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Probing the large-scale structure of the Universe with the Sunyaev-Zel'dovich Effect Tanimura, Hideki 2017

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Probing the Large-Scale Structureof the Universewith the Sunyaev-Zel’dovich EffectbyHideki TanimuraM.Sc., The University of British Columbia, 2013Ph.D., The University of British Columbia, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Astronomy)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2017c© Hideki Tanimura 2017AbstractThe Sunyaev-Zeldovich (SZ) effect is a spectral distortion in the CosmicMicrowave Background (CMB), due to up-scattering of CMB photons byhigh energy electrons in clusters of galaxies or any cosmic structure. ThePlanck satellite mission has measured the spectral distortion with greatsensitivity and has produced a full-sky SZ (y) map, which can be used totrace the large-scale structure of the Universe.In this dissertation, I construct the average SZ (y) profile of ∼ 65,000Luminous Red Galaxies (LRGs) from the Sloan Digital Sky Survey DataRelease 7 (SDSS/DR7) using the Planck y map and compare the measuredprofile with predictions from the cosmo-OWLS suite of cosmological hydro-dynamical simulations. This comparison agrees well for models that includefeedback from active galactic nuclei (AGN feedback).In addition, I search for the SZ signal due to gas filaments between∼260,000 pairs of LRGs taken from the Sloan Digital Sky Survey DataRelease 12 (SDSS/DR12), lying between 6-10 h−1Mpc of each other in thetangential direction and within 6h−1Mpc in the radial direction. I find astatistically significant SZ signal between the LRG pairs. This is the firstdetection of gas plausibly located in filaments, expected to exist in the large-scale structure of the universe. I compare this result with the BAHAMASsuite of cosmological hydrodynamical simulations and find that it predictsa slightly lower, but marginally consistent result.As an extension of my MSc. thesis work, I study CMB polarization.The B-mode component of CMB polarization is an important observable totest the theory of inflation in the early universe. However, foreground emis-sions in our own galaxy dominates the B-mode signal and therefore multi-frequency observations will be required to separate any CMB signal fromthe foreground emission. I assess the value of adding a new low-frequencychannel at 10 GHz for the foreground removal problem by simulating real-istic experimental data. I find that such a channel can greatly improve ourdetermination of the synchrotron component which, in turn, significantlyimproves the reliability of the CMB separation.iiLay SummaryThe evolution of structure in the Universe poses many challenges to ourunderstanding. On the very largest scales, the evolution is relatively simplebecause it is dominated by gravity. But on smaller scales, the structurebecomes more complicated because baryonic (gas and plasma) physics beginsto become important. One of the key tracers of large scale structure isclusters of galaxies. These objects are the most massive bound systems in theUniverse and they roughly mark the transition from the simple larger scalesto the more complicated smaller scales. Observational probes of baryonicmatter on these scales are difficult to come by. The primary focus of my PhDresearch has been to study the properties of baryonic gas in and between theclusters of galaxies using the Sunyaev-Zel’dovich(SZ) effect, which allows usto trace the gas distribution through high energy electron distributions inclusters of galaxies or any cosmic structure.iiiPrefaceThis dissertation is original, submitted or unpublished, mostly independentwork by the author, Hideki Tanimura with the dedicated support from mysupervisor, Gary Hinshaw. The images and plots from other materials arere-used based on the Creative Commons Attribution license (CC BY 4.0) inarXiv (https://arxiv.org/help/license).In Chapters 2, 3, 4, and 5, I conducted all the data analysis includingthe observational and simulation data. The observational data are publiclyavailable, obtained from the Sloan Digital Sky Survey and Planck satellitemission. The simulation data from the cosmo-OWLs and BAHAMAS suiteof cosmological hydrodynamical simulations are kindly provided by Ian Mc-Carthy at Liverpool John Moores University.The work in Chapters 6 and 7 is an extension of my master’s thesis,“10GHz Sky Survey to probe Inflation with CMB Polarization” (Tanimura,2013), which has been accepted by University of British Columbia. In thisdissertation, the study is extended to the polarization mode, while a major-ity of the text in the chapters has been extracted from the manuscript in(Tanimura, 2013).ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Modern cosmology . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Expanding universe . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Friedmann equations . . . . . . . . . . . . . . . . . . 31.2.2 The Hubble diagram . . . . . . . . . . . . . . . . . . 51.2.3 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . 71.2.4 Cosmic microwave background . . . . . . . . . . . . . 81.3 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Horizon problem . . . . . . . . . . . . . . . . . . . . . 131.3.2 Solution to the horizon problem . . . . . . . . . . . . 141.3.3 Primordial power spectrum . . . . . . . . . . . . . . . 151.4 Structure formation . . . . . . . . . . . . . . . . . . . . . . . 161.4.1 Linear structure evolution . . . . . . . . . . . . . . . 161.4.2 Non-linear structure evolution . . . . . . . . . . . . . 191.4.3 Evolution of baryon . . . . . . . . . . . . . . . . . . . 201.4.4 Clusters of galaxies . . . . . . . . . . . . . . . . . . . 221.5 Sunyaev-Zel’dovich effect . . . . . . . . . . . . . . . . . . . . 23vTable of Contents1.5.1 Kompaneetz equation . . . . . . . . . . . . . . . . . . 231.5.2 Sunyaev-Zeldovich effect . . . . . . . . . . . . . . . . 252 Construction of Compton parameter y map . . . . . . . . . 302.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Construction of y map with internal linear combination tech-nique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3 Comparison with Planck y map . . . . . . . . . . . . . . . . 342.3.1 Compton y parameter profile of Luminous red galaxies 342.3.2 Relation between integrated Compton y parameterand stellar masses . . . . . . . . . . . . . . . . . . . . 342.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Probing hot gas in halos through the Sunyaev Zel’dovicheffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 AGN feedback effects . . . . . . . . . . . . . . . . . . . . . . 403.3 Luminous red galaxies (LRGs) . . . . . . . . . . . . . . . . . 423.3.1 Luminous red galaxies . . . . . . . . . . . . . . . . . 423.3.2 LRG catalog . . . . . . . . . . . . . . . . . . . . . . . 433.4 Compton y parameter profile of the LRGs . . . . . . . . . . 443.5 Comparison to cosmo-OWLs hydrodynamic simulations toprobe AGN feedback effect . . . . . . . . . . . . . . . . . . . 443.5.1 cosmo-OWLS hydrodynamic simulations . . . . . . . 443.5.2 Comparison to cosmo-OWLs hydrodynamic simula-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6 Comparison to prediction from halo model and universal pres-sure profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.6.1 The Stacked y profile with cross-correlation of the tSZand distribution of galaxy clusters . . . . . . . . . . . 483.6.2 Universal Pressure Profile . . . . . . . . . . . . . . . . 513.6.3 Estimating halo masses of LRGs . . . . . . . . . . . . 523.6.4 Comparison to the prediction . . . . . . . . . . . . . 533.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 Probing hot gas in the cosmic web through the SunyaevZel’dovich effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Missing baryon problem . . . . . . . . . . . . . . . . . . . . . 59viTable of Contents4.3 Pair stacking of LRG pairs . . . . . . . . . . . . . . . . . . . 614.3.1 LRG pair catalog . . . . . . . . . . . . . . . . . . . . 614.3.2 Stacking on LRG pairs . . . . . . . . . . . . . . . . . 624.3.3 Subtracting the halo contribution . . . . . . . . . . . 634.3.4 Null tests and error estimates . . . . . . . . . . . . . 664.4 Interpretation of the detected tSZ signal between the LRGpairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.5 Comparison to BAHAMAS hydrodynamic simulations . . . . 744.5.1 BAHAMAS hydrodynamic simulations . . . . . . . . 744.5.2 Comparison with the hydrodynamic simulations . . . 754.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 Probing hot gas in the cosmic web between galaxy groupsand clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Pair stacking of galaxy group/clusters . . . . . . . . . . . . . 805.2.1 Galaxy groups and clusters for SDSS DR10 galaxies . 805.2.2 SDSS DR10 group pair catalog . . . . . . . . . . . . . 815.2.3 Stacking on group pairs . . . . . . . . . . . . . . . . . 825.2.4 Subtracting the halo contribution . . . . . . . . . . . 825.2.5 Null tests and error estimates . . . . . . . . . . . . . 825.3 Comparison to BAHAMAS hydrodynamic simulations . . . . 875.4 Interpretation of the detected tSZ signal between the grouppairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 CMB polarization: A probe of the early universe . . . . . 916.1 CMB polarization . . . . . . . . . . . . . . . . . . . . . . . . 916.1.1 E/B decomposition . . . . . . . . . . . . . . . . . . . 926.1.2 Observable predictions and current observational con-straints . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2 Foreground emission in the microwave bands . . . . . . . . . 956.2.1 Synchrotron emission . . . . . . . . . . . . . . . . . . 956.2.2 Free-Free emission . . . . . . . . . . . . . . . . . . . . 976.2.3 Thermal dust emission . . . . . . . . . . . . . . . . . 976.2.4 Spinning dust emission . . . . . . . . . . . . . . . . . 98viiTable of Contents7 A 10GHz polarization sky survey . . . . . . . . . . . . . . . . 997.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.2 Field of view . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.3 Simulations of the observational data in polarization . . . . . 1027.3.1 Simulations of the polarized skies . . . . . . . . . . . 1027.3.2 Noise estimate . . . . . . . . . . . . . . . . . . . . . . 1067.4 Markov chain monte carlo simulation . . . . . . . . . . . . . 1077.4.1 Model function . . . . . . . . . . . . . . . . . . . . . . 1077.4.2 Improvement by the 10GHz data (without spinningdust) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.4.3 Improvement by the 10GHz data (with spinning dust) 1097.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124viiiList of Tables1.1 History of Modern Cosmology . . . . . . . . . . . . . . . . . . 21.2 ΛCDM cosmological parameters . . . . . . . . . . . . . . . . . 132.1 Band data for the Planck y maps (Van Waerbeke et al., 2014) 323.1 The baryon feedback models in the cosmo-OWLS simulation.Each model has been run in both Planck and WMAP7 cos-mology (McCarthy et al., 2014). . . . . . . . . . . . . . . . . 477.1 Mean noises of stokes Q/U map at each frequency. . . . . . . 1067.2 Uncertainty of the fit parameters by MCMC [uK] (withoutspinning dust). . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.3 Uncertainty of the fit parameters by MCMC [uK] (with spin-ning dust). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115ixList of Figures1.1 A recent Hubble diagram of a large combined sample of galax-ies using SNIa as standard candles . . . . . . . . . . . . . . . 71.2 Constraints on the baryon density from Big Bang Nucleosyn-thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 CMB Spectrum measured by COBE/FIRAS . . . . . . . . . 101.4 CMB temperature fluctuation map from Planck satellite mis-sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Power spectrum of CMB temperature fluctuations from Plancksatellite mission . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 The linear matter power spectrum P (k) versus wavenumberextrapolated to z=0 . . . . . . . . . . . . . . . . . . . . . . . 201.7 Structure formation of the universe at z = 0 in a N-bodysimulation box 100 Mpc/h on side. . . . . . . . . . . . . . . . 211.8 The mass function of dark matter halos . . . . . . . . . . . . 231.9 The CMB spectrum and the distorted spectrum by the SZeffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.10 Spectral distortion of the CMB radiation due to the Sunyaev-Zel’dovich effect . . . . . . . . . . . . . . . . . . . . . . . . . . 281.11 Cleaned images of Abell 2256 at z = 0.058 observed by Planckat 100, 143, 217, 353, and 545 GHz . . . . . . . . . . . . . . . 281.12 Planck all-sky Compton parameter maps . . . . . . . . . . . 292.1 Maps of the Compton parameter, y, formed from linear com-binations of the Planck HFI maps . . . . . . . . . . . . . . . 332.2 The average y profile of 63,398 LRGs using the Planck y map(blue) is compared with the y profile using our y maps . . . . 352.3 Mean SZ signal vs. stellar mass for locally brightest galaxiesderived with our y map version D . . . . . . . . . . . . . . . . 373.1 The luminosity function of galaxies . . . . . . . . . . . . . . . 413.2 First measurement of the MBH-σ relation . . . . . . . . . . . 42xList of Figures3.3 Left: The stellar mass distribution of SDSS DR7 LRGs. Right:The redshift distribution of the LRGs. . . . . . . . . . . . . 443.4 The average Planck y map stacked against 74,681 LRGs . . . 453.5 Compton y maps simulated by cosmo-OWLS hydrodynamicsimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.6 The average y profile of LRGs is compared with the beam-convolved y profile of the simulated central galaxies in Planckcosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.7 The average y profile of LRGs is compared with the y profileof the simulated central galaxies in WMAP7 cosmology . . . 493.8 Universal pressure profile . . . . . . . . . . . . . . . . . . . . 523.9 The relation between the stellar masses of central galaxiesand halo masses . . . . . . . . . . . . . . . . . . . . . . . . . . 533.10 The average y-profile of LRGs is compared to the predictionsbased on the halo model and UPP . . . . . . . . . . . . . . . 554.1 Evolution of the four cosmic baryon components in mass frac-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 The baryon census in the local Universe . . . . . . . . . . . . 624.3 The distribution of redshift and tangential separations for theselected LRG pairs . . . . . . . . . . . . . . . . . . . . . . . . 634.4 The average Planck y map stacked against 262,864 LRG pairs 644.5 The best-fit circular halo profiles fit to the map . . . . . . . . 654.6 The residual y-map after the best-fit radial halo signals aresubtracted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.7 A sample null map obtained by stacking the y map against theLRG pairs that were rotated in galactic longitude by randomamounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.8 An average y map stacked against a catalog of LRG pseudopairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.9 The result from 1000 rotated null stacks . . . . . . . . . . . . 704.10 The result from 1000 pseudo-pair null stacks . . . . . . . . . . 704.11 The single-halo model y map is stacked against the same262,864 LRG pairs . . . . . . . . . . . . . . . . . . . . . . . . 734.12 The stacked y map of the central galaxy pairs from the BA-HAMAS simulations, at 10 arcsecond angular resolution (un-smoothed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1 The distribution of redshift and tangential separations for thegroup pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81xiList of Figures5.2 The average Planck y map stacked against 34,955 group pairs 835.3 The best-fit circular halo profiles fit to the map . . . . . . . . 845.4 The residual y-map after the best-fit radial halo signals aresubtracted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.5 The result from 1000 rotated null stacks . . . . . . . . . . . . 865.6 The result from 1000 pseudo-pair null stacks . . . . . . . . . . 866.1 E-mode and B-mode patterns of polarization . . . . . . . . . 926.2 Angular power spectra of CMB with varying tensor-to-scalarratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.3 Brightness temperature rms as a function of frequency andastrophysical component for temperature and polarization . . 967.1 The image of observation strategy at Penticton. . . . . . . . . 1007.2 Hit map of one-year of observations, in galactic coordinates(Nside = 64) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.3 The simulated stokes Q maps . . . . . . . . . . . . . . . . . . 1047.4 The simulated stokes U maps . . . . . . . . . . . . . . . . . . 1057.5 Correlation maps among the fit parameters (without spinningdust) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.6 Stokes Q spectra (without spinning dust) . . . . . . . . . . . 1117.7 Stokes U spectra (without spinning dust) . . . . . . . . . . . 1117.8 Input, output(best-fit) and residual maps of Qs estimated byMCMC, without spinning dust . . . . . . . . . . . . . . . . . 1127.9 Input, output(best-fit) and residual maps of Us estimated byMCMC, without spinning dust . . . . . . . . . . . . . . . . . 1137.10 Input, output(best-fit) and residual maps of βs estimated byMCMC, without spinning dust . . . . . . . . . . . . . . . . . 1147.11 Correlation maps among the fit parameters (with spinningdust) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.12 Stokes Q spectra (with spinning dust) . . . . . . . . . . . . . 1177.13 Stokes U spectra (with spinning dust) . . . . . . . . . . . . . 1177.14 Input, output(best-fit) and residual maps of Qs estimated byMCMC, with spinning dust . . . . . . . . . . . . . . . . . . . 1187.15 Input, output(best-fit) and residual maps of Us estimated byMCMC, with spinning dust . . . . . . . . . . . . . . . . . . . 1197.16 Input, output(best-fit) and residual maps of βs estimated byMCMC, with spinning dust . . . . . . . . . . . . . . . . . . . 1207.17 Input, output(best-fit) and residual maps of Qsp estimatedby MCMC, with spinning dust . . . . . . . . . . . . . . . . . 121xiiList of Figures7.18 Input, output(best-fit) and residual maps of Usp estimated byMCMC, with spinning dust . . . . . . . . . . . . . . . . . . . 122(*The images and plots from other materials are re-used based on theCreative Commons Attribution license (CC BY 4.0) in arXiv (https://arxiv.org/help/license))xiiiList of AcronymsACT Atacama Cosmology TelescopeAGN Active Galactic NucleiBAO Baryonic Acoustic OscillationBBN Big Bang NucleosynthesisCMB Cosmic Microwave BackgroundCDM Cold Dark MatterDec DeclinationDR7 Data Release 7DR12 Data Release 12FOF Friends-Of-FriendsFOV Field Of ViewFRW Friedmann-Robertson-WalkerFWHM Full Width Half MaximumHMF Halo Mass FunctionIC Inverse ComptonICM Intra Cluster MediumIGM Inter Galactic MediumILC Internal Linear CombinationISM Inter Stellar MediumLRG Luminous Red GalaxiesLSS Last Scattering SurfacePSF Point Spread FunctionRA Right AscensionSDSS Sloan Digital Sky SurveySHM Stellar-to-Halo Mass (relation)SMBH Super Massive Black HoleSNe SuperNovaeSPT South Pole TelescopeSZ Sunyaev-Zel’dovich (effect)UPP Universal Pressure ProfileWHIM Warm Hot Intergalactic MediumxivList of AcronymsWMAP Wilkinson Microwave Anisotropy ProbeΛCDM Lambda Cold Dark MatterxvAcknowledgementsThe research topic was completely new and challenging for me and in ad-dition, this is my first experience to study abroad. I would like to expressmy sincere gratitude to my supervisor, Professor Gary Hinshaw for provid-ing me with this precious research opportunity. I also would like to expressmy deepest appreciation to him for his elaborated guidance, considerableencouragement and invaluable discussion that helps me to understand thischallenging topic and leads to a fascinating discovery. I am also very gratefulto all the staff and students who helped my studies at UBC for their valuablecooperation in my studies, especially, Professor Ludovic Van Waerbeke whoinvited me his group meeting and gave me lots of advice.Finally, I would like to extend my indebtedness to my wife and son,Hitomi and Jirokichi, for their endless love, understanding, support, encour-agement and sacrifice throughout my study. (Also to my dog, Mametaro forhis cute face and behavior that always refreshed me so much.)xviChapter 1Introduction1.1 Modern cosmologyEinstein’s theory of General Relativity, developed almost 100 years ago, hasdramatically changed the view of our universe. The Einstein’s equationswere applied to the universe and it was noticed that the universe is dynamicalby Friedmann and LeMaˆıtre. At the time, it was believed that the universewas static, thus, a constant term called “cosmological constant” was addedto enforce a static universe. However, after the Hubble’s discovery, thenon-static universe was widely accepted and the cosmological constant wasdropped. The Hubble’s observation implied that the universe is expanding,and was presumably much hotter and denser in the past. The idea of anexpanding universe, called the Big Bang theory, is now supported by threeimportant observational results: Hubble’s observed expansion, light elementabundances implying primordial nucleosynthesis, and the cosmic microwavebackground (CMB) left over from 380,000 years after the creation of theuniverse, which are described in the next section.Once the cosmological constant was dropped, however, recent observa-tions strongly support a non-vanishing cosmological constant (Λ). The non-zero cosmological constant results in an acceleration of the expansion in theFriedman equations derived from the Einstein’s equations. The nature ofthis exotic component, called “dark energy”, has been studied to explain theacceleration of the universe discovered by measurements of distant type Iasupernovae. In addition, the existence of non-baryonic matter, called “(cold)dark matter” (CDM) is supported by observations such as rotation curve ofour galaxy, gravitational lensing effects, CMB and so on. Including thesemysterious dark components, currently, the standard model of cosmologyis called “ΛCDM model”. The model is discussed further in this chapter.The main historical events supporting modern cosmology are summarizedin Tab. 1.1.11.2. Expanding universe1916 General Relativity Einstein1922 The expanding universe solution to general relativity Friedmann1929 The discovery of Hubble’s law Hubble1948 The Big Bang theory and Primodial Nucleosynthesis Gamow, Alpher, Herman1965 The discovery of Cosmic Microwave Background Penzias and Wilson1981 The development of Inflationary theory Guth, Linde1992 The discovery of CMB anisotropy by COBE Smoot, et al.1998 The discovery of accelerating expansion of the universe Supernova Projects2003 Establishment of ΛCDM WMAP Team2013 Precise measurements of six cosmological parameters Planck TeamTable 1.1: History of Modern Cosmology (Dodelson, 2003).1.2 Expanding universeModern cosmology is constructed based on the assumption that the universeis homogeneous and isotropic and no special place does not exist, which iscalled ”Cosmological Principle”. It is an important basis that the laws ofphysics are universal. Homogeneity means that the Universe looks the sameanywhere, while isotropy states that the Universe looks the same in all di-rections. It seems contradictory to the fact that the distribution of planets,stars and galaxies in our nearby universe is highly inhomogeneous. However,the principle is consistent with the recent observations extended over a largescale. For isotropy, (i) radio galaxies are randomly distributed across theentire sky: NRAO VLA Sky Survey (NVSS) detected nearly 2×106 sourcesstronger than S = 2.5 mJy at 1.4 GHz. The distribution of the discretesources on the sky is extremely isotropic (Condon et al., 1998), (ii) distantgalaxies with the same distance(known by standard candles such as SNIa)are moving away from us with the same speed(redshift) in any direction ofthe sky, implying that the expansion of the universe is isotropic, (iii) thetemperature of the cosmic microwave background radiation is same in alldirections within a fractional precision better than 10−4, after subtractingthe contribution from the CMB dipole component due to the proper mo-tion of our solar system and also the contribution from point sources suchas clusters of galaxies causing the distortion of the CMB spectra. For ho-mogeneity, as a result of large galaxy surveys, it is found that their spatialdistribution consists of a tangled cosmic web structure up to 400 Mpc (1pc = 3.26 light year), but on scales larger than 400 Mpc, little structure is21.2. Expanding universefound. Therefore, the cosmological principle is currently supported by theobservational facts.1.2.1 Friedmann equationsFriedmann solved the Einstein’s field equations assuming the cosmologicalprinciple. A homogeneous and isotropic expanding universe is expressed bythe Friedmann-Robertson-Walker (FRW) metric as c = 1 (c is the speed oflight),ds2 = −dt2 + a2(t)[dr21−Kr2 + r2(dθ2 + sin2 θdφ2)], (1.1)where ds is a geodesics, dt is a time, a(t) is a time-dependent scale(expansion)factor, and K is a constant representing a curvature of the space (flat or pos-itive/negative curvature space) in polar coordinates (r, θ and φ). The Ein-stein’s field equations