Controlling intrinsic-shear alignment in three-point weak lensing statistics
aa r X i v : . [ a s t r o - ph . C O ] D ec Astronomy&Astrophysicsmanuscript no. final c (cid:13)
ESO 2018October 30, 2018
Controlling intrinsic-shear alignment in three-point weak lensingstatistics
X. Shi , , B. Joachimi , and P. Schneider Argelander-Institut f¨ur Astronomie (AIfA), Universit¨at Bonn, Auf dem H¨ugel 71, 53121 Bonn, Germany International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and CologneReceived 3 February 2010 / Accepted 5 July 2010
ABSTRACT
Three-point weak lensing statistics provide cosmic information that complements two-point statistics. However, both statistics su ff erfrom intrinsic-shear alignment, which is one of their limiting systematics. The nulling technique is a model-independent methoddeveloped to eliminate intrinsic-shear alignment at the two-point level. In this paper we demonstrate that the nulling technique canalso be naturally generalized to the three-point level, thereby controlling the corresponding GGI systematics.We show that under the assumption of exact redshift information the intrinsic-shear alignment contamination can be completelyeliminated. To show how well the nulling technique performs on data with limited redshift information, we apply the nulling techniqueto three-point weak lensing statistics from a fictitious survey analogous to a typical future deep imaging survey, in which the three-point intrinsic-shear alignment systematics is generated from a power-law toy model.Using 10 redshift bins, the nulling technique leads to a factor of 10 suppression of the GGI / GGG ratio and reduces the bias on cos-mological parameters to less than the original statistical error. More detailed redshift information allowing for finer redshift bins leadsto better bias reduction performance. The information loss during the nulling procedure doubles the statistical error on cosmologicalparameters. A comparison of the nulling technique with an unconditioned compression of the data suggests that part of the informa-tion loss can be retained by considering higher order nulling weights during the nulling procedure. A combined analysis of two- andthree-point statistics confirms that the information contained in them is of comparable size and is complementary, both before andafter nulling.
Key words. cosmology: theory – Methods: data analysis – gravitational lensing – large-scale structure of the Universe – cosmologicalparameters
1. Introduction
Weak gravitational lensing refers to the mild distortion of thelight from distant sources by the large-scale matter inhomogene-ity between the source and the observer. One observable e ff ect ofweak gravitational lensing is the coherent shape distortion of thelight sources, known as cosmic shear. Cosmic shear is sensitiveto all cosmological parameters which have influence on the den-sity perturbations and / or the geometry of the universe, includingthose concerning properties of dark energy, which have been akey concern after the discoveries made by observations of super-novae, the cosmic microwave background, and the large-scalestructure (for a review see e.g. Munshi et al. 2008).Since its first detection in 2000 (Bacon et al. 2000;Kaiser et al. 2000; Van Waerbeke et al. 2000; Wittman et al.2000), cosmic shear has been developed into a competitive cos-mological probe. Its constraining power on cosmological param-eters is now comparable to other probes (e.g. Spergel et al. 2007;Fu et al. 2008).With forthcoming large field multicolor imaging surveys(e.g. DES , KIDS , EUCLID , etc), photometric redshift andshape information of a huge number of galaxies will be avail- Send o ff print requests to : X. Shi,e-mail: [email protected] able, rendering cosmic shear even greater statistical power.In particular, cosmic shear is considered to be one of themost promising dark energy probes (Albrecht et al. 2006;Peacock et al. 2006) when the results of these surveys becomeavailable.Such constraining power will be further enhanced by the useof higher-order statistics. Second-order statistics measure onlythe Gaussian signature of a random field. Even if the primordialcosmic density field is Gaussian, non-Gaussianity will be gener-ated due to the nonlinear nature of gravitational clustering. Suchnon-Gaussianity in the cosmic density field will then show upvia its lensing e ff ect, leading to non-Gaussian signals in the cos-mic shear field. A common way of measuring non-Gaussianityis to use higher-order statistics. In cosmic shear studies, severalauthors have shown that the lowest order of them, i.e. the third-order statistics, already provide a valuable probe for cosmologi-cal parameter estimates; in particular it can break the near degen-eracy between the density parameter Ω m and the power spectrumnormalization σ (Bernardeau et al. 1997; Jain & Seljak 1997;van Waerbeke et al. 1999; Hui 1999). A more recent study byTakada & Jain (2004) (TJ04 afterwards) showed that includingthird-order statistics can improve parameter constraints signifi-cantly, typically by a factor of three.But the ultimate performance of these future surveys stilllargely depends on the how well the systematic errors can becontrolled (e.g. Huterer et al. 2006). In this paper we focus on aparticularly worrisome systematic error in cosmic shear studies:
1. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics the intrinsic-shear alignment, and demonstrate a way to controlit for shear three-point statistics.In the weak lensing limit the observed ellipticity of a galaxy ǫ obs can be written as the sum of the intrinsic ellipticity ǫ I of thegalaxy, and the shear γ which is caused by gravitational lensingof the foreground matter distribution. Here ǫ obs , ǫ I and γ are com-plex quantities. Intrinsic-shear alignment is defined in two-pointcosmic shear statistics as the correlation between the intrinsicellipticity of one galaxy and the shear of another galaxy (the GIterm, Hirata & Seljak 2004). Three-point statistics D ǫ i obs ǫ j obs ǫ k obs E ,a correlator of ellipticities of three galaxy images i , j and k , canalso be expanded into lensing (GGG), intrinsic-shear (GGI andGII), and intrinsic (III) terms: D ǫ i obs ǫ j obs ǫ k obs E = GGG + GGI + GII + III , with (1)GGG = D γ i γ j γ k E , (2)GGI = D ǫ i I γ j γ k E + D ǫ j I γ k γ i E + D ǫ k I γ i γ j E , (3)GII = D ǫ i I ǫ j I γ k E + D ǫ j I ǫ k I γ i E + D ǫ k I ǫ i I γ j E , (4)III = D ǫ i I ǫ j I ǫ k I E . (5)Physically, if one assumes that galaxies are randomly ori-ented on the sky, only the desired GGG term remains on theright-hand side of (1). However, when these galaxies are sub-ject to the tidal gravitational force of the same matter structure(e.g. they formed under the influence of the same massive darkmatter halo), their shapes can intrinsically align and become cor-related, giving rise to a nonvanishing III term. Furthermore, GGIand GII terms can be generated when a matter structure tidallyinfluences close-by galaxies and at the same time contributes tothe shear signal of background objects, leading to correlationsamong them.In two-point statistics, the corresponding intrinsic (II)and intrinsic-shear (GI) terms have been subject to de-tailed studies both theoretically (e.g. Catelan et al. (2001);Croft & Metzler (2000); Heavens et al. (2000); Hui & Zhang(2002); Mackey et al. (2002); Jing (2002); Hirata & Seljak(2004); Heymans et al. (2006); Bridle & Abdalla (2007);Schneider & Bridle (2010)) and observationally (Brown et al.2002; Heymans et al. 2004; Mandelbaum et al. 2006, 2009;Hirata et al. 2007; Fu et al. 2008; Brainerd et al. 2009;Okumura et al. 2009; Okumura & Jing 2009). Althoughthe results of these studies show large variations, most ofthem are consistent with a 10 % contamination by both II andGI correlations for future surveys with photometric redshiftinformation. Especially, neglecting these correlations can biasthe dark energy equation of state parameter w by as much as50 % (Bridle & King 2007) for a “shallow” survey describedin Amara & R´efr´egier (2007). For three-point shear statistics,there have been few measurements up to now (Bernardeau et al.2002b; Pen et al. 2003; Jarvis et al. 2004). However the poten-tial systematics level in these studies is found to be high. Arecent numerical study by Semboloni et al. (2008) showed thatintrinsic alignments a ff ect three-point weak lensing statisticsmore strongly than at the two-point level for a given surveydepth. In particular, neglecting GGI and GII systematics wouldlead to an underestimation of the GGG signal by 5 −
