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Probe the universe with PIXIE experiment and tSZ-lensing cross-correlation Yan, Ziang 2017

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Probe the Universe with PIXIEExperiment and tSZ-LensingCross-correlationbyZiang YanB.Sc., Tsinghua University, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Astronomy)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Ziang Yan 2017AbstractThe polarization of Cosmic Microwave Background can help us probe theearly universe. The polarization pattern can be classified into E-mode and B-mode. The B-mode polarization is a smoking gun of cosmological inflation.PIXIE is an in-proposal space telescope observing CMB polarization. Itis extremely powerful to extract CMB polarization signal from foregroundcontamination. The second chapter of this thesis summarizes my work onoptimizing the optical system of PIXIE. I run a Monte-Carlo Markov Chainfor the instrument parameters to maximize the value ”Good” which judgesthe behavior of the instrument. For the optimized instrument, with all kindsof noises from inside instrument and wrong polarization taken into account,good rays from the sky make up of 15.27% of all the rays received by thedetector. The instrument has a 1.1◦ top-hat beam response.The third chapter summarizes my work on studying the potential con-tamination in the reconstructed y map by doing cross-correlation betweentSZ signal and weak lensing. The weak lensing data is the convergence mapfrom the Red Sequence Cluster Lensing Survey. I reconstruct the tSZ mapwith a Needlet Internal Linear Combination method with 6 HFI sky mapsmade by Planck satellite. The reconstructed cross correlation is consistentwith Planck NILC SZ map. I take Cosmic Infrared Background (CIB) andgalactic dust as two potential source of contamination in the reconstructedmap. I find that κ×CIB contributes (5.8±4.6)% in my reconstructed NILCy map for 500 < ` < 2000 with 2.2σ significance. Dust residuals only changethe error bar of the cross correlation signal. I find the best value for dustindex is βd = 1.57. I then introduce a piecewise power spectrum for the CIBand make a NILC CIB map to make a CIB-nulled NILC y map. κ×y signalfrom this y map differs by only ∼ 0.08σ to the CIB-uncleaned y map.iiLay SummaryChapter 2 of my thesis presents research towards the development of a newspace-borne telescope, called PIXIE, aimed at testing whether or not cosmicinflation occurred in the first fraction of a second of our universe. Specificallywe investigate the optical design parameters for this telescope.Chapter 3 aims to better understand the connection between dark matterand atomic matter in the large-scale distribution of galaxies in our universe.I investigate whether emission from dusty galaxies is contaminating previousmeasures of the the dark matter - gas correlation.iiiPrefaceChapter 2 is a summary of my work on PIXIE instrument simulation underthe instruction of Prof. Gary Hinshaw, Alan Kogut and Dale Fixsen fromGCSC/NASA. The original IDL code for the instrument simulation waswritten by Dale Fixsen and I translated and modified it in python. TheMCMC work is entirely executed by myself.Chapter 3 of this thesis is based on a discussion between myself andAlireza Hojjati. Section 3.3, 3.4, 3.5 are my original, independent work withthe instruction of Prof. Ludovic van Waerbeke, Prof. Gary Hinshaw indiscussion with Alireza Hojjati. A paper on this work is in progress.The computation of both projects are executed on our group server jade.The figures in this manuscript, if not stated in the caption, are plotted bymyself. The manuscript is written entirely by myself with feedback fromGary Hinshaw and Ludovic van Waerbeke. The third chapter also receivedfeedback from Alireza Hojjati.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . 11.2 The Big Bang Cosmology . . . . . . . . . . . . . . . . . . . . 41.3 The Evolution of Large Scale Structure . . . . . . . . . . . . 71.4 Anisotropy in the Cosmic Microwave Background . . . . . . 101.5 The ΛCDM Model . . . . . . . . . . . . . . . . . . . . . . . . 151.6 Brief Introduction to Cosmological Inflation . . . . . . . . . 181.6.1 Problems in Standard Cosmology Model . . . . . . . 181.6.2 The Inflation Solution . . . . . . . . . . . . . . . . . . 201.6.3 The Physics of Inflation . . . . . . . . . . . . . . . . . 222 Observing CMB Polarization: The PIXIE Experiment . . 272.1 Studying the Inflation Era with CMB Polarization . . . . . . 272.1.1 The Stokes Parameters . . . . . . . . . . . . . . . . . 272.1.2 Thomson Scattering . . . . . . . . . . . . . . . . . . . 282.1.3 Angular Power Spectrum of Polarization . . . . . . . 30vTable of Contents2.1.4 From Temperature Fluctuations to Polarization Fluc-tuations . . . . . . . . . . . . . . . . . . . . . . . . . 322.1.5 CMB Polarization Observations . . . . . . . . . . . . 342.2 Overview of the PIXIE Experiment . . . . . . . . . . . . . . 372.3 Instrument Simulation . . . . . . . . . . . . . . . . . . . . . 432.3.1 Code Realization of the PIXIE Instrument . . . . . . 442.3.2 Parameters and Criteria . . . . . . . . . . . . . . . . 472.4 MCMC for Instrument Parameters . . . . . . . . . . . . . . 512.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Observing the Gas Distribution in Galaxy Clusters: The y-κCross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . 613.2 Studying the Large Scale Structure with Weak Lensing andtSZ Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Reconstruction of the y signal . . . . . . . . . . . . . . . . . 663.4 A Worked Example: κ× y Cross Correlation . . . . . . . . . 733.4.1 The CIB Contamination . . . . . . . . . . . . . . . . 733.4.2 The Galactic Dust Contamination . . . . . . . . . . . 783.5 An Attempt to Reconstruct the CIB Signal . . . . . . . . . . 783.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96A The Spin-Weighted Spherical Harmonic Function . . . . . 96B The Needlet ILC . . . . . . . . . . . . . . . . . . . . . . . . . . 98viList of Tables1.1 Thermal history of the universe. Data are from and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 ΛCDM independent parameters given by Planck Collabora-tion [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 The state of some current and future CMB polarization ex-periments. Part of the data is from . . . . . 362.2 Optical Parameters. . . . . . . . . . . . . . . . . . . . . . . . 422.3 A summary of instrument elements to be optimized. Allthe functions take (R,D,P,G,B,L,K) as input and output(R,D,P,G,B). . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4 The output for a rays track with the optimized parameters.The numbers of living rays and missing rays have been weightedby cos θ0. The ’Missing rays’ column shows the number of raysthat miss the corresponding element. Note that according to(2.25), the Good value is calculated by taking the sum of thelast column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58B.1 The ILC coefficients for the fiducial NILC y map in differentneedlet scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 100viiList of Figures1.1 Density of each content of the universe. Source: 61.2 The distribution of galaxies in part of the 2dF sky survey.Source: 81.3 Linear matter power spectrum P(k) versus wavenumber ex-trapolated to z = 0, from various measurements of cosmolog-ical structure. The black line is the best-fit ΛCDM model.Source: . . . . . . . . . . . . . . . . . 111.4 Sensitivity of the angular power spectrum to four fundamen-tal cosmological parameters. (a) The curvature as quantifiedby Ωtot = 1 − Ωk . (b) The dark energy as quantified bythe cosmological constant ΩΛ(wΛ = −1) . (c) The physicalbaryon density Ωbh2. (d ) The physical matter density Ωmh2.All are varied around a fiducial model of Ωtot = 1, ΩΛ = 0.65,Ωbh2 = 0.02, Ωmh2 = 0.147. Image is from [45] . . . . . . . . 141.5 68.3%, 95.4%, and 99.7% confidence regions of the (Ωm, w)plane from supernovae data combined with the constraintsfrom BAO and CMB. This image is from [14] . . . . . . . . . 161.6 A potential of the inflaton field that can give rise to inflation.This figure comes from [72]. . . . . . . . . . . . . . . . . . . . 232.1 E and B-mode polarization patterns. (left panel) A represen-tative Fourier mode of a density perturbation. (middle) E-mode polarization pattern resulting from Thomson scatteringof this mode (growing amplitude). (right) B-mode polariza-tion pattern. Figure is from [21] . . . . . . . . . . . . . . . . . 312.2 Local quadrupole perturbation field. Red color representsredshift and blue is blueshift. Figures are from . . . . . . . . 32viiiList of Figures2.3 Angular power spectra of EE,BB and ΘE generated by CAMB.Reionization and gravitational lensing are taken into account.The cosmology is: Ωk = 0, Ωbh2 = 0.02, Ωmh2 = 0.16,ns =1, r = 0.1, TCMB = 2.7255K. The dashed line representsnegatively correlated. . . . . . . . . . . . . . . . . . . . . . . . 332.4 Marginalized joint 68% and 95% CL regions for ns and r0.002from Planck in combination with other data sets comparedto the theoretical predictions of selected inflationary mod-els. r0.002 is the tensor-to-scalar ratio at a pivot scale k∗ =0.002Mpc−1.; N∗ is the number of e-fold. This image is from[8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 The RMS on angular scales of 1 for the polarized CMB withdifferent r value compared with that from foregrounds ex-tracted from the WMAP data at ` = 90.[73] . . . . . . . . . . 372.6 Theoretical angular power spectra for the unpolarized, E-mode, and B-mode polarization in the CMB. The dashed redline shows the PIXIE sensitivity to B-mode polarization.Redpoints and error bars show the response within ` bins to aB-mode power spectrum with amplitude r = 0.01[51]. . . . . 382.7 Upper panel: Schematic view of the PIXIE optical signalpath. As the dihedral mirror moves, the detectors measure afringe pattern proportional to the Fourier transform of the dif-ference spectrum between orthogonal polarization states fromthe two input beams (Stokes Q in instrument coordinates).A full-aperture blackbody calibrator can move to block ei-ther input beam, or be stowed to allow both beams to viewthe same patch of sky; Lower panel: Instrument physical lay-out showing the beam-forming optics and Fourier TransformSpectrometer[51]. . . . . . . . . . . . . . . . . . . . . . . . . . 402.8 A 2-D sketch of the PIXIE instrument. Black curves and linesshows the mirrors. Red line is the track of a ray generatedperpendicular to the detector. . . . . . . . . . . . . . . . . . . 432.9 The side-view of the horn in the horn coordinate system witha nominal focus. The shaded part is the horn, the short leftside is the detector and long right side the mouth. The blackline is the top wall and the purple line is the bottom wall. Thefoci of the horn are denoted by points with corresponding colors. 49ixList of Figures2.10 A 2-D sketch of the SFP system . Grey curves and lines showsthe mirrors. Red arrows track the central ray from T1 to theaperture. The blue arrow is the normal vector of the flat.Points S, F, P are the center of the corresponding elements.Blue points are focus of primary and Second. . . . . . . . . . 502.11 The MCMC result for f t2,fb2 and fT51 . The colors show thenumber of rays reaching T1 with corresponding parameters.The crossing dashed lines labels the position of maximum onthe parameter space. In each panel, the black points withlabels on it shows the center of corresponding instrument el-ements as reference points. . . . . . . . . . . . . . . . . . . . . 532.12 The MCMC result for iris angle and size. Upper and bot-tom right panels show the histogram for θiris and riris. Thebottom left plot shows the chain points in the 2-D parameterspace color-coded by Good. The crossing dashed lines labelsthe position of maximum on the parameter space. Contoursshows the 68.3%, 95.4% and 99.7% level of confidence. . . . . 542.13 Upper panel: 2-D projection of the optimized HIT5 system.Red lines shows part of the rays from iris to T5. Blue point isfT51 . Note that here we use the instrument coordinate. Lowerpanel: the horn mouth with iris represented by a circle. Thecolor plot shows the 2-D histogram for number of rays landingon the horn mouth. To show a more smooth histogram, I takeNside = 64 for this plot. . . . . . . . . . . . . . . . . . . . . . 552.14 The MCMC result for FfPri and FfSec2 . Upper and bottomright panels show the histogram for FfPri and FfSec2 . Thebottom left plot shows the chain points in the 2-D parameterspace color-coded by Good. The crossing dashed lines labelsthe position of maximum on the parameter space. Contoursshows the 68.3%, 95.4% and 99.7% level of confidence. . . . . 562.15 Upper right panel: the 2-D histogram in ~D space of the out-coming rays. x and y axis are two components of the polarangle of ~D. The color bar shows the weighted rays number.Upper left and bottom right panels: marginalized 1-D his-togram for θy and θx. . . . . . . . . . . . . . . . . . . . . . . . 592.16 Left right panel: the 2-D histogram in ~D space of the co-polarization of out-going rays. Right right panel: the 2-Dhistogram in ~D space of the cross-polarization of out-goingrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60xList of Figures3.1 Effect of κ and γ ≡ γ1 + iγ2 on a spherical source. . . . . . . 643.2 Flow chart for our NILC procedure. . . . . . . . . . . . . . . 693.3 y signal of a small region of the sky for Planck NILC mapand our y map. . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4 Comparison between the measured tSZ flux of the Planckcluster sample measured in Planck NILC map and our y map. 703.5 Footprint of RCSLenS field in galactic coordinate. . . . . . . 733.6 Cross correlation between CIB signal and κ for three differentCIB maps in ` space. The cross correlation signal is binnedto 5 ` bins centered at 290, 670, 1050, 1430, 1810. . . . . . . 743.7 Upper panel: Cross correlation between y signal and κ forthree different y maps in ` space. The cross correlation sig-nal is binned to 5 ` bins centered at 290, 670, 1050, 1430,1810. Blue, green and red points are corresponding to PlanckNILC y map, our NILC y map, our CIB-subtracted y map (seeEq.3.32); lower panel: The bootstrap estimation of 〈∆C`〉 / 〈C`〉for each ` bin. The error bars correspond to a 68% C.L. . . . 773.8 Histogram for Td (upper panel) and βd (lower panel) in RCSfield. The dust model we use here is the Planck COMMAN-DER thermal dust map [2]. . . . . . . . . . . . . . . . . . . . 793.9 Upper panal: κ × y cross correlation for the fiducial y map(βd = 1.57) and four non-standard y maps. Lower panel:standard derivation for cross correlation signal in each ` bin. 803.10 Histogram for CIB indices β1 and β2 in the unmasked domain. 813.11 κ × CIB cross correlation signal for our NILC CIB map andPlanck CIB map. Three panels are corresponding to threefrequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.12 Upper panel: cross correlation signals between κ and three ymaps; lower panel: Difference of κ×y cross correlation signalsNILC y maps and NILC-(CIB-nulled) y map. . . . . . . . . 85B.1 Needlet windows acting as bandpass filters in ` space. . . . . 99xiList of AbbreviationsBAO Baryon Acoustic OscillationBBN Big Bang NucleosynthesisCMB Cosmic Microwave BackgroundFRW metric Friedmann-Robertson-Walker metricFTS Fourier Transform SpectrometerGUT Grand Unified TheoriesHIT5 horn-iris-T5 systemMCMC Markov Chain Monte CarloPDF Probability Distribution FunctionPIXIE Primordial Inflation ExplorerSFP secondary-flat-primary systemSNe SupernovaetSZ effect thermal Sunyaev-Zeldovich effectΛCDM Λ Cold Dark MatterxiiAcknowledgementsI would like to express my great acknowledgement to my supervisor, Prof.Gary Hinshaw, who supervises me with instructive guidance, kind encour-agement and fruitful discussion. He is a great scientist who greatly bringsup my interest in the topics of cosmology. My co-supervisor, Prof. Ludovicvan Waerbeke also gives me much helpful advice.I’m thankful to all the faculty and students who helped my studies. Espe-cially Alireza Hojjati, who encouraged me to work on κ×y cross-correlation.I have been obtaining great experience working with him. Tanimura Hidekiand Tilman Troester also help me a lot with multiple discussions.I thank Alan Kogut and Dale Fixsen from NASA/GSFC who providegreat guidance on PIXIE work.I also acknowledge the Department of Physics and Astronomy at UBCfor financially supporting me during these two year. I appreciate my courseprofessors for their inspiring teaching.Finally, I would like to express my endless love and gratitude to myparents, for their selfless love, outstanding education, and support of mystudies.xiiiChapter 1IntroductionCosmology is the study of our universe as a whole. Currently we use physicsto describe its origin, evolution and content. Though it might not be asbeautiful as from the literature or arts, it is exciting due to its highly logicaland precise language, and also the capability to know the past, learn thecurrent and predict the future in a convincing way.Like other disciplines in astronomy, cosmology depends critically on ob-servations. Instead of single celestial objects, cosmologists observe the struc-ture of the whole universe. The improving observational techniques haveboosted the development of cosmology in the past half century. And wehave reasons to believe that our understanding of the universe will continueto advance.This chapter is an introduction to cosmology. The first section introducesthe history of cosmology. The second to the fifth sections discuss topics inthe standard cosmological model. The sixth section introduces cosmologicalinflation.The calculations in this chapter, unless otherwise noted, follows [28],[72], [86], [88] and [53].1.1 Historical IntroductionHuman beings have never stopped thinking about the universe. In ancienttime, there already existed several theories to describe our universe. Eachcivilization has its own interpretation of the universe. The Chinese word foruniverse is yu zhou. Its definition is described in an ancient Chinese textShizi from 2400 years ago:All the directions named ’yu’; all the ages named ’zhou’.So yu zhou can be directly translated into English as ’In all the directionand through all the age’, or ’overall spacetime’, which coincidences with thephysical definition of