describe the relation between the space-time geometryand energy-momentum,Rµν − 12gµνR + Λgµν = 8piGTµν , (1.2)where Rµν is the Ricci tensor, which depends on the metric (gµν) and itsderivatives; R is the Ricci scalar (R ≡ gµνRµν); G is Newton’s constant;Λ is the cosmological constant; and Tµν is the energy-momentum tensor.By applying the FRW metric to the Einstein’s equations, the equations ofmotion called “Friedmann equations” can be derived, which determine theevolution of the scale factor,(a˙a)2=8piG3ρ− Ka2+Λ3(1.3)a¨a= −4piG3(ρ+ 3P ) +Λ3, (1.4)where ρ is the energy density and P is the pressure of an isotropic fluid. Inaddition, the energy-momentum conservation law in an expanding universe,Tµν;µ = 0, givesρ˙+ 3a˙a(ρ+ P ) = 0, (1.5)which can be also acquired from the above two equations.To solve for the scale factor as a function of cosmic time, an equation ofstate for the fluid, which is the relationship between the energy density and31.2. Expanding universepressure, is necessary. It takes the form of P = wρ, using a dimensionlessnumber w. Eq. 1.5 can be solved for a constant w asρ ∝ a−3(1+w). (1.6)The universe is filled with a mixture of different ingredients characterizedby different equations of state. The essential components in the universeare radiation, matter and dark energy (cosmological constant). Since eachcomponent reacts to the expansion differently, w should be also a functionof time. However, one has a period when one component is dominant overthe others, thus, the constant w would be valid for most of the cosmic time.The early universe was filled with radiation. In the radiation dominatedera, P = 13ρ, and the evolution of the scale factor is ρr ∝ a−4. The energydensity of radiation decreased quickly and it was overcome by the one ofmatter. In the matter dominated era, the pressure is much smaller than theenergy density, |P |  ρ, and P ' 0 gives ρm ∝ a−3, so that matter dilutesby following 1/V = 1/a3.The energy density of matter also decreased with time and now the uni-verse is dominated by dark energy (cosmological constant). Without thecosmological constant, Eq. 1.4 states that both the energy density and pres-sure cause a deceleration in the expanding universe. However, according toobservations, the total matter density is only one third of the critical den-sity (ρcr = 3H20/8piG: characteristic density to make the geometry of theUniverse flat) and a(t) is accelerating, a¨ > 0. Thus, the cosmological con-stant, which was once proposed by Einstein to achieve a stationary universe,has been revived to explain the acceleration of the universe. As in Eq. 1.4,the cosmological constant can cause an acceleration in the expansion of theuniverse. For the cosmological constant to be constant, w = −1 in Eq. 1.6.In this case, the scale factor grows exponentially. Note that the expansionof the universe accelerates for any equation of state of w < −1/3.Considering the density evolution of the various cosmic components,ρm(t) = ρm,0 a−3(t), (1.7)ρr(t) = ρr,0 a−4(t), (1.8)ρΛ(t) = ρΛ = Λ/8piG = const, (1.9)where ρr,0 is the radiation energy density today, ρm,0 is the matter energydensity today and ρΛ is the dark energy density, and the dimensionlessdensity parameters for matter, radiation, and dark energy can be defined asΩm =ρm,0ρcr; Ωr =ρr,0ρcr; ΩΛ =ρΛ,0ρcr. (1.10)41.2. Expanding universeThe density parameters influence the expansion of the universe. With thedensity parameters and ρ =∑ρi, the expansion equation (Hubble parame-ter) can be written asH2(t) = H20 [a(t)−4Ωr + a(t)−3Ωm + a(t)−2Ωk + ΩΛ], (1.11)for H(t0) = H0 and a(t0) = 1, including the curvature density parameterΩk = K/H20 .The radiation energy density today is very small and can be neglectedwhen compared to that of matter. However, since ρr evolves faster than ρm,the radiation and matter had the same energy density at an epoch,ρr(t)ρm(t)=ρr,0ρm,0 a(t)=ΩrΩm a(t)' 1. (1.12)From the radiation density today, aeq ∼ 4.2 × 10−5(Ωmh2)−1 is an epochwhen the radiation and matter had an equal value of energy density, andthe radiation energy density was dominant before the time. These equationsare the basis of the standard Big Bang cosmological model.1.2.2 The Hubble diagramThe expansion of the universe causes galaxies to move away from each other.Therefore, we should see all the galaxies receding from us. However, we cannot see the recession of distant galaxies directly. Otherwise, it is observedas a stretch of light wavelength, such as Doppler shift. This stretch factoris defined as redshift z:1 + z =λobsλemit, (1.13)where λemit is the wavelength of light at emission and λobs is the wavelengthof light at observation. Hubble found that distant galaxies are in fact re-ceding from us and, in addition, the velocity increases with distance. Thisis exactly expected result in a homogeneously expanding universe, which iscalled Hubble’s law:v = H0 d, (1.14)where H0 is called “Hubble constant”, v is the receding velocity of an objectfrom us and d is the distance to the object.The redshift is actually caused by the expansion of the universe. Theequation for a light wave in the isotropic universe is given from Eq. 1.1,ds2 = −dt2 + a2(t)dr21−Kr2 , (1.15)51.2. Expanding universeand ds = 0 provides a straight line in a curved space-time for the light wave.Assuming a flat space (K = 0), the total path is given by the integral inboth space and time, ∫dta(t)=∫dr. (1.16)Suppose a crest of the light wave was emitted at t = t1 in the past andthe subsequent crest was emitted at t = t1 + δt1, and for an observer, thefirst crest was observed at t = t0 and the subsequent crest was observedat t = t0 + δt0. Since the both crests travel through the same (comoving)distance, ∫ t0t1dta(t)=∫ t0+δt0t1+δt1dta(t). (1.17)Note that the comoving distance is a distance that does not change in timedue to the expansion and the physical(proper) distance can be obtained bymultiplying it by the scale factor, a(t). Over the period of the light wave,the scale factor is essentially constant. This yieldsδt1a(t1)=δt0a(t0). (1.18)The wavelength at the source and observer are given by λ1 = c δt1 andλ0 = c δt0. Therefore, using Eq. 1.13 and a(t0) = 1 (normalization of thescale factor), the redshift is obtained by1 + z =λ0λ1=1a(t1). (1.19)It is called “cosmological redshift”, which means that the wavelength emittedat the time with the scale factor, a(t), is stretched by 1/a(t) (a(t) ≤1) dueto the expansion.The relation between the distance and redshift of distant objects is shownin the Hubble diagram. Redshifts are obtained by observations directly,but distances are difficult. One of the most useful techniques to measuredistances is find a object with the same luminosity, called standard candle.Currently Type Ia supernovae (SNIa) is used to extend the Hubble diagramout to very large redshifts, z ∼ 1.7. Fig. 1.1 is the recent Hubble diagramof a large combined sample of galaxies using SNIa as standard candles fordistance measurement (Betoule et al., 2014). Using the dataset, they findΩm = 0.295 ± 0.034 for a flat ΛCDM cosmology. The result is consistentwith the CMB measurements from the WMAP and Planck experiments.The best-fit ΛCDM cosmology for a fixed H0 = 70 km s−1 Mpc−1 is shown61.2. Expanding universein black line. The Hubble diagram is still the most direct evidence of theexpanding universe.Figure 1.1: Top: A recent Hubble diagram of a large combined sample ofgalaxies using SNIa as standard candles for distance measurement (Betouleet al., 2014) (arXiv:1401.4064). The graph presents distance (as distancemodulus; proportional to log of distance) vs. redshift z. The different SNIasamples are denoted by different colors [low-z sample; Sloan SDSS sample;SN legacy survey, SNLS; and Hubble Space Telescope SNIa, HST]. Thedistance modulus redshift relation of the best-fit ΛCDM cosmology for afixed H0 = 70 km s−1 Mpc−1 is shown in black line. Bottom: residuals fromthe best-fit ΛCDM cosmology as a function of redshift.1.2.3 Big Bang NucleosynthesisIn the middle of 20th century, it was considered that all the elements heavierthan hydrogen were formed in stellar interiors by nuclear fusions or by su-pernova explosions. However, according to observations, it was found that∼25 % of (baryonic) matter in the universe comprises of helium in mass andit was much larger than the prediction of the stellar theory. A similar mys-tery was present also for deuterium. Even worse, it is destroyed inside starsrather than produced. To solve for the problem, George Gamow suggested a71.2. Expanding universetheory that the light elements should have been produced in the very earlyuniverse, which is now called Big Bang Nucleosynthesis (BBN).The Big Bang model implies that the universe was much hotter anddenser, and most of baryons existed as protons and neutrons at the temper-ature of ≥ 109 [K]. At such a high temperature, atomic nuclei are quicklydestroyed by high energy photons. Along with the expansion of the universe,the temperature cooled down less than binding energies of typical atomicnucleus and then nucleosynthesis began. In the nucleosynthesis, deuteriumformed first, and subsequently, light elements such as helium and lithiumfollowed. However, the atomic nuclei heavier than lithium were not pro-duced because stable elements do not exist at the atomic number of 5 and8, which would have been necessary to produce further heavier elements.The production of atomic nuclei stopped when the universe cooled down tothe temperature of ∼ 4 × 108 [K]. Through the process, most of neutronswere captured in the atomic nuclei and the leftover decayed into protons.The BBN theory predicts the amount of the light elements produced in theearly universe and it is consistent with observations. Thus, the abundanceof light elements is the important basis to support the Big Bang model. Nowit is recognized that light elements (deuterium, helium, and lithium) weremainly produced in the very early universe after the Big Bang, while heavyelements were produced inside of stars much later than that.Fig. 1.2 is the prediction from BBN for the light element abundances,which depend on the amount of baryons at the time of BBN. Therefore,observations of the light elements can constrain the baryon density in theuniverse. In particular, the primordial deuterium has a high sensitivity tothe baryon density and the result is consistent with the outcome from theCMB measurements, though the primordial 7Li abundance derived from ob-servations is lower than the expected. The derived baryon density indicatesthat the baryon contribution is only 5 % of the critical density. However,the total matter density is estimated to be ∼30 %. This result supports theexistence of non-baryonic matter, called “dark matter”.1.2.4 Cosmic microwave backgroundThe discovery of the cosmic microwave background (CMB) strongly sup-ports the Big Bang theory. In the Big Bang model, the early universe wasfilled with a hot and dense plasma consisting of photons, electrons, baryonsand dark matter. The expansion of the universe caused an adiabatic cool-ing of the plasma until the temperature allowing electrons to combine withprotons and form neutral hydrogen atoms stably. This event is called “re-81.2. Expanding universeFigure 1.2: Constraints on the baryon density from Big Bang Nucleosynthe-sis (Fields et al., 2014) (arXiv:1412.1408). The abundances of 4He, D, 3Heand 7Li as predicted by the standard model of the BBN. The bands showthe 95% CL range. Boxes indicate the observed light element abundances.The narrow vertical band indicates the CMB measure of the cosmic baryondensity by WMAP, while the wider band indicates the BBN concordancerange (both at 95% CL).91.2. Expanding universecombination”. The recombination happened approximately 380,000 yearsafter the Big Bang when the temperature went down to around 3,000 K.After the recombination, the universe became neutral and also transparentfor photons since most of free electrons were captured in hydrogen atoms.This resulted in “decoupling” of matter and radiation, and the radiation be-gan to travel freely through space. This ancient radiation is called “cosmicmicrowave background radiation” and reaches us today isotropically from aspherical surface called “last scattering surface”.The Big Bang theory predicts a nearly blackbody spectrum of the CMBradiation with a lower temperature since it must have been emitted from avery optically thick plasma and the temperature should have been cooleddown due to the expansion of the universe. It also predicts its isotropy sincethe radiation source should be very far and the last scattering surface isreceding from us in the same way. The observations are completely con-sistent with these predictions and can not be explained by other sources.Currently the CMB radiation is measured very precisely (Fig. 1.3) and thetemperature is found to be 2.725 ± 0.002 (Fixsen et al., 1996). This is oneof the greatest successes of the Big Bang theory.Figure 1.3: CMB Spectrum measured by COBE/FIRAS (Fixsen et al.,1996). The FIRAS instrument aboard COBE measured the spectrum ofthe CMB precisely, and found a blackbody spectrum with deviations lim-ited to 50 ppm of the peak brightness, with a peak wavelength of 1.869 mm,corresponding to a temperature of T = 2.725 ± 0.002 K.101.3. InflationThe Big Bang theory also predicts a temperature anisotropy in the CMB.The temperature anisotropy was first detected by COBE (Smoot et al.,1992). It is believed to be caused by inhomogeneities in the matter distri-bution at the epoch of recombination. The discovery provides an evidencethat density fluctuations existed since a very early universe, perhaps causedby quantum fluctuations in the scalar field of inflation or by some othermechanisms. It is now widely accepted that the primordial density fluctua-tions grew with a gravitational collapse and seeded the formation of currentcosmic structure such as galaxies and clusters of galaxies.The CMB anisotropies have been measured more precisely by WMAP(Hinshaw et al., 2013) and Planck (Planck Collaboration, 2016c) (Fig. 1.4).In particular, the Planck mission, with high sensitivity and small angularresolution, measured the CMB anisotropies up to smaller angular scales andseveral acoustic peaks were detected in the power spectrum of the CMBtemperature fluctuations (Fig. 1.5). Note that the power spectrum charac-terizes the amplitude of fluctuations as a function of angular scale, l = pi/θ.The acoustic oscillations arise from a balance in the photon-baryonplasma. The pressure of photons erase the anisotropies, whereas the gravi-tational attraction of baryon and dark matter raises the anisotropies. Thesetwo effects compete and create acoustic oscillations in the CMB. The CMBphotons decouple when a particular spatial wavelength is at its peak ampli-tude.The ΛCDM theory predicts the acoustic peaks, depending on cosmolog-ical parameters. The angular scale of the first peak probes the curvatureof the universe and the WMAP and Planck results demonstrate that thegeometry of the Universe is flat, rather than curved. The ratio of the oddpeaks to the even peaks determines the baryon density. The third peakcan be used to get information about the dark matter density. The best-fitΛCDM parameters are summarized in Tab. 1.2.1.3 InflationThe Big Bang model explains many observations such as the Hubble dia-gram, the abundance of the light elements, and CMB. However, this modelraises new questions.111.3. InflationFigure 1.4: CMB temperature fluctuation map by Planck satellite mission(Planck Collaboration, 2016a) (arXiv:1502.01582).Figure 1.5: Power spectrum of CMB temperature fluctuations from Plancksatellite mission (Planck Collaboration, 2016a) (arXiv:1502.01582).121.3. InflationLabel Definition ValueΩb Baryon fraction 0.0486± 0.0010ΩCDM Dark matter fraction 0.2589± 0.0057ΩΛ Cosmological constant 0.6911± 0.0062τ Optical depth 0.066± 0.012ns Scalar index 0.9667± 0.0040109∆2R Scalar amplitude 2.441+0.088−0.092H0 (km/s/Mpc) Hubble constant 67.74± 0.46t0 (Gyr) Age of universe 13.799± 0.021zeq Radiation to matter time 3371± 23zreion Reionization period 8.5+1.0−1.1Table 1.2: ΛCDM cosmological parameters from Planck CMB power spec-tra, in combination with CMB lensing reconstruction and external data in-cluding BAO(6dFGRS, SDSS, BOSS and WiggleZ), SNe(JLA) and H0(HSTCepheid+SNe) (Planck Collaboration, 2016c).1.3.1 Horizon problemIn the theory of relativity, the speed of light is finite, meaning that we canonly observe a limited part of the entire universe, which is called “hori-zon”. The horizon also existed at an earlier time. The horizon scale at therecombination (z ∼ 1, 100) can be calculated as follows.In a time interval dt, light can travel a distance of c dt. In a comovingdistance, it corresponds to dχ = c dt/a(t). Therefore, the horizon at a timet from the Big Bang isχhor =∫ t0c dta(t), (1.20)in comoving distance. This can be expressed with a scale parameter, a(t),using dt = da/a˙ = da/(aH),χhor =∫ (1+z)−10c daa2H(a), (1.21)and H(a) can be expressed with the density parameters,H2(a) = H20 [a−4Ωr + a−3Ωm + a−2Ωk + ΩΛ]. (1.22)Since zeq(∼ 3, 400)  zrec(∼ 1, 100) at the recombination and only a smallfraction of time is in the radiation dominated era, it would be valid that131.3. Inflationmost of the time have been in the matter dominated era, which leads toH(a) ≈ H0√Ωma−3/2. Substituting it to Eq. 1.21 givesχhor ≈ 2 cH01√(1 + z)Ωm, (1.23)where zeq ∼ 3400 is the redshift(epoch) when radiation and matter had anequal value of energy density. Since the physical distance can be calculatedbyxhor ≈ 2 cH0Ω−1/2m (1 + z)−3/2, (1.24)the angular size of the horizon on the sky in case of ΩΛ = 0 can be simplygiven byθhor,rec =xhor(zrec)DA(zrec)≈√Ωmzrec∼ 1◦, (1.25)where DA is an angular diameter distance. It means that CMB radiationsseparated by more than about one degree were not causally connected beforethe recombination. However, due to the observations, the CMB tempera-tures are same from all the direction of the sky within only fluctuations of∆T/T ∼ 10−5, which is called “Horizon problem”.The second problem is called “Flatness problem”. The WMAP andPlanck observations suggest that the geometry of our universe is nearlyflat, which means the energy density of the universe is nearly equal to thecritical density and our universe is at a peculiar point between an eternalexpansion and eventual collapse. Since the curvature of spacetime wouldgrow with time, the probability of such a situation is very unlikely and itwould requires an extreme fine-tuning in the past. However, the (traditional)Big Bang theory can not provide a solution for it.1.3.2 Solution to the horizon problemA. Guth came up with a solution for these questions (Guth, 1981). Hepostulates that our universe experienced a very rapid accelerated expansion,called “Inflation”, in a very early stage. It can be realized if the universe isdominated by a potential energy of slowly rolling scalar field.In the Friedmann equation, the slow rolling scalar filed takes the form ofH2 =8piG3ρ =8piG3(12φ˙2 + V (φ))' 8piG3V (φ), (1.26)where φ˙2/2 is the kinetic and V (φ) is the potential term of the scalar field,and the potential term is dominant over the kinetic term. For a constant141.3. Inflationφ, 8piGV (φ)/3 = H2 = const and the solution is simply a(t) ∝ exp(Ht),resulting in an exponential expansion.How much inflation is required to solve the horizon problem? For infla-tion to be valid, so-called “comoving Hubble radius” at the start of inflationhad to be larger than the current comoving Hubble radius. Note that thecomoving Hubble radius is the radius of the observable universe in comvingdistance unit, which is 1/aH. Inflation causes the decrease of comovingHubble radius due to the increase of the scale factor.Most inflationary models typically assume the energy scale at inflationto be order 1015 GeV or larger, thus,a0H0aeHe' T01015GeV' 10−28, (1.27)where ae and He are the scale factor and Hubble constant at the end ofinflation, and T0 is the CMB temperature today. Thus, inflation can solvethe horizon problem if the universe expands exponentially for more than 60e-folds (ln(1028) ∼ 64).Inflation can also solve the flatness problem. Inflation and its subsequentexpansion has essentially flattened the curvature of the universe, just as theworld appears flat for a observer on the surface of the earth.1.3.3 Primordial power spectrumInflation also explains the origin of structure in the universe. Before theinflation, the size of the universe observed today was microscopic and thequantum fluctuations in the microscopic scales expanded to astronomicalscales during the inflation. The quantum fluctuations grow over a longperiod and build up a large-scale structure today such as stars, galaxies,and clusters of galaxies.The slow-roll inflation produces a spectrum of curvature perturbations(PR) that is almost scale-invariant since no characteristic length-scale ex-isted then. The small deviation from the scale-invariance can be quantifiedby forming a spectral index ns(k) as a function of wavenumber, k = 2pi/L,ns − 1 ≡ d lnPRd ln k, (1.28)where ns = 1 is a scale-free spectrum, called Harrison-Zel’dovich spectrum.A constant ns is a power-law spectrum,PR(k) = As(k∗)(kk∗)ns−1, (1.29)151.4. Structure formationwhere k∗ is a pivot scale.The Planck observation finds ns = 0.9603 ± 0.0073 with a significanceof more than 5 standard deviations (Planck Collaboration, 2016a) and itis considered as a great success of the inflation theory. This scenario alsoexplains several important observations about flatness, isotropy and homo-geneity of the universe. Thus, the inflation model is the best theory toexplain the earliest moment of the universe so far.1.4 Structure formation1.4.1 Linear structure evolutionThe formation and evolution of the large-scale structure induced by inhomo-geneities. In the early universe, the inhomogeneities were subtle. The smallinhomogeneities can be treated in a perturbation theory. In addition, New-tonian gravity is an adequate description for non-relativistic matter such ascold dark matter and baryons after recombination (Schneider, 2006). Con-sider a non-relativistic fluid in an element of mass density ρ, pressure P  ρand velocity u with a position r and time t. For the fluid, the equations ofmotion are given by the continuity equation and Euler equation.