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A new approach to photometric redshift contamination, providing critical insight for weak lensing cosmology Benjamin, Jonathan Remby Embro 2013

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A new approach to photometric redshift contamination, providing criticalinsight for weak lensing cosmology.byJonathan Remby Embro BenjaminB.Sc Physics, Bishop?s University, 2004M.Sc Physics, University of British Columbia, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Astronomy)The University of British Columbia(Vancouver)August 2013? Jonathan Remby Embro Benjamin, 2013AbstractLight travelling through the Universe is deflected by the presence of mass, this effect will distort the sizeand shape of observed galaxies. Weak gravitational lensing measures the amount of mass in the Universeby observing these subtle changes in the shapes of distant galaxies. In order to properly interpret theobserved shapes of galaxies their distances must be accurately known, this information is encoded inthe redshift distribution.A detailed spectroscopic observation is the most reliable way to measure the redshift of a galaxy.Unfortunately this is a time-intensive process and weak lensing surveys are composed of millions ofgalaxies many of which are too faint for spectroscopic observation. For this reason photometric redshiftsare used. Photometric redshifts are less accurate than spectroscopic redshifts but are easier to obtainsince they rely on only a few measurements over large ranges of wavelength.Thorough knowledge of uncertainties in the photometric redshifts is vital to weak lensing becausephotometric redshifts provide the distances necessary to understand the weak lensing signal. In thisthesis we present a new technique to measure the reliability of photometric redshifts with the goal of im-proving the estimated redshift distribution for use in weak lensing studies. Mock observational surveysare used to test the technique before applying it to two surveys: the Deep component of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS-Deep) and the CFHT Lensing Survey (CFHTLenS).We demonstrate our ability to construct both the true redshift distribution and the true average redshiftof galaxies in a given photometric redshift range. Furthermore, we show that the photometric redshiftprobability distribution function can be used as an accurate measure of the true redshift distributionwhen summed for an ensemble of galaxies.Using our tested redshift distribution we present cosmological constraints for CFHTLenS from aweak lensing analysis. We present constraints on cosmological parameters for a model of the Universewith dark energy and cold (non-interacting) dark matter (?CDM). We find that our weak lensing analy-sis, combined with other cosmological probes, improves the precision of these measurements by a factorof 1.5 to 2.iiPrefaceA version of Chapter 2 has been published: J. Benjamin, L. Van Waerbeke, B. Me?nard, and M. Kil-binger, Photometric redshifts: estimating their contamination and distribution using clustering infor-mation, 2010, MNRAS1, Volume 408, Issue 2, pp.1168-1180. The author of this thesis was the primaryinvestigator and was solely responsible for authoring the paper. The collaborators provided valuablediscussions in the formulation and development of the research and in critiquing the written work. Thiswork also resulted in the author contributing a related analysis in another published work: T. Erben, H.Hildebrandt, M. Lerchster, P. Hudelot, J. Benjamin, L. Van Waerbeke, T. Schrabback, F. Brimioulle, O.Cordes, J. P. Dietrich, K. Holhjem, M. Schirmer, P. Schneider, CARS: the CFHTLS-Archive-ResearchSurvey. I. Five-band multi-colour data from 37 sq. deg. CFHTLS-wide observations, Astronomy andAstrophysics, Volume 493, Issue 3, 2009, pp.1197-1222. In this case the thesis author was responsiblefor the analysis and writing contained in Section 4.1.1. with the exception of Figure 10 which wasproduced by Hendrik Hildebrandt.A version of Chapter 3 has been published: J. Benjamin, L. Van Waerbeke, C. Heymans, M. Kil-binger, T. Erben, H. Hildebrandt, H. Hoekstra, T. D. Kitching, Y. Mellier, L. Miller, B. Rowe, T. Schrab-back, F. Simpson, J. Coupon, L. Fu, J. Harnois-De?raps, M. J. Hudson, K. Kuijken, E. Semboloni, S.Vafaei, M. Velander, CFHTLenS tomographic weak lensing: Quantifying accurate redshift distribu-tions, 2013, MNRAS, Volume 431, Issue 2, pp.1547-1564. The author of this thesis was the primaryinvestigator and was solely responsible for authoring the paper with the exception of Section 3.4 whichwas first drafted by Catherine Heymans. All collaborators provided comments on the manuscript, andparticipated in discussions concerning the direction of the research.1Monthly Notices of the Royal Astronomical SocietyiiiTable of contentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Proper and comoving distances . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Cosmological parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Weak gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Weak lensing by galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Weak lensing by large scale structure . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Cosmological dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.4 Intrinsic alignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Photometric redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Photometric redshifts: estimating their contamination and distribution using clusteringinformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Angular correlation function and estimators . . . . . . . . . . . . . . . . . . . . . . . 222.3 Analytic development of the contamination model . . . . . . . . . . . . . . . . . . . . 232.3.1 Multi-bin analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Pairwise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25iv2.4 Application to a simulated galaxy survey . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Null test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.2 Artificial contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.3 Effect of galaxy density and redshift bin width . . . . . . . . . . . . . . . . . 312.4.4 Global pairwise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Application to a real galaxy survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.1 Applying the global pairwise analysis . . . . . . . . . . . . . . . . . . . . . . 382.6 Covariance and likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.7 Solving the three-bin case analytically . . . . . . . . . . . . . . . . . . . . . . . . . . 442.8 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 CFHTLenS tomographic weak lensing: quantifying accurate redshift distributions . . . 483.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Galaxy redshift distributions determined from the photometric redshift PDF . . . . . . 523.2.1 Comparison with spectroscopic redshifts . . . . . . . . . . . . . . . . . . . . 533.2.2 Comparison with COSMOS photometric redshifts . . . . . . . . . . . . . . . 543.2.3 Redshift contamination from angular correlation functions . . . . . . . . . . . 563.3 Weak lensing tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.1 Overview of tomographic weak lensing theory . . . . . . . . . . . . . . . . . 623.3.2 The tomographic weak lensing signal . . . . . . . . . . . . . . . . . . . . . . 643.3.3 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4 Impact of non-linear effects and baryons on the tomographic cosmological constraints . 753.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 Conclusions and prospects for future research . . . . . . . . . . . . . . . . . . . . . . . 814.1 Photometric redshift contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Tomographic weak lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.1 A brief look at Planck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2.2 Future observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86vList of tablesTable 3.1 Details of the model dependent cosmological parameters for each of the consideredcosmologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Table 3.2 Constraints on the combination of parameters ?8(?m0.27)?. . . . . . . . . . . . . . . 68Table 3.3 Constraints on cosmological parameters. . . . . . . . . . . . . . . . . . . . . . . . 69Table 4.1 Comparison of constraints on ?8 from CFHTLenS, Planck, ACT, and SPT. . . . . . 85viList of figuresFigure 1.1 Sketch of lensing geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 1.2 Effect of lensing on the size and shape of galaxies. . . . . . . . . . . . . . . . . . 8Figure 1.3 The shear decomposed into the tangential ?t and cross-component ?x. . . . . . . . 9Figure 1.4 The effect of cosmology and non-linear modelling on the mass density power spec-trum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 1.5 The effect of cosmology and non-linear modelling on the convergence power spec-trum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 1.6 Spectral energy distribution for different galaxy types and broadband transmissionfilters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 1.7 Comparison of photometric and spectroscopic redshifts for CFHTLenS. . . . . . . 18Figure 1.8 Several photometric redshift probability distribution functions. . . . . . . . . . . . 19Figure 2.1 Upper bounds on the contamination fraction, f , as a function of the number ofredshift bins n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.2 Two redshift bin tests of the contamination method with the Millennium Simulation. 29Figure 2.3 Test for biases of the angular cross-correlation function in the Millennium Simulation. 30Figure 2.4 The effect of redshift bin width on constraints of the contamination fractions. . . . 32Figure 2.5 Angular correlation functions and the corresponding constraints on the contamina-tion fractions for 4 redshift bins of the Millennium Simulation. . . . . . . . . . . . 33Figure 2.6 The true number of galaxies within each photometric redshift bin using the Millen-nium Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.7 The true average redshift within a photometric redshift bin using the MillenniumSimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.8 Finely binned redshift distribution for each of the four deep fields. . . . . . . . . . 39Figure 2.9 Angular correlation functions and the corresponding constraints on the contamina-tion fractions for 4 redshift bins of CFHTLS-Deep. . . . . . . . . . . . . . . . . . 39Figure 2.10 The true number of galaxies within each photometric redshift bin estimated fromCFHTLS-Deep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 2.11 The true average redshift within a photometric redshift bin estimated fromCFHTLS-Deep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43viiFigure 2.12 Covariance matrix for the angular cross-correlation function. . . . . . . . . . . . . 44Figure 3.1 Details of the CFHTLenS fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.2 Comparison of the predicted redshift distribution from the summed PDFs and spec-troscopic redshifts for i? < 23.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.3 Comparison of the predicted redshift distribution from the summed PDFs and re-sampled COSMOS30 redshifts for i? < 24.7. . . . . . . . . . . . . . . . . . . . . 54Figure 3.4 Comparison of the predicted redshift distribution from the contamination method,summed PDFs, and spectroscopic redshifts for i? < 23.0. . . . . . . . . . . . . . . 59Figure 3.5 Comparison of the predicted redshift distribution from the contamination method,summed PDFs, and resampled COSMOS30 redshifts for i? < 24.7. . . . . . . . . . 60Figure 3.6 Redshift distributions used in the weak lensing analysis. . . . . . . . . . . . . . . 64Figure 3.7 The shear correlation functions measured from the CFHTLenS data. . . . . . . . . 65Figure 3.8 Marginalised parameter constraints in the ?m??8 plane for a flat ?CDM model. . 70Figure 3.9 Comparison of constraints in the ?m??8 plane for 2D lensing and 2-bin tomography. 72Figure 3.10 Parameter constraints for each combination of redshift bins. Demonstrating that themeasured shear scales with redshift according to ?CDM. . . . . . . . . . . . . . . 73Figure 3.11 Marginalised parameter constraints in the ?m??8 and ?m??? planes for a curved?CDM cosmology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.12 Marginalised constraints on the deceleration parameter. . . . . . . . . . . . . . . . 76Figure 3.13 The effect of removing highly non-linear scales on the ?m??8 parameter constraints. 77Figure 4.1 Constraints in the ?m-?8 plane from CFHTLenS and Planck. . . . . . . . . . . . . 84viiiGlossaryACFAngular correlation function. The two-point correlation function of galaxies on thesky. The ACF is positive if the amount of clustering is in excess of what is expectedfrom a random distribution of points.BAOBaryon Acoustic Oscillations. In the early Universe baryons and photos were cou-pled. When they decoupled a characteristic scale was imprinted on the baryonicdensity fluctuations. BAO studies attempt to recover this information by looking atthe distribution of galaxies.BOSSBaryon Oscillation Spectroscopic Survey. A survey designed to study BAOs anddescribed in Anderson et al. (2012).BPZBayesian Photometric redshift. A publicly available code used to measure photo-metric redshifts. BPZ uses Bayesian statistical methods to estimate a posterior PDFfor each galaxy?s photometric redshift.CARSCFHTLS Archive Research Survey. A data-mining survey carried out byErben et al. (2009).CCDCharge Coupled Device. A silicone chip technology capable of electronic imaging.CCDs and related technologies make digital cameras possible.CDMCold dark matter. There is evidence that dark matter has an extremely small cross-section for self-interaction. Such fluids are referred to as ?cold?. Most cosmologicalmodels assume that dark matter is cold.CFHTCanada-France-Hawaii Telescope. CFHT is an 8 meter class telescope with a 1square degree camera known as MEGACAM. It uses 5 optical filters u?, g?, r?, i?,and z?.CFHTLenSCFHT Lensing Survey. Applies data-reduction and data-processing techniques tai-lored to the study of weak lensing to the CFHTLS-Wide data.ixCFHTLSCFHT Legacy Survey. A large survey conducted by CFHT. Composed of a deepcomponent CFHTLS-Deep and a wide component CFHTLS-Wide.CFHTLS-DeepThe deep component of the CFHTLS survey. CFHTLS-Deep is composed of 4fields each subtending 1 square degree on the sky.CFHTLS-WideThe wide component of the CFHTLS survey. CFHTLS-Wide is composed of 4fields with varying sky coverage. The total sky area observed is 179 square degrees.CMBCosmic Microwave Background. Primordial photons which have been free-streaming since they decoupled from the matter in the early Universe. As suchthey are a direct probe of the early Universe.COSMOSCosmic Evolution Survey. COSMOS was observed with the Hubble space tele-scope?s Advanced Camera for Surveys. COSMOS is a deep survey covering 1.4square degrees on the sky.EuclidEuclid is a planned space-based mission which plans to survey 15,000 square de-grees with optical and infrared photometric bands from 550nm to 200nm.FLRWThe Friedmann-Lema??tre-Robertson-Walker metric can be derived from assump-tions of homogeneity and isotropy. It relates the interval between events in space-time (ds) to intervals in one or both of time (dt) and 3-dimensional space (d?)HSTHubble Space Telescope. The HST distance ladder is a suit of observations meantto establish cosmic distances in order to measure the Hubble constant with highprecision (Riess et al., 2011).?CDMThe standard model of cosmology which requires a cosmological constant ? (iden-tified as dark energy) and cold dark matter.LSSLarge Scale Structure. The arrangement of matter on the large scales in the Uni-verse. N-body simulations such as the Millennium Simulation predict that darkmatter is arranged in filamentary structures. The Universe is homogeneous only onvery large scales ? 10 Mpc.LSSTLarge Synoptic Survey Telescope. A planned ground based 8 meter class telescope.It will image 30,000 square degrees in 6 photometric bands from 320nm to 1050nm.xMSMillennium Simulation. A large N-body simulation starting from small densityperturbations and evolving them forward in time simulating the formation of largescale structure (LSS) in the Universe.PDFProbability Distribution Function. In this thesis the photometric redshift PDFs de-scribe the probability that a galaxy has a given redshift.PhotometricredshiftA redshift estimate from broadband photometric observations. Observed broad-band magnitudes are compared to template spectral energy distributions convolvedwith the transmission curves of the braodband filters. Less accurate than spectro-scopic redshifts but they require less observing time making them essential for largesurveys.PlanckAn all-sky survey to measure the cosmic microwave background over the wholesky. It significantly improves the resolution of the earlier survey conducted byWMAP.R11Constraints on the Hubble constant from the HST distance ladder as presented inRiess et al. (2011).SEDSpectral Energy Distribution. The energy distribution as a function of wavelength(or frequency) of light emitted from a galaxy. If individual emission or absorptionfeatures can be identified the redshift of the galaxy can be determined.SNAPSuperNova Acceleration Probe. A planned space telescope optimized to detect Su-pernovae and conduct a deep wide field survey ideal for weak gravitational lensing.The survey component will map ?10,000 square degrees in 6 photometric bandsfrom 350nm to 1700nm.SpectroscopicredshiftVery precise redshift measurements that require long observing times. Individualemission or absorption features are identified and their shift in wavelength is at-tributed to cosmological redshift.VVDSVIRMOS VLT Deep Survey. VIRMOS is the VIsible imaging Multi-Object Spec-trograph allowing up to 600 simultaneous spectra to be obtained. VLT is the VeryLarge Telescope located at the European Southern Observatory. VVDS is a deepimaging and redshift survey of more than 150,000 redshifts covering 16 squaredegrees.Weak gravita-tional lensingThe distortion of light from background galaxies by intervening mass. The dis-tortions have an isotropic term ? , the convergence, which changes the observedsize of the galaxy, and an anisotropic term ? , the shear, which induces an observedellipticity.xiWMAPWilkinson Microwave Anisotropy Probe. WMAP is a space mission to measurethe CMB temperature anisotropies over the entire sky. Data obtained after sevenyears of observation (and the measured cosmological parameters) are referred to asWMAP7.Symbols symbolsa(t) Scale factor. Defined to be unity today t = 0 and a(t)< 1 in the past.c Speed of light in a vacuum.?Comoving distance. Due to the expansion of the Universe proper distance is afunction of time. The Comoving distance is defined such that it is constant withtime.fijThe contamination fraction. The number of galaxies which originate in redshift bini but are found to be in redshift bin j divided by the true (uncontaminated) numberof galaxies in redshift bin i.fK(?) Comoving angular diameter distance.? The shear. The anisotropic distortion caused by weak gravitational lensing whichcauses a galaxy to have an induced ellipticity.H0Hubble constant. The slope of the linear relationship between the distance to agalaxy and the rate at which it is moving away from us v = H0r. Typically given inthe units of kms?1 Mpc?1.h The dimensionless Hubble constant h = H0100kms?1 Mpc?1 .?The convergence. Describes the isotropic distortion caused by weak gravitationallensing which changes the observed size of a galaxy.ns Slope of the primordial mass density power spectrum.?b Energy density of baryons.?m Energy density of matter (baryons + dark matter).xii?? Energy density of dark energy.P?The mass density power spectrum. Quantifies on which scales mass over-densitiesexist.P?The convergence power spectrum. Quantifies on which scales weak gravitationallensing correlates the shapes of galaxies.?8 Normalisation of the mass density power spectrum.w0 Constant term in the dark energy equation of state, w(a) = w0.?ij(?)The angular correlation function. This describes the excess probability, comparedto a random distribution, of finding pairs of galaxies between samples i and j at sep-aration ? . If i=j this is the auto-correlation function, if not it is the cross-correlationfunction. In this thesis i and j represent different redshift bins.xiiiChapter 1Introduction1Cosmology is the study of the large scale properties of the Universe, including its origin, evolution, andultimate fate. Cosmology as a physical science is a relatively young field, which began sometime afterAlbert Einstein published the theory of General Relativity in 1917. At that time, the existence of othergalaxies was controversial. The large scale implications of the theory of General Relativity were beyondthe scope of the scientific establishment.The scale of our perceived Universe increased dramatically in the subsequent dozen years. EdwinHubble made two historic observations that challenged our understanding of the cosmos. First, in 1923,Hubble made observations of novae, bright stellar events with well characterised brightness, in theMilky Way?s closest neighbour, the Andromeda galaxy. The distances measured from these observationsestablished that Andromeda was not within the Milky Way, but was an analogous system with trillionsof stars. Hubble?s second historic observation came in 1929, when he demonstrated that galaxies moredistant from the Milky Way were receding at increased speeds.Georges Lema??tre had already published a theoretical study in 1927 that used the equations of Gen-eral Relativity to describe an expanding fabric of space-time, which could explain Hubble?s observa-tions. The equations of General Relativity depend on the amount of mass and other forms of energy inthe Universe. Large amounts of mass would mean that the expansion of space would eventually stop andreverse. With small amounts of mass, the expansion of space would continue indefinitely. The contentof the Universe is key to understanding which of these scenarios will come to pass.After conducting detailed observations of Milky Way stars, Jan Oort concluded in 1932 that thevisible matter in our Galaxy could not account for the orbital motions observed. There was a significantundetected mass component. Dubbed dark matter, this form of matter does not emit or absorb light,making it impossible to detect except by the gravitational force it exerts on visible matter. The influenceof dark matter in galaxy clusters was observed by Fritz Zwicky in 1933, suggesting that dark matter wasthe dominant gravitational mass in all locations and over all scales in the Universe. Modern observationsestimate that about 85 per cent of the mass in the Universe is in the form of dark matter (for e.g.,Dodelson, 2003). Therefore, the amount of dark matter in the Universe is critical in cosmology.1Versions of Sections 1.2 and 1.3 (excluding 1.3.4) of this chapter have been published. J. Benjamin. Cosmologicalconstraints from the 100 square degree weak lensing survey, 2007, M.Sc. Thesis, University of British Columbia, Canada1The final pillar of modern cosmology became well established in the late 1990s through the obser-vations of distant supernovae by two groups: Riess et al. (1998) and Perlmutter et al. (1999). It wasdiscovered that the expansion of the Universe is accelerating. The easiest way to explain this in GeneralRelativity is through the cosmological constant, ?, which acts as a repulsive form of energy. This isreferred to as dark energy due to its unintuitive behaviour.Modern observations indicate that dark energy accounts for about 70 per cent of the energy densityof the Universe. Matter accounts for about 30 per cent, and of that 30 per cent only about 4 per centis ordinary matter, and the rest is dark matter (for e.g., Dodelson, 2003). This model of the Universeis called ? Cold Dark Matter (?CDM) and has been extremely successful at explaining observations.However, many mysteries remain. We have yet to determine the true nature of either dark matter or darkenergy, despite our ability to measure their impact on cosmological observations. Cosmologists? contri-bution on this frontier is to increase the precision with which we can test the ?CDM model as well asalternative models. Precise measurements can be used to rule out certain particle physics candidates fordark matter, theoretical models for dark energy, and modifications to General Relativity. Cosmologicalobservations may hold the key to unlocking the mysteries of the so-called dark sector of the Universe.Future survey missions such as the Large Synoptic Survey Telescope (LSST), the SuperNova Accel-eration Probe (SNAP), and Euclid2 are predicted to measure the contributions from dark matter and darkenergy to unprecedented precision. A central observational tool in these missions is weak gravitationallensing, which is distinguished by its ability to detect dark matter and ordinary matter simultaneouslythrough the gravitational distortion of light from distant galaxies. With new levels of precision in obser-vations comes the need for a more precise theoretical understanding. In order to take full advantage ofweak lensing from these future surveys, work must be done to improve our ability to model the weaklensing signal in order to properly interpret these observations. This thesis presents a novel technique toimprove our understanding of the distances to galaxies. This is critical in the modelling of weak lensingand the future of precision cosmology.The following sections cover some fundamental background theory and concepts from cosmology,weak gravitational lensing, and photometric redshifts. These three topics are relevant to the researchpresented in this thesis. In the broadest context, the work presented in this thesis is important becauseit improves our understanding of the content of the Universe. Specifically, this work improves weakgravitational lensing measurements which are critical to understanding both dark matter and dark energy.1.1 CosmologyThis section introduces some essential equations and parameters that will be used in this thesis. Formore comprehensive works on cosmology, the reader may be interested in the following textbooks:Ryden (2003), Dodelson (2003) and Weinberg (2008). The goal here is to introduce key equations andparameters in order to gain some insight into their importance and physical meaning.2See the glossary for more details of these missions.21.1.1 Proper and comoving distancesDistance measures are extremely important to cosmology. The fact that the Universe is expanding meansthat distances must be carefully defined.The Friedmann-Lema??tre-Robertson-Walker (FLRW) metric can be derived from assumptions ofhomogeneity and isotropy. It relates the interval between points (more generally called events) in space-time (ds) to intervals in one or both of time (dt) and 3-dimensional space (d?):ds2 =?c2dt2 +a(t)2d?2, (1.1)where a(t) is the scale factor. The scale factor is defined such that a(tnow) = 1 is today. At earliertimes, the Universe had expanded less and the scale factor was less than unity. In this way, a(t) encodesthe expansion history of the Universe. Most cosmological models have 0 < a(t) ? 1 for t ? tnow. It isconvenient to use spherical coordinates and to write the 3-dimensional term with an explicit dependenceon the curvature of space:d?2 = d?2 + fK(?)2(d? 2 + sin2?d? 2), (1.2)where ? is the radial comoving distance (which will be defined in a moment), ? and ? are the polar andazimuthal angles respectively, and fK(?) is a piecewise defined factor that depends on the curvature ofspace (K):fK(?) =?????????sin(??K)?K for K > 0? for K = 0sinh(??|K|)?|K|for K < 0.