the universe. Around 300 A.D, Chinese philosophersbelieve that the universe is like an egg and the earth is yolk. The sun11.1. Historical Introductionand stars are moving in the murky ’egg white’. Chinese Taoists believedthat there was an underlying natural order of the universe called ’Tao’ thatgenerates and gauges everything. The Greek philosopher Aristotle thoughtour earth was fixed in the universe surrounded by concentric celestial sphereof planets and stars. There also exists some seemingly funny model likethe ’Turtles all the way down’ from ancient Indian philosophy. Since thesethoughts are mainly based on thought, they are more like philosophy thanscience. Scientists and philosophers like Newton, Laplace and Kant all havetheir own idea, but observation was very limited to judge them.That cosmology became a science dates back to 1917, when Einsteinpublished his paper Cosmological Considerations in the General Theory ofRelativity [29]. In this paper, Einstein firstly applied his General Relativityto describe the universe as cylindrical space-time manifold. It is the firstattempt to describe the geometric structure of the universe. In the sameyear, de Sitter [26] developed a ’spherical’ model for cosmological geome-try. The first work to study the dynamics of the universe is by AlexanderFriedmann [33] who generalized Einstein and de Sitter’s cosmological met-ric and plugged it into the Einstein Equation. Thus he discovered the firstform of Friedmann equations. In 1935 and 1936, H.P.Roberson [71] andA.G.Walker [85] introduced the curvature k and completed the mature formof the Friedmann-Roberson-Walker metric and the Friedmann equations.The FRW metric is based on the Cosmo Principle which states thatthe universe is isotropic and homogeneous. Beyond the Cosmo Principle,people studied inhomogeneities and anisotropy of the universe which tellsus how the large scale structure forms and evolves. From 1970s to 1990s, thegeneral theories to describe the evolution of fluctuations of each ingredient(matter, photon, neutrino and so on) on the ’stage’ of an evolving universehas been developed (see, for example, P.Peebles and J.Yu [61]; R. Sunyaevand Zeldovich [78]; M. Wilson and J. Silk [87]). These theories describe thephysics of those fluctuations guide observation.In the early 20th century, studies of the rotation curves of galaxies [93]suggested that there exists some dark matter in the galaxies. Dark matterdoes not significantly interact with electro-magnetic field so they neitheremit photons nor absorb or scatter photons. One way to detect it is toobserve its gravitational effect. From cosmological observations like weaklensing [83] and the CMB [42], it is realized that dark matter makes up84.5% of the total matter of the universe. Dark matter must be accountedfor in the study of large scale structure. It is still an open question as towhat the dark matter is. Particle physicists come up with many modelsfor dark matter, like WIMPs, sterile neutrinos, axions, etc, which are open21.1. Historical Introductionto test. For cosmologists, the current generally-accepted phenomenologicalmodel for dark matter is cold dark matter (CDM, see Peebles [63]). CDMparticles move at a speed which are much lower than the speed of light.In 1998, Adam Riess [70], Brian Schmidt, Saul Perlmutter [65] studiedthe Hubble diagram of distant supernovae and found that the expansion ofuniverse is actually accelerating. This suggests the existence of dark energywhich has a negative pressure and is responsible for cosmological accelera-tion. Based on CMB and Baryonic Acoustic Oscillation (BAO) data, theamount of dark energy is constrained to be 70% of the critical density. Likethe dark matter, people know very little about the dark energy. Observa-tions suggest that dark energy is likely to be the cosmological constant Λwith the equation of state p = −ρ.A model that treats dark energy to be Λ and dark matter to be cold (themovement of dark matter particle is much less than the speed of light) iscalled the Λ-Cold Dark Matter model (ΛCDM). The ΛCDM model now es-tablished as the standard model for cosmology. Yet there are some problemsthat can’t be resolved under this work. Among them the most famous onesare the horizon problem, flatness problem and magnetic monopole problem.These questions can be solved by cosmological inflation. See section 1.6 fordetails.The first observational evidence for modern cosmology is the discoveryof Hubble’s Law [47]. Hubble’s law states that distant galaxies are movingapart from us with a velocity proportional to its distance.v = H0d (1.1)where H0 is the Hubble constant, v is the speed that an object is movingaway from us, d is the distance of that object. This law can be directlyderived from Friedmann equations. Hubble’s law states that our universe isexpanding.The cosmic microwave background was first predicted in 1948 by GeorgeGamow [34], Ralph Alpher and Robert Herman [13]. R. Alpher and R.Herman estimated the temperature of the blackbody radiation should bearound 5K based on cosmological nucleosynthesis. This radiation, oftenquoted as the Cosmic Microwave Background (CMB) was detected in 1965by Penzias and Wilson [64]. The CMB radiation is significantly consistentwith a blackbody spectrum with temperature 2.7K and is highly isotropic.As an isotropic blackbody radiation is hard to be produced from a nearbyprocess, CMB should be from very far away. It is the relic radiation of hotplasma during the early universe. So it serves as a robust evidence for the31.2. The Big Bang Cosmologythermal big bang theory of cosmology.In 1989 the Cosmic Background Explorer (COBE) was launched to ob-serve the CMB. It made the first discovery of the anisotropy in the CMB[77]. In 2001, the second generation space-based CMB detector, the Wilkin-son Microwave Anisotropy Probe (WMAP) began to take data with higherprecise [42]. In 2013, the third generation of detector, the Planck satelliteput forward the precision further [6] .Another observational evidence for physical cosmology is the Big BangNucleosynthesis. It studies the nucleosynthesis of light elements (H, He, Li)in the early phase of the universe. The theory outline was first proposedin the famous AlpherBetheGamow paper[12]. The BBN theory gives a pre-diction of the relative abundance of light elements, which have been testedwith multiple observations. See [80] as a review.Other types observations like surveys of distant galaxy clusters (for ex-ample, 2dS survey, SDSS), weak lensing, BAO, Cosmic Infrared Backgroundalso provide large amounts of data for different scales, spectral frequen-cies and objects. The improving quality and quantity of observational datagreatly enhance the power of model testing. Cosmology is now a preciseobservational science.1.2 The Big Bang CosmologyWe can use classical mechanics to describe the movements of local planetsand stars. But since classical mechanics deals with action at a distance, itis not appropriate for studying the universe as a whole.The general theory of relativity describes the geometrical and dynamicalproperty of spacetime manifold. It can deal with the large scale spacetimeprecisely. Since the universe is the largest spacetime that contains everythingwe know of, we need to use general relativity to describe it.The standard model of cosmology is based on two basic assumption: 1.The universe is homogeneous; 2. The universe is isotropic. These assump-tion asserts that the metric of the universe is invariant under translationand rotation. To satisfy this two assumption, the metric should be writtenas:ds2 = −dt2 + a2(t)[dr21−Kr2 + r2dθ2 + r2 sin2 θdφ2](1.2)This is called the Friedmann-Roberson-Walker (FRW) metric. ds is theinfinitesimal interval, dt is the infinitesimal time change and dr, dθ,dφ are41.2. The Big Bang Cosmologythe infinitesimal difference of spherical coordinate. a(t) is the scale factorwhich describes the time evolution of the scale of universe. By convention,the value of a today is 1. K is the curvature of the universe, which canhave values −1, 0,+1. For K = 0 the universe is flat, and K = ±1 showsthe positive (spherical) and negative (saddle surface-like) curvature of theuniverse.Inserting the FRW metric into the Einstein Field equation, we can derivethe dynamical equations of universe.a¨ = −4pi3(ρ+ 3p)a(t) (1.3)a˙2 +K =8pi3ρa2(t) (1.4)where ρ and p are the density and pressure contained in the energy-momentumtensor. Their relation is defined by the equation of state:pi = wiρi (1.5)Here the suffix i denotes different physical components in the universe. Formassive matter (including dark matter) wm = 0; radiation wr =13 , darkenergy (cosmological constant Λ) wΛ = −1. Combining Eq.1.5 and Eq.1.3we get:ρi ∝ a−3(1+wi) (1.6)By inserting different values of wi, we can determine the evolution ofdensity for different components in the universe. For matters, ρ ∝ a−3,radiation ρ ∝ a−4, dark energy ρ is constant. So every era of the universeis dominated by one component. The early universe was dominated byradiation. Matter dominated the universe until recently being surpassed bydark energy.Use suffix 0 to represent the current value for all the variables and sett=0 in Eq.1.4. Note that a0 = 1, we have3H208pi−∑iρi0 =3K8pi(1.7)H is defined as H ≡ a˙a and H0 is the Hubble constant. In the above equation,if∑iρi0 = ρc ≡ 3H208pi(1.8)51.2. The Big Bang CosmologyFigure 1.1: Density of each content of the universe. Source: the curvature of the universe is zero. We call this ρc the critical density.Cosmologists often normalize the densities to be Ωi0 ≡ ρi0ρc .The Friedmann equations Eq.1.3 and Eq.1.4 are the dynamical equationfor the scale of the universe. With knowledge of w and Ωi0, we can derivethe time dependence of the scale factor a. Under each cosmic epoch, thecontent and temperature are different. Combining with thermal dynamicsand particle physics we can study the cosmological thermal history and BigBang Nucleosynthesis. Table.1.1 is a summary of the thermal history of theuniverse.A crucial object for modern cosmology is to constrain these cosmologicalparameters with observational data. The current constraints on the densitiesis shown in Table.1.1The hot big bang model has been tested with several observations in-cluding Hubble expansion, CMB spectrum and Big Bang Nucleosynthesis.It is the foundation for the standard model of cosmology.61.3. The Evolution of Large Scale StructureTime Event10−43s Planck time. Unknown physics10−38s GUT phase transition10−34s Cosmological inflation10−14s EM and weak interaction decoupling3min Big-Bang nucleosynthesis60kyr matter-radiation equality380kyr Recombination and decoupling100-400Myr Reionization13.8Gyr PresentTable 1.1: Thermal history of the universe. Data are from and The Evolution of Large Scale StructureThe homogeneous and isotropic assumption of the universe is an importantpillar of big bang cosmology and gives precise description for the cosmo-logical spacetime. But the spatial distribution of matter in the universe isby no means homogeneous. According to large scale sky surveys, galax-ies in the universe tend to concentrate to be clusters while leaving somespace as nearly empty voids (see Fig.1.2). COBE also find that the CMB isnot perfectly isotropic [77], which tells us that the early universe is slightlyinhomogeneous.According to inflation theory, the seed of large scale structure is thequantum fluctuation during the inflation era. The quantum fluctuation per-turbs the matter-radiation fluid and the metric of the universe. Althoughthe universe is continuously diluting and cooling down, some place is fasterthan the other. So there is fluctuations around the average temperatureand density. When the universe cools down to about 1eV, the photonsand baryons decouples. Photons propagates freely and becomes CMB. Theanisotropy observed by COBE reflects perturbation in the early universe. Iwill discuss the CMB anisotropy in the next section.Baryons and dark matter continue to form the large scale structure. Wecan treat them to be continuous fluid governed by gravity and write downthe classical fluid equations:71.3. The Evolution of Large Scale StructureFigure 1.2: The distribution of galaxies in part of the 2dFsky survey. Source:∂ρ∂t+∇ · (ρv) = 0∂v∂t+ (v · ∇)v = −1ρ∇p−∇Φ∇2Φ = 4piρ(1.9)The equations above have a stationary solution, that isv = 0and {ρ, p,Φ} are constant {ρ0, p0,Φ0}. The stationary solution is the aver-aged density. What we are interested in is the overdensity δ ≡ ρ−ρ0ρ0 . Sincewe have 3 equations and 4 variables, we only need to find the solution for δand the other variables can be derived from it.For weak fluctuations, we only need to keep the first-order term in theequation. We are considering perturbations in the background of an ex-panding universe, neglect peculiar velocity and according to Hubble’s law:v =a˙ar (1.10)81.3. The Evolution of Large Scale StructureCombining (1.10) with (1.9) and considering the equation of state Eq.1.5,we find the evolution equation for δ:δ¨k + 2a˙aδ˙k +(c2sk2 − 4pi(1 + w)(1 + 3w)ρ) δk = 0 (1.11)cs is the adiabatic sound speed defined as cs ≡√(∂p∂ρ)s. This is the mostsimplified equation for the perturbation δ. But we also need to considerthe perturbation of the spacetime metric itself and the interaction betweendifferent contents. At later times and at small scales,where gravity is verystrong, we also need to consider the nonlinear effect. which are beyond ourscope of discussion. For more detailed discussion, see [28], [89].Practically, we measure the power spectrum for perturbations:P (k, t) ≡ 〈δk(t)〉2 (1.12)For linear perturbation, the later-time power spectrum can be formallywritten as:P (k, tf ) = P (k, t0)T (k)2D(t0, tf )2 (1.13)Here P (k, t0) is the primordial power spectrum from the end of inflation.The transfer function T (k) accounts for the scale-dependent evolution duringthe epoch of horizon-passing and radiation/matter transition. The growthfactor D(t0, tf ) describes the scale-independent growth during the later pe-riod.According to inflation theory, the spatial distribution of primordial fluc-tuation is Gaussian, in case of both amplitude and phase [62]. And is alsoa power law primordial power spectrum:P (k, t0) = A∗(kk∗)ns(1.14)where A∗ is the amplitude and k∗ is chosen to be a characteristic scale, andns is the power-law index. The primordial fluctuations evolves to form theobserved large scale structure. A∗ and power index ns are undeterminedwhich can be constrained with data. From Eq.1.11, the large scale structureis closely relative to the density of cosmological material and their equa-tions of state. So the observation of large scale structure can also constrain{Ωi, wΛ}.The primordial power-law index is directly related to the inflationaryphysics [28]. By constraining this parameter we can study inflation. The91.4. Anisotropy in the Cosmic Microwave Backgroundtransfer function helps us to study the epoch of horizon-passing and radia-tion/matter transition [30]. While growth factor describe the late evolutionof fluctuation. Since the late universe becomes dominated by dark energy,growth factor contains the information of it [56].Another quantity is often used to describe large scale structure is thetwo-point correlation function:ξ(r) ≡ 〈δ(x+ r)δ(x)〉 (1.15)which is the Fourier transformation of the power spectrumP (k) =∫exp(−ik · r)ξ(r)d3r (1.16)Observations of large scale structure cover a broad range of wavelengthsand different kind of objects. Sky surveys like SDSS[24] and 2dF[90] projectsmake redshift surveys of distant galaxies. CMB observations like WMAPand Planck measures the secondary effects of large scale structure on CMB[79].The Canadian Hydrogen Intensity mapping Experiment is going to map thedistribution of neutral Hydrogen over the redshift range from 0.8 to 2.5[15].CFHTLens, RCSLens and KiDS are measuring the weak lensing effect. Thecross analysis of different observations give us more precise knowledge aboutlarge scale structure.In this thesis, I will discuss the role of cross correlation between thermalSunyaev-Zeldovich effect, weak lensing and the Cosmic Infrared Backgroundin the study of large scale structure.1.4 Anisotropy in the Cosmic MicrowaveBackgroundIn the very early universe, when temperature is much higher than the ioniza-tion energy of hydrogen, electrons are free. The optical depth is extremelyhigh so photons are tightly coupled with baryon, and they behave like a sin-gle fluids. When the temperature cools down to about 10eV, electrons aretrapped by protons, and photons decouple from the baryons. These earliestphotons propagate nearly freely to us and comprise the Cosmic MicrowaveBackground.We can treat fluctuations of the photon fluid the same way as the for-mer section. In order to relate the calculation with observation, it is moreconvenient to use the temperature fluctuation Θ ≡ δTT instead of density todescribe the fluctuation. Here the average temperature scales proportional101.4. Anisotropy in the Cosmic Microwave BackgroundFigure 1.3: Linear matter power spectrum P(k) versus wavenumber ex-trapolated to z = 0, from various measurements of cosmological structure.The black line is the best-fit ΛCDM model. Source: Anisotropy in the Cosmic Microwave Backgroundto 1/a, with the current value 2.728K [31]. To study CMB anisotropy, wemainly consider a single-wave Θ(k) in ` space, that is:Θ`(k) ≡ 1(−i)`∫dµ2P`(µ)Θ(k, µ) (1.17)where µ is the cosine of the polar angle of k, and P`(µ) is the Legendrepolynomial.The complete dynamical equations for the photon-baryon fluid are de-rived from Boltzmann equation. The dominate term is Θ0 which satisfies:Θ¨0 +a˙aR1 +RΘ˙0 + k2c2sΘ0 = F (k, η) (1.18)The derivative Θ˙0 shows the derivative with respect to the comovingtime η. Here R ≡ 3ρb4ργ denotes the ratio between baryon density and thephoton density. cs is the sound speed of the baryon-photon fluid:cs =√11 +R(1.19)The right side of the equation F (k, t) denotes the driving force fromthe metric. So this equation looks exactly like the equation of a dampingoscillator with a driving force. If we neglect the damped term and thedriving force, the solution would be:Θ0(k) ∼ cos[krs(η)] (1.20)where the comoving sound horizon at the time η is defined as:rs(η) ≡∫ η0cs(η′)dη′ (1.21)The peaks will appear at kp = npi/rs where n is an integer. After therecombination, the sound horizon freezes to be rs(η∗) and kp