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Cosmological constraints from the 100 square degree weak lensing survey Benjamin, Jonathan Remby Embro 2007

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Cosmological Constraints From the 100 Square Degree Weak Lensing Survey by Jonathan Remby Embro Benjamin A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics) The University of British Columbia April 2007 © Jonathan Remby Embro Benjamin, 2007 Abstract We present a cosmic shear analysis of the LOO square degree weak lensing survey, combining data from the CFHTLS-Wide, RCS, VIRMOS-DESCART and GaBoDS surveys. Spanning ~ 100 square degrees, with an average source redshift z ~ 0.8, this combined survey allows us to place tight joint constraints on the matter density parameter fim, and the amplitude of the matter power spectrum as, finding ^ (g^) 0 5 9 = 0.84 ± 0.07. Tables of the measured shear correlation function and the calculated covariance matrix for each survey are included. The accuracy of our results are a marked improvement on previous work owing to three important differ-ences in our analysis; we correctly account for cosmic variance errors by including a non-Gaussian contribution estimated from numerical simulations; we correct the measured shear for a calibration bias as estimated from simulated data; we model the redshift distribution, n(z), of each survey from the largest photometric redshift catalogue currently available from the CFHTLS-Deep. This catalogue is randomly sampled to reproduce the magnitude distribution of each survey with the resulting survey dependent n(z) parametrised using two differ-ent models. While our results are consistent for the n(z) models tested, we find that our cosmological parameter constraints depend weakly (at the 5% level) on the inclusion or exclusion of galaxies with low confidence pho-tometric redshift estimates (z > 1.5). These high redshift galaxies are relatively few in number but contribute a significant weak lensing signal. It will therefore be important for future weak lensing surveys to obtain near-infra-red data to reliably determine the number of high redshift galaxies in cosmic shear analyses. ii T a b l e o f C o n t e n t s Abstract h Table of Contents J i i List of Tables v List of Figures v u Acknowledgements x Statement of Co-Authorship x l 1 Introduction ^ LL Gravitational Lensing ^ 1.2 Weak Gravitational Lensing 2 1.2.1 Weak Lensing by Galaxy Clusters 2 1.2.2 Weak Lensing by Large Scale Structure 2 1.2.3 Cosmological Dependence 3 Bibliography 9 2 100 sq. deg. WL Survey 1 1 2.1 Introduction 1 1 2.2 Cosmic Shear Theory 1 2 2.3 Data description 1 4 2.3.1 CFHTLS-Wide 1 4 2.3.2 GaBoDS • • • 1 6 2.3.3 VIRMOS-DESCART L6 2.3.4 RCS • 1 6 2.4 Cosmic shear signal ^ 2.5 Redshift Distribution 1 7 2.5.1 Magnitude Conversions • • • • 20 iii Table of Contents 2.5.2 Modeling the redshift distribution n(z) 20 2.6 Parameter estimation 23 2.6.1 Maximum likelihood method 23 2.6.2 Joint constraints on ag and fim 25 2.7 Discussion and Conclusion 30 Bibliography 3 3 3 Prospects for Future Research 3 7 Bibliography 3 ^ Appendices A Additional data 3 9 List o f Tables 2.1 Summary of the published results from each survey used in this study. The values for as corre-spond to Qm = 0.24, and are given for the Peacock & Dodds (1996) method of calculating the non-linear evolution of the matter power spectrum. The statistics listed were used in the previ-ous analyses; the shear correlation functions £E,B (E<L 2.6), the aperture mass statistic (Mfp) (Eq. 2.4) and the top-hat shear variance ( i l B) ( s e e V a n Waerbeke et al. (2005)) 15 2.2 The best fit model parameters for the redshift distribution, given by either Eq.(2.9) or Eq.(2.10). Models are fit using the full calibration sample of Ilbert et al. (2006) 0.0 < zp < 4.0, and using only the high confidence region 0.2 < zp < 1.5. xi l s t n e reduced x 2 statistic, (z), and zm are the average and median redshift respectively, calculated from the model on the range 0.0 < z < 4.0 19 2.3 Data used for each survey to calculate the Gaussian contribution to the cosmic variance covari-ance matrix. Aeg is the effective area, nes the effective galaxy number density, and ae the intrinsic ellipticity dispersion 24 2.4 Joint constraints on the amplitude of the matter power spectrum as, and the matter energy density Clm, parametrised as <r8 = f3(§ffc)~a 28 A. 1 E and B modes of the shear correlation function for the CFHTLS-Wide survey, the error (<5£) is statistical only, and given as the standard deviation 40 A.2 E and B modes of the shear correlation function for GaBoDS, the error (SQ is statistical only, and given as the standard deviation 40 A.3 E and B modes of the shear correlation function for RCS, the error (<5£) is statistical only, and given as the standard deviation 41 A.4 E and B modes of the shear correlation function for the VIRMOS-DESCART survey, the error (<5£) is statistical only, and given as the standard deviation 41 A.5 The correlation coefficient matrix (r) for the CFHTLS-Wide survey, the scales 9{ correspond to those given in Table A.L. The column (£?) is given in units of 10 - 1 0 , and can be used to reconstruct the covariance matrix C 42 A.6 The correlation coefficient matrix (r) for GaBoDS, the scales 0i correspond to those given in Table A.2. The column is given in units of 10~10, and can be used to reconstruct the covariance matrix C 43 v List of Tables A.7 The correlation coefficient matrix (r) for RCS, the scales 6{ correspond to those given in Ta-ble A.3. The column (£?) is given in units of 10~10, and can be used to reconstruct the covari-ance matrix C 44 A.8 The correlation coefficient matrix (r) for the VIRMOS-DESCART survey, the scales 6i corre-spond to those given in Table A.4. The column {£?) is given in units of 10~10, and can be used to reconstruct the covariance matrix C 45 vi L i s t o f F i g u r e s l. 1 Sketch of the lensing geometry, the dashed line is the optical axis defined by the line connecting the observer and the center of the lensing mass. The circular source in the source plane has a 1.2 Schematic of the first order effects weak lensing has on a circular background galaxy of radius Ro. The convergence is an isotropic distortion that increases the radius, scaling Ro by a factor of (1 — The shear is an anisotropic distortion, creating a major (a) and minor (b) axis which 1.3 Top: How the mass density power spectum changes with various CDM cosmologies. ACDM (Solid): A flat cosmology (ilm + O A = 1) with a non-zero cosmological parameter, flm = 0.24, and fiA = 0.76. OCDM (Dashed): An open cosmology (Qm + ftA < 1) with Qm = 0.24, = 0.0. EdS (Dash-dot): An Einstein de-Sitter cosmology with Clm = 1.0, f2A = 0.0. For all cases as = 0.76 and the non-linear prescription of Smith et al. (2003) is used. Bottom: The linear power spectrum is given by the dash-dot line, two methods are used to account for the non-linear power, Peacock & Dodds (1996) (dashed) and Smith et al. (2003) (solid) 6 1.4 Top: How the convergence power spectum changes with various cold dark matter cosmologies. ACDM (Solid): Qm = 0.24, fiA = 0.76. OCDM (Dashed): Qm = 0.24, QA = 0.0. EdS (Dash-dot): fim = 1.0, ^ A = 00. For all cases as = 0.76 and the non-linear prescription of Smith et al. (2003) is used. Bottom: The linear power spectrum is given by the dash-dot line, two methods are used to account for the non-linear power, Peacock & Dodds (1996) (dashed) postion described by j5, it's lensed image in the lens plane has a position described by 0. 3 are scalings of the new radius by factors of 1 + 7 and 1 — 7 respectively. 4 and Smith et al. (2003) (solid), 8 vii List of Figures 2.1 E and B modes of the shear correlation function £ (filled and open points, respectively) as measured for each survey. Note that the la errors on the .E-modes include statistical noise, non-Gaussian cosmic variance (see §2.6) and a systematic error given by the magnitude of the .B-mode. The la error on the J3-modes is statistical only. The results are presented on a log-log scale, despite the existence of negative jB-modes. We have therefore collapsed the infinite space between IO - 7 and zero, and plotted negative values on a separate log scale mirrored on 10 - 7. Hence all values on the lower portion of the graph are negative, their absolute value is given by the scaling of the graph. Note that this choice of scaling exagerates any discrepancies. The solid lines show the best fit ACDM model for Qm = 0.24, h = 0.72, V = hClm, as given in Table 2.4, and n(z) given in Table 2.2. The latter two being chosen for the case of the high confidence redshift calibration sample, an n(z) modeled by Eq.(2.9), and the non-linear power spectrum estimated by Smith et al. (2003) 18 2.2 Normalised redshift distribution for the CFHTLS-Wide survey, given by the histogram, where the error bars include Poisson noise and cosmic variance of the photometric redshift sample. The dotted curve shows the best fit forEq.(2.9), and the solid curve for Eq.(2.10). Left: Distribution obtained if all photometric redshifts are used 0.0 < z < 4.0, and x 2 is calculated between 0.0 < z < 2.5. Right: Distribution obtained if only the high confidence redshifts are used 0.2 < 2 < 1.5, and x 2 is calculated on this range. The existence of counts for z > 1.5 is a result of drawing the redshifts from their full probability distributions 22 2.3 Joint constraints on as and Clm from the 100 deg2 WL survey assuming a flat ACDM cosmology and adopting the non-linear matter power spectrum of Smith et al. (2003). The redshift distri-bution is estimated from the high confidence photometric redshift catalogue (Ilbert et al., 2006), and modeled with the standard functional form given by Eq.(2.9). The contours depict the 0.68, 0.95, and 0.99% confidence levels. The models are marginalised, over h = 0.72 ± 0.08, shear calibration bias (see §2.4) with uniform priors, and the redshift distribution with Gaussian priors (see §2.6.1). Similar results are found for all other cases, as listed in Table 2.4 26 2.4 Joint constraints on a$ and Qm for each survey assuming a flat ACDM cosmology and adopting the non-linear matter power spectrum of Smith et al. (2003). The redshift distribution is esti-mated from the high confidence photometric redshift catalogue (Ilbert et al., 2006), and modeled with the standard functional form given by Eq.(2.9). The contours depict the 0.68, 0.95, and 0.99% confidence levels. The models are marginalised, over h = 0.72 ± 0.08, shear calibra-tion bias (see §2.4) with uniform priors, and the redshift distribution with Gaussian priors (see §2.6.1). Similar results are found for all other cases, as listed in Table 2.4 27 List of Figures 2.5 Values for as when f2m is taken to be 0.24, filled circles (solid) give our results with la error bars, open circles (dashed) show the results from previous analyses (Table 2.1). Our results are given for the high confidence photometric redshift catalogue, using the functional form for n(z) given by Eq.