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Galaxy cluster studies with weak lensing magnification & shear Ford, Jessica Lange 2015

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Galaxy Cluster Studieswith Weak Lensing Magnification & ShearbyJESSICA LANGE FORDB.Sc. Physics, The University of Nevada Reno, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2015c© JESSICA LANGE FORD, 2015AbstractClusters of galaxies offer a unique window for studying the Universe on the largest scales. As the most massivegravitationally bound systems to have formed, they serve as probes of the large-scale distributions of dark matter,the underlying cosmology, and the complicated intracluster physics that characterizes the evolution of thesemassive systems. Gravitational lensing is the deflection of light coming from distant sources, by gravitationalpotentials along its path. Being sensitive to all mass regardless of type or dynamical state, lensing is a valuabletool for studying dark matter and characterizing galaxy clusters. In the weak lensing regime, the very slightapparent distortion of galaxy shapes is referred to as the shear, while the focusing and amplification of light isreferred to as the magnification. The former has become a well-developed and robust technique in astronomyover the past decade, but the latter has been largely overlooked until now.The work embodied in this thesis includes the first-ever significant detection of magnification by galaxygroups, and the first comparison between masses measured with weak lensing magnification and shear (Chap-ter 2). This is followed by an application to an enormous sample of galaxy clusters, yielding ground-breakingsignal-to-noise for magnification and an analysis of redshift-dependent systematic effects. This project alsoprovides measurements of the cluster mass-richness scaling relation, and is a milestone in moving from magni-fication detection to useful science (Chapter 3). Finally, a comprehensive gravitational lensing shear analysis isperformed on the previous cluster sample, allowing for a critical comparison between cluster masses measuredwith the independent techniques, as a function of both richness and redshift. These shear measurements alsoallow for important constraints on a new sample of galaxy clusters, including the distribution of cluster centroidoffsets, the mass-richness relation, and cluster redshift evolution (Chapter 4).This thesis details unprecedented measurements using a new technique – weak lensing magnification – andcomparisons with the well-studied shear approach. The final product exemplifies the promise of the new methodfor measuring galaxy cluster masses, and also points to likely issues that will need to be addressed in futureexperiments.iiPrefaceThe body of this thesis is composed of three separate studies that have all been published in peer-reviewedjournals. They have been subject to minor edits to fit with the required form of this thesis.Chapter 2 is an adaptation of the article – Magnification by Galaxy Group Dark Matter Halos by J. Ford,H. Hildebrandt, L. Van Waerbeke, A. Leauthaud, P. Capak, A. Finoguenov, M. Tanaka, M. R. George, andJ. Rhodes, published in The Astrophysical Journal, Volume 754, Issue 2, article id.∼143, 6 pp. (2012). Theauthor of this thesis led the analysis, performed all calculations, wrote all code used in the work, and drafted thepublished manuscript. H. Hildebrandt and L. Van Waerbeke (thesis advisor) provided regular discussions andguidance that shaped the formulation of the research. A. Leauthaud and P. Capak provided data and astronomicalcatalogs for analysis. All authors gave comments and edits on the final manuscript.Chapter 3 is an adaptation of the article – Cluster Magnification and the Mass-Richness Relation in CFHTLenSby J. Ford, H. Hildebrandt, L. Van Waerbeke, T. Erben, C. Laigle, M. Milkeraitis, and C. B. Morrison, publishedin Monthly Notices of the Royal Astronomical Society, Volume 439, Issue 4, p.3755–3764 (2014). The author ofthis thesis led the analysis, performed all calculations, wrote or heavily modified all code used in the work, anddrafted the published manuscript. H. Hildebrandt and L. Van Waerbeke (thesis advisor) provided regular discus-sions and guidance that shaped the formulation of the research. T. Erben was heavily involved in producing theCFHTLenS data products used in this work. C. Laigle wrote the first version of the code for calculating clusterrichness, which was adapted by the thesis author for this work. M. Milkeraitis compiled the 3D-Matched-Filtergalaxy cluster catalog. C. B. Morrison performed a study of Lyman-break galaxy low-redshift contamination,upon which this manuscript relied. All authors gave comments and edits on the final manuscript.Chapter 4 is an adaptation of the article – CFHTLenS: A Weak Lensing Shear Analysis of the 3D-Matched-Filter Galaxy Clusters by J. Ford, L. Van Waerbeke, M. Milkeraitis, C. Laigle, H. Hildebrandt, T. Erben, C.Heymans, H. Hoekstra, T. Kitching, Y. Mellier, L. Miller, A. Choi, J. Coupon, L. Fu, M. J. Hudson, K. Kuijken,N. Robertson, B. Rowe, T. Schrabback, and M. Velander, published in Monthly Notices of the Royal Astronom-ical Society, Volume 447, Issue 2, p.1304–1318 (2015). The author of this thesis led the analysis, performed allcalculations, wrote or heavily modified all code used in the work, and drafted the published manuscript. Theauthorship list reflects the lead authors of this paper followed by two alphabetical groups. L. Van Waerbeke(thesis advisor) and H. Hildebrandt provided regular discussions and guidance that shaped the formulation ofthe research. M. Milkeraitis compiled the 3D-Matched-Filter galaxy cluster catalog. C. Laigle wrote the firstversion of the code for calculating cluster richness, which was adapted by the thesis author for this work. Thenext group includes key contributors to the science analysis and interpretation in this paper, the founding coreteam and those whose long-term significant effort produced the final CFHTLenS data product. The final groupcovers members of the CFHTLenS team who made a significant contribution to the project and/or this paper. Allauthors gave comments and edits on the final manuscript.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Our Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 Cosmic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.5 Distances in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Weak Lensing Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 Weak Lensing Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.3 Magnification vs. Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.1 Finding Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 Clusters for Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.3 Clusters for Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.4 Cluster Mass Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.4 Impact of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25iv2 Magnification by Galaxy Group Dark Matter Halos . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.1 Measuring α(m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 Optimally Weighted Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.3 Halo Mass Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Cluster Magnification and the Mass-Richness Relation in CFHTLenS . . . . . . . . . . . . . . . 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.1 3D-Matched-Filter (3D-MF) Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Magnification Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 The Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.2 The Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.1 The Mass-Richness Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4.2 Redshift Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 CFHTLenS: A Weak Lensing Shear Analysis of the 3D-Matched-Filter Galaxy Clusters . . . . . 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 The Canada-France-Hawaii Telescope Legacy Survey Wide . . . . . . . . . . . . . . . 564.2.2 Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) Shear Catalog . . . . . . 574.2.3 3D-MF Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.1 Stacking Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.2 Measuring ∆Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.3 Miscentering Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.4 The Halo Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4 Galaxy Cluster Weak Lensing Shear Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.1 Fits to ∆Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.2 The Mass-Richness Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4.3 Results of Binning Clusters in Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.5.1 Interpretation of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73v4.5.2 Comparisons of Cluster Catalog Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 744.5.3 Comparison with other Mass-Richness Relations . . . . . . . . . . . . . . . . . . . . . 744.5.4 Comparisons with other Cluster Centroid Analyses . . . . . . . . . . . . . . . . . . . . 764.5.5 Comparison with Magnification Results . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Final Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87viList of TablesTable 1.1 Cosmological Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Table 2.1 Luminosity Function Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Table 3.1 Centroid Offset Fit Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Table 3.2 Magnification Results for Richness-Binned Clusters . . . . . . . . . . . . . . . . . . . . . . 53Table 3.3 Magnification Results for Redshift-Binned Clusters . . . . . . . . . . . . . . . . . . . . . . 53Table 4.1 Shear Results for Richness-Binned Clusters (Full Model) . . . . . . . . . . . . . . . . . . . 70Table 4.2 Shear Results for Richness-Binned Clusters (Perfectly Centered Model) . . . . . . . . . . . 70Table 4.3 Shear Results for Redshift-Binned Clusters (Full Model) . . . . . . . . . . . . . . . . . . . . 71Table 4.4 Shear Results for Redshift-Binned Clusters (Perfectly Centered Model) . . . . . . . . . . . . 71viiList of FiguresFigure 1.1 Gravitational Lensing Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.2 Strong Lensing Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.3 κ and γ Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 1.4 Tangential Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 1.5 Magnification Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.1 Masses and Redshifts of Cosmological Evolution Survey (COSMOS) Groups . . . . . . . . . 30Figure 2.2 Magnitude-Binned Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 2.3 Optimally-Weighted Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.1 Centroid Offset Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 3.2 3D-MF Cluster Richness Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.3 Redshift Distributions of Clusters and Sources . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.4 Magnification for all Stacked 3D-MF Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.5 Magnification for Richness-Binned Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.6 Mass-Richness Relation from Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.7 Magnification for Redshift-Binned Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.8 Richness Distributions in Redshift Bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.1 Mass-Significance Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.2 Richness and Redshift Distributions of 3D-MF Clusters . . . . . . . . . . . . . . . . . . . . 59Figure 4.3 Richness-Significance Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 4.4 Example of Miscentering Effect on Shear Profile . . . . . . . . . . . . . . . . . . . . . . . 63Figure 4.5 Shear for Richness-Binned Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 4.6 Cluster Mass Distributions for each Richness Bin . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.7 Mass-Richness Relation from Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.8 Shear for Redshift-Binned Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.9 Redshift Dependence of Mass-Richness Normalization . . . . . . . . . . . . . . . . . . . . 73Figure 4.10 Comparison of Magnification and Shear Masses . . . . . . . . . . . . . . . . . . . . . . . . 76viiiGlossaryAGN Active Galactic Nuclei. The extremely bright and energetic inner region of some galaxies,powered by the central supermassive black hole.BCG Brightest Cluster Galaxy. In many rich clusters, this obvious and very bright elliptical galaxy sitsat the center of the gravitational potential (the acronym is sometimes interpreted as BrightestCentral Galaxy).CFHT Canada-France-Hawaii Telescope. A 3.6-m optical/infrared telescope on the summit of MaunaKea, in Hawaii.CFHTLenS Canada-France-Hawaii Telescope Lensing Survey. The gravitational lensing survey incorporatingthe CFHTLS data. The CFHTLenS team has produced the only publicly available shear catalog todate.CFHTLS Canada-France-Hawaii Telescope Legacy Survey. Astronomical survey optimized for weaklensing, covering 154 deg2 of sky in 5 filters from the optical to near-infrared wavelengths.CMB Cosmic Microwave Background. Thermal radiation consisting of photons that were released fromThomson scattering when neutral atoms first formed in the early Universe.COBE COsmic Background Explorer. The first satellite dedicated to CMB measurements, launched in1989.COSMOS Cosmological Evolution Survey. A very deep 2 deg2 survey aimed at cosmological studies,incorporating data from many different space and ground-based telescopes (including Hubble,Spitzer, Chandra, and many others).Euclid Euclid. A planned 1.2-m space-based telescope that will rely partly on weak lensing to achieve itsmain goal of constraining dark energy.ICM Intracluster Medium. The gas and ionized plasma that fills the space between galaxies in a cluster.LBG Lyman-break galaxy. High-redshift star forming galaxy that emits only at wavelengths longer thanthe Lyman limit (rest frame 912A˚), allowing its detection through observation in multiple filters.LF Luminosity Function. The number of objects (e.g. galaxies) as a function of luminosity ormagnitude.ixLSST Large Synoptic Survey Telescope. An 8.4-m ground-based telescope, currently under constructionin Chile, with wide ranging science goals, including weak lensing.MACHO MAssive Compact Halo Object. A term encompassing black holes, planets, and other compactobjects, MACHOs were once a serious dark matter candidate.NFW Navarro-Frenk-White. A halo mass model based on simulations of dissipationless gravitationalcollapse. Density is proportional to 1/r for small radii, transitioning to 1/r3 for large radii.Planck Planck. The most recent generation of CMB satellites, providing the tightest cosmologicalconstraints to date.SDSS Sloan Digital Sky Survey. Ongoing multicolor imaging and spectroscopic survey of one third ofthe sky, using a 2.5-m telescope at Apache Point, New Mexico.SIS Singular Isothermal Sphere. A very simple model of halo mass, with density proportional to 1/r2.WFIRST Wide-Field InfraRed Survey Telescope. A planned 2.4-m space-based telescope, with wideranging science goals including weak lensing.WIMP Weakly-Interacting Massive Particle. A popular theoretical candidate for the dark matter particle.WMAP Wilkinson Microwave Anisotropy Probe. The second generation of CMB satellites, WMAP createddetailed maps of temperature fluctuations in the early Universe.3D-MF 3D-Matched-Filter. Optical galaxy cluster finding algorithm, which recovered over 18,000 clustercandidates in the CFHTLS.Unitsarcmin Arcminute. Unit of angular separation on the sky. 1 degree = 60 arcmin.M Solar mass. A common measure of mass in astronomy, equal to 1.989×1030 kg.pc Parsec. A unit of distance in astronomy, equal to 3.086×1016 m. Mpc = 106 pc, kpc = 103 pc.Symbolsa(t) Scale factor of the Universe, defined to be unity today.A Amplification matrix, describing the gravitational lens mapping from source to image plane.α(m) LF slope. The logarithmic slope of the source luminosity function, controlling the direction ofthe number count effect of magnification.b Bias factor. Dimensionless number quantifying how much an object clusters relative to the darkmatter in the Universe.β Slope of the mass-richness power-law relation.c Speed of light in a vacuum. c = 2.998×108 m s−1.c200 Concentration parameter for an NFW dark matter halo profile.xγ Shear. The two component pseudo-vector quantifying the anisotropic focusing of gravitationallylensed light rays.γt Tangential shear. Component of the shear oriented tangential to the direction of the lens.dA Angular diameter distance. The cosmological distance equal to length divided by angle sub-tended.dL Luminosity distance. The cosmological distance for which flux drops as luminosity divided bythe square of that distance.dp Proper distance. The cosmological distance that would be measured between two objects if youcould lay down a very large ruler at a specified epoch.Dl Angular diameter distance from the observer to the gravitational lens.Dls Angular diameter distance between the gravitational lens and source.Ds Angular diameter distance from the observer to the gravitationally lensed source.D(z) Linear growth function.δc Characteristic overdensity of a halo. An NFW parameter that is a function of c200.δµ Magnification contrast. δµ ≡ µ−1.fclustering Fraction of the background source sample (for magnification) that is at the same redshift as thelenses, and therefore leads to a clustering signal.flensing Fraction of the background source sample that is at high redshift, and therefore can be lensed.G Newton’s gravitational constant. G = 6.673×10−11 N m2 kg−2.g Reduced shear. g = γ/(1−κ).h Hubble parameter. h≡ H0/(100 km s−1 Mpc−1).H(z) Hubble rate. The redshift (or time) dependent expansion rate of the Universe.H0 Hubble constant. The present-day Hubble rate. H0 ≈ 70 km s−1 Mpc−1.κ Convergence. The part of gravitational lensing composed of the isotropic focusing of light rays.m Apparent magnitude. Logarithmic measure of flux relative to a reference value.M Absolute magnitude. Equal to the apparent magnitude a source would have if it was located at aluminosity distance of 10 pc.M Mass (generic usages of mass in the text, to be distinguished from specific definitions, such asM200).M0 Normalization of the mass-richness relation, defined to be the average mass of clusters withrichness N200 = 20.M200 Mass of a dark matter halo within the radius R200.µ Magnification. Simply proportional to κ in the weak lensing limit, µ is a function of the gravi-tational lens mass.n Observed source number counts as a function of magnitude or flux and redshift.n0 Unlensed (intrinsic) source number counts.N200 Richness. The number of galaxies brighter than i-band absolute magnitude −19.35, within theradius R200.P(k) Power spectrum of density fluctuations in the Universe.P(Roff) Probability distribution of cluster centroid offsets.xiP(z) Probability distribution of redshifts.pcc Fraction of 3D-MF clusters that are correctly centered on their dark matter halos.r Comoving distance, which grows along with the expansion of space.R Projected physical distance on the sky.R200 Radius of a dark matter halo (at z) within which the average density is 200ρcrit(z).Roff The distance between the 3D-MF identified center of a cluster, and the true center of the darkmatter halo.ρcrit(z) Critical energy density of the Universe. Current value is ρcrit(0)≈ 9.2×10−27 kg m−3.Σ Surface mass density. Mass density projected onto the 2D plane of the sky.Σcrit Critical surface mass density. A function of the angular diameter distances between observer,lens and source, this is the minimum surface mass density of a lens for it to produce stronglensing features.Σsm Smoothed surface mass density of a stack of clusters that are offset, as described by the distribu-tion P(Roff).Σ2halo The two halo term. The contribution to the measured surface mass density of a dark matter halo,at large scales, caused by neighboring halos.∆Σ Differential surface mass density. Defined as the difference between the surface mass density atsome radius from the center of a lens, and the average inside of that radius.σcl The detection significance of a 3D-MF cluster candidate.σoff The radius where the P(Roff) distribution peaks.σ8 Normalization of the matter power spectrum. σ8 ≈ 0.8.te Time of emission of light.t0 Present time, current age of the Universe.w Correlation function.wdm Auto-correlation function of dark matter.wopt Optimally-weighted correlation function for magnification.z Redshift. A common distance measure in cosmology.Φ(M) Schechter function. A common parameterization of the LF.χ2 Chi-squared statistic, quantifying the goodness-of-fit of a model to the data.χ2red Reduced chi-squared statistic. χ2red is χ2 divided by the number of degrees of freedom in themodel.ψ(θ) Two-dimensional analogue to the Newtonian gravitational potential.ω Size of a galaxy image.Ωb Baryon density parameter. The fraction of the Universe consisting of baryons. Ωb ≈ 0.05.Ωc Cold dark matter density parameter. The fraction of the Universe consisting of dark matter.Ωc ≈ 0.27.ΩΛ Dark energy density parameter. The fraction of the Universe consisting of dark energy. ΩΛ≈ 0.7.Ωm Matter density parameter. The fraction of the Universe consisting of matter, including darkmatter and baryons. Ωm ≈ 0.3.Ωr Radiation density parameter. The fraction of the Universe consisting of relativistic particles.Ωr ≈ 8×10−5.xiiDedicationThis thesis is dedicated toJody, for aweDave, for persistanceAlison, for joyLee, for lovexiiiChapter 1IntroductionThis thesis is concerned with weak gravitational lensing studies of galaxy clusters, using two complementaryapproaches, shear and magnification. This introductory chapter provides the necessary background and contextfor understanding the novel research presented in the subsequent chapters. The basics of cosmology, whichis the larger field within which this thesis research resides, is given in Section 1.1, with a particular focus ondistances, which will be important for the presentation of gravitational lensing in Section 1.2. Galaxy clustersare discussed in Section 1.3, from the cosmological as well as the intracluster physics perspective. Section 1.4outlines the novelty and importance of the research contained in this thesis. A brief overview of the body ofwork that is presented in the main chapters of this thesis is given in Section 1.5.1.1 CosmologyCosmology is the study of our Universe as a whole. It can be easy to take for granted the simple fact that we,as scientists, can even do cosmology at all – that is, we can make quantitative and testable predictions about thephysical nature of the vast Universe we inhabit. At the same time, as we cosmologists forge ahead, caught upin the day-to-day struggles that concern some minute detail of a model, a prediction, or an idea, it is easy tooverlook the sheer beauty of what we are so deeply invested in. The introduction of this thesis serves both to laythe requisite theoretical foundation, upon which the thesis research relies, while also giving an honest depictionof the big picture ideas for which this work strives to be relevant.“The Cosmos is all that is or was or ever will be. Our feeblest contemplations of the Cosmos stir us – thereis a tingling in the spine, a catch in the voice, a faint sensation, as if a distant memory, of falling from a height.We know we are approaching the greatest of mysteries” (Sagan, 1980).1.1.1 Our UniverseLooking out into the night sky from our vantage point on planet earth, the Universe appears full of structure onmany scales: planets orbiting stars; stars bound into star clusters and galaxies; and galaxies themselves organizedinto clusters ranging from small associations like our local group, to enormous conglomerates of many thousandsof galaxies. But if we adopt a holistic mindset on the scale of around 100 Megaparsecs (Mpc), and ignore thesmaller scale density fluctuations that are so crucial to our own existence, we observe two remarkable apparentrealities of our Universe:1. Homogeneity. On large scales, the Universe is the same in all locations. There are no preferred places.12. Isotropy. On large scales, the Universe is the same in all directions. There are no special or unique partsof the sky.These two postulates, which are supported by observations, form the foundation of the current standard cosmo-logical model (e.g. Peebles, 1993; Dodelson, 2003; Ryden, 2003).A third important fact about our Universe on large scales, is that it is expanding. Galaxies are moving awayfrom all other galaxies, at a rate proportional to their distance.1 This simple relationship was discovered byHubble (1929), and can be expressed asv≈ H0d, (1.1)where v is recessional velocity, d is distance, and H0 is known as the Hubble constant (Ryden, 2003). Ourcurrent best estimate is H0 = (67.8± 0.77) km s−1Mpc−1 (Planck Collaboration et al., 2014); this and othercosmological constants are listed in Table 1.1. The discovery of the expansion of the Universe simultaneouslyabolished any notion that our Universe was static, while also giving rise to the idea of a Big Bang origin.Extrapolating back in time, it seems that the Universe was once much smaller, denser, and hotter.Several pieces of evidence, including the remarkable success of Cosmic Microwave Background (CMB) ex-periments such as the COsmic Background Explorer (COBE) (Fixsen et al., 1996), the Wilkinson MicrowaveAnisotropy Probe (WMAP) (Hinshaw et al., 2013) and Planck (Planck Collaboration et al., 2014), provide verystrong support for the Big Bang theory. First detected by Penzias and Wilson (1965), the CMB is an isotropicbackground of microwave photons that have been essentially free-streaming since the Universe was a denseopaque cloud. At around 380,000 years of age, the density had dropped sufficiently, and the Universe hadcooled enough for neutral atoms to form, freeing the photons from constant Thomson scattering. The CMB pho-tons match a blackbody spectrum with temperature 2.73 K and have a number density of 4.11×108 m−3. Thetiny fluctuations in CMB temperature on the sky, which are of order 1 part in 105, are the seeds of structure for-mation in the Universe. Measurements of these anisotropies by WMAP and Planck have provided cosmologicalparameter constraints of incredible precision (see Table 1.1).One consequence of the expansion of the Universe (and also the way it was first discovered, Hubble, 1929) isthat light from distant objects is redshifted as it travels to us. This shifts the spectrum of light emitted by galaxies,so that known absorption lines will appear at different wavelengths than they are observed in laboratories onEarth. This cosmological redshift can be expressed in terms of the observed and the emitted wavelengths:z≡λobs−λemλem. (1.2)Redshift is often a convenient means of indicating cosmological distances, and will be used frequentlythroughout this thesis. Since the Universe is expanding, it is convenient to express its growth in terms of a scalefactor a(t). We define a to be unity today (a(t0) = 1), and say that a(t) < 1 in the past. In this framework wecan directly convert the cosmological redshift of an object to the scale factor when its light was emitted (Hogg,1999):1+ z =a(t0)a(te)=1a(te). (1.3)The expansion history, given by the scale factor a(t), depends upon the constituents of the energy density1Note: the linear Hubble relation only holds for relatively small cosmic distances, below a few hundred Mpc (Ryden, 2003). Onlarger scales we cannot ignore the acceleration of the Universe – see Section 1.1.3.2of the Universe. Numerous studies show that, in addition to obvious stuff like normal matter2 and radiation,the Universe also contains copious amounts of cold dark matter and some form of dark energy, possibly acosmological constant (e.g. Dodelson, 2003). These components will be discussed in more detail below. Thisdark sector actually makes up the majority of the present energy density of the Universe, but that was not alwaysthe case because of the way the density of each component has evolved differently with the scale factor.1.1.2 Dark MatterThe notion of an invisible dark matter has been around for a surprisingly long time. The idea was first proposedby Zwicky (1933) to explain the fact that the mass of the Coma Cluster estimated from radial velocities ofmember galaxies greatly exceeded the mass estimated from luminosity (see also Zwicky, 1937). Several yearslater a similar mismatch was observed between the rotation curves of individual galaxies and the amount ofvisible matter they contained (Babcock, 1939), indicating that an enormous amount of invisible mass mustextend far beyond the visible range of the galaxies. This gave rise to the concept of the dark matter halo, a muchlarger spherical halo of invisible dark matter existing around every galaxy and galaxy cluster (see van den Bergh,1999, for a review of dark matter’s discovery).Current measurements confirm the existence of dark matter, refining its abundance to around 30% of theenergy content of the entire Universe – an order of magnitude greater average density than ordinary matter.For much of the latter part of the 20th century it was thought that the dark matter could simply be normalmatter that was not giving off light. Faint stars, brown dwarfs, black holes, rocky bodies, and an abundance oflight weight neutrinos, were all candidates for the non-luminous material (van den Bergh, 1999). The strongestevidence for the non-baryonic nature of dark matter comes from the tight constraints on baryon density duringthe period of Big Bang Nucelosynthesis in the early universe (Alpher et al., 1948). The now mainstream ideaof a non-baryonic cold dark matter was first introduced by Peebles (1982), followed closely by Bond et al. atthe Third Moriond Astrophysics Meeting (Audouze and Tran Thanh Van, 1984). Cold dark matter is supportedby requirements for structure formation in the Universe (Peebles, 1982), and the amplitude and shape of CMBclustering (White et al., 1995), as well as more “direct” evidence of the non-interaction of dark matter studiedin several merging galaxy cluster systems, including the now famous Bullet Cluster (Clowe et al., 2006, see alsoSection 1.3.3).Dark matter seems to interact only through gravity, and certainly not via the electromagnetic force (it doesnot emit, absorb, or reflect light). The most plausible contender for the actual dark matter particle is called aWeakly-Interacting Massive Particle (WIMP), which, as the name implies, is a massive particle that interactsonly through gravity and the weak nuclear force. Common supersymmetric extensions to the Standard Model ofparticle physics include particles that could be candidates for WIMPs (Schumann, 2014). Unfortunately, directdetection of these particles has proved extremely difficult, despite the fact that numerous groups are pursuingdetection using a variety of approaches. One group has claimed a WIMP detection (this is the DAMA/LIBRAexperiment which has observed an annual signal modulation), but this remains controversial and appears to beruled out by other experiments (particularly the XENON-100 experiment) (Cushman et al., 2013).2Normal matter consists of atoms and other standard model particles, which, in cosmology, are typically all lumped together underthe label “baryonic matter,” despite the fact that they are not all strictly baryons in the particle physics sense (Roos, 2010).31.1.3 Dark EnergyProbably the biggest mystery in all of physics is the nature of dark energy. Like dark matter, the idea hasbeen around for many decades, but unlike dark matter it was not initially based upon observations, but rathera preconceived bias. Since this was before Hubble had discovered the expansion of space, Einstein (and manyothers) assumed the Universe was static. But a matter-dominated Universe described by Einstein’s theory ofrelativity could not be static – it would contract and fall back on itself due to the gravitational potential of all themass in it (Einstein, 1917; Weinberg et al., 2013).Einstein added a term to his field equations known as the cosmological constant, which was supposed to bea repulsive term that perfectly balanced the Universe against gravitational collapse. These equations describedhow the geometry (left-hand-side) was related to the energy content (right-hand-side):Rµν −12gµνR =−8piGc4Tµν +Λgµν . (1.4)Here Rµν and R are the Ricci tensor and scalar. G is Newton’s gravitational constant and c is the speed of light ina vacuum. The metric tensor is given by gµν , and the energy-momentum tensor is Tµν (Einstein, 1917). Whenthe Hubble expansion was discovered, Einstein famously abandoned the cosmological constant term, Λgµν , butit has since reappeared (Weinberg et al., 2013).In the late 1990s, two teams of astronomers were trying to measure the deceleration of the Universe, sincethe current expansion was expected to be slowing under the gravitational attraction of all the matter. Both theHigh-z Supernovae Search Team (Riess et al., 1998) and the Supernovae Cosmology Project (Perlmutter et al.,1999) used type Ia supernovae as standard candles, relating the peak luminosity to the width of the light-curve,and found evidence for a non-zero cosmological constant. The Universe’s expansion was actually accelerating.This Nobel-prize-winning discovery has been a paradigm shift for the field of cosmology.The name “dark energy” refers to the unknown force causing the Universe to accelerate. The simplestpossibility would be a cosmological constant, perhaps a vacuum energy that is uniform throughout space.3Other more complicated theories have been invented to explain the acceleration, so dark energy is the moregeneral term encompassing them all, but currently the data are consistent with a cosmological constant makingup almost 70% of the energy-density of the Universe (Planck Collaboration et al., 2014, 2015b).1.1.4 Cosmic DynamicsGeneral Relativity stipulates that space and time are part of a single fabric, known as space-time. On the largestscales, events separated in this 4-dimension space-time can be described using the Robertson-Walker metric.This particular form of the metric is a direct consequence of the assumptions of homogeneity and isotropy of theUniverse (e.g. Bertone et al., 2005):ds2 =−c2dt2 +a(t)2[dr2 +Sk(r)2dΩ2]. (1.5)Here dt is an interval of proper time, dΩ2 = dθ 2 + sin2θdφ 2, and (r,θ ,φ) are the set of comoving positioncoordinates (comoving coordinates grow along with the Hubble expansion – see Section 1.1.5).The Sk(r) term is specified by the curvature of the Universe, which can be flat (zero curvature, Euclidean),3Unfortunately, calculating the expected energy density of the vacuum from particle physics gives a value about 120 orders ofmagnitude higher than observed. This is a major unsolved issue in theoretical physics and cosmology (Weinberg et al., 2013).4closed (positively curved, analogous to the surface of a sphere in 2-dimensions), or open (negatively curved, likethe surface of a saddle in 2-dimensions). Explicitly,Sk(r) =Rsin(r/R), for positive curvaturer, for zero curvatureRsinh(r/R), for negative curvature.