also fixes to bekp = npi/rs(η∗).For other angular scales, we can always write the dynamic equation andsolve for Θ`(k, η). These equations hold until recombination that happens atη∗. The first free photons propagates to us today and we need to consider theimprint of inhomogeneities of large scale structure including gravitationalwells and barriers; scattering by high energy electron (thermal Sunyaev-Zeldovich effect). As a result, the anisotropy today Θ`(k, η0) is a mixture ofΘ`(k, η∗) and dominated by Θ0(k, η∗).121.4. Anisotropy in the Cosmic Microwave BackgroundA perturbation with wavenumber k contributes predominantly to an an-gular scale of order ` ∼ kη0, where η0 is the comoving time today. Since thedominant term Θ0(k, η∗) peaks at kp = npi/rs(η∗), Θ`(k, η0) has the maxi-mum at `p = npiη0/rs(η∗), and the corresponding angle is θp = rs(η∗)/nη0,which is a fraction of the angle corresponding to sound horizon at recombi-nation.The observed CMB map is Θ(nˆ). We can expand the signal in terms ofspherical harmonics:Θ(nˆ) =∞∑`=0∑`m=−`a`mY`m(nˆ) (1.22)According to inflation, the a`m are expected to be Gaussian randomvariables with mean value zero, and variance:〈a`ma∗`′m′〉 = δ``′δmm′C` (1.23)where C` is the angular power spectrum of CMB. The angular power spec-trum may be estimated from data by.Cˆ` =12`+ 1∑`m=−`a`ma∗`m (1.24)For each `, we have 2` + 1 samples of the a`m, so there will be aninevitable uncertainty for our estimated Cˆ` even from an all-sky observation,which is called the cosmic variance. For a Gaussian sample, the variance isproportional to one over square root of number of samples, so:∆C`C`∼√22`+ 1(1.25)At low ` the cosmic variance is relatively high and cannot be improved byany observation from earth since it is due to the limited number of samplesthat we have.The relation between C` and Θ`(k) isC` =2pi∫ ∞0P (k)∣∣∣∣Θ`(k, η0)δ(k)∣∣∣∣2 k2dk (1.26)where δ(k) is the overdensity and P (k) is the mass power spectrum definedin Eq.1.12. By solving the equation for δ and Θ` we can know the angularspectrum of CMB.131.4. Anisotropy in the Cosmic Microwave BackgroundFigure 1.4: Sensitivity of the angular power spectrum to four fundamentalcosmological parameters. (a) The curvature as quantified by Ωtot = 1− Ωk. (b) The dark energy as quantified by the cosmological constant ΩΛ(wΛ =−1) . (c) The physical baryon density Ωbh2. (d ) The physical matterdensity Ωmh2. All are varied around a fiducial model of Ωtot = 1, ΩΛ =0.65, Ωbh2 = 0.02, Ωmh2 = 0.147. Image is from [45]In the equations governing δ and Θ` depend many cosmological param-eters. The observed shape of the angular power spectrum helps us to con-strain these cosmological parameters. For example, based on observationwe can find the angular distance of the sound horizon at recombination byidentifying the first peak of the angular power spectrum. If the universeis not flat, then the angular distance of the sound horizon will not equalto the comoving distance which can be derived with the knowledge of Rand η∗. So the cosmic curvature can in fact change the position of theacoustic peaks. An illustration of sensitivity of angular power spectrum tocosmological parameters is shown in Fig. The ΛCDM ModelThe decoupled baryons will also contain the information of the soundhorizon at recombination. Before recombination, a single density peak prop-agates with the sound speed cs. The baryons are pushed by photons sincethey are tightly coupled. After recombination, photons propagate away asthe CMB and leave the acoustic peak of baryon concentrating the soundhorizon r(η∗). This concentration makes a higher overdensity called acous-tic peak. When we analyze the two-point correlation function of galaxies,we can see this peak at a scale of ∼ 150h−1Mpc. This effect is called theBaryon Acoustic Oscillation (BAO). As the sound horizon is a fixed value(irrespective of what object we observe), the scale of the BAO peak serves asa standard ruler. And its position can probe the distance-redshift relation.1.5 The ΛCDM ModelThe current standard model of cosmology is called the ΛCDM model. It is aparametrized model of the big bang cosmology. Λ stands for the cosmologicalconstant which serves as the dark energy. Its state of equation is w = −1 soit has an effective negative pressure. CDM is the abbreviation of ’cold darkmatter’, which means that the dark matter particles, no matter what theyare, move with a low speed compared to speed of light.In the framework of the ΛCDM model, there are 6 free parameters tobe fixed by observation. They are: physical baryon density Ωb, the darkmatter density Ωc, the age of the universe t0, the scalar spectral indexns, and the fluctuation amplitude As. The other model values, includingthe Hubble constant and age of the universe can be derived from theseparameters assuming a flat universe.The estimation of ΛCDM parameters from observations is one of the fore-most tasks in modern cosmology. In the context of statistics, it is basicallya process to maximize P (M |Dˆ), the probability distribution of parameters(M) given the condition of observational data (Dˆ). According to Bayes’theorem:P (M |Dˆ) = P (Dˆ|M)P (M)P (Dˆ)(1.27)The prior for D is generally set such that∫P (M |D)dnM = 1 and theprior for model parameters is set by prior information. P (M |Dˆ) dependson P (Dˆ|M), which is also called the likelihood L(M,D). The simplestlikelihood predicted by inflation is the Gaussian likelihood:151.5. The ΛCDM ModelFigure 1.5: 68.3%, 95.4%, and 99.7% confidence regions of the (Ωm, w) planefrom supernovae data combined with the constraints from BAO and CMB.This image is from [14]L(M,D) ≈ 1√(2pi)ND detCDexp[−12(Dˆ −D(M))TC−1D (Dˆ −D(M))](1.28)Here D(M) is the theoretical value for the data given model the parametersM and CD is the data covariance matrix. Since P (M |Dˆ) is proportionalto likelihood, what we need to do is to maximize the likelihood to find thebest-fit values for M . Moreover, we need to estimate the error of M toevaluate the goodness of fit.A quick and dirty estimate of parameter covariance matrix is providedby Fisher matrix:Fij ≡ ∂Da∂MiC−1D,ab∂Db∂Mj(1.29)161.5. The ΛCDM ModelParameter ValueΩbh2 0.0223± 0.00014Ωch2 0.1188± 0.001t0 13.799± 0.021Gyrln(1010As) 3.064± 0.023ns 0.9667± 0.004τ 0.066± 0.012Table 1.2: ΛCDM independent parameters given by Planck Collaboration[5]If the model is purely linear, then the parameter covariance matrix isexactly the Fisher matrix. For cosmology it is typically not. A commonly-used method to estimate parameter errors is the Markov-Chain-Monte-Carlomethod (MCMC).An MCMC chain is a sampler in parameter space. The chain is generatedby comparing the likelihood of a randomly-picked new point with the lastpoint in the existing chain to decide whether keep the new point in thechain or discard it. After many iterations, the Markov Chain will samplethe parameter distribution. Then we can draw the contour correspondingto different confidence level to present the parameter error.There are data from many different experiments including distant super-novae, sky survey, CMB observations etc. Different data set can be combinedto make better constraints on ΛCDM parameters. Take Fig.1.5 as an ex-ample. In this figure blue, orange and green contours are confidence regionsof the (Ωm, w) plane from supernovae data combined with the constraintsfrom BAO and CMB. SNe data is from the Supernova Cosmology Project;BAO data is from SDSS DR7 and 2dFGRS; CMB data is from WMAP7.The grey contour shows the combined confidence region of these three ob-servations which gives a much better constrain on (Ωm, w) than any singleexperiment.Table.1.2 gives the ΛCDM parameters given by Planck Collaboration.Other experiments may give a slightly different value but most of them arecompatible. It is expected that with the continuously improving observa-tional precision, we can make better constraints on these parameters in thefuture.171.6. Brief Introduction to Cosmological Inflation1.6 Brief Introduction to Cosmological Inflation1.6.1 Problems in Standard Cosmology ModelThe standard ΛCDM model for cosmology has become a successful modelto describe our universe. However, there are some puzzles in our universethat cannot be explained by the ΛCDM model. The 3 important puzzlesare:1. The horizon problem: the standard cosmology model cannot explainthe fact that the sky looks quite similar between two largely separated pointsin space. In the past these two points should have interacted with each otherto reach equilibrium. But information could only travel at most light speedor lessThe particle horizon dH at time t is defined as the proper distance thatlight can travel from the beginning of the universe to t. For light we haved2s = 0. According to the definition of FRW metric:dH(t) ≡ a(t)∫ t0dt′a(t′)(1.30)If we assume that the universe is dominated by radiation at early times,it can be easily calculated that at the time of last scattering the particlehorizon is dH(tls) = 0.251Mpc where tls is the time of last scattering. TheCMB is at an angular distance from our position to the last scattering surfacedA ≈ 12.8Mpc. Assuming a flat universe (which is shown to be true withmultiple observations), points on the last scattering surface separated by ahorizon distance will have an angular separation:θH =dH(tls)dA≈ 0.251Mpc12.8Mpc≈ 1.1◦ (1.31)This means that two points in a CMB map which are separated by anangle larger than 1.1 ◦ should had never interacted with each other. Thisconflicts with the fact that the temperature of CMB is highly isotropic toone part in 105.2. The flatness problem: The spatial curvature of the universe at time tcan be defined as:1− Ωtot(t) = −K(H(t)a(t))2=H20 (1− Ωtot(t0))H(t)2a(t)2(1.32)With the current data, we have181.6. Brief Introduction to Cosmological Inflation1− Ωtot(t0) ≤ 0.005 (1.33)Combine Eq.1.4, 1.5 and the definition of H. In the context of big bangcosmology, the early universe is dominated by matter and radiation, so that:H(t)2H20=Ωr,0a4+Ωm,0a3(1.34)so the curvature parameter evolves as:1− Ωtot(t) = 1− Ωtot(t0)a2Ωr,0 + aΩm,0(1.35)We can see that this parameter is always increasing as a function of time.Given the values of Ωr,0 and Ωm,0 and 1−Ωtot(t0) we can calculate the valueat the earliest time we dare to describe the universe using general relativity,the Planck time:|1− Ωtot(tP )| ≤ 2× 10−62 (1.36)This is an extremely tiny value, which means that the early universeshould be extremely flat. It is unnatural to require such a fine-tuned uni-verse.3. The magnetic monopole problem: Earlier than tGUT ∼ 10−36s, theuniverse is thought to be in the the Grand Unified Theories (GUT) epoch,when strong, weak and the electromagnetic interaction cannot be distin-guished from one another. When t = tGUT, the strong interaction decouplesfrom electroweak interaction, which is called the GUT phase transition. Thisphase transition is associated with a loss of symmetry and it gives rise toflaws known as topological defects. GUT theories predict that the GUTphase transition creates point-like topological defects that act as magneticmonopoles.From GUT theory, the number density of monopoles is n(MM)(tGUT ) ∼1082m3. Given the mass of magnetic monopoles, it can be directly calculatedthat the energy density of magnetic monopoles isM (tGUT) ∼ 1094Tev m−3 (1.37)which is 10 orders of magnitude less than the energy density of radiation. Asmagnetic monopoles are massive particles, their energy density evolves atthe rate ∝ a−3 while for radiation ∝ a−4. So at the time around t ∼ 10−16s,191.6. Brief Introduction to Cosmological Inflationthe energy density of magnetic monopoles and radiation should be equaland the current universe should be dominated by magnetic monopoles.However, there is no strong evidence that they exist now at all. Obser-vations show that the upper bound of the density of the magnetic monopoleis ΩMM,0 < 5× 10− The Inflation SolutionAll of the three problems are based on the standard model of cosmologyfor which the early universe is dominated by massive matter and radiation.This suggests that the solution to these problem may be that the very earlyuniverse experiences a different expansion history.In 1980, Alan Guth proposed the cosmological inflation on a SLAC sem-inar primarily to solve the magnetic monopole problem. The idea is thatat some time after the GUT time, the universe experienced a extremelydramatic expansion, which diluted the density of monopoles to a very smallnumber. Guth first proposed an accelerating expansion of the early uni-verse which is called ’cosmological inflation’[37] and then showed that itsuccessfully solved the horizon and flatness problems.During the inflation epoch, a¨ > 0. According to (1.3), this means thatP < −13ρ (1.38)So there existed something with negative pressure during the inflationepoch of the universe. The simplest implementation of inflation states thatthe universe was temporarily dominated by a positive cosmological constantρΛ. According to 1.4, the scale factor scales asa(t) ∝ eHit (1.39)where Hi is the Hubble constant during inflation, which is defined as:Hi ≡ 8pi3ρΛ (1.40)It remains a constant during inflation and keeps the universe expandsexponentially. Thus, between the time ti, when the exponential inflationbegan, and the time tf , when the inflation terminated, the scale factorincreased by a factora(tf )a(ti)= eN (1.41)201.6. Brief Introduction to Cosmological Inflationwhere N, the number of e-foldings of inflation, wasN ≡ Hi(tf − ti) (1.42)The number of e-foldings of inflation is a very important parameter de-scribing inflation. The value can be constrained by studying how inflationcan solve the three problems. Let’s first look at the flatness problem. As-sume that inflation happens right after the GUT time, so ti = 10−36s withHi ≈ t−1i and lasts N e-foldings, ending at tf ≈ (N + 1)ti. Putting 1.39 into1.32 yields:|1− Ωtot(t)| ∝ e−2Hit (1.43)Suppose that the universe is strongly curved (which seems not that un-natural) before inflation, so at the time tf after N e-foldings|1− Ωtot(tf )| ∼ e2N (1.44)After inflation, the universe became dominated by radiation. So we canextrapolate the scale factor back to the time tf ≈ (N + 1) · 10−36sa(tf ) ≈ 2× 10−28√N + 1 (1.45)Given the measured value |1−Ωtot(t0)| ≤ 0.005 today, the flatness factorhad the value|1− Ωtot(tf )| ≤ 2× 10−54(N + 1) (1.46)Comparing (1.44) and (1.46) gives an estimate N ≥ 60. This meansthat if the universe inflated this number of e-foldings, the curvature density|1− Ωtot| can be of order of 1, which solves the flatness problem.To resolve the horizon problem, we first calculate the horizon distanceat the beginning of inflation using (1.30) and assume that the universe isdominated by radiation at this epoch:dH(ti) = a(ti)∫ ti0dta(t)= 2ti (1.47)Then the horizon size at the end of inflation wasdH(tf ) = a(ti)eN(∫ ti0dta(t)+∫ tftidta(ti) exp[Hi(t− ti)])≈ 3eN ti (1.48)211.6. Brief Introduction to Cosmological InflationGiven ti ∼ 10−36s, the horizon size immediately before inflation was:dH(ti) = 2ti ≈ 6× 10−28m (1.49)After 65 e-foldings of inflation (we take this value for concreteness. Thisvalue is compatible with the requirement to solve the flatness problem, forwhich N ≥ 60.), the horizon size immediately after inflation wasdH(tf ) ≈ 3eN ti ∼ 15m (1.50)Given the current radius of the last scattering surface dp(tls) ≈ 2 ×14000Mpc, at the end of inflation, this radius wasdp(tf ) = a(tf )dp(t0) ∼ 3× 10−23Mpc ∼ 0.9m (1.51)This value is less than the horizon size at the end of inflation, whichmeans that particles in this radius can interact with each other during thatepoch. So any point on the last scattering surface could have interactedduring inflation. So the horizon problem is resolved.For the monopole problem, the inflationary universe can greatly dilutethe number density of monopoles. Too see how it works, let’s take thenumber density of monopoles at ti to be nMM(ti) ∼ 1082m−3. After 65e-foldings of expansion, the number density would have been nMM(tf ) =e−195nMM(ti) ∼ 0.002m−3. The number density today, after the additionalexpansion from a(tf ) ∼ 2 × 10−27 to a0 = 1, would then be nMM(t0) ∼2×10−83m−3 ∼ 5×10−16Mpc−3, which means that within the last scatteringsurface, the total number of monopole is of order 10−11. It is absolutelyunlikely to be detected.1.6.3 The Physics of InflationThe first success of inflation is that it provides solutions to these threeproblems. But the physics of inflation remains a question both for cosmologyand for high energy physics. The first theory that describes the physics ofinflation was proposed by Alan Guth. The idea is that there exists a scalarfield φ (Guth thought it was actually the Higgs field) that is responsiblefor inflation. During the cooling process of the very early universe, thisfield was trapped in a local minimum of its potential V0 (the false vacuum).When the other contents cooled to an energy density lower than V0, thenthe universe was dominated by this field. During the inflation epoch, theuniverse was actually in a metastable state with a constant energy V0, soit underwent an exponential expansion (inflation). At the end of inflation221.6. Brief Introduction to Cosmological InflationFigure 1.6: A potential of the inflaton field that can give rise to inflation.This figure comes from [72].epoch, the scalar field tunneled through the potential barrier and rolledtoward the true minimum of the potential (true vacuum). The energy of thescalar field (latent heat) reheated the universe.There is a problem with this model called ’graceful exit problem’. Dur-ing inflation, the universe was in a false vacuum state everywhere. At theend of inflation, the universe did not tunnel into the true vacuum simultane-ously. Instead, there appeared many true vacuum ’bubbles’ and they expandquickly and the walls of bubbles collapsed with each other then made theuniverse decay into the true vacuum. The problem is that reheating onlyhappened around the walls while the interior of the bubble is completelyempty. which means that at the end of inflation, reheating was very inho-mogeneous. So the universe would be highly inhomogeneous and anisotropictoday.The original version of inflation model by Guth was soon supplanted bya new version proposed by Linde[55] and by Albrecht and Steinhardt[11].The new inflation model is called the slow-roll inflation. It assumes thatthe potential of the inflaton field was