10 % fora moderately deep survey like the CFHTLS Wide. Therefore, to match the statistical power expected for cosmic shear in thefuture surveys, it is essential to control these systematics.The intrinsic alignment, II (III) in the two- (three-) pointcase, is relatively straightforward to eliminate, since it re-quires that the galaxies in consideration are physically closeto each other, i.e. have very similar redshifts and angularpositions (King & Schneider 2002, 2003; Heymans & Heavens2003; Takada & White 2004). The control of intrinsic-shear sys-tematics, GI for the two-point case and GGI in the three-pointcase (GII also requires that two of the three galaxies are phys-ically close and thus can be eliminated in the same way as IIand III), turns out to be a much greater challenge. However, asalready pointed out by HS04, the characteristic dependence ongalaxy redshifts is a valuable piece of information that helps tocontrol the intrinsic-shear alignments.Several methods for this have already been constructed inthe context of two-point statistics. They can be roughly classi-fied into three categories: modeling (King 2005; Bridle & King2007), nulling (Joachimi & Schneider 2008, JS08 hereafter;Joachimi & Schneider 2009) and self-calibration (Zhang 2008;Joachimi & Bridle 2009). Modeling separates cosmic shear fromthe intrinsic-shear alignment e ff ect by constructing templatefunctions for the latter. It su ff ers from uncertainties of the modeldue to the lack of knowledge of the angular scale and redshiftdependence of the intrinsic-shear signal. The nulling techniqueemploys the characteristic redshift dependence of the intrinsic-shear signal to “null it out”. It is a purely geometrical method andis model-independent, but su ff ers from a significant informationloss. Self-calibration intends to solve the problem of informationloss by using additional information from the galaxy distributionto “calibrate” the signal. The original form of self-calibration,proposed by Zhang (2008), is model-independent but strong as-sumptions have been made. Joachimi & Bridle (2009) then de-velop it into a modeling method, by treating intrinsic alignmentsand galaxy biasing as free functions of scale and redshift.All these methods have the potential of being generalizedto three-point statistics. In this paper we focus on the nullingtechnique, and establish it as a method to reduce the three-pointintrinsic-shear alignments GGI and GII. Since GII can be re-moved by discarding close pairs of galaxies as in the case ofII controlling (e.g. Heymans & Heavens 2003), we focus on thecontrol of GGI systematics.As known from the case of two-point statistics, the nullingtechnique introduces significant information loss while (in prin-ciple) completely removing the intrinsic-shear alignment fromthe signal. In this work we compare the nulling technique to anunconditioned linear compression of the data, distinguish di ff er-ent sources of such information loss, and discuss the possibleways of reducing it. We also study the combined constraints oncosmological parameters with both two- and three-point cosmicshear statistics.In Sect. 2 we demonstrate why and how the nulling techniquecan be applied to three-point lensing statistics. We then apply thenulling technique to the modeled lensing bispectrum which wecontaminate by intrinsic-shear alignment. The modeling detailsare described in Sect. 3. The method of nulling weights construc-tion and the corresponding results are shown in Sect. 4, while theresults concerning the constraints on cosmological parametersare presented in Sect. 5. We conclude in Sect. 6.We will work in the context of a spatially flat CDM cosmol-ogy with a variable dark energy whose equation of state w isparameterized as w = w + w a (1 − a ), with a the cosmic scalefactor. The adopted fiducial values for cosmological parametersare Ω m = . Ω b = . Ω de = . w = − . w a = .
2. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics h = . n s = .
0, and σ = .
8. Here, Ω m , Ω b and Ω de arethe density parameters of the matter (including cold dark matterand baryons), baryons and the dark energy at present time, n s is the spectral index of the primordial power spectrum of scalarperturbations, h is the dimensionless Hubble parameter definedby H = h km / s / Mpc, and σ is the rms mass fluctuation inspheres of radius 8 h − Mpc.
2. The nulling technique applied to three-pointshear tomography
The shear on the image of a distant galaxy is a result of grav-itational distortion of light caused by the inhomogeneous three-dimensional matter distribution in the foreground of that galaxy.For notational simplicity, we will use the dimensionless surfacemass density (the convergence) κ instead of the shear γ as a mea-sure for the lensing signal throughout the paper, although in re-ality the signal is based on the measurement of the shear. Thiswill not a ff ect our results since κ and γ are linearly related oneach redshift plane while our method is dealing with the redshiftdependence of them (the same reason justifies the turning to theFourier domain in the next subsection).When one measures the shear γ , the direct observable is thegalaxy ellipticity ǫ obs = ǫ I + γ . The shear γ is a signal caused bygravitational distortion which is a deterministic process, wherethe intrinsic ellipticity ǫ I can be further written as the sum ofa deterministic part ǫ detI which is caused by intrinsic alignment,and a stochastic part ǫ ranI which does not correlate with any otherquantity. There is no correlation between ǫ ranI of di ff erent galax-ies either.We define κ obs and κ I which are the correspondences of ǫ detI + γ and ǫ detI . We remove the stochastic part since κ is deter-ministic. Note that κ obs and κ I are analogs of the dimensionlesssurface mass density κ but do not have any direct physical mean-ing as κ does. They are complex quantities in general and canlead to a B-mode signal. To better distinguish the real measur-able κ from them, we denote it as κ G in the rest of the paper sinceit is the physical quantity which is related to the gravitationallensing signal. Keeping the dominating linear term, the conver-gence κ G can be written as (details see e.g. Schneider 2006): κ G ( θ , χ s ) = Ω m H c Z χ s d χ χ ( χ s − χ ) χ s δ ( χ θ , χ ) a ( χ ) , (6)where δ is the three-dimensional matter density contrast, χ s isthe comoving distance of the background galaxy which is actingas a source, and a ( χ ) is the cosmic scale factor at the comovingdistance χ of δ which is acting as a lens.Equation (6) clearly shows that the contribution of the mat-ter inhomogeneity δ at comoving distance χ i to the cosmic shearsignal of background galaxies can be considered as a functionof the source distance χ s , and this function is proportional to1 − χ i /χ s . The nulling technique takes advantage of this char-acteristic dependence on source distance χ s by constructing aweight function T ( χ i , χ s ) such that the product of T ( χ i , χ s ) and1 − χ i /χ s has an average of zero on the range between χ i and thecomoving distance to the horizon χ hor : Z χ hor χ i d χ s T ( χ i , χ s ) − χ i χ s ! = . (7) One then uses this weight function as a weight for integratingover the source distance:ˆ κ G ( χ i , θ ) : = Z χ hor χ i d χ s T ( χ i , χ s ) κ G ( θ , χ s ) . (8)The resulting new measure of shear signal ˆ κ G ( χ i , θ ) is then freeof contributions from the matter inhomogeneity at distance χ i .Note that although the weight function T has two arguments χ i and χ s here, we consider it as a function of χ s for a particular χ i .Consider a correlator D κ i obs κ j obs E with comoving distances χ i < χ j . With a similar decomposition as (1), it is straightfor-ward to see that the GI term in it is D κ i I κ j G E . The term D κ i G κ j I E vanishes since the lensing signal at χ i is correlated only withmatter with χ ≤ χ i , whereas κ j I originates solely from physicalprocesses happening at χ j . If we integrate D κ i I