Seeing in the dark -- II. Cosmic shear in the Sloan Digital Sky Survey
Eric M. Huff, Tim Eifler, Christopher M. Hirata, Rachel Mandelbaum, David Schlegel, Uros Seljak
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 17 July 2018 (MN L A TEX style file v2.2)
Seeing in the dark – II. Cosmic shear in the Sloan DigitalSky Survey
Eric M. Huff , Tim Eifler , Christopher M. Hirata , Rachel Mandelbaum , ,David Schlegel , Uroˇs Seljak , , , Department of Astronomy, University of California at Berkeley, Berkeley, CA 94720, USA Center for Cosmology and Astro-Particle Physics, The Ohio State University, 191 W. Woodruff Avenue, Columbus, OH 43210, USA Department of Astronomy, Caltech M/C 350-17, Pasadena, CA 91125, USA Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Space Sciences Lab, Department of Physics and Department of Astronomy, University of California, Berkeley, CA 94720, USA Institute of the Early Universe, Ewha Womans University, Seoul, Korea Institute for Theoretical Physics, University of Zurich, Zurich, Switzerland
17 July 2018
ABSTRACT
Statistical weak lensing by large-scale structure – cosmic shear – is a promising cos-mological tool, which has motivated the design of several large upcoming surveys.Here, we present a measurement of cosmic shear using coadded Sloan Digital SkySurvey (SDSS) imaging in 168 square degrees of the equatorial region, with r < . i < .
5, a source number density of 2.2 per arcmin and median redshift of z med = 0 .
52. These coadds were generated using a new method described in the com-panion Paper I (Huff et al. 2011) that was intended to minimise systematic errors inthe lensing measurement due to coherent PSF anisotropies that are otherwise preva-lent in the SDSS imaging data. We present measurements of cosmic shear out to an-gular separations of 2 degrees, along with systematics tests that (combined with thosefrom Paper I on the catalogue generation) demonstrate that our results are domi-nated by statistical rather than systematic errors. Assuming a cosmological modelcorresponding to WMAP7 (Komatsu et al. 2011) and allowing only the amplitude ofmatter fluctuations σ to vary, we find a best-fit value of σ = 0 . +0 . − . (1 σ ); with-out systematic errors this would be σ = 0 . +0 . − . (1 σ ). Assuming a flat ΛCDMmodel, the combined constraints with WMAP7 are σ = 0 . +0 . − . (1 σ ) +0 . − . (2 σ )and Ω m h = 0 . +0 . − . (1 σ ) +0 . − . (2 σ ); the 2 σ error ranges are respectively 14 and17 per cent smaller than WMAP7 alone. Aside from the intrinsic value of such cosmo-logical constraints from the growth of structure, we identify some important lessonsfor upcoming surveys that may face similar issues when combining multi-epoch datato measure cosmic shear. Key words: cosmology: observations – gravitational lensing: weak – surveys.
As a result of gravitational lensing, large scale inhomo-geneities in the matter density field produce small but sys-tematic fluctuations in the sizes, shapes, and fluxes of dis-tant objects that are coherent across large scales. This effectwas first suggested as a tool for constraining the form of themetric in 1966 by Kristian & Sachs (1966). In a more mod-ern context, the two-point statistics of lensing fluctuationsallow the only truly direct measurement of the matter powerspectrum and the growth of structure at late times, when dark energy has caused an accelerated expansion of the uni-verse (Riess et al. 1998; Perlmutter et al. 1999) and affectedthe growth of structure. Many studies have pointed outthat high signal-to-noise cosmic shear measurements wouldbe extraordinarily sensitive probes of cosmological param-eters (e.g., Huterer 1998; Benabed & van Waerbeke 2004),which led to its being flagged as one of the most promis-ing probes of dark energy by the Dark Energy Task Force(Albrecht et al. 2006). Direct measurements of the growth c (cid:13) E. M. Huff et al. of structure also offer the opportunity to test alternativemodels of gravity (e.g., Laszlo et al. 2011).Cosmic shear measurements were attempted as early as1967 (Kristian 1967), but until the turn of the millennium(Bacon et al. 2000; Kaiser et al. 2000; van Waerbeke et al.2000; Wittman et al. 2000), no astronomical survey had thestatistical power to detect it. The difficulty of the measure-ment is a consequence of the near-homogeneity and isotropyof the universe. An order unity distortion to galaxy imagesrequires an integrated line-of-sight matter over-density of:Σ crit = c πG d S d L d LS (1)where d S , d L , and d LS are the angular diameter distancesfrom the observer to the background source, from the ob-server to the lens, and from the lens to the backgroundsource, respectively. A fluctuation in the surface density ∆Σleads to a shear distortion γ ∼ ∆Σ / Σ crit .Averaged over large ( ∼
100 Mpc) scales, typical line-of-sight matter fluctuations are only 10 − Σ crit . The primarysource of noise in the shear measurement, the random in-trinsic dispersion in galaxy shapes, is orders of magnitudelarger; typically the shape noise results in a dispersion inthe shear of σ γ = 0 .
2. Worse, even in modern ground-based astronomical imaging surveys, the coherent distor-tions – or point-spread function (PSF) – induced by effectsof the atmosphere, telescope optics, and detectors are typi-cally several times larger than the cosmological signal (e.g.,Heymans et al. 2011 and Paper I in this series). Estimat-ing the distances to the background sources is both cru-cial (Ma et al. 2006) and difficult (Ma & Bernstein 2008;Bernstein & Huterer 2010); errors there will modulate theamplitude of the signal through Σ crit , biasing inference ofthe growth of structure.These obstacles define the observational problem. Whilethe existence of cosmic shear has been established bythe first studies to detect the effect, the full potential ofcosmological lensing remains to be exploited. Few datasets capable of achieving the signal strength for a cos-mologically competitive measurement presently exist – theCanada-France-Hawaii Telescope Legacy Survey (CFHTLS;Hoekstra et al. 2006; Semboloni et al. 2006; Benjamin et al.2007; Fu et al. 2008), the Cosmological Evolution Survey(COSMOS; Massey et al. 2007a; Schrabback et al. 2010),and the subset of the SDSS imaging studied here. How-ever, several large surveys are planned for the immediateand longer-term future that will substantially expand theamount of available data for cosmological weak lensing stud-ies. In the next few years, these include Hyper Suprime-Cam (HSC, Miyazaki et al. 2006), Dark Energy Survey(DES , The Dark Energy Survey Collaboration 2005), theKIlo-Degree Survey (KIDS ), and the Panoramic SurveyTelescope and Rapid Response System (Pan-STARRS ,Kaiser et al. 2010). Further in the future, there are evenmore ambitious programs such as the Large Synoptic Sur-vey Telescope (LSST , LSST Science Collaborations et al. http://pan-starrs.ifa.hawaii.edu/public/ , and the Wide-Field Infrared Survey Tele-scope (WFIRST ).For this work, we have combined several methods dis-cussed in the literature as viable techniques for measuringcosmic shear while removing common systematic errors. InPaper I (Huff et al. 2011), we began with the PSF modelgenerated by the Sloan Digital Sky Survey (SDSS) pipelineover ∼
250 deg that had been imaged many times, and em-ployed a rounding kernel method similar to that proposedin Bernstein & Jarvis (2002). The result, after appropriatemasking of problematic regions, was 168 square degrees ofdeep coadded imaging with a well controlled, homogeneousPSF and sufficient galaxy surface density to measure a cos-mic shear signal. The usable area in r band was only 140square degrees because of a PSF model error problem on thecamcol 2 charge-coupled device (CCD), which is suspectedto be an amplifier non-linearity problem.In this work, we use the catalogue from Paper I toproduce a cosmic shear measurement that is dominated bystatistical errors. Section 3 enumerates the primary sourcesof systematic error when measuring cosmic shear using ourcatalogue (the properties of which are summarized briefly inSec. 2), and describes our approaches to constraining each ofthem. In Section 4, we outline our correlation function esti-mator and several transformations of it that are used for sys-tematics tests. Our methods for estimating covariance ma-trices for our observable quantities (both due to statisticaland systematic errors) are described in Sec. 5. Finally, sec-tion 6 presents the constraining power of this measurementalone for a fiducial cosmology, and in combination with the7-year Wilkinson Microwave Anisotropy Probe (WMAP7,Komatsu et al. 2011) parameter constraints to produce aposterior probability distribution over Ω m h , Ω b h , σ , n s ,and w . We show that in addition to its value as an inde-pendent measurement of the late-time matter power spec-trum, this measurement provides some additional constrain-ing power over WMAP7 within the context of ΛCDM. Weconclude with some lessons for the future in Sec. 7.While this work was underway, we learned of a paralleleffort by Lin et al. (2011). These two efforts use differentmethods of coaddition, different shape measurement codes,different sets of cuts for the selection of input images andgalaxies, and analyze their final results in different ways;what they have in common is their use of SDSS data (notnecessarily the same sets of input imaging) and their useof the SDSS Photo pipeline for the initial reduction of thesingle epoch data and the final reduction of the coaddeddata (however, they use different versions of
Photo ). Us-ing these different methods, both groups have extracted thecosmic shear signal and its cosmological interpretations. Wehave coordinated submission with them but have not con-sulted their results prior to this, so these two analysis effortsare independent, representing versions of two independentpipelines. http://wfirst.gsfc.nasa.gov/ c (cid:13) , 000–000 osmic shear in SDSS Paper I (Huff et al. 2011) describes a coadd imaging dataset,optimised for cosmic shear measurement, constructed fromsingle-epoch SDSS images in the Stripe 82 equatorial region,with right ascension (RA) − ◦ < RA < +45 ◦ , and decli-nation − . ◦ < Dec < +1 . ◦ . In that work, we apply anadaptive rounding kernel to the single-epoch images to nullthe effects of PSF anisotropy and match to a single homo-geneous PSF model for the entire region, and show that inthe resulting shear catalogues, the amplitude of the galaxyshape correlations due to PSF anisotropy at angular sepa-rations greater than 1 arcminute is negligible compared tothe expected cosmic shear statistical errors.The final shape catalogue described in that work con-sists of 1 067 031 r -band and 1 251 285 i -band shape measure-ments with characteristic limiting magnitudes of r < . i < .
5, over effective areas of 140 and 168 squaredegrees, respectively.