very flat around V0 in the beginningand then goes down to the vacuum state at φ = φ0. The field started atφ = 0 where V (0) = V0 and very slowly rolled toward φ0 (see Fig.1.6). Theuniverse cooled down enough that the inflaton field dominated the energydensity to cause inflation. At the end of inflation φ rolled down to φ0 andthe reached its true vacuum state. The phase transition also happened by231.6. Brief Introduction to Cosmological Inflationforming bubbles, but under this scenario the interior of the bubble startedat φ ∼ 0 and slow-rolled to φ0. Eventually the field energy was convertedinto radiation by oscillating around the minimum of the potential. Thehorizon is assumed to be within one bubble so the universe homogeneousand isotropic.For the inflaton field φ, its energy density and pressure take the form:ρ =12φ˙2 + V (φ), P =12φ˙2 − V (φ) (1.52)I drop the suffix φ for ρ and P here for simplicity. But reader should bear inmind that in this subsection all the density and pressure terms are associatedwith the inflaton field.The energy conservation equation ρ˙ = −3H(ρ+ P ) takes the formφ¨+ 3Hφ˙+ V ′(φ) = 0 (1.53)The Hubble parameter during inflation is given byH =√8piρ3=√8pi3(12φ˙2 + V (φ))(1.54)Combining (1.53) and (1.54), we haveH˙ = −4piφ˙2 (1.55)In order to have exponential expansion, the fractional change of H in aHubble time, 1/H, must be much less than unity: ≡ |H˙|H1H 1 (1.56)With (1.55), this requires thatφ˙2  |V (φ)| (1.57)This has the consequence that P ' −ρ, and alsoH '√8piV (φ)3(1.58)Notice that the Hi value discussed in the last section is nothing but theH value at φ = 0. We require that φ changes very slowly so that duringinflation H is approximately constant.241.6. Brief Introduction to Cosmological InflationUsually it is also assumed that the fractional change of φ˙ during a Hubbletime is also much less than unity, that is:η ≡ |φ¨||φ˙|1H 1 (1.59)This means that in (1.53), we can drop the term φ¨ and the equation becomesφ˙ = −V′(φ)3H= − V′(φ)√24piV (φ)(1.60)Then (1.56) becomes ∣∣∣∣V ′(φ)V (φ)∣∣∣∣ √16pi (1.61)Taking time derivative of (1.60) and combining with inequality (1.61), wehave ∣∣∣∣V ′′(φ)V (φ)∣∣∣∣ 24pi (1.62)(1.61) and (1.62) are two flatness conditions that insure the slow rollof φ. Also we have defined two slow-roll parameters  and η. The twoflatness conditions are equivalent to the requirement that the two slow-rollparameters are both much less than 1. With these two conditions, we canestimate the length of inflation to be tf − ti ' φ0/φ˙. and the number ofe-foldings isN = Hi(tf − ti) '√8piV03φ0φ˙= 8piV0φ0V ′(φ)(1.63)The last equation comes from taking into account (1.60). Large values of φ0and V0 (that is, a broad, high plateau) and small values of V′(φ) (that is, aflat plateau) lead to more e-foldings of inflation.After rolling off the plateau, the inflaton field φ oscillates about theminimum at φ0. But the oscillation are damped by the term Hφ˙ in (1.53)and energy is carried away by particles. These particles reheat the universeafter the inflation.So far we have only discussed the zero-th order term of the inflaton fieldφ. Actually it has fluctuations, δφ, during the inflation era. Such fluctuationactually serves as the seed of cosmological structure. For detailed calculation251.6. Brief Introduction to Cosmological Inflationplease see [28]. There is a relation between the matter power spectrum indexns and the slow-roll parameters:ns = 1− 4− 2η (1.64)So the observation of a ’tilt’ in the matter power spectrum, ns < 1, canprovide evidence of inflation.A ’smoking-gun’ of inflation is the existence of tensor perturbations, orso-called primordial gravitational waves. There are generated from quantumfluctuation during inflation. I will discuss this process in the next section.The three phenomenological parameters N, , η of inflation can be con-strained by observation of large scale structure and the CMB (especiallyCMB polarization). We have not discuss the origin of the inflaton field andwhat the exact form of V (φ): these are still open questions. There are manymodels for inflation and most (if not all) of them need to be ruled out withobservation. See [54] for a summary of recent developments in the study ofinflation.26Chapter 2Observing CMBPolarization: The PIXIEExperimentThe frequency spectrum and temperature anisotropy of the CMB have con-tributed immensely to our understanding of the universe. It’s polarizationanisotropy also carries unique information from the early universe, especiallythe polarization anisotropy can tell us more. Especially about the physicsof inflation.Research on CMB polarization started in the 1980s [18] and maturedaround 1997. [75][49] give a clear mathematical description of CMB polar-ization and how to relate it to observations. Reviews of CMB polarizationmay be found in [91][46].In the first section I will some theoretical and some observational issuesin CMB polarization. The calculation follows [28] and [66]. The secondsection briefly introduces the PIXIE experiment and the last section discussmy work on instrument simulation for PIXIE.2.1 Studying the Inflation Era with CMBPolarization2.1.1 The Stokes ParametersThere are several ways to describe the polarization of photons, for example,the x and y components (when taking z as the propagation direction) andthe Jones matrix. For the CMB, we often use the Stokes parameters. Formonochromatic light propagating along the z direction, its electric field canbe written ~E = (Exxˆ+Eyyˆ)eiωt+iφ(~r) where xˆ, yˆ are unit vector along x andy direction, and φ(~r) is an arbitrary phase factor which depends only onposition. Define the Stokes parameters as:272.1. Studying the Inflation Era with CMB PolarizationI ≡ 〈E2x〉+ 〈E2y〉Q ≡ 〈E2x〉− 〈E2y〉U ≡2Re 〈E∗xEy〉V ≡2Im 〈E∗xEy〉(2.1)where 〈·〉 represents expectation values. I only contains information on theintensity but not polarization. V only appears when there is a circularpolarization component. When the light is monochromatic and linearlypolarized 1, the Q and U parameters can be simplified as:Q ≡E2x − E2yU ≡2ExEy(2.2)The polarized intensity and polarization angle can be denoted as:P =√Q2 + U2ψ =12arctan(UQ)(2.3)The information is fully included in Q and U parameters, which agrees withthe fact that a linearly polarized light has 2 degrees of freedom.If we rotate the light wave around z axis by an angle of α, then Q andU will change into: (Q′U ′)=(cos 2α − sin 2αsin 2α cos 2α)(QU)(2.4)So the Q and U parameters change like a spin-2 particle. Note that ifwe set α = pi2 then Q and U interchange.2.1.2 Thomson ScatteringCMB photons that we receive today are the photons which were last-scatteredat the last scattering surface. Thomson scattering leaves footprint on CMBpolarization. For Thomson scattering, the cross section of ith component isproportional to2∑j=1|ˆi(nˆ) · ˆ′j(nˆ′)|2. nˆ′ is the incoming direction and nˆ is theoutcoming direction, while ˆ and ˆ′ are the polarizations for incoming and1CMB photons are linearly polarized. And we can always deal with a ’template’ of Qand U then multiply it by the black body spectrum.282.1. Studying the Inflation Era with CMB Polarizationoutcoming rays. According to (2.2), the Q and U parameters for scatteredlight are:Q = A∫dΩ′f(nˆ)2∑j=1(|xˆ · ˆ′j |2 − |yˆ · ˆ′j |2) (2.5)U = 2A∫dΩ′f(nˆ)2∑j=1|xˆ · ˆ′j |2 ·2∑j=1|yˆ · ˆ′j |2 (2.6)where A is a normalization factor which we are not interested in. Ω′ is thesolid angle of the incoming light. f(nˆ′) is the amplitude of light incomingfrom nˆ′ direction. In xyz coordinate, assume nˆ′ = (sin θ′ cosφ′, sin θ′ sinφ, cos θ′).Choose xˆ′ and yˆ′ direction so that the z component of yˆ′ is zero. Soxˆ′ = (cos θ′ cosφ′, cos θ′ sinφ′,− sin θ′)yˆ′ = (− sinφ′, cosφ′, 0) (2.7)plug this into (2.5) and (2.6), we have:(QU)= A∫dΩ′f(nˆ) sin2 θ′(cos 2φ′sin 2φ′)(2.8)In terms of spherical harmonic functions, (2.8) reads:(QU)= A∫dΩ′f(nˆ) sin2 θ′(Y 22 (Ω′) + Y −22 (Ω′)1i[Y 22 (Ω′)− Y −22 (Ω′)]) (2.9)We are only interested in the fluctuation of the amplitude, which isproportional to the temperature fluctuation, so(QU)∝ A∫dΩ′Θ(nˆ) sin2 θ′(Y 22 (Ω′) + Y −22 (Ω′)1i[Y 22 (Ω′)− Y −22 (Ω′)]) (2.10)So the perturbation on Q and U only depends on the quadrupole of thetemperature anisotropy Θ. We need to solve the Boltzmann equation forphotons to get Θ(nˆ), then integrate (2.10) to get the exact form for Q andU .292.1. Studying the Inflation Era with CMB Polarization2.1.3 Angular Power Spectrum of PolarizationFirst define Q± ≡ Q ± iU . From (2.4) the transformation of Q± is Q± →e±2iαQ±. So Q± can be decomposed into spin 2-weighted spherical harmon-icsQ±(nˆ) =inf∑`=2∑`m=−`Q±`m±2Ym` (nˆ) (2.11)The definition of spin2-weighted spherical harmonics ±2Y m` (nˆ) will be dis-cussed in the Appendix A.The E and B mode of the polarization anisotropy are defined as:E`m ≡ −Q+`m +Q−`,−m2, B`m ≡ iQ+`m −Q−`,−m2(2.12)Under parity transformation, Q does not change while U changes into−U . Also ±Y m` → (−1)`±Y m` , thus E`m → (−1)`E`m andB`m → (−1)`+1B`m.E is a scalar field while B is a pseudo-scalar field. E has the same parity asΘ but opposite to B, so EB and ΘB cross-correlations are both zero.Define the angular power spectra:〈E∗`mE∗`′m′〉 = δ``′δmm′CEE`〈B∗`mB∗`′m′〉 = δ``′δmm′CBB`〈Θ∗`mE∗`′m′〉 = δ``′δmm′CΘE`(2.13)The EE signal is relatively strong and was first detected by DASI in 2002[21].It can be used to constrain cosmological parameters when combined withother observations. ΘE at low ` contains information about the reionizationera. BB is what this chapter mainly concern about because it serves as agood method to study the inflation.At small angular scales (high `), the spin 2-weighted spherical harmonicscan be approximated as ±Y m` (nˆ) → e±2iφ`ei~`·nˆ. The spherical harmonicdecomposition is then approximately equal to a Fourier transformation in2-d.(Q± iU)(nˆ) =∑`∑m(E`m + iB`m)±2Y m` (nˆ)→∫ [E(~`) + iB(~`)]e±2iφ`ei~`·nˆd~l(2.14)Transform (Q± iU)(nˆ) into ~` space:302.1. Studying the Inflation Era with CMB PolarizationFigure 2.1: E and B-mode polarization patterns. (left panel) A representa-tive Fourier mode of a density perturbation. (middle) E-mode polarizationpattern resulting from Thomson scattering of this mode (growing ampli-tude). (right) B-mode polarization pattern. Figure is from [21](Q± iU)(nˆ) =∫(Q± iU)(ˆ`)ei~`·nˆd~l (2.15)plug into (2.14) we get:(E(~`)B(~`))=(cos 2φ` sin 2φ`− sin 2φ` cos 2φ`)(Q(~`)U(~`))(2.16)Here ~` is the wave vector in 2-D plane and φ` is the angle between ` andpolar axis. Note that this wave vector is for the perturbation pattern butnot for the light wave. For a single `, E and B distribute just like a single-mode wave. If we choose x axis as the polar axis and let φ` = 0, we haveE = Q and B = U . So the pure E polarization is parallel or perpendicularto ` while pure B mode is pi4 or3pi4 off ` (see Fig.2.1).So the small-scale power spectra for E and B mode are defined as:〈E∗(~`)E(~`′)〉= (2pi)2δ(~`− ~`)CEE`〈B∗(~`)B(~`′)〉= (2pi)2δ(~`− ~`)CBB`(2.17)ΘE power spectrum can be similarly defined. The early papers like [75][44][91]use 2.13, while [74] and [28] use 2.17 as the definition of E and B modes.The definition 2.13 shows the parity property of E and B modes while 2.17has a more clear picture and easy to calculate. [49] decomposes the rank-2312.1. Studying the Inflation Era with CMB PolarizationFigure 2.2: Local quadrupole perturbation field. Red color represents red-shift and blue is blueshift. Figures are from polarization matrix into a curl part (C) and gradient part (G).They are equivalent to E and B modes.2.1.4 From Temperature Fluctuations to PolarizationFluctuationsThe perturbation of the spacetime metric can be denoted as:g00 = −1− 2Ψg0i = Vigij = a2δij(1 + 2Φ) + a2hij(2.18)where Ψ and Φ are called the scalar perturbations; Vi is the vector pertur-bation and the symmetric-traceless matrix hij is the tensor perturbation.Plug this metric into Einstein’s Field Equation to obtain the dynamicalequations for the three modes of perturbation. The scalar fluctuations actlike a harmonic oscillator, vector perturbations act a vortex field and tensorperturbations are gravitational waves.These three modes of perturbation have different origins. The scalarperturbations are driven by matter fluctuations; the vector modes cannotbe generated by inflation and decay quickly to zero; tensor modes originate inthe quantum fluctuations during inflation. Their amplitude is proportionalto the energy scale of inflation. So tensor perturbation can probe the physicsbehind inflation. The tensor-to-scalar ratio r represents the energy level ofinflation. ns (see (1.64)) and r are two most important parameters for theinflation model.The perturbations of the metric affects the photon-baryon fluid becauseit perturbs the movement of photons and baryons. See Fig.2.2. We can322.1. Studying the Inflation Era with CMB Polarizationsolve the exact Boltzmann equation for the photon polarization combiningwith the Einstein Field Equation with a perturbed metric. The scalar andtensor perturbation are decoupled and we can treat them separately.101 102 103`10-410-310-210-1100101102103104`(`+1)C`/2pi[νK2] ↓  reionization↑  gravitational waves↑  weak lensingΘΘEEBBΘEFigure 2.3: Angular power spectra of EE,BB and ΘE generated by CAMB.Reionization and gravitational lensing are taken into account. The cosmol-ogy is: Ωk = 0, Ωbh2 = 0.02, Ωmh2 = 0.16,ns = 1, r = 0.1, TCMB = 2.7255K.The dashed line represents negatively correlated.For scalar perturbation, one can solve the Boltzmann equation and findthatCEE` ≈ CP`CBB` = 0(2.19)CP` is the angular power spectrum for the polarization intensity. Theseequations means that scalar field can only produce E mode perturbation.Tensor perturbation can generate both E and B mode:332.1. Studying the Inflation Era with CMB PolarizationCEE` = (2pi)2∫ {[D1,+` (~k)]2+[D1,×` (~k)]2}d3kCBB` = (2pi)2∫ {[D2,+` (~k)]2+[D2,×` (~k)]2}d3k(2.20)Where D is defined as:D1,` =22`+ 1[(`+ 1)Θ`+1(~k) + `Θ`−1(~k)]D2,` =22`+ 1[(`+ 1)(`+ 2)2`+ 3Θ`+2(~k) + 26`3 + 9`2 − `− 2(`− 1)(2`+ 3) Θ`(~k)+`(`− 1)2`− 3 Θ`−2(~k)] (2.21)where Θ` are the two modes of CMB temperature perturbation generated bytensor metric perturbation. From (2.20) we can see thatB mode polarizationcan be only generated by tensor perturbation. So it is a ’smoking gun’ forinflation.The previous discussion only considers the ’primordial’ effect on CMBpolarization. However, secondary scattering from foreground also leavestrace on CMB polarization. Reionization causes a boost on large scales(` < 10) because it introduces Thomson scattering from nearby reionlizedclouds. Weak lensing can displace CMB polarization and cause leakagefrom E mode to B mode. Fortunately this effect only happens at smallscale (` & 100). See Fig.2.3 for an example of angular power spectra ofCMB temperature and polarization. The B mode is very weak compare tothe E mode signal.Polarized dust emission from hot galactic dust contaminate the CMBpolarization. It is a crucial task for all the CMB polarization experimentsto separate the dust component from true CMB signal.2.1.5 CMB Polarization ObservationsThe first detection of polarized CMB signal was given by DASI in 2002[21].It detected a 4.9σ E mode signal and obtained EE and ΘE power spectra.In 2003 WMAP also detected ΘE signal. Some following experiments (likeCBI[57], CAPMAP[16], BICEP1[50], QUaD[20]) gave more precise observa-tions of E mode polarization, but none of them has made a detection of Bmode polarization yet.342.1. Studying the Inflation Era with CMB PolarizationIn 2013, Planck provided constraints on ns and r by combining theirdata with some other observations [8]. Fig.2.4 shows their constraints for nsand r. Since no B mode had been detected, the confidence region only givesupper limits on r, but it is enough to rule out some of the inflation models.Figure 2.4: Marginalized joint 68% and 95% CL regions for ns and r0.002from Planck in combination with other data sets compared to the theoreticalpredictions of selected inflationary models. r0.002 is the tensor-to-scalar ratioat a pivot scale k∗ = 0.002Mpc−1.; N∗ is the number of e-fold. This imageis from [8]Secondly, in March, 2014, BICEP2 announced a first 5σ detection of Bmode in 30 < ` < 150 at 150GHz. From this data, they derived constraintson r of r = 0.2+0.07−0.05. r = 0 was ruled out in 7σ confidence level.However, the BICEP2 interpretation was soon be questioned. Firstly,their constraint on r was larger than the upper limit given by Planck anWMAP. In May 2014, [59] and [32] pointed out that BICEP2 might haveunderestimated the polarized emission from galactic dust. In September,2014, Planck released its first measurement of dust polarization at 353GHz[4]. After extrapolating to 150GHz in BICEP2 region, the Planck Collabora-tion found a similar dust B-mode signal. A joint analysis between BICEP2and Planck showed a highly significant cross correlation suggesting that theCBB` signal detected by BICEP2 was indeed due to dust. Taking this cor-relation into account, the new constraint on r is an upper limit r < 0.12 at352.1. Studying the Inflation Era with CMB Polarization95% confidence.There are currently many experiments aiming to detect CMB B-modepolarization. Most of them are ground-based experiments. These experi-ments use a variety of polarizers to separate the different polarization com-ponents and bolometers or HEMT amplifiers to measure the brightness ofthe signal. Table.2.1 gives a summary of current and future CMB polariza-tion experiments.Project Name Year Status ` range Frequency(GHz) TypePOLARBEAR 2012-date Active 50-2000 150 GroundKECKArray 2010-date Active 21-335 95, 150, 220 GroundACTPol 2013-date Active 225-8725 90, 146 GHz GroundSPTpol 2012-date Active 501-5000 95, 150 GHz GroundQUIJOTE 2012-date Active 10-30011, 13, 17,19, 30, 40GroundAMiBA 2007-date Active n/a-4300 90 GroundCOMPASS 2003-date Active 200-600 26-36 GroundPOLAR 2000 Active 2-30 26-46 GroundBEAST 2000-date Active 10-1000 100 and 150Balloon,GroundKUPID 2003-date Active 100-600 12-18 GroundABS 2011-date Active 25-200 145 GroundSPIDER - Active 10-300 90, 150, 280 BalloonCLASS 2016-date Active 2-20040, 90, 150,220GroundBICEP3/Keck Array 2016-date Active degree scale 95,150,220 GroundMBI-B - Future 360-16000 90 GroundEBEX - Future 25-1000 150-450 BalloonPIPER - Future -200, 270, 350,800BalloonPIXIE - Future 30GHz-6THz SatelliteQUBIC Future - 150,220 GroundTable 2.1: The state of some current and future CMB polarization experi-ments. Part of the data is from Overview of the PIXIE Experiment2.2 Overview of the PIXIE ExperimentThe misinterpretation of the BICEP2 data was a reminder to pay closemuch attention to foreground contamination. The B-mode signal is faintcompared to the polarized Galactic synchrotron and dust foregrounds [73].Fig.2.5 shows the frequency spectra for the CMB B-mode with different rvalues, compared with synchrotron and dust foregrounds. The best windowto observe the CMB B-mode is around 80GHz. But reliably separating CMBemission from foreground emission based on their different frequency spectrarequires observations at multiple frequency channels.Figure 2.5: The RMS on angular scales of 1 for the polarized CMB withdifferent r value compared with that from foregrounds extracted from theWMAP data at ` = 90.