(1.3)The curvature is given by K > 0 for an open universe, K < 0 for a closed universe, and K = 0 for afamiliar flat Euclidean space. The 3-dimensional distance between two points at a given time dt = 0 (i.e.the distance one would measure if one could lay out measuring sticks between two locations) is calledthe proper distance, and is given by dp = a(t)?d?. If we consider the distance from Earth to somedistant object, the proper distance has only a radial component: dp = a(t)? . The comoving distance isdefined as dp/a(t); this distance is constant for all values of the scale factor (i.e. it does not change withtime). In the purely radial case, the comoving distance is ? .Consider a yardstick of known proper length L, which subtends an angle of ?? . The distance tosuch a yardstick can be deduced from the angle it subtends. For small angles (?? << 1), the angulardiameter distance is defined by:dA ?L?? . (1.4)If we label the two ends of the yardstick ?1 and ?2, then ?? = |?2??1|. If we use the FLRW metric,the interval between the two ends of the yardstick isds = a(t) fK(?)?? . (1.5)Identifying ds as the length of the yardstick L gives us an expression for the proper angular diameter3distance,dA = a(t) fK(?). (1.6)In analogy to the proper and comoving distances, the term fK(?) can be understood as the comoving an-gular diameter distance. For flat space, the comoving angular diameter distance is equal to the comovingdistance ? , and the proper angular diameter distance is equal to the proper distance a(t)? .1.1.2 Cosmological parametersIn 1929, Edwin Hubble observed that galaxies are moving away from Earth (and the Milky Way) withradial velocities (v), which are linearly dependent on their proper distance, r, from Earth,v = H0r, (1.7)where H0 is Hubble?s constant today. Using a typical value from contemporary literature and conven-tional units,H0 = 70kms?1Mpc?1, (1.8)where Mpc represents a megaparsec, which is equal to ? 3.09?1019km. For convenience, the dimen-sionless Hubble constant is defined to beh ? H0100kms?1Mpc?1 . (1.9)More generally, it is possible to relate a time dependent Hubble parameter to the scale factor and itstime derivative (a?),H(t) = a?(t)a(t) . (1.10)The evolution of the Hubble parameter is governed by the Friedmann equation. Conceptually, thisequation establishes how the Universe will expand (or contract) as a function of time given the energydensity of its constituents. The Friedmann equation can be written as:H(t)2 = H20[?ma(t)3 +?? +1??0a(t)2], (1.11)where the current energy density of matter is given by ?m, the current energy density of dark energy isgiven by ??, and the current total energy density is ?0 = ?m +??. For a flat Universe, ?0 = 1, theopen and closed cases are described by ?0 > 1 and ?0 < 1 respectively. The ? parameters describe theenergy density of each component as a fraction of the current critical energy density. The critical energydensity is the average energy density of the Universe required for spatial flatness. Taking the energydensity of matter as an example,?m =Em,0Ecrit,0= Em,08piG3c2H20, (1.12)where G is Newton?s gravitational constant and c is the speed of light.4Cosmological probes such as weak gravitational lensing allow us to measure these cosmologicalparameters. The values of these parameters are integral to understanding the past, present, and futureevolution of the Universe. The subsequent sections will show how weak gravitational lensing studiesare capable of putting constraints on cosmology.1.2 Gravitational lensingGravitational lensing ? the deflection of a light bundle?s trajectory by the presence of mass ? is oneof the most fundamental results of general relativity. The phenomenon was experimentally verified in1919 during a solar eclipse in which background stars were seen to have shifted with respect to theirusual positions due to the gravity of the Sun. This experiment, carried out by Sir Arthur Eddington(Eddington, 1920), served as a resounding affirmation of Einstein?s theory of general relativity.The deflection of star light due to the Sun is very small, owing to the small mass of the Sun andthe relative distances of the observer-Sun-star system. The strength of a gravitational lens ? its abilityto alter the light coming from background sources ? depends on the mass of the lens and the ratio ofdistances,? ?DolDlsDos, (1.13)where ? is the angular deflection of the light, Dol is the distance between the observer and the lens, Dosis the distance between the observer and the source, and Dls is the distance between the lens and thesource. Thus the effect is quite small for the Sun lensing background stars, since Dos ? Dls and Dol isrelatively small.Little work was done on the theory of gravitational lensing at that time. Chwolson (1924) publisheda work wherein he considered a perfectly co-aligned source and foreground mass, concluding that theresulting image would be a ring around the lensing mass. Einstein (1936) wrote a paper considering thelensing effects by stars. He derived equations for image locations, separation, and magnification, andconcluded that the separation of the images would be far too small (on the order of milliarcseconds)to be resolved. Inspired by Einstein, Zwicky published two papers in 1937 (Zwicky 1937a, 1937b),which focused on the potential lensing signal from extragalactic nebulae (i.e. galaxies). He concludedthat the separation of images would be about 10 arcseconds and hence easily resolved. In addition,the magnification of the source would allow the observation of distant galaxies. He calculated theprobability of a distant source galaxy being lensed, concluding that about 1 in 400 would be affected,practically guaranteeing the existence of gravitationally lensed galaxies.The field lay dormant until the 1960s, when several works (Klimov, 1963; Liebes, 1964; Refsdal,1964a,b) extended the theory of gravitational lensing and outlined its usefulness to astronomy. Whenthe first quasars were detected in 1963, it was realised that they were far more distant than most galaxiesand therefore could be lensed by the galaxies. However, it was not until 1979 that the first multiplyimaged quasar was detected by Walsh et al. (1979) with a redshift of z?1.4.Aided by the recent development of the Charge Coupled Device (CCD), gravitational lensing sawmuch advancement during the 1980s. Of particular interest here is the first detection of giant luminous5arcs in galaxy clusters (Lynds & Petrosian, 1986; Soucail et al., 1987). These faint arcs are stretchedtangentially with respect to the cluster center, extending about 10 times further in this direction thanin the radial direction. These arcs were later identified as highly distorted and magnified images ofbackground galaxies (Mellier et al., 1991; Paczynski, 1987), whose light was being strongly lensed bythe gravitational field of the galaxy cluster. Less distorted arcs than these were named arclets (Fort et al.,1988), and are visible in many galaxy clusters.Gravitational lensing has become a large field with many branches of research, including strong lens-ing, micro lensing, galaxy-galaxy lensing, and weak lensing. This thesis is concerned only with the lastapplication; for more detailed reviews of gravitational lensing, the reader is referred to Schneider et al.(1992) and Petters et al. (2001).1.3 Weak gravitational lensing1.3.1 Weak lensing by galaxy clustersUnlike giant luminous arcs, or the less pronounced arclets, galaxies affected by weak gravitationallensing do not exhibit an immediately identifiable distortion. That is, the distortion of their ellipticityis not easily distinguishable from the intrinsic ellipticity. However, if the distortions vary slowly withposition then nearby galaxies will be distorted in a similar way. Assuming that there is no correlationof the intrinsic ellipticities, a local ensemble average of galaxy ellipticities will provide a measure ofthe distortion due to weak lensing. This signal was first detected in 1990 around two galaxy clusters(Tyson et al., 1990). The affect of intrinsic alignments of galaxy ellipticities is discussed below inSection 1.3.4.One of the most appealing aspects of weak lensing is its ability to measure mass in an unbiased way;it is not sensitive to the form of matter (baryonic or dark matter) or its state. Strong lensing can alsobe used to measure the mass of clusters, provided that there are enough giant luminous arcs present.However, weak lensing can probe the matter distribution to much larger radii and can be applied toclusters that do not have a strong lensing signal. The theoretical edifice of weak lensing by clusters wasdetailed by Kaiser & Squires (1993), wherein it was shown that the measurement of galaxy distortionscan be used to construct a parameter-free map of the two-dimensional projected mass distribution. Thefirst mass reconstruction of a cluster was carried out a year later by Fahlman et al. (1994).For a review of weak lensing by galaxy clusters, the reader is referred to Fort & Mellier (1994).1.3.2 Weak lensing by large scale structureArguably the most remarkable application of weak lensing is the measurement of mass on cosmic scales.Light travelling through the inhomogeneous large scale structure (LSS) of the Universe is deflected,causing shape distortions of background galaxies, aptly named ?cosmic shear?. A statistical treatmentof cosmic shear reveals details of the matter distribution of the Universe, which can be compared withreliable theoretical models of structure growth. Hence, it can be used to constrain cosmology.6Figure 1.1: Sketch of the lensing geometry, the dashed line is the optical axis defined by the lineconnecting the observer and the center of the lensing mass. The circular source in the sourceplane has a position described by ?, its lensed image in the lens plane has a position describedby ?. (Source: Bartelmann & Schneider (2001))The first mention of light deflection by LSS is often credited to Gunn (1967). However, likemany great ideas it can be traced back to Richard Feynman, specifically a lecture given by him atCaltech in 19643. The theory of light propagation in an inhomogeneous universe and the devel-opment of weak lensing as a statistical treatment of galaxy distortions were explored by severaltheorists (e.g., Babul & Lee, 1991; Blandford et al., 1991; Jaroszynski et al., 1990; Kaiser, 1992;Kristian & Sachs, 1966; Lee & Paczynski, 1990; Schneider & Weiss, 1988; Villumsen, 1996). The-oretical studies concerned with the measurement of cosmological parameters via weak lensing weresoon to follow (Bernardeau et al., 1997; Hu & Tegmark, 1999; Jain & Seljak, 1997; Kaiser, 1998;Kamionkowski et al., 1998; Van Waerbeke et al., 1999). Finally, after decades of theoretical study,weak lensing by LSS was observationally detected by several groups (Bacon et al., 2000; Kaiser et al.,2000; Van Waerbeke et al., 2000; Wittman et al., 2000). The following section will briefly introduce theconcepts of weak lensing. For more detailed reviews, the reader is referred to Bartelmann & Schneider(2001), Van Waerbeke & Mellier (2003), and Schneider (2005).1.3.3 Cosmological dependenceA light bundle observed at a position ? has been deflected by the LSS of the Universe and has a positionin the source plane given by ? (see Figure 1.1). The distortion of images can then be described by the3Refregier (2003), who cites a personal communication with J.E. Gunn7Figure 1.2: Schematic of the first order effects weak lensing has on a circular background galaxyof radius R0. The convergence is an isotropic distortion that increases the radius, scaling R0by a factor of (1? ?)?1. The shear is an anisotropic distortion, creating a major ?a? andminor ?b? axis which are equal to the new radius (R0) scaled by the factors 1+ ? and 1? ?respectively. (Source: Van Waerbeke & Mellier (2003))Jacobian matrix (or amplification matrix), which describes how changes in position in the source plane(?) are related to changes in the observed position (?):A (?) = ???? =(1?? ? ?1 ??2??2 1?? + ?1), (1.14)where ? is identified as the convergence and ? = ?1 + i?2 = |? |e2i? as the shear. The shape of a galaxy inthe source plane gets distorted through this relationship and is observed to have a different shape. Theshape distortion of galaxies by weak lensing can be quantified by the convergence, which is a uniformscaling of the galaxy image, and the shear, which is an anisotropic distortion. In the weak lensing regimeboth of these quantities are much smaller than unity, hence the amplification matrix is approximatelyequal to the identity matrix, and the image distortions are small.Consider a circular source with radius R0 (see Fig.1.2). In the absence of shear, (? = 0), the imagewill be circular with a modified radius given by R0(1? ?)?1. Shearing will cause the image to beelliptical with |? |= (1? r)(1+ r)?1 , where r = b/a is the ratio of the minor (b) to major (a) axis, andthe direction of the shear is given by its phase ? . Note that in the simplified case of a circular source,the ellipticity ? of the image is a direct measure of the shear:|? |= |? |= 1? r1+ r . (1.15)In general, the intrinsic ellipticity of the source ?s is non-zero, yielding (in the weak lensing limit,? << 1 and ? << 1 )? = ?s + ? . (1.16)Although the intrinsic ellipticity of a given source is unknown, it is safe to assume that the average oversource ellipticities will tend to zero; that is, there is no preferred direction for the intrinsic ellipticities.8Figure 1.3: The shear decomposed into the tangential ?t and cross-component ?x. The shearedimage has an ellipticity |? |= |? |= 0.3. (Source: Schneider (2005))Hence, a sufficiently large sample of background galaxies will provide an accurate measure of the shear4???= ???. This assumption can break down in certain regimes, which is discussed in Section 1.3.4.The ellipticity is the primary observable in weak lensing analyses. Its interpretation as analogous tothe shear will allow us to relate the measured shear correlation functions to the modelled matter densitypower spectrum. The shear ? is typically decomposed into two components: the tangential shear ?t andthe cross component ?x, as shown in Figure 1.3. The shear components are defined relative to the sepa-ration vector for each pair of galaxies, with ?t describing elongation and compression of the ellipticityalong the separation vector and ?x describing elongation and compression along a direction rotated 45?from the separation vector. The sign convention for ?t is such that a galaxy sheared tangentially, as anarc around a central mass, has a positive ?t. A negative ?t describes a radially stretched ellipticity. Theshear components are presented visually in Figure 1.3.Pairs of galaxies are used because it is impossible to determine from individual galaxies whetherlensing has occurred, since the weak lensing shape distortions are relative to the position of the lensingmass. We do not know where the lensing masses are on the sky, but we do know that they will inducecharacteristic shape distortions. Galaxies will tend to align their ellipticity tangentially to the center ofthe lensing mass. By considering the two-point correlation function, the relative alignment of an averagepair of galaxies is quantified and can be compared to expectations from models. In the absence of weaklensing, one would expect there to be no preferential alignment and therefore no correlation of galaxyellipticities except for intrinsic alignment, which is discussed in Section 1.3.4.The two-point correlation functions are defined for each component ??t?t?(?) and ??x?x?(?), whereall pairs separated by an angle ? are averaged. The following combination of these statistics is veryuseful since they can be measured directly from the data. Other second order statistics are often derived4The practical details of accurately measuring the shapes of galaxies from data is beyond the scope of this work. TheCFHTLenS data used in Chapter 3 has been analysed with a new Bayesian technique known as lensfit (Miller et al., 2013).9from these shear-shear correlation functions:??(?) = ??t?t?(?)???x?x?(?)= ?i,j[?t,i(?i)?t,j(?j)? ?x,i(?i)?x,j(?j)]?ijNp, (1.17)where galaxy pairs labelled i and j are separated by angular distance ? = |?i ??j|. Here ?i is a two-dimensional vector given by the angular position of galaxy i. If ? falls in the angular bin given by? , then ?ij=1; otherwise, ?ij=0. The number of pairs summed over is given by Np. These correlationfunctions can be related to the convergence power spectrum, P?(?) (e.g. Kaiser, 1992)?+(?) = 12pi??0d??J0(??)P?(?)??(?) = 12pi??0d??J4(??)P?(?), (1.18)where Jn represents the nth order Bessel function of the first kind, and ? is the modulus of the two di-mensional wave vector. The convergence power spectrum (defined mathematically below) quantifieson which scales weak lensing has the largest signal. In order to use the observable quantity in Equa-tion (1.17) to constrain cosmology, we must find a cosmologically dependent model for the convergencepower spectrum P?(?). This is the goal of the rest of this subsection.Defining mass density as ? and the average mass density of the Universe as ?? , the mass densitycontrast ? is? (?) = ?(?)? ???? . (1.19)The mass density power spectrum P? (k,?) is defined as? ?? (k) ?? ?(k?)?= (2pi)3?D(k?k?)P? (k,?), (1.20)where ?? (k) is the Fourier transform of ? , the asterisk denotes complex conjugation, ?D is the Diracdelta function, ? is the comoving distance, and k is the modulus of the three dimensional wave vector k.The Dirac delta function arises due to the assumption of homogeneity and isotropy of the Universe onlarge scales. The mass density power spectrum describes the scales on which density fluctuations occur.If it is large for a particular value of k, then mass tends to cluster on the corresponding scales. Note thatk is a reciprocal distance measure and large k correspond to small distances. The mass density powerspectrum can be modelled from our understanding of the initial density fluctuations and how they evolveover time. The evolution is analytical in the linear regime of structure formation (where ? << 1), butbecomes highly non-linear at small scales and later times. Methods to account for non-linear structuregrowth have been devised by Peacock & Dodds (1996) and Smith et al. (2003a), following the work ofHamilton et al. (1991). Combining the linear and non-linear models provides a cosmology-dependentdescription of the shape of the power spectrum. The overall normalisation is a free parameter. Thenormalisation parameter is defined as the mass density variance within a sphere of 8 h?1Mpc radius at10zero comoving distance (Peebles, 1993):? 28 = ?? 2R?=12pi3?dkP? (k,? = 0)|W (kR)|2, (1.21)where W (kR) is the Fourier transform of the top-hat window function. In real space, the top-hat windowfunction is constant for |x|< R and zero otherwise; its Fourier transform isW (kR) = 3(kR)2(sin(kR)kR ? cos(kR)). (1.22)Figure 1.4 shows how the mass density power spectrum changes with cosmological parameters and withdifferent prescriptions for the non-linear growth of density perturbations.The convergence power spectrum P?(?) is defined to be:???(?)???(??)?= (2pi)2?D(?? ??)P?(?), (1.23)where ? is the modulus of the two-dimensional wave vector perpendicular to the line of sight ?. We canexpress the convergence power spectrum in terms of a two-dimensional wave vector instead of a three-dimensional wave vector because the series of deflections a light bundle undergoes can be approximatedby a single lensing plane. In the weak lensing regime, the correlation function of the shear in Fourierspace is identical to the correlation function of the convergence:???(?)???(??)?= ???(?)???(??)?. (1.24)This equivalence is why the shear correlation functions in Equation (1.17) can be expressed in terms ofthe convergence power spectrum. The convergence power spectrum can also be written asP?(?) =9H40 ?2m4c4? ?h0d? g2(?)a2(?)P?(k = ?fK(?) ,?), (1.25)where ?h is the comoving distance to the horizon5, a(?) is the scale factor, fK(?) is the comoving angu-lar diameter distance (Equation 1.3), and g(?) is the lensing efficiency which depends on the distributionof sources n(?)g(?) =? ?h?d? ? n(? ?) fK(??? ?)fK(? ?) . (1.26)It is evident from the above relationships that the strength of the weak lensing signal depends oncosmology. The convergence power spectrum can be modelled (Equation 1.25) for different cosmolo-gies. A best fit cosmology can be determined through a standard likelihood analysis, with the likelihoodgiven byL = 1?(2pi)n|C|exp[?12(d?m)C?1(d?m)T], (1.27)5The furthest comoving distance from which light has had enough time to reach us.1110-310-210-110010110210310410510-4 10-3 10-2 10-1 100 101 102P ?(k)k [Mpc-1]?CDMOCDMEdS10-410-310-210-110010110210310410510-4 10-3 10-2 10-1 100 101 102P ?(k)k [Mpc-1]Smith (2003)PD (1996)LinearFigure 1.4: Top: How the mass density power spectrum (Equation 1.20) changes with variousCold Dark Matter (CDM) cosmologies. ?CDM (solid): A flat cosmology (?m +?? = 1)with a non-zero cosmological parameter, ?m = 0.27, and ?? = 0.73. OCDM (dashed): Anopen cosmology (?m +?? < 1) with ?m = 0.27, ?? = 0.0. EdS (dotted): An Einsteinde-Sitter cosmology with ?m = 1.0, ?? = 0.0. For all cases ?8 = 0.78 and the non-linearprescription of Smith et al. (2003a) is used. Bottom: The linear power spectrum is given bythe dotted line, two methods are used to account for the non-linear power, Peacock & Dodds(1996) (dashed) and Smith et al. (2003a) (solid).12where d is the data vector ??(?) measured at n angular scales (Equation 1.17), m is the model vectorfor the same scales calculated from Equations (1.25) and (1.18), and C is the n? n covariance matrix.As seen from Eq.(1.25), weak lensing is most sensitive to a combination of the mass energy density atredshift zero ?m, the redshift distribution of the sources n(?), the Hubble parameter at redshift zeroH0, and the normalisation of the mass density power spectrum ?8. However, the most useful constraintsare those placed on ?m and ?8 since they compliment constraints from other cosmological probes (seeChapter 3). Figure 1.5 shows how the convergence power spectrum changes with cosmology and non-linear modelling of the mass density power spectrum.The above framework considers the gravitational lensing by LSS for one sample of source galaxieswith a redshift distribution given by n(?). In Chapter 3, we perform such an analysis on data fromthe Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS), which is currently the largest weaklensing survey having an effective area on the sky of 154 square degrees. We extend this theoreticalframework to include weak lensing from several redshift bins, known as weak lensing tomography. Themain advantage of a tomographic analysis is to probe cosmological parameters as a function of red-shift, which leads to better constrains on cosmological models. Breaking the sample into redshift binsrequires well-measured redshifts. Unfortunately, spectroscopic redshifts for large and deep surveys arenot practical, requiring the use of photometric redshifts. The dependence of weak lensing on photomet-ric redshifts motivates the work presented in Chapter 2, which aims to better characterise photometricredshift distributions.1.3.4 Intrinsic alignmentsWe saw in Equation (1.16) that the observed ellipticity of a galaxy (?) depends on the source ellipticity(?s) and the shear (?). The two point shear correlation function is measured by averaging over all galaxypairs (i,j) separated by angle ? :?? i? j? = ?(? is + ? i)(? js + ? j)?,= ?? is? js?+ ?? is? j?+ ?? js? i?+ ?? i? j?. (1.28)Most weak lensing analyses can safely assume that the first three terms on the right hand side aresufficiently small to be ignored since they contain the source ellipticity term. Therefore, the two-pointshear correlation function ?? i? j? is equal to the correlation function of the observed galaxy ellipticities?? i? j?. The three terms are known as intrinsic alignments since they all involve a correlation of thesource ellipticity.Early studies investigating the impact of intrinsic alignments focused on the so-called ?II? term re-sulting from a correlation of source ellipticities ?? is? js?. This term can be significantly non-zero if galaxiesevolve in a common gravitational field that correlates their shapes. Therefore, the effect is most pro-nounced between physically nearby galaxies. Numerical studies (Croft & Metzler, 2000; Heavens et al.,2000), analytical studies (Catelan et al., 2001; Crittenden et al., 2001; Lee & Pen, 2001), and low red-shift observational constraints (Brown et al., 2002a; Pen et al., 2000) of II were in close agreement.1310-1510-1410-1310-1210-1110-1010-910-8100 101 102 103 104 105P ?(?)? [Mpc-1]?CDMOCDMEdS10-1610-1510-1410-1310-1210-1110-1010-910-8100 101 102 103 104 105P ?(?)