Sources of contamination to weak lensing three-point statistics: constraints from N-body simulations
Elisabetta Semboloni, Catherine Heymans, Ludovic van Waerbeke, Peter Schneider
aa r X i v : . [ a s t r o - ph ] F e b Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 11 November 2018 (MN L A TEX style file v2.2)
Sources of contamination to weak lensing three-point statistics:constraints from N-body simulations
Elisabetta Semboloni ⋆ , Catherine Heymans , , Ludovic van Waerbeke ,Peter Schneider Argelander-Institut f¨ur Astronomie, Auf dem H¨ugel 71, Bonn, D-53121, Germany. University of British Columbia, 6224 Agricultural Road, Vancouver, V6T 1Z1 B.C.,Canada. Institut d’Astrophysique de Paris, 81bis Bd. Arago, F-75014, Paris, France.11 November 2018
ABSTRACT
We investigate the impact of the observed correlation between a galaxies shape and its sur-rounding density field on the measurement of third order weak lensing shear statistics. Usingnumerical simulations, we estimate the systematic error contribution to a measurement of thethird order moment of the aperture mass statistic (GGG) from three-point intrinsic ellipticitycorrelations (III), and the three-point coupling between the weak lensing shear experienced bydistant galaxies and the shape of foreground galaxies (GGI and GII). We find that third orderweak lensing statistics are typically more strongly contaminated by these physical systemat-ics compared to second order shear measurements, contaminating the measured three-pointsignal for moderately deep surveys with a median redshift z m ∼ . by ∼ . It has beenshown that accurate photometric redshifts will be crucial to correct for this effect, once amodel and the redshift dependence of the effect can be accurately constrained. To this endwe provide redshift-dependent fitting functions to our results and propose a new tool for theobservational study of intrinsic galaxy alignments. For a shallow survey with z m ∼ . wefind III to be an order of magnitude larger than the expected cosmological GGG shear signal.Compared to the two-point intrinsic ellipticity correlation which is similar in amplitude to thetwo-point shear signal at these survey depths, third order statistics therefore offer a promisingnew way to constrain models of intrinsic galaxy alignments. Early shallow data from the nextgeneration of very wide weak lensing surveys will be optimal for this type of study. Key words: cosmology: theory - gravitational lenses - large-scale structure
Weak gravitational lensing represents a powerful tool to investi-gate the large-scale distribution of matter. The majority of lensingresults to date have focused on using two-point shear statistics toconstrain the matter density parameter Ω m and the matter powerspectrum normalisation σ (van Waerbeke et al. 2001; Hoekstra etal. 2002; Bacon et al. 2003; Jarvis et al. 2003; Hamana et al. 2003;Rhodes et al. 2004; Heymans et al. 2005; van Waerbeke et al. 2005;Massey et al. 2005; Semboloni et al. 2006; Hoekstra et al. 2006;Massey et al. 2007; Benjamin et al. 2007; Fu et al. 2008). As thesetwo parameters are strongly degenerate however there is great in-terest in measuring higher order statistics as their combination withthe two-point shear statistics can effectively break the degeneracybetween Ω m and σ (Bernardeau et al. 1999). To date, there havebeen few measurements of three-point shear statistics (Bernardeauet al. 2002; Jarvis et al. 2004; Pen et al. 2003). These results found ⋆ [email protected] that the three-point shear statistics were significantly affected bysystematics even when the two-point statistics showed a very lowsystematic level. The aim of this paper is to investigate, by using Λ CDM N-body simulations, whether the intrinsic alignment of thesources (see for example Heavens et al. 2000) and the correlationbetween the shear field and the intrinsic ellipticity of the sources(Hirata & Seljak 2004; Heymans et al. 2006; Mandelbaum et al.2006; Hirata et al. 2007) can explain the presence of some system-atics.In the weak lensing regime the observed ellipticity e o of asource galaxy is related to the original ellipticity e s through: e o ≃ e s + γ, (1)where γ = γ + i γ is the complex shear, and e is the complexellipticity defined as e = 1 − β β exp(2i φ ) R , (2)where φ is the angle between the semi-major axis and the x -axis, β is the ratio between the semi-major and semi-minor axis and R c (cid:13) is the response of a galaxy to weak lensing shear field. Note that inthe commonly used KSB method (Kaiser et al. 1995) the respon-sivity R is expressed by the polarizability tensor which is com-puted for each object and relates the measured weighted ellipticityto the shear field. In this analysis where no weight is applied we areable to calculate the average responsivity for our sample finding R = 0 . (see Rhodes et al. 2000, Bernstein & Jarvis 2002).The two- and three-point ellipticity correlation functions be-tween the observed galaxies a , b and c are given by : h e a o e b o i = h e a s e b s i + GI + h γ a γ b i , (3) h e a o e b o e c o i = h e a s e b s e c s i + GGI + GII + h