[73]The Primordial Inflation Explorer (PIXIE)[51] is an Explorer-class mis-sion to detect the primordial CMB polarization signal. The proposed instru-ment combines multi-moded optics with a Fourier Transform Spectrometer(FTS) to provide breakthrough sensitivity for CMB polarimetry using onlyfour semiconductor detectors. The FTS system synthesizes 400 channelsacross 2.5 decades in frequency (30GHz to 6THz). This frequency range isbroader than any operating and proposed CMB polarization experiment and372.2. Overview of the PIXIE Experimentthe channels are continuous, which provides extraordinarily strong capabil-ity to separate CMB from Galactic foregrounds. In addition, PIXIEs highlysymmetric design enables operation as a nulling polarimeter to provide thenecessary control of instrumental effects.Figure 2.6: Theoretical angular power spectra for the unpolarized, E-mode,and B-mode polarization in the CMB. The dashed red line shows thePIXIE sensitivity to B-mode polarization.Red points and error bars showthe response within ` bins to a B-mode power spectrum with amplituder = 0.01[51].Fig.2.6 shows the sensitivity of PIXIE to the B-mode polarization signalin the CMB. PIXIE is sensitive to a B-mode signal on a relatively largescale.Fig.2.7 shows the instrument concept. Two off-axis primary mirrors 550mm in diameter produce twin beams co-aligned with the spacecraft spinaxis. A folding flat and 50 mm secondary mirror route the beams to theFTS. A set of six transfer mirror pairs (also called the Totem mirrors), eachimage the previous mirror to the following one and shuttles the radiationthrough a series of polarizing wire grids. Polarizer A transmits vertical po-382.2. Overview of the PIXIE Experimentlarization and reflects horizontal polarization, separating each beam intoorthogonal polarization states. A second polarizer (B) with wires oriented45◦ relative to grid A mixes the polarization states. A Mirror TransportMechanism moves back-to-back dihedral mirrors to inject an optical phasedelay. The phase-delayed beams re-combine (interfere) at Polarizer C. Po-larizer D (oriented the same as A) splits the beams again and routes themto two multi-moded concentrator feed horns. Each concentrator is square topreserve linear polarization and contains a pair of identical bolometers, eachsensitive to a single linear polarization but mounted at 90◦ to each otherto measure orthogonal polarization states. To control stray light, all inter-nal surfaces except the active optical elements are coated with a microwaveabsorber, forming a blackbody cavity isothermal with the sky.392.2. Overview of the PIXIE ExperimentFigure 2.7: Upper panel: Schematic view of the PIXIE optical signal path.As the dihedral mirror moves, the detectors measure a fringe pattern propor-tional to the Fourier transform of the difference spectrum between orthog-onal polarization states from the two input beams (Stokes Q in instrumentcoordinates). A full-aperture blackbody calibrator can move to block eitherinput beam, or be stowed to allow both beams to view the same patch ofsky; Lower panel: Instrument physical layout showing the beam-formingoptics and Fourier Transform Spectrometer[51].402.2. Overview of the PIXIE ExperimentEach of the four detectors measures an interference fringe pattern be-tween orthogonal linear polarizations from the two input beams. For thisand the following sections, we use the convention that z axis is the directionof the optical axis and the FTS lies in xz plane. Let ~E = Exxˆ+Eyyˆ repre-sent the electric field incident from the sky. The power at the detectors asa function of the dihedral mirror position l may be written:PLy =∫(E2Ax + E2By) + (E2Ax − E2By) cos(4lω/c)dωPLz =∫(E2Ay + E2Bx) + (E2Ay − E2Bx) cos(4lω/c)dωPRy =∫(E2Ay + E2Bx) + (E2Bx − E2Ay) cos(4lω/c)dωPRz =∫(E2Ax + E2By) + (E2By − E2Ax) cos(4lω/c)dω(2.22)where ω is the angular frequency of incident radiation, L and R refer to thedetectors in the left and right concentrators, and A and B refer to the twoinput beams (Fig.2.7).The term modulated by the mirror scan is proportional to the Fouriertransform of the frequency spectrum for Stokes Q linear polarization ininstrument-fixed coordinates. Rotation of the instrument about the beamaxis interchanges xˆ and yˆ on the detectors. The sky signal (after the Fouriertransform) then becomes:S(ν)Ly =14[I(ν)A − I(ν)B +Q(ν)sky cos 2γ + U(ν)sky sin 2γ]S(ν)Lz =14[I(ν)A − I(ν)B −Q(ν)sky cos 2γ − U(ν)sky sin 2γ]S(ν)Ry =14[I(ν)B − I(ν)A +Q(ν)sky cos 2γ + U(ν)sky sin 2γ]S(ν)Rz =14[I(ν)B − I(ν)A −Q(ν)sky cos 2γ − U(ν)sky sin 2γ](2.23)where γ is the spin angle and S(ν) denotes the synthesized frequency spec-trum with bins ν set by the fringe sampling.PIXIE operates as a nulling polarimeter: when both beams view thesky, the instrument nulls all unpolarized emission so that the fringe pat-tern responds only to the sky polarization. The resulting null operationgreatly reduces sensitivity to systematic errors from unpolarized sources.Normally the instrument collects light from both co-aligned telescopes. A412.2. Overview of the PIXIE ExperimentParameter Value NotesPrimary Mirror Diameter 55 cm Sets beam size onskyEtendu 4 cm2 sr 2.7 times largerthan FIRASBeam Diameter 2.◦6 tophat Equivalent 1.◦6Gaussian FWHMThroughput 82.00% Excludes detectorabsorptionDetector Absorption 54.00% Reflective back-shortMirror Stroke ±2.6 mm peak-peak Phase delay ±10mmSpectral Resolution 15 GHz Set by longestmirror strokeHighest Effective Frequency 6 THz Spacing in polar-izing gridsDetector NEP 0.7× 10−16 W Hz−1System NEP 2.7× 10−16 W Hz−1 Background limitTable 2.2: Optical Parameters.full-aperture blackbody calibrator can move to block either beam, replac-ing the sky signal in that beam with an absolute reference source, or bestowed to allow both beams to view the same sky patch. The calibratortemperature is maintained near 2.725 K and is changed ±5mK every otherorbit to provide small departures from null as an absolute reference signal.When the calibrator blocks either beam, the fringe pattern encodes infor-mation on both the temperature distribution on the sky (Stokes I) as wellas the linear polarization. Interleaving observations with and without thecalibrator allows straightforward transfer of the absolute calibration scale tolinear polarization, while providing a valuable cross-check of the polarizationsolutions obtained in each mode.Table.2.2 summarizes the instrument optics. For detailed informationabout PIXIE instrument performance, please check the white paper [51].422.3. Instrument SimulationDetectorHornIrisT5T4T3T2T1Secondary FlatPrimaryApertureGDGCGBGAFigure 2.8: A 2-D sketch of the PIXIE instrument. Black curves and linesshows the mirrors. Red line is the track of a ray generated perpendicular tothe detector.2.3 Instrument SimulationMy job on PIXIE is to optimize its optical system. For now, we only con-centrate on the right half of the telescope and treat the half-transparent gridto be flat mirrors, and we temporarily remove the movable dihedral mirror.Fig.2.8 is a 2-d sketch of the simulated elements.The set of instrument elements we want to optimize is summarized asfollows:The detector is a 12.7 x 12.7 mm square.These rays rattle around in the horn. The horn is rotated 45 deg aboutthe optic axis so that the two polarizations are treated symmetrically. Each432.3. Instrument Simulationhorn surface is a section of an elliptic cylinder. In order to make a symmetrichorn, the top two walls and bottom two walls share the same shape.There is a round iris at the mouth of the horn.Next is the totem mirror-grid mirror system. There are 5 totem mirrorsstaggered with 4 round grid mirrors. The totem mirrors are a round segmentof ellipsoids around the vertex. They are set up so that each totem mirroris at one of the foci of the previous (and following) totem mirror. This is a”periscope” structure.Once we get to T1 (Totem mirror No.1, we refer to Totem mirror No.nas Tn hereafter) we change focal length to get to the secondary mirror. Thesecondary, folding flat and primary have several constraints. The polariza-tion on the sky depends on the angles of the secondary, folding flat andprimary. This is critical so that the polarization from the left is orthogonalto the polarization from the right one. The folding flat also needs to avoidblocking of the beam from the sky to the primary and the beam from thesecondary to T1.The basic logic of the optimization is to set a bunch of rays launch fromthe detector, let them travel through the whole instrument until they leavethe aperture or go stray. In operation, the stray rays are thermal radia-tion from the instrument that could be received by the detector as excessnoise. We define a parameter called ’Good’ to evaluate the performance ofthe instrument. Our goal is to adjust the parameters (like sizes, foci andpositions) to maximize Good.2.3.1 Code Realization of the PIXIE InstrumentWe use a custom Python code to do the optimization. Firstly we package allthe information of each element into a 3-D array L. For each element (exceptthe horn, which I will discuss later), we need to consider its position, shapeand size. All the ellipsoid mirrors (including totem mirror and secondarymirror) are rotationally symmetric so the positions are represented by thevertex, and the shapes are determined by their foci. For round flat mirrors(including Grid mirrors and Folding flat), their position is their center andtheir shape is determined by the direction of normal vector. For the primarymirror, its position is the vertex and its shape is determined by the focus.The size of all round elements are naturally described by their radius. Theinstrument coordinate system is set up so that the z axis is parallel to theoptic axis and the FTS part (see 2.7 is in xz-plane (see Fig.2.7(a)).The detector and horn are rotated around the x axis by 45◦. We setup a horn coordinate with the origin point at the center of the detector.442.3. Instrument SimulationIn the following text, I will use a ′ to denote the coordinate in the horncoordinate system. The x′ axis points in the same direction as the x axisof the instrument coordinate system. The y′ and z′ axes are parallel to theedge of the detector.In the code, there are 3 vectors for each ray: the positions ~R, directions~D and polarizations ~P . ~D and ~P are normalized. We originate rays on thedetector. The detector is divided into 23× 23 grids which covers the wholearea. The lowest distance between two grid vertices is 12.7/23 = 0.55mm.So a photon with wavelength larger than 2×0.55 = 1.1mm (corresponding tofrequency 271GHz) is well modeled. Since the intensity for rays coming withan angle θ0 to the normal of detector is weighted by cos2 θ0, we assign cos2 θ0weight for a evenly distributed ~D on each point of the grid. The uniformlydistributed ~D is generated by calling a Healpix function. Healpix is analgorithm to pixelize a sphere [36]. The finess of pixelization is characterizedby a factor called Nside. The pixels are denoted by a normalized directionvector. The number of pixels on a whole sphere is: Np = 12N2side. So thetotal number of rays from the detector is M ≡ 23× 23× 6N2side = 3174N2side2. So Nside actually counts the number of rays we want to track in theinstrument. Based on the limit of our computer, we typically take Nside = 8(corresponding to M = 203136 rays). The original polarization ~P ’s aredefined as:~P = (− Dz′Dx′√D2x′ +D2y′,− Dz′Dy′√D2x′ +D2y′,√D2x +D2y) (2.24)This definition satisfies ~P · ~P = 1 and ~P · ~D = 0, and it gives thelargest value of Pz′ . If we treat it as the polarization of a ’detected’ ray,then the detected Q2 value is maximized. Note that this definition is inthe horn coordinate system, when we rotate to the instrument coordinate,then U2 is maximized. Ideally, the output ray should have a polarization~P =(−√22 ,√22 , 0).The simulation is initialized by a function detector(L,Nd) which out-puts the initial R,D,P,G.Once the rays are launched from the detector, they travel from elementto element using specular reflection until leaving the aperture or going stay.For detector-horn-iris system, ray tracking is executed in the horn coordi-nate system. After getting out of the iris, the code will transfer ~R ~D and2I will use the same letter to label parameters as in my code.452.3. Instrument SimulationInst # Inst Name Position Function0 detector (3,0,-679.5) detector1 horn - horn2 iris (103,0,-651) iris3 T5 (244,0,-617) elp4 GD (0,0,-553.5) flat5 T4 (240,0,-490) elp6 GC (0,0,-426.5) flat7 T3 (240,0,-363) elp8 GB (0,0,-299.5) flat9 T2 (240,0,-236) elp10 GA (0,0,-172.5) flat11 T1 (244,0,-109) elp12 secondary (65,45,-60.5) elp13 flat (80,200,-5) flat14 primary (320,480,-776.5) par15 Aperture (320,480,0) apertureTable 2.3: A summary of instrument elements to be optimized. All thefunctions take (R,D,P,G,B,L,K) as input and output (R,D,P,G,B).~P into the instrument coordinate system, then carry out the subsequentpropagation.For each reflective element, there is a function which outputs the fallingpoint ~R on that element, and the corresponding reflected ~D and ~P . The rayswith an incident point outside of the element are assumed stray and removedfrom the system. The input is ~R, ~D and ~P from the previous element, a1-D vector G is an auxiliary array to record the index of ’surviving’ rays,the element index K, and the element information L. Another 1-D vector Bis defined with size M which record where the rays end their tracks. Forexample, if the second ray ends up on T5, then we assign B[1] = 3. Theoutput for each function is the incident point ~R, reflected ~D and ~P , G andB.The horn is more complicated than the subsequent elements. Rays canbounce several times in the horn, so I wrote a loop in horn function tocall a function HitA that calculates the incident point of the rays after onebounce. In each loop, I rule out the rays that bounced back to the detectorand record the rays that reaches the horn mouth. The loop ends when alarge fraction (99.99%) of rays leaving the mouth.462.3. Instrument SimulationExcept the horn, we define three kinds of functions for 3 different mirrorshape: elp for ellipsoid mirrors; flat for flat mirrors, par for paraboloidmirrors.For the transparent elements, iris and aperture, the function iris andaperture simply record dead rays.Table 2.3 summarizes the instrument element to be optimized. The posi-tion coordinates here are in the instrument coordinate system. Some of theposition coordinates are parameters that need to be optimized. The valuesshown here are nominal values. The optimized value should be similar.2.3.2 Parameters and CriteriaA key problem for optimization is how to define Good to weight the variousrays. Suppose after one run of ray tracking, indices set for out-coming raysis Gout. The index set of rays stray element a (a is the element number, see2.3) is Ba.Take an on axis ray on the sky with the right polarization as the standard,i.e Good = 1, for this ray. Now consider a ray that gets lost in the horn (oriris). It contributes to neither signal nor noise so its Good = 0.Consider rays that get lost in the FTS. These contribute noise but nosignal. I define a penalty number, pa, for each ray that misses element a.For T5,GD,T4,GC,T3, I set a penalty pa = −0.8. For T2 to the secondary,pa = −1.2, flat to Aperture pa = −1.5. For late instrument elements thepenalty is higher as they will be modulated and add signal as well as noise.Rays that get to the sky with the wrong polarization are really bad.Not only do they add noise but they subtract signal from the correct po-larization. So they get Good = -4. Since the correct polarization should beperpendicular to ~V ≡(√22 ,√22 , 0), the penalty for the wrong polarization isdefined as −4×(~V · ~P)2Finally there are still rays with the correct polarization, but which areoff axis: 1− (D2x+D2y)/α2 gives a gentle nudge to points a little off axis andpushes harder far off axis. α is a tolerance parameter which describes thelargest off axis angle we could bear. I take α = 0.035 which corresponds to2◦In summary, Good is calculated as:472.3. Instrument SimulationGood ≡{ ∑i∈Goutcos2 θ0i[1− (D2xi +D2yi)/α2 − 4×(~V · ~Pi)2]+15∑a=3pa∑i∈Bacos2 θ0i}/Mw × 100%(2.25)where Mw ≡M∑i=1cos2 θ0i gives the total number of rays, where M is the totalray number. θ0i is the angle between ith ray and the normal of the detectorwhen it is launched. I need to weight each ray with cos2 θ0i because the initialrays have uniformly generated ~D. There is a function good(R,D,P,G,B) tocalculate Good after the calling of aperture.We are mainly interested in two parts of the instrument, the first is thehorn-iris-T5 system (HIT5 hereafter) because it is the most complicated andit is also nearest the detector so there should be more stray photon comingaround T5. The top two walls of the horn share the same shape and so dothe bottom walls. There are 4 foci to be optimized. In horn coordinates,each focus is represented by two coordinates (note that the horn walls areelliptical cylinders which are parallel to z′ or y′ axis, so there are only twocoordinates). The first coordinate is x′ and the second is either y′ or z′depending on which wall it is.We fix the first focus for both top and bottom walls to be on the con-tralateral edge of the detector, so the only free parameters to be optimizedare the four coordinates of f t2 and fb2 . See the side-view of the horn inFig.2.9.The position of the iris is also free. Considering symmetry, its centershould be in the xz plane of the instrument coordinate system and rightat the mouth of the horn so there is only one degree of freedom. We setthe elevation angle of the iris center θiris to be a free parameter. Also thecomplicated optical path through the horn makes the first focus of T5 fT51un-determined. It should be in the xz plane.The other part of the instrument to be optimized is the secondary-flat-primary system (SFP system hereafter). A 2-D sketch of this system isshown in Fig.2.10. The first focus of the secondary ~fSec1 is fixed at the imageof T1. The normal vector of the flat should be the bisector of flat-secondaryand flat-primary link:482.3. Instrument Simulation50 0 50 100 150 200 250 300 350x ′10050050100150200y′  or z′f b1f t1 fb2f t2Figure 2.9: The side-view of the horn in the horn coordinate system witha nominal focus. The shaded part is the horn, the short left side is thedetector and long right side the mouth. The black line is the top wall andthe purple line is the bottom wall. The foci of the horn are denoted bypoints with corresponding colors.