? [Mpc-1]Smith (2003)PD (1996)LinearFigure 1.5: Top: How the convergence power spectrum (Equation 1.23) changes with variousCDM cosmologies. ?CDM (solid): ?m = 0.27, ?? = 0.73. OCDM (dashed): ?m = 0.27,?? = 0.0. EdS (dotted): ?m = 1.0, ?? = 0.0. For all cases, ?8 = 0.78 and the non-linearprescription of Smith et al. (2003a) is used. Bottom: The linear power spectrum is given bythe dotted line, two methods are used to account for the non-linear power, Peacock & Dodds(1996) (dashed) and Smith et al. (2003a) (solid).14Deep weak lensing surveys could expect contamination of the measured shear correlation of at most afew per cent. However, King & Schneider (2003) and Heymans et al. (2004) were quick to point outthat this effect could be much more significant if the galaxy sample was split into narrow redshift bins,which would enhance the relative number of close pairs. Heymans et al. (2004) also advocated for thedown-weighting of close pairs to lessen the impact of the II term.The second and third terms on the right hand side of Equation (1.28) are referred to as ?GI? since theyinvolve the correlation of the intrinsic shape with the gravitational shear ??s??. Hirata & Seljak (2004a)first demonstrated that the GI contribution can be significant. They showed that lensed source galaxiesthat get sheared by a foreground matter over-density may be correlated with foreground galaxies whoseshapes are also correlated with the over-density. This effect cannot be eliminated by removing physicallyclose pairs since the spurious correlation is between galaxies well separated in redshift.Observations of low-redshift galaxies were used to constrain the magnitude of the II and GI terms,by cross-correlating the number density of galaxies (a proxy for the mass density) and the shape ofgalaxies. The first such study probed galaxies at z?0.1 in the Sloan Digital Sky Survey (SDSS) spectro-scopic sample (Mandelbaum et al., 2006). The SDSS Luminous Red Galaxy sample (LRG, Hirata et al.,2007) and the MegaZ-LRG sample (Joachimi et al., 2011) extended this method to z?0.6. These stud-ies discovered a dependence on galaxy type, with II and GI being more significant for red (late-type)galaxies, as opposed to blue (early-type). The atypical selection function of these works meant that theIA terms were not well characterised for a typical weak lensing survey, which contains far fewer LRGs.The results most applicable to deep weak lensing surveys come from Wiggle-Z (Mandelbaum et al.,2011) with a depth of z?0.7. Mandelbaum et al. (2011) report a null detection of the IA signal forblue galaxies but estimate that the error on the normalisation of the matter power spectrum (?8) for aCFHTLS-type survey6 is ?0.03, well below the statistical error.For tomographic analyses, intrinsic alignment contamination from both II and GI is more pro-nounced. This results from the splitting of galaxies into redshift bins, which increases the density ofcontaminating pairs compared to a 2D analysis. In the case of GI, the cross-correlation of distant redshiftbins will enhance the GI signal, whereas these pairs would be diluted by including all redshift bins in a2D analysis. Similarly, the auto-correlation of narrow redshift slices will enhance the II contamination,which would be diluted if a broad range of redshifts were included. In order to perform finely binnedtomographic analyses, these IA contaminations needs to be addressed. King (2005) propose a modelfitting technique to reduce the impact of IA contamination. They model the II and GI terms simultane-ously with the cosmic shear signal and marginalise over the nuisance parameters introduced to modelthe intrinsic alignments. This method was extended by Bridle & King (2007a), who demonstrated thatthe predicted constraining power when marginalising over IAs was degraded with a severity dependingon assumed prior knowledge of the intrinsic alignment signal, the accuracy of the photometric redshifts,and the flexibility of the IA model. This can be mitigated by simultaneously fitting galaxy cluster datawhich can calibrate the IA signal (Joachimi & Bridle, 2010; Zhang, 2010). This approach was firstused by Kirk et al. (2010) combining lensing information from the 100 square degree lensing survey6See Chapter 3, where the fully marginalised error on ?8 for CFHTLenS is ?0.2315(Benjamin et al., 2007) and shear-shape clustering data from SDSS (Mandelbaum et al., 2006).1.4 Photometric redshiftsWeak lensing surveys require knowledge of the radial distribution of galaxies (see equations 1.25 and1.26). The statistical nature of the lensing signal requires large surveys with large galaxy number den-sities, leading to millions of detected galaxies. It is not possible to measure spectroscopic redshifts(zspec) for all galaxies in weak lensing surveys since faint galaxies require long periods of observationto secure reliable spectroscopic redshifts. The alternative is the use of photometric redshifts (zp), whichare far less accurate but are the only tool capable of estimating the redshifts of millions of galaxies in areasonable amount of time.Spectroscopic redshifts are obtained by diffraction of light from a galaxy. This allows the intensityof light to be measured as a function of wavelength, and fine details such as individual absorption andemission lines can be detected. The amount of light energy per unit wavelength is referred to as thespectral energy distribution (SED). Figure 1.6 shows examples of SEDs for several types of galaxies.If individual features in an SED can be identified, then the redshift can be determined by the observedwavelength (?o) of these features,z = ?o ??e?e, (1.29)where ?e is the wavelength of emission for this feature. This is nothing more than the familiar DopplerEffect, which causes the observed wavelength of light to increase if the source emitting that light ismoving away from the observer.7Measuring the SEDs of galaxies is time consuming since the energy in small intervals of wavelengthis measured. This becomes worse for faint galaxies. For this reason, large surveys measure the flux ofgalaxies in large intervals of wavelength. These intervals are called broadband filters or photometricbands. A common set of filters and the one used in Chapters 2 and 3 is presented in the bottom rightpanel of Figure 1.6. The bands are labelled u?, g?, r?, i?, and z?. The transmission curves in Figure 1.6quantify what percentage of light energy will reach the detector for a given filter. The transmissioncurves are combined with the quantum efficiency of the detector to fully describe the fraction of incidentlight that is detected as a function of wavelength. Observing a galaxy in these broadband filters takesfar less time than obtaining a detailed spectra.Photometric redshifts can be estimated by comparing the observed photometry of a galaxy againsta template SED that has been convolved with the filter transmission curves. The template spectrumcan be shifted in wavelength to simulate a higher redshift galaxy. Model fitting techniques are usedto estimate the best-fitting template. The result is an estimate of the redshift of the galaxy. Typically,spectroscopic redshifts are used either as a training set to iteratively improve the model fitting, or as atest sample with which to quantify errors in the derived photometric redshifts. The spectroscopic samplewill usually only cover a small fraction of the entire survey area and have a much brighter magnitudelimit. An example of such a comparison is presented in Figure 1.7. Note that these are the redshifts used7Note that the redshift and scale factor introduced in Section 1.1 are closely related: a(t) = (1+ z)?1 .16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 100  200  300  400  500  600  700  800  900 1000Energy? (nm)Eliptical 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 100  200  300  400  500  600  700  800  900 1000Energy? (nm)Barred Spiral 0 5 10 15 20 25 30 100  200  300  400  500  600  700  800  900 1000Energy? (nm)Star Burst 0 0.1 0.2 0.3 0.4 0.5 0.6 100  200  300  400  500  600  700  800  900 1000Transmission (%)? (nm)u?g?r?i?z?Figure 1.6: Three template spectral energy distributions are shown for different galaxy types.These templates are taken from Ben??tez et al. (2004). Each template shows the energy distri-bution as a function of wavelength (? ) for a galaxy of the indicated type at redshift zero (i.e.the wavelength of emission). The panel in the bottom right shows the transmission curvesof five broad band filters. The convolution of the broad band filters with an SED, which canbe shifted to higher wavelengths to simulate a higher redshift galaxy, yields the observedcolours of the Chapter 3.In this work, we will discuss only the template fitting method (described above) for obtaining pho-tometric redshifts; however, several other methods exist. For an overview of photometric redshift meth-ods and codes used in the literature, the reader is referred to the PHoto-z Accuracy Testing programme(PHAT, Hildebrandt et al., 2010). PHAT was undertaken in order to quantify the performance of themany different photometric redshift methods and codes that are used today. For purposes of this thesis,it is important to remember that regardless of the method used, all photometric redshifts contain errors.The type of scatter seen in Figure 1.7 is generic to all photometric redshift methods.The scatter in Figure 1.7 is due primarily to uncertainties in the observed photometry and subsequentimperfect model fitting. A galaxy may not be well described by the template SEDs, or there may be adegeneracy between two or more template SEDs at different redshifts. The scatter is often described bythe standard deviation in the quantity?z = zphot ? zspec1+ zspec, (1.30)17Figure 1.7: Comparison of the photometric redshifts (zphot) to the spectroscopic redshifts (zspec)for CFHTLenS (Hildebrandt et al., 2012). If the photometric redshifts were perfect, all thepoints would fall on the plotted zphot = zspec line.denoted by ??z. The scatter in Figure 1.7 is noticeably worse at very low redshift zphot . 0.1 and veryhigh redshift zspec & 1. A detailed analysis in Hildebrandt et al. (2012) determines a high confidencephotometric redshift range of 0.1 ? zphot < 1.3 for which the scatter is 0.03 < ??z < 0.06.There are several outliers in Figure 1.7; for example, there is a group of galaxies predicted to beat zphot < 0.5 but with zspec > 1. In Chapter 2, we present a method to measure contamination inthe photometric redshifts well suited to detecting this type of catastrophic error. This is importantto weak lensing analyses since low redshift galaxies are not strongly lensed. If there are high redshiftgalaxies contaminating the low redshift sample, the observed lensing signal will be higher than expected.Therefore, the derived cosmological parameters will be biased. Understanding the scatter betweenphotometric redshifts in order to improve the accuracy of cosmological parameter constraints from weaklensing is one of the central topics of this thesis.In Chapter 3, we investigate the accuracy of the posterior probability distribution function (PDF) ofthe photometric redshifts. For each galaxy, a PDF can be estimated, which quantifies the probabilityof finding the galaxy as a function of redshift (see Figure 1.8). This is achieved by using a Bayesianphotometric redshift technique described by Ben??tez (2000) and applied to the CFHTLenS data set byHildebrandt et al. (2012). We show several representative PDFs in Figure 1.8. The legend displays thebest-fitting redshift, which is taken as the peak of the PDF. Note that some galaxies have significant18Figure 1.8: The posterior probability distribution functions (PDF) for several characteristic galax-ies selected from the CFHTLenS catalogue (Hildebrandt et al., 2012). The label provides thebest-fitting redshift estimate, which is taken to be the peak of the PDF.degeneracies between high and low redshifts. In Chapter 3, we investigate the accuracy of using theentire PDF, showing that this provides a better estimate of the redshift distribution of the galaxies.1.5 Thesis overviewThis thesis is composed of two related research projects. The first, presented in Chapter 2, derives anovel method to measure contamination between photometric redshift bins by employing the angularcorrelation function. The theoretical details of the method are derived and it is tested on both simulateddata and survey data. The importance of accurate photometric redshifts for weak lensing cosmologyis highlighted in Chapter 2 and, in fact, provided the motivation for the research project. The secondproject is presented in Chapter 3, and comprises two scientific objectives:1. Use the method presented in Chapter 2 to analyse the photometric redshifts of the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) and determine if the photometric redshifts? Prob-ability Distribution Functions (PDF) accurately account for redshift contamination.2. Use two redshift bins to perform a tomographic weak lensing analysis of CFHTLenS in order toobtain cosmological parameter constraints and verify that the redshift scaling of the shear signalagrees with expectations.We conclude and briefly discuss future work in Chapter 4.19Chapter 2Photometric redshifts: estimating theircontamination and distribution usingclustering information12.1 IntroductionThe time-intensive nature of spectroscopy and the immense number of galaxies in current and futuresurveys have secured the place of photometric redshifts among the most valuable astronomical tools.Nearly all cosmological probes are sensitive to distance, making redshifts particularly important tocosmology. While spectroscopy is the most accurate and precise way to measure redshift, it is far tootime-intensive to apply to current or future surveys which require redshift estimates for millions ofgalaxies. Hence, photometric redshifts and a thorough understanding of their uncertainties are vital tothe study of large galaxy surveys.Degeneracies in fitting multi-band photometric data can result in significant misidentification caus-ing contamination between all redshifts (see Section 1.4). In this chapter we investigate contaminationbetween photometric redshift bins and present a method for estimating the contamination using the an-gular correlation function. This method relies only on the photometric redshifts; it does not require aspectroscopic sample. However, it is likely that the photometric redshifts will have been calibrated witha spectroscopic training sample.Quantifying photometric redshift contamination is essential to exploit the full potential of futurephotometric surveys, such as the Large Synoptic Survey Telescope (LSST) or the Supernova Accelera-tion Probe (SNAP), which aim to constrain dark energy. Weak lensing and baryon acoustic oscillations(BAO) are both sensitive to the mean redshift of a bin, requiring an unbiased measurement-accuracyon the order of 0.001? 0.005 in redshift so that the constraints on dark energy are not significantly1A version of this chapter has been published. J. Benjamin, Ludovic Van Waerbeke, Brice Me?nard, and Martin Kilbinger.Photometric redshifts: estimating their contamination and distribution using clustering information, 2010, MNRAS, Volume408, Issue 2, pp. 1168-118020degraded (Huterer et al., 2004).Several studies have investigated the use of the angular correlation function in determining the trueredshift distribution of galaxies binned by photometric redshift (Newman, 2008; Schneider et al., 2006;Zhang et al., 2009). These works have adopted LSST-like survey parameters and focused on the Fisherinformation matrix as a means of forecasting constraints. Schneider et al. (2006) show that the angularcorrelation function in the linear regime can be used to measure the mean redshift to an accuracy of 0.01,noting that there is further constraining power at smaller scales. The method investigated by Newman(2008) relies on an overlapping spectroscopic sample so that the angular cross-correlation between thespectroscopic and the photometric redshifts can be exploited. They find that the desired accuracy onthe mean redshift can be reached provided a spectroscopic sample of 25,000 galaxies per unit redshift.Zhang et al. (2009) investigate a self-calibration method, where the mean redshift in a bin is estimatedfrom angular cross-correlations between photometric redshift bins, very similar to the one presented inthis work and Erben et al. (2009). They demonstrate that self-calibration can reach the required accuracyon the mean redshift if additional information from weak lensing shear is used to help break parameterdegeneracies.We focus here on the practical application of measuring photometric contamination in both simu-lated and real data without the use of spectroscopic redshifts. This is of interest not only for meeting thestringent requirements of future surveys as mentioned above, but also for other applications such as: adiagnostic tool for photometric redshifts; determining background samples for cluster lensing and esti-mating the true redshift distribution for 2-D cosmic shear measurements. The details of our method fora strict two-bin analysis of the CFHTLS-Archive-Research-Survey (CARS) are presented in Erben et al.(2009), where we demonstrate that the contamination present in the bright spectroscopic training sam-ple is consistent with the contamination seen in the much deeper photometric redshift sample. Thisaddresses a central concern when calibrating photometric redshifts with spectroscopic redshifts.This chapter is organised as follows: The angular correlation function and the estimators we use arepresented in Section 2.2. The details of the analytic method for estimating redshift bin contaminationare discussed in Section 2.3, which first addresses the general problem of contamination between anarbitrary number of redshift bins before focusing on the two-bin case and its extension to multipleredshift bins. The analytic model is tested using mock observational catalogues in Section 2.4, wherewe show that contamination between redshift bins can be accurately determined with our model. InSection 2.5 we measure contamination in a real galaxy survey demonstrating the ability of the methodto constrain the true (uncontaminated) redshift distribution and measure the average redshift of eachphotometric redshift bin. Section 2.6 presents details of the maximum likelihood method, including adiscussion of the covariance matrix. The contamination model for three redshift bins is discussed inSection 2.7. Concluding remarks, including limitations of the method and ideas for future work, arepresented in Section Angular correlation function and estimatorsThe two-point angular correlation function describes the amount of clustering in a distribution of galax-ies relative to what would be expected from a random distribution. The angular correlation function ?can be interpreted as the excess probability of finding an object in the solid angle d? a distance ? fromanother object. The total probability is given bydP = N[1+?(?)]d?, (2.1)where N is the density of objects per unit solid angle (Peebles, 1980). Note that Nd? is the Poissonprobability that the solid angle element is occupied by a galaxy.Many estimators have been devised for the correlation function. Kerscher et al. (2000) present acomparison of the most widely used estimators. The estimators primarily differ in the handling ofedge effects, and for arbitrarily large number densities they all converge. In Section 2.3 we employ thesimplest estimator,?ij =(DiDj)?(RR)?NRNRNiNj?1, (2.2)where (DiDj)? is the number of pairs separated by a distance ? between data sets i and j, (RR)? is thenumber of pairs separated by a distance ? for a random set of points, NR is the number of points inthe random sample and Ni (Nj) is the number of points in data sample i (j). In this work we consideri and j to be non-overlapping redshift bins. The auto-correlation is described by the case i=j, and thecross-correlation by the case i6=j.A more robust estimator (Landy & Szalay, 1993) is?ij =(DiDj)?(RR)?NRNRNiNj? (DiR)?(RR)?NRNi? (DjR)?(RR)?NRNj+1. (2.3)This is used when measuring the correlation function from data, either the Millennium Simulation inSection 2.4 or the CFHTLS-Deep fields in Section 2.5.Galaxies cluster in over-dense regions, leading to an excess number of pairs when compared toa random distribution of points. On small scales, the angular auto-correlation function of galaxiesis positive for all redshifts, though the shape and amplitude vary as a function of redshift due to theevolution of structure formation. The angular cross-correlation between two distant redshift bins shouldbe zero since galaxies that are physically separated by large distances do not cluster with one another.When considering galaxies binned in redshift, neighbouring bins may have a significant cross-correlation, especially if the bin width is narrow, since a significant number of galaxies in each bin couldbe clustered with each other. In the case of photometric redshifts the typical redshift error (?z ? 0.05)will result in a large cross-correlation if the width of the bins is not much larger than this error.Photometric redshift bins that are not neighbouring should not be physically clustered with oneanother and their cross-correlation should be zero. Deviations from zero indicate that the two binscontain galaxies that are physically clustered. These galaxies may result from contamination between22the two bins or from the mutual contamination of both bins from other redshifts.Weak lensing magnification can cause galaxies at high redshift to cluster near lower redshift galax-ies. This effect can be calculated and accounted for ?we discuss this in Section Analytic development of the contamination modelThe goal of the contamination model is to measure the level of contamination between all photometricredshift bins by measuring the angular cross-correlation. We have presented the details of the two-binmodel in Erben et al. (2009) and applied it to the CFHTLS-Archive-Research Survey (CARS). Here wedevelop a fully consistent multi-bin approach in Section 2.3.1 before revisiting the two-bin case andextending it to a global pairwise analysis in Section Multi-bin analysisWe first address the case of an arbitrary number of redshift bins, where each bin is potentially contami-nating every other bin. The number of galaxies observed to be in the ith bin is Noi which, by virtue of themixing between bins, is not equal to the true number of galaxies in that redshift bin NTi . We define fij tobe the number of galaxies from bin i contaminating bin j as a fraction of the true number of galaxies inbin i. Therefore fijNTi is the number of galaxies from bin i present in bin j. The fraction of galaxies inbin i which do not contaminate other bins is taken to be fii which is convenient shorthand forfii = 1?n?k6=ifik, (2.4)where n is the number of redshift bins. The summation convention here, and throughout, assumes thesum is begins at one.The observed number of galaxies in bin i will be those galaxies which contaminate the bin, plusthose galaxies from bin i which do not contaminate other bins,Noi =n?kNTk fki, (2.5)where the k=i term accounts for those galaxies that remain in bin i while all other terms account forgalaxies that have contaminated bin i. This results in a system of n equations which can be written as:??????No1No2. .Non??????=??????f11 f21 . . . fn1f12 f22 . . . fn2. . . . . . . . . . . . . . . . .f1n f2n . . . fnn????????????NT1NT2. . .NTn??????. (2.6)If the n?n matrix is invertible, then we can solve the system for the true number of galaxies in eachbin. In the limit of zero contamination the off-diagonal elements will tend to zero while the diagonalelements tend to unity. In this limit the matrix is trivially non-singular. If a matrix is strictly diagonally23dominant then it follows from the Gershgorin circle theorem that it is non-singular. Therefore as longasfii >n?k6=ifik, (2.7)the matrix is strictly diagonally dominant and therefore is invertible. This condition is simply statingthat a solution exits if the majority of the galaxies from the ith true redshift bin do not contaminate otherbins. The case of uniform contamination, where each bin sends NTi /n galaxies to each other bin, resultsin a singular matrix where all rows are identical.The observed correlation functions can be derived by investigating which pairs contribute when cor-relating bin i and bin j. The observed number of data pairs between bins can be related to contributionsfrom the true number of pairs,(DiDj)o? =n?kn?l(DkDl)T? fki flj. (2.8)Multiplying both sides by 1(RR)?NRNRNiNj and using Equation (2.2) to relate the pair counts to the correlationfunction yields?oij =n?kn?l?TklNTk NTlNoi Nojfki flj, (2.9)which can also be derived by considering?Noi ,Noj?.We assume that the true cross-correlation between any two redshift bins is zero. Hence, it is usefulto rewrite Equation (2.9) as?oij =n?k?Tkk(NTk )2Noi Nojfki fkj. (2.10)Note that this allows us to express the observed auto- and cross-correlations as linear combinations ofthe true auto-correlation functions. The following matrix equation follows from the above when weconsider only the equations for the observed auto-correlations (i.e, when i=j):???????o11No21?o22No22. . . . . .?onnNo2n??????=??????f 211 f 221 . . . f 2n1f 212 f 222 . . . f 2n2. . . . . . . . . . . . . . . . .f 21n f 22n . . . f 2nn?????????????T11NT21?T22NT22. . . . . . .?TnnNT2n??????. (2.11)We are again left with an n? n matrix which, when inverted, lets us express the true auto-correlationsin terms of the observed auto-correlations. Once they are known, we can use Equation (2.10) to expressthe observed cross-correlations as a linear combination of observed auto-correlation functions,?oij =n?k?okk(Nok )2Noi Nojgk( f ), (2.12)where i 6= j and gk( f ) is a complicated function of the contamination fractions. Note that the true numberof objects from Equation (2.10) has cancelled with that from Equation (2.11).The n? n matrix in Equation (2.11) above is very similar to that found in Equation (2.6). The24diagonal elements tend to unity as the contamination between bins tends to zero, resulting in a nearlydiagonal matrix. The matrix will be strictly diagonally dominant and hence non-singular under the samecondition found above, and for uniform contamination the matrix is singular.The total number of unknown parameters ( fij, where i 6= j) is n(n?1). This is clear since, in general,each of the n bins can contaminate each of the other bins (n?1). Our goal is to constrain these parame-ters with Equation (2.12), which yields only n(n?1)2 equations since ?oij = ?oji . If we could only measurethe angular correlation function at one scale then it would be impossible to constrain the parameters.However, if we have two independent measurements of the cross-correlations at different scales, wedouble the number of equations and are able to constrain the problem. In practice the measurements ofthe angular correlation function at different scales are not independent, and we should strive to includeas large a range of scales as possible.Equation (2.12) relates the amplitude of the cross-correlation to a weighted sum of the auto-correlations. If the shape of all the auto-correlations were the same, then it would be impossible todistinguish their contributions to the cross-correlation, which would also have the same shape. Forexample if the auto-correlations were well described by power laws with the same slope. In such casesthis method would yield completely degenerate parameter constraints.2.3.2 Pairwise analysisHere we first consider the case of exactly two redshift bins. This provides a simple case with whichto test our method using the Millennium Simulation in Section 2.4. We then expand this to multipleredshift bins by considering each pair in turn. We show that this global pairwise analysis will yieldaccurate estimates for the entire set of contamination fractions if the contamination is sufficiently small.The case of exactly three redshift bins is presented in Section 2.7.We consider two redshift bins, labelled 1 and 2. From Equation (2.6), we can express the observednumber of galaxies in each bin in terms of the true number of galaxies and the contamination fractions,No1 = NT1 (1? f12)+NT2 f21,No2 = NT2 (1? f21)+NT1 f12. (2.13)Each observed redshift bin contains those galaxies which do not contaminate the other bin (e.g. NT1 (1?f12)) plus those galaxies which are contaminating from the other bin (e.g. NT2 f21). Note that the totalnumber of galaxies, No1 +No2 = NT1 +NT2 , is conserved.Inverting Equation (2.11) allows us to express the true auto-correlations in terms of the observedauto-correlations. The resulting equalities can be plugged into Equation (2.10) yielding?o12 =?o11No1No2f12(1? f21)+?o22 No2No1f21(1? f12)((1? f12)(1? f21)+ f12 f21) . (2.14)As long as the two bins considered comprise the entire sample of galaxies, this formalism is con-sistent. Considering a sub-sample allows for leakage to and from the region exterior to the two bins.25 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 2  4  6  8  10  12  14  16  18  20Contamination fractionNumber of binsFigure 2.1: Upper bounds on the contamination fraction, f , as a function of the number of redshiftbins n. The upper bound represents the limit at which the pairwise approach breaks down. Aglobal contamination at these levels will cause at most a 10 per cent error in the estimate of?Tii (Equation 2.15).This can induce a cross-correlation due to mutual contamination of the considered bins from the exteriorregion and breaks the implicit assumption of galaxy number conservation.The two-bin case has practical applications such as background selection in cluster lensing, whereone seeks a background population that does not share members with the selected foreground cluster.We now address the global pairwise analysis which ?if the off diagonal contamination fractions aresmall? is a good approximation of the full-matrix approach detailed at the beginning of this section. Inthis case considering each pair of redshift bins in turn will yield estimates for the entire set of contam-ination fractions. We start by assuming that second order terms in the cross-contaminations are small,such that the off-diagonal elements of the matrix in Equation (2.11) satisfy f 2ij ? f 2ii . This results in thefollowing simple relationship between the observed and true auto-correlations:?Tii = ?oii(NoiNTi)2 1f 2ii. (2.15)When combined with Equation (2.10) this yields the following equation for the observed cross-correlation:?oij = ?oiiNoiNojfijfii +?ojjNojNoifjifjj +n?k,k6=i,k6=j?okk(Nok )2Noi Nojfki fkj( fkk)2 , (2.16)All terms that are first order in f have been taken out of the summation operator. This equation isidentical to the two bin case of Equation (2.14) if we assume that second order terms are negligible.26This approximation will hold as long as the eigenvalues of the n?n matrix of Equation (2.11) arewell approximated by the diagonal elements. To attempt to understand the limitation of this approxi-mation we assume the severe case of uniform cross contamination. In this case all fij equal f , thereforefii = (1? (n?1) f ) yielding the following contamination matrix:??????(1? (n?1) f )2 f 2 . . . f 2f 2 (1? (n?1) f )2 . . . f 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .f 2 f 2 . . . (1? (n?1) f )2??????. (2.17)The eigenvalues are (1?(n?1) f )2? f 2 with multiplicity n?1 and (1?(n?1) f )2 +(n?1) f 2. The lat-ter eigenvalue, being the larger deviation from the case of a purely diagonal matrix, gives us a constrainton f . If we require that the deviation be at most 10 per cent, we find(n?1) f 2(1? (n?1) f )2 = 0.1, (2.18)f ={1?