(2.9), and the Smith et al. (2003) prescription for the non-linear power spectrum. The literature values use the Peacock & Dodds (1996) prescription for non-linear power, and are expected to be ~ 3% higher than would be the case for Smith et al. (2003). The forward slashed hashed region (enclosed by solid lines) shows the la range allowed by our combined result, the back slashed hashed region (enclosed by dashed lines) shows the la range given by the WMAP 3 year results 29 2.6 Redshift distributions for the RCS survey. The solid line shows the best fit rc(z) from Hoek-stra et al. (2002a), the dotted curve is our best fit n(z), the average redshifts are 0.6 and 0.78 respectively 31 Acknowledgements I would like to thank Ludovic Van Waerbeke for being an excellent supervisor, not only for offering ample motivation, inspiration, and guidance, but for knowing when they were needed. I would also like to thank Catherine Heymans who was always available to offer help and critical insight. This research was performed with infrastructure funded by the Canadian Foundation for Innovation and the British Columbia Knowledge Development Fund (A Parallel Computer for Compact-Object Physics). This research has been enabled by the use of WestGrid computing resources, which are funded in part by the Canada Foundation for Innovation, Alberta Innovation and Science, BC Advanced Education, the participating research institutions. WestGrid equipment is provided by IBM, Hewlett Packard and SGI. This work is based, in part, on observations obtained with MegaPrime equipped with MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. x Statement of Co-Authorship The research presented in this thesis was an international collaboration involving members of several astronomy departments, including the University of British Columbia, the Universite Pierre & Marie Curie, the University of Victoria, the Universitat Bonn, the University of Chicago, and the University of Toronto. This statement is a confirmation that the author of this thesis was primarily responsible for the research contained herein. The initial design of the research program was identified by Ludovic Van Waerbeke. The work presented herein builds upon code and methodology developed by Ludovic Van Waerbeke, Elisabetta Semboloni, and Catherine Heymans. The modification, extension and application of these resources to this research program was performed by the author. All of the observational data used in this work were provided by collaborators. Marco Hetterscheidt and Thomas Erben provided a weak lensing catalogue for the Garching-Bonn Deep Survey (GaBoDS). Henk Hoek-stra, Howard K.C. Yee and Michael D. Gladders provided a weak lensing catalogue for the Red-Sequence Cluster Survey (RCS). Henk Hoekstra and Yannik Mellier provided a weak lensing catalogue for the Canada-France-Hawaii telescope legacy survey's wide synopic survey (CFHTLS-Wide). Ludovic Van Waerbeke, Yannik Mellier and Henk Hoekstra provided a weak lensing catalogue for the VIRMOS-DESCART survey. Preparation of the paper resulting from this work was a collaborative effort involving all co-authors; the author of this thesis assumed the primary role in preparation of that publication. The research, including data analysis, production of figures and writing of the manuscript, was the work of the author. xi Chapter 1 Introduction 1.1 Gravitational Lensing Gravitational lensing - the deflection of a light bundle's trajectory by the presence of mass - is one of the most fundamental results of general relativity. The phenomenon was experimentally verified in 1919 during a solar eclipse in which background stars were seen to have shifted with respect to their usual positions due to the gravity of the Sun. This experiment, carried out by Sir Arthur Eddigton (Eddigton, 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 and geometry. The strength of a gravitational lens, its ability to alter the light coming from background sources, depends on the mass of the lens and the ratio of distances, D]Pla, where Di is the distance between the observer and the lens, Ds is the distance between the observer and the source and Dis is the distance between the lens and the source. Hence the effect is quite small for the Sun lensing background stars, since Ds ~ Dis and D\ is relatively small. Little work was done on the theory of gravitational lensing at this time. Chwolson (1924) published a work wherein he considered a perfectly co-aligned source and foreground mass, concluding that the resulting image would be a ring around the lensing mass. Einstein (1936) wrote a paper considering the lensing effects by stars, he derived equations for image locations, separation and magnification. He concluded 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 concluded that the separation of images would be about 10 arcseconds, hence easily resolved, in addition the magnification of the source would allow the observation of distant galaxies (often referred to as a poor man's telescope). He went on to calculate the probability of a distant source being lensed, concluding that about 1 in 400 would be affected, practically guaranteeing their observation. 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 usefullness to astronomy. When the first quasars were detected in 1963, it was realised that they were far more distant than most galaxies and therefore could be lensed by them. However it wasn't until 1979 that the first multiply imaged quasar was detected by Walsh, Carswell, & Weymann (1979). Aided by the recent development of the Charge Coupled Device (CCD), gravitational lensing saw much advancement during the 1980s. Of particular interest here is the first detection of giant luminous arcs in galaxy clusters (Lynds & Petrosian, 1986; Soucail, Fort, Mellier, & Picat, 1987). These faint arcs are stretched tan-1 Chapter 1. Introduction gentially with respect to the cluster center, extending about 10 times further in this direction than in the radial direction. They would be identified as highly distorted and magnified images of background galaxies (Mellier et al., 1991; Paczynski, 1987), whose light was being strongly lensed by the 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 lensing, micro lensing, galaxy-galaxy lensing, and weak lensing. This thesis is concerned only with the latter application, for more detailed reviews of gravitational lensing the reader is referred to Schneider, Ehlers, & Falco (1992) and Petters, Levine, & Wambsganss (2001). 1.2 Weak Gravitational Lensing 1.2.1 Weak Lensing by Galaxy Clusters Unlike giant luminous arcs, or the less pronounced arclets, galaxies affected by weak gravitational lensing do not exhibit an immediately identifiable distortion. That is, the distortion of their ellipticity is not easily distinguishable from the intrinsic ellipticity. However, if the distortions vary slowly with position then nearby galaxies will be distorted in a similar way. Assuming that there is no correlation of the intrinsic ellipticities, a local ensemble average of galaxy ellipticities will provide a measure of the distortion due to weak lensing. This signal was first detected in 1990 around two galaxy clusters (Tyson, Valdes, & Wenk, 1990). One of the most appealing aspects of weak lensing is its ability to measure the mass in an unbiased way, it is not sensitive to the form of matter (baryonic or dark matter) or its state. Strong lensing can also be used to measure the mass of clusters, provided there are enough giant luminous arcs present. However, weak lensing can probe the matter distribution to much larger radii and can be applied to clusters that don't have a strong lensing signal. The theoretical edifice of weak lensing by clusters was detailed by Kaiser & Squires (1993), wherein it is shown that the measurement of galaxy distortions can be used to construct a parameter-free map of the two-dimensional projected mass distribution. The first 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). 1.2.2 Weak Lensing by Large Scale Structure Arguably 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. A statistical treatment of this so-called cosmic shear reveals details of the matter distribution of the Universe, which can be compared with reliable theoretical models of structure growth, hence it can be used to constrain cosmology. The first mention of light deflection by LSS is often credited to Gunn (1967), however, like many great 2 Chapter 1. Introduction Figure 1.1: Sketch of the lensing geometry, the dashed line is the optical axis defined by the line connecting the observer and the center of the lensing mass. The circular source in the source plane has a postion described by (3, it's lensed image in the lens plane has a position described by 0. ideas it can be traced back to Richard Feynman, specifically a lecture given by him at Caltech in 19641. The theory of light propagation in an inhomogeneous universe, and the development of weak lensing as a statistical treatment of galaxy distortions were explored by several theorists (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). Theoretical studies concerned with the measurement of cosmological parameters via weak lensing were soon to follow (Bemardeau 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). 1.2.3 Cosmological Dependence A light bundle which is observed at a position 9 has been deflected by the LSS of the Universe, and has a position in the source plane given by /3 (see figure 1.1). The distortion of images can then be described by the Jacobian matrix (or amplification matrix) 'Refregier (2003, who cites a personal communication with J .E. Gunn) 3 Chapter 1. Introduction Figure 1.2: Schematic of the first order effects weak lensing has on a circular background galaxy of radius Ro. The convergence is an isotropic distortion that increases the radius, scaling Ro by a factor of (1 — The shear is an anisotropic distortion, creating a major (a) and minor (b) axis which are scalings of the new radius by factors of 1 + 7 and 1 — 7 respectively. A ( e ) = ?P = ( 1 - K - ' n de \ - 7 2 1 - K + 7 1 where K is identified as the convergence, and 7 = 7 1 + 1 7 2 = \l\^l<p as the shear. The shape distortion of galaxies by weak lensing can be quantified by the convergence which is a uniform scaling of the galaxy image, and the shear which is an anisotropic distortion. In the weak lensing regime both of these quantities are much smaller than unity, hence the amplification matrix is approximately equal to the identity matrix, and the image distortions are small. Consider a circular source with radius Ro (see Fig. 1.2), in the absence of shear the image will be circular with a modified radius given by Shearing will cause the image to be elliptical with, |-yj = j^p, where r = b/a is the ratio of the minor (b) to major (a) axis, and the direction of the shear is given by its phase ip. Note that in the simplified case of a circular source the ellipticity (e) of the image is a direct measure of the shear, l7l = kl = (1-2) In general the intrinsic ellipticity of the source is non-zero yielding, £ = £ ( S ) + 7 , (1-3) where eW is the intrinsic source ellipticity. Although the intrinsic ellipticity of a given source is unknown, it is safe to assume that the average over source ellipticities will tend to zero, that is, there is no preferred direction for the intrinsic ellipticities. Hence a sufficiently large sample of background galaxies will provide an accurate 4 Chapter 1. Introduction measure of the shear2 (e) = (7). Taking 5 to be the mass density contrast C2^), the mass density power spectrum PZD(^) W) is defined as (6(£)F(l!)) = (2ir)36D(£-£')P3D(e,w). (1.4) S(£) is the Fourier transform of <5, 5® is the Dirac delta