(1.6)If curvature is non-zero, then R gives the radius of curvature (Ryden, 2003). Strong limits have been placed oncurvature, and it turns out that our Universe is flat, or at least extremely close to flat. The fraction of the energydensity of the Universe contained in curvature is less than about 0.005 and is consistent with zero (PlanckCollaboration et al., 2015a).Applying the Robertson-Walker metric (Equation 1.5) to the Einstein field equations (Equation 1.4), weobtain the Friedmann Equation:(a˙a)2=8piG3ρtotal. (1.7)Here a = a(t) is the scale factor, and a˙ is its first order derivative with respect to time. The total energy density ofthe Universe ρtotal appears to be equal to the critical energy density ρcrit, which is exactly the case if the Universehas zero curvature. The critical density is given by (Ryden, 2003):ρcrit(z)≡3H(z)28piG , (1.8)and its current value is:ρcrit(0)≈ 9.2×10−27 kg m−3 ≈ 1.4×1011 M Mpc−3. (1.9)Cosmologists frequently express the density of each component of the Universe as a fraction of the total orcritical energy density, using the notation Ωi = ρi/ρcrit. ΩΛ represents dark energy, Ωc is the cold dark matter,Ωb is the baryonic matter, and Ωr is for radiation. Since the Universe is flat ∑Ωi =ΩΛ+Ωc+Ωb+Ωr = 1. Eachof these components evolves differently with the scale factor (i.e. with time, or redshift). The present values ofthese density parameters are given in Table 1.1, along with other cosmological constants relevant to this thesis.The evolution of matter (both normal matter and dark matter) is the most intuitive, since its energy density issimply inversely proportional to the volume of space, ρm(t)∝ a(t)−3 = (1+z)3. The energy density of radiation(massless particles such as photons) scales as ρr(t)∝ a(t)−4 = (1+z)4, because the energy of the particles dropsoff as the cosmological expansion increases their wavelength, yielding an extra factor of a(t)−1 over the case formatter. Dark energy appears to be consistent with a cosmological constant, which, as the name implies, wouldhave constant energy density ρΛ ∝ a(t)0 = (1+ z)0.It is useful to introduce the Hubble rate H(t) ≡ a˙/a, which, at the present time t0, is equal to the Hubbleconstant H0. Thus we can re-write the Friedmann Equation, describing the evolution of the Universe, in thisform:H(z)2 = H20[ΩΛ+Ωm(1+ z)3 +Ωr(1+ z)4] . (1.10)5Symbol Value DescriptionΩbh2 0.02205±0.00028 Fraction of the present day energy density of the Universe that iscomposed of baryons (times h2).Ωch2 0.1199±0.0027 Fraction of the present day energy density of the Universe that iscomposed of cold dark matter (times h2).τ 0.089+0.012−0.014 Optical depth due to reionization.ns 0.9603±0.0073 Scalar spectrum power-law index.ln(1010As) 3.089+0.024−0.027 Log power of the primordial curvature perturbations.100θMC 1.04131±0.00063 θMC ≈ θ∗ [radians], the ratio of the comoving size of the soundhorizon at τ = 1 to the angular diameter distance of the redshiftat τ = 1.H0 (67.3±1.2) km s−1Mpc−1 Present day value of the Hubble constant, the ratio of recessionalvelocity to distance.Ωm 0.315+0.016−0.018 Fraction of the present day energy density of the Universe that iscomposed of pressureless matter.ΩΛ 0.685+0.018−0.016 Fraction of the present day energy density of the Universe that iscomposed of dark energy.σ8 0.829±0.012 Normalization of the matter power spectrum.t0 (13.817±0.048) Gyr The age of the Universe.Table 1.1: Cosmological Constants. Current best values of the Cosmological Constants using a combina-tion of CMB data from the Planck and WMAP missions (Planck Collaboration et al., 2014). It is quiteremarkable that our entire model for the current state and evolution of the Universe can be fully en-capsulated by a mere 6 parameters – the top 6 rows. The constants in the lower portion of the table arederived from these top 6 values, and are more relevant for the topics explored in this thesis. Note: Thedimensionless Hubble parameter “little h” is just the Hubble Constant in units of 100 km s−1Mpc−1.h≈ 0.7 is used throughout this thesis.The curvature term, which is so close to zero to be negligible, is ignored.4 In weak lensing, it is usually practicalto ignore Ωr as well, and approximate the Universe as being flat and composed of only matter and a cosmologicalconstant. That will be the case in most of this thesis, and we will use the matter density Ωm≈Ωc+Ωb, absorbingthe small fraction of the Universe’s baryon fraction into the cold dark matter term, and use a single term for thefractional energy density that is contributed by matter, Ωm.1.1.5 Distances in CosmologyDescribing distances between objects in an expanding and accelerating Universe is no simple task. Since physi-cal distances are growing with the Universe’s expansion, a natural coordinate system to use is that of comovingcoordinates, which grow along with the Universe. The comoving distance between two galaxies is a constant, aslong as they have no velocity relative to the Hubble expansion, while the physical distance between them grows.This physical distance, usually called the proper distance, is simply given bydp(t) = a(t)r, (1.11)4If included, the curvature term would scale proportionally to (1+ z)2.6and at the time of observation t0 is equal todp(t0) = r = c∫ t0tedta(t). (1.12)Here te is the time of emission, and r is the comoving radial distance, the same r used in the Robertson-Walkermetric (Equation 1.5) (Hogg, 1999). Since actually measuring a cosmological scale proper distance wouldrequire us to pause the Universe’s expansion while we extend an enormous tape measurer, we have to rely onother forms of distance, discussed below.Frequently we want to relate the apparent brightness of a distant astronomical object to its distance. Thisis especially crucial for objects of known intrinsic luminosity (often called standard candles), such as the typeIa supernovae discussed in Section 1.1.3. For everyday distances here on Earth, we observe that the bolometricflux f of an object with luminosity L falls off as 1/(distance)2. We can therefore define an analogous luminositydistance in cosmology asdL ≡√L4pi f , (1.13)with the understanding that this is different than proper or comoving distance, because the Universe has beenexpanding during the time it took for the light to travel to us. In fact, this implies the relationshipdL = (1+ z)r = (1+ z)dp(t0). (1.14)Similar to the notion of a standard candle, we can imagine a standard ruler of a fixed physical length `.We can define the angular diameter distance to be the distance at which this object would have to be, in orderto conform to our everyday experience of the relationship between distance, length, and angle subtended (θ[radians]):dA ≡`θ . (1.15)Here we have invoked the small angle approximation, and the angular diameter distance dA is simply related tothe other distance definitions (Ryden, 2003):dA =r(1+ z)=dL(1+ z)2. (1.16)Angular diameter distance is the distance measure relevant for gravitational lensing, which will be discussed inSection 1.2.1.2 Gravitational LensingPhotons follow null geodesics on the journey from a distant light source to our telescopes. Due to inhomo-geneities in the gravitational potentials along their path, the light rays are deflected. In particular, large overden-sities, such as galaxies and galaxy clusters, will cause light rays to be bent and focused, altering the images ofthe background objects. Einstein’s theory of General Relativity predicts this effect, and specifically requires theangle of deflection to be twice that of in Newtonian gravity. During a solar eclipse in 1919, Sir Arthur Eddingtonmeasured the shifted apparent positions of stars being gravitationally lensed by the Sun, providing experimentalevidence for the new theory of gravity and paving the way for the future of gravitational lensing as a field (e.g.7Figure 1.1: Gravitational Lensing Diagram. Diagram showing the geometry of gravitational lensing.Light from the background source is bent by an angle αˆL when it passes near the gravitational lens(gray oval) on its way to the earth. While the actual source (black star) is at an angle βL relative tothe horizontal, its image (gray star) appears to be at an angle θ . The angular diameter distances to thelens (Dl), to the source (Ds) and between the lens and source (Dls) are labeled.Blandford and Narayan, 1992; Narayan and Bartelmann, 1996; Bartelmann and Schneider, 2001; Schneider,2006b).The lensing geometry is displayed (not to scale) in Figure 1.1. For most lensing studies the distances betweenastronomical objects involved is far greater than the size of the gravitational lens itself. It is therefore reasonableto approximate the path of the light ray as being bent at a sharp angle (as opposed to gradually arcing throughthe gravitational potential) (Bartelmann and Schneider, 2001). In analogy to refraction of light by an opticallens, this is known as the “thin lens approximation.” When light passes nearby an object of mass M, at impactparameter b, its path will be bent by the angle αˆL (Ryden, 2003):5αˆL =4GMc2b. (1.17)This causes the background light source to appear as if it is at an angle θ , when it is really at angular positionβL, as shown in Figure 1.1. Note that the thin lens approximation can also be applied to a succession of multipledeflections along the path of a light ray.The fact that gravitational lensing directly probes the underlying density field along the line of sight is whatmakes the technique extremely valuable. Most other methods for probing the matter distribution of the Universedo not probe the mass itself (which is mostly dark matter), but rather the baryonic component of the mass –stars, and interstellar gas and dust. While we expect the baryons to trace the underlying density of dark matter,5Note that this thesis uses a similar notation to lensing reviews such as Bartelmann and Schneider (2001) and Schneider (2006b),but we add the subscript “L” to some symbols in this chapter, pertaining to the lensing equations, to avoid confusion with the use of thesame symbols in later sections of the thesis (e.g. β will be the slope of the cluster mass-richness relation in Chapters 3 and 4).8Figure 1.2: Strong Lensing Image. Abell 2218, a galaxy cluster at z≈ 0.18, yields a magnificent exampleof strong gravitational lensing. Many large arcs, which are highly distorted images of backgroundgalaxies, are easily visible in this image [Source: NASA].there are many complicating factors (see Section 1.3) that render the tracking to be inexact. Additionally, othermeans of measuring masses (such as using radial velocities) rely on assumptions about the virial equilibrium ofa system, which may not be satisfied.Gravitational lensing is broadly divided into several branches, depending upon the strength of the lensingeffect. “Strong lensing” refers to the rarest and most obvious distortions, leading to images of giant arcs, Ein-stein rings, and multiple images of the same source. Strong lensing features are generally apparent to the eye(see Figure 1.2 for an example), whereas “weak lensing” and “microlensing” are not. The first observation ofgravitational lensing producing multiple images was made by Walsh et al. (1979) using a lensed quasar. Thefirst gravitationally lensed arcs were discovered nearly a decade later by Lynds and Petrosian (1986) and Soucailet al. (1987, 1988). Currently, over 500 strong gravitational lenses are known (More et al., 2015). Strong lensingis very useful for accurately probing the dark matter halo mass profiles of galaxies and clusters of galaxies, oftenyielding detailed information on lens concentration (Auger et al., 2010) and substructure (Mao and Schneider,1998; Dalal and Kochanek, 2002).Weak lensing, just as the name implies, leads to much less significant distortions. The hallmark of weaklensing is that it is a statistical effect, only measurable using ensembles of many background sources and fore-ground lenses. Unlike the rarity of a strong lensing event, however, weak lensing is everywhere. Essentiallyall light rays are distorted at least slightly while traveling to us through the inhomogeneous gravitational fieldsof the Universe. Weak lensing itself, and different approaches to measuring it, will be discussed in much moredetail in Sections 1.2.1 and 1.2.2 below.Finally, microlensing is the even weaker signature of gravitational lensing, wherein stars are lensed by low9mass compact objects, like black holes, brown dwarfs, and planets. Probably the most important use of mi-crolensing has been the search for a significant MAssive Compact Halo Object (MACHO) population in theMilky Way. MACHOs were once considered a serious dark matter candidate until various microlensing experi-ments demonstrated that the mass density of MACHOs was strongly insufficient to explain the missing mass inour Galaxy (Paczynski, 1996; Wyrzykowski et al., 2011; Sumi et al., 2013).1.2.1 Weak Lensing ShearWeak lensing shear is the component of weak lensing that deals with shape distortion of galaxy images. If allgalaxies were intrinsically circular, or of known shape, then each individual background source would provideinformation on the gravitational field through which its light had propagated. Instead, however, galaxies takeon a variety of shapes and orientations, and their unlensed representations are impossible to know. In order toproceed, weak lensing astronomers make two critical assumptions: (1) galaxy shapes can be approximated aselliptical; and (2) the orientation of these ellipses are random in the absence of gravitational lensing (Bartelmannand Schneider, 2001). The second point follows from the isotropy of the Universe, although on small scalesintrinsic galaxy-galaxy alignment is an issue. Though only a few percent effect, characterization of any intrinsicalignments is an active area of research (see e.g. Hirata and Seljak, 2004; Heymans et al., 2013).As illustrated in Figure 1.1, the lens equation is given byβ L = θ −αL, (1.18)where we now use bold face to indicate angular positions with two components on the sky. The reduced deflec-tion angle αL is related to the deflection angle of Equation 1.17 by αL = (Dls/Ds)αˆL. The reduced deflectionangle αL can be expressed as the gradient of the lensing (or deflection) potential αL =∇ψ . The lensing potentialψ(θ ) is the two-dimensional analogue to the Newtonian gravitational potential Φ, and is given by (Narayan andBartelmann, 1996):ψ(θ ) = 2c2DlsDlDs∫Φ(Dlθ ,z)dz =1pi∫R2κ(θ ′)ln|θ −θ ′|d2θ ′. (1.19)Here κ is known as the convergence, which encapsulates the magnification information to be described in Sec-tion 1.2.2 below. Explicitly, the convergence is given byκ(θ ) = 12(∂ 2ψ(θ )∂θ 21+∂ 2ψ(θ )∂θ 22), (1.20)and another useful expression is the ratioκ(θ ) = Σ(θ )Σcrit. (1.21)Here Σ(θ ) is the two-dimensional surface mass density (relative to the average density, with units of mass perarea on the sky), and Σcrit is the critical surface mass density of the lens (Wright and Brainerd, 2000). The latterdemarcates the separation between strong and weak gravitational lenses, depending critically on the geometryof the angular diameter distances between objects, given byΣcrit =c24piGDsDlDls. (1.22)10Figure 1.3: κ and γ Diagram. Demonstration of the effect of convergence κ and shear γ on a circularsource. The diagram shows the unlensed source (solid gray circle) and the final lensed image (blackoutline) for positive and negative values of κ and each of the two components of γ .Strong lenses (i.e. those capable of forming multiple images) must have Σ≥ Σcrit (Schneider, 2006a).The transformation of background objects from source (unlensed) to image (lensed) is described by theJacobian (or amplification) matrix A (e.g. Dodelson, 2003):A (θ ) = ∂β L∂θ =(δi j−∂ 2ψ(θ )∂θi∂θ j)=(1−κ− γ1 −γ2−γ2 1−κ+ γ1). (1.23)The two components of the shear γ(θ ) can be expressed as derivatives of the lensing potential:γ1 =12(∂ 2ψ(θ )∂θ 21−∂ 2ψ(θ )∂θ 22); (1.24)γ2 =∂ 2ψ(θ )∂θ1∂θ2. (1.25)The shear is often written as a complex number, γ ≡ γ1+ iγ2 = |γ|e2iϕ . Here |γ| and ϕ indicate the amplitude anddirection of distortion, respectively, which is unchanged when rotated by 180◦. An illustration of the meaningof the real and imaginary components of γ , as well as κ , is given in Figure 1.3We do not observe the true shear, but rather the quantity g(θ ) = γ(θ ) [1−κ(θ )]−1, which is called the“reduced shear.” We can thus rewrite the Jacobian matrix as:A (θ ) = (1−κ)(1−g1 −g2−g2 1+g1). (1.26)In the regime of weak lensing, the convergence and shear are small, κ  1 and |γ|  1. Therefore it is oftensafe to assume that γ ≈ g (Schneider, 2006b). Near the centers of massive lenses, such as galaxy clusters, theweak lensing approximation is no longer valid, and the full lensing formalism must be accounted for (as inSection 3.3.2), or corrections applied (as in Section 4.3.2).11The surface brightness, I(θ ), is a conserved quantity in pure gravitational lensing, I(θ ) = I(s)(β (θ )). Theconservation of surface brightness can be obtained from the second law of thermodynamics, as well as otherarguments, such as the fact that photons are neither created nor destroyed in pure lensing processes. The bright-ness distribution of a galaxy is not in general perfectly elliptical, but we can approximate it as an ellipse in thefollowing manner. The center of the brightness distribution of the image isθ¯ ≡∫d2θ I(θ )qI(I(θ ))θ∫d2θ I(θ )qI(I(θ )), (1.27)where qI(I(θ )) is some weight function, and we assume that the galaxy image of interest is isolated on the sky.To describe ellipticity we will be interested in the second brightness moments of the source, contained in thetensorQi j =∫d2θ I(θ )qI(I(θ ))(θi− θ¯i)(θ j− θ¯ j)∫d2θ I(θ )qI(I(θ )), (1.28)where i, j ∈ (1,2) (Schneider, 2006b). The size (area) ω of the galaxy image is just a function of the diagonalcomponents of this matrix,ω = (Q11Q22−Q212)1/2, (1.29)while the shape, or ellipticity, of the image involves the off-diagonal elements:χ ≡ Q11−Q22 +2iQ12Q11 +Q22; (1.30)ε ≡ Q11−Q22 +2iQ12Q11 +Q22 +2(Q11Q22−Q212)1/2. (1.31)The complex ellipticity can be characterized by either of χ or ε , which are simply related and interchangeable(in different situations one may be easier to work with, Bartelmann and Schneider, 2001).In analogy with the image center θ¯ and tensor of second brightness moments Qi j, one can define the samequantities for the unlensed source center β¯ and tensor of second brightness moments Q(s)i j . The relation betweenthe source and image tensors isQ(s) =A QA T =A QA , (1.32)whereA ≡A (θ ) is the Jacobian defined in Equations 1.23 and 1.26. The observed source ellipticity ε is relatedboth to the shear caused by weak lensing, and to the intrinsic ellipticity ε(s) of the unlensed background source:ε(s) =ε−g1−g∗ε , for|g| ≤ 1;1−gε∗ε∗−g∗ , for|g|> 1.(1.33)If we average over many background sources that are randomly oriented, then the average intrinsic ellipticity〈ε(s)〉= 0 and we can apply the weak lensing approximation to conclude that γ ≈ g≈ 〈ε〉 ≈ 〈χ〉/2 (Bartelmannand Schneider, 2001).In this thesis we are concerned with a particular manifestation of gravitational lensing – lensing by galaxyclusters. If we consider a circularly symmetric mass density on the sky (an idealized galaxy cluster), then weexpect the shear distortion to be oriented tangential to the center of the lens. It is therefore useful in clusterlensing (and also in galaxy-galaxy lensing) to express the shear in terms of tangential and rotated (or cross)12Figure 1.4: Tangential Shear. Diagram illustrating the components of shear, when decomposed into atangential and cross-component relative to the center of a gravitational lens, as given in Equations1.34 and 1.35. The angle φ measures the azimuthal position of the source about the center of thegravitational lens.components:γt =−Re[γe−2iφ]; (1.34)γr =−Im[γe−2iφ]. (1.35)Here the angle φ is the azimuthal angle measured about the center of the lens (Schneider, 2006b). The rotatedshear (which would represent a curl component) should be consistent with zero, and is often used as a checkof systematic effects. Even though any single galaxy cluster (or other lens) is likely not perfectly azimuthallysymmetric, we expect a stack of many galaxy clusters to yield a symmetric profile on average.Similar to the surface mass density representation of the convergence (Equation 1.21), we can then relatethe tangential shear to the differential surface mass density of the lens (Kaiser and Squires, 1993; Kaiser, 1994;Fahlman et al., 1994):γt(θ) =∆Σ(θ)Σcrit, (1.36)where θ now specifies the radial angle of separation between the lens center and the source image. The differ-ential surface mass density is defined to equal the difference between the average surface mass density interior13to θ and the surface mass density at θ (Wright and Brainerd, 2000):∆Σ(θ)≡ Σ(< θ)−Σ(θ). (1.37)Given an expression for the mass density profile of a gravitational lens, and angular diameter distances involved,the expected tangential shear profile can be calculated. Useful models for galaxy cluster masses, and the resultingγt(θ) and ∆Σ(θ) profiles, are given in Section 1.3.4.1.2.2 Weak Lensing MagnificationGravitational lensing causes the magnification of background galaxies due to the isotropic focusing of the lensedlight rays (whereas shear arises from the anisotropic component). Magnification µ(θ ) is related to the determi-nant of the Jacobian, µ = 1/detA . It is commonly expressed asµ = 1(1−κ)2− γ2 , (1.38)where γ ≡√γ21 + γ22 is the magnitude of the shear distortion. In the weak lensing limit (κ,γ 1), the magnifica-tion is simply a measure of the convergence of a lensing mass, µ ≈ 1+2κ , where κ is defined in Equations 1.20and 1.21 (Schneider, 2006a). For the case of circularly symmetric sources we can write µ = (θ/βL)(dθ/dβL)(Narayan and Bartelmann, 1996). Conceptually, magnification can be understood as the stretching of solid an-gle on the sky, which causes the amplification of source flux, since lensing conserves surface brightness. Thisdirectly leads to a change in source size, ω = µ(θ )ω(s), where size is defined in Equation 1.29 (Bartelmann andSchneider, 2001).In general, two different approaches can be taken to measure magnification: (1) quantifying sizes of galaxiesbehind lenses and measuring lensing-induced changes; or (2) detecting the effect on background source numberdensity that ensues as a result of amplification in a flux-limited survey. This thesis focuses on the latter approach.The former method suffers from many of the limitations facing shear analysis, and will be discussed briefly inSection 1.2.3 below.Define n0( f ,z)d f dz to be the intrinsic (unlensed) number of galaxies per solid angle within d f of flux fand dz of redshift z. The cumulative lensed number density detectable (i.e. brighter than f ) in an astronomicalsurvey will then be:n(> f ,θ ,z) = 1µ(θ ,z)n0(> fµ(θ ,z) ,z). (1.39)In this equation we can see that magnification affects the source number densities in two ways. The prefactor1/µ represents the stretching of apparent solid angle, and decreases the number density on the sky. Meanwhile,the denominator in the argument of n0 implies that sources will be detectable to a fainter intrinsic magnitude,effectively increasing the observed source number density if there exist sources fainter than the detection limitof the survey (Schneider, 2006b). An illustration of the two effects of magnification is given in Figure 1.5.Since magnification affects the background source number densities in these two opposing ways, we requireknowledge of the intrinsic source number densities as a function of brightness, in order to predict or interpret ourmagnification observations. Typically, astronomical sources such as galaxies contain relatively few very brightmembers and are much more numerous towards the faint end. If we approximate the slope of the number density14Figure 1.5: Magnification Illustration. Highly exaggerated illustration of the two effects of magnification(dilution and amplification) by a massive foreground galaxy. The left panel shows an idealized setof unlensed sources, the right panel shows the effect of gravitational lensing. Dilution refers to thestretching of solid angle on the sky, which reduces background source number density. Since lensingconserves surface brightness, a consequence is that source flux is amplified. [Source: Joerg Colberg,Ryan Scranton, Robert Lupton, SDSS].in a narrow flux bin as a power law n0( f ) ∝ f−α , then we haven(> f )n0(> f )= µα−1. (1.40)Unfortunately, instead of using flux, astronomers frequently characterize source brightness using the ancient-Greek-inspired system of magnitudes. The apparent (observed) magnitude of an object with flux f ism≡−2.5log10( f/ fx), (1.41)where fx = 2.53×10−8 Wm−2 is a reference flux. Bright sources have smaller apparent magnitudes. Absolutemagnitude M can be defined similarly, but relative to a reference luminosity. Its relation to apparent magnitudecan be expressed as M = m− 5log10(dL/10pc), where dL is the luminosity distance in Equation 1.13 (Ryden,2003). In terms of apparent magnitudes we can relate the lensed and intrinsic number densities using the equationn(m,z)dm = µα−1n0(m,z)dm, (1.42)where we now define α explicitly according toα ≡ α(m,z) = 2.5 ddmlogn0(m,z). (1.43)The form of Equation 1.42 was first demonstrated by Narayan (1989), who applied it to lensed quasar number15densities, but it can be generalized to any galaxy type as long as one has a means of obtaining the slope of thenumber counts α .Returning to the question of whether gravitational lensing magnification will cause an increase or decreasein observed number densities of background sources, we find that the answer depends precisely on the value ofα . Sources for which α−1 > 0 will appear to be correlated on the sky with a lens position, while sources withα − 1 < 0 will be anti-correlated, as a dearth of objects will be observed in the vicinity of a lens. The numberdensity of galaxies for which the intrinsic number count slope gives α − 1 ≈ 0 will essentially be unaffectedby lensing magnification, as the dilution and amplification effects will cancel, and no correlation signal willbe observed for these objects. The brightest sources, which usually have steep number counts, will exhibit anincrease in number density when lensed, as the amplification allows more objects to be detected, while thenumber density of the faintest sources, having relatively shallow number counts, will decrease (Narayan, 1989;Scranton et al., 2005).Lensing magnification has had an interesting history. During the 1970s and ’80s several studies showed thatbright quasars were correlated on the sky with foreground galaxies (Seldner and Peebles, 1979; Arp, 1981; Web-ster et al., 1988; Narayan, 1989). The reason for this association was hotly debated, with some astronomers evenarguing for physical associations of the objects (e.g. Arp, 1987). The proponents of the lensing interpretationwere frequently obtaining results discrepant with each other as well as with the expected lensing signal strength(see e.g. Schneider, 1992) through the end of the 20th century. The first convincing magnification detection wasmade by Scranton et al. (2005), using 200,000 quasars lensed by 13 million galaxies in the Sloan Digital SkySurvey (SDSS). The dominant reasons for the previous discrepancies between theory and observation were a lackof homogeneous and well-characterized source populations, leading to systematic effects such as the inability toobtain clear redshift separation between foreground and background samples (Scranton et al., 2005).Quasars are appealing background sources for magnification studies for two reasons. First of all, their highredshifts ensure that they are well separated from the foreground lenses, which is important so that physicalassociations due to gravitational attraction do not contaminate the lensing signal. Secondly, their steep num-ber counts yield high values for the slope α(m,z), which leads to strong positive correlations with the lenses.Submillimeter galaxies have been studied behind cluster lenses for nearly two decades, exploiting the strongmagnification effect on their steep number counts (Blain and Longair, 1996; Smail et al., 1997). More recently,Lyman-break galaxy (LBG) sources have also been used successfully in magnification studies by Hildebrandtet al. (2009b), Morrison et al. (2012), and Hildebrandt et al. (2013) and by the author of this thesis in Ford et al.