m aps Q{x) and U{x) can be expanded into Fourier com ponents as well Q{k) = j ( f x Q{x) (5.53) U{k) = J ( f x U{x) (5.54) and Q{k) and U{k) also obey the relation in E quation 5.52. If we w ish to decom pose the polarization m aps into a curl free com ponent, E{x), and divergence free com ponent, B{x), th en [15] gives a sim ple relation in Fourier space. T he Fourier com ponents are related by Q(k) = E{k)cos{2(j)i^) — B{k)sm{2(l)j^) (5.55) U(k) - E{k)sin{2(l>j^) + B{k)cos{2(f>j^), (5.56) where (j)k is th e angle a specific k m ode makes w ith th e horizontal axis. Analogous to the spherical harm onic case, th e E and B com ponents are linear com binations of th e Q and U com ponents in Fourier space, and vice versa. Prom E quations 5.55 an d 5.56 its clear th a t for pure E ty p e fluctuations Q will be m axim um a t 0 and vr degrees w ith th e horizontal axis, and m inim um a t 7t / 2 and 3n/2. Analogously th e U m ap for p u re E ty p e fluctuations will be m axim um a t 7t / 4 and 57t / 4 degrees w ith th e horizontal axis, an d m inim um a t 7t / 2 a n d 37t / 2 . Its easy to see th a t a transform ation to pure B type fluctuations, E Q U and U —> —Q. T his is illu strated explicitly in Figure 5.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B , causes 84 t 50 100 150 200 10 250 (a) A Gaussian hot spot with a peak in tensity of 20 /xK t 0 50 100 150 200 1 0 -3 50 250 (b) A Q map assuming the fluctuations are pure E from Figure 1.1.a. 100 150 20 0 250 (c) A U map assuming the fluctuations are pure E from Figure 1.1. a. 1 1 » 50 100 150 200 250 (d) A Q map assuming the fluctuations are pure B from Figure 1.1.a. 50 100 150 20 0 250 (e) A U map assuming the fluctuations are pure B from Figure 1.1. a. Figure 5.2; Various real space intensity maps for the Stokes vectors Q and U for either E ox B type fluctuations. This illustrates the fact that as E ^ B, then Q ^ U and U —t —Q. The vertical and horizontal axis axe pixel number. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 85 Therefore, to m ake sim ulated G aussian CMB m aps in th e sm all sky approxim ation we first choose G aussian random num bers in Fourier space such th at: {ff*) = c r ; {fE*) = C r (5.57) ; {B B *)^C ^^, (5.58) w here we’ve neglected unphysical cross correlation term s (which are straightforw ard to include if desired). In analogy w ith E quation 5.21, for a specific k m ode th is is done by letting Ir^TT f{ky = f{ky = xi\ (5.59) fiTT (5.60) w here x i a n d X2 are G aussian random num bers w ith m ean zero and variance one. T he E{k) a n d B{k ) m odes can be form ed in th e same fashion. Since we have included cross correlation, the random variables m ust again be form ed by taking th e Cholesky decom position of th e covariance m atrix as shown in Section 5.2 5 .2 .3 T w o P o in t C o rrela tio n M e th o d Yet a th ird m ethod exists for com puting G aussian random fields which, uses th e real space two point correlation function betw een pixel pairs to generate sim ulated m aps. T his m ethod is com putationally expensive for generating a few m aps because it involves form ing th e full real space covariance m atrix. However it is useful for generation a large num ber (several th ousand or m ore) m aps. T his was th e m ethod used in [103] to generate ~ 10® sim ulations to test th e effect of asym m etric beam s in a CM B receiver w hen m apping th e tem p eratu re anisotropy. Given a set of Np pixel locations of th e sky {9,4>) we can form a SiVp x ZNp signal covariance m atrix: ■ (T T ) (TQ ) {TU) ■ M (TQ ) {QQ) {QU) - , (5.61) . {TU) {QU} {UU) . w here each block covariance m atrix is N p X Np. T he covariance m atrix for th e tem p eratu re elem ents (derived in C h ap ter 1.4.3) is given by {TT)ij = ^ ^ ^ c rP ,{c o s9 ij). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.62) 86 T he other polarization and cross polarization tem p e ra tu re block covariance m atrices are given in [97], a n d are derived in C h ap ter 6. T hus, given a set a tem p e ra tu re an d polarization power sp ectra and a set of sky coordinates we can form th e signal covariance m atrix M . To com pute T , Q, and U m aps w ith the desired statistical pro p erties we next form a 3Np colum n vector of G aussian random variables w ith m ean zero an d variance unity, called X. T hen d = Lx , (5.63) w here L is Cholesky decom position of M , and T (5.64) Q U Form ing th e m atrix M is com putationally expensive, so this m ethod is not useful for gener atin g only a few realizations. However, once it is form ed m any realizations can be generated efficiently w ith E q uation 5.63. It was found th a t in practice Cholesky decom posing th e m a trix M for a MAXIMA-I size m ap w ith a sta n d a rd ACDM m odel again led com putational difficulties. R em em ber th a t in order to perform Cholesky decom position th e m atrix m ust be positive definite, i.e. all the eigenvalues m ust be greater th a n zero. However, due to num erical instabilities some very sm all eigenvalues m ay ap p ear as zero in th e code, causing th e code to crash. We therefore add a sm all am ount of noise in order to slightly increase th e eigenvalues which enables the code to run. T his is done by replacing M w here M + N, is a ZNp x 3A/p diagonal m atrix, w here th e N a diagonal elem ents represents the noise variance in pixel i. T he resulting sim ulated m aps th en are signal a n d noise (though th e noise com ponent is sm all depending on N ) . For th e specific case of sim ulating a noise variance of m a x i m a -I pointing w ith a ACDM m odel we add a T he resulting m aps are shown in Figure 5.3. T his sim ulation took ~ 20 m inutes to ru n on 224 processors on th e IBM SP2 a t N ERSC (seaborg). Each m aps have 5,972 pixels (the covariance m atrices have 17,916 x 17,916). Form ing th e m atrices are 0 { 9 N p ) operations, so its clear th a t a sim ulation w ith 65,536 pixels, like those form ed from th e previous two m ethods, are beyond the capabilities of th is m ethod. 5.3 Statistical Properties To test th e sta tistic a l properties of the sim ulated m aps generated using th e m ethods de scribed in C h ap ter 5.2 we construct estim ators for th e tem p e ra tu re an d p olarization power spectra of th e m aps. We can th en com pare the estim ated sp ectra w ith th e sp ectra used Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 M a x im a —I S i m u l a t e d M a p (T ) o M a x i m a - ] S i m u l a t e d M a p (Q ) t 56 14.8 15.0 15.2 15.4 15.6 15.8 18.0 1 0 16 16.2 (a )T (b)Q M a x i m a - I S i m u l a t e d M a p (U ) 14.6 15.0 15.2 15.4 (c) 15.6 15.8 16.0 16.2 U Figure 5.3: Simulated CMB anisotropy maps using the M A X I M A - I pointing. These maps were generated using the two point correlation, and contain only T T, E E type fluctuations, and T E correlation. The each pixel contains 1/iK^ noise. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 to construct th e m aps to determ ine if th e two are consistent. In th is section we construct power sp ectra estim ators using F F T ’s, an d assum e th e m aps are pure signal. Therefore, we can only apply these estim ators to sm all m aps generated from th e spherical harm onic transform m ethod, and F F T m ethod. T he reason we cannot apply th is technique to the m aps generated from th e two point correlation m ethod is because we found in Section 5.2.3 th a t a sm all am ount of diagonal noise had to be included in order to Cholesky decompose th e covariance m atrix. In C hapter 6 we describe a m ethod for estim ating th e tem p e ra tu re a n d polarization power sp ectra from CM B m aps w ith noise using the full spherical harm onic form alism . We also estim ate th e power sp ectra from th e m aps shown in Figure 5.3 We now construct an estim ator for and It is easy generalize th is estim ato r for CT'^ Given a flat sky tem p eratu re m ap T ( x ) , the each Fourier com ponent T{ k) can be com puted w ith F F T . E ach Fourier com ponent is a G aussian random num ber w ith m ean zero and variance k . We can therefore w rite down th e likelihood w here th e vector k has been broken into bins, k s is a. k vector in b in B , an d N b is the num ber of jfc| values in bin B . Also, the superscript T T has been rem oved from for simplicity. T hen up to an uninteresting constant -2 1 n £ = iV B ln C s + X ^ j ^ ^ ^ ^ . To find th e m axim um likelihood estim ate for the bin power spectrum (5.66) we set th e first derivative of th e log likelihood w ith respect to C b to zero, and find th a t Cb = (5.67) kB T he variance in th e estim ator can be found by taking th e inverse of th e second derivative of th e log likelihood a t th e m axim um likelihood value. T h e curvature of th e likelihood a t th e m axim um likelihood bin power spectrum value is dC^\CB-CB ^ q 2 ^ kB QZ ^ (5.68) Nb ^ (5.70) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 ^ 6000 e S- 0 500 1000 1500 500 2000 1000 1500 (b) T E (a) T T — 0.6 i I 30 tt tt * f- 0 500 1000 (c) E E 1500 2000 (d) B B F ig u r e 5.4: The T T , T E , E E , and B B power spectra estimated from simulated maps generated using the spherical harmonic expansion method. The solid lines are the theoretical models used to generate the maps, and the solid circles with error bars are computed using the FFT method described in the text. There is a residual B B component at small scales comparable with the pixel size of the maps. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 t 0 so 100 150 200 0 250 SO 100 200 250 (b)B (a) E Figure 5.5: The real space ISO E and B maps computed from the maps shown in Figure 5.1. From this we can com pute th e “error” (the square root of th e variance, ctb) in our m axim um likelihood estim ate for th e bin power spectrum , Cb , which is (5.71) T he error bars in our estim ator is sim ply the value of th e estim ato r divided by th e square root of th e num ber of values in b in B . We use th is technique to estim ate th e T T , E E , B B and T E power sp ectra w ith error bars for th e sim ulated T , Q, and U m aps shown in Figure 5.1. To estim ate a cross power spec tru m , for exam ple C j ^ , sim ply replace th e \T{kB)\ ^ in E quation 5.65 w ith ( r { k B ) E * { k B ) ^ ■ T h e results are shown in Figure 5.4. N ote th a t there is some residual B B signal in th e m aps. T his is due to edge effects and th e fact th a t a full sky m easurem ent is needed for a perfect E and B decom position. These effects are discussed in detail in [14]. Figure 5.5 shows the real space E and B m aps com puted from th e m aps in Figure 5.1. M ost of th e fluctuations are in the E m ap, b u t there are edge effects which show up in th e B m ap. To confldently say th a t the m aps generated do indeed have th e desired sta tistic a l prop erties we com pute a for each com puted power sp ectra w ith th e m odel used to generate th e maps: (cb X = E where the sum is over bin power and retical m ode. O f course this aB (5.72) is th e average power in bin B from th e theo is not X^ because the likelihood function is not G aussian Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 Table 5.1: Values for Simulated Maps Power Spectrum DoF TT TE EE BB 13 13 13 13 10.3 14.4 12.3 945 The of the power spectra estimated from the simulated maps with the models used to generate the maps shown in Figure 5.4. as a function of C b (it is G aussian as a function of T). We still com pute th e sta tistic in E quations 5.72 to determ ine how consistent the estim ate sp ectra are w ith th e m odels used to generate th e m aps. T he x2 is com puted for all four power sp ectra for all points w ith 50 < £ < 1500. T he T T , T E , a n d E E power sp ectra give values of 10.3, 14.4, an d 12.3, respectively, which represent a good fit to 13 DoF. T here is B leakage in th e sim ulated m aps pure E E and T E m aps from Figure 5.1, as evidenced by b o th the non-zero power B B power spectrum in Figure 5.4. T his fact yields a very high of 945 of th e estim ated B B power spectrum w ith the model. 5.4 Cross Polarization In th is section we use sim ulations of CM B polarization m aps to exam ine how cross polar ization in th e receiver m ay affect th e final d eterm ination of the p olarization power spectra. W hen polarized rad iatio n is incident on th e receiver th e polarization vector m ay be ro ta ted by some am ount due to im perfections in th e system . T his is called in stru m en tal polar ization. For exam ple, th e am ount of instru m en tal polarization in th e MAXIPOL receiver is approxim ately 5° ± 1° [77]. A ro tatio n of th e incom ing polarization vector would cause m ixing betw een E an d B polarization com ponents. In fact, we know from Section 5.2 th a t a ro tatio n of 45° converts E fluctuations com pletely into B type fluctuations, an d vice versa. T his is a serious problem for experim ents designed to m easure th e gradient free polarization com ponent. T he E signal is expected to be an order of m agnitude large th a n th e B signal [106]. A n y u n c e r ta in ty in t h e a m o u n t o f cr o ss p o la r iz a t io n in t h e r e c e iv e r w o u ld c o n v e r t a significant p o rtio n of the dom inant E signal into B type fluctuations, th u s m asking any real B signal evident in th e CMB. We use sim ulations to determ ine would m uch E —y B m ixing is induced for various am ounts of cross polarization. To determ ine how m uch po ten tial cross polarization will affect th e E an d B m easure m ents we use th e sim ulated m aps generated using th e spherical harm onic expansion m ethod. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 60 50 4’ degi 40 30 20 to 0 (a) E E (b) B B Figure 5.6: The E E and B B power spectra estimated of cross polarization. For each set of power spectra the in the upper right hand corner of the figure. The solid the pure signal maps with no cross polarization. There from simulated Q and U maps with various amounts corresponding amount of cross polarization is given line in the E E figure is the model used to generate are no B B fluctuations in the simulated maps. We generate sim ulated Q and U m aps th a t contain 65,536 6' square pixels, an d cover ~ 650 deg^ on th e sky. We use pure E polarization w hen generating these m aps, i.e. no B fluctu ations are present in th e m aps. At each pixel the m agnitude and angle of th e polarization vector can be com puted from the Q and U m aps using th e relation a y/O ^T lP (5.73) - a rc tan (U/Q) (5.74) A T h e angle of th e polarization vector, a, is ro ta ted by some angle, a n d th e new ’co n tam in ated ’ Q a n d U m aps can be recom puted. From th e contam inated polarization m aps th e E E and B B power sp e ctra can be com puted using th e F F T m ethod, an d these power sp ectra can be com pared w ith th e uncontam inated power spectra. Exam ples of th e polarization power sp ectra w ith various am ounts of cross polarization is shown in Figure 5.6. T he red points are the are power sp e ctra e stim ated from th e sim ulated m aps w ith no cross polarization input. T he red E E power sp ectra is consistent w ith the in p u t model, given by th e solid line. T he B B power spectrum w ith no cross polarization is slightly above zero, even though no B type fluctuations were used to generate these m aps. T his is due to th e incom plete sky coverage of th e sim ulations used. It is im possible to com pletely separate p a rtial sky polarization m aps perfectly into E a n d B com ponents [14]. Therefore, there is some E B m ixing sim ply due to im perfect com ponent separation. M ethods to decrease th e am ount oi E B m ixing from com ponent separation are given in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 [14]. We therefore a ttrib u te th e non zero B B signal for th e zero cross p olarization case to im perfect com ponent separation. As the am ount of cross polarization increases (an increase in th e angle used to ro ta te th e incident polarization vector) the am ount of E ^ B increases, as shown in Figure 5.6. T he larger th e am ount of cross polarization, the greater the am ount of E ^ B mixing. T he results are also sum m arized in Table 5.2, where th e am ount of RM S fluctuations in th e E a n d B m aps are shown. Also shown in th e final colum n of th e tab le is th e percent change in th e E m ap RMS. For only 0.5° th e percent change in th e E m ap RM S is 0.09%, which is clearly below th e noise sensitive of any realistic experim ent. A 5° cross polarization yields a 2.3% decrease in th e E m ap RM S (which of course im plies a 2.3% increase in the B RM S). However, th is is a worst case scenario for m a x ip o l . If m a x ip o l m ade no cross polarization correction th en th e net effect on th e E E power sp ectru m would be of a few percent level. F urtherm ore, a sim ple correction can be im plem ented by reverse ro ta tin g the polarization vectors in th e m aps by th e known cross polarization in th e system . However, for experim ents atte m p tin g to constrain the B B power spectrum th e few percent cross fluctuations are crucial, and it is im perative th a t such experim ents accurately constrain the cross polarization in their system to m inimize such effects. Table 5.2: RMS fluctuations from cross polarization angle (degrees) E RMS (pK) B RMS (pK) % change RMS (pK) 0.0 0.5 1.0 2.0 5.0 10 15 20 30 45 6.404 6.398 6.389 6.366 6.253 5.914 5.401 4.728 3.005 0.8619 0.8619 0.9093 0.9671 1.106 1.630 2.602 3.548 4.405 5.720 6.404 0.000 0.0933 0.2351 0.5893 2.365 7.644 15.66 26.17 53.07 86.54 The total amount of RMS fluctuations in the E and B maps for various amounts of input cross polarization given by the angle in the first column. The map simulations used to generate these numbers are described in the text. Note that for 45° rotation there is a complete E B conversion in RMS fluctuations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Power Spectrum E stim ation T his chapter discusses how to estim ate the tem p e ra tu re and p olarization power sp ectra from m easurem ents of th e CM B anisotropy using m axim um likelihood techniques. We first begin w ith an idealized CMB experim ent which m easures th e T , an d U anisotropy over th e entire sky w ith negligible noise and beam sm oothing. T hen noise a n d sim ple experi m ental beam s are included. T his enables us to construct estim ators for th e tem p e ra tu re and polarization power spectrum for a full sky m easurem ent, and derive some basic prop erties of th e tem p e ra tu re and polarization power sp e ctra we expect to com pute. We also derive th e expected variances, covariances, and correlations for our full sky tem p e ra tu re a n d polarization power sp ectra estim ators. T his is done in Section 6.1. We next discuss how to estim ate th e power sp ectra for an experim ent which has only p a rtia l sky coverage. F irst th e entire likelihood function in the spacial dom ain is derived in Section 6.2. For an experim ent th a t does span the entire sky th e m aps cannot be de com posed into spherical harm onic coefficients, so we m ust com pute th e likelihood function in th e real space dom ain. For G aussian fluctuations (som ething we assum e a t every stage) th e real space com ponents are correlated, which m eans we have to deal w ith a large, dense covariance m atrix in th e likelihood function. We therefore employ th e use of high perfor m ance super com puters when estim ating C f ^ , C f ^ , and power spectra. Even though we expect no T B or E E signal because those com ponents have opposite parity, we still construct estim ators for those cross power sp ectra as well. E stim atin g these com po nents can provide for a strong consistency check th a t a given d a ta set does not contain any spurious signals. Once th e likelihood in th e real space pixel dom ain has been w ritte n down we wish to m axim ize it as a function of th e tem p eratu re and polarization power spectra. T his is done using th e q u ad ratic estim ator technique, which is discussed in Section 6.3. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T his is an 95 extension to polarization from th e m axim um likelihood m ethods for th e tem peratxme only case discussed in [9, 11]. T his requires m atrix inversion of m atrix -m atrix m ultiplication of large dense m atrices. We next apply th e q u ad ratic estim ator technique to sim ulated tem p e ra tu re a n d polarization m aps using the MAXIMA-I pointing. It is shown th a t the estim ated power sp ectra agree w ith th e m odel used to generate th e sim ulations. We end this chapter by investigating the correlations betw een power sp ectra estim ates, an d show th a t these correlations are expected given w hat is derived in Section 6.1. 6.1 Full Sky A pproxim ation T he following argum ents are sim ilar to those given in C h a p te r 4.2.1, except th a t in th a t case we considered two T m aps, and here we have a T , Q, and U m ap. Suppose we have a full sky m easurem ent of th e CM B anisotropy in th e T , Q, and U stokes p aram eters. Because CM B photons T hom son scattered off electrons a t last scattering we expect no circular polarization (i.e. V = G). We m ake th e assum ption th a t the m aps from each experim ent have negligible noise and neglect any beam sm oothing. Prom each experim ent we have a pixelized m ap of th e CMB anisotropy, which we call T ( 0 ,0), Q(0, (^), an d 1/(0, </>). Prom these m aps we can calculate aL J T { 0, 4>)Ytm{e,<l>)dn, = (6 . 1 ) a n d from E quations 5.39 and 5.40 we get «fm =- j = Q{e,(j>)Wtm(eA) + iu{e,<i))Xi,^{e,(f)d^ - J Q{6, m u e . 0) - iU{6, 4>)d^. (6 .2 ) (6.3) A ssum ing G aussianity and statistical isotropy implies th a t a j ^ , a f ^ , and a f ^ and have th e sta tistic properties given in E quations 5.14, 5.15, an d 5.16 . E ach of these spherical harm onic coefficients can be estim ated for every i and m from th e th e full sky m aps using E quations 6.1, 6.2, and 6.3. Por each I and m we can w rite a com bined experim ent d a ta vector as (6.4) dim — which has th e property ^ m ,= \ c r c r c r o r c r c r . c r c r c ] r . Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (6.5) 96 Now th e log likelihood (up to an uninteresting constant), £ , is -2 1 n £ ^f(2 £ + l)ln lM il-l- = £V (6.6) / m=— i Solving for th e m axim um likelihood tem p eratu re and polarization power sp e ctra an d cross sp ectra for a given I yields (6.7) m—— l o r = ^ m=—£ E ; o r = m=— t o r =^ e m=— i E lotai' : o r =~ E «■ m = — t T his is exactly w hat was found in [48]. Using th e m ethods described in Section 4.2.1 we can com pute th e variances, covariances, and correlations for the various tem p eratu re and polarization auto- and cross-spectra, es tim ators. T h e variances are 9 // JT'I*'/’\\ 2 Var(Cr) = V ar((Jf® ) V a r(C fB ) = - ^ ^ o r ) " (6.11) = V ii(c r ) = (6.12) ^^{^{ory+crory (6.i3) T his is sim ilar to w hat was [48], except th a t here we considering th e full sky negligible noise case. To get th e expected error bars on th e auto- an d cross-spectra estim ates we sim ply take th e square root of the variances. These represent th e “b est possible” error bars one could hope to get w hen m easuring the tem p e ra tu re and polarization power spectra. To get th e expected error bars for a m ore realistic experim ent, one replaces th e au to spectra in E quations 6.10- 6.13 w ith + (6.14) w here w is th e noise weight (inverse variance) in each pixel [48], a n d Be is th e spherical harm onic beam filter function. T he resulting expressions th en m atch w h at [48]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was found in 97 T he covariances for our estim ates of th e tem p eratu re and p olarization power spectra (again going back to th e idealized full sky negligible noise case) are C o v ( c r .< 7 ,“ ) = ^ ( c T C o v ( c fB , c D ) ' 2 / = ^ ( c 2^ + 1 V W C o v (C f * , C D = 2^ + 1 ^ f ^ ; C o v ( C r .c r ) = ^ ( c , ™ ) ' ; ’ 2 -^TT/^TE C o v ( c r , c P ) = ^ c r <'i 5 , ’ — J 2£ + 1 ;; ov((7f C ov(C 'f B, C D = , \ i ^ t ) 2^+1 where we have only given th e correlations for estim ators which we expect to be non-zero. T hus, the correlations betw een estim ates are C o r r iC D C D = Q Covv{Cf^, Cf^) = , ; C o rr(C r, C D = Q s iS-L ; C orr(C r,C P ) = -----------------------J { c p ) C o rrlC fB .C fB ) = ; C o n { C f > . C D = ---------- V ^ C fB C g B ---------^ (c p ) cr^icq --^) + c p cfB N ote th e com plicated expression for th e C j^ = + cj^cf^ + C f^O D and C j ^ estim ators, which should be zero if = 0 as expected. At the end of this C h ap ter we show how these correlations do exist for actu al estim ates of th e power spectra. 6.2 R eal Space Likelihood Function Up to now we have been considering full sky m aps in our toy m odels of CM B m easurem ents. Since in practice this is near im possible we now consider the case of p a rtia l sky coverage. W ith o u t com plete sky coverage th e individual spherical harm onic coefficients cannot be determ ined. We are therefore forced to deal w ith th e real space likelihood functions, which is m ore com putationally challenging due to off diagonal correlations in th e correlation m atrix. We can w rite th e d a ta vector in pixel space as T Q U + n, (6.15) w here d and n (the pixel noise) are now colum n vectors of length 3Ap. A ssum ing th e signal and noise w ithin are uncorrelated we find th a t {dd^) = M = S +N Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.16) 98 ■ {TT) {TQ) {TU) {TQ) {QQ) {QU) + N, {QU) {UU) _ . {TU) (6.17) w here each Np x Np block m atrix is a function of th e power spectra. These are derived in subsequent Section 6.2.1. A gain up to an uninteresting constant, we can w rite down our likelihood for d as -2 1 n £ 6.2.1 {3Np)ln\M \+ d^M -^d. = (6.18) T h e S ign al C ovariance M a trix In this section we derive th e signal covariance m atrix for th e CMB polarization and tem per a tu re /p o la riz a tio n experim ents. These are th en used to form th e signal covariance m atrix, which in tu rn is used to construct th e likelihood function for a CM B d a ta set com prised of pixelized T , Q, a n d U m aps. In C h ap ter 1 we derived the tem p eratu re two point correlation function {TT). In this section we derive (QQ), (UU), (TQ), (TU), and (QU). T his follows w hat was derived in [48] and [106], who provide a com plete form alism of CM B polarization. Also, for explicit form ulation of th e tem p e ra tu re and polarization covariance m atrix see [97]. As sta te d in C h ap ter 5, these functions are not only im p o rtan t for d a ta analysis, b u t can also be used to generate sim ulated CM B m aps. To sim plify our argum ents we explicitly derive only (TQ) and (QQ). T h e following can be easily applied to derive (UU), (TU), and {QU). From E quations 5.7 and 5.39, and w riting everything out explicitly we have {T{B,4,)Q{e',<!>')) = Im t'm' - E 4,'). E (6.19) im i'm' Using E quations 5.14 and 5.15 yield {T{6,<I>)Q{9',<!>')) = - Y ^ C r ( j 2 " ^ i m { 0 , m L i ^ ' , <!>')] i \ m / ( i' £ Y e m { e , 4 > ) X i m { 0 ' , <!>')) . i \ m (6.20) J In principle th is is all we need to com pute th e two p oint correlation function. However, E quation 6.20 can be sim plified w ith th e aid of the spin spherical harm onic ad d itio n theorem . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 E quation B.6. T his yields the following relations ( 6 . 22 ) m which gives { T { e A ) Q { 0 ' , <!>')) = - ^ ^ ^ ^ W M { C F ^ o s { 2 P ) + Cj^sin{2l3)), (6.23) w here ( is th e angle betw een {9,4>)and {O', <j)'), and see A ppendix B for th e in te rp re ta tio n of th e angle /3.T his is equivalent to w hat was found in [48, 97]. T o derive th e (QQ) function we begin w ith E quation 5.39, which yields {Q{e,ci>)Q{e',ct>')) = J2T.