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Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed11September 2012 (MNLATEXstylefilev2.2) Size magnification as a complement to Cosmic Shear B. Casaponsa,1,2⋆, A. F. Heavens3†, T. D. Kitching4‡, L. Miller5, R.B. Barreiro1, E. Mart´ınez-González1 1 Institutode F´ısica de Cantabria, CSIC-Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain. 2 2 Dpto. de F´ısica Moderna, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain. 1 3 Imperial Centre for Inference and Cosmology, Imperial College, Blackett Laboratory, Prince Consort Road, London SW7 2AZ U.K. 0 4 SUPA, Institute for Astronomy, Universityof Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, U.K. 2 5 Department of Physics, Universityof Oxford, DenysWilkinson Building, Keble Road, Oxford OX13RH, U.K. p e S Accepted ;Received;inoriginalform 7 ABSTRACT ] O We investigate the extent to which cosmic size magnification may be used to complement cosmic shear in weak gravitational lensing surveys, with a view to obtaining C high-precision estimates of cosmological parameters. Using simulated galaxy images, h. we find that size estimation can be an excellent complement, finding that unbiased p estimation of the convergence field is possible with galaxies with angular sizes larger - than the point-spread function (PSF) and signal-to-noise ratio in excess of 10. The o statisticalpowerissimilarto,butnotquiteasgoodas,cosmicshear,anditissubject r t to different systematic effects. Application to ground-based data will be challeng- s ing, with relatively large empirical corrections required to account for with biases for a [ galaxieswhichare smallerthan the PSF, but for space-baseddata with 0.1 arcsecond resolution, the size distribution of galaxies brighter than i ≃ 24 is ideal for accurate 1 estimation of cosmic size magnification. v 6 Key words: data analysis - weak lensing- size magnification 4 6 1 . 9 1 INTRODUCTION redshift relation and the growth of density perturbations 0 (Huterer2002;Munshi & Wang2003).Thereforelensingef- 2 GeneralRelativitypredictsthatthepathoflightisdistorted fects are a source of valuable information for three of the 1 when interacting with a gravitational field. This modifica- : importantopenissuesinmoderncosmology,namelythedis- v tion of the light paths is called gravitational lensing and tribution of dark matter, the properties of dark energy and i is a powerful tool for probing the distribution of mass in X thenatureof gravity. the Universe. The variation of the light path depends on r the position in the sky of the emitting object, the distance a Theamountofgravitationallensinginducedontheim- from the emitting object to the lens and the observer and age of a background source is characterized by an image on the potential along the light path. As the Universe is distortion matrix, which may be written in terms of the in permanent evolution photons that come from an ear- convergence κ, that is the field responsible for the changes lier stage of the Universe will be deflected in a different in the image size and brightness, and the complex shear γ way from those emitted in later stages (for a detailed read- that, in the limit of small κ, parameterises the distortion ing see Schneideret al. 1992; Narayan & Bartelmann 1996; of the shape. Observationally there are three main effects Mellier 1999; Munshi et al. 2008). Hence by studying the on the background sources: changes in the ellipticity, am- lensing observable distortion, a reconstruction of the three- plification in terms of flux, and magnification of the size. dimensionalunbiaseddistributionofmatter(bothlightand For large densities of (dark) matter the effects are all very dark) can be performed. The result is a picture of the Uni- strong, and multipleimages of thebackgroundgalaxies can verseovertime,thereforeboththepowerspectrumandthe be produced with large distortion. The studies of this dis- growth of density perturbations with redshift can be in- tortion have led to local reconstruction of the distribution ferred. Dark energy and modifications to Einstein gravity of non-baryonic matter for many years (Tyson et al. 1984). act to modify the lensing effect by changing the distance- Whentheconcentrationofmatterisnotashigh,theeffects are not as obvious, and more difficult to detect. The lensed background source is observed with changes in the elliptic- ⋆ e-mail:[email protected] † e-mail:[email protected] ities, flux and size, but the effects are so small that, for a ‡ e-mail:[email protected] single galaxy, it is impossible to know if they are intrinsic (cid:13)c 0000RAS 2 B. Casaponsa et