function, w is the radial distance parameter, and I is the three dimensional wave vector. The mass density power spectrum can be modeled from our understanding of the initial density fluctuations and how they evolve over time. The evolution is analytical in the linear regime of structure formation (where S << 1), but becomes highly non-linear at small scales and later times. Methods to account for non-linear structure growth have been devised by Peacock & Dodds (1996) and Smith et al. (2003), following the work of Hamilton et al. (1991). Combining the linear and non-linear models provides a cosmology dependant description of the shape of the power spectrum, however the overall normalisation is a free parameter. The normalisation parameter is defined as the mass density variance within a sphere of 8 h^Mpc radius at redshift zero: <4 = (4) = 2^ 3 / d*Vu>V, o)\w(eR)\2, (i.5) where W(£R) is the Fourier transform of some window function. If the window function is taken to be a so-called top-hat in real space, which is constant for \x\ < R and zero otherwise, one finds W ^ = i ^ ( ^ - C ° S m ) - ( L 6 ) Figure 1.3 shows how the mass density power spectrum changes with cosmological parameters and with different prescriptions for the non-linear growth of density perturbations. The convergence power spectrum PK(s) is defined to be: («(k)«*(k')> = (27r) 2JD(k - k') PK(k), (1.7) where k is the two-dimensional wave vector perpendicular to the line of sight. We can express 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 approximated by a single lensing plane. In the weak lensing regime the correlation function of the shear in Fourier space is identical to the correlation function of the convergence <7(k)7*(k')) = <A(k)«*(k')>, (1.8) 2The practical details of accurately measuring the shapes of galaxies from data is beyond the scope of this introduction. The most extensively used method, and the one employed in Chapter 2, was developed by Kaiser, Squires & Broadhurst (1995), with later additions by Luppino & Kaiser (1997) and Hoekstra et al. (1998) 5 Chapter 1. Introduction CO I I CO I O E 1 1 1—i i i i i | 1 1 1— I i Figure 1.3: Top: How the mass density power spectum changes with various CDM cosmologies. ACDM (Solid): A flat cosmology (Clm+i~l\ — 1) with a non-zero cosmological parameter, fi7n = 0.24, and fiA = 0.76. OCDM (Dashed): An open cosmology (Om + ftA < 1) with f2m = 0.24, fiA = 0.0. EdS (Dash-dot): An Einstein de-Sitter cosmology with Clm — 1.0, fiA = 0.0. For all cases as = 0.76 and the non-linear prescription of Smith et al. (2003) is used. Bottom: The linear power spectrum is given by the dash-dot line, two methods are used to account for the non-linear power, Peacock & Dodds (1996) (dashed) and Smith et al. (2003) (solid). 6 Chapter 1. Introduction hence by measuring the shear one can estimate PK(k). The convergence power spectrum can also be written as: PK{k) = I r du, ^ lp3D ( * ; W) , (1.9) 4 \ c / Jo a2(w) \fK(w) J where tun is the radial distance to the horizon, a(w) is the scale factor, is the angular diameter distance which depends on the curvature K ( K-1'2 sm(s/Kw) for K > 0 , fK{w)=\w for i f = 0, (1.10) I (-K)-V2 sinh(V^Kw) for K < 0 , and g(w) is a weight function which depends on the distribution of sources n(w) g(W)= r d . ' n y ) ^ ' ; ' (1.11) Jw JK(W) It is evident from the above relationships that weak lensing depends on cosmology. A measurement of the shear from surveys is a direct estimate of the convergence power spectrum, as seen from equations 1.7 and 1.8. The convergence power spectrum can be modeled (Eq.1.9) for many different cosmologies, and a best fit cosmology can be determined through a standard likelihood analysis. As seen from Eq.(1.9) weak lensing is most sensitive to the mass energy density (fim), the redshift distribution of the sources, the Hubble parameter at redshift zero (Ho), and the normalisation of the mass density power spectrum (as, which normalises P3D)-Figure 1.4 shows how the convergence power specrum changes with cosmology and non-linear modeling of the mass density power spectrum. The work presented in Chapter 2 represents the best cosmological constraints currently available from a weak lensing analysis. Data from four of the largest weak lensing surveys performed are combined to yield an effective survey area of approximately 100 square degrees. 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The unambiguous interpretation of the weak lensing signal makes it a powerful tool for measuring cosmological parameters which complement those from other probes such as, CMB anisotropics (Spergel et al., 2006) and type la supernovae (Astier et al., 2006). Cosmic shear has only recently become a viable tool for observational cosmology, with the first measure-ments reported simultaneously in 2000 (Bacon et al., 2000; Kaiser et al., 2000; Van Waerbeke et al., 2000; Wittman et al., 2000). The data these early studies utilised were not optimally suited for the extraction of a weak lensing signal. Often having a poor trade off between sky coverage and depth, and lacking photometry in more than one colour, the ability of these early surveys to constrain cosmology via weak lensing was limited, but impressive based on the data at hand. Since these early results several large dedicated surveys have detected weak lensing by large-scale structure, placing competitive constraints on cosmological parameters (Bacon et al., 2003; Brown et al., 2003; Hamana et al., 2003; Heymans et al., 2005; Hoekstra et al., 2006, 2002a; Jarvis et a l , 2003; Massey et al., 2005; Rhodes et al., 2004; Schrabback et al., 2006; Semboloni et al., 2006; Van Waerbeke et al., 2005). With the recent first measurements of a changing lensing signal as a function of redshift Bacon et al. (2005); Massey et al. (2007b); Semboloni et al. (2006); Wittman (2005), and the first weak lensing con-straints on dark energy Hoekstra et al. (2006); Jarvis et al. (2006); Kitching et al. (2006), the future of lensing is promising. There are currently several 'next generation' surveys being conducted. Their goal is to provide multi-colour data and excellent image quality, over a wide field of view. Such data will enable weak lensing to place tighter constraints on cosmology, breaking the degeneracy between the matter density parameter fim and the amplitude of the matter power spectrum a& and allowing for competitive constraints on dark energy. In this paper, we present a weak lensing analysis of the 100 deg2 W L survey that combines data from four of the largest sur-veys analysed to date, including the wide component of the Canada-France-Hawaii Telescope Legacy Survey 3 A version of this chapter has been submitted for publication. J. Benjamin, C. Heymans, E. Semboloni, L. Van Waerbeke, H. Hoekstra, T. Erben, M.D. Gladders, M. Hetterscheidt, Y. Mellier & H.K.C. Yee. Cosmological Constraints From the 100 Square Degree Weak Lensing Survey, 2007, MNRAS 11 Chapter 2. 100 sq. deg. WL Survey (CFHTLS-Wide, Hoekstra et al., 2006), the Garching-Bonn Deep Survey (GaBoDs, Hetterscheidt et al., 2006), the Red-Sequence Cluster Survey (RCS, Hoekstra et al., 2002a), and the VIRMOS-DESCART survey (VIR-MOS, Van Waerbeke et al., 2005). These surveys have a combined sky coverage of 113 deg2 (96.5 deg2, after masking), making this study the largest of its kind. Our goal in this paper is to provide the best estimates of cosmology currently attainable by weak lensing, through a homogeneous analysis of the major data sets available. Our analysis is distinguished from previous work in three important ways. For the first time we account for the effects of non-Gaussian contributions to the analytic estimate of the cosmic variance covariance matrix (Schneider et al., 2002), as described in Semboloni et al. (2007). Results from the Shear TEsting Programme (STEP, see Heymans et al., 2006a; Massey et al., 2007a) are used to correct for calibration error in our shear measurement methods, a marginalisation over the uncertainty in this correction is performed. We use the largest deep photometric redshift catalogue currently available (Ilbert et al., 2006), which provides redshifts for ~ 500,000 galaxies in the CFHTLS-Deep. Addition-ally, we properly account for cosmic variance in our calculation of the redshift distribution, an important source of error discussed in Van Waerbeke et al. (2006). This paper is organised as follows: in §2.2 we give a short overview of cosmic shear theory, and outline the relevant statistics used in this work. We describe briefly each of the surveys used in this study in §2.3. We present the measured shear signal in §2.4. In §2.5 we present the derived redshift distributions for each survey. We present the results of the combined parameter estimation in §2.6, closing thoughts and future prospects are discussed in §2.7. 2.2 Cosmic Shear Theory We briefly describe here the notations and statistics used in our cosmic shear analysis. For detailed reviews of weak lensing theory the reader is referred to Munshi et al. (2006), Bartelmann & Schneider (2001), and Schneider et al. (1998). The notation used in the latter are adopted here. The power spectrum of the projected density field (convergence K) is given by 9 H ^ 2 fw" dw P ( k ) - i ^ k r ^ p Z D ( ~ 4c * Jo ai(wf3D\fK(wy' W" ^.J^^fKiw' -W) &w'n(w') 2 (2.1) where H 0 is the Hubble constant, is the comoving angular diameter distance out to a distance w (wH is the comoving horizon distance), and n[w(z)} is the redshift distribution of the sources (see §2.5). P3D is the 3-dimensional mass power spectrum computed from a non-linear estimation of the dark matter clustering (see for example Peacock & Dodds, 1996; Smith et al., 2003), and k is the 2-dimensional wave vector perpendicular to the line-of-sight. 12 Chapter 2. 100 sq. deg. WL Survey In this paper we focus on the shear correlation function statistic £. For a galaxy pair separation 9, we define where the shear 7 = ("ft,Jr) is rotated into the local frame of the line joining the centres of each galaxy pair separated by 0, The shear correlation function | + is related to the convergence power spectrum through A quantitative measurement of the lensing amplitude and the systematics is obtained by splitting the signal into its curl-free (i?-mode) and curl (B-mode) components respectively. This method has been advocated to help the measurement of the intrinsic alignment contamination in the weak lensing signal (Crittenden et al, 2001, 2002), but it is also an efficient measure of the residual systematics from the PSF correction (Pen et al, 2002). The E and B modes derived from the shape of galaxies are unambiguously defined only for the so-called aperture mass variance (M a p), which is a weighted shear variance within a cell of radius 9C. The cell itself is defined as a compensated filter, such that a constant convergence K gives M a p = 0. (M a p) can be rewritten as a function of the tangential shear 7 4 if we express 7 = ( 7 4 , 7 , . ) in the local frame of the line connecting the aperture centre to the galaxy. (M a p) is given by: The jB-mode (M a p)x is obtained by replacing jt with 7 R . The aperture mass is insensitive to the mass sheet degeneracy, and therefore it provides an unambiguous splitting of the E and B modes. The drawback is that aperture mass is a much better estimate of the small scale power than the large scale power which can be seen from the function Jn(k6^) in Eq.