(2012) and Ford et al. (2014) (see Chapters 2 and 3). These galaxies tend to also have high α(m,z) values, butmore care must be taken to ensure redshift separation from the lenses involved.1.2.3 Magnification vs. ShearThe shear approach to weak lensing has dominated the field for the last several decades. Early work by Schnei-der et al. (2000) showed that shape information had less intrinsic scatter than sizes or number counts, andconcluded that very little was gained by including magnification along with shear. A review article by Bartel-mann and Schneider (2001) compared the signal-to-noise ratios of shear and magnification, finding the shearsignal-to-noise to be at least 5 times larger than for magnification. This comparison included several simplifyingassumptions, like equal number densities of sources for shear and magnification, and a number count slope ofα = 0.5. However, LBGs and quasars can have α values of several at the bright end of the luminosity function,16and magnification with number counts can certainly include many times the number of sources relevant for ashear analysis (since shapes are not required).One of the really attractive aspects of magnification is its ability to be applied in regimes where the sheartechnique starts to fail. Since shear studies require accurate measurements of galaxy shapes, in order for asource to be used at all it must necessarily be well resolved. Specifically, for lenses at high redshift, and forground-based surveys facing the extra complication of blurring due to atmospheric seeing, the number densityof background sources for which shear can be well-determined is greatly reduced (Van Waerbeke, 2010). Thisis in stark contrast to magnification studies using source number densities, which have no such requirement forthe sources to be resolved at all. In principle only source magnitudes, redshifts, and positions relative to a lensmust be known. This simple fact makes it possible to extend weak lensing magnification analyses to a muchhigher redshift than possible for shear, and allows a much higher source density to be included in the analysis(Van Waerbeke et al., 2010).Shear is susceptible to an issue known as the mass sheet degeneracy. This is the fact that, since shear probesdifferential mass profiles, ∆Σ(θ), adding a constant mass sheet across an entire lens does not change the mea-sured shear (Falco et al., 1985; Schneider and Seitz, 1995). Any weak lensing shear measurement is thus leftwith some ambiguity (although in practice, if the survey area is large enough, this is usually not a concern).Magnification, which directly probes the surface mass density, Σ(θ), can be used to break this mass sheet de-generacy (Broadhurst et al., 1995). External knowledge of the unmagnified sources must be available, however,to compare to the lensed observations. The combination of shear and magnification, in order to circumvent thismass sheet degeneracy has been demonstrated in a series of cluster analyses by Umetsu et al. (2011), Umetsu(2013), and Umetsu et al. (2014).Some progress has been made in terms of measuring magnification using source size information (Huffand Graves, 2014; Schmidt et al., 2012). Unfortunately, the approach of measuring size magnification suf-fers from many of same limitations as the shear technique, since it also requires quantifying the spatial extentof sources, which must necessarily be well resolved. Although back-of-the-envelope calculations by Bartel-mann and Schneider (2001) showed a larger signal-to-noise ratio for size than for number density magnification,Schneider (2006b) explains how the Point Spread Function circularizes the sources, and that seeing-convolvedimage sizes are even more difficult to measure than shapes. A related alternative approach to measuring magni-fication, by using the modified redshift distributions of lensed background sources, was originally proposed byBroadhurst et al. (1995), and recently demonstrated on observational data by Coupon et al. (2013).Because the constraint on source resolution is considerably relaxed, ground-based observations can be in-corporated to a greater degree with magnification (using number counts) than for shear (or size magnification),as we are not concerned with correcting for the smearing of the image due to the atmospheric Point SpreadFunction. From the financial perspective, these types of magnification studies are therefore extremely cheap tocarry out, since the excellent resolution of space-based telescopes is not required (Hildebrandt et al., 2009b).The bottom line regarding magnification is that it provides gravitational lensing information that is inde-pendent of, and complementary to, the information obtained from a weak lensing shear analysis. Since magni-fication with source number densities (which is the approach taken in Chapter 2 and Chapter 3 of this thesis)essentially imposes no additional constraints on a weak lensing shear survey, the ability to perform a magnifi-cation analysis comes along for free. Any costless source of cosmological information ought to be investigatedand exploited in full. See Van Waerbeke (2010), Rozo and Schmidt (2010), and Umetsu et al. (2011), for more17detailed discussions of the benefits of combining magnification with shear in gravitational lensing studies.1.3 Galaxy ClustersClusters of galaxies represent the largest and most massive gravitationally-bound systems to have formed thusfar in our Universe. They range from associations of only a few nearby galaxies, called groups, all the way upto very rich clusters containing many thousands of members. The distinction between groups and clusters isnot well defined, and this thesis will largely use the word “clusters” as a blanket term covering all groupings ofgalaxies. Clusters themselves are observed to clump together into enormous superclusters.Galaxies were first recognized to cluster together by Charles Messier in 1784 (who noted the concentrationof “nebulae,” as they were called, in what we now know as the Virgo galaxy cluster), and then independently byF. Wilhelm Herschel in the early 19th century, long before they were actually recognized as very distant galaxiessimilar to our own (Biviano, 2000). The first cluster masses were estimated by Zwicky (1933), but it was notuntil the first comprehensive catalog of over two thousand clusters was produced by Abell (1958) that the studyof galaxy clusters really took off. Today hundreds of thousands of galaxy clusters have been cataloged andstudied (see e.g. Wen et al., 2012).Because they harbor the deepest large-scale gravitational potentials in the Universe, clusters are uniquelaboratories for studying the highest energy phenomena since the big bang. They can be used as testing groundsfor general relativity, and gravitational structure formation. Baryonic processes of the intergalactic medium,high energy plasma physics, and the interplay with member galaxies and galaxy evolution, can all be exploredusing galaxy clusters (Kravtsov and Borgani, 2012).This section begins with an overview of the main techniques used to discover galaxy clusters in astro-nomical surveys (Section 1.3.1). The main focus of the section is then the discussion of the two key uses forstudying galaxy clusters – to constrain cosmology (Section 1.3.2) and to understand astrophysical processes(Section 1.3.3) – with an attempt to highlight the significant aspects of these areas that are relevant to this thesis.This is followed by a discussion of the density profiles that are employed to model the distribution of dark matterin a galaxy cluster (Section 1.3.4).1.3.1 Finding Galaxy ClustersThe first step in any galaxy cluster study is to identify these objects in an astronomical survey. This can bedone using a variety of techniques, applicable in different wavelength ranges. For the most massive systems, itis possible to detect the X-ray (also known as Bremsstrahlung) emission from the hot diffuse cluster gas in theIntracluster Medium (ICM). The ICM also upscatters CMB photons to higher energies, through inverse-Comptonscattering, leaving an essentially redshift-independent imprint upon the microwave light of the CMB sky. Thisthesis employs galaxy clusters discovered in the optical part of the spectrum, which is a region where oneprimarily detects light from the individual galaxies, rather than diffuse emission from the cluster halo.Constructing a galaxy cluster catalog from the galaxies detected in a survey involves several potential sourcesof systematic bias. A major source of concern includes projection effects, which can cause galaxies at differentdistances, but distributed along a similar line-of-sight, to appear to be in physical proximity. Projection effectsare especially pernicious if galaxy redshifts are poor or unknown. However, even for state of the art photometricredshifts calibrated on a spectroscopic sample, outliers exist and can lead to false positives in the catalogedclusters. There are two common approaches to circumventing this effect. The first is to use color information,18as galaxies in clusters tend to be redder than field galaxies (galaxies not bound to a cluster). The well-knownred sequence technique (Gladders and Yee, 2000) and variants such as redMaPPer (Rykoff et al., 2014), use theobserved tight correlations between cluster galaxy magnitudes and colors to limit false detections. However,recently formed or lower mass galaxy groups (e.g. our own Local Group) do not necessarily have a strong redsequence, and would be missed by these techniques.An alternative approach to using color information is to employ a matched-filer technique. Pioneered byPostman et al. (1996), the original matched-filter algorithm attempted to match an expected radial and luminosityprofile for galaxies in clusters, across a survey. Without redshift information, however, line-of-sight projectionswere still a significant concern. The inclusion of galaxy redshifts into a matched-filter approach was first doneby Milkeraitis et al. (2010), and the application of this algorithm to the Canada-France-Hawaii Telescope LegacySurvey (CFHTLS) generated the cluster catalog used in this thesis, in Chapters 2 and 3.Whatever the algorithm used to identify clusters of galaxies, it is safe to assume that the catalog generatedwill be neither 100% complete nor 100% pure, and that the completeness and purity will have some redshiftand magnitude dependance. Characterizing these aspects of a cluster-finder (the so-called selection function ofa technique) through extensive tests using simulations, is crucial for correcting for observational biases and forultimately deriving cosmological constraints from a cluster sample. Large scale simulations, for which the exactproperties of galaxies, including halo membership, are known, allow one to connect observed cluster propertiesto the simulated halos. This can, for example, illuminate the extent to which detected cluster candidates aremerely projection effects or even blended systems of multiple distinct halos (Cohn et al., 2007).1.3.2 Clusters for CosmologyClusters of galaxies represent the high-mass end of structure formation in the Universe, and the most recentobjects to have collapsed gravitationally. Quantifying cluster number density as a function of mass can be asensitive probe of cosmology. Two cosmological parameters that are of particular importance for the study ofgalaxy clusters are Ωm and σ8. The first, already discussed in Section 1.1, is the matter content of the Universeexpressed as a fraction of the total energy density. The second parameter, σ8, is known as the normalizationof the matter power spectrum. It can also be described as the variance in the mass contained within spheresrandomly located, with comoving radius 8h−1Mpc (hence the subscript “8”) (Davis and Peebles, 1983; Whiteet al., 1993).The fact that our Universe even contains galaxies and other structures, means that it is not perfectly homo-geneous. The canonical description is that, at early times there existed slight perturbations in the density field,which underwent gravitational collapse to eventually form the varied structures we observe today. If the averagedensity was 〈ρm〉, then these density perturbations as a function of position can be written asδ (x) = ρm(x)−〈ρm〉〈ρm〉. (1.44)The Fourier components of the density perturbations are δk(k) =∫δ (x)eik·xd3x.If we believe that the Universe is indeed isotropic, and that these perturbations δ (x) can be described as aGaussian random field (the simplest scenario), then we can fully characterize them using the isotropic powerspectrum P(k)≡ 〈|δk|2〉 (Peebles, 1993; Kravtsov and Borgani, 2012). The primordial power spectrum is oftenapproximated as a power law P(k) ∝ kn, where n = 1 is the special scale-invariant case proposed around the19same time by Harrison (1970), Peebles and Yu (1970), and Zeldovich (1972). Recent measurements by PlanckCollaboration et al. (2014) have confirmed that n≈ 0.96.Defining a spherical window function W (r), and its Fourier transform Wk, we can write the variance on massscale M asσ2 ≡〈∣∣∣∣δMM∣∣∣∣2〉=1(2pi)3∫P(k)|Wk|2d3k. (1.45)Thus σ8 is simply a special case of the square root of the above expression, wherein a top hat window functionof radius 8h−1Mpc is used.6 This particular radius is historical in nature, simply chosen because (δM/M)∼ 1in spheres of this volume (Davis and Peebles, 1983), but it remains widely used in the cosmological literaturetoday. Better constraints now give σ8 ≈ 0.83 (Planck Collaboration et al., 2014).At first, when the density constrast was small δ (x) 1, the perturbations simply expanded along with theUniverse. The modes grew independently, according to the linear growth function (Peebles, 1993; Voit, 2005):D(a) ∝δρρ ∝a˙a∫ a0daa˙3. (1.46)During the era of radiation domination, perturbations grew as δ ∝ a2, transitioning to δ ∝ a when the uni-verse became matter dominated. As the different wavelength modes entered each other’s horizon, the overdenseregions grew by attracting surrounding matter. Gravitationally collapsing regions eventually reached virial equi-librium, and decoupled from the global expansion. (Schneider, 2006a). Smaller overdensities merged togetherto form larger structures, in what is known as hierarchical structure formation. A full mathematical descriptionof the growth of structure is beyond the scope of this introduction, and the reader is referred to important earlypapers (e.g. Press and Schechter, 1974; Gott and Rees, 1975) or more recent excellent reviews on the subject(e.g. Voit, 2005; Schneider, 2006a; Kravtsov and Borgani, 2012).A common approach to extracting cosmological information from galaxy clusters involves measuring the(comoving) number density of clusters at a given mass. This important quantity nM(M,z) is known as thecluster mass function. In the original formalism of Press and Schechter (1974) the cluster mass function is givenbynM(M,z) =Ωmρcrit(z = 0)Merfc[δcrit√2σ(M,z)], (1.47)where δcrit is the critical linear overdensity for collapse. Improvements have been made on the Press-Schechterformalism, notably by Sheth and Tormen (1999) and Jenkins et al. (2001), generalizing to ellipsoidal perturba-tions and improving parameterizations based on cluster simulations.Observers usually express the cluster mass function in differential form (Borgani, 2008):dnM(M,z)dM=√2piρ¯mM2δcritσ(M,z)∣∣∣∣d logσ(M,z)d logM∣∣∣∣exp(−δ 2crit2σ(M,z)2). (1.48)For the purpose of fitting universal functions to data from observations or simulations, it is common to simplifythe above expression into the formdnM(M,z)dM= f (σ) ρ¯mMd lnσ−1dM, (1.49)6The top hat window function is constant inside the 8h−1Mpc radius, zero outside of it, and is normalized so it integrates to one.20where the function f (σ) is some parameterization that fits the data (see e.g. Tinker et al., 2008). The cluster massfunction nM depends very specifically on the meaning of the mass of a cluster. A common choice, which will beused throughout this thesis, is the mass parameter M200. This is the total mass interior to a radius R200, withinwhich the average density is 200 times the critical energy density of the Universe, 〈ρ(r < R200)〉= 200ρcrit(z).There are several difficulties in connecting galaxy cluster observations to theories of structure formation incosmology. Clusters evolve over time, but we cannot observe the collapse of an individual halo because of thetime scales involved. Instead we must make inferences by observing ensembles of clusters at different redshifts.Other issues are the observational difficulties associated with compiling pure and complete samples of galaxyclusters, using techniques that nearly always probe some visible tracer of the cluster dark matter halo. Finally,it is very difficult to accurately know the true mass M that appears in the cluster mass function. Gravitationallensing is the most promising method for obtaining accurate cluster masses, and the work in this thesis is aimedat improving these estimates further, especially for higher redshift clusters. Future experiments may build uponthis work to use galaxy clusters to improve constraints on cosmology.1.3.3 Clusters for AstrophysicsGalaxy clusters live in deep gravitational potentials, containing (in addition to the galaxies and dark matter)copious amounts of hot ionized plasma and cooler intracluster gas and dust, collectively known as the ICM. Amultitude of astrophysical studies are made possible by investigating individual galaxy clusters, merging galaxycluster systems, or ensemble-average properties of galaxy clusters. A full review of cluster astrophysics isbeyond the scope of this introduction, and the reader is referred to reviews by Kravtsov and Borgani (2012),Voit (2005), and references within, for further information. The goal of this section is to highlight areas ofastrophysics that particularly benefit from galaxy cluster studies, in order to demonstrate that accurate massmeasurements – as sought in this thesis – can have a wide range of implications beyond cosmology.Clusters are defined by the fact that they contain galaxies, and the properties and evolution of these clustermember galaxies is itself a very large field in astronomy. On average, galaxies in clusters are redder in colorthan field galaxies (non-cluster galaxies). The fact that all known rich7 clusters exhibit a very tight relationshipbetween galaxy colors and magnitudes allows for cluster-finding algorithms specifically based on detecting thisso-called red-sequence (Gladders and Yee, 2000; Rykoff et al., 2014). The nature of exactly how cluster galaxiesco-evolve with their parent cluster, as measured e.g. by their Luminosity Function (LF) or stellar mass, as well asthe role of galaxy interactions and mergers in this process, are important for understanding the cluster evolutionas a whole and the relative contributions from the dark matter potential and baryonic interactions.An important piece of this picture is the evolution of the Brightest Cluster Galaxy (BCG), a giant red ellipticalgalaxy that sits near the center of the potential in a large fraction of very massive systems. BCGs are thoughtto form early in the history of a cluster and then build up their stellar mass hierarchically by absorbing smallercompanion galaxies (De Lucia and Blaizot, 2007). The properties (and presence) of a BCG may be redshift-dependent, and some studies suggest that the evolutionary processes might be more complex or varied (Oliva-Altamirano et al., 2015). Best et al. (2007) used SDSS data to show that BCGs were much more likely to host aradio-loud Active Galactic Nuclei (AGN)8 than other galaxies of similar stellar mass.Studies of the ICM are of interest to plasma physicists, due to the high energies involved. The ICM tem-7In the context of galaxy clusters, rich means having many galaxy members.8AGN are extremely bright and energetic central regions of some galaxies, presumed to be powered by the central supermassive blackhole, leading in some cases to the formation of energetic jets of relativistic particles.21peratures are typically in the range of 107 to 108 K, and the particle number density ranges from around 10−2to 10−4 cm−3, consisting mainly of ionized hydrogen, with some helium and other trace elements (Markevitchand Vikhlinin, 2007). Various areas of study within the ICM involve general properties of gas flow, includingturbulence and shock propagation, and the presence and properties of large-scale magnetic fields.One longstanding problem has been explaining the observed rates of ICM cooling in cluster cores. In theabsence of other interactions, the ICM would be expected to condense and form stars at much higher rates thanactually observed (Fabian, 1994). The catch-all term “feedback” has been used describe how any of manypossible processes redistribute energy to the ICM to prevent its collapse. Candidate mechanisms for feedbackinclude supernovae explosions, high mass X-ray binaries (massive stars orbiting black holes or neutron stars),and AGN. The latter are undoubtedly crucial components of cluster astrophysics, as these accreting black holescan generate jets of relativistic particles extending up to 100 kpc or more (Fabian, 2012). These jets interact withthe ICM, possibly driving shock waves and generating turbulence, which distributes energy to other regions ofthe cluster. Recently Voit et al. (2015) have proposed a model that appears to well explain the rates of coolingand regulation of feedback required, wherein cool clouds condense due to thermal instability and accrete ontothe central black hole.The hot plasma that makes up the ICM can be detected by its X-ray radiation, which is due to Bremsstrahlung(free-free) emission. The total mass of the ICM is much higher than the stellar mass of the cluster membergalaxies (up to 20 times higher for very massive clusters), but the two are positively correlated (Kravtsov andBorgani, 2012). Scaling relations that seek to allow for simple conversions from one type of observable toanother, such as stellar mass or X-ray luminosity to total cluster mass, are valuable for understanding physicalprocesses but must be carefully calibrated (Leauthaud et al., 2010). To this end, recent gravitational lensinganalyses by Applegate et al. (2014) and Hoekstra et al. (2015) have quantified existing biases between differentmethods of mass estimation. The limiting factor for accurate masses from weak lensing shear is currently thequality of photometric redshifts (Hoekstra et al., 2015).Currently, a particularly promising area of cluster astrophysics involves mergers – 2 clusters of galaxiescolliding – which are the most energetic events in the current-day Universe, reaching 1058 J (Markevitch andVikhlinin, 2007). These systems often exhibit an offset between the position of the collisionless dark matter(and galaxies) and the collisional diffuse ICM, which gets left behind when the clusters pass through each other.Mergers are proving to be exciting laboratories for constraining the particle nature of dark matter. The canonicalexample is the well-known Bullet Cluster, from which the ratio of the dark matter particle self-interaction cross-section to its mass has been constrained to . 1cm2 g−1 (Markevitch et al., 2004). Other merging systems haveyielded similar upper limits (see e.g. Clowe et al., 2012; Bradacˇ et al., 2008). For these measurements, weaklensing is crucial for creating accurate mass maps that yield detections of the separation between the dark matterdensity peaks and the hot ICM.1.3.4 Cluster Mass ProfilesThe most well-known and oft-employed model for the mass distribution in a dark matter halo is the Navarro-Frenk-White (NFW) profile. High-resolution N-body simulations carried out by Navarro et al. (1996) demon-strated that this universal profile well-described the spherically-averaged matter distribution of gravitationallycollapsed objects, independent of cosmological parameters and the spectrum of initial density fluctuations. Theapplicability of the model spans a wide range in mass, from the scale of individual dwarf galaxies to rich clusters22of galaxies, and fits over two decades in radius. The density of an NFW halo is given by:ρNFW(r) =δcρcrit(z)(r/rs)(1+ r/rs)2. (1.50)Here rs is known as the scale radius, and δc is the characteristic overdensity of a halo,9 which is interchangeablewith the halo concentration parameter c200 throughδc =(2003)c3200ln(1+ c200)− c200/(1+ c200). (1.51)In the NFW prescription, the virial radius of a halo is R200, which is the radius within which the averagedensity is 200ρcrit. This can be related to the scale radius and the concentration parameter though the equationrs = R200/c200. A common definition for the mass is the total mass interior to a sphere of radius R200 (Wrightand Brainerd, 2000):M200 =800pi3ρcritR3200. (1.52)Equivalently, one can define masses interior to other radii (e.g. M500 and M2500 are often used in X-ray clusterstudies), but M200 is a common choice for weak lensing, and is used in this thesis.As expected in the scenario of hierarchical structure formation, more massive halos are less centrally con-centrated (i.e. higher M200 implies lower c200). This is a reflection of the density of the Universe when each halowas collapsing, so that recently formed massive halos are less dense. Strong correlations exist between the massand concentration, both as determined in simulations (Duffy et al., 2008; Prada et al., 2012; Dutton and Maccio`,2014) and more recently confirmed by observations (Okabe et al., 2013; Merten et al., 2014).Following from the NFW formalism, Wright and Brainerd (2000) performed calculations of the two-dimensionalmass profiles, which are relevant for weak lensing. The surface mass density Σ and differential surface mass den-sity ∆Σ (discussed in Section 1.2.1) can be derived by integrating the NFW mass density along the line-of-sight,ΣNFW(R) = 2∫ ∞0ρNFW(R,y)dy, (1.53)where R is the projected radial distance (in the plane of the sky) and y is the line-of-sight dimension, both relativeto the halo center. Adopting the dimensional radius x≡ R/rs, we have:ΣNFW(x) =2rsδcρcrit(x2−1)[1− 2√1−x2arctanh√1−x1+x], for x < 1;2rsδcρcrit3 , for x = 1;2rsδcρcrit(x2−1)[1− 2√x2−1arctan√x−11+x], for x < 1.(1.54)The differential surface mass density ∆Σ, probed by shear, can be obtained from Equation 1.37. One mustintegrate as follows to obtain the mean interior surface mass density:ΣNFW(< x) =2x2∫ x0ΣNFW(x′)x′dx′. (1.55)This yields the final required set of equations for the 2-dimensional NFW profiles for weak gravitational lensing9Note that δc should not be confused with δcrit in Equations 1.47 and 1.48.23(Wright and Brainerd, 2000):ΣNFW(< x) =4rsδcρcritx2[2√1−x2arctanh√1−x1+x + ln(1/2)], for x < 1;4rsδcρcrit [1+ ln(1/2)] , for x = 1;4rsδcρcritx2[2√x2−1arctan√x−11+x + ln(1/2)], for x < 1.(1.56)Several other commonly-used halo density models exist. The generalized-NFW, or gNFW, profile simplyadds another free parameter to the NFW model. While the NFW density goes as r−3 at large radii, and r−1at small radii, the gNFW formalism allows the latter exponent (the inner logarithmic density slope) to vary.Alternatively, the Einasto profile has been shown in some recent work by Dutton and Maccio` (2014), to be abetter description of dark matter halos than the NFW or gNFW profiles. The Einasto profile was first introducedby Einasto (1965), and is a 3-parameter model with density profile ρ(r) ∝ exp(−ArB).Another commonly-used mass profile is the Singular Isothermal Sphere (SIS) model (which is employed inChapter 2 of this thesis). The SIS is fully characterized by a constant velocity dispersion σv, implying a densityprofile that goes as ρ(r)∝ r−2. The surface mass density for an SIS model is given by Bartelmann and Schneider(2001) to beΣSIS(R) =σ2v2GR. (1.57)Although unphysical – the total mass diverges – the SIS has been popular historically because of its simplicityas a one-parameter model, and its success in describing halos reasonably well at larger radii (Schneider, 2006b).1.4 Impact of this ThesisThis work contained in this thesis has pushed the boundaries of the knowledge that can be extracted from weaklensing surveys, and used to understand galaxy cluster properties. By including intrinsically smaller and fainterbackground sources, which cannot be used in conventional weak lensing studies, we pave the way for a moreoptimal use of survey data. These gravitationally-lensed sources, which are too small for reliable shape or sizemeasurements, can still be included in a lensing analysis by using the flux magnification formalism describedin Section 1.2.2. The utility of measuring magnification has been proven through several key publications thatappear as chapters in this work, and include thorough studies of the systematic effects that provide limitations.Prior to this thesis research, weak lensing was dominated by the shear method. This was originally motivatedby some early work showing that the signal-to-noise for shear was several times larger than for magnification(Schneider et al., 2000). While it is true that, for a fixed sample of galaxies, there is less scatter in galaxy shapesthan in galaxy positions, the latter is far easier to measure. This simple fact has motivated the research herein.As lensing studies push to higher redshift, and increasingly rely on blurry ground-based data, we have elevatedconfidence in our measurements of source positions over difficult shape determinations.When this thesis work began, only a handful of magnification studies had been completed. The first ground-breaking theoretical formulation of how number densities of sources could be used to measure masses of clusterswas laid out in 1995 by Broadhurst et al. (1995), but it took another 10 years before the first convincing observa-tional detection was made (Scranton et al., 2005). Following this significant 8σ detection of galaxy-magnifiedquasars in the 3800 deg2 of the SDSS, several studies followed, achieving magnification detections for lensing ofnormal galaxies (Hildebrandt et al., 2009b), and of blue galaxies behind strong lensing clusters (Umetsu et al.,242011). These studies all stopped short of deriving scientifically useful results from the magnification measure-ments – they either represented proof-of-concept studies for a new technique, or they demonstrated consistencywith a lensing interpretation of the signal.The work in this thesis made major steps forward in the area of lensing magnification. The author performedthe first-ever measurement of magnification by stacked galaxy groups in 2012 (See Chapter 2). This particularwork was also the first time that shear and magnification mass estimates and signal-to-noise had been compared(Ford et al., 2012). Following this influential work in the Cosmological Evolution Survey (COSMOS) survey,the thesis transitioned to focus on the much larger astronomical survey known as the Canada-France-HawaiiTelescope Lensing Survey (CFHTLenS).Two important studies resulted from magnification analyses of the CFHTLenS for this thesis (See Chapter 3and Chapter 4). First of all, the most significant magnification detection thus far (9.7σ – similar in significance tothe SDSS measurement, with only 4% the sky survey area) was published in Ford et al. (2014). More importantly,however, that work moved beyond simple magnification-detection to actual physical constraints. Masses ofstacked galaxy clusters were measured in bins as a function of different attributes (redshift and richness), andthe dependence of a magnification signal on these parameters was seen for the first time. A mass-richness scalingrelation was determined solely from the magnification results, providing a useful tool for making cosmologicalinferences from optical cluster surveys, as discussed in Section 1.3.This work contained the important inclusion of a means of accounting for one of the dominant systematiceffects for magnification, namely the contamination of the background sources with low-redshift objects. Theformalism was extended from earlier work by Hildebrandt et al. (2013), but allowing for different contaminationfractions and models for the halo occupation distribution of the galaxy contaminants. This was the first timethat galaxy cluster lenses could be used for magnification in a redshift range where there was known sourcecontamination. Prior to this work, the redshift ranges of overlap had to be avoided because the physically-induced cross-correlations of lens and source objects overwhelmed the magnification signal, and could not beseparated from it.Arguably the most important magnification result in all the literature to date is contained in Chapter 4 ofthis thesis. After the semi-blind magnification analysis of Ford et al. (2014), Ford et al. (2015) followed with anidentical treatment of the same cluster sample, but this time using the weak lensing shear approach. This studycontained a detailed comparison between cluster masses measured with the two independent techniques, as afunction of different cluster attributes, and contained valuable insights regarding systematic effects that are stillimportant to resolve for magnification. Moving forward, this work frames the case for including magnification,and also pin-points some important issues that must be addressed in future work (see Chapter 5).1.5 Thesis OverviewThe body of this thesis is composed of three published studies that develop the weak gravitational lensingmagnification technique, particularly for the study of galaxy clusters, and compares with results using the com-plementary and much more ubiquitous weak lensing shear approach:• Chapter 2 contains the first magnification study of galaxy groups and first comparison with shear forstacked lens samples. The data are X-ray selected groups, and high-redshift LBGs in the COSMOS field.• Chapter 3 represents the highest-significance magnification detection, and the first magnification study25that could be binned as a function of cluster parameters. The data are optically-selected galaxy clustersand high-redshift LBGs in the CFHTLenS field.