Dark Energy Survey Year 1 Results: The Impact of Galaxy Neighbours on Weak Lensing Cosmology with im3shape
S. Samuroff, S.L. Bridle, J. Zuntz, M.A. Troxel, D. Gruen, R.P. Rollins, G.M. Bernstein, T.F. Eifler, E.M. Huff, T. Kacprzak, E. Krause, N. MacCrann, F.B. Abdalla, S. Allam, J. Annis, K. Bechtol, A. Benoit-Levy, E. Bertin, D. Brooks, E. Buckley-Geer, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, M. Crocce, C.B. D'Andrea, L.N. da Costa, C. Davis, S. Desai, P. Doel, A. Fausti Neto, B. Flaugher, P. Fosalba, J. Frieman, J. Garcia-Bellido, D.W. Gerdes, R.A. Gruendl, J. Gschwend, G. Gutierrez, K. Honscheid, D.J. James, M. Jarvis, T. Jeltema, D. Kirk, K. Kuehn, S. Kuhlmann, T.S. Li, M. Lima, M.A.G. Maia, M. March, J.L. Marshall, P. Martini, P. Melchior, F. Menanteau, R. Miquel, B. Nord, R.L.C. Ogando, A.A. Plazas, A. Roodman, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, I. Sevilla-Noarbe, E. Sheldon, M. Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, G. Tarle, D. Thomas, D.L. Tucker
DDES 2017-0727FERMILAB-PUB-17-291-E
MNRAS , 000–000 (0000) Preprint 7 August 2017 Compiled using MNRAS L A TEX style file v3.0
Dark Energy Survey Year 1 Results: The Impact of GalaxyNeighbours on Weak Lensing Cosmology with IM SHAPE
S. Samuroff (cid:63) , S. L. Bridle , J. Zuntz , M. A. Troxel , , D. Gruen , † , R. P. Rollins , G. M. Bernstein ,T. F. Eifler , E. M. Huff , T. Kacprzak , E. Krause , N. MacCrann , , F. B. Abdalla , , S. Allam ,J. Annis , K. Bechtol , A. Benoit-L´evy , , , E. Bertin , , D. Brooks , E. Buckley-Geer , A.Carnero Rosell , , M. Carrasco Kind , , J. Carretero , M. Crocce , C. B. D’Andrea , L. N. daCosta , , C. Davis , S. Desai , P. Doel , A. Fausti Neto , B. Flaugher , P. Fosalba , J. Frieman , ,J. Garc´ıa-Bellido , D. W. Gerdes , , R. A. Gruendl , , J. Gschwend , , G. Gutierrez ,K. Honscheid , , D. J. James , , M. Jarvis , T. Jeltema , D. Kirk , K. Kuehn , S. Kuhlmann ,T. S. Li , M. Lima , , M. A. G. Maia , , M. March , J. L. Marshall , P. Martini , , P. Melchior ,F. Menanteau , , R. Miquel , , B. Nord , R. L. C. Ogando , , A. A. Plazas , A. Roodman , ,E. Sanchez , V. Scarpine , R. Schindler , M. Schubnell , I. Sevilla-Noarbe , E. Sheldon , M. Smith ,M. Soares-Santos , F. Sobreira , , E. Suchyta , G. Tarle , D. Thomas , D. L. Tucker (DES Collaboration) Author affiliations are listed at the end of the paper.
ABSTRACT
We use a suite of simulated images based on Year 1 of the Dark Energy Survey to explore theimpact of galaxy neighbours on shape measurement and shear cosmology. The H
OOPOE im-age simulations include realistic blending, galaxy positions, and spatial variations in depthand PSF properties. Using the IM SHAPE maximum-likelihood shape measurement code,we identify four mechanisms by which neighbours can have a non-negligible influence onshear estimation. These effects, if ignored, would contribute a net multiplicative bias of m ∼ . − . in the DES Y1 IM SHAPE catalogue, though the precise impact will bedependent on both the measurement code and the selection cuts applied. This can be reducedto percentage level or less by removing objects with close neighbours, at a cost to the effectivenumber density of galaxies n eff of 30%. We use the cosmological inference pipeline of DESY1 to explore the cosmological implications of neighbour bias and show that omitting blend-ing from the calibration simulation for DES Y1 would bias the inferred clustering amplitude S ≡ σ (Ω m / . . by σ towards low values. Finally, we use the H OOPOE simulations totest the effect of neighbour-induced spatial correlations in the multiplicative bias. We find theimpact on the recovered S of ignoring such correlations to be subdominant to statistical errorat the current level of precision. Key words: cosmological parameters - cosmology: observations - gravitational lensing: weak- galaxies: statistics
A standard and well tested prediction of General Relativity is thata concentration of mass will distort the spacetime around it, andthus produce a curious phenomenon called gravitational lensing. (cid:63) [email protected] † NASA Einstein Fellow
The most obvious manifestation is about massive galaxy clusters,where background galaxies can be elongated into cresent-shapedarcs. So-called strong lensing of galaxies was first observed in thelate 1980s and has been confirmed many times since. A subtler, butfrom a cosmologist’s perspective more powerful, consequence ofgravitational lensing is that background fluctuations in the densityof dark matter will induce coherent distortions to photons’ paths.This effect is known as cosmic shear, and it was first detected c (cid:13) a r X i v : . [ a s t r o - ph . C O ] A ug S. Samuroff, S. L. Bridle, J. Zuntz et al by four groups at around the same time close to two decades ago(Bacon et al. 2000; Van Waerbeke et al. 2000; Kaiser et al. 2000;Wittman et al. 2000). Cosmic shear has the potential to be the sin-gle most powerful probe in the toolbox of modern cosmology. Thespatial correlations due to lensing are a direct imprint of the largescale mass distribution of the Universe. Thus it allows one to studythe total mass of the Universe and the growth of structure withinit (Maoli et al. 2001; Jarvis et al. 2006; Massey et al. 2007; Kil-binger et al. 2013; Heymans et al. 2013; Abbott et al. 2016; Jeeet al. 2016; Hildebrandt et al. 2017; K¨ohlinger et al. 2017), or tomap out the spatial distribution of dark matter on the sky (eg Kaiser1994; Van Waerbeke et al. 2013; Chang et al. 2015). As a probe ofboth structure and geometry, cosmic shear is also attractive as amethod for shedding light on the as yet poorly understood compo-nent of the Universe known as dark energy (Albrecht et al. 2006;Weinberg et al. 2013). Alternatively, lensing will allow us to placeever more stringent tests of our theories of gravity (Simpson et al.2013; Harnois-D´eraps et al. 2015; Brouwer et al. 2017). Is also the-oretically very clean, responding directly to the power spectrum ofdark matter, which is affected by baryonic physics only on smallscales, and avoids recourse to poorly-understood phenomenologi-cal rules. Indeed galaxy number density enters only at second orderas a weighting of the observed shear due to the fact that one canonly sample the shear field where there are real galaxies (Schmidtet al. 2009).Though well modelled theoretically, cosmic shear is techni-cally highly challenging to measure; as with all these probes it isnot without its own sources of systematic error. It also cannot bereiterated too many times that the shear component of even themost distant galaxy’s shape is subdominant to noise by an order ofmagnitude. Indeed, the ambitions of the current generation of cos-mology surveys will require sub-percent level uncertainties (bothsystematic and statistical) on what is already a tiny cosmologicalellipticity component g ∼ . .It was realised early on how significant the task of translat-ing photometric galaxy images into unbiased shear measurementswould be. In response came a series of blind shear measurementchallenges, designed to review, test and compare the best methodsavailable. The first of these, called STEP1 (Heymans et al. 2006)grew out of a discussion at the 225th IAU Symposium in 2004.The exercise was based around a set of simple S KY M AKER sim-ulations (Bertin & Fouqu´e 2010), which were designed to mimicground based observations but with analytic galaxies and PSFs andconstant shear. The algorithms at this point represented a first waveof shear measurement codes and included several moments-basedalgorithms (Kaiser et al. 1995; Kuijken 1999; Rhodes et al. 2001),some early forward modelling methods (Bridle et al. 2002), as wellas a technique called shapelets, which models a light profile as a setof 2D basis functions (Bernstein & Jarvis 2002; Refregier & Bacon2003).The simulations and the codes themselves steadily grew incomplexity. STEP2 was followed by series of GREAT challenges(Massey et al. 2007; Bridle et al. 2009; Kitching et al. 2010; Man-delbaum et al. 2014), which focused on different aspects of shapemeasurement bias and have been essential in quantifying a num-ber of significant effects. In recent years the drive to find evermore accurate ways to measure shear has intensified, with manynovel approaches being suggested. For example Fenech Conti et al.(2017) use a form of self-calibration, which repeats the shape mea-surement on a test image based on the best-fitting model for eachgalaxy. A related approach, named metacalibration, involves de-riving corrections to the galaxy shape measurements directly from the data, using modified copies of the image with additional shear(Huff & Mandelbaum 2017; Sheldon & Huff 2017). More advancedmoments-based approaches include the BFD method (Bernstein &Armstrong 2014), which derives a prior on the ensemble elliptic-ity distribution using deeper fields, and SNAPG (Herbonnet et al.2017), a similar approach which builds ensemble shear estimatesusing shear nulling.This paper is intended as a companion study to Zuntz et al.(2017) (Z17), where we present two shear catalogues derived fromDES Y1 dataset. It is also presented alongside a raft of other pa-pers, which use both catalogues and show them to be consistent ina number of different scientific contexts (Troxel et al. 2017; Pratet al. 2017; Chang et al. 2017; DES Collaboration et al. 2017) Con-taining 22 million and 35 million galaxies respectively, these cat-alogues are the product of two independent maximum likelihoodcodes. The first, called IM SHAPE , implements simultaneous fitsusing multiple models and we calibrate externally using simula-tions. The second implements a Gaussian model fitting algorithm,supplemented by shear response corrections using
METACALIBRA - TION . Whereas in Z17 we focus on the catalogues themselves, pre-senting a raft of calibration tests and a broad overview of the value-added data products, here we use the same resources to explorea narrower topic: the impact of image plane neighbours on shearmeasurement. Specifically we use the image simulations describedin Z17, from which the Y1 IM SHAPE calibration is derived, to ex-plore the mechanisms for neighbour bias, and then propagate the re-sults to mock shear two-point data to investigate the consequencesfor weak lensing cosmology. The results presented in this paper willbe somewhat dependent on the choice of measurement algorithm,selection cuts and the configuration of the object detection code.Unlike previous studies on this subject, however, we make use of ahighly realistic simulation and measurement pipeline. Our choiceson each of aspects are realistic, if not unique, for a leading-edgecosmology analysis.It is worth remarking, however, that the tests described in thispaper make use of IM SHAPE only, and should not be assumed toapply generically to its sister Y1
METACALIBRATION catalogue.A complementary set of tests using
METACALIBRATION are pre-sented in § OOPOE sim-ulator. We test the earlier predictions under more typical observingconditions in Section 5, and extend them into a quantitative set ofresults using the more extensive Y1 H
OOPOE dataset. Section 6then presents a numerical analysis designed to test the cosmolog-ical implications of neighbour bias of the nature and magnitudefound in our simulations. We conclude in Section 7.