We model the observed galaxy shape field as the sumof a cosmic shear component, an independent systemat-ics field produced by anisotropies in the effective PSF e psf , and a systematics field produced by the intrinsicspatial correlations of galaxy shapes e int (intrinsic align-ments; e.g., Hirata & Seljak 2004). We allow for a shear cal-ibration factor that depends on the shear responsivity R (Bernstein & Jarvis 2002) of the ensemble of galaxy surfacebrightness profiles to the underlying gravitationally-inducedshear γ . Typically R ≈ − e , however we consider it tobe a more general factor that also includes any biases dueto effects such as uncorrected PSF dilution, noise-related bi-ases, or selection biases. We assume that the galaxy shaperesponse to PSF anisotropies R psf is not a priori known, butrather suffers from a similar set of ‘calibration’ uncertaintiesas the response of the ensemble of galaxy images to gravita-tional lensing shear. Thus we define our model for the twoellipticity components e = ( e , e ) as e = R γ + R psf e psf + e int . (2)We assume that the two-point statistics of the under-lying (cosmological) shear field h γγ i consist entirely of E -modes, e γ,E (which is a good enough approximation giventhe size of our errors; Crittenden et al. 2002; Schneider et al.2002), and are statistically independent of the PSF whenaveraged over large regions. We also assume that the PSFand the intrinsic alignments are independent – but not thatthe lensing shear and intrinsic alignments are independent(Hirata & Seljak 2004). The two-point correlation of thegalaxy shapes contains terms resulting from gravitationallensing and from systematic errors: h e e i = 4 R ξ γ,E + R ξ psf + ξ int + h γe int i . (3)Here, ξ psf is the auto-correlation of the PSF ellipticity field.Errors in the determination of the galaxy redshift distribu-tion will enter as a bias in the predicted ξ γ,E .Our goal is to carry out a statistics-limited measure-ment of ξ γ,E . This will entail showing that the combined amplitudes of R ξ psf , ξ int , h γ e int i , the uncertainty in thetheoretically-predicted ξ γ,E arising from redshift errors, andthe uncertainty in the shear calibration (via the responsivity R ) contribute less than 20 per cent to the statistical errorsin h e e i .Our approach to handling of systematic error is as fol-lows: we attempt to reduce each systematic to a term thatcan be robustly and believably estimated from real data (ei-ther the data here or in other, related work), and we then ex-plicitly correct for it. These corrections naturally have someuncertainty associated with them, which we use to derive asystematic error component to the covariance matrix. Theexception to the rule given here is if there is a systematicerror for which there is no clear path to estimating its mag-nitude, then we do not attempt any correction, and simplymarginalize over it by include an associated uncertainty inthe covariance matrix. Foreground anisotropies in the matter distribution along theline of sight to a galaxy will generically distort the galaxyimage. For weak lensing, the leading order lensing contribu-tion to galaxy shapes can be thought of as arising from a lin-ear transformation of the image coordinates A x true = x obs ,where A = (cid:18) κ + γ γ γ κ − γ (cid:19) . (4)The convergence κ causes magnification, whereas theshear components γ and γ map circles to ellipses. Theshear is related to the projected line-of-sight matter distri-bution, weighted by the lensing efficiency:( γ , γ ) = ∂ − Z ∞ W ( χ, χ i ) (cid:0) ∂ x − ∂ y , ∂ x ∂ y (cid:1) δ ( χ ˆ n i ) d χ. (5)Here we integrate along the comoving line-of-sight distance χ (where χ i is the distance to the source), and the matterover-density δ = ( ρ − ρ ) /ρ . The window function in a flatuniverse is W ( χ, χ i ) = 32 Ω m H (1 + z ) χ (cid:18) χ − χ i (cid:19) . (6)The two-point correlation function of the shear can becalculated by identifying pairs of source galaxies, and defin-ing shear components ( γ t , γ x ) for each one to be the shearin the coordinate system defined by the vector connectingthem, and in the π/ P δ averaged over the line ofsight to the sheared galaxies: ξ ± = h γ t γ t i ± h γ × γ × i = 12 π Z ∞ d ℓ ℓ P κ ( ℓ ) J , ( ℓθ ) (7)and P κ = (cid:18) m d H (cid:19) Z ∞ d χa ( χ ) P δ (cid:18) ℓd ( χ ) (cid:19) × (cid:20)Z ∞ χ d χ ′ n (cid:0) χ ′ (cid:1) d ( χ ′ − χ ) d ( χ ′ ) (cid:21) , (8) c (cid:13)000
We model the observed galaxy shape field as the sumof a cosmic shear component, an independent systemat-ics field produced by anisotropies in the effective PSF e psf , and a systematics field produced by the intrinsicspatial correlations of galaxy shapes e int (intrinsic align-ments; e.g., Hirata & Seljak 2004). We allow for a shear cal-ibration factor that depends on the shear responsivity R (Bernstein & Jarvis 2002) of the ensemble of galaxy surfacebrightness profiles to the underlying gravitationally-inducedshear γ . Typically R ≈ − e , however we consider it tobe a more general factor that also includes any biases dueto effects such as uncorrected PSF dilution, noise-related bi-ases, or selection biases. We assume that the galaxy shaperesponse to PSF anisotropies R psf is not a priori known, butrather suffers from a similar set of ‘calibration’ uncertaintiesas the response of the ensemble of galaxy images to gravita-tional lensing shear. Thus we define our model for the twoellipticity components e = ( e , e ) as e = R γ + R psf e psf + e int . (2)We assume that the two-point statistics of the under-lying (cosmological) shear field h γγ i consist entirely of E -modes, e γ,E (which is a good enough approximation giventhe size of our errors; Crittenden et al. 2002; Schneider et al.2002), and are statistically independent of the PSF whenaveraged over large regions. We also assume that the PSFand the intrinsic alignments are independent – but not thatthe lensing shear and intrinsic alignments are independent(Hirata & Seljak 2004). The two-point correlation of thegalaxy shapes contains terms resulting from gravitationallensing and from systematic errors: h e e i = 4 R ξ γ,E + R ξ psf + ξ int + h γe int i . (3)Here, ξ psf is the auto-correlation of the PSF ellipticity field.Errors in the determination of the galaxy redshift distribu-tion will enter as a bias in the predicted ξ γ,E .Our goal is to carry out a statistics-limited measure-ment of ξ γ,E . This will entail showing that the combined amplitudes of R ξ psf , ξ int , h γ e int i , the uncertainty in thetheoretically-predicted ξ γ,E arising from redshift errors, andthe uncertainty in the shear calibration (via the responsivity R ) contribute less than 20 per cent to the statistical errorsin h e e i .Our approach to handling of systematic error is as fol-lows: we attempt to reduce each systematic to a term thatcan be robustly and believably estimated from real data (ei-ther the data here or in other, related work), and we then ex-plicitly correct for it. These corrections naturally have someuncertainty associated with them, which we use to derive asystematic error component to the covariance matrix. Theexception to the rule given here is if there is a systematicerror for which there is no clear path to estimating its mag-nitude, then we do not attempt any correction, and simplymarginalize over it by include an associated uncertainty inthe covariance matrix. Foreground anisotropies in the matter distribution along theline of sight to a galaxy will generically distort the galaxyimage. For weak lensing, the leading order lensing contribu-tion to galaxy shapes can be thought of as arising from a lin-ear transformation of the image coordinates A x true = x obs ,where A = (cid:18) κ + γ γ γ κ − γ (cid:19) . (4)The convergence κ causes magnification, whereas theshear components γ and γ map circles to ellipses. Theshear is related to the projected line-of-sight matter distri-bution, weighted by the lensing efficiency:( γ , γ ) = ∂ − Z ∞ W ( χ, χ i ) (cid:0) ∂ x − ∂ y , ∂ x ∂ y (cid:1) δ ( χ ˆ n i ) d χ. (5)Here we integrate along the comoving line-of-sight distance χ (where χ i is the distance to the source), and the matterover-density δ = ( ρ − ρ ) /ρ . The window function in a flatuniverse is W ( χ, χ i ) = 32 Ω m H (1 + z ) χ (cid:18) χ − χ i (cid:19) . (6)The two-point correlation function of the shear can becalculated by identifying pairs of source galaxies, and defin-ing shear components ( γ t , γ x ) for each one to be the shearin the coordinate system defined by the vector connectingthem, and in the π/ P δ averaged over the line ofsight to the sheared galaxies: ξ ± = h γ t γ t i ± h γ × γ × i = 12 π Z ∞ d ℓ ℓ P κ ( ℓ ) J , ( ℓθ ) (7)and P κ = (cid:18) m d H (cid:19) Z ∞ d χa ( χ ) P δ (cid:18) ℓd ( χ ) (cid:19) × (cid:20)Z ∞ χ d χ ′ n (cid:0) χ ′ (cid:1) d ( χ ′ − χ ) d ( χ ′ ) (cid:21) , (8) c (cid:13)000 , 000–000 E. M. Huff et al. where the last expression makes use of Limber’s approx-imation and d ( χ ) is the distance function, i.e. χ in aflat universe, K − / sin K / χ in a closed universe, and( − K ) − / sinh( − K ) / χ in an open universe. In the expres-sion in brackets, n ( χ ′ ) represents the source distribution asa function of line-of-sight distance (normalised to integrateto 1). This statistic ( P κ ) is sensitive both to the distributionof matter δ and to the background cosmology, via both theexplicit Ω m dependence and the distance-redshift relations. Many studies have discussed intrinsic alignments of galaxyshapes due to effects such as angular momentum alignmentsor tidal torque due to the large-scale density field (for pi-oneering studies, see Croft & Metzler 2000; Heavens et al.2000; Catelan et al. 2001; Crittenden et al. 2001; Jing 2002;Hopkins et al. 2005). While these effects can generate co-herent intrinsic alignment 2-point functions, Hirata & Seljak(2004) pointed out that the large-scale tidal fields that cancause intrinsic alignments are sourced by the same large-scale structure that is responsible for producing a cosmicshear signal. Thus, in this model, the intrinsic alignmentsdo not just have a nonzero auto-correlation, they also havea significant anti-correlation with the lensing shear whichcan persist to very large transverse scales and line-of-sightseparations. If left uncorrected, this coherent alignment ofintrinsic galaxy shapes suppresses the lensing signal, sincethe response of the intrinsic shape to an applied tidal fieldhas the opposite sign from the response of the galaxy imageto a shear with the same magnitude and direction. We gener-ally refer to the intrinsic alignment auto-correlation as the“ II ” contamination and its correlation with gravitationallensing as the “ GI ” contamination. This can be comparedto the pure gravitational lensing auto-correlation (“ GG ”).To address the uncertainty related to intrinsicalignments, we rely on empirical measurements thatconstrain the degree to which they might affect ourmeasurement. Several studies using SDSS imaging andspectroscopic data (e.g., Mandelbaum et al. 2006a;Hirata et al. 2007; Okumura et al. 2009; Joachimi et al.2011; Mandelbaum et al. 2011b) have demonstrated theexistence of intrinsic alignments of galaxy shapes oncosmological distance scales. Hirata et al. (2007) used theluminosity and colour-dependence of intrinsic alignmentsfor several SDSS galaxy samples to estimate the contamina-tion of the cosmic shear signal due intrinsic alignments forlensing surveys as a function of their depth. These estimateswere a function of the assumptions that were made, forexample about evolution with redshift. The “central” modelgiven in that paper leads to a fractional contamination of C ℓ =500 ,GI C ℓ =500 ,GG ≈ − .
08 (9)for a limiting magnitude of m R, lim = 23 .
5, which is closeto the limiting magnitude of our sample. Subsequent work(Joachimi et al. 2011; Mandelbaum et al. 2011b) providedmore information about redshift evolution; primarily thoseresults were in broad agreement with the previous ones, andwere sufficient to rule out both the “optimistic” and the“very pessimistic” models in Hirata et al. (2007).We thus adopt the “central” model, and apply the cor- rection given in Eq. (9) to our theory predictions for the C ℓ due to cosmic shear, multiplying the predicted cosmic shearpower spectrum by 0 .