(0.1(n?1))?0.5n?1?10 : n 6= 11(2(n?1))?1 : n = 11The limit on the contamination becomes more stringent when more redshift bins are included in theanalysis. Figure 2.1 gives the upper limit on f as a function of the number of redshift bins. Thecontamination must be less than or equal to these values so that the maximum error made in determining?Tii via Equation (2.15) is 10 per cent.With these limitations in mind we adopt the pairwise analysis throughout the rest of this work. Astandard maximum likelihood procedure is used to estimate the contamination fractions by measuringthe angular correlation functions at multiple scales and fitting them with Equation (2.14). Measure-ments of the angular correlation functions at different scales are not independent, thus the errors mustbe described by a covariance matrix. We use a bootstrapping technique to construct the covariancematrix, and include an additional source of error which we refer to as the clustering covariance matrix(Van Waerbeke, 2010). We refer the reader to Section 2.6 for the details of the covariance matrix andthe likelihood method.2.4 Application to a simulated galaxy surveyThe Millennium Simulation tracks the hierarchical growth of dark matter structure from a redshift of127 to the present. The simulation volume is a periodic box of 500 Mpc h?1 on a side, containing 21603particles each with a mass of 8.6? 108M?. The simulation assumes a concordance model ?CDMcosmology, ?m = ?dm +?b = 0.25, ?b = 0.045, ?? = 0.75, h = 0.73, n = 1 and ?8 = 0.9, though thedetails of the cosmology do not affect any of the results presented here. A complete description of theSimulation is presented in Springel et al. (2005) and references therein.We employ mock observational catalogues constructed from the Millennium Simulation to test27the method described in Section 2.3. The catalogues consist of six square pencil beam fields of 1.4degrees on a side, containing a total of 28.7 million galaxies at redshifts less than 4.0. A full de-scription of the catalogues is given by Kitzbichler & White (2007). To identify the mock catalogues,Kitzbichler & White (2007) give each a label ?a? through ?f?. We maintain this notation throughout thecurrent work when distinguishing between the catalogues.It is important to note that these mock observational catalogues do not include the effects of lensing,and in particular they do not account for weak lensing magnification. Since magnification will create anangular correlation between high and low redshift, it is relevant to the current work and will be addressedin Section 2.8.We first present the results when applying the two-bin analysis to the mock observational catalogues,and discuss the effects of bin width and galaxy density. We then apply the global pairwise analysisdemonstrating its ability to recover the true number of galaxies in a redshift bin, and the true averageredshift of galaxies in each redshift bin.2.4.1 Null testSince there is no contamination between redshift bins in the simulated data, the angular cross-correlationfunction between any two redshift bins ought to be consistent with zero. To test this, two redshift bins arechosen so that there is, approximately, an equal number of galaxies in each; the bins are 0.3 < z1 < 0.5which contains 69,139 galaxies per field on average and 0.8 < z2 < 0.85 which contains 50,575 galaxiesper field on average.The top left panel of Figure 2.2 shows the result of measuring the angular correlation functions oneach of the six fields. The auto-correlation functions for a given redshift slice have different amplitudesdue to the presence of different structures in each of the fields. The insert shows a zoom of the cross-correlation functions, and the errors include a bootstrapping term as well as the clustering term (seeSection 2.6). The clustering term dominates the error at all but the smallest scales, providing a relativelyuniform source of noise which behaves like a constant shift in the cross-correlation function. As seen inthe top left panel of Figure 2.2; the six fields provided by the mock catalogues are slightly biased towardsa positive shift. However, the clustering term is symmetric about zero and with enough measurementsshould have no bias. To test this we measured the cross-correlation between all combinations of high-and low-redshift bins, so that the high-redshift bin from field ?a? is cross-correlated with the low redshiftbin of fields ?a?, ?b?, ?c?, ?d?, ?e? and ?f?. The resulting 36 cross-correlations are displayed in Figure 2.3,where the solid (red) line shows the average value which is consistent with zero at all scales.The contamination fractions are estimated from the measured angular correlation functions as de-scribed in Section 2.6. The resulting likelihood is presented in the bottom left panel of Figure 2.2. Zerocontamination is consistent with the results, and contamination in excess of ?2 per cent can be ruledout at the 99.9 per cent confidence level.We conclude that, as expected, there is no significant angular cross-correlation between widelyseparated redshift bins in the Millennium Simulation. Furthermore our method of estimating the con-tamination performs well in this case, indicating that there is less than ?1 per cent contamination at the28Figure 2.2: The left side panels are for the case of no contamination, the right side panels arefor a contamination of f12 = 0.08 and f21 = 0.04. Top row: The angular auto- and cross-correlation functions for each test contamination of the mock catalogues. Each of the sixfields is plotted with a different line style (colour). The higher redshift bin, 0.8 < z2 < 0.85,has a larger amplitude than the lower redshift bin, 0.3 < z1 < 0.5. A zoom of the cross-correlation is given in the insert, and the legend therein identifies which line style applies toeach of the six mock catalogues. The error bars on the auto-correlation functions come frombootstrapping the catalogue. The cross-correlation functions also include the contributionfrom the clustering covariance (refer to Section 2.6 for details). The cross-correlation isconsistent with zero for the case of zero contamination (left panel) and deviates significantlyfrom zero for the contaminated case (right panel) Bottom row: The likelihood region of thecontamination fractions. The shaded regions denote the 68, 95 and 99.9 per cent confidenceareas, with increasing darkness indicating increasing significance. The contours result fromsumming the log-Likelihoods for the six individual fields. The insert shows the result foreach of the individual fields. The input contamination in the right panel is marked with a dot.29-0.1-0.05 0 0.05 0.1 0.0001  0.001  0.01  0.1Angular correlationScale (deg)36 swapped cross correlationsaverage cross correlationFigure 2.3: The 36 dotted (gray) lines are all of the possible cross-correlations found by swappingthe high and low redshift bins between fields. The dashed (green) line shows y=0. Thesolid (red) line is the average of the 36 cross-correlations, and the error bar is the standarderror. The average is consistent with zero, showing that the clustering term does not bias thecross-correlation.68 per cent confidence level.2.4.2 Artificial contaminationContamination was added to each field by randomly shifting galaxies between the redshift bins. Thecontamination fractions were taken to be f12 = 0.08 and f21 = 0.04. The measured angular correlationsare shown in the top right panel of Figure 2.2 where a cross-correlation signal is clearly seen. Therecovered likelihood for the contamination fractions is shown in the bottom right panel of Figure 2.2.The input value lies in the 68 per cent confidence region, and zero contamination ( f12 = f21 = 0.00) isruled out well beyond the 99.9 per cent confidence level.The parameter space is highly degenerate; contamination in either direction (low to high redshift,or vice versa) increases the cross-correlation amplitude. Degeneracy breaking comes from the abil-ity to probe the shape of the auto-correlation functions. If only one scale is used, the parameters arecompletely degenerate. Likewise if the auto-correlation of each redshift bin has the same shape (forexample, if they are power laws with the same slope), then the parameters are completely degenerate.Each field has independent structure, causing variations in the measured correlation functions. It isimportant that this is not considered a source of sample variance. Any features in the auto-correlation30function will also be present in the cross-correlation function if contamination is present. The insertsin the bottom panels of Figure 2.2 show the parameter constraints obtained by analysing each fieldindependently. These contours are combined, yielding a better measurement of the contamination.We have repeated this procedure for several different contamination fractions ( f12, f21) including(0.0,0.02), (0.15,0.0), (0.0,0.15) and (0.0,0.45). In all cases the input contamination was recoveredwithin the 68 per cent confidence region.2.4.3 Effect of galaxy density and redshift bin widthEach of the six mock catalogues of Kitzbichler & White (2007) contain on average 4.79?106 galaxiesout to redshift 4. The two redshift bins used thus far, 0.3 < z < 0.5 and 0.80 < z < 0.85, were chosento be well separated in redshift and contain roughly equal numbers of galaxies. On average there are? 6?104 galaxies per redshift bin per field, corresponding to a density of 8.5 arcmin?2.In order to test the effects of object density, galaxies were randomly removed from the mock cata-logues which were then contaminated with f12 = 0.08 and f21 = 0.04. Densities of 4.3 and 1.7 arcmin?2were tested. We found that lower densities yielded only marginally weaker constraints compared tothose in the bottom right panel of Figure 2.2.Reducing object density has a greater effect on constraints from individual fields; much of the con-straining power results from the combination of the six fields. As long as there are sufficient galaxiesto measure the angular correlation function accurately, the contamination can be constrained. There is adisproportionately small change to the contours when the density is reduced since we are not dominatedby Poisson noise.If more narrow redshift bins are taken, for example 0.3 < z < 0.4 and 0.8 < z < 0.82, then theconstraints on the contamination improve considerably. Each bin contains 24,451 galaxies per field onaverage with a corresponding density of 3.5 arcmin?2. Figure 2.4 shows the parameter constraints whena contamination of f12 = 0.08 and f21 = 0.04 is used, the constraints for wider bins are over-plotted forcomparison. The narrow bins offer much tighter constraints despite the density being lowered as a resultof the smaller bin size.The angular correlation function for narrower redshift bins is more distinct since a larger fractionof galaxies are physically clustered with each other. Making the bins wider dilutes the sample withgalaxies that are not clustered, reducing the correlation. Arbitrarily narrow bins will cluster strongly,providing a good measure of the correlation function, but will suffer due to low number densities. Morework is needed to determine an optimal trade-off between these parameters.2.4.4 Global pairwise analysisWe divide the data into the following redshift bins: [0.0,0.5], (0.5,0.8], (0.8,1.1] and (1.1,1.5], whichwe label z1, z2, z3 and z4 respectively. The average number of galaxies in each bin from low to highredshift is: 60804, 94907, 63707, 55290. In order to test the global pairwise analysis, contaminationfractions between all redshift bins must be specified. In lieu of a completely random contaminationmatrix we have taken one which is similar to the contamination we measure for the CFHTLS-Deep31Figure 2.4: Constraints on the contamination fractions for an input model of f12 = 0.08 and f21 =0.04, which is marked with a dot. The lined contours show the results for the wider redshiftbins; 0.3 < z < 0.5 and 0.80 < z < 0.85. The filled contours give the constraints for morenarrow redshift bins; 0.3 < z < 0.4 and 0.8 < z < 0.82. Narrow redshift bins provide tighterconstraints despite the decreased number density.fields in Section 2.5. Since the measured contamination in the Deep fields appears to be on the extremeend of the global pairwise approximation we have reduced some of the contamination values, thoughthe adopted contamination matrix remains a strong test of the global pairwise approximation. Thecontamination matrix used is,fij =??????0.70 0.01 0.10 0.100.10 0.925 0.15 0.040.05 0.06 0.70 0.080.15 0.005 0.05 0.78??????, (2.19)where fij refers to the entry in the ith column and the jth row. Galaxies are chosen randomly to be movedbetween redshift bins such that Equation (2.6) is satisfied.The result of measuring the angular auto-correlation function for each redshift bin and the cross-correlation function for each pair of redshift bins, as measured from field ?a?, is presented in the leftpanel of Figure 2.5. One hundred bootstraps of the contaminated catalogues were constructed, from32Figure 2.5: The redshift bins are: z1 =[0.0,0.5], z2 =(0.5,0.8], z3 =(0.8,1.1] and z4 =(1.1,1.5].Left panel: The angular correlation functions as measured for field ?a?. The subplots alongthe diagonal contain the auto-correlation for each redshift bin. The off-diagonal subplotscontain the cross-correlation function between two bins. The redshift bin labels at the topof each column and to the left of each row denote which bins are involved in the givencorrelation function. The vertical scale on the top-left subplot applies to all subplots on thediagonal, the scale on the top-right subplot applies to all off-diagonal subplots. Right panel:The filled contours depict the parameter constraints estimated for field ?a?. The shading fromlight to dark represents the 68, 95 and 99.9 per cent confidence levels. The lined contours arethe result of combining the parameter constraints from each individual field, for clarity weonly show the 68 and 99.9 per cent confidence levels. Each subplot contains the constraintsfor a pair of contamination fractions. The x-axis is taken to be the contamination fractionfrom the high redshift bin to the low redshift bin and the y-axis is the reverse. The two binscontributing to each subplot are indicated by redshift bin labels at the top of each column andto the left of each row.which the angular correlation functions were measured. These hundred measurements were used toestimate the covariance matrix, applying the correction described in Hartlap et al. (2007). For the auto-correlation functions (subplots on the diagonal) this is the only source of error, the cross-correlationfunctions (off-diagonal subplots) have the clustering term as an additional source of error.The pairwise analysis is applied to each pair of redshift bins yielding constraints on the contami-nation between each pair. The constraints on the contamination fractions for field ?a? are given by thefilled contours in the right panel of Figure 2.5. Each of the six fields is considered independent, andtheir constraints on the contamination fractions are combined by multiplying the likelihoods together.These combined constraints are given as lined contours, over-plotted in Figure 2.5.The pairwise analysis does not impose global constraints since each pair of bins is considered in-33dependently. Not all combinations of contamination fractions suggested by Figure 2.5 will produce aconsistent picture. For example, it is possible to select points from each plot in the right most columnsuch that more than 100 per cent of galaxies from the highest redshift bin are contaminating other bins.To get a sense of which solutions are globally consistent, and what a typical global solution looks like,we employ a Monte-Carlo method. We first randomly sample 1000 points from each of the pairwiselikelihood regions such that the density of points reflects the underlying probability distribution. Takinga random point from each pairwise analysis will uniquely specify a realization of the global contam-ination, i.e. the matrix of Equation (2.6). Thus a realization of the true redshift distribution can bedetermined. We then impose two constraints: the true number of galaxies in a bin cannot be less thanzero, and no more than 100 per cent of galaxies can be scattered from a single bin. This procedure isrepeated until 200,000 admissible realizations are found.We have verified that the admissible realizations of the contamination fractions are representative ofthe full probability distributions from which they are drawn. It is not the case that the global constraintsexclude particular regions of parameter space. This is checked by constructing a probability distributionfor each contamination fraction from the 200,000 measurements and comparing it to the marginalisedPDF obtained from the likelihood contours. In all cases the two agree with each other. For the combinedcase, the estimated contamination fractions (taken to be the average value) with 68 per cent confidencelevels are as follows,fij =??????0.60+0.04?0.05 0.03+0.02?0.02 0.11+0.03?0.03 0.11+0.05?0.060.12+0.04?0.03 0.89+0.04?0.03 0.18+0.05?0.04 0.09+0.05?0.050.10+0.02?0.02 0.07+0.02?0.02 0.66+0.06?0.06 0.16+0.07?0.030.18+0.02?0.01 0.01+0.00?0.01 0.05+0.02?0.05 0.65+0.09?0.09??????. (2.20)This result agrees extremely well with the input contamination matrix of Equation (2.19), with 11 ofthe 16 contamination fractions, 69 per cent, agreeing within the 68 per cent error estimate. As ex-pected the pairwise analysis slightly overestimates the level of contamination, which is probably morepronounced in this case since the contamination matrix is so aggressive. The contamination fractionson the off-diagonal are overestimated since the pairwise method assumes that the observed angularcross-correlation function is only due to contamination between the two considered redshift bins. How-ever, the pair of redshift bins considered will each be contaminated from other redshift bins, leading to across-correlation between the redshift bins which is not dependent on the scattering of redshifts betweenthem. Note that the diagonal elements in Equation (2.20) are smaller than for the input contaminationmatrix, which is also a result of overestimating the contamination fractions (see Equation 2.4).Using Equation (2.6) allows us to estimate the true number of galaxies in each redshift bin for eachof the 200,000 realizations of the contamination matrix. This is done for each of the six fields as well asthe combined case, the probability distributions for the estimated true number of galaxies are presentedin Figure 2.6. The cross hashed regions denote the 68 per cent confidence level. Since this is simulateddata with artificial contamination we know what the actual uncontaminated number of galaxies is for agiven bin, this is over-plotted as a dashed vertical line, for clarity we will refer to this as the true number34of galaxies, and to our attempt to recover this as the estimated true number of galaxies. The solid verticalline shows the observed number of galaxies; i.e., the number of galaxies after contamination. For thecombined case the number of observed galaxies is taken to be the average number of galaxies fromthe 6 fields. The global pairwise analysis does a good job of recovering the true number of galaxiesin each bin, although in many cases the observed number is also an acceptable fit, owing to the smallcontamination fractions and the fact that the redshift distribution was nearly flat to begin with. Focusingon the combined result for z2 we see that the true number of galaxies is a significantly better fit to theestimated true number than the observed number.It is also possible to estimate the true average redshift of a redshift bin. The true average redshift isdefined as the average redshift of galaxies in a redshift bin which has contamination. With real worlddata it is the same as asking what the average spectroscopic redshift is in a given photometric redshiftbin. This is straightforward to calculate for a simulated survey since we know for each redshift bin thefraction of galaxies that came from each other redshift bin. The true average redshift is given by:z?Ti =1Noin?k=1z?uncontamk fkiNTk , (2.21)where z?uncontamk is the uncontaminated average redshift of a galaxy in bin k, and fkiNTk is the number ofgalaxies in bin i from bin k. In general the uncontaminated average redshift of a bin is not known sinceit requires knowledge of the true redshift of each galaxy. If we assume that the contamination does notsignificantly alter the shape of the redshift distribution at the sub-redshift bin level, then the averageredshift of a bin will not be changed by contamination. Therefore we estimate the average redshift of anuncontaminated redshift bin (z?uncontamk ) by the average redshift of the contaminated redshift bin.For a given field, each of the 200,000 realizations of the contamination matrix yields an estimate ofthe true average redshift for each redshift bin. Thus a probability distribution function is constructedfor each bin and each field, including the combined case, this result is presented in Figure 2.7. Thesolid vertical lines show the average redshift with no contamination, this is what one would measureto be the average redshift if the effects of contamination were ignored. The dashed vertical lines showthe true average redshift for each bin, which we can measure directly here because we?re working withsimulated data with known contamination. Our method does a very good job of recovering the trueaverage redshift. The lowest and highest redshift bins suffer the most, this is expected since they canonly be contaminated by galaxies that are either higher or lower in redshift respectively, whereas themiddle bins are contaminated by galaxies which are both higher and lower than the average, allowingfor a cancellation of the effect.We have demonstrated that the global pairwise analysis can be used to reliably recover small con-taminations between redshift bins. Using the Monte Carlo approach detailed in this section we haveshown that we are able to estimate the true redshift distribution, and the true average redshift within abin.35Figure 2.6: Millennium Simulation: This demonstrates the ability of the global pairwise anal-ysis to estimate the true (uncontaminated) redshift distribution. The redshift bins are:z1 =[0.0,0.5], z2 =(0.5,0.8], z3 =(0.8,1.1] and z4 =(1.1,1.5]. The x-axis is the number ofgalaxies in units of 1? 104. The y-axis is the probability, which has been scaled differentlyin each subplot for clarity. The histograms are the probability distribution of the true numberof galaxies, and the cross-hashing denotes the 68 per cent confidence region. Each row ofsubplots is the result from one of the Millennium Simulation fields. The bottom row is theresult when the constraints on the contamination fractions for each field are combined (seeFigure 2.5). Please note that the bottom row is not a direct combination of the results fromthe other rows. Each column represents a redshift bin, as labelled. The solid vertical line ineach subplot indicates the observed number of galaxies. The dashed vertical line is the truenumber of galaxies.36Figure 2.7: Millennium Simulation: The global pairwise analysis is used to estimate the trueaverage redshift of each photometric redshift bin. The redshift bins are: z1 =[0.0,0.5],z2 =(0.5,0.8], z3 =(0.8,1.1] and z4 =(1.1,1.5]. The x-axis is the average redshift. The y-axis is the probability, which has been scaled differently in each subplot for clarity. Thehistograms are the probability distribution of the average redshift, and the cross-hashing de-notes the 68 per cent confidence region. Each row of subplots is the result from one of theMillennium Simulation fields. The bottom row is the result when the constraints on the con-tamination fractions for each field are combined (see Figure 2.5). Please note that the bottomrow is not a direct combination of the results from the other rows. Each column represents aredshift bin, as labelled. The solid vertical line in each subplot indicates the average redshiftas measured from the uncontaminated catalogue. The dashed vertical line is the true averageredshift.372.5 Application to a real galaxy surveyThe Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) is a joint Canadian-French programdesigned to take advantage of Megaprime, the CFHT wide-field imager. This 36-CCD mosaic camerahas a 1? 1 degree field of view and a pixel scale of 0.187 arcseconds per pixel. The deep componentconsists of four one-square-degree fields imaged with five broad-band filters: u?, g?, r?, i?, and z?. Thefields are designated D1, D2, D3 and D4, and are centred on RA;DEC coordinates of 02 26 00; -04 3000, 10 00 28; +02 12 21, 14 19 28; +52 40 41 and 22 15 32; +17 44 06 respectivelyWe use the deep photometric redshift catalogues from Ilbert et al. (2006) who have estimated red-shifts for the T0003 CFHTLS-Deep release. The full photometric catalogue contains 522286 objects,covering an effective area of 3.2 deg2. A set of 3241 spectroscopic redshifts with 0 ? z ? 5 from theVIRMOS VLT Deep Survey (VVDS) were used as a calibration and training set for the photometricredshifts. The resulting photometric redshifts have an accuracy of ?