(2.4) which peaks at k0c ~ 5. Essentially, all scales larger than a fifth of the largest survey scale remain inaccessible to M a p . The large-scale part of the lensing signal is lost by M a p , while the remaining small-scale fraction is difficult to interpret because the strongly non-linear power is difficult to predict accurately (Van Waerbeke et al, 2002). It is therefore preferable to decompose the shear correlation function into its E and B modes, as it is a much deeper probe of the linear regime. Following Crittenden et al. (2001, 2002), we define Z+(0) = (ft(rht(r + 9)) + (lT(r)lT(r + 9)). £-(0) = {lt(rht(r + 0))-(lr(rhr(r + e)), (2.2) (2.3) (2.4) (2.5) The E and B shear correlation functions are then given by *+(<>)+m (2.6) 2 2 13 Chapter 2. 100 sq. deg. WL Survey In the absence of systematics, £g = 0 and £E = £+ (Eq.2.3). In contrast with the M a p statistics, the separation of the two modes depends on the signal integrated out with the scales probed by all surveys. One option is to calculate Eq.(2.6) using a fiducial cosmology to compute £_ on scales 6 —» oo. As shown in Heymans et al. (2005), changes in the choice of fiducial cosmology do not significantly affect the results, allowing this statistic to be used as a diagnostic tool for the presence of systematics. This option does however prevent us from using £ E to constrain cosmology. As the statistical noise on the measured is ~ \f2 smaller than the statistical noise on the measured £ + , there is a more preferable alternative. If the survey size is sufficiently large in comparison to the scales probed by we can consider the unknown integral to be a constant that can be calibrated using the aperture mass 5-mode statistic (Mfp (A0))j_. A range AO of angular scales where (Mfp (A0))± ~ 0 ensures that the S-mode of the shear correlation function is zero as well (within the error bars), at angular scales ~ In this analysis we have calibrated £E,B f ° r o u r survey data in this manner. This alternative method has also been verified by re-calculating £E,B using our final best-fit cosmology to extrapolate the signal. We find the two methods to be in very close agreement. 2.3 Data description In this section we summarise the four weak lensing surveys that form the 100 deg2 WL survey: CFHTLS-Wide (Hoekstra et al., 2006), GaBoDs (Hetterscheidt et al., 2006), RCS (Hoekstra et al., 2002a), and VTRMOS-DESCART (Van Waerbeke et al., 2005), see Table 2.1 for an overview. Note that we have chosen not to include the CFHTLS-Deep data (Semboloni et al., 2006) in this analysis since the effective area of the analysed data is only 2.3 deg2. Given the breadth of the parameter constraints obtained from this preliminary data set there is little value to be gained by combining it with four surveys which are among the largest available. 2.3.1 CFHTLS-Wide The Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) is a joint Canadian-French program designed to take advantage of Megaprime, the CFHT wide-field imager. This 36 CCD mosaic camera has a 1 x 1 degree field of view and a resolution of 0.187 arcseconds per pixel. The CFHTLS has been allocated ~ 450 nights over a 5 year period (starting in 2003), the goal is to complete three independent survey components; deep, wide, and very wide. The wide component will be imaged with five broad-band filters: «*, g', r', /', z'. With the exception of u* these filters are designed to match the SDSS photometric system (Fukugita et al., 1996). The three wide fields will span a total of 170 deg2 on completion. In this study, as in Hoekstra et al. (2006), we use data from the Wl and W3 fields that total 22 deg2 after masking, and reach a depth of i' = 24.54. For detailed information concerning data reduction and object analysis the reader is referred to Hoekstra et al. (2006), and Van Waerbeke et al. (2002). 4 14 Table 2.1: Summary of the published results from each survey used in this study. The values for og correspond to f2 m = 0.24, and are given for the Peacock & Dodds (1996) method of calculating the non-linear evolution of the matter power spectrum. The statistics listed were used in the previous analyses; the shear correlation functions £E,B (Eq. 2.6), the aperture mass statistic (Mfp) (Eq. 2.4) and the top-hat shear variance ill B) ( s e e V a n Waerbeke et al. (2005)). CFHTLS-Wide GaBoDS RCS VIRMOS-DESCART Area (deg2) 22/0 UJ6 53 N f i e i d s 2 52 13 4 Magnitude Range 21.5 < i' < 24.5 21.5 < R < 24.5 22 < Rc < 24 21 < 7AB < 24.5 < «source > 0.81 0.78 ~ 0.6 0.92 08 (An = 0.24) 0.99 ± 0 . 0 7 0.92 ± 0 . 1 3 0.98 ± 0 . 1 6 0.96 ± 0 . 0 8 Statistic tejV) {M^M {M2ap){e) (jlB)(6) Chapter 2. 100 sq. deg. WL Survey 2.3.2 GaBoDS The Garching-Bonn Deep Survey (GaBoDS) data set was obtained with the Wide Field Imager (WFI) of the MPG/ESO 2.2m telescope on La Silla, Chile. The camera consists of 8 CCDs with a 34 by 33 arcminute field of view and a resolution of 0.238 arcseconds per pixel. The total data set consists of 52 statistically independent fields, with a total effective area (after trimming) of 13 deg2. The magnitude limits of each field ranges between RVEGA — 25.0 and RVEGA = 26.5 (this corresponds to a 5a detection in a 2 arcsecond aperture). To obtain a homogeneous depth across the survey we follow Hetterscheidt et al. (2006) performing our lensing analysis with only those objects which lie in the complete magnitude interval R 6 [21.5,24.5]. For further details of the data set, and technical information regarding the reduction pipeline the reader is referred to Hetterscheidt et al. (2006). 2.3.3 VIRMOS-DESCART The DESCART weak lensing project is a theoretical and observational program for cosmological weak lensing investigations. The cosmic shear survey carried out by the DESCART team uses the CFH12k data jointly with the VIRMOS survey to produce a large homogeneous photometric sample in the B V R l broad band filters. CFH12k is a L2 CCD mosaic camera mounted at the Canada-France-Hawaii Telescope prime focus, with a 42 by 28 arcminute field of view and a resolution of 0.206 arcseconds per pixel. The VIRMOS-DESCART data consist of four uncorrelated patches of 2 x 2 deg2 separated by more than 40 degrees. Here, as in Van Waerbeke et al. (2005), we use /AB-band data from all four fields, which have an effective area of 8.5 deg2 after masking, and a limiting magnitude of 7 A B = 24.5 (this corresponds to a 5a detection in a 3 arcsecond aperture). Technical details of the data set are given in Van Waerbeke et al. (2001) and McCracken et al. (2003), while an overview of the VIRMOS-DESCART survey is given in Le Fevre et al. (2004). 2.3.4 RCS The Red-Sequence Cluster Survey (RCS) is a galaxy cluster survey designed to provide a large sample of opti-cally selected clusters of galaxies with redshifts between 0.3 and 1.4. The final survey covers 90 deg2 in both Rc and z', spread over 22 widely separated patches of ~ 2.1 x 2.3 degrees. The northern half of the survey was observed using the CFH12k camera on the CFHT, and the southern half using the Mosaic II camera mounted at the Cerro Tololo Inter-American Observatory (CTIO), Victor M. Blanco 4m telescope prime focus. This camera has an 8 CCD wide field imager with a 36 by 36 arcminute field of view and a resolution of 0.27 arcseconds per pixel. The data studied here, as in Hoekstra et al. (2002a), consist of a total effective area of 53 deg2 of masked imaging data, spread over 13 patches, with a limiting magnitude of 25.2 (corresponding to a 5a point source depth) in the Rc band. 43 deg2 were taken with the CFHT, the remaining 10 deg2 were taken with the Blanco telescope. A detailed description of the data reduction and object analysis is described in Hoekstra et al. (2002b). 16 Chapter 2. 100 sq. deg. WL Survey 2.4 Cosmic shear signal Lensing shear has been measured for each survey using the common KSB galaxy shape measurement method (Hoekstra et al, 1998; Kaiser et al., 1995; Luppino & Kaiser, 1997). The practical application of this method to the four surveys differs slightly, as tested by the Shear TEsting Programme; STEP (Heymans et al., 2006a; Massey et al., 2007a), resulting in a small calibration bias in the measurement of the shear. Heymans et al. (2006a) define both a calibration bias (m) and an offset of the shear (c), 7meas - 7true = «27true + c ; " ( 2 - 7 ) where 7 m e a s is the measured shear, 7t r Ue is the true shear signal, and m and c are determined from an analysis of simulated data. The value of c is highly dependent on the strength of the PSF distribution because an unquantified uniform component of the PSF will behave as an offset in the shear. We take c = 0, which for our purposes is a very good approximation since the methods used here have very small shear offsets when recovering the shear from simulated data (Massey et al., 2007a). Eq.(2.7) then simplifies to 7 t r U e = 7meas(?7i + I) - 1- The calibration bias for the CFHTLS-Wide, RCS and VIRMOS-DESCART (HH analysis in Massey et al. (2007a)) is m = -0.0017 ± 0.0088, and m = 0.038 ± 0.026 for GaBoDS (MH analysis in Massey et al. (2007a)). We measure the shear correlation functions £E(0) and (Eq. 2.6) for each survey, shown in Figure 2.1. As discussed in §2.2, these statistics are calibrated using the measured aperture mass 5-mode (M 2 p) j _ - The lcr errors on £E include statistical noise, computed as described in Schneider et al. (2002), non-Gaussian cosmic variance (see §2.6 for details), and a systematic error added in quadrature. The systematic error for each angular scale 6 is given by the magnitude of £B(0). The lcr errors on £B are statistical only. We correct for the calibration bias by scaling £E by (1 + m)~2, the power of negative 2 arising from the fact that fjj is a second order shear statistic. Tables containing the measured correlation functions and their corresponding covariance matrix can be found in appendix A. When estimating parameter constraints in §2.6.2 we marginalise over the error on the calibration bias, which expresses the range of admissible scalings of £E. 2.5 Redshift Distribution The measured weak lensing signal depends on the redshift distribution of the sources, as seen from Eq.(2.1). Past weak lensing studies (e.g., Hoekstra et al. (2006); Semboloni et al. (2006); Massey et al. (2005); Van Waerbeke et al. (2005)) have used the Hubble Deep Field (HDF) photometric redshifts to estimate the shape of the redshift distribution. Spanning only 5.3 arcmin2, the HDF suffers from cosmic variance as described in Van Waerbeke et al. (2006), where it is also suggested that the HDF fields may be subject to a selection bias. This cosmic variance on the measured redshift distribution adds an additional error to the weak lensing analysis that is not typically accounted for. In this study we use the largest deep photometric redshift catalogue in existence, from Ilbert et al. (2006), who have estimated redshifts on the four Deep fields of the T0003 CFHTLS release. The redshift catalogue 17 Chapter 2. 