The problem of shape measurement is far more intricate than itmight first appear. Any cosmological analysis based on cosmicshear is reliant on a series of technical choices, which can havea non-trivial impact on measurement biases, precision and cosmo-logical sensitivity. Specifically we must choose (a) how to parame-terize each galaxy’s shape, and which measurement method to useto estimate it, (b) what selection criteria are needed to obtain dataof sufficiently high quality for cosmology and (c) how biased is the
MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours measurement and what correction is needed? These choices shouldbe made on a case-by-case basis, since the optimal solutions aredependent on a number of survey-specific factors. We discuss eachbriefly in turn below. IM SHAPE
The shape measurements upon which the following analyses arebased make use of the maximum likelihood model fitting code IM SHAPE (Zuntz et al. 2013). It is a well tested and understoodalgorithm, which has since been used in a range of lensing studies(Abbott et al. 2016; Whittaker et al. 2015; Kacprzak et al. 2016;Clampitt et al. 2017). It was also one of two codes used to produceshear catalogues in the Science Verification (SV) stage and Year1 of the Dark Energy Survey. We refer the reader to Jarvis et al.(2016) (hereafter J16) and Z17 for the most recent modifications tothe code.We use the definition of the flux signal-to-noise ratio of Z17,J16 and Mandelbaum et al. (2015): S/N ≡ (cid:32) N pix (cid:80) i =1 f m i f im i /σ i (cid:33)(cid:32) N pix (cid:80) i =1 f m i f m i /σ i (cid:33) . (1)The indices i = (1 , ...N pix ) run over all pixels in a stack of im-age cutouts at the location of a galaxy detection. The model pre-diction and observed flux in pixel i are denoted f m i and f im i re-spectively and σ i is the RMS noise. This signal-to-noise measureis maximised when the differences between the model and the im-age pixel fluxes are small. Note that if the best-fitting model f m is identical for two different postage stamps, S/N will favour theimage with the greater total flux.A useful size measure, referred to as R gp /R p is defined asthe measured Full Width at Half Maximum (FWHM) of the galaxyafter PSF convolution, normalised to the PSF FWHM. Real galaxyimages are are not perfectly symmetric (i.e. size is not independentof azimuthal angle about a galaxy’s centroid), and single-numbersize estimates are obtained by circularising (azimuthally averag-ing) the galaxy profile and computing the weighted quadrupole mo-ments of the resulting image. For each galaxy we take the meanmeasured size across exposures. There are many ways bias can enter an ensemble shear estimatebased on a population of galaxies. Although the list is not exhaus-tive, a handful of mechanisms are particularly prevalent, and havebeen extensively discussed in the literature. • Noise Bias:
On addition of pixel noise to an image, the best-fitting parameters of a galaxy model will not scale linearly withthe noise variance. This is as an estimator bias as much as a mea-surement bias, and results in an asymmetric, skewed likelihood sur-face (Hirata & Seljak 2003; Refregier et al. 2012; Kacprzak et al.2012; Miller et al. 2013). Any code which uses the point statis-tics of the distribution (either mean or maximum likelihood) as asingle-number estimates of the ellipticity results in a bias. This is https://bitbucket.org/joezuntz/im3shape-git true even in the idealised case where the galaxy we are fitting canbe perfectly decribed by our analytic light profile. The bias is sen-sitive to the noise levels and also the size and flux of the galaxy,and thus is specific to the survey and galaxy sample in question.For likelihood-based estimates one solution would be to imposea prior on the ellipticity distribution and propagate the full poste-rior. However, the results can become dependent on the accuracy ofthat prior, and such codes require cautious testing using simulations(Bernstein & Armstrong 2014; Simon & Schneider 2016) • Model Bias:
In reality galaxies are not analytic light profileswith clear symmetries. For the purposes of model-fitting, however,we are constrained to use models with a finite set of parameters.A model which does not allow sufficient flexibility to capture therange of morphological features seen in the images will producebiased shape measurements (Lewis 2009; Voigt & Bridle 2010;Kacprzak et al. 2014). • Selection Bias:
Even if we were to devise an ideal shape mea-surement algorithm, capable of perfectly reconstructing the his-togram of ellipticities in a certain population of galaxies, our at-tempts to estimate the cosmological shear could still be biased. Ifa measurement step prefers rounder objects or those with a partic-ular orientation, the result would be a net alignment that could bemistaken as having cosmological origin. In practice selection biascommonly arises from imperfect correction of PSF asymmetries(eg Kaiser et al. 2000; Bernstein & Jarvis 2002), and the fact thatmany detection algorithms fail less frequently on rounder galaxies(Hirata & Seljak 2004). It is such effects that make post facto qual-ity cuts on quantities such as signal-to-noise or size (both of whichcorrelate with ellipticity) particularly delicate. • Neighbour bias:
In practice, galaxies in photometric surveyslike DES are not ideal isolated objects. Rather, they are extractedfrom a crowded image plane using imperfect deblending algo-rithms. The term “neighbour bias” refers to any biases in the re-covered shear arising from the interaction between galaxies in theimage plane. This can include both the direct impact on the per-galaxy shapes (e.g. Hoekstra et al. 2017) and changes in the selec-tion function (e.g. Hartlap et al. 2011). Neighbour bias is the subjectof relatively few previous studies, and is the focus of this paper.
To develop a picture of how image plane neighbours affect shearestimates with IM SHAPE , we build a simplified toy model. Using
GALSIM we generate a × pixel postage stamp containing asingle exponential disc profile convolved with a tiny sphericallysymmetric PSF (though we confirm that our results are insensi-tive to the exact size of the PSF). We can then apply a small shearalong one coordinate axis prior to convolution and use IM SHAPE to fit the resulting image. In the absence of noise or model biasthe maximum of the likelihood of the measured parameters coin-cides exactly with the input values. The basic setup then has fouradjustable parameters: the flux and size of the galaxy plus two el-lipticity components, denoted f c , r c , g tr and g tr . Unless otherwisestated we fix these to the median values measured from the DES Y1 IM SHAPE catalogue. We do not model miscentering error betweenthe true galaxy centroid and the stamp centre.It is worth noting that neither this basic model nor the more https://github.com/GalSim-developers/GalSimMNRAS , 000–000 (0000) S. Samuroff, S. L. Bridle, J. Zuntz et al complex simulations that follow attempt to model spatial correla-tions in shear. Even at different redshifts, a real neighbour-centralpair share some portion of their line of sight. These spatial cor-relations will amplify the impact of blending, and are worthy offuture investigation. This is, however, likely a second-order effectof neighbours, and we postpone such study to a future date.
To explore the interaction in single neighbour-galaxy instances weintroduce a second galaxy into the postage stamp, convolved withthe same nominal PSF. This adds four more model parameters:neighbour size r n , flux f n , radial distance from the stamp centre d gn and azimuthal rotation angle relative to the x coordinate axis θ . At this stage the neighbour has zero ellipticity.We show this setup at three neighbour positions in Fig. 1. Un-der zero shear, the system has perfect rotational symmetry, and themeasured ellipticity magnitude ˜ g ( θ | g tr = 0) is independent of θ As a first exercise, we generate a circular central galaxy with a cir-cular Gaussian neighbour, which is gradually shifted outwards fromthe stamp centre. Following the usual convention for galaxy-galaxylensing, tangential shear is defined such that it is negative whenthe major axis of the measured shape is oriented radially towardsthe neighbour. The measured two-component ellipticity shown bythe solid and dot-dashed lines in Fig. 2. The decline in the mea-sured tangential shear to zero at small separations is understand-able, as there is no reason to expect drawing one circular profiledirectly atop another should induce spurious non-zero ellipticity. Inthe regime of a few pixels, however, strong blending can increasethe flux of the best-fitting model.Next, we repeat the calculation, now applying a moderatecross-component shear to the neighbour ( g = 0 . ). The resultis shown by the blue lines in Fig. 3. Unsurprisingly the measuredtangential shear is unaffected by a true shear along an orthogonalaxis. In cases where the objects share a large portion of their half-light radii, we are fitting a strongly blended pair with a single pro-file, and the neighbour/central distinction becomes difficult to de-fine. The best-fitting ellipticity recovered from the blended imageis not a pure measurement of either galaxy’s shape; rather it is alinear combination of the two. We repeat the zero-offset measure-ment using a range of neighbour fluxes and find that the best-fitting e i follows roughly as a flux-weighted sum over the two galaxies ˜ g i ≈ ( f c g tri,c + f n g tri,n ) / ( f n + f c ) . While useful for understanding what follows, the impact of neigh-bours on individual galaxy instances is not particularly informativeabout the impact on cosmic shear measurements. Even significantbias in the per-object shapes could average away over many galax-ies with no residual impact on the recovered shear. More impor-tant is the collective response to neighbours. To explore this webuild on the toy model concept. To estimate the ensemble effect,we measure a neighbour-central image at 70 positions on a ringof neighbour angles. Again, under zero shear g tr = 0 the mea-sured shape is constant in magnitude, and simply oscillates about0 with peaks of amplitude | ˜ g ( θ | , d gn ) | . This sinusoidal variationis shown by the dotted lines in Fig. 3b at two values of d gn (7 and Unless otherwise stated we fix the other model parameters to their fidu-cial values. θ , which results in an unbiased mea-surement of the shear (cid:104) ˜ g ( θ | g tr = 0 , d gn ) (cid:105) θ = g tr = 0 . A non-zero shear g tr (cid:54) = 0 , however breaks the symmetry of the system. Agalaxy sheared along one axis will not respond to a neighbour in thesame way irrespective of θ , which can result in a net bias. To showthis we fix g tr = − . and proceed as before. The solid lines inFig. 3b show the periodicity in the measured shear at two d gn . Themean value averaged over θ is shifted incrementally away from theinput shear, shown by the horizontal dot-dashed line. Specificallywe should note that the peaks below g tr at π/ and π/ radi-ans are deeper and narrower than those above it. The cartoon inFig. 4 shows how this arises. The purple lines are iso-light con-tours in a strongly sheared S´ersic disc profile ( g = − . ). Clearlyrotating the neighbour from position A to C carries it from the rel-atively flat low wings of the central galaxy’s light profile closer tothe core. Perturbing an object about C by a small angle results ina much greater change in the local gradient, (cid:53) f c ( x, y ) than doingthe same about A. All other parameters fixed, an incremental shiftalong the blue tangent vector will have a larger impact at θ = 0 than at π/ , resulting in asymmetry in the width of the positive andnegative peaks in Fig. 4. The depth of the peak can be explainedqualititatively by similar arguments. At C a neighbour of given fluxis closer to the centre of the light distribution and thus has a greaterflux overlap with the central galaxy than at A. Naturally, then, onemight expect neighbour A to have less impact than C. Returning toFig. 3, we can see that the two effects are in competition. Depend-ing on the exact neighbour configuration, the simultaneous narrow-ing and deepening the negative peaks can result in a bias in theneighbour-averaged ellipticity towards large or small values.The level of this effect will clearly correlate with the magni-tude of the shear, and so induce a multiplicative bias. To illustratethis point the above exercise is repeated with a range of differentinput shears. The results for our fiducial setup are shown in in Fig.5. Each point on these axes corresponds to a ring of neighbour po-sitions for a given input shear. The equivalent measurements with-out the neighbour are indistinguishable from the x axis. At smallshears, the neighbour induced bias ˜ g − g tr is well aproximatedas a linear in g tr . We leave exploration of the possible nonlinearresponse at large ellipticities for future investigations. Though theabove numerical exercise demonstrates that it is possible for signif-icant multiplicative bias to arise as a result of neighbours, it doesnot make a clear prediction of the magnitude or even the sign. In-deed, our toy model is effectively marginalised over θ , but there isnothing to guarantee that fixing the other neighbour parameters tothe median measured values is representative of the real level ofneighbour bias in a survey like DES. Motivated by this observa-tion we add a final layer of complexity to the model, as follows.A single neighbour-central realisation is created as before, definedby a unique set of model parameters. Now, however, the valuesof those parameters p = ( d gn , f n , r n , f c , r c ) are drawn randomlyfrom the DES data. As these quantities will, in reality, be corre-lated we sample from the 5-dimensional joint distribution ratherthan each 1D histogram individually. We then fit the model at 70neighbour angles and two input shears g ± = ± . (a total of 140measurements), and estimate the multiplicative bias as a two-pointfinite-difference derivative: m + 1 = (cid:104) ˜ g ( θ | g + ) (cid:105) θ − (cid:104) ˜ g ( θ | g − ) (cid:105) θ g + − g − . (2)This process is repeated to create 1.33M unique toy model reali- MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours Figure 1.