γ a γ b γ c i , (4)where the GI, GII and GGI terms are: GI = h e a s γ b i + h e b s γ a i , (5) GII = h γ a e b s e c s i + h γ b e c s e a s i + h γ c e a s e b s i , (6) GGI = h γ a γ b e c s i + h γ b γ c e a s i + h γ c γ a e b s i . (7)One assumes that galaxies are intrinsically randomly oriented sothe measurement of the observed ellipticity is an unbiased estima-tor of the shear. In a similar way one assumes that the measurementof the correlation between ellipticity of pairs and triplets of galax-ies is an unbiased estimator of the second and third order momentsof the weak lensing shear field. However, this assumption is notcorrect for galaxies that are physically close as an intrinsic align-ment between the galaxies can be induced by a tidal field whichacts on cluster scales as has been observed by Brown et al. (2002)and Mandelbaum et al. (2006). These results imply that the firstterm of Equation (3), hereafter referred as to II, and the first termof Equation (4), hereafter referred as to III, are non-zero and theysystematically affect the value of the two- and three-point weaklensing shear statistics estimated through the observed ellipticityalignment. It has been shown (King & Schneider 2002, 2003, Hey-mans & Heavens 2003, King 2005) that this bias can be removedif close pairs of galaxies are downweighted in the lensing analy-sis, something which is possible only when the redshift for eachindividual galaxy can be reliably estimated. For moderately deepsurveys such as CFHTLS Wide (Fu et al. 2008), the intrinsic align-ment does not significantly affect the estimation of the normalisa-tion of power spectrum σ . However, once the redshift informationis added it is possible to exploit all the information carried by theshear signal, by using three-dimensional measurements. The mea-surement of the two-point shear statistics in redshift bins, i.e. to-mography (Semboloni et al. 2006) and the 3D-lensing (Heavens etal. 2006, Kitching et al. 2007), are particularly promising to con-strain the equation of state of dark energy, but are also more suscep-tible to stronger contamination by intrinsic alignment (Heymans etal. 2004, King 2005).A more subtle effect is the correlation between the inducedweak lensing shear and the foreground intrinsic ellipticity field, alsocalled “shear-shape” correlation (Hirata & Seljak 2004). This effectcan be explained as follows. Consider a background galaxy withredshift z a whose light is deflected by a foreground over-density at z b . Consider now a galaxy at redshift z b stretched by the tidal forcesgenerated by the dark matter over-density. Then the term h γ a e b s i 6 =0 . Similarly, for the three-point shear statistics, one can considertwo sources with redshift z a and z b whose light is deflected by thesame halo hosting a galaxy at redshift z c < z a , z b : in this case theterm h γ a γ b e c s i 6 = 0 . Finally, one can imagine a background galaxywith redshift z a being sheared by an over-density hosting galaxiesat redshifts z b and z c with the result h γ a e bs e cs i 6 = 0 .The first measurement of the shear-shape systematic effect on the Sloan Digital Sky Survey (SDSS) data by Mandelbaum etal. (2006) has been refined by Hirata et al. (2007), which usessubsamples from the SDSS and 2SLAQ surveys to check the de-pendence of the shear-shape correlation on the color, morphologyand luminosity of galaxies. Hirata et al. (2007) predict that theshear-intrinsic alignment coupling could cause an underestimate ofthe normalisation of the power spectrum σ of up to ten percent.Heymans et al. 2006 (hereafter H06) provide a simple toy modelto investigate this effect using N-body simulations which we de-scribe in more detail in section 3. This simple model also predictsa shear-intrinsic alignment of of the expected weak lensingshear signal, and provides a good fit to the measurements of Hi-rata et al. (2007). In contrast to the intrinsic alignment systematicthat affects the lensing measured in the same tomographic redshiftslice, the shear-shape systematic affects galaxies in very differenttomographic redshift bins. The shear-shape systematic strongly re-duces the precision with which one is able to constrain the dark en-ergy equation of state by tomography, as shown by Bridle & King(2007).Adopting the same strategy as H06, we analyse a set of Λ CDMN-body simulations, in order to compare the amplitude of the in-trinsic ellipticity and the shear-shape terms to the three-point weaklensing shear statistics.The paper is organised as follows. In the section 2 we definethe quantities and we describe the method used in this work to mea-sure three-point statistics. In section 3 we describe the simulationsused in this work. In section 4 we present measurements of thethree-point intrinsic alignment, and in section 5 we show the evolu-tion of the three-point coupling between shear and intrinsic elliptic-ity fields as a function of the redshift distribution of the lenses andsources. We also provide a fitting formula for the three-point shear-ellipticity correlation from different galaxy models. We concludein section 6.