92 before transforming into the statis-tics that are used for the actual cosmological constraints .We also assume this correction has a conservative system-atic uncertainty of 50 per cent, which amounts to an overall4 per cent uncertainty in the theory prediction (see Sec. 5for a quantitative description of how we incorporate this andother systematic uncertainties into the covariance matrix).Since the GI correlation is first order in the intrinsicalignment amplitude, while the II power is second order, weexpect the first to be the dominant systematic. In principle,the GI effect could be smaller than II if the correct align-ment model is quadratic in the tidal field rather than lin-ear (Hirata & Seljak 2004). However, in the aforementionedcases in which intrinsic alignment signals are detected athigh significance (i.e. for bright ellipticals) the linear modelfor intrinsic alignments appears to be valid (Blazek et al.2011). Therefore we attempt no correction for II . Another source of systematic error for weak lensing mea-surements is uncertainty in the shear calibration factor. Thegalaxy ellipticity ( e + , e × ) observed after isotropizing thePSF need not have unit response to shear: in general, aver-aged over a population of sheared galaxies, we should have h ( e + , e × ) i = R ( γ + , γ × ) , (10)where R is the shear responsivity. It depends on both theshape measurement method and the galaxy population (e.g.Massey et al. 2007b; Bernstein 2010; Zhang 2011).For this work, we used the re-Gaussianization method(Hirata & Seljak 2003), which is based on second momentsfrom fits to elliptical Gaussians, and has been previouslyapplied to SDSS single-epoch imaging (Mandelbaum et al.2005; Reyes et al. 2011). For this class of methods, in theabsence of selection biases and weighting of the galaxies,perfectly homologous isophotes, and no noise, there is ananalytic expectation (Bernstein & Jarvis 2002): R = 2(1 − e ) , (11)where e rms is the root-mean-square ellipticity per component(+ or × ).The calibration errors for re-Gaussianization and otheradaptive-weighting methods are well-studied in the liter-ature (e.g., Hirata et al. 2004b; Mandelbaum et al. 2005,2011a; Reyes et al. 2011). They arise from all of the devia-tions from the assumptions of Eq. (11). Higher-order depar-tures from non-Gaussianity in the galaxy light profile causeerrors in the PSF dilution correction. Errors in the mea-surement of the PSF model will cause a similar error in thedilution correction. The resolution factor of an individualgalaxy depends on its ellipticity, so any resolution cut onthe galaxy sample will introduce a shear bias in the galaxy While the intrinsic alignment contamination is in principlescale-dependent, the plots in Hirata et al. (2007) suggest that thisscale dependence is in fact quite weak for the scales used for ouranalysis, so we ignore it here.c (cid:13) , 000–000 osmic shear in SDSS selection function. Due to the non-linearity of the shear in-ference procedure, noise in the galaxy images causes a biasin the shears (rather than just making them noisier). The es-timation of the shear responsivity, or even of e rms , is anotherpotential source of error, as the response of the galaxies tothe shear depends on the true, intrinsic shapes, rather thanthe gravitationally sheared, smeared (by the PSF), noisyones that we observe.Past approaches to this problem have included detailedaccounting for these effects one by one. In this paper, weinstead use detailed simulations of the image processingand shape measurement pipelines, including real galaxy im-ages, to estimate both the shear calibration and the red-shift distribution of our catalogue. The advantage is thatthis includes all of the above effects and avoids uncertain-ties associated with analytic estimates of errors. The Shera (SHEar Reconvolution Analysis) simulation package hasbeen previously described (Mandelbaum et al. 2011a) andapplied to single-epoch SDSS data for galaxy-galaxy lensing(Reyes et al. 2011), but this is its first application to cosmicshear data.To simulate our images, we require a fair, flux-limitedsample of any galaxies that could plausibly be resolved in ourcoadd imaging, including high-resolution images with realis-tic morphologies . For this purpose we use a sample of 56 662galaxy images drawn from the COSMOlogical e volutionSurvey (COSMOS: Koekemoer et al. 2007; Scoville et al.2007a,b) imaging catalogues. The deep Hubble Space Tele-scope (HST) Advanced Camera for Surveys/Wide FieldCamera (ACS/WFC) imaging in F W (“broad I ”) in this1.6 deg field is an ideal source of a fairly-selected galaxysample with high resolution, deep images . These imagesconsist of two samples – a “bright” sample of 26 116 galax-ies in the magnitude range I < .
5, and a “faint” sampleconsisting of the 22 . < I < . ′′ ) galaxy postage stamp images have beenselected to avoid CCD edges and diffraction spikes frombright stars, and have been cleaned of any other nearbygalaxies, so they contain only single galaxy images withoutimage defects. The bright sample is used for ground-basedimage simulations in Mandelbaum et al. (2011a); the faintsample is selected and processed in an identical way . Eachpostage stamp is assigned a weight to account for the rel-ative likelihoods of generating postage stamps passing allcuts (avoidance of CCD edges and bright stars) for galax-ies of different sizes in the COSMOS field; this weight iscalculated empirically, by comparing the size distribution ofgalaxies with postage stamps to the size distribution of apurely flux-limited sample of galaxies. ∼ rmandelb/shera/shera.html Simple models with analytic radial profiles and ellipticalisophotes are not adequate to measure all sources of systematicerror such as under-fitting biases or those due to non-ellipticalisophotes (Bernstein 2010). Admittedly there may be some sampling variance that affectsthe morphological galaxy mix. We thank Alexie Leauthaud for kindly providing these pro-cessed images.
Each of these postage-stamp images has several proper-ties associated with it that are of interest for this analysis.The COSMOS photometric catalogues (Ilbert et al. 2009)contain HST F W magnitudes as well as photometricredshifts and Subaru r − i colours based on PSF-matchedaperture magnitudes.In order to simulate our observations, we first select acoadd ‘run’ consisting of five adjacent frames in the scandirection at random from the list of completed runs. Wedraw 1250 galaxies (exactly 250 per frame) at random fromthe list of COSMOS postage stamps according to the weightsdescribed above, up-weighting the probability of drawing thefaint galaxies by a factor of 1.106 to account for the fact thatwe have sampled the faint population at a lower rate thanthe bright one in constructing the image sample.Once a list of postage-stamp images is selected, we as-sign r - and i -band magnitudes by re-scaling each image; eachgalaxy image is inserted into the coadded imaging with theflux it would have been observed to have in SDSS beforethe addition of pixel noise. The i -band is chosen to be 0 . F W ( I ) band MAG AUTO values; this small offset is based on empirical com-parison with SDSS magnitudes for brighter galaxies, to ac-count for slight differences in the F W and i passbands(Mandelbaum et al. 2011a). The r -band is chosen so as tomatch the Subaru PSF-matched aperture colours for eachobject. Each postage stamp is assigned a random, uniformly-sampled position in the coadd run, with the postage stampsdistributed equally among the frames.We use the shera code to pseudo-deconvolve the HSTpoint-spread function, apply (if necessary; see below) a shearto each galaxy, reconvolve each image with the known coaddpoint-spread function, renormalise the flux appropriately,and resample from the COSMOS pixel scale to the coaddpixel scale before adding that postage stamp to the coaddimage. This procedure, suggested by Kaiser (2000) and im-plemented to high precision in Mandelbaum et al. (2011a),can be used to simulate ground-based images with a shearappropriately applied, despite the space-based PSF in theoriginal COSMOS images, and with a user-defined PSF.The normal coadd masking algorithm is then applied,and shear catalogues are generated as in Paper I by run-ning the SDSS object detection and measurement pipeline, Photo-frames , followed by the shape measurement codedescribed in Sec. 3.3. The output catalogues are matchedagainst the known input object positions, and a simulationcatalogue of the matches is created. We employ these simu-lations below to determine the shear calibration and as anindependent validation of our inferred redshift distribution.For each suite of simulation realisations, we use thesame random seed (i.e., we select the same galaxies fromour catalogue and place them at identical locations in thecoadded image) but with different applied shears per com-ponent ranging from − .
05 to +0 .
05. We measure the meanweighted shape of the detected simulation galaxies producedby our pipeline, and fit a line to the results. Since the samegalaxies are used without rotation, only the slope and notthe intercept is meaningful. The shear response in each com-ponent for each applied shear is shown in Fig. 1. The re-sponsivities in the two components are consistent, which isexpected on oversampled data with a rounded PSF. (Theunequal size of the error bars reflects the number of runs c (cid:13) , 000–000 E. M. Huff et al. -0.1-0.05 0 0.05 0.1 -0.04 -0.02 0 0.02 0.04 < e > γ Shear calibration simulationsd
The response of the mean ellipticities h e i and h e i to applied shear, as determined in the shera -based simulations.Poisson error bars are shown. The additive offset to the responsecurve is not shown in the fit; these simulations do not accuratelymeasure an additive shear bias. that we were able to process by the time the shear calibra-tion solution was frozen.) The total number of galaxies inthe final simulated catalogues was 130 063. The response ap-pears to be linear for small applied shears. Based on theseresults, we adopt a shear responsivity for this galaxy popu-lation of 1 . ± . The explicit dependence of the shear signal in Eqs. (5)and (8) on the distribution of lensed galaxy redshifts, com-bined with the practical impossibility of acquiring a spectro-scopic redshift for the millions of faint galaxies statisticallynecessary for a cosmic shear measurement, can be a trouble-some source of bias and systematic uncertainty for cosmicshear measurements.An error in the estimated redshift distribution leadsto an incorrect prediction for the amplitude of the shearsignal at a given cosmology. This is similar in principle tothe bias arising in the amplitude of the galaxy-galaxy lens-ing signal due to photometric redshift biases explored inNakajima et al. (2011); uncorrected, standard photometricredshift estimation techniques can lead to biases in the pre-dicted lensing signal at the ∼
10 per cent level. For cosmicshear measurements, an imperfect estimate of the redshiftdistribution leads to biases in σ and Ω m that are compara- ble in amplitude to the errors in the estimated mean of theredshift distribution (van Waerbeke et al. 2006).As a fiducial reference, the redshift distribution of thesingle-epoch SDSS imaging catalogue is established to ap-proximately 1 per cent (Sheldon et al. 2011); for deeper sur-veys over a smaller area, this becomes a more difficult prob-lem, as the spectroscopic calibration samples available forinferring the redshift distribution are limited in their red-shift coverage and widely dispersed across the sky. We em-ploy a colour-matching technique similar to that employedby Sheldon et al. (2011); in what follows, we describe thetechnique, our estimate of its uncertainty, and several cross-checks on the results. The source redshift distribution used in our analysis is de-rived following Lima et al. (2008) and Cunha et al. (2009),and is similar in spirit to Sheldon et al. (2011); the princi-ple is that, for two galaxy samples that span broadly similarranges in redshift, colour, and limiting magnitude, matchedcolour samples correspond to matched redshift distributions.Our spectroscopic calibration sample is composed of12 360 galaxies, from the union of the VIMOS VLT DeepSurvey (Le F`evre et al. 2005, VVDS) 22h field, the DEEP2Galaxy Redshift Survey (Davis et al. 2003; Madgwick et al.2003), and portions of the PRism MUlti-object Survey(PRIMUS; Coil et al. 2011, Cool et al. 2011 in prep. ). Wefollow the procedures outlined in Nakajima et al. (2011) forselecting good quality spectroscopic redshifts, and avoidingduplicate galaxies in samples that overlap (such as DEEP2and PRIMUS). Each of these samples has a redshift distri-bution that is likely to differ substantially from the redshiftdistribution of our lensing catalogue: the DEEP2 cataloguein the fields we use at 23 h m and 02 h m is heavily colour-selected (in non-SDSS bands) towards objects at z > .
7; thePRIMUS catalogue includes several fields, some of which areselected from imaging with a shallower limiting magnitude;and the VVDS catalogue is selected in the I band ( I < . × galaxies with replacement from thefull area (not just in these regions), with sampling proba-bility proportional to the mean of the weights assigned inthe r and i bands to that galaxy for the correlation analy-sis (Eq. 17). Note that this procedure does not incorporatethose galaxies in the excluded camcol 2 region.We use the Lima et al. (2008) code to solve for a setof weights over the calibration sample, such that the re-weighted 5-dimensional magnitude distributions of the cal-ibration sample match those of the representative randomsubset of the lensing catalogue. The histogram of the cali-bration sample redshifts reweighted in this manner is shownas a solid line in Fig. 2. The inferred mean redshift is 0.51; in http://kobayashi.physics.lsa.umich.edu/ ∼ ccunha/nearest/ c (cid:13) , 000–000 osmic shear in SDSS Figure 2.