(^frn4m')wue,m;,^,{9',<t>') im I'm' + {af^af,l,)X,m{0.4>)Xi'm' (^', 4>') = (6-24) Y.Cf^(j2^im{e,ct>)WimiO'A')] i \ m / i ^ X t ^ { 9 , < l > ) X l ^ { e ' , < t> ' ^ . (6.25) A gain using th e spin spherical harm onic ad dition theorem gives = jnp I 1 y - ^ ( l E ,2 ( ^ ) c o s ( 2 « ) c o s ( 2 / 3 ) -X ^ 2 (a s in (2 « )s in (2 /3 )) m (6.26) Y ,X im {eA )X lm {G 'A ') = y ^ ^ ( W , 2 ( 0 s i n ( 2 « ) s i n ( 2 ^ ) - X ,2 ( O c o s ( 2 a ) c o s ( 2 ^ ) ) . m (6.27) Again, see A ppendix B for th e physical in te rp re ta tio n of th e angles a an d We can now w r ite o u t t h e fu ll tw o p o in t c o r r e la tio n f u n c tio n for t h e Q s to k e s p a r a m e te r b y c o m b in in g E quation 6.25, 6.26, and 6.27. T his is a som ew hat com plicated expression an d is a function of th e angles a and T hings simplify considerably if we first assum e th e two points (0, (f)) an d {O' ^(j)') have an orientation such th a t a great circle passing th ro u g h th e two points is parallel w ith th e equator (i.e. . they are connected by w hat we define as th e horizontal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 axis). In th a t case a = /3 = tt/ 2, and we get {Q{e, 4,)Q{0\ </>'))n = {crw t2 (0 - C f^X n iO ), 2 (6-29) w here the subscript H represents th e fact th a t this is only valid for points connected along th e horizontal. T he o ther p o lariza tio n /te m p era tu re two point correlation functions are { Q (0 ,0 )C /(0 ',0 '))h - ^J ^^C riW e 2 {0 + X i2 i0 ) (6.31) {U{9,<t>)U{9\4>'))u = X; t (6.32) 47T For pixel pairs w here th a t do not lie on the horizontal one sim ply calculates th e angles а, and P, th e value of the tem p e ra tu re /p o la riz a tio n two point correlation functions in the horizontal fram e, and th en perform the ap p ro p riate rotations. T his is described explicitly in [97], a n d in A ppendix C.2 we give C source code for com puting th e angles a and /0, and th e two p oint correlation functions in th e horizontal fram e. These are th e n used to form th e entire signal covariance m atrix as w ritten in E q uation 6.17 б.3 Q uadratic E stim ator To m axim ize th e likelihood function given E quation 6.18 as a function of th e tem p e ra tu re and polarization power sp ectra we employ a q u ad ratic estim ator. T his technique is described in detail in [9] for estim ating tem p eratu re power sp ectra alone. However, generalizing from th e tem p e ra tu re to th e tem p e ra tu re /p o la riz a tio n case is straightforw ard. In stead of the d a ta being sim ply one tem p eratu re m ap th e d a ta vector is w ritte n as in E q uation 6.15. Also, if each m ap contains Np pixels, th e signal and noise covariance m atrices an d now 3 N p X 3Np. We now briefly review the q u ad ratic estim ator algorithm as described in [9]. Suppose ou r likelihood function £ is a function of a set of p aram eters C b - Given some in itial guess a t th e param eter values, Co, th e correction SC q iCo = - I dC l J ^ 1 ^ / Cb =Co Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (6.33) 101 would m axim ize th e log likelihood function if th e log likelihood is a q u ad ratic function (the likelihood is a G aussian function) of the param eters C b - O f course th is is not exactly the case, so we by letting th e next guess C\ — Co + SC q, and th en ite rate u n til some convergence criteria is reached. To construct SC q we m ust calculate the first and second derivatives of th e log likelihood w ith respect to th e param eters. Using E quation 6.18 we see th a t = dCe d'^lnC iiv 2 (6.34) rr.r/j,T { Ar-l OCb C b ' 1(m 2 I ~^ — ~ — dCBdCB' + - Tr 2 dS dS 3Cb OCb ' (6.35) T he second derivative of the log likelihood w ith respect to th e p aram eters, or curvature m atrix, is quite cum bersom e to calculate. Things sim plify considerably if we com pute the ensem ble average of th e curvature m atrix, the F isher m atrix F b b ' ■,which is = (e ) r- i dS 5 5 3Cb 9Cb' (6.37) T his is because (dd^) = M . Henceforth th e Fisher m atrix quan tity is assum ed to be the inverse covariance m atrix (i.e. to th e errors in th e bin powers you take th e inverse square root of th e F isher m atrix), an approxim ation which is justified in [9]. T hus, to m axim ize the likelihood given in E quation 6.18 as a function of bin tem p e ra tu re an d polarization power spectra we first form th e two quantities given in E quations 6.34 a n d 6.37. These quantities are com puted using w hat was derived in Section 6.2.1. A parallel im plem entation of this algorithm for finding the peak of th e m axim um in likelihood space for the tem p eratu re case is given in [11]. Here, we b u ilt upon th e software developed in [11], an d extended this to include polarization a n d cross p olarization power spectra. T he fundam ental difference change in including polarization is th e calculation of th e signal covariance m atrix, which was described in detail in Section 6.2.1. A dditionally, we provide C source code for com puting th e ro tatio n angles necessary for form ing polarization correlation term s in A ppendix C. T his software is fully scalable a n d designed to ru n on a m ultiple processor d istrib u ted m em ory supercom puters. It uses th e message passing Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 102 lib rary (M PI) for com m unication, and th e ESSL^, LAPACK^, an d ScaLAPACK^ num erical libraries for th e linear algebra com putations. An exam ple of th e p o lariza tio n /te m p era tu re q u ad ratic estim ator is given in Figure 6.1. We used th e sim ulated MAXIMA-I T , Q, and U m aps shown in Figure 5.3, added l^ K ^ noise per pixel, an d estim ated the T T , T E , E E , and B B power spectra. As is evident by the figure, th e estim ated power sp ectra agrees w ith th e theoretical m odels used to generate the m aps, as expected. We also show th e correlations betw een th e tem p e ra tu re and p olarization estim ates in Table 6.1. These were com puted assum ing th e Fisher m atrix is th e covariance m atrix, w hich is exactly how th e correlations where com puted in C h ap ter 4 for th e cross-spectrum case. T he correlations com puted agree w ith w hat was derived in Section 6.1 It was shown th a t C o rr(C ^ ^ , C § ^ ) is positive definite, whereas C orr((7^^, C q ^ ) an d C o r i ( C § ^ , C g ^ ) are not. Also, it was derived th a t C o rr(C ^ ^ , and C o r r { C g ^ , are proportional to C g ^ , which is also consistent w ith th e results from th e q u ad ratic estim ato r presented in Table 6.1. For bins where C g ^ is negative Co T r{ C g^ ,C g^ ) a n d C o r r ( C g ^ , C g ^ ) are negative, an d th e opposite is tru e for bins w here C g ^ is positive. Finally, it was derived th a t CoTT(Cg^, C g ^ ) = C o r r { C g ^ , C g ^ ) . However, th is is not tru e for th e lower £ bins. T his is because th e m a x im a - i sim ulations used to generate these results do not contain inform ation a t large angular scales. For larger £ bins it is clear th a t th is relationship holds. Specifically, for all bins I > 261 we find th a t Covr{Cg^, C g ^ ) — C or r ( C g^ , C g ^ ) w ithin two percent accuracy. ^h ttp ://w w w -l.ibm.com /servers/eserver/pseries/library/sp_books/essl.htm l ^ http;//www.netlib.org/lapack/ ®http://w w w .netlib.org/scalapack/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 150 100 „ 50 6000 0. 4000 -50 100 2000 150 •200 0 200 600 800 1000 1200 (a) The T T power spectrum estimated from the temperature map shown in Figure 5.3 using the quadratic estimator. (b) The T E power spectrum estim ated from the polarization maps shown in Figure 5.3 using the quadratic estimator. 60 50 a ->0 P A 30 10 0 1000 1200 (c) The E E power spectrum estim ated from the temperature maps shown in Figure 5.3 using the quadratic estimator. F ig u r e 6.1: Estimated power spectra using the quadratic estimator. The circles with error bars are the estimated power spectra, and the solid lines are the underlying models of the maps. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 Table 6.1: Correlations between Temp/Polarization Bin Spectra Estimates t bin [B] [ 2, 35] [ 36, 110] [ 111, 185] [ 186, 260] [ 261, 335] [ 336, 410] [ 411, 485] [ 486, 560] [ 561, 635] [ 636, 710] [711, 785] [ 786, 1500] C o r r ( C r ,^ r ) 0.27 0.28 0.28 0.04 0.19 0.00 0.11 0.00 0.00 0.12 0.14 0.01 C o rr(C g ^,(7 |^) C o rr(C |^ ,C 'g ^ ) -0.71 -0.69 -0.68 0.30 0.57 0.10 -0.45 -0.01 -0.10 -0.47 -0.51 -0.14 -0.49 -0.46 -0.65 0.27 0.55 0.12 -0.44 0.01 -0.11 -0.46 -0.51 -0.11 The correlations between temperature and polarization spectra estimates shown in Figure 6.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 7 MAXIPOL Sim ulations In this ch ap ter we develop a d a ta analysis pipeline designed to estim ate m axim um likelihood (ML) CMB tem p e ra tu re and polarization m aps from a MAXIPOL like experim ent. We first dem onstrate in Section 7.1 how to generate a sim ulated m a x ip o l like tim e stream , which is a com bination of b o th signal and noise. In Section 7.2 th e entire ML m ap m aking form alism is developed for th e specific case of a polarim eter w ith a ro ta tin g H W P and fixed linear polarizer. We th en test th e m ap m aking software w ith two different sim ulations. first test uses th e actu al pointing inform ation from the realizations. m a x ip o l - i We use sim ulated polarization m aps in th e T he fiight, an d various noise m a x ip o l - i region an d a sm all am ount of noise to generate a sim ulated tim e stream . T his tim e stream w ith known noise properties is used as in p u t to th e ML m ap m aking software. It is shown th a t th e fluctuations in th e in p u t m aps are reproduces in th e estim ated ML m aps. T h e second test involves only a sm all num ber of pixels, which allows us to com pare pixel by pixel th e in p u t an d o u tp u t values of th e m aps and pixel noise covariance m atrix. We d em onstrate how to calculate th e pixel noise covariance m atrix elem ents in th e presence of pure w hite noise in the tim e dom ain. It is shown th a t variance in th e polarization m aps is approxim ately twice th a t of th e tem p e ra tu re m ap, and the noise variance for all m aps decreases linearly w ith the num ber of tim e sam ples in a given pixel. 