al. properties of the galaxy or they are produced by lensing. a slightly a smaller S/N for lensing measurements based on However in this weak lensing limit on large scales, the dis- size rather than ellipticity, but not markedly so. Further- tortions may be observed statistically using a large sample more, Bertin & Lombardi (2006) proposed a method based of sources. on the tight relation between sizes and the central veloc- In weak lensing, the most studied effect is the modi- ity dispersion (the fundamental plane relation) to reduce fication to the galaxy shape (complex ellipticity), a mea- the observable size variance. Huff& Graves (2011) applied sure of the shear. The shape distortion has the main ad- a similar method to 55,000 galaxies of the SSDS catalogue, vantage that the intrinsic distribution of galaxy elliptici- andfindconsistencywithshearusingthesamesample.Also ties is expected to be random, according to the cosmolog- adetectionwith COSMOSHSTsurveyusingarevised ver- ical principle. Therefore under this assumption the aver- sionoftheKSBmethodisclaimedinSchmidtet al.(2012). age complex ellipticity is zero. Weak lensing effects using Revisiting the size magnification in detail is the aim of galaxy ellipticities havebeendetectedbyseveralgroupsus- this paper. As we will show, to use size-magnification we ingdifferentsurveysanddifferentmethods,seeforexample require i) a large area survey to result in sufficiently high Wittman et al.(2000);Semboloni et al.(2006);Jarvis et al. numberdensities of galaxies to overcome the intrinsic scat- (2006);Benjamin et al.(2007);Schrabback et al.(2010).An ter, and ii) a consistently small PSF that does not destroy important effort has been made to include and test many thesize information of theobserved galaxies. Both of these possible systematic effects on shape measurement (includ- requirements can be met with a wide-area space-based sur- ing the point spread function [PSF], instrumental noise, vey,althoughsomesciencemaybepossiblefromtheground. pixelization for example), and there are several algorithms Space-based surveys such as Euclid1 (Laureijs et al. 2011) that can measure shapes with varying degrees of accuracy shouldmeettheserequirements(largesampleswillbeavail- such as KSB (Kaiser et al. 1995), KSB+ (Hoekstra et al. able, and the PSF size is smaller than typical galaxies), so 1998) and its variants (Rhodes et al. 2000; Kaiser 2000) the size information should be considered as a complemen- and shapelets (Bernstein & Jarvis 2002; Refregier & Bacon tarycosmological probetoweaklensingellipticitymeasure- 2003;Kuijken2006)amongstothers.Totest,inablindway, ments. the ability of methods to measure the shapes of galaxies a Oneadvantageofusingthesizeinformation isthatthe series of simulations have been created: STEP, GREAT08 magnification is directly related to the convergence field κ, and GREAT10, where several methods have been tested which is in turn directly related to the weighted surface andcomparedsystematically,anexplanationofthemethods mass density, while γ is related to the differential surface and their performance are shown in Heymanset al. (2006); massdensity.Thosedifferentradialdependencescanbevery Massey et al. (2007); Bridle et al. (2010); Kitching et al. useful to lift any degeneracies (Rozo & Schmidt 2010), and (2012)respectively.AnovelBayesianmodelfittingapproach combiningthesizemagnificationinformationwiththeshear lensfit was presented in Miller et al. (2007), that includes will reduce the uncertainties on the reconstruction of the PSF and pixelization effects, and was successfully applied distribution of matter (Jain 2002; Rozo & Schmidt 2010; by Kitching et al. (2010) to the STEP challenge images. Vallinotto et al. 2011). All of the shear estimation meth- Moving beyond measuring the shapes of galaxies, the am- ods referred to above already estimate the size of galaxies plification effect is usually quantified using the number of when calculating the ellipticities, so we expect to measure counts of a given flux (see for example Broadhurst et al. this additional information for free, given an accurate el- 1995;Hildebrandt et al.2009,2011);thenumberofobserved lipticity measurement, but theaccuracy of size information galaxies at agiven fluxincreases duetotheweak lensing of shouldnotbetakenforgranted:itisimportanttoknowthe theforegroundgalaxies,wedonotstudythisfurtherinthis uncertaintiesin size measurement, and how theypropagate paper. to a convergence field estimation. It is this question of how In contrast to galaxy ellipticity measurement, the size- accurately one can measure the sizes of galaxies, that this magnification effect