100 sq. deg. WL Survey Figure 2.1: E and B modes of the shear correlation function £ (filled and open points, respectively) as measured for each survey. Note that the la errors on the _E-modes include statistical noise, non-Gaussian cosmic variance (see §2.6) and a systematic error given by the magnitude of the B-mode. The la error on the B-modes is statistical only. The results are presented on a log-log scale, despite the existence of negative B-modes. We have therefore collapsed the infinite space between 10~7 and zero, and plotted negative values on a separate log scale mirrored on 10~7. Hence all values on the lower portion of the graph are negative, their absolute value is given by the scaling of the graph. Note that this choice of scaling exagerates any discrepancies. The solid lines show the best fit ACDM model for Vtm = 0.24, h = 0.72, T = hUm, a& given in Table 2.4, and n(z) given in Table 2.2. The latter two being chosen for the case of the high confidence redshift calibration sample, an n(z) modeled by Eq.(2.9), and the non-linear power spectrum estimated by Smith et al. (2003). 18 Table 2.2: The best fit model parameters for the redshift distribution, given by either Eq.(2.9) or Eq.(2.10). Models are fit using the full calibration sample of Ilbert et al. (2006) 0.0 < zp < 4.0, and using only the high confidence region 0.2 < zp < 1.5. xl i s the reduced \ 2 statistic, (z), and z m are the average and median redshift respectively, calculated from the model on the range 0.0 < z < 4.0. Eq.(2.9) lki.(2.10) a P zo (z) z^ xl a b c N (z) z^ xT~ CFHTLS-Wide 0.836 3.425 1.171 0.802 0.788 1.63 0.723 6.772 2.282 2.860 0.848 0.812 1.65 GaBoDS 0.700 3.186 1.170 0.784 0.760 1.96 0.571 6.429 2.273 2.645 0.827 0.784 2.80 - Z P - 1 - 5 RCS 0.787 3.436 1.157 0.781 0.764 1.41 0.674 6.800 2.095 2.642 0.823 0.788 1.60 VIRMOS-DESCART 0.637 4.505 1.322 0.823 0.820 1.48 0.566 7.920 6.107 6.266 0.859 0.844 2.06 CFHTLS-Wide 1.197 1.193 0.555 0.894 0.788 8.94 0.740 4.563 1.089 1.440 0.945 0.828 2.41 GaBoDS 1.360 0.937 0.347 0.938 0.800 7.65 0.748 3.932 0.800 1.116 0.993 0.832 2.40 _ « P _ 4.0 R C S 1 4 2 3 L Q 3 2 Q 3 9 l o g 8 g 0 7 ? 2 5 6 ? Q 8 1 9 4 4 ^ g Q 8 0 0 L 2 Q 1 Q 9 3 9 o g o g l g 4 VIRMOS-DESCART 1.045 1.445 0.767 0.909 0.816 9.95 0.703 5.000 1.763 2.042 0.960 0.864 2.40 Chapter 2. 100 sq. deg. WL Survey is publicly available at terapix:iap.fr. The full photometric catalogue contains 522,286 objects, covering an effective area of 3.2 deg2. A set of 3241 spectroscopic redshifts with 0 < z < 5 from the VIRMOS VLT Deep Survey (VVDS) were used as a calibration and training set for the photometric redshifts. The resulting photo-metric redshifts have an accuracy of o-| 2 p- Z B|/i+ 2. = 0.043 for i ' A B = 22.5 - 24, with a fraction of catastrophic errors of 5.4%, where a catastrophic error is dened to be those objects with |zp — zs\/l + zs > 0.15. Ilbert et al. (2006) demonstrate that their derived redshifts work best in the range 0.2 < z < 1.5, having a fraction of catastrophic errors of ~ 5% in this range. The fraction of catastrophic errors increases dramatically at z < 0.2 and 1.5 < z < 3 reaching ~ 40% and ~ 70% respectively. This is explained by a degeneracy between galax-ies at 2 s p e c < 0.4 and 1.5 < z p hot < 3 due to a mismatch between the Balmer break and the intergalactic Lyman-alpha forest depression. Van Waerbeke et al. (2006) estimate the expected sampling error on the average redshift for such a pho-tometric redshift sample to be ~ 3%/\/4 = 1.5%, where the factor of \fi comes from the four independent CFHTLS-Deep fields, a great improvement over the ~ 10% error expected for the HDF sample. 2.5.1 Magnitude Conversions In order to estimate the redshift distributions of the surveys, we calibrate the magnitude distribution of the photometric redshift sample (hence forth the z p sample) to that of a given survey by converting the CFHTLS filter set u"g'r'i''z' to 7 A B magnitudes for VIRMOS and - R V E G A magnitudes for RCS and GaBoDS. We employ the linear relationships between different filter bands given by Blanton & Roweis (2007): IAB = i' - 0.0647 - 0.7177 {{%' - z') - 0.2083] , (2.8) - R V E G A = r' - 0.0576 - 0.3718 [(/ - i') - 0.2589] - 0.21. These conversions were estimated by fitting spectral energy distribution templates to data from the Sloan Digital Sky Survey. They are considered to be accurate to 0.05 mag or better, and are thus good enough for our purposes. 2.5.2 Modeling the redshift distribution n(z) The galaxy weights used in each survey's lensing analysis result in a weighted source redshift distribution. To estimate the effective n(z) of each survey we draw galaxies at random from the z p sample, using the method described in §6.5.1 of Wall & Jenkins (2003) to reproduce the shape of the weighted magnitude distribution of each survey. A Monte Carlo (bootstrap) approach is taken to account for the errors in the photometric redshifts, as well as the statistical variations expected from drawing a random sample of galaxies from the z p sample to create the redshift distribution. The process of randomly selecting galaxies from the z p sample, and therefore estimating the redshift distribution of the survey, is repeated 1000 times. Each redshift that is selected is drawn from its probability distribution defined by the ±1 and ±3cr errors (since these are the error bounds given in 20 Chapter 2. 100 sq. deg. WL Survey Ilbert et al. (2006)'s catalogue). The sampling of each redshift is done such that a uniformly random value within the la error is selected 68% of the time, and a uniformly random value within the three sigma error (but exterior to the la error) is selected 32% of the time. The redshift distribution is then defined as the average of the 1000 constructions, and an average covariance matrix of the redshift bins is calculated. At this point cosmic variance and Poisson noise are added to the diagonal elements of the covariance matrix, following Van Waerbeke et al. (2006). They provide a scaling relation between cosmic (sample) variance noise and Poisson noise where < T s a m p i e / o p o i s s o n is given as a function of redshift for different sized calibration samples. We take the curve for a 1 deg2 survey, the size of a single CFHTLS-Deep field, and divide by \/4 since there are four independent 1 deg2 patches of sky in the photometric redshift sample. Based on the photometric redshift sample of Ilbert et al. (2006) we consider two redshift ranges, the full range of redshifts in the photometric catalogues 0.0 < z < 4.0, and the high confidence range 0.2 < z < 1.5. The goal is to assess to what extent this will affect the redshift distribution, and — in turn — the derived parameter constraints. The shape of the normalised redshift distribution is often assumed to take the following form: n ( . ) = — ^ ( ± ) Q e x P [ - ( - ^ ] , (2.9) where a, /3, and zo are free parameters. If the full range of photometric redshifts is used the shape of the redshift distribution is poorly fit by Eq.(2.9), this is a result of the function's exponential drop off which cannot accommodate the number of high redshift galaxies in the tail of the distribution (see Fig.(2.2)). We therefore adopt a new function in an attempt to better fit the normalised redshift distribution, n { z ) = N^Tc> ( 2 - 1 0 ) where a, b, and c are free parameters, and N is a normalising factor, roo z l a x N=(L dzWz) • (2-u)-The best fit model is determined by minimising the generalised chi square statistic: X2 = {di-mi)C-\di-mi)T, (2.12) where C is the covariance matrix of the binned redshift distribution described above, di is the data vector, and mi is the model vector. For each survey we determine the best fit for both Eq.(2.9) and Eq.(2.10) in either case we consider both the full range of redshifts and the high confidence range, the results are presented in Table 2.2. Figure 2.2 shows the best fit models to the CFHTLS-Wide survey, when the full range of photometric red-shifts is considered (left panel) Eq.(2.10) clearly fits the distribution better than Eq.(2.9), having reduced x2 statistics of 2.41 and 8.94 respectively. When we consider only the high confidence redshifts (right panel of Fig-Figure 2.2: Normalised redshift distribution for the CFHTLS-Wide survey, given by the histogram, where the error bars include Poisson noise and cosmic variance of the photometric redshift sample. The dotted curve shows the best fit for Eq.(2.9), and the solid curve for Eq.(2.10). Left: Distribution obtained if all photometric redshifts are used 0.0 < z < 4.0, and \ 2 is calculated between 0.0 < z < 2.5. Right: Distribution obtained if only the high confidence redshifts are used 0.2 < z < 1.5, and x2 is calculated on this range. The existence of counts for 2 > 1.5 is a result of drawing the redshifts from their full probability distributions. to to Chapter 2. 100 sq. deg. WL Survey ure 2.2) both functions fit equally well having reduced x 2 statistics of 1.65 and 1.63 for Eq.(2.10) and Eq.(2.9) respectively. Note that the redshift distribution is non-zero at z > 1.5 for the high confidence photometric red-shifts (0.2 < zp < 1.5) because of the Monte Carlo sampling of the redshifts from their probability distributions. 2.6 Parameter estimation 2.6.1 Maximum likelihood method We investigate a six dimensional parameter space consisting of the mean matter density flm, the normalisation of the matter power spectrum ag, the Hubble parameter h, and n(z) the redshift distribution parametrised by either a, /?, and zn (Eq.2.9) or a, b, and c (Eq.2.10). A flat cosmology (fim + SIA = 1) is assumed throughout, and the shape parameter is given by T = f2m h. The default priors are taken to be fim 6 [0.1, l},<Jg € [0.5,1.2], and h 6 [0.64, 0.8] with the latter in agreement with the findings of the HST key project (Freedman et al., 2001). The priors on the redshift distribution were arrived at using a Monte Carlo technique. This is necessary since the three parameters of either Eq.(2.9) or Eq.(2.10) are very degenerate, hence simply finding the 2 or 3 a levels of one parameter while keeping the other two fixed at their best fit values does not fairly represent the probability distribution of the redshift parameters. The method used ensures a sampling of parameter triplets whose number count follow the 3-D probability distribution; that is 68% lie within the lcr volume, 97% within the 2cr volume, etc. We find 100 such parameter trios and use them as the prior on the redshift distribution, therefore this is a Gaussian prior on n(z). Given the data vector fi, which is the shear correlation function (£e of Eq.2.6) as a function of scale, and the model prediction rrii(Qm, ag, h, n(z)) the likelihood function of the data is given by L=7mmexp -(ti-mjC-^fi-mi)1 (2.13) where n is the number of angular scale bins and C is the n x n covariance matrix. The shear covariance matrix can be expressed as Cij = <(&-wf&-^)>> (2-14) where m is the mean of the shear at scale i, and angular brackets denote the average over many independent patches of sky. To obtain a reasonable estimate of the covariance matrix for a given set of data one needs many independent fields, this is not the case for either the CFHTLS-Wide or VIRMOS-DESCART surveys. We opt to take a consistent approach for all 4 data sets by decomposing the shear covariance matrix as C = C n + CB + C s , where C n is the statistical noise, CB is the absolute value of the residual J3-mode and CS is the cosmic variance covariance matrix. CN can be measured directly from the data, it represents the statistical noise inherent in a finite data set. CB is diagonal and represents the addition of the 5-mode in quadrature to the uncertainty, this provides a conservative limit to how well the lensing signal can be determined. For Gaussian cosmic variance the matrix CS can be computed according to Schneider et al. (2002), assuming 23 Chapter 2. 