∂tρ = −∇r · (ρu), (1.30)∂t + u · ∇r = −∇rPρ−∇rΦ, (1.31)where the gravitational potential, Φ, is determined by the Poisson equation,∇2rΦ = 4piGρ. (1.32)In the expanding universe, physical coordinates, r, can be replaced withcomoving coordinates, x by r(t) = a(t)x. The velocity field is also replacedby u(t) = r˙ = Hr + v where Hr is the Hubble flow and v is the propervelocity.Introducing the density perturbation ofδ ≡ ρ− ρ¯ρ¯(1.33)in the continuity equation, Euler equation and Poisson equation, the firstorder equations result inδ˙ = −1a∇ · v, (1.34)161.4. Structure formationv˙ +Hv = − 1aρ¯∇δP − 1a∇δΦ, (1.35)∇2δΦ = 4piGa2ρ¯δ, (1.36)where the overdot is the derivative with respect to time. Combining theseequations findsδ¨ + 2Hδ˙ − c2sa2∇2δ = 4piGρ¯δ, (1.37)where δP = c2sδρ and cs is the sound speed of the fluid. These equations arevalid in the linear regime (δ  1).For a static space (H=0), the solution is δ ∝ ei(wt−k·r) with w2 = c2sk2−4piGρ¯. It has a critical wavenumber at kJ ≡ 4piGρ¯cs called “Jeans’ length”,where the frequency of the oscillation becomes zero. On small scales (k <kJ), the pressure dominates and the density fluctuations oscillate. On largescales (k > kJ), the gravity dominates and the density fluctuations growexponentially.In an expanding universe (H 6= 0), the equation includes a friction term,2Hδ˙, and the growth of density fluctuations depends on time via ρ(t) andcs(t). Below the Jeans length, the density fluctuations oscillate with decreas-ing amplitude (decaying mode) and, above it, the density fluctuations growwith a power-law (growing mode). The decaying mode does not contributeto the structure formation, and only the growing mode does. The solutionof the growing mode isδ(x, t) = D+(t)δ(x), (1.38)whereD+(t) ≈ 52a(t) Ωm[Ω4/7m − ΩΛ +(1 +Ωm2)(1 +ΩΛ70)]−1. (1.39)The approximation in the equation is good to a few percent for plausibleΩm and ΩΛ (Lahav et al., 1991) (Carroll et al., 1992).To describe the inhomogeneous universe quantitatively, the power spec-trum P (k) can be used:δ(x) =∫d3k(2pi)3δ(k) eik·x, (1.40)〈δ(k)δ(k′)〉 = (2pi)3δ3D(k − k′)P (k), (1.41)where δD denotes a delta function and 〈〉 does an ensemble average for allthe pairs of k and k′. Note that the power spectrum P (k) describes the171.4. Structure formationlevel of structure as a function of the length-scale k = 2pi/L, where k is acomoving wavenumber. Using the power spectrum, the density fluctuationscan be described byP (k, t) = D2+(t)P0(k), (1.42)where P0(k) is a primordial power spectrum.At early times, no natural length-scale existed in the universe as de-scribed in §1.3.3, hence, one should expect the primordial power spectrumto be a power-law form ofP0(k) ∝ kns . (1.43)Using the primordial power spectrum, the power spectrum after the infla-tionary epoch can be written asP (k) = A knsT 2(k), (1.44)where T (k) is called “transfer function”. The transfer function provides theevolution of density perturbations through the epochs of horizon crossingand radiation/matter transition. The evolution of density perturbationsfreezes out after horizon crossing and remains approximately constant. Afterthe end of inflation, the Hubble horizon grows and the macroscopic densityperturbation from inflation re-enters the horizon from modes of small scales.However, if the density perturbation enters the horizon in the radiation-dominated era, the density fluctuation cannot grow due to the radiationpressure. On the other hand, if the density perturbation enters the horizonin the matter-dominated era, the density perturbations grow as δ ∝ D+(t).According to the quantitative consideration of these effects, the transferfunction can be computed numerically, or analytically for two limiting cases,T (k) ∝ 1 (k  keq) (1.45)∝ k−2 (k  keq). (1.46)Therefore, the power spectrum for the cold dark matter with the Harrison-Zel’dovich spectrum (ns = 1) is derived to beP (k) ∝ kT 2(k) ∝ k (k  keq) (1.47)∝ k−3 (k  keq). (1.48)Fig. 1.6 shows the observed linear matter power spectrum by the CMBobservation, extrapolated to z=0. The normalization of the power spectrum181.4. Structure formationcannot be determined from the theory but has to be determined by obser-vations. For the normalization factor, a variance in a smoothed density fieldin R = 8h−1Mpc is often used,σ28(R) =∫dkkk3P (k)2pi3|W (kR)|2, (1.49)where W (kR) is a window function and if it is a tophat in real space,W (kR) = [3/(kR)3](sin(kR) − (kR) cos(kR)). In the framework of ΛCDMmodel, σ8 can be predictable from the CMB observation at scale of k ' 10−3and currently constrained to be σ8 = 0.8159± 0.0086 by the (Planck + ext)data.In addition, the matter power spectrum can be determined from thegalaxy distribution today. However, the formation and evolution of galaxiesare not understood sufficiently to predict the relation between galaxies anddark matter. The connection between them is parametrized by so-calledlinear bias factor b,δg =∆nn¯= b∆ρρ¯= b δ, (1.50)where n¯ is the average number density of the galaxy population, and ∆n isthe deviation from the average. The linear relation is not strictly justifiedfrom theories, however, it is plausible on scales where the density field islinear.The best fit ΛCDM model is shown in Fig. 1.6. As one can see, the fit isquite good. The fact that the CMB and LSS data agree over a substantialregion of overlap gives a confidence in the correctness of the concordancemodel.1.4.2 Non-linear structure evolutionThe linear perturbation theory is applicable on large scales, but the evolutionof structures like clusters of galaxies can not be treated within the frameworkof linear perturbation theory. Instead, the power spectrum, P (k), turnsnon-linear around k ∼ [0.1, 1.0]h/Mpc. Higher-order perturbation theorycan be used to follow slightly larger values of the density fluctuations, butthe achievements do not justify the large mathematical effort in general.Additionally, the fluid approximation is no longer valid if gravitationallybound systems form.For this reason, numerical simulations have become a main tool to predictthe structure formation. Since the matter distribution in the universe isdominated by dark matter, it is often sufficient to compute the behavior of191.4. Structure formationFigure 1.6: Linear matter power spectrum P (k) versus wavenumber extrapo-lated to z=0, from various measurements of cosmological structure (Tegmarket al., 2004) (arXiv:astro-ph/0310725). The best fit ΛCDM model is shownin solid line.the dark matter and thus to consider only gravitational interactions. Theresults of the simulations have contributed very substantially to establishthe standard model of cosmology.In recent years, the computational power has increased and baryons canbe treated with their hydrodynamic processes such as radiative transfer andother baryonic feedback processes. Using the hydrodynamical simulations,one can predict the distribution of both baryon and dark matter. Thesimulations can be compared directly with most of the observables and theinfluence of the heating and cooling of the baryonic effects can be examined,however, it is still challenging on cosmological scales.1.4.3 Evolution of baryonThe matter distribution is not directly observable since it is mostly darkmatter. What we see is mainly baryon through photon. However, it wouldhave grown in a different manner from dark matter. Before recombination,201.4. Structure formationFigure 1.7: Structure formation of the universe at z = 0 in a N-body simu-lation box 100 Mpc/h on side, simulated by GADGET-2 software (Springel,2005).baryons, electrons and photons existed as one plasma fluid. The densityperturbations in the baryon-electron-photon plasma did not grow due to thehigh pressure from photons. Instead, the perturbations caused sound wavespropagating in the plasma with time-independent amplitudes, which is called“Baryon Acoustic Oscillations” (BAO). Hence, the density perturbations ofbaryons only began to grow after recombination when baryons decoupledfrom photons.In the matter dominated universe, the gravitational instability growswith (δρ/ρ)(t) ∝ a(t). Hence, in a universe without dark matter, the growthfactor of baryon density perturbations would have been at most δρ/ρ ∼ 10−3since zrec ∼ 1, 100 (1 + z = 1/a). However, the amplitude of baryon densityperturbations at recombination is known to be δρ/ρ ∼ 10−5 from the CMBanisotropy measurements.After recombination, the universe became neutral and the baryonic mat-ter was no longer influenced by photons. It induced the baryons to behavelike dark matter, only subject to gravitational forces. However, since darkmatter decoupled from the plasma much earlier than baryons, the perturba-tions of the dark matter had begun to grow much earlier. The baryons fell211.4. Structure formationinto the potential wells already formed by the dark matter, then the pertur-bations of both dark matter and baryon have developed together soon afterrecombination. It induced the baryon perturbations to grow faster than ex-pected and galaxies formed in the regions where dark matter was overdenseoriginally.1.4.4 Clusters of galaxiesThe large-scale structure of the universe has been developed by the gravita-tional instability originated from the primordial density fluctuations. Smallsub-clumps form first, then undergo a merging process to form larger struc-tures, up to clusters of galaxies.The galaxy clusters trace fields of dark matter density fluctuations, andtheir number density as a function of mass and redshift, so-called “halomass function” (HMF) n(M, z), has a dependance on cosmological param-eters. Therefore, by comparing the theoretical one with observations, onecan constrain Ωm especially, however, degenerate with σ8. The degeneracycan be solved by studying the redshift evolution of the mass function (e.g.,(Borgani, 2008)).In theory, the HMF can be described by the “Press-Schechter model”(Press and Schechter, 1974), under the assumption of initial Gaussian per-turbations. It provides the mass function ofdn(M, z)dM=√2piρ¯M2δcσM (z)∣∣∣∣dlnσM (z)dlnM∣∣∣∣ exp(− δ2c2σM (z)2), (1.51)where ρ¯ = M/VM , σM (z) is the variance at the mass scale M at redshift z,and δc is the critical overdensity given by δc ' 1.69 under the assumptionof spherical collapse in the Einstein-de Sitter universe. It assumes that thefraction of matter ending up in objects of a given mass M can be found bylooking at the portion of the initial density field, smoothed on the mass-scaleM , lying at an overdensity exceeding a given critical threshold value, δc.With the advent of N-body simulations such as Millennium simulation(Springel et al., 2005), significant deviations from the Press-Schechter modelhave been found as seen in Figure. 1.8. Thus, analytical descriptions havebeen improved along with simulations. More realistic ellipsoidal collapsemodel has been investigated, and in addition, universal analytic fitting func-tions are proposed. These advanced models are found to be in very goodagreement with numerical simulations as demonstrated in Figure. 1.8, pro-viding a good description of n(M, z).221.5. Sunyaev-Zel’dovich effectHowever, the accuracy of the cosmological parameter measurements iscurrently limited by uncertainties to trace the masses, in the relation be-tween the cluster masses and observables such as luminosity or temperature.In simulations, the mass and radius of galaxy clusters are estimated witha spherical overdensity (SO) algorithm. The virial radius is defined suchthat within a sphere of radius rvir, the average mass density of the clusteris about 200 times the critical density ρcr of the universe. The mass withinrvir is called the virial mass Mvir (' M200). However, it is still challengingfor the “real” galaxy clusters.Figure 1.8: The mass function of dark matter halos (pants with errorbars)identified at different redshifts in the Millenniun simulation (Springel et al.,2005) (arXiv:astro-ph/0504097). Solid lines are predictions from an analyticfitting function. The dashed lines give the Press-Schechter model.1.5 Sunyaev-Zel’dovich effect1.5.1 Kompaneetz equationAfter recombination, CMB photons free-stream, but they are influenced byfree electrons in the middle of the path, if baryons become ionized in theirevolution. For example, through X-ray observations, it is found that galaxy231.5. Sunyaev-Zel’dovich effectclusters are filled with large clouds of hot ionized gas. It changes the phasespace density of the CMB photons, n(ν). The evolution of the photon phasespace density n(ν, t) in the presence of electron gas can be described by theBoltzmann equation,∂n(ν, t)∂t=∫d3p∫dΩdσdΩ[fe(p1)n(ν1, t)(1+n(ν, t))−fe(p)n(ν, t)(1+n(ν1, t))],(1.52)where fe(p) is an electron phase space density with momentum p, σ is a crosssection and Ω is a solid angle. This equation describes an energy transfer viascattering events p + ν ↔ p1 + ν1. The first term describes the populationof ν state by incoming photons with ν1, while the second term describes thede-population. (1 + n) factors are due to the stimulated absorption.For thermal non-relativistic electrons,fe(E) = ne(2pimeTe)−3/2e−E/Te with E =p22m, (1.53)where ne is the number density of electrons, me is the mass of electron andTe is the temperature of electrons. Assuming the energy transfer is smallcompared to the electron kinetic energy, it holds∆ ≡ ν1 − νTe 1. (1.54)The energy conservation holds E1 = E − Te∆, thereforefe(E1) ≈(1 + ∆ +∆22)fe(E). (1.55)For photons scattered by electrons,n(ν1) ≈ n(ν) + Te∆∂νn(ν) + 12(Te∆)2∂2νn(ν). (1.56)By definingx˜ =νTe, (1.57)so that Te∂νn = ∂x˜n ≡ n′, the Boltzmann equation can be simplified to∂yn(x˜, y) = x˜−2∂x˜[x˜4(n′ + n+ n2)] (1.58)with time replaced by so-called “Compton y parameter” (y =∫dtneσTTeme ).This equation is called “Kompaneetz equation” (Kompaneetz, 1956).241.5. Sunyaev-Zel’dovich effect1.5.2 Sunyaev-Zeldovich effectThe Sunyaev-Zeldovich (SZ) effect is inverse Compton scattering of the CMBphotons off electrons in clusters of galaxies or any cosmic structure withunbound electrons. X-ray observations show that galaxy clusters are filledwith large clouds of hot gas with the temperature of Te ∼ 107−108 [K]. Thehigh energy electrons in the hot gas inject energy into the CMB photonsthrough the inverse Compton scattering and the photons are pumped tooccupy higher energy states.Figure 1.9: The CMB spectrum (dashed line) and the distorted spectrumby the SZ effect (solid line) (Carlstrom et al., 2002).The Kompaneetz equation (Eq. 1.58) provides the distortion of the CMBspectrum due to the SZ effect. Since the CMB spectrum is initially Planckfunction with temperature TCMB  Te, the term proportional to the radia-tion phase space density of n(n+ 1) can be neglected,∂yn(x, y) = x−2∂x[x4∂xn(x, y)], (1.59)where x is the dimensionless photon frequency of x = hpν/kBTCMB, hp isthe Planck constant and kB is the Boltzmann constant. (Note that x˜ =hpνkBTeis replaced with x =hpνkBTCMB. ) Substituting the Planck function of n =251.5. Sunyaev-Zel’dovich effect(ex − 1)−1 gives∂n∂y=xex(ex − 1)2(x cothx2− 4), (1.60)and∆nn0=∆IνIν,CMB=xexex − 1(x cothx2− 4)y, (1.61)or equivalently,∆TCMBTCMB=(x cothx2− 4)y, (1.62)in thermodynamic temperature unit. This is called (thermal) “Sunyaev-Zeldovich” (tSZ) effect (Sunyaev, 1980; Sunyaev and Zeldovich, 1970, 1972;Zeldovich and Sunyaev, 1969) and the amplitude of the distortion is definedby the Compton y parameter.It is amongst the major sources of secondary anisotropies of the CMBon sub-degree angular scales. When the CMB photons pass through a cloudof free electrons in equilibrium with number density ne, they are subject toscattering with a probability characterized by the optical depth,τe =∫σTnedl ∼ 2× 10−3( ne10−3cm−3)( lMpc), (1.63)where σT is the Thomson scattering constant andy =∫σTnekBTemec2dl ∼ 4× 10−5( ne10−3cm−3)( Te108K)(lMpc), (1.64)which is essentially the dimensionless electron pressure integrated along theline of sight and only a tiny effect even for a massive galaxy cluster.The thermal SZ effect is the spectral distortion in order of (ve/c)2 (ve iselectron velocity). To the lowest order in ve/c, a small change in TCMB iscaused by the Doppler effect, which is called kinetic SZ (kSZ) effect,∆TCMBTCMB=∫σTnev‖cdl ∼ 7× 10−6( ne10−3cm−3)( v‖103km s−1)( lMpc),(1.65)where v‖ is the line-of-sight component of ve. Note that random velocitiescancel out, so the coherent motion with respect to the CMB is responsiblefor the kSZ effect. In the non-relativistic limit, the spectral shape of thekinetic SZE is still described by a Planck spectrum, but at a slightly differenttemperature, lower (higher) for positive (negative) peculiar velocities. Eventhough the tSZ effect is in order of (ve/c)2, it is dominant over the kSZ261.5. Sunyaev-Zel’dovich effecteffect, typically by one order of magnitude for galaxy clusters, because thethermal velocity of electrons is much larger than the bulk motions of . 103km s−1.In the non-relativistic limit, the spectral shapes of the kSZ and tSZ ef-fects depend only on the frequency. Corrections due to higher-order termsare non-negligible once electrons become relativistic (e.g., Challinor andLasenby (1998); Itoh et al. (1998); Nozawa et al. (1998); Sazonov and Sun-yaev (1998)). In the relativistic case, the spectral shape of the tSZ effectbegins to depend on Te and that of the kSZ effect on both Te and the bulkvelocity. However, for a massive cluster with kBTe ∼ 10 keV (M ∼ 1015M),the relativistic correction to the SZ effect is of order a few percent in theRayleigh-Jeans (RJ) portion of the spectrum. In any case, the observedamplitude and spectral shape of the SZ effect are both independent of z, be-cause T and Iν are redshifted in exactly the same way as TCMB and Iν,CMB.The characteristic spectral shape helps to separate it from other compo-nents such as emissions from radio galaxies and primary CMB anisotropies.Recent developments of wide-field surveys by the South Pole Telescope(SPT) (Reichardt et al., 2013; Staniszewski et al., 2009; Vanderlinde et al.,2010; Williamson et al., 2011), the Atacama Cosmology Telescope (ACT)(Hasselfield et al., 2013; Hincks et al., 2010; Marriage et al., 2011), andthe Planck satellite (Planck Collaboration, 2014b,c, 2016f) have boosted thenumber of galaxy clusters observed through the SZ effect by more than anorder of magnitude over the last decade.Especially, the Planck team constructed all-sky y-maps 1 (Planck Col-laboration, 2016d) as one of the dataset from the Planck 2015 data releaseand provided in HEALpix 2 format with a resolution of Nside = 2048. Twotypes of y-maps are publicly available: MILCA and NILC y-map, both ofwhich are obtained by the Internal Linear Combination (ILC) approach, butdifferent algorithm (Fig. 1.12).The y maps can be used for a variety of cosmological and astrophysicalpurposes. The Planck team have performed a blind search for objects andidentified about 1,600 galaxy clusters such as Fig. 1.11 (Planck Collabora-tion, 2011). They also shows that the sensitivity of the y map is sufficient todetect faint and diffuse structures such as bridges between merging clusters.Moreover, they have proved via a stacking analysis that even low signal-to-noise regions in the y map preserve the tSZ signal for small galaxy groups(with tens of galaxies) (Planck Collaboration, 2016d).1Planck y-maps, http://pla.esac.esa.int/pla/#results2http://healpix.sourceforge.net/271.5. Sunyaev-Zel’dovich effectFigure 1.10: Spectral distortion of the CMB radiation due to the Sunyaev-Zel’dovich effect. The thick solid line is the thermal SZE and the dashedline is the kinetic SZE. For reference the 2.7 K thermal spectrum for theCMB intensity scaled by 0.0005 is shown by the dotted line. The clusterproperties used to calculate the spectra are an electron temperature of 10keV, a Compton y parameter of 10−4, and a peculiar velocity of 500 km s−1(Carlstrom et al., 2002).Figure 1.11: Cleaned images of Abell 2256 at z = 0.058 observed by Planckat 100, 143, 217, 353, and 545 GHz (left to right) over a size of 1 squaredegree. Blue, green, and red colors indicate negative, null, and positiveintensities with respect to the CMB, respectively (Planck Collaboration,2011) (arXiv:1101.2024).281.5. Sunyaev-Zel’dovich effectFigure 1.12: Planck all-sky Compton parameter maps for NILC (top) andMILCA (bottom) in orthographic projections (Planck Collaboration, 2016d)(arXiv:1502.01596).29Chapter 2Construction of Comptonparameter y map2.1 IntroductionIn this chapter, all-sky Compton parameter (y) maps are constructed fromthe individual Planck frequency maps. The Compton y parameter is theamplitude of the thermal Sunyaev-Zeldovich (tSZ) effect, produced by theinverse Compton scattering of cosmic microwave background (CMB) pho-tons by high energy electrons along the line of sight. It provides a directmeasurement of thermal pressure due to electrons, allowing to study thebaryonic (gaseous) physics of clusters of galaxies as well as structure for-mation in the universe. The Planck satellite mission has made sensitivemeasurements at 9 frequency bands with all-sky coverage, which provides aunique data set to produce all-sky Compton parameter (y) maps.“Internal Linear Combination” (ILC) technique, described in the nextsection, is applied to make y maps that extract only the thermal SZ signaland remove other emissions. Amongst the various emissions, the y mapis primarily focusing on removing dust emission. The dust emission is themost dominant emissions in the band maps, thus, its contamination in they map is carefully checked by using a variety of dust models with varyingspectral index.2.2 Construction of y map with internal linearcombination techniqueThe ILC method (e.g., (Eriksen et al., 2004; Remazeilles et al., 2011)) as-sumes little about the properties of the data. It simply assumes that ob-served temperature maps T (ν) at frequency ν can be written in a combi-nation of different components. The ILC provides coefficients to extracta component of interest by forming the linear combination of the observedmaps, so that it has a unit response to the target based on “known” spectral302.2. Construction of y map with internal linear combination techniqueshape from theories or observations.In the frequency range of Planck observations, the dominant emissioncomponents are the CMB, thermal Sunyaev-Zel’dovich distortion and ther-mal dust emission from interstellar dust grains in our galaxy and from dustyexternal galaxies including the Cosmic Infrared Background (CIB). There-fore, the band maps can be expanded toT (ν) = SSZ(ν)Tν′SZ + SCMB(ν)Tν′CMB + Sdust(ν)Tν′dust, (2.1)where Si is the spectral shape of i component and ν′ is a pivot frequency,which can be different for each component. Here, we assume that the extra-galactic dust emission consist of the same spectral shape with the galacticdust emission.The spectra of the CMB and tSZ are well known. The CMB has aconstant frequency spectrum in thermodynamic unit and the tSZ distortionshave a frequency dependence of ∆TCMB(ν)/TCMB = y SSZ (x), where TCMB= 2.725 K is the monopole temperature, y is the Compton parameter alongthe line-of-sight, and SSZ(x) = x coth(x/2) − 4, with x ≡ hν/kBTCMB, asdescribed in §1.5.However, the spectra of the dust emission is less known. It varies frompixel to pixel in the maps. The Planck team finds that it spans a range ofspectra in the vicinity of the nominal spectra index of βd ≈ 1.6 based onthe greybody spectrum (sd ∝ νβd B(ν)) where B(ν) is the blackbody spec-trum (Planck Collaboration, 2016g). Here, we simply assume that the dustspectra is a power-law with a constant index at all the pixels (sd ∝ νβd).Though the power-law model would be valid for given frequencies we use,thus, the y maps we construct will inevitably have residual contaminationfrom the dust emission and the level of contamination varies among the dif-ferent estimates we form. We check the contamination by assigning differentspectral indices for the dust emission. Given the band coefficients used inour estimates, we can predict the fractional foreground residual as a functionof the “true” spectral index, as discussed below. In practice, we find a veryweak dependence on the constructed y maps from varying βd.We construct y map by forming linear combinations of four lowest-frequency HFI maps: 100 GHz, 143 GHz, 217 GHz, and 353 GHz. Theband maps are provided in thermodynamic CMB temperature unit from thePlanck release 2 products. We smooth the band maps to a common Gaus-sian FWHM beam of 10 arcmin since the beams at different band maps aredifferent. The band coefficients are chosen to extract the tSZ distortion, butto project out the CMB and thermal dust emission.312.2. Construction of y map with internal linear combination techniqueWe form a given y map, y, from a linear combination of the band mapswith the common beam,y =∑νbνT (ν)/TCMB, (2.2)where the sum is over frequency band, T (ν) is the map at frequency ν, and bνis the coefficients for y map. The coefficients for various βd values are givenin Table. 2.1 and the resulting maps are shown in Figure. 2.1. In all cases,these coefficients satisfy the following constraints, 1)∑ν bν SSZ(ν) = 1 toproduce a map in unit of the Compton y parameter y ; 2)∑ν bν SCMB(ν) = 0to null the primary CMB; 3)∑ν bν Sdust(= cν (ν/ν0)βd) = 0 to null thedust emission with a spectral index βd (the factor cν given in Table. 2.1converts an antenna temperature to thermodynamic temperature). One lastfreedom is used to minimize the noise in the y map, ∂σ2y/∂bν = 0, whereσ2y =∑b2νσ(ν)2 and σ(ν) is the mean noise of T (ν).The last column in Table. 2.1 gives an indication of how much dust signalmight remain in a given y map. The quoted value is the factor by whicha dust signal with true index βd = 2.0 would be suppressed (or amplified)relative to its 100 GHz amplitude, in the given y map. For example, in mapB where we assume βd = 1.4, a dust signal with βd = 2.0 would survive withan amplitude of -2 times it 100 GHz amplitude. Similar results are readilytabulated for other dust indices.Table 2.1: Band data for the Planck y maps (Van Waerbeke et al., 2014)Map βd b100 b143 b217 b353 r2.0A ... -0.1707 -0.1148 0.0085 0.2770 44.42B 1.4 -0.7089 -0.1372 0.9388 -0.0927 -1.99C 1.6 -0.6952 -0.1378 0.9169 -0.0839 -3.13D 1.8 -0.6826 -0.1385 0.8969 -0.0758 -0.95E 2.0 -0.6710 -0.1393 0.8787 -0.0684 0.00F 1.2 -0.7235 -0.1367 0.9624 -0.1022 -4.4G 1.0 -0.7389 -0.1364 0.9876 -0.1124 -5.67cν 1.288 1.657 3.003 13.012SSZ(x) -1.506 -1.037 -0.001 2.253322.2. Construction of y map with internal linear combination techniqueFigure 2.1: Maps of the Compton parameter, y, formed from linear com-binations of the Planck HFI maps, shown on a scale 0 < y < 1 × 10−4. The residual contamination from foreground signals, primarily thermaldust emission, varies widely among the different maps. In all cases, the pri-mary CMB fluctuations have been projected out by enforcing the condition∑ν bν · 1 = 0. Very top: A (no dust rejection), Top left: y map version B(reject βd = 1.4), top right: version C (reject βd = 1.6), middle left: versionD (reject βd = 1.8), middle right: version E (reject βd = 2.0), bottom left:version F (reject βd = 1.2), and bottom right: version G (reject βd = 1.0).332.3. Comparison with Planck y map2.3 Comparison with Planck y map2.3.1 Compton y parameter profile of Luminous redgalaxiesTo compare our y maps with the one constructed by the Planck team, weuse Compton y parameter profile of luminous red galaxies (LRGs), studiedin Chapter 3. The detail is described in Chapter 3, so we describe thesummary here.Luminous red galaxies (LRGs) are early-type and massive galaxies, con-sist mainly of old stars with little ongoing star formation, and typicallyreside in the centers of galaxy groups and clusters. We construct the meantSZ Compton y profile of ∼ 65,000 LRGs from the Sloan Digital Sky SurveyData Release 7 using the Planck y map. The y profile shows the pressureprofile in galaxy groups and clusters, and can be used to probe the baryoniceffects (effect of gas and plasma) inside them.In Figure. 