hasnot been studiedin detail, possibly paperwill address. becausethecomplicatingeffectsofthePSFandpixellisation Thepaperisorganizedasfollows. First,insection2we were thought to betoo challenging. However, thereare two willpresenttheestimatorthatwewilluse,thenabriefcom- reasons for revisiting size magnification as a potential tool mentonthemethodandthecharacteristicsofthesimulated forcosmology:oneisthataccurateshearestimationisitself images. In section 3 the analysis and results are explained very challenging, and size could add useful complementary and finally we will summarize the conclusions in section 4. information;thesecondisthatmethodsdevisedforelliptic- ityestimationmustdealwiththePSFandpixellisation,and as a byproduct provide a size estimate, or a full posterior 2 METHOD probabilitydistributionforestimatedsize,whichiscurrently ignored or marginalised over. In terms of signal-to-noise of A good algorithm for ellipticity weak lensing analysis must shearorconvergenceestimation,therelativestrengthsofthe be able to measure shapes or sizes with an accuracy of a methodsdependonthepriordistributionsofellipticity and fewpercent,takingintoaccountanypossiblesystematicer- size. The former has an r.m.s. of around 0.3-0.4; for bright rors as the distortion introduced by the PSF, shot noise or galaxies, the Sloan Digital Sky Survey (SDSS) found that pixelization. Another requirement is that it should be fast the size distribution of massive galaxies is approximately because the statistical analysis will be made on large sam- log-normal with σlnR ∼ 0.3, and for less massive galax- ples. This means that the algorithm development is chal- ies σ(lnR)∼0.5, where R is thePetrosian half-light radius (Shen et al. 2003). For deeper space data the dispersion is modestlylarger(Simard et al.2002).Thusonemightexpect 1 http://www.euclid-ec.org (cid:13)c 0000RAS,MNRAS000,000–000 Size magnification as a complement to Cosmic Shear 3 lenging because of the dissonant requirements of both in- ties but that shear (a statistical change in ellipticity) is the creased accuracy and increased speed as the required sys- quantityofinterest.A similar correction isrequiredforsize tematic level decreases. Several methods have been pro- measurement,wherebywemeasurethesizebutitisthecon- posed and applied to weak lensing surveys, in particular vergence that is the quantity of interest; this correction is several of these methods are decribed in the challenge re- neededbecauseforasinglegalaxy thepriorinformation for ports of STEP, GREAT08 and GREAT10 (Heymanset al. theconvergenceisnotknown,andweassumeitiszero.With 2006; Bridle et al. 2010; Kitching et al. 2012). These blind aBayesianmethodwecanestimatethemagnitudeofthisef- challengeshavebeencriticalindemonstratingthatmethods fectforeachgalaxy,afurtherreasontouseaBayesianmodel can achieve the required accuracy for upcoming surveys by fitting code in these investigations. Consider the Bayesian creating simulations with controlled inputs against which estimate ofthesizeof galaxy iandwrite itsdependenceon results can be tested. Here we propose a very similar ap- κ as a Taylor expansion: proach astheonepresented intheGREAT10challenge, we haveusedsimulatedgalaxyimageswithdifferentproperties sî=ssi +κddsˆκi, (2) to measure the response of the size/convergence measurement under different conditions (corresponding to changes Inthesimplistic casewherethelikelihood L(s)isdescribed in thePSF, S/N and bulge fraction). by a Gaussian distribution with variance b2, with an ex- pectedvalues,andapriorP(s)thatalsofollowsaGaussian distributioncentredons¯withvariancea2,thentheposterior 2.1 Estimator probabilitywillfollowaGaussiandistributionwithexpected Iftheintrinsicsizeofagalaxyssismagnified,s=µss,where value: tliokefirµst=ord1er+thκe,mthaegnnifiκca=tionsssis−re1l.atWedetocatnheccoonnstvreurcgtenacne hsi= s¯ba22++sbd2a2 estimator forκin theweak fieldlimit byassuming thatthe mean size value is not modified by lensing, i.e., hssi=hsi: and variance s 2 a2b2 κˆ= −1. (1) σ = hsi a2+b2 Thisassumptionshouldbevalidbecausehκi=0.Wecannot Note that here s stands for the fitted model parameter for obtainaccurateestimatesofκforasinglegalaxy,wecansee thesizeexplainedinSec.2.3,andsdisthedata.Theseequa- that from the definition of the estimator, smaller galaxies tionsillustratethattheposteriorisdriventowardstheprior than the mean will always give a negative κˆ, while larger inthelowS/Nlimit (b→∞),andthusrequirescorrection. galaxieswillproducepositiveκˆ.Whatisimportantistotest Differentiating the expression sd = ss(1+κ) we find that ifourestimatorisunbiasedoverapopulationtoasufficient theκ sensitivity correction