Following the approach of Pen et al. (2003) and Jarvis et al.(2004) we define the complex ‘natural components’ of the three-point shear correlation functions (Schneider & Lombardi 2003)for a triangle of vertex X , X and X and separations vectors r = X − X r = X − X , r = X − X : Γ ( r , r , r ) = h γ ( r ) γ ( r ) γ ( r )e [ − φ + φ + φ )] i , (8) Γ ( r , r , r ) = h γ ⋆ ( r ) γ ( r ) γ ( r )e [ − − φ + φ + φ )] i , (9) Γ ( r , r , r ) = h γ ( r ) γ ⋆ ( r ) γ ( r )e [ − φ − φ + φ )] i , (10) Γ ( r , r , r ) = h γ ( r ) γ ( r ) γ ⋆ ( r )e [ − φ + φ − φ )] i , (11)where γ is the complex shear and we indicate with γ ⋆ the com-plex conjugate of γ . The choice of the directions φ i along whichone projects the shear are free as long as the value of Γ i does notdepend on the triangles orientation. Assuming that the Universe ishomogeneous and isotropic, the eight real components of the shearcorrelation function depend on the side-lengths of the triangle, r , r , and r . Thus we prefer to use a derived statistic which is eas-ier to visualize and interpret, namely the third order moment of theaperture mass statistic.The complex aperture mass is defined as : M ϑ = M ap + i M × = Z d r Q ϑ ( r ) γ ( r ) e ( − φ ) , (12)where Q ϑ ( r ) is a filter of characteristic size ϑ . Using the aperture c (cid:13) , 000–000 mass statistics allows one to uniquely separate the E-mode ( M ap )and B-mode ( M × ) component of the measured shear (Crittenden etal. 2002), providing a powerful test for systematics. Indeed, weaklensing shear fields are E-type fields; thus the presence of othersources of distortion can be revealed by measuring M × statisticswhich are non-zero only for B-mode fields.In practice, the variance and the third order moment of theaperture mass can be measured as a function of the two- and three-point shear statistics, respectively (Crittenden et al. 2002; Pen etal. 2002). Estimating the third order moment of the aperture massthrough three-point correlation functions is preferred however, asit yields a higher signal-to-noise ratio than the direct measurementof the moments of the integral of Equation (12) (Schneider et al.2002). In addition, in the case of real data, where masked regionsare present, the estimate of the aperture mass statistic through themeasurement of the correlation function is unbiased, whilst the di-rect measurement of the moments is biased. By choosing the fol-lowing filter, Q ϑ ( r ) = 12 πϑ “ − r ϑ ” exp “ − r ϑ ” , (13)the four components of the third order moment h M i , h M M × i , h M ap M × i and h M × i can be obtained throughintegration of the three-point shear statistics Γ i (see for exampleequations (45), (50) of Jarvis et al. 2004 and equations (61-71) ofSchneider et al. 2005). There are other advantages of using aperturemass statistics defined by using Equation (13). This filter has infi-nite support but the exponential cutoff allows one to assume a finitesupport for real calculations. Furthermore, the Fourier transformedfilter, I ( η ) = 12 π η − η ) (14)were we defined η = sϑ , is a very narrow window filter, probingessentially modes with s ∼ /ϑ .We measure the four complex Γ i components using a binarytree code built following the model suggested by Pen et al. (2003).We assign to each box an ellipticity ( e , e ) and a position ( x, y ) which are the average of the ellipticity and positions of the galaxiescontained in the box. To each box we assign also a weight which isthe sum of the weights assigned to each galaxy. The characteristicsize l i of each box is chosen to be the distance between the centreof the box and the furthest galaxy contained in the box. The cor-relation between triplets is computed for triangle of sizes l , l , l satisfying the conditions: l + l r < . , l + l r < . , l + l r < . . (15)Each triangle is ordered such that r r r . We measure Γ i for triangles with r between and
80 arcmin , rangingin logarithmic bins of width log( r ) = 0 . . With this choice, wecan assume that the distance between each pair of galaxies withinthe boxes that we are correlating has a maximum error of one bin.Finally we integrate Γ i using equation (45) and equation (50) ofJarvis et al. (2004). As we already anticipated, the filter defined byEquation (13) for a given size ϑ has infinite support so we ideallywould need to measure Γ i at all scales to perform the integral giv-ing the components of the the third-order moment of the aperturemass. However the integral is significant only for triangles up to r ≃ ϑ , implying that our measurement of the third order momentis reliable up to an angular scale ϑ ≃
20 arcmin . We tested our al-gorithm against the direct measurement of the third-order momentof the aperture mass on Λ CDM simulations in fields of
25 deg and we found indeed a good agreement between the two methodsat these scales.Throughout the paper we use only the aperture mass statisticsand we call the systematic produced by the intrinsic alignments onthe second and third order moments of the aperture mass II andIII, respectively. Similarly, we call GI the second order momentmeasured by using the filter defined by the Equation (13) producedby the shear-shape coupling and we call the third order momentsGGI and GII. Finally, we call the second and third order momentsof aperture mass produced by a weak lensing field GG and GGG.In order to know the importance of the GGI, GII and III whenestimating the third order moment of the aperture mass GGG usingthe observed ellipticity of galaxies we have to compare those termswith the third order moment of the aperture mass produced by aweak lensing shear field. In the quasi-linear perturbation theory,the perturbation of the density field δ is considered to be small sothat it can be developed in a series δ = δ (1) + δ (2) + .... , where δ (1) is the linear evolving density field and δ ( n ) ∝ O (( δ (1) ) n ) . Inthis approximation the h M i ϑ weak lensing signal is (Fry 1984,Schneider et al. 1998) : GGG ≡ h M i ϑ = 12 π H c Ω Z w H dw g ( w ) a ( w ) f κ ( w ) (16) × Z ∞ d s P “ s f κ ( w ) , w ” I ( s ϑ ) × Z ∞ d s P “ s f κ ( w ) , w ” I ( s ϑ ) × I ( | s + s | ϑ ) F ( s , s ) , with g ( w ) = Z w H w dw ′ p s ( w ′ ) f κ ( w − w ′ ) f κ ( w ′ ) . (17) f κ ( w ) is the comoving angular diameter distance, P ( s, w ) is the3 dimensional power spectrum of matter fluctuations, p s ( w ) is thecomoving distance distribution of the sources, I ( sϑ ) is the Fouriertransform of the filter as defined by Equation (14) and F ( s , s ) is the coupling between two different modes of density fluctua-tions characterised by the wave vectors s and s . We compute F ( s , s ) using the fitting formula suggested by Scoccimarro &Couchman (2001). In this analysis we use the same set of simulations used as in H06.We recall here only the main characteristics and refer the reader toH06 for more detailed information. The N-body simulation is a boxof h − Mpc realized using a Λ CDM cosmology. The matterdensity parameter is Ω m = 0 . , the cosmological constant is Ω Λ =0 . , the normalisation of the power spectrum of matter fluctuationsis σ = 0 . , the reduced Hubble constant is h = 0 . and thebaryon density parameter is Ω b h = 0 . . The simulation startedat redshift z = 60 and evolved to z = 0 . Twelve lines of sightwere built by stacking boxes back along the z -axis with randomshifts between the boxes in the x and y direction in order to avoidartificial correlations, producing (almost) independent realisations.The over-densities are identified using a ‘friends of friends’group finder, which allows one to identify the halos. The particlemass of . × h − M ⊙ allows us to find bound halos withmasses larger than few h − M ⊙ . The halos are then populatedwith galaxies. The luminosity is assigned following the conditional c (cid:13) , 000–000 Figure 1.