100 sq. deg. WL Survey Table 2.3: Data used for each survey to calculate the Gaussian contribution to the cosmic variance covariance matrix. AeB is the effective area, n e fj the effective galaxy number density, and ae the intrinsic ellipticity disper-sion. neg (deg2) (arcmin-2) CFHTLS-Wide 22 12 0.47 RCS 53 8 0.44 VIRMOS-DESCART 8.5 15 0.44 GaBoDS 13 12.5 0.50 an effective survey area Aes, an effective number density of galaxies roefr, and an intrinsic ellipticity dispersion ae (see Table 2.3). However, as shown by Semboloni et al. (2007), non-linear effects at small scales lead to significant deviations from Gaussian behaviour. They use ray tracing simulations to determine the effect of non-Gaussian contributions to the cosmic shear covariance matrix, calculating the following: m,^ ) = § = # ^ , (2.i5) ^Gaus(?+i V I , ^2) where ,F(#i,i?2) is the ratio of the covariance measured from N-body simulations (C m e a s (£ + ; ' i?i , i?2)) to that expected from Gaussian effects alone ( C G a u s ( £ + ; i ? 2 ) ) - It is found that T^i,^) increases significantly above unity at scales smaller than 10 arcminutes, the parametrised fit as a function of mean source redshift is given by, W,02) = P 1 ( Z ) „ , (2.16) 16 9 pa(z) = 1.62 z " 0 6 8 exp( -z - ° - 6 8 ) - 0.03. A fiducial model is required to calculate the Gaussian covariance matrix, it is taken as J7m = 0.3, A = 0.7, <T8 = 0.8, h = 0.72, and the best fit n(z) model (see Table 2.2). We then use the above prescription at scales less than ~ 10 arcminutes to account for non-Gaussianities. For each survey the total covariance matrix (C = C n + Cs + Cg) is presented in appendix A. To test this method we compare our analytic covariance matrix for GaBoDS with that found by measuring it from the data (Hetterscheidt et al, 2006). Since GaBoDS images 52 independent fields it is possible to obtain an estimate of C directly from the data. The contribution from the B-modes (Cs) is not added to our analytic estimate in this case, since its inclusion is meant as a conservative estimate of the systematic errors. We find a median percent difference along the diagonal of ~ 15%, which agrees well with the accuracy obtained for 24 Chapter 2. 100 sq. deg. WL Survey simulated data (Semboloni et al, 2007). 2.6.2 Joint constraints on CT8 and n m For the combined survey we place joint constraints on as and f i m as shown in Figure 2.3. Fitting to the maximum likelihood region we find a 8 = (0.84 ± 0 . 0 7 ) ( ^ ) ~ ° • 5 9 ) where the error is the la level for Qm = 0.24. This result assumes a flat ACDM cosmology and adopts the non-linear matter power spectrum of Smith et al. (2003). The redshift distribution is estimated from the high confidence CFHTLS-Deep photometric redshifts using the standard n(z) model given in Eq.(2.9). Marginalisation was performed over h 6 [0.64,0.801 w ' t n flat priors, n(z) with Gaussian priors as described in §2.6.1, and a calibration bias of the shear signal with flat priors as discussed in §2.4. The corresponding constraints from each survey are presented in Figure 2.4 and tabulated in Table 2.4. Results in Table 2.4 are presented for two methods of calculating the non-linear power spectrum; Peacock & Dodds (1996) and Smith et al. (2003). We find a difference of approximately 3% in the best-fit a 8, where results using Smith et al. (2003) are consistently the smaller of the two. The contribution to the error budget due to the conservative addition of the B-modes to the covariance matrix is small, amounting to at most 0.01 in the la error bar on as f° r a n of 0.24. We present a comparison of the measured ag values with those previously published in Figure 2.5. The quoted ag values are for an ttm of 0.24, and all error bars denote the la region from the joint constraint contours. Our error bars (filled circles) are typically smaller than those from the literature (open circles) mainly due to the improved estimate of the redshift distribution. Our updated result for each survey agrees with the previous analysis within the error bars, this remains true if the literature values are lowered by ~ 3% to account for the difference in methods used for the non-linear power specrum. Also plotted are the la limits for the combined result and WMAP 3 year constraints (solid and dashed lines respectively), our result is consistent with WMAP at the la level. In addition to our main analysis we have investigated the impact of using four different models for the redshift distribution. These models are dependent on which functional form is used (Eq.(2.9) or Eq.(2.10)) and which range is used for the photometric redshift sample (0.0 < zp < 4.0 or 0.2 < zp < 1.5). Table 2.4 gives the best fit joint constraints on as and fim, for each survey as well as the combined survey result. Comparing the best fit as values with the average redshifts listed in Table 2.2, we find, as expected, that higher redshift models result in lower values for as. Attempting to quantify this relation in terms of mean redshift fails. Changes in mean redshift are large (~ 14%) between the photometric samples, compared to ~ 6% between the two n(z) models, however a ~ 5% change in as is seen for both. The median redshift is a much better gauge, changing by ~ 4% between both photometric samples and n(z) models. Precision cosmology at the 1% level will necessitate roughly the same level of precision of the median redshift. Future cosmic shear surveys will require thorough knowledge of the complete redshift distribution of the sources, in particular to what extent a high redshift tail exists. 25 Chapter 2. 100 sq. deg. WL Survey OO b 1.20 1 . 1 0 1 1.00 0 .90 0 .80 0 .70 0 .60 0 .50 E ~i 1 1 1 1 1 1 1 1 1 1 1 1 1 r J I L 0.2 0.4 0.6 Q 0.8 1.0 M Figure 2.3: Joint constraints on og and fim from the 100 deg2 WL survey assuming a fiat ACDM cosmology and adopting the non-linear matter power spectrum of Smith et al. (2003). The redshift distribution is estimated from the high confidence photometric redshift catalogue (Ilbert et al., 2006), and modeled with the standard functional form given by Eq.(2.9). The contours depict the 0.68, 0.95, and 0.99% confidence levels. The models are marginalised, over h = 0.72 ± 0.08, shear calibration bias (see §2.4) with uniform priors, and the redshift distribution with Gaussian priors (see §2.6.1). Similar results are found for all other cases, as listed in Table 2.4. 26 Chapter 2. 100 sq. deg. WL Survey CO b oo b 0.50 Figure 2.4: Joint constraints on as and ftm for each survey assuming a fiat ACDM cosmology and adopting the non-linear matter power spectrum of Smith et al. (2003). The redshift distribution is estimated from the high confidence photometric redshift catalogue (Ilbert et al, 2006), and modeled with the standard functional form given by Eq.(2.9). The contours depict the 0.68, 0.95, and 0.99% confidence levels. The models are marginalised, over h ' 0.72 ± 0.08, shear calibration bias (see §2.4) with uniform priors, and the redshift distribution with Gaussian priors (see §2.6.1). Similar results are found for all other cases, as listed in Table 2.4. 27 Table 2.4: Joint constraints on the amplitude of the matter power spectrum CT8. and the matter energy density Qm, parametrised as ag Smith et al. (2003) 0.0 < zp < 4.0 0. 2 < zp < 1.5 Eq.(2.9) Eq.(2.10) Eq.(2.9) Eq.(2.L0) CFHTLS-Wide 0.84 ±0.07 0.55 0.81 ± 0.06 0.54 0.86 ± 0.07 0.56 0.84 ± 0.07 0.55 GaBoDS 0.93 ±0.11 0.60 0.89 ± 0.09 0.59 1.01 ±0.11 0.65 0.98 ±0 .13 0.63 RCS 0.75 ±0.07 0.55 0.73 ±0.08 0.55 0.78 ±0.07 0.57 0.76 ± 0.07 0.56 VIRMOS-DESCART 0.99 ±0.07 0.59 0.95 ±0.08 0.58 1.02 ±0.08 0.60 1.00 ±0 .09 0.59 Combined 0.80 ±0.06 0.57 0.77 ±0.05 0.56 0.84 ±0.07 0.59 0.82 ±0.06 0.58 CFHTLS-Wide 0.87 ± 0.08 0.57 0.83 ± 0.07 0.56 0.89 ± 0.08 0.58 0.87 ± 0.08 0.57 GaBoDS 0.96 ± 0.10 0.62 0.93 ± 0.10 0.60 1.04 ± 0.12 0.66 1.01 ± 0.11 0.64 Peacock & Dodds (1996) RCS 0.77 ± 0.07 0.57 0.74 ± 0.07 0.56 0.81 ± 0.09 0.58 0.79 ± 0.08 0.58 VIRMOS-DESCART 1.02 ± 0.08 0.60 0.98 ± 0.08 0.59 1.05 ± 0.08 0.62 1.03 ± 0.09 0.61 Combined 0.82 ± 0.06 0.58 0.79 ± 0.06 0.57 0.86 ± 0.07 0.60 0.84 ± 0.07 0.59 to 00 Chapter 2. 100 sq. deg. WL Survey Figure 2.5: Values for as when Qm is taken to be 0.24, filled circles (solid) give our results with la error bars, open circles (dashed) show the results from previous analyses (Table 2.1). Our results are given for the high confidence photometric redshift catalogue, using the functional form for n(z) given by Eq.(2.9), and the Smith et al. (2003) prescription for the non-linear power spectrum. The literature values use the Peacock & Dodds (1996) prescription for non-linear power, and are expected to be ~ 3% higher than would be the case for Smith et al. (2003). The forward slashed hashed region (enclosed by solid lines) shows the la range allowed by our combined result, the back slashed hashed region (enclosed by dashed lines) shows the la range given by the WMAP 3 year results. 29 Chapter 2. 100 sq. deg. WL Survey 2.7 Discussion and Conclusion We have performed an analysis of the 100 square degree weak lensing survey that combines four of the largest weak lensing datasets in existence. Our results provide the tightest weak lensing constraints on the amplitude of the matter power spectrum as and matter density fim and a marked improvement on accuracy compared to previous results. Using the non-linear prediction of the cosmological power spectra given in Smith et al. (2003), the high confidence region of the photometric redshift calibration sample, and Eq.(2.9) to model the redshift distribution, we find a8 = (0.84 ± 0.07)(^) - 0- 5 9. Our analysis differs from previous weak lensing analyses in three important aspects. We correctly account for non-Gaussian cosmic variance using the method of Semboloni et al. (2007), thus improving upon the purely Gaussian contribution given by Schneider et al. (2002). Using the results from STEP (Massey et al, 2007a) we correct for the shear calibration bias and marginalise over our uncertainty in this correction. In addition we use the largest deep photometric redshift catalogue in existence (Ilbert et al, 2006) to provide accurate models for the redshift distribution of sources, we also account for the effects of cosmic variance in these distributions (Van Waerbeke et al, 2006). Accounting for the non-Gaussian contribution to the shear covariance matrix, which dominates on small scales, is very important. At a scale of 10 arcminutes the full non-Gaussian contribution is twice that of the Gaussian contribution alone, this discrepancy increases with decreasing scale, reaching an order of magnitude by 2 arcminutes. This increases the errors on the shear correlation function at small scales, leading to slightly weaker constraints on cosmology. Ideally we would estimate the shear covariance directly from the data, as is done in the previous analyses for both GaBoDS (Hetterscheidt et al, 2006) and RCS (Hoekstra et al, 2002a). However, since we can not accomplish this for all the surveys in this work, due to a deficit of independent fields, we opt for a consistent approach, using the analytic treatment along with the non-Gaussian calibration as described in Semboloni et al. (2007). Accurate determination of the redshift distribution of the sources is crucial for weak lensing cosmology since it is strongly degenerate with cosmological parameters. Past studies using small external photometric catalogues (such as the HDF) suffered from cosmic variance, a previously unqualified source of error that is taken into account here using the prescription of Van Waerbeke et al. (2006). In most cases the revised redshift distributions were in reasonable agreement with previous results, however this was not the case for the RCS survey. We found that the previous estimation was biased toward low redshift (see Figure 2.6), resulting in a significantly larger estimation of as (0.86tQQ5 for f2m = 0.3) than is presented here (0.69 ± 0.07). This difference is primarily due to a problem in the filter set conversion, between those used to image HDF (F300W, F450W, F606W, and F814W) and the Cousins Rc filter used by RCS. In addition to using the best available photometric redshifts, we marginalise over the redshift distribution by selecting parameter triplets from their full 3D probability distribution, instead of fixing two parameters and varying the third as is often done. Thus the marginalisation is representative of the full range of n(z) shapes, which are difficult to probe by varying one parameter due to degeneracies. 