Postage stamp snapshots of the basic two-object toy model described in Section 3. The overlain ellipse shows the maximum likelihood fit to theimage. The panels show three neighbour positions in the range θ = [0 , π/ rad. The best fit ellipticity and half light radius are shown above each image. Inall cases the input values are e = (0 , , r = 0 . arcseconds. Neighbour Distance d gn / pixels . . . . . T a n g e n t i a l o r C r o ss S h e a r ˜ g (round neighbour) ˜ g (round neighbour) ˜ g ( g tr ,n = 0 . ) ˜ g ( g tr ,n = 0 . ) Figure 2.
Tangential shear measured using the numerical toy model de-scribed in Section 3.1 as a function of radial neighbour distance. The solidpurple line shows the shape component aligned with the central-neighbourseparation vector and the dot-dashed line is measured along axes rotatedthrough ◦ . Note that the latter is smaller than − at all points on thisscale. The dashed and dotted black lines show the same ellipticity compo-nents when the neighbour is sheared in the e direction by g = 0 . . sations. Binning by neighbour distance we can then make a roughprediction for the level of neighbour-induced bias and the angularscales over which it should act. The result is shown in Fig. 6, wherefull results using all model realisations are indicated by the dashedblue line. The majority of cases yield a negative bias, particularlyat low neighbour separation (referring back to Fig. 4, the broad-ening of the peak around position A dominates over the increasedflux overlap at C). In the real data, of course, we apply a qualitybased selection and ¨uberseg object masking (J16), both of whichare neglected here. We can, however, test the impact of selectingon fitted quantities that respond to neighbour bias. Imposing a flatprior on the centroid offset ∆ r = ( x + y ) (i.e. discardingrandomly generated model realisations where the galaxy centroidis displaced from the stamp centre by more than a fixed number ofpixels) changes the shape of this curve significantly, as illustratedby the thick purple line.We can understand the difference between the results with andwithout the centroid cut as a form of selection bias, whereby the cutpreferentially removes toy model realisations in which the neigh-bour is bright relative to the central galaxy. At any given d gn we areleft with a relative overrepresentation of galaxies with f n /f c (cid:28) . Figure 3.
Best-fit galaxy ellipticity as a function of neighbour position angleat fixed neighbour distance d gn from the toy model described in the text.The two panels (left, right) show the same central-neighbour system ( g tr = − . ), but with different d gn (7 and 8 pixels) and biases m (shown atopeach panel). The solid line in each case is the recovered galaxy shape at each θ , and the integrated mean along this range is shown by the horizontal dot-dashed line. The dotted lines show the zero-shear shape (ie. the ellipticitythat would be measured if the input shear were zero), but shifted downwardssuch that the mean is at − . . Finally, to illustrate the (a)symmetry of thesystem we show the solid line flipped about y = g tr and shifted by π/ radians as a dashed curve. Faint neighbours, which in reality tend to be compact high redshiftobjects, have little impact when they sit on the outskirts of the cen-tral profile (A in the cartoon picture in Fig. 4; the regime whichproduces negative m ). The same faint galaxy has a stronger im-pact if it is rotated to a position closer to the centre of the central’sflux profile. Thus one might expect a selection on ∆ r to makethe mean m in a particular bin less negative (or even positive) bypreferentially removing brighter galaxies. OOPOE
IMAGE SIMULATIONS
In this section we provide a brief overview of the simulationpipeline. The process is the same as that described in § MNRAS , 000–000 (0000)
S. Samuroff, S. L. Bridle, J. Zuntz et al
ABC
Figure 4.
Cartoon diagram of a neighbour-central system. The purple con-tours show the lines of constant flux in a S´ersic disc profile with extremenegative ellipticity ( g = − . ). The blue crosses labelled A, B and C arepoints on a ring of equal distance from the centre of the profile. The bluearrows show the local unit vector along a tangent to the ring. Figure 5.
Measured shear minus input shear plotted as a function of inputshear. The purple points show the recovered ˜ g from averaging over ringof 70 neighbour positions. The dark blue lines show the linear relation ˜ g − g tr = mg tr at m = ( − . , − . , − . . The dotted line shows whatwould be measured using the same central profile in the absence of theneighbour, and is near indistinguishable from the x axis line on all pointswithin this range of g tr . ation and noise maps measured from the progenitor data. Each sim-ulated galaxy is then inserted into a subset of overlapping exposuresand into the coadd at the position of a real detection in the DES Y1data. Object detection is rerun on the new coadd images and galaxycutouts and new segmentation masks are extracted and stored in theMEDS format decribed by J16. The mock survey footprint is shownin Fig. 7. In the lower panels we show an example of a simulatedcoadd (left) and the spatial variation in PSF orientation within thesame image (right). We use reduced images from Year One of the Dark Energy Survey(DES Y1; Diehl et al. 2014) as input to the simulations discussedin this paper. The Dark Energy Survey is undertaking a five year
Neighbour Distance d gn / pixels . . . . . . M u l t i p li c a t i v e B i a s m No r Cut r < pixels Figure 6.
Multiplicative bias estimated using the Monte Carlo toy modeldescribed in the text. For each neighbour realisation, defined by a particulardistance, flux and size we compute the average of the measured ellipticitycomponents over 70 rotations on a ring of neighbour angles. To estimate thebias we perform this averaging twice at two non-zero shears, g + and g − ,and compute the finite-difference deriviative using equation 2. The dashedthin blue line shows the result of using all measurements, while the boldpurple line has a cut based on the offset between the centroid position of thebest-fitting model and the stamp centre. programme with the ultimate aim of observing ∼ square de-grees of the southern sky to ∼ th magnitude in five optical bands, grizY , covering . − . microns. The dataset is recorded using a570 megapixel camera called DECam (Flaugher et al. 2015), whichhas a pixel size of . arcseconds. In full it will consist of ∼ interwoven sets of exposures in the g , r , i , z and Y bands.The Y1 data were collected between August 2013 and Febru-ary 2014, and cover a substantially larger footprint than the prelim-inary Science Verification (SV) stage at 1500 square degrees, albeitto a reduced depth. Details of the reduction and processing are pre-sented in Z17. Our H OOPOE simulations use a selection of the total3000 . × . degree coadded patches known as “tiles”. For populating the mock survey images a sample of real galaxy pro-files from the HST COSMOS field, imaged at significantly lowernoise and higher resolution than DES by the Hubble Space Tele-scope Advanced Camera for Surveys (HST ACS) (Scoville et al.2007). The COSMOS catalogue extends significantly deeper thanthe Y1 detection limit of M r , lim = 24 . , extending to roughly . mag in the SDSS r -band. A main sample for our DES Y1 simula-tions is defined by imposing a cut at < . mag.Since the DES images do not cut off abruptly at 24th magni-tude, in reality they contain a tail of fainter galaxies that contributeflux are not identifiable above the pixel noise. To assess the impactof these objects on shape measurements in Y1, we simulate a pop-ulation of sub-detection galaxies in addition to the main sample.In brief we use the full histogram of COSMOS magnitudes to esti-mate the number of faint galaxies within a given tile. The requiredprofiles are selected randomly from the faint end of the COSMOSdistribution. Each undetected galaxy is paired with a detection, andinserted at a random location within the overlapping bounds of the MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours . . . . . . . . n g / arcmin (a) (b) Figure 7. Top:
The projected footprint of the simulated survey, visualised using the
SKYMAPPER package a . The colour indicates the local raw number densityin HEALPIX cells of nside = 1024 . The axes shown are right ascension and declination in units of degrees. The full simulation comprises 1824 . × . degree tiles drawn randomly from the DES Y1 area. The solid blue line indicates the bounds of the planned area to be covered by the complete Y5 dataset. Bottom:
A random tile (DES0246-4123) selected from the H
OOPOE area. The left panel (a) shows a square subregion of approximately × arcminutes.The right hand panel (b) shows a PSF whisker plot covering the full . × . tile. The length and orientation of each line represents the magnitude andposition angle of the spin-2 PSF ellipticity at that position. Only galaxies which pass IM SHAPE quality cuts are shown. The white patches show the spatialmasking inherited from the
GOLD catalogue, and correspond to the positions of bright stars in the parent data. a https://github.com/pmelchior/skymapper same (subset of) single-exposure images. A more detailed descrip-tion of this process can be found in Z17.If these galaxies were present in the data they would enterthe background flux calculation, and thus the subtraction appliedwould change due to their presence. Since the simulation pipelineproduces images effectively in a post-background subtraction statethis is not captured. To test this we rerun the SE XTRACTOR back-ground calculation on a handful of tiles drawn with and withoutthe faint galaxies. The impact was found to be well approximatedas a uniform shift in the background correction. A flux correctionequal to the pixel-averaged flux of the sub-detection galaxies overeach image plane is, then, applied to postage stamps prior to shapemeasurement.In reality the overdensity of sub-threshold galaxies will be coupled to the density of detectable objects, which is clearly notthe case in our simulations. To gauge the impact of this we per-form the following test. Each tile is divided into a × grid, andthe mean multiplicative bias is calculated in each sub-patch. Webin sub-patches according to the ratio f faint ≡ N faint /N det , or thetotal number of faint galaxies relative to the number of detectableones. The impact is significant, but not leading order; excludingpatches outside the range . < f faint < . induces a shift of ∆ m ∼ − . .An independent noise realisation is generated for each expo-sure using the weight map from the parent data. We simulate thenoise in each pixel by drawing from a Gaussian of correspondingwidth. The coaddition process is not rerun, but rather we computean independent noise field by drawing the flux in each pixel from a MNRAS , 000–000 (0000)
S. Samuroff, S. L. Bridle, J. Zuntz et al
Figure 8.
An example of an object in the main DES Y1 calibration simula-tion and the neighbour-free resimulation. The upper panels show the coaddcutout in the original simulated images (left, labelled H
OOPOE ) and in theneighbour-subtracted version (right, labelled W
AXWING ). The lower panelsare the segmentation masks for the same galaxy. A number of neighbours,both masked (upper left and centre left) and unmasked (lower right) arevisible within the stamp bounds. zero-centred Gaussian of width determined by the measured vari-ance in that pixel.
For the purpose of untangling the impact of image plane neighbourswe use the simulated H
OOPOE images to create a new spin-offdataset. In a subset of a little over 500 tiles we store the (convolved)input profile for each object and the noise-only cutout, taken fromthe same position in the image plane prior to objects being drawn.By adding together these two components we can generate a suiteof spin-off MEDS files, which are equivalent to the results of asimpler neighbour-free simulation (eg Miller et al. 2013, J16). Thepixel noise realisation, COSMOS selection and input shears, how-ever, are identical to the progenitor H
OOPOE simulations.We will call this process “resimulating”, and the basic conceptis illustrated in Fig. 8. The 506-tile set of neighbour-free data arenamed the W
AXWING resimulations. Finally the (now empty) seg-mentation masks corresponding to the subtracted neighbours arealso removed. In subsequent IM SHAPE runs on these data we ig-nore the SE
XTRACTOR flags obtained from the main simulations.
OOPOE
Equipped with qualitative predictions from Section 3, we now turnto the question of neighbour bias in the more complete simulationsdescribed in Section 4. The mock survey was designed to captureas much of the complexity of shape measurements on real photo-metric data as possible. We refer to Section 4 of this paper for ashort overview and to § IM SHAPE catalogue, including quality cuts andselection masks.