κ j G E over χ j witha weight function that eliminates the contributions to κ j G by thematter inhomogeneity at distance χ i , this correlator will also van-ish, Z χ hor χ i d χ j T ( χ i , χ j ) D κ i I κ j G E = , (9)since it is just the matter inhomogeneity at distance χ i that givesrise to the correlation between κ i I and κ j G . Thus, when we inte-grate over D κ i obs κ j obs E with the same weight function, the GI con-tamination in it will be “nulled out”. Equation (7) is the conditionthat the weight function T should satisfy in order to “null” theintrinsic-shear alignment terms, so we call it “the nulling condi-tion”.The same applies to three-point statistics. Consider a cor-relator D κ i obs κ j obs κ k obs E with χ i being the smallest comoving dis-tance of the three. Both GII and GGI systematics containedin it also originate from the matter inhomogeneity at distance χ i . Typically, the generation of GII systematics requires that χ i ≈ χ j < χ k , while the generation of GGI requires χ i < χ j and χ i < χ k . For both cases, the dependence of GII or GGI sys-tematics on χ k is also just 1 − χ i /χ k . So new measures built as R χ hor χ i d χ k T ( χ i , χ j , χ k ) D κ i obs κ j obs κ k obs E with T satisfying the nullingcondition for three-point statistics Z χ hor χ i d χ k T ( χ i , χ j , χ k ) − χ i χ k ! = T ( χ i , χ j , χ k ) here should be seen as a function of χ k whose formdepends on χ i and χ j .Note that this method only depends on the characteristic red-shift dependencies of the lensing signal and intrinsic-alignmentsignals, and is not limited to E-mode fields. This is a reassur-ing feature since while the κ G field is a pure E-mode field tofirst order, the κ I field can have a B-mode component. However,if parity-invariance is assumed, any correlation function whichcontains an odd number of B-mode shear components vanishes(Schneider 2003), thus there should be no B-mode component inthe GGI signal. Since the nulling technique relies on the distinct redshift de-pendence of the intrinsic-alignment signal, redshift informationis crucial for it. With the help of infrared bands, forthcoming
3. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics multicolor imaging surveys can provide rather accurate photo-metric redshift information for the galaxies (e.g. Abdalla et al.2008; Bordoloi et al. 2010), allowing tomographic studies ofcosmic shear statistics. We base our study on cosmic shear bis-pectrum tomography, and outline the corresponding formalismof the nulling technique in the following.Given the galaxy redshift probability distribution of redshiftbin i which we denote as p ( i )s ( z ) = p ( i )s ( χ s ) d χ s / dz , one can de-fine the average convergence field in redshift bin i by integrating κ ( θ , χ s ) in (6) over p ( i )s ( χ s ). We turn to angular frequency spacenow and define˜ κ ( i )G ( ℓ ) : = Z χ hor d χ s p ( i )s ( χ s ) ˜ κ G ( ℓ , χ s ) , (11)where ˜ κ G ( ℓ , χ s ) is the Fourier transform of κ G ( θ , χ s ). To bettershow the relation between ˜ κ ( i )G ( ℓ ) and three-dimensional matterinhomogeneity in Fourier space ˜ δ ( k , χ ), one can write, basingon (6) and (11),˜ κ ( i )G ( ℓ ) = Z χ hor d χ W ( i ) ( χ ) ˜ δ ( ℓ /χ, χ ) , (12)by defining a lensing weight function W ( i ) ( χ ) as W ( i ) ( χ ) : = Ω m H χ a ( χ ) c Z χ hor χ d χ s p ( i )s ( χ s ) χ s − χχ s . (13)The tomographic lensing bispectrum is defined via D ˜ κ ( i )G ( ℓ )˜ κ ( j )G ( ℓ )˜ κ ( k )G ( ℓ ) E = (2 π ) B ( i jk )GGG ( ℓ , ℓ , ℓ ) δ D ( ℓ + ℓ + ℓ ) , (14)where the Dirac delta function ensures that the bispectrum isdefined only when ℓ , ℓ , and ℓ form a triangle. This fact arisesfrom statistical homogeneity, while that the bispectrum can bedefined as a function independent of the directions of the angularfrequency vectors arises from statistical isotropy.In a survey, the convergence field ˜ κ obs is determined from theobserved galaxy ellipticities, and the corresponding bispectrum B obs su ff ers from intrinsic-shear alignments. As we did with thethree-point correlator in Sect. 1, we separate the observed lens-ing bispectrum into the four terms: B obs = B GGG + B GGI + B GII + B III . (15)Among them, B GGI , B GII and B III can be linked to the conver-gence in a similar way as (14), for example D ˜ κ ( i )I ( ℓ )˜ κ ( j )G ( ℓ )˜ κ ( k )G ( ℓ ) E = (2 π ) B ( i jk )GGI ( ℓ , ℓ , ℓ ) δ D ( ℓ + ℓ + ℓ ) . (16)Here we assume disjunct redshift bins and let i to be the red-shift bin with the lowest redshift, so D ˜ κ ( i )G ( ℓ )˜ κ ( j )I ( ℓ )˜ κ ( k )G ( ℓ ) E and D ˜ κ ( i )G ( ℓ )˜ κ ( j )G ( ℓ )˜ κ ( k )I ( ℓ ) E both vanish due to the same reason as ex-plained in Sect. 2.1 for the two-point statistics.The purpose of the nulling technique is to filter B obs in sucha way that the GGI term is strongly suppressed in comparisonwith the GGG term. The GII and III terms can be removed byignoring the signal coming from bispectrum B ( i jk )obs ( ℓ , ℓ , ℓ ) withtwo or three equal redshift bins.To fulfill this purpose, we construct our new measures as Y ( i j ) ( ℓ , ℓ , ℓ ) : = N z X k = i + T ( i j ) ( χ k ) B ( i jk )obs ( ℓ , ℓ , ℓ ) χ ′ k ∆ z k , (17) where N z is the total number of redshift bins, χ ′ k is the deriva-tive of comoving distance with respect to redshift, and ∆ z k is thewidth of redshift bin k . The weight function is written now as T ( i j ) ( χ k ) since i and j indicate two redshift bins, i.e. two popu-lations of galaxies, rather than two comoving distances as in theprevious subsection. The weight T ( i j ) is required to satisfy thenulling condition (7) in its discretized form, O ( i j ) : = N z X k = i + T ( i j ) ( χ k ) − χ i χ k ! χ ′ k ∆ z k = , (18)for all j > i . Here, χ i and χ k should be chosen such that theyrepresent well the distance to redshift bins i and k . In this paperwe choose them to be the distances corresponding to the medianredshift of the bin. The summation over index k runs from i + i since we consider only bispectrum measures with j > i and k > i to avoid III and GII systematics. In this case B ( i jk )obs in (17) can be written as a sum of B ( i jk )GGG and B ( i jk )GGI , and Y ( i j ) canbe expressed as Y ( i j ) ( ℓ , ℓ , ℓ ) = N z X k = i + T ( i j ) ( χ k ) B ( i jk )GGG ( ℓ , ℓ , ℓ ) χ ′ k ∆ z k + N z X k = i + T ( i j ) ( χ k ) B ( i jk )GGI ( ℓ , ℓ , ℓ ) χ ′ k ∆ z k . (19)Suppose one has infinitely many redshift bins, then the lens-ing signal in bin k caused by the matter inhomogeneity in bin i is exactly proportional to 1 − χ i /χ k , which means B ( i jk )GGI ( ℓ , ℓ , ℓ )can be written as a product of 1 − χ i /χ k and some function of theparameters other than χ k : B ( i jk )GGI ( ℓ , ℓ , ℓ ) = F ( χ i , χ j , ℓ , ℓ , ℓ ) − χ i χ k ! . (20)Then we have N z X k = i + T ( i j ) ( χ k ) B ( i jk )GGI ( ℓ , ℓ , ℓ ) χ ′ k ∆ z k = F ( χ i , χ j , ℓ , ℓ , ℓ ) N z X k = i + T ( i j ) ( χ k ) − χ i χ k ! χ ′ k ∆ z k = . (21)This suggests that only the GGG contribution is left in the nulledmeasure Y ( i j ) , the GGI contribution has been “nulled out” due tothe nulling condition. If only a limited number of redshift binsis available, (20) holds only approximately, leading to a residualin (21).Since the nulling condition is the only condition that theweight T ( i j ) must satisfy in order to “null”, there is much free-dom in choosing the form of it. We would like to further specifyits form such that it preserves as much Fisher information in Y ( i j ) as possible. The method we have adopted for the nulling weightconstruction will be detailed in Sect. 4.Note that for each ( i , j ) combination, one can in principleapply more than one nulling weight to the original bispectrum,and obtain more nulled measures. If one retains the conditionof maximizing the Fisher information and demands that all theweight functions built for one ( i , j ) combination are orthogonalto each other, one arrives at higher-order modes that have thesecond-most, third-most, etc., information content (higher-orderweights, see JS08). The total number of such linearly indepen-dent nulled measures for a certain ( i , j ) equals the possible val-ues of k ≥ i +
1. In this schematic study we will only use theoptimum, i.e. the first-order nulling weights. We will assess theinformation loss due to this limitation in Sect. 5.4.
4. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics
3. Modeling
We set up a fictitious survey with a survey size of A = which is similar to the survey size of DES. This can beeasily scaled to any survey size using the proportionality of sta-tistical errors to A − / . We assume a galaxy intrinsic ellipticitydispersion σ ǫ = σ ( ǫ ranI ) = .
35. As galaxy redshift probabil-ity distribution we adopt the frequently used parameterization(Smail et al. 1994), p s ( z ) ∝ zz ! α exp − zz ! β , (22)and use z = . α = β = .
5. The distribution is cut at z max = z m = .
9, which is compatibleto a survey like EUCLID. We adopt an average galaxy numberdensity ¯ n g =
40 arcmin − which is again EUCLID-like.Disjunct redshift bins without photo-z error are assumed,which means that the galaxy redshift probability distribution inredshift bin i takes the form p ( i )s ( z ) ∝ p s ( z ) if and only if theredshift that corresponds to comoving distance χ s is within theboundaries of redshift bin i . A number of 10 redshift bins areused by default. The boundaries of the redshift bins are set suchthat each bin contains the same number of galaxies.We adopt 20 angular frequency bins spaced logarithmicallybetween ℓ min =
50 and ℓ max = ℓ . Within this range thenoise properties of the cosmic shear field are still not too far inthe non-Gaussian regime, allowing a more realistic theoreticalestimation of the bispectrum and its covariance. Whether thisnumber of angular frequency bins can reconstruct the angularfrequency dependence of the bispectrum is tested, and 20 binsare found to be su ffi cient for our requirements on precision. Thisis also expected since the bispectrum is rather featureless as afunction of angular frequency. We show the modeling of B GGG and its covariance in thissection. We will only consider the tomographic bispectrum atredshift bins satisfying z i < z j and z i < z k , which already ensuresan elimination of B III and B GII systematics in our case.Applying Limber’s equation, it can be shown that the tomo-graphic convergence bispectrum can be written as a projectionof the three-dimensional matter bispectrum B δ ( k , k , k ; χ ) (seee.g. TJ04): B ( i jk )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) = Z χ hor d χ W ( i ) ( χ ) W ( j ) ( χ ) W ( k ) ( χ ) χ B δ ¯ ℓ χ , ¯ ℓ χ , ¯ ℓ χ ; χ ! . (23)To compute B δ , we employ the fitting formula byScoccimarro & Couchman (2001), which is based on hyper-extended perturbation theory (Scoccimarro & Frieman 1999). Acomparison of this formula with the halo model results can befound in Takada & Jain (2003a,b).Estimating the bispectrum covariance is often done withina flat-sky spherical harmonic formalism (Hu 2000, Hu00 here-after), which su ff ers - at least formally - from drawbacks sinceits basis functions Y lm are only defined for discrete angular fre-quency values and full sky coverage. In Joachimi et al. (2009) another approach exclusively based on the two-dimensionalFourier formalism was constructed, which we use here.The bispectrum covariance is a six-point correlation functionwhich can be expanded into its connected parts as outlined in e.g.Bernardeau et al. (2002a). As argued in TJ04, for angular scales ℓ ≤ (cid:16) B ( i jk )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) , B ( lmn )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) (cid:17) = (2 π ) A ¯ ℓ ¯ ℓ ¯ ℓ ∆ ¯ ℓ ∆ ¯ ℓ ∆ ¯ ℓ Λ − (cid:16) ¯ ℓ , ¯ ℓ , ¯ ℓ (cid:17) × (cid:16) ¯ P ( il ) ( ¯ ℓ ) ¯ P ( jm ) ( ¯ ℓ ) ¯ P ( kn ) ( ¯ ℓ ) δ ¯ ℓ ¯ ℓ δ ¯ ℓ ¯ ℓ δ ¯ ℓ ¯ ℓ + (cid:17) , (24)in which ∆ ¯ ℓ i is the bin width of the angular frequency bin withtypical value ¯ ℓ i , and ¯ P ( i j ) ( ¯ ℓ ) is the observed power spectrumwhich contains the intrinsic ellipticity noise (e.g. Kaiser 1992;Hu 1999; Joachimi et al. 2008):¯ P ( i j ) ( ¯ ℓ ) = P ( i j ) ( ¯ ℓ ) + δ i j σ ǫ n i , (25)where ¯ n i is the galaxy number density in redshift bin i . The term Λ ( ℓ , ℓ , ℓ ) is defined as Λ ( ℓ , ℓ , ℓ ) ≡ (cid:26) q ℓ ℓ + ℓ ℓ + ℓ ℓ − ℓ − ℓ − ℓ (cid:27) − if | ℓ − ℓ | < ℓ < ℓ + ℓ , . (26)When | ℓ − ℓ | < ℓ < ℓ + ℓ is satisfied, Λ − ( ℓ , ℓ , ℓ ) is thearea of a triangle with side lengths ℓ , ℓ and ℓ . We use theEisenstein & Hu (1998) transfer function to evaluate the linearthree-dimensional matter power spectrum, and the Smith et al.(2003) fitting function for the nonlinear power spectrum.Note that there is no intrinsic ellipticity noise in the observedbispectrum, since the galaxy intrinsic ellipticity distribution isassumed to be skewless.Under the assumptions of a compact survey geometry andscales much smaller than the extent of the survey area, (24)provides a bispectrum covariance that naturally incorporates thescaling with survey size, is not restricted to integer angular fre-quencies, and allows for any appropriate binning. In terms ofFisher information, the result given by this approach and theHu00 one agree to high accuracy (Joachimi et al. 2009). In this section we present a toy model for generating GGIsystematics. Since the physical generation of intrinsic-shearalignments concerns nonlinear growth of structure and complexastrophysical processes which are not easy to quantify, a realisticmodel is not yet available. Current simulations involving bary-onic matter also have some way to go before they can simulatethe generation of the GGI systematics reliably.Up to now there has not been any attempt to measure GGIand GII in galaxy surveys. Semboloni et al. (2008) studied thesesystematics using ray-tracing simulations. They provided fits inreal space to projected GII and GGI signals, but the results arestill too crude to lead to su ffi cient constraints on an intrinsic-shear alignment model.This situation emphasizes the importance of a methodintended to control intrinsic-shear alignment to be model-independent, especially at the three-point level. Since this is the
5. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics case for the nulling technique, for this work we only require asimple model for B ( i jk )GGI which satisfies the characteristic redshiftdependence and leads to a reasonable bias.Based on the observation that the lensing bispectrum ex-pression (23) comes directly from (12) and the definition ofthe tomography bispectrum (14), we link B ( i jk )GGI also to a three-dimensional bispectrum B δ I δδ via B ( i jk )GGI ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) = Z χ hor d χ p ( i )s ( χ ) W ( j ) ( χ ) W ( k ) ( χ ) χ B δ I δδ ¯ ℓ χ , ¯ ℓ χ , ¯ ℓ χ ; χ ! . (27)Similar to B δ ( k , k , k ) which is given by D ˜ δ ( k , χ )˜ δ ( k , χ )˜ δ ( k , χ ) E = (2 π ) δ D ( k + k + k ) B δ ( k , k , k ; χ ) , (28) B δ I δδ is defined via D ˜ δ I ( k , χ )˜ δ ( k , χ )˜ δ ( k , χ ) E = (2 π ) δ D ( k + k + k ) B δ I δδ ( k , k , k ; χ ) , (29)where ˜ δ I ( k ) is the three-dimensional density field which is re-sponsible for the intrinsic alignment, and it satisfies˜ κ ( i )I ( ℓ ) = Z χ hor d χ p ( i )s ( χ ) ˜ δ I ℓ χ , χ ! , (30)The definition of both ˜ κ ( i )I and ˜ δ I originates from the deter-ministic part of galaxy intrinsic ellipticity ǫ detI . We have as-sume the existence of these underlying smooth fields. Similarquantities have been defined in Joachimi & Bridle (2009), seealso Hirata & Seljak (2004) and Schneider & Bridle (2010). Wewould like to point out again that, although we introduce thesequantities for the clarity of our model, we do not need them forthe main purpose of this paper. What we need to model is theprojected GGI bispectrum B ( i jk )GGI .Note that in (27), the weight for the lowest redshift bin i isthe source distribution function p ( i )s which is zero outside red-shift bin i , rather than the lensing weight W ( i ) which is a muchbroader function. Since ˜ κ ( i )I depends only on physical processesat redshift bin i and is inferred from ellipticity measurements inthis bin, and ˜ κ ( j )G is linked to the three-dimensional matter den-sity through the lensing weight W ( j ) , this assignment of weightfunctions will ensure the correct redshift dependence of B ( i jk )GGI .When the redshift bins are not disjunct, however, the intrinsicalignment signal can no longer be associated with bin i . Therewill be two permutations in both the left-hand side of (16) andthe right-hand side of (27), similar to the two-point case, e.g.Eq. 11 in Hirata & Seljak (2004).The modeling of B δ I δδ is then a pure matter of choice. Webuild a simple three-dimensional GGI bispectrum with power-law dependence on both redshift z and spatial frequency k : B δ I δδ ( k , k , k ; χ ) : = −A B δ ( k ref , k ref , k ref ; χ ( z med )) + z + z med ! r − × k k ref ! s − + k k ref ! s − + k k ref ! s − , (31) where z med is the median redshift of the whole survey, and A , k ref , r , s are free parameters. Among them the parameter k ref is designed to be a characteristic wave number, whose valuewe set to be a weakly nonlinear scale of 10 h Mpc − here. Theminus sign ensures that the contamination of GGI systematicsleads to an underestimation of the GGG signal, as found bySemboloni et al. (2008).Little is known about the redshift and angular scale depen-dence of B δ I δδ . However one can roughly estimate how it com-pares to the B δδδ signal. A linear alignment model suggests δ I ∝ δ lin ¯ ρ ( z ) / ((1 + z ) D + ( z )) (see e.g. Hirata & Seljak 2004),in which ¯ ρ ( z ) is the mean density of the universe, D + ( z ) isthe growth factor, and δ lin is the linear matter density contrast.Thus we have, very roughly, δ I ∝ (1 + z ) δ lin which suggests B δ I δδ ∝ (1 + z ) B δδδ . The linear alignment model assumes thatthe intrinsic alignment is linearly related to the local tidal gravi-tational field (e.g. Catelan et al. 2001; Hirata & Seljak 2004). Ifthis holds true, we also expect B δ I δδ to have a stronger angularscale dependence than B δδδ since tidal gravitational interactionfollows the inverse cube law rather than the inverse square lawwhich gravity itself follows. For a Λ CDM model, in the weaklynonlinear regime where perturbation theory holds, the depen-dence of B δδδ on (1 + z ) has a negative power shallower than − k has a power of around −
2. In this paperwe choose r = s = r = − r =
2, and s = A of the GGI signal, the only directstudy up to now is Semboloni et al. (2008), which suggests anoverall GGI / GGG ratio of 10 % for a z m = . A such that theamplitude of the tomographic GGI bispectrum is limited to bewithin 10 % of the amplitude of the lensing GGG signal, i.e.GGI / GGG .
10 % at redshift bin combinations with z i ≪ z j and z i ≪ z k where the GGI signal is expected to be most signifi-cant. This leads to a relatively modest overall GGI / GGG ratioat percent level. We will show examples of the generated GGIand GGG signals in Fig. 3. As an order-of-magnitude estimate,one can also relate the GGI / GGG ratio to that of GI / GG by ex-panding three-point signals to couples of two-point signals usingperturbation theory, in analogy to the Scoccimarro & Couchman(2001) fitting formula. For the case of z i ≪ z j ≈ z k , the leadingorder terms would give that the GGI / GGG ratio approximatesthat of GI / GG evaluated at redshifts z i and z j . This suggests thatour adopted GGI / GGG ratio is also consistent with available ob-servational studies of the GI signal (Mandelbaum et al. 2006,2009; Hirata et al. 2007; Fu et al. 2008; Okumura et al. 2009;Okumura & Jing 2009), although the results of these studies varya lot according to di ff erent median redshift, color and luminosityof the selected galaxy sample.
4. Construction of nulling weights
As mentioned in Sect. 2.2, we would like to construct a sin-gle first-order weight function T ( i j ) ( χ ) for each ( i , j ) combinationwhich preserves the maximum of information. This can be seenas a constrained optimization problem. The constraining condi-tion here is the nulling condition and the quantity to be optimizedis the Fisher information after nulling. In JS08, several practicalmethods were developed to solve this optimization problem atthe two-point level, and very good agreement was found amongthe di ff erent methods.
6. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics (cid:0) ( (cid:1) ) F ( i, j ) o i=1, j=2i=1, j=6i=5, j=6i=5, j=10 0 100 200 300 400 500 600 (cid:2) F ( i, j ) o Fig. 1.