The redshift distribution inferred from matching thecolours of the spectroscopic calibration sample to those of thelensing catalogue (solid black line, Sec. 3.4.1) shown alongsidethe noisier redshift distribution inferred from the shear calibrationsimulations (dashed red line, Sec. 3.4.3). The best-fit distributionfor the single-epoch SDSS lensing catalogue from Nakajima et al.(2011) is shown for reference as the blue dot-dashed line. contrast to the redshift distribution for single-epoch imag-ing, there is a non-negligible fraction of the galaxy sampleabove z > .
7. We use the solid curve based on the colour-matching techniques to calculate the shear covariance ma-trix, and to predict the shear correlation function for anygiven cosmology.
We expect that the primary source of error in the red-shift distribution as estimated from the combined calibra-tion sample is sample variance, resulting from the finite vol-ume of the calibration sample. To estimate its magnitude,we use the public code of Moster et al. (2011) for estimatingthe cosmic variance of number counts in small fields.Our redshift binning scheme has 19 bins between 0 5. For a collection of disparate calibration fields, weuse the Moster et al. (2011) code to produce a fractionalerror in the number counts σ gg ,i,j for the j th redshift bin inthe i field (where fields are distinguished by their coveragearea) in bins of stellar mass.The redshift sampling rate of each distinct survey inthe calibration sample differs, and so the balance of contri-butions to the final redshift distribution will change as well.To account for this, we sum over every calibration field’scontribution to the reweighted redshift distribution in the j bin to estimate an absolute (not relative) overall error: σ j = X i ( σ gg ,i,j n eff ,i,j ) (12)where the effective number of galaxies contributed in the j bin by the i survey is just the sum over the nearest-neighbourderived weights assigned to calibration sample galaxies k inthat field i and bin j : n eff ,i,j = X k w nn,i,j,k (13)To propagate these errors into the covariance matrix for ξ E , we first fit a smooth function of the form n z ( z ) ∝ z a e − ( z/z ) b (14)to the nearest neighbour weighting-derived redshift dis-tribution shown in Figure 2; the best fit parameters are a = 0 . z = 0 . b = 2 . n z ( z j ) (normalised tothe weighted number of calibration galaxies in that bin) andstandard deviation σ j at the location of the j th redshift bin.We then renormalise the perturbed distribution to unity, andcompute the predicted cosmic shear signal. The covariancematrix of 402 realisations of this procedure is added to thestatistical covariance matrix. As an independent check on the redshift distribution, wealso use the shear calibration simulations (Sec. 3.3) to con-strain the redshift distribution of our sources. The COSMOSphotometric redshifts, inferred as they are from many moreimaging bands (typically with deeper imaging) than for theSDSS data discussed here, are very accurate. For example,Ilbert et al. (2009) finds a photo- z scatter of σ z / (1 + z ) ∼ . 01 for a galaxy sample with the flux limit of the SDSScoadds. In contrast, Nakajima et al. (2011) found that inthe SDSS single-epoch imaging, the scatter defined in thesame way was ∼ . n ( z ) inferred from the COSMOS-based sim-ulations is also shown in Fig. 2, and agrees extremely wellwith the fiducial n ( z ) derived from colour-matching.A final (but obviously not independent) sanity check isto compare to the COSMOS Mock Catalogue (Jouvel et al.2009), which is being used extensively to plan future darkenergy programmes, using the cuts r eff > . 47 arcsec, limit-ing magnitudes r < . 5, and i < . h z i = 0 . 51, identical to that obtained viathe re-weighting procedure. Given the crudeness of the pro-cedure for comparing the results, this is an excellent val-idation of the COSMOS Mock Catalogue as a forecastingtool. Stellar contamination of the galaxy catalogue reduces theapparent shear by diluting the signal with round objects that c (cid:13) , 000–000 E. M. Huff et al. are not sheared by gravitational lensing. Because the imagesimulations described in Sec. 3.3 only included galaxies, theresulting shear responsivities do not include signal dilutiondue to accidental inclusion of stars in the galaxy sample. InPaper I, we estimated the stellar contamination by compar-ison with the DEEP2 target selection photometry (which isdeeper and was acquired at the Canada-France-Hawaii Tele-scope under much better seeing conditions than typical forSDSS), and found a contamination fraction of 0.017. We alsoargued that the mean stellar density in the stripe must belarger than in the high-latitude DEEP2 fields, by a factoras large as 2.8. We therefore conservatively take the stellarcontamination fraction f star to be f star = 0 . . ± . 9) = 0 . ± . . (15)The resulting suppression of the cosmic shear signal istreated in much the same way as for intrinsic alignments: wereduce the theory signal by a factor of (1 − . = 0 . Among the most worrying systematics in the early detec-tions of cosmic shear was additive power. This comes fromany non-cosmological source of fluctuations in shapes suchas PSF anisotropy that add to the ellipticity correlationfunction of the galaxies. Such power was clearly detectedin Paper I in the form of systematic variation of both starand galaxy e as a function of declination. The sense ofthe effect – a negative contribution to e (in r band wehave h e i = − . h e i = +0 . i band h e i = − . h e i = − . mask-ing bias , in which the selection of a galaxy depends on itsorientation, with galaxies aligned in the along-scan direction( e < 0) being favoured, and with no effect on e (consis-tent with zero mean over the whole survey). The reasonfor this particular sign is seen in Figure 2 of Paper I; asshown, bad columns along the scan direction tend to be re-peated at the same location in multiple images, resultingin significant (non-isotropic) masks with that directional-ity. Direct evidence for masking bias comes from the changein mean ellipticity due to increased masking: when we re-moved from the coadded image pixels that were observed infewer than 7 input runs and reran Photo-Frames , the h e i signal became worse : − . r band and − . i band, whereas h e i was essentially unchanged. This increaseis difficult to explain in terms of spurious PSF effects, so weconclude that our galaxy catalogue likely contains a mixtureof masking bias as well as possible additive systematics fromPSF ellipticity in the coadded image.The mean e signal as a function of declination is shownin Fig. 3 in bins of width 0.05 degrees. We take this as a tem-plate for mask-related selection biases (combined with anysystematic uncorrected PSF variation as a function of dec-lination, which in west-to-east drift-scan observations is ahighly plausible type of position dependence). Before com- The 1 σ Poisson uncertainty in these numbers is 0.0005 (0.0004)per component in r ( i ) band. -25-20-15-10-5 0 5 10 15-1.5 -1 -0.5 0 0.5 1 1.5 < e > / - Declination (deg)Mean ellipticity: r band-25-20-15-10-5 0 5 10 15-1.5 -1 -0.5 0 0.5 1 1.5 < e > / - Declination (deg)Mean ellipticity: i band Figure 3. The mean ellipticity h e i as a function of declinationin the r and i bands. This signal was removed from the galaxycatalogue prior to computing the final correlation function. The r band data between declination − . ◦ and − . ◦ were rejecteddue to the known problems with camcol 2. The error bars arePoisson errors only. puting the correlation function, we subtracted this mean sig-nal from the galaxy ellipticity catalogue. One danger in this procedure to remove spurious h e i is that some real power could be removed – that is, evenin the absence of any systematic error, some of the actualgalaxy shape correlation function signal could be suppressedsince the method determines the mean e of the real galax-ies and by subtracting it introduces a slight artificial anti-correlation. The best way to guard against this is with simu-lations. Using the Monte Carlo simulation tool of Sec. 5.1.2,we generated simulated realisations of our ellipticity cata-logue and either implemented the h e i projection or not.The difference in the correlation functions is a measure ofhow much power was removed. The result is shown in Fig. 4,and shows that the loss of real power is insignificant com-pared to our error bars. Convolution with an elliptical PSF will induce a spuri-ous ellipticity in observed galaxy surface-brightness profiles.While the effective PSF for these coadds is a circular doubleGaussian to quite high precision, the tests in Paper I indi-cate a low level of residual anisotropy that we must considerhere. c (cid:13) , 000–000 osmic shear in SDSS -1-0.5 0 0.5 1 1.5 1 10 100 θ ξ s g ( θ ) ( a r c m i n ) θ (arcmin)Star-galaxy ellipticity correlations: rr -1-0.5 0 0.5 1 1.5 1 10 100 θ ξ s g ( θ ) ( a r c m i n ) θ (arcmin)Star-galaxy ellipticity correlations: ri-1-0.5 0 0.5 1 1.5 1 10 100 θ ξ s g ( θ ) ( a r c m i n ) θ (arcmin)Star-galaxy ellipticity correlations: ir -1-0.5 0 0.5 1 1.5 1 10 100 θ ξ s g ( θ ) ( a r c m i n ) θ (arcmin)Star-galaxy ellipticity correlations: ii Figure 5. The star-galaxy ellipticity correlation functions. Shown are the rr , ri (i.e. star r × galaxy i ), ir , and ii correlation functions,reduced to 10 bins. The solid points, which are offset to slightly lower θ -values for clarity, are the ++ correlation functions, and thedashed points are the ×× functions. All error bars are Poisson only. -2-1.5-1-0.5 0 0.5 1 1.5 2 1 10 100 θ [ ξ s ub t r ( θ )- ξ o r i g ( θ ) ] ( a r c m i n ) θ (arcmin)Change in signal from e projection [ww] Figure 4. The loss of actual power due to e projection. Us-ing 36 realizations from the Monte Carlo simulation, we find thedifference in post-projection ellipticity correlation function ξ ( θ )and original ξ ( θ ). These are shown as the solid points ( ξ ++ ) anddashed points ( ξ ×× ) in the figure, re-binned to 10 bins in angularseparation θ . The dashed lines at top and bottom are the ± σ statistical error bars of our measurement. The reduction of actualpower is detectable by combining many simulations, but is verysmall compared to the error bars on the measurement. Possible sources of this issue include: (i) inaccuraciesin the single-epoch PSF model used to determine the ker-nel to achieve the desired PSF; (ii) colour-dependence ofthe PSF that means the single-epoch PSF model from thestars is not exactly the PSF for the galaxies; or (iii) thefact that we determine the rounding kernel on a fixed grid,so that smaller-scale variations in PSF anisotropy might re-main uncorrected. All of these must be present at some level,although the last two cannot be the full solution: (ii) doesnot explain the residual stellar ellipticity , and (iii) doesnot explain why there is structure in the declination direc-tion on the scale of an entire CCD (0.23 degrees).For a galaxy and a PSF that are both well-approximated by a Gaussian, the PSF-correction givenabove produces a measured ellipticity of: e obs = R psf e PSF = 1 − R R e PSF ; (16)see e.g. Bernstein & Jarvis (2002). The weighted (by the We have searched for a g − i dependence in the stellar elliptic-ities in the coadded image. We only found effects at the ∼ . (cid:13)000 The loss of actual power due to e projection. Us-ing 36 realizations from the Monte Carlo simulation, we find thedifference in post-projection ellipticity correlation function ξ ( θ )and original ξ ( θ ). These are shown as the solid points ( ξ ++ ) anddashed points ( ξ ×× ) in the figure, re-binned to 10 bins in angularseparation θ . The dashed lines at top and bottom are the ± σ statistical error bars of our measurement. The reduction of actualpower is detectable by combining many simulations, but is verysmall compared to the error bars on the measurement. Possible sources of this issue include: (i) inaccuraciesin the single-epoch PSF model used to determine the ker-nel to achieve the desired PSF; (ii) colour-dependence ofthe PSF that means the single-epoch PSF model from thestars is not exactly the PSF for the galaxies; or (iii) thefact that we determine the rounding kernel on a fixed grid,so that smaller-scale variations in PSF anisotropy might re-main uncorrected. All of these must be present at some level,although the last two cannot be the full solution: (ii) doesnot explain the residual stellar ellipticity , and (iii) doesnot explain why there is structure in the declination direc-tion on the scale of an entire CCD (0.23 degrees).For a galaxy and a PSF that are both well-approximated by a Gaussian, the PSF-correction givenabove produces a measured ellipticity of: e obs = R psf e PSF = 1 − R R e PSF ; (16)see e.g. Bernstein & Jarvis (2002). The weighted (by the We have searched for a g − i dependence in the stellar elliptic-ities in the coadded image. We only found effects at the ∼ . (cid:13)000 , 000–000 E. M. Huff et al. -0.5 0 0.5 1 1.5 2 1 10 100 θ ∆ ξ ( θ ) ( a r c m i n ) θ (arcmin)Contamination implied by star-galaxy correlation Figure 6. The implied contamination to the galaxy ellipticitycorrelation function if the star-galaxy correlation function is usedas a measure of the additive PSF power. The solid points arethe ++ correlation functions, and the dashed points are the ×× functions. All error bars are propagated from the Poisson errorsassuming correlation coefficient +1 (a better assumption than in-dependent errors, but likely an overestimate). The dotted curvesshow the 1 σ errors in each radial bin from the Monte Carlo sim-ulations (see Sec. 5.1.2) which include both Poisson and cosmicvariance uncertainties. Note also that the shapes and normalisa-tions of the ++ and ×× signals are nearly identical. same weights used for the correlation function; see Eq. 17)average of the PSF anisotropy response defined in Eq. (2)over the sample of galaxies considered in this work is R psf =0 . 