Left panel: h M i signal generated by the intrinsic alignments (III) as a function of the angular size ϑ . By comparing the E-modes (black diamond)with the expected shear signal (black solid line) one can determine the level of contamination produced by the existence of intrinsic alignments betweengalaxies. The shear signal has been computed using Equation (16) for the same redshift distribution used to compute the intrinsic alignment term III. Themedian redshift of this distribution is z m ∼ . . The error-bars are the dispersion of the average signal measured in twelve independent realizations. TheB-mode signal, h M × i , and the E/B components, h M M × i and h M ap M × i , are shown dashed and for clarity we chose not to plot the error bars. The rightpanel shows the same results as the left panel for a lower redshift sample of galaxies < z < . . For this redshift distribution characterised by z m ≃ . ,the predicted shear signal GGG ∼ − is indistinguishable from zero in this plot. luminosity function (CLF) of Cooray & Milosavljevic (2005) andthe ellipticity is assigned using a toy model that has been showneffective at reproducing the observations of Mandelbaum et al.(2006) and Hirata et al. (2007). In this model, elliptical galaxiesare given the same ellipticity as their parent halos. Spiral galaxiesare modeled as a thick disk oriented almost perpendicular to the an-gular momentum vector of the halo with a mean random misalign-ment of ∼
20 deg (van den Bosch et al. 2002, Heymans et al. 2004). At each identified dark matter halo in the simulations we generateboth a spiral and an elliptical galaxy so that we can investigate theresults as a function of morphology. We define the ellipticity of themodel galaxies using Equation (2). The resulting ellipticity distri-bution has a zero mean and dispersion of σ e = 0 . .In the results that follow we present three models: one com-posed exclusively of spiral galaxies, one composed exclusively ofelliptical galaxies and one containing both the morphologies de- c (cid:13) , 000–000 noted ‘mixed’. The mixed model has been built following Cooray& Milosavljevic (2005) and it contains ∼ elliptical galaxies.The mixed model is the most realistic model out of the three tested;the GI and II components predicted using the mixed model agreewith the ones measured using the SDSS data (Mandelbaum et al.2006, Heymans et al. 2006, Hirata et al. 2007). The number densityof the final catalogue, which contains galaxies between < z < and is complete up to r = 25 . , is about 5 per arcmin , which isconsiderably lower than what is expected for such a survey. Thisis due on one hand to the fact that the low resolution of the sim-ulations implies the loss of low-mass halos (only halos with masslarger than few M ⊙ are identified) and that each halo is pop-ulated with only one galaxy. The lack of the low-mass halos andsatellite galaxies is the main limitation of our model and this pointis discussed further in the conclusions.Finally, these same N-body simulation which have been popu-lated with galaxies are ray-traced to produce twelve projected massdistributions κ for a source plane at z s = 0 . and for z s = 1 . .We use these simulated κ maps in section 5 to study the shear-shapeeffect. Each projected mass map covers an area of × in × pixels. In this section we report the third order moment components of theaperture mass given by the intrinsic alignment of the sources. Wemeasure the third order moment of the aperture mass for a surveywith maximum depth z ∼ and compare the measured intrinsicalignment to the expected weak lensing GGG signal from Equation(16). The redshift distribution of the sources p s ( w ) is computeddirectly from the catalogues used for the III measurement so thatboth III and GGG terms have been computed for the same survey.Figure 1 shows the three-point intrinsic alignment for ourthree toy models: elliptical (upper panel), spiral (middle panel) andmixed (lower panel) for two survey types; one with a median red-shift of z m ∼ . (left panels) and the other with z m ∼ . (rightpanels). The error-bars on the measurements have been computedas the dispersion in the average signal measured in the twelve real-isations, thus they include statistical and cosmic variance. For themoderately deep survey ( z m ∼ . ) the III term is consistent withzero for all the three models. This result is in agreement with H06who find that the two-point intrinsic ellipticity term II, is consistentwith zero for the same survey characteristics. As expected, how-ever, we find that when the redshift depth of the survey decreasesthe III term becomes significant due to the fact that for lower red-shift galaxies, a given angular distance corresponds to a physicallycloser triplet. Comparing now the results for the three different toymodels for this shallow survey we find very different results. Fig-ure 1 shows that the III term is significantly positive for ϑ > arcmin for elliptical galaxies (upper right panel) meanwhile it isslightly negative for the spiral model (middle right panel) and sig-nificantly negative for the mixed model (lower right panel). Themixed model contains roughly of spiral galaxies and ofelliptical galaxies, such that the most frequent triangle in the three-point measurement contains two spiral galaxies and one ellipticalgalaxy. As this type of correlation does not exist when one consid-ers only elliptical or only spiral morphologies, the mixed result isnot a weighted average of the result containing only elliptical orspiral galaxies. We verified that the third order moment of the aper-ture mass is indeed negative when we correlate triplets containingtwo spirals and one elliptical. Figure 1 shows that for shallow sur- Figure 2.