7.1 T im e Stream Sim ulations In this section we describe how to generate a sim ulated tim e stream for a m a x ip o l like ex perim ent. T h e in p u ts to th e sim ulation are a pointing strategy, pure signal CM B anisotropy m aps generated using one th e m ethods described in C h ap ter 5, a n d a noise power spectrum . T he sim ulations m ethods discussed here are general enough to apply to different polarim eter techniques, b u t here we show how one would apply these m ethods to th e specific MAXIPOL 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 situation. T he MAXIPOL receiver is com prised of 12 photom eters w ith b a n d w id th centered a t 140 GHz, and 4 bolom eters w ith a t 420 GHz. For a thorough discussion of bolom eters see [100], and for m ore inform ation on the experim ental aspects of MAXIPOL see [46, 47]. For sim plicity we consider only one channel. G eneralizing to m ultiple channels is straightforw ard because d a ta is com bined in th e m ap dom ain. In other words, th e ML m aps for individual detectors are com puted, th e final m ap is sim ply th e m ean of each individual m ap, and the elem ents of th e noise covariance m atrix are th e sam ple variances an d covariances. In our sim ulations we use th e pointing for th e B45 channel. T his pointing pixelized in 3' square pixels in galactic coordinates is shown in Figure 7.1. Using this pointing we direct it a t a pure signal sim ulated m ap, and th e tim e stream signal is com puted as discussed in th e next Section. 7 .1.1 S ign al T he m ost sim ple m odel th e tim e stream is th a t it is a com bination of b o th signal and noise d = s + n, (7.1) where each variable is a vector w ith Nt tim e stream elem ents. We first wish to determ ine the signal com ponent s of th e tim e stream d a ta for a polarization experim ent. T he relationship betw een th e in p u t and o u tp u t polarization for a polarizer can be determ ined using stokes vectors a n d th e M uller m atrix of the polarizer [82]. For an a rb itra ry polarization system w ith M uller m atrix M , if we let S and S' represent th e in p u t and o u tp u t polarization Stokes vectors, then S' = MS. (7.2) T he MAXIPOL instru m en t consists of a ro ta tin g h alf wave plate (H W P) a n d a fixed grid. T he M uller m atrix for an ideal HW P, H , is [82] H = 1 0 0 0 cos(4a) sin(4a) 0 sin(4o!) - cos(4o:) (7.3) where a is th e angle th e H W P makes w ith th e reference direction. MAXIPOL has adopted the convection th a t the reference direction (i.e. the axis w ith which th e Stokes vector are defined) is a great circle on the sphere passing th rough th e n o rth galactic point (N G P) [16]. T he linear grid on th e MAXIPOL system is fixed w ith respect to th e in stru m en t, so the axis of transm ission is a constant in the in strum ental fram e. However we are defining Q Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 and U w ith respect to th e sky, so th e axis of transm ission of th e grid will vary as a function of tim e (as th e sky rotates). We call th e angle of transm ission of th e grid w ith respect to T hen the M uller m atrix for the grid, G, is th e reference direction cos26 1 sin 26 cos 25 (cos 25)^ cos 25 sin 25 sin 25 cos 25 sin 25 (sin 25)^ (7.4) Therefore th e out-com ing Stokes vectors from the MAXIPOL system is given b y S' = O H S . (7.5) O f course th e detectors only see the intensity of th e in p u t radiation, so we are only concerned w ith the / ' com ponent. Therefore ^ (Tk + Qk cos(4o!i = 26i) + Uk sin(4aj - 25j)) ^ { T k + Qk cos 6i + U k s m e i ) , (7.6) (7.7) w here we have defined th e angle 6 — 4a - 25, and k represents th e pixel elem ent th a t tim e stream sam ple i is pointing at. T hus, by com bining a pointing strategy, H W P and grid angles projected on th e sky, and sim ulated CM B m aps E q uation 7.7 allows us to generate a sim ulated pure signal tim e stream . 7.1.2 Noise We make th e assum ption th a t the random noise in th e detectors is a random realization of a G aussian probability d istribution. T his assum ption is key not only in generating the noise, b u t also in th e m ap m aking procedure, which is described in Section 7.2. We begin w ith th e noise power spectrum , P{k). It is easy to show th a t for an idealized tim e stream th e power spectrum is sim ply th e Fourier transform of th e noise correlation m atrix, Aftt' = (n (t)n (t')). Let n{k) be th e Fourier transform of n (t), We assum e th e noise in th e Fourier dom ain is uncorrelated. T hen (n (t)n (t')) = j { n { k ) n i k ' ) ) e ‘^^^^'^^-'^'^'Ukdk' (7.8) = J (7.9) In the ideal case th e tim e noise correlation m atrix is circulant. M atrix inversion and m atrixvector m ultiplication can be perform ed efficiently using F F T ’s for a circulant m atrix. T his will become extrem ely useful when generating m axim um likelihood m aps from th e tim e stream . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 To generate a sim ulated tim e stream we first generate th e random variables in Fourier space th a t have th e properties {h{k)) = 0 (7.10) {n{k)h{k')) = P{k)Skk' (7.11) O f course n(t) is real, so we have th e condition n(k) — n ( —k)*. Therefore, if x i and X2 are G aussian random num bers w ith m ean zero and variance one th en let n ( k ) ’’ - xix/(F(k)/2 (7.12) h(ky X2y^(P{m , (7.13) = w here th e superscripts r and i sta n d for th e real and im aginary p a rts, respectively. T his is done for th e positive m odes only, from which th e negative m odes are determ ined trivially. T h en n(t) is obtained efficiently from h(k) via an F F T . 7.2 M ap M aking To determ ine th e pixelized anisotropy m aps from the tim e stream we use m axim um likeli hood techniques. For an exhaustive list of m ap m aking algorithm s w ith an application to th e MAXIMA-I d a ta see [90]. T he m ap m aking techniques are also described in detail in, for exam ple, [11, 93]. Those papers discuss ML m ap m aking for experim ents th a t m easure tem p e ra tu re fluctuations only. We rew rite E q uation 7.7 w ith noise as (7.14) A m + n. w here m are th e m aps and A is called th e pointing m atrix. A ssum ing each m ap contains Np pixels, th en m is a colum n vector of length 3Np, and A is a n Nt x 3Np. H ereinafter we m ake th e assum ption th a t th e m aps in m are stored in th e order T , Q, a n d th en U. For a MAXIPOL like experim ent th e pointing m atrix is sparse, w ith only th ree non zero entries for each row. C onsider an experim ent th a t consists of 4 tim e stream sam ples. In th is exam ple th e first th ree tim e sam ples point a t three different pixels, and th e last sam ple retu rn s to th e first pixel. If we take a MAXIPOL type polarim eter, th en the pointing m atrix would look like 1 2 0 0 sin 0 0 0 0 cos 0 0 0 COS 01 0 0 sin 0 1 0 1 0 0 cos 0 2 0 0 sin 0 2 0 0 cos 0 3 0 0 sin 0 3 0 0 ' 1 0 0 0 1 0 1 0 (7.15) Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 109 where 9i is dependent on th e transm ission axis of th e grid and H W P w ith respect to some reference direction on th e sky. It is easy to see th a t for a m ore realistic m a x ip o l type experim ent {Nt ~ 10® and Np ~ 10®), A will be very sparse. We assum e th a t th e tim e dom ain noise n obeys a statio n ary G aussian probability dis trib u tio n . Therefore, we can w rite down the likelihood A ") = T he noise correlation m atrix A/’t can be estim ated from th e tim e dom ain d a ta using iterative techniques described in [22]. Using E quation 7.14 we can rew rite th e likelihood in term s of th e tim e ordered d a ta d (som ething we know), and th e m ap m (w hat we w ant to estim ate). T his gives, C(m\d) - ^___. (7 17) { -i(r f- A m ) T A 7 (d - A m ) } We therefore wish to m axim ize the likelihood £ as a function of th e m aps m . T his is done w ith expression m = {A ^A ff^A y^ (7.18) Afp = {A ^A fyA y\ (7.19) w here Afp is th e pixel noise correlation m atrix. T he m ap m aking algorithm can be broken into two steps. F irst, form the pixel noise correlation m atrix w ith E q uation 7.19, and second com pute the m axim um likelihood m aps w ith E quation 7.18. Im m ediately one can see com putational difficulties in b o th storage and com pute tim e. For a realistic MAXIPOL like scan, w ith Nt = 10® a n d Np = 10®, th en A would have 3 x 10®elem ents and Aft GB and 4 T B of storage would have 10®^. These m atrices would require 12 respectively, assum ing 4 byte precision. W hile reading in th e first m atrix is possible (but not trivial) w ith m odern day supercom puters, th e second is not. Secondly, inverting a 10® x 10® m atrix is clearly com putationally prohibitive, as is all of the m atrix -m atrix m ultiplications in E quations 7.18 and 7.19. To reduce th e storage space required by th e m ap m aking algorithm s we m ake use of the fact th a t th e m atrix Aft is circulant, which m eans th e entire m atrix is specified by its first row. Therefore we’ve reduced th e storage of Aft from 4 T B down to 4 MB. Secondly we can significantly reduce the storage of A by m aking use o f sparseness an d sym m etry. For each row only one integer is required to specify the location of th e non-zero entries, so A can be broken into a iVj x 3 a n d a Nt x 1 “location” m atrix. T his reduces th e storage of A from 12 GB to 16 MB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 To reduce th e com putational requirem ents of th e m ap m aking algorithm we again make use of the fact th a t Mt is circulant. A circulant m atrix C has th e p ro p erty th a t its eigenvalues are given by th e Fourier transform of th e first row of th e m atrix, and it can be diagonalized as follows^ C = (7.20) w here F if th e Fourier transform m atrix, and A is a diagonal m atrix w here each diagonal entry is th e Fourier transform of the first row of C. T he m atrix F has th e p roperty th a t it is u n itary ( F ^ = F “ ^), and if x is an a rb itra ry complex colum n vector w ith some finite length, th en F x is th e F F T of x. T hus, com puting = F ^ A -^ F (7.21) requires only 