30 Chapter 2. 100 sq. deg. WL Survey 2.5 0 0.5 1 1.5 2 2.5 3 Z Figure 2.6: Redshift distributions for the RCS survey. The solid line shows the best fit n(z) from Hoekstra et al. (2002a), the dotted curve is our best fit n(z), the average redshifts are 0.6 and 0.78 respectively. We have conducted our analysis using two different functional forms for the redshift distribution, the stan-dard form given in Eq.(2.9) and a new form given by Eq.(2.10). The new function is motivated by the presence of a high-z tail when the entire photometric catalogue (0.0 < zp < 4.0) is used to estimate the redshift distribu-tion, Eq.(2.9) does a poor job of fitting this distribution (Figure 2.2). However when we restrict the photometric catalogue to the high confidence region (0.2 < zp < 1.5) both functions fit well, and the tendency for Eq.(2.10) to exhibit a tail towards high-z increases the median redshift resulting in a slightly lower estimate of erg. Though the tail of the distribution has only a small fraction of the total galaxies, they may have a significant lensing signal owing to their large redshifts. The influences of the different redshift distributions on cosmology is consistent within our lcr errors but as survey sizes grow and statistical noise decreases, such differences will become sig-nificant. As the CFHTLS-Deep will be the largest deep photometric redshift catalogue for some years to come, this posses a serious challenge to future surveys attempting to do precision cosmology. To assess the extent of any high redshift tail, future surveys should strive to include photometric bands in the near infra-red, allowing for accurate redshift estimations beyond z = 1.5. For fim = 0.24, the combined results using Eq.(2.9) are 0.80 ± 0.06 and 0.84 ± 0.07 for the unrestricted and high confidence photometric redshifts respectively, using Eq.(2.10) we find 0.77 ± 0.05 and 0.82 ± 0.06 (these results use the non-linear power spectrum given by Smith et al. (2003)). For completeness we provide results for both the Smith et al. (2003) and Peacock & Dodds (1996) non-linear power spectra, the resulting ag values differ by ~ 3% of the Smith et al. (2003) value which is consistently the smaller of the two. As the Smith et al. (2003) 31 Chapter 2. 100 sq. deg. WL Survey study has been shown to be more accurate than Peacock & Dodds (1996), we prefer the results of this model, however, given the magnitude of variations resulting from different redshift distributions, this difference is not an important issue in the current work. However, accurately determining the non-linear matter power spectra is another challenge for future lensing surveys intent on precision cosmology. Alternatively, surveys focusing only on the large scale measurement of the shear (Fu et al, in prep.) are able to avoid complications arising from the estimation of non-linear power on small scales. Surveys of varying depth provide different joint constraints in the Qm — as plane, thus combining their likeli-hoods produces some degeneracy breaking. Taking the preferred prescription for the non-linear power spectrum, and using Eq.(2.9) along with the high confidence photometric redshifts to estimate the redshift distribution, we find an upper limit of f2m <~ 0.4 and a lower limit of as >~ 0.6 both at the la level (see Figure 2.3). Our revised cosmological constraints introduce tension, at the ~ 2a level, between the results from the VIRMOS-DESCART and RCS survey as shown in Figure 2.5. In this analysis we have accounted for systematic errors associated with shear measurement but have neglected potential systematics arising from correlations between galaxy shape and the underlying density field. This is valid in the case of intrinsic galaxy alignments which are expected to contribute less than a percent of the cosmic shear signal for the deep surveys used in this analysis (Heymans et al, 2006b). What is currently uncertain however is the level of systematic error that arises from shear-ellipticity correlations (Heymans et al, 2006b; Hirata et al, 2007; Hirata & Seljak, 2004; Mandelbaum et al, 2006) which could reduce the amplitude of the measured shear correlation function by ~ 10% (Heymans et al, 2006b). It has been found from the analysis of the Sloan Digital Sky Survey (Hirata et al, 2007; Mandelbaum et al, 2006), that different morphological galaxy types contribute differently to this effect. It is therefore possible that the different R-band imaging of the RCS and the slightly lower median survey redshift make it more suceptible to this type of systematic error. Without complete redshift information for each survey, however, it is not possible to test this hypothesis or to correct for this potential source of error. In the future deep multi-colour data will permit further investigation and correction for this potential source of systematic error. Weak lensing by large scale structure is an excellent means of constraining cosmology. The unambiguous interpretation of the shear signal allows for a direct measure of the dark matter power spectrum, allowing for a unique and powerful means of constraining cosmology. Our analysis has shown that accurately describing the redshift distribution of the sources is vital to future surveys intent on precision cosmology. 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E., Albert J., Berge J., Heymans C , Johnston D., Kneib J.-R, Mellier Y, Mobasher B., Semboloni E., Shopbell P., Tasca L., Van Waerbeke L., 2007b, preprint astro-ph/0701480 McCracken H. J., Radovich M., Bertin E., Mellier Y, Dantel-Fort M., Le Fevre O., Cuillandre J. C , Gwyn S., Foucaud S., Zamorani G., 2003, A&A, 410, 17 Munshi D., Valageas P., Van Waerbeke L., Heavens A., 2006, preprint astro-ph/0612667 Peacock J. A., Dodds S. J., 1996, MNRAS, 280, L19 Pen U.-L., Van Waerbeke L., Mellier Y, 2002, ApJ, 567, 31 Rhodes J., Refregier A., Collins N. R., Gardner J. P., Groth E. J., Hill R. S., 2004, ApJ, 605, 29 Schneider P., Van Waerbeke L., Jain B., Kruse G., 1998, MNRAS, 296, 873 Schneider P., Van Waerbeke L., Kilbinger M., Mellier Y, 2002, A&A, 396, 1 Schrabback T., Erben T., Simon P., Miralles J. ., Schneider P., Heymans C , Eifler T., Fosbury R. A. E., Freudling W:, Hetterscheidt M., Hildebrandt H., Pirzkal N., 2006, preprint astro-ph/0606611 Semboloni E., Mellier Y, Van Waerbeke L., Hoekstra H., Tereno I., Benabed K., Gwyn S. D. J., Fu L., Hudson M. J„ Maoli R., Parker L. C , 2006, A&A, 452, 51 Semboloni E., Van Waerbeke L., Heymans C , Hamana T., Colombi S., White M., Mellier Y, 2007, MNRAS, 375, L6 Smith R. E., Peacock J. A., Jenkins A., White S. D. M., Frenk C. S., Pearce F. R., Thomas P. A., Efstathiou G., Couchman H. M. P., 2003, MNRAS, 341, 1311 Spergel D. N., Bean R., Dore O., Nolta M. R., Bennett C. L., Dunkley J., Hinshaw G., Jarosik N., Komatsu E., Page L., Peiris H. V, Verde L., Halpern M., Hill R. S., Kogut A., Limon M., Meyer S. S., Odegard N., Tucker G. S., Weiland J. L., Wollack E., Wright E. L„ 2006, ArXiv Astrophysics e-prints 35 Bibliography Van Waerbeke L , Mellier Y, Erben T, Cuillandre J. C , Bernardeau F, Maoli R, Bertin E , Mc Cracken H. J , Le Fevre O, Fort B , Dantel-Fort M , Jain B , Schneider P, 2000, A&A, 358, 30 Van Waerbeke L , Mellier Y, Hoekstra H , 2005, A&A, 429, 75 Van Waerbeke L., Mellier Y, Pello R, Pen U.-L., McCracken H. J., Jain B., 2002, A&A, 393, 369 Van Waerbeke L , Mellier Y, Radovich M , Bertin E , Dantel-Fort M , McCracken H. J , Le Fevre O, Foucaud S, Cuillandre J.-C, Erben T„ Jain B , Schneider P., Bernardeau F, Fort B„ 2001, A&A, 374, 757 Van Waerbeke L , White M , Hoekstra H , Heymans C , 2006, Astroparticle Physics, 26, 91 Wall J. V , Jenkins C. R, 2003, Practical Statistics for Astronomers. Princeton Series in Astrophysics Wittman D, 2005, ApJ, 632, L5 Wittman D. M., Tyson J. A., Kirkman D., Dell'Antonio I., Bernstein G , 2000, Nature, 405, 143 36 Chapter 3 Prospects for Future Research The research presented in Chapter 2 represents a significant milestone in weak lensing cosmology. The con-straints presented are the tightest of any such study, and incorporate several important updates to the analysis. An immediate extension of this work would increase the constrained parameter space to include dark energy, specifically its equation of state. Such constraints would be particularly valuable when combined with results from super nova studies (e.g., Astier et al., 2006). With redshift information for each survey 3D lensing (or tomography) would be possible, allowing for more robust dark energy constraints. Photometric redshifts will be attainable for the CFHTLS-Wide which is being imaged in 5 colours, 4-band photometry has already produced a redshift catalogue for RCS (Hsieh et al., 2005), the VIRMOS-DESCART project has planned multi-band observations including bands in the near infra-red, and photometric redshifts could be estimated for a large fraction of GaBoDS. As redshift estimates become available for these surveys the constraints on cosmology will improve - in the recent 3D lensing analysis of the COSMOS survey Massey et al. (2007) found that their parameter constraints improved by a factor of three. One of the most interesting results of Chapter 2 is the dependace of the estimated cosmology on the source redshift distribution. The photometric catalogue used (Ilbert et al., 2006) will likely be the largest deep catalogue of its kind for many years, making the uncertainty of the high redshift tail particularly concerning. Near infra-red data for a subset of this catalogue will be available from WIRCam shortly, this will help in determining photometric redshifts in the high-z tail. Until deep spectroscopic redshifts can be obtained for a large number of galaxies on a reasonable time scale weak lensing analyses will have to make do with photometric redshifts. In the research presented in Chapter 2 those redshifts were part of an external catalogue that was then used to calibrate the redshift distribution of each survey. Hence, even estimating photometric redshifts for each survey will improve our estimates of n(z). 