• Chapter 4 is the follow-up shear analysis of the same cluster sample presented in the previous chapter,using the CFHTLenS shear catalog for background source shape measurements.Finally, Chapter 5 wraps up with conclusions on the topic of cluster studies using both magnification and shear,and briefly outlines future directions for progress within the field.26Chapter 2Magnification by Galaxy Group Dark MatterHalosWe report on the detection of gravitational lensing magnification by a population of galaxy groups, at a signif-icance level of 4.9σ . Using X-ray selected groups in the Cosmological Evolution Survey (COSMOS) 1.64 deg2field, and high-redshift LBGs as sources, we measure a lensing-induced angular cross-correlation between thesamples. After satisfying consistency checks that demonstrate we have indeed detected a magnification signal,and are not suffering from contamination by physical overlap of samples, we proceed to implement an optimallyweighted cross-correlation function to further boost the signal to noise of the measurement. Interpreting thisoptimally weighted measurement allows us to study properties of the lensing groups. We model the full distri-bution of group masses using a composite-halo approach, considering both the Singular Isothermal Sphere (SIS)and the Navarro-Frenk-White (NFW) profiles, and find our best fit values to be consistent with those recoveredusing the weak lensing shear technique. We argue that future weak lensing studies will need to incorporatemagnification along with shear, both to reduce residual systematics and to make full use of all available sourceinformation, in an effort to maximize scientific yield of the observations.2.1 IntroductionWeak gravitational lensing is a unique tool for probing the mass distribution of the Universe and for constrainingdark matter halo properties of galaxies and clusters. In contrast to alternative mass estimate methods (employinge.g. X-ray temperatures, radial velocities, or mass-to-light ratios), weak lensing does not rely on any assumptionsabout virial equilibrium and is sensitive to all mass along the line of sight, making no distinction betweenluminous and dark matter.Over the past decade, an enormous international effort has been invested in improving the reliability of weaklensing analysis (Heymans et al., 2006; Massey et al., 2007a; Bridle et al., 2009; Kitching et al., 2010), seekingto remove biases and systematic effects that limit the accuracy of the method. By far most of the work hasbeen focused on measuring the shear signal, the coherent stretching and distortion of distant galaxy shapes by aforeground lensing mass, but recently the magnification signal has begun to attract attention as well (Scrantonet al., 2005; Hildebrandt et al., 2009b, 2011; Van Waerbeke et al., 2010; Umetsu et al., 2011; Huff and Graves,2014).Weak lensing magnification is, to first order, a measure of the convergence of a lensing mass. It can be27detected through the stretching of solid angle on the sky, which leads to the amplification of source flux, sincelensing conserves surface brightness (i.e. photons are neither created nor destroyed in purely lensing processes).In general, two different approaches can be taken to measure magnification. The method we employ hereinvolves observing the effects on source number densities; an interesting alternative method is being exploredby Schmidt et al. (2012), which makes use of source size and flux information, and employs the same COSMOSX-ray groups used in this study.Magnification affects the source number densities in two ways, and the one that dominates is determinedby the intrinsic magnitude number counts of the sources in question. Simply put, the brightest sources, whichusually have steep number counts, will exhibit an increase in number density when lensed, as the amplificationallows more objects to be detected, while the number density of the faintest sources, having relatively shallownumber counts, will decrease (Narayan, 1989).Compared to shear measurements, magnification exhibits a slightly lower signal-to-noise ratio (S/N), thereason it has been largely ignored until recently. However, what magnification lacks in signal strength, it makesup for in terms of its ability to be applied to lenses at higher redshift and to poorly resolved sources (Van Waer-beke, 2010). Since shear studies require measurements of galaxy shapes, in order for a source to be used it mustnecessarily be well resolved. This is in stark contrast to magnification studies using source number densities,which have no such requirement for the sources to be resolved at all! In principle, only source magnitudes,redshifts, and positions relative to a lens must be known. This simple fact makes it possible to extend weaklensing magnification analyses to a much higher redshift than possible for shear, and allows a much highersource density to be included in the analysis. See Van Waerbeke (2010), Rozo and Schmidt (2010), and Umetsuet al. (2011) for more detailed discussions of the benefits of combining magnification with shear in gravitationallensing studies.In Section 2.2 we review the equations describing the effects of weak lensing magnification on source num-ber densities. Section 2.3 gives the properties of the X-ray groups and LBGs that are used in this study. ThenSection 2.4 describes the steps of our analysis, and results of the composite-halo model fitting. We summarizethe results in Section 2.5, and compare with weak lensing shear measurements that have previously been madeon populations of galaxy groups. We use the WMAP7 ΛCDM cosmological parameters H0 = 71 km s−1 Mpc−1and ΩΛ = 0.734 (Larson et al., 2011), and set Ωm = 1−ΩΛ.2.2 TheoryThe amplification matrix A maps the image deformation from the source to observer frame, and describes thefirst order effects of gravitational lensing.A =(1−κ− γ1 −γ2−γ2 1−κ+ γ1)(2.1)It is a function of the convergence κ , and the shear γ , which define the isotropic and anisotropic focusing of lightrays, respectively. The magnification factor µ is the inverse determinant of this matrix, so thatµ = 1detA=1(1−κ)2−|γ|2(2.2)(Bartelmann and Schneider, 2001).28The cumulative number counts of distant unlensed sources N0 are related to the observed lensed numbercounts N, up to some flux f , by the equationN(> f ) =1µ N0(>fµ). (2.3)Here the two distinct effects of weak lensing magnification, on source number counts, are made explicit. Theprefactor of 1/µ is the dilution of source density, as the observed solid angle on the sky is stretched by aforeground massive lens. The modification to the flux f/µ inside the argument of N0 represents the effect ofsource amplification by a lens, such that one is able to detect intrinsically fainter objects due to gravitationallensing.Switching from working in fluxes to magnitudes m, the differential number count relationship was demon-strated by Narayan (1989) to ben(m)dm = µα−1n0(m)dm, (2.4)where α is defined according toα ≡ α(m) = 2.5 ddmlogn0(m). (2.5)Thus, distant source galaxies, lensed by an intervening concentration of mass, may have their observed numbercounts increased or decreased depending on the sign of the quantity (α−1). Sources for which (α−1)> 0 willappear to be correlated on the sky with a lens position, while sources with (α − 1) < 0 will be anti-correlated,as a dearth of objects will be observed in the vicinity of a lens. The number density of galaxies for whichthe intrinsic number count slope gives (α − 1) ≈ 0 will essentially be unaffected by lensing magnification, asthe dilution and amplification effects will cancel, and no correlation signal will be observed for these objects(Scranton et al., 2005).2.3 Data2.3.1 LensesThe lenses in this study consist of X-ray selected galaxy groups in the COSMOS Field. See Leauthaud et al.(2010) for the detailed properties of these groups. From the full sample of 206 groups investigated in theaforementioned study, we use the shear-calibrated mass estimates to construct the most massive subsample ofgroups for this magnification study. Here masses are characterized by the parameter M200, the total mass interiorto a sphere of radius R200, within which the average density is 200 times the critical.Any groups that have less than 4 member galaxies, that appear to be undergoing mergers, that have uncertaincentroids, or that raise concerns about projection effects, are excluded from the analysis. These restrictions fol-low from the group catalog requirement FLAG INCLUDE=1, discussed in George et al. (2011). The remaining44 most massive groups have shear determined masses in the range 3.56×1013 ≤M200/M ≤ 1.70×1014, andwe employ stacking to increase the S/N of the magnification measurement. The redshift range of the groups is0.32≤ z≤ 0.98. Figure 2.1 displays these lens properties.Choosing an optimal lens centroid about which to construct angular bins is an area of ongoing research, andcommon choices include the Brightest Cluster Galaxy (BCG) or the X-ray emission peak. If the location of thedark matter density peak were known a priori, then it would obviously be the ideal choice, but instead we must29Figure 2.1: Masses and Redshifts of COSMOS Groups. Masses and photometric redshifts of the groupsin this study. We select the most massive groups in our sample, M200/M ≥ 3.56×1013. Using onlythe cleanest groups (characterized by having ≥ 4 members, well-defined centroids, and no flags onpossible mergers or projection effects), and applying appropriate masking, we are left with a sampleof 44 groups for this lensing magnification analysis.rely upon some combination of observables to approximate this position. In this paper, we define lensing masscenters by the location of the group galaxy with the highest stellar mass (MMGGscale) lying within a distance(Rs +σx) of the X-ray center, where Rs is the group scale radius and σx is the uncertainty in the X-ray centerposition (George et al., 2011). In order to be very confident about the locations of group centers, we excludegroups for which this galaxy is not the most massive member of the group. This choice of centroid has beenshown to accurately trace the centers of halos in this sample by optimizing the shear signal on small scales(George et al., 2012).2.3.2 SourcesBackground sources are LBGs, a type of high-redshift star-forming galaxy that has been used successfully inprevious magnification studies (see Hildebrandt et al., 2009b, 2011). These LBGs were selected using the typicalthree color dropout technique. For the U-, G-, and R-dropouts the selections described in Hildebrandt et al.(2009a) were used, however the COSMOS Subaru g+ and r+ data were used instead of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) g∗ and r∗ data (see Capak et al. (2007) for the filter definitions). Forthe B-dropouts the selection from Ouchi et al. (2004) was used.The appeal of using LBGs for magnification is rooted in the fact that their Luminosity Function (LF) hasbeen extensively studied and their redshift distributions are fairly narrow and accurate. After all quality cuts andimage masking, we are left with 45,132 LBGs in total. The four distinct sets are comprised of 12,980 U-, 22,520G-, 4,870 B-, and 4,762 R-dropouts, located at redshifts of ∼ 3.1, 3.8, 4.0, and 4.8, respectively.We first test our data selection by cross-correlating the foreground groups with LBGs separated into discretemagnitude bins. Here we use the basic Landy and Szalay (1993) estimator,w(θ) = D1D2−D1R−D2R+RRRR, (2.6)30Figure 2.2: Magnitude-Binned Cross-Correlation. Angular cross-correlation of the X-ray groups withLBGs, the latter separated into 3 magnitude-selected samples. The Bright sample contains U , G,B, R-dropouts in the magnitude ranges 23 < r < 25, 23.5 < i < 25, 23.5 < i < 25, 24 < z < 25.5,respectively. Similarly, the Medium ranges are 25< r < 25.5, 25< i< 26, 25< i< 26, 25.5< z< 26.The Faint ranges are r > 25.5, i > 26, i > 26, z > 26. These magnitude ranges are selected tocontain LBGs for which (α−1)> 0, ≈ 0, and < 0. The measured correlations for each LBG sampleare simply averaged here (weighting by the number counts) in order to more clearly display thisdiagnostic check. The dashed curves are calculated from the composite-NFW fit, using weighting bythe appropriate 〈α − 1〉 factor, which is given in each panel. The negative correlation observed forthe faintest sample is a good indication that no redshift overlap exists between foreground lenses andbackground sources.to simply compute cross-correlations between groups and background sources. D1 and D2 represent the datasets of lenses and sources, and R are the random objects from a mock catalog we create, containing pointsuniformly distributed throughout the COSMOS survey area. Each product of terms is the number of pairs ofthose objects found to lie within some angular bin, normalized by the total number of pairs found at all angularseparations. This cross-correlation estimator has been shown to be both robust and unbiased (Kerscher et al.,2000).In any lensing study, care must be taken to ensure that regions of an image containing artifacts such assaturated pixels, satellite tracks, or other spurious effects, are masked out of the investigation. We consistentlyapply the same masks to the group, source, and random catalogs, prior to the correlation analysis. Using a largenumber (584,586) of objects in this random catalog serves to reduce shot noise.We expect that the faintest (brightest) magnitude bins should yield a negative (positive) cross-correlationwith the group centers, and this is exactly what we find. Figure 2.2 displays this anticipated result, where wesimply use a number count weighted average to combine the signal of the distinct LBG samples. As discussed inHildebrandt et al. (2009b), this negative correlation is one of the strongest verifications that no redshift overlapexists between lens and source populations, for no viable reason other than lensing magnification can be givenfor such a signal to exist. Redshift overlap between samples must be avoided in magnification studies, as positivecross-correlations due to physical clustering would overwhelm any lensing-induced signal.312.4 Analysis and Results2.4.1 Measuring α(m)Along with the mass of the lens itself, the slope of the source number counts as a function of magnitude, pa-rameterized by the quantity α ≡ α(m), controls the amplitude and sign of the expected magnification signal. Tointerpret the correlations that we measure, and to implement an optimally-weighted procedure, we must deter-mine the value of this quantity for every source galaxy that we intend to use in the measurement. Fortunately,LBGs have been extensively studied and many measurements of their LF have been published.For the U-, G-, and R-dropouts, we use the recent measurements by van der Burg et al. (2010). For theB-dropouts we use the results of Sawicki and Thompson (2006). These two sets of measurements both involvedfitting a Schechter Function (Schechter, 1976) to their galaxy number counts, and their best fit parameters thatwe use here, are displayed in Table 2.1. The Schechter Function is given byΦ(M) = 0.4ln(10)Φ∗100.4(αLF+1)(M∗−M) exp[−100.4(M∗−M)], (2.7)where Φ∗, M∗, and αLF are the normalization, characteristic magnitude, and faint-end slope of the LF. Note thatthe α(m) which we want to calculate is not the same as the constant parameter αLF, but approaches it in the limitof very faint magnitudes.Solving this equation for α(m), we obtainα(m) = 2.5 ddmlogn0(m) = 2.5ddMlogΦ(M) = 100.4(M∗−M)−αLF−1. (2.8)We convert the observed apparent magnitudes m of the LBGs to absolute magnitudes M via the relationshipM = m−DM+2.5log(1+ z), where DM and z are the distance modulus and redshift of the galaxy in question.Since we select apparent magnitudes in the r, i, and z bands for the U-, G- and B-, and R-dropouts, we probe verysimilar restframe wavelengths and the K-correction between the samples is negligible. Thus we ignore it here.Using the LF parameters in Table 2.1, combined with the conversion to absolute magnitudes, we then obtain arobust measure of α(m) for every LBG in the sample.It is important to assess how uncertainties in the quantity α(m) can affect the interpretation of the magnifi-cation measurement. Since we will rely on the quantity (α−1) as a weight factor in this analysis, using a wrongα could potentially lead to a bias in the mass measurement. For very faint objects the observed magnitudesbecome less certain due to shot noise. We propagate these magnitude errors through Equation 2.8 to obtain anuncertainty on α(m), which is used to find a magnitude-based cut on the sources. We find that cutting ∼10% ofthe very faintest sources, largely R-dropouts, gives us a good balance between removing the most uncertain αvalues, but still retaining a significant number of sources for the analysis.Here we consider two possible sources of systematic error, which are combined in quadrature to yield the to-tal systematic error, reported in Section 2.4.3. The first source is uncertainty on the LF parameters (see Table 2.1),which includes the effects of cosmic variance. We repeat the composite-halo fit, detailed in Section 2.4.3, forthe range of permitted values of M∗ and αLF, finding a maximum variation in the mass measurement of up to∼40%. Second, we consider the possibility for a small photometric offset to exist between the various surveysused in this work. Assuming a maximum offset of ±0.05 magnitudes between surveys, we vary all observedsource magnitudes uniformly by offsets in the range−0.5≤ δm≤ 0.5, and find the maximum effect on the mass32Table 2.1: Luminosity Function Parameters. LF (Schechter) parameters from external LBG measure-ments. a LF parameters from van der Burg et al. (2010). b LF parameters from Sawicki and Thompson(2006).LBG Sample M∗ αLF NumberU (z∼ 3.1)a −20.84+0.15−0.13 −1.60+0.14−0.11 12,980G (z∼ 3.8)a −20.84+0.09−0.09 −1.56+0.08−0.08 22,520B (z∼ 4.0)b −21.00+0.40−0.46 −1.26+0.40−0.36 4,870R (z∼ 4.8)a −20.94+0.10−0.11 −1.65+0.09−0.08 4,762measurement to be ∼15%.2.4.2 Optimally Weighted Cross-CorrelationWe implement a modified version of the Landy and Szalay (1993) estimator for the angular cross-correlationfunction, in which pair counts are weighted by their expectations from the differential source number counts as afunction of magnitude. This weighted correlation function has been shown to optimally boost the magnificationsignal (Me´nard et al., 2003):wopt(θ) =Sα−1L−Sα−1R−〈α−1〉LRRR+ 〈α−1〉. (2.9)Optimal-weighting was first implemented by Scranton et al. (2005) and, apart from notation, this equation isidentical to the estimator used in Hildebrandt et al. (2009b). As with the original basic estimator, each termrepresents the number of pairs of objects found in a given angular θ bin, normalized by the total number of pairsat all angular separations.S stands for the sources, or background lensed galaxies, L are the lenses, or X-ray groups, and once againR are the random objects. The superscript α−1 on the S indicates that pair counts involving sources are to beweighted by this factor. After removing masked objects from the catalogs, and satisfying the above selectioncriteria, we are left with 39,710 LBG sources, 44 X-ray group lenses, and 584,586 random objects for theanalysis.The brightest source galaxies, which are observationally found to lie in the steepest part of the LF, areexpected to be positively correlated with the group centers, have the largest value of α − 1, and so receive arelatively large weight in this correlation study. In contrast, the faintest background galaxies are expected tobe anti-correlated, on average, with the group positions, because the effects of magnification dilution shouldbe greater than the amplification of flux can compensate for, and these galaxies thus receive a negative weight.Sources for which α − 1 ≈ 0, ought to have the effects of dilution and amplification cancel out overall, andreceive very little to no weight in this analysis (Scranton et al., 2005).The optimally weighted correlation function is given in Figure 2.3, and shows the measured radial profilesfor this stack of massive galaxy groups. Error bars are 1σ uncertainties, obtained by jackknife resamplingof the source population. To do this, we create 50 jackknife samples of data, each with a different 1/50 ofsources removed from it. Then we measure the optimal correlation function for each, and from these estimate33the covariance matrix throughC(θ1,θ2) =(NN−1)2×1NN∑j=1[w j(θ1)− w¯(θ1)]× [w j(θ2)− w¯(θ2)], (2.10)where the index j runs over the N = 50 jackknife measurements.2.4.3 Halo Mass ProfilesMeasuring the magnification-induced effects on source number counts behind massive lenses allows one toestimate properties of the lens, such as the mass profile. In this paper, we use a composite-halo approach whichallows us to fit for the full range of both group masses and redshifts. The horizontal axes in Figure 2.2 andFigure 2.3 are therefore actual transverse distances obtained by taking account of the angular diameter size ateach unique group redshift. We incorporate both the SIS and the NFW (Navarro et al., 1997) density profiles intothe composite-halo modeling.The magnification contrast is δµ(θ)≡ µ(θ)−1, and for an SIS halo it is simply given byδµSIS(θ) =θEθ −θE, (2.11)where θE = 4pi(σvc )2 DlsDsis the Einstein radius of the lens, and Dls and Ds are angular diameter distances betweenlens and source, and observer and source, respectively. The velocity dispersion of the lens, σv, can be expressedin terms of the mass and critical energy density of the Universe at lens redshift z:σv =[pi6200ρcrit(z)M2200G3] 16. (2.12)For the NFW halo, the magnification contrast takes a slightly more complicated form. From Equation 2.2,we haveδµNFW(θ) =[(1−κNFW)2−|γNFW|2]−1−1. (2.13)We use the analytical NFW expressions for κ and γ derived in Wright and Brainerd (2000) to evaluate δµNFWfor every lens-source pair in the study. The two NFW fit parameters are the scale radius rs and the concentrationc200, which together determine the massM200 =4pi3(200)ρcrit(z)c3200r3s . (2.14)As we do not find this magnification measurement precise enough to provide meaningful two-parameter con-straints, we use the mass-concentration relation given in Mun˜oz-Cuartas et al. (2011) to reduce this to a single-parameter fit.We perform a composite-halo fit for both lens models, similar to the multi-SIS used in Hildebrandt et al.(2011). This allows us to fit for a range of masses and redshifts, thereby avoiding any biases that would beintroduced by simply fitting to a stacked average lens profile. The optimally weighted correlation function is34Figure 2.3: Optimally-Weighted Cross-Correlation. Composite-halo fits to the optimally weighted cor-relation function, using the LBG background source sample. The significance of the magnificationdetection is 4.9σ . The dashed line is the composite-SIS and the solid line is the composite-NFW. Wefind the best-fit relative scaling relations for each to be a = Mmag/Mshear = 1.2±0.4±0.4sys (SIS) anda = 1.8± 0.5± 0.4sys (NFW). The dotted line shows the prediction from the shear measured valuesof M200 (A. Leauthaud 2011, private communication).related to the magnification contrast throughw(a)optimal =1NlNl∑i=1〈(α−1)2〉iδµ(zi,aMshear,i), (2.15)where i runs over all lenses. Here the fit parameter a characterizes the scaling relation between the M200 previ-ously measured from the shear, and the best fit M200 from magnification, so that a≡Mmagnification/Mshear.We use the generalized minimum-χ2 method to fit the composite-halo profiles to the magnification measure-ments (see Figure 2.3), using the full unbiased inverse covariance matrix, according to the prescription in Hartlapet al. (2007). The χ2/dof is 1.5 for the composite-SIS and 0.8 for the composite-NFW (χ2SIS = 7.5, χ2NFW = 4.2,dof=5 in both cases). For the composite-SIS, we obtain a best-fit value of a = 1.2± 0.4± 0.4sys, and with thecomposite-NFW we get a = 1.8±0.5±0.4sys. These results indicate consistency with the previous shear massmeasurements, albeit with large uncertainties.352.5 Summary and ConclusionsWe report a 4.9σ detection of weak lensing magnification from a population of X-ray-selected galaxy groups.This is the first magnification measurement using source number densities successfully performed on the groupscale. Schmidt et al. (2012) have recently explored the magnification of these groups using source sizes andfluxes. For comparison, the shear detection significance is 11σ on the same selection of 44 groups (A. Leauthaud2011, private communication).1To improve S/N in this measurement, we stack the lenses, consisting of 44 massive X-ray detected galaxygroups in the COSMOS 1.64 deg2 field. We measure an optimally weighted cross-correlation between the X-raygroups and high-redshift LBGs, with 1σ error bars determined from jackknife resampling of the sources. Per-forming composite-halo fits to this optimally weighted signal yields a measurement of the relative scaling be-tween shear- and magnification-derived masses. Our magnification measurement yields a mass Mmag = aMshearwhere the best-fit parameter a = 1.2± 0.4± 0.4sys (SIS), and a = 1.8± 0.5± 0.4sys (NFW), demonstrating arough consistency with the shear measurement.As discussed in Section 2.4.1, a central issue is the importance of having correct (α−1) measures for everysource galaxy, to ensure that the optimal weighting truly is optimal. We perform a thorough error analysisthat includes measurement uncertainties from the full covariance matrix, and investigate possible sources ofsystematic errors from both photometry and externally calibrated LF parameters.We claim that LBGs are a preferred source sample when it comes to performing lensing magnification anal-yses using source number counts. A few reasons for the superiority of the LBG sample include more reliableredshift determinations, as well as greater lensing efficiencies and generally much higher values of the quantity(α−1). The single most significant reason to choose LBGs for this type of analysis, however, is for the ease ofobtaining a reliable measure of α(m). Previous deep measurements of LBG LFs allow us to perform calculationsyielding the optimal weight factor (α−1), as well as its associated uncertainty.Although the S/N of shear is superior to magnification in general, the latter probes the surface mass densityof the lens directly, while the shear measures the differential mass density. Thus the combination of these twoindependent measurements is desirable, and breaks the lens mass-sheet degeneracy. In fact, Rozo and Schmidt(2010) demonstrated that joining magnification into shear analyses, independent of survey details, can improvestatistical precision by up to 40−50%. Magnification using source number densities is also far less sensitive tothe effects of atmospheric seeing than either shear or magnification using source sizes. Both of these methodsrequire quality source images which, for very high redshift sources, can currently only be obtained from space-based data.Improving the overall weak-lensing-derived constraints on cosmological and astrophysical parameters isnot the only benefit to incorporating magnification into our analyses, however. Measurements of magnificationare sensitive to completely different systematics than shear, and therefore uniquely positioned to help improvecalibration of these residual effects on shear measurements. For example, magnification (using number counts)is not at all sensitive to the possible intrinsic alignment of source galaxies, since it does not use any shapeinformation. Magnification can also be used as a simultaneous probe of intergalactic dust extinction, a smallbut measurable effect through its wavelength dependence (Me´nard et al., 2010), and as a direct way to measuregalaxy bias (Van Waerbeke, 2010).1The significance quoted for the shear does not take into account the full covariance matrix, as we have done for the magnificationmeasurement. Therefore this shear significance might be a bit optimistic.36As one proceeds to investigate dark matter structures at increasingly high redshift, it becomes more andmore important to include the magnification component of the signal. This is a direct consequence of the factthat higher redshift lenses necessitate more distant sources, which are in turn much harder to measure shapes for.Proceeding exclusively with shear necessarily means that a high fraction of detected sources are being eliminatedfrom the lensing analysis, and information is therefore being lost, simply because we lack the capabilities torobustly determine their shapes. With photometric redshifts available, the possibility to do magnification studieson our shear catalogs really comes along free of charge. Upcoming projects will survey the entire extragalacticsky, and the inclusion of magnification will be a necessary component of any robust weak lensing study.37Chapter 3Cluster Magnification and the Mass-RichnessRelation in CFHTLenSGravitational lensing magnification is measured with a significance of 9.7σ on a large sample of galaxy clustersin the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). This survey covers ∼154 deg2 and con-tains over 18,000 cluster candidates at redshifts 0.2 ≤ z ≤ 0.9, detected using the 3D-Matched-Filter (3D-MF)cluster-finder of Milkeraitis et al. (2010). We fit composite-NFW models to the ensemble, accounting forcluster miscentering, source-lens redshift overlap, as well as nearby structure (the 2-halo term), and recovermass estimates of the cluster dark matter halos in range of ∼ 1013 M to 2× 1014 M. Cluster richnessis measured for the entire sample, and we bin the clusters according to both richness and redshift. A mass-richness relation M200 = M0(N200/20)β is fit to the measurements. For two different cluster miscentering mod-els we find consistent results for the normalization and slope, M0 = (2.3± 0.2)× 1013 M, β = 1.4± 0.1 andM0 = (2.2±0.2)×1013 M, β = 1.5±0.1. We find that accounting for the full redshift distribution of lenses andsources is important, since any overlap can have an impact on mass estimates inferred from flux magnification.3.1 IntroductionClusters of galaxies are the most massive gravitationally bound structures in the Universe today. As such theycan be useful cosmological probes, as well as laboratories for all kinds of interesting physics including galaxyevolution, star formation rates, and interactions of the intergalactic medium. There are several methods com-monly used to estimate cluster masses (e.g., mass-to-light ratios, X-ray luminosities, the Sunyaev-Zeldovicheffect), but among them gravitational lensing is unique in being sensitive to all mass along the line of sight,irrespective of its type or dynamical state (Bartelmann and Schneider, 2001).There are multiple ways to measure the signature of gravitational lensing, and each has its own specificadvantages and limitations. Observation of strong lensing arcs and multiple images is extremely useful forstudying the innermost regions of clusters, and getting precise mass estimates, but can only be applied to verymassive objects which are observationally limited in number. On the other hand, weak lensing shear, whichmeasures slight deformations in background galaxy shapes, can be applied across a much wider range of lensmasses. Shear studies have been used with much success to map large scale mass distributions in the nearbyUniverse (Van Waerbeke et al., 2013; Massey et al., 2007b). However, because they rely on precise shapemeasurements, shear faces the practical limitation of an inability to sufficiently resolve sources for lenses more38distant than a redshift of about one (Van Waerbeke et al., 2010).A third approach to measuring gravitational lensing is through the magnification of background sources,observable either through source size and flux variations (Schmidt et al., 2012; Huff and Graves, 2014), or theresultant modification of source number densities (Ford et al., 2012; Morrison et al., 2012; Hildebrandt et al.,2013, 2011, 2009b; Scranton et al., 2005). Magnification has been recently measured using quasar variabilityas well (Bauer et al., 2011). Although relative to the shear, magnification will tend to have a lower signal-to-noise for typical low-redshift lenses, the requirement for source resolution is completely removed. This makesmagnification competitive for higher redshift lenses, and especially for ground based surveys where atmosphericseeing has a strong influence on image quality.In this work we adopt the number density approach, known as flux magnification, using LBGs for the lensedbackground sources. The observed number density of LBGs is altered by the presence of foreground structure,due to the apparent stretching of sky solid angle, and the consequential amplification of source flux. Because ofthe variation in slope of the LBG LF, magnification can either increase or decrease the number densities of LBGs,depending on their intrinsic magnitudes. By stacking many clusters we can overcome the predominant sourceof noise - physical source clustering (Hildebrandt et al., 2011).In Section 3.2 we describe the cluster and background galaxy samples. Section 3.3 lays out the methodologyfor the measurement and modeling of the magnification signal. We discuss our results in Section 3.4 and con-clude in Section 3.5. Throughout this paper we give all distances in physical units, and use a standard ΛCDMcosmology with H0 = 70 km s−1 Mpc−1, Ωm = 0.3, and ΩΛ = 1−Ωm = 0.7.3.2 DataFor the magnification results presented in this paper, we are fortunate to work with a very large sample ofgalaxy cluster candidates and background galaxies in CFHTLenS.1 (Erben et al., 2013; Hildebrandt et al., 2012).CFHTLenS is based on the Wide portion of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS), withdeep 5 band photometry. The survey is composed of four separate fields, in turn divided into 171 individualpointings, covering a total of 154 deg2.3.2.1 3D-MF Galaxy ClustersThe 3D-MF cluster finding algorithm of Milkeraitis et al. (2010), essentially creates likelihood maps of the sky(in discrete redshift bins) and searches for peaks of significance above the galaxy background. The likelihood isestimated assuming that clusters follow a radial Hubble profile as well as a Schechter LF. A significance peak of>3.5σ is considered a cluster detection. The reader is referred to Milkeraitis et al. (2010) for the details of the3D-MF algorithm; here we discuss only the essential points relevant to our purposes.The radial component of the 3D-MF likelihood employs a cutoff radius of 1 Mpc, which was chosen toroughly correspond to the radius R200 of an M200 ∼ 3× 1013M cluster. Milkeraitis et al. (2010) motivatesthis choice by the desire to optimally search for relatively high mass clusters, but notes that this radius willbe less ideal for low mass clusters. One should expect that random galaxy interlopers may contaminate theestimation of significance for likelihood peaks