is: degree to be useful for real data. dsˆ = a2 dsd = a2 ss, (3) dκ a2+b2 dκ a2+b2 2.2 lensfit substitutinginto eq. 2 Throughoutweuselensfit(Miller et al.2007;Kitching et al. a2 2008; Miller et al. 2012)to estimate the galaxy size; we use sî =ssi +κssia2+b2 thisbecause:1)ithasbeenshownthatlensfitperformswell on ellipticity measurement; 2) it is a model fitting code we find the estimator for κ will be the same as in eq. 1, such that it measures the sizes of galaxies (that are part corrected by thesensitivity factor: of the model); 3) it allows for the consistent investigation sˆ a2+b2 of the intrinsic distribution of galaxy sizes through the in- κ= −1 . (4) clusion of a prior on size, and 4) it includes the effects of (cid:18)hsî (cid:19) a2 PSF and pixellisation. lensfit was proposed in Miller et al. Inthiswork wehaveusedthisapproximation for simplicity (2007)andhasbeenprovedtobeasuccessfultoolforgalaxy butingeneralanormaldistributionshouldnotbeassumed. ellipticity shape measurements (Kitchinget al. 2008). Al- A more general estimation of the κ correction can be done though model-fitting is the optimal approach for this type in the same way as with the shear and can be evaluated of problem if the model used is an accurate representation numerically,withouttheneedofusingexternalsimulations. of the data, the main disadvantage is that is usually com- To calculate the sensitivity correction in the general putationallydemandingtoexplorealargeparameterspace. case we consider the response of the posterior to a small lensfitsolvesthisproblembyanalyticallymarginalizingover κ, L(s−ss) 7→ L(s−ss−κss) and expand it as a Taylor some parameters that are not of interest for weak lensing series: ellipticity measurements, such as position, surface bright- dL ness and bulge-fraction. The size reported by lensfit is also L(s−ss−κss)≃L(s−ss)−ssκ . ds marginalised overthegalaxy ellipticity. Wethen substituteinto 2.2.1 Sensitivity correction sP(s)L(s)ds hsi= P(s)L(s)ds Miller et al.(2007)introducedtheshearsensitivity,afactor R that corrects for the fact that the code measures elliptici- and differentiate to obtainR the analytic expression for the (cid:13)c 0000RAS,MNRAS000,000–000 4 B. Casaponsa et al. κ sensitivity (for more details of this applied to ellipticity thatthepostage-stamp.Thegalaxysizesexploredherehave measurement see Miller et al. 2007; Kitching et al. 2008) a somewhat smaller range (σ(lnR) ∼ 0.18) than found by Shenet al.(2003)withtheSSDScatalogue, whereinterms ds (hsi−s)P(s)ssdLds ≃ ds . (5) of pixelsthemean valueof thefullsample is around 5with dκ R P(s)L(s)ds σ(lnR)∼0.3(seeFig.1ofShen et al.(2003)).Thereforethe If the prior and likelihoodR are described by a normal dis- sensitivitycorrectionsareconsequentlylargerthanwouldbe tribution,thisexpressioncan beanalytically computedand needed for real data. Besides a wider distribution of galax- thesensitivitycorrectionisthesameasbefore.Asimilarem- ies, the important change from the original images for the piricallymotivatedcorrectionontheestimatorexpressionis GREAT10 challenge is the addition of a non-zero kappa- used in Eq.5 of Schmidtet al. (2012), where the factor is field that creates a size-magnification effect (in GREAT10 computed with simulations. onlyashearfieldwasusedtodistorttheintrinsicgalaxyim- ages).AGaussianconvergencefieldwithasimplepower-law powerspectrumofPk ∝10−5l−1.1 hasbeenappliedtoeach 2.3 Simulations image using the flat-sky approximation. The power-law is good approximationtothetheoreticalpowerspectrumover In order to test the estimation of sizes with lensfit we thescales 106l61000 (see i.e. Huterer2002). have generated the same type of simulations used in the GREAT10 challenge (Bridle et al. 2010; Kitching et al. 2010,2012),butwithnon-zeroκ.Multipleimagesweregen- erated,eachcontaining10,000simulatedgalaxiesinagridof 3 RESULTS 100x100postagestampsof48x48pixels;eachpostagestamp contains one galaxy. Before trying to estimate the convergence field of our most Each galaxy is composed of a bulge and a disk, each realistic image, we have tested the dependence on differ- modelled with Sérsic light profiles: ent aspects separately: S/N, PSF size and bulge-fraction. We expect these to be the observable effects that have the 1 r n largestimpactontheabilitytomeasurethesizeofgalaxies. I(r)≃I0exp −K −1 (6) ( "(cid:18)rd(cid:19) #) Lower S/N will cause size estimates to become more noisy and possibly biased (in a similar way as for ellipticity, see where I0 is the intensity at the effective radius rd that en- Melchior & Viola2012);alargerPSFsizewillacttoremove closes half of the total light and K =2n−0.331. The disks information on