(zphot?zspec)/(1+zspec) = 0.043 for i?AB= 22.5 - 24, with a fraction of catastrophic errors of 5.4 per cent.In this work we consider galaxies with 21.0 < i? < 25.0 and divide the data into the followingredshift bins: [0.0,0.2], (0.2,1.5], (1.5,2.5] and (2.5,6.0], which we label z1, z2, z3 and z4 respectively.The average number of galaxies in each bin from low to high redshift is: 5772, 96019, 15546, 5315.These redshift bins are chosen to isolate the low confidence redshifts and to probe the bump in theredshift distribution near redshift 3 (see Figure 2.8). We would like to know if this feature is an artefactof the photometric redshifts or if it corresponds to a physical over-density of galaxies at this redshift.Adopting the definition of catastrophic error used by Ilbert et al. (2006), ?z > 0.15(1+ z), we mea-sure the fraction of galaxies with catastrophic errors in each bin. From low to high redshift we find:23.2, 12.4, 33.2 and 23.1 per cent. It is difficult to interpret these values in the context of the contam-ination fractions. Photometric redshifts with large errors do not necessarily contaminate other redshiftbins. Furthermore ?z does not take into account degeneracies in the spectral template fitting that al-low for alternative solutions to the galaxy?s redshift. These catastrophic errors give some indication ofthe leakage between adjacent redshift bins but cannot account for those galaxies which are completelymisclassified.2.5.1 Applying the global pairwise analysisWe apply the two-bin analysis between each pair of redshift bins, and for each of the four Deep fields.The measured angular correlation functions for D1 are presented in Figure 2.9. The covariance matri-ces are estimated by bootstrapping the catalogues 100 times, and applying the correction described byHartlap et al. (2007). For the cross-correlation covariance matrix we also calculate the clustering co-variance described in Section 2.6. Parameter constraints are estimated for each cross-correlation, thosefor D1 are presented in Figure 2.9. The other three fields have very similar angular correlation functionsand parameter constraints. The parameter constraints for all of the fields can be combined by treatingthem as statistically independent and multiplying their likelihoods together, yielding tighter constraints(lined contours in Figure 2.9).We constructed 200,000 realizations of a globally consistent contamination matrix, as detailed in38 0 0.2 0.4 0.6 0.8 1 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5normalized frequencyz (redshift)D1D2D3D4Figure 2.8: Finely binned redshift distribution for each of the four deep fields. The verticallines denote the binning adopted in our global pairwise analysis: The redshift bins are:z1 =[0.0,0.2], z2 =(0.2,1.5], z3 =(1.5,2.5] and z4 =(2.5,6.0]. Note the bump near redshift3.Figure 2.9: Same as caption to Figure 2.5 but with field D1 in place of field ?a?. The redshift binsare: z1 =[0.0,0.2], z2 =(0.2,1.5], z3 =(1.5,2.5] and z4 =(2.5,6.0].39Section 2.4.4, and verified that the admissible realizations are representative of the full probability dis-tributions they are drawn from. The contamination matrix estimated from the combined constraintsisfij =??????0.50+0.17?0.17 0.006+0.004?0.003 0.13+0.04?0.02 0.13+0.07?0.040.18+0.06?0.18 0.979+0.014?0.006 0.31+0.08?0.04 0.04+0.01?0.040.16+0.05?0.16 0.006+0.001?0.006 0.53+0.05?0.10 0.09+0.03?0.090.11+0.03?0.11 0.001+0.000?0.001 0.07+0.04?0.02 0.74+0.10?0.10??????, (2.22)where the entries represent the average value calculated from their probability distribution. The maxi-mum contamination fraction for four redshift bins presented in Figure 2.1 is ?0.12 which is smaller thansome of the contamination fractions found here (or seen in Figure 2.9). It is possible that this is an indi-cation that the pairwise analysis does not hold for these data; however, several assumptions were madein deriving the maximum contamination fraction which do not hold here. We assumed that all contami-nation fractions have the same value; this is not the case here, and although some tend to be larger than0.12, some are very close to zero, and several have large errors encompassing zero. We also assumedthat there were equal numbers of galaxies in each bin. The ratio of the number of galaxies between thepair of bins enters into the expression for the observed cross-correlation. Since the first and last redshiftbins have far fewer galaxies, a large contamination fraction from one of these bins represents only asmall number of galaxies. We have also demonstrated our ability to recover the contamination fractionsfrom a similarly aggressive contamination matrix in Section 2.4.4. For these reasons we believe that theglobal pairwise analysis remains a good approximation here.The probability distribution of the true number of galaxies for each redshift bin and each field is pre-sented in Figure 2.10; the cross-hatched regions indicate 68 per cent confidence. The observed numberof galaxies in each bin is denoted by a vertical line. The bottom row contains the result when the con-straints on the contamination fractions for each of the four fields are combined. The smallest fractionalchange is for z2 which is the high confidence photometric redshift bin as defined by Ilbert et al. (2006).The peak of the probability distribution indicates about a factor of two fewer galaxies in the highestredshift bin than are observed, suggesting that the bump seen in the photometric redshift distribution isan artefact of contamination. However, with only four square degrees of data, we are unable to rule outthe existence of this feature.The set of globally consistent realizations of the contamination can also be used to estimate theaverage redshift for each photometric redshift bin. We use Equation (2.21), and estimate the uncontam-inated average redshift of each bin (z?uncontamk ) by the average of the photometric redshifts. Which is agood approximation as long as the shape of the observed redshift distribution within each bin is similarto that of the true redshift distribution.The results are presented in Figure 2.11 which shows the probability distribution of the average red-shift for each redshift bin and each field. Vertical lines show the average redshift for each bin measuredfrom the photometric catalogue. It is clear that the smallest, and largest, redshift bins (z1, and z4) con-tain galaxies whose true average redshifts deviate significantly from the average redshift expected forthose redshift bins. This suggests that many galaxies in bin z1 are in fact from much higher redshifts.40Similarly galaxies in bin z4 have a lower than expected average redshift.2.6 Covariance and likelihoodHere we present the details of the maximum-likelihood method, and the covariance matrix used.Throughout this chapter we fit the observed angular cross-correlation between two redshift bins withthe model described by Eq.(2.14). Since there are observational errors associated with the angular auto-and cross-correlation functions, we have grouped these quantities on the left hand side of Eq.(2.14),yielding:?oij(?)( fii fjj + fij fji)??oii(?)NoiNojfij fjj ??ojj(?)NojNoifji fii = 0, (2.23)where the angular correlation functions are written as a function of scale ?. For simplicity let F representthe left hand side of the equation. We therefore seek to calculate the likelihood,L = 1?(2pi)s|C|exp[?12(F?m)C?1(F?m)T], (2.24)where s is the number of angular scale bins, m is the model which is zero for all scales and C is the s? scovariance matrix. The covariance matrix isCkl = ?FkFl?, (2.25)where k and l denote the scales at which the angular correlation functions are measured. Expanding theabove yieldsCkl = ??oij(?k)?oij(?l)?( fii fjj + fij fji)2 (2.26)+ ??oii(?k)?oii(?l)?(NoiNojfij fjj)2+ ??ojj(?k)?ojj(?l)?(NojNoifji fii)2+ ??oij(?k)?oii(?l)?2( fii fjj + fij fji)NoiNojfij fjj+ ??oij(?k)?ojj(?l)?2( fii fjj + fij fji)NojNoifji fii+ ??oii(?k)?ojj(?l)?2 fij fjj fji fii.Ideally the covariance matrix can be estimated directly from the data, but this requires many fields. Itis not possible to do this for either the Millennium Simulation or the CFHTLS-Deep data sets whichwe consider in this work. An alternative is to use a bootstrapping method, wherein the data catalogueis resampled multiple times, and each resampled catalogue is used to measure the angular correlation41Figure 2.10: CFHTLS-Deep: This demonstrates the ability of the global pairwise analysisto reconstruct the true (uncontaminated) redshift distribution. The redshift bins are:z1 =[0.0,0.2], z2 =(0.2,1.5], z3 =(1.5,2.5] and z4 =(2.5,6.0]. The x-axis is the number ofgalaxies in units of 1?104. The y-axis is the probability, which has been scaled differentlyin each subplot for clarity. The histograms are the probability distribution of the true num-ber of galaxies, and the cross-hashing denotes the 68 per cent confidence region. Each rowof subplots is the result from one of the CFHTLS-Deep fields. The bottom row is the re-sult when the constraints on the contamination fractions for each field are combined. Eachcolumn represents a redshift bin, as labelled. The vertical line in each subplot indicates theobserved number of galaxies.42Figure 2.11: CFHTLS-Deep: The global pairwise analysis is used to estimate the true averageredshift of each photometric redshift bin. The redshift bins are: z1 =[0.0,0.2], z2 =(0.2,1.5],z3 =(1.5,2.5] and z4 =(2.5,6.0]. The x-axis is the average redshift. The y-axis is the prob-ability, which has been scaled differently in each subplot for clarity. The histograms arethe probability distribution of the average redshift, and the cross-hashing denotes the 68 percent confidence region. Each row of subplots is the result from one of the CFHTLS-Deepfields. The bottom row is the result when the constraints on the contamination fractions foreach field are combined. Each column represents a redshift bin, as labelled. The verticalline in each subplot indicates the average redshift as measured from the photometric redshiftcatalogue.43Figure 2.12: The covariance matrix of the cross-correlation ??oij(?k)?oij(?l)? has two contribu-tions. We take field ?a? of the Millennium Simulation and two redshift bins as describedin Section 2.4.1. The LEFT panel shows the clustering term calculated as described inVan Waerbeke (2010). The matrices are 6? 6 and increase in scale from bottom to topand left to right. The RIGHT panel is the total covariance which includes the bootstrapcovariance matrix in addition to the clustering covariance. The scale on the right is in unitsof 10?4. Note that the clustering term results in a very flat covariance between all scales,whereas the bootstrap covariance adds relatively little and only on the smallest scales.functions. These angular correlation functions can then be used to calculate the covariance.This procedure suffices for all contributions to the covariance matrix Equation (2.26), except thefirst term ??oij(?k)?oij(?l)?. As shown in Van Waerbeke (2010), the covariance of the cross-correlationfunction has a contribution due to the intrinsic clustering of the background and foreground populations.This so-called clustering term can be calculated analytically given the auto-correlation functions andthe survey geometry. We add the covariance due to the clustering term to our bootstrap covariancematrix. The left panel in Figure 2.12 shows the covariance from the clustering term and the right panelshows the total covariance of the cross-correlation function between two redshift bins of the MillenniumSimulation (see Section 2.4.1). This constitutes the largest contribution to the covariance matrix ofEquation (2.26).2.7 Solving the three-bin case analyticallyFor three bins it is easy to derive equations for the three observed cross-correlation functions, analogousto what is presented in Equation (2.14) for the two-bin case. Let the n?n matrix in Equation (2.11) becalled F. The inverse can be calculated using the adjoint,F?1 = adj(F)det(F) . (2.27)44Both the adjoint (adj(F)) and the determinant (det(F)) can be calculated easily:adj(F) =???f 222 f 233 ? f 232 f 223 f 231 f 223 ? f 221 f 233 f 221 f 232 ? f 231 f 222f 232 f 213 ? f 212 f 233 f 211 f 233 ? f 231 f 213 f 231 f 212 ? f 211 f 232f 212 f 223 ? f 222 f 213 f 221 f 213 ? f 211 f 223 f 211 f 222 ? f 221 f 212???det(F) = f 211 f 222 f 233 + f 221 f 232 f 213 + f 231 f 212 f 223? f 211 f 232 f 223 ? f 221 f 212 f 233 ? f 231 f 222 f 213 (2.28)With the inverse of of the matrix in hand we can now use Equation (2.11) to write the true auto-correlations in terms of the observed auto-correlations,?Tii det(F) = ?oii(NoiNTi)2( f 2jj f 2kk ? f 2kj f 2jk)+?ojj(NojNTi)2( f 2ki f 2jk ? f 2ji f 2kk)+?okk(NokNTi)2( f 2ji f 2kj ? f 2ki f 2jj ),(2.29)where i6=j6=k. The three true auto-correlation functions are found by permutations of the indices(i,j,k)=(1,2,3), (2,3,1) and (3,1,2). Note that the equation is symmetric in the last two indices yieldingthe same result for (i,j,k) and (i,k,j). Substituting these equations into Equation (2.10) for the observedcross-correlations we find,?oijdet(F) = ?oiiNoiNoj[ fii fij( f 2jj f 2kk ? f 2kj f 2jk)+ fji fjj( f 2kj f 2ik ? f 2ij f 2kk)+ fki fkj( f 2ij f 2jk ? f 2jj f 2ik)]+ ?ojjNojNoi[ fii fij( f 2ki f 2jk ? f 2ji f 2kk)+ fji fkk( f 2ii f 2kk ? f 2ki f 2ik)+ fki fkj( f 2ji f 2ik ? f 2ii f 2jk)]+ ?okkNo2kNoi Noj[ fii fij( f 2ji f 2kj ? f 2ki f 2jj )+ fji fjj( f 2ki f 2ij ? f 2ii f 2kj)+ fki fkj( f 2ii f 2jj ? f 2ji f 2ij )](2.30)Permuting the indices as above yields equations for the three observed cross-correlation functions. Thereare three equations and six unknowns ?note that fii, fjj and fkk depend only on fij, fik, fji, fjk, fki andfkj. By considering more than one scale we can double the number of equations making the systemconstrained.2.8 Conclusion and discussionWe have presented an analytic framework for estimating contamination between photometric redshiftbins, without the need for any spectroscopic data beyond those used to train the photometric redshiftcode. To measure the contamination between redshift bins we exploit the fact that mixing between binswill result in a non-zero angular cross-correlation between those bins. We have shown how the con-tamination will affect the observed angular correlation functions for the general case of contaminationbetween an arbitrary number of bins. For the case of two- and three-bins we explicitly work out theequations.The case of two-bins is given special attention since it is the simplest case. We note that if thecontamination between bins is small enough, then each pair of bins can be considered independently,45yielding an accurate measure of contamination between all bins. We refer to this as a global pairwiseanalysis.We test our formalism with mock galaxy catalogues created from the Millennium Simulation. Weverify that there is no evidence of contamination, finding an upper limit of ?2 per cent at the 99.9per cent confidence level. The catalogues are then contaminated by moving galaxies between redshiftbins. We demonstrate that the two-bin analysis is able to recover input contamination between redshiftbins. The effects of galaxy density and bin-width are investigated. We find that our ability to constrainthe contamination fractions is not very sensitive to object density, whereas narrower bins offer betterconstraints.We split the mock catalogues into four redshift bins and apply artificial contamination between allpairs. The global pairwise analysis is used to constrain the contamination fractions between all pairsof redshift bins. A Monte-Carlo method is then used to estimate the true (uncontaminated) redshiftdistribution, and the true average redshift of galaxies in each bin. This is valuable information for thecosmological interpretation of galactic surveys, and in particular weak lensing by large scale structure.We demonstrate the ability of the method to accurately recover the input contamination as well asreconstruct the true redshift distribution and average redshift of each bin.The formalism is applied to a real galaxy survey; the four square degree deep component of theCanada-France-Hawaii Telescope Legacy Survey for which there are photometric redshift catalogues(Ilbert et al., 2006). We divide the data into four redshift bins and apply the global pairwise analysis.This yields constraints on the contamination fractions, the true redshift distribution, and the true averageredshift of galaxies in each bin. We demonstrate here the feasibility of the method with only foursquare degrees of sky coverage; future application to large galaxy surveys will significantly improveconstraining power.This work has focused on the application of the two-bin and global pairwise methods. For a smallnumber of redshift bins, with sufficiently small contamination, the global pairwise analysis offers aquick and easy means of assessing the contamination between redshift bins. The benefit is largelycomputational, it is very fast to constrain a model with only two free parameters. A more sophisticatedmethod (such as a Monte-Carlo-Markov-Chain (MCMC)) will be needed to implement the full multi-bin approach. With only three redshift bins there is a total of six free parameters which already rendersthe simple maximum-likelihood approach impractical. Although the full multi-bin approach will yieldthe most accurate results, for some applications the more simplistic pairwise analysis should suffice.Future work will need to take into account the effect of weak lensing magnification, which causesan angular cross-correlation between background galaxies and foreground lenses. Since foregroundlenses boost the magnitude of background galaxies there are more close pairs detected between theseredshift slices then one would expect from random placement. This effect is well understood and canbe easily modelled and accounted for (Scranton et al., 2005), however, since it depends on cosmologyand the redshifts of the lens and background galaxies it cannot be removed in a model-independentway. Dust extinction of the background sources is also important, reducing the brightness of lensedgalaxies (Me?nard et al., 2009). This de-magnification is wavelength dependant, and in visible passbands46is comparable in magnitude to lensing magnification, therefore, it will need to be accounted for alongwith magnification.The expected amplitude of the angular cross-correlation due to magnification is small. Using theCFHTLS-Deep fields, and a magnitude cut similar to that used in this work, Van Waerbeke (2010) findthe amplitude of the angular cross-correlation between the redshift bins z = [0.1,0.6] and z = [1.1,1.4]to be about 0.01 on scales smaller than 1 arcmin. While this is clearly an important contribution to theangular cross-correlations measured here it can only account for about 10 per cent of the observed signalon these scales. This is similar to the uncertainty introduced by the pairwise approximation, and withinthe error budget of this analysis.Photometric redshifts are better measured for red galaxy types. Furthermore red and blue galaxiescluster differently, resulting in distinct angular correlation functions. Therefore, it is likely the case thatgalaxies doing the contamination are predominantly blue and exhibit a systematically different angularcorrelation than the red galaxies which do not contaminate. More work is needed to understand theseverity of this bias. However, since any cross-correlation signal (above that expected from magnifica-tion) indicates contamination it is always possible to use this technique as a null test.We have presented a method of measuring contamination between photometric redshift bins usingthe angular correlation function, and without any need for spectroscopically determined redshifts. Themethod is able to constrain the true redshift distribution and the true average redshift in a photometricbin, both of which are of keen interest to cosmological use of these data. The accuracy of this methodwill need to be improved to address the needs of high precision cosmology. The inclusion of the galaxy-shear correlation function to break parameter degeneracies has been investigated by Zhang et al. (2009),showing that the stringent requirements of future surveys can be reached if this information is included.Without the need for accurate weak lensing shear measurements, the method we present here is moreaccessible and provides valuable information.47Chapter 3CFHTLenS tomographic weak lensing:quantifying accurate redshiftdistributions13.1 IntroductionWeak gravitational lensing by large-scale structure provides valuable cosmological information that canbe obtained by analysing the apparent shapes of distant galaxies that have been coherently distortedby foreground mass (Bartelmann & Schneider, 2001). Since weak lensing is sensitive to the distance-redshift relation and the time-dependent growth of structure, it is a particularly useful tool for constrain-ing models of dark energy (Albrecht et al., 2009, 2006; Peacock et al., 2006). To measure the contribu-tion of dark energy over time, the lensing signal must be measured at several redshifts, this is knownas weak lensing tomography (see for example, Hu, 1999; Huterer, 2002). Several observations of weaklensing tomography have been completed (Bacon et al., 2005; Massey et al., 2007; Semboloni et al.,2006). Most recently a study of the Cosmic Evolution Survey (COSMOS) by Schrabback et al. (2010)found evidence for the accelerated expansion of the Universe from weak lensing tomography.Redshift information is vital to weak lensing interpretation since the distortion of light bundles isa geometric effect and the growth of structure is redshift-dependent. Weak lensing data sets necessi-tate the use of photometric redshifts due to the large number of galaxies they contain. Spectroscopicredshifts typically exist for a small and relatively-bright fraction of galaxies, providing a training setfor photometric redshifts at brighter magnitudes. Several approaches for determining the redshift dis-tribution of galaxies have been used in past weak lensing studies. Many early studies (see for example,Bacon et al., 2003; Benjamin et al., 2007; Fu et al., 2008; Hamana et al., 2003; Hoekstra et al., 2006;Jarvis et al., 2003; Van Waerbeke et al., 2005, 2002), lacking multi-band photometry, relied on external1A version of this chapter has been published. J. Benjamin, L. Van Waerbeke, C. Heymans, M. Kilbinger, T. Erben, H.Hildebrandt, H. Hoekstra, T. D. Kitching, Y. Mellier, L. Miller, B. Rowe, T. Schrabback, F. Simpson, J. Coupon, L. Fu, J.Harnois-De?raps, M. J. Hudson, K. Kuijken, E. Semboloni, S. Vafaei, and M. Velander. CFHTLenS tomographic weak lensing:Quantifying accurate redshift distributions, 2013, MNRAS, Volume 431, Issue 2, pp.1547-156448photometric redshift samples such as the Hubble Deep Field North and South, and the CFHTLS-Deepfields. Due to the small area of these fields, sampling variance was an important, but often neglected,source of error in these studies, as presented by Van Waerbeke et al. (2006).Current and planned weak lensing surveys have multi-band photometry enabling photometric red-shift estimates for all galaxies. Methods for measuring photometric redshifts use various model-fittingtechniques with the goal of finding a match between the observed photometry and template galaxy spec-tra which are displaced in redshift and convolved with the optical response of the filter set, telescope, andcamera. Depending on the set of photometric filters, degeneracies can exist between different templatespectra at different redshifts. We refer to large errors in the best-fitting parameters due to mismatchesunder these degeneracies as catastrophic errors. The effect of catastrophic errors on weak lensing pa-rameter constraints has been investigated in several studies, for example Bernstein & Huterer (2010);Ma et al. (2006) and Hearin et al. (2010). Using a detailed Fisher matrix analysis, Hearin et al. (2010)show the importance of properly characterising catastrophic errors to dark energy parameter constraintsusing weak lensing tomography. The implication of neglecting these errors is not well known, althoughHearin et al. (2010) argue that there are many factors governing the final impact on dark energy param-eters and each survey needs to be carefully considered to make any definitive statement. It is clear thatcatastrophic errors will become increasingly important in the next generation of weak lensing cosmicshear surveys.In this chapter we present a tomographic weak lensing analysis of the Canada-France-Hawaii Tele-scope Lensing Survey2 (CFHTLenS), with redshifts measured in Hildebrandt et al. (2012) using theBayesian photometric redshift code (BPZ, Ben??tez, 2000). The BPZ analysis of the CFHTLenS pho-tometry uses a set of 6 recalibrated spectral energy distribution galaxy templates from Capak et al.(2004) and a magnitude dependent prior on the redshift distribution (see Hildebrandt et al., 2012, forfurther details). If the galaxy template set and prior used are an accurate and complete representationof the true galaxy population at all redshifts, then the probability distribution function (PDF) calculatedusing BPZ determines the true error distribution. The redshift distribution of a galaxy sample can thenbe calculated from the sum of the PDFs to determine an accurate redshift distribution that includes theeffects of both statistical and catastrophic errors. This is in contrast to the standard method of usinga histogram of photometric redshifts taken from the maximum of the posterior. We test the accuracyof the summed PDFs with overlapping spectroscopic redshifts at bright magnitudes and with resam-pled COSMOS-30 redshifts (Ilbert et al., 2009) at faint magnitudes. In both cases we also assess thelevel of contamination between redshift bins using the angular cross-correlation technique presented inChapter 2.Demonstrating the accuracy of the summed PDFs is of particular interest to those using theCFHTLenS data products. However, the results in this chapter also contribute to a broader un-derstanding of the use of redshift PDFs as measures of the redshift distribution. The CFHTLenSphotometric redshifts are measured using BPZ (Ben??tez, 2000), however, other methods also produceposterior probability distribution functions such as the Zurich Extragalactic Bayesian Redshift Analyzer2http://www.cfhtlens.org49(ZEBRA) photometric redshift code (Feldmann et al., 2006) and a photometric redshift-independentmethod presented in Lima et al. (2008) and extended in Cunha et al. (2009). The Cunha et al. (2009)method estimates a probability distribution based on the redshifts of nearest neighbour galaxies inmulti-dimensional phase-space. There have been several studies focused on using redshift PDFs asestimates of the redshift distribution. Brodwin et al. (2006) show that the summed PDFs of galaxiescan be used instead of the maximum likelihood values, as a better estimate of the redshift distributionof galaxies. This was tested on simulated galaxy samples using a Monte Carlo technique. Using thefull PDF for galaxies has been shown to dramatically reduce the weak lensing calibration bias forgalaxy-galaxy lensing (Mandelbaum et al., 2008). Wittman (2009) presents a technique to estimate anunbiased redshift using the PDF of a galaxy, effectively correcting the maximum likelihood value. Thework in this chapter builds on these results and most notably we will use two methods to assess theaccuracy of the summed PDFs at faint magnitudes where spectroscopic redshift coverage does not exist.Previous CFHT Legacy Survey (CFHTLS) results were found to be biased, underestimatingthe shear at high redshifts, and requiring the addition of a nuisance parameter when model fitting(Kilbinger et al., 2009). Furthermore, the field selection excluded fields based on cosmology dependentcriterion which possibly biased cosmological constraints. The CFHTLenS catalogues we use in thischapter have been thoroughly tested for systematic errors. These tests are cosmology insensitive andwere completed without any cosmological analysis of the data (Heymans et al., 2012b). One of theprimary goals of this chapter is to demonstrate that the redshift scaling of the shear is consistent withexpectations. We limit our cosmic shear analysis to two broad redshift bins in order to obtain param-eter constraints that do not depend on the modelling of intrinsic alignment (Croft & Metzler, 2000;Heavens et al., 2000; Hirata & Seljak, 2004b). A study of cosmological constraints from CFHTLenSwith several redshift bins, accounting for intrinsic alignment is presented in Heymans et al. (2012a).Kilbinger et al. (2013) present a thorough investigation of 2D cosmic shear, including a comparisonof all popular second order shear statistics. Simpson et al. (2013) use the tomographic shear signalpresented in this chapter to constrain deviations from General Relativity on cosmological scales.CFHTLenS has an effective area of 154 square degrees with deep photometry in five broad bandsu?, g?, r?, i?, and z? and a 5? point source limiting magnitude in the i?-band of i?AB?25.5. These data wereobtained as part of the CFHTLS, which completed observations in early 2009. Details of the fields arepresented in Figure 3.1. Heymans et al. (2012b) present an overview of the CFHTLenS analysis pipelinesummarizing the weak lensing data processing with THELI (Erben et al., 2012), shear measurementwith lensfit (Miller et al., 2013) and photometric redshift measurement from PSF-matched photometry(Hildebrandt et al., 2012) using BPZ. Each galaxy in the CFHTLenS catalogue has a shear measurement?obs, an inverse variance weight w, a PDF giving the posterior probability as a function of redshift, and aphotometric redshift estimate from the peak of the PDF zp. The shear calibration corrections describedin Miller et al. (2013) and Heymans et al. (2012b) are applied and we limit our analysis to the 129of 171 pointings that have been verified as having no significant systematic errors through a series ofcosmology-insensitive systematic tests described in Heymans et al. (2012b).This chapter is organized as follows, in Section 3.2, we use a series of tests to determine whether50Figure 3.1: CFHTLenS is composed of the four wide fields from CFHTLS labelled W1, W2, W3,and W4. The number of pointings in each field varies with 72 in W1, 33 in W2, 49 in W3,and 25 in W4 for a total of 179 pointings. Full photometry was not obtained for the 8 greypointings in W2 and they are not used in CFHTLenS. Overlap between fields and masking(Erben et al., 2012) brings the effective area of the remaining 171 pointings to 154 squaredegrees. The enclosed areas in W1 and W4 denote regions with spectroscopic coverage(Hildebrandt et al., 2012).51the PDFs are sufficiently accurate to determine the redshift distributions for the many different scienceanalyses of the CFHTLenS data set, and then apply our findings to the first tomographic analysis of theCFHTLenS data set in Section 3.3. We investigate the effect of non-linear modelling of the mass powerspectrum and baryons on our tomographic weak lensing results in Section 3.4. Section 3.5 contains ourconcluding remarks.3.2 Galaxy redshift distributions determined from the photometricredshift PDFWhen considering the redshift of an individual galaxy, a best-fitting redshift is often measured from thepeak of the PDF (e.g. Hildebrandt et al., 2012; Ilbert et al., 2006). If many galaxies are considered, thesum of their PDFs can be used as an estimate of the redshift distribution instead of the distribution ofbest-fitting redshifts. We show in this section that, by using information from the entire PDF, we achievean accurate model of the redshift distribution. The accuracy of the PDFs is not known a priori since thisdepends on whether the template spectral energy distributions and prior information are a representativeand complete description of the galaxies in the survey.We compare the summed PDFs against several other methods of measuring the redshift distribution.These methods include a comparison with the overlapping VVDS and DEEP2 spectroscopic redshifts(see Section 3.2.1), statistical resampling of the CFHTLenS photometric redshifts using the COSMOS-30 redshifts (see Section 3.2.2), and a photometric redshift contamination analysis (see Section 3.2.3).We divide the data into six redshift bins and measure the redshift distribution of each. We arelimited in the total number of bins by the pairwise contamination analysis, which breaks down for largernumbers of bins (see Section 3.2.3 for a more detailed discussion). The redshifts are most reliable in therange 0.1 < zp < 1.3 where comparison to spectroscopic redshifts, for i? < 24.5, shows the scatter to be0.03 < ??z < 0.06, with an outlier rate of less than 10 per cent (Hildebrandt et al., 2012). Here ? 