2.2, the y profile of the LRGs using the Planck y map iscompared to the ones using our y maps assuming different dust spectralindices. The y profile with Planck y map is completely consistent with they profiles with our y maps. In addition, the y profiles using our different ymaps show little difference even for widely varying dust spectral indices. Itsuggests that the ILC technique works well to extract the tSZ signal and thecontamination due to the galactic dust emission should not be significant.2.3.2 Relation between integrated Compton y parameterand stellar massesAdditionally, to compare our y maps with the one constructed by the Planckteam, we follow the analysis in (Planck Collaboration, 2013c)(PCXI), butusing our y map (version D) instead, and compare the results.First, we select “locally brightest galaxies” (LBG) as central galaxies intheir dark matter halos using the same criterion used in Planck Collabora-tion (2013c) from the SDSS DR7 galaxy catalog (Abazajian et al., 2009).The SDSS DR7 galaxy catalog provides the positions, magnitudes, spectro-scopic redshifts and stellar masses in the New York University Value-AddedCatalog (NYU-VAGC) 3 (Blanton et al., 2005). The stellar masses are es-timated with the K-correct software 4 of Blanton and Roweis (2007) byfitting the five-band SDSS photometry to more than 400 spectral templates,3http://sdss.physics.nyu.edu/vagc/4http://howdy.physics.nyu.edu/index.php/Kcorrect342.3. Comparison with Planck y mapFigure 2.2: The average y profile of 63,398 LRGs using the Planck y map(blue) is compared with the y profiles using our y maps constructed withdifferent dust spectral index models. Green: y map version B (βd = 1.4),red: version C (βd = 1.6), yellow: version D (βd = 1.8) and cyan: versionE (βd = 2.0). The 1 σ statistical uncertainties are complemented for the yprofiles from the Planck and y(D) maps as their width.most of which are based on stellar evolution synthesis models of Bruzualand Charlot (2003) assuming the stellar initial mass function of Chabrier(2003).From the SDSS DR7 galaxy catalog, we reject a galaxy if a brightergalaxy in r-band resides within a tangential distance of 1.0 Mpc and withina radial velocity difference of |c∆z | < 1000 km s−1. we also use SDSS pho-tometry to eliminate all objects with a companion that is close and brightenough that it might violate the above criteria using the “photometric red-shift 2” catalogue (photoz2 (Cunha et al., 2009)) from the SDSS DR7 web-site to search for additional companions. We then eliminate any candidatewith a companion in this catalog of equal or brighter r-magnitude and pro-jected within 1.0 Mpc, unless the photometric redshift distribution of the“companion” is inconsistent with the spectroscopic redshift of the candidate(inconsistent means that the total probability for the companion to have aredshift equal to or less than that of the candidate is less than 0.1). Thisprocedure leaves us with a sample of 248,643 locally brightest galaxies.352.4. ConclusionFor each LBG, we extract from our y map, a circular aperture of angularradius θc ≡ 5 × θ500 and an annular aperture of inner radius θc and outerradius θc + FWHM, where FWHM is the Planck beam of 10 arcmin. Wedefine the observed signal to be the sum of all pixels inside the circularaperture minus the mean pixel value inside the annular aperture (Y cylc ).This subtraction is meant to remove large-scale foreground contamination,assuming it is roughly constant over the extracted aperture. Then Y cylc isconverted to Y sph500 (the Comptonization parameter integrated over a sphereof radius R500) using the knowledge of the electron pressure profile in darkmatter halos. We use the Universal Pressure Profile (UPP) from (Arnaudet al., 2010).Finally, we scale it by E−2/3(z)×(dA(z)/500Mpc)2 (E(z) = (Ωm(1+z)3+Ωd)1/2) to account for the effects of comparing similar objects at differentredshifts. They are stacked in 10 stellar mass bins between 1011−1012M toestimate the binned averages and uncertainties. Our result is compared withthe Planck result (PCXI) in Figure. 2.3. The results are consistent with eachother, though the uncertainties are relatively larger in our result especiallyin lower-mass range. It also appears in the fact that the Planck result isstable in the power-law relation, though our result slightly fluctuates aroundthe relation.The Planck (MILCA) y map is constructed using more Planck bandmaps including 545 GHz and 857 GHz, and ILC coefficients are allowedto vary as a function of multipole ` and are computed independently ondifferent sky regions (a maximum of 3072 regions at high resolution map)to incorporate spatially varying foreground emissions such as the spectralindex of galactic dust emission and dust temperature. It can reduce theuncertainties in the Planck y map and have a better sensitivity even forlow-mass objects.2.4 ConclusionWe construct all-sky Compton parameter (y) map from the individual Planckfrequency maps with the Internal Linear Combination (ILC) technique, usedto make the y map explicitly to remove contamination from the CMB andpreserve the thermal SZ signal. Amongst the various other sources of con-tamination, our y maps are primarily focused on removing contaminationdue to galactic dust emission by assuming several dust models with varyingspectral index, βd. Our y maps are compared with the y map constructedby the Planck team.362.4. ConclusionFigure 2.3: Mean SZ signal vs. stellar mass for locally brightest galaxiesderived with our y map version D, compared to the original Planck resultin (Planck Collaboration, 2013c)(PCXI). Error bars show 1 σ statisticaluncertainty in each mass bin of 0.1 dex.• The y profile of the LRGs are constructed using the Planck y mapand our y maps. They are completely consistent with each other, andmoreover the y profiles using our different y maps show little differenceeven for widely varying dust spectral indices. It suggests that the ILCtechnique works well to extract the tSZ signal and the contaminationdue to the galactic dust emission should not be significant.• We follow the analysis in (Planck Collaboration, 2013c)(PCXI) andcompare the relation between the integrated y signals of central galax-ies from SDSS DR7 galaxies and their stellar masses. The resultsare consistent with each other, though the uncertainties are relativelylarger in our results especially in lower-mass range. It also appearsin the fact that the Planck result is stable in the power-law relation,though our result slightly fluctuates around the relation. These im-ply that our y maps might have more uncertainties than the Planck ymap.The comparison of our y maps with Planck y map provides the consistentresults, which allows to use either of them. However, in the study of this372.4. Conclusionchapter, the Planck y map should have a better sensitivity with less uncer-tainties, therefore we use the Planck y map in the studies in the followingchapters.38Chapter 3Probing hot gas in halosthrough the SunyaevZel’dovich effect3.1 IntroductionThe prominent advances in observational cosmology have provided six keycosmological parameters within a few percent, where in the current ΛCDMcosmology, most of the energy density of the universe is due to the darkenergy and dark matter and the contribution of the baryonic matter is only∼4.6% according to the CMB observation by WMAP and Planck satellite(Hinshaw et al., 2013; Planck Collaboration, 2016c).However, the evolution of the Universe from the very primitive initialstate to the current state is not well understood such as the evolution ofgalaxies and large scale structure of their distribution. One of the importanttracers of them is clusters of galaxies. They are the most massive boundstructures and therefore mark the most prominent density peaks of the largescale structure in the universe. Their cosmological evolution is thereforedirectly related to the growth of cosmic structures.X-ray observations have discovered that galaxy clusters are intense X-ray sources which are emitted by hot gas (T ∼ 107 [K]) located betweengalaxies. This intergalactic gas (intracluster medium, ICM) contains morebaryons than the stars seen in the member of galaxies. It indicates thatthe evolution of the ICM is more complex and regulated by the radiativecooling and also non-gravitational heating sources such as the active galacticnucleus (AGN).AGN feedback has a wide range of impacts on galaxies and galaxy clus-ters: the observed relation between the black hole mass and bulge veloc-ity dispersion, the regulation of cool cores and, the suppression of produc-ing massive galaxies predicted by N-body simulations (Gitti et al., 2012;Schneider, 2006). Thus, the interplay of hot gas with the relativistic plasma393.2. AGN feedback effectsejected by the AGN is key for understanding the growth and evolution ofgalaxies and the formation of large-scale structures. It has thus becomeclear that AGN feedback effects on the ICM must be incorporated in anymodel of galaxy evolution (e.g., (Battaglia et al., 2010; McCarthy et al.,2014; Schaye et al., 2010; Sijacki et al., 2007; Steinborn et al., 2015; Vogels-berger et al., 2013)). However, the non-gravitational processes beyond sim-ple gravity, gas dynamics and radiative cooling are not well understood. Tostudy the effect of non-gravitational processes, particularly, galaxy groupsand low-mass clusters are ideal laboratories since their relatively shallowpotentials compared to massive clusters would have a noticeable impact ofnon-gravitational effects on their formation and evolution (e.g., (Battagliaet al., 2012; Dong et al., 2010; Giodini et al., 2010; Johnson et al., 2009;Le Brun et al., 2014)).In this chapter, we probe the hot gas in the luminous red galaxies(LRG’s) halos from the Sloan Digital Sky Survey seventh data release (Abaza-jian et al., 2009) (SDSS DR7 LRG (Kazin et al., 2010)) using the PlanckCompton parameter (y) map from the 2015 Planck data release (PlanckCollaboration, 2016c).3.2 AGN feedback effectsOne of the main problems of the current cosmological model is why so fewbaryons have formed stars (Cole, 1991; White and Frenk, 1991). Only 10%have been observed in the form of stars (e.g., (Balogh et al., 2001)), which ismuch less than predicted by numerical simulations of cosmological structureformation including the hydrodynamics of baryons. Especially, simulationsincluding only gravitational heating predict an excessive cooling of baryonsand result in a population of too massive and bright galaxies with respectto the ones observed, thus fail to reproduce the truncation of the high-luminosity end of the galaxy luminosity function (Benson et al., 2003; Sijackiand Springel, 2006).This problem can be solved by the non-gravitational heating suppliedby supernovae (SNe). According to simulations, the heating due to SNesuppresses the low-luminosity side of galaxy luminosity function, however,it is not enough to quench cooling in massive galaxies (Borgani et al., 2002).Energetically, AGN heating appears to be the most likely mechanism toexplain the quenching of massive galaxies (Voit, 2005; Voit and Donahue,2005) in Figure 3.1.AGNs are powered by accretion of material onto supermassive black403.2. AGN feedback effectsFigure 3.1: The luminosity function of galaxies from observations is com-pared schematically to the one predicted from the CDM-motivated theory.holes (SMBH), which are located at the centers of galaxies. For a SMBHwith M ∼ 109M, the energy release during the formation and growth isof the order of EBH ∼ 2 × 1062 erg s−1. Even a tiny fraction (. 1%) ofthe released energy could heat and blow away the gas in the bulge, thuspreventing the star formation in the region. In particular, the correlationbetween the mass of central black hole (MBH) and the stellar velocity dis-persion (σ) in the galaxy’s bulge is found as in Figure 3.2. The data showsthat the formation of SMBHs and bulges are closely linked (Ferrarese andMerritt, 2000; Gebhardt et al., 2000; Magorrian et al., 1998), and the for-mation and evolution of galaxies are largely influenced by the SMBHs. Thephysical process regulating these phenomena is called “feedback”, and theunderstanding of its origin and influence in detail is still an important openquestion in extragalactic astrophysics.413.3. Luminous red galaxies (LRGs)Figure 3.2: First measurement of the MBH-σ relation. Panel (a) showsthe data points and fit from (Ferrarese and Merritt, 2000) (arXiv:astro-ph/0006053), while panel (b) shows the results from (Gebhardt et al., 2000)(arXiv:astro-ph/0006289). Both relations show a tight correlation betweenblack hole masses and velocity dispersions of host galaxies.3.3 Luminous red galaxies (LRGs)3.3.1 Luminous red galaxiesIt has long been known that the most luminous galaxies in clusters area very homogeneous population (e.g., (Postman and Lauer, 1995)). Forexample, Hogg et al. (2005) tests the homogeneity with the luminous redgalaxy (LRG) spectroscopic sample of the Sloan Digital Sky Survey. Thesurvey sky area is divided into 10 disjoint regions of each ∼ 2×107h−3Mpc3,and the fractional rms number density variations of the LRG sample in theredshift range 0.2 < z < 0.35 among the regions is found to be 7% of themean density.Because these objects are intrinsically very luminous, they can be ob-served to great distance and used as powerful tracers of large-scale structureof the universe. In particular, luminous red galaxies are early-type and mas-sive galaxies, consist mainly of old stars with little ongoing star formation,and typically reside in the centers of galaxy groups and clusters. Therefore,LRGs have been used to detect and characterize the remnants of baryonacoustic oscillations (BAO) at low to intermediate redshift (Anderson et al.,2014; Eisenstein et al., 2005; Kazin et al., 2010).In the Sloan Digital Sky Survey, LRGs are selected using a variant of423.3. Luminous red galaxies (LRGs)the photometric redshift technique and are meant to comprise a uniform,approximately volume-limited sample of objects with the reddest colors inthe rest frame. The sample is selected via cuts in the (g-r, r-i, r) color-color-magnitude cube. The resulting LRG sample has nearly constant comovingnumber density at low to intermediate redshift with a passively-evolved lu-minosity threshold that is close to constant 5.3.3.2 LRG catalogThe SDSS DR7 LRG catalog (N=105,831) by (Kazin et al., 2010) 6 providesthe positions, magnitudes and spectroscopic redshifts. The stellar massesof the LRGs are provided in the New York University Value-Added Catalog(NYU-VAGC) 7 (Blanton et al., 2005). They are estimated with the K-correct software 8 of Blanton and Roweis (2007) by fitting the five-bandSDSS photometry to more than 400 spectral templates, most of which arebased on stellar evolution synthesis models of Bruzual and Charlot (2003)assuming the stellar initial mass function of Chabrier (2003).Not all LRG’s are central galaxies in massive halos. (Parejko et al., 2013)estimates that 89 % of LRG’s are central based on the shape of the two-pointcorrelation function of LRGs, but Hoshino et al. (2015) find that, at a halomass of 1014.5M, only 73% of LRG’s are central based on comparing LRGpositions to cluster centers, as measured by X-rays in the redMaPPer clustercatalog, however, this is a relatively small sample compared to the full LRGcatalog. To minimize the fraction of satellite LRG’s in our sample (whichcould bias our results in chapter 3 and 4), we add an additional constraintand select the locally most-massive LRG’s (based on stellar mass) using acriterion that is analogous to that used in Planck Collaboration (2013c): wereject a galaxy if a more massive galaxy resides within a tangential distanceof 1.0 Mpc and within a radial velocity difference of |c∆z | < 1000 km s−1,which leaves ∼ 101,000 LRGs. Using an analogous selection based on r-band magnitude in the DR7 galaxy catalog, Planck Collaboration (2013c)assesses that the fraction of the central galaxies is at least ∼ 85 % forM∗ > 1011.3M and reaches up to ∼ 90 % in higher stellar masses basedon their simulation studies. It suggests that the LRGs we use are mostlycentrals. The distributions of redshift and stellar masses of the LRGs isshown in Figure. 3.3.5http://classic.sdss.org/6http://cosmo.nyu.edu/ eak306/SDSS-LRG.html7http://sdss.physics.nyu.edu/vagc/8http://howdy.physics.nyu.edu/index.php/Kcorrect433.4. Compton y parameter profile of the LRGsFigure 3.3: Left: The stellar mass distribution of SDSS DR7 LRGs. Right:The redshift distribution of the LRGs.3.4 Compton y parameter profile of the LRGsIn this section, we describe our procedure for stacking a y map against theLRGs and construct the mean y profile. For each LRG in the catalog, weplace one LRG at the center in a 2-dimensional angular coordinate systemof −40′ < ∆l < 40′ and −40′ < ∆b < 40′ divided in 80 × 80 bins. The meantSZ signal in the annular region between 30 and 40 arcmin is subtracted asan estimate of the local background signal.The left panel in Figure. 3.4 shows the average y map stacked againstthe 74,681 LRGs in which the tSZ signal at each pixel is smoothed by takingthe average of 5 × 5 square pixels nearby. The right panel in Figure. 3.4is the average y profile of the LRGs (blue) and the width of the blue linerepresents 1σ statistical uncertainty of the y profile. The Planck beam of10 arcmin in FWHM is shown as the black dashed line for comparison, theamplitude of which is normalized to the central peak of the data y profile.The average y profile has a central peak of y ∼ 1.8×10−7 and we detect thetSZ emission out to ∼ 30’ well beyond the Planck beam.3.5 Comparison to cosmo-OWLs hydrodynamicsimulations to probe AGN feedback effect3.5.1 cosmo-OWLS hydrodynamic simulationsThe cosmo-OWLS suite is an extension of the OverWhelmingly Large Sim-ulations project (Schaye et al., 2010) designed with cluster cosmology and443.5. Comparison to cosmo-OWLs hydrodynamic simulations to probe AGN feedback effectFigure 3.4: Left: The average Planck y map stacked against 74,681 LRGsin an angular coordinate system of −40′ < ∆l < 40′ and −40′ < ∆b < 40′divided in 80 × 80 bins. The tSZ signal at each pixel is smoothed by takingthe average of 5 × 5 square pixels nearby. Right: The average y profile ofthe LRGs (blue). The 1σ statistical uncertainty is represented in a widthof the blue line. The Planck beam of 10 arcmin in FWHM is supplementedin black dashed line for comparison, the peak of which is normalized to thecenter of the LRGs’ y profile.large scale-structure surveys in mind (Le Brun et al., 2014; McCarthy et al.,2014; van Daalen et al., 2014). The cosmo-OWLS suite consists of box-periodic hydrodynamical simulations, the largest of which have volumes of(400h−1Mpc)3 and contain 10243 each of baryonic and dark matter parti-cles. The suite employs two different cosmological models: Planck release1 cosmology (Planck Collaboration, 2014a) with {Ωm,Ωb,ΩΛ, σ8, ns, h} ={0.3175, 0.0490, 0.6825, 0.834, 0.9624, 0.6711} and WMAP7 cosmology (Ko-matsu et al., 2011) with {Ωm,Ωb,ΩΛ, σ8, ns, h}= {0.272, 0.0455, 0.728, 0.81, 0.967, 0.704}respectively. Each simulation is run with 5 different models of baryon sub-grid physics: “NOCOOL”, “REF”, “AGN 8.0”, “AGN 8.5” and “AGN 8.7”,which is summarized in Table. 3.1. Earlier studies demonstrate that theAGN 8.0 model reproduces a variety of observed gas features in local groupsand clusters of galaxies by optical and X-ray data. For each model of sim-ulation, 10 almost-independent mock galaxy catalogues are generated ona light cone and 10 corresponding y maps are generated from the hot gas(McCarthy et al., 2014). Each of these light cones contain about one milliongalaxies out to z ∼ 3, and each spans a 5◦ × 5◦ patch of sky. To comparewith data, we convolve the simulated y maps with a Gaussian kernel of 10arcmin, FWHM, corresponding to the Planck beam.453.5. Comparison to cosmo-OWLs hydrodynamic simulations to probe AGN feedback effectFigure 3.5: Compton y maps simulated by cosmo-OWLS hydrodynamicsimulations compared to different AGN feedback models (McCarthy et al.,2014). (Top: AGN 8.0, middle left: NOCOOL, middle right: REF, bottomleft: AGN 8.5, and bottom right: AGN 8.7)463.5. Comparison to cosmo-OWLs hydrodynamic simulations to probe AGN feedback effectTable 3.1: The baryon feedback models in the cosmo-OWLS simulation.Each model has been run in both Planck and WMAP7 cosmology (Mc-Carthy et al., 2014).SimulationUV/X-raybackground CoolingStarformationSNfeedbackAGNfeedback ∆TheatNOCOOL Yes No No No No ...REF Yes Yes Yes Yes No ...AGN 8.0 Yes Yes Yes Yes Yes 108.0 KAGN 8.5 Yes Yes Yes Yes Yes 108.5 KAGN 8.7 Yes Yes Yes Yes Yes 108.7 K3.5.2 Comparison to cosmo-OWLs hydrodynamicsimulationsWe compare the average y profile to simulations. To do so, we analyze lightcones from the cosmo-OWLS suite of simulations (§3.5.1) as we did the realdata. As LRGs, we select simulated “central” galaxies with the same stellarmass and redshift distribution as the real data. To match the distributions,the average y profile in each stellar mass and redshift bin is constructed inthe simulations and they are summed up using the number weight of theLRGs,y(θ)sim =∑M∗,z[y¯(θ,M∗, z)sim × w(M∗, z)LRG] , (3.1)where y¯(θ,M∗, z)sim is the average y profile of simulated central galaxies ina stellar mass and redshift bin and w(M∗, z)LRG is a normalized number ofactual LRGs in the same stellar mass and redshift bin. However, since thefield of view of each light cone (25 [deg2]) is much smaller than the overlap-ping region of SDSS-Planck survey (∼ 8000 [deg2]), massive central galaxiesare scarce in the simulations. Therefore, we restrict the maximum stellarmass of LRGs to 1011.7M in order to match the stellar mass distributionand also avoid a Poisson noise due to a few massive objects. As a result,the number of LRGs is reduced to 63,398, but this would be an appropriateprocedure considering that we probe the baryonic effects, which would bemore evident in low-mass group/clusters.The average y profile of 63,398 LRGs is compared to the simulationswith different AGN feedback models in Figure. 3.6, where the gray linesshow the average y profiles of the simulations. In the comparison, the cleardifference between the data and REF model can be seen. In general, theenergy release around the center of halos heats the gas and prevents cooling,473.6. Comparison to prediction from halo model and universal pressure profilethus the star formation around the center. Therefore, in the same system(the same total mass of halos), the stellar mass of central galaxy is less witha powerful energy release from AGN. Since we select the central galaxiesbased on the stellar mass, lower-mass halos are selected in the REF modelcompared to the models including AGN feedback. This appears in the lowercentral peak of the y profile in the REF model than others.This effect is not significant among the AGN feedback models in thesimulations. However, a visible trend is seen that the higher the power offeedback, the lower the peaks of y profile. This is due to the fact that theAGN feedback ejects the gas around the center of halos outward and thegas density is lowered.As a result of this comparison, the REF model is strongly rejected. Thissuggests that the model without AGN feedback can not account for they profile of the LRGs. Furthermore, the AGN 8.0 and AGN 8.5 modelsagree well with the real data, but AGN 8.7 does not, which constrains theAGN feedback models. In our study, a similar result is obtained using thesimulations of WMAP7 cosmology. Our result is consistent with the otherstudies that the AGN 8.0 model reproduces a variety of observed gas featuresin local groups and clusters of galaxies, using optical and X-ray data.3.6 Comparison to prediction from halo modeland universal pressure profile3.6.1 The Stacked y profile with cross-correlation of thetSZ and distribution of galaxy clustersFor the calculation of the stacked y profile, we follow the method in Fanget al. (2012) and work in the flat-sky and Limber approximation (Limber,1954).The cross power spectrum for the tSZ signal and the distribution ofgalaxy clusters is given by the sum of the one-halo term (contribution fromown halos) and two-halo term (contribution from the correlated environmentsurrounding the halos):Cyh` = Cyh,1h` + Cyh,2h` . (3.2)The one-halo term is calculated byCyh,1h` =f(x)n¯2D∫dzd2VdzdΩ∫dMdndM(M, z)× S(M, z)y˜`(M, z), (3.3)483.6. Comparison to prediction from halo model and universal pressure profileFigure 3.6: The average y profile of LRGs (blue) is compared with the beam-convolved y profile of the simulated central galaxies in Planck cosmology(grey), with the same stellar mass and redshift distributions in differentAGN feedback models respectively. Top left: AGN 8.0 model, Top right:AGN 8.5 model, Bottom left:: AGN 8.7 model and Bottom right: REFmodel.Figure 3.7: The average y profile of LRGs (blue) is compared with the yprofile of the simulated central galaxies in WMAP7 cosmology (grey).493.6. Comparison to prediction from halo model and universal pressure profilewhere d2V/dzdΩ is the comoving volume element and dn/dM is the halomass function. We adopt the halo mass function of Tinker et al. (2010),calculated using a code called ’HMFcalc’ (Murray et al., 2013). In a clustersurvey, galaxy clusters are selected depending on the observational strat-egy. The selection function can be defined by their observed redshift andestimated mass, which is given byS(M, z) = Θ(z − zmin)Θ(zmax − z)×Θ(M −Mmin)Θ(Mmax −M), (3.4)where Θ stands for the Heaviside step function. The average 2-dimensionalangular number density of the selected galaxy clusters is calculated byn¯2D =∫dzd2VdzdΩ∫dMdndM(M, z)S(M, z). (3.5)y˜`(M, z) is 2D Fourier transform of y profile for a cluster with a pressureprofile Pe(x,M, z), which is calculated byy˜`(M, z) =σTmec24pirs`2s∫dxx2sin(`x/`s)`x/`sPe(x,M, z), (3.6)wherex =rrs, `s =dArs, (3.7)where rs is a characteristic scale radius of ICM pressure profile where x =r/rs is defined to be a dimensionless radial scale, and dA is an angulardiameter distance where `s = dA/rs is defined to be an associated multipolemoment.The two-halo term is calculated byCyh,2h` = f(x)∫dzd2VdzdΩPLm(k =`χ, z)W h(z)W y` (z), (3.8)where PLm(k, z) is a linear matter power spectrum. The function Wh(z) isdefined to beW h(z) =1n¯2D∫dMdndM(M, z)S(M, z)b(M, z), (3.9)and W y` (z) isW y` (z) =∫dMdndM(M, z)b(M, z)y˜`(M, z), (3.10)503.6. Comparison to prediction from halo model and universal pressure profilewhere b(M, z) is a halo bias. We use the halo bias from Tinker et al. (2010).By putting them together, a Fourier-transform of the stacked y profile,Cyh` , is calculated. In an angular space, the stacked y profile is given byy¯(θ) =1f(x)∫`d`2piJ0(`θ)Cyh` , (3.11)where J0 is the zeroth order Bessel function. Finally, we convolve it with apoint spread function of an observation,y¯(θ)obs =1f(x)∫`d`2piJ0(`θ)Cyh` Bl, (3.12)where Bl = exp[l(l + 1)σ2/2] and σ = θFWHM/√8ln(2) with θFWHM = 10arcmin, corresponding to the Planck beam.3.6.2 Universal Pressure ProfileFor the pressure profile, we adopt the universal pressure profile (UPP). TheUPP has an analytical formulation given by Nagai et al. (2007) for thegeneralised Navarro-Frenk-White (GNFW) profile (Navarro et al., 1997),P(x) =P0(c500x)γ [1 + (c500x)α](β−α)/γ, (3.13)where x = r/R500, and the model is defined by the following parameters:P0, normalization; c500, concentration parameter defined at a characteristicradius R500; and the slopes in the central (x  1/c500), intermediate (x ∼1/c500) and outer regions (x  1/c500), given by γ, α and β, respectively.The scaled pressure profile for a halo with M500 and z is represented byP (r)P500= P(x), (3.14)withP500 = 1.65× 10−3h(z)8/3×[M5003× 1014h−170 M]2/3h270 keV cm−3,(3.15)using a characteristic pressure, P500, reflecting the mass variation expectedin the standard self-similar model, purely based on gravitation (Arnaudet al., 2010). The deviation from the standard self-similar scaling appears513.6. Comparison to prediction from halo model and universal pressure profileas a variation of the scaled pressure profile. As in Arnaud et al. (2010), thisvariation can be expressed as a function of total mass,P (r)P500= P(x)[M5003× 1014h−170 M]αp, (3.16)where αp can be approximated by αp = 0.12. For the parameters in theGNFW, we adopt the best-fit values of [P0, c500, γ, α, β] = [6.41, 1.81, 0.31, 1.33, 4.13],estimated using Planck tSZ and XMM-Newton X-ray data in Planck Col-laboration (2013a), which is shown in Figure. 