galaxy size from the image and a change in were modelled as galaxies with an exponential light pro- galaxytypeorbulge-fractionmaycausebiasesbecausenow file (n = 1), and the bulges with a de Vaucouleurs pro- twocharacteristicsizesarepresentintheimages(bulgeand file (n = 4). Ellipticities for bulge and disk were drawn disk lengths). In order to study carefully the sensitivity of from a Gaussian distribution centred on zero with disper- our estimator to systematic noise, PSF or galaxy proper- sion σ =0.3. Both components were centred in thepostage ties,westartedfromthesimplestcaseandaddedincreasing stamp with a Gaussian distribution of σ = 0.5 pixels. The levels of complexity. The number of galaxies used for the galaxy image was then created adding both components. analysis is 200,000 for the first three sets and we increased TheS/Nwas fixedfor all galaxies of theimage andtheim- thenumberto500,000 forthelast test togivesmaller error plementationisthesameasinKitching et al.(2012).Finally bars. thePSFwasmodeledwithaMoffatprofilewithβ=3,with WecomparetheestimatedκˆcomputedasinEq.1with FWHM fixed for all galaxies on the image, with different the input field κ, and fit a straight line to the relationship ellipticies, drawn from a uniform distribution, with ranges to estimate a multiplicative bias m and an additivebias c: given in Table. 1. Wehavegenerated4differentsetsofimages,whosede- κˆ=(m+1)κ+c. (7) tailedinformationisgiveninTable.1.Thedifferenttypesof To obtain m and c values, we estimate the observed size of image were generated to studytheeffectsof thebulge frac- eachgalaxy,andcomputeκûsingtheestimatorinEq.1.The tion(fractionofthetotalfluxconcentratedinthebulge),the sensitivity correction is applied as explained in Sec. 2.2.1. S/NandthePSFseparately.Insummary,themaincharac- Then we compare the corrected estimated κ of each galaxy teristics of the considered sets are: with the original convergence field introduced. Finally we • Set1.Disk-onlygalaxies,(bulgefraction=0),negligible bin the data2 and a linear fit is done to compute m and c. PSF effect (FWHM PSF= 0.01 pixels) and different S/N. This process is shown in Fig. 1. • Set 2. Disk-only galaxies, (bulge fraction = 0), with Wenow discuss each of thecategories in turn. S/N=20 and different sizes of PSF. • Set3.NegligiblePSFeffect,S/N=20anddifferentbulge fractions. 3.1 Signal-to-noise • Set 4. Bulge fraction of 0.5, FWHM of PSF 1.5 times As a first approach to the problem, disk-only galaxies with smallerthanthecharacteristicsizeofthedisk,anddifferent anegligiblePSF(FHWM=0.01 pixels)andzeroellipticity S/N. weregeneratedtotestthesensitivityontheS/Nonlygiven To characterize thesize of the galaxy thehalf-light disk radius,s=rd isused.WehavedrawnaGaussiandistribution for rd, with expected value of 7 pixels and dispersion of 1.2 2 Note that the differences between the results before and after pixels, to keep disk sizes of at least 2 pixels and not larger thebinningofκarewithintheerrorbars. (cid:13)c 0000RAS,MNRAS000,000–000 Size magnification as a complement to Cosmic Shear 5 SetName S/N fwhm PSF ePSF B/Dfraction rd rb e Set1 [10,40] 0.01 0 0 hrdi=7,σ=1.2 - 0 Set2 20 [0.01,10] 0 0 hrdi=7,σ=1.2 - hei=0,σ=0.3 Set3 20 0.01 0 [0,0.95] hrdi=7,σ=1.2 rd/2 hei=0,σ=0.3 Set4 [10,40] 4.5 [-0.1,0.1] 0.5 hrdi=7,σ=1.2 hrbi=3.5,σ=0.6 hei=0,σ=0.3 Table 1. Majorcharacteristicsofthedifferentsetsusedinthisanalysis.Inboldaremarkedthevariablesexploredineachsetandthe range ofvariation. InSet 4, PSFellipticities aredrawnfrom auniformdistributioninthe range specified. Note that ri corresponds to thehalf-lightradius.Lastcolumnisthegalaxyellipticityandisthesameforbothcomponents, bulgeanddisk. 12 0.5 data data 0.02 data 10 bc==00..1919 bc==00..0908 bc==01..0001 0.01 8 rout kout 0 kout 0 6 −0.01 4 −0.02 2 −0.5 2 4 6 8 10 12 −0.1 −0.05 0 0.05 0.1 −0.02 −0.01 0 0.01 0.02 r k k in in in Figure1.Sequenceofstepstoobtainmandcvalues.Firstpanelshowsthelensfitoutputsizecomparedtotheinputsize,inthesecond paneltheestimatedκcomparedtotheinputconvergenceateachgalaxy,andinthethirdpanelisshownthesameplotusingbins.Slope andinterceptvaluesofthefittingareshownineachplot(Throughout,wefitgenericallyy=bx+c,withb=m+1andc=cofeq.7). ThisisforgalaxiesofSet1withsignal-to-noise40. otherwise perfect data. In Fig. 2 we can see that as we in- S/N=10 S/N=20 creasetheS/Ntheaccuracyofthesizeestimationgrows,as 14 14 data data expected. In Fig. 3 we show the estimation of the conver- 12 b=0.84 12 b=0.96 gencefield,correctedbythesensitivitycorrection.Although 10 c=1.16 10 c=0.35 tohneeroarndgereooffmthaegnoiuttupduet,tahnedreinispuatclveaalruecsordriefflaetrisonbybeatlwmeoesnt rout 68 rout 68 4 4 the inputs and the outputs, and the slope is close to unity 2 2 forallS/Nexplored.InFig.4theestimatesformandcare schasoewnth,ewictohrrtehcetisoennsditoievsitynoctorarletcetriomnuacnhdtwhiethroeusutlitts.eInxctehpist 00 5 rin 10 00 5 rin 10 S/N=30 S/N=40 forlowS/N,becausethesizesarelessaccuratelyestimated. 