Left side: ratio between the intrinsic alignment (II) and the aper-ture mass variance (GG) for three surveys. Right side: ratio between III theexpected weak lensing signal (GGG) for the same surveys as the left panel.The galaxy model is a mixed model. The expected second and third orderaperture mass moment are computed for each survey by using the samesource redshift distribution used to measure the III and II terms. These dis-tributions are characterized by: z m ∼ . for source distribution between < z < . , z m ∼ . for source distribution < z < . and sourcedistribution z m ∼ . for < z < . . veys ( z m < ∼ . ) the intrinsic alignment dominates the signal andsuggests that also for deeper surveys the intrinsic alignment couldaffect the measured the three-point weak lensing signal.In order to quantify the effect of the intrinsic alignment onthe weak lensing shear statistics we compare the amplitude of thetwo- and three-point intrinsic alignment signal II and III with theexpected weak lensing signal, respectively GG and GGG, for dif-ferent shallow surveys. Figure 2 shows the amplitude of the II/GG(left panel) and III/GGG (right panel) ratios for three different sur-veys depth using the mixed model as an example. We find that fora given survey depth, the ratio between the III term and expectedweak lensing third order moment GGG, is generally higher than theratio between the II term and the variance of the weak lensing aper-ture mass GG. One can think to reduce significatively the intrinsicellipticity contamination to the two- and three-point shear statisticsby removing very low redshift sources ( z < ∼ . ) which contributeto the intrinsic alignment signal but not to the weak lensing signal.We checked that the ratio II/GG and III/GGG drops on small scales ϑ < ∼ arcmin if galaxies with z < . are rejected, but it is almostunchanged at larger scales.Figure 1 and Figure 2 show that without a technique to cor-rect for the intrinsic alignment it will be impossible have a precisemeasurement of the three-point shear statistics in shallow surveys( z m < ∼ . ). For moderately deep surveys like the CFHTLS Wide,the ratio III/GGG is consistent with zero as shown by the left panelsof Figure 1. However, for larger surveys, where the statistical andcosmic variance are small the intrinsic alignment could still play arole in the precision with which one can constrain the cosmologi-cal parameters. This can be seen in Figure 2 where for a relativelydeep survey ( z < . ) the intrinsic alignment contribution is likelyto be non-zero, specially for the third order moment of the aperturemass.This result strongly supports the need for reliable photometricredshift for weak lensing studies. Moreover, Figure 2 demonstratesthat if the redshift is known, one can select sources in order to en-hance the II and III signal relative to the contribution of the GGand GGG terms to the measured two and three-point shear statis- c (cid:13) , 000–000 tics. This is an important result since one needs a good intrinsicalignment model to be able to remove this contribution from thetwo-point (three-point) weak lensing shear statistics (King 2005).The fact that the III/GGG signal is more significant than the II/GGsignal implies that it may be easier to study intrinsic alignments us-ing three-point statistics. In this respect Figure 2 shows that largemulti-band surveys like SDSS and KIDS or PanSTARRS, representexcellent surveys to investigate and model the intrinsic alignmentof elliptical and spiral galaxies. In this section we study the behavior of the three-point shear-shapecoupling which, as it has been detailed in equations (6) and (7), canbe divided in two terms, namely GGI and GII. One would expectthese two terms to have a different behavior as a function of theredshift distribution of the lenses and of the sources. The GII termdepends on the intrinsic ellipticity correlation between two pairs ofgalaxies thus it should be significant only for triplets which containforeground galaxies closer than the scales on which tidal forces act.The GGI term correlates the shape of lensed background galaxieswith the intrinsic shape of a foreground galaxy so it is expected tobe significant also when correlating well-separated redshift slices,similarly to what has been found for the GI term (Bridle & King2007). Using both ray tracing simulations with redshift z s = 0 . and z s = 1 . and dividing the foreground galaxies into redshiftbins, we study the evolution of the GII and GGI terms as a func-tion of the source and lens redshift distribution. Figure 3 showsthe amplitude of the GGI (left panels) and GII (right panels) termsfor elliptical (upper panels), spiral (middle panels) and the mixedmodel (lower panels) for different redshift bins. We note that theamplitude of the GGI and GII terms for a given survey dependson the morphology of the galaxies. The elliptical galaxies show asignificant negative GGI signal whereas the spiral model shows aGGI signal which is consistent with zero. Finally, the