86 ( r band) or 0.95 ( i band); in what follows we take avalue of 0.9.A nonzero star-galaxy correlation function ξ sg resultingfrom systematic PSF anisotropy (as estimated in Paper I) in-dicates the presence of a spurious contribution to the shear-shear correlation function with amplitude ≈ . ξ sg . We willnot determine this response to high enough accuracy to sub-tract the effect with small residual error: doing so would notrequire just a simulation, but a simulation that knows thecorrect radial profile of the PSF errors. In our case, thestar-galaxy correlation function is detectable but below theerrors on the galaxy-galaxy ellipticity auto-correlation (al-though not by very much), so a highly accurate correctionis unnecessary.We constrain the PSF anisotropy contribution by com-puting the star-galaxy correlation function. This was donein Paper I, but some of the star-galaxy signal is due to thesystematic variation of PSF ellipticity with declination andis removed by the subtraction procedure above. The star-galaxy ellipticity correlation function with the corrected cat-alogue is shown in Fig. 5. The implied contamination to thegalaxy ellipticity correlation function, appropriately averag-ing the bands and applying the factor of R psf = 0 . 9, is shownin Fig. 6.These measured star-galaxy correlations can be used This might be an option in future space-based surveys ifthe type of error can be traced to the source of ellipticity(astigmatism × defocus, coma, or jitter). In either space or ground-based data, one could imagine doing cross-correlations of higher-order shapelet modes (Refregier 2003) to extract the particularform of the errors. None of these options are pursued here. to construct a reasonable systematics covariance matrix forthis systematic. We take the amplitude of the diagonal ele-ments of the PSF systematic covariance to be equal to theamplitude of the measured contamination. We also assumethat the off-diagonal terms are fully-correlated between bins,which is equivalent to fixing the scaling of this systematicwith radius, and saying that only the overall amplitude ofthe systematic is uncertain.Since there are a number of uncertainties in this pro-cedure, we do not apply any correction for these additivePSF systematics as we do for ones that are previously dis-cussed, such as intrinsic alignments or stellar contamina-tion. Instead, we simply include a term in the systematicscovariance matrix to account for it. We also will present aworst-case scenario for the impact of this term on cosmo-logical constraints; in Sec. 6 we will show what happens tothe cosmology constraints if we assume that the systematicerror is +2 σ from its mean, i.e. 40 per cent of the statisticalerrors. This should be taken as a worst-case scenario for thisparticular systematic. We compute the ellipticity correlation functions definedin Eq. (7) on scales from 1–120 arcminutes. For the cos-mological analysis, we start by computing the correlationfunction in 100 bins logarithmically spaced in separation θ to avoid bin width artifacts. For the cosmological pa-rameter constraints, we project these onto the CompleteOrthogonal Sets of E -/ B -mode Integrals (COSEBI) basis(Schneider et al. 2010) to avoid the instabilities of invertinga large covariance matrix estimated via Monte Carlo simu-lations (we will describe our implementation of COSEBIs inSec. 4.3). However, for display purposes, it is more conve-nient to reduce the θ resolution to only 10 bins so that thereal trends are more visually apparent. The correlation functions used here are weighted by the in-verse variance of the ellipticities, where the “variance” in-cludes shape noise. Specifically, we define a weight for agalaxy w i = 1 σ e + 0 . , (17)where σ e is the ellipticity uncertainty per component de-fined by our shape measurement pipeline. As demonstratedby Reyes et al. (2011), these may be significantly underesti-mated in certain circumstances; however, this will only makeour estimator slightly sub-optimal, so we do not attempt tocorrect for it. The value of 0 . 37 for the root-mean-square(RMS) intrinsic ellipticity dispersion per component comesfrom the results of Reyes et al. (2011), for r < 22, and there-fore we are implicitly extrapolating it to fainter magnitudes.Given that Leauthaud et al. (2007) found a constant RMS c (cid:13) , 000–000 osmic shear in SDSS -4-2 0 2 4 6 8 10 1 10 100 θ ξ ( θ ) ( a r c m i n ) θ (arcmin)Ellipticity correlations: rr (proj) -4-2 0 2 4 6 8 10 1 10 100 θ ξ ( θ ) ( a r c m i n ) θ (arcmin)Ellipticity correlations: ri (proj)-4-2 0 2 4 6 8 10 1 10 100 θ ξ ( θ ) ( a r c m i n ) θ (arcmin)Ellipticity correlations: ii (proj) -4-2 0 2 4 6 8 10 1 10 100 θ ξ ( θ ) ( a r c m i n ) θ (arcmin)Ellipticity correlations: ww (proj) Figure 7. The ellipticity correlation functions in the rr , ri , ii and ww (combined) band combinations. The solid points denote the ++and the dashed points denote the ×× components of the correlation function. The points have been slightly displaced horizontally forclarity. The Monte Carlo errors are shown. ellipticity to far fainter magnitudes in the COSMOS data,we consider this extrapolation justified . A direct pair-count correlation function code was used forthe cosmological analysis. It is slow ( ∼ × galaxies on a modern laptop) but robust and well-adapted tothe Stripe 82 survey geometry. The code sorts the galaxiesin order of increasing right ascension α ; the galaxies areassigned to the range − ◦ < α < +60 ◦ to avoid unphysicaledge effects near α = 0. It then loops over all pairs with Note that we do not use the actual value of RMS ellipticityfrom Leauthaud et al. (2007) – only the trend with magnitude– because, as demonstrated by Mandelbaum et al. (2011a), theRMS ellipticity value in Leauthaud et al. (2007) is not valid forour adaptively-defined moments, which use an elliptical weightfunction matched to the galaxy light profile. | α − α | < θ max . The usual ellipticity correlation functionscan be computed, e.g. ξ ++ ( θ ) = P ij w i w j e i + e j + P ij w i w j , (18)where the sum is over pairs with separations in the rele-vant θ bin, and the ellipticity components are rotated tothe line connecting the galaxies. The direct pair-count codeworks on a flat sky, i.e. equatorial coordinates ( α, δ ) areapproximated as Cartesian coordinates. This is appropri-ate in the range considered, | δ | < . ◦ , where the maxi-mum distance distortions are δ = 2 . × − . The directpair-count code is applicable to either auto-correlations ofgalaxy shapes measured in a single filter ( rr , ii ) or cross-correlations between filters or between distinct populationsof objects ( ri and all of the star-galaxy correlations).Simple post-processing allows one to compute the ξ + and ξ − correlation functions, defined by ξ + ( θ ) ≡ ξ ++ ( θ ) + ξ ×× ( θ ) (19) c (cid:13) , 000–000 E. M. Huff et al. and ξ − ( θ ) ≡ ξ ++ ( θ ) − ξ ×× ( θ ) . (20) Finally, the different band correlation functions rr , ri , and ii must be combined according to some weighting scheme: ξ ww ++ ( θ ) = w rr ξ rr ++ ( θ ) + w ri ξ ri ++ ( θ ) + w ii ξ ii ++ ( θ ) , (21)where the label “ ww ” indicates that the bands were com-bined. The relative weights were chosen according to thefraction of measured shapes in r - and i -bands, i.e. w rr = f r , w ri = 2 f r f i , and w ii = f i where the weights are f r = 0 . f i = 0 . θ resolution reduced to 10 bins) are shown in Fig. 7. We implement several null tests on the correlation functionto search for remaining systematic errors.The first test, shown in Fig. 8, constructs the differencebetween the cross-correlation function of r and i band galaxyellipticities versus the rr and ii auto-correlations. The dif-ferences in the two types of correlation functions are smallcompared to the statistical uncertainty in the signal. This isconsistent with our expectations, as the true cosmic shearsignal should be independent of the filters in which galaxyshapes are measured.The second test, shown in Fig. 9, compares the (bandaveraged or ww ) correlation function computed using galaxypairs separated in the cross-scan (north-south) direction ver-sus pairs separated in the along-scan (east-west) direction.This difference should be zero if the signal we measure isdue to lensing in a statistically isotropic universe. The er-ror bars shown are Poisson errors, so they may be slightunderestimates at the larger scales, where cosmic variancebecomes important. Visual inspection shows no obvious off-set from zero, but the error bars are larger for this test thanin Fig. 8 because the null test includes no cancellation ofgalaxy shape noise. As a final check for systematics, we decompose the 2-pointcorrelation function (2PCF) into E - and B -modes, where,to leading order, gravitational lensing only creates E -modes.The B -modes can arise from the limited validity of the Bornapproximation (Jain et al. 2000; Hilbert et al. 2009), red-shift source clustering (Schneider et al. 2002), and lensing(magnification) bias (Schmidt et al. 2009; Krause & Hirata2010), however the amplitude of B -modes from these sourcesshould be undetectable with our data. At our level of sig-nificance, a B -mode detection would indicate remaining sys-tematics, e.g. due to spurious power from an incomplete PSFcorrection.Formerly used methods to decompose E -and B -modes,such as the aperture mass dispersion h M i ( θ ) = Z θ d ϑ ϑ θ (cid:20) ξ + ( ϑ ) T + (cid:18) ϑθ (cid:19) + ξ − ( ϑ ) T − (cid:18) ϑθ (cid:19)(cid:21) , (22) -1-0.5 0 0.5 1 1.5 2 2.5 3 1 10 100 θ ξ ++ ( θ ) ( a r c m i n ) θ (arcmin)Colour difference plot, 0.5(rr+ii)-ri: ++-1-0.5 0 0.5 1 1.5 2 2.5 3 1 10 100 θ ξ xx ( θ ) ( a r c m i n ) θ (arcmin)Colour difference plot, 0.5(rr+ii)-ri: xx Figure 8. The difference between the galaxy ellipticity cross-correlations ( ri ) and the auto-correlations ( rr + ii ) / 2, with errorbars determined from the Monte Carlo simulations. The upperpanel shows the ++ correlations and the lower panel shows the ×× correlations. The dashed line is the 1 σ statistical error baron the actual signal. with the filter functions T ± as derived in Schneider et al.