Distribution of the nulled Fisher information as defined in (40) per ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) bin and per redshift bin combination amongdi ff erent angular frequency triangle shapes and sizes. Results for four redshift bin combinations ( i , j ) are presented. Left panel :Distribution of the nulled Fisher information among di ff erent triangle configurations. We consider triangles with the commonshortest side length ¯ ℓ =
171 which corresponds to the 7th angular frequency bin. Due to our logarithmic binning and the constraintthat the three side lengths must be able to form a triangle, only 8 such triangle configurations exist. Plotted is the nulled Fisherinformation contained in these 8 triangles against α , which is the angle opposite to the shortest side length in that triangle. Smaller α correspond to more elongated triangles, and larger α correspond to almost equilateral triangles. Right panel : Distribution of thenulled Fisher information contributed by each ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) bin over di ff erent triangle sizes. A fixed triangle shape with ¯ ℓ : ¯ ℓ : ¯ ℓ = .
64 : 4 .
52 (corresponds to the leftmost points in the left panel) is chosen. The nulled Fisher information contained in one( ¯ ℓ , ¯ ℓ , ¯ ℓ ) bin is plotted against the shortest side length ¯ ℓ of each triangle.We adopt the simplified analytical approach as described inJS08, and reformulate it for three-point statistics here. For con-venience we introduce the following notations:the bispectrum covariance matrix C ovB, whose elements areCovB( i jk ¯ ℓ , ¯ ℓ , ¯ ℓ ; lmn ¯ ℓ , ¯ ℓ , ¯ ℓ ) : = Cov (cid:16) B ( i jk )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) , B ( lmn )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) (cid:17) ;(32)the covariance matrix C ovY of the nulled bispectra Y , whoseelements areCovY( i j ¯ ℓ , ¯ ℓ , ¯ ℓ ; lm ¯ ℓ , ¯ ℓ , ¯ ℓ ) : = Cov (cid:16) Y ( i j ) ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) , Y ( lm ) ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) (cid:17) = N z X k = i + N z X n = l + Cov (cid:16) B ( i jk )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) , B ( lmn )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) (cid:17) × T ( i j ) ( χ k ) T ( lm ) ( χ n ) χ ′ k χ ′ n ∆ z k ∆ z n ; (33)a vector B , µ whose elements are partial derivatives of the bispec-trum with respect to the cosmological parameter p µ B , µ ( i jk ¯ ℓ , ¯ ℓ , ¯ ℓ ) : = ∂ B ( i jk )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) ∂ p µ ; (34)and a corresponding vector Y , µ for nulled bispectra Y , whoseelements are Y , µ ( i j ¯ ℓ , ¯ ℓ , ¯ ℓ ) : = ∂ Y ( i j ) ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) ∂ p µ . (35) Then the Fisher information matrix from the original bispec-tra can be written as (following TJ04)F i µν = B , µ C ovB − B , ν , (36)and that from the nulled bispectra can be written asF f µν = Y , µ C ovY − Y , ν . (37)Here the matrix multiplication is a summation of all possi-ble angular frequency combinations ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) and redshift bincombinations, ( i jk ) for the original bispectra and ( i j ) for thenulled bispectra. In (36) and (37), C ovB − and C ovY − indi-cate the inverse of the covariance matrix. When the covari-ance is approximated by triples of power spectra, the covari-ance between two di ff erent angular frequency combinations( ¯ ℓ , ¯ ℓ , ¯ ℓ ) , ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) is zero, see (24), which means that thecovariance matrix is block diagonal. In this case the matrix in-version can be done separately for each block specified by anangular frequency combination ( ¯ ℓ , ¯ ℓ , ¯ ℓ ).According to the idea of the simplified analytical approach,we consider the Fisher information on one cosmological parame-ter contained in bispectrum measures B ( i jk )GGG ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) with a sin-gle ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) combination and with redshift bin ( i , j , k ) combi-nations having common ( i , j ) indices. For every ( i , j ) combina-tion we build nulling weights T ( i j ) which maximizes the nulledFisher matrix using the method of Lagrange multipliers. Sincehere the nulled Fisher matrix receives contribution only fromcertain angular frequency and redshift combinations, we denoteit as F ( i j )o to avoid ambiguity. F ( i j )o has only one component sinceonly one cosmological parameter is taken into consideration. As
7. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics only a single ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) combination is involved, we will omitthe ¯ ℓ -dependence in all variables in the rest of this subsection tokeep a compact form.Again for notational simplicity, we follow JS08 and intro-duce a vector notation as follows. For each ( i , j ) in considera-tion, let the values of the weights T ( i j ) ( χ k ) form a vector T = T k ,and define another vector ρ and a matrix ¯ C with elements ρ k : = B , µ ( i jk ) χ ′ k ∆ z k , (38)¯ C kn : = CovB( i jk ; i jn ) χ ′ k χ ′ n ∆ z k ∆ z n . (39)Thus F ( i j )o can be expressed, according to (37), as F ( i j )o : = Y , µ ( i j ) C ovY − ( i j ; i j ) Y , µ ( i j ) = ( T · ρ ) T τ ¯ C T . (40)We further define a vector f with elements f k = − χ i χ k ! χ ′ k ∆ z k (41)to write the nulling condition (18) as O ( i j ) = T · f = . (42)The problem of finding nulling weights T which maximize F ( i j ) o under the constraint given by the nulling condition can besolved with the method of Lagrange multipliers by defining afunction G : = F ( i j )o + λ O ( i j ) = ( T · ρ ) T τ ¯ C T + λ T · f (43)with λ being the Lagrange multiplier, and setting the gradient of G with respect to T to zero, ∇ T G = ρ ( T · ρ ) T τ ¯ C T − C T ( T · ρ ) T τ ¯ C T ! + λ f = . (44)The solution to this equation is (for more details see JS08) T = N ( ¯ C − ρ − f τ ¯ C − ρ f τ ¯ C − f ¯ C − f ) , (45)with the normalization N adjusted to give | T | = ℓ , ¯ ℓ , ¯ ℓ ) combination is considered and with respect towhich cosmological parameter we optimize the information con-tent. In this paper the default cosmological parameter to optimizeis Ω m , and we choose for each ( i , j ) combination the ( ¯ ℓ , ¯ ℓ , ¯ ℓ )combination which maximizes F ( i j )o . However one needs to beaware that this serves only as a clear choice of a ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) com-bination and is not necessarily the best in terms of informationpreservation considering all angular frequency bins and all cos-mological parameters.To show which triangle shapes and sizes contain more in-formation, we plot F ( i j )o against the ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) triangle shape andsize for four typical ( i , j ) combinations in Fig. 1. In the left panel,the nulled information F ( i j )o contained in di ff erent triangles witha common shortest side length ¯ ℓ =
171 is plotted against α ,which is the angle opposite to ¯ ℓ . Due to our logarithmic bin-ning in angular frequency, only eight ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) combinations z (cid:3) T ( i j ) w i t h i = , j = (cid:4) m (cid:5) hn s (cid:6) b w w a Fig. 2.