14 14 Inthispaperthefactor a2/(a2+b2) isestimated bythein- 12 data 12 data b=0.98 b=0.99 verse of the slope of the size estimation fitting (see Fig. 2). 10 c=0.16 10 c=0.10 Using 200,000 galaxies for this test, the values found for m ut 8 ut 8 and c are consistent with zero, typically m ≃ 0.02±0.05, ro 6 ro 6 and c≃(5±5)×10−4. 4 4 2 2 0 0 0 5 10 0 5 10 3.2 PSF effect rin rin To study the uncertainties on the size estimation due to Figure 2.Comparisonoftheestimatedsizesbylensfitwiththe the PSF size, we generated images with different FWHM input galaxy size for different S/N in the range [10,40]. Disk- PSFvalues,withanintermediatesignaltonoise(S/N=20), onlycirculargalaxieswithanegligiblePSFeffectareconsidered maintaining the same properties as before, except that we (Set 1). Slope and intercept of the fitting are shown (b and c, considered here a Gaussian distribution of ellipticities with respectively).Notethattheinputsizeisthelensedone. meanvalueofe=0andσe =0.3(percomponent).Thesize estimatesaregoodforsmallPSFs,butbecomeprogressively morebiasedasthePSFsizeincreases beyondthediskscale tribution,someofthesmaller galaxies areconvolvedwith a length (see Fig. 5). A PSF with a FWHM larger or similar PSF larger than its size, and this could produce an overall tothesizeofthedisk,tendstomakethegalaxylooklarger, bias in κ, but if the number is not very large, this will not and the estimator for κ is biased. This effect can be seen affecttheglobalestimationoftheconvergencefield.Similar in the slope and intercept of κˆ vs κ plot (Fig. 6). Fig. 7 biases exist with shear for large PSFs, but the biases are shows the variation of the parameters m and c with the larger here. Fora space-based experiment, with a relatively ratio between the scale-length of the PSF and the galaxy bright cutat i∼24, suchas planned for Euclid, thelimita- (ratio=rd/PSFFWHM). tiononPSFsizewillnotbeaproblem,asthemediangalaxy Wefindnoevidenceforanadditivebias,butwedofind size is 0.24 arcsec (Simard et al. 2002; Miller et al. 2012), a multiplicative bias for large PSFs. With a wide size dis- comfortably larger than the PSF FWHM of 0.1 arcsec. For (cid:13)c 0000RAS,MNRAS000,000–000 6 B. Casaponsa et al. S/N=10 S/N=20 fwhm=0.0 fwhm=1.0 14 14 0.02 data 0.02 data data data b=1.00 b=1.01 12 b=0.96 12 b=0.94 0.01 c=0.00 0.01 c=0.00 10 c=0.35 10 c=0.52 kout 0 kout 0 out 8 out 8 r r −0.01 −0.01 6 6 −0.02 −0.02 4 4 −0.02 −0.01 k0in 0.01 0.02 −0.02 −0.01 k0in 0.01 0.02 20 5 r 10 20 5 r 10 S/N=30 S/N=40 in in fwhm=7.0 fwhm=17.5 0.02 data 0.02 data 16 16 b=0.98 b=1.01 data data 0.01 c=0.00 0.01 c=0.00 14 b=0.89 14 b=0.67 kout−0.010 kout−0.010 rout1102 c=2.47 rout1102 c=6.18 8 −0.02 −0.02 −0. 02 −0.01 0 0.01 0.02 −0. 02 −0.01 0 0.01 0.02 6 8 k k in in 4 6 0 5 10 15 0 5 10 15 r r in in Figure 3.Comparisonofthebinnedestimatedconvergence and theinputvalueforSet1withdifferentS/Nintherange[10,40]. Figure 5. Sizes estimates vs input sizes for four different PSF Slope and intercept of the fitting are shown (b and c, respec- scale-lengths between 0.1 and 7 pixels. Galaxies are disks with tively).Forerrors,seetext. S/N=20andmeansize7pixels.Slopeandinterceptofthefitting areshown(bandc,respectively). x 10−3 0.3 3 m c 0.2 m corrected 2 c corrected fwhm=0.0 fwhm=1.0 0.02 0.1 1 0.02 data data b=1.01 0.01 b=1.01 m 0 c 0 0.01 c=0.00 c=0.00 −0.1 −1 kout 0 kout 0 −0.01 −0.2 −2 −0.01 −0.02 0 10 20 30 40 50 −30 10 20 30 40 50 −0. 02 −0.01 0 0.01 0.02 −0.−002. 02 −0.01 0 0.01 0.02 S/N S/N k k in in fwhm=7.0 fwhm=17.5 Figure 4. m and c values computed with 200,000 galaxies of 0.02 0.02 data data Set 1. Triangles are for the values obtained with the sensitivity 0.01 b=0.78 0.01 b=0.55 correctionandsquareswithoutit. c=0.00 c=0.00 kout 0 kout 0 ground-basedsurveys,suchasCFHTLenSandfutureexper- −0.01 −0.01 imentsthesituationisnotsoclear,themeasurementwillbe −0.02 −0.02 morechallenging,andlargeempiricalbiascorrectionsofthe −0.02 −0.01 k0 0.01 0.02 −0.02 −0.01 k0 0.01 0.02 in in order of m ≃ −0.5 will be needed (see the first point of Fig. 7). Figure6.