corresponding to lower mass clusters. This may be a key factorin explaining the wide range of cluster significances conferred upon low mass clusters from simulations, whilehigh mass simulated clusters were assigned significances that correlated strongly with mass (see figure 10 in1www.cfhtlens.org; Data products available at http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/CFHTLens/query.html390.0 0.5 1.0 1.5 2.0 2.5Roffset [arcmin]0.00.20.40.60.81.01.21.41.61.8P(Roffset)(1.5-5)*1013M(5-10)*1013M(1-2.5)*1014M2.5*1014MAll ClustersFigure 3.1: Centroid Offset Model. Modeled Rayleigh distribution of radial offsets between defined3D-MF centers and simulated cluster centers. The black points and solid curve is the combined dataand best fit for all CFHTLenS clusters combined. The colored curves show the best fit distributionsfor separate mass bins, and colors match the empirical offsets measured and presented in Figure 13of Milkeraitis et al. (2010). As each of these colored curve fits is consistent with the solid blackcurve for the entire combined sample, we choose to use this single Rayleigh distribution to modelmiscentering for all clusters.Milkeraitis et al., 2010). Because peak significance may therefore not be an ideal mass proxy to use for the fullcluster ensemble, in this work we rely upon a measure of the cluster richness, which is discussed in Section 3.2.1below. Using cluster richness has the added benefit that the mass-richness relation can be measured and used asa scaling relation.Using the 3D-MF method, a total of 18,036 galaxy cluster candidates (hereafter clusters) have been detectedin CFHTLenS, at a significance of >3.5σ above the background. In contrast to previous cluster magnificationstudies, which have been limited by small number statistics, this huge sample of clusters allows us to pursuemultiple avenues of investigation. In particular, we bin the clusters according to both richness and redshift, to re-cover trends in physical characteristics such as the mass-richness relation, and also investigate halo miscenteringas a function of these parameters.Cluster CentersDue to the nature of the method, the defined centers of the 3D-MF clusters, which are located at peaks inthe likelihood map, do not necessarily coincide with member galaxies. Hence the defined centers are notablydifferent from many other cluster finders, which commonly choose the BCG, the peak in X-ray emission, sometype of (possibly luminosity-weighted) average of galaxy positions, or a combination of these, as a measure ofthe center of a dark matter halo.The choice of cluster center is always ambiguous, both observationally and in simulations. One wants to40Table 3.1: Centroid Offset Fit Parameters. Best Fit Rayleigh Distributions for the Cluster Miscenteringin Figure 3.1.Mass Range [M] Color Best fit σoff [arcmin] χ2red(1.5-5)×1013 red 0.37±0.06 1.8(5-10)×1013 green 0.42±0.06 1.4(1-2.5)×1014 blue 0.42±0.06 1.2≥2.5×1014 purple 0.45±0.06 1.4≥1.5×1013 black 0.40±0.06 1.1know the center of the dark matter distribution, as the point around which to measure a radially-dependent signal.Obviously the dark matter cannot be directly seen, so an observable such as galaxies or X-ray emission must beused (see George et al. (2012) for an excellent review and analysis of cluster centroiding). The chosen center ofthe cluster can be wrong for several reasons. The observable chosen (e.g. the BCG) may simply be offset fromthe true center of the dark matter potential. Misidentification of the BCG can be a significant problem for thisparticular example as well (Johnston et al., 2007).Perhaps a more interesting source for miscentering comes from the fact that clusters halos are not perfectlyspherical, and exhibit substructure and irregularities caused by their own unique mass assembly histories. Espe-cially for very massive halos, which have formed more recently and in many cases are still undergoing mergersand have yet to virialize, we really should not expect a clear center to exist. Following visual inspection, Man-delbaum et al. (2008a) chose to exclude the most massive clusters from their weak lensing analysis for this veryreason. Instead of throwing away the highest mass halos in our sample, we include them in this study, but takecare to account for possible miscentering effects.Milkeraitis et al. (2010) tested for centroid offsets in 3D-MF by running the cluster-finder on simulationsand comparing detected cluster centers to known centers. The simulations used were the mock catalogs ofKitzbichler and White (2007), which were created from a semi-analytic galaxy catalog (De Lucia and Blaizot,2007) derived from the Millenium Simulation (Springel et al., 2005). Figure 13 of that work shows the numberof clusters detected as a function of distance from true cluster center. Because 3D-MF was optimized to producecluster catalogs that are as complete as possible (in contrast to, e.g. Gillis and Hudson (2011), which is designedto maximize purity), the trade-off is the presence of some contamination with false-detections, especially at thelow mass end.We use the numbers of clusters at each offset, and the contamination from Milkeraitis et al. (2010), toestimate the probability of radial offsets P(Roff). We fit the result for each mass bin with a Rayleigh distribution,which is the radial representation of a 2-dimensional Gaussian:P(Roff) =Roffσ2offexp[−12(Roffσoff)2]. (3.1)This resulting curves are presented in Figure 3.1 (colors are selected to match those in Figure 13 of Milkeraitiset al. (2010)), and for clarity we only show the data points for the bin that combines all clusters. We findconsistent fits for the separate mass bins, which we list in Table 3.1, and therefore use the combined distribution(black curve) to model the effects of miscentering in our measurements.410 50 100 150 200 250 300 350 400 450N200100101102103104# of clustersFigure 3.2: 3D-MF Cluster Richness Distribution. Distribution of richness (N200) values for clusters inthis study.Cluster RichnessWe define the richness parameter N200 in this work to be the number of galaxies within a radius R200, and redshift∆z, of a cluster candidate center (both points discussed below). Member galaxies are also required to be brighterthan i-band absolute magnitude -19.35. This cut-off is chosen to correspond to the limiting apparent magnitudeof CFHTLenS (i ∼ 24) at the highest redshift clusters that we probe, z ∼ 1. So at the expense of removingmany galaxies from the richness count, we hope to largely avoid the effect of incompleteness on the number ofgalaxies per cluster. Then clusters of the same intrinsic richness at high and low redshift should have comparableobserved N200, within the expected scatter of the mass-richness relation.For the line-of-sight dimension, we require galaxies to fall within ∆z < 0.08(1+ z) of the cluster redshift.This ∆z is the 2σ scatter of photometric redshifts in the CFHTLenS catalog, chosen so that we reduce the proba-bility of galaxies in a cluster being missed due to errors in their photo-z estimation. Of course this comes at theexpense of counting galaxies within a quite broad line-of-sight extent, especially for the higher redshift clusters.This effect should cancel out though, since we also use the same ∆z range in calculating the galaxy backgrounddensity, which is subtracted to yield N200 as an overdensity count of cluster galaxy members.In the plane of the sky, galaxies must lie within a projected radius R200 of the cluster center (defined above).R200 is defined as the radius within which the average density is 200 times the critical energy density of theUniverse, ρcrit(z), evaluated at the redshift of the cluster. However, since R200 itself is unknown, we requiresome kind of assumption about radius or mass in order to proceed with the galaxy counting. There is no uniqueway to do this. We begin by making an initial approximation of the masses using a best fit power-law relationbetween mass and cluster peak significance (σcl), for 3D-MF clusters (Milkeraitis, 2011):log[M2001014 M]= 0.124σcl−1.507. (3.2)42As discussed in Section 3.2.1, 3D-MF tests on simulations suggested that peak significance was a good massproxy for high, but not low, mass clusters. In light of this, we merely employ the above relation as a startingpoint for calculating the radii from mass,R200 =[3M2004pi(200)ρcrit(z)]1/3. (3.3)These R200 estimates are then used for counting galaxies for cluster richness. Richness N200 is the variable ofchoice used as a mass proxy for binning the magnification measurement. The distribution of these richnessvalues is shown in Figure 3.2.3.2.2 SourcesWe use LBGs as the magnified background sources. LBGs are high-redshift star-forming galaxies (Steidel et al.,1998), that have been succesfully employed in past magnification studies (see Hildebrandt et al., 2009b, 2011;Morrison et al., 2012; Ford et al., 2012) due to the fact that their redshift distributions and LF are reasonablywell understood. Knowledge of the intrinsic source LF allows for an interpretation of the magnification signal,which depends sensitively on the slope of the number counts as a function of magnitude. In addition, the high-redshift nature of LBGs is important to reduce redshift overlap between lenses and sources. Any source galaxiesin the redshift range of the cluster lenses will contaminate the lensing-induced cross-correlation signal, withcorrelations due to physical clustering.The LBG sample is selected with the color selection criteria of Hildebrandt et al. (2009a) (see Sect. 3.2 ofthat paper). It is composed of 122,144 u-dropouts with 23 <r ≤ 24.5, located at redshift ∼3.1. We choose thismagnitude range to avoid as much potential low-redshift contamination as possible. See Section 3.3.2 for ourmodeling of the residual contamination. The detailed properties of this LBG population will be described in aforthcoming paper (Hildebrandt et al. in prep.).3.3 Magnification Methodology3.3.1 The MeasurementThe magnification factor, µ , of a gravitational lens can be expressed in terms of the change from intrinsic (n0)to observed (n) differential number counts of background sources:n(m)dm = µα−1n0(m)dm (3.4)(Narayan, 1989). Here m is apparent magnitude, and α ≡ α(m) is proportional to the logarithmic slope of thesource LF. Depending on the LF’s slope in a given magnitude bin, it is possible to observe either an increase ora decrease in source number counts, as demonstrated in Figure 2.2 of Chapter 2 (equivalently, Figure 2 of Fordet al., 2012).In practice the magnification signal is easily measured using the optimally-weighted cross-correlation func-tion of Me´nard et al. (2003):wopt(R) =Sα−1L−Sα−1R−〈α−1〉LRRR+ 〈α−1〉. (3.5)43In this expression, the terms are normalized pair counts in radial bins, where L stands for the lenses, and Sα−1are the optimally-weighted sources. R represents objects from a random catalog more than ten times the size ofthe source catalog, with the same masks applied.In order to determine the optimal weight factor α−1, for both the measurement and the interpretation, werequire knowledge of the source LF. As done in Chapter 2 (Ford et al., 2012) we determine the LBG LF slopefrom the Schechter Function (Schechter, 1976), givingα = 2.5 ddmlogn0(m) = 100.4(M∗−M)−αLF−1, (3.6)and rely on externally measured LFs for the characteristic magnitude M∗ and faint end slope αLF. We use theLBG LF of van der Burg et al. (2010), measured using much deeper data from the CFHTLS Deep fields. For u-dropouts M∗ is -20.84 and αLF is -1.6. Thus every source galaxy is assigned a weight factor of α−1 accordingto its absolute magnitude M.The magnification signal, wopt(R), is measured in logarithmic radial bins of physical range 0.09− 4 Mpc(in contrast to angle), so that we can stack clusters at different redshifts without mixing very different physicalscales. Each cluster’s signal is measured separately before stacking the measured wopt(R), and full covariancematrices are estimated from the different measurements.3.3.2 The ModelingThe magnification is a function of the halo masses, and to first order it is proportional to the convergence κ . Inthis work, however, we will use the full expression for µ to account for any deviations from weak lensing in theinner regions of the clusters:µ = 1(1−κ)2−|γ|2(3.7)(Bartelmann and Schneider, 2001).We assume a spherical Navarro-Frenk-White (NFW) model (Navarro et al., 1997) for the dark matter halos,along with the mass-concentration relation of Prada et al. (2012). The convergence is modeled as the sum ofthree terms,κ = [pccΣNFW +(1− pcc)ΣsmNFW +Σ2halo]/Σcrit, (3.8)where pcc is the fraction of clusters correctly centered (i.e. with Roff = 0), and Σcrit(z) is the critical surface massdensity at the lens redshift. The expression for the shear, γ , is identical with κ → γ , and Σ→ ∆Σ. Note that thefirst term in Equation 3.8 is equivalent to adding a delta function to the miscentered distribution of Figure 3.1,to represent clusters with perfectly-identified centers. As discussed in Section 3.4, the fits do not give strongpreference to miscentering in the measurement, but in future work (in particular with weak lensing shear) it willbe useful to constrain the degree of miscentering using the data, instead of relying solely on simulations.We assume both lenses and sources are located at known discrete redshifts. This is z ∼ 3.1 for the LBGs.Since they are at very high redshift the effect of any small offsets from this has negligible effect on the angulardiameter distance, the relevant distance measure for lensing. The clusters, on the other hand, have redshiftuncertainties of 0.05 (due to the shifting redshift slices employed by 3D-MF (Milkeraitis et al., 2010)). Thistranslates into an uncertainty on the mass estimates ranging from less than a percent up to ∼17% (depending oncluster z), and is included in the reported mass estimates.44ΣNFW is the standard surface mass density for a perfectly centered NFW halo, calculated using expressionsfor κ (and γ) in Wright and Brainerd (2000). ΣsmNFW on the other hand, is the expected surface mass densitymeasured for a miscentered NFW halo:ΣsmNFW(R) =∫ ∞0ΣNFW(R|Roff)P(Roff)dRoff. (3.9)The distribution of offsets P(Roff) is given by Equation 3.1, and the other factor in the integrand isΣNFW(R|Roff) =12pi∫ 2pi0ΣNFW(R′)dθ , (3.10)where R′ =√R2 +R2off +2RRoff cosθ (Yang et al., 2006).The 2-halo term Σ2halo accounts for the fact that the halos we study do not live in isolation, but are clusteredas all matter in the Universe is. We account for neighboring halos following the prescription of Johnston et al.(2007):Σ2halo(R,z) = bl(M200,z)Ωmσ28 D(z)2Σl(R,z), (3.11)Σl(R,z) = (1+ z)3ρcrit,0∫ ∞−∞ξ((1+ z)√R2 + y2)dy, (3.12)ξ (r) = 12pi2∫ ∞0k2P(k)sinkrkrdk. (3.13)Here small r is comoving distance, D(z) is the growth factor, P(k) is the linear matter power spectrum, and σ8 isthe amplitude of the power spectrum on scales of 8 h−1Mpc. For the lens bias factor bl(M200,z) we use Equation5 of Seljak and Warren (2004).Composite-Halo FitsThe part of the optimal correlation function which is caused by gravitational lensing is related to the magnifica-tion contrast δµ ≡ µ−1 throughwlensing(R) =1NlensNlens∑i=1〈(α−1)2〉iδµ(R,M200)i. (3.14)Here the sum is over the number of lenses in a given stacked measurement, and 〈(α−1)2〉i refers to the averageof the weight factor squared in the pointing of a given cluster.We perform composite-halo fits using the above prescription, in which we allow for the fact that the clustersin a given measurement have a range of masses and redshifts. We do not fit a single average mass. Instead, wecalculate δµ(R,M200)i for each individual cluster using a scaling relation between mass and richness,M200 = M0(N20020)β. (3.15)The fit parameters are the normalization, M0, and (log-) slope, logβ , of the assumed power-law relation. From45012 34 5z0.00.51.01.52.0P(z)ClustersLBGsFigure 3.3: Redshift Distributions of Clusters and Sources. Redshift probability distribution functionsfor the clusters and the LBG sources. Low-redshift contamination of the LBGs will lead to physicalclustering correlations where overlap with the cluster redshifts occurs.this we calculate the optimal correlation wopt(R) according to Equation 3.14. The best-fit relation is determinedby minimizing χ2, which is calculated using the full covariance matrix. We apply the correction factor fromHartlap et al. (2007) to the inverse covariance matrix; this corrects for a known bias (related to the number ofdata sets and bins) which would otherwise lead to our error bars being too small.LBG ContaminationAn important source of systematic error for magnification comes from low-redshift contamination of the sources,leading to physical clustering between the lens and source populations. The cross-correlation that results fromcontamination can easily overwhelm the measurement of magnification, making redshift overlap far more im-portant for magnification than for shear. Past studies sought to minimize this effect, for example by checkingfor the negative cross-correlation that should exist between lenses and very faint sources with shallow numbercount (Ford et al., 2012; Hildebrandt et al., 2009b). Here we incorporate this clustering into the model, using asimilar approach to Hildebrandt et al. (2013).Figure 3.3 shows the redshift probability distributions, P(z), for the clusters and the LBGs. The LBG redshiftdistribution is based on the stacked posterior P(z) put out by the BPZ redshift code (for details on the CFHTLenSphoto-z see Hildebrandt et al., 2012). Since the BPZ prior is only calibrated for a magnitude limited sample ofgalaxies we can not expect the stacked P(z) to reflect the real redshift distribution of the color-selected LBGs.Hence we use the location and shape of the primary (high-z) and secondary (low-z) peaks but adjust their relativeheights separately. This can be done with a cross-correlation technique similar to Newman (2008). Details ofthis technique will be presented in Hildebrandt et al. (in prep.).Despite our efforts to avoid contamination, there is obviously some redshift overlap with the clusters. Weuse the products of the lens and source P(z) to define selection functions, and calculate the expected angularcorrelations using the code from Hamana et al. (2004). The weighted correlation function that we measure isthe sum of the correlations due to lensing magnification and clustering contamination:wopt(R,z) = flensingwlensing + fclusteringwclustering. (3.16)Note that flensing + fclustering ≤ 1, since some of the contaminants may be neither in the background and lensed,nor close enough in redshift to be clustered with the lenses.46The clustering contamination fraction fclustering(z) for each cluster redshift is defined as the fraction of eachsource P(z) that lies within 0.1 in redshift (twice the cluster redshift uncertainty). The part of the source P(z)that lies at higher redshift than the lens is then the lensed fraction flensing(z), and the part at lower redshift (i.e.in front of the lens) has no contribution to the signal.The factor fclustering(z) itself is generally very small for the LBGs used in this work, only really non-negligiblefor cluster redshifts z ∼ 0.2-0.3, which can be seen in Figure 3.3. The more significant effect on the estimatedmasses is that flensing(z) ∼0.9 across all redshift bins, because about 10% of the sources are not really beinglensed. We tested our results for robustness against uncertainties in the contamination fraction. When we varythe total low-z contamination fraction by±1σ (∼4%), the best fit cluster mass estimates remain within the statederror bars.We explore three ways of determining wclustering. Because of the weighting applied to LBGs in our measure-ment (which is optimal for the lensed sources, and should suppress contributions from redshift overlap), therewill always be a prefactor of 〈α − 1〉 in each estimation of clustering. The first method uses the dark matterangular auto-correlation, wdm, and estimates of the galaxy and cluster bias to calculate:wclustering(R,z) = 〈α−1〉blbswdm(R,z). (3.17)We set the bias factor for the galaxy contaminants bs=1 for this analysis, which is reasonably consistent with thebias relation of Seljak and Warren (2004) that is employed for the cluster bias (bl).We also calculate both the 1- and 2-halo terms for NFW halos, w1halo and w2halo (again using the code andmethods described in Hamana et al., 2004). Here the expression for physical clustering takes the form:wclustering(R,z) = 〈α−1〉bs [w1halo(R,z)+w2halo(R,z)] . (3.18)This method requires knowledge of the occupation distribution of the low-z galaxy contaminants in the clusterdark matter halos, which is not well determined. As a first approximation we use the simple power-law formdescribed in Hamana et al. (2004),Ng(M200) =(M200/M1)A for M200 > Mmin0 for M200 < Mmin. (3.19)Since these parameters are unknown, we use the values for M1 and A measured for galaxies in the SDSS (seeTable 3 of Zehavi et al., 2011). We choose Mmin to correspond to the minimum mass measured for cluster halos,and assume that halos above this mass always host a detected cluster. As a final check, we also ask what theclustering signal would be if every halo above Mmin hosted both a cluster and a single low-z galaxy contaminant(so that Ng = 1 for M200 > Mmin).This final method yields the largest estimates of wclustering, and therefore a smaller estimate of cluster masses.The former (using SDSS parameters) gives the highest mass estimates, and the simple biasing approach of Equa-tion 3.17 yields intermediate results. We use the range of these results to estimate an uncertainty in mass esti-mates coming from lack of knowledge about the nature of the low-z galaxies that contaminate our LBG sourcesample. This additional systematic error affects only the clusters at low redshift, where the source and lens P(z)distributions overlap, and is reported on the mass estimates given in Table 3.3. All best fit mass values reportedin the tables of this work are calculated using the contamination approach of Equation 3.17, since this method470.11R [Mpc]0.00.10.20.30.40.5wopt(R)All Clusterspcc = 1pcc = 0Figure 3.4: Magnification for all Stacked 3D-MF Clusters. Optimal cross-correlation signal measuredfor the entire stacked sample of 18,024 clusters. The model fits are both composite-NFW (see text forall terms in the fit). The solid line assumes the clusters are perfectly centered on the peak likelihoodof the 3D-MF cluster detection, while the dashed line includes the effects of cluster miscentering.relied on the fewest assumptions about the nature of the galaxy contaminants.Accounting for redshift distributions in this particular source sample effectively means that cluster massesare higher than one would naively guess by fitting for only the magnification signal. However, note that in acase with more significant redshift overlap, so that fclustering was large, the opposite statement would be true, andmass estimates that included the full P(z) distribution would be smaller than than the naive magnification-onlyapproach. These are important effects to consider, and future flux magnification studies should be careful to usefull redshift distributions in modeling the measured signal.3.4 ResultsStacking the entire set of 18,036 clusters gives a total significance of 9.7σ for the combined detection, shownin Figure 3.4. The perfectly centered model is a better fit to the overall measurement, with χ2red ∼ 1.2, whilethe miscentered model gives χ2red of 2.3. For both models, there are two free parameters (M0 and logβ ), leadingto 8 degrees of freedom. To investigate miscentering and mass-richness scaling, we divide the clusters into sixrichness bins, and measure the optimal cross-correlation in each.We measure the characteristic signature of magnification in every richness bin with significances between4.6 and 5.9σ . These results are shown in Figure 3.5, where we try fitting both a perfectly centered model(pcc = 1) and a model where every cluster is affected by centroid offsets (pcc = 0). Details of the fits, includingreduced χ2 and the average of the best fit mass values 〈M200〉, are given in Table 3.2.The lowest mass (richness) bin is not well fit by either model. Overall there is not a strong preference for480.00.10.20.30.40.50.6wopt(R)2 < N200  10pcc = 1pcc = 010 < N200  20pcc = 1pcc = 00.00.20.40.60.81.01.21.4wopt(R)20 < N200  30pcc = 1pcc = 030 < N200  40pcc = 1pcc = 00.11R [Mpc]0.01.02.03.04.0wopt(R)40 < N200  60pcc = 1pcc = 00.11R [Mpc]60 < N200pcc = 1pcc = 0Figure 3.5: Magnification for Richness-Binned Clusters. Optimal cross-correlation signal measured foreach N200 (richness) bin. Two composite-NFW fits are shown. The solid curve assumes clusters areperfectly centered, while the dashed curve accounts for cluster miscentering, using the gaussian offsetdistribution modeled in Figure 3.1 and discussed in Section 3.3.2.either perfectly centered (pcc = 1) or miscentered (pcc = 0) clusters, and both are reasonably good fits. Generally,the miscentered model yields slightly higher masses for the clusters (though it is sensitive to the shape of thedata), due to the smoothing applied, which lowers the model amplitude in the innermost regions. However thisis easily within the uncertainty on the mass estimates, so the results are in agreement.The issue of cluster miscentering is interesting in its own right as discussed in Section 3.2.1. It is temptingto try and fit for the parameter pcc, describing the fraction that are actually correctly centered, or else for themiscentering Rayleigh distribution width σoff, as done in Johnston et al. (2007). The issue here is a strongdegeneracy between pcc, σoff, and cluster concentration. Increasing the number of clusters that have offsetcenters produces essentially the same results as leaving pcc fixed and increasing σoff, an effect that can bemimicked by a lower concentration in the NFW model. We run the risk of overfitting to the results.In fact, Johnston et al. (2007) found very little constraining power on the miscentering width and the frac-tion of miscentered MaxBCG clusters, and applied strong priors to these distributions. George et al. (2012)performed an extensive weak lensing miscentering study of groups in the COSMOS Field, and chose to forgo theadditional parameter pcc, as they achieved sufficiently good fits without it. Mandelbaum et al. (2008a) performeda lensing analysis of the MaxBCG clusters, and found that including miscentering effects with the Johnston et al.(2007) prescription strongly affected the resultant fits for concentration, again asserting the degeneracies of theseparameters. Mandelbaum et al. (2008a) conclude that this method of accounting for miscentering depends heav-49101102<N200>10131014<M200> [M]pcc = 1 (no miscentering)pcc = 0 (with miscentering)Figure 3.6: Mass-Richness Relation from Magnification. Cluster mass-richness relation, using the sameN200 bins as in Figure 3.5. Power law fits to the data are presented for both cases of with (bluediamonds and dashed line) and without (green squares and solid line) the effects of miscentering.Points are slightly offset horizontally for clarity.ily on the mock catalogs from which the input parameters are generated, and in the case of MaxBCG clusterslikely overcompensates.In a forthcoming paper, we will present weak lensing shear measurements of these clusters, as well as amore detailed investigation of the centroiding. Shear, being proportional to the differential surface mass density,is more affected by offset centers than magnification (Johnston et al., 2007), and will be a better probe ofmiscentering.3.4.1 The Mass-Richness RelationWe observe a prominent scaling of best fit mass to richness, across the six richness bins (although the first twobins do generally have overlapping error bars). We plot this trend in Figure 3.6, showing the average of the fitmasses as a function of average cluster richness in each bin. Note that the distribution of N200 in a bin is notuniform, and in the case of the highest richness bin the distribution is highly skewed (see Figure 3.2).We fit a simple power-law, Equation 3.15, to these points, using the same plotted color and line schemes forperfectly centered and miscentered clusters. For this cluster sample, we find the best fit gives the normalizationand slope of the mass-richness relation to beM0 = (2.3±0.2)×1013 M,β = 1.4±0.1 (3.20)50for the perfectly centered pcc = 1 case, andM0 = (2.2±0.2)×1013 M,β = 1.5±0.1 (3.21)for the miscentered pcc = 0 case. The χ2red are 0.9 and 1.7, respectively (4 degrees of freedom), and there is goodagreement between the two different centering scenarios explored here.It is difficult to directly compare the results for the mass-richness relation in this work to other studies. Themain reason is that the richness N200 we use is different than other definitions, which often count only red-sequence galaxies. Some uncertainty exists in the measure of richness as well, which we do not include in theanalysis. Alternative measures of cluster richness would yield different scaling relations. Another factor is thecluster sample, which was compiled using a novel cluster-finder, and may well have different characteristics thanother samples in the literature. In a follow-up paper we will present a shear analysis of the CFHTLenS clusters,and compare the mass-richness relation obtained using that complementary probe of halo mass.3.4.2 Redshift BinningFinally, we investigate the magnification as a function of redshift. We stack clusters of all richnesses, at eachredshift in the catalog, 0.2 ≤ z ≤ 0.9, and measure the optimal correlations in each. This is displayed in Fig-ure 3.7. We observe a steady decrease in measured signal as the cluster redshift increases from z ∼0.2 to 0.5,then roughly consistent measurements from 0.6≤ z≤0.8, followed by rather low signal at z∼0.9.The N200 distributions in Figure 3.8 show that these trends cannot be caused by deviations in richness be-tween these different cluster redshifts. This is difficult to reconcile with the clear mass-richness scaling observedwhen all redshifts are combined. Table 3.3 shows that detection significance for each redshift bin is more tightlylinked to mass than the 〈N200〉. Perhaps the richness estimates used in this work are not optimized for use asa mass proxy at all redshifts. Another possibility is that we have not correctly accounted for redshift overlapbetween samples. If the contamination fraction is higher than estimated, this could lead to a boost in correlationstrength at low redshift, as well as a depletion at higher redshift. However it is still very difficult to explain theanomalously low measurement at intermediate redshift, z∼0.5, with this reasoning.One factor that we have not accounted for is possible cluster false detections in our sample. Since 3D-MFwas optimized to produce cluster catalogs that are as complete as possible, false detection rates could be quitehigh. In particular, we would expect these rates to increase at high redshift, which would also weaken thosemeasured correlations. We note in particular that cluster redshift bins z ∼ 0.5 and 0.9, which yield relativelylow cluster masses, are seen in Figure 3.3 to have excess numbers of detected clusters, possibly an indication ofhigher false detection rates at these redshifts. In future work, false-detection rates for 3D-MF could be examinedwith more extensive tests on simulations, and through cross-matching with independent galaxy cluster catalogs.510.00.20.40.60.81.01.2wopt(R)z  0.2pcc = 1pcc = 0z  0.3pcc = 1pcc = 0z  0.4pcc = 1pcc = 0z  0.5pcc = 1pcc = 00.11R [Mpc]0.00.10.20.30.40.50.60.7wopt(R)z  0.6pcc = 1pcc = 00.11R [Mpc]z  0.7pcc = 1pcc = 00.11R [Mpc]z  0.8pcc = 1pcc = 00.11R [Mpc]z  0.9pcc = 1pcc = 0Figure 3.7: Magnification for Redshift-Binned Clusters. Optimal correlation for clusters binned in red-shift.z  0.2 z  0.3 z  0.4 z  0.510 20 30 40N200z  0.610 20 30 40N200z  0.710 20 30 40N200z  0.810 20 30 40N200z  0.9Figure 3.8: Richness Distributions in Redshift Bins. N200 distributions as a function of cluster redshift.52Table 3.2: Magnification Results for Richness-Binned Clusters. Details of fits for richness-binned measurements in Figure 3.5. We list the richnessrange selected, the number of clusters in that bin, the detection significance, the average richness of the bin, and the mass estimates and reducedχ2 for both the centered and miscentered models fit to the data. Note that the average mass given is not the value fit itself, but the average of allresulting masses fit using the composite-halo approach discussed in Section 3.3.2.Richness # Clusters Significance 〈N200〉 pcc=1: 〈M200〉 χ2red pcc=0: 〈M200〉 χ2red2 < N200 18,036 9.7σ 17 (2.0±0.3)×1013M 1.2 (1.8±0.3)×1013M 2.32 < N200 < 10 4,453 5.3σ 8 (0.9±0.5)×1013M 3.0 (0.7±0.4)×1013M 3.210 < N200 < 20 9,398 5.9σ 15 (1.3±0.3)×1013M 1.6 (1.0±0.3)×1013M 2.220 < N200 < 30 2,967 5.4σ 24 (2.9±0.7)×1013M 0.7 (3.3±0.8)×1013M 0.330 < N200 < 40 695 5.0σ 35 (7±2)×1013M 0.3 (7±2)×1013M 0.540 < N200 < 60 351 4.6σ 47 (1.0±0.2)×1014M 0.4 (1.1±0.2)×1014M 0.360 < N200 172 5.5σ 99 (2.0±0.4)×1014M 0.5 (2.1±0.4)×1014M 0.6Table 3.3: Magnification Results for Redshift-Binned Clusters. Details of fits