mixed modelshows a slightly negative GGI signal.The GII component for the spiral galaxies show an angulardependence similar to the one of the elliptical galaxies and inter-estingly the signal at intermediates scales ( < ∼ ϑ < ∼ arcmin) isstronger than for the elliptical galaxy model. Similarly to what wepreviously found for the III term, the mixed model GII term is dif-ferent from the one obtained using only elliptical or spiral galaxies.This is likely to be a consequence of the fact that for the two mor-phologies the ellipticity depends differently on the properties of theparent halos, which also could explain the difference in results forthe III term. To understand this phenomena would require build-ing a model for the three-point correlation function between weaklensing shear and tidal field, in a similar analysis as that of the two-point correlation function presented in Hirata & Seljak (2004). Thishowever is beyond the scope of this paper.In order to allow a more quantitative comparison between theweak lensing three-point shear signal and the systematics GII andGGI, we report in Figure 4 the ratio between the terms GII andGGG (left panel) and GGI and GGG (right panel) for the mixedmodel. To aid future comparisons we provide fits to our results. Wemake the assumption that for a given triplet the GGI and GII termscan be factorized in two functions: one depending on the comovingdistances of lenses and sources modeled as in King (2005) and theother depending on the angular scale. We rewrite the GGI term as: GGI( w s , w s , w L , ϑ ) = E GGI ( w s , w s , w L ) F ( ϑ ) , (18) with: E GGI ( w s , w s , w L ) = Z w L dw l f k ( w s − w l ) f k ( w s ) (19) × f k ( w s − w l ) f k ( w s ) p l ( w l ) , where w L is the maximal comoving distance of the lenses with thecondition w L < min( w s , w s ) , f k ( w ) is the angular diameterdistance, p l ( w l ) is the radial lens distribution and F ( ϑ ) is a genericfunction of the angular scale.Similarly we factorize GII with the assumption that the intrin-sic alignment acts only for pairs of galaxies with the same redshift.Thus we write: GII( w s , w L , ϑ ) = E GII ( w s , w L ) F ( ϑ ) , (20)with: E GII ( w s , w L ) = Z w L dw l f k ( w s − w l ) f k ( w s ) p l ( w l ) , (21)with the condition w L < w s . For a broad source distribution oneshould integrate Equation (19) and Equation (21) over the sourcedistribution to obtain the average effect. However in our case theray-tracing, source planes follow a Dirac distribution centered at z s = 0 . or z s = 1 . . The lens distribution p l ( w l ) is measureddirectly from the galaxy catalogues. We find that these scaling fac-tors provide a good description of the redshift dependence both forthe GII and GGI term. With this redshift dependence model we tryto model the behavior of the GII and GGI terms as a function of theangular scale. We use a two parameter function F ( ϑ ) : F ( ϑ ) = A exp( − ϑ/ϑ ) (22)where A and ϑ are free parameters.In order to avoid bias from the limiting resolution of the simu-lations we chose to perform the fit using only angular scales ϑ > arcmin. In Figure 3 we compare the measurements of the GGI andGII terms with the best-fit model (solid lines) obtained by using theredshift rescaling defined by the equations (19) and (21) and an an-gular scale dependence defined by Equation (22) for the elliptical,spiral and mixed galaxies. Figure 5 shows the best fit parameters A and ϑ for the GGI (left panel) and the GII (right panel) terms forseveral redshift slices, for elliptical, spiral and mixed model. Thesevalues are also summarized in the table 1 which includes also thereduced χ of the fit for both the GGI and GII terms. The smallvalues of the χ show that the model we suggest is a good fit tothe measured GGI component. However, for the elliptical modelthe dispersion of the best-fit A parameter between the four red-shift slices (see left panel of Figure 5) is larger than the error bars.This suggests that the redshift-dependent rescaling described by theEquation (19) could require some modification. The GII compo-nent is generally more noisy than the GGI component, and for thisreason it is hard to establish whether the model defined by equa-tions (21) and (22) is a good one. However, for the mixed model,which is our most realistic model, the GII term can be fairly welldescribed by our fitting function. Increasing the size of simulationsin future analyses will allow us to improve upon these models.Because of the different dependence on the morphology ofgalaxies and on the redshift distribution of the sources and lensesfor the III, GGI and GII terms, one may be interested in knowingthe total effect of the intrinsic alignments and of the shear-couplingon the three-point shear for realistic surveys. Figure 6, shows theratio between the sum of GGI, GII and III terms and the expectedweak lensing signal for several redshift distributions for both theelliptical (left panel) and mixed (right panel) model. c (cid:13) , 000–000 Figure 3.