2 F F T ’s and inverting A “ \ which is trivial. Also, circulant m atrix-vector m ultiplication C x is done w ith 3 F F T ’s and com puting Arc, which is also trivial. W ith these benefits th e m ost com putationally expensive step becom es form ing th e in verse pixel noise covariance m atrix, By E xploiting th e sparseness of A these operation can be perform ed w ith 0 { N f ) operations. T he o peration count for th is algorithm can be fu rth er reduced w ith th e “M A D C A P” approxim ation [11, 90], w here we assum e th a t no noise correlations exist for tim e stream elem ents separated by m ore th a n some tim e length A. If th e num ber of tim e stream elem ents w here correlations exist is Nx, th en the operation count is reduced to 0 { N t N \ ) . We can decrease th e operatio n count even further if th e algorithm is ru n on a supercom puter w ith M PI using A^proc processors. No com m uni cation is necessary betw een tim e elem ent pairs, so th e tim e stream can be broken into ATproc segm ents, an d th e o peration count is 0{NtN\/Nproc)- T h e C source code for form ing the pixel noise correlation m atrix is given in A ppendix C .l, which m akes use of all of th e above approxim ations. T he code also uses the fftw package^, a highly optim ized F F T package available on b o th th e IBM SP a t MSI and on Seaborg a t NERSC. To provide some exam ples of the com putational requirem ents of th e m ap m aking algo rith m we ru n th e code for various exam ples, and present th e results in Table 7.1. N ote th a t for sim ulations w ith N x « Nt th e lim iting calculation of th e algorithm is inverting the inverse pixel noise correlation m atrix, which requires 0 { N ^ ) operations. For th is reason, th e sim ulations w ith Nx = 1 used only one processor because there was no speed up in adding addition processors. T his is because th e only parallel p o rtio n of th e program was in form ing the pixel noise correlation m atrix, which is done very fast when N x — 1Actually these two properties of Circulant Matrices are equivalent. ’^http://www.fftw.org Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Table 7.1: Map Making computational requirements Nt Nx Np l^proc Run Time 100,000 100,000 100,000 5,505,024 5,505,024 1 50,000 100,000 1 5,505,024 330 330 330 1788 1788 1 10 10 1 160 1 min 3 min 20 min 1 hour 8 hour The computational requirements of the map making algorithm described in the text for various M AXIPO L simulations. We assume a sample rate of .0048 seconds, simulate a rotating grid, and that the maps are pixelized with square 3' pixels. The simulations with Nt = 100,000 were run on the IBM SP at MSI, and those with Nt = 5,505,024 (the full 7.5 hour M A XIPO L BUM scan) were run on Seaborg at NERSC. 7.3 Tests In this section we describe two tests of th e m ap m aking code. In th e first test we generate an entire m a x ip o l sim ulated tim e stream using th e techniques described in Section 7.1. Prom this sim ulated tim e stream the in p u t and o u tp u t m aps can be com pared to see if the stru c tu re is reproduced in the o u tp u t m aps. T he second test is based on a sim ple, albeit unrealistic situation, w here only a sm all num ber of pixels are used in th e sim ulation. T his allows us to determ ine exactly w hat the num erical values of th e o u tp u t m aps and pixel noise correlation m atrix should be. We call this the sm all m ap test. 7.3.1 B 45 M A X IPO L sim u la tio n For a qualitative test of th e m ap m aking code a sim ulated MAXIPOL tim e stream was generated using th e actu al pointing from a single detector from th e MAXIPOL-1 flight. T here are exactly 5,505,024 pointing sam ples. For each pointing sam ple a sim ulated tim e stream sam ple, which is com prised of b o th signal and noise, is generated. T he sim ulated tim e stream is th en pixelized in 3' square pixels in G alactic coordinates, an d th e m axim um likelihood m aps are com puted. W hite noise w ith an am plitude of l{p,K)'^ was added in the tim e dom ain. Figure 7.1 shows th e num ber of hits per pixel for th e B45 pointing, and a com parison of th e in p u t and o u tp u t m aps is given in Figure 7.2. T h e pixels a t th e center o f t h e s c a n c o n t a in t h e m o s t h its (a n d th u s w ill h a v e t h e b e s t s ig n a l- to - n o is e p r o p e r tie s ), while the pixels on th e edges have less samples. T he in p u t m aps (shown in th e left panels of Figure 7.2) were used to generate a sim ulated tim e stream , an d th e o u tp u t m aps (shown on th e right of Figure 7.2) are th e m axim um likelihood m aps estim ated from th e sim ulated tim e stream . By a sim ple visual com parison it is clear th a t th e stru c tu re s contained in in p u t T , Q, an d U m aps are reproduced in the o u tp u t m aps. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 Figure 7.1: The number of hits per 3' pixel for the m a x i p o l - i pointing solution for the B45 channel. This pointing was used to generate the simulations shown in Figure 7.2. 7.3 .2 S m a ll M ap te s t For a sim pler test a non-realistic sim ulated pointing strategy is employed. T his test is com prised of 1,000 tim e stream sam ples spread evenly over five pixels. T he signal (T, Q, and U) is known a t each pixel, and a low am ount of w hite noise is added a t each pixel. T he noise power spectrum used for th is test is P{k) = 1 x 10“ ®//K^, which m eans th a t th e tim e noise correlation m atrix is (7.22) (7.23) w here we have defined P q = ( l x 10“ ®yuK^). Thus, for w hite noise (equal am ount of power for all frequencies) th e tim e noise correlation m atrix is diagonal. We can cross check the o u tp u ts from th e m ap m aking code, th e pixel noise correlation m atrix a n d th e m aps, w ith expected results. T he expected inverse noise pixel correlation m atrix is K p' (7.24) = (7.25) T he inverse noise correlations in th e tem p eratu re m ap is (7.26) Po ^ tt i€p = Pr0 1 -^hitsGp Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (7.27) 113 m 111 112 113 t 114 (b) T map recovered (a) T map input 1 " t 0 (d) Q map recovered (c) Q map input t » t 0 (e) U map input (f) U map recovered F ig u r e 7.2: An example of the map-making code using the pointing information from the B45 channel of the M A X I P O L - I flight. The left panel contains the pure signal input maps to the simulation. On the right panel are the recovered maps. Only a small amount of noise was input in the tim e domain. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 114 Figure 7.3: The pixel noise covariance matrix for the small map making test described in the text. The test is for a simulation with 1,000 total time stream samples spread evenly over 5 pixels. T he sum in equation is non-zero only for elem ents w here tim e stream elem ents i is in pixel p. T h e inverse noise correlations in th e polarization m aps is , —l . ^ h i t s e p pp' QQ (7.28) iep 1 .^ hits£p (7.29) t£p where we have used th e fact th a t (cos0^) = (sin0^) = 1/2. We can also com pute the cross term s. These are _ i y ^ cos6>j K p' tq — -^0 K p' t u ~ ^ (7.30) "pp iep _ p-i-^ sin ^ i ^0 iep COS 9 i s i n 9 i qu n -‘E (7.31) A °PP Sppi. (7.32) i&p T he cross noise betw een m aps is non-zero only for corresponding pixels, can be either positive or negative, a n d should be close to zero because averaging over m any angles gives (cos0) = (sin0) = (cos 0 sin 0) = 0. A ssum ing th a t th e off diagonal cross term s are close to zero, we can calculate th e expected noise variance in each tem p e ra tu re and polarization pixel by inverting th e inverse noise pixel Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 m atrix entry. T his gives — Jj •‘''hitsep W w ]„ ^ W pp ' ] uu - ^pp' (7.33) ( m -‘''hitsep C7-35) N u itsep ^ ^ '' T he noise variance in a tem p e ra tu re pixel is h alf th a t of th e corresponding polarization pixel, and th e noise in each m ap decreases as th e num ber of hits increases, b o th of which m ake sense physically. Each incident polarization m ap only uses h alf of th e to ta l num ber of photons on th e receiver, whereas th e tem p e ra tu re m aps use all of th e photons. For the specific case a t hand we have Pq = I x 1 0 “ ®//K^, and iVhitsep = 2 0 0 , so we expect to get ^ 2 X 1 0 “ ^/iK ^, and [Upp']^^ — [-^pp']„„ 4 x 1 0 “ ®/.tK^. T he com puted noise covariance m atrix from th e m ap m aking code for th is test is shown in Figure 7.3, which is consistent w ith th e expect results. T he noise variance p olarization pixels is twice th a t of th e tem p e ra tu re pixels, and the cross noise is non-zero for corresponding pixels. T he diagonal elem ents are equal to th e expected values w ith .3%. T he in p u t and o u tp u t values for th e 5 pixels in th e T , Q, an d U m aps are given in Table 7.2. T he value in each o u tp u t pixel agrees w ithin the noise variance to th e input value, as expected. 