2?z isthe variance in the value of ?z, which is given by?z = zp ? zs1+ zs, (3.1)where zp and zs are the photometric and spectroscopic redshifts, respectively.The redshift bins are chosen such that each bin is approximately four times wider than the photo-metric redshift error 0.04(1+ zp). This is done to avoid excessive contamination between adjacent bins.Since the high confidence photometric redshift range ends at z = 1.3 we ensure this is a bin edge. Forzp > 1.3 there are only a small number of galaxies, making a subdivision of this range difficult. Simi-larly, a lack of objects at low redshift prevents us from selecting z = 0.1 as a bin edge, despite this beingthe boundary of the high confidence region. The six bins are as follows:Bin 1: 0.00 < z1 ? 0.17Bin 2: 0.17 < z2 ? 0.38Bin 3: 0.38 < z3 ? 0.62Bin 4: 0.62 < z4 ? 0.905200 0.1 0.4 0.9 1.6 2.5 3.5nzz4=(0.62,0.90]P-value=0.90.1 0.4 0.9 1.6 2.5 3.5zz5=(0.90,1.30]P-value=0.70.1 0.4 0.9 1.6 2.5 3.5zz6=(1.30,7.00]P-value=4x10-80nz1=(0.00,0.17]P-value=0.35spec-zsummed PDFsz2=(0.17,0.38]P-value=0.35z3=(0.38,0.62]P-value=0.6Figure 3.2: Comparison of the predicted redshift distributions within each broad photometric red-shift bin, labelled zi. A magnitude cut of i? < 23.0 is used for comparison with spectroscopicredshifts. Solid lines (pink) show the summed PDFs for all galaxies within a given photomet-ric redshift bin. Dashed lines (green) show the spectroscopic redshift distribution. The listedP-values are the result of a two-sample Kolmogorov-Smirnov test of the distributions, weadopt a significance level of ? = 0.05 rejecting the null hypothesis that the two distributionsare drawn from the same population for the highest redshift bin.Bin 5: 0.90 < z5 ? 1.30Bin 6: 1.30 < z63.2.1 Comparison with spectroscopic redshiftsWe begin by investigating the redshift distribution given by spectroscopic redshifts. Spectroscopic red-shifts from the VIMOS3 VLT4 Deep Survey (VVDS, Le Fe`vre et al., 2005) and the DEEP2 galaxyredshift survey (Newman et al., 2012) overlap with CFHTLenS and were used to test the photometricredshifts. For a given photometric redshift bin we can select those galaxies that have spectroscopicredshifts and examine their redshift distribution. The spectroscopic sample is complete for i? . 22.0,dropping to ?90 per cent completeness for i? < 23.0. We adopt the latter cut to ensure that there are asufficient number of galaxies for our analysis. The catalogues are also cut to exclude objects on maskedregions and those that are flagged as stars. Stars are selected with star flag (see Erben et al., 2012,for more details). Due to the dithering pattern, which ensures that exposures exist between individualCCD chips, there is a variable number of exposures over a single pointing. This changing photometricdepth is difficult to account for when constructing a random catalogue with the same properties, whichis necessary for the contamination analysis presented in Section 3.2.3. To avoid this complexity, a finalcut is made to select galaxies on areas of the sky that were detected during every exposure, and random3VIsible imaging Multi-Object Spectrograph4Very Large Telescope array5300 0.1 0.4 0.9 1.6 2.5 3.5nzz4=(0.62,0.90]P-value=0.90.1 0.4 0.9 1.6 2.5 3.5zz5=(0.90,1.30]P-value=0.250.1 0.4 0.9 1.6 2.5 3.5zz6=(1.30,7.00]P-value=0.0050nz1=(0.00,0.17]P-value=0.2resampled zsummed PDFz2=(0.17,0.38]P-value=0.4z3=(0.38,0.62]P-value=0.99Figure 3.3: Comparison of the predicted redshift distributions with a magnitude cut of i? < 24.7.Solid lines (pink) show the summed PDFs for all galaxies within a given photometric redshiftbin. Dot-dashed histogram (cyan) shows the result of resampling the CFHTLenS redshiftsusing the constructed conditional probability P(z30|zp). The P-values are the result of a KStest, we reject the null hypothesis for the highest redshift bin at ? = 0.05.objects are placed only in these areas. We do not expect this to bias our results as there is no correlationbetween the physical properties of a galaxy and which part of the CCD mosaic it was observed on.A comparison of redshift distributions for i? < 23.0 is presented in Figure 3.2. For each redshiftbin we show the redshift distribution predicted by the summed PDF (solid line), and the spectroscopicredshift distribution (dashed line). The PDFs of all galaxies within a given redshift bin are summed andthe resulting distribution normalised to obtain the solid line. If the summed PDF is a good representationof the true error distribution, then we would expect this distribution to agree with the redshift distributionmeasured with the spectroscopic redshifts.We use the Kolmogorov-Smirnov two-sample test (KS test) to determine if the two distributionsin Figure 3.2 are consistent with being drawn from the same population (details of this test can befound in, for example, Wall & Jenkins, 2003). Before performing the test we adopt as a discriminatingcriterion a significance level of ? = 0.05. The P-values found from the KS test are presented as labels inFigure 3.2. We find that the distributions for the first five redshift bins are consistent with having beendrawn from the same population at a significance level of ? = 0.05. However, we can reject the nullhypothesis for the last redshift bin at the same level of significance, indicating that the two distributionsare significantly different. This is indicative of the large uncertainties in the photometric redshifts atzp > 1.3 and confirms the choice of this cut-off made by Hildebrandt et al. (2012).3.2.2 Comparison with COSMOS photometric redshiftsThe agreement at bright magnitudes shown in Figure 3.2 is encouraging; however, the majority oflensing studies include fainter galaxies, for example a magnitude limit of i? < 24.7 is adopted for the54measurement of CFHTLenS galaxy shapes (Miller et al., 2013). Therefore, we wish to investigate theredshift distribution with this deeper magnitude cut. The spectroscopic redshift sample described inSection 3.2.1 cannot be used for comparison since the completeness of this sample drops sharply beyondi? ? 23. Instead, we use the COSMOS-30 photometric redshift catalogue (Ilbert et al., 2009), which isaccurate to ??z ? 0.012 due to 30 bands of wavelength coverage from the ultraviolet to the mid-infrared.Quoting values for the Subaru i-band, the COSMOS-30 data are 99.8 percent complete for i < 25.5, andhave a 5? point source limiting magnitude of i ? 26.2 (Ilbert et al., 2009). A resampling procedureis used to estimate the redshift distribution of deep CFHTLenS galaxies based on the distribution ofCOSMOS-30 redshifts with CFHTLS overlap.Although the 1.6 square degree COSMOS field contains the one square degree CFHTLS-Deep fieldD2, there are no overlapping CFHTLS-Wide fields. Therefore, it is not possible to directly matchCFHTLenS galaxies to objects in the COSMOS-30 catalogue. This issue can be circumvented in anovel way using the photometric catalogue of D2 provided in Hildebrandt et al. (2009). Using thefact that the photometric systems for the CFHTLS-Wide and Deep data are identical we add randomGaussian noise, scaled to simulate data taken at CFHTLS-Wide depth, to the magnitude estimates in theD2 photometric catalogue. Using artificially-degraded catalogues generated in this way, we calculate aWide-like photometric redshift estimate zp using the maximum of the posterior distribution as describedin Hildebrandt et al. (2012). This is done for each D2 object in a catalogue matched to the COSMOS-30catalogue of Ilbert et al. (2009), employing an association radius of 1.0 arcsecond. The COSMOS-30redshifts in the matched catalogue are labelled z30.This matched catalogue of z30 and noise-degraded Wide-like zp estimates can then be used to ac-quire information about the joint probability distribution of COSMOS-30 and CFHTLenS redshifts. Wegenerate 100 realisations of the artificially degraded Wide-like catalogues, running the Bayesian pho-tometric redshift estimation of Hildebrandt et al. (2012) for each realisation. Using this ensemble of(z30,zp) pairs, we construct a two-dimensional histogram of galaxy number counts in square bins ofwidth 0.0025 in redshift for both z30 and zp. This histogram is then used as an empirical estimate of theconditional probability density function P(z30|zp) and allows us to estimate the corresponding cumula-tive probability distribution function P(< z30|zp) for each zp bin. Then, using inversion sampling fromP(< z30|zp) with a uniform pseudo-random number generator, samples of redshifts distributed accordingto P(z30|zp) can be drawn.With the assumption that P(ztrue|zp) = P(z30|zp), the contamination in tomographic redshift binscan be estimated by resampling CFHTLenS redshifts according to P(z30|zp). The resulting redshiftdistributions predicted from this method are given as dot-dash lines in Figure 3.3, and the summedPDFs are presented as solid lines. Note that the fine structure seen in the resampled redshifts is due tostructures in the COSMOS field and does not represent real structures in the distribution of CFHTLenSgalaxies. The small size of COSMOS means that it is limited by sample variance and individual clustersare able to leave an imprint in the resampled galaxies. This only affects the fine details of the resampledredshifts leading to a breakdown of the assumption P(ztrue|zp) = P(z30|zp) for small redshift intervals.We again adopt the null hypothesis that the two distributions are drawn from the same population.55Using a KS test we find that the null hypothesis can be rejected at a significance level of ? = 0.05 forthe zp > 1.3 redshift bin, but not for any of the other bins. The P-values found from the KS test arepresented as labels in Figure 3.3. Our results again confirm that the CFHTLenS photometric redshifts ofthe zp > 1.3 galaxies are unreliable. However, we find no evidence that the galaxies at zp < 0.1, which isthe lower limit for the high-confidence redshift range (Hildebrandt et al., 2012), are unreliable. This islikely because our lowest redshift bin extends to zp = 0.17 and is therefore dominated by galaxies withwell-measured photometric redshifts.3.2.3 Redshift contamination from angular correlation functionsIn order to further test the accuracy of the photometric redshift PDFs, we measure the redshift contami-nation using the angular cross-correlation technique detailed in Chapter 2. This method does not rely onmany assumptions and is sensitive to any contamination between redshift bins. Since it only relies onthe angular correlation function of the galaxies it is independent of the other methods used and servesas a critical test.Overview of methodGalaxies cluster in over-dense regions, leading to an excess in the number of pairs found at a separation? when compared to a random distribution of points. The two-point angular correlation function ?(?)quantifies this excess probability of finding pairs. A common estimator (Landy & Szalay, 1993) is?ij =(DiDj)?(RR)?NRNRNiNj? (DiR)?(RR)?NRNi? (DjR)?(RR)?NRNj+1, (3.2)where (DiDj)? is the number of pairs separated by a distance ? between data sets i and j, (RR)? is thenumber of pairs separated by a distance ? for a random set of points, (DiR)? is the number of pairsseparated by a distance ? between data set i and a random set of points, NR is the number of pointsin the random sample, and Ni (Nj) is the number of points in data sample i (j). The auto-correlation isdescribed by the case i=j, and the cross-correlation by the case i6=j. Our analysis would hold for anyestimator of the angular correlation function.In the ?CDM model, galaxies in well-separated non-overlapping redshift bins are not significantlyclustered with each other. Therefore, clustering between these bins should be consistent with a randomdistribution of points, resulting in ?ij = 0. Adjacent redshift bins will have a small positive ?ij owing togalaxy clustering at their shared edge, which becomes more pronounced for narrow redshift bins. If anynon-zero angular cross-correlation is detected between the photometric redshift bins, they must sharegalaxies with similar redshifts.As shown in Chapter 2, this simple realisation can be exploited to estimate contamination betweenphotometric redshift bins. The reader is referred to that work for the full details of the method. Here wepresent only a few key equations and concepts before we apply the method to the CFHTLenS data.The contamination fraction, fij is defined as the number of galaxies contaminating bin j from bin i asa fraction of the total number of galaxies NTi which have a spectroscopic redshift that lies within redshift56bin i. If there is no overlap or contamination between redshift bins, fij = 0 when i 6= j, and NTi = Noi ,where Noi is the total number of galaxies which have a photometric redshift that lies within redshiftbin i. In the standard case of overlapping photometric redshift bins, the contamination fraction relatesthe observed number of galaxies in each photometric redshift bin Noi to the true underlying number ofgalaxies NTi as follows;??????No1No2. . .Nom??????=??????f11 f21 . . . fm1f12 f22 . . . fm2. . . . . . . . . . . . . . . . . . .f1m f2m . . . fmm????????????NT1NT2. . .NTm??????, (3.3)where m is the number of redshift bins and fii = 1??mk6=i fik. We determine the contamination fractionsfij from measurements of ?oij , the observed two-point correlation function between photometric redshiftbins i an j. For two redshift bins it can be shown that,?o12 =?o11(No1No2)f12(1? f21)+?o22(No2No1)f21(1? f12)(1? f12)(1? f21)+ f12 f21 . (3.4)Since ?o12 is measured at multiple angular scales it is possible to determine the contamination fractionsf12 and f21. Degeneracy exists between f12 and f21, which can be broken if the angular auto-correlationfunctions ?o11 and ?o22 have significantly different shapes. We expect this to be the case for the angularcorrelation functions of galaxy samples at different redshifts.When considering more than two redshift bins we measure the contamination fractions using Equa-tion (3.4) for each pair of bins in turn. This pairwise approximation assumes that higher-order con-tamination can be safely ignored, that is, the angular cross-correlation is not affected by the mutualcontamination of the pair of bins by another redshift bin. As the number of bins increases or if thecontamination fractions become large, this assumption is no longer valid and the method breaks down.Once the contamination fractions fij have been measured, we can invert5 the contamination matrixin Equation (3.3) to determine the true underlying number of galaxies in each redshift range NTi from theobserved number of galaxies in each photometric redshift bin Noi . The true redshift distribution ni(zj) foreach photometric redshift bin i is then calculated over the full redshift range, sampled at each redshift zjfromni(zj) = fijNTi . (3.5)Contamination analysisIn order to estimate the contamination fractions in the CFHTLenS data the angular correlation functionsmust be measured. Brute-force pair counting algorithms are O(N2), where N is the number of galaxies,which for our data-set results in prohibitively large computation times. The publicly available codeATHENA6 employs a tree data structure to increase the speed of pair counting to O(Nlog(N)) at the5Since we expect the non-diagonal contamination fractions to be small the matrix should be diagonally dominant andtherefore invertible.6 of accuracy. The level of approximation is parametrized by the opening angle. Larger valuesindicate larger approximations with an opening angle of zero representing no approximation. Galaxiesare grouped together into nodes in the tree data structure based on angular position. The structure isa hierarchy with the nodes on top containing more galaxies. The opening angle determines when todescend to lower nodes and higher spacial resolution. Tests of ATHENA against a more simplistic androbust algorithm are used to determine that with an opening angle of 0.03 we are making at most a oneper cent error on the angular correlation function. This value of opening angle is used when measuringthe angular correlation function.For each pointing we measure the angular correlation function in six angular bins spaced logarith-mically on the range 0.15 < ? < 30 arcmin. Above 30 arcmin the signal is very small providing littleadditional information. For each pointing the contamination fractions are estimated via the angular cor-relation function as outlined in Section 3.2.3. The covariance is estimated via a bootstrap technique, withan additional contribution coming from the field-to-field variance for the angular cross-correlations. Thedetails of the maximum likelihood technique and covariance matrix are presented in Section 2.6. Thelikelihoods for the contamination fractions from each field are then combined with equal weighting.The following matrices contain the measured contamination fractions with 68 per cent confidenceregions. All values are multiplied by one hundred for ease of viewing. For the bright sample, i? < 23.0,we find,fij =???????????65?5 4?1 < 1 < 1 < 1 6?628?4 87?3 8?2 < 1 < 1 7?71?1 7?2 85?3 9?2 1?1 7?71?1 < 1 6?2 85?2 38?6 5?5< 1 < 1 < 1 4?1 56?7 29?123?1 < 1 < 1 < 1 3?1 18?16???????????. (3.6)With a cut of i? < 24.7 we measure,fij =???????????42?8 3?2 < 1 1?1 < 1 9?632?4 75?3 8?2 < 1 < 1 1?1< 1 18?2 79?3 7?2 < 1 1?14?4 < 1 11?2 78?3 20?3 3?3< 1 < 1 < 1 9?2 73?4 42?518?4 2?1 < 1 4?1 5?3 36?6???????????. (3.7)Many of the contamination fractions are one per cent deviations from zero, which is expected since wehave this level of uncertainty in our estimation of the angular correlation functions. Note that the ith col-umn contains the location of all bin i galaxies. Due to the pairwise treatment of redshift bins columns donot sum to exactly 100 per cent. These matrices are extremely well conditioned with condition numbersof 8.72 and 6.35 for the i? < 23.0 and i? < 24.7 cases respectively, indicating that matrix inversion isnumerically stable and does not contribute a significant uncertainty to the solution of Equation (3.3).With the contamination fractions measured, the true number of galaxies in each redshift bin can5800 0.1 0.4 0.9 1.6 2.5 3.5nzz4=(0.62,0.90]0.1 0.4 0.9 1.6 2.5 3.5zz5=(0.90,1.30]0.1 0.4 0.9 1.6 2.5 3.5zz6=(1.30,7.00]0nz1=(0.00,0.17]spec-zcontaminationsummed PDFsz2=(0.17,0.38] z3=(0.38,0.62]Figure 3.4: Comparison of the predicted true redshift distribution within each broad photometricredshift bin, labelled zi. A magnitude cut of i? < 23.0 is used for comparison with spectro-scopic redshifts. All horizontal error bars denote the width of the redshift bin and points areoffset horizontally for clarity. Crosses with solid lines (pink) denote the summed PDFs whenintegrated within a given broad redshift bin, the error is calculated as the standard deviationfrom 1000 bootstrap samples. Filled circles with dotted lines (blue) show the result from ourcontamination analysis with 68 per cent confidence region. Filled squares with dashed lines(green) show the spectroscopic redshift data integrated within each broad redshift bin. Theerror is the standard deviation of 1000 bootstrap calculated from Equation (3.3). The redshift distribution is then found from Equation (3.5). This isdone with a Monte Carlo procedure for finding global solutions to the contamination matrix presented inSection 2.4.4. We can now compare our contamination results with those found from the spectroscopicredshifts and the COSMOS-30 photometric redshifts. However, since our contamination results exist inonly six redshift bins we must also sum the distributions shown in Figures 3.2 and 3.3 within these sixredshift bins.For i? < 23.0 we present our contamination results in Figure 3.4. For each redshift bin we showthe redshift distribution predicted by the PDFs (crosses with solid lines), the contamination analysis(filled circles with dotted lines), and the spectroscopic redshift distribution (filled squares with dashedlines). The horizontal error bars on all points denote the width of the redshift bins. The vertical scaleis proportional to the number of galaxies but uses arbitrary units. For the summed PDF, the verticalerror bar is calculated as the standard deviation of the summed PDFs for 1000 bootstraps of the galaxieswithin each bin zi (this error bar is smaller than the points in Figure 3.4). Note that given the largenumber of galaxies in each bin (?40,000? 600,000) the statistical error of the summed PDF is verysmall. The vertical error bars on the contamination results enclose the 68 per cent confidence regionwhich comes from a procedure for finding global solutions to the contamination (Section 2.4.4). Thevertical error on the spectroscopic redshift distribution is taken as the standard deviation from 1000bootstraps of the spectroscopic catalogue. For both cases where bootstraps are used we verified that5900 0.1 0.4 0.9 1.6 2.5 3.5nzz4=(0.62,0.90]0.1 0.4 0.9 1.6 2.5 3.5zz5=(0.90,1.30]0.1 0.4 0.9 1.6 2.5 3.5zz6=(1.30,7.00]0nz1=(0.00,0.17]resampled zcontaminationsummed PDFsz2=(0.17,0.38] z3=(0.38,0.62]Figure 3.5: Same as Figure 3.4 except for the following differences. A magnitude cut of i? < 24.7is used. Filled squares with dot-dashed line (cyan) show the resampled COSMOS-30 dataintegrated within each broad redshift bin. The error is given as the standard deviation of the100 low-resolution reconstructions (see Section 3.2.2).1000 bootstraps yields stable error estimates. For each bootstrap, objects are sampled with replacementand the resulting redshift distributions measured, the total number of galaxies sampled is equal to thenumber in the original catalogues.Figure 3.4 shows the predicted redshift distribution for each of the six redshift bins used. The con-tamination points for a given sub-plot are contained within the corresponding row of the contaminationmatrix in Equation (3.6). For example, the top row shows that the majority of galaxies from bin 1 remainin bin 1, with f11 = 65?5 per cent. Contamination from other bins is less than the per cent level exceptfor the neighbouring bin f21 = 4? 1 per cent and the highest redshift bin f61 = 6? 6. Keep in mindthat the relative heights of points in the z1 sub-plot do not follow these contamination values becausethe contamination fij represents the number of galaxies in bin j from bin i divided by the true numberin bin i. However, investigating the matrix in relation to Figure 3.4 can help in grasping the presentedinformation. We expect that the spectroscopic redshift distribution is the true distribution, assuming thatthe limited area of the spectroscopic samples does not bias the results, which is a reasonable assump-tion for our purposes. The contamination model is in poor agreement with the spectroscopic samplein the z1 and z6 sub-plots. For z1 the contamination model underpredicts the contamination from bin 2to bin 1, underestimating f21 as evidenced by the discrepancy between the contamination point and thespectroscopic point in the second bin of the z1 sub-plot. Similarly the contamination is overpredictedfor f12 which is seen in the first bin of the z2 sub-plot. This represents a fundamental degeneracy in theangular cross-correlation method. Although an angular cross-correlation is detected between these twobins, unless the angular auto-correlations have significantly different slopes, the method cannot distin-guish easily between bin 1 galaxies contaminating bin 2 or vice versa. A similar degeneracy explainsthe discrepancies in the z6 sub-plot, there we see that f56 predicted by the angular cross-correlation is60too low and f65 in the z5 sub-plot is too high. The contamination between these bins is detected but thedirection of scatter is misidentified.We use a KS test to determine if the distributions in Figure 3.4 are consistent with being drawn fromthe same population. Since there are three distributions and the KS test is a two-sample test we applyit to each pair of distributions. Furthermore, due to the small number of bins we must rely on tabulatedcritical values which exist for very few significance levels, therefore we are not able to list the P-valuesfor each redshift bin. For each pair of distributions we find that we cannot reject the null hypothesis(drawn from the same population) at a significance level of ? = 0.05 for any of the redshift bins.We present the results of the contamination analysis for i? < 24.7 in Figure 3.5. The summedPDF and contamination results are presented similarly to Figure 3.4. The resampled redshifts usingCOSMOS-30 are given as the dot-dash line and filled squares. The vertical error on the COSMOS-30points is taken as the standard deviation of the 100 low-resolution resamplings (see Section 3.2.2). If wecompare the contamination results to the resampled redshifts the greatest discrepancies are for the z1,z5, and z6 sub-plots. The f12 ? f21 and f56 ? f65 degeneracies noted for the bright sample above appearagain in Figure 3.5. Additionally the resampled redshifts predict a larger f61 and smaller f16 than dothe contamination results which can be seen in the first and last bins of the z1 and z6 sub-plots respec-tively. The contamination analysis predicts that a significantly lower number of galaxies belong in bin5 compared to the other methods, see bin 5 of the z5 sub-plot. However, this is not due to scatteringof bin 5 galaxies elsewhere (note that f55 = 73?4 per cent) instead the contamination analysis simplypredicts fewer galaxies occupying this bin. To determine if these differences are statistically significantwe use a KS test. We again have three distributions and apply the test between each pair. For each pairof distributions we find that we cannot reject the null hypothesis (drawn from the same population) at asignificance level of ? = 0.05 for any of the redshift bins.When using the finely binned spectroscopic and resampled redshifts in Sections 3.2.1 and 3.2.2 wewere able to reject the null hypothesis for the high redshift bin zp > 1.3. The smallest P-value foundwas in the high redshift bin when comparing the summed PDF with the spectroscopic distribution inFigure 3.2. Performing the same comparison with these distributions when summed within the six red-shift bins of the contamination analysis we are not able reject the null hypothesis. The coarse binningrequired by the contamination analysis has reduced the statistical power of the test. However, the con-tamination analysis provides a complementary estimation of the redshift distribution, which agrees wellwith the other estimates and strengthens our confidence in the summed PDF as an accurate measure ofthe redshift distribution.We conclude that the summed PDF can be used to estimate the redshift distribution for the high con-fidence redshift range 0.1 < zp < 1.3 determined by Hildebrandt et al. (2012). The comparison with theresampled COSMO-30 redshifts and the contamination analysis for i? < 24.7 suggest that an accurate es-timate of the redshift distribution, including statistical and catastrophic errors, can be obtained from thesum of the PDFs. This result suggests that the model galaxy spectra and priors used in Hildebrandt et al.