GGI (left panels) and GII (right panels) components for different redshift slices and three galaxy models: elliptical (upper panels), spiral (middlepanels) and mixed (lower panel). We show three sets of measurements corresponding to: a lens distribution < z l < and a source distribution z s = 1 . (black diamonds), < z l < . and z s = 1 . (red triangles) and < z l < . and z s = 0 . (blue squares). For the GGI component we also includea third model which contains lenses < z l < . and sources in two different planes: one at z s = 0 . and the other at z s = 1 . . Error bars representthe dispersion in the average value between the twelve simulations. For each measurement of the GGI and GII components we show the best-fit model (solidlines) obtained by rescaling the redshift distribution dependence given by equations (19) and (21) with the angular dependence described by Equation (22). For shallow surveys ( z l < ∼ . ), the ratio is largely dominatedby the III term. This is true both for elliptical and mixed modelwith the difference that for the elliptical galaxies the intrinsic align-ment enhances the weak lensing shear signal whereas for the mixedmodel it suppresses the weak lensing signal. For a deep survey z m ≃ . the ratio is dominated by the GGI term, whose amplitudeis of the GGG signal for elliptical galaxies and few percentfor mixed model.In order to compare our results to observations we present in in Figure 7 the h M i results by Jarvis et al. (2004) and the h M i that we would expect for this same survey from weak gravitationallensing and the III, GGI and GII terms. For this comparison wehave used all of the three galaxy models. We compute the GGGsignal using a redshift plane at z s = 0 . as done by Jarvis et al.(2004). We add the III model using galaxies with z < . so thatthe redshift distribution is characterized by the a median redshiftsimilar to the one of the CTIO galaxy catalogue. We use Equation(19) and Equation (21) to rescale the GII and GGI to a survey with c (cid:13) , 000–000 Table 1.
Summary table of the best fit parameters of the GGI (first four columns) and GII (last three columns) terms for elliptical (upper lines), spiral (middlelines) and mixed (lower lines) galaxies using Equation (19) and Equation (21). The values the parameters ϑ are given in arcmin, the values of the parameter A are given in units of − h Mpc − . For each model we report the value of the reduced χ .GGI GII z s = 1 z s = 1 z s = 0 . z s = 1; 0 . z s = 1 z s = 1 z s = 0 . z l < z l < . z l < . z l < . z l < z l < . z l < . ϑ . ± .
38 4 . ± .
33 4 . ± .
45 4 . ± .
42 0 . ± .
02 0 . ± .
34 0 . ± . elliptical A − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . χ .
11 0 .
59 0 .
31 0 .
52 4 .
34 0 .
62 0 . ϑ . ± .
92 4 . ± .
75 3 . ± .
73 0 . ± .
73 0 . ± .
66 1 . ± .
70 0 . ± . spiral A − . ± . − . ± . − . ± . − . ± .
12 0 . ± .
11 0 . ± .
19 1 . ± . χ .
14 0 .
36 0 .
23 0 .
24 0 .
74 0 .
48 0 . ϑ . ± .
40 1 . ± .
24 1 . ± .
83 1 . ± .
52 1 . ± .
88 4 . ± .
90 4 . ± . mixed A − . ± . − . ± . − . ± . − . ± .
22 0 . ± .
07 0 . ± .
02 0 . ± . χ .
55 0 .
51 0 .
30 0 .
49 0 .
29 0 .
55 0 . Figure 4.
Left side: ratio between the GII term and the expected signal(GGG) for three shallow surveys. Right side: ratio between the GGI termand the GGG signal for the same surveys as the left panel. The galaxy modelis a mixed model. We used the same surveys as Figure 3. The GGG signalhas been computed by taking z s = 1 . (black diamonds and red triangles)and z s = 0 . (blue squares). z s = 0 . and z l < . . For this survey we find that the dominantcontribution is given by the GGI which slightly decreases the esti-mated the weak lensing signal. We find that all three galaxy modelsare consistent with the Jarvis et al. (2004) results. Using a set of realistic N-body Λ CDM simulations we have ex-plored the effect of intrinsic galaxy alignments and the couplingbetween weak lensing and the foreground ellipticity field on thethird order moment of the aperture mass. We find that the intrin-sic alignment dominates the three-point shear signal for shallowsurveys. For deeper surveys the intrinsic alignment is less signif-icant, as a result of projection effects. Nevertheless, if not takeninto account properly, it will still limit the accuracy of tomographymeasurement and affect the predictions of cosmological values forthe next generation of very large surveys, where the signal to noiseratio is high.We found that for a given survey depth intrinsic alignments affect the three-point weak lensing statistics more strongly thanthe two-point shear statistics. In other words, in order to achievethe same level of accuracy in the three-point shear statistics mea-surement as the two-point weak lensing shear statistics one needsdeeper surveys.Overall, this result shows once more the importance of theknowledge of redshift for each individual galaxy for high-precisioncosmology. Knowing the redshift of each source allows one to re-move the bias from intrinsic galaxy alignments by discarding phys-ically close triplets (pairs) when computing the three-point (two-point) shear statistics. Moreover the knowledge of redshift allowsone to model the intrinsic alignment and remove its effect on thetwo-point weak lensing shear statistics (Joachimi & Schneider, inprep.).In this perspective, the measurement of the three-point shearstatistics from a shallow survey offers a particularly effective wayto test intrinsic alignment models. As we showed in this work,the three-point shear statistics in low redshift bins are dominatedby the intrinsic alignment term, permitting accurate measurementson intrinsic alignment models. As future dark energy surveys willrely heavily on the good modeling of these effects so they can bemarginalised out, this result provides an important new route toconstrain and model this physical systematic effect. It is indeedpossible to choose low-redshift bins in order to enhance the intrin-sic alignment signal so that the shear signal becomes negligible; forexample, by selecting galaxies at z < . . In this case the III/GGGratio is around fifty, allowing one to study the intrinsic alignment,essentially without contamination from the weak lensing shear.Using projected mass maps we have studied the shear-shapecoupling effect which is also likely to bias the measurement of thethird order moment of weak lensing shear. We showed that thissystematic can be described by two terms, GGI and GII, which af-fects the third order moment measurement in a non-trivial way andis dependent on the distance between the lenses and sources andon the morphology of galaxies. Even for a moderately deep surveylike the CFHTLS Wide the amplitude of the third order momentof the shear estimation could be underestimated by ∼ − .For shallower surveys such as KIDS or PanSTARRS-1 the bias isexpected to be higher. Our results show that it will not be possibleto carry out precise three-point cosmic shear measurements withthese surveys without modeling the coupling between weak lens-ing shear and intrinsic alignment. Also for the next generation of c (cid:13) , 000–000 Figure 5.