for redshift-binned measurements in Figure 3.7. We list the same binproperties and fits given in Table 3.2, as well as fclustering, which is the total fraction of LBGs expected to lie within ∆z∼0.1 of the cluster z.Redshift fclustering Clusters Significance 〈N200〉 pcc=1: 〈M200〉 χ2red pcc=0: 〈M200〉 χ2redz ∼ 0.2 0.07 1,157 12.5σ 11.6 (9±2±2sys)×1013M 3.6 (9±2±2sys)×1013M 3.4z ∼ 0.3 0.02 1,515 8.0σ 14.4 (6±1±1sys)×1013M 2.2 (6±1±1sys)×1013M 2.1z ∼ 0.4 3×10−3 2,242 4.6σ 15.2 (1.9±0.7)×1013M 1.4 (1.6±0.7)×1013M 1.6z ∼ 0.5 4×10−4 2,932 4.0σ 15.9 (0.3±0.4)×1013M 1.9 (0.2±0.5)×1013M 1.9z ∼ 0.6 1×10−4 2,455 4.6σ 18.0 (2.2±0.8)×1013M 1.5 (2.0±0.8)×1013M 1.6z ∼ 0.7 2×10−5 2,331 4.5σ 19.3 (1.2±0.7)×1013M 1.7 (1.1±0.7)×1013M 1.9z ∼ 0.8 2×10−5 2,364 4.9σ 19.9 (2.5±0.9)×1013M 1.5 (2.2±0.9)×1013M 1.7z ∼ 0.9 2×10−5 3,040 2.6σ 17.6 (0.5±0.5)×1013M 0.6 (0.3±0.6)×1013M 0.8533.5 ConclusionsWe present the most significant magnification-only cluster measurement to date, at 9.7σ . A sample of 18,036cluster candidates has been detected using the 3D-MF technique in the ∼154 deg2 CFHTLenS survey. In thisanalysis we have investigated the mass of cluster dark matter halos, from flux magnification, as a function ofboth richness and redshift. A forthcoming paper will present the weak lensing shear analysis of these clusters aswell.We fit a composite-NFW model that accounts for the full redshift and mass ranges of the cluster sample, aswell as redshift overlap with low-z source contaminants, cluster halo miscentering, and the 2-halo term. We findthat the entire cluster sample is marginally better fit by the model that does not include miscentering, but donot see a strong preference either way across richness bins. In the future, shear measurements, which are moresensitive to miscentering, may illuminate this aspect of the investigation.We observe a strong scaling between measured mass and cluster richness, and fit a simple power-law relationto the data. The two miscentering models explored in this work yield consistent values for the normalization andslope of the mass-richness relation.We have attempted to account for the contamination of our background sources with low-z galaxies. This isa serious systematic effect for magnification, as redshift overlap between lenses and sources will lead to physi-cal clustering correlations, swamping the lensing-induced correlations that we want to measure. We use the fullstacked redshift probability distributions for the lens and source populations, and include the expected cluster-ing contribution in our model. In spite of this we see unexpected features in the redshift-binned measurements.Part of the reason could come from cluster false detections, which can be high for the 3D-MF method, whichis optimized for completeness. Another contribution could come from errors in the source redshift distribu-tions. Accounting for redshift overlap is imperative if significant overlap exists between the lens and sourcedistributions, or else mass estimates can end up very biased.This is the first analysis presented of the 3D-MF clusters in CFHTLenS, but much more science is left todo with the sample. In particular, a more thorough investigation of the miscentering problem will be carriedout in the forthcoming shear analysis, where it will be possible to compare different candidate centers. Anotherinteresting question is whether dust can be detected on cluster scales by simultaneously measuring the chromaticextinction along with flux magnification. Finally different background source samples may be employed toimprove signal-to-noise, but only if their redshift distributions can be well determined. We leave these tasks tofuture work.This work has been an important step in the development of weak lensing magnification measurements,and the progression from signal detection to science. Many upcoming surveys will benefit from the inclusionof magnification in their lensing programs, as the technique offers a very complimentary probe of large scalestructure. Since measuring flux magnification is not a strong function of image quality, it is especially usefulfor ground-based surveys which must deal with atmospheric effects. Next generation surveys like the LargeSynoptic Survey Telescope (LSST), the Wide-Field InfraRed Survey Telescope (WFIRST), and Euclid, will havegreater numbers of sources, and improved redshift probability distribution estimates, so we can expect futuremagnification studies to yield important contributions to weak lensing science and cosmology.54Chapter 4CFHTLenS: A Weak Lensing Shear Analysisof the 3D-Matched-Filter Galaxy ClustersWe present the cluster mass-richness scaling relation calibrated by a weak lensing analysis of >18,000 galaxycluster candidates in the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). Detected using the3D-Matched-Filter (3D-MF) cluster-finder of Milkeraitis et al., these cluster candidates span a wide range ofmasses, from the small group scale up to ∼ 1015 M, and redshifts 0.2 ≤ z ≤ 0.9. The total significanceof the stacked shear measurement amounts to 54σ . We compare cluster masses determined using weak lensingshear and magnification, finding the measurements in individual richness bins to yield 1σ compatibility, but withmagnification estimates biased low. This first direct mass comparison yields important insights for improving thesystematics handling of future lensing magnification work. In addition, we confirm analyses that suggest clustermiscentering has an important effect on the observed 3D-MF halo profiles, and we quantify this by fitting forprojected cluster centroid offsets, which are typically ∼ 0.4 arcmin. We bin the cluster candidates as a functionof redshift, finding similar cluster masses and richness across the full range up to z∼ 0.9. We measure the 3D-MFmass-richness scaling relation M200 = M0(N200/20)β . We find a normalization M0 ∼ (2.7+0.5−0.4)×1013 M, anda logarithmic slope of β ∼ 1.4± 0.1, both of which are in 1σ agreement with results from the magnificationanalysis. We find no evidence for a redshift-dependence of the normalization. The CFHTLenS 3D-MF clustercatalog is now available at cfhtlens.org.4.1 IntroductionThe evolution of large scale structure is overwhelmingly driven by the invisible components which make up themajority of the present day energy density of the Universe. In order to probe these structures we are forcedto rely on biased tracers of the underlying density field that we can actually observe, such as galaxies. Largegalaxy cluster surveys are invaluable in providing sufficient statistics for classifying and analysing the mostmassive gravitationally bound systems that have had time to form in our cosmic history. In addition to providinga cosmological probe, they are interesting laboratories for the evolution of individual galaxies and the ICM (Voit,2005).Several methods have been developed for identifying clusters in optical galaxy surveys, including the redsequence technique (Gladders and Yee, 2000), density maps (Adami et al., 2010), redMaPPer (Rykoff et al.,2014), and matched-filter methods (Postman et al., 1996). An extension of the latter, 3D-MF, is described in55Milkeraitis et al. (2010) and used in this work. This cluster finder attempts to circumvent the common issue ofline-of-sight projections by using photometric redshift information to identify clusters in redshift slices. Beyondthe use of photometric redshifts, 3D-MF does not apply any additional color-selection criteria for identifyingclusters (e.g. that cluster members must fall on the red sequence). A similar algorithm tuned for galaxy groupswas introduced by Gillis and Hudson (2011). Every cluster-finding technique will pick out clusters with some-what distinct characteristics because of different assumptions that are made in the algorithm, and it is thereforeimportant to characterize and contrast independent samples of clusters (Milkeraitis et al., 2010).Among the broad array of analysis tools employed by the galaxy cluster research community, gravitationallensing is a crucial technique for obtaining masses and density profiles, independent of assumptions regardingcluster dynamical state. In the weak regime, lensing provides robust measurements of stacked cluster sam-ples (and individual masses for very massive clusters), affording a statistical view of average galaxy clusterproperties (Hoekstra et al., 2013). The majority of weak lensing studies measure the shear, or shape distor-tion, of lensed source galaxies. The complementary magnification component of the lensing signal has morerecently been measured with increasing precision (Scranton et al., 2005; Hildebrandt et al., 2009b; Ford et al.,2012, 2014; Morrison et al., 2012; Hildebrandt et al., 2013; Bauer et al., 2014), and has been combined withshear in joint-lensing analyses (Umetsu et al., 2011, 2014). When combined with other cluster observables,lensing yields useful scaling relations that can be extrapolated with some caution to wider cluster populations,or cross-examined to characterize intrinsic disparities that may distinguish catalogs compiled using differentcluster-finding techniques (Hoekstra, 2007; Johnston et al., 2007; Leauthaud et al., 2010; Hoekstra et al., 2012;Covone et al., 2014; Oguri, 2014).Section 4.2 of this paper describes the data, Section 4.3 gives the formalism of the weak lensing measure-ment, and Section 4.4 presents the results. We then discuss and compare our findings to other results, includingour previous magnification measurements of the same lens sample, in Section 4.5. We finish with conclusionsin Section 4.6. Throughout this work we use a concordance Λ cold dark matter cosmology with Ωm = 0.3, ΩΛ =0.7, and H0 = 70 km/s/Mpc.4.2 Data4.2.1 The Canada-France-Hawaii Telescope Legacy Survey WideThe Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) is a multi-component optical survey conductedover more than 2,300 h in 5 yr (∼ 450 nights) using the wide field optical imaging camera MegaCam on theCanada-France-Hawaii Telescope (CFHT)’s imaging system MegaPrime. The Wide survey is composed of fourpatches ranging from 25-72 deg2, together totaling an effective survey area of ∼ 154 deg2. The data wereacquired through five filters: u*, g′, r′, i′, z′, and has a 5σ point source i′−band limiting magnitude of 24.5.The breadth of CFHTLS-Wide was intended for the study of large scale structure and matter distribution in theUniverse.The CFHTLS-Wide optical multi-color catalogs used in this work were created from stacked images of theaforementioned Wide fields (see Erben et al., 2009; Hildebrandt et al., 2009a, 2012; Erben et al., 2013, fordetails on the data processing and multi-color catalog creation). Basic photometric redshift (zphot) statistics weredetermined by Hildebrandt et al. (2012). In this work we restrict ourselves to a redshift range of 0.1 ≤ z≤ 1.2,which has outlier rates ≤ 6% and scatter σ ≤ 0.06.564.2.2 CFHTLenS Shear CatalogThe CFHTLenS reduced CFHTLS-Wide data for weak lensing science applications (Heymans et al., 2012; Erbenet al., 2013). Many factors affect high-precision weak lensing analyses, including correlated background noise,PSF measurement, and galaxy morphology evolution for example (for a more detailed list and study, see Masseyet al., 2007a; Heymans et al., 2012). The efforts of CFHTLenS have led to new reduction methodologies withreduced systematic errors and a more thorough understanding of the PSF and its variation in the CFHTLS-Wideimages. As part of this pipeline, lensfit was used to measure galaxy shapes (Miller et al., 2013), which weretested for systematics in Heymans et al. (2012). The galaxy shear measurements and photometric redshifts usedin this work are publicly available.14.2.3 3D-MF ClustersHere we give a brief overview of the 3D-MF galaxy cluster-finding algorithm. For additional background anddetails on the algorithm, including extensive testing on the Millennium Simulation data set, and information onthe completeness and purity of a 3D-MF derived galaxy cluster catalog, the reader is directed to Milkeraitis et al.(2010).3D-MF searches survey data for areas that maximally match a given LF and radial profile for a fiducialgalaxy cluster, similar to the technique used by Postman et al. (1996). For the LF we use an integrable Schechterfunction, given byΦ(M) = 0.4 ln(10) Φ∗100.4(α+1)(M∗−M) exp[−100.4(M∗−M)], (4.1)where Φ is the galaxy LF, Φ∗ sets the overall normalization, M is absolute magnitude, M∗ is a characteristicabsolute magnitude, and α is the faint end slope of the LF. As discussed in Milkeraitis et al. (2010), the multi-plicative term, exp[−100.4(M∗−M)], keeps this function from diverging when α <−1 and M < M∗. For the radialprofile we use a truncated Hubble profile, given byP(RRc)=1√1+(RRc)2−1√1+(RcoRc)2, (4.2)where Rc is the cluster core radius, and Rco Rc is the cutoff radius. In an attempt to match both of the aboveprofiles, 3D-MF creates likelihood maps of the sky survey area. Peaks in this map are possible cluster detections,and are each assigned a significance σcl relative to the background signal (σcl is calculated using Equation 5 ofMilkeraitis et al., 2010, which the reader is referred to for more details). The cluster centers are defined to bethe locations of the likelihood peaks; see Section 4.3.3 for how uncertainties in the centers are dealt with.An important characteristic of this cluster-finding algorithm is the fact that the described process is carriedout in discrete redshift bins to avoid spurious false-detections due to line-of-sight projections. 3D-MF was runon the CFHTLS-Wide catalogs with redshift slices of width ∆z = 0.2, which are then shifted by 0.1, and thefinder is run again on the overlapping redshift slices. Clusters are assigned a final redshift estimate (of bin width∆z = 0.1) by using the center of the slice that maximizes cluster detection significance. 3D-MF was run usingthe same run-time parameters listed in table 2 in Milkeraitis et al. (2010), with the exception of an absolutei′−band magnitude of M∗i′−band =−23.22±0.01 and slope of the Schechter LF, α =−1.04±0.01, derived from1www.cfhtlens.org; Data products available at http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/CFHTLens/query.html57246 8 10 1214〈σcl〉10131014〈M200〉[M]Figure 4.1: Mass-Significance Relation. Scaling of shear-measured mass M200 with the 3D-MF clusterdetection significance σcl. Since we find significance to be a good proxy for mass, we use the derivedmass-significance relation to estimate a radius R200 for each cluster candidate, within which we countgalaxies for richness N200, as described in the Section 4.2.3.the Wide data (Milkeraitis, 2011).Excluding possible multiple detections, a total of 22,694 galaxy cluster candidates were found in the CFHTLS-Wide data set with detection significance σcl≥ 3.5. Using 3D-MF’s multiple detection criteria, there were 34.4%additional duplicate detections of galaxy clusters. This is comparable to the∼ 36% multiple detection rate foundfrom Millennium Simulation tests and 37.6% found in the CFHTLS-Deep galaxy cluster catalog in Milkeraitiset al. (2010). Using the Millennium Simulation, Milkeraitis et al. (2010) determined that there are potentially∼ 16%−24% false positives in 3D-MF-derived galaxy cluster catalogs, distributed mostly in the lower signifi-cance ranges (see Table 3 in Milkeraitis et al., 2010).Following the 3D-MF methodology for galaxy cluster catalog generation, the significance of galaxy clusterdetections was used to select the best galaxy cluster candidate among multiple detections, and the remainingmultiple detections were rejected from the analysis. A single detection of each cluster candidate then makes upthe CFHTLS-Wide galaxy cluster candidate catalog. We restrict our analysis herein to a cluster redshift range of0.2 ≤ z≤ 0.9, where 3D-MF detections are the most reliable.In Chapter 3 (Ford et al., 2014), we described our method of calculating richness for each of these can-didate clusters. N200 is defined to be the number of member galaxies brighter than absolute magnitude Mi ≥−19.35, which is chosen to match the limiting magnitude at the furthest cluster redshift that we probe (N200is background-subtracted; there is no correction for passive evolution). To be considered a cluster member, agalaxy must lie within a projected radius R200 of a cluster center, and have ∆z < 0.08(1+ z) (based on the pho-tometric errors of the CFHTLenS catalog; for details regarding N200 see Chapter 3 or, equivalently, Ford et al.,2014). R200 is defined as radius within which the average density is 200 times the critical energy density of the580 150 300 450 600 750 900N2001001011021031040.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9z100020003000Figure 4.2: Richness and Redshift Distributions of 3D-MF Clusters. Number of 3D-MF cluster candi-dates as a function of richness N200 and redshift z.Universe (M200 is the total mass inside R200), and in this work has been re-estimated from the data as follows.Initially cluster candidates were stacked in bins of cluster detection significance σcl, which was found tocorrelate well with the amplitude of the measured shear profiles, and therefore with mass (see Figure 4.1). Thesepreliminary masses were estimated using the same method described in Section 4.3. A new mass-significancerelationship,log[Mprelim200M]=(0.161+0.006−0.009)σcl +12.39+0.05−0.08, (4.3)was derived from this result and the preliminary mass values converted into the corresponding radii, which wereused to count galaxies for richness (σcl→Mprelim200 → R200→ N200). Compared with the richness estimates usedin Chapter 3 (Ford et al., 2014), which were based on a preliminary shear analysis using a more basic clustermodeling approach (Milkeraitis, 2011), the updated richnesses are larger in most cases (see the Full Modeldescription in Section 4.3.4 for improvements). For the curve plotted in Figure 4.1, as well as for all models fitin this work, the best fit is the curve that minimizes χ2, using a downhill simplex algorithm to search parameterspace.Cluster candidates used in this work are required to have at least N200 > 2, and a detection significance≥ 3.5.The richness and redshift distributions are summarized in Figure 4.2. Figure 4.3 shows the relative scalingbetween richness and detection significance. The final catalog contains the same 18,036 cluster candidates usedin Chapter 3 (Ford et al., 2014), now with updated richness estimates based on the shear mass-significancescaling just described. There are also 20 additional low-significance cluster candidates whose revised N200 nowsurvive the cuts – these systems have negligible impact on the overall results, but do increase the total numberof clusters to 18,056. The full 3D-MF catalog is available at cfhtlens.org.59246 8 10 1214〈σcl〉20406080100〈N200〉Figure 4.3: Richness-Significance Scaling. Scaling of richness N200 with the 3D-MF cluster detectionsignificance σcl. Error bars denote the standard deviation of the ensemble of N200 values in eachσcl bin. Since N200 is estimated using individual cluster radii calculated from the mass-significancerelation (Equation 4.3), this figure confirms what we would expect – a strong scaling between richnessand significance.4.3 Method4.3.1 Stacking Galaxy ClustersThe mass of a galaxy cluster can be determined by measuring shear in binned annuli out from the clustercenter, and fitting this with a theoretical density profile. For the most massive galaxy clusters, this is relativelystraightforward. However, for most galaxy clusters (especially given the high number of lower mass galaxycluster candidates explored in this work), the background noise overwhelms the measurable shear. Fortunately,stacking many individual galaxy clusters together improves the signal-to-noise ratio, enabling the measurementof a statistically significant signal, averaged over a cluster ensemble.To obtain a meaningful average for a property of an ensemble of galaxy clusters, similar clusters must clearlybe chosen for a stack. It is desirable to stack clusters of very similar mass (and thus clusters of roughly the samesize and profile), as an average mass measurement of the cluster stacks is the goal. In fitting models to thestacked weak lensing measurements in this work, we assume that the haloes are spherical on average. However,recent studies have explored halo orientation bias in simulations, demonstrating that optically-selected clusterswill tend to be aligned along the line-of-sight, and this effect could lead to our mass estimates being biased highby 3−6% (Dietrich et al., 2014).For this analysis, the cluster candidates are stacked in bins of richness N200 as well as redshift, identical tothose used in Chapter 3 (Ford et al., 2014). The overall approach is conceptually very similar to that used in60galaxy-galaxy lensing (see Velander et al., 2014), except we replace the galaxy lenses with cluster lenses.4.3.2 Measuring ∆ΣWe measure the radial profile of the tangential shear, γt(R), around each cluster candidate in bins of projectedphysical distance R, extending from 0.09 to 5 Mpc. The logarithmically-spaced radial bins are chosen to matchthose used in Chapter 3 (Ford et al., 2014), which we compare results to in Section 4.5.5, and the resulting massmeasurements are insensitive to small adjustments in the innermost radii. To select background galaxies formeasuring shear, we use their redshift probability distributions P(zs), where zs is the source redshift. Relativeto a given cluster redshift (zl), we require both that (1) the peak of a galaxy’s P(zs) distribution is at higherredshift, and (2) at least 90% of a galaxy’s P(zs) is at higher redshift. The second requirement is designed toaccount for the occasional galaxy with an odd P(zs), which may peak at high redshift (and so would be includedin many conventional shear analyses), but could perhaps have a non-negligible tail extending to low z, or evenbe bimodal.From the individual shear profiles we construct ∆Σ, the differential surface mass density, for each stackedcluster candidate sample:∆Σ(R)≡ Σ(< R)−Σ(R) = 〈γt(R)〉Σcrit. (4.4)Here Σ(R) is the surface mass density of a lens, and Σcrit is the critical surface mass density, which depends onthe geometry of the lens-source pairs. It is given byΣcrit =c24piGDsDlDls, (4.5)where c is the speed of light and Ds, Dl, and Dls, are the angular diameter distances to the source, to the lens,and between the lens and source, respectively.In computing Σcrit for each lens-source pair, we treat the individual lens zl as fixed, and integrate over thefull source P(zs), for zs > zl, to compute the distances:Ds =∫ ∞zldA(0,zs)P(zs)dzs; (4.6)Dls =∫ ∞zldA(zl,zs)P(zs)dzs. (4.7)Here dA is the angular diameter distance between two redshifts (and Dl is simply dA(0,zl)). The source redshiftprobability distribution is renormalized behind the lens, so that∫ ∞zlP(zs)dzs = 1. Using the full P(zs) distributionshould improve any residual photo-z calibration bias in the lensing measurement (Mandelbaum et al., 2008b).We follow the same procedure described in detail in Velander et al. (2014), wherein we combine shearprofiles using the lensfit source weighting (Equation 8 of Miller et al., 2013), and apply a correction for mul-tiplicative bias (Miller et al., 2013), so that the 〈γt(R)〉 appearing in Equation 4.4 is the average calibratedtangential shear. We estimate a covariance matrix for each stacked sample, by running 100 sets of bootstrapped61cluster measurements, and calculating the covariance asC(Ri,R j) =[NN−1]2 1NN∑k=1[∆Σk(Ri)−∆Σ(Ri)]×[∆Σk(R j)−∆Σ(R j)].(4.8)Here N is the number of bootstrap samples, Ri and R j denote specific angular bins, and ∆Σ(Ri) is the differentialsurface mass density at Ri, averaged across all bootstrap realizations. The square-root of the diagonal of thismatrix yields the error bars displayed on the weak lensing measurements in Section 4.4. We confirm that N = 100bootstrap realizations of the data is sufficient by tracking the covariance estimated from different numbers ofbootstrapped samples and checking for convergence, which typically occurs at around 40 realizations. We usethe full covariance matrices when fitting to the data, as will be described in Section 4.4.1.We test our ∆Σ measurements for systematics by measuring the rotated shear γr(R) (where each galaxyellipticity is rotated by 45◦), finding a signal consistent with zero. We also check that masked areas and edgeeffects are not affecting our measurement, by measuring ∆Σ around many randomly chosen points (> 50 timesthe number of cluster candidates), and we find no significant signal here either.The NFW modelWe use the Navarro-Frenk-White (NFW) dark matter density profile (Navarro et al., 1997) for modeling ∆Σ.As demonstrated by numerical simulations, the dissipationless collapse of density fluctuations under gravityproduces overdensities that are approximated well by the NFW profileρNFW(r) =δcρcrit(z)(r/rs)(1+ r/rs)2, (4.9)where δc is the characteristic overdensity of a halo, and ρcrit(z) is the critical energy density of the Universe atthat redshift. The scale radius is rs = R200/c200, where c200 is the concentration parameter (not to be confusedwith the speed of light c in Equation 4.5). R200 is the cluster radius, and the total mass within that radius isknown as M200. Wright and Brainerd (2000) derived the NFW forms of the projected mass density profiles inEquation 4.4, which we make use of in this work.In general, the NFW profile is a two parameter model for the halo density, commonly parametrized in termsof M200 and c200. However, there is a well-established correlation between these two parameters, and it iscommon to introduce a mass-concentration relation to reduce the dimensionality of the problem (note that con-centration itself may be degenerate with cluster centroid offsets, which will be discussed in Section 4.3.3). Inthis work we invoke the mass-concentration relation recently presented by Dutton and Maccio` (2014) for thePlanck cosmological parameters, which successfully characterizes the profiles of simulated halos spanning awide range of masses and redshifts. Given a cluster mass, the concentration is then fixed, and we have just asingle mass-related fit parameter to deal with.Non-weak shear correctionsThe gravitational lensing observable is galaxy shapes. From these, we measure the reduced shear g = γ/(1−κ)about the lens, where γ is the true shear and κ = Σ/Σcrit is the convergence (as before, calculated using theNFW halo formalism in Wright and Brainerd, 2000). At the innermost radii that we probe (∼ 0.1 Mpc) the620.0 0.5 1.01.52.0Roff [arcmin]P(Roff)0.11R [Mpc]050100150200250∆Σ,∆Σsm∆Σ∆ΣsmFigure 4.4: Example of Miscentering Effect on Shear Profile. An illustrative example of typical ∆Σ(R)and ∆Σsm(R) profiles, to demonstrate the effects of cluster miscentering (Equations 7−11) on mea-sured shear density profiles. The left-hand panel shows a typical probability distribution of centroidoffsets, P(Roff), modeled via a Rayleigh distribution (which represents the radial component of a 2DGaussian) with σoff = 0.4 arcmin. The right-hand panel demonstrates the effect of this offset distri-bution on the measured shear profile (in vertical axis units of [M/pc2]) of a fiducial halo of massM200 = 1014 M, located at z = 0.5. The dashed black curve shows the perfectly centered ∆Σ(R)profile, and the solid blue curve shows the miscentered profile ∆Σsm(R). In both panels, the verticaldotted line marks the location of the miscentering offset σoff, to guide the eye in the comparison.common weak lensing assumption that g ≈ γ may break down for the more massive clusters. We account forthe difference between true and reduced shear using the correction factor from Johnston et al. (2007), which wasworked out in detail in Mandelbaum et al. (2006). The differential surface mass density corrected for non-weakshear is given by∆̂Σ= ∆Σ+∆Σ ΣLz, (4.10)where Lz = 〈Σ−3crit〉/〈Σ−2crit〉 is calculated for each cluster redshift, using the full distribution of background galax-ies satisfying the same redshift requirements outlined in Section 4.3.2. Similar to Leauthaud et al. (2010), weignore any radial variations of Lz, but do account for the variation with redshift, as our cluster sample spans alarge z range. The entire correction term ∆Σ ΣLz is negligible at all radii except for the innermost bin, where ittypically makes up a few percent (at most ∼10%) of the measured signal.4.3.3 Miscentering FormalismAs was shown in Milkeraitis et al. (2010), 3D-MF does not always determine the exact correct center for a galaxycluster, and clusters may not always have a well-defined center. This is a problem with all galaxy cluster findersand dealing with it properly involves understanding and quantifying its effects, such as including the uncertaintyof the center in calculations. The amplitude of measured shear profiles is absolutely dependent on the declaredcenter of the profile, so miscentering can potentially have a large impact on results. Offset cluster centers thatare mistakenly modeled as being the true centers of the gravitational potentials will lead to underestimates in theinferred lens masses.In our first analysis of the 3D-MF cluster candidates, we found modest evidence for cluster centroid errors(see Section 3.4 or, equivalently, Section 4 of Ford et al., 2014). However, that work relied on the lensingmagnification technique, which is less sensitive to these effects than the shear, since magnification directly63probes Σ(R), while it is ∆Σ(R) that is more drastically reduced by a misplaced center. See, for example, figure4 in Johnston et al. (2007), for a nice illustration of the comparative effect of miscentering on these two lensingprofiles.In this work, we are able to directly quantify the presence of cluster miscentering by fitting for the offsets inour measurements of ∆Σ. As will be shown in Section 4.4, we find that the best-fitting distribution of centroidoffsets is in agreement with the following distribution based on simulations, which we assumed in Chapter 3(Ford et al., 2014).The distribution of cluster offsets can be modeled as a two-dimensional Gaussian, by using a uniform angulardistribution and a Rayleigh distribution for the radial profile:P(Roff) =Roffσ2offexp[−12(Roffσoff)2 ]. (4.11)Here, Roff is the projected offset of the 3D-MF derived galaxy cluster center from the true galaxy cluster center,and σoff is the width of the distribution and one of the miscentering parameters which we fit to the stacked shearmeasurement. An example P(Roff) curve is plotted in the left-hand panel of Figure 4.4, for σoff=0.4 arcmin. Notethat we use physical units (e.g. Mpc) for most distances in this work, the exception being σoff which we reportin angular size (arcmin). The reason for this choice is that we believe a significant contribution to miscenteringderives from 3D-MF’s cluster characterization, which does not for example select a member galaxy as the center(this choice of angular size is a matter of taste, since complex cluster physics certainly contributes to ambiguoushalo centers).The effect of this offset distribution P(Roff) is to reduce the ideal Σ(R) to a smoothed profile (see e.g.Johnston et al., 2007; George et al., 2012)Σsm(R) =∫ ∞0Σ(R|Roff) P(Roff) dRoff, (4.12)which is illustrated in the right-hand panel of Figure 4.4. Equation 4.12 is an integration over all possible valuesof Roff in the distribution. The expression for the surface mass density at a single Roff isΣ(R|Roff) =12pi∫ 2pi0Σ(r)dθ , (4.13)where r =√R2 +R2off−2RRoff cos(θ) and θ is the azimuthal angle (Yang et al., 2006). From the smoothedΣsm(R) profile, we can obtain the smoothed shear profile:∆Σsm = Σsm(< R)−Σsm(R); (4.14)Σsm(< R) =2R2∫ R0Σsm(R′)R′dR′. (4.15)See George et al. (2012) for a discussion of the effects of cluster miscentering on measured shear profiles.There are several different approaches in the literature for actually applying this formalism to data. For example,in some work authors apply the same smoothing to all clusters in a stack (George et al., 2012), whereas othersapply a two-component smoothing profile (Oguri, 2014), or chose a uniform distribution of offsets instead of64the Gaussian (Sehgal et al., 2013). In our previous analysis of this cluster candidate sample, the magnificationtechnique did not give significant constraining power for additional parameters, so we simply compared fitsfor both a perfectly centered and miscentered model, using estimates of σoff obtained from running 3D-MF onsimulations (see Chapter 3 or Ford et al., 2014). Both Johnston et al. (2007) and Covone et al. (2014) applieda combination of perfectly centered and miscentered haloes, thus fitting for the fraction of offset clusters inaddition to the magnitude of the offset distribution σoff. We follow this latter approach in the current analysis.As a caveat, we note that the degree of miscentering is fairly degenerate with the cluster concentrationparameter, as both can have an effect on the amplitude of the inner shear profile. For example, we tried using themass-concentration relation of Prada et al. (2012), which yields higher concentration for a given mass than theDutton and Maccio` (2014) relation used here, and results in a best fit with larger