3.8.Figure 3.8: Universal pressure profile derived from (Planck Collaboration,2013a) (red line) compared to the one in (Arnaud et al., 2010) (blue line).3.6.3 Estimating halo masses of LRGsIn order to construct the y profile of the LRGs using the halo model andUPP, we estimate the halo masses of the LRG halos using the SHM relationfrom Planck Collaboration (2013c) (P13-SHM), Coupon et al. (2015) (C15-SHM) and Wang et al. (2016) (W16-SHM). In P13-SHM, the halo massesare estimated using a semi-analytic N-body simulation of galaxy populationevolution from Guo et al. (2013). This simulation is tuned to reproduce theobserved stellar mass function of the SDSS galaxies and the SHM relation isobtained by the abundance-matching method. In W16-SHM, the P13-SHM523.6. Comparison to prediction from halo model and universal pressure profilerelation is recalibrated with the gravitational lensing measurements fromReyes et al. (2012). In C15-SHM, it is estimated in the CFHTLenS/VIPERSfield by combining deep observations from the near-UV to the near-IR, sup-plemented by ∼70 000 secure spectroscopic redshifts, and analyzing galaxyclustering, galaxy-galaxy lensing and the stellar mass function. The threeSHM relations are shown in Figure. 3.9 on top of the SHM relations be-tween central galaxies and corresponding halos in the AGN 8.0 simulation:P13-SHM in cyan, C15-SHM in magenta and W16-SHM in yellow. In thestellar mass range of the LRGs, the halo mass estimates from C15-SHM andW16-SHM are consistent and the one from P13-SHM is estimated higher by∼0.1 dex than the others.Figure 3.9: Black points show the relation between the stellar masses ofcentral galaxies and halo masses in 0.16 < z < 0.47 from the AGN 8.0simulation and its mean relation is shown in red. The three other SHMrelations are shown on top of it: P13-SHM (Planck Collaboration, 2013c)in cyan, C15-SHM (Coupon et al., 2015) in magenta and W16-SHM (Wanget al., 2016) in yellow.3.6.4 Comparison to the predictionUsing the estimated halo masses, we can calculate the average y profile ofthe LRG halos, using the halo model and UPP, via the procedure describedin §3.6.1. To match the mass and redshift distributions, we apply the same533.7. Discussionmethod described in §3.5.2; the average y profile in each mass and redshiftbin is calculated, and convolved with the Planck beam, and then they aresummed up using the number weight of the LRGs. The constructed y profileswith different halo mass estimates are shown in Figure. 3.10 as well as the yprofile of the LRGs and the one from the AGN 8.0 simulation. Note that theAGN 8.0 simulation is customized to a larger field of view of 10◦ × 10◦ [deg2]but a limited redshift of z < 1 in this figure to increase the number of objects.It shows a better agreement with the y profile of the LRGs.The predictions from C15-SHM + UPP (magenta) and W16-SHM +UPP (yellow) agree well with the y profile of the LRGs. Assuming the C15-SHM or W16-SHM relation is correct, it implies that the UPP, estimatedfor galaxy clusters in the mass range of 1014−1015M, is applicable even forlow-mass halos, as suggested by the scaling relation of Y500-M∗ (Y500: theComptonization parameter integrated over a sphere of radius R500) reportedin Planck Collaboration (2013c)(P13). However, we adopt the “average”SHM relation for all the LRGs and the scatter is not included. For example,applying the Gaussian scatter of halo mass in the C15-SHM relation fromMoster et al. (2010) raises the peak of the y profile by ∼25 %. In addition,the distribution of the scatter is not guaranteed to be symmetric and it wouldalso affect the model prediction of the y profile. Furthermore, according toLe Brun et al. (2015), the AGN 8.0 simulation, which also agrees with they profile of the LRGs, predicts lower and more extended pressure profilesin low-mass halos than the UPP. Therefore, it is still hard to conclude adefinitive statement about pressure profiles in low-mass halos because ofthe degeneracy between pressure profiles and SHM relations in the modelprediction. For the conclusion, we need a better understanding of the SHMrelation including its scatter, which is not well understood so far.3.7 DiscussionThis study is partially motivated by the contradictory result between (PlanckCollaboration, 2013c)(P13) and (Anderson et al., 2015)(A15) on the stateof hot gas in galaxy group/clusters through the scaling relation. The self-similar scaling relation is valid under the assumption that the process isdominated by gravity and the deviation from the relation points to the pres-ence of more complex processes such as baryonic effects. Using the LocallyBrightest Galaxies (LBGs) in SDSS DR7, P13 finds self-similar scaling inY -Mh, implying the hot gas represents roughly the mean cosmic fraction ofthe mass even in low-mass systems. On the other hand, A15 finds a steeper543.7. DiscussionFigure 3.10: The average y profile of LRGs (blue) is compared to the predic-tions using the halo model (Tinker et al., 2010) and UPP (Planck Collabora-tion, 2013a) when the halo masses of the LRGs are estimated using the SHMrelation of P13-SHM (cyan), C15-SHM (magenta) and W16-SHM (yellow).The y profile of the simulated central galaxies in the AGN 8.0 simulation isshown in grey. Note that the AGN 8.0 simulation is customized to a largerfield of view of 10◦ × 10◦ [deg2] but a limited redshift of z < 1 in this figureto improve the number of objects. As a comparison, we also apply the UPPto the halo mass distribution of the AGN 8.0 simulation, which is shown ingrey dashed line.scaling than the self-similar relation in LX-Mh, suggesting the importance ofnon-gravitational heating such as AGN feedback, as numerous X-ray stud-ies of galaxy groups find a deficit of baryons (e.g., (Gastaldello et al., 2007;Gonzalez et al., 2013; Pratt et al., 2009; Sun et al., 2009)). These results canbe reconciled by the idea that the low-mass halos may contain the cosmicfraction of baryons, just like galaxy clusters, but the density profile of thegas in the halos may be less concentrated.We find that the measured y profile of the LRGs agree well with theone from AGN 8.0 simulation, but does not without the AGN feedback.The result is consistent with other studies in which the AGN 8.0 modelreproduces a variety of observed gas features in local groups and clustersof galaxies by optical and X-ray data. The AGN 8.0 model predicts lower553.8. Conclusionpressure profile than the UPP in low-mass halos (Le Brun et al., 2015), andthus a deviation from the self-similar scaling relation in Y -Mh.On the other hand, we also demonstrate that the measured y profileof the LRGs agree well with the predictions using the UPP with the SHMrelation from C15-SHM (Coupon et al., 2015) or W16-SHM (Wang et al.,2016), estimated by gravitational lensing measurements. This implies thatthe UPP, estimated for galaxy clusters in the mass range of 1014-1015M,can be applied to low-mass systems down to Mh ∼ 2 × 1013M such asLRG halos, and thus little deviation from the self-similar scaling relation inY -Mh.However, for the model prediction, the degeneracy between pressure pro-files and SHM relations remains. Especially, the scatter in the SHM is notwell understood and has a negligible impact on the model prediction of yprofile. Because of the degeneracy, it is still hard to conclude a definitivestatement about pressure profiles in low-mass halos, in turn, a deviation fromself-similar relation due to non-gravitational heating such as AGN feedback.For the conclusion, a better understanding of the SHM relation is importantin our analysis.3.8 ConclusionIn this chapter, using the the Planck y map, we present a stacking analysisof the SDSS DR7 LRGs, which are considered to be mostly central galaxiesin dark matter halos. We construct the average y profile centered on theLRGs and study the thermodynamic state of the gas in groups and low-massclusters. The major results of our analysis are summarized as follows:• The central tSZ signal is y ∼ 1.8 × 10−7 and we detect tSZ emissionout to ∼ 30 arcmins well beyond the extent the Planck beam of the10 arcmin.• We compare the average y profile of LRGs with the predictions fromthe cosmo-OWLS suite of cosmological hydrodynamical simulations.This comparison agrees well with the models including the AGN feed-back (AGN 8.0 and AGN 8.5), but not with the model not includingit (REF). This result suggests that an additional heating mechanismis required over and above SNe feedback and star formation, which isconsistent with other studies showing that the AGN 8.0 model repro-duces a variety of observed gas features in optical and X-ray data.563.8. Conclusion• The average y profile of LRGs is also compared with the predictionsusing the halo model (Tinker et al., 2010) and UPP (Planck Collabora-tion, 2013a). The predicted y profile is consistent with the data, if oneaccounts for the two-halo clustering term in the model, and assumingthe halo mass distribution by the C15-SHM and W16-SHM estimatedwith gravitational lensing measurements. This may imply that theUPP, estimated for galaxy clusters in the mass range of 1014−1015M,is applicable even in low-mass halos down to Mh ∼ 2 × 1013M asimplied in P13. However, for a definitive statement, a better under-standing of the SHM relation is needed.In our analysis, the dominance of the two-halo term in the low-masssystems is partially due to the coarse Planck beam. We emphasize thatmore precise measurements with a better angular resolution and sensitivitysuch as ACTPol (Niemack et al., 2010) and SPTpol (Austermann et al.,2012) will shed light further light on the issue and help to clarify the impactof AGN feedback on the formation and evolution of galaxies. Moreover, asan extension of our studies, the newer data release of the SDSS DR13 LRGcatalog will help to probe the evolution of the gas inside halos to higherredshift, however, a better understanding of the SHM relation is needed.57Chapter 4Probing hot gas in thecosmic web through theSunyaev Zel’dovich effect4.1 IntroductionIn the distant Universe (z & 2), most of the expected baryons are found inthe Lyα absorption forest: the diffuse, photo-ionized intergalactic medium(IGM) with a temperature of 104-105 K (e.g., (Rauch et al., 1997; Weinberget al., 1997)). However, in the local Universe (z . 2), the observed baryonsin stars, the cold interstellar medium, residual Lyα forest gas, OVI and BLAabsorbers, and hot gas in clusters of galaxies account for only ∼50% of theexpected baryons (e.g., (Fukugita and Peebles, 2004; Nicastro et al., 2008;Shull et al., 2012)). Hydrodynamical simulations suggest that 40-50% ofbaryons could be in the form of shock-heated gas in a cosmic web betweenclusters of galaxies. This so-called Warm Hot Intergalactic Medium (WHIM)has a temperature range of 105-107 K (Cen and Ostriker, 2006). The WHIMis difficult to observe due to its low density: several detections in the far-UVand X-ray have been reported, but few are considered definitive (Yao et al.,2012). It appears that ∼30% of the baryons are still unaccounted for in thelocal universe (Shull et al., 2012).Recently, (Clampitt et al., 2016) searched for evidence of massive fila-ments between proximate pairs of LRG’s taken from the Sloan Digital SkySurvey seventh data release (SDSS DR7). Using weak gravitational lensingsignal, stacked on 135,000 pairs of LRG’s, they find evidence for filamentmass at ∼4.5σ confidence. Similarly, Epps and Hudson (2017) detects theweak lensing signal using the Canada France Hawaii Telescope Lensing Sur-vey (CFHTLenS) mass map from stacked filaments between SDSS-III/BOSSLRG’s at 5σ confidence.The tSZ signal provides an excellent tool for probing baryonic gas at lowand intermediate redshifts. Atrio-Barandela and Mu¨cket (2006) and584.2. Missing baryon problemAtrio-Barandela et al. (2008) suggest that electron pressure in the WHIMwould be sufficient to generate potentially observable tSZ signals. The mea-surement is challenging due to the morphology of the source and the relativeweakness of the signal, however, the Planck team reports a significant tSZsignal in the inter-cluster region between the merging clusters of A399−A401(Planck Collaboration, 2013b). In conjunction with ROSAT X-ray data,they estimate the temperature and density of the inter-cluster gas to bekT = 7.1± 0.9 keV with a baryon density of (3.7± 0.2)× 10−4 cm−3.Van Waerbeke et al. (2014), Ma et al. (2015) and Hojjati et al. (2015)report correlations between gravitational lensing and tSZ signals in thefield using the CFHTLenS mass map and Planck tSZ map. Similarly, Hilland Spergel (2014) reports a statistically significant correlation between thePlanck CMB lensing potential and the Planck tSZ map. These results showclear evidence for hot gas tracing dark matter. Further, in the context of ahalo model, there is clear evidence for contributions from both the one- andtwo-halo terms, but there is no statistically significant evidence for contri-butions from diffuse, unbound gas not associated with (correlated) collapsedhalos.In this chapter, we search the Planck data for a tSZ signal due to gasfilaments between pairs of Luminous Red Galaxies taken from the Sloan Dig-ital Sky Survey Data Release 12 (SDSS/DR12) (Alam et al., 2015; Prakashet al., 2016).4.2 Missing baryon problemThe energy density of baryons in the Universe, Ωb, can be measured by theCMB power spectrum, especially through the second peak. In addition, anindependent measurement can be obtained from the observed abundancesof light elements (D, He, Li), which constrain the baryon fraction throughthe Big Bang Nucleosynthesis model. The result from the BBN is currentlyΩbh2 = 0.021 - 0.025 (e.g. the review by (Fields et al., 2014)) and consistentwith the WMAP (0.02264 ± 0.00050) (Hinshaw et al., 2013) and Planck(0.02225± 0.00016) (Planck Collaboration, 2016c) CMB measurements.After reionization, most of the baryons in the Universe are ionized. Eventhough the remaining neutral fraction is tiny (∼ 10−4-10−5), it produces theLy-α forest. For example, a continuum spectra from a quasar emitted atλ < 1215.7 A˚ is absorbed by intervening neutral gas when it is redshiftedto the energy of Lyman-α. Therefore, the absorption lines trace the columndensity of the neutral gas along the time of the universe. At high redshift594.2. Missing baryon problem(z & 2), most of the expected baryons are found in the Lyα absorption forest(e.g., (Rauch et al., 1997; Weinberg et al., 1997)).However, at low redshift (z . 2), the observed baryons in stars, the coldinterstellar medium, residual Lyα forest gas, OVI and BLA absorbers, andhot gas in clusters of galaxies account for only ∼50% of the expected baryons– the remainder has yet to be identified (e.g., (Fukugita and Peebles, 2004;Nicastro et al., 2008; Shull et al., 2012)).Where is the remainder of the baryons in the local Universe? Alongwith the structure formation, accretion shocks heat the intergalactic medium(IGM), roughly to the virial temperature of the structures onto which thematerial is accreting. This changes the ionization equilibrium of the IGMand causes the neutral fraction to drop quickly with the temperature in-crease. In addition, feedback from star formation and AGN activity ingalaxies heats the IGM as well. With these processes, a large amount of IGMhas been converted into so-called warm hot intergalactic medium (WHIM).The WHIM no longer shows up in the Lyman-α forest. Hydrodynamicalsimulations suggest that 40-50% of baryons could be in the form of WHIMlocated in a cosmic web between clusters of galaxies with a temperaturerange of 105-107 K as seen in Fig. 4.1 (Cen and Ostriker, 2006).The effort to detect the WHIM has been focused on metal absorptionlines such as OVII and OVI corresponding to a series of soft X-ray linesaround 0.6-0.8 keV. A few detections of WHIM filaments have been claimedby XMM-Newton and Chandra observations, but they have little sensitivityto the WHIM and most have not been confirmed significantly. The WHIMalso produces the diffuse X-ray emission, however, the emissivity is pro-portional to the square of the gas density and drops significantly for thelow-density plasma. Fig. 4.2 from Shull et al. (2012) shows the current cen-sus of baryons including the detected WHIM. The sum of the known baryonsfalls short of the expected and ∼30% of the baryons are still missing.On the other hand, the tSZ effect has a linear response to the gas densityand higher sensitivity to low density plasma. The optical depth of the SZeffect is subtle and the detection is challenging. For example, in a filamentof length 1 Mpc and electron density 2 × 10−6 cm−6 (δ ∼ 10 at z ∼ 0),the optical depth is only τ ∼ 4 × 10−5, however this is potentially observ-able through a stacking method or cross-correlation with other tracers ofstructure.604.3. Pair stacking of LRG pairsFigure 4.1: Evolution of the four cosmic baryon components in mass fraction(Cen and Ostriker, 2006) (arXiv:astro-ph/0601008). It shows that about halfof all baryons at redshift zero are in the WHIM. Galactic superwind (GSW):ionizing UV photons, the effects of the cumulative supernova explosions andthe output from active galactic nuclei.4.3 Pair stacking of LRG pairs4.3.1 LRG pair catalogSDSS Data Release 12 (DR12) is the final release of data from SDSS-III.The catalog provides the position, spectroscopic redshift, and classificationtype for each object. We extract objects identified as sourcetype=LRG.The stellar masses of these objects have been estimated by three differ-ent groups9. We use the estimate based on a principal component anal-ysis method by (Chen et al., 2012), which uses stellar evolution synthesismodels from (Bruzual and Charlot, 2003), and a stellar initial mass func-tion from (Kroupa, 2001). For the stacking analysis, we select LRG’s with9http://www.sdss.org/dr12/spectro/galaxy614.3. Pair stacking of LRG pairsFigure 4.2: The baryon census in the local Universe (Shull et al., 2012).M∗ > 1011.3M. According to the scaling relation of Y500-M∗ (Y500: theComptonization parameter integrated over a sphere of radius R500) reportedin (Planck Collaboration, 2013c), these LRG’s should have a central tSZsignal-to-noise ratio of order unity. Since our analysis requires us to esti-mate and subtract the tSZ signal associated with the halos of the individualLRG’s, this cut enhances the reliability of that estimation, via the proceduredescribed in §4.3.3.As we discussed in §3.3, not all LRG’s are central galaxies in massivehalos. We reject a given galaxy if a more massive galaxy resides within atangential distance of 1.0 h−1Mpc and within a radial velocity difference of|c∆z | < 1000 km s−1. We construct the LRG pair catalog from this centralLRG catalog by finding all neighbouring LRG’s within a tangential distanceof 6-10 h−1Mpc and within a proper radial distance of ± 6h−1Mpc. Theresulting catalog has 262,864 LRG pairs to redshifts z ∼ 0.4. Their redshiftand separation distributions are shown in Figure 4.3.4.3.2 Stacking on LRG pairsThe angular separation between LRG pairs in our catalog ranges between 27and 203 arcmin. For each pair in the catalog, we follow Clampitt et al. (2016)624.3. Pair stacking of LRG pairsFigure 4.3: Left: The redshift distribution of LRG pairs peaks at z ∼ 0.35.Top right: The distribution of tangential separations between LRG pairs.Bottom right: The distribution of radial separations between LRG pairs.and form a normalized 2-dimensional image coordinate system, (X,Y ), withone LRG placed at (−1, 0) and the other placed at (+1, 0). The correspond-ing transformation from sky coordinates to image coordinates is also appliedto the y map and the average is taken over all members in the catalog. Themean tSZ signal in the annular region 9 < r < 10 (r2 ≡ X2 + Y 2) is sub-tracted as an estimate of the local background signal.The top panel of Figure 4.4 shows the average y map stacked against262,864 LRG pairs over the domain −3 < X,Y < +3, and the lower panelshows a slice of this map at Y = 0. Not surprisingly, the average signalis dominated by the halo gas associated with the individual LRGs in eachpair. The peak amplitude of this signal is ∆y ∼ 1.4× 10−7, consistent withthe study in §3.4 in the previous chapter.4.3.3 Subtracting the halo contributionWe estimate the average contribution from single LRG halos as follows.Since we have selected central LRGs for our pair catalog, we assume thatthe average single-halo contribution is circularly symmetric about each LRGin a pair. To determine the radial profile of each single halo, we fit the map(indexed by pixel p) to a model of the formyh(p) = yL,i(p) + yR,j(p), (4.1)where yL,i is the single-halo signal in the ith radial bin centred on the “left”LRG at (−1, 0), yR,j is the single-halo signal in the jth radial bin centred634.3. Pair stacking of LRG pairsFigure 4.4: Top: The average Planck y map stacked against 262,864 LRGpairs in a coordinate system where one LRG is located at (X,Y ) = (−1, 0)and the other is at (X,Y ) = (+1, 0). The square region, −3 < X,Y < +3,comprises 151 × 151 pixels. Bottom: The corresponding y signal along theX axis.644.3. Pair stacking of LRG pairson the “right” LRG at (+1, 0), and p (=p(X,Y )) is a pixel on the map.When performing the fit, we choose radial bins of size ∆r = 0.02, and weweight the map pixels uniformly. To avoid biasing the profiles with non-circular filament signal, we exclude the central region −1 < X < +1 and−2 < Y < +2 from the fit. Figure 4.5 shows the resulting best-fit profilesfor the LRG halos.Figure 4.5: Top: The best-fit circular halo profiles fit to the map in Fig-ure 4.4. Bottom: The best-fit radial profile of the left and right halos shownabove.Figure 4.6 shows the residual y map after subtracting the best-fit circularprofiles shown in Figure 4.5 (note the change in color scale from Figures 4.4and 4.5). The bright halo signals appear to be cleanly subtracted, whilea residual signal between the LRGs is clearly visible. The lower panelsof Figure 4.6 show the residual signal in horizontal (Y = 0) and vertical(X = 0) slices through the map. The shape of the signal is consistent654.3. Pair stacking of LRG pairswith an elongated filamentary structure connecting average pairs of centralLRGs. The mean residual signal in the central region, −0.8 < X < +0.8and −0.2 < Y < +0.2, is found to be ∆y = 1.31× 10−8.Figure 4.6: Top: The residual y-map after the best-fit radial halo signals aresubtracted. Bottom: The residual tSZ signal along the X and Y axes.4.3.4 Null tests and error estimatesTo assess the reality of the residual signal and estimate its uncertainty, weperform two types of Monte Carlo-based