(2002), or the shear E -mode correlation function, suffer from E/B -mode mixing (Kilbinger et al. 2006), i.e. B -modes af-fect the E -mode signal and vice versa. These statistics canbe obtained from the measured 2PCF, for an exact E/B -mode decomposition, however they require information onscales outside the interval [ θ min ; θ max ] for which the 2PCFhas been measured.The ring statistics (Schneider & Kilbinger 2007;Eifler et al. 2010; Fu & Kilbinger 2010) and more recentlythe COSEBIs (Schneider et al. 2010) perform an EB-modedecomposition using a 2PCF measured over a finite angularrange. COSEBIs and ring statistics can be expressed asintegrals over the 2PCF as EB = Z θ max θ min d θ θ [ T log+ n ( θ ) ξ + ( θ ) ± T log − n ( θ ) ξ − ( θ )] (23)and R EB ( θ ) = Z θθ min d θ ′ θ ′ [ ξ + ( θ ′ ) Z + ( θ ′ , θ ) ± ξ − ( θ ′ ) Z − ( θ ′ , θ )] . (24)For the ring statistics, we use the filter functions Z ± spec-ified in Eifler et al. (2010). The derivation of the COSEBIfilter functions T ± n is outlined in Schneider et al. (2010),where the authors provide linear and logarithmic filter func- c (cid:13) , 000–000 osmic shear in SDSS -10-5 0 5 10 1 10 100 θ ξ ++ ( θ ) ( a r c m i n ) θ (arcmin)Null test, [N/S]-[E/W] pairs: rr++-10-5 0 5 10 1 10 100 θ ξ xx ( θ ) ( a r c m i n ) θ (arcmin)Null test, [N/S]-[E/W] pairs: rrxx -10-5 0 5 10 1 10 100 θ ξ ++ ( θ ) ( a r c m i n ) θ (arcmin)Null test, [N/S]-[E/W] pairs: ri++-10-5 0 5 10 1 10 100 θ ξ xx ( θ ) ( a r c m i n ) θ (arcmin)Null test, [N/S]-[E/W] pairs: rixx -10-5 0 5 10 1 10 100 θ ξ ++ ( θ ) ( a r c m i n ) θ (arcmin)Null test, [N/S]-[E/W] pairs: ii++-10-5 0 5 10 1 10 100 θ ξ xx ( θ ) ( a r c m i n ) θ (arcmin)Null test, [N/S]-[E/W] pairs: iixx Figure 9. The null test of the correlation functions measured using galaxy pairs whose separation vector is within 45 ◦ of the north-southdirection, minus that measured using galaxy pairs whose separation vector is within 45 ◦ of the east-west direction. The error bars shownare the Poisson errors only. The dashed curve shows the 1 σ error bars of the actual signal (all colour combinations and separation vectorsaveraged). The 6 panels show the three colour combinations ( rr , ri , and ii ) and the 2 components (++ or ×× ). tions indicating whether the separation of the roots of thefilter function is distributed linearly or logarithmically in θ .Note that whereas the ring statistics are a function of an-gular scale, the COSEBIs are calculated over the total an-gular range of the 2PCF, condensing the information fromthe 2PCF naturally into a set of discrete modes. The lin-ear T -functions can be expressed conveniently as Legendrepolynomials, however T log ± n compresses the cosmological in-formation into significantly fewer modes; we therefore choosethe logarithmic COSEBIs as our second-order shear statis-tic in the likelihood analysis in Sec. 6. The COSEBI filterfunctions are displayed graphically in Fig. 10.Figure 11 shows three different E/B -mode statistics de-rived from our measured shear-shear correlation function,i.e. the COSEBIs, the ring statistics, and the aperture massdispersion. The error bars are obtained from the square rootof the corresponding covariances’ diagonal elements (statis-tics only). Note that the COSEBIs data points are signifi-cantly correlated. Slightly smaller is the correlation for theaperture mass dispersion, and the ring statistics’ data pointshave the smallest correlation.From the COSEBIs, we find a reduced χ for the E -modes to be consistent with zero of 6.395, versus 1.096 forthe B -modes (5 degrees of freedom each). The latter is con-sistent with purely statistical fluctuations. The covariance matrix of the ellipticity correlation functionestimated via Eq. (21) was computed in several ways. Thepreferred method for our analysis is a Monte Carlo method(Sec. 5.1.2) but we compare that covariance matrix with anestimate of the Poisson errors (Sec. 5.1.1) as a consistencycheck. The direct pair-count correlation function code can computethe Poisson error bars, i.e. the error bars neglecting the cor-relations in e i + e j + between different pairs. This estimate ofthe error bar is σ [ ξ ++ ( θ )] = P ij w i w j | e i | | e j | hP ij w i w j i . (25)Equivalently, this is the variance in the correlation functionthat one would estimate if one randomly re-oriented all ofthe galaxies. The Poisson method is simple, however it is notfully appropriate for ri cross-correlations (since the same in-trinsic shape noise is recovered twice for pairs that appearin both ri and ir cross-correlations). Moreover, at scales of c (cid:13) , 000–000 E. M. Huff et al. − . − . − . − . − mode C O SEB I s E−modeB−mode 20 40 60 80 100 120 − − − − + − − θ [arcmin] r i ng s t a t i s t i cs E−modeB−mode 0 20 40 60 80 100 120 − − + − θ [arcmin] ape r t u r e m a ss d i s pe r s i on E−modeB−mode Figure 11. The measured COSEBIs, ring statistics, and aperture mass dispersion from the combined cosmic shear signal. The errorbars equal the square root of the corresponding covariances’ diagonal elements (statistics only). Note that the COSEBIs data points aresignificantly correlated. Slightly smaller is the correlation for the aperture mass dispersion, and the ring statistics’ data points have thesmallest correlation. − ϑ [arcmin] T n + T T T T T − ϑ [arcmin] T n − T T T T T Figure 10. The COSEBI filter functions T n + (upper panel) and T n − (lower panel) for the first 5 modes. tens of arcminutes and greater there is an additional contri-bution because the cosmic shear itself is correlated betweenpairs. Therefore the Poisson error bars should be used onlyas a visual guide: they would underestimate the true uncer-tainties if used in a cosmological parameter analysis. We used a Monte Carlo method to compute the covariancematrix of ξ ++ ( θ ) and ξ ×× ( θ ). The method is part theoret-ical and part empirical: it is based on a theoretical shear power spectrum, but randomizes the real galaxies to cor-rectly treat the noise properties of the survey. The advan-tages of the Monte Carlo method – as implemented here– are that spatially variable noise, intrinsic shape noise in-cluding correlations between the r and i band, and the sur-vey window function are correctly represented. The principaldisadvantages are that the cosmic shear field is treated asGaussian and a particular cosmology must be assumed (seeEifler et al. 2009, for alternative approaches). However, solong as this cosmology is not too far from the correct one(an assumption that can itself be tested!), the Monte Carloapproach is likely to yield the best covariance matrix.The Monte Carlo approach begins with the generationof a suite of 459 realizations of a cosmic shear field in har-monic space according to a theoretical spectrum. For ouranalysis, the theoretical spectrum was that from the WMAP b h = 0 . m h = 0 . n s = 0 . H = 71 . − Mpc − ; and σ = 0 . E -mode shear harmonic space coefficients a E lm .The full power spectrum is used at l < l < l > E -mode power spec-trum is C EE = 3 . × − , as compared to a shot noiseof γ / ¯ n ∼ . × − .) No B -mode shear is included.The particle-mesh spherical harmonic transform code ofHirata et al. (2004a) with a 6144 × L ′ = 6144)and a 400-node interpolation kernel ( K = 10) was used totransform these coefficients into shear components ( γ , γ )at the position ˆ n j of each galaxy j . The use of a full-sky approach for the Monte Carlo realisationsc (cid:13) , 000–000 osmic shear in SDSS A synthetic ellipticity catalogue was then generated asfollows. For each galaxy, we generated a random positionangle offset ψ j ∈ [0 , π ) and rotated the ellipticity in both r and i bands by ψ j . We then added the synthetic shearweighted by the shear responsivity to the randomised ellip-ticity to generate a synthetic ellipticity: e syn j = e ψ j e true j + 1 . γ (ˆ n j ) . (26)The 1.73 prefactor was estimated from Eq. (11), which weexpected to be good enough for use in the Monte Carloanalysis, so that the Monte Carlos could be run in parallelwith the shear calibration simulations. The latter gave a finalresult of 1 . ± . 04, which is not significantly different.The direct pair-count correlation function code, in allversions ( rr , ri , and ii ) was run on each of the 459 MonteCarlo realisations, before combining the different correla-tions to get the weighted value via Eq. (21).The Monte Carlo and Poisson error bars are comparedin Fig. 12. The correlation coefficients of the correlationfunctions in different bins are plotted graphically in Fig. 13.From each Monte Carlo correlation function we com-pute the COSEBIs via Eq. (23) and use their covariancematrix in our subsequent likelihood analysis. In order totest whether our covariance has converged, meaning thatthe number of realisations is sufficient to not alter cosmo-logical constraints, we perform 3 likelihood analyses in σ vs. Ω m space varying the numbers of realisations from whichwe compute the covariance matrix (see Sec. 6 for detailedmethodology; for now we are just establishing convergenceof the covariance matrix). In Fig. 14 we show the 68 and95 per cent likelihood contours, i.e. the contours enclose thecorresponding fraction of the posterior probability (withinthe ranges of the parameters shown). We see that the con-tours hardly change when going from 300 to 400 realizationsand show no change at all when going from 400 to 459 reali-sations, hence the 459 Monte Carlo realizations are sufficientfor our likelihood analysis. The following additional contributions are added to theMonte Carlo covariance matrix (and if appropriate the the-ory result) described in Sec. 5.1.2.(i) The intrinsic alignment error was included followingSec. 3.2: the theory shear correlation function was reducedby a factor of 0.92, and an uncertainty of 4 per cent of thetheory was added to the covariance matrix, i.e. we add anintrinsic alignment contributionCov[ ξ i , ξ j ](intrinsic alignment) = 0 . ξ (th) i ξ (th) j , (27)where the theory curve (th) is obtained at the fiducial was not necessary for the SDSS Stripe 82 project, but was thesimplest choice given legacy codes available to us. To simplify bookkeeping, the actual implementation was thata sequence of 10 random numbers was generated, and a galaxywas assigned one of these numbers based on its coordinates in afine grid with 0.36 arcsec cells in ( α, δ ). σ [ ξ ( θ ) , M C ]/ σ [ ξ ( θ ) , P o i ss on ] θ (arcmin)Error bar comparison: MC/Poissonrr++rrxxii++iixx Figure 12. The ratio of error bars obtained by the Monte Carlomethod to those obtained by the Poisson method, for 10 angularbins. The four curves show either rr or ii band correlation func-tions, and either the ++ or ×× component. Note the rise in theerror bars at large values of the angular separation, due to modesampling variance. Correlation matrix: ww, 10 bins 0 5 10 15 20bin b i n -0.2 0 0.2 0.4 0.6 0.8 1 Figure 13. The matrix of correlation coefficients for the com-bined ( ww ) correlation functions in the 10 angular bins for whichthe correlation function is plotted in the companion figures. Thebin number ranges from 0–9 for ξ ++ ( θ ) and from 10–19 for ξ ×× ( θ ); all diagonal components are by definition equal to unity.Based on 459 Monte Carlo realisations. WMAP7 point. This covariance matrix includes perfect cor-relation between radial bins, implying that we treat this sys-tematic as being an effect with a fixed scaling with separa-tion, so the only degree of freedom is its amplitude.(ii) The stellar contamination was included followingSec. 3.5: the theory shear correlation function was reducedby a factor of 0.936, and an uncertainty of 3 per cent of the c (cid:13)000 The matrix of correlation coefficients for the com-bined ( ww ) correlation functions in the 10 angular bins for whichthe correlation function is plotted in the companion figures. Thebin number ranges from 0–9 for ξ ++ ( θ ) and from 10–19 for ξ ×× ( θ ); all diagonal components are by definition equal to unity.Based on 459 Monte Carlo realisations. WMAP7 point. This covariance matrix includes perfect cor-relation between radial bins, implying that we treat this sys-tematic as being an effect with a fixed scaling with separa-tion, so the only degree of freedom is its amplitude.(ii) The stellar contamination was included followingSec. 