centroid offsets. For the lowermass (richness) clusters this change is negligible, but for the most massive clusters in this study, the choice ofconcentration-mass relation can affect the miscentering fit parameters by as much as 40%. Importantly, however,the best-fitting cluster mass is the same in both cases (within the stated 1σ uncertainties). The degeneracy ofcluster concentration and miscentering would be important to consider in a study seeking to constrain clustermass-concentration relations. The measured concentrations will be biased low if cluster centroid offsets aresignificant and not fully accounted for.4.3.4 The Halo ModelWeak lensing measurements are sensitive to the fact that structures in the Universe are spatially correlated. Weaccount for this large scale clustering using the halo model, which provides a useful framework for model-ing the clustered and complex dark matter environments that we probe in gravitational lensing studies. Thisphenomenological approach places all the matter in the Universe into spherical haloes, which are clustered ac-cording to their mass. Observables such as galaxies and clusters are considered biased tracers of the underlyingdark matter distribution, with a bias factor that has been constrained in many numerical simulations (e.g. Moand White, 1996; Sheth and Tormen, 1999; Tinker et al., 2010). See Cooray and Sheth (2002) for an extensivereview of the halo model.We follow an approach similar to Johnston et al. (2007), in considering a two-halo term in addition to themain NFW halo fit to our weak lensing shear measurement. Calculation of the two-halo term is identical to ourapproach in Chapter 3 (Ford et al., 2014), and we refer the reader there for explicit details. The two-halo term isproportional to a cluster bias factor which depends on mass, and for this we continue to use the bias relation ofSeljak and Warren (2004). The full model including the two-halo term is∆Σ(R) = pcc∆ΣNFW +(1− pcc)∆ΣsmNFW +∆Σ2halo. (4.16)The fraction of cluster candidates that is correctly centered on their parent dark matter haloes, pcc, is aparameter that we fit to the data. pcc is a continuous variable, bounded between 0 and 1, fit separately for eachstacked weak lensing measurement. Thus we have two cluster-centering-related parameters (pcc and σoff), aswell as one mass-related parameter (M0), in the final modeling of the data.650204060802 < N200 ≤ 10Full Modelpcc≡110 < N200 ≤ 20050100150∆Σ(R)[M/pc2]20 < N200 ≤ 30 30 < N200 ≤ 400.11R [Mpc]05010015020025040 < N200 ≤ 600.11R [Mpc]60 < N200Figure 4.5: Shear for Richness-Binned Clusters. Best-fitting models for each richness-binned stack ofcluster candidates. The solid green curves are the best fits to the full model given by Equation 4.16.The dashed purple curves are the best-fitting models, which assumes that every cluster center identi-fied by 3D-MF is perfectly aligned with the dark matter halo center. With the exception of the lowestrichness bin, where the best-fitting curves coincide, the perfectly centered model does not provide agood fit to the data at small R. Table 4.1 and Table 4.2 summarize the results of both fits.4.4 Galaxy Cluster Weak Lensing Shear Results4.4.1 Fits to ∆ΣWe divide our cluster candidate catalog into six richness bins, and measure the differential surface mass densityas described in Section 4.3.2. The significances of the separate stacked measurements of ∆Σ(R) shown inFigure 4.5 range from 14.2σ to 25.6σ , calculated using the full covariance matrices to include correlationbetween radial measurement bins. Error bars are calculated as the square root of the diagonal of the covariancematrices. These values, along with details of the richness bins and fits are given in Table 4.1. This yields a total3D-MF cluster shear significance of ∼54σ .In modeling the halo mass, we use a composite-halo approach, which allows for the fact that the clustercandidates in a given stacked measurement may have a range of individual masses and redshifts. We emphasizethat instead of fitting a single average mass (and also avoiding a single effective cluster redshift), we actually fitto the normalization of the mass-richness relation, M0. We convert the array of cluster N200 values into masses6620 160 300 44060 < N200111315 171940 < N200 ≤ 606.0 6.8 7.6 8.430 < N200 ≤ 403.3 3.94.5 5.1             M200 [1013 M]20 < N200 ≤ 301.1 1.51.9 2.310 < N200 ≤ 200.2 0.4 0.6 0.81101001000100002 < N200 ≤ 10Figure 4.6: Cluster Mass Distributions for each Richness Bin. Underlying distribution of cluster can-didate masses, within each of the six richness bins in Figure 4.5, for the full miscentered model(note that each panel has a different horizontal axis range). Because the parameters fit to the shearmeasurements are the normalization of the mass-richness relation (Equation 4.18) and the miscen-tering parameters pcc and σoff, the full (not binned) set of cluster N200 values are each converted toan individual cluster mass. We bin these masses for presentation in the above histograms only, butemphasize that the composite-halo modeling approach in this work treats every cluster candidate ashaving an individual mass (richness) and redshift. This figure is also a visual representation of the3D-MF cluster mass function, as obtained from weak lensing shear.101102〈N200〉10131014〈M200〉[M]Full ModelFigure 4.7: Mass-Richness Relation from Shear. Powerlaw best fit to mass-richness relation (Equa-tion 4.18) obtained from average masses measured for the individual N200 bins in Figure 4.5 andTable 4.1, for the full model which accounts for miscentering, and including the (very small) cor-rection for intrinsic scatter. The dotted lines show the 1σ limits on this relation. As discussed inSection 4.4.2 the simple pcc ≡ 1 model, which assumes perfect cluster centers, yields the same slope,but a slightly lower overall normalization.67with the equationM200 = M0(N20020)1.5. (4.17)In each separate stacked weak lensing measurement, we keep the slope of this mass-richness relation fixed,to avoid over-fitting to each stack with parameters that are quite degenerate within a narrow cluster bin. The NFWmass of each individual cluster is given by Equation 4.17, with the fixed slope of 1.5 from Chapter 3 (Ford et al.,2014), which will be shown to be consistent with the global mass-richness relation, measured and discussed inSection 4.4.2 of this current work. We note that because of the free normalization M0, this approach does neitherimpose the form of the richness distribution (Figure 4.2) nor does it set a prior on the individual mass.We fit the halo model given in Equation 4.16 to the data, employing the downhill simplex method to minimizethe generalized χ2, using the full covariance matrices estimated from bootstrap resampling. The results aredisplayed as the green curves in Figure 4.5 (labeled “Full Model”), and summarized in Table 4.1. The numberof degrees of freedom for the model is 7 (10 radial bins minus 3 fit parameters).To emphasize the importance of cluster miscentering, we also plot the best-fitting model where pcc ≡ 1 (i.e.perfect cluster centers) for comparison. This is shown as the dashed purple curves in Figure 4.5 (with a single fitparameter, M0, this model has 9 degrees of freedom). Visual inspection reveals poor fits to the data at small radiifor this model, and this fact is quantified by the reduced generalized χ2 statistic (χ2red) values in Table 4.2. Theseresults imply that cluster centroiding is an important component in the modeling of the 3D-MF weak lensingshear mass profiles, especially at the high mass (richness) end. For the majority of the rest of this work we willfocus our attention on the results of the full model, which accounts for offset cluster centers.The ensemble of cluster masses that result from the composite-halo modeling approach are displayed inFigure 4.6, where each panel represents a single stacked weak lensing measurement, congruent with Figure 4.5.This visual representation of the cluster mass function is largely distinct from the N200 histogram in Figure 4.2,because these masses are dependent upon the mass-richness normalization, as well as the miscentering parame-ters, which are fit to the measurements.4.4.2 The Mass-Richness RelationThe results of the previous section demonstrate a strong scaling of mass with richness. In Figure 4.7 we plot theaverage mass M200 measured in each richness bin, as a function of richness N200, and fit the power law scalingrelation:M200 = M0(N20020)β. (4.18)This is similar to Equation 4.17, but the slope β is now a free parameter, and the mass-richness normalizationM0 is fit across the full distribution of clusters. We note that the choice of β = 1.5 in Equation 4.17 does nothave a significant effect on the β measured here. Because of the degeneracy between β and M0 in each narrowcluster bin, a different choice of slope for the measurements in Section 4.4.1 still yields essentially the samemass estimates M200, and thus the same global mass-richness relation.Since galaxy clusters exhibit a natural intrinsic scatter between halo mass and richness (or other mass proxy),a bias in scaling relations can result if this scatter is ignored (Rozo et al., 2009a). The idea here is that whilegalaxy clusters at a given richness will scatter randomly with regard to their average mass, because of the shapeof the cluster mass function, the net effect is to scatter from low to high mass. This can lead to a biased mass68estimate in a given richness bin, as well as affect the global result for the mass-richness relation. We correctfor intrinsic scatter using the data itself, following a procedure inspired by Velander et al. (2014), which is asfollows.We first fit Equation 4.18 to the uncorrected raw mass estimates from each richness bin, and use this powerlaw relation to assign an individual mass to each cluster, based on its value of N200. We then draw many“simulated” clusters from the observed cluster mass function (i.e. the N200 histogram in Figure 4.2), taking1,000 times as many “simulated” as observed clusters. We then scatter their masses by values drawn from aGaussian in ln(M200), with width σlnM|N , centered on the particular N200. For the width of the intrinsic scatter,we use values estimated by Rozo et al. (2009a) for the MaxBCG clusters in the SDSS. This is σlnM|N ∼ 0.45,which is the scatter in the natural logarithm of mass, at fixed richness.The resulting mass estimates are then used to calculate the corrected arithmetic mean mass in each of therichness bins, which are plotted in Figure 4.7 and used to re-fit Equation 4.18, yielding the final mass-richnessrelation reported below. The corrections applied to the mass estimates are at the sub-percent level, and thereforenegligible compared to other sources of uncertainty in this work. Nevertheless, we include these small correc-tions when fitting for the mass-richness relation. We note that increasing σlnM|N up to the 95% confidence limitreported by Rozo et al. (2009a) still does not affect the conclusions drawn in this work. A glance at Figure 4.6justifies the low-impact of the intrinsic scatter correction, as most richness bins do not exhibit a very strongslope, which would otherwise lead to a larger effect on average mass in each bin.In this work we measure M0 = (2.7+0.5−0.4)×1013 M and β = 1.4±0.1 for the full model (Figure 4.7), with aχ2red of 0.9. For the perfectly centered model, we get M0 = (2.2±0.2)×1013 M and β = 1.4±0.1, with a χ2redof 1.0. (Note that uncertainties are larger on parameters estimated from the full model, both here and throughoutthis work, since there are simply more parameters than the perfectly-centered model). These results demonstratethat not including the centroid uncertainty in our analysis would lead us to systematically underestimate thecluster masses, as well as the mass-richness normalization, because a miscentered stack of halos has a lowershear profile amplitude at small projected radii. Section 4.5.5 contains a thorough comparison of these resultswith our previous magnification measurements of these cluster candidates.69Table 4.1: Shear Results for Richness-Binned Clusters (Full Model). Details of the “Full Model” fits for the richness-binned measurements (Equa-tion 4.16, green curves in Figure 4.5). This model has 7 degrees of freedom. We list the richness range selected, the number of cluster candidates inthat bin, the shear detection significance, and the average richness and redshift of clusters in the bin. Fitted parameters include the centering-relatedparameters pcc and σoff, and the normalization of the mass-richness relation M0, from which the average mass in each bin 〈M200〉 is derived. Notethat the average mass given is not the value fit itself, but the average of all resulting masses fit using the composite-halo approach discussed inSection 4.3.2. See Figure 4.6 for a summary of the mass distributions within each N200 bin. Reduced generalized χ2 are given for each bin, andshould be compared with the corresponding fits listed in Table 4.2, for the simple one-parameter model assuming perfect centers.Richness Clusters Significance 〈N200〉 〈zl〉 pcc σoff M0[1013 M]〈M200〉[1013 M]χ2red2 < N200 ≤ 10 3,745 14.2σ 8 0.45 1.0−0.2 — 2.4+0.9−1.0 0.6+0.2−0.3 2.110 < N200 ≤ 20 9,034 22.8σ 15 0.63 0.5±0.1 (0.40+0.06−0.2 )′ 2.4±0.6 1.6±0.4 2.320 < N200 ≤ 30 3,409 25.6σ 24 0.67 0.5±0.1 (0.4+0.2−0.1)′ 2.9±0.5 3.9±0.7 0.830 < N200 ≤ 40 986 23.4σ 35 0.65 0.5±0.2 (0.4±0.1)′ 3.0±0.7 7±2 2.640 < N200 ≤ 60 568 22.2σ 48 0.60 0.54±0.08 (1.3+0.5−0.4)′ 3.6+0.8−1.0 14+3−4 0.360 < N200 314 22.5σ 114 0.55 0.5±0.2 (0.4+0.2−0.1)′ 1.6+0.4−0.5 26+6−7 3.4Table 4.2: Shear Results for Richness-Binned Clusters (Perfectly Centered Model). This table is a companion to Table 4.1, giving details of thepcc ≡ 1 model fits for the richness-binned measurements (purple dashed curves in Figure 4.5). This model has 9 degrees of freedom. We list therichness range selected (the reader can refer to Table 4.1 for the number of clusters, shear significance, and average richness and redshift). For thismodel, there is a single fit parameter, the normalization of the mass-richness relationM0, from which 〈M200〉 is derived (again see Figure 4.6 for thefull distribution of masses in each richness bin).Richness M0[1013 M]〈M200〉[1013 M]χ2red2 < N200 ≤ 10 2.4+0.4−0.6 0.6±0.1 1.610 < N200 ≤ 20 1.8±0.2 1.2±0.2 4.820 < N200 ≤ 30 2.2+0.2−0.3 3.0+0.3−0.4 5.330 < N200 ≤ 40 2.4±0.3 5.5±0.8 4.440 < N200 ≤ 60 2.1±0.3 8±1 4.760 < N200 1.4±0.2 23±3 4.470Table 4.3: Shear Results for Redshift-Binned Clusters (Full Model). Details of the “Full Model” fits for the redshift-binned measurements (greencurves in Figure 4.8). This model has 7 degrees of freedom. We list the same bin properties and fits given in Table 4.1. The systematic errors listedon some cluster masses stem from uncertainties on the exact redshift of the cluster candidate. The fits in this table should be compared with thecorresponding values in Table 4.4, which represents the perfectly centered model.Redshift Clusters Significance 〈N200〉 pcc σoff M0[1013 M]〈M200〉[1013 M]χ2redz ∼ 0.2 1,161 13.8σ 14 0.3±0.3 (0.4+0.3−0.1)′ 3±1 2.3+0.9−1.0±0.4sys 0.6z ∼ 0.3 1,521 15.7σ 17 0.8+0.2−0.3 (0.4+1−0.4)′ 2.3+0.7−0.9 2.6+0.8−0.9±0.2 0.4z ∼ 0.4 2,248 17.0σ 18 0.7±0.2 (0.4+0.3−0.2)′ 2.6±0.9 3±1±0.1sys 0.8z ∼ 0.5 2,935 20.2σ 18 0.8±0.2 (0.4+0.2−0.3)′ 2.5+0.6−0.8 3.0+0.7−1.0 1.7z ∼ 0.6 2,456 14.7σ 20 0.4±0.2 (0.4±0.1)′ 3±1 4±1 1.1z ∼ 0.7 2,331 11.9σ 22 0.7±0.3 (0.4+0.6−0.4)′ 2.1+0.9−1.0 3±1 0.8z ∼ 0.8 2,364 8.7σ 22 0.2±0.2 (0.4±0.2)′ 3+1−3 4+2−3 1.9z ∼ 0.9 3,040 6.8σ 19 0.6±0.4 (0.4+1−0.4)′ 1.8+0.8−1.7 1.9+0.9−1.8 0.5Table 4.4: Shear Results for Redshift-Binned Clusters (Perfectly Centered Model). This table is a companion to Table 4.3, giving details of thepcc ≡ 1 model fits for the redshift-binned measurements (purple dashed curves in Figure 4.8). This model has 9 degrees of freedom. For this model,there is a single fit parameter, the normalization of the mass-richness relationM0, from which 〈M200〉 is derived.Redshift M0[1013 M]〈M200〉[1013 M]χ2redz ∼ 0.2 2.6±0.6 2.0±0.5±0.3sys 2.1z ∼ 0.3 2.1±0.4 2.4±0.4±0.2sys 0.4z ∼ 0.4 2.2±0.4 2.7±0.5±0.1sys 1.4z ∼ 0.5 2.2±0.3 2.7±0.4 1.6z ∼ 0.6 2.4±0.6 2.9±0.7 4.5z ∼ 0.7 1.9+0.4−0.5 2.4+0.6−0.7 0.8z ∼ 0.8 1.4±0.6 1.8±0.8 3.3z ∼ 0.9 1.3±0.6 1.4±0.6 0.671020406080100z ∼ 0.2Full Modelpcc≡1z ∼ 0.3 z ∼ 0.4 z ∼ 0.50.11R [Mpc]020406080100               ∆Σ(R)[M/pc2]z ∼ 0.60.11R [Mpc]z ∼ 0.70.11R [Mpc]z ∼ 0.80.11R [Mpc]z ∼ 0.9Figure 4.8: Shear for Redshift-Binned Clusters. Best-fitting models for each stack of cluster candidates,this time binned in redshift. As in Figure 4.5, the solid green curves are the best fits to the fullmodel given by Equation 4.16. The dashed purple curves are the best-fitting models which assumesthat every cluster center identified by 3D-MF is perfectly aligned with the dark matter halo center.Table 4.3 and Table 4.4 summarize the results of these fits.4.4.3 Results of Binning Clusters in RedshiftWe also investigate the weak lensing shear measurement of 3D-MF cluster candidates as a function of clusterredshift. 3D-MF sorts candidate clusters into bins of width ∆z ∼ 0.1, so these are natural bin choices, and thesame used in our previous analysis (Chapter 3, Ford et al., 2014). Figure 4.8 shows the measurements and fits to∆Σ, with error bars again obtained from the covariance matrices (Section 4.3.2). The significance of the shearmeasurements reaches ∼ 20σ at z∼ 0.5, where there is an abundance of 3D-MF cluster candidates, and drops to∼ 7σ at the highest redshifts, where shear signal-to-noise is depleted.In Figure 4.8 (similar to Figure 4.5), we plot the full model in solid green, and the perfectly centered model indashed purple. Table 4.3 and Table 4.4 display the results and fit parameters for these two models, respectively.The measurements at lower redshifts have an additional systematic error listed, which stems from uncertaintieson the cluster redshifts, due to the way the 3D-MF method slices in redshift space (Chapter 3, Ford et al., 2014).The 3D-MF cluster candidates are found to be quite similar in average mass across the range of redshift probed– we consistently obtain measurements of a few 1013 M. The best-fitting miscentering parameter pcc variessomewhat erratically as a function of redshift, but the error bars are too large to infer any significance from this.The width of the offset distribution on the other hand remains squarely at σoff ∼ 0.4 arcmin. We discuss thisresult in relation to other cluster miscentering studies in Section 4.5.4.We investigate possible redshift evolution of the mass-richness relation (given by Equation 4.18) in Fig-ure 4.9, which shows the normalization of this scaling relation, M0, as a function of redshift (with β = 1.5 fixed),as listed in Table 4.3. We fit a powerlaw relation of the formM0(z) = M0(z = 0) · [1+ z]η . (4.19)720.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0z12345M0[1013M]Figure 4.9: Redshift Dependence of Mass-Richness Normalization. Normalization of the mass-richnessrelation M0 as a function of redshift z. The evidence for redshift evolution is not significant: the mildlynegative slope is consistent with zero.We find a normalization M0(z = 0) = (3.0± 0.6)× 1013 M, and a powerlaw slope η = −0.4+0.5−0.6. The slopeis consistent with zero, so no significant redshift-evolution is detected for the 3D-MF mass-richness scalingrelation.4.5 Discussion4.5.1 Interpretation of the ResultsThe 3D-MF clusters represent a wide range of halo masses and impose a significant shear signal on backgroundgalaxies. The measured ∆Σ profiles from different stacked subsamples of clusters yield an important glimpse atthe state of the dark matter haloes. We fit a model that includes parameters designed to distinguish the fractionof well-centered versus offset haloes, and the width of the offset distribution. The latter is consistently measuredto peak at an offset of ∼ 0.4 arcmin, except for the richness bin 40 < N200 ≤ 60, for which we find a largerbest fit of 1.3 arcmin (this much larger offset is puzzling, and will require follow-up to determine whether it isphysical or perhaps a spurious effect of overfitting). The fraction of clusters that are not correctly centered isgenerally about 50% across richness bins, but has large error bars that do not allow us to distinguish interestingfeatures at a statistically significant level. Nonetheless, we do find overall that the 3D-MF cluster halo profilesare better fitted by not enforcing perfect centroiding.This study comprises several novel components, which will be discussed in more detail below. The largenumber of clusters, and the fact that 3D-MF does not assume anything about cluster galaxy colors, makes theuniqueness of the data set valuable in its own right. Evolution of the normalization of the mass-richness relationacross a wide span of redshift has only been constrained previously by van Uitert (2012) and Andreon andCongdon (2014). The direct comparison between shear and magnification measured masses is a first for acluster catalog of this volume. There are several caveats to the implications of this work, notably the very likelypresence of false-detections at the low-significance (low-richness) end of the cluster candidate spectrum.734.5.2 Comparisons of Cluster Catalog VolumeThe most noteworthy aspect of the CFHTLS-Wide 3D-MF cluster catalog is its sheer size. With over 100 clustercandidates per square degree (18,056 clusters in 154 deg2), spanning redshifts up to z∼ 0.9, this compilation ofcluster candidates is one of the most complete available. We encourage others to utilize this catalog, availablefrom cfhtlens.org, as there are an abundance of scientific investigations now possible with it.The current widest survey with a galaxy cluster catalog is SDSS. The SDSS collaboration found 13,823 galaxygroups and clusters spread over 7,500 deg2, using their maxBCG method (Koester et al., 2007). This amountsto less than two clusters per deg2, and is restricted to lower redshifts (0.1 < z < 0.3). The maxBCG techniquerelies on finding potential bright galaxies and searching around them for the presence of a red sequence in color-magnitude space (which would indicate the presence of red, elliptical galaxies, common in galaxy clusters).Interestingly, visual inspection of 3D-MF galaxy cluster candidates shows that the lower redshift clustersoften do have a BCG, but this is less true at higher redshifts. It would be interesting to quantify this aspectin future work, especially when an opportunity presents itself to compare 3D-MF to other algorithms directly,by running both on the same optical data set. Galaxy clusters do not always have one BCG, and if they dohave one, it is not always exactly in the center of the galaxy cluster, so comparing the biases of both methodscould ultimately result in a more complete cluster list, or could potentially show the limitations of methods likemaxBCG.Several cluster catalogs have been compiled in the CFHTLS-Wide. Durret et al. (2011) used photometricredshift information to construct galaxy density maps in CFHTLS, building upon earlier work by Adami et al.(2010) and Mazure et al. (2007). They found 4,061 cluster candidates in the Wide fields, with masses greaterthan about 1014 M, spanning redshifts 0.1 ≤ z ≤ 1.15. Shan et al. (2012) used a 3D-lensing approach, withconvergence maps and galaxy photometric redshifts, to detect 85 clusters at 〈z〉 ∼ 0.36 in the W1 field of theCFHTLS-Wide.Wen et al. (2012) compiled an optical cluster catalog from SDSS-III, using galaxy photometric redshiftsand a friends-of-friends algorithm. They found an impressive 132,684 clusters over 14,000 deg2 in a redshiftrange 0.05 ≤ z < 0.8. A recent cluster shear analysis was done by Covone et al. (2014), using the overlappingportion of the Wen et al. (2012) catalog, with the CFHTLenS shear catalog. To date, this is the most completecluster catalog analyzed in the context of CFHTLenS, but still the cluster density is ≤ 1/10th of that achievedwith 3D-MF. A comparison of 3D-MF with the cluster catalogs compiled using these different techniques willbe presented in a future analysis.4.5.3 Comparison with other Mass-Richness RelationsThe 3D-MF cluster finder presents us with a sample of cluster candidates which, like every other cluster-finder,are drawn from a somewhat unique distribution defined by its particular selection function. Despite the difficul-ties inherent to making exact comparisons between scaling relations measured on disparate cluster samples, weattempt a broad look at how the 3D-MF mass-richness scaling compares to other relations in the literature.Wen et al. (2009) defined a measure of richness R for their SDSS clusters, which is somewhat similar to theN200 used in this work. They counted all galaxies brighter than absolute magnitude Mr ≤ −21, within a 1 Mpcradius and ∆z < 0.04(1+ z). Converting their mass-richness relation to the form of ours (Equation 4.18), theyobtained a somewhat steeper slope β ∼ 1.9, and a higher normalization M0 ∼ 2.5×1014 M than the best-fittingmodels presented in this work (Full Model: M0 ∼ 2.7× 1013 M, β ∼ 1.4). We tried measuring richness for74the 3D-MF clusters following the same prescription as Wen et al. (2009), but found the one-size-fits-all radiusto be a serious limitation for our sample, since the 3D-MF cluster candidates span a wide range of masses andtherefore characteristic radii. The resulting richness estimates had greatly enhanced scatter and did not scalewell with mass at the more massive end of the cluster catalog.In a follow-up paper, Wen et al. (2012) defined a new richness RL∗ – the total r-band luminosity within R200in units of L. For the portion of clusters with previously measured masses (weak lensing or X-ray), a scalingbetween the radius R200 derived from these masses and the luminosity within 1 Mpc was measured, and this wasused to estimate radii for calculating RL∗ for the full sample of 132,684 clusters. For the subsample with existingmass estimates, Wen et al. (2012) found a mass-richness relation with normalization M0 ∼ 1.1× 1014 M andslope β ∼ 1.2 (again converting to the form of our Equation 4.18). Covone et al. (2014) measured weak lensingmasses for 1,176 of the clusters from Wen et al. (2012), which overlapped with CFHTLenS. They found a verysimilar mass-richness scaling, with M0 ∼ 1014 M and β ∼ 1.2.The mass-richness slope of the 3D-MF cluster candidates sits squarely between the results of Wen et al.(2009), using the R richness, and Wen et al. (2012) and Covone et al. (2014), which used the RL∗ measure. The3D-MF normalization is lower than the other cluster catalogs, which could partly be a result of 3D-MF detectingmore lower mass clusters missed by other finders. However, the different definition of richness, namely thefainter limit on galaxies contributing to N200, means that the same mass cluster will have a larger measuredrichness in this work, implying a lower mass-richness normalization. Finally, the presence of false detectionsin the 3D-MF catalog (estimated from simulations to be at the level of 16−24%) would certainly bias the massestimates low.Johnston et al. (2007) used a quite different definition of richness for the maxBCG clusters, counting onlyred-sequence galaxies brighter than 0.4L∗, within an Rgals200 that was estimated from the number of galaxies within1 Mpc (following a prescription in Hansen et al., 2005). Weak lensing masses were used to find a normalizationM0 ∼ 1.3×1014 M and slope β ∼ 1.3. Rozo et al. (2009b) created updated richness estimates of the maxBCGclusters by applying an improved color modeling of cluster members, and allowing individual cluster radii tovary until the scatter between richness and X-ray luminosity was minimized.Andreon and Hurn (2010) defined a measure of richness for the Cluster Infall Regions in SDSS catalog,for which masses M200 and radii R200 were already available (from application of the caustic technique). Theystudied a sample of 53 low-redshift clusters, in the range 0.03 < z < 0.1, and their N200 included all red galaxiesbrighter than MV = −20 within the radius R200. In a follow-up analysis they measured a tight mass-richnessscaling relation with normalization M0 ∼ 1.4×1011 M and slope β ∼ 2.1 (Andreon and Berge´, 2012).The addition of the galaxy color information in the richness estimate of the previous three examples, inparticular, creates difficulty in drawing meaningful comparisons between their mass-richness scaling relationand the 3D-MF scaling relation. We emphasize that the value of any mass-richness relation is limited to theparticular cluster sample for which it was derived, which in turn depends on the cluster-finding algorithm anddetails of the survey on which the catalog was compiled. As discussed in Rozo et al. (2009b), the simple fact thatestimates of richness are readily available in an optical cluster survey, and that they can be applied to clusters ofvirtually any mass, makes richness a worthwhile parameter to measure. So although richness has many differentdefinitions, and some unavoidable scatter in its scaling relations with various cluster mass estimates, it remainsa useful tool for characterizing galaxy clusters.75101102〈N200〉10131014〈M200〉[M]0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9zShear (this work, full model)Magnification (Ford et al. 2014)Figure 4.10: Comparison of Magnification and Shear Masses. Comparison of mass measurements ob-tained for the 3D-MF cluster candidates using weak lensing shear (i.e. this work) with the resultsobtained measuring the masses with the lensing magnification technique (the N200 estimates fromthat work, Chapter 3 or Ford et al., 2014, are used in this plot for the purposes of comparison). Thefirst panel compares mass measurements when cluster candidates are binned in richness N200, andthe second panel shows the redshift z binning. Bins are identical for magnification and for shear, butthe points are slightly offset horizontally for clarity. Blue diamonds represent the shear, and orangesquares are for magnification.4.5.4 Comparisons with other Cluster Centroid AnalysesWe find the distribution of centroid offsets to be well characterized by a Rayleigh distribution of width σoff ∼0.4 arcmin,2 and that this miscentering has an effect on a significant portion of the candidate clusters (up to ∼80% of them are affected, see Table 4.1). Interestingly, previous studies applying 3D-MF to simulations yieldedan average σoff = 0.40± 0.06 arcmin (see Figure 3.1 or, equivalently, Figure 1 in Ford et al., 2014), which iseasily consistent with the best-fitting offset measured on the real 3D-MF cluster candidates in this work.The maxBCG clusters were found to have centroid offsets around 0.42 h−1Mpc, based on simulations (John-ston et al., 2007), which is several times larger than the ones measured for the 3D-MF cluster candidates. Therewere large uncertainties associated with the probability of a cluster having a correct centroid selected, but thiswas determined to be approximately ≥50% (see Figure 5 in Johnston et al., 2007), which is similar to pcc foundin this work. George et al. (2012) performed a miscentering analysis of X-ray groups in the COSMOS field.They found offsets of∼ 20−70 kpc, for different candidate centers, which are smaller than measured for 3D-MFclusters.It is worth noting that candidate cluster centers that are coincident with a member galaxy have been foundto better trace the halo’s center of mass, relative to other types of centroids such as X-ray, or various weightedcenters of galaxy positions (George et al., 2012). See also Bildfell et al. (2008) for a study of massive X-rayclusters. 3D-MF centers (peaks in the likelihood map) do not necessarily coincide with a cluster galaxy member,so future work should investigate various possible candidate centers to find the one that best traces the center ofmass for 3D-MF cluster haloes.2For comparisons, 0.4 arcmin ∼ 147 kpc at redshift 0.5.764.5.5 Comparison with Magnification ResultsOne of the most interesting aspects of this work is the direct comparison between magnification and shearmass estimates, which is now made possible in the context of a very large lens sample. Prior to this work, theonly observational magnification–shear direct cluster mass comparison in the literature was Ford et al. (2012)(equivalently, Chapter 2). That study demonstrated a 1σ consistency between masses measured with the twotechniques, but applied to a small sample of just 44 galaxy groups, so any trends in cluster size or redshiftwere unable to be explored. Huff and Graves (2014) compared magnification and shear masses for SDSS galaxylenses, using a different and novel approach to measuring lensing magnification, and found mass profiles to bewithin a factor 3 of agreement.Important work related to the joint analysis of shear and magnification has been developed in Umetsu et al.