The most straightforward way to assess the impact of neighbourson individual shape measurements in our simulations is to rotatethe measured shapes into a frame defined by the central-neighbourseparation vector. Whereas in the earlier toy model we had only oneneighbour per galaxy, we now have a crowded image plane contain-ing many objects simultaneously. For simplicity, in the earlier casewe included no masking. For H
OOPOE we wish to mimic the pro-cess of shape measurement on real data as closely as possible. Wegenerate new segmentation maps by running SE
XTRACTOR on thesimulated images, and incorporate them into our shape measure-ments using the ¨uberseg algorithm (J16). Each simulated galaxy isallocated a nearest neighbour using a k -d tree matching algorithmconstructed on the coadd pixel grid using every galaxy simulated at r -band magnitude M r < . . The quantities d gn and θ are nowredefined slightly as nearest-neighbour distance and angle. We de-fine the tangential shear of a galaxy relative to its nearest neighbourusing the standard convention, e + = − [ e cos( θ ) + e sin( θ )] , (3)and the cross shear e × = − [ e cos( θ ) − e sin( θ )] . (4)Note that negative values of e + imply a net tangential alignmentof the measured shapes towards neighbours. By analogy, we de-fine e , n and e , n , which are the measured ellipticity components,rotated into a reference frame defined by the major axis of theneighbour. Non-zero e i, n would indicate leakage of the neighbour’sshape into the measurement, which might conceivably be inducedby inadequate deblending of very close neighbours or by extensivenon-circular masking. We first divide the main simulated catalogueinto bins according to d gn , and measure the tangential shear aboutnearest neighbours in each bin. The result is shown by the pur-ple curve in Fig. 9. Note that the statistical uncertainty is withinthe width of the line in all bins. The results here show qualitativeagreement with the numerical predictions in Fig. 3. As we foundearlier, the exact shape of this curve is sensitive to the propertiesof both the neighbour and the central galaxy. Despite small dif-ferences, the range of variation is comfortably within the scale ofthe postage stamp for the bulk of galaxies in DES Y1. Repeatingthe measurement, rotated into the plane of the neighbour shape re-sults in the dotted and dot-dash lines in this figure. As noted above,there are not necessarily reliable ellipticity measurements for eachneighbour, so we instead use the sheared input ellipticities. Bothcomponents of e i, n are seen to be negligible over all scales. To explore the more practical question of how neighbours impactshear estimates we divide the catalogue into bins according toneighbour distance. Within each d gn bin, the galaxies are furthersplit into twelve bins of input shear, which are fitted to estimate themultiplicative and additive bias. We show the result as the purple MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours Nearest Neighbour Distance d gn / pixels − . − . − . − . − . − . − . . . T a n g e n t i a l o r C r o ss S h e a r Tangential Shear e + Cross Shear e × Tangential Shear e + , n Cross Shear e × , n − . − . . . Figure 9.
Tangential shear around image plane neighbours in the fullH
OOPOE simulation. The purple solid line shows the mean component ofthe measured galaxy shapes radial to the nearest image plane neighbour.Dashed blue shows the component rotated by ◦ , which we have no rea-son to expect should be non-zero. The dotted and dot-dash lines show themeasured ellipticity components when rotated into a coordinate frame de-fined by the major axis of the neighbour. The inset shows the same rangein d gn (the x-axis tick markers are the same), but with a magnified verticalaxis. points in Fig. 10, which can be compared with the earlier numericalmodel prediction in Fig. 6. The horizontal band on these axes showsthe σ mean m measured using all galaxies in the H OOPOE cat-alogue, and sits at m ∼ − . . We note a steeper decline than inthe bold line (without the centroid cut), more akin to the case withthe centroid cut ( ∆ r < arcsec). This is not surprising giventhat the quality selection implemented by IM SHAPE includes ex-actly this cut. We do not report a local peak at ∼ pixels, whichwe saw before in Fig. 6. We suggested previously that effect wasthe result of positive m in galaxies where the nearest neighbour isrelatively faint and at middling distance. It is likely that many ofthese objects manifest themselves as large changes in other quan-tities to which IM SHAPE ’s INFO FLAG (see Z17) is sensitive suchas ellipticity magnitude and fit likelihood, or are flagged by theSE
XTRACTOR deblending cuts.When divided into broad bins according to the r -band magni-tude of the nearest neighbour M r , neigh (the coloured stripes in Fig.10) we find the surviving objects show relatively weak dependenceon neighbour brightness, except at the neighbour distances, wherebright neighbours have a slightly stronger (negative) impact thanfaint ones.We measure the additive bias components in the same bins,but find no systematic variation with d gn above noise.Finally we show the analogous measurement in bins of galaxymagnitude in Fig. 11. The steep inflation of | m | at the faint endof this plot has been seen elsewhere (e.g. Zuntz et al. 2017; FenechConti et al. 2017), and is easily understandable as the result of noisebias. We find that splitting by neighbour magnitude does not revealany obvious trend here. A central plank of this analysis rests on a comparison of the mainH
OOPOE simulations with the neighbour-free W
AXWING resimu-
Neighbour Distance d gn / pixels − . − . − . − . . M u l t i p li c a t i v e B i a s m M r , neigh = 21 . M r , neigh = 23 . M r , neigh = 23 . M r , neigh = 24 . all galaxies Figure 10.
Multiplicative bias as a function of separation from the nearestimage plane neighbour. The purple points show the bias calculated in binsof neighbour distance using the main H
OOPOE simulated shape catalogue.The coloured bands show the same dataset divided into four equal-numberbins according to the r -band magnitude of the neighbour. As shown in thelegend, the median values in the four bins are 21.5, 23.0, 23.5 and 24.0.The mean bias and its uncertainty across all distance bins is indicated bythe horizontal band. . . . . . . . . r -band Magnitude − . − . − . − . . M u l t i p li c a t i v e B i a s m M r , neigh = 21 . M r , neigh = 23 . M r , neigh = 23 . M r , neigh = 24 . all galaxies Figure 11.
Multiplicative bias as a function of r -band magnitude. As in Fig.10 the four coloured bands represent equal number bins of neighbour mag-nitude. Purple points show the full catalogue, with no magnitude binning.The mean bias and its uncertainty are shown by the purple horizontal band. lations described in Section 4.3. The simplest comparison wouldbe between multiplicative bias values, calculated using all galax-ies in each catalogue after cuts. These values are shown by the twoupper-most lines (purple) in Fig. 12. The difference is an indicatorof the net impact of neighbours through any mechanism, which wefind to be ∆ m ∼ − . .To untangle the various contributions to this shift, we con-struct a matched catalogue. Galaxies in the overlap betweenH OOPOE and W
AXWING (12M galaxies over 183 square degrees)are matched by ID; quality cuts are calculated for each set of mea-surments (see Appendix E from Z17). Geometric masking from theDES Y1 G
OLD catalogue (Drlica-Wagner et al. 2017) and SE X - MNRAS , 000–000 (0000) S. Samuroff, S. L. Bridle, J. Zuntz et al m ≡ ( m + m ) / hoopoe , own cuts waxwing , own cutsMatched hoopoe , own cutsMatched waxwing , own cutsMatched waxwing , hoopoe cutsMatched hoopoe , own cuts & d gn > pixMatched hoopoe , both cuts & d gn > pixMatched waxwing , both cuts & d gn > pixNo sub-detection, own cutsMatched hoopoe , own cuts hoopoe , cuts from no sub-detection Figure 12.
Graphical illustration of the measured multiplicative bias in the various scenarios considered in this paper. The upper two lines show the mean m inthe main DES Y1 H OOPOE simulations and a spin-off neighbour-free resimulation named W
AXWING , as described in Section 4.3. The middle section (green)shows results using only galaxies which appear in both the H
OOPOE and W
AXWING simulations. The matching process alone does not imply any quality-based selection function. The final three lines in red are from a similar matching between a smaller rerun of the simulation with and without sub-detectionlimit galaxies. See the text for details about each of these cases.
TRACTOR deblending flags are included for H
OOPOE . Since thelatter are irrelevant to W
AXWING , we omit them from quality flagson that dataset. For conciseness we will refer to the two measure-ments as “matched H
OOPOE ” and “matched W
AXWING ”, and theircuts as “H
OOPOE cuts” and “W
AXWING cuts”. Since the imagesare identical in all respects, but for the presence of neighbours, thestatistical noise on the change in measured quantities should besmaller than the face-value uncertainties.The appropriate cuts are first applied to each catalogue, thenthe results are divided into equal number signal-to-noise bins andfitted for the multiplicative bias in each. The result is shown bythe points in the upper left-hand part of Fig. 13. The equivalent inbins of PSF-normalised size is shown on the right. The differencebetween the blue and the purple points gives an indication of thetotal effect of all neighbour-induced effects on m , indicated by thesolid purple line in the lower panel. The generic shift attributed to“neighbour bias” is in reality a collection of distinct effects. Bycomparing the matched catalogues we identify four main mecha-nisms: direct contamination, selection bias, S/N bin shifting andneighbour dilution. Each of these components that we describe is shown by one of the lines in Fig. 13. For a visual summary of thevarious tests designed to isolate them see Fig. 12.
The most intuitive form of neighbour bias arises from the factthat, even after masking, neighbours contribute some flux to thecutout image of a galaxy. To gauge its impact we take the com-mon sample of galaxies, which pass cuts in both datasets. The com-parison is complicated somewhat by binning in measured
S/N or R gp /Rp ; for this test, we divide both sets of galaxies using theW AXWING -derived quantities. The resulting m measured using theH OOPOE galaxies is unrealistic in the sense that we are binningmeasurements made in the presence of neighbours by quantities de-rived from neighbour-subtracted images. This exercise does, how-ever, isolate the impact of the neighbour flux on the measured el-lipticity. The result is shown by the purple dotted and purple dot-dashed lines in the upper and lower panels of Fig. 13. The effectscales significantly with signal-to-noise and size. Faint small galax-ies are affected strongly by neighbour light, while larger brighterones are relatively immune.
MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours − . − . − . − . − . . M u l t i p li c a t i v e B i a s m HOOPOE , matched per galaxy
HOOPOE , matched per galaxy (diluted)
HOOPOE , matched per binW
AXWING , matched per galaxy
HOOPOE , full selectionW
AXWING , full selection . . . . . . . Signal-to-Noise log(
S/N ) − . − . − . − . . . ∆ m Direct Neighbour BiasSelection BiasNeighbour DilutionBin ShiftingTotal Neighbour Bias − . − . − . . M u l t i p li c a t i v e B i a s m . . . . . . R gp /R p − . − . − . − . . . . . ∆ m Figure 13. Top half of each panel
Multiplicative bias as a function of signal-to-noise and size. The purple circles show the measured bias using the mainH
OOPOE simulation, and the blue diamonds are the resimulated neighbour-free version. The lines show permutations of the same measurements to highlightthe neighbour-induced effects causing the two to differ. The dot-dashed blue and dashed purple lines show the impact of applying the H
OOPOE selection maskto W
AXWING and vice versa. The impact of bin shifting is shown by the purple dotted line, which is calculated from the same matched galaxies, using theH
OOPOE shape measurements for the bias and W
AXWING size and
S/N for binning. The pink curve is the same as the dashed purple, but with a fraction ofheavily blended galaxies added back with randomised shear (see Section 5.3.4).
Bottom half of each panel
The change in bias due to the effects describedabove. The green (dashed) line shows the impact of selection effects (the difference between the blue diamonds and the dashed line in the top panel). The directneighbour bias due to light contamination is shown by the purple dash-dotted line (purple dotted minus blue dash-dot top). The impact of shifting betweenbins is shown by the blue dotted (dashed minus purple dotted, top). The pink dot-dot-dashed line illustrates the impact of adding back randomised shears, asdescribed. Finally the solid line represents the total neighbour bias, which includes all these effects (circles minus diamonds, top).