(2012) are a fair and sufficiently complete representation for the population of galaxies studied here.613.3 Weak lensing tomographyIn this chapter we present an analysis of the CFHTLenS tomographic weak lensing signal using twobroad redshift bins and compare our results with a 2D analysis over the same redshift range. Setting ouranalysis in a flat ?CDM cosmology framework, the initial aim is to use the consistent results we findbetween successive tomographic bins as a demonstration that the CFHTLenS catalogues are not subjectto the redshift-dependent systematic biases that were uncovered in an earlier analysis of CFHTLS data(Kilbinger et al., 2009). This cosmology-dependent demonstration is the last in an extensive series oftests, which investigate the robustness and accuracy of the CFHTLenS catalogues. We stress, however,that this analysis was performed after the conclusion of a series of cosmology-insensitive tests presentedin Heymans et al. (2012b) and the photometric redshift accuracy analysis presented in Section 3.2. Mostimportantly, no feedback loop existed between this cosmology-dependent test and the systematics andimage simulation tests that determined the calibration corrections and the subset of reliable data that weuse from the survey.We choose to use two broad mid-to-high redshift bins for our tomographic analysis in order toreduce the potential contamination to the signal from intrinsic galaxy alignments (see, for example,Heavens et al., 2000; Heymans et al., 2012a, and references therein). We estimate the expected con-tamination of the measured weak lensing signal using the linear tidal field intrinsic alignment modelof Hirata & Seljak (2004b), and following Bridle & King (2007b) by fixing its amplitude to the ob-servational constraints obtained by Brown et al. (2002b). By limiting the redshift bins to photometricredshifts 0.5 < zp ? 0.85 and 0.85 < zp ? 1.3, we estimate that any contamination from intrinsic align-ments is expected to be no more than a few per cent for each redshift bin combination. We thereforeignore any contributions from intrinsic alignments in this analysis as they are expected to be small incomparison to our statistical errors. Note that a low level of contamination would not be expected ifwe instead used the 6 narrow redshift bins that were analysed in the redshift contamination analysis inSection 3.2. We present a fine 6-bin tomographic analysis of the data in Heymans et al. (2012a) wherethe impact of intrinsic galaxy alignments is mitigated via the simultaneous fit of a cosmological modeland an intrinsic alignment model. The findings of Heymans et al. (2012a) support the approach taken inthis chapter to neglect the contribution of intrinsic alignments for our choice of redshift bins.The 2D lensing analysis presented here is restricted to the same redshift range used in our tomo-graphic analysis 0.5 < zp ? 1.3. We measure the shear correlation function on angular scales from ?1to ?40 arcmin. The upper limit is set by our ability to measure the covariance matrix from simulations,see Section Overview of tomographic weak lensing theoryThe complex weak lensing shear ? = ?1+ i?2, which is directly analogous to the complex galaxy elliptic-ity, can be decomposed into two components: the tangential shear ?t and the cross component ?x. Theseare defined relative to the separation vector for each pair of galaxies, with ?t describing elongation andcompression of the ellipticity along the separation vector and ?x describing elongation and compressionalong a direction rotated 45? from the separation vector. The following shear-shear correlation functions62can then be computed:? k,l? (?) =?i,j[?kt,i(?i)? lt,j(?j)? ?kx,i(?i)? lx,j(?j)]wiwj?ij?i,jwiwj?ij, (3.8)where galaxy pairs labelled i,j are separated by angular distance ? = |?i ??j|. If ? falls in the angularbin given by ? then ?ij=1, otherwise ?ij=0. The labels k,l identify redshift bins. The summation isperformed for all galaxies i in bin k and all galaxies j in bin l. The contribution of each galaxy pairis weighted by its inverse variance weight wiwj. This gives greater significance to galaxy pairs withwell-measured shapes.Shear calibration is performed as described in Miller et al. (2013) and Heymans et al. (2012b). Thissignal-to-noise (S/N) and size-dependent calibration includes an additive (c) and a multiplicative (m)correction term as follows;?obs = (1+m)? true + c. (3.9)An average additive correction of 2?10?3 is found for ?2. The additive correction for ?1 is found to beconsistent with zero. The multiplicative correction to ?? is found by calculating the weighted correlationfunction of 1+m (Miller et al., 2013),1+Kk,l(?) =?i,j(1+mki )(1+mlj)wiwj?ij?i,jwiwj?ij. (3.10)The shear correlation functions ?? are then corrected by dividing them by 1+K.The shear-shear correlations can also be expressed as filtered functions of the convergence powerspectra? k,l+/?(?) =12pi??0d??J0/4(??)Pk,l? (?), (3.11)where Jn is the nth order Bessel function of the first kind and ? is the modulus of the two-dimensionalwave vector. These can be related to line-of-sight integrals of the three-dimensional matter power spec-trumPk,l? (?) =9H40 ?2m4c4? ?h0d? gk(?)gl(?)a2(?) P?( ?fK(?) ,?), (3.12)where c is the speed of light, ?m is the matter energy density, H0 is the Hubble constant, fK(?) is thecomoving angular diameter distance out to a distance ? , ?h is the comoving horizon distance, a(?) is thescale factor, and P? is the 3-dimensional mass power spectrum computed from a non-linear estimationof dark matter clustering (Smith et al., 2003b). The two terms, gk(?), are the geometric lens-efficiency,which depend on the redshift distribution of the sources, nk(? ?),gk(?) =? ?h?d? ?nk(? ?)fK(? ?? ?)fK(? ?) . (3.13)Given a cosmological model, matter power spectrum, and redshift distribution of the sources we63 0 0.5 1 1.5 2 2.5 3 3.5 0  0.5  1  1.5  2normalised frequencyzlow-zhigh-zlow+highFigure 3.6: Redshift distributions used in the weak lensing analysis. Low and high redshift binscorrespond to zp = (0.5,0.85] and zp = (0.85,1.3] respectively. Smooth curves show theresult of summing the photometric redshift probability distribution functions (PDFs) of allgalaxies within the respective redshift bin. The smooth solid and dashed curves are used inthe tomographic analysis and the sum of the PDFs over the entire redshift range is given bythe smooth dot-dashed line which is used in the 2D lensing analysis. For comparison, thehistograms show the redshift distribution obtained from the photometric redshifts.can model the shear correlation functions. Bayesian model fitting techniques are then used to obtain theposterior probability on the model vector given the observed shear correlation functions. We discussthis further in Section The tomographic weak lensing signalBased on the results presented in Section 3.2, the redshift distribution in each bin is taken to be the sumof the PDFs determined from the photometric redshift analysis of Hildebrandt et al. (2012). We referto the maximum posterior photometric redshift estimate as the ?photometric redshift?. The histogramof photometric redshifts and the sum of the PDFs for each redshift bin are presented in Figure 3.6.Note that the summed PDFs extend to lower and higher redshifts then the photometric redshifts do,broadening the range below z = 0.5 and above z = 1.3. The summed PDFs for the two redshift bins alsooverlap considerably with one another. The average redshift from the summed PDFs is 0.7 for the lowredshift bin and 1.05 for the high redshift bin. For the photometric redshifts we find 0.69 and 1.03 forthe low and high redshift bins respectively. The average redshift for both bins taken together is found tobe 0.87 from the summed PDFs and 0.84 from the photometric redshifts.We use ATHENA with an opening angle of 0.02 to measure the shear-shear correlation function.We have tested that the difference to the shear-shear correlation function when using an opening angle6410-610-510-410-3?1,1low-low ?+?-10-610-510-4?1,2low-high10-710-610-510-41 10?2,2? (arcmin)high-highFigure 3.7: The filled circles with solid lines and filled squares with dashed lines show the mea-sured signal for ?+ and ?? respectively. Each panel shows the shear correlation functionsfor a unique pairing of redshift bins. The top, middle and bottom panels correspond to lowredshift correlated with low redshift (low-low), low with high redshift (low-high), and highwith high redshift (high-high). Error bars are the square-root of the diagonal of the covari-ance matrix measured from mock catalogues (see Section 3.3.3). Theoretical predictions fora fiducial (WMAP7, Komatsu et al., 2011) cosmology are presented as lines; these are not thebest-fitting models. There are two negative data points for ?? in the top panel, their valuesare ?2.3?10?6 and ?4.9?10?6 for scales 1.34 and 2.18 arcminutes respectively.65Table 3.1: Details of the model dependent cosmological parameters for each of the consideredcosmologies. Parameter ranges denote hard priors. A flat distribution is used throughout therange. The bottom three parameters are constrained by WMAP7, and are required in order todeduce ?8.Parameter flat ?CDM curved ?CDM description?m [0,1.0] [0,1.2] Energy density of matter (baryons + dark matter).?8 [0.2,1.5] [0.2,1.5] Normalisation of the matter power spectrum.h [0.4,1.2] [0.4,1.2] The dimensionless Hubble constant h = H0100kms?1 Mpc?1 .?b [0,0.1] [0,0.1] Energy density of baryons.ns [0.7,1.3] [0.7,1.3] Slope of the primordial matter power spectrum.?? 1??m [0,2] Energy density of dark energy.w0 ?1 ?1 Constant term in the dark energy equation of state, w(a) = w0.? [0.04,0.20] [0.04,0.20] Reionisation optical depth.?2R [1.8,3.5] [1.8,3.5]Amplitude of curvature perturbations.Units of 10?9 times the amplitude of density fluctuations.ASZ [0.0,2.0] [0.0,2.0] Sunyaev-Zel'dovich template amplitude.of 0.02 compared to a brute force calculation is negligible, approximately 8 per cent of the size ofthe errors. The signal is first measured on each of the four wide mosaics: W1, W2, W3, and W4,applying the shear calibration described in Section 3.3.1. The correlation functions are then combinedby calculating the weighted average. The weight for a given angular bin and wide mosaic is the sumof the inverse variance weight terms for each pair of galaxies. We present ?+ and ?? for each redshiftbin combination in Figure 3.7. The error bars correspond to the diagonal elements from the covariancematrix, discussed in more detail in Section 3.3.3. The lines are the theoretical prediction for a fiducialcosmological model using the Wilkinson Microwave Anisotropy Probe 7-year (WMAP7) best-fittingresults (Komatsu et al., 2011), hence the following parameter vector is used: (?m = 0.271, ?8 = 0.78,h = 0.704, ?b = 0.0455, ns = 0.967, ?? = 0.729, w0 = ?1). Descriptions of each parameter can befound in Table 3.1. To compute the theoretical models we employ the halo-model of Smith et al. (2003b)to estimate the non-linear matter power spectrum and the analytical approximation of Eisenstein & Hu(1998) to estimate the transfer function.Emphasizing that no cosmology-dependent systematic tests were used to vet the catalogues(Heymans et al., 2012b), Figure 3.7 demonstrates the robustness of the CFHTLenS catalogues. Thetomographic shear signal shows no evidence of a redshift-dependent bias as was seen in earlier CFHTLSdata analyses (Kilbinger et al., 2009). We discuss further tests of the redshift scaling of the shear inSection CosmologyFrom the signal measured in Section 3.3.2, cosmological parameters are estimated using COSMOPMC.COSMOPMC is a freely available7 Population Monte Carlo (PMC) code, which uses adaptive impor-7http://cosmopmc.info66tance sampling to explore the posterior (Kilbinger et al., 2011). COSMOPMC documentation can befound in Kilbinger et al. (2011); discussion of Bayesian evidence and examples of its application tovarious cosmological data sets can be found in Kilbinger et al. (2010) and Wraith et al. (2009). ThePMC method is detailed in Cappe? et al. (2007). The non-linear matter power spectrum is estimatedusing the halo-model of Smith et al. (2003b). The transfer function is estimated using the analyticalapproximation of Eisenstein & Hu (1998).We explore two cosmologies: a flat ?CDM universe and a curved ?CDM universe. Themodel-dependent data vector used with COSMOPMC contains the following seven parameters:(?m,?8,h,?b,ns,??,w0). Physical descriptions and priors are presented in Table 3.1. For the flat?CDM model we have a five-parameter fit, where we fix ?? = 1??m and w0 = ?1. For the curved?CDM model we have six free parameters as ?? is allowed to vary, while w0 remains fixed.Three other cosmological data sets are used to provide complementary constraining power. Con-straints from the cosmic microwave background (CMB) are taken from the 7-year results of WMAP(Komatsu et al., 2011, WMAP7). To obtain parameter constraints we use the publicly released WMAPlikelihood code. Baryon acoustic oscillation (BAO) data is taken from the Baryon Oscillation Spectro-scopic Survey (Anderson et al., 2012, hereafter referred to as BOSS). We consider the ratio DV/rs =13.67?0.22 of the apparent BAO at z = 0.57 to the sound horizon distance to be Gaussian distributed.The Hubble constant is constrained with the results from the HST distance ladder (Riess et al., 2011,hereafter referred to as R11). Following R11, we use a Gaussian prior of mean value h = 0.738 andstandard deviation ? = 0.024. For more details of these data sets see Kilbinger et al. (2013). WithWMAP7 the parameter set is expanded to include ? , ASZ, and ?2R , from which we deduce ?8. Priorranges and brief descriptions are given in Table 3.1. For further details see Komatsu et al. (2011) andreferences therein. Throughout this section when stating parameter values we quote the 68.3 per centconfidence level as the associated uncertainty with all other parameters marginalised over.Covariance matrixIn order to estimate a covariance matrix for our measured shear correlation functions in Equation (3.8),we analyse mock CFHTLenS surveys constructed from the three-dimensional N-body numerical lens-ing simulations of Harnois-De?raps et al. (2012). The 10243 particle simulations have a box size of147.0h?1 Mpc or 231.1h?1 Mpc, depending on the redshift of the simulation, and assume a flat ?CDMcosmology parametrized by the best-fitting constraints from Komatsu et al. (2009). There are a totalof 184 fully independent lines of sight spanning 12.84 square degrees with a resolution of 0.2 arcminsampled at 26 redshift slices between 0 < z < 3. The two-point shear statistics match the theoretical pre-dictions of the input cosmology from 0.5 < ? < 40 arcmin scales at all redshifts (Harnois-De?raps et al.,2012), this sets the upper angular limit for our tomographic analysis. See Heymans et al. (2012a)for a detailed discussion of covariance matrix estimation from the N-body simulations presented inHarnois-De?raps et al. (2012), including the required Anderson (2003) correction that we apply to de-bias our estimate of the inverse covariance matrix used in the likelihood analysis that follows.67Table 3.2: Constraints orthogonal to the ?m??8 degeneracy for a flat ?CDM cosmology. Resultsare shown with and without highly non-linear scales which are potentially biased due to non-linear modelling and the effects of baryons (see Section 3.4). ?All scales? refers to scales thecorrelation functions are measured on: 1 < ? < 40 arcmin. We remove scales correspondingto ?? < 10 arcmin in the case labelled ?removed: ?? < 10 arcmin?.Data ?8(?m0.27)??tomography:all scales 0.771?0.040 0.553?0.016removed: ?? < 10 arcmin 0.776?0.041 0.556?0.0182D Lensing:all scales 0.785?0.036 0.556?0.018removed: ?? < 10 arcmin 0.780?0.043 0.611?0.015Flat ?CDMWe present marginalised two-dimensional likelihood constraints in the ?m ??8 plane in Figure 3.8.The best constraint from weak lensing alone is for a combination of ?m and ?8, which parametrizes thedegeneracy. We find ?8(?m0.27)?= 0.771?0.040 with ? = 0.553?0.016.When combining CFHTLenS with WMAP7, BOSS, and R11 data sets, we find ?m = 0.2762?0.0074 and ?8 = 0.802?0.013. The precision is ?20 times better than for CFHTLenS alone where wefind ?m = 0.27? 0.17 and ?8 = 0.67? 0.23. Constraints on the full set of parameters are presentedin Table 3.3. We show the results for CFHTLenS tomography, CFHTLenS combined with WMAP7,BOSS and R11, and, to assess the contribution of our data set to these constraints, we include results forWMAP7 combined with R11 and BOSS. The most valuable contribution from CFHTLenS is for ?m,?8, and ?b, where we improve the precision of the constraints by an average factor of 1.5.For comparison we perform the analysis with a single redshift bin spanning the range of our 2-bin analysis, 0.5 < zp ? 1.3. We refer to this as the 2D lensing case, in contrast to the tomographiccase where we split the galaxies into two redshift bins. Figure 3.9 shows the marginalised parameterconstraints in the ?m ? ?8 plane for both 2D lensing and tomography, and the two cases result invery similar constraints. For 2D lensing we find ?8(?m0.27)?= 0.785? 0.036 and ? = 0.556? 0.018,which is in agreement with what we find for tomography (Table 3.2). When combining the 2D lensingresults from CFHTLenS with WMAP7, BOSS, and R11 data sets, we find ?m = 0.2774? 0.0074 and?8 = 0.810 ? 0.013, which are nearly identical to those found for tomography (listed above and inTable 3.3). This level of agreement is also found for all other parameters when combining CFHTLenSwith the other data sets. We note that all the parameter estimates agree within the 68.3 per cent errorsand the size of the error bars from 2D lensing are very similar to those found with tomography whencombining CFHTLenS with WMAP7, BOSS, and R11 data sets.For CFHTLenS alone the parameter estimates for 2D lensing and tomographic lensing agree witheach other within their 68.3 per cent uncertainties, however, the parameter constraints do not improve.68Table 3.3: Parameter constraints with 68.3 per cent confidence limits. The following parametersare deduced for CFHTLenS: ?K and q0. When combining data sets the deduced parametersare: ?8, ??, and q0. The label CFHTLenS+Others refers to the combination of CFHTLenS,WMAP7, BOSS, and R11.Parameter flat ?CDM curved ?CDM Data?m0.27?0.17 0.28?0.17 CFHTLenS0.288?0.010 0.285?0.014 WMAP7+BOSS+R110.2762?0.0074 0.2736?0.0085 CFHTLenS+Others?80.67?0.23 0.69?0.29 CFHTLenS0.828?0.023 0.819?0.036 WMAP7+BOSS+R110.802?0.013 0.795?0.013 CFHTLenS+Others??1??m 0.38?0.36 CFHTLenS1??m 0.717?0.019 WMAP7+BOSS+R111??m 0.7312?0.0094 CFHTLenS+Others?K0 0.19?0.43 CFHTLenS0 ?0.0020?0.0061 WMAP7+BOSS+R110 ?0.0042?0.0040 CFHTLenS+Othersh0.84?0.25 0.81?0.24 CFHTLenS0.692?0.0088 0.694?0.012 WMAP7+BOSS+R110.6971?0.0081 0.693?0.011 CFHTLenS+Others?b0.030?0.029 0.031?0.030 CFHTLenS0.0471?0.0012 0.0472?0.0016 WMAP7+BOSS+R110.04595?0.00086 0.0470?0.0015 CFHTLenS+Othersq0?0.57?0.27 ?0.29?0.40 CFHTLenS?0.568?0.016 ?0.574?0.025 WMAP7+BOSS+R11?0.585?0.011 ?0.594?0.014 CFHTLenS+Othersns0.93?0.17 0.91?0.17 CFHTLenS0.965?0.012 0.969?0.014 WMAP7+BOSS+R110.960?0.011 0.972?0.012 CFHTLenS+Others?0.086?0.014 0.086?0.015 WMAP7+BOSS+R110.081?0.013 0.085?0.015 CFHTLenS+Others?2R2.465?0.086 2.45?0.13 WMAP7+BOSS+R112.429?0.081 2.361?0.094 CFHTLenS+OthersASZ0.97?0.62 1.35?0.61 WMAP7+BOSS+R111.33?0.60 1.39?0.57 CFHTLenS+Others69Figure 3.8: Marginalised parameter constraints (68.3, 95.5, and 99.7 per cent confidence levels)in the ?m ??8 plane for a flat ?CDM model. Results are shown for CFHTLenS (blue),WMAP7 (green), CFHTLenS combined with WMAP7 (black), and CFHTLenS combinedwith WMAP7, BOSS and R11 (pink).With two broad overlapping redshift bins of average redshift 0.7 and 1.05, there appears to be insuffi-cient additional information to tighten parameter constraints. Previous estimates of the improvement inconstraints from weak lensing tomography use non-overlapping redshift bins, Gaussian covariance, andestimate errors using a Fisher matrix analysis (see Simon et al., 2004). For two redshift bins with z < 3and divided at z = 0.75 they find the ratio of the error on individual parameters from tomography tothose from a 2D analysis to be ?tomo/?2D = 0.88. With our overlapping redshift bins and non-Gaussiancovariance it is not surprising that this marginal improvement is significantly degraded. Additionally,our 2D lensing result is for redshifts 0.5< zp < 1.3, this removes low redshift galaxies with small signal-to-noise improving constraints and weakening the gains from dividing the redshift range. We thereforeexpect a modest improvement at best. We find that the ratio of our tomographic errors to our 2D lensingerrors is ?tomo/?2D = 1.16. The fact that we find larger errors for tomography is surprising and warrantsfurther discussion.70To test our covariance matrices we perform the analysis again, replacing the measured shear corre-lation function with that predicted from our model using a WMAP7 cosmology. In this case we find thatthe tomographic errors are a factor of 0.98 of the 2D errors. Therefore the increase in the tomographicerrors compared to the 2D errors is not an inherent product of the covariance matrices used. This re-sult confirms that for the case of overlapping redshift bins and non-Gaussian covariance the expectedimprovement from tomography is marginal, at best.As discussed in detail in Section 3.4 we have also analysed the data after removing small scaleswhich could be affected by errors in the non-linear modelling of the matter power spectrum and baryoniceffects. When removing these scales (?? < 10 arcmin) from both the tomographic and 2D lensinganalyses we see an improvement in the errors for tomography finding ?tomo/?2D = 1.04. The remainingdiscrepancy could be due to several factors. The Smith et al. (2003b) non-linear prescription couldeasily be biased at the few per cent level. Residual errors in the redshift of galaxies or other per centlevel systematics could be present. In addition we expect some degradation of the tomography errors dueto the bias correction of the inverse covariance matrix (Anderson, 2003). The covariance is estimatedfrom a finite number of mock catalogues (see Section 3.3.3), since the tomographic covariance containsthree times the number of elements as the 2D covariance, measuring it from the same number of mockcatalogues results in a noisier measure. Hartlap et al. (2007) predict an erroneous increase in likelihoodarea of three per cent given our number of mock catalogues 184 and the size of the data vector for 2Dlensing 16 and tomography 48.Finally we note that our two redshift bins are chosen based on concerns of intrinsic alignment con-tamination, as such, they are not optimised for constraining cosmology. With more carefully selectedredshift bins it may be possible to overcome the issues discussed above and obtain improved cosmolog-ical constraints.Redshift scaling of the cosmic shear signalPrevious CFHTLS data were found to underestimate the shear signal at high redshift necessitating ad-ditional calibration parameters when performing cosmological fits to the data (Kilbinger et al., 2009).We demonstrate here that the CFHTLenS data have a redshift dependent shear-signal which agrees withexpectations from the modelled ?CDM cosmology.The excellent agreement between the 2D and tomographic lensing results (Figure 3.9) suggests thatthe shear signal across our two redshift bins is scaling as expected. This is also observed in the excellentagreement between the measured shear and the shear prediction based on a fiducial WMAP7 cosmologyshown in Figure 3.7.The shear correlation function for each pair of tomographic redshift bins is analysed separately,corresponding to the shear correlation functions shown in each panel of Figure 3.7. In Figure 3.10 wepresent marginalised parameter constraints (68.3 per cent confidence level) in the ?m ??8 plane foreach redshift bin combination. Since each contour is obtained from a sub-sample of the full data-setthe degeneracy between the parameters is more pronounced and the area of the contours is larger thanwhen analysing the full data set (Figure 3.8). The agreement between the contours in Figure 3.10 is a710.0 0.2 0.4 0.6 0.8 lensing CFHTLenStomography CFHTLenS2D lensing CFHTLenS+WMAP7+BOSS+R11tomography CFHTLenS+WMAP7+BOSS+R11Figure 3.9: Marginalised parameter constraints (68.3 per cent confidence level) in the ?m ??8plane for a flat ?CDM cosmology. We compare the results for 2D lensing (blue) and 2-bintomography (green). We combine CFHTLenS with WMAP7, BOSS, and R11. Results areshown for 2D lensing (black) and 2-bin tomography (pink).convincing demonstration that the redshift scaling of the shear in the CFHTLenS data is consistent withexpectations from the modelled ?CDM cosmology.The power-law fits to the degenerate parameter constraints in Figure 3.10 for each case are?8(?m0.27)?= 0.820 ? 0.067, 0.753 ? 0.053, and0.753 ? 0.050 with ? = 0.662 ? 0.020, 0.621 ?0.016, and 0.535?0.013 for the low-low, low-high, and high-high redshift bin pairings respectively.We reiterate that the cosmological model-dependent verification of redshift scaling presented hereis completely independent of the calibration of the data, and the rejection of bad fields, that were donewith tests which are not sensitive to cosmology (Heymans et al., 2012b).Curved ?CDMA curved ?CDM cosmology is modelled, for the full details of parameters and priors used see Table 3.1.We present constraints in the ?m??8 and ?m??? plane in Figure 3.11. One-dimensional marginalisedresults when combining CFHTLenS with WMAP7, BOSS, and R11 are ?m = 0.2736?0.0085, ?? =720.0 0.2 0.4 0.6 0.8 3.10: Marginalised parameter constraints (68.3 per cent confidence level) in the ?m ??8plane for a flat ?CDM cosmology. The results are shown for each combination of the tworedshift bins. The low and high redshift bins correspond to 0.5 < zp ? 0.85 and 0.85 <zp ? 1.3 respectively. The excellent agreement shows that redshift scaling of the signal isconsistent with the modelled ?CDM cosmology.0.7312?0.0094, ?K = ?0.0042?0.0040, and ?8 = 0.795?0.013. The constraints on ?m and ?8 donot change significantly from the flat ?CDM case. Parameter constraints for both models are presentedin Table 3.3. The addition of CFHTLenS to WMAP7, BOSS, and R11 is most helpful at constraining?m, ?8, ?K, and ??. The precision for these parameters improves, on average, by a factor of two.We again find excellent agreement with the 2D lensing analysis. When combining the 2D lensing ofCFHTLenS with WMAP7, BOSS, and R11 data sets, we find ?m = 0.2766? 0.0082, ?? = 0.7273?0.0089, ?K = ?0.0035?0.0035 and ?8 = 0.804?0.016. We do not show the complete details of our2D lensing parameter estimations. However, we note that in all cases, either with CFHTLenS alone orcombined with the other cosmological probes, the 2D results agree with the tomographic results withinthe 68.3 per cent errors and the size of the error bars are similar for both cases. We again find that forCFHTLenS alone the ratio of individual parameter uncertainties from the tomographic analysis to thoseof the 2D lensing analysis is ?tomo/?2D = 1.16.73Figure 3.11: Marginalised parameter constraints (68.3, 95.5, and 99.7 per cent confidence levels)for a curved ?CDM cosmology. Results are shown for CFHTLenS (blue), WMAP7 (green),CFHTLenS combined with WMAP7 (black), and CFHTLenS combined with WMAP7,BOSS, and R11 (pink). Top panel: Constraints in the ?m ??8 parameter space. Bottompanel: Constraints in the ?m??? parameter space.74Constraining the deceleration parameterThe deceleration parameter q0 parametrizes the change in the expansion rate of the Universe. We cal-culate this as a deduced parameter for both the flat and the curved ?CDM models. The decelerationparameter depends on the energy density parametersq0 ? ?a?