3.5: the theory shear correlation function was reducedby a factor of 0.936, and an uncertainty of 3 per cent of the c (cid:13)000 , 000–000 E. M. Huff et al. . . . . . . N(realizations)=459N(realizations)=400N(realizations)=300 σ Ω m Figure 14. Convergence test of the σ vs. Ω m parameter con-straints as a function of the number of Monte Carlo realizationsused to compute the covariance. The plot shows the 68 and 95 percent likelihood contours (however, the lower 95 per cent contoursare not visible). The covariance includes statistical errors only. theory was added to the covariance matrix, i.e. we add astellar contamination contributionCov[ ξ i , ξ j ](stellar contamination) = 0 . ξ (th) i ξ (th) j , (28)where the theory curve (th) is obtained at the fiducialWMAP7 point.(iii) The implied error from the redshift distribution un-certainty is derived from 402 realisations of the samplingvariance simulations as described in Sec. 3.4.2. We constructthe covariance matrix of the predicted E -mode COSEBIs.(iv) The shear calibration uncertainty was conservativelyestimated in Sec. 3.3 to be ± . ξ i , ξ j ](shear calibration) = 0 . ξ (th) i ξ (th) j . (29)(v) In Sec. 3.6, we described a procedure for includinguncertainty due to additive PSF contamination. Accordingto this procedure, the relevant systematics covariance matrixis related to the amplitude of the measured contaminationsignal:Cov[ ξ i , ξ j ](PSF contamination) = 0 . ξ sg ,i ξ sg ,j , (30)again assuming a fixed scaling with radius for this system-atic uncertainty. Since all entries scale together, we do notspuriously “average down” our estimate of the systematicerror by combining many bins.The final data vector and its covariance matrix (includ-ing all the statistical and systematic components) are givenin Tables A1 and A2. Note that given our procedure of ap-plying the systematic corrections to the theory, the datavector is the observed one without any such corrections forthe stellar contamination and intrinsic alignments contami-nation. With this in hand, we can estimate the significance of the E - and B -mode signals described in section 4.3. Theprobability that the COSEBI E -mode signal that we ob-serve is due to random chance given the null hypothesis (nocosmic shear) is 6 . × − . The probability of measuringour B -mode signal due to random chance given the null hy-pothesis of zero B modes is . 36, evidence that there is nosignificant B-mode power. Having described the measured cosmic shear two-pointstatistics, and shown that the systematic bias in this mea-surement is small compared with the statistical constraints,we now turn to the cosmological interpretation. We work inthe context of the flat ΛCDM parametrisation, taking wherenecessary the WMAP7 (Komatsu et al. 2011) constraints forour fiducial parameter values. To produce a cosmological interpretation of our measuredcosmic shear signal from our model framework, we requirea method to convert a vector of cosmological parametersinto a prediction of the observed cosmic shear signal. Due toprojection effects, we expect that a significant fraction of theobserved cosmic shear signal is produced by the clustering ofmatter on nonlinear scales, so a suitably accurate predictionalgorithm must ultimately rely on numerical simulations ofstructure formation.The prediction code used in our likelihood analysis is amodified version of the code described in Eifler (2011). Wecombine Halofit (Smith et al. 2003), an analytic approachto modeling nonlinear structure, with the Coyote UniverseEmulator (Lawrence et al. 2010), which interpolates the re-sults of a large suite of high-resolution cosmological simu-lations over a limited parameter space, to obtain the den-sity power spectrum. The derivation is a two-step process:First, we calculate the linear power spectrum from an ini-tial power law spectrum P δ ( k ) ∝ k n s employing the dewig-gled transfer function of Eisenstein & Hu (1998). The non-linear evolution of the density field is incorporated usingHalofit. In order to simulate w CDM models we follow thescheme implemented in icosmo (Refregier et al. 2011), in-terpolating between flat and open cosmological models tomimic Quintessence cosmologies (see Schrabback et al. 2010for more details). In a second step, we match the Halofitpower spectrum to the Coyote Universe emulator (version1.1) power spectrum, which emulates P δ over the range0 . k . h/ Mpc within 0 z k and z . However, evenoutside the range of the Emulator, we rescale the Halofitpower spectrum with a scale factor P δ (Coyote)/ P δ (Halofit)calculated at the closest point in parameter space (cosmo-logical parameters, k , and z ) where the Emulator gives re-sults. Outside the range of the Emulator, the accuracy of this“Hybrid” density power spectrum is of course worse than 1per cent, however it should be a significant improvement c (cid:13) , 000–000 osmic shear in SDSS . . . . . . Coyote Universe calibratedHalofit σ Ω m Figure 15. The 68 and 95 per cent likelihood contours of thecombined data vector including a full treatment of systematicswhen using the Halofit prediction code (dashed) and when usingthe Coyote Universe-calibrated prediction code (solid). The redlines correspond to the best-fitting value of σ for a given Ω m .The dot indicates the WMAP7 best-fitting values. over a density power spectrum from Halofit only. From theso-derived density power spectrum we calculate the shearpower spectrum via Eq. (8) and the shear-shear correlationfunction via Eq. (7). As a final step, we transform thesepredicted correlation functions to the COSEBI basis as de-scribed above in Sec. 4.3.For our final results in the (Ω m , σ ) likelihood analysis,we used both prediction codes; the results are compared inFig. 15, where they are seen to agree to much better than1 σ . We therefore conclude that uncertainty in the theorypredictions is sub-dominant to the other sources of system-atic error, and to the statistical error. For our primary science results, we use the measured 5COSEBI modes (see Fig. 11, left panel). As a first step wewant to determine the number of COSEBI modes that needto be included in our likelihood analysis. In Fig. 16 we showa likelihood analysis in the σ -Ω m parameter space varyingthe number of modes in the data vector. We find that thereis hardly a change in the likelihood contours when goingfrom 4 to 5 modes; we therefore conclude that 5 modes isa sufficient number to capture the cosmological informationencoded in our data set.As shown in Eifler et al. (2008), the information con-tent of the aperture mass dispersion can be greatly improvedwhen including 1 data point of the shear-shear correlationfunction ξ + into the data vector; here we adopt this conceptfor the COSEBIs. The basic idea is that the data point ofthe correlation function is sensitive to scales of the powerspectrum to which the COSEBIs are insensitive. We incor-porate only a single data point of the correlation functionas this is sufficient to capture the bulk of the additional in-formation while simultaneously minimising possible B-modecontamination. . . . . . . σ Ω m Figure 16. Convergence test of the σ vs. Ω m parameter con-straints as a function of number of COSEBI modes in the datavector. The plot shows the likelihood contours enclosing 68 and95 per cent of the posterior distribution. (The lower boundingcurve for the 95 per cent contours is not visible on the plot.) Thecovariance contains statistical errors only. The dot indicates theWMAP7 best-fitting values. In order to determine the optimal scale of the data pointthat is to be included, we consider 10 bins of ξ + ranging from1.3 to 97.5 arcmin and perform 10 likelihood analyses for acombined data vector consisting of 5 COSEBI modes andone additional data point of ξ + . We quantify the informationcontent through the so-called q figure of merit ( q -FoM) q = p | Q | , where Q ij = Z d π p ( π | d ) ( π i − π f i )( π j − π f j ) , (31) π = (Ω m , σ ) is the parameter vector, p ( π | d ) is the posteriorlikelihood at this parameter point, and π f i denotes the fidu-cial parameter values. If the likelihood in parameter space(i.e. the posterior probability) is Gaussian, the q -FoM cor-responds to the more common Fisher matrix based figure ofmerit f = 1 / |√ F | . The Fisher matrix F can be interpretedas the expectation value of the inverse parameter covarianceevaluated at the maximum likelihood estimate parameterset, which in our ansatz corresponds to the fiducial param-eters. Mathematically we can express this equivalence as f = 1 p | F | = p | C π | = p | Q | = q . (32)Since the assumption of a Gaussian posterior is clearly vio-lated in the σ -Ω m parameter space, we perform a full like-lihood analysis and calculate q to quantify the size of thelikelihood. Note that smaller q -FoM is “better.”We varied the angular scale (in arcmin) of the added ξ + ( θ ) data point, and found a minimal q -FoM at θ = 37 . ξ + datapoint henceforth. Note that this analysis uses a simulatedinput data vector in order to avoid biases from designing astatistical test based on the observed data. The constraintscoming from the various possible data vectors – the COSE-BIs, the COSEBIs supplemented with a single ξ + point, and c (cid:13)000 Convergence test of the σ vs. Ω m parameter con-straints as a function of number of COSEBI modes in the datavector. The plot shows the likelihood contours enclosing 68 and95 per cent of the posterior distribution. (The lower boundingcurve for the 95 per cent contours is not visible on the plot.) Thecovariance contains statistical errors only. The dot indicates theWMAP7 best-fitting values. In order to determine the optimal scale of the data pointthat is to be included, we consider 10 bins of ξ + ranging from1.3 to 97.5 arcmin and perform 10 likelihood analyses for acombined data vector consisting of 5 COSEBI modes andone additional data point of ξ + . We quantify the informationcontent through the so-called q figure of merit ( q -FoM) q = p | Q | , where Q ij = Z d π p ( π | d ) ( π i − π f i )( π j − π f j ) , (31) π = (Ω m , σ ) is the parameter vector, p ( π | d ) is the posteriorlikelihood at this parameter point, and π f i denotes the fidu-cial parameter values. If the likelihood in parameter space(i.e. the posterior probability) is Gaussian, the q -FoM cor-responds to the more common Fisher matrix based figure ofmerit f = 1 / |√ F | . The Fisher matrix F can be interpretedas the expectation value of the inverse parameter covarianceevaluated at the maximum likelihood estimate parameterset, which in our ansatz corresponds to the fiducial param-eters. Mathematically we can express this equivalence as f = 1 p | F | = p | C π | = p | Q | = q . (32)Since the assumption of a Gaussian posterior is clearly vio-lated in the σ -Ω m parameter space, we perform a full like-lihood analysis and calculate q to quantify the size of thelikelihood. Note that smaller q -FoM is “better.”We varied the angular scale (in arcmin) of the added ξ + ( θ ) data point, and found a minimal q -FoM at θ = 37 . ξ + datapoint henceforth. Note that this analysis uses a simulatedinput data vector in order to avoid biases from designing astatistical test based on the observed data. The constraintscoming from the various possible data vectors – the COSE-BIs, the COSEBIs supplemented with a single ξ + point, and c (cid:13)000 , 000–000 E. M. Huff et al. . . . . . . COSEBIs + ξ + (37.8 arcmin)shear−shear correlation functionCOSEBIs σ Ω m Figure 17. The likelihood contours of the combined data vec-tor (solid), the shear-shear correlation function (dashed), and theCOSEBIs (dotted) data vector to illustrate how much informa-tion is gained when including the additional data point. Note thatthe COSEBIs’ lower 95 per cent contour is outside the consideredregion. The dot indicates the WMAP7 best-fitting values. COSEBI/ ξ + correlations [stat] E E E E E ξ + ( ’ ) E1E2E3E4E5 ξ + (38’) -0.2 0 0.2 0.4 0.6 0.8 1 COSEBI/ ξ + correlations [stat+sys] E E E E E ξ + ( ’ ) E1E2E3E4E5 ξ + (38’) -0.2 0 0.2 0.4 0.6 0.8 1 Figure 18. The correlation matrix of the COSEBI modes 1–5(“E1...E5” in the figure) and ξ + (38 ′ ). The left panel shows onlythe statistical (Monte Carlo) errors, and the right panel includesthe systematics as well. the full shear correlation function – are compared in Fig. 17.They are not identical, which is expected since they weightthe data in different ways, but are consistent with each other.The COSEBI modes are highly correlated with eachother, and they are correlated to a lesser extent with ξ + at 38 arcmin. The correlation matrix is shown in Fig. 18,and the corresponding covariance matrix is tabulated in theAppendix in Table A2. We perform all of our fits to a standard five-parameterΛCDM model . For the initial likelihood analysis, we fix n s ,Ω b h , Ω m h , and w at their fiducial best-fit WMAP7 val-ues (Komatsu et al. 2011), and vary σ . The upper panel of The optical depth to reionization τ is a sixth parameter im-plicitly included in the WMAP7 chains, but with no effect on thelensing shear correlation function. Fig. 19 shows the likelihood of σ with all other parametersfixed, with a value at the peak and 68 per cent confidenceinterval of 0 . +0 . − . . For a survey of this size and depth,the constraints are comparable to the statistically achievableconfidence limits.We also perform a likelihood analysis fixing three pa-rameters, and varying Ω m and σ simultaneously, as thesetwo parameters are much more sensitive to the measuredcosmic shear signal than the others. The resulting two-dimensional constraints are shown in the bottom panel ofFig. 