~nflat = FˆS + PˆSnˆflat =~nflat|nflat|(2.26)Where FˆS is the unit vector pointing from the flat to the secondary centerand FˆP is similarly defined. nˆflat is the normal vector to the flat mirror.The secondary and primary mirrors should see each other from the flatmirror, so ~fSec2 , the second focus of the secondary mirror should be on theaxis between the secondary mirror and the flat mirror and it should coincidewith the image of the primary and vice visa. We can parametrize these focias:~fPri = ~rF + FfPri × FˆP~fSec2 = ~rF + FfSec2 × FˆS(2.27)where FfPri is the distance between fPri and F and FfSec2 is the distancebetween fSec2 and F. ~rF is the position vector for F. The center and size of492.3. Instrument SimulationSecFlatPriApertureSFP~nFlatfSec2fPriFfPriFfSec2Figure 2.10: A 2-D sketch of the SFP system . Grey curves and lines showsthe mirrors. Red arrows track the central ray from T1 to the aperture. Theblue arrow is the normal vector of the flat. Points S, F, P are the center ofthe corresponding elements. Blue points are focus of primary and Second.the mirrors are fixed. Ideally FfPri =SF and FfSec2 =PF, but practically welet FfPri and FfSec2 be free parameters.In summary, there are 11 independent parameters to be optimized:• 5 for the horn: f t2x′ , f t2y′ or z′ , f b2x′ , f b2y′ or z′ ; horn length lhorn;• 2 for the iris: iris size riris; iris elevation angle θiris• 2 for T5: fT51x , fT51z ;• 2 for SFP system: FfPri and FfSec2 .In the code, I wrap all these parameters into an array called Z and con-struct the instrument model L with it. The main function raystrack(Z,Nd)502.4. MCMC for Instrument Parameterstraces the rays from the detector down to the aperture. First it callsinstrument(Z) to generate the instrument model and then all the func-tions one by one in Table.2.3, and then Good. Our task is to maximizeraystrack(Z,Nd = 8).2.4 MCMC for Instrument ParametersAs discussed above, the optimization is executed separately for HIT5 andSFP systems. Both of them are implemented with a Markov-Chain MonteCarlo method. I use the python package emcee to run the MCMC procedure.The code runs a 300 step burn in. Each chain contains 3000Nd pointswhere Nd is the dimension of parameter space. As we only want to findthe best set of parameters, we do not really care about the confidence levelof the parameters. However, we can use them to better understand theperformance of the instrument.We first apply MCMC for the HIT5 system3, then keep the optimizedHIT5 parameters fixed for the SFP optimization. The HIT5 system is verycomplicated. First, the rays bounce several times in the horn and are dis-persive at the iris. So it is likely that many rays will miss T5. Also, the firstfocus of T5 fT51 is not easily identified. A displacement of fT51 will result ina lot of rays missing T4.Intuitively, the horn should point directly towards T5. The iris shouldbe placed on the center of the horn with a proper size. If it is too big, thenthe rays from the corner of the horn will be likely to miss T4; if it is toosmall, then we will not have enough out-going rays. fT51 should be on thedetector-T5 link, and near the mouth of the horn. However, if I include allof these parameters into an MCMC chain, it is very likely to converge into asolution which yields a very large horn mouth and a very small iris blockingabout half of the rays. This configuration gives a high Good value becausethe penalty for iris is 0.0 while the penalty for T5,GD,T4 is -0.8. So theMCMC chain will easily go to a small iris on a large horn mouth to blockmany rays on the iris instead of T5-GD-T4.To recover from this, I run an MCMC with only the 6 parameters of thehorn and T5. The output to be maximized is the number of rays endingup on T1 instead of Good. This chain will give a very small horn mouthpointing directly towards T5, and fT51 should be near the center of the horn.3All the result presented in this section is from the most recent run. Actually I haverun MCMC for many times, each time taking the best-fit point from the last run as thestarting point of the chain.512.4. MCMC for Instrument ParametersThen I fix the horn parameters and optimize riris,θiris by maximizing Good.The result for horn and T5 is shown in Fig.2.11. It is a scatter plotshowing the MCMC chain color-coded by the number of rays landing onT1. The number of rays have been weighted by cos2 θ0. The optimizedvalues for these parameters are labeled by crossing dashed lines. I thenfixed the shape of horn and T5 with these best-fit parameters and run anMCMC for the iris to maximize Good. The result is shown in Fig.2.12.522.4. MCMC for Instrument Parametersf t150 0 50 100 150 200 250 300 350 40040302010010y'(mm)f t2 (In horn coordinate)f b1200 0 200 400 600 800 1000x'(mm)1000100200300400500600y'(mm)f b2  (In horn coordinate)DetectorT550 0 50 100 150 200 250 300x(mm)690680670660650640630620610y(mm)fT5 (In instrument coordinate)34400 35200 36000 36800 37600 38400 39200 40000 40800Number of rays on T1Figure 2.11: The MCMC result for f t2,fb2 and fT51 . The colors show thenumber of rays reaching T1 with corresponding parameters. The crossingdashed lines labels the position of maximum on the parameter space. In eachpanel, the black points with labels on it shows the center of correspondinginstrument elements as reference points.532.4. MCMC for Instrument Parameters5 10 15 20θiris(◦)12182430r iris(mm)12 18 24 30riris(mm)3.0 4.5 6.0 7.5 9.0 10.512.013.515.0GoodFigure 2.12: The MCMC result for iris angle and size. Upper and bottomright panels show the histogram for θiris and riris. The bottom left plotshows the chain points in the 2-D parameter space color-coded by Good.The crossing dashed lines labels the position of maximum on the parameterspace. Contours shows the 68.3%, 95.4% and 99.7% level of confidence.The left panel of Fig.2.13 shows the 2-D projection of the HIT5 system inxz plane. With the optimized parameters, the horn is pointing towards T5and fT51 is very close to the center of the horn mouth, just as expected. Theright panel is the 2-D histogram of weighted rays number on the horn mouth.The circle is the iris with optimized θiris and riris. And not surprisingly, theiris centers at the center of the horn mouth.542.4. MCMC for Instrument Parameters0 50 100 150 200 250 300x(mm)700650600550z(mm)DetectorHornIrisT5fT5120 10 0 10 20y(mm)680670660650640z(mm)708090100110120130140150Weighted Number of rays$Figure 2.13: Upper panel: 2-D projection of the optimized HIT5 system.Red lines shows part of the rays from iris to T5. Blue point is fT51 . Notethat here we use the instrument coordinate. Lower panel: the horn mouthwith iris represented by a circle. The color plot shows the 2-D histogramfor number of rays landing on the horn mouth. To show a more smoothhistogram, I take Nside = 64 for this plot.552.4. MCMC for Instrument ParametersGiven the optimized parameters for HIT5, I run another MCMC for SFPsystem. The result is shown in Fig.2.14120150180210FfPri(mm)200040006000800010000FfSec2(mm)200040006000800010000Ff Sec2 (mm)8 9 10 11 12 13 14 15GoodFigure 2.14: The MCMC result for FfPri and FfSec2 . Upper and bottomright panels show the histogram for FfPri and FfSec2 . The bottom left plotshows the chain points in the 2-D parameter space color-coded by Good.The crossing dashed lines labels the position of maximum on the parameterspace. Contours shows the 68.3%, 95.4% and 99.7% level of confidence.The MCMC result shows that FfPri is well constrained to be 167.11mm,which is very near to SF(163mm), as expected. FfSec2 is very loosely con-strained. Given the fact that sec is very small comparing with Pri, this is562.5. Discussionnot too surprising. Because if FfPri is very near the center of Sec, thenno matter what shape the sec is, the rays reflected by Pri will likely to beco-aligned.I summarize the value for the best-fit MCMC result for these 11 param-eters in the followingf t2 = (376.24mm,−36.18mm)f b2 = (919mm, 522.9mm)θiris = 11.79◦riris = 15.54mmFfPri = 167.11mmFfSec2 = 9730.52mm(2.28)I then run raystrack(Z,Nd = 8) one time with Z chosen to be theoptimized parameters. The output is summarized in Table.2.4 .We can see that most of the rays got lost at the iris and T5, whichindicates that they are two most important noise source. However, the irisdoes not contribute to the total gain. The total fraction of out-coming raysis 42% which shows the efficiency of the instrument. Since the worst parthappens at the HIT5 system, there might still be space to further improveit.2.5 DiscussionFrom the optimized instrument model I run a ray track with Nside = 64 andcollect all the out-going rays. PIXIE is not a perfect optical system, so theout-going rays are scattered about the optical axis.Note that the ray tracking is a time-reversed process from the detectorto the aperture. In practice, the distribution of this scattered out-goingrays means that an in-coming ray in that direction can be detected. So thisscatter actually shows the beam behavior. The number of rays in each pixelreflects the strength of radiation from that direction.The upper right panel of Fig.2.15 shows a 2-D histogram of weightedrays number in the ~D space. The upper left and bottom right panels are themarginalized 1-D histogram for θy and θx. This plot defines the shape of thebeam function. The beam is a 1.1◦ top-hat, slightly better than the originaldesign shown in Table:2.2. However, this beam is not perfectly symmetric.Fig.2.16 shows the distribution of co-polarization and cross-polarizationin the ~D space. We see that the co-polarization distributes very much like572.5. DiscussionIndex (a) Inst Name Living rays Missing rays pa∑i∈Bacos2 θ0i/Mw0 Det 67613.73 0 01 horn 67041.52 572.21 02 iris 44243.23 22798.29 03 T5 33265.1 10978.13 -12.99%4 GD 33265.1 0 05 T4 33211.61 53.49 -0.06%6 GC 33211.61 0 07 T3 33211.61 0 08 GB 33211.61 0 09 T2 33211.61 0 010 GA 33211.61 0 011 T1 33211.61 0 012 Sec 32668.62 542.99 -0.96%13 Flat 32668.62 -0.00069 014 Pri 32395.89 272.73 -0.61%15 Aper 32394.24 1.65 -0.0037%∑i∈Goutcos2 θ0i[1− (D2xi +D2yi)/α2]/Mw = 40.53%∑i∈Goutcos2 θ0i[−4×(~V · ~Pi)2]/Mw = -10.63%Good = 15.27%Fraction of out-coming rays: = 42%Table 2.4: The output for a rays track with the optimized parameters. Thenumbers of living rays and missing rays have been weighted by cos θ0. The’Missing rays’ column shows the number of rays that miss the correspondingelement. Note that according to (2.25), the Good value is calculated bytaking the sum of the last column.582.5. Discussion1 0 1θx(◦)101θ y(◦)0. 3 ◦0. 6◦0. 9 ◦1. 2 ◦ 1. 2◦1.2◦ 1. 2 ◦020406080100120140160180Number of Rays1.5 1.0 0.5 0.0 0.5 1.0 1.5θy(◦)1.5 1.0 0.5 0.0 0.5 1.0 1.5θx(◦)Figure 2.15: Upper right panel: the 2-D histogram in ~D space of the out-coming rays. x and y axis are two components of the polar angle of ~D. Thecolor bar shows the weighted rays number. Upper left and bottom rightpanels: marginalized 1-D histogram for θy and θx.the beam, and the cross-polarization signal is very weak. This means thatthe polarization of the out-going rays are not distorted evidently.It is clear that this set of instrument parameters are not the best one,because: 1. they are only a fraction of all the instrument parameters in-cluded in the fit; 2. the MCMC chain has a limited number of points; 3. wedecouple the HIT5 and SFP system, but they could be correlated with eachother; 4. We optimized the horn first then add iris and fT51 ; 5. xFP and xSPare not necessarily on the link of Pri-flat or Sec-flat.There are a couple of directions we can go from here. One is that weshould allow other parameters to float maybe all of the foci of T1-5 andsmall tweaks on the primary, flat and secondary.A second direction is to look more closely at the polarization on variouselements: the mouth of the horn, on T5, and on the sky.592.5. Discussion1 0 1θx(◦)101θ y(◦)0.3◦0. 6 ◦0. 9 ◦1. 2 ◦ 1. 2◦1.2◦1. 2 ◦Co-Pol0 15 30 45 60 75 90105120135150Number of Rays1 0 1θx(◦)θ y(◦)0.3◦0. 6 ◦0. 9 ◦1. 2 ◦ 1. 2◦1.2◦1. 2 ◦Cross-Pol0 2 4 6 8 10 12 14Number of RaysFigure 2.16: Left right panel: the 2-D histogram in ~D space of the co-polarization of out-going rays. Right right panel: the 2-D histogram in ~Dspace of the cross-polarization of out-going rays.A third direction is to reconsider the definition of Good, especially thepenalty for stray rays.In my simulation I discuss only a little about the polarization. It isdefinitely a crucial task for the future. We can study the pattern for polar-ization on each element to get an insight of how they affect the polarization.Then we can add the dihedral mirror between GC and GB and move itback and forth to see how it changes the output. Our ultimate task for thiswork is replace the Grid mirrors with polarizers and add the left part of theinstrument to simulate a true differential FTS.60Chapter 3Observing the GasDistribution in GalaxyClusters: The y-κCross-Correlation3.1 General IntroductionThe Sunyaev-Zeldovich effect [92] is the inverse Compton scattering of theCosmic Microwave Background (CMB) photon by high energy electrons.CMB photons get an energy boost through this effect and thus the energyspectrum gets distorted. This observable effect provides a useful tool toobserve distant clusters of galaxies. The thermal SZ effect (tSZ effect here-after) arises from the scattering of the CMB photons by electrons that havehigh energies due to their high temperature. It it mainly occurs in the hotintracluster gas in galaxy clusters. The effect is independent of redshift be-cause it is a scattering effect. So high redshift clusters can be observed aseasily as nearby ones. Besides searching for new clusters, SZ effect can alsobe used to constrain cosmological parameters by providing information onthe abundance of galaxy clusters, which depends on Ωm [58].With current observational precision and angular resolution, it is possi-ble to detect tSZ signal from galaxy clusters [e.g. 38] after filtering out othercomponents like CMB, dust components and point sources. Moreover, sincethe frequency dependence of the tSZ is well understood, it is possible to ex-tract the demensionless tSZ template from multi-frequency sky signal data.In 2015, the Planck group constructed two full-sky tSZ maps [9] using 30 to857 GHz frequency channel maps from the Planck satellite survey with twospecifically tailored component separation algorithms (NILC [27]: NeedletInternal Linear Combination and MILCA[48]: A Maximum Internal LinearComponent Analysis). Several subsequent analyses have been using fromthese tSZ maps (e.g.[3] and [84]).613.1. General IntroductionThe tSZ effect offers a unique method to observe the diffuse baryoniccomponent in galaxy clusters. In those clusters, only about 10% of thebaryons are in compact objects like stars and dust while 90% are in theform of diffuse gas [82]. A comparison of group and cluster masses de-rived from dynamical and x-ray data reveals that baryons are missing at allscales, especially at galactic halo scales. This is likely related to the “miss-ing baryon” problem occurring at redshift z < 2, where the intercluster gasbecomes ionized in a warm phase that is particularly difficult to observe.Recently, it has also been realized that missing baryons poses a problem forthe interpretation of gravitational lensing because baryonic processes couldimpact the dark matter distribution, even on large scales, via gravitationalfeedback [81]. To solve these question, we need an unbiased tracer of largescale structure which can be feasibly observed. Unlike x-ray luminosity, thetSZ signal is linearly proportional to the baryon density, which makes iteasier to be detected low-density gas. Since it does not depend on redshift,it is useful to cross correlate with signals more localized in redshift. But forarea with very low density, the signal of tSZ can be lower than the noise.So it is useful to cross-correlate it with some other probe to extract it out.Gravitational lensing provides an unbiased tracer of the projected mass, in-dependent of its dynamical and physical state. Cross correlating tSZ andgravitational lensing data is a potential method to help us understand themissing baryon problem and the interaction between baryons and dark mat-ter. [82] presented the first detection of a correlation signal between the tSZand weak lensing with a confidence level of 6σ . This discovery traces thespatial distribution of the missing baryons and has also been used to con-strain the feedback mechanism of AGN in the host galaxies. Other studies,like [41], detect the cross correlation signal between tSZ and CMB lensing,which also reveals the information about intracluster gas.However, residual of noise in the tSZ map can cause contamination onthe cross correlation results. The Cosmic Infrared Background (CIB) andthe thermal galactic dust emission are the two potential sources of contam-ination. CIB [39] is the redshifted starlight from distant galaxies. We findthat the Planck CIB map shows a nonzero cross correlation with weak lens-ing, which remind us that the CIB residual in the y map we used mightcause significant contamination signal in the cross correlation result of theprevious studies. Galactic dust emission is not correlated with weak lens-ing because it is a local effect, but it contributes to the noise in the y mapsince it dominates the high frequency channels. [82] made a set of y mapswith a uniform power-law model for dust