κestimatesvsinputvaluesforfourdifferentPSFscale- lengths.GalaxiesarediskswithS/N=20andmeansize7pixels. Dashedlineisκout=κinandthesolidlineistheleastsquaresfit, 3.3 Bulge fraction withslopeandintercept shownintheplots.Notethatb=m+1. In this test we generated galaxy images with bulges with different fractions of the total flux, to test the response to iments have bulge fractions lower than 0.5 (Schadeet al. thegalaxytype.InFig.8weshowthatgalaxysizeestimates 1996),thentheusefulpopulationcanbelargeenoughtodo for a bulge fraction of 0.2 is much better than for galaxies a successful analysis. with bulge-fraction of 0.95. This is because for bulge domi- nated models the central part of the galaxy becomes under sampled due to a limiting pixel scale. The poor estimation 3.4 Most Realistic Set of sizes is reflected in the κ estimation (Fig. 9). The parameters m and c for this set are shown in The last set includes realistic values for all the effects we Fig. 10, where we can see that for bulge-fraction greater investigate. We have generated elliptical galaxies with a than 0.8, the results are clearly biased (m =-0.25). For all bulgefractionof0.5convolvedwithananisotropicPSFwith bulge-fractions theerror bars are around 10%. FWHM of 4.5 pixels (1.5 times smaller than the character- Although for bulge-only galaxies, the κ estimates are istic scale-length of the disk), and again we investigate the poor, most of the galaxies used for weak lensing exper- dependenceonS/N.Thisisalsoachallengingtestforlensfit, (cid:13)c 0000RAS,MNRAS000,000–000 Size magnification as a complement to Cosmic Shear 7 x 10−3 x 10−3 4 4 0.6 m c 0.6 m c 0.4 m corrected 2 c corrected 0.4 m corrected 2 c corrected 0.2 0.2 m 0 c 0 m 0 c 0 −0.2 −0.2 −0.4 −2 −0.4 −2 −0.6 −0.6 −0.8 −4 −0.8 −4 −2 0 2 4 6 8 −2 0 2 4 6 8 −0.5 0 0.5 1 1.5 −0.5 0 0.5 1 1.5 ln(<r >/r ) ln(<r >/r ) bulge fraction bulge fraction d PSF d PSF Figure 7. m and c values computed with 200,000 galaxies of Figure 10. m and c values computed with 200,000 galaxies of Set2.Trianglesrepresentthevaluesobtainedwiththesensitivity Set3.Valuesobtainedwiththesensitivitycorrectionaremarked correctionandsquareswithoutit. bytrianglesandbysquaresarewithoutthecorrection. 12 BF=0.20 14 BF=0.95 1 5x 10−3 data data 10 bc==00..7857 12 bc==30..4773 0.5 mm corrected mm corrected 8 10 rout 6 rout 8 m 0 c 0 4 2 6 −0.5 0 4 0 5 r 10 0 5 r 10 −10 20 40 60 −50 20 40 60 in in S/N S/N Figure 8.Sizes estimates vs inputsizes fordifferent bulgefrac- tions in the range [0.2,0.95]. Bulge+Disk elliptical galaxies are Figure 11. m and c parameters for 500,000 galaxies of Set 4. used,withS/N=20andnegligibleeffectofthePSF(Set3).Slope Squares are raw m and c values; triangles have the sensitivity andinterceptofthefittingareshown(bandc,respectively). correctionincluded. thecurrentversion ofwhich usesasimplified parameterset 4 CONCLUSIONS AND DISCUSSIONS where the bulge scale length is assumed to be half the disk Inthispaperwepresentthefirstsystematicinvestigationof scalelength.Here,weincludeadispersioninthebulgescale theperformanceofaweaklensingshapemeasurementmeth- length of 0.6 pixels around a mean value of 3.5 pixels. The ods’ ability to estimate the magnification effect through an analysis was done with 500,000 galaxies, to keep the error estimateofobservedgalaxysizes.Weperformedthistestby bars smaller than 10%. In Fig. 11 are shown the values of creatingasuiteofsimulations,withknowninputvalues,and m and c for this set, with and without the sensitivity cor- byusingthemostadvancedshapemeasurementavailableat rection. If we compare it with the previous plots we can thecurrent time, lensfit. see that as the galaxies get more realistic the importance A full study of the magnification effect using sizes was ofthecorrection increases. Resultsforthisset areshownin performedtestingthedependenceonS/N,PSFsizeandtype Fig.11,showingunbiasedresultsexceptforS/N=10,which of galaxy. The requirements on biases on shear (or equiva- has m = −0.19±0.1. For the higher S/N points, we find lently convergence) for Euclid, so that systematics do not |m|<0.06 with errorbars of ±0.09. dominate the very small statistical errors in cosmological parametersarestringent:themultiplicativeandadditivebi- 0.03 BF=0.20 0.02 BF=0.95 ases are required to be |m| 6 2×10−3 and |c| 6 10−4. A data data much larger study will be required to determine whether 0.02 b=0.98 b=0.74 c=0.00 0.01 c=0.00 these requirements can be met for size, but we find in this 0.01 studynoevidenceforadditivebiasesatall,andnoevidence kout 0 kout 0 formultiplicativebiasprovidedthatthePSFissmallenough (<galaxy scalelength/1.5), the S/N high enough (> 15), −0.01 −0.01 and the bulge not too dominant (bulge/disk ratio < 4). −0.02 We find that if the disk is faint in comparison to the bulge −0.03 −0.02 (bulge/diskratio>1.5),theestimationofthehalf-lightdisk −0.02 −0.01 0 0.01 0.02 −0.02 −0.01 0 0.01 0.02 k k radiusisbiased.Thiseffectincreaseswiththebulge/diskra- in in tio,andwefindthatthebiasin theconvergenceestimation Figure 9. κ estimates vs input values for two different bulge can beimportant for bulge fraction > 0.8. fractions[0.2,0.95].Bulge+DiskellipticalgalaxieswithS/Nof20 Besides instrumental and environmental issues, there andnegligibleeffectofthePSF(Set3).Dashedlineisκout=κin can be astrophysical contaminants associated with weak andthesolidlineisthefitoftheoutputvalues.Notethatb=m+1 lensing. In the case of shape distortion, the first assump- ofeq.7. tion that galaxy pairs have no ellipticity correlation is (cid:13)c 0000RAS,MNRAS000,000–000 8 B. Casaponsa et al. not entirely accurate. There are intrinsic alignments of byMiller et al.