To gauge the neighbour-induced selection effect, we take theW
AXWING catalogue but now impose, in addition to its own qual-ity cuts, the selection function derived from the with-neighbourH
OOPOE dataset. The double masking removes an additional 0.5Mgalaxies, which survive cuts in W
AXWING but would be cut fromthe H
OOPOE catalogue. The resulting change in m is shown by thedot-dash blue lines in the upper panels of Fig. 13 (dashed green inthe lower). The multiplicative bias arising from this cut is less thanone percent in all but the faintest and smallest galaxies, where itcan reach up to m ∼ − . . The above two tests encapsulate the impact on the measured ellip-ticities, and the selection flags from neighbour flux. An additionalsubtlety arises from the fact that the measured quantities used tobin galaxies (
S/N and R gp /R p ) are themselves affected by thepresence of neighbours. To test this we recalculate m using thesame galaxy selection as in Section 5.3.1 (i.e. passing both sets ofcuts), but now binned by the appropriate measured S/N . For clar-ity, the bin edges are unchanged, defined to contain equal numbersof W
AXWING galaxies. The result is shown by the dashed lines inFig. 13. The difference compared with the case using fixed binningis purely the result of galaxies moving between bins. This shift-ing contributes multiplicative bias if one bins galaxies by observedquantities such as
S/N , as we do in order to calibrate IM SHAPE ’sshear estimates. The amplitude of this is illustrated by the blue dot-ted line in the lower panels. Such neighbour-induced shifting isnoticable if we plot out the
S/N of objects in H
OOPOE against the
S/N of the same objects in W
AXWING . Objects which are stronglyshifted in
S/N are more likely to scatter upwards than downwards. A similar skew can be seen in the R gp /R p plane; when galaxies arescattered in size they tend to be thrown further and more often up-wards than downwards. Small galaxies (which we know already aremore strongly affected by noise bias) are shifted strongly upwardsby the presence of neighbour flux in the H OOPOE images. The re-sult is a net negative m added to the upper R gp /R p bins, and asimultaneous positive shift in the lowest bins from which galaxiesare lost. In the case of galaxy size we see the effects of bin scatterand direct neighbour bias almost negate each other, although thedegree of cancellation is likely dependent on the specifics of themeasurement code and the dataset. A final point can be gleaned from Fig. 13: that applying theW
AXWING cuts to H
OOPOE induces a shift in m . Naively onemight expect the H OOPOE selection function, which includesneighbours, to remove the same galaxies as the W
AXWING selec-tion, plus some extra strongly blended galaxies. It is true that asizeable number of galaxies are cut in the presence of neighbours,but would otherwise not be. There is also, however, a smaller popu-lation that survive cuts because they have image plane neighbours.We can see this clearly from the fact that the purple points andthe dashed purple lines Fig. 13 are non-identical. We identify threeseparate (but partially overlapping) galaxy selections in this figure:(a) galaxies passing both sets of cuts, (b) galaxies passing cuts inthe absence of neighbours, but cut by the H
OOPOE selection and (c)galaxies which pass cuts in the presence of neighbours, but cut bythe W
AXWING selection. We find that populations (b) and (c) havemuch smaller mean neighbour separation than the full population(the histograms of d gn show a sharp peak at under 10 pixels). In MNRAS , 000–000 (0000) S. Samuroff, S. L. Bridle, J. Zuntz et al contrast, both the full catalogue and population (a) objects a muchbroader distribution ( ¯ d gn ∼ pixels).Based on the toy model predictions in Section 3 we set out aworking proposal: that population (c), objects cut out only whenneighbours are removed, are extreme blends dominated by a super-posed neighbour. We will assume these objects are boosted con-siderably in size, S/N or both, such that what would otherwisebe a small faint galaxy is now sufficiently bright to pass qualitycuts. In these cases the measured shape of a simulated galaxy mightbe expected to be only weakly linked with the input ellipticity. Toapproximate this effect we take population (a) H
OOPOE galaxies,subject to both sets of cuts, and add back some of the population(c) galaxies. Specifically, we include any objects shifted in
S/N or R gp /R p by more than . The true shears associated with thesegalaxies are now randomised to eliminate any correlation with themeasured ellipticity. The result is shown as a pink dot-dot-dashedline in Fig. 13. We can see that this effect, which we call neigh-bour dilution, to good approximation accounts for the residual dif-ference between the population (a) and (c) samples. Particularlyin the upper size bins of the right hand panel the differences arenot eliminated entirely. This is thought to be the result of resid-ual (albeit weakened) covariance between the measured shapes ofstrongly blended objects and the input shears. Clearly the scenarioin which a neighbour totally overrides the original shape of a galaxyis extreme, and there will be an indeterminate number of moderateblends which are boosted sufficiently to survive cuts but which re-tain some correlation with their original unblended shapes. Suchcases are, however, extremely difficult to model accurately with theresources available for this investigation. A handful of previous studies have attempted to quantify the impactof galaxies below, or close to, a survey’s limiting magnitude. Forexample, Hoekstra et al. (2015) and Hoekstra et al. (2017) suggestthey can induce a non-trivial multiplicative bias, which is depen-dent on the exact detection limit. They recommend using a shearcalibration sample at least by 1.5 magnitudes deeper than the sur-vey in question (which ours does). Their findings, however, makeexclusive use of the moments-based KSB algorithm (see Kaiseret al. 1995); such techniques are known to probe a galaxy’s elliptic-ity at larger radii than other methods, which could in principle makethem more sensitive to nearby faint galaxies. It is thus a worthwhileexercise to to gauge their impact in our case with IM SHAPE . We first compare our H
OOPOE simulations with the neighbour-freeW
AXWING resimulations. Since W
AXWING postage stamps con-sist of only a single profile added to Gaussian pixel noise, they areunaffected by neighbours of any sort (faint or otherwise). We haveseen that the impact of neighbours is strongly localised, with theexcess m converging within a nearest neighbour distance d gn of adozen pixels or so. Thus selecting galaxies that are well separatedfrom their nearest visible neighbour will isolate the impact of the undetected ones.A further cut is thus imposed on d gn < pixels. Relative tothe case with quality cuts only, the global multiplicative bias nowshifts from m ∼ − . to m = − . ± . (the first andsecond lines in green on Fig. 12). This measurement is in mild ten-sion with the value measured from W AXWING (again under its own
Nearest Subdetection Galaxy d gf / pixels With Faint, Own CutsOnly Without FaintOnly With Faint
Figure 14.
Histogram of radial distances between galaxies in our measuredshape catalogues (the full H
OOPOE simulations) and the nearest object be-low the DES detection limit. The dotted line includes all objects prior toquality cuts, while the solid line shows the impact of applying IM SHAPE ’s INFO FLAG cuts (see J16). The dashed blue line shows the population ofgalaxies which survive cuts only in the presence of the faint galaxies. cuts, with the selection on d gn ). This difference, which amounts toa negative shift in m of ∼ . is, we suggest, the net effect of thesub-detection galaxies. From these numbers alone we cannot tell ifthis is a result of selection effects, flux contamination, bin shiftingor some combination thereof.Interestingly we find that imposing both the H OOPOE andW
AXWING selection functions in addition to the cut on d gn brings m into agreement to well within the level of statistical precison(compare the final and penultimate lines in green in Fig. 12). Thatis, when restricted to a subset of galaxies that pass quality cuts inboth simulations the flux contributed by the faint objects has littleimpact. Their main impact is rather that they allow a populationof marginal faint galaxies which would otherwise be flagged andremoved by quality cuts to pass into the final H OOPOE catalogue.To test this idea further we rerun a subset of 100 random tilesfrom the simulated footprint, without the final step of adding sub-detection galaxies. To minimise the statistical noise in this com-parison we enforce the same COSMOS profiles, shears and ro-tations as well as the per-pixel noise realisation as before. SE X - TRACTOR source detection is applied and the blending flags arepropagated into the postprocessing cuts.The raw m values calculated from the rerun and the mainH OOPOE simulations, matched to the same tiles, are shown by theupper two red lines on Fig. 12. The downward shift of ∼ . isconsistent with the previous result based on the main simulation.This comparison should encapsulate the full effect of the faint ob-jects (since there are no other differences between these datasets).For each galaxy we next measure the distance to the nearestfaint object d gf , the distribution of which is shown under vari-ous selections in Fig. 14. Like in the comparison in Section 5.3.4,there is a population of galaxies that survive cuts only in the sim-ulation with the sub-detection objects, and these galaxies tend tobe ones with extremely close faint neighbours. Interestingly theinverse population surviving only when they are removed do notpreferentially have small d gf . This is intuitively understandable: afaint object might boost its neighbour’s apparent size or S/N if it
MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours were centred within a few pixels. Otherwise it would act as a sourceof background noise, which would reduce the quality of the fit.Finally we find that if we apply both selection functions tothe with-faint galaxies, the measured biases become roughly con-sistent. These findings, combined with the observations in the pre-vious section lead us to an interesting conclusion: the major effectof the faint galaxies in the DES Y1 IM SHAPE catalogue is to al-low a population of small faint galaxies to pass quality cuts, whereotherwise they would have been removed. This is analogous to theneighbour dilution effect described above, but is subdominant tothe influence of visible neighbours.