null tests. In the first test, we rotatethe center of each LRG pair by a random angle in galactic longitude (while664.3. Pair stacking of LRG pairskeeping the galactic latitude fixed, in case there is a systematic galacticbackground signal). We then stack the y map against the set of rotated LRGpairs, and we repeat this stacking of the full catalog 1000 times to determinethe rms fluctuations in the background (and foreground) sky. Figure 4.7shows one of the 1000 rotated, stacked y maps: the map has no discerniblestructure. We can use this ensemble of maps to estimate the uncertaintyof the filament signal quoted above. Taking the same region used before(−0.8 < X < +0.8 and −0.2 < Y < +0.2), we find that the ensemble ofnull maps has a mean and standard deviation of ∆y = (−0.03±0.24)×10−8in Figure 4.10. Since the average signal in this null test is consistent withzero, we cautiously infer that our estimator is unbiased, however, we presentanother test in the following.The second null test is to stack the y map against “pseudo-pairs” ofLRGs: that is, pairs of objects that satisfy the transverse separation cri-terion, but which have a large separation along the radial direction. Suchpairs are not expected to be connected by filamentary gas. We generatea pseudo-pair catalog as follows: for each pair in the original catalog, wepick one of the two members at random, then pick a new partner LRG thatis located within 6-10 h−1Mpc of it in the transverse direction, but whichis more than 30 h−1Mpc away in the radial direction. We select the samenumber of pairs meeting this criterion as in default LRG pair catalog, sothat the stacked image is of approximately the same depth. We repeat thisselection 1000 times to generate an ensemble of pseudo-pair catalogs.The top panel of Figure 4.8 shows an average y maps stacked againstone of the pseudo pair catalog realizations. This map is similar to thegenuine pair-stacked map shown in Figure 4.4, but with less apparent signalbetween the LRGs. We perform the same single-halo model fit describedabove and subtract it from the map with the result is shown in the middlepanel of Figure 4.8. As with the rotated null test above, this map shows nodiscernible structure. To generate statistics, we repeat this test 1000 times:we find that the ensemble of null maps has a mean and standard deviationof ∆y = (0.00± 0.25)× 10−8 in Figure 4.10, virtually the same as with therotated ensemble. We adopt this standard deviation as the final uncertaintyof the mean filament signal due to instrument noise, sky noise (i.e., cosmicvariance and foreground rejection errors), and halo subtraction errors.674.3. Pair stacking of LRG pairsFigure 4.7: Top: A sample null map obtained by stacking the y map againstthe LRG pairs that were rotated in galactic longitude by random amounts.Bottom: The tSZ signal along the X and Y axes of the y map shown above.684.3. Pair stacking of LRG pairsFigure 4.8: Top: An average y map stacked against a catalog of LRG pseudopairs (see text for a definition). Middle: The residual y map after subtractingthe best-fit circular halos from the above map, using the same procedure thatwas applied to the genuine pair stack. Bottom: The tSZ signal along the Xand Y axes of the residual map shown in the middle panel.694.3. Pair stacking of LRG pairsFigure 4.9: The result from 1000 rotated null stacks. The data y value(∆y = 1.31× 10−8) is expressed in red-dash line.Figure 4.10: The result from 1000 pseudo-pair null stacks. The data y value(∆y = 1.31× 10−8) is expressed in red-dash line.704.4. Interpretation of the detected tSZ signal between the LRG pairs4.4 Interpretation of the detected tSZ signalbetween the LRG pairsWe can estimate the physical conditions of the gas we detect by considering asimple, isothermal, cylindrical filament model of electron over-density with adensity profile proportional to rc/r, at redshift z. The Compton y parameterproduced by the tSZ effect is given byy =σTkBmec2∫ne Te dl. (4.2)In general, the electron density at position x may be expressed asne(x, z) = n¯e(z)(1 + δ(x)), (4.3)where δ(x) is the density contrast, and n¯e(z) is the mean electron densityin the universe at redshift z,n¯e(z) =ρb(z )µemp, (4.4)where ρb(z) = ρcΩb(1 + z)3 is the baryon density at redshift z, ρc is thepresent value of critical density in the universe, Ωb is the baryon density inunits of the critical density, µe =21+χ ' 1.14 is the mean molecular weightper free electron for a cosmic hydrogen abundance of χ = 0.76, and mp isthe mass of the proton.We can express the profile in the Compton parameter as a geometricalprojection of a density profile with ne(r, z):y(r⊥) =σTkBTemec2∫ Rr⊥2r ne(r, z)√r2 − r2⊥dr, (4.5)where r⊥ is the tangential distance from the filament axis on the map andR is the cut-off radius of the filaments. Assuming negligible evolution of thefilaments and constant over-density, δc, at the core since z = 0.4,ne(r = 0, z) =ne(r = 0, z) n¯e(z)n¯e(z)= δc n¯e(0) (1 + z)3. (4.6)We consider three density profiles,ne(r) = constant (r < rc), (4.7)ne(r) =ne(0)√1 + (r/rc)2(r < 5rc), (4.8)ne(r) =ne(0)1 + (r/rc)2(r < 5rc), (4.9)714.4. Interpretation of the detected tSZ signal between the LRG pairswhere rc is the core radius. To regularize the profiles, we adopt a cutoffradius of 5rc for the second and third profile. Applying the profiles to thesimulations described in §4.5, we find the best-fit density profile to follow(rc/r).For this model, the predicted tSZ signal in the region (−0.8 < X < +0.8,−0.2 < Y < +0.2) for the 262,864 filaments can be written as∆y¯ = 4.9× 10−8 ×(δc10)(Te107 K)(rc0.5h−1 Mpc). (4.10)Applying the observational constraint on mean ∆y, we have,δc(Te107 K)(rc0.5h−1 Mpc)= 2.7± 0.5. (4.11)Assuming the same temperature and morphology estimates from the simula-tions in §4.5 apply to the observational data, the mean filament over-densitybetween LRG pairs is δ ∼ 3.2 ± 0.7. This implies that the gas in the fil-aments between the LRGs can be widely spread over the large scale withvery low density.Is the signal we detect due to unbound diffuse gas outside of halos orbound gas in halos between the LRG pairs? To investigate this, we simu-late a model y map using only bound gas in the SDSS DR12 LRGs with1010M < M∗ < 1012M and 0. < z < 0.8 and compare the result withPlanck y map. To make the single-halo model y map, we select “central“LRGs described in §4.3.1, which leaves ∼ 1,100,000 LRGs, and estimate thehalo masses of the LRGs with the stellar-to-halo mass relation of Couponet al. (2015) (§3.6.3). Then we locate y profiles within the virial radius(r < r200) of the LRG halos on the map using the universal pressure profile(UPP) (§3.6.2). For the model y map, we perform the same analysis. Thepeak y valus of the LRG halos is dimmer than the y map in Figure 4.4since we only include the contribution within the virial radius of the LRGhalos and in addition, no sub-halos are included. After the circular halosubtraction, we obtain ∆y = 0.29× 10−8 from the bridge region. Moreover,we simulate a model y map including y profiles within r < 3 × r200 of theLRG halos and the result is ∆y = 0.36 × 10−8. These results suggest thatmost of the y signal we detect between the LRGs should originate in un-bound diffuse gas, although the contributions from other types of galaxies,less massive galaxies and galaxies in higher redshift should be present atsome level.Some systematic effects that might enhance or diminish the tSZ signal inthe filaments may exist. For example, in our analysis, although we assume724.4. Interpretation of the detected tSZ signal between the LRG pairsFigure 4.11: Left: The single-halo model y map described in the text isstacked against the same 262,864 LRG pairs as in the data analysis. Right:The residual model y map after subtracting the best-fit circular halos fromthe map at left, using the same procedure that was applied to the genuinepair stack.that the average single-halo contribution is circularly symmetric about eachLRG, one might speculate that the signal we detect between nearby LRGpairs is a result of tidal effects, in which single-object halos are elongatedalong the line joining the two objects. However, considering that the ∼260,000 LRG pairs are made from ∼ 220,000 LRGs and each LRG hasroughly 2 pairs (One is aligned between our LRG pairs and the other is notaligned), the direction of the elongation is not obvious. While tidal effectsmust be present at some level, if they were the dominant explanation forthe residual signal we see, we would expect the elongated halo structure toextend in both directions along the line joining the objects. The fact that wesee no significant excess signal outside the average pair suggests that tidaleffects are not significant.On the other hand, there are possible systematic effects that might lowerthe tSZ signal in the filaments. For example, some of the LRG pairs arenot connected by filaments, or some of the filaments may not straight butcurved. However, in a study of N -body simulations, Colberg et al. (2005)found that cluster pairs with separations < 5h−1 Mpc are always connectedby dark matter filaments, mostly straight filaments. Further, they foundfilaments connecting ∼85% of pairs separated by 5-10h−1 Mpc and ∼70%of pairs separated by 10-15h−1 Mpc. This effect would lower the averagey-value in the stacked filament and/or broaden the shape at some level, butthe simulation study implies that most LRG pairs should be connected by734.5. Comparison to BAHAMAS hydrodynamic simulationsdark matter (and presumably gas) filaments, and that dilution is unlikelyto be significant.In addition, there may be effects from asymmetric feature outside halos,due to the filaments extending out to different direction other than be-tween our LRG pairs. We estimate the effect assuming circularly symmetricdistribution of unaligned filaments around the halos. The aligned-filamentsbetween the LRG’s occupy roughly 10% with ∆y ∼ 1.0×10−8 region arounda circular halo, so one unaligned filament makes only a few % extra excesson top of the circular halo profile and the effect should be negligible.Finally, there might be an effect due to the Planck beam. We study thebeam effect for the filaments using the BAHAMAS simulations. Using thesmoothed y maps, the result is ∆y = 0.84×10−8 and it is ∆y = 1.00×10−8with the unsmoothed y maps. With the study, we find that the beam dilutesthe amplitude of y-value by ∼ 15%, although the mean separation anglebetween the LRGs is ∼ 0.7 deg, therefore, the beam effect should not besignificant in our study.4.5 Comparison to BAHAMAS hydrodynamicsimulations4.5.1 BAHAMAS hydrodynamic simulationsTo compare our results with theory, we analyze the BAHAMAS suite of cos-mological smoothed particle hydrodynamics (SPH) simulations (McCarthyet al., 2017) in the same manner as the data. The BAHAMAS suite isa direct descendant of the OWLS (Schaye et al., 2010) and cosmo-OWLSprojects (Le Brun et al., 2014; McCarthy et al., 2014; van Daalen et al.,2014). The simulations reproduce a variety of observed gas features in groupsand clusters of galaxies in the optical and X-ray bands. The BAHAMASsuite consists of box-periodic hydrodynamical simulations, the largest ofwhich have volumes of (400h−1Mpc)3 and contain 10243 each of baryonic anddark matter particles. The suite employs two different cosmological mod-els: WMAP9 cosmology (Hinshaw et al., 2013) with {Ωm,Ωb,ΩΛ, σ8, ns, h}= {0.2793, 0.0463, 0.7207, 0.821, 0.972, 0.700}, and Planck 2013 cosmology(Planck Collaboration, 2014a) with {Ωm,Ωb,ΩΛ, σ8, ns, h} ={0.3175, 0.0490, 0.6825, 0.834, 0.9624, 0.6711}. We have four realizations withthe WMAP9 cosmology and one with the Planck 2013 cosmology.From each realization, 10 almost-independent mock galaxy cataloguesare generated on a light cone and 10 corresponding y maps are generated744.5. Comparison to BAHAMAS hydrodynamic simulationsfrom the hot gas (McCarthy et al., 2014). Each of these light cones containabout one million galaxies out to z ∼ 1, and each spans a 10◦ × 10◦ patchof sky.4.5.2 Comparison with the hydrodynamic simulationsWe compare our results to these simulations. To do so, we analyze lightcones from the BAHAMAS suite of simulations (§4.5.1) as we did the realdata. For each cosmology, we construct simulated LRG pairs by selectingcentral galaxies with the same separation criteria as the real data. We in-voke a stellar mass threshold such that the mean stellar mass of the samplematches the mean of the data. The resulting catalog has 242,669 pairs.Prior to stacking, we also convolve the simulated y map in each light conewith a 10 arcmin FWHM Gaussian kernel to match the Planck map. Afterstacking and radial halo subtraction, we find the residual tSZ signal betweencentral galaxy pairs to be ∆y = (0.84± 0.24)× 10−8 with the WMAP9 cos-mology. The uncertainty is estimated by drawing 1000 bootstraps samplesfrom among the 40 light cones. We have also analyzed the simulations basedon the Planck 2013 cosmology and find ∆y = (1.14±0.33)×10−8. However,this model only has one realization of the initial conditions, instead of four,so it has a larger uncertainty than the WMAP9 estimate.The comparison of the simulations to the data is not entirely straight-forward because of possible selection effects. In particular, the methods forestimating stellar mass in these two systems are different. The data esti-mates we use are based on the principal component method in Chen et al.(2012), which are, on average, ∼0.2 dex higher than those based on spectro-photometric model fitting (Maraston et al., 2013). The simulation estimateswe use are based on directly counting the baryonic mass within 30 kpc ofa given central galaxy. As noted in §4.3.1, we adopt a stellar mass thresh-old of 1011.3 M for the data. In order to match the mean stellar mass ofthe simulation population, we must adopt a stellar mass threshold of 1011.2M. This procedure produces the same peak y values at the center of eachmean halo: data and simulation. We believe this selection should producecomparable filament amplitudes.In addition, we examine four independent realizations of the WMAP9cosmology and find that the mean residual tSZ signal between central galaxypairs varies from ∆y = 0.26 × 10−8 to 1.37 × 10−8 by factor of ∼ 5. Thissuggests the cosmic variance has a large impact on the simulations consid-ering the limited volumes in the simulations compared to the data coveringalmost one quarter of the sky.754.6. DiscussionWith the caveats noted above, we can use the simulations to furtherprobe the physical conditions in the observed filaments using the “unsmoothed“maps. In Figure 4.12, we separately examine the electron over-density andtemperature in the stacked simulation data. For each light cone in oursimulation box, we form optical depth and electron temperature maps,τ = σT∫ne dl, (4.12)〈Te〉 = 〈y〉〈τ〉 × kBmec2, (4.13)where σT is the Thomson scattering cross section, kB is the Boltzmannconstant, me is the electron mass, c is the speed of light, ne is the electronnumber density, Te is the electron temperature, and the integral is takenalong the radial direction. Using the mean over-density map, we fit a varietyof density profile models (§4.4) to the stacked τ map, and we find the best-fit profile to follow (rc/r), where r is the perpendicular distance from thecylinder axis, and rc = 0.5h−1Mpc, where rc is the core radius of the densityprofile. Assuming this profile, the best-fit central over-density is δ = 1.5 ±0.4. From the 〈Te〉 map, the mean (electron density-weighted) temperatureof the electron gas in the filament region (−0.8 < X < +0.8, −0.2 < Y <+0.2) is found to be (0.82± 0.06)× 107 K.4.6 DiscussionOther groups have studied filamentary gas in the large scale structure. Wecompare and contrast those results to ours as follows.The Planck Team (Planck Collaboration, 2013b) studied the gas be-tween the merging Abell clusters A399 and A401, which have a tangentialseparation of 3h−1Mpc. Using a joint analysis of Planck tSZ data andROSAT X-ray data, they estimate a gas temperature of kT = 7.1± 0.9 keV(T ∼ 8.2×107 K), and a central electron density of ne = (3.72±0.17)×10−4cm−3 (δ ∼ 1500). Assuming a filament diameter of 1.0 Mpc with a cylindri-cal shape, it corresponds to y ∼ 10−5. This high density and temperaturemay be because the filaments in merger systems have been shock-heated andcompressed more than normal.Using XMM-Newton observations, Werner et al. (2008) study the gasproperties in a filament connecting the massive Abell clusters A222 andA223, at redshift z ∼ 0.21. Assuming a separation of 15 Mpc, they findkT = 0.91±0.25 keV (T ∼ 1.1×107 K) and ne = (3.4±1.3)×10−5 cm−3 (δ ∼764.6. DiscussionFigure 4.12: Top left: The stacked y map of the central galaxy pairs from theBAHAMAS simulations, at 10 arcsecond angular resolution (unsmoothed).Top right: The same y map after the best-fit circular halos are subtracted.Middle left: The stacked τ map for the same pair sample as above. Middleright: The same τ map after circular halo subtraction. Bottom: The electrondensity-weighted temperature (Te) map on a log10 scale.774.6. Discussion150). In addition, Eckert et al. (2015) find filamentary structures aroundthe galaxy cluster Abell 2744, at z ∼ 0.3, and estimate a gas temperatureof T ∼ 107 K, and an over-density of δ ∼ 200 on scales of 8 Mpc. Assuminga filament diameter of 1.0 Mpc with a cylindrical shape, it corresponds toy ∼ 10−7, which is one order of magnitude higher than our result.Interestingly, as a study with similar targets, but using the CFHTLenSmass map, Epps and Hudson (2017) studied the weak lensing signal of fil-aments between SDSS-III/BOSS LRG’s and find a mass of (1.6 ± 0.3) ×1013M for a stacked filament region of 7.1 h−1Mpc long and 2.5 h−1Mpcwide. Assuming a uniform-density cylinder, they estimate δ ∼ 4 in thefilaments, consistent with our result.Considering the limited number of X-ray detections, it could be that ourresult of y ∼ 10−8 is typical of filaments between lower-mass galaxy groupsand clusters, but systematic effects may also contribute to the difference aswe discussed.A similar study using simulations is presented in Colberg et al. (2005),wherein 228 filaments between pairs of galaxy clusters are studied usingthe ΛCDM N -body simulation of Kauffmann et al. (1999). They identifystraight mass filaments longer than 5h−1 Mpc, normalize the length of eachfilament to unity, and find the average density of matter contained within2h−1 Mpc of the (normalized) filament axis to be δ ∼ 7. This is somewhathigher than our estimate of δ = 3.2±0.7, however, the following factors maycompromise this comparison.1) They select halos with masses larger than 1014M, whereas we selectLRGs with the stellar masses larger than 1011.3M. According to the SHMrelation used in Planck Collaboration (2013c) (see also Wang et al. (2016)),this corresponds to halo masses with M200 ∼ (5 − 7) × 1013M, and mayinclude lower mass systems, given the scatter in the SHM relation. Themass of filaments can be correlated with the mass of the halos associatedwith the filaments as West et al. (1995) suggests the cluster formation alongfilaments. This selection of low-mass halo pairs can result in the lower tSZsignal from the filaments between the LRGs.2) The ΛCDM simulation tracks dark matter, whereas we analyze hydro-dynamic simulations which include baryonic effects such as radiative cooling,star formation, SN feedback and AGN feedback. The baryonic gas in thefilaments may not trace the dark matter faithfully.3) They examine filaments between cluster pairs separated by 5h−1 to25h−1 Mpc, whereas we study smaller separations of 6-10h−1 Mpc.784.7. Conclusion4.7 ConclusionIn this chapter, using the Planck Sunyaev-Zeldovich (tSZ) map and theSDSS DR12 catalog of Luminous Red Galaxies (LRGs), we search for warm/hotfilamentary gas between pairs of LRGs by stacking the y map on a gridaligned with the pairs. We detect a strong signal associated with the LRGhost halos and subtract that using a best-fit, circularly symmetric model.We detect a statistically significant residual signal and draw the followingconclusions.• The residual tSZ signal in the region between LRG pairs is ∆y =(1.31 ± 0.25) × 10−8, with a 5.3σ significance. Assuming a simple,isothermal, cylindrical filament model of electron over-density with aradial density profile proportional to rc/r (as determined from simula-tions), we constrain the physical parameters of the gas in the filamentsto beδc(Te107 K)(rc0.5h−1 Mpc)= 2.7± 0.5. (4.14)• We apply the same analysis to the BAHAMAS suite of cosmologi-cal hydrodynamic simulations (McCarthy et al., 2017). The resultsare marginally consistent, but the simulations predict a slightly lowermean tSZ signal of ∆y = (0.84± 0.24)× 10−8.• Out result is comparable with the result in Epps and Hudson (2017)for the overdensity in the filaments. They study the weak lensingsignal of filaments between SDSS-III/BOSS LRG’s and estimate δ ∼ 4assuming a uniform-density cylinder of filaments. We also compare ourresults to complementary X-ray results, such as Werner et al. (2008)and Eckert et al. (2015). Our result is lower by a factor of ∼ 10, butour systems are much different.Our investigation can be extended with larger spectroscopic surveys suchas extended BOSS (eBOSS) in SDSS-IV and the Dark Energy SpectroscopicInstrument (DESI). Their larger samples would improve the signal-to-noiseand allow for a more detailed study of the physical conditions as a functionof LRG properties, such as stellar mass and redshift. Large-area experi-ments with higher tSZ angular resolution, such as the Atacama CosmologyTelescope and the South Pole Telescope, would also help to ascertain thestate of filament gas.79Chapter 5Probing hot gas in thecosmic web between galaxygroups and clusters5.1 IntroductionIn the previous chapter, we detect the tSZ signal due to gas filaments be-tween the ∼ 260,000 LRG pairs using the Planck y map. In this chapter,we search for the tSZ signal of the gas filaments between galaxy group andclusters, identified by (Tempel et al., 2014) based on the spectroscopic sam-ple of the galaxies of SDSS data release 10 (DR10) by following the sameanalysis procedure in chapter 4.The SDSS DR10 groups and clusters catalog is constructed from SDSSDR10 galaxies by friends-of-friends (FoF) algorithm, which has been themost frequently applied method of identifying groups and clusters in galaxyredshift data. The advantage to use the group catalog is that they areclearly identified as galaxy group/clusters in their dark matter halos. Thedrawback is that the number of available systems is limited to massive onesin the local Universe so far since the galaxy survey has to be complete in aredshift range, otherwise the identification can be biased.5.2 Pair stacking of galaxy group/clusters5.2.1 Galaxy groups and clusters for SDSS DR10 galaxiesGalaxies tend to gather in groups of several members, or even larger compan-ions since gravitationally bound galaxy systems are linked by an underlyingdark matter halo. However, there is no straightforward way to identify them.The friends-of-friends (FoF) algorithm has been the most frequently appliedto identify groups and clusters in galaxy redshift data. The FoF methoduses galaxy distances as the main basis of grouping, and thus it is relatively805.2. Pair stacking of galaxy group/clusterssimple and straightforward. (Tempel et al., 2014) uses the FOF method forthe SDSS DR10 galaxies (Ahn et al., 2014) with a variable linking lengthin the transverse and radial directions to identify as many realistic groupsas possible and constructed a flux-limited FoF group and cluster catalog,consisting of 82,458 galaxy groups.Dynamical masses of the galaxy group are estimated based on radialvelocity dispersions and group extent in the sky to the extracted groups. Forthe stacking analysis, we select the groups with Mdyn > 1013M. Accordingto the scaling relation of Y500-M∗ (Y500: the Comptonization parameterintegrated over a sphere of radius R500) reported in (Planck Collaboration,2013c), these groups should have a central tSZ signal-to-noise ratio of orderunity. Since our analysis requires us to estimate and subtract the tSZ signalassociated with the halos of the individual groups, this cut enhances thereliability of that estimation.5.2.2 SDSS DR10 group pair catalogWe construct the pair catalog from the SDSS DR10 group catalog by find-ing all neighboring groups within a tangential distance of 6-10 h−1Mpc andwithin a proper radial distance of ± 6h−1Mpc. The resulting catalog has34,955 group pairs to redshifts z ∼ 0.2. Their redshift and separation dis-tributions are shown in Figure 5.1.Figure 5.1: Left: The redshift distribution of group pairs peaks at z ∼ 0.08.Top right: The distribution of tangential separations between group pairs.Bottom right: The distribution of radial separations between group pairs.815.2. Pair stacking of galaxy group/clusters5.2.3 Stacking on group pairsWe follow the same analysis in the previous chapter and form a normalized 2-dimensional image coordinate system, (X,Y ), with one galaxy group placedat (−1, 0) and the other placed at (+1, 0). The corresponding transformationfrom sky coordinates to image coordinates is also applied to the y map andthe average is taken over all members in the catalog. The mean tSZ signalin the annular region 6 < r < 8 (r2 ≡ X2 +Y 2) is