(2011) and Umetsu (2013). Umetsu et al. (2014) combined shear and magnification to measure the mass pro-files of 20 massive X-ray-selected clusters. This work demonstrated that the geometric mean mass of theshear+magnification measurement was consistent with the shear-only measurement, but did not show magnifica-tion results on their own. Earlier work in Umetsu et al. (2011) compared the signal-to-noise of the magnificationand shear, but did not present mass estimates from separate analyses.In this work we exploit the volume of the 3D-MF cluster catalog to fully compare masses determined witheach of the independent techniques, as a function of both candidate cluster richness and redshift. The findingsare summarized in Figure 4.10. For consistency in the comparison, this plot uses the original N200 estimatesfrom Chapter 3 (Ford et al., 2014), so that the cluster candidate stacks in each richness bin are identical. Also,in this section only, we use the mass-concentration relation of Prada et al. (2012), in identical fashion to themagnification work. The Prada et al. (2012) relation is in excellent agreement with recent measurements byCovone et al. (2014) of the masses and concentrations of a cluster sample in CFHTLenS, although it is in tensionwith other measurements such as Merten et al. (2014).The left-hand panel of Figure 4.10 displays the results when 3D-MF cluster candidates are stacked across allredshifts. The average of the composite-halo masses fit to each stack is comparable between the two methods,but the magnification estimates are systematically lower than the shear estimates. This yields a mass-richnessnormalization which is about 2σ higher for the shear method, although the slope of the relation recovered withthe two techniques is essentially identical. The magnification measurements yielded M0 = (2.2±0.2)×1013 Mand β = 1.5±0.1 (see the miscentered model in Chapter 3 or Ford et al., 2014), while the shear measurementshere give M0 = (3.1± 0.5)× 1013 M and β = 1.5± 0.2. We note that the mass-richness relation parametersobtained from the shear measurements in Figure 4.10 are consistent within 1σ with the new mass-richnessparameters obtained with shear in Figure 4.7 and discussed in Section 4.4.2. We reiterate that the slight differencebetween the shear measurements in Figure 4.7 and Figure 4.10 is due to a recalibration of the cluster N200estimates (see Section 4.2.3) and a different choice of mass-concentration relation.It is important to note that in both of the aforementioned magnification studies (Chapters 2 and 3, or Fordet al., 2012, 2014), the background source sample is completely distinct from the background sources usedto measure shear. Indeed, both magnification results used magnified LBGs, which are point-like sources whosenegligible apparent size would not permit a measurement of the shear. In this sense, the magnification results arelargely independent from the shear measurements to which they are compared, having only the lens populationin common. We note that alternative methods of measuring magnification using source size information wouldinstead tend to use the same source sample employed for measuring shear.77The comparison becomes more interesting as a function of redshift, shown in the right-hand panel of Fig-ure 4.10. Here we see that the shear-measured average mass of cluster candidates does not vary as a function ofredshift, while the magnification masses fluctuate. In Chapter 3 (Ford et al., 2014), we discussed this behavior ofthe magnification signal, but without an alternative mass determination were not able to conclude whether thisvariation communicated an intrinsic property of the 3D-MF cluster candidates, or was an artifact of the magnifi-cation measurement. We are still unable to say with certainty whether the masses of the 3D-MF cluster sampletruly are constant or evolving across the redshift range, as suggested by the conflicting shear and magnificationmeasurements. Here, we discuss several possible reasons for these discrepant redshift-binned results.First of all, the distributions of richness values for the separate z slices are very similar, with the lower redshiftslices containing relatively higher fractions of low-richness cluster candidates (see Figure 3.8 or, equivalently,Figure 7 in Ford et al., 2014). So if (1) richness is a good estimator for mass, which it appears to be giventhe strong scaling, and (2) the mass-richness relation does not evolve strongly with redshift over the rangez ∼ 0.5→ 0.2, then we would expect similar masses across this range, or for masses to actually decrease atlower redshift in concordance with the lower mean cluster N200 (i.e. the opposite of the trend suggested bymagnification).As discussed in detail in Chapter 3 (Ford et al., 2014), at z ∼ 0.2−0.3 the magnification measurement isexpected to be affected by some low-z contamination in the LBG source sample. In that work we attemptedto compensate for this effect by including a term in the modeling of the measured signal, to account for phys-ical clustering where the populations overlap. A crucial assumption was the actual fraction of contaminatedsources, which was estimated using a cross-correlation technique with foreground galaxies (Hildebrandt et al.,in preparation). If these fractions were biased low, then much of the physical clustering signal would have beeninterpreted as due to magnification, leading to mass estimates that were too high (at low-z).Currently, we are also investigating the influence of several other systematic effects on our magnificationmeasurements. In particular, we are studying how the varying depth and the varying seeing of the surveyaffect different lens and source samples, and how stellar contamination (or also just the light halos of stars) andgalactic dust can alter the magnification signal. This in-depth analysis of systematic effects will be presented in aforthcoming paper (Morrison et al., in preparation) and might provide additional insight into the apparent redshiftdependence of the cluster magnification signal reported in Chapter 3 (Ford et al., 2014). Another possibilitymay be related to the masking effect of cluster galaxy members, which is survey dependent and can affect bothmagnification and shear measurements (Simet and Mandelbaum, 2014). This sky obscuration could lead to ourmagnification masses being biased low, but our shear measurements should be robust because of the stringentcriteria used for selecting background galaxies (Section 4.3.2).In order for magnification to yield robust results that encourage its employment in the next generation of largesurveys, this discrepancy needs to be addressed. Studies that compare shear and magnification measurementsfor large binned lens and source samples are crucial for teasing out these underlying systematics.4.6 ConclusionsThis work has presented weak lensing shear results, measured at 54σ significance, for a new catalog of clustercandidates detected by the 3D-MF algorithm. 3D-MF is a three-dimensional advancement of older matched-filter techniques, which automatically searches wide and deep optical data for galaxy clusters across a rangeof redshifts. Given a sensible luminosity and radial profile, 3D-MF is able to search within data for a range78of galaxy cluster masses. By construction, 3D-MF has allowed us to find lower mass cluster candidates (andgroups) which other popular techniques, such as the red sequence and maxBCG, may not be capable of finding.3D-MF was run on the CFHTLS-Wide fields using galaxy photometric redshifts and i′−band data for clus-ter luminosity profiles, producing one of the largest and most complete cluster catalogs currently available.18,056 cluster candidates were detected with a significance ≥ 3.5 and richness N200 > 2, out to a redshift of0.9 (>100/deg2). Many of these cluster candidates are in the lower mass ranges (down to ≤ 1013 M), whichis notably a larger low mass sample than currently exists from deep, wide surveys in the literature, offering anenormous opportunity for further study.The CFHTLS-Wide 3D-MF catalog was investigated to learn more about candidate cluster properties, suchas masses and centroiding, as well as to follow up on previous results applying the less developed technique oflensing magnification to this cluster sample (Chapter 3 or Ford et al., 2014). Shear profiles were measured aroundcluster candidates, which were stacked as a function of richness and redshift, and we focused on presentingcomposite-halo model fits to measurements of the differential surface mass density ∆Σ.Careful consideration of potential miscentering of galaxy clusters by 3D-MF had to be taken into accountin the analysis. We fit the data with smoothed shear profiles, ∆Σsm, that describe a cluster whose halo is offsetfrom its assumed center. The fraction of clusters that are affected by miscentering, as well as the probabilitydistribution of the offsets, were both allowed to vary in the modeling. We found the inclusion of these parametersto significantly improve the χ2 of the cluster profile fits, relative to a perfectly centered model for ∆Σ, whichwe also demonstrated for comparison. The stacked cluster shear measurements were well fitted by a model inwhich about half the clusters are affected by miscentering (pcc ∼ 0.5), with the distribution of centroid offsetspeaking at ∼ 0.4 arcmin.The large sample of cluster candidates in this work allowed us to bin the shear measurements as a functionof both richness and redshift. The average cluster candidate masses were found to be relatively constant withredshift, estimated at 2 to 4 ×1013 M. The masses scaled strongly with richness, ranging from ∼ 6×1012 Mto∼ 3×1014 M. We measured the normalization and slope of the mass-richness relation for the 3D-MF clustercandidates, finding M0 = (2.7+0.5−0.4)×1013 M and β = 1.4±0.1. The redshift dependence of the normalizationM0(z) was not significant, yielding a powerlaw slope in (1+ z) of −0.4+0.5−0.6.The masses of individual cluster candidates were found to range from a small group scale, with stackedaverage masses of less than 1013 M, all the way up to a few very massive clusters, at several 1015 M. Sincethe 3D-MF catalog has not been followed up spectroscopically, we expect some fraction of false-detections(estimated between ∼ 16 and 24% from simulations), which would lead to these mass estimates being biasedlow, and would especially affect the low-richness stacked measurements. We note, however, that the impact offalse detections may be less severe than implied, if line-of-sight projections are significant. Chance alignmentsof low-mass structures would have a similar effect on a shear measurement (which probes surface mass density)as it would on an estimate of optical cluster properties like richness.By design, we binned cluster candidates in an identical fashion to the previous magnification study (Chap-ter 3 or Ford et al., 2014), and compared the results obtained. This is the first large study directly comparingthe outcomes of magnification and shear on the same lens sample. When stacked across all redshifts, we foundthat the average masses derived within a given richness bin were similar (within 1σ ), but magnification masseswere systematically lower, yielding a 2σ difference in the normalization of the mass-richness relation derivedfrom the two techniques. The mass-richness slope was essentially identical for magnification and for shear. The79comparison across redshift slices yielded very interesting insights into problems that may still exist for mag-nification. The fact that the shear-determined masses were roughly constant across redshift led us to concludethat the magnification measurement (using magnification-biased number counts of LBG sources) may still sufferfrom residual systematics at low-z. Notably, however, this occurs at very predictable lens redshifts, so if one hasaccurate photometric redshift distributions for the sources, these contaminated redshift zones could potentiallybe avoided.In future work, it would be interesting to apply various cluster finding algorithms to the same large data setin order to compare the capabilities of the finders, potentially increase the overall cluster sample and reduceits biases, or even just to compare how different search algorithms perform. This could ideally lead to morecomplete and unbiased cluster samples. Current surveys, such as the Kilo-Degree Survey (de Jong et al., 2013),the Subaru Hyper Suprime-Cam Project (Takada, 2010), and the Dark Energy Survey (The Dark Energy SurveyCollaboration, 2005) for example, are large enough that cluster masses and concentrations will be measuredquite accurately as a function of redshift and richness. The area of these surveys is an order of magnitude higherthan the CFHTLS-Wide, and high precision cluster profiling will naturally continue to evolve alongside thesesurveys.The CFHTLS-Wide 3D-MF galaxy cluster catalog contains 18,056 cluster candidates, over a wide range ofmass and redshift, and is now publicly available at cfhtlens.org. We encourage others to make use of the richscience opportunities afforded by this catalog.80Chapter 5Conclusions5.1 Thesis SummaryThis thesis has presented novel results in the study of galaxy clusters through the use of weak gravitationallensing techniques. The scientific domain is the field of cosmology, which was discussed in Section 1.1. Cos-mologists seek to understand the nature of the Universe on the largest scales – its history, evolution, and fate,its large scale structure, and all the components of its energy density. This therefore requires knowledge of thematter distribution in the Universe, and how that has changed over time.Matter makes up a significant portion of the energy density of the Universe (∼ 30% today), and almost all ofit is composed of dark matter, which appears to interact only through the gravitational force (see Section 1.1.2).Matter was the dominant component of the Universe for much of its history – from the time of matter-radiationequality, when the scale factor was just a(t) ≈ 2.9×10−4, until the recent transition to the current dark energydomination (Planck Collaboration et al., 2015a). A very useful way to probe the dark matter distribution isthrough the study of gravitationally collapsed halos, such as those that house galaxies and galaxy clusters.The specific importance and usefulness of galaxy cluster studies was discussed in Section 1.3. The main goal,regarding galaxy clusters in this thesis, was to improve measurements of clusters masses, which are so importantfor both cosmological and astrophysical purposes. The methods employed in this thesis for improving galaxycluster mass estimates consisted of two different and complementary approaches to weak gravitational lensing.Weak lensing is the subfield of gravitational lensing (see Section 1.2), wherein light from background sourcesis subtly bent by gravitational potentials along its path, leading to the focusing and distortion of backgroundgalaxies. Unlike strong lensing, where lensing-induced features like giant arcs and multiple images are usuallyvisible to the eye in images, weak lensing is not strong enough to produce obvious effects, but instead is used ina statistical sense, by averaging over many galaxies.The first of the two weak lensing techniques, weak lensing shear, was explained in Section 1.2.1. Shearis ubiquitous in the weak lensing literature, and involves measuring slight distortions in image shapes. Thesecond technique, magnification, was explained in Section 1.2.2 and is the dominant focus of the thesis. Magni-fication is a relatively underused technique, compared to weak lensing shear, and its complementarity with thelatter provided the motivation for this thesis research. Both types of weak lensing are statistical in nature, onlymeasurable using many thousands of background galaxies behind many galaxy clusters or other gravitationallenses. Magnification can be measured in several different ways, but the approach explored in this thesis usesthe lensing-induced modifications to the source number densities that are detectable behind a gravitational lens.81The body of this thesis was composed of three published articles, sandwiched between an introduction tothe relevant topics in Chapter 1, and the current concluding chapter. These three peer-reviewed journal articleswere all scientifically-led by the thesis author. The publication details can be found either in the Thesis Prefaceor in the Bibliography references for Ford et al. (2012), Ford et al. (2014), and Ford et al. (2015).Chapter 2 was comprised of the first magnification study applied to galaxy groups. Prior magnificationstudies were of galaxy lenses or of very massive (i.e. strongly lensing) clusters. This study yielded the firsttest of the technique on intermediate mass scales (several times 1013 M), and also included the first directcomparison between shear-measured masses and magnification-measured masses. These mass estimates werefound to be in agreement, albeit with rather large uncertainties. The signal-to-noise from each technique wasquantified and compared – magnification yielded a 4.8σ detection, whereas shear achieved 11σ .In Chapters 3 and 4, a much larger galaxy cluster sample was explored – the 3D-MF cluster catalog from theCFHTLS-Wide fields. This is one of the largest galaxy cluster catalogs that has been compiled (with over 18,000clusters), notable containing many lower mass galaxy groups, and covering a wide range of redshifts, up to z∼ 1.Optimized with the goal of finding as many clusters as possible, the 3D-MF cluster finder produced a catalogthat is 100% complete for clusters with mass greater than 3×1014 M and 88% complete above 1014 M (seeSection 4.2.3 or the original 3D-MF paper Milkeraitis et al., 2010, for more details). Compared with the smallsample of X-ray selected galaxy groups analyzed in Chapter 2, the 3D-MF clusters in Chapter 3 and Chapter 4allow for superior signal-to-noise, even when the clusters are binned according to different attributes. Thisallows for important studies of clusters as a function of redshift and cluster richness (number of galaxies).Chapter 3 focuses on the magnification signal of the 3D-MF clusters, which was detected at a significanceof 9.7σ . This chapter described the first magnification analysis with a large enough cluster sample to be able tobin as a function of richness (however, note that the publication of this work was followed in quick successionby another magnification study of clusters and luminous red galaxies in the SDSS by Bauer et al., 2014). Usingthe richness-binned cluster mass estimates, a power-law scaling relation between cluster mass and richness wasdetermined. This work compared the goodness of fit for two different types of cluster miscentering models – themodel assuming that 3D-MF’s selected cluster centers were perfectly accurate did not always fit the data as wellas the model that assumed an offset distribution, with offsets based on simulations.Additionally, the masses of clusters binned as a function of redshift were obtained, yielding unexpectedvariation of mass across redshift bins that had very similar richness distributions. This finding inspired furtherefforts to model a potential systematic effect for magnification – physical overlap between lenses and sourcesin redshift space. Assuming some fraction of source contamination, the expected physical clustering signal wascalculated and included in the models that were fit to the magnification signal. This was the first time suchan approach had been attempted – previous magnification studies had chosen to simply avoid interpreting anymeasurements in redshift regions where contamination is expected to be significant. Judgment regarding thesuccess of this approach was withheld until the follow-up shear study of Chapter 4 was completed, in order tofollow a semi-blind approach and avoid applying confirmation bias to the magnification results.Chapter 4 paralleled the previous chapter in many respects. This included a shear analysis in the samespirit as the magnification one in Chapter 3, using the publicly available shear measurements from CFHTLenS.The binning of clusters was identical in each analysis, in order to produce a side-by-side comparison of theresults from each of the two techniques. At 54σ , the shear signal measured from the cluster lenses is verystrong, and this study was able to go beyond what was possible with magnification. In addition to measuring82the mass-richness scaling relation, this analysis also searched for any evidence of this relation’s evolution withcluster redshift. No significant evolution was detected, which is in line with other recent studies (Andreon andCongdon, 2014).This work also placed constraints on the 3D-MF cluster centroid offsets, which was an improvement over themagnification analysis of Chapter 3, and was possible because shear is more sensitive to offset halos. Insteadof comparing a perfectly centered model with an offset model, the shear analysis actually fit for the distributionof offsets, finding it to be reasonably well described by a Rayleigh distribution, peaking at a radial offset ofσoff ∼ 0.4 arcmin (although see Table 4.1 and Table 4.3 for exact offset values, as well as other parameters).A thorough comparison of the results obtained for the 3D-MF clusters with other cluster analyses in theliterature was presented in Section 4.5. Briefly, the slope of the mass-richness relation determined in this thesiswas in line with other similar work (Wen et al., 2012; Covone et al., 2014), while the normalization is less easilycompared, because it is highly sensitive to the definition of richness used for the sample (which varies widely inthe literature). The cluster miscentering offsets measured in Chapter 4 of this thesis were intermediate betweenother results in the literature, which ranged from about half the radial offsets of 3D-MF (George et al., 2012) toseveral times larger than 3D-MF (Johnston et al., 2007).The research in this thesis pushed the limits of maximizing the extraction of weak lensing information tolearn about cluster dark matter halos in our Universe. Two major analyses incorporating magnification informa-tion were completed, with some success, but also generated some doubt about the reliability of magnificationwhen redshift contamination of sources is not well known. Prospects for improving the systematics-handling offuture magnification studies will be discussed in Section 5.3. Regardless of any shortcomings of the technique,the undertaking of the detailed magnification studies in this thesis has added value to the weak lensing literature.This body of work provides a framework for accounting for systematic effects in magnification, and highlightsissues and important considerations for future studies to build upon.5.2 Final ConclusionsThe overall important findings produced by this thesis research can be summarized as a list of key take-awaymessages.• Magnification can be successfully measured for galaxy groups and clusters, achieving up to half the signal-to-noise as the more commonly-measured shear technique (Chapters 2, 3, and 4).• It is possible to obtain consistent cluster mass estimates using weak lensing magnification and shear, atleast when low-redshift contamination of sources (for magnification) is minimal (Chapters 2 and 4).• The 3D-MF galaxy cluster catalog is publicly available as a result of this work (Chapter 4).• The 3D-MF galaxy clusters exhibit a robust scaling between mass and richness (Chapters 3 and 4).• The mass-richness scaling relation does not evolve significantly with redshift (Chapter 4).• The cluster detection significance of the 3D-MF clusters scales with mass and can be used as an alternativemass proxy (Chapter 4).• The miscentering of the 3D-MF clusters is significant (though less severe than for some other clustercatalogs) and must be accounted for to avoid biased lensing mass estimates (Chapters 3 and 4).83• The effects of cluster miscentering are degenerate with cluster concentration (Chapter 4).• It is confirmed that magnification is highly sensitive to source contamination. Source redshifts and con-tamination fractions must be known very accurately, and underestimation may have devastating effects onrecovered magnification-based masses (Chapters 3 and 4).• Additional unconsidered systematic effects may plague magnification as well. Specific redshift intervalsyielded anomalous magnification mass estimates that were difficult to blame on source contamination(Chapter 4).5.3 Future ProspectsThere are a number of avenues of future research that could build upon the work in this thesis. The possibilitiescan be broken roughly into three major areas, which will each be discussed below. They include: galaxy clustercatalog comparisons for different cluster-finding techniques; galaxy cluster halo characterization and study; andthe future of weak lensing magnification measurements in the era of upcoming large surveys. One commonendeavor that will increase the progress of scientific results in general, will be encouragement of open sciencethrough making software and data products public and accessible for others to build upon.The 3D-MF cluster catalog has many unique and important characteristics, and future work should comparethe cluster sample to catalogs compiled using different cluster-finding techniques. For example, 3D-MF does notuse any color information (aside from photometric redshifts) nor does it require a galaxy to be colocated withthe cluster center. Many other cluster-finders rely on the red-sequence of member galaxies and assume that aluminous red galaxy lies at the center. This may allow 3D-MF to pick up less massive or evolved clusters, similarto our own Local Group. The careful comparison of different catalogs would help quantify real differencesbetween the cluster samples recovered, and importantly would assist in removing false detections from the3D-MF catalog (which are expected to be a significant fraction of the low-mass cluster candidates). Running the3D-MF cluster-finder on the same optical data set, alongside other cluster-finders, and analyzing the differentobjects recovered, would also illuminate the overlap and complementarity of different techniques. Additionally,investigating the properties of galaxies in the 3D-MF clusters could be useful for constraining the halo occupationdistribution, a framework for describing how galaxies occupy dark matter halos (Coupon et al., 2015).The issue of halo miscentering is interesting for a couple of reasons: (1) weak lensing mass estimates willbe biased if miscentering is significant and not accounted for in modeling the lensing profile; (2) clusters thatare poorly centered may represent a population of newly forming clusters, some of which might be undergoingmergers, and quantifying their presence and characteristics could be a useful probe of cluster and large-scalestructure evolution. Future research should investigate the nature of the offset clusters in the 3D-MF catalog,to discover whether their centers are merely misidentified, or whether interesting cluster morphology exists.Additionally, alternative center definitions should be explored for the 3D-MF clusters. Side-by-side shear profilecomparisons could demonstrate better center finding for some or all of the clusters, over the original definitionemployed by the 3D-MF algorithm.More broadly, all weak lensing cluster studies need to consider the effects of miscentering. Justificationshould be given if the distribution of offsets is not accounted for in a weak lensing shear analysis. Studies thatpurport to measure cluster concentration should be careful to rule out miscentering, the effect of which mimicsa low-concentration dark matter halo, by reducing the amplitude of the shear profile at small radii.84Future magnification studies will need to carefully address the serious systemic effect of the contaminationof background sources with objects that overlap with the low-redshift lens population. This can be approachedeither by improving redshift estimates for sources, so that a pure background sample can be obtained, or bysimply avoiding regions of redshift overlap. Additional possible sources of systematics need to be quantified,and some work is being done in this regard by Hildebrandt et al. (private communication). Possible issues mayinclude variations across the images in depth, atmospheric seeing, and contamination by the light halos of stars.Within the next decade several important large astronomical surveys will begin collecting data. The LargeSynoptic Survey Telescope (LSST) is an 8.4-m telescope, currently under construction in Chile. Starting around2021, LSST will spend 10 years surveying 30,000 deg2 of sky (nearly 200 times the area of CFHTLenS) in6 optical to infrared wavelength bands, providing unprecedented wide field data for weak lensing studies andmany other astronomical pursuits (LSST Science Collaboration et al., 2009). Euclid is a European Space Agency1.2-m space-based telescope, planned for launch in 2020, with the goal of improving dark energy constraints. Itwill spend 6 years obtaining deep imaging of 15,000 deg2 of sky, and weak gravitational lensing is one of thecentral focuses (Laureijs et al., 2011). By performing tomographic weak lensing (redshift-binned weak lensing),the growth of structure and the influence of dark energy on the expansion will be measured. The Wide-FieldInfraRed Survey Telescope (WFIRST) is a NASA 2.4-m space-based telescope, planned for launch by 2024.Among its other scientific goals, WFIRST will perform a weak lensing survey of 2,200 deg2, employing sixfilters in the near-infrared wavelength range (Spergel et al., 2015).Weak gravitational lensing is a common central focus of all major upcoming missions and surveys, as it hasbecome an indispensable tool for probing the dark matter, geometry, and growth of structure in our Universe.Key questions that remain prominent in cosmology include the nature of dark matter and dark energy, andthe complicated connection between baryonic physics and this dark sector. Gravitational lensing provides ameans to accurately mapping the dark matter structures and distribution, and will continue to yield constraintson the dark matter particle self-interaction cross-section through the study of merging clusters (Dawson, 2013;Wittman, 2013). By measuring the redshift dependence of the cross-correlations of matter in the universe, thenature of dark energy – as a cosmological constant or a more complex phenomena – will be elucidated. Theunique ability of gravitational lensing to measure total mass (including dark matter) is immensely important forimproving models that seek to connect the state of visible matter – various types of galaxies and the ICM – withenvironment and evolutionary history.Magnification, despite some limitations discussed in this thesis, offers great benefits and additional opportu-nities for these surveys. Especially for ground-based surveys like LSST, which will be affected by atmosphericseeing, the ability to use unresolved sources increases the number of gravitationally lensed galaxies that can beincluded in an analysis. A few simple calculations demonstrate the enormous improvements in signal-to-noisethat will be possible for magnification with LSST. Assuming that signal-to-noise scales roughly with the squareroot of the number of sources, which depends on survey area and limiting magnitude, we can project that thesignal-to-noise for magnification with LSST will be more than 100 times larger than for CFHTLenS.11This estimate assumes that the source flux at limiting magnitude is inversely proportional to the square root of observing time, andtime is proportional to number of sources detected, as in Equation 20 of Chang et al. (2013). Additionally, LSST will be about 100 timeslarger area than CFHTLenS, which will increase the number of sources by roughly that factor as well:(S/N)LSST(S/N)CFHTLenS∼√18,000 deg2171 deg2×10−0.4(mCFHTLenS−mLSST) ∼ 135.The first factor is the ratio of unmasked areas, the second gives the contribution from the relative limiting magnitudes of each survey. Inreality, various factors will complicate this calculation, including telescope seeing, accuracy of galaxy redshifts, purity of the lens andsource selection, among other observational and technical challenges associated with a ground-breaking new telescope like LSST.85Importantly, for any of these surveys, the opportunity to perform magnification studies at all is a free sourceof additional lensing information. Regardless of whether the final deliberation for these surveys is to improvecontamination modeling, to use only unaffected redshift regimes, or even to focus on alternative measures ofmagnification using sizes or redshift distributions, magnification information can and will be exploited, becauseit is free information. The work contained in this thesis has played an important role in laying the foundation forfuture studies that will maximize the use of weak gravitational lensing survey data.86BibliographyG. O. Abell. The Distribution of Rich Clusters of Galaxies. ApJS, 3:211, May 1958. doi:10.1086/190036. →pages 18C. Adami, F. Durret, C. Benoist, J. Coupon, A. Mazure, B. Meneux, O. Ilbert, J. Blaizot, S. Arnouts, A. Cappi,B. Garilli, L. Guennou, V. Lebrun, O. Lefe`vre, S. Maurogordato, H. J. McCracken, Y. Mellier, E. Slezak,L. Tresse, and M. P. Ulmer. Galaxy structure searches by photometric redshifts in the CFHTLS. 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