Left panel: best-fit values for the parameters ϑ and A of Equation (22) used to fit the angular dependence of GGI for the four redshift distributions;we chose to indicate each source/redshift distribution using the same symbols as Figure 3. Right panel: shows the same parameters as left panel but now forthe GII component. Figure 6.
Ratio between the sum of the terms GII, GGI and III and the weak lensing third order moment signal GGG, for the elliptical (left panel) and mixedmodel (right panel). Three cases are shown: z s = 1 , < z l < (black diamonds), z s = 1 , < z l < . (red triangles) and z s = 0 . , < z l < . (bluesquares). For the first and second case the GGG component is computed assuming z s = 1 . . For the third case z s = 0 . . deep large surveys, such as SNAP, DUNE or LSST the precisionone can achieve on the cosmological constraints relies in the abilityto model and marginalize out the shear-shape and intrinsic correla-tions.It is possible to use simple models to describe both the angularand redshift dependence of the GGI and GII terms. These modelscan be used to correct the effect of the coupling between shear andintrinsic alignment on the third order moment of the aperture mass.Unfortunately, the sample available for this work is not bigenough to give a definitive answer; more precisely, the parametricmodels we used to fit the GGI and GII components are marginallyconstrained, due to the large statistical and sampling variance af-fecting our sample. A more detailed study of the redshift depen-dence of the shear-ellipticity correlation will be required. Such astudy should use a larger sample of simulations in order to bet-ter constrain our models. Furthermore, we still need to develop amethod to include satellite galaxies and account for environmen-tal effects when determining the ellipticity of galaxies in a singleparent halo. In addition, an improved mixed galaxy population as afunction of redshift evolution would also play an important role inestablishing the correct average effects, as we have shown in thispaper the net effect depends on the morphology of the galaxy pop-ulation. We also think that a comparison between results from sim-ulations with results from real data is the best way to validate our results. Concerning this point, the agreement between simulationsand real data shown in the study of the II (H06) and GI (Hirata etal. 2007) components is a good indication of the fact that the sim-ulation used in this work, even if it represents a simplified model,shows the correct dependence of the intrinsic alignment effect onmorphology of the galaxies, luminosity and redshift.Like other previous works on intrinsic alignment and shear-shape coupling, this paper demonstrates the great importance of re-liable redshifts for future and current galaxies surveys, which couldbe used, in parallel with simulated catalogues, to study and correctfor intrinsic alignment and shear-shape coupling on the two- andthree-point weak lensing statistics. We thank Alan Heavens and Ismael Tereno for helpful sugges-tions on this project and Asantha Cooray for making code basedon the CLF publicly available. We also would like to thank MartinWhite for generating the N-body simulations used for this work.These simulations were performed on the IBM-SP at NERSC. ESis supported by the Humboldt Foundation. CH is supported by theEuropean Commission Programme in the framework of a MarieCurie Fellowship under contract MOIF-CT-2006-21891. LW ac- c (cid:13) , 000–000 Figure 7.
Third order moment of the aperture mass h M i ϑ mea-sured by Jarvis et al. (2004) on the CTIO data (pink triangles) com-pared with the expected measured third order aperture mass statistics, i.e.GGG+III+GII+GGI, for the three models of galaxies used in this paper. Foreach model we show a lower and upper value of the total third order momentgiven by the average ± σ error bars. This error is computed as a quadraticsum of the error affecting each term. The GGG weak lensing signal is thesame used in Jarvis et al. (2004), i.e. the one for a single source redshiftat z s = 0 . . The III signal has been computed using a galaxy redshiftdistribution with z l < . which is characterized by the similar medianredshift z m ∼ . used to compute the GGG signal. The GII and GGIterms have been rescaled by using the equations (19) and (21) for a surveywith z s = 0 . and z l < . . For comparison we plot the expected weaklensing h M i ϑ (black solid line). knowledges support from NSERC and CIAR. This work has beensupported by the RTN Network DUEL and the DFG through theTR33 ‘The Dark Universe’ and the project SCHN 342/6–1. REFERENCES
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