subtracted as an estimateof the local background signal.The top panel of Figure 5.2 shows the average y map stacked against34,955 group pairs over the domain −3 < X,Y < +3, and the lower panelshows a slice of this map at Y = 0. The average signal is dominated bythe halo gas associated with the individual groups in each pair. The peakamplitude of this signal is ∆y ∼ 2.5 × 10−7, which is higher than ∆y ∼1.4×10−7 of the LRG pairs. It suggests that more massive halos are selectedfor this study than the LRG halos in the previous chapter.5.2.4 Subtracting the halo contributionWe estimate the average contribution from single group halos by assumingthat the average single-halo contribution is circularly symmetric about eachgroup in a pair. When determining the radial profile of each single haloand performing the fit, we mask the central region −1 < X < +1 and−0.5 < Y < +0.5 from the fit. Figure 5.3 shows the resulting best-fitprofiles for the group halos.Figure 5.4 shows the residual y map after subtracting the best-fit circularprofiles shown in Figure 5.3. The bright halo signals appear to be cleanlysubtracted, while a residual signal between the groups is clearly visible. Thelower panels of Figure 5.4 show the residual signal in horizontal (Y = 0) andvertical (X = 0) slices through the map. The shape of the signal is consistentwith an elongated filamentary structure connecting average pairs of centralLRGs. The mean residual signal in the central region, −0.8 < X < +0.8and −0.2 < Y < +0.2, is found to be ∆y = 2.28× 10−8.5.2.5 Null tests and error estimatesTo assess the reality of the residual signal and estimate its uncertainty, weperform two types of Monte Carlo-based null tests: rotated null stacks andpseudo-pair null stacks as we did in §4.3.4. We find that the ensemble of nullmaps has a mean and standard deviation of ∆y = (0.01± 0.33)× 10−8 from825.2. Pair stacking of galaxy group/clustersFigure 5.2: Top: The average Planck y map stacked against 34,955 grouppairs in a coordinate system where one LRG is located at (X,Y ) = (−1, 0)and the other is at (X,Y ) = (+1, 0). The square region, −3 < X,Y < +3,comprises 151 × 151 pixels. Bottom: The corresponding y signal along theX axis.835.2. Pair stacking of galaxy group/clustersFigure 5.3: Top: The best-fit circular halo profiles fit to the map in Fig-ure 5.2. Bottom: The best-fit radial profile of the left and right halos shownabove.1000 rotated null stacks and ∆y = (0.07 ± 0.36) × 10−8 from 1000 pseudo-pair null stacks, which are shown in Figure 5.6. The average signals in thesenull tests are consistent with zero and the standard deviations are consistentwith each other. We adopt the standard deviation from the pseudo-pair nullstacks as the final uncertainty of the mean filament signal due to instrumentnoise, sky noise (i.e., cosmic variance and foreground rejection errors), andhalo subtraction errors.845.2. Pair stacking of galaxy group/clustersFigure 5.4: Top: The residual y-map after the best-fit radial halo signals aresubtracted. Bottom: The residual tSZ signal along the X and Y axes.855.2. Pair stacking of galaxy group/clustersFigure 5.5: The result from 1000 rotated null stacks. The data y value(∆y = 2.28× 10−8) is expressed in red-dash line.Figure 5.6: The result from 1000 pseudo-pair null stacks. The data y value(∆y = 2.28× 10−8) is expressed in red-dash line.865.3. Comparison to BAHAMAS hydrodynamic simulations5.3 Comparison to BAHAMAS hydrodynamicsimulationsWe compare our results to simulations. To do so, we analyze light conesfrom the BAHAMAS suite of simulations. However, the number of groupsin z < 0.2 is limited in the original light cones, therefore the simulations arecustomized for this study to span a 25◦×25◦ patch of sky for each light coneby limiting the redshift up to z ∼ 0.13. To hold the independence amongthe generated light cones, only 3 y maps are generated for each realizationof four, instead.For WMAP9 cosmology, we construct simulated group pairs by selectinggroup pairs with the same separation criteria as the real data. We invoke amass threshold such that the mean mass of the sample matches the meanof the data. For the simulations, we use M200m (mass enclosed within asphere of radius R200 such that the enclosed density is 200 times the mean“matter” density). The resulting catalog has 175,844 pairs.Prior to stacking, we also convolve the simulated y map in each light conewith a 10 arcmin FWHM Gaussian kernel to match the Planck map. Afterstacking and radial halo subtraction, we find the residual tSZ signal betweencentral galaxy pairs to be ∆y = (0.85± 0.37)× 10−8 with the WMAP9 cos-mology. The uncertainty is estimated by drawing 1000 bootstraps samplesfrom among the 12 light cones.The result from the BAHAMAS simulations is not consistent with thedata. However, the comparison of the simulations to the data is not en-tirely straightforward because of possible selection effects. In particular, themethods for estimating mass in these two systems are different. The dataestimates we use are based on radial velocity dispersions, which is only 1-dimensional information for 3-dimensional velocity. On the other hand, thesimulation estimates we use are based on directly counting the particle masswithin R200 of a given galaxy group. As noted, we adopt a mass thresholdof 1013 M for the data. We check that the mean mass as well as luminosityand velocity dispersions are consistent between the data and simulations.We believe this selection should produce comparable filament amplitudes.In addition, the galaxy groups are identified by the FOF algorithm forboth, but the group catalog in the data is identified by a varying linkinglength, on the other hand, the simulations use one typical value of linkinglength, which might be causing the difference in the identification of galaxygroups, thus physical parameters such as mass of the groups.Moreover, we examine four independent realizations of the WMAP9 cos-875.4. Interpretation of the detected tSZ signal between the group pairsmology and find that the mean residual tSZ signal between central galaxypairs varies from ∆y = 0.15 × 10−8 to 1.22 × 10−8 by factor of ∼ 10. Thissuggests the cosmic variance has a large impact on the simulations consid-ering the limited volumes in the simulations compared to the data coveringalmost one quarter of the sky.5.4 Interpretation of the detected tSZ signalbetween the group pairsWe can estimate the physical conditions of the gas we detect by consideringthe same model used in §4.4 with a density profile proportional to rc/r, atredshift z. For the 34,955 group pairs, the predicted tSZ signal in the region(−0.8 < X < +0.8, −0.2 < Y < +0.2) can be written as∆y¯ = 3.4× 10−8 ×(δc10)(Te107 K)(rc0.5h−1 Mpc). (5.1)Applying the observational constraint on mean ∆y, we have,δc(Te107 K)(rc0.5h−1 Mpc)= 6.7± 1.1. (5.2)Since the simulations are not consistent with the data unlike the case of theLRG pairs, we can not assume the temperature or density from the simu-lations. As we discussed, the comparison between the data and simulationsis not straightforward. Since we are limited to access the entire simulationsso far, it is probably fair to say that the issue is still under consideration.5.5 DiscussionOur result from the group pairs is y ∼ 2.28 × 10−8, which is a slightlyhigher than the result from the LRG pairs (y ∼ 1.31 × 10−8), however, itis still lower than the results from the Planck team (Planck Collaboration,2013b)(y ∼ 10−5) and X-ray observations (y ∼ 10−7) such as (Eckert et al.,2015; Werner et al., 2008). Therefore, the same discussion in §4.6 appliesto the group pairs. Here, we focus on the comparison in our results (LRGpairs and group pairs) as follows.For the difference, one of the possible reasons is the correlation betweenthe mass of filaments and the mass of halos associated with the filaments.The mean mass of halos consisting of the group pairs (y ∼ 2.5 × 10−7) is885.6. Conclusionclearly higher than the mass of the LRG halos (y ∼ 1.4 × 10−7) accordingto the peak y signal. Along with the higher SZ signal from the filamentsbetween the group pairs, this might imply the correlation between the massof filaments and the mass of halos associated with the filaments as Westet al. (1995) suggests the cluster formation along filaments.Another possibility is that the evolution of the filaments. The meanredshift of the LRGs is z ∼ 0.3, on the other hand, the mean of the groupsis z ∼ 0.1. During the time, the overdensity in the filaments grow and it maycause the difference. The time evolution of the mean density for filaments isstudied in (Cautun et al., 2014) using cosmological simulations and they findthat it is only a little in 0.3 < z < 0.1. However, to study these hypothesisusing the data, we run short of statistics for filaments (or sensitivity of theSZ effect) so far.On the other hand, it could be explained by some systematic effects.The tidal effect due to the group pairs enhances the tSZ signal in the regionbetween the pairs and it would have more impact for more massive pairs.In addition, the unconnected pairs by filaments lower the average tSZ signalbetween the pairs and it would have more impact for the LRG pairs sincethey are not clearly identified as “central galaxies” in galaxy groups. Theseeffects would mitigate the difference, although we believe that these effectsare not significant from our studies in §4.6.5.6 ConclusionIn this chapter, using the Planck Sunyaev-Zeldovich (tSZ) map and theSDSS DR10 groups and clusters catalog constructed by (Tempel et al., 2014),we search for warm/hot gas filamentary gas between pairs of groups bystacking the y map on a grid aligned with the pairs. We detect a strongsignal associated with the group halos and subtract that using a best-fit,circularly symmetric model. We detect a statistically significant residualsignal and draw the following conclusions.• The residual tSZ signal in the region between group pairs is ∆y =(2.28 ± 0.36) × 10−8, with a 6.1σ significance. Assuming a simple,isothermal, cylindrical filament model of electron over-density with aradial density profile proportional to rc/r (same model used for thefilamentary gas between the LRG pairs in the previous chapter 4), weconstrain the physical parameters of the gas in the filaments to beδc(Te107 K)(rc0.5h−1 Mpc)= 6.7± 1.1. (5.3)895.6. Conclusion• We apply the same analysis to the BAHAMAS suite of cosmologicalhydrodynamic simulations (McCarthy et al., 2017). The simulationspredict a lower mean tSZ signal of ∆y = (0.85 ± 0.37) × 10−8 andit is not consistent with our result. However, there are some factorsmaking the comparison hard, especially mass.• Our estimate of δ = 6.7 ± 1.1 from the group pairs we use assumingthe temperature of T ∼ 107 [K] is somehow higher than another ourestimate of δ = 3.2 ± 0.7 from the LRG pairs we use in the previouschapter 4. The mean mass of halos associated with the group pairs(y ∼ 2.5 × 10−7) is clearly higher than the mass of the LRG halos(y ∼ 1.4 × 10−7) according to the peak y signal. It might imply thecorrelation between the mass of filaments and the mass of the halosassociated with the filaments, but some of the systematics has to betaken into account.Our investigation can be extended with larger spectroscopic surveys suchas extended BOSS (eBOSS) in SDSS-IV and the Dark Energy SpectroscopicInstrument (DESI). Their larger samples would improve the signal-to-noiseand allow for a more detailed study of the physical conditions as a functionof group properties, such as mass and redshift. Large-area experiments withhigher tSZ angular resolution, such as the Atacama Cosmology Telescopeand the South Pole Telescope, would also help to ascertain the state offilament gas.90Chapter 6CMB polarization: A probeof the early universe6.1 CMB polarizationIn the very early universe, a tiny second after the Big Bang, it is hypoth-esized that the universe experienced a huge expansion of the space, called“Inflation”. The theory explains several important observations such asflatness, isotropy and homogeneity of the universe. Currently, it is the besttheory to explain what happened at the earliest moment of the universe,however, no direct evidence for inflation has been observed yet.One of the key observables is the so-called “B-mode polarization” inthe CMB. The accelerated expansion of space during inflation would havecreated ripples of gravitational waves in the space, and these gravitationalwaves could have left a distinctive polarization pattern in the CMB. There-fore, CMB polarization is one of the most important tools to probe inflation.A linear polarization field is specified with two types of CMB polariza-tion called “E-mode” and “B-mode”. They are analogous to the E andB vector fields of electromagnetism because they have similar behavior un-der parity inversion. The E-mode polarization has an even-parity and it ischaracterized by a curl-free mode with polarization vectors that are radialaround cold spots (E < 0) and tangential around hot spots (E > 0) onthe sky. The Planck stack more than 11,000 cold and 10,000 hot spots inthe CMB and measure the E-mode polarization patterns to high precision.They actually oscillates from the radial pattern to tangential pattern (orvice versa) around the cold(hot) spots as a function of scale. On the otherhand, the B-mode polarization has an odd-parity and it is characterized bya divergence-free (curl) mode.Both E-mode and B-mode are invariant under rotations, but when re-flected with a line through the center, the E-patterns remain unchanged,while B-patterns change the sign (Figure 6.1). The E-mode polarization isproduced by scalar perturbations (density fluctuations) and tensor perturba-916.1. CMB polarizationtions (primordial gravitational waves). However, the B-mode polarization isonly produced by tensor perturbations, therefore, the B-mode polarizationcan be used to probe inflation.A key cosmological parameter is characterized by the tensor-to-scalarratio, r, which determines the energy scale of inflation. Some inflationarymodels predict the energy scale to be near the GUT scale (the energy scaleabove which, it is believed, the electromagnetic force, weak force, and strongforce are unified to one force) of ∼ 1015 GeV. Such high energies can not begenerated in laboratory, so the early universe provides an environment totest theories at these energy scales. Therefore, many experiments are underway to search for B-mode polarization in the CMB.Figure 6.1: E-mode and B-mode patterns of polarization. Note that E-mode patterns are symmetric, while theB-mode patterns are anti-symmetricunder parity (Hu and White, 1997).6.1.1 E/B decompositionThe mathematical description of CMB temperature and polarization anisotropiesare summarized as follows. The linear polarization can be described bythe Stokes parameters Q and U , and the magnitude and angle are P =√Q2 + U2 and α = 12 tan−1 (U/Q). The CMB temperature anisotropy is926.1. CMB polarizationexpanded in terms of scalar (spin-0) spherical harmonicsT (nˆ) =∑l,maTlmYlm(nˆ), (6.1)where nˆ denotes the direction on the sky. The quantity T is invariant under arotation in the plane perpendicular to nˆ. On the other hand, the quantitiesQand U transform under rotation by an angle ψ as a spin-2 field (Q±iU)(nˆ)→e∓2iψ(Q ± iU)(nˆ). Therefore, the harmonic analysis of Q ± iU requiresexpansion on the sphere in terms of tensor (spin-2) spherical harmonics(Q± iU)(nˆ) =∑l,ma(±2)lm (±2)Ylm(nˆ). (6.2)The linear combination of multipole coefficient a(±2)lm can be defined,aElm ≡ −12(a(+2)lm + a(−2)lm ), aBlm ≡ −12i(a(+2)lm − a(−2)lm ). (6.3)Then one can define two scalar (spin-0) fields instead of the spin-2 quantitiesQ and U,E(nˆ) =∑l,maElmYlm(nˆ), B(nˆ) =∑l,maBlmYlm(nˆ), (6.4)to specify the linear polarization field of E and B modes (Baumann et al.,2009). Usually, angular power spectra of the E-mode and B-mode fields areused to evaluate their amplitude,CEEl = 〈aElmaE∗lm 〉, CBBl = 〈aBlmaB∗lm 〉. (6.5)Cross-correlations among the T -mode, E-mode and B-mode of harmoniccoefficients can be also evaluated mathematically, but only CTTl , CTEl , CEEl ,and CBBl are non-zero because the T -mode and E-mode have a positiveparity and the B-mode has a negative parity (Baumann et al., 2009).Fig. 6.2 shows the CMB angular power spectra of T -mode, E-mode andB-mode with varying values of the tensor-to-scalar ratio r based on thePlanck cosmology. The B-mode polarization in the CMB is generated onlyby the primordial gravitational waves. However, the CMB is distorted bythe cosmic structure between us and the last scattering surface, due to thelensing effect (CMB lensing), and it produces “apparent” B-mode. Theintrinsic CMB B-mode polarization is dominant over the B-mode due toCMB lensing for ` < 100, but if r < 0.001, it would be less dominant evenfor the scales.936.1. CMB polarizationFigure 6.2: Angular power spectra of CMB with varying tensor-to-scalarratio. The angular power spectra shows total intensity, E modes, primordialB modes and lensing B modes.6.1.2 Observable predictions and current observationalconstraintsIn 2014, the BICEP2 experiment announced the detection of a B-mode po-larization signal with an amplitude of r = 0.2 (BICEP2 Collaboration, 2014).This result requires an inflationary energy scale of about Einf = V1/4 '2×1016 GeV. However, as will be described in §6.2, foreground componentsfrom our Galaxy also produce E- and B-mode polarization signals and theyare much stronger than the CMB polarization. After the result was pub-lished, the Planck team together with BICEP and Keck teams reanalyzedthe data using the detailed galactic dust §6.2.3 measurements made possiblewith the large frequency coverage of Planck’s since the BICEP2 data comefrom only one frequency. The conclusion was that the BICEP findings arecompatible with a pure dust signal, yielding an upper limit for the tensorto scalar ratio of r < 0.11 (BICEP2/Keck Collaboration, 2015).The current constraint on r mainly driven by the measurements of thePlanck satellite mission:r < 0.09 (95%CL with TT,TE,EE + lowP), (6.6)where lowP denotes the polarization power spectrum in ` < 30 and the946.2. Foreground emission in the microwave bandssmaller scales of the foreground dominated region are not used. The Planckmeasurement of the B-mode polarization contributes little to the limit on rso far. The current limit on r is constrained by T -mode, and it is somewhatdegenerate with ns. Thus, better limits on ns provide better limits on r.6.2 Foreground emission in the microwave bandsIn the microwave range, four foreground emission components from ourGalaxy are known besides the CMB signal: synchrotron, free-free (unpo-larized), thermal dust, and spinning dust emission (Fig. 6.3). The BB po-larization power spectra due to foregrounds (dust and synchrotron) is muchhigher than the CMB polarization spectrum. The spinning dust emissionis very uncertain and hard to distinguish with the synchrotron emission.Therefore, new data at lower frequencies will provide important informa-tion on the galactic emissions and can be used to to subtract the polarizedforeground emissions from the CMB. In the next chapter, we simulate newdata at 10 GHz and evaluate its ability for the foreground removal. Beforethat, in this chapter, we summarize emission mechanisms of the foregroundsources (Rybicki and Lightman, 1979) and their current constraints (Dick-inson, 2016).6.2.1 Synchrotron emissionSynchrotron radiation is emitted by relativistic cosmic ray electrons accel-erated by the Galactic magnetic field. In the magnetic field, the electronsspiral around the field lines and emit radiation as traversing the circularpath. The emission reflects the number and energy spectrum of the CRelectrons and also the strength of the magnetic field. Therefore, it can varyacross the sky, but the spectrum is well-approximated by power-law.At GHz frequencies, the typical spectral indexes are β ≈ −2.7 withvariations ∆β ≈ ±0.2 (Platania et al., 1998, 2003). At higher frequencies,the spectrum appears to steepen further, presumably due to radiative losses,with β ≈ −3.0 around WMAP and Planck frequencies (Davies et al., 2006).The polarization of synchrotron is less known. Below a few GHz, it is affectedby the Faraday Rotation (Wolleben et al., 2006) and difficult to probe theoriginal state. It can be mapped at frequencies above a few GHz. WMAPand Planck have polarization measurements at low frequencies, however, themeasurements are limited in S/N ratio and the accurate spectral indexesare not derived yet. It is considered that the synchrotron emission can bepolarized at as high as 75 % in a uniform and regular magnetic field and956.2. Foreground emission in the microwave bandsFigure 6.3: Brightness temperature rms as a function of frequency and astro-physical component for temperature (top) and polarization (bottom) (PlanckCollaboration, 2016b) (arXiv:1502.01588). For temperature, each compo-nent is smoothed to an angular resolution of 1◦ FWHM, and the lower andupper edges of each line are defined by masks covering 81 and 93 % of thesky, respectively. For polarization, the corresponding smoothing scale is 40′, and the sky fractions are 73 and 93 %.966.2. Foreground emission in the microwave bandsseems to be polarized at a level of 10 - 40 % (Planck Collaboration, 2016e;Vidal et al., 2015).6.2.2 Free-Free emissionFree-free (bremsstrahlung) emission is the radiation by free electrons accel-erated due to the interaction with ions in an ionized gas (usually protons).Its spectrum is well understood. At frequencies above a few GHz, it has aspectral index of β = -2.1 with very little variations with electron tempera-tures. At higher frequencies around 100 GHz, the spectral index is steepenedslightly to β = -2.13 (Planck Collaboration, 2014d).Free-free emission is intrinsically unpolarized. Coulombic interactionsare by their nature random and no significant alignment with the magneticfield. Residual polarization can be generated on sharp edges due to Thomsonscattering, however, the polarization is expected to be very low ( 1 %) withcurrent upper limit at < 3 % for diffuse emission and  1 % for compactHII regions (Macellari et al., 2011). Therefore, the free-free emission wouldnot be a major foreground for CMB polarization measurements.6.2.3 Thermal dust emissionThermal dust emission is produced by dust grains, absorbing ultra-violetphotons from the exciting radiation field and re-radiating thermally. Themajority of the dust emission is radiated from star forming regions, wherethe dust is heated by nearby young stars. Dust grains are made of silicateand graphite, or coated with ices in cold regions. The distribution of thegrain size shows more small grains and fewer large grains with an averagesize of ∼ 0.1 mm. Dust is usually located with H2 in molecular clouds,with the mass ratio of M(dust)/M(H2) ∼ 0.01. Thermal dust emission iscommon in the Milky Way as well as other galaxies.Thermal dust emission is represented by a modified black body functionof the form in antenna temperature:TA(ν) ∝ νβdB(ν, Td), (6.7)where βd is the dust spectral index and Td is the equilibrium tempera-ture of dust grains. According to the Planck measurements combined withIRAS/COBE data, the thermal dust emission is well modeled by a singlemodified blackbody with mean temperature Td ≈ 19 K and index βd ≈ 1.6(Planck Collaboration, 2016g). Thermal dust emission can be significantlypolarized. Elongated dust grains emit preferentially along their shortest976.2. Foreground emission in the microwave bandsaxes while large dust grains can align efficiently by the Galactic magneticfield, causing a net polarization. The Planck measurements find the meanpolarization fraction of ≈ 10 % at high latitude and it is polarized up to ≈20 % (Planck Collaboration, 2015, 2016h).6.2.4 Spinning dust emissionIn our own galaxy and other galaxies, there is recent evidence of so-called“anomolous radio emission”, plausibly from spinning dust grains. Draineand Lazarian (1998) proposed that very small grains containing 10 - 100carbon atoms could be spun up to tens of GHz frequencies in the ISM(Inter Stellar Medium). Electric dipole moments from the spinning dustgrains likely produce radio emission at these frequencies with the spectrumdepending on their local conditions. Thus, in addition to free-free, thermaland synchrotron emission, spinning dust emission is a fourth componentthat is present in radio emission from other external galaxies and our ownMilky Way.According to Ali-Hamoud et al. (2009) and Silsbee et al. (2011), currentspinning dust models can predict the spectral shape including a variety ofphysical processes such as collisions with neutral and ionized gas, plasmadrag, absorption of photons etc. The peak frequency of the spinning dustspectrum is determined by the size of the smallest grains, and the total powerfollows the same dependence, the power emitted by the smallest grains. In afew Galactic clouds, observations show a good fit to the model spectrum witha typical peak frequency of ≈ 30 GHz. However, at high Galactic latitudes,the spectrum of spinning dust has not been measured clearly due to thedifficulty of component separation. The theoretical work also suggests thatthe spinning dust emission is not highly polarized at frequencies above a fewGHz. Measurements show that the level of polarization is only a few percentfor