(t0)a(t0)a?2(t0)= ?m2??? (curved ?CDM), and= 3?m2?1 (flat?CDM), (3.14)where the scale factor at present time is a(t0) and derivatives with respect to time are denoted with adot. For the flat case q0 is simply a transformation of our results for the matter density parameter ?m.We present marginalised constraints for q0 in Figure 3.12. The pink line is for the curved case where wefind q0 =?0.29?0.40, and the blue line is for the flat case where we find q0 =?0.57?0.27. Negativevalues indicate acceleration of the expansion of the Universe. Summing the posterior for q0 < 0 tellsus the confidence level at which we have measured an accelerating Universe. For the curved and flatmodels we find that q0 < 0 at the 82 and 89 per cent confidence level, respectively.Schrabback et al. (2010) constrain q0 with a six-bin tomographic analysis of the COSMOS-30 data.Besides having more tomographic bins the redshift range probed is also greater, extending to z = 4. Fora curved ?CDM cosmology, holding ?b and ns fixed and using a Gaussian prior on the Hubble constantof h = 0.72? 0.025, they find q0 < 0 at 96 per cent confidence. If we do a similar analysis with ?band ns held fixed and using the a Gaussian prior of h = 0.738? 0.024 (R11), we find q0 < 0 at 84 percent confidence. The difference in constraints on the deceleration parameter can be understood as aresult of the much larger values of the dark-energy density preferred by COSMOS-30 ?? = 0.97+0.39?0.60,which lead to smaller values of q0. Whereas the dark-energy density found here from CFHTLenS is?? = 0.38?0.36, resulting in larger values of q0 for CFHTLenS.With CFHTLenS alone we are not able to put a strong constraint on the acceleration of the expansionof the Universe. With the addition of the other cosmological probes the entire posterior distribution ofq0 is less than zero. For a curved model with CFHTLenS combined with the other probes, we findq0 =?0.594?0.014 (see Table 3.3). An accelerating Universe is unambiguously detected.3.4 Impact of non-linear effects and baryons on the tomographiccosmological constraintsWe have presented cosmological parameter constraints from an analysis of the tomographic two-pointshear correlation function ? k,l? (?) (Equation 3.11), incorporating the non-linear dark matter only powerspectrum from Smith et al. (2003b) as our theoretical model of P? (k,z) in Equation (3.12). Note thatk = ?( fK(?))?1. This halo-model prescription for the non-linear correction has been calibrated onnumerical simulations and shown to have an accuracy of 5? 10 per cent over a wide range of scales(Eifler, 2011). The N-body simulations used to estimate the covariance matrices used in this analysissuggest that the accuracy is even better than this for a WMAP5 cosmology (Harnois-De?raps et al., 2012)75?2.0 ?1.5 ?1.0 ?0.5 0.0 LCDMcurved LCDMFigure 3.12: Marginalised constraints on the deceleration parameter using the CFHTLenS 2-bintomographic weak lensing results. An accelerating universe (q0 < 0) is found at the 82 percent confidence level for a curved ?CDM model (pink), and at the 89 per cent confidencelevel for a flat ?CDM model (blue).over the redshift range covered in this analysis. While these comparisons give us confidence in ourresults, and suggest that any error from the non-linear correction will be small in comparison to ourstatistical error, it is prudent to assess how errors in the non-linear correction will impact our results.A fully 3D weak lensing analysis of the CFHTLenS data is presented in Kitching et al. (2012). Thispower spectrum analysis allows for exact redshift dependent cuts in the wave-vector k, which can bemotivated by either the comparison of the power spectrum measured from N-body simulations to thenon-linear prescription, or the selection of linear scales where the non-linear correction is negligible. Forreal-space statistics, as used in this chapter and in Kilbinger et al. (2013), it is not possible to make anunambiguous separation of scales. The two-point correlation function ? k,l? (?) is related to the underlyingmatter power spectrum P? (k,z) through integrals over k and z, modulated by the lensing efficiency g(z)(Equation 3.13) and Bessel functions J0/4(??) for ?+/?(?) (see Equations 3.11 and 3.12). The measuredtomographic two-point shear correlation function ? k,l? at a fixed scale ? is therefore probing a range ofk in the underlying matter power spectrum. In addition, owing to the different Bessel functions, ?+ is76Figure 3.13: Marginalised parameter constraints (68.3, 95. 5, and 99.7 per cent confidence levels)in the ?m ? ?8 plane for a flat ?CDM cosmology. The pink contours show the resultwhen all eight scales are included, this is the same as the result for CFHTLenS shown inFigure 3.8. The blue contours show the result of removing highly non-linear scales, whichare possibly biased due to the non-linear correction to the matter power spectrum or theeffect of baryons. We remove the 5 smallest scales of ??, corresponding to ? < 10 arcmin.The contours are only slightly different, indicating that we are not sensitive to these effectsgiven the level of precision of our results.preferentially probing much smaller k, and hence larger physical scales, than ??.Kilbinger et al. (2013) present an analysis of the 2D shear correlation function out to large angularscales ? < 350 arcmin. The consistent constraints obtained from the large quasi-linear regime ? > 53arcmin in comparison to the full angular range, analysed using the Smith et al. (2003b) non-linear powerspectrum, give us confidence that the accuracy of this correction is sufficient, falling within our statisticalerrors.Comparing the theoretical expectation of ? k,l? (?) (Equation 3.11) for a WMAP7 cosmology, calcu-lated using a non-linear and a linear power spectrum, we determine the angular scale below which thenon-linear and linear models differ in amplitude by greater than 10 per cent. For ?+, this quasi-linear77limit ranges from 10?14 arcmin for the three different tomographic combinations (the lowest redshiftbin requiring the largest ? cut). For ??, the quasi-linear limit ranges from 100? 140 arcmin. In thisanalysis, we limit our angular range to scales with ? . 40 arcmin where we can accurately assess acovariance matrix from the lensing simulations (Harnois-De?raps et al., 2012). We are therefore unableto follow Kilbinger et al. (2013) by limiting our real-space analysis to this quasi-linear regime as we donot probe sufficiently large angular scales. We can however make an assessment of how an error on thenon-linear correction would impact our results. We first compare the WMAP7 theoretical expectation of? k,l? (?) calculated using a non-linear correction boosted by 7 per cent, with a model calculated with thenon-linear correction decreased by 7 per cent. We chose the value of 7 per cent from the average errorover the range of k tested in Eifler (2011). We find that these two limits on the non-linear correctioncause at least a 10 per cent change in the amplitude of ? k,l? (?) for scales ? . 1 arcmin for ?+ and ? . 10arcmin for ??. Applying these cuts in angular scale corresponds to removing the first 5 angular scalesfor ?? for each tomographic bin shown in Figure 3.7. All ?+ scales remain in the analysis. With thesescales removed any remaining uncertainty due to the non-linear modelling is well within our statisticalerror.For reference, we calculate the approximate wavenumber corresponding to the physical separation ofthe source galaxies when the small scale cuts are applied. Using the relation k=2pi(? fK(?))?1, and as-suming a flat ?CDM cosmology, the angular scale cuts correspond to wavenumbers k?=1??12hMpc?1and k?=10??1.2hMpc?1. Where we have calculated the angular diameter distance to the mean redshiftof the low-redshift bin fK(?)=1800h?1 Mpc. To determine what wavenumber the shear correlationfunctions are sensitive to we must account for the effect of the Bessel functions, power spectrum, andlensing efficiency in Equations (3.11) and (3.12). To get a rough estimate we take the power spectrumto be a pure power-law with exponent ?2, and take the mid-point of the angular diameter distance sincethis is approximately where the lensing efficiency will peak fK(?)?900h?1 Mpc. We perform the inte-gral in Equation (3.11) for both ?+ and ?? and note for which ? 90 per cent of the final value is reached.This results in an approximate wavenumber of k?2.5hMpc?1 for both ?+ and ??.Figure 3.13 compares cosmological parameter constraints in the ?m ? ?8 plane for this limitednumber of scales in comparison to the full data set analysed in Section 3.3. The removal of small scalesresults in a slight change to the degeneracy of the parameters. This test gives us confidence that the non-linear correction used is sufficiently accurate given the statistical error of the survey. This is unlikely tobe true for future surveys, where the increased statistical accuracy will require better knowledge of thenon-linear correction to the power spectrum (Eifler, 2011).Finally we turn to the impact of baryons on our results. In our analysis we assume the underlyingmatter power spectrum is sufficiently well represented by the non-linear dark matter only power spec-trum, neglecting the role of baryons. The impact of baryons on the power spectrum is sensitive to thebaryonic feedback model used. Therefore, the magnitude of the impact of baryons remains uncertain.Semboloni et al. (2011) present an analysis of cosmological hydrodynamic simulations to quantify theeffect of baryon physics on the weak gravitational lensing shear signal using a range of different bary-onic feedback models. Their work suggests that a conservative weak lensing analysis should be limited78to those scales where k . 1.5hMpc?1. We implement such a conservative scheme in the 3D powerspectrum analysis of Kitching et al. (2012). As discussed above, our real-space analysis mixes k and zscales, leaving us unable to perform a similarly clear test here.Semboloni et al. (2011) present a comparison of ? k,l? (?) measured for both the cosmological hydro-dynamic simulations and a dark matter only simulation for different redshifts, which we use to judgethe level of error we should expect baryons to introduce. Assuming the realistic active galactic nucleusfeedback model, and considering the scales used in the conservative analysis of Figure 3.13 (? ? 1.34arcmin and ? ? 15.4 arcmin for ?+ and ?? respectively), we expect baryons to cause a decrease tothe modelled signal of less than ten per cent. The fact that we see very little difference between theconservative and full analysis presented in Figure 3.13 demonstrates that the underlying matter powerspectrum is indeed sufficiently well represented by the non-linear dark matter only power spectrum forour statistical accuracy. It also indicates that the impact of baryons on the non-linear dark matter onlypower spectrum is unlikely to be larger than that predicted by Semboloni et al. (2011). However, bary-onic effects will have to be carefully considered for the next generation of weak lensing surveys thatwill have significantly smaller statistical errors. Semboloni et al. (2012) show the importance of bary-onic effects on three-point shear statistics and propose a modification to the modelling of the non-linearmatter power spectrum to account for these effects.3.5 ConclusionThe most important result of this study is that the sum of the photometric redshift probability distributionfunctions (PDF) within a redshift bin provides an accurate measure of the true redshift distribution ofthose galaxies; accounting for the scatter due to catastrophic as well as statistical errors. To demonstratethe accuracy of the PDFs we have compared the summed PDFs with the redshift distribution predictedby spectroscopic redshifts, resampled COSMOS-30 redshifts, and predictions from a redshift contam-ination analysis using the angular correlation function. We find excellent agreement for the redshiftrange zp < 1.3. This result indicates that the priors and spectral templates used in Hildebrandt et al.(2012) to derive the photometric redshifts provide an accurate and complete description of the galaxiesat zp < 1.3. This also motivates our use of the summed PDF as a measure of the redshift distributionsin our tomographic weak lensing analysis. Furthermore, the proven accuracy of the summed PDFsprovides a reliable method for estimating the source redshift distribution in future weak lensing studies.We have performed a cosmological analysis of the CFHTLenS data on angular scales 1 < ? < 40arcmin, using two broad redshift bins, 0.5 < zp ? 0.85 and 0.85 < zp ? 1.3, that are not significantly af-fected by the intrinsic alignment of galaxy shapes. We model two cosmologies; flat and curved ?CDM.Due to complementary degeneracies our results add valuable constraining power when combined withthose from the cosmic microwave background (Komatsu et al., 2011, WMAP7), baryon acoustic os-cillations (Anderson et al., 2012, BOSS), and a prior on the Hubble constant (Riess et al., 2011, R11).The addition of our weak lensing results to these other cosmological probes increases the precision ofindividual marginalised parameter constraints by an average factor of 1.5?2.For a flat ?CDM model the joint parameter constraints for CFHTLenS, WMAP7, BOSS, and R1179are ?m = 0.2762? 0.0074 and ?8 = 0.802? 0.013. For a curved ?CDM model, combining the samedata sets, we find ?m = 0.2736? 0.0085, ?? = 0.7312? 0.0094, ?K = ?0.0042? 0.0040, and ?8 =0.795?0.013. Full details of our parameter estimates for both cosmologies are presented in Table 3.3.Our results are consistent with those presented in other studies of the CFHTLenS data: a 2D lensinganalysis probing much larger scales where linear theory provides a more accurate model to the matterpower spectrum (Kilbinger et al., 2013); and a fine-binned tomographic analysis with six redshift binsaccounting for intrinsic alignments (Heymans et al., 2012a).We compare the tomographic constraints with those from a 2D lensing analysis spanning the samerange of redshift 0.5 < zp ? 1.3. We find the two analyses to be completely consistent with all parameterestimates agreeing within their 68.3 per cent confidence levels. We note that the ratio of uncertaintieson individual parameters from tomography to those from 2D lensing is on average ?tomo/?2D = 1.16.This statistic is 0.98 if we replace our data vectors with a fiducial model, indicating that our covariancematrices do show a slight improvement for tomography. We argue that our non-Gaussian covarianceand broad overlapping redshift bins degrade the modest improvement (?tomo/?2D = 0.88) expectedfrom idealised Fisher matrix calculations (Simon et al., 2004). We identify small scales as being largelyresponsible for the observed increase, finding ?tomo/?2D = 1.04 when these scales are removed fromthe analyses. These scales could be biased due to uncertainties in the modelling of the non-linearities inthe matter power spectrum and baryonic effects. Although small scales have inflated our uncertaintiesfrom tomography we show in Section 3.4 that they do not significantly affect our results.Previous analyses of CFHTLS data were hindered by a strong redshift dependent bias in the weaklensing shear, necessitating additional nuisance parameters when analysing the tomographic shear(Kilbinger et al., 2009). We demonstrate that the redshift scaling of the CFHTLenS cosmic shear signalagrees with expectations from the modelled ?CDM cosmology. The strongest test of this is presented inFigure 3.10, which shows the agreement of cosmological constraints measured for each combination ofredshift bins in the ?m ??8 plane for a flat ?CDM cosmology. This demonstrates the effectiveness ofthe cosmology-independent tests of residual systematics presented in Heymans et al. (2012b), includingthe rejection of 25 per cent of the MegaCam pointings which failed to pass these tests. Note also that theshear calibration performed on numerical simulations (Miller et al., 2013) was completed before anycosmological analysis was performed on the data. The two-bin analysis presented here is sensitive toredshift dependent cosmology without introducing additional parameters to model intrinsic alignments,as such it is an excellent final test of the CFHTLenS data product.80Chapter 4Conclusions and prospects for futureresearch4.1 Photometric redshift contaminationThe analytical framework developed in Chapter 2 is a remarkably useful tool. As outlined in thatchapter, there have been several proposals along similar lines (Newman, 2008; Schneider et al., 2006;Zhang et al., 2009). The method developed in this thesis is unique in that only the photometric redshiftdata are needed. This makes the method applicable to all multi-band optical surveys without the needfor extensive spectroscopic overlap (Newman, 2008) or reliance on shear measurements (Zhang et al.,2009). The application of the method in Chapter 3 beautifully demonstrates the utility of this ap-proach. The verification that the sum of the photometric redshifts? Probability Distribution Functions(PDF) provide an accurate measure of the true redshift distribution is extremely useful. Accurate red-shift distributions were used in the subsequent weak lensing analysis. This result has been critical inother CFHTLenS analyses that require accurate redshift distributions, including: weak lensing tomog-raphy with 6 redshift bins (Heymans et al., 2012a), 2-dimensional weak lensing shear measurements(Kilbinger et al., 2013), tests of general relativity (Simpson et al., 2013), and 2-dimensional mass re-constructions (Van Waerbeke et al., 2013).Further work can be done to improve the estimation of photometric redshift contamination fromthe angular correlation function. The most obvious improvement would be to abandon the pair-wiseapproach wherein the contamination is measured between each pair of bins. A full multi-bin analysisis possible and the necessary theoretical framework is already developed in Chapter 2. The difficulty isthe alarming number of free-parameters (Np) in the analysis as a function of the number of redshift bins(Nz),Np = Nz(Nz ?1), (4.1)where the number of free-parameters are the contamination fractions fij. For example, the 6-bin analysispresented in Chapter 3 has 30 free-parameters! Simultaneously fitting all of these parameters in finitetime becomes a serious challenge. This should be possible with Monte-Carlo Markov Chain (MCMC)81techniques, especially if prior knowledge (e.g. the true number of galaxies within each bin NTi ) can beused to constrain parameter space. Such an analysis should result in significantly tighter constraints onthe parameter space.In Chapter 3 the sum of the PDFs is used as an estimate of the redshift distribution. No error wasassigned to this estimate, and therefore the impact of this uncertainty is missing from the weak lensingerror budget. The problem is not in estimating the error, which could easily be done by bootstrappingthe summed PDF or by using the standard deviation of the PDF from each galaxy. The largest barrieris in encoding the redshift distribution error in such a way that it is easy to marginalise over. Due tothe shape of the summed PDF, simple 3-parameter fitting functions (which have been used in past weaklensing studies) do not fit well. For example, Van Waerbeke et al. (2013) use a 6-parameter model tofit the summed PDF for the redshift bin 0.4 < z < 1.1. For a tomographic analysis, one would need toadd 6 parameters per redshift bin and define ranges over which to allow them to vary. That in itself isproblematic, since the derived error distribution will likely not yield a single set of parameter ranges foreach of the 6 parameters. An approach like the one used in Benjamin et al. (2007) would be ideal. Inthat work, a set of 100 redshift distributions, whose parameters were sampled from the full probabilitydistribution, were created and then all 100 redshift distributions were marginalised over. Unfortunately,it is not obvious how to adapt this technique for use with Population Monte-Carlo (PMC) or MCMCcodes.Future work on this method will also need to incorporate the effect of weak lensing magnification,which can cause an angular cross-correlation between redshifts. As discussed in Me?nard et al. (2009),dust extinction can also cause an angular cross-correlation on the same order of magnitude as weaklensing magnification; therefore, this effect will also need to be properly accounted for.4.2 Tomographic weak lensingThe weak lensing tomography analysis presented in Chapter 3 is important for several reasons:? The scaling of the shear correlation function with redshift is found to agree well with a ?CDMcosmology. Previous analysis of CFHT Legacy Survey data had to introduce an extra nuisanceparameter when fitting the data (Kilbinger et al., 2009). That the redshift scaling works inCFHTLenS is a testament to the thorough systematic testing of the data (Heymans et al., 2012b).? Removal of the highly non-linear scales is shown to have little impact on the cosmological con-straints. Since the interpretation of the weak lensing signal depends on the non-linear modellingof the mass density power spectrum, it is reassuring that the results do not change significantlywhen they are removed.? The cosmological constraints demonstrate the power of tomography without the need to model in-trinsic alignments (IA). With only two redshift bins, the constraining power is similar to a 6-bin to-mographic analysis which is sensitive to potential biases in the modelling of IAs (Heymans et al.,2012a).82? Probing only the quasi-linear regime (0.1 < ? < 40 arcmin), the tomographic analysis is similar insensitivity to a non-tomographic analysis extending to highly linear scales (0.9 < ? < 300 arcmin,Kilbinger et al., 2013).? It is shown that a single redshift cut to remove low-z galaxies is extremely powerful, and furthersub-dividing the redshift range (0.5 < z < 1.3) into two bins slightly decreases parameter con-straints. The potency of this ?1-bin tomography? was unexpected and future surveys will need tocarefully consider how to include these galaxies in their analyses.One avenue of future work is to determine the impact of low-redshift cuts on cosmologicalconstraints from cosmic shear. A Fisher-information matrix analysis, similar to that presented inSimon et al. (2004), could be done using realistic redshift distributions. The gain in cosmologicalconstraints as a function of the number of redshift bins, and with various low-redshift cuts, would bevery informative.4.2.1 A brief look at PlanckRecent results from the Planck mission (Planck Collaboration XVI et al., 2013) have significantly im-proved the precision of cosmological constraints from the Cosmic Microwave Background (CMB).Planck surpasses WMAP in its ability to probe small scale anisotropies (i.e. large wavenumbers ?).Two complementary data sets that extend to larger ? than Planck are the Atacama Cosmology Telescope(ACT, Hasselfield et al., 2013) and the South Pole Telescope (SPT, Reichardt et al., 2013). Planck?sconstraining power comes from its ability to measure the CMB over the entire sky (but note that galacticmasks reduce the total usable area significantly). Although a detailed study of the Planck results cannotbe addressed in this thesis, given the importance of the Planck survey a brief discussion is warranted.The current analysis does not include polarization data measured from Planck, yet the size of the errorbars are 85 per cent of those from WMAP7, which does include polarization data. The figure of 85 percent is calculated from the 6 primary parameters of WMAP7, excluding optical depth, which WMAP7is able to constrain much more tightly due to the inclusion of polarization data.Figure 4.1 shows the constraints from the Planck-CMB data, ?m = 0.314?0.02 and ?8 = 0.834?0.027, over-plotted with the 2-bin tomographic constraints from Figure 3.8. The Planck contours arenot derived from their MCMC chains. The contours are unofficial and have been calculated, under theassumption of Gaussianity, from multiples of the 68 per cent confidence limits quoted above. This isa poor substitute for the full likelihood information; however, the degeneracy of Planck?s constraintsshould be very similar to that of WMAP7 (also plotted), suggesting that the extension of the boxesorthogonally to the CFHTLenS constraints is not terribly exaggerated. While the agreement is notperfect, there is not significant tension between the Planck results and those presented in Chapter 3 forCFHTLenS alone. In Figure 4.1 the 95 per cent confidence level of either CFHTLenS or Planck overlapswith the 68 percent confidence level of the other. Note that comparing fully marginalised constraintscan be misleading. The tomographic results, ?m = 0.27?0.17 and ?8 = 0.67?0.23, suggest much lesstension than is evident in Figure 4.1.83Figure 4.1: Constraints from Figure 3.8 of Chapter 3. The results from Planck are shown tooverlap significantly with the 2-bin tomographic results from CFHTLenS. The Planck con-tours are not official results; they are multiples of the 68 per cent confidence limits given inPlanck Collaboration XVI et al. (2013); ?m = 0.314?0.02 and ?8 = 0.834?0.027.Table 4.1 summarizes the parametrized constraint ?8 (?m/0.27)? for all three CFHTLenS cosmol-ogy papers, the Planck-CMB result discussed above, and the Planck constraint from Sunyaev-Zel'dovich(SZ) cluster counts (Planck-SZ, Planck Collaboration XX et al., 2013). Also included are two other SZcluster count constraints, one from ACT and the other from SPT. What is striking is the tension betweenPlanck-CMB and all other constraints. In Planck Collaboration XX et al. (2013), it is noted that the SZcluster constraints can be reconciled if a mass bias parameter of (1?b) = 0.55?0.06 is adopted. Thisvalue is typically found to be (1?b) = 0.8. It is hard to understand how numerical simulations, X-raycluster studies, and weak lensing cluster studies could all be similarly biased toward large values of(1?b). No explanation for the discrepancy between Planck-CMB and CFHTLenS results is offered inPlanck Collaboration XVI et al. (2013).Tension between cosmological probes is exciting, and is possibly a hint of physics beyond the stan-dard ?CDM model. Future work is needed in understanding the tensions that exist between these data84Table 4.1: Constraints on ?8 for a fixed ?m = 0.27, errors are 68 per cent confidence limits. Thetop three results are from the three CFHTLenS cosmology papers submitted to date. Themiddle results are from Planck, both the temperature anisotropies of the Cosmic MicrowaveBackground (Planck-CMB) and the Sunyaev-Zel'dovich cluster counts (Planck-SZ). The bot-tom set of results are from two other Sunyaev-Zel'dovich cluster count measurements madeby ACT and SPT. The value of ? is not relevant for this comparison but is included for com-pleteness.Data ?8(?m0.27)??2-bin tomography (Chapter 3) 0.771?0.040 0.553?0.0166-bin tomography (Heymans et al., 2012a) 0.774+0.032?0.041 0.46?0.022D lensing (Kilbinger et al., 2013) 0.787?0.032 0.59?0.02Planck-CMB (Planck Collaboration XVI et al., 2013) 0.89?0.03 0.46Planck-SZ (Planck Collaboration XX et al., 2013) 0.782?0.010 0.30ACT (Hasselfield et al., 2013) 0.768?0.025 0.30SPT (Reichardt et al., 2013) 0.767?0.037 0.30sets. If we are fortunate, this tension will not be due to systematic errors in one or several analysesand will instead be a genuine discrepancy due to the different physical processes responsible for thesecosmological probes.4.2.2 Future observationsFuture surveys such as the Large Synoptic Survey Telescope (LSST Ivezic et al., 2008), the SuperNovaAcceleration Probe (SNAP SNAP Collaboration: G. Aldering et al., 2004) and Euclid (Laureijs et al.,2011) will provide unprecedented precision for weak lensing measurements. LSST is a planned groundbased telescope which will survey 30,000 square degrees (20 times the area of CFHTLenS) in 6 pho-tometric bands spanning the optical spectrum. SNAP is a space-based mission optimized to detectSupernovae while conducting a deep wide-field survey. SNAP will survey 10,000 square degrees in 6photometric bands including three bands in the infrared which will significantly improve the ability tomeasure photometric redshifts. Finally, Euclid is a planned space-based mission which plans to survey15,000 square degrees with optical and infrared photometric bands. These surveys will revolutionise theprecision of weak lensing measurements. In order to take full advantage of these data, work is neededin understanding non-linear and baryonic effects on the mass density power spectrum. 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