emission, which is imprecise athigh frequencies. The Planck NILC and MILCA y map only take CMB as623.2. Studying the Large Scale Structure with Weak Lensing and tSZ Effectforeground signal but include dust as noise. Thus the dust signal is notcompletely removed.In this chapter I summarize my study of the residual CIB and dustcontamination in y maps by calculating the lensing-tSZ cross correlationin collaboration with Alireza Hojjati. We reconstruct the NILC y map.The weak lensing data is from the Red Sequence Cluster Lensing Survey(RCSLens) mass map. We carefully examine the CIB and dust residualin our y map and calculate the cross correlation. The structure of thischapter is as following: Section 2 introduces the formalism for our study;Section 3 discuss our reconstruction of the y map; Section 4 provides ourcross correlation results; Section 5 is an attempt to introduce an all-sky CIBmodel. The last section is a brief discussion and conclusion.3.2 Studying the Large Scale Structure withWeak Lensing and tSZ EffectGravitational lensing is the phenomenon that light rays get deflected bythe gravitational field generated by some mass between the observer and asource. The shape of the source will be magnified and distorted and can bedescribed with the following formula:[θixθiy]=[1− κ− γ1 −γ2−γ2 1− κ+ γ1] [θsxθsy](3.1)where θi denotes the angular coordinate of the image and θi is the angularcoordinate of the source. The convergence κ describes the magnification ofthe source while the shear γ1 and γ2 describe the distortion (see Fig.3.1).Lensing by mass fluctuations in large scale structure is usually too weakto generate noticeable distortions of a source galaxy, but it can leave foot-print on the statistical properties of a large number of source galaxies. Thisis called the Weak Lensing. We can study the power spectrum and two-pointcorrelation functions for κ, γ1 and γ2. In this thesis, I focus on κ.A κ map can be modeled as a projected mass density along the lineof sight specified by position angle θ on the sky. It can be formulated asthe integral of the density fluctuation δm(fK(w)(θ), w) weighted by a kernelW κ(w), that is:κ(θ) =∫ wH0dwW κ(w)δm(fK(w)(θ), w) (3.2)633.2. Studying the Large Scale Structure with Weak Lensing and tSZ EffectFigure 3.1: Effect of κ and γ ≡ γ1 + iγ2 on a spherical source.643.2. Studying the Large Scale Structure with Weak Lensing and tSZ EffectThis is called the Limber approximation. Here w(z) is the radial comov-ing distance, wH is the comoving distance to the horizon and fK(w) is thecorresponding angular diameter distance. For κ, the kernel W κ(w) isW κ(w) =32Ω0(H0c)2g(w)fK(w)a(3.3)g(w) is a function which depends on the redshift distribution of the sourcespS(w):g(w) =∫ wHwdw′pS(w′)fK(w′ − w)fK(w′)(3.4)The angular power spectrum of κ is calculated from a κ map and a ymap via:Cˆκ×κ` ≡12`+ 1∑maκ`m∗aκ`m (3.5)where aκ`m is the spherical harmonic decomposition of κ. In theory, thecorrelation signal should be:Cκ×κ` =∫ wH0dw[W κ(w)2f2K(w)]Pm(`fK(w), w)(3.6)where Pm is the matter power spectrum:〈δm(k, z)δm(k′, z)〉 = (2pi)3δ(k − k′)Pm(k, z) (3.7)So the angular power spectrum of κ contains information on cosmologicalparameters via Pm.The thermal Sunyaev-Zeldovich (tSZ) effect is the distortion of the CMBfrequency spectrum through inverse Compton scattering by energetic elec-trons in cluster gas. These electrons move quickly and pass energy to theCMB photons.The tSZ-induced temperature change at frequency ν is characterized bythe Compton parameter y:∆T (θ, x)TCMB= y(θ)SSZ(x) (3.8)where SSZ(x) = x coth(x/2) − 4 is the tSZ spectral dependence in termsof x ≡ hν/kBTCMB. Here h is the Planck constant, kB is the Boltzmannconstant and TCMB is the mean temperature of CMB.653.3. Reconstruction of the y signalThe Compton parameter y is given by the line-of-sight integral of theelectron pressure:y(θ) =∫ wH0adwkBσTmec2ne(fK(w)(θ), w)Te(w) (3.9)where σT is the Thomson cross section, ne(fK(w)(θ), w) and Te(w) are thenumber density of electrons and their temperature, respectively. The elec-tron number density depends both on angular position and radial distance.For the electron gas, it can be written as ne(fK(w)(θ), w) = n¯eδgas(fK(θ), w)where n¯e is the mean electron number density and δgas is the gas density con-trast, which is given by bgas(z)δm with bgas ∝ (1+z)−1 the gas bias [35]. Theelectron temperature depends only on radial distance by Te(w) ∝ (1 + z)−1.So the y can also be estimated by the Limber approximation with the kernel:W SZ = bgas(0)n¯eσTkBTe(0)mec211 + z(3.10)So the tSZ effect probes the electron temperature Te and baryon biasbgas which reveals the baryon physics in clusters.The κ× y cross correlation can be calculated from:Cˆκ×y` ≡12`+ 1∑maκ`m∗ay`m (3.11)where ay`m is the spherical harmonic decomposition of y. Theoretically, thecorrelation signal should be[25]:Cκ×y` =∫ wH0dw[W SZ(w)W κ(w)f2K(w)]Pm(`fK(w), w)(3.12)3.3 Reconstruction of the y signalOur y map is made from 6 Planck HFI all-sky temperature maps: 100GHz, 143 GHz, 217 GHz, 353 GHz, 545 GHz and 857 GHz. They are fromPlanck’s 2nd data release [1]. To evaluate the contamination by CIB andgalactic dust, we also use the Planck CIB map [10] and the all-sky dustmodel maps [6]. The CIB maps cover the 3 highest frequencies: 353GHz,545GHz and 857GHz, with about 40% of the sky near the galactic planemasked out. Dust model maps contain a map for the dust temperature anddust spectral index, which are used to make our y map. All of these mapsare in HealPix format [36] with Nside = 2048.663.3. Reconstruction of the y signalThe raw temperature map is a superposition of different emission com-ponents, including CMB, galactic dust emission, free-free radiation, syn-chrotron radiation, CIB, y signal etc. A well-known method to extract oneof those components and null other components is the Internal Linear Com-bination (ILC) technique, which is used by WMAP and Planck to makeCMB and other component maps [52]. The basic idea of the ILC methodis to use a linear combination to keep the target component and to projectother components out.y, CMB and galactic dust intensity for certain frequency can be modeledas an isotropic frequency dependence multiply by a template map whichis independent of frequency. Let Iα(θ) be the template of component α(α = y, CMB or dust). The signal of component α at frequency ν andangular coordinate θ is Iαν = fα(ν)Iα(θ), where fα(ν) is the frequencydependence for component α at frequency ν. The observed frequency mapdi is a combination of all the components plus noise:di(θ) =∑αIανi(θ) + ni(θ)=∑αfα(νi)Iα(θ) + ni(θ)=∑αMiαIα + ni(θ)(3.13)Miα ≡ fανi is the mixing matrix which shows the frequency dependencefor the αth component in ith frequency (we use Latin letters i, j, k... forfrequency channels and Greek letters α, β, γ... for components). In ouranalysis, νi ∈ {100GHz, 143GHz, 217GHz, 353GHz, 545GHz, 857GHz} andα ∈ {CMB, y,dust}. ni(θ) is the sum of some other components that arenot included in Iα (like CIB, whose frequency dependence is not uniformacross the sky) and systematic plus statistical errors.To extract the α component while nulling the other components, we haveto solve the linear equations:∑icαifα(νi) = 1∑icαifβ(νi) = 0, β 6= α(3.14)or more concisely: ∑icαiMiβ = δαβ (3.15)673.3. Reconstruction of the y signalcαi is the ILC coefficient for α component at frequency νi. We use Iˆα todenote the estimated template for component α which is calculated by su-perposing the observed sky maps with the ILC coefficients:Iˆα(θ) =∑icαidi(θ) = Iα(θ) +∑icαini (3.16)The number of components (Nc) cannot be larger than the number ofchannels (Nf ), or we will run out of degrees of freedom. For Nc < Nf , theremaining degrees of freedom are used to minimize the noise residual byminimizing the χ2:χ2(θ) ≡∑ij(di(θ)−∑αMiαIˆα(θ))(N−1)ij(dj(θ)−∑αMjαIˆα(θ)) (3.17)where N is the signal covariance matrix. Taking the partial derivative withrespect to Iˆα(θ):∂χ2(θ)∂Iˆα(θ)= −2∑ijMαi(N−1)ij(dj(θ)−∑βMjβ Iˆβ(θ)) (3.18)To minimize χ2, set this to zero, then∑ijMαi(N−1)ijdj(θ) =∑ijMαi(N−1)ij∑βMjβ Iˆβ(θ) (3.19)This leads toIˆα(θ) =∑αβkl[(MTN−1M)−1]αβMβk(N−1)kldl(θ) (3.20)Comparing with Eq:3.16, the coefficient for component α at frequency chan-nel l iscαl =∑αβk[(MTN−1M)−1]αβMβk(N−1)kl (3.21)The frequency dependence for each component is contained in the mix-ing matrix Miα. Free-free scattering and synchrotron only effect the lowfrequency data, so we ignore it here and take CMB, galactic dust as thecontaminating components to be projected out. The frequency dependencefor each component is as following:683.3. Reconstruction of the y signalInput: 6 Planck HFIAll-sky maps, di,rawUnify the angu-lar resolutionsand units, get diNeedlet filter themaps, get d(a)ia = 1Calculate Cov ma-trices with Eq.3.26Calculate needlety maps y(a)(θ)with Eq.3.20a ≥ 10?a = a+ 1Stack needlet y maps,y(θ) =∑ay(a)(θ)Output: NILCy map y(θ)yesnoFigure 3.2: Flow chart for our NILC procedure.693.3. Reconstruction of the y signal20 10 0RA(◦)506070Dec(◦ )Planck NILC y Map−3. 5× 10−6 5. 0× 10−620 10 0RA(◦)506070Dec(◦ )This work−3. 5× 10−6 5. 0× 10−6Figure 3.3: y signal of a small region of the sky for Planck NILC map andour y map.10-4 10-3 10-2Our Work YR500[arcmin2]10-410-310-2Planck NILC YR500[arcmin2]Figure 3.4: Comparison between the measured tSZ flux of the Planckcluster sample measured in Planck NILC map and our y map.703.3. Reconstruction of the y signal1. Primary CMB fluctuation is a black body spectrum with monopoletemperature 2.725K [31]:ICMBν (θ) =2kBν2c2x2ex(ex − 1)2 ICMB(θ) (3.22)ICMB(θ) is the CMB template which depends only on position θ. x is definedas hνkBTCMB .2. For thermal galactic dust we use a grey body spectrum [6]:Idustν (θ) ∝ νβdBν(Td)Idust(θ) (3.23)where Td is the dust temperature and βd is the dust spectral index takenfrom the Planck dust model map. Since we only use a very limited RCSfield, we use a spatially independent Td and βd by choosing them as themode in this field. The values we use here are Td = 20.5K and βd = 1.57.3. For SZ signal [19]:ISZν (θ) = SSZ(x)ISZ(θ) = SSZ(x)I0y(θ) (3.24)where SSZ = x coth(x/2)− 4 and I0 ≡ 2kBT 3CMB/(hc)2 is the average inten-sity of the CMB. The Compton-y parameter y(θ) is a dimensionless param-eter which describes the strength the SZ effect. We need to first solve forISZ(θ) with Eq.3.20 then divide it by I0 to get the y map.In practice, before we form the ILC map, we need to prepocess the rawtemperature maps as follows:1. The units of the first four maps (100GHz, 143GHz, 217GHz and 353GHz)are µK and for the last two are MJy/sr. We convert all maps to MJy/sr.2. The corresponding angular resolutions are FWHM0 = {9.2, 7.1, 5.0, 5.0,5.0, 5.0} arcmin. To first order we can take the Planck beam function tobe Gaussian[23]. We smooth the maps to a common angular resolution of10 arcmin by convolving each map with a Gaussian beam with FWHM =√102 − FWHM20 arcmin.At different spatial scales, the noise may have different frequency depen-dence. We use the Needlet Internal Linear Combination (NILC) method togenerate the y map. The raw temperature map is first transformed into `space and multiplied by a needlet filter {h(a)(`)} then transformed back intoreal space. The output map is called a needlet-filtered map. {h(a)(`)} peaksat a certain scale `a, so a needlet-filtered map corresponding to {h(a)(`)}preserves intensity around scale `a. We form ILC maps independently foreach needlet-filter, so noise is minimized for each angular scales713.3. Reconstruction of the y signalBased on [17], we use 10 Gaussian window functions each peaked at adifferent scale as {h(a)(`)}. We first filter the raw intensity maps correspond-ing to 6 frequencies with these 10 needlet filters. We then have 10 sets ofintensity maps d(a)i , 1 ≤ a ≤ 10 each corresponds to a needlet window . Eachset has 6 intensity maps at different frequencies. For details about needletILC procedure, please check Appendix.B.The ILC is formed independently with each set of filtered maps. Firstcalculate the covariance matrix within each set:N(a)ij =〈d(a)i (θ)d(a)j (θ)〉θ∈D(3.25)Here a is the number of needlet window. D is the domain in the real spacewhich we are interested in, typically a masked map. In practice, the covari-ance matrix is calculated by multiplying together signals of the same pixelin ith and jth map, then summing over pixels in the domain D.N(a)ij =1Np∑p∈Dd(a)i (p)d(a)j (p) (3.26)Where Np is the number of pixels in domain D. For our analysis, we mask40% of sky, around the Milky Way and point sources.The weight for component separation is calculated independently in do-mains of a needlet decomposition (in ` space) and then added together tomake the whole component map. Our Needlet process is summarized asa flow chart in Fig.3.2. This method is used by Planck collaboration tomake their y maps [9](Planck NILC map hereafter.) and CMB maps [67].Our reconstructed y map (labeled as yˆrec hereafter) is different from PlanckNILC y map (labeled as yˆPlanck hereafter) in that we take galactic dust as aforeground component while Planck NILC map only considered CMB. Weonly use 6 HFI maps while Planck NILC uses LFI maps at large angularscales. Also the Planck’s NILC map only masked the most central part ofthe Milky Way, which is about 2% of the skyFig.3.3 shows the y signal in the same field for yˆrec and yˆPlanck. Thesignals look close to each other. We also calculate integrated y signal withinR500 for 858 Planck tSZ clusters [68] on each maps. The signal-to-signalscatter plot is shown in Fig.3.4. From Fig.3.4 we can see that the y signalfrom both maps agree well with each other. A paired Student t-test showsthat the SZ flux in our map agree with that from Planck NILC map to aconfidence level of 7σ. The difference is due to the different ILC model andcovariance matrices.723.4. A Worked Example: κ× y Cross Correlation0 1Figure 3.5: Footprint of RCSLenS field in galactic coordinate.3.4 A Worked Example: κ× y Cross CorrelationOur analysis focused on calculating the cross correlation between the weaklensing convergence κ and y. The lensing data is from Red Sequence ClusterLensing Survey (RCSLenS) which is part of the second Red sequence ClusterSurvey [40]. Data was acquired from the MegaCAM camera from 14 separatefields and covers a total area of 785 deg2 of the sky. For our analysis weuse the reconstructed projected mass map (convergence map). This map isprovided as a HealPix map with Nside = 2048.We use the PolSpice package [22] to calculate the cross correlation func-tion. It is a HealPix-based package calculating angular power spectrum (see(3.12)) for masked and weighted sky map.3.4.1 The CIB ContaminationThe CIB signal comes from the thermal radiation emitted by dust in earlygalaxies. It is actually discretely distributed in the sky. It depends onfrequency, redshift and spatial distribution. For a single galaxy cluster, itis characterized by the integrated luminosity within 500 times virial radius[76]:733.4. A Worked Example: κ× y Cross Correlation200 400 600 800 1000 1200 1400 1600 1800 2000``3C`(MJr / Sr)2× ICIB353× ICIB545× ICIB857Figure 3.6: Cross correlation between CIB signal and κ for three differentCIB maps in ` space. The cross correlation signal is binned to 5 ` binscentered at 290, 670, 1050, 1430, 1810.ICIB500 (ν, z) = L0[M5001014M]CIBΨ(z)Θ[(1 + z)ν, Td(z)] (3.27)where L0 is the normalization parameter, Td = Td0(1 + z)αCIB and Θ[ν, Td]is the spectral energy distribution for a typical galaxy that contributes tothe total CIB emission,Θ[ν, Td] ={νβCIBBν(Td), if ν < ν0ν−γCIB , if ν ≥ ν0(3.28)with ν0 being the solution of d log[νβCIBBν(Td)]/d log(ν) = −γCIB. Theredshift dependence is assumed to be the form:Ψ = (1 + z)δCIB (3.29)whereTd0, αCIB, βCIB, γCIB, δCIB are the CIB model parameters given in[69].The Planck collaboration made 3 CIB maps corresponding to 353GHz,545GHz and 857GHz [10]. The maps are made by disentangling CIB sig-nal from a galactic dust emission map. The galactic dust emission map is743.4. A Worked Example: κ× y Cross Correlationgenerated with a Generalized ILC method. The CIB covariance matrix isacquired from simulated CIB maps [7]. The units of the maps are MJy/srand their angular resolution is 5 arcmin.Since the CIB traces the spatial distribution of distant galaxy clusters,it should also have a non-zero cross correlation with the weak lensing bylarge scale structure. We first estimate the cross correlation between the 3Planck CIB maps and the RCSLens κ map in the same field. All three CIBmaps show a non-zero cross correlation signal (Fig.3.6) with a confidencelevel of {3.6,3.7,4}σ respectively. The uncertainties are derived from the χ2statistics. It is therefore possible that CIB residual in our reconstructed ymap can contaminate the κ× y cross correlation signal.Our reconstructed NILC y map is calculated as:yˆ(θ)rec =∑i,ac(a)i d(a)i (θ)= y(θ) +∑i,ac(a)i n(a)i (θ)= y(θ) +∑i,ac(a)i ICIB,(a)νi (θ) +∑i,ac(a)i n′(a)i (θ)(3.30)Where, in the last line, we single out the residual CIB contributions to they map explicitly. Here (a) denotes the ath needlet window. c(a)ν is the ILCcoefficient for y (I omit the component label α for Eq.3.21 because we areonly concerned about y now). yˆ is the estimated y signal and y is the truey signal. CMB and galactic dust are removed while noise is minimized butnot completely removed. The noise term ni(θ) contains both CIB signaland other noise either from the sky (CO emission) or from the instrument(photon noise). Cross correlating both side of Eq.3.30 with κ, we get:Cκ×yˆ` = Cκ×y` +∑i,ac(a)i h(a)(`)Cκ×ICIBνi`+∑i,ac(a)i h(a)(`)Cκ×n′i`(3.31)where Cκ×yˆ` is the estimated κ×y cross correlation directly from the y map.It consists of the true κ × y signal as well as contamination from CIB andother noise.To correct for the CIB contamination, we make a CIB-subtracted y map.CIB signal is not isotropically dependent on frequency (see Eq.3.27). Itdepends on redshift and individual galaxy clusters. So we do not include it as753.4. A Worked Example: κ× y Cross Correlationan ILC foreground. We subtract the CIB map from the original temperaturemap and make the y map using the same NILC procedure:yˆ(θ)CIB−subtracted =∑i,ac(a)i (d(a)i (θ)− ICIB,(a)νi (θ))= y(θ) +∑i,ac(a)i n′(a)i (θ)(3.32)Here yˆ(θ)CIB−subtracted is the CIB-subtracted y map. We subtract the 3Planck CIB maps from the 3 highest frequency maps. Since most CIBmodels give a grey body spectrum with peak higher than 857GHz [7], weassume that the CIB contribution in the 3 lowest frequencies (100, 143,217GHz) is negligible.We estimate the CIB contamination in our cross correlation function bycalculating κ× yˆrec and κ× yˆCIB-subtracted. The cross correlation signals areshown in the upper panel of Fig.3.7. The cross correlation values are calcu-lated as the average signals in 5 ` bins centered at ` = {290, 670, 1050, 1430, 1810}.The error bars are the standard error for each bin.err` bin ≡ 1∆`√ ∑`∈`bin(C` − C¯`)2 (3.33)where ∆` is the length of each ` bin, C¯` is the mean cross correlation signalin that bin. The three sets of points with 3 different color are correspondingto 3 different y maps cross correlated with a same κ map: the Planck y map,out fiducial NILC map, and our CIB-corrected NILC map.All the 3 cross correlation functions show a nonzero signal with > 5σconfidence level. yˆrec has an 8σ signal which is consistent with [43].To evaluate the fractional of CIB contamination in the κ × yˆ cross cor-relation, we performed a bootstrap resampling for all of the five ` bins. Theinherent covariance between C`’s are negligible compare to the sample noise.The mode coupling covariance can be calculated by PolSpice by given theRCS field mask and the galactic mask. The covariance matrix shows a cou-pling within δ` ∼ 5, so we make a ’blocked’ bootstrap resampling to recoverit. Specifically, we first divide the 380 `’s into 76 blocks and calculate theaverage C`’s and ∆C`’s in each block.For each ` bin, we made 10000 realizations of 75 block-averaged C`’s and∆C`’s from the measured κ× yˆrec and κ× yˆrec−κ× yˆCIB-subtracted. Then wecalculate 〈∆C`〉 / 〈C`〉 for each realization. 