Nulling weights T ( i j ) for redshift bins i = , j = k .Remarkable consistency is found between nulling weights opti-mized on di ff erent parameters, shown with di ff erent line styles.with ¯ ℓ =
171 can form triangles. One sees that the more elon-gated triangles (small α ) contain much more Fisher informationthan the almost equilateral triangles (large α ). The small separa-tion between the 3rd and the 4th points from the left is caused bythe degeneracy of di ff erent triangle shapes with respect to α , e.g.two equal and very long side lengths can result in the same valueof α as two shorter side lengths with a length di ff erence close tothe length of the shortest side length. The right panel shows thedistribution of the Fisher information contained in one ( ¯ ℓ , ¯ ℓ , ¯ ℓ )bin over the triangle size. When the redshift in consideration ishigher, the peak of the information distribution moves to higherangular frequencies. The figure suggests that most informationcomes from high redshifts and small angular scales.To explore the sensitivity of nulling weights on the choice ofthe cosmological parameter, we construct seven sets of weightfunctions, each optimizing the information content in terms ofone parameter. For all ( i , j ) combinations we find that the nullingweights are not very sensitive to the choice of parameter. As anexample, the weights for ( i , j ) = (1 ,
2) are shown in Fig. 2. Thisresult is rather surprising at first sight, since for di ff erent parame-ters the distribution of information (contained in the bispectrum)over redshift bins is quite di ff erent. However, such insensitivitysuggests that the shapes of nulling weights are already stronglyconstrained under our construction scheme. One constraint is,evidently, the nulling condition. Moreover, considering the factthat we optimize the nulling weights for each ( i , j ) combinationwith respect to the information content they preserve, we havealready required the shapes of these first order nulling weightsto be as smooth as possible.The fact that these two conditions have already imposedstrong constraints on the nulling weights also suggests thatnulling weights can be robustly and e ffi ciently constructed, i.e.it is not critical to construct the “best” nulling weights.
5. Performance of the nulling technique
What the nulling technique “nulls” is the GGI signal B GGI ,so the GGI / GGG ratio is the most direct quantification of its per-formance. We plot the modeled GGI and GGG bispectra before
8. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics -4 -6 -8 -10 (cid:7) B (i=1, j=4, k=3) (i=1, j=4, k=5) (i=1, j=4, k=7) (i=1, j=4, k=9) 10 -6 -8 -10 -12 (cid:8) Y (i=1, j=4)10 (cid:9) -4 -6 -8 -10 (cid:10) B (i=1, j=8, k=3) (i=1, j=8, k=5) (i=1, j=8, k=7) (i=1, j=8, k=9) 10 -6 -8 -10 -12 (cid:11) Y (i=1, j=8)10 (cid:12) -4 -6 -8 -10 (cid:13) B (i=4, j=8, k=5) 10 (cid:14) (i=4, j=8, k=7) 10 (cid:15) (i=4, j=8, k=9) 10 (cid:16) -6 -8 -10 -12 (cid:17) Y (i=4, j=8) GGGGGI
Fig. 3.
Tomographic convergence bispectrum (GGG, solid curves) and intrinsic-shear alignment (GGI, dashed curves) for equilateraltriangles are plotted against triangle side length. Measures both before (left panel) and after (right panel) applying the nullingtechnique ( B and Y respectively) are shown for three typical redshift bin ( i , j ) combinations in the three rows.and after nulling in Fig. 3. The original GGI signal is shown inthe left panels by dashed lines. For comparison the GGG signalsare shown as solid curves. The results are shown for equilateraltriangle configurations for the convenience of presenting. Onesees that when the redshift bin number j and / or k increase, thechanges in GGG and GGI signals are di ff erent, which shows theexpected di ff erent redshift dependence. For all redshift bin com-binations the GGI signal is modeled to be subdominant to theGGG signal. In the nulled measures shown in the right panels,the GGI / GGG ratio is suppressed by a factor of 10 over all an-gular scales, which reflects the success of the nulling technique.
We further evaluate the performance of the nulling techniqueby looking at the constraining power of cosmic shear bispectrumtomography on cosmological parameters, as well as the biasescaused by the GGI systematics before and after nulling.The full characterization of the bispectrum involves three an-gular frequency vectors which form a triangle. In some worksconcerning three-point statistics, only equilateral triangle con-figurations i.e. ℓ = ℓ = ℓ = ℓ are used for simplicity reasons(e.g. Pires et al. 2009). But as several authors have pointed out(e.g. Kilbinger & Schneider 2005; Berg´e et al. 2010), only a lowpercentage of information is contained in equilateral triangles. Thus, to calculate the full information content, we use generaltriangle configurations but limit our calculation to triangles withthree di ff erent side lengths, again for reasons of simplicity (fordetails see Appendix A ).We will use the figure of merit (FoM, Albrecht et al. 2006)to quantify the goodness of parameter constraints. Here the FoMfor constraints in the parameter plane p α − p β is defined to beproportional to the inverse of the area of the parameter constraintellipses: FoM(p α , p β ) ≡ (cid:16) ( F − ) αα ( F − ) ββ − ( F − ) αβ (cid:17) − . (46)To compute biases, we adopt a method based on a simple ex-tension of the Fisher matrix formalism (e.g. Huterer et al. 2006;Amara & R´efr´egier 2008). Then one needs to define a bias vec-tor B GGI which in our case reads: B GGI ν, i = B GGI C ovB − B , ν , (47) B GGI ν, f = Y GGI C ovY − Y , ν , (48)with B GGI ( i jk ¯ ℓ , ¯ ℓ , ¯ ℓ ) : = B ( i jk )GGI ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) , (49)
9. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics Y GGI ( i j ¯ ℓ , ¯ ℓ , ¯ ℓ ) : = Y ( i j )GGI ( ¯ ℓ , ¯ ℓ , ¯ ℓ ) . (50)The bias of the parameter estimator ˆ p µ is given by the di ff er-ence between its ensemble average and the fiducial value of theparameter p fid µ : b µ = h ˆ p µ i − p fid µ = X ν (cid:16) F − (cid:17) µν B GGI ν . (51)The information content before and after nulling can be seenin Fig. 6. On the cost of increasing the error on each parameterto about twice its original value, GGI systematics are reduced tobe within the original statistical error. The relative informationloss in terms of FoM can be found in Table 1. The constraintsshown in Fig. 6 do not represent the best constraints obtainablefrom a cosmic shear bispectrum analysis since we consider onlythe triangles with angular scale ¯ ℓ , ¯ ℓ , ¯ ℓ . Also note that thenulling technique can in principle remove the GGI systematicscompletely. But as shown in Fig. 6, the systematics still causesome residual biases on cosmological parameters after nulling,due to the finite number of redshift bins. The GGI systematicswill be reduced to a lower level when more redshift bins areavailable. We will discuss this further in the following subsec-tion. Analyzing the cosmic shear signal in a tomographic waywas originally meant to maximize the information. For this pur-pose alone, a crude redshift binning will su ffi ce (Hu 1999).However, to control intrinsic-shear alignment, which is aredshift-dependent e ff ect, much more detailed redshift infor-mation is required (e.g. King & Schneider 2002; Bridle & King2007; Joachimi & Schneider 2008). Thus, for a method intendedto eliminate intrinsic-shear alignment, it is necessary to show itsrequirement on the redshift precision. In the case of nulling, de-tailed redshift information is not only needed for the method tobe able to eliminate the bias, but also for the preservation of areasonably large amount of information through the nulling pro-cess. JS08 examined the number of redshift bins required forthe nulling technique in the two-point case, and showed that10 redshift bins already ensure that parameters are still well-constrained after nulling.To re-assess this problem at the three-point level, we con-sider two di ff erent situations to address the requirements comingfrom control of the intrinsic-shear alignment and preservation ofthe information content separately. In both cases we split the red-shift range between z = z = Ω m , to befree and study the biases introduced by the GGI signal on Ω m both before and after nulling. We use only equilateral triangleconfigurations to reduce the amount of calculation. The resultsare shown in Fig. 4. Within the range of consideration, the ratioof the nulled and the original biases drops quickly with the in-crease of the number of redshift bins for all GGI models. Formost of the models, 5 redshift bins seem to be not su ffi cientfor the nulling technique to control the bias induced by GGIdown to a percent level. Going from 5 redshift bins to 10 red-shift bins is very rewarding in terms of bias reduction. However,we note that a decrease | b f / b i | doesn’t neccessarily indicate a N Z | b f b i | s=1, r=-2s=1, r=0s=1, r=2s=0, r=-2s=0, r=0s=0, r=2 Fig. 4.