(2012),whoanalysedthefitstogalaxieswith nearby galaxies due to the alignment of angular momen- i.25ofSimard et al.(2002)andestimatedβ≃0.29.Thus tumproducedbytidalshearcorrelations(IIcorrelations,see we expect this effect in a real survey to dilute the lensing for detections Brown et al. (2002); Heymanset al. (2004); magnificationsignalbyafactor0.42,butstillallowingdetec- Mandelbaum et al.(2011);Joachimi et al.(2011,2012),and tionoflensingmagnification.Inpractice,thedilutionfactor fortheoryHeavenset al.(2000);Catelan & Porciani(2001); could beevaluated by fittingto the size-flux relation in the Crittenden et al. (2001); Heymans& Heavens (2003)). In lensing survey. addition there can be correlations between density fields Lensingnumbermagnification surveysarealsoaffected and ellipticities (GI correlations, Hirata & Seljak (2004); by the problem that varying Galactic or extragalactic ex- Mandelbaum et al.(2006)).Theseintrinsiccorrelationshave tinction reduces the flux of galaxies and thus may cause a been studied in detail and it is not trivial to account for or spurious signal (e.g. Ménard et al. 2010). Such extinction remove them when quantifying theweak lensing signal. wouldalsoaffectthesizemagnificationofgalaxies,butwith Theintrinsiccorrelation of sizesand itsdependenceon a different sign in its effect. Thus a combination of lensing the environment, are still open issues. In fact, the corre- numbermagnification andsize magnification might bevery lation of sizes and density field, is known to play an im- effectiveat removing theeffects of extinction from magnifi- portantroleindiscriminatingbetweenmodelsofsizeevolu- cation analyses. tion;recentworkfindsasignificantcorrelationbetweensizes Space-basedsurveysasEuclidshouldovercomethelim- and the density field using around 11,000 galaxies drawn itationsthatwehaveexposedhere,havingalargenumberof fromthejointDEEP2/DEEP3data-set(Cooper et al.2012; galaxies,withS/N>10,andimportantlyaPSFatleast1.5 Papovich et al. 2012), while earlier studies with smaller smaller than the average disk size. The addition of the size samples have been in disagreement. Using 5,000 galax- information to the ellipticity analysis is expected to reduce ies of STAGES data-set, Maltby et al. (2010) find a pos- the uncertainties in the estimation of weak lensing signal, sible anti-correlation between density field and size for and therefore improve theconstraints of the distribution of intermediate/low-mass spiral galaxies. Clustered galaxies matter and dark energy properties. seemtobe15%smallerthantheonesinthefield,whilethey do not find any correlation for high-mass galaxies. Also for massive elliptical galaxies from ESO Distant Clusters Sur- ACKNOWLEDGMENTS vey, Retturaet al. (2010) do not find any significant correlation, while using the same data set Cimatti et al. (2012) We thank Catherine Heymans for interesting discussions. claims a similar correlation as in Cooper et al. (2012). In BC thanks the Spanish Ministerio de Ciencia e Innovación Park & Choi(2009)theystudythecorrelationbetweensizes for a pre-doctoral fellowship. We acknowledge partial fi- andseparationwithlateandearly-typegalaxiesfromSSDS nancial support from the Spanish Ministerio de Economa catalogue, at small and large scales. They compare the size y Competitividad AYA2010-21766-C03-01 and Consolider- ofthenearestneighbourwiththeseparationbetweenthem, Ingenio2010CSD2010-00064projects.TDKissupportedby andfindlargergalaxiesatsmallerseparations.Thiscorrela- a Royal Society University Research Fellowship. The au- tionisfoundforearly-typegalaxiesiftheseparationbetween thorsacknowledgethecomputerresourcesattheRoyalOb- the galaxies is smaller than the merging scale, but not for servatory Edinburgh. The author thankfully acknowledges larger separations. The size of late-type galaxies seems not the computer resources, technical expertise and assistance to have a correlation with the separation in any scale. 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