〈·〉 is the mean value of the 76bootstrap samples. The error bar is estimated by calculating the 68% C.L.763.4. A Worked Example: κ× y Cross Correlation200 400 600 800 1000 1200 1400 1600 1800 2000`10123456`3C`×10 6× yˆPlanck× yˆrec× yˆCIB− subtracted200 400 600 800 1000 1200 1400 1600 1800 2000`∆C`/C`× yˆrec − × yˆCIB− subtracted× yˆrecFigure 3.7: Upper panel: Cross correlation between y signal and κ for threedifferent y maps in ` space. The cross correlation signal is binned to 5 `bins centered at 290, 670, 1050, 1430, 1810. Blue, green and red points arecorresponding to Planck NILC y map, our NILC y map, our CIB-subtractedy map (see Eq.3.32); lower panel: The bootstrap estimation of 〈∆C`〉 / 〈C`〉for each ` bin. The error bars correspond to a 68% C.L.773.5. An Attempt to Reconstruct the CIB SignalFor the last ` bin where the signal is very weak, the probability distributionof 〈∆C`〉 / 〈C`〉 is highly non-Gaussian, so the standard deviation is largerthan the 68% C.L.The resulting 〈∆C`〉 / 〈C`〉 is shown in the lower panel of Fig.3.7. Thefirst ` bin has a 3.6σ non-zero fraction. From a Pearson Null test with χ2statistics, the contamination fraction is only around ∼ 2σ significant for thelast 4 ` bins which corresponds to an ` range of 600∼2000. By calculatingthe average number of bootstrap realizations for the last four ` bins, wefound the contamination fraction is 7.75%± 5.21%.3.4.2 The Galactic Dust ContaminationThermal radiation from galactic dust does not correlate with weak lens-ing since weak lensing is extragalactic in origin. However, since the dustradiation dominates the highest frequency maps, an improper estimate ofthe dust signal may result in additional noise in the y map. The improperestimation of dust emission could result from the fact that both the dusttemperature and its spectral index are spatially dependent (see Fig.3.8).Toanalyze the impact of galactic dust contamination, we also make some othernon-standard y maps with different dust residual by slightly varying βd valuein Eq(22). We use βd = 1.30, 1.43, 1.76, 1.85 to generate the more y maps.These numbers correspond to ∼ ±1σ and ∼ ±2σ around our fiducial βdvalue, which is 1.57. We then calculate the cross correlation of these y mapsand the weak lensing mass map.Fig.3.9(a) shows that the change of dust index does not significantlyaffect the cross correlation signal. But as is shown in Fig.3.9(b), differentdust residual do have an effect on the error. Our fiducial dust index givesthe lowest errorbar, which suggests that this model removes the dust signalmost completely.In our analysis, we take the model for dust to be isotropic in the RCSfield, but this is not precise. Both the dust temperature and dust index varyspatially. Our analysis of the cross correlation points out that this variationdoes contribute to the cross correlation signal. So a spatially dependent dustmodel is needed for a more precise y map.3.5 An Attempt to Reconstruct the CIB SignalOne notable feature is that the frequency dependence of the three crosscorrelation signals in Fig.3.6 seems to be independent of scale, which suggeststhat there might be a template for the κ × CIB cross correlation signal783.5. An Attempt to Reconstruct the CIB Signal10 15 20 25 30 35 40Td(K)0. 00. 20. 40. 60. 81. 01. 21. 4Number of Pixels×1051.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6βd0. 00. 51. 01. 52. 02. 53. 03. 54. 04. 5Number of Pixels×104Figure 3.8: Histogram for Td (upper panel) and βd (lower panel) in RCSfield. The dust model we use here is the Planck COMMANDER thermaldust map [2].793.5. An Attempt to Reconstruct the CIB Signal200 400 600 800 1000 1200 1400 1600 1800 2000`10123456`3C`1e 6βd =1. 30βd =1. 43βd =1. 57βd =1. 76βd =1. 85200 400 600 800 1000 1200 1400 1600 1800 2000`σ(`3C`)1e 6βd =1. 30βd =1. 43βd =1. 57βd =1. 76βd =1. 85Figure 3.9: Upper panal: κ×y cross correlation for the fiducial y map (βd =1.57) and four non-standard y maps. Lower panel: standard derivation forcross correlation signal in each ` bin.803.5. An Attempt to Reconstruct the CIB Signal0 1 2 3 4 5β0. of Pixels1e6β1β2Figure 3.10: Histogram for CIB indices β1 and β2 in the unmasked domain.which is independent of frequency. We also want a CIB map that is freeof y signal. According to [7], the Planck CIB maps were made by takingthe difference between two dust maps. One dust map contains CIB andwith CIB suppressed in another dust map. The tSZ signal in the CIBmap is suppressed but not nulled. So the nonzero κ× CIB signal might becontaminated by κ × y signal. So we propose another way to construct aCIB map with tSZ signal completely removed.Our method to reconstruct CIB signal is again based on NILC. Thistime we take CMB, tSZ and galactic dust as foregrounds. From Fig.3.6, itis appealing to try a homogeneous CIB frequency spectrum fCIB(ν). TheCIB intensity map for frequency ν is the template ICIB(θ) multiplied by thespectrum:ICIBν (θ) = fCIB(ν)ICIB(θ) (3.34)Here we use a piecewise power law function for CIB:fCIB(ν) ∝{νβ1 , if ν ≤ 545GHzνβ2 , if ν > 545GHz(3.35)813.5. An Attempt to Reconstruct the CIB SignalWe first mask the CIB maps with the 40% galactic mask, then the power-law indices are calculated withβ1 =ln(ICIB545GHz/ICIB353GHz)ln(545/353)β2 =ln(ICIB857GHz/ICIB545GHz)ln(857/545)(3.36)where β1 and β2 are calculated for each pixel in the unmasked domain, andthe estimated value is chosen to be the mode, which is β1 = 2.2 and β2 = 1.3(Fig.3.10). We make the NILC CIB template map ICIB(θ) following thesame procedure in Section 3.3. and the CIB intensity maps for 353GHz,545GHz and 857GHz by multiplying the template map with fCIB(ν).We calculate and compare the κ× CIB cross correlation signal betweenour NILC CIB map and the Planck CIB map. The κ data is the same asthat for κ × y.From Fig.3.11 we see that the shapes of two cross correla-tion functions are different, and our CIB map produces larger errors and anegligible cross correlation. Little can be told from our CIB map about theκ× CIB cross correlation signal because it is buried in noise.823.5. An Attempt to Reconstruct the CIB Signal200 800 1400 20000.`3C`353GHz× ICIB,Planck353× ICIB,This work353200 800 1400 20000.`3C`545GHz× ICIB,Planck545× ICIB,This work545200 800 1400 2000``3C`857GHz× ICIB,Planck857× ICIB,This work857Figure 3.11: κ × CIB cross correlation signal for our NILC CIB map andPlanck CIB map. Three panels are corresponding to three frequencies.833.5. An Attempt to Reconstruct the CIB SignalWe also make a NILC y map using this CIB spectrum model by includingfCIB(ν) in the mixing matrix Miα. We perform exactly the same analysisas in Section 3.3. We call this map the CIB-nulled NILC y map. Thedifference between an CIB-nulled map and a CIB-subtracted map is thatthe CIB-subtracted map is made by subtracting the CIB signal from theraw intensity map before the NILC procedure, while the CIB nulled mapemploys a CIB frequency model in the NILC analysis. We compute theκ× y cross correlation function just like Section 3.3. Only this time we usethe CIB-nulled NILC y map.The difference κ× yˆPlanck,NILC−κ× yˆCIB nulled and κ× yˆrec−κ× yˆCIB nulledare each with 0.03σ and 0.06σ significance. So our CIB-nulled NILC y mapis not significantly different from the Planck NILC y map.843.5. An Attempt to Reconstruct the CIB Signal200 400 600 800 1000 1200 1400 1600 1800 2000`10123456`3C`×10 6× yˆPlanck,NILC× yˆrec× yˆCIB−nulled200 400 600 800 1000 1200 1400 1600 1800 2000`1. 51. 00. 50. 00. 51. 0∆(`3C`)×10 6× yˆPlanck,NILC − × yˆCIB−nulled× yˆrec − × yˆCIB−nulledFigure 3.12: Upper panel: cross correlation signals between κ and three ymaps; lower panel: Difference of κ×y cross correlation signals NILC y mapsand NILC-(CIB-nulled) y map.853.6. Discussion3.6 DiscussionWe estimate the contamination from CIB and galactic dust signal in theNILC y map. We make a new y map using the NILC method. Our methoddiffers from the Planck NILC y map in that we only use the 6 HFI maps andinclude a grey body spectrum for dust, while the Planck group only nulls theCMB component. We find a nonzero cross correlation signal between PlanckCIB map and the weak lensing κ, which suggest that a CIB-contaminated ymap may produce a false cross correlation signal. To estimate this contami-nation, we then make a CIB-subtracted y map by subtracting the CIB signalfrom the raw 353GHz, 545GHz and 857GHz temperature map. We estimatethe contamination from CIB by taking the cross correlation between the ymap and the RCSLenS mass map. Our NILC y map shows very similarcross correlation signal as the Planck NILC y map, which means that our ymap is reliable for cross correlation analysis. The difference between κ×yrecand κ× yCIB−subtracted is (5.8± 4.6)% within 600 < ` < 2000.[82] presented a detection of κ × y cross correlation and constrainedbgas(Te(0)/0.1keV)(ne/m−3) = 2.01± 0.31± 0.21, where bgas is the gas bias,ne is the mean electron number density and Te(0) is the electron tempera-ture. The first error is statistical and the second is systematic. This combi-nation of parameters is proportional to the amplitude of the cross correlationsignal (see Eq.3.10 and Eq.3.11). So a shift of (5.8 ± 4.6)% causes a com-parable shift in the constrained result. But this is not a very significantcontamination. The largest uncertainty in the cross correlation still comesfrom the noise.Our study does not take the 100-217GHz CIB signal into account. Soin our CIB-subtracted y map, some CIB still remains which depends bothon the CIB amplitude and the ILC coefficients for those frequencies. In afuture study, it would be fruitful to construct an all sky CIB map for morefrequencies so that we can remove the CIB more completely. In addition, aswe find a significant κ × CIB result, it is helpful to understand this signaland investigate how it can constrain cosmological parameters. Our study isbased on the assumption that the Planck CIB map is not contaminated bytSZ signal. If it is, then the CIB-subtracted y map is not accurate becausewe subtracted some y signal as well. Future studies should estimate theleakage of tSZ signal in the CIB maps.Galactic dust contamination affects the noise in the output y map. Wecontrol the dust residual in our y map by varying the dust spectral index βdin the dust model. The cross correlation result Fig.3.9 shows that this changedoes not affect the amplitude of the κ×y cross correlation but indeed changes863.6. Discussionthe error bar for the signal. The error bar for βd = 1.57 is the lowest one,which means that this is our best estimate of dust contamination. However,it needs to be pointed out that our dust model is an isotropic model, so it isbiased because both βd and Td are expected to vary spatially. Taking ILCfor each pixel is time consuming and will also introduce more noise from theunequal zero point for different frequency maps. It’s not clear how to solvethis problem.The noise term n′ in Eq.3.30 contains photon noise which is proportionalto the square root of photon number. So it depends on the spatial distribu-tion of SZ signal and could also be correlated with weak lensing. We don’tconsider it in this work.Finally, we come up with a piecewise power spectrum for the CIB andmake a CIB-nulled y map by introducing the CIB spectrum in the mixingmatrix. Cross correlation of the CIB-nulled y map and κ map makes littledifference with the cross correlation of the y map and the κ map. With thisCIB spectrum, we make a NILC CIB map. But the κ×CIB cross correlationsignal is noisy and not significant.87Chapter 4Conclusions• PIXIE: We wrote a code model for a simplified optical system for thePIXIE telescope and simulated ray propagation in the instrument. Welet some instrument parameters to be free and optimize them by max-imizing a judging parameter Good (defined in (2.25)). The optimizedparameters are summarized in (2.28) and the output with this parame-ters Table.2.4 The optimized instrument yields a good value of 15.27%and the telescope gets a 1.1◦ beam resolution. Future work for thisinstrumental simulation includes:1. Looking at the polarization performance of the instrument: di-vide the area of interest into many squares, and sum P‖ and P⊥(parallel and perpendicular to the desired polarization) weightedby cos2 θ0. Then plot a little line showing the average polariza-tion direction in each square. This shows the curvature of thepolarization across the field.2. Include more free parameters to be optimized: for example, thefoci of other mirrors.3. Include the moving dihedral mirror in future simulations, andoptimize Good with moving dihedral mirror.• κ× y cross-correlation: we estimate the contamination from CIB andgalactic dust signals in the reconstructed κ× y cross-correlation. Theresult shows that CIB contributes only (5.8 ± 4.6)% with only 2.2σconfidence within 600 < ` < 2000. The dust does not change the signalamplitude but only the noise level. A dust spectral index βd = 1.57minimizes the standard variance in κ × y. There are couple of thefuture plans:1. 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They can be decomposedby spin-weighted spherical harmonic function which satisfy the normaliza-tion relation: ∫ 2pi0dφ∫ pi0sY∗`m(θ, φ)sY`′m′(θ, φ)dθ = δ``′δmm′∑`msY∗`m(θ, φ)sY(`mθ′, φ′) = δ(φ− φ′)δ(cos θ − cos θ′)(A.2)As what we do for angular momentum in quantum mechanics, we candefine the upper and lower operator ∂± which can transform an s spin-weighted function to a s+1 or s−1 spin-weighted function. Under sphericalcoordinate, ∂± writes:∂± = − sin±s θ(∂∂θ± isin θ∂∂φ)sin±s θ (A.3)when acting on sY`m, repeat the calculation for angular momentum, weget a familiar relation:∂±sY`m = ±√(`∓ s)(`+ 1± s)s±1Y`m (A.4)So the spherical harmonic function with spin weight = s can be obtainedby acting ∂± s times on Y`m (for which s = 0):sY`m =√(`− s)!(`+ s)!(∂±)sY`m (A.5)96Appendix A. The Spin-Weighted Spherical Harmonic Functionfor s = 2:2Y`m =√(`− 2)!(`+ 2)!(∂±)2Y`m (A.6)For a regular spherical harmonic function, under flat plane approxima-tion (`→ inf), Y`m → (e)i~`·nˆ. So for s = 2:2Y`m → 1`2(∂±)2ei~`·nˆ = e±2i(φ−φ`)ei~`·nˆ (A.7)Thus we prove (2.14)97Appendix BThe Needlet ILCOur ILC process is performed in a Needlet frame. Needlet is first introducedby Narcowich et al. [60] as a particular construction of a wavelet frame on asphere. The most distinctive property of the needlets is their simultaneousperfect localization in the spherical harmonic domain (actually they arespherical polynomials) and potentially excellent localization in the spatialdomain.Basically, the raw temperature maps are first filtered into needlet win-dows by first make spherical harmonic transforms of the maps x`m, thenmultiplied by the needlet window h(a)(`) and transformed back into realspace. The result is called a needlet map, characterized by a given rangeof angular scales given in h(a)(`). ILC is performed for each needlet scale,and the synthesized map is obtained by co-adding the ILC estimates foreach needlet scale. In this work,the needlet bandpass windows are definedfollowing Aghanim et al. [10], which is a set of successive Gaussian beamtransfer functions in harmonic space.h(1)(`) =√b1(`)2,h(a)(`) =√ba+1(`)2 − ba(`)2,h(10)(`) =√1− b10(`)2,(B.1)whereba(`) = exp(−`(`+ 1)σ2a/2) (B.2)andσa =(1√8 ln 2)(pi180× 60′)FWHM[a] (B.3)with FWHM = [300′, 120′, 60′, 45′, 30′, 15′, 10′, 7.5′, 5′]. So we have10∑a=1(h(a)(`))2= 1 (B.4)98Appendix B. The Needlet ILC100 101 102 103 104` WindowsFigure B.1: Needlet windows acting as bandpass filters in ` space.So the signal for the output synthesized map from different needlet isconserved.To calculate the needlet filtered map d(a)i , we first calculate the sphericalharmonic transformation of di:di(θ) =∑`mxi,`mY`m(θ) (B.5)xi,`m is the spherical harmonic coefficient for map of the ith channel.Multiply it by the needlet filter h(a)(`) and transform back, we get theneedlet filtered map:d(a)i (θ) =∑`mh(a)(`)xi,`mY`m(θ) (B.6)With all the information given in section 3, we present the ILC coeffi-cients for our fiducial NILC y map at different needlet scales in Table ??.99Appendix B. The Needlet ILCTable B.1: The ILC coefficients for the fiducial NILC y map in differentneedlet scaleshi(`) 104 × c100 104 × c143 104 × c217 104 × c353 104 × c545 104 × c8571 -23.414 12.546 1.395 0.642 -0.696 0.1122 -10.955 -0.177 5.736 -0.188 -0.615 0.1213 -7.235 -2.745 5.577 0.414 -0.855 0.1544 -4.942 -4.268 5.416 0.802 -0.962 0.1605 -4.682 -4.263 5.194 0.947 -0.966 0.1526 -4.772 -3.943 4.911 1.062 -0.915 0.1307 -3.561 -4.540 4.602 1.353 -0.894 0.1058 -2.537 -5.110 4.418 1.551 -0.857 0.0819 -1.387 -5.690 4.143 1.808 -0.816 0.05210 -0.139 -5.915 3.388 2.305 -0.727 -0.007100


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