Ratio of the nulled and the original biases for cosmolog-ical parameter Ω m as a function of number of redshift bins N z .Results for di ff erent GGI models are shown. Parameter s and r are the slopes of angular frequency and redshift dependence ofour power-law model (31). N Z F O M Fig. 5.
Figure of merit (FoM) as defined in (46) in the Ω m - σ plane as a function of number of redshift bins N z . FoM fromtwo-point (2p) measures, three-point measures (3p) and com-bined (2p + / GGG signal that is directly controlled byany of these methods. Between | b f / b i | and the original unbinnedGGI / GGG signal lies the binning process as well as the sum-mation over angular frequency bins and redshift bins. Since thesigns of the biases contributed by di ff erent angular frequenciesand redshifts can be di ff erent, there can be bias cancellation dur-ing these processes. In another word, | b f / b i | can depend on bin-ning choices.We then vary two cosmological parameters ( Ω m and σ ) andinvestigate how the original and the nulled parameter constraintschange with respect to the number of redshift bins available. For
10. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics
A = 4000 deg n g = 40 arcmin (cid:18) (cid:19)(cid:20) = 0.35 originalnulledcompressed0.760.800.84 (cid:21) (cid:22) (cid:23) (cid:24) w (cid:25) m (cid:26) w a (cid:27) (cid:28) (cid:29) (cid:30) w Fig. 6.
Projected 1-sigma(68 % CL) parameter con-straints from cosmic shearbispectrum tomography.Hidden parameters aremarginalized over. Theblack solid and blue dash-dotted ellipses correspondto the original constraintsand those after nulling,respectively. The black crossin the center of each panelrepresents the fiducial valuesadopted for the parameters,and the distance from thecenter of one ellipse tothe black cross reflects thebias caused by intrinsic-alignment GGI systematicson the corresponding param-eter. As nulling can be seenas a linear data compressionunder the constraint of thenulling condition, we alsoplot the constraints and bi-ases after an unconditionedlinear data compression asmagenta dashed ellipses forcomparison (see Sect. 5.4).this case we use all triangle shapes to enable a comparison withresults for two-point statistics.Our result (Fig. 5) shows that a further increase of the num-ber of redshift bins beyond 10 is not very rewarding in termsof information preservation as characterized by the FoM, in ei-ther 2p, 3p, or 2p +
3p cases. This suggests, when the possibilityof more redshift bins exists, the choice of redshift bin numbershould be based mainly on the requirement of bias reductionlevel in case of negligible photometric errors. When there arenon-negligible photometric errors, however, the information losswill probably be more severe, as found by Joachimi & Schneider(2009) for the two-point case.
The necessity of carrying out data compression in cosmol-ogy has long been recognized (e.g. Tegmark et al. 1997) and hasbeen ever increasing due to the increasing size of the data sets.In cosmic shear studies the survey area of next generation multi-color imaging surveys will be an order of magnitude larger thanthe current ones. The study of three-point statistics also implies ahuge increase in the amount of data directly entering the Fisher-matrix / likelihood analysis, compared to the two-point case.The basic principle of data compression is to reduce theamount of data while preserving most of the information. This isalready naturally encoded in the nulling technique. If one keepsonly the first-order weights for nulling, as we do in this paper,the nulling procedure reduces the number of data entries in eachangular frequency bin from the number of redshift bin ( i , j , k )combinations, to the number of ( i , j ) combinations, which meansroughly from N z to N z . The nulling transformation is linear Table 1.
Change of cosmic shear bispectrum statistical powerafter nulling (null) and linear data compression (compress). i null null / i compress compress / i Ω m - σ Ω m - w
637 123 19.3 % 428 67.2 % Ω m - w a
145 33 23.0 % 110 75.9 % σ - w
434 87 20.0 % 299 68.9 % σ - w a
101 26 25.4 % 72 71.3 % w - w a Notes.
Presented are FoM on two-dimensional parameter planes be-tween cosmological parameters Ω m , σ , w and w a . The cosmologicalparameters h , Ω b and n s are marginalized over. The second column isthe FoM from the original bispectrum; the third and fifth columns areFoM from the nulled and the compressed measures, respectively; thefourth (sixth) column shows the percentage of the third (fifth) columncompared to the first column, which reflects the relative informationloss through the nulling (the unconditioned compression) procedure. since the resulting nulled entry is a linear combination of k orig-inal entries weighted by the nulling weight (17). In the sense thatan “optimum” set of nulling weights is constructed, the nullingtechnique also intends to preserve as much information as pos-sible. But there is yet another additional constraining conditionin the nulling procedure: the nulling condition (7), which largelyconfines the shape of the nulling weights by requiring the ex-istence of at least one zero-crossing (see Fig. 2). In short, thenulling technique can be seen as a conditioned linear compres-sion of data.It is then interesting to know how much of the informationloss during the nulling process actually comes from the nullingcondition, and how much just comes from the fact that a data
11. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics
A = 4000 deg n g = 40 arcmin (cid:31) ! = 0.35 powerspectrum, nulledbispectrum, nulledcombined, nulledcombined, original0.780.800.82 " $ % w & m ’ w a ( ) * + w Fig. 7.
The thick green(gray) solid, thick blue(black) solid and thin blackdashed ellipses indicate 1-sigma (68 % CL) parameterconstraints from the nulledpower spectrum measures,bispectrum measures, andcombined. Hidden parame-ters are marginalized over.The distance from the centerof an ellipse to the blackcross reflects the nulled biason the corresponding pa-rameter. The original biasesfrom bispectrum measurescan be seen in Fig.6. Thethin black solid ellipsesover-plotted on to the cen-ters of the nulled combinedconstraint ellipses indicatethe statistical power (68%CL) of combined constraintsbefore nulling. Note the dif-ferent ranges of parameterscompared to Fig.6.
Table 2.
FoM before (‘ i ′ ) and after (‘ f ′ ) nulling and their ratio, using the cosmic shear power spectrum (2pt), bispectrum (3pt), andcombined (2pt + + / f 3pt, f 3pt, i / f 2pt + + / f Ω m - σ Ω m - w Ω m - w a
517 145 872 69 13.3 % 33 23.0 % 121 13.9 % σ - w
864 434 3832 132 15.2 % 87 20.0 % 488 17.2 % σ - w a
326 101 709 47 14.4 % 26 25.4 % 107 15.1 % w - w a
45 11 184 7.4 16.4 % 2.3 20.2 % 27 14.5 % compression process is naturally involved in nulling. To explorethis, we perform an unconditioned linear data compression, bysimply ignoring the nulling condition in the whole nulling pro-cedure i.e. dropping the Lagrange multiplier term in (43), butotherwise keeping the simplifications inherent to the analyticalapproach. The results are shown in Fig. 6. A summary of theFoM from the original and the nulled