As a test of the robustness of this result we recompute our IM SHAPE fits on the faint-free images, with and without the cor-rection for the shift in the background flux that would have beenapplied if the sub-detection galaxies had been drawn. The meanper-tile correction is ∆ f ∼ . , against typical noise fluctua-tions σ n ∼ . . Matching galaxies and examining the histogramsof ∆ S/R and ∆ R gp /R p reveals weak downwards scatter in bothquantities (i.e. the flux subtraction alone makes galaxies appearsmaller and fainter). The magnitude of the shift is, however, tiny,peaking at ∼ − . and − . respectively. This is logical giventhe definition in equation 1. If the change is small enough suchthat the best-fitting model is stable, then an incremental reductionin flux will reduce the signal-to-noise of the measurement. Lookingat the best-fit shapes, we find a small shift towards high ellipticities,which can likewise be understood as a numerical effect; imposinga flat positive field of zero ellipticity will dilute the measured shear,producing a bias towards round | e | . The reverse logic applies withthe flux correction, and subtracting a flat value from all pixels willmake galaxies appear slightly more elliptical. In practice we find asharp peak at ∆ e ∼ . . There is no universal definition for the shape-weighted effectivenumber density commonly used as proxy for cosmological con-straining power in a shear catalogue. One which is particularly use-ful in the context of weak lensing, and which has been adpoted inDES Y1 is the prescription of Chang et al. (2013), which is de-signed to account for shape noise and fitting error (see equation7.5 in Zuntz et al. 2017). A second useful definition is set out byHeymans et al. (2012) in terms of the (see also Z17). We com-pute a neighbour distance d gn for every object in the real data,which allows us to cut on this quantity. Removing any galaxy witha neighbour detected within a radius of d gn = 20 pixels reducesthe effective number density of sources using either definition toabout of its initial value, from n H13eff = 5 . to n H13eff = 3 . arcmin − using Heymans et al. (2012)’s definition. Using the pre-scription of Chang et al. (2013), the equivalent density drops from n C13eff = 3 . prior to cuts and n C13eff = 2 . arcmin − afterwards.This cut is stringent, as we have shown that beyond ∼ pixelsthe multiplicative bias becomes insensitive to further selection on d gn . There are, however, a number of limitations in our analysis,including the fact that d gn is defined using the true input positions,and indeed that we are using only the detected positions in DES todraw our simulated M r < . galaxies. We thus judge that a levelof conservatism is appropriate here. Relaxing the cut to d gn > pixels leaves n eff at of its full value. As we have shown in the previous sections, if ignored completelyimage plane neighbours can induce negative calibration biases in IM SHAPE of a few per cent or more. The earlier part of the inves-tigation focused on when and how neighbour bias can arise, firstin the context of single-galaxies and then on ensemble shear esti-mates. We now turn to a more pressing question from the generalcosmologist’s perspective: how far should I be concerned aboutthese effects in practice ? We present a set of numerical forecasts us-ing the M
ULTI N EST nested sampling algorithm (Feroz et al. 2013)to sample trial cosmologies. Each of the likelihood analyses pre-sented in this paper has been repeated using a Markov Chain MonteCarlo sampler (
EMCEE ). Although we see the same small shift incontour size noted by DES Collaboration et al. (2017) (see theirAppendix A), which diminishes as the length of the MCMC chainsincreases, we find our conclusions are robust to the choice of sam-pler. Our basic methodology here follows previous numerical fore-casts (e.g. Joachimi & Bridle 2010; Krause & Eifler 2016; Krauseet al. 2017a). We construct mock DES Y1 cosmic shear measure-ments using a matter power spectum derived from the Boltzmanncode CAMB with late-time modifications from HALOFIT . Thecosmic shear likelihood surface is sampled at trial cosmologies us-ing C
OSMO
SIS . The final data used for the likelihood calcula-tion have the form of real-space ξ ± correlations. For the photo-metric redshift distributions we use the measured estimates in fourtomographic bins, obtained from runs of the BPZ code on the Y1 IM SHAPE catalogue, as described by Hoyle et al. (2017). Sincethis analysis was completed before the details of the photometricredshift calculation for DES Y1 had been finalised, these distribu-tions differ marginally from (but are qualitatively the same as) thefinal version used in Troxel et al. (2017) and DES Collaborationet al. (2017). In all chains which follow we maginalise over twonuisance parameters (an amplitude and a power-law in redshift) forintrinsic alignments, photo- z bias and shear calibration bias. In to-tal this gives 10 extra free parameters in addition to six for cosmol-ogy ( Ω m , Ω b , A s , n s , h , Ω ν h ), which are also allowed to vary.Apart from the difference in redshift distributions remarked uponabove, our analysis choices match the DES Y1 cosmic shear anal-ysis of Troxel et al. (2017). We refer the reader to that paper fordetails of the priors and scale cuts, and their derivation. Finally,the following adopts shear-shear covariance matrices derived fromthe analytic halo model calculations of Krause et al. (2017b). Weassume a fiducial Λ CDM cosmology σ = 0 . , Ω m = 0 . , Ω b = 0 . , n s = 0 . , h = 0 . , τ = 0 . , with non-zerocomoving neutrino density Ω ν h = 0 . . We first seek to quantify the bias that would be present in acosmic shear analysis in a survey like DES, if we were to usea simple postage stamp simulation of the sort presented in J16and Miller et al. (2013). To this end we use the neighbour-freeW
AXWING dataset to construct an alternative shear calibration. InZ17 we compared three methods for shear calibration using theH
OOPOE simulations and found our results to be robust to thedifferences. We now use the fiducial (grid-based) scheme to de-rive an alternative set of bias corrections from W
AXWING . These http://camb.info https://bitbucket.org/joezuntz/cosmosisMNRAS , 000–000 (0000) S. Samuroff, S. L. Bridle, J. Zuntz et al Angular Scale ✓ / arcseconds ⇠ a b ( ✓ ) / ⇥ h mm i ¯ m h g m i -0.10.00.1 (4, 1) (3, 1) (2, 1)
10 100 (1, 1) waxwing
CalibrationSpatial Correlations -0.10.00.1 (4, 2) (3, 2)
10 100 (2, 2) -0.10.00.1 (1, 1) -0.10.00.1 (4, 3)
10 100 (3, 3) -0.10.00.1 (2, 2) (1, 2) θ / arcmin-0.2-0.10.00.1 (4, 4) -0.10.00.1 (3, 3) (2, 3) (1, 3)
10 100 θ / arcmin-0.2-0.10.00.1 (4, 4)
10 100 (3, 4)
10 100 (2, 4)
10 100 (1, 4) ξ o b s + ( θ ) / ξ + ( θ ) − ξ o b s − ( θ ) / ξ − ( θ ) − Figure 15. Top : The observed two point correlation of multiplicative bias, as measured from the main H
OOPOE simulation set presented in this paper. Sub-patches are used to compute m in spatial patches of dimension . × . degrees and the correlation function calculated as described in the text. Thedashed vertical line shows the diagonal scale of the sub-patches, below which we do not attempt to directly measure spatial correlations. The shaded bluebands show the minimum and maximum scales used in the DES Y1 cosmic shear analysis of Troxel et al. (2017). Bottom : Residuals between the mock twopoint shear-shear data used in this paper, before and after different forms of bias have been applied. The upper-left and lower-right triangles show the ξ + and ξ − correlations respectively, calculated using the redshift distributions of Hoyle et al. (2017). The dotted black lines, which are flat across all scales butvary between panels, show the result of calibrating our Y1 shear measurements with a simple postage stamp simulation without image plane neighbours. Thedashed lines illustrate the impact of ignoring scale-dependent selection effects, which are not captured by our simulation-based calibration. The shaded blueregions of each panel show the excluded scales for each particular tomographic bin pairing. are then applied to the same galaxies in the matched H OOPOE sim-ulation, and residual biases are measured in four DES-like tomo-graphic bins. The process is very similar to the diagnostic tests in § AXWING simulations. In the four tomographic
MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours Figure 16.
Expected cosmology constraints from DES Y1 cosmic shearonly. The purple (solid) contours show the results of calibrating using a sim-ulation which fully encapsulates all biases in the data, leaving no residual m in the final catalogue. In blue (dash-dotted) we show the result of cal-ibrating with an insufficiently realistic simulation, which leaves a residualbias between − . and − . in each of the redshift bins. For referencewe mark the input cosmology with a black cross. bins used in DES Y1 we find (∆ m (1) , ∆ m (2) , ∆ m (3) , ∆ m (4) ) =( − . , − . , − . , − . , and apply these biases to ourmock data. The resulting shift in the shear two-point correlations isshown by the black dotted lines in the lower panel of Fig. 15. Sincethe calibration scheme does not explicitly include neighbour dis-tances, but rather orders galaxies into cells of S/N and R gp /R p ,this test does not include any scale dependent neighbour effects.The calibration effectively marginalises out d gn , and the residualbiases are an average over the survey. For the moment we will as-sume this mean shift in m is sufficient, and return to the questionof scale dependence in the following section.Our predicted cosmology constraints with weak lensing alonein DES Y1 are shown in Fig. 16. In purple we show the results ofthe fiducial analysis, in which the shear calibration fully capturesall neighbour effects and leaves no residual multiplicative bias. Theblue (solid) contours then show the impact of residual neighbourbiases per bin at the level described. As we can see, even whenmarginalising over m i with an (erroneously) zero-centred Gaussianprior of width σ m = 0 . , our cosmology constraints are shiftedenough to place the input cosmology outside the σ confidencebounds. We reiterate here that this calculation highlights the biasthat would arise were we to naively apply a calibration of the sortused in DES SV based on neighbour-free simulations to the Y1data. Since we are confident that the H OOPOE code captures theeffects of image plane neighbours correctly (at least to first order)this is a hypothetical scenario only and not a prediction of actualbias in DES Y1.
It is not trivial that including an mean neighbour-induced compo-nent to m over the entire survey will be sufficient to mitigate allforms of neighbour bias. The local mean m on a patch of sky is sen-sitive to spatial fluctuations in source density, which could inducescale dependent bias on arcminute scales. Clearly, when correlat- Figure 17.
The same as Fig. 16, but now showing the impact of residualscale dependent selection bias. The two sets of confidence contours repre-sent different assumptions about the small scale extrapolation of the ξ mm correlation, as outlined in the Section 6.2. In green (dashed) we show amildly optimistic case, using the linear fit shown in Fig. 15. The pink dot-ted contours show a (strongly pessimistic) power law extrapolation. Thedark blue solid line makes identical assumtions to the pink, but incorporatessmall-scale information, to a minimum separation of θ +min = 0 . arcmin-utes in ξ + ( θ ) and θ − min = 4 . arcminutes in ξ − ( θ ) . As in Fig. 16 the inputcosmology is shown by a black cross. ing galaxy pairs on small scales one can expect a larger fraction inwhich the objects come from a similar image plane environment,and more often than not that enviroment will be densely populated.Thus the true multiplicative bias should become more negative onsmall scales.Two subtly different effects emerge from this thought experi-ment. First, the multiplicative bias of galaxies will be spatially cor-related i.e. a correlation involving two galaxy populations (cid:10) m i m j (cid:11) is not just the product of the means ¯ m i ¯ m j . Second, in the small θ bins one is selecting galaxies with close partners with which tocorrelate, and thus oversampling the dense parts of the image. Togauge the level of these effects, we divide each simulated coadd tileinto a grid of 25 square sub-patches with dimension . × . degrees. We fit for m using the galaxies in each sub-patch and as-sign the resulting value to these objects. While this only allows anoisy measurement of m , it should capture the spatial variationsin number density to the level of a few percent. We next measurethe two-point correlation function of multiplicative bias values as-signed in this way using T REE C ORR , excluding galaxy pairs atangular separation smaller than the scale of the sub-patches. We re-fer to this bias-bias autocorrelation as ξ mm , which we show as afunction of angular scale in the upper panel of Fig. 15. analogouslyone could use the sub-patches to construct correlations between m and galaxy number density ξ gm or density with density ξ gg . Thestatistical noise on these correlations is significantly lower than thaton the individual sub-patches by virtue of the large simulation foot-print. Note that in Fig. 15 we subtract ¯ m , measured from all galax-ies in the simulation, from the measured ξ mm . If there were no θ dependence the correlation (cid:104) m i m j (cid:105) should simply average to thesquare of the global mean in all scale bins. As we can see from http://rmjarvis.github.io/TreeCorrMNRAS , 000–000 (0000) S. Samuroff, S. L. Bridle, J. Zuntz et al the circular points in this figure, scales larger than the diagonal sizethe sub-patches (shown by the vertical dashed line) exhibit non-negligible excess ξ mm . One obvious question is whether this couldbe the result of finite binning error, which scatters galaxies in thesame sub-patch into different θ bins. To verify this is not the casewe repeat the measurement as before, but halve