7.4 D iscussion In this ch ap ter we have dem onstrated how to generate tim e stream sim ulations for the MAXIPOL CM B experim ent, and developed a corresponding MAXIPOL m ap m aking code th a t com putes th e m axim um likelihood CMB tem p e ra tu re an d p olarization anisotropy m aps given a tim e stream an d noise power spectrum . T he tim e stream sim ulations were built upon earlier sim ulations discussed in C h ap ter 5. T he m a x ip o l m ap m aking code is a sim ple extension of m axim um likelihood m ap m aking algorithm for CM B tem p e ra tu re experim ents. Using th e tim e stream sim ulations th e m ap m aking code was tested using two different tests. In th e first test th e entire pointing from th e actu al MAXIPOL-I flight for th e B45 c h a n n e l w a s u se d . A lo w n o is e s im u la te d t im e s tr e a m w a s b a s e d o n th is p o in t in g , and it was shown th a t th e stru c tu re in all three m aps was recovered th ro u g h th e m ap m aking code. In th e second test we took a very sim ple test case where 1,000 tim e stream samples were evenly d istrib u ted am ong five pixels. An approxim ate expression based on pure w hite noise in th e tim e dom ain was derived for th e pixel noise covariance m atrix . T he derived expression shows th a t th e noise in the polarization m aps is twice th a t of th e tem p e ra tu re Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 Table 7.2: m a x ip o l Map Making software test Map Pixel Number Input Output T T T T T 1 2 3 4 5 -13.21 123.2 31.96 54.46 -89.18 -13.22 123.2 31.95 54.45 -89.19 Q Q Q Q Q 1 2 3 4 5 4.712 -1.431 2.333 -1.575 1.916 4.713 -1.438 2.342 -1.567 1.921 u u u u u 1 2 3 4 5 11.20 4.896 3.592 0.3717 1.668 11.19 4.896 3.586 0.3777 1.670 A simple test of the m a x i p o l map making software. Five different pixels were observed 200 times with a negligible amount of noise. The input values and output values agree withing the signal to noise expected. m ap, and th a t th e noise variance in all three m aps decreases linearly w ith th e num ber of tim e sam ples in a given pixel. It was shown th a t th e com puted pixel noise covariance m atrix agrees w ith derived result, and th a t th e in p u t and o u tp u t T , Q, a n d U values for each pixel agree. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A P robability D istribution o f th e E xperim ental D ata B JK have shown th a t th e band-pow ers are w ell-approxim ated by an offset log-norm al distrib n tio n . Specifically, they showed th a t th e probability d istrib u tio n | Zf) is approxi m ately G aussian as a function of Z\. Furtherm ore, it is possible to com pute th e covariance m atrix o f these G aussian random variables. T he calculation in B JK was perform ed in a Bayesian fram ework. For th e frequentist analysis we need to know th e probability d istrib u tio n p { Z f | Z |) as a function of Z f , not as a function of Z\. (T his is the heart of th e difference betw een th e two approaches: for a Bayesian th e d a ta are fixed and th e theoretical quantities are described probabilistically; a frequentist tre a ts th e d a ta as a random variable for fixed values of th e param eters.) We therefore m ake th e ansatz th a t th e probability d istrib u tio n is G aussian in Z f as well. If th e z f are indeed G aussian d istrib u ted , th en th e entries of th e weight m atrix M (inverse covariance m atrix) should be exactly th e sam e for independent observations of universes which have the same underlying CM B power spectrum . We test th e assum ption of G aussianity using sim ulations. We generate CM B m aps using a p articu la r cosmological Moo Mqi M il Sm all M ap 1.5 X 10^ ± 1.2 X 10*^ 2.0 X 10® =fc 7.4 X 10^ 2.1 X 10^ ± 2 .6 X 10^ Large M ap 1.2 X 10« ± 2 .8 x 1 0 ^ 4.5 X lO*’ ± 7.9 X 1Q4 1.2 X 10*^ ± 7 .3 X 10*^ Table A .l: The average values with sample standard deviations of the marginalized Zi weight matrix entries for both large and small map simulations. The large map simulations were based on pointing from the M A X I M A -I 8 ' map, and the small map pointing was based on a center patch of the map. Units are dimensionless MADCAP units 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 m odel, com pute th e power spectrum and M for each m ap and assess th e variance in the entries of M betw een sim ulations. A sm all variance would indicate th a t th e assum ption th a t th e z f are G aussian-distributed is adequate. We generated 100 sm all-area and 30 large-area m ap sim ulations using ao as th e cosmolog ical m odel a n d com puted th e M m atrix for each m ap (the num ber of sim ulations is lim ited by th e com putational resources required to estim ate th e power sp ectru m for each m ap). T he small- a n d large-area m aps contain 542 and 5972 8' pixels respectively. Power sp ectra and Zi were com puted in four bins of I ={2,300},{301,600},{601,900},{901,1500}, and M was obtained by m arginalizing over the first and last bins. T he results are sum m arized in Table A l. T h e average value of the diagonal entries increases for th e larger-area m aps because for those th e b an d powers have sm aller errors and hence larger values in th e weight m atrix. T h e percent fluctuation in th e m atrix entries of th e large-area m aps are 2% for the first diagonal entry, 6% for th e second diagonal entry, and 2% for th e off diagonal entries. We consider th is variance to be sm all enough to indicate th a t th e assum ption of G aussianity of th e z f is acceptable. We also note th a t th e variance of the m atrix elem ents decreases as a function of increasing m ap size. If such a tre n d continues to m aps of th e size of the MAXIMA-I m ap, which has m ore th a n 15,000 pixels, th en th e assum ption of G aussianity is well satisfied. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Appendix B Spin Spherical H arm onics In this appendix we present useful form ulas and identities involving spin spherical harm on ics. We also show th a t polarization is a spin ± 2 quantity. For a com plete discussion of spin weighted spherical harm onics see [30] and [64], and for a m odern tre a tm e n t [48, 106]. Any location on a sphere can be described w ith polar coordinates 9 an d 0. At each point on th e sphere we can describe a coordinate system w ith three orthonorm al vectors, h, which points radially outw ard, and ci and C2 , which are tan g en tial to th e sphere. A counter-clockwise ro ta tio n by th e angle a defines two new u n it vectors e'^ e'g, which obey th e relation (see Figure B .l) e[ = Cl c o s ( q : ) -h 6 2 sin (a ) ^ 2 = - 6 1 sin (a )-f - 6 2 (B .l) cos(a). (B.2) A function on th e sphere f{ 6, (j)) is a spin-s quan tity if it transform s as f ' { 6 , 0) = / ( 0 , where a is a right handed ro tatio n ab o u t h. A spin-s function s f {6■,<!)) defined on the sphere can be expanded in term s of spin-s spherical harm onics s f{ e A ) = Y .^ s M s y U 9 ,4 > ) (B.3) Im T he spin-zero analogy of th e spin spherical harm onics are the usual spherical harm onics Y^jjj(0, (/>), which can be for s used to expand scalar functions on the sphere b u t are inappropriate 0. E xam ples of th e real v a lu e d s p in -0 sp h e r ic a l h a r m o n ic s a re s h o w n in F ig u r e B .2 . T he spin-s spherical harm onics obey the orthogonality and com pleteness relations 27T r1 / d(j) J d c o s 9 (0, (/>) gYijjfi(^9, Y, sYil{9,4>) gYi^{9',<t>') = S{(j)-(P')6icose-cose'). Im 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B .4 ) 120 \ \ ^ A'"' Figure B .l: Two coordinate systems which differ by a rotation of the angle 9. O th er useful relations are th e conjugation relation (B.5) and the spin spherical harm onic addition theorem 2£ + l 47T S2 Yi-sii^,a)e iS20 (B. 6 ) where ^ is th e angle betw een the points (0, (p) and (0', 4>') on the sphere. T h e angles a and represent how m uch th e locale coordinate system (6 1 ,6 2 ) m ust be ro ta te d a t (0 , <^) and {9\ (p'), respectively, to becom e aligned w ith a great circle connecting th e two points [39]. We now show th a t Q ± i U are spin-± 2 quantities. C onsider a n incident rad ia tio n field, and assum e for sim plicity th a t it is perfect linearly polarized rad iatio n w ith intensity I. Let ax and Cy be th e x and y com ponents of the polarization vector: ax — I cos(^) (B.7) Oy = (B. 8 ) I sin{(p) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 w here ip is th e angle th e polarization vector makes w ith th e x axis. Prom E quations 5.3-5.5 Q — al U - a,y = cos(2<^) = 2uxay = (B.9) sin{2(p). (B.IO) Now assum e we m ake a right handed coordinate ro ta tio n by an angle ip. T h e com ponents of th e polarization vector in the new coordinate system are 4 = a,y Icos{<p-ip) ( B .ll) Ism{ip — ip). (B.12) Recall th a t a spin 2 quan tity is one which transform s as f = f e Q' + C onsider th e q uantity iU': Q' + iu' 4 2 ) - ( 4 2 2*44 = ( ) + = /2 cos(2v? —2ip) + i f i sin(2i/? — 2ip) (B.14) = /2 (cos(2</j) cos(2'0) + sin(2(/5) sin(2'0)) + (B.15) */2 (sin(2(/5) cos{2ip) - sin(2V') cos(2(/?)) = Qcos{2ip) + U sin(2'0) + iU cos(2'i/’) - *Qsin(2<^) (B.16) = {Q + iU)e~‘^ ^ . (B.17) A sim ilar proof shows th a t Q — ill is spin —2. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 122 (c) £ = 2 m = 2 (d) £ — 10 m — 0 II (e) £ = 10 m = 10 (f) ^ = 50 m = 10 Figure B.2: Examples of the spin-0 spherical harmonics Vim for various multipole coefficients £ and m. These figures show only the real component (spherical harmonics are complex functions). Note that the m = 0 harmonics have no azimuthal dependence, whereas the m ^ 0 (and especially the m = £) harmonics have strong azimuthal dependence. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix C C Source Code T his A ppendix has C source code for various algorithm s discussed in th is docum ent. C .l P ix el N oise Correlation M atrix T his is section gives th e C code used to form the noise correlation m atrix using. T he algorithm is described in detail in C h ap ter 7. T his code runs on m ultiple processors using th e message passing interface (M PI) subroutines. T his code actually form s th e inverse noise correlation m atrix. It is th en inverted in a subsequent subroutine in th e m ap m aking code. T he in p u ts are th e inverse tim e noise correlation m atrix (N tt), th e pointing m atrix (A), th e location vector (L), th e to ta l num ber of processors (no.pe), an d a processor identifier (my_pe). T he o u tp u t is th e inverse pixel noise correlation m atrix (Npp). /♦ Form the 3 N_p by 3 N_p n o is e c o r r e la t io n matrix. */ #include < std lib .h > #include < std io.h > #include <math.h> #include <fftw.h> #include <rfftw.h> #include "mpi.h" long m in (lo n g ,lo n g ); 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 long m a x (lo n g ,lo n g ); void make_Npp(double *Npp,fftw_real N t t [ ] , f l o a t A [ ] [ 3 ] , i n t L [] , in t n o _ p e,in t my_pe) FILE * p t , * t l , * t 2 ; double *Nppt, *my_Npp; in t n u m ,i,j.c o u n t,n p ix 3 ,n p ix 2 ; in t ro w ,co l; in t t i , t j , a i j ; in t my_s,my_n; long lambda=N/2+l; npix3=3*npix; npix2=2*npix; my_n=N/no_pe; i f (my_pe==no_pe-l) my_n+=N*/,no_pe; my_s=my_pe*(N/no_pe); MPI_Barrier(MPI_COMM_WORLD); for(ti=m y_s ; ti<my_s+my_n ; t i + + ) { for(tj= m a x(0,ti-lam b d a) ; tj<min(N,ti+lambda) ; tj++)-[ a ij= a b s(ti-tj); /♦ <T T> * / Npp[L[ti] *npix3+L[tj]] += A [ti] [0] *Ntt [ a i j ] *A [t j] [0] ; / * <T Q> * / Npp[L[ti] *npix3+L[tj]+npix] += A [ti] [0] *Ntt [ a i j ] *A [tj] [ 1 ] ; /* <T U> * / Npp[L[ti]*npix3+L[tj]+npix2] += A[ t i ] [0] *Ntt [ a i j ] *A [t j] [2] ; /* <Q Q> ♦/ Npp[(npix+L[ti])*npix3+L[tj]+npix] += A[ti] [1] *Ntt [ a i j] *A[t j ] [ 1 ] ; /♦ <Q U> * / N pp[(npix+L[ti])*npix3+L[tj]+npix2] += A [ t i ] [ 1 ] * N t t [ a i j ] * A [ t j ] [2]; /* <U U> * / Npp[(npix2+L[ti])*npix3+L[tj]+npix2] += A [ti] [2] * N t t [ a i j ] * A [ t j ] [2]; >} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 MPI.Barrier(MPI_COMM_WORLD); MPI.Reduce(Npp, Npp, npix3*npix3, MPI_DOUBLE, MPI_SUM, 0 , MPI_COMM_WORLD); if(my_pe==0){ f o r (1=0 ; i<npix3 ; i++) f o r ( j = 0 ; j<npix3 ; j++) N pp[j*npix3+i]=Npp[i*npix3+j];} r e tu r n ; } /* u t i l i t y fu n c tio n s * / long m indong a ,lo n g b ){ if(a < b ) return a; e l s e return b; } long maxdong a ,lo n g b ){ if(a > b ) return a; e l s e return b; > C.2 P ixel Signal Covariance M atrix In this section we give th e C source code for form ing th e pixel signal covariance m atrix. T his is a subroutine in the Bayesian m axim um likelihood te m p e ra tu re /p o la riz a tio n power spectra estim ation code. It is im p o rtan t to note th a t th e code presented in th is section is based on M A D CA P v2.1 [11], software developed by Dr. Ju lia n Borrill. T h a t software com putes only th e (T T ) covariance m atrix. Therefore m uch of th is code uses th e same variables, conventions, and subroutines from th e M A D CA P package. T his includes error checking an d i/o routines. T he code which com putes th e entire 3Np x 3Np pixel signal covariance m atrix is too long to p r e s e n t in it s e n tir e ty . I n s te a d w e sh o w t h e c o m p o n e n t s o f t h e so ftw a r e c o m p u t e s th e ro tatio n angles for a pixel p air (0,(f>) and {O', </)'), which is a crucial com ponent to form ing th e tem p erature-polarization signal correlation m atrix. Also shown are th e calculations for th e ( T E ) subm atrices. T h e entire signal covariance m atrix contains a {T E) dependence in only four of th e nine subm atrices. T he in p u ts to th e entire code are sim ply a pixelization Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 (it assum es th e T , Q, and U m aps all have th e same pixelization on th e sky), an d a binning stru c tu re for all six power spectra. R otation A ngles A=(double *)m alloc(no_p ix*no_p ix*8)); f o r ( i= 0 ,n=0; i<no_pix ; i++) fo r (j= 0 ; j<no_pix ; j++,n++){ r a d e c to r (p ix [i].r a ,p ix [i].d e c ,r i); r a d e c tc r (p ix [j].r a ,p ix [j].d e c .r j); / * compute cro ss products * / c r o s s (z ,r i,r is ); c r o s s (r i,r j,r ij); /* cos Angle * / c o s_ a = d o t(r ij,r is); / * s e t A matrix * / i f (co s_ a < -1 .0 ) cos_a=-1.0; e l s e i f (cos_a> 1 .0 ) cos_a=1.0; A [ i* n o _ p ix + j] = ( d o t( r ij,z ) > 0 ? a cos(cos_a) : - 1 .O * a c o s (c o s _ a )); /* check li m i t i n g ca ses * / i f ( i = = j ) A [i*no_pix+j]=0.0; } T he above code uses the following subroutines /* cro ss product of two R3 v ecto r s * / void cross(d oub le a [] .double b[] .double c [ ] ) { double n; in t i ; c [0] =a [1] *b [2] - a [2] *b [1] ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 c [1] =a [2] *b [0] -a [0] *b [2] ; c [2] =a [0] *b [1] - a [1] *b [0] ; n=sqrt (c [0] *c [0] +c [1] *c [1] +c [2] *c [2 ]); if(n>TOL) f o r ( i= 0 ; i<3 ; i++) c [ i ] / = n ; r e tu r n ; } /* dot product of two R3 v e c to r s */ double dot (double a [ ] , double b [ ] ) { return a [0] *b [0] +a [1] *b [1] +a [2] *b [2] ; } /* convert ra and dec t o normal v e c to r , ra and dec are in radians */ void radector(double ra,double dec,double r [ ] ) in t i ; double th e ta ; t h e t a = 9 0 .0 - dec; r[0 ] = sin (th e ta )* c o s(r a ); r[l] = sin (th e ta )* sin (r a ); r[2 ] = cos ( t h e ta ) ; r e tu r n ; > {T E) correlation subm atricies / * C alcu late t - e c o r r e la t io n s ig n a l convariance matrix d e r i v a t i v e s * / if(my_pe==0&&no_bin_te) p r i n t f C computing TE . . . \ n " ) ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 /* I n i t i a l i z e ♦ / error_check(my_pe, "malloc", dSdCte_b= (double >i‘)malloc(my_no_matrix_elm*8)); fo r (n=0; n<2; n++) error_check(my_pe, "malloc", LP[n]=(double *)malloc(my_no_matrix_elm*8)); /* I n i t i a l i z e legendre polynomial recu rsio n v a r ia b le s * / f o r (n=0; n<my_no_matrix_elm; n++) { LP[0] [n] = 0 .0 ; LPCl] [n] = 1 .0 ; } /* For each b in * / fo r (b=0; b<no_bin_te; b++) { /* I n i t i a l i z e * / Imin = b in _ te[b ].m in ; Imax = bin_te[b].m ax; 10 = (b) ? b in _te[b -1].m ax + 1 : 1 ; fo r (n=0; n<my_no_matrix_elm; n++) dSdCte_b[n] = 0 .0 ; /* For each l o c a l matrix entry . . . * / fo r (j= 0 , n=0; j<my_no_matrix_col; j++) { fo r (i=0; i<my_no_matrix_row; i++, n++) { / * f in d g lo b a l c o o r id in a te s eind p i x e l se p e r a tio n * / l o c a l _ t o _ g l o b a l ( i , j,m y_pe_row,m y_pe_col,blocksize, no_pe_row,no_pe_col,&pi,&pj); cos_ch i = s i n ( p i x [ p i ] . d e c ) * s i n ( p i x [ p j ] . d e c ) + c o s C p ix E p i] .d e c ) * c o s ( p ix [ p j ] .d e c ) * c o s ( p ix [ p i] .r a - p i x [ p j ] . r a ) ; x = l . 0 / ( 1 .0 - c o s _ c h i* c o s _ c h i) ; / * only i f in correct noctant * / n o c = n o c t a n t ( p i,p j ,n o _ p ix ) ; if(n o c = = l II noc==2 I I noc==3 I I noc==6){ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 /* I n i t i a l i s e legendre polynomial recu rsion v a r ia b le s * / pO = LP [0] [n] ; p i = LP [1] [n] ; /* Continue r ec u r sio n up t o Imin * / f o r (1=10 ; K lm in ; 1++) { ld = (d o u b le)1; p2 = 2 .0 * c o s _ c h i* p l- p 0 - (c o s _ c h i* p l- p 0 ) /ld ; pO = p i; p i = p2; } /* Continue recu rsio n up t o Imax, adding t o dSdCt_b * / fo r ( ; K=lmax; 1++) { ld = (d o u b le )l; p2 = 2 .0 * c o s _ c h i* p l- p 0 - (c o s _ c h i* p l-p 0 )/ld ; fiO = 2 . 0 * l d e n _ s q r t [ l ] * ( l d * x * ( c o s _ c h i * p l - p 2 ) - p 2 * l d * ( l d - l ) / 2 . 0 ) ; dSdCte_b[n] -= (2 *ld + l)*C S _ te[ 1 ] *f10*MW_t[ 1 ] *MW_e[1]; pO = p i; p i = p2; } dSdCte_b[n] *= inv_four_pi; /* same or d ia m e tr ic a lly o p p o site p i x e l * / if(fa b s(co s_ ch i)> l-T O L ) dSdCte_b[n]=0; /* Save recu rsio n endpoint terms ♦/ LP [0] [n] = pO; LP[1] [n] = p i; /* multiply by angle matrix A */ a i j =A [ (p j 7,no_pix) *no_pix+ (pi*/,no_pix) ] ; a j i=A [ (piy,no_pix) *no_pix+ (p j %no_pix) ] ; if(noc==3) d S d C te_ b [n ]* = co s(2 .0 * a ji); if(n o c = = l) dSdCte_b[n]*=cos( 2 . 0 * a i j ); if(noc==6) d S d C te _ b [n ]* = -s in (2 .0 * a ji); Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 if(noc==2) dSdC te_b[n]*=-sin(2. 0 * a i j ); } } /* Write t o d is c * / s p r in t f (m y _ file , "files/dSdCy,d_*/,d.7,d", b+no_bin_t, my_pe_row, m y_pe_col); error_check(my_pe, "fopen", f = fo p en (m y _ file, "w")); output_dtof(my_pe, f , dSdCte_b, my_no_matrix_elm); f c lo s e C f ); > T he above code uses the following subroutines /* convert g lo b a l array v a r ia b le t o l o c a l p rocessor * / void g lo b a l_ t o _ l o c a l ( i n t g_row ,int g _ c o l , i n t nprow.int n p c o l . i n t b lo c k s iz e , in t * p r ,in t * p c ,in t * l_ r o w ,in t * l_ c o l) { i n t l ,m ,x ,y ; l=g_row /(n p row *block size); m = g _ c o l/(n p c o l* b lo c k siz e ); x=g_row7,blocksize; y=g_coT/,blocksize; *pr= (g _ ro w /b lo ck siz e) 7,nprow; *pc=(g_col/b lock size)7,np col; * l_row = l*b locksize+ x; *l_col=m *blocksize+y; r e tu r n ; } / * convert l o c a l processor array v a r ia b le t o g lo b a l value * / void l o c a l_ t o _ g lo b a l( i n t l_ r o w ,in t l _ c o l , i n t myrow,int m y c o l,in t b lo c k s iz e , i n t nprow.int n p c o l , i n t *g_row,int ♦g_col) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 in t x , y ,l, m ; x=l_row7,blocksize; y=l_col*/,blocksize; 1=( l_ r o w ) /b l o c k s i z e ; m = ( l_ c o l) / b l o c k s iz e ; *g_row=l*blocksize*nprow+x+myrow*blocksize; *g_col=m*blocksize*npcol+y+mycol*blocksize; r e tu r n ; } /* f in d which noctant you are in * / in t n o c t a n t ( in t i , i n t j . i n t no_pix) { return j /n o _ p ix + 3 * ( i/n o _ p ix ) ; > Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R eferences [1] Abroe, M. E., et al. 2002, M NRAS, 334, 11 [2] Abroe, M. E., et al. 2004, A pJ, 605, 607 [3] Balbi, A., et al., 2000, A pJ, 545, L I [4] B ennett, C. L., et al. 2003a, A pJS, 148, 1 [5] B ennett, C. 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