an upper limit (Dickinson et al., 2011; Rubin˜o Mart´ın et al., 2012) andthus, the spinning dust emission should not be a major foreground for CMBpolarization measurements.98Chapter 7A 10GHz polarization skysurvey7.1 IntroductionB-mode polarization in the CMB is a direct tracer of tensor perturbationscaused by gravitational waves in the inflationary period of the early universe.However, galactic foreground sources also produce E- and B-mode polar-ization, and are much stronger than the CMB polarization. The dominantsource of the polarized emission is galactic synchrotron and dust emission,but especially synchrotron emission is currently characterized with low sig-nal/noise. We are developing a project called the Canadian Galactic Emis-sion Mapper (CGEM) to produce half-sky maps of total intensity and linearpolarization at 10 GHz to constrain synchrotron emission as well as spinningdust emission.In this chapter, we simulate polarized skies using version 1.0 of PythonSky Model (PySM) software package (Thorne et al., 2016). The software isused to generate full-sky simulations of Galactic foregrounds in intensity andpolarization at microwave frequencies. The components simulated are ther-mal dust, synchrotron, AME (Spinning dust) and CMB based on publiclyavailable data from the WMAP and Planck satellite missions. Small-scalerealizations of these components at resolutions greater than current all-skymeasurements are also added.Using this software, we simulate the 10 GHz polarized sky and then ob-servations with estimated noise appropriate to one-year observation fromPenticton BC in Canada. The simulated 10 GHz data is combined withcurrently available data from WMAP + Planck and data from the futuresatellite mission of LiteBIRD (Matsumura et al., 2014) (Their estimatedinstrumental noises are also simulated) and then the improvement of fore-ground estimates with/without the new data is evaluated. We use a Markovchain Monte Carlo (MCMC) algorithm to derive the probability distribu-tions of parameters for the foreground emissions and CMB such as their997.2. Field of viewamplitudes and spectral indexes.7.2 Field of viewHere, we will verify the field of view of the telescope in galactic coordinates.In CGEM, the telescope is proposed to rotate with an opening angle 40◦from the zenith at Penticton at a rotation rate of 1 rpm, see in Fig. 7.1.To simulate this, we transform the line of sight from telescope coordi-nates to Penticton coordinates. With the Euler angles, the coordinates aretransformed by xtytzt = Rz(ψ)Rx(θ)Rz(φ) xpypzp (7.1)= cosψ sinψ 0− sinψ cosψ 00 0 1 1 0 00 cos θ sin θ0 − sin θ cos θ cosφ sinφ 0− sinφ cosφ 00 0 1 xpypzp ,(7.2)where (xt,yt,zt) = (0, 0, 1) is in telescope coordinates and (xp,yp,zp) is inPenticton coordinates, and where a precession opening angle is ψ = 40◦, aprecession angle is θ = 360◦ × t(sec)/60(sec) and a spin angle is φ = 0◦.Figure 7.1: The image of observation strategy at Penticton. The red arrowis the direction of observation, which rotates once a minute with an openingangle 40◦ and blue arrow is the direction to NCP.1007.2. Field of viewNext, we transform the line of sight from Penticton coordinates to Earthcoordinates. Since Penticton is located at the latitude, 49◦29′ N, the direc-tion of the telescope in Earth coordinates is given by xpypzp = Ry(90◦ − θ) xeyeze (7.3)= cos (90◦ − θ) 0 − sin (90◦ − θ)0 1 0sin (90◦ − θ) 0 cos (90◦ − θ) xeyexe , (7.4)where (xe,ye,ze) is in Earth coordinates and θ = 49.5◦.The third step is to transform the line of sight from Earth coordinatesto celestial coordinates. The earth spins around its axis once a day based onthe sidereal time, which is the time it takes the earth to make one rotationrelative to the vernal equinox. A mean sidereal day is 23 hours, 56 minutes,4.0916 seconds, a little shorter than a solar day. xeyeze = Rz(LST ) xcyczc (7.5)= cos (LST ) sin (LST ) 0− sin (LST ) cos (LST ) 00 0 1 xcycxc , (7.6)where (xc,yc,zc) is in celestial coordinates, and LST , a local sidereal time,is given byLST (deg) = 360◦×frac(0.671262+1.00273790935×(MJD−40000)+longitude360◦)(7.7)where longitude is −119.6◦ at Penticton, MJD stands for Modified JulianDate, and ”frac” represents the decimal part of the result.The final step is to transform the line of sight from celestial coordinatesto galactic coordinates. The right ascension and declination of the galacticcenter is 266.40500 deg and -28.93617 deg, so the rotation along the z-axis by266.40500◦ and along the y-axis by 28.93617◦ points to the galactic center.In addition, the rotation along the x-axis by 58.59866◦ determines the angle1017.3. Simulations of the observational data in polarizationof the galactic place, so xgygzg = Rx(ψ)Ry(θ)Rz(φ) xcyczc (7.8)= 1 0 00 cosψ sinψ0 − sinψ cosψ cos θ 0 − sin θ0 1 0sin θ 0 cos θ cosφ sinφ 0− sinφ cosφ 00 0 1 xcyczc ,(7.9)where (xg,yg,zg) is in galactic coordinates, and where ψ = 58.59866◦, θ =28.93617◦ and φ = 266.40500◦.With this sequence of transformations, we can obtain the direction ofthe telescope in galactic coordinates at any time. Fig. 7.2 show the hit mapsof one-year observation in galactic coordinates. 42% of all the sky is coveredwith this observing strategy.Figure 7.2: Hit map of one-year of observations, in galactic coordinates(Nside = 64).7.3 Simulations of the observational data inpolarization7.3.1 Simulations of the polarized skiesThe PySM software (Thorne et al., 2016) has template maps and spectralindex maps of four known components in the microwave range: synchrotron,1027.3. Simulations of the observational data in polarizationthermal dust, spinning dust and CMB. The CMB map is simulated froman input power spectrum defined by a set of cosmological parameters (Ωb =0.0486, Ωm = 0.3075, Ωk = 0.0, H0 = 67.74 km s−1 Mpc−1, As = 2.14×10−9,ns = 0.9667, τ = 0.066) derived by Planck+BAO+JLA+H0. We use r =0.05 for the tensor-to-scalar ratio, just below the current upper limits ofr < 0.09 (BICEP2/Keck Collaboration, 2015). Note that we use a pivotscale of k0 = 0.05 Mpc1 but the results are very weakly dependent on thisbecause we have no tilt in the tensor spectrum (nT = 0.0). The inputtheoretical power spectrum is calculated using the CAMB software (Lewiset al., 2000). We do not include the effects of gravitational lensing, whichwould contribute significant power at scales ` > 100.Using these maps, the polarized sky at 10 GHz and other frequencyranges can be simulated withQmodel(ν, nˆ) = Qs(nˆ)×( ν23)βs(nˆ)+Qd(nˆ)×( ν353)βd(nˆ)Bν(Td(nˆ)) +QCMB(nˆ),(+Qsp(nˆ)× (ν, νpeak(nˆ)))(7.10)Umodel(ν, nˆ) = Us(nˆ)×( ν23)βs(nˆ)+ Ud(nˆ)×( ν353)βd(nˆ)Bν(Td(nˆ)) + UCMB(nˆ),(+Usp(nˆ)× (ν, νpeak(nˆ)))(7.11)where the subscripts, s, d, sp and CMB, represent the synchrotron, ther-mal dust, spinning dust and CMB respectively at a direction, nˆ, and  isthe emissivity function of spinning dust emission calculated using Spdust2(Ali-Hamoud et al., 2009; Silsbee et al., 2011), evaluated for a cold neutralmedium, where νpeak is the peak frequency of the emission varying spa-tially. We set Td(nˆ) = 18 [K] and νpeak(nˆ) = 30 [GHz] in our simulation asRemazeilles et al. (2016) did.For the frequencies, ν, we simulate the sky at 10 GHz, at the frequenciesof LiteBIRD (60, 78, 100, 140, 195, 280 [GHz]) and at lower frequencies cur-rently available from WMAP and Planck (23, 30, 33, 41, 44 [GHz]), whichwould be profitable to reconstruct the low frequency components such assynchrotron and spinning dust emissions, which have not been well under-stood. The simulated polarization Q and U maps are shown in Fig. 7.3 andFig. 7.4.1037.3. Simulations of the observational data in polarizationFigure 7.3: The simulated stokes Q maps at10,23,30,33,41,44,60,78,100,140,195,280 GHz in the galactic coordinates(Nside = 128).1047.3. Simulations of the observational data in polarizationFigure 7.4: The simulated stokes U maps at10,23,30,33,41,44,60,78,100,140,195,280 GHz in the galactic coordinates(Nside = 128).1057.3. Simulations of the observational data in polarization7.3.2 Noise estimateWe estimate the noise of CGEM’s 10 GHz observations assuming Tsys (sys-tem temperature) = 50 (K), ∆ν (band width) = 2×109 (Hz) and τ (samplinginterval) = 0.05 (sec),σ0(@10GHz) =Tsys√∆ντ=50√2× 109 ∗ 0.05 = 5(mK). (7.12)This give a noise per observation of 5 mK. The noise for the map is calculatedby σ(ν, nˆ) = σ0(ν)/√Nobs(ν, nˆ), where σ0(ν) is the noise per observation ateach frequency and Nobs(ν, nˆ) is the number of observations at each pixeland frequency. Nobs(ν, nˆ) of the 10 GHz data is obtained from the hit mapof one-year observation (Fig. 7.2). With the given pixel noise, σ(ν, nˆ), wecan simulate random noise at each pixel using a Gaussian random numbergenerator, N(0, 1), followed by Q(U)noise(ν, nˆ) = N(0, 1)× σ(ν, nˆ).Here, we simply estimate the mean noise at each frequency, σ¯(ν) =σ0(ν)/√Nobs(ν), where N¯obs(ν) is the mean number of observations at apixel, estimated by N¯obs(ν) =∑nˆNobs(ν, nˆ)/Npix. We distribute the ran-dom noise to each pixel by Q(U)noise(ν, nˆ) = N(0, 1) × σ¯(ν) and simulatethe observational data,Q(U)obs(ν, nˆ) = Q(U)model(ν, nˆ) +Q(U)noise(ν, nˆ). (7.13)The data now include three foreground emissions, CMB and noise.For the WMAP data, the noise per observation at each frequency isdescribed in (Jarosik et al., 2011) and the hit maps are provided in 10. Forthe Planck data, the covariance matrices of I, Q and U maps, and hit mapsare provided in 11. The mean noise at each frequency for this study issummarized in Table 7.1.Table 7.1: Mean noises of stokes Q/U map at each frequency.Frequency [GHz] 10 23 30 33 41 44 60 78 100 140 195 280Mean noise [uK] 15 15 10 15 15 12 0.375 0.236 0.170 0.135 0.113 0.138The mean noise of stokes Q/U map at each frequency is given in CMBthermodynamic unit at Nside = 128.10https://lambda.gsfc.nasa.gov/product/map/dr5/m products.cfm11http://pla.esac.esa.int/pla/#results1067.4. Markov chain monte carlo simulation7.4 Markov chain monte carlo simulationIn this section, we assess the utility of 10 GHz data to separate galactic fore-ground emissions and CMB from the simulated observational maps. Theseemission components are expressed with 12 parameters, { Qs, Qd, Qsp,QCMB, Us, Ud, Usp, UCMB } (the amplitude of each emission componentat a reference frequency) and { βs, βd, Td, νpeak } (the spectral index andshape parameter of the relevant emission component). A Markov ChainMonte Carlo (MCMC) method is used to acquire a probability distributionof each parameter. We show that 10 GHz data helps to determine these pa-rameters more precisely by comparing the result with/without the 10 GHzdata.For this analysis, we adopt ”emcee” software (Foreman-Mackey et al.,2013), the affine invariant ensemble sampler proposed by Goodman andWeare (2010). The software is provided as open source and it has been usedfor many astrophysics projects.7.4.1 Model functionMCMC can be used to find a set of parameters that best fits data for aproposed model function. For our model, we use the same function used tosimulate data (Eq. 7.10, 7.11).y1(ν, nˆ) = p0(nˆ)×( ν23)p2(nˆ) ×+p3(nˆ)× ( ν353)p5(nˆ)Bν(p6(nˆ))×+p7(nˆ)+p9(nˆ)× (ν, p11(nˆ)),y2(ν, nˆ) = p1(nˆ)×( ν23)p2(nˆ) ×+p4(nˆ)× ( ν353)p5(nˆ)Bν(p6(nˆ))×+p8(nˆ)+p10(nˆ)× (ν, p11(nˆ)),(7.14)where y1,2 is a function for each Q/U component.The χ2 of our fit isχ2(nˆ) =∑ν{(Q(ν, nˆ)obs − y1(ν, nˆ))2σ(ν)2+(U(ν, nˆ)obs − y2(ν, nˆ))2σ(ν)2}, (7.15)and the likelihood function is calculated by assuming that the probabilityfollows Gaussian distribution,Probability(nˆ) = exp(−χ2/2). (7.16)1077.4. Markov chain monte carlo simulationLikelihood(nˆ) = −0.5× χ. (7.17)The likelihood is maximal when the sum of square errors is minimal. Wefind that the dust temperature can not be well constrained with the chosenfrequency range, therefore, we adopt a Gaussian prior for the dust tempera-ture Td = 18±0.05 K. With this prior, we confirm that the input parametervalues can be recovered successfully by the MCMC fitting, before includinginstrumental noises.7.4.2 Improvement by the 10GHz data (without spinningdust)Next we simulate the observational data, Q(U)obs(ν, nˆ), including instru-mental noises. For the noises, we consider two cases at 10 GHz, 62 uK andalso 15 uK, which can be reached by recent polarimeters. (In this paper,the main results are shown with the noise of 15 uK.) First, we consider asimple case for the foregrounds, assuming that the spinning dust emission isnegligible. Later, we include the spinning dust emission in the foregroundsboth for the input data and model function assuming it is polarized at the2% level.With the emcee package, the Markov chain provides a probability distri-bution of each parameter in the model function, y(ν, nˆ). Fig. 7.5 shows thecorrelation matrix of the fit parameters from the MCMC fitting at a samplepixel. The red lines are the result with the 10 GHz data (w/10GHz) and bluelines are the one without the 10 GHz data (w/o10GHz). The histograms onthe diagonal show the marginalized distribution for each parameter and thegreen lines show the input (“true”) values. The off-diagonal figures repre-sent the correlation between any two components and the three circles show1,2,3 σ confidence level. The figure shows that the parameters for the syn-chrotron emissions are constrained better with the 10 GHz data as expected,however, the improvement is marginal for the other parameters.With the best-fit (“output”) values, the spectra of each emission com-ponent and the total emission in the target range of frequency can be recon-structed with the 10 GHz data and without it. One example, at a samplepixel, is shown in Fig. 7.6 and Fig. 7.7. The solid lines are the input spectraand the dash lines are the output spectra from the MCMC fitting. As seenin the figures, the input synchrotron spectra (green solid line) can be recov-ered with the MCMC fitting well when including the 10 GHz data, howeverit can not without it.In other pixels, since each emission component contributes to the totalwith a different amplitude, the improvement should be dependent on it.1087.4. Markov chain monte carlo simulationTherefore, we run MCMC for all the pixels to generate output parametermaps that can be compared to the input parameter maps. We also producethe residual maps, which are made by subtracting the output parametermaps from the input parameter maps. Fig. 7.8, Fig. 7.9 and Fig. 7.10 showthe input, output, residual maps for Qs, Us, βs. In the figures, the w/10GHzcase and the w/o10GHz case are also compared. In the w/o10GHz case, largeresiduals are seen, but they are not clear in the w/10GHz case.Finally, we show the uncertainties of the fit parameters in Table 7.2.The uncertainty is defined by one sigma standard deviation as a result ofGaussian fit to the probability distribution for each parameter. The uncer-tainty of the synchrotron amplitude is improved by a factor of ∼ 2.7 and ∼3.1 for the spectral index. However, it is ∼ 1.3 for the CMB and little forthe dust emission because the CMB and dust emission are already relativelywell constrained by the sensitive LiteBIRD data at higher frequencies.According to current observations, the B-mode polarization in the CMBis expected to be extremely faint and many “sensitive” experiments areunder way to search for it. Therefore, we need to remove all the contami-nating signals accurately, otherwise it biases the view of the early universe.Our result shows that new 10 GHz data will allow the contamination tobe subtracted more accurately by combining with proposed sensitive high-frequency measurements such as LiteBIRD, and play an essential role inquesting for the tiny signal in the CMB and understanding the origin of ouruniverse.Table 7.2: Uncertainty of the fit parameters by MCMC [uK] (without spin-ning dust).Qs Us βs Qd Ud βd Qcmb Ucmbw/o 10GHz 3.6 2.9 0.58 0.069 0.064 0.26 0.16 0.14w/ 10GHz(62uK) 2.1 1.8 0.30 0.065 0.060 0.22 0.14 0.13w/ 10GHz(15uK) 1.3 1.1 0.19 0.060 0.055 0.19 0.12 0.117.4.3 Improvement by the 10GHz data (with spinning dust)Next we include the spinning dust emission both in the input data and modelfunction, and repeat the study.Fig. 7.11 shows the correlation matrix of the fit parameters from theMCMC fitting at a sample pixel. The figure tells that the fit parameters forsynchrotron and spinning dust emission are constrained better with the 101097.4. Markov chain monte carlo simulationFigure 7.5: Correlation maps among the fit parameters (without spinningdust) obtained by the MCMC fitting at a sample pixel. Red lines are theresult with the 10 GHz data and blue lines are the result without the 10GHz data. The histograms at the diagonal are projections of the samples byMCMC to each parameter axis and the green lines show the input values.The figures at the off-diagonal represent the correlation between any twocomponents and the three circles show 1,2,3 σ confidence level. It is clearlyseen that the synchrotron parameters are well constrained with the 10 GHzdata. The CMB is better constrained modestly along with it.1107.4. Markov chain monte carlo simulationFigure 7.6: (Left): Stokes Q spectra (without spinning dust) estimated bythe MCMC fitting with the 12 band data including 10 GHz in a sample pixel.The solid lines are the input spectra and the dash lines are the output spectraby the MCMC fitting. (Right): Spectra obtained by the MCMC fitting withthe 11 band data, not including 10 GHz, in the same pixel.Figure 7.7: (Left): Stokes U spectra (without spinning dust) estimated bythe MCMC fitting with the 12 band data including 10 GHz in a sample pixel.The solid lines are the input spectra and the dash lines are the output spectraby the MCMC fitting. (Right): Spectra obtained by the fitting with the 11band data, not including 10 GHz, in the same pixel.1117.4. Markov chain monte carlo simulationFigure 7.8: Input, output(best-fit) and residual maps of Qs estimated byMCMC, without spinning dust. Top: Input Qs map at 23 GHz, middleleft: Output map with 10 GHz data, middle right: Residual map with 10GHz data, bottom left: Output map without 10 GHz data, and bottom right:Residual map without 10 GHz data.1127.4. Markov chain monte carlo simulationFigure 7.9: Input, output(best-fit) and residual maps of Us estimated byMCMC, without spinning dust. Top: Input Us map at 23 GHz, middleleft: Output map with 10 GHz data, middle right: Residual map with 10GHz data, bottom left: Output map without 10 GHz data, and bottom right:Residual map without 10 GHz data.1137.4. Markov chain monte carlo simulationFigure 7.10: Input, output(best-fit) and residual maps of βs estimated byMCMC, without spinning dust. Top: Input βs map, middle left: Outputmap with 10 GHz data, middle right: Residual map with 10 GHz data,bottom left: Output map without 10 GHz data, and bottom right: Residualmap without 10 GHz data.)1147.4. Markov chain monte carlo simulationGHz data as expected, however, the improvement is marginal for the otherparameters.Fig. 7.12 and Fig. 7.13 show the spectra of each emission component andthe total emission reconstructed with the output values at a sample pixel.As seen in the figures, the input synchrotron spectra (green solid line) andspinning dust spectra (cyan solid line) can be reconstructed by the MCMCfitting well with the 10 GHz data, however it can not without it.We run MCMC for all the pixels to estimate the average improvement.Fig. 7.14, Fig. 7.15 and Fig. 7.16 show the input, output, residual maps forQs, Us, βs. In the figures, the w/10GHz case and w/o10GHz case are alsocompared. In the w/o10GHz case, large residuals are seen, but that is notclear in the w/10GHz case. We also show the input, output, residual mapsfor Qsp and Usp in Fig. 7.17 and Fig. 7.18. Unlike the correlation matrix ina sample pixel, the spinning dust emission is not improved very much withthe 10 GHz data. The reason is because spinning dust emission is negligiblyweak in any frequency for most of the pixels. Such a weak signal is by nomeans recovered well, however the effect to the CMB would be also limited.Finally, we show the uncertainties of the fit parameters in Table 7.3. Theuncertainty of the synchrotron amplitude is improved by a factor of ∼ 2.4and ∼ 2.7 for the spectral index. This is a little worse than the case withoutspinning dust. This suggests that some of the contribution of the 10 GHzdata are used to improve the spinning dust emission. The improvementfactor is ∼ 1.3 for the CMB and ∼ 1.5 for the spinning dust emission, andlittle for the dust emission. The marginal average improvement of spinningdust emission is due to the same reason as mentioned above.Table 7.3: Uncertainty of the fit parameters by MCMC [uK] (with spinningdust).Qs Us βs Qd Ud βd Qcmb Ucmb Qsp Usp νspw/o 10GHz 3.4 2.7 0.54 0.074 0.068 0.24 0.16 0.14 0.21 0.17 8.5w/ 10GHz(62uK) 2.1 1.8 0.31 0.070 0.064 0.21 0.14 0.13 0.18 0.14 7.2w/ 10GHz(15uK) 1.4 1.1 0.20 0.065 0.060 0.19 0.12 0.11 0.14 0.12 6.21157.4. Markov chain monte carlo simulationFigure 7.11: Correlation maps among the fit parameters (with spinning dust)obtained by the MCMC fitting at a sample pixel. Red lines are the resultwith the 10 GHz data and blue lines are the result without the 10 GHz data.The histograms at the diagonal are projections of the samples by MCMC toeach parameter axis and the green lines show the input values. The figuresat the off-diagonal represent the correlation between any two componentsand the three circles show 1,2,3 σ confidence level. It is clearly seen thatthe synchrotron parameters are well constrained with the 10 GHz data. Thespinning dust emission and CMB are better constrained along with it.1167.4. Markov chain monte carlo simulationFigure 7.12: (Left) Stokes Q spectra (with spinning dust) estimated by theMCMC fitting with the 12 band data including 10 GHz, in a sample pixel.The solid lines are the input spectra and the dash lines are the output spectraby the MCMC fitting. (Right) Spectra obtained by the MCMC fitting withthe 11 band data, not including 10 GHz, in the same pixel.Figure 7.13: (Left) Stokes U spectra (with spinning dust) estimated by theMCMC fitting with the 12 band data including 10 GHz, in a sample pixel.The solid lines are the input spectra and the dash lines are the output spectraby the MCMC fitting. (Right) Spectra obtained by the fitting with the 11band data, not including 10 GHz, in the same pixel.1177.4. Markov chain monte carlo simulationFigure 7.14: Input, output(best-fit) and residual maps of Qs estimated byMCMC, with spinning dust. Top: Input Qs map at 23 GHz, middle left:Output map with 10 GHz data, middle right: Residual map with 10 GHzdata, bottom left: Output map without 10 GHz data, and bottom right:Residual map without 10 GHz data.1187.4. Markov chain monte carlo simulationFigure 7.15: Input, output(best-fit) and residual maps of Us estimated byMCMC, with spinning dust. Top: Input Us map at 23 GHz, middle left:Output map with 10 GHz data, middle right: Residual map with 10 GHzdata, bottom left: Output map without 10 GHz data, and bottom right:Residual map without 10 GHz data.1197.4. Markov chain monte carlo simulationFigure 7.16: Input, output(best-fit) and residual maps of βs estimated byMCMC, with spinning dust. Top: Input βs map, middle left: Output mapwith 10 GHz data, middle right: Residual map without 10 GHz data, bottomleft: Output map with 10 GHz data, and bottom right: Residual map without10 GHz data.1207.4. Markov chain monte carlo simulationFigure 7.17: Input, output(best-fit) and residual maps of Qsp estimated byMCMC, with spinning dust. Top: Input Qsp map at 22.8 GHz, middleleft: Output map with 10 GHz data, middle right: Residual map with 10GHz data, bottom left: Output map without 10 GHz data, and bottom right:Residual map without 10 GHz data.1217.4. Markov chain monte carlo simulationFigure 7.18: Input, output(best-fit) and residual maps of Usp estimated byMCMC, with spinning dust. Top: Input Usp map at 22.8 GHz, middle left:Output map with 10 GHz data, middle right: Residual map with 10 GHzdata, bottom left: Output map without 10 GHz data, and bottom right:Residual map without 10 GHz data.1227.5. Conclusion7.5 ConclusionUsing version 1.0 of the Python Sky Model (PySM) software package (Thorneet al., 2016), we simulate the 10 GHz polarized maps and observations withestimated noise appropriate to one-year observation from Penticton BC withthe CGEM telescope. The simulated 10 GHz data is combined with cur-rently available data from WMAP + Planck and simulated data from theproposed satellite mission LiteBIRD (Matsumura et al., 2014). The improve-ment of foreground reconstruction with/without the new data is evaluatedusing the Markov chain Monte Carlo (MCMC) algorithm.• Assuming the spinning dust emission is negligible, the uncertainty ofthe synchrotron amplitude is improved by a factor of ∼2.7 and thespectral index is by a factor of ∼3.1, on average.• By including spinning dust emission in the simulation, the uncertaintyof the synchrotron amplitude is improved by a factor of ∼2.4 and thespectral index is by a factor of ∼2.7, on average• The improvement in the recovery of the spinning dust, thermal dust,and CMB signals are modest (less than a factor of 2) over most ofthe sky. In the case of the CMB and thermal dust components, thedata are rather well constrained by the more-sensitive higher frequencydata, especially from the proposed LiteBIRD mission. 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