the parameter con-trolling binning error tolerance (“bin slop”) and obtain the sameresults.To extend this measurement down to scales below the sub-patch size we must make some assumptions about the functionalform of the mm correlation. We fit a power law, ∆ ξ mm ( θ ) ≡ ξ mm ( θ ) − ¯ m : ∆ ξ mm ( θ ) = βθ − α , (5)which is shown by the dotted purple line in this figure. This pro-vides a qualitiatively good fit to the measured points, but as we cansee implies a rather dramatic inflation on small scales.In the limited range over which we have a nonzero measuredcorrelation, however, a linear function of θ (truncated at θ = 27 arcminutes) also provides a reasonable by-eye fit. The small-scaleextrapolation in this case is significantly milder. The (cid:104) mδ g (cid:105) and (cid:104) δ g δ g (cid:105) measurements are linear with θ to good approximation, andso we use linear fits to extrapolate them below the patch size.Assuming the bias per tomographic bin can be written as thesum of a redshift dependent contribution (i.e. a scale invariant meanin each bin), and a scale dependent term, one can write the corre-lation per bin as m i m j = ¯ m i ¯ m j + ∆ ξ mm ( θ ) . A more completederivation of this expression can be found in Appendix A. The firstpart can be extracted from the DES Y1 calibration, and we can fitfor ∆ ξ mm ( θ ) as described above. A set of modified ξ ij ± are thuscomputed. These appear in the lower panels of Fig. 15 as dashedlines. As we can see, the scale cuts of Troxel et al. (2017) (ex-cluded scales are shaded in blue) are sufficiently stringent to re-move almost all of the visible scale dependence. Though reassur-ing for the immediate prospects of DES Y1, this will not triviallybe true for all future (or indeed ongoing) lensing surveys. It is thusimportant that the effects we identify here are properly understoodat a level beyond the resources of the current paper. These biaseddata are then passed through our likelihood pipeline to gauge thecosmological impact, which is shown in Fig. 17. In the linear case(dashed green) there is no discernable bias in the σ Ω m pair; eventhe much harsher power-law extrapolation (pink dotted) inducesonly an incremental shift along the degeneracy direction. In bothcases the input cosmology still sits comfortably within the σ con-fidence contour.Finally we test the impact of relaxing the stringency of ourscale cuts. The minimum scales used for ξ ij + and ξ ij − are shifteddownwards to . and . arcminutes respectively, irrespective ofbin pair, which are the cut-off values used in fiducial cosmic shearanalysis of Hildebrandt et al. (2017). This increases the size of ourdatavectors considerably. Incorporating smaller angular scales willclearly improve the constraining power of the data to an extent. Pri-marily the effect is to shorten the lensing degeneracy ellipse, cuttingout much of the peripheral curvature, but it also reduces the widthin the S direction. These scales, however, contain biased informa-tion, which induces tension between the small and large angularscales. With the strongest (power law) scale dependence consid-ered, the input cosmology is displaced marginally along the degen-eracy curve, though it remains comfortably within the σ confi-dence bound. The Dark Energy Survey is the current state of the art in cosmo-logical weak lensing. Multi-band imaging down to 24th magnitudeacross 1500 square degrees of the southern sky has yielded hith-erto unparalleled late-time constaints on the basic parameters ofthe Universe (see Troxel et al. 2017 and DES Collaboration et al.2017).In this paper we have used one of two DES Y1 shear cata-logues, and large-area simulations based upon them, to quantifythe impact of image plane neighbours on both ensemble shear mea-surements, and on the inferred cosmological parameters.In order to properly mitigate the influence of galaxy neigh-bours, and thus avoid drawing flawed conclusions about cosmologyfrom the data, it is important to first understand the mechanismsby which they enter the shape measurement. Using a simple toymodel of the galaxy-neighbour system we have shown that shearbias can arise even when the distribution of neighbours is isotropic(i.e. there is no preferred direction). This is the result of a smalldifference in the impact of the same neighbour, when it is placedon or away from the axis of the shear. We have furthermore shownthat the resulting multiplicative shear bias m can be either positiveor negative, depending on the model parameters. With slight mod-ifications to the toy model, whereby we Monte Carlo sample inputparameters from the joint distribution of the equivalent propertiesmeasured in DES Y1, we have shown that a mild negative m isdominant when marginalising over a realistic ensemble of neigh-bours. This was seen to be strongly dependent on the distance ofthe neighbour, and to be mitigated but not eliminated by basic cutson the centroid position of the best-fitting model.Using the DES Y1 H OOPOE simulations, which werealso used to derive shear calibration corrections for the Y1 IM SHAPE catalogue of Z17, we have presented a detailed study ofthe ensemble effects of galaxy neighbours. In this analysis we haveidentified four mechanisms for neighbour bias, which we call fluxcontamination, selection effects, bin shifting and neighbour dilua-tion. All can be understood in intuitive terms, resulting from close-by or moderately close neighbours. Our results from the full sim-ulation are consistent with the toy model calculation. Though wehave shown strong dependence on distance to the nearest neigh-bour (and thus on number density) we found only weak sensitiv-ity to neighbour brightness, when averaged across broad bins ofmagnitude. In addition to this, cuts on the DES Y1 catalogue suf-ficient to null the impact of the detectable neighbours would resultin a degradation of over 20% in source number density. We can-not recommend such measures for a code like IM SHAPE , in partbecause the data contains correlations between shear and numberdensity. Unless the link is preserved in the calibration simulations,such selection could conceivably induce additional bias towardslow shear .Our investigation also assessed the impact of the faintestgalaxies, which are not reliably detected but nonetheless contributeflux to the survey images. Via two different routes, first using aspin-off neighbour-free resimulation, and also using a subset of im-ages simulated again with the sub-detection galaxies missing, ourfindings suggested a net contribution to the multiplicative bias bud-get of m ∼ − . .Unlike most earlier works on shear measurement, we have Although the sister catalogue to Y1 IM SHAPE uses a form of internalcalibration, which should allow one to correct for the additional selectionbias. MNRAS , 000–000 (0000) osmic Shear & Galaxy Neighbours propagated these findings to the most meaningful metric for cos-mic shear: bias on the inferred cosmological parameters. The studywe have presented here uses the DES Y1 cosmology pipeline, aswell as real non Gaussian shear covariance matrices and photo-metric redshift distributions to implement MCMC forecasts. In thefirst case considered, the data included a (different) multiplica-tive bias in each redshift bin, designed to approximate the resid-ual m that would arise were we to calibrate DES Y1 with a simpleneighbour-free simulation. Even marginalising over m with a priorof N (0 , . this scenario was demonstrated to result in a shift inthe favoured cosmology towards low clustering amplitude of morethan σ .Finally, we have explored a second possible source of mea-surement bias arising from the link between number density andneighbour bias. This enters two-point measurements as an addi-tional correlation between the multiplicative bias in galaxy pairsat small angular separation. In the final section we have measuredsuch a correlation from the H OOPOE mock images. With the mostpessimistic small-scale extrapolation, this was found to result ina shift in the best-fitting cosmology of under σ in the negative S direction, which is not remedied by marginalising over m . Aless dramatic, though still considerable, increase in the correlationstrength on small scales was demonstrated to result in no discern-able cosmological bias.Both of these effects are of primary concern for the next gen-eration of cosmological surveys. By the end of their lifetime KiDS,DES and HSC are set to offer lensing-based cosmological con-straints comparable to the CMB. The first, dominant, effect can beremedied relatively easily by calibrating our shear measurementswith sufficiently complex image simulations. Indeed, the most re-cent shear constraints of Hildebrandt et al. (2017), K¨ohlinger et al.(2017) and Troxel et al. (2017) have done just that. Unfortunately,the correct treatment of scale dependent bias is not as clear, thoughit should be captured at some level by the per-galaxy responsesupon which METACALIBRATION relies. Though further statementsabout the likely small scale dependence of the mm correlation arebeyond the scope of the present study, understanding this intricatetopic will be crucial for future surveys if we are to fully exploit theconstraining power of the data. The massive simulation efforts ofLSST and Euclid, combined with advancement in neighbour miti-gation using techniques such as multi-object fitting will be invalu-able in this task. With the enhanced understanding these will pro-vide and the exquisite data of the next generation surveys, the com-ing decade will be an exciting time for cosmology. We thank Nicolas Tessore, Catherine Heymans and Rachel Man-delbaum for various insights that contributed to this work. We arealso indebted to the many DES “eyeballers” for lending their holi-day time to help us understand and validate our simulations. TheH
OOPOE simulations were generated using the National EnergyResearch Scientific Computing Center (NERSC) facility, which ismaintained by the U.S. Department of Energy. The likelihood cal-culations were performed using NERSC and the Fornax computingcluster, which was funded by the European Research Council. SLBacknowledges support from the European Research Council in theform of a Consolidator Grant with number 681431. Support for DGwas provided by NASA through Einstein Postdoctoral Fellowshipgrant number PF5-160138 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatoryfor NASA under contract NAS8-03060.Funding for the DES Projects has been provided by theU.S. Department of Energy, the U.S. National Science Founda-tion, the Ministry of Science and Education of Spain, the Sci-ence and Technology Facilities Council of the United Kingdom, theHigher Education Funding Council for England, the National Cen-ter for Supercomputing Applications at the University of Illinois atUrbana-Champaign, the Kavli Institute of Cosmological Physics atthe University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institutefor Fundamental Physics and Astronomy at Texas A&M Univer-sity, Financiadora de Estudos e Projetos, Fundac¸˜ao Carlos ChagasFilho de Amparo `a Pesquisa do Estado do Rio de Janeiro, Con-selho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico andthe Minist´erio da Ciˆencia, Tecnologia e Inovac¸˜ao, the DeutscheForschungsgemeinschaft and the Collaborating Institutions in theDark Energy Survey.The Collaborating Institutions are Argonne National Labora-tory, the University of California at Santa Cruz, the University ofCambridge, Centro de Investigaciones Energ´eticas, Medioambien-tales y Tecnol´ogicas-Madrid, the University of Chicago, Univer-sity College London, the DES-Brazil Consortium, the Universityof Edinburgh, the Eidgen¨ossische Technische Hochschule (ETH)Z¨urich, Fermi National Accelerator Laboratory, the University ofIllinois at Urbana-Champaign, the Institut de Ci`encies de l’Espai(IEEC/CSIC), the Institut de F´ısica d’Altes Energies, LawrenceBerkeley National Laboratory, the Ludwig-Maximilians Univer-sit¨at M¨unchen and the associated Excellence Cluster Universe, theUniversity of Michigan, the National Optical Astronomy Observa-tory, the University of Nottingham, The Ohio State University, theUniversity of Pennsylvania, the University of Portsmouth, SLACNational Accelerator Laboratory, Stanford University, the Univer-sity of Sussex, Texas A&M University, and the OzDES Member-ship Consortium.The DES data management system is supported by the Na-tional Science Foundation under Grant Numbers AST-1138766and AST-1536171. The DES participants from Spanish institu-tions are partially supported by MINECO under grants AYA2015-71825, ESP2015-88861, FPA2015-68048, SEV-2012-0234, SEV-2016-0597, and MDM-2015-0509, some of which include ERDFfunds from the European Union. IFAE is partially funded by theCERCA program of the Generalitat de Catalunya. Research leadingto these results has received funding from the European ResearchCouncil under the European Union’s Seventh Framework Pro-gram (FP7/2007-2013) including ERC grant agreements 240672,291329, and 306478. We acknowledge support from the AustralianResearch Council Centre of Excellence for All-sky Astrophysics(CAASTRO), through project number CE110001020.This manuscript has been authored by Fermi Research Al-liance, LLC under Contract No. DE-AC02-07CH11359 with theU.S. Department of Energy, Office of Science, Office of High En-ergy Physics. The United States Government retains and the pub-lisher, by accepting the article for publication, acknowledges thatthe United States Government retains a non-exclusive, paid-up, ir-revocable, world-wide license to publish or reproduce the publishedform of this manuscript, or allow others to do so, for United StatesGovernment purposes.Based in part on observations at Cerro Tololo Inter-AmericanObservatory, National Optical Astronomy Observatory, which isoperated by the Association of Universities for Research in As-
MNRAS , 000–000 (0000) S. Samuroff, S. L. Bridle, J. Zuntz et al tronomy (AURA) under a cooperative agreement with the NationalScience Foundation.