37 Bibliography Astier, P. and Guy, J. and Regnault, N. and Pain, R. and Aubourg, E. and Balam, D. and Basa, S. and Carlberg, R. G. and Fabbro, S. and Fouchez, D. and Hook, I. M. and Howell, D. A. and Lafoux, H. and Neill, J. D. and Palanque-Delabrouille, N. and Perrett, K. and Pritchet, C. J. and Rich, J. and Sullivan, M. and Taillet, R. and Aldering, G. and Antilogus, P. and Arsenijevic, V. and Balland, C. and Baumont, S. and Bronder, J. and Courtois, H. and Ellis, R. S. and Filiol, M. and Goncalves, A. C. and Goobar, A. and Guide, D. and Hardin, D. and Lusset, V. and Lidman, C. and McMahon, R. and Mouchet, M. and Mourao, A. and Perlmutter, S. and Ripoche, P. and Tao, C. and Walton, N., 2006, A&A, 447, 31 Hsieh, B. C., Yee, H. K. C., Lin, H., & Gladders, M. D. 2005, VizieR Online Data Catalog, 215, 80161 Ilbert O., Arnouts S., McCracken H. J., Bolzonella M., Bertin E., Le Fevre O., Mellier Y., Zamorani G., Pello R., Iovino A., Tresse L., Le Brun V., Bottini D., Garilli B., Maccagni D., Picat J. P., Scaramella R., Scodeggio M., Vettolani G., Zanichelli A., Adami C , Bardelli S., Cappi A., Chariot S., Ciliegi P., Contini T., Cucciati O., Foucaud S., Franzetti P., Gavignaud I., Guzzo L., Marano B., Marinoni C , Mazure A., Meneux B., Merighi R., Paltani S., Polio A., Pozzetti L., Radovich M., Zucca E., Bondi M., Bongiorno A., Busarello G., de La Torre S., Gregorini L., Lamareille F, Mathez G., Merluzzi P., Ripepi V., Rizzo D., Vergani D., 2006, A&A, 457, 841 Massey, R. and Rhodes, J. and Leauthaud, A. and Capak, P. and Ellis, R. and Koekemoer, A. and Refregier, A. and Scoville, N. and Taylor, J. E. and Albert, J. and Berge, J. and Heymans, C. and Johnston, D. and Kneib, J.-P. and Mellier, Y. and Mobasher, B. and Semboloni, E. and Shopbell, P. and Tasca, L. and Van Waerbeke, L., 2007, ArXiv Astrophysics e-prints, arXiv:astro-ph/0701480 38 Appendix A Additional data Here we present both the measured shear correlation function and the covariance matrix for each survey. The shear correlation function, given in Tables A.l , A.2, A.3 and A.4 for the CFHTLS-Wide, GaBoDS, RCS and VIRMOS-DESCART surveys respectively, has been calibrated on large scales where ( M A P ( A0))_L is consistent with zero as described in §2.4. Tables A.5, A.6, A.7 and A.8 for the CFHTLS-Wide, GaBoDS, RCS and VIRMOS-DESCART surveys respectively tabulate the correlation coefficient matrix: C- • r i j = <a i / 2 fe 1 / 2 ' ( A ' 1 } where C is the covariance matrix, as described in §2.6 for each survey. We also tabulate (£f) so that the covariance matrix may be calculated from the correlation coefficient matrix. Note that (£?) is the variance of the ith scale, equivalent to Ca. The calculation of the non-gaussian contribution requires an average redshift (see §2.6), here we have taken the average redshift obtained for the high confidence photometric redshifts using Eq.(2.9) to model n(z). 39 Appendix A. Additional data Table A.l: E and B modes of the shear correlation function for the CFFTTLS-Wide survey, the error (5£) is statistical only, and given as the standard deviation. fe fe 0.74 I. 44 2.37 3.54 5.41 7.28 8.69 II. 72 18.74 28.09 37.44 49.12 62.92 1.53060e-04 1.00161e-04 5.82357e-05 3.11136e-05 2.89396e-05 3.08030e-05 1.41053e-05 1.66208e-05 1.50680e-05 1.07719e-05 8.72227e-06 8.1U50e-06 8.9707 le-06 -4.16557e-05 1.79443e-05 -1.60557e-05 7.14429e-06 -2.0557 le-06 -3.47571e-06 6.14429e-06 -5.15714e-07 4.14286e-07 -7.95715e-07 4.14286e-07 -2.18571e-06 -3.0157 le-06 2.31000e-05 1.22000e-05 9.64000e-06 6.66000e-06 4.29000e-06 4.76000e-06 4.43000e-06 2.16000e-06 1.25000e-06 1.03000e-06 9.10000e-07 6.64000e-07 6.25000e-07 Table A.2: E and B modes of the shear correlation function for GaBoDS, the error (<5f) is statistical only, and given as the standard deviation. fe fe 0.36 0.50 0.70 0.98 1.36 1.90 2.65 3.69 5.16 7.19 10.03 13.99 2.80637e-04 1.72930e-04 1.15363e-04 1.20006e-04 8.75079e-05 7.4323 le-05 9.55859e-05 6.48524e-05 4.30325e-05 2.73408e-05 9.97772e-06 1.22154e-05 1.38754e-04 1.24754e-04 6.74540e-05 -1.59760e-05 2.76540e-05 8.35400e-06 -2.30860e-05 3.054O0e-06 5.25400e-06 -7.24600e-06 -7.46000e-07 -6.54600e-06 9.54000e-05 6.92000e-05 4.99000e-05 3.81000e-05 2.80000e-05 2.15000e-05 1.71000e-05 1.40000e-05 1.19000e-05 1.03000e-05 8.65000e-06 6.55000e-06 40 Appendix A. Additional data Table A.3: E and B modes of the shear correlation function for RCS, the error (<5£) is statistical only, and given as the standard deviation. 9 IE IB <5| 1.12 9 71718e-05 2.78320e-05 1 99000e 05 1.75 3 38178e-05 1.67320e-05 1 91000e •05 2.50 3 74410e-05 4.00200e-06 1 16000e -05 3.24 4 27205e-05 6.63200e-06 1 43000e -05 4.00 1 49772e-05 1.58320e-05 9 21000e -06 5.75 1 52878e-05 -8.80000e-08 5 13000e -06 8.50 1 50807e-05 -3.89600e-06 3 95000e -06 12.25 6 89234e-06 2.15200e-06 2 85000e -06 16.49 4 81573e-06 1.83200e-06 2 65000e -06 20.74 4 69151e-06 1.82200e-06 2 30000e -06 26.49 5 29607e-06 3.59200e-06 1 75000e -06 34.99 5 33747e-06 1.11200e-06 1 36000e -06 46.49 6 91304e-06 -4.88000e-07 1 08000e -06 56.51 6 69565e-06 -4.88000e-07 1 34000e -06 Table A.4: E and B modes of the shear correlation function for the VIRMOS-DESCART survey, the error (<S£) is statistical only, and given as the standard deviation. e IE IB <5| 0.73 2 26695e-04 3.91133e-05 3 98000e 05 1.05 1 69759e-04 -1.59667e-06 3 90000e -05 1.51 1 30421e-04 5.31333e-06 2 20000e -05 2.29 9 60525e-05 2.82133e-05 1 57000e -05 3.20 6 60318e-05 -1.38767e-05 1 35000e -05 4.11 5 56798e-05 6.41333e-06 1 22000e -05 5.02 5 29883e-05 3.71333e-06 1 12000e -05 6.39 3 50794e-05 -1.26667e-06 7 46000e -06 8.21 1 94893e-05 2.11333e-06 6 75000e -06 11.41 2 72119e-05 -8.46667e-07 3 99000e -06 15.97 1 98620e-05 4.81333e-06 3 50000e -06 20.53 1 67357e-05 5.41333e-06 3 23000e -06 25.25 1 05659e-05 4.51333e-06 2 95000e -06 30.23 9 77399e-06 1.31333e-06 2 73000e -06 35.10 1 06073e-05 -1.05667e-06 2 71000e -06 39.78 1 17771e-05 -2.08667e-06 2 61000e -06 41 Table A.5: The correlation coefficient matrix (r) for the CFHTLS-Wide survey, the scales &i correspond to those given in Table A.l. The column is given in units of 10 - 1 0 , and can be used to reconstruct the covariance matrix C. ($) 01 02 03 04 0s 06 07 0a 09 0io 0ii 012 #13 01 29.92 1.00 0.26 0.18 0.20 0.21 0.11 0.06 0.07 0.07 0.06 0.05 0.04 0.03 02 6.60 0.26 1.00 0.22 0.24 0.25 0.16 0.11 0.15 0.14 0.13 0.12 0.08 0.06 03 4.26 0.18 0.22 1.00 0.22 0.23 0.16 0.12 0.19 0.18 0.16 0.15 0.10 0.07 04 1.35 0.20 0.24 0.22 1.00 0.35 0.25 0.21 0.34 0.32 0.28 0.26 0.19 0.13 05 0.44 0.21 0.25 0.23 0.35 1.00 0.42 0.37 0.61 0.56 0.48 0.45 0.32 0.22 06 0.56 0.11 0.16 0.16 0.25 0.42 1.00 0.34 0.55 0.49 0.42 0.40 0.29 0.19 07 0.81 0.06 0.11 0.12 0.21 0.37 0.34 1.00 0.47 0.41 0.35 0.33 0.24 0.16 08 0.25 0.07 0.15 0.19 0.34 0.61 0.55 0.47 1.00 0.75 0.63 0.60 0.43 0.29 09 0.18 0.07 0.14 0.18 0.32 0.56 0.49 0.41 0.75 1.00 0.78 0.71 0.51 0.35 010 0.14 0.06 0.13 0.16 0.28 0.48 0.42 0.35 0.63 0.78 1.00 0.82 0.57 0.39 011 0.11 0.05 0.12 0.15 0.26 0.45 0.40 0.33 0.60 0.71 0.82 LOO 0.69 0.45 012 0.13 0.04 0.08 0.10 0.19 0.32 0.29 0.24 0.43 0.51 0.57 0.69 1.00 0.42 013 0.16 0.03 0.06 0.07 0.13 0.22 0.19 0.16 0.29 0.35 0.39 0.45 0.42 1.00 > •a •a n> 3 g. > > CL £*. 5" 3 EL a. ET 4^ Table A.6: The correlation coefficient matrix (r) for GaBoDS, the scales 6i correspond to those given in Table A.2. The column is given in units of 10 - 1 0, and can be used to reconstruct the covariance matrix C. 01 02 03 04 05 06 07 08 09 010 0ii 012 323.76 1.00 0.12 0.13 0.17 0.14 0.16 0.10 0.14 0.11 0.09 0.09 0.06 223.36 0.12 1.00 0.12 0.16 0.12 0.15 0.10 0.13 0.11 0.09 0.09 0.06 03 80.29 0.13 0.12 1.00 0.21 0.16 0.19 0.13 0.18 0.16 0.13 0.14 0.09 04 22.13 0.17 0.16 0.21 1.00 0.25 0.30 0.20 0.29 0.25 0.22 0.23 0.15 05 18.26 0.14 0.12 0.16 0.25 1.00 0.28 0.19 0.28 0.25 0.22 0.23 0.17 06 6.91 0.16 0.15 0.19 0.30 0.28 1.00 0.28 0.42 0.38 0.33 0.37 0.27 07 9.21 0.10 0.10 0.13 0.20 0.19 0.28 1.00 0.34 0.32 0.28 0.32 0.23 08 2.67 0.14 0.13 0.18 0.29 0.28 0.42 0.34 1.00 0.58 0.53 0.59 0.42 09 2.20 0.11 0.11 0.16 0.25 0.25 0.38 0.32 0.58 1.00 0.58 0.64 0.46 010 2.01 0.09 0.09 0.13 0.22 0.22 0.33 0.28 0.53 0.58 1.00 0.66 0.47 0n 1.13 0.09 0.09 0.14 0.23 0.23 0.37 0.32 0.59 0.64 0.66 1.00 0.61 012 1.20 0.06 0.06 0.09 0.15 0.17 0.27 0.23 0.42 0.46 0.47 0.61 1.00 > *a c» 3 & x' > > a. & o' 3 S-a. Table A.7: The correlation coefficient matrix, (r) for RCS, the scales 6{ correspond to those given in Table A.3. The column (£t2) is given in units of IO - 1 0 , and can be used to reconstruct the covariance matrix C. (g) 01 02 03 04 05 06 07 08 09 0io 0ii 012 013 014 0i 12.95 1.00 0.10 0.13 0.08 0.05 0.11 0.07 0.07 0.06 0.06 0.04 0.05 0.05 0.04 02 7.03 0.10 1.00 0.14 0.08 0.06 0.12 0.08 0.09 0.08 0.08 0.06 0.07 0.07 0.05 03 1.84 0.13 0.14 1.00 0.14 0.10 0.21 0.16 0.17 0.16 0.15 0.11 0.14 0.13 0.10 04 2.74 0.08 0.08 0.14 1.00 0.08 0.17 0.13 0.14 0.13 0.13 0.09 0.12 0.10 0.08 05 3.56 0.05 0.06 0.10 0.08 1.00 0.15 0.11 0.12 0.12 0.11 0.08 0.10 0.09 0.07 06 0.42 0.11 0.12 0.21 0.17 0.15 1.00 0.35 0.36 0.34 0.32 0.23 0.29 0.27 0.21 07 0.44 0.07 0.08 0.16 0.13 0.11 0.35 1.00 0.38 0.34 0.32 0.23 0.29 0.26 0.21 08 0.24 0.07 0.09 0.17 0.14 0.12 0.36 0.38 1.00 0.49 0.42 0.31 0.40 0.36 0.29 09 0.19 0.06 0.08 0.16 0.13 0.12 0.34 0.34 0.49 1.00 0.49 0.35 0.45 0.41 0.33 010 0.17 0.06 0.08 0.15 0.13 0.11 0.32 0.32 0.42 0.49 1.00 0.38 0.48 0.44 0.35 011 0.23 0.04 0.06 0.11 0.09 0.08 0.23 0.23 0.31 0.35 0.38 1.00 0.45 0.38 0.30 012 0.08 0.05 0.07 0.14 0.12 0.10 0.29 0.29 0.40 0.45 0.48 0.45 1.00 0.66 0.51 013 0.06 0.05 0.07 0.L3 0.10 0.09 0.27 0.26 0.36 0.41 0.44 0.38 0.66 1.00 0.64 014 0.05 0.04 0.05 0.10 0.08 0.07 0.21 0.21 0.29 0.33 0.35 0.30 0.51 0.64 1.00 Table A.8: The correlation coefficient matrix (r) for the VIRMOS-DESCART survey, the scales Oi correspond to those given in Table A.4. The column ((?) is given in units of IO - 1 0 , and can be used to reconstruct the covariance matrix C. <#> 01 02 03 04 05 06 07 08 09 0io 0ii 012 013 014 015 016 01 46.55 1.00 0.35 0.37 0.19 0.21 0.22 0.20 0.19 0.16 0.14 0.08 0.06 0.06 0.07 0.06 0.06 02 23.02 0.35 1.00 0.40 0.20 0.22 0.24 0.22 0.21 0.17 0.15 0.10 0.09 0.09 0.10 0.09 0.08 03 8.92 0.37 0.40 1.00 0.26 0.28 0.28 0.26 0.26 0.22 0.21 0.17 0.15 0.14 0.15 0.15 0.13 04 12.17 0.19 0.20 0.26 1.00 0.19 0.19 0.17 0.18 0.17 0.18 0.14 0.13 0.12 0.13 0.13 0.11 05 4.76 0.21 0.22 0.28 0.19 1.00 0.26 0.25 0.28 0.27 0.29 0.23 0.20 0.20 0.21 0.20 0.18 06 2.69 0.22 0.24 0.28 0.19 0.26 1.00 0.35 0.39 0.36 0.38 0.30 0.27 0.26 0.28 0.27 0.24 07 2.11 0.20 0.22 0.26 0.17 0.25 0.35 1.00 0.45 0.42 0.43 0.33 0.30 0.29 0.31 0.30 0.27 08 1.26 0.19 0.21 0.26 0.18 0.28 0.39 0.45 1.00 0.55 0.56 0.44 0.39 0.38 0.41 0.39 0.35 09 1.08 0.16 0.17 0.22 0.17 0.27 0.36 0.42 0.55 1.00 0.64 0.49 0.43 0.41 0.44 0.42 0.38 0io 0.68 0.14 0.15 0.21 0.18 0.29 0.38 0.43 0.56 0.64 1.00 0.64 0.52 0.52 0.56 0.54 0.48 0ii 0.77 0.08 0.10 0.17 0.14 0.23 0.30 0.33 0.44 0.49 0.64 1.00 0.49 0.49 0.53 0.51 0.46 012 0.80 0.06 0.09 0.15 0.13 0.20 0.27 0.30 0.39 0.43 0.52 0.49 1.00 0.50 0.53 0.51 0.45 013 0.62 0.06 0.09 0.14 0.12 0.20 0.26 0.29 0.38 0.41 0.52 0.49 0.50 1.00 0.63 0.60 0.52 014 0.41 0.07 0.10 0.15 0.13 0.21 0.28 0.31 0.41 0.44 0.56 0.53 0.53 0.63 1.00 0.74 0.65 015 0.34 0.06 0.09 0.15 0.13 0.20 0.27 0.30 0.39 0.42 0.54 0.51 0.51 0.60 0.74 1.00 0.70 016 0.35 0.06 0.08 0.13 0.11 0.18 0.24 0.27 0.35 0.38 0.48 0.46 0.45 0.52 0.65 0.70 1.00 > T3 O a £-> > a. a. o' 3 a. 4^ 

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