19. Our 68 per cent confidence limits on the degen-erate product σ (cid:0) Ω m . (cid:1) . are 0 . +0 . − . for the CoyoteUniverse prediction code (see Fig. 19, solid red line), and σ (cid:0) Ω m . (cid:1) . = 0 . +0 . − . for the Halofit prediction code (seeFig. 19, dashed red line).We show the effects of removing each systematic errorcorrection, Fig. 19 also shows, for both the one- and two-dimensional analyses, the impact of systematic error correc-tions. The combined effects of these uncertainties are clearlysubstantially smaller than the statistical error on the ampli-tude of the shear signal.Finally, we adopt the WMAP7 likelihoods as priors, andevaluate our likelihood at each link in the WMAP7 Markovchain. For each chain element, we assign a weight equal toour likelihood function evaluated at the parameter vectorfor that chain element. For each of the parameter constraintplots shown here, we first assign each Markov Chain MonteCarlo (MCMC) chain element to a point on a regular grid inthe parameter space; the value of the marginalised likelihoodat each grid-point, H i,j is then the sum of our likelihoodweights over the MCMC chain elements at the ( i, j ) grid-point, H i,j = X k I k ( i, j ) L k , (33)where the indicator function I k ( i, j ) is equal to unity whenthe ( i, j ) grid-point in parameter space is nearest the k thchain element, and zero otherwise. The likelihood L k foreach chain element is evaluated in the usual way as: L k = exp − ¯ d Tk C − ¯ d k ! . (34)Here C is the full covariance matrix for the measurement,incorporating both the statistical and systematic uncertain-ties, and the normalization is arbitrary. The data vector ¯ d k isthe extended COSEBI vector described above; where shown,the WMAP7 priors are simply this sum with L k = 1 for eachpoint.We estimate the detection significance for the finalsignal, the difference √− 2∆ log L between the highest-likelihood Markov Chain element for both the ΛCDM and w CDM models and the likelihood evaluated with no sig-nal. The 1 σ detection significances for these two models are2.64 and 2.88, respectively. This is not the significance ofthe detection of cosmic shear (as in Sec. 5.2), but rather ameasurement of the likelihood of these two models given thecombination of WMAP7 priors with this experiment.In Fig. 20, we show marginalized posterior likelihoodsin the case of fixed ΛCDM (i.e., w = − 1) for Ω m h , Ω b h , n s , and σ . The results with a free equation of state of darkenergy (wCDM) are in Fig. 21. Our measurement provides c (cid:13) , 000–000 osmic shear in SDSS . . . . . σ po s t e r i o r p r obab ili t y IA+ShearCal+PSF+Masking systematicsIA+ShearCal+PSFIA+ShearCalNo systematics . . . . . . IA+ShearCal+PSF+Masking systematicsIA+ShearCal+PSFIA+ShearCalNo systematics σ Ω m Figure 19. The effect of systematic errors in the 1-D likelihood of σ (upper panel) and in the 2-D constraints (68 per cent likelihoodcontours only) in the σ − Ω m plane (lower panel). The solidcurve shows our final analysis, while the other curves show resultsincluding subsets of the systematic errors. The dot-dashed curvelabeled “no systematics” shows only the statistical errors, withoutany systematic error corrections either to the theory or to thecovariance matrix. The dot indicates the WMAP7 best-fittingvalues. some additional constraints beyond those from WMAP7 onthese parameters. In particular, the low amplitude of themeasured shear signal rules out some of the previously al-lowed volume of Ω m h and σ WMAP7 constraints. Using coadded imaging constructed from SDSS Stripe 82data, we constructed a weak lensing catalogue of 1 328 885galaxies covering 168 square degrees (Paper I), and showedthat the additive shear systematics arising from the PSFare negligible compared to the cosmic shear signal. In thispaper, we carried out a cosmic shear measurement that re-sulted in a 20 per cent constraint on σ (with all other cos-mological parameters fixed). This adds constraining powerbeyond that from WMAP7, and serves as an importantindependent data point on the amplitude of the matterpower spectrum at late times. In particular, the primary CMB anisotropies presently provide only a modest con-straint on Ω m h , and (due to the effect of matter den-sity on the growth of structure) there is then an elon-gated allowed region in the (Ω m h , σ ) plane; see Fig. 20.The WMAP7-allowed region is ideally oriented for lens-ing to play a role: the lensing signal at the high-Ω m h ,high- σ end of the ellipse leads to a much higher lensingsignal than low Ω m h , low σ . The low amplitude of cos-mic shear observed in this paper eliminates the high-Ω m h ,high- σ solutions, and leads to a WMAP7+SDSS lensingsolution of σ = 0 . +0 . − . (1 σ ) +0 . − . (2 σ ) and Ω m h =0 . +0 . − . (1 σ ) +0 . − . (2 σ ); the 2 σ error ranges are respec-tively 14 and 17 per cent smaller than for WMAP7 alone.We have also carefully evaluated other sources of un-certainty such as the source redshift distribution, intrinsicalignments, and shear calibration, to ensure that our mea-surement is dominated by statistical errors rather than sys-tematic errors. This achievement is important when consid-ering that (i) the SDSS data were never designed with thisapplication in mind, and indeed includes several features(e.g. the minimal amount of cross-scan dithering) that causesignificant difficulty, and (ii) with the multitude of upcom-ing multi-exposure lensing surveys in the next few years, itis important to cultivate new data analysis techniques (suchas the one used here) that are capable of producing homo-geneous data with tight control over PSF anisotropies. Asa quantitative measure of the extent of PSF correction pos-sible with SDSS data, we take the RMS residual spuriousshear at a particular scale estimated from the star-galaxycorrelations, γ rms , eq ( θ ) = p R psf ξ + , sg ( θ ) R . (35)From Fig. 6, we see that this is ∼ × − at the small-est scales (1–6 arcmin), is < − at scales θ > . . × − in the final bin (1.2–2.0 degrees). There is almost no difference between the ++ and ×× sig-nals, suggesting that the spurious additive ellipticity signalcontains similar amounts of E - and B -modes ; somethingsimilar was seen in the SDSS single-epoch data via run-by-run comparisons of ellipticity measurements on the samegalaxies (Mandelbaum et al. 2006b, Fig. 8). This is goodnews for the use of the B -mode as a diagnostic of PSF sys-tematics, although an understanding of the generality of thispattern remains elusive.A major lesson learned from this project is the impor-tance of masking bias , in which the intrinsic orientation of agalaxy affects whether it falls within the survey mask. Thisis likely the main reason why we had to implement the h e i projection. While we have clearly not exhausted the range ofoptions for removing this bias at the catalogue level, futuresurveys should be designed to produce more uniform dataquality via an appropriate dithering strategy and suppressthe masking bias at the earliest stages of the analysis.Our major limitation in the end was the source numberdensity, which was driven by the fact that our PSF-matchingprocedure was limited by the worst seeing in the images that We used R psf = 0 . R = 1 . Recall that ξ ++ ( θ ) − ξ ×× ( θ ) and P E ( ℓ ) − P B ( ℓ ) are J Hankeltransforms of each other.c (cid:13) , 000–000 E. M. Huff et al. Figure 20. The cosmological parameter constraints using the extended COSEBI data vector, fixing the dark energy equation of state w at − 1, but allowing all other parameters to vary. Off-diagonal panels show joint two-dimensional constraints after marginalization overall the other parameters, which are shown. For these, the red contours show the WMAP7 priors containing 68.5 and 95.4 per cent ofthe posterior probability. The black contours are the same but for WMAP7+SDSS lensing. Diagonal panels show the fully-marginalizedone-dimensional posterior distribution for each parameter; for these panels, the red (dashed) contours show the marginalized WMAP7constraints. we use, and therefore we had to eliminate the images withseeing worse than the median. This means that the coaddswere not as deep as they could have been, and the final ef-fective seeing was 1.31 arcsec (full-width half maximum). Inprinciple this will be an obstacle to applying this techniquein the future, but in fact, that statement depends on context.For example, for a survey such as HSC or LSST where weexpect typically ∼ . r and i -band imagingthat will be used for shape measurement, it is conceivablethat nearly all images intended for lensing will have seeingin the 0.6–0.8 arcsec range. In that context, a PSF-matchedcoadd that has the rounding kernel applied may actuallynot result in much loss of information about the shapes ofmost useful galaxies, and will have the advantage of the re-moval of PSF anisotropies. Moreover, even for surveys forwhich the loss of information that results from this methodmay not be suitable for the final cosmological analysis, thismethod may still serve as a useful diagnostic of the additivePSF systematics. ACKNOWLEDGMENTS We thank Alexie Leauthaud for providing faint COS-MOS galaxy postage stamp images for simulation purposes.E.M.H. is supported by the US Department of Energy’s Of-fice of High Energy Physics (DE-AC02-05CH11231). Dur- ing the period of work on this paper, C.H. was supportedby the US Department of Energy’s Office of High EnergyPhysics (DE-FG03-02-ER40701 and de-sc0006624), the USNational Science Foundation (AST-0807337), the Alfred P.Sloan Foundation, and the David & Lucile Packard Foun-dation. R.M. was supported for part of the duration of thisproject by NASA through Hubble Fellowship grant c (cid:13) , 000–000 osmic shear in SDSS Figure 21. The cosmological parameter constraints using the extended COSEBI data vector, varying all five parameters. Off-diagonalpanels show joint two-dimensional constraints after marginalization over all the other parameters, which are shown. 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A., Kirkman D., Dell’Antonio I.,Bernstein G., 2003, MNRAS, 341, 1311Zhang J., 2011, MNRAS, 414, 1047 c (cid:13) , 000–000 osmic shear in SDSS Table A1. Our data vector. The first five elements are COSEBImode amplitudes; the final is the correlation function averaged inthe range 29 . θ . Table A2. The covariance matrix for the data vector shown intable A1.Data vector index Data vector index Covariance0 0 3.37161E-200 1 4.67637E-200 2 4.00484E-200 3 2.49916E-200 4 9.84257E-210 5 3.01770E-171 1 1.06383E-191 2 1.19226E-191 3 8.39508E-201 4 3.86519E-201 5 1.82344E-162 2 1.99923E-192 3 1.87469E-192 4 1.12196E-192 5 5.07790E-163 3 2.56568E-193 4 2.13363E-193 5 8.02118E-164 4 2.67774E-194 5 5.67797E-165 5 3.68112E-11 APPENDIX A: THE DATA VECTOR ANDCOVARIANCE MATRIX. Here we reprint the data vector and covariance matrix usedin this measurement. The code used to project the correla-tion function onto the COSEBI basis functions is availablefrom the authors upon request. c (cid:13)000