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APPENDIX A: DERIVATION OF A TWO-POINTMODIFIER FOR SCALE DEPENDENT BIAS
In the following we set out a brief derivation of the analytic modifi-cations to account for scale-dependent neighbour effects the shear-shear two-point correlations used in the earlier section. We do notclaim that this is a precise calculation of the sort that could be usedto derive a robust calibration. Rather it is an order of magnitudeestimate to allow us to assess the approximate size of the cosmo-logical bias these effects could induce in the data.First, with complete generality it is possible to write the i com-ponent of the measured shear at angular position θ as γ obs i ( θ ) = [1 + m i ( θ )] γ i ( θ ) , (A1)where γ i is the underlying true shear, which is sensitive to cos-mology only. Extending this to the level of a two-point correlationbetween two populations α and β this implies: ξ obs ,αβi ( θ ) ≡ (cid:68) γ obs ,αi ( θ (cid:48) ) γ obs ,βi ( θ (cid:48) + θ ) (cid:69) θ = (cid:68) [1 + m αi ( θ (cid:48) )][1 + m βi ( θ (cid:48) + θ )]˜ γ αi ( θ (cid:48) )˜ γ βi ( θ (cid:48) + θ ) (cid:69) θ . (A2)Note that the observed shear used in a particular bin correlationis now weighted by the overdensity of galaxies in the image, inaddition to the calibration bias, such that ˜ γ αi ( θ ) ≡ (cid:2) δ αg ( θ ) (cid:3) × γ αi ( θ ) . (A3)Expanding each of the terms one finds: MNRAS000
In the following we set out a brief derivation of the analytic modifi-cations to account for scale-dependent neighbour effects the shear-shear two-point correlations used in the earlier section. We do notclaim that this is a precise calculation of the sort that could be usedto derive a robust calibration. Rather it is an order of magnitudeestimate to allow us to assess the approximate size of the cosmo-logical bias these effects could induce in the data.First, with complete generality it is possible to write the i com-ponent of the measured shear at angular position θ as γ obs i ( θ ) = [1 + m i ( θ )] γ i ( θ ) , (A1)where γ i is the underlying true shear, which is sensitive to cos-mology only. Extending this to the level of a two-point correlationbetween two populations α and β this implies: ξ obs ,αβi ( θ ) ≡ (cid:68) γ obs ,αi ( θ (cid:48) ) γ obs ,βi ( θ (cid:48) + θ ) (cid:69) θ = (cid:68) [1 + m αi ( θ (cid:48) )][1 + m βi ( θ (cid:48) + θ )]˜ γ αi ( θ (cid:48) )˜ γ βi ( θ (cid:48) + θ ) (cid:69) θ . (A2)Note that the observed shear used in a particular bin correlationis now weighted by the overdensity of galaxies in the image, inaddition to the calibration bias, such that ˜ γ αi ( θ ) ≡ (cid:2) δ αg ( θ ) (cid:3) × γ αi ( θ ) . (A3)Expanding each of the terms one finds: MNRAS000 , 000–000 (0000) osmic Shear & Galaxy Neighbours ξ obs ,αβi ( θ ) = (cid:68) γ αi ( θ (cid:48) ) γ βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) m αi ( θ (cid:48) ) γ αi ( θ (cid:48) ) γ βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) m βi ( θ (cid:48) + θ ) γ αi ( θ (cid:48) ) γ βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) δ αg ( θ (cid:48) ) γ αi ( θ (cid:48) ) γ βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) δ βg ( θ (cid:48) + θ ) γ αi ( θ (cid:48) ) γ βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) m αi ( θ (cid:48) ) m βi ( θ (cid:48) + θ ) γ αi ( θ (cid:48) ) γ βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) δ αg ( θ (cid:48) ) δ βg ( θ (cid:48) + θ ) γ αi ( θ (cid:48) ) γ βi ( θ (cid:48) + θ ) (cid:69) θ . (A4)The terms contributing to the measured two-point shear correla-tion, then, is sensitive to both spatial correlations between the m in different galaxies and to the correlations with the source den-sity. Note that we’ve chosen to neglect a higher-order (six-point)term. In reality there will also be a connection between galaxy den-sity and shear, but we will follow the normal convention and as-sume the contribution is small enough to be neglected. In simpleterms, an excess in the (cid:104) mm (cid:105) term above the product of the mean m values indpendently could arise because galaxy pairs separatedon small scales tend to come from similar image plane environ-ments. In contrast the density weighted correlations (cid:104) δ g m (cid:105) wouldbe zero, but for a simple observation; selecting a random galaxywith a suitable correlation pair at a distance θ is not the same as un-conditionally selecting a random galaxy. In the small scale bins wewill over-sample the dense regions, where m tends to be larger (seeSection 6.2). The angular brackets here indicate averaging over allgalaxy pairs separated by θ . If we can assume the bias is indepen-dent of the underlying cosmology the above expression simplifiessignificantly: ξ obs ,αβi ( θ ) = (1 + ¯ m αi + ¯ m βi + (cid:68) m αi ( θ (cid:48) ) m βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) δ αg ( θ (cid:48) ) m βi ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) m αi ( θ (cid:48) ) δ βg ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) δ αg ( θ (cid:48) ) δ βg ( θ (cid:48) + θ ) (cid:69) θ ) × ξ αβi ( θ | p ) , (A5)with ξ αβi being the true correlation function of cosmological shears (cid:104) γ i γ i (cid:105) , which is contingent on the underlying cosmological param-eters p . It can be shown that ξ + ( θ ) ≡ (cid:10) γ + ( θ (cid:48) ) γ + ( θ (cid:48) + θ ) (cid:11) θ ± (cid:10) γ × ( θ (cid:48) ) γ × ( θ (cid:48) + θ ) (cid:11) θ = (cid:10) γ ( θ (cid:48) ) γ ( θ (cid:48) + θ ) (cid:11) θ ± (cid:10) γ ( θ (cid:48) ) γ ( θ (cid:48) + θ ) (cid:11) θ = ξ ( θ ) + ξ ( θ ) , (A6)and so one can use equation A5 to construct the observed ξ ± cor-relation functions ξ obs ,αβ ± ( θ ) = (cid:18) m α + ¯ m β + (cid:68) m α ( θ (cid:48) ) m β ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) δ αg ( θ (cid:48) ) m β ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) m α ( θ (cid:48) ) δ βg ( θ (cid:48) + θ ) (cid:69) θ + (cid:68) δ αg ( θ (cid:48) ) δ βg ( θ (cid:48) + θ ) (cid:69) θ (cid:19) ξ αβ ± ( θ | p ) . (A7)The i subscript has been discarded here under the assumption that m and m are approximately equal for a given set of galaxies.Next, let’s say imagine that we have a measured datavector.Our measurements are biased, but we’ll assume it is possible to devise a correction that recovers the true cosmological signal pre-cisely. Our observed datavector is then just, ξ obs ,αβ ± ( θ ) = Υ tr ,αβ ξ αβ ± ( θ | p ) , (A8)which follows trivially from equation A7. Since we do not triviallyknow Υ tr ,αβ ab initio (that’s why we need simulations!) we canonly construct a best-estimate approximation. By applying a cor-rection factor to the raw measurements we construct a best-estimatedatavector: ξ BE ,αβ ± ( θ ) = 1Υ BE ,αβ ξ obs ,αβ ± ( θ ) = Υ tr ,αβ Υ BE ,αβ ξ αβ ± ( θ | p ) . (A9)Of course, if our best correction is perfect then the ratio goes tounity, and we recover the underlying cosmology. Since we applycorrections to the single-galaxy shears we will assume Υ BE ,αβ in-cludes the (cid:104) δ g δ g (cid:105) term, but neglects the correlations involving m .We then can write: Υ BE ,αβ = (cid:16) m α + ¯ m β + ¯ m α ¯ m β + (cid:68) δ αg ( θ (cid:48) ) δ βg ( θ (cid:48) + θ ) (cid:69) θ (cid:17) . (A10)We can measure the mean bias in each bin that would be obtainedfrom the calibration directly. As we show in Z17, using the fullDES Y1 H OOPOE catalogues, these biases are ∼ − . to − . .Finally, assume that although m clearly varies betweenredhshift bins, the strength of the correlation does not That is, thebias-bias term is the product of the mean m s (which varies between z bins) plus a scale dependent shift (which doesn’t). One then has: (cid:68) m α ( θ (cid:48) ) m β ( θ (cid:48) + θ ) (cid:69) θ = ¯ m α ¯ m β + ∆ ξ mm ( θ ) . (A11)The additive part can be measured directly from the simulation us-ing sub-patches, as described earlier. The density-density correla-tion can be obtained in the same way. This, then, leaves only the m × δ g cross-correlation. This should vanish in the case of zerocorrelation, but it also seems reasonable to assume that the magni-tude should be proportional to the mean bias ¯ m α in a particular bin.This allows the scale dependent (non-tomographic) cross correla-tion measured from H OOPOE to be rescaled appropriately for eachbin pair: (cid:68) δ α ( θ (cid:48) ) m β ( θ (cid:48) + θ ) (cid:69) θ = (cid:18) ¯ m β ¯ m (cid:19) ξ gm ( θ ) , (A12)where ¯ m is the global multiplicative bias and ξ gm ( θ ) ≡ (cid:104) mδ g (cid:105) ,each measured using all simulated galaxies. Using the above equa-tions, with our fiducial calibration and three measured correlations,one can derive a scale dependent modification to shear-shear twopoint correlation data using equation A8. AFFILIATIONS Jodrell Bank Centre for Astrophysics, School of Physics andAstronomy, University of Manchester, Oxford Road, Manchester,M13 9PL, UK Institute for Astronomy, University of Edinburgh, EdinburghEH9 3HJ, UK Center for Cosmology and Astro-Particle Physics, The OhioState University, Columbus, OH 43210, USA Department of Physics, The Ohio State University, Columbus,
MNRAS , 000–000 (0000) S. Samuroff, S. L. Bridle, J. Zuntz et al
OH 43210, USA Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box2450, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025,USA Department of Physics and Astronomy, University of Pennsylva-nia, Philadelphia, PA 19104, USA Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Dr., Pasadena, CA 91109, USA Department of Physics, ETH Z¨urich, Wolfgang-Pauli-Strasse 16,CH-8093 Z¨urich, Switzerland Department of Physics & Astronomy, University CollegeLondon, Gower Street, London, WC1E 6BT, UK Department of Physics and Electronics, Rhodes University, POBox 94, Grahamstown, 6140, South Africa Fermi National Accelerator Laboratory, P. O. Box 500, Batavia,IL 60510, USA LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014,Paris, France Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095,Institut d’Astrophysique de Paris, F-75014, Paris, France Laborat´orio Interinstitucional de e-Astronomia - LIneA, RuaGal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio deJaneiro, RJ - 20921-400, Brazil Department of Astronomy, University of Illinois, 1002 W.Green Street, Urbana, IL 61801, USA National Center for Supercomputing Applications, 1205 WestClark St., Urbana, IL 61801, USA Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Insti-tute of Science and Technology, Campus UAB, 08193 Bellaterra(Barcelona) Spain Institute of Space Sciences, IEEC-CSIC, Campus UAB, Carrerde Can Magrans, s/n, 08193 Barcelona, Spain Department of Physics, IIT Hyderabad, Kandi, Telangana502285, India Kavli Institute for Cosmological Physics, University of Chicago,Chicago, IL 60637, USA Instituto de Fisica Teorica UAM/CSIC, Universidad Autonomade Madrid, 28049 Madrid, Spain Department of Astronomy, University of Michigan, Ann Arbor,MI 48109, USA Department of Physics, University of Michigan, Ann Arbor, MI48109, USA Astronomy Department, University of Washington, Box351580, Seattle, WA 98195, USA Cerro Tololo Inter-American Observatory, National OpticalAstronomy Observatory, Casilla 603, La Serena, Chile Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064,USA Australian Astronomical Observatory, North Ryde, NSW 2113,Australia Argonne National Laboratory, 9700 South Cass Avenue,Lemont, IL 60439, USA Departamento de F´ısica Matem´atica, Instituto de F´ısica, Univer-sidade de S˜ao Paulo, CP 66318, S˜ao Paulo, SP, 05314-970, BrazilStation, TX 77843, USA Department of Astronomy, The Ohio State University, Colum-bus, OH 43210, USA Department of Astrophysical Sciences, Princeton University,Peyton Hall, Princeton, NJ 08544, USA Instituci´o Catalana de Recerca i Estudis Avanc¸ats, E-08010Barcelona, Spain Centro de Investigaciones Energ´eticas, Medioambientales yTecnol´ogicas (CIEMAT), Madrid, Spain Brookhaven National Laboratory, Bldg 510, Upton, NY 11973,USA School of Physics and Astronomy, University of Southampton,Southampton, SO17 1BJ, UK Instituto de F´ısica Gleb Wataghin, Universidade Estadual deCampinas, 13083-859, Campinas, SP, Brazil Computer Science and Mathematics Division, Oak RidgeNational Laboratory, Oak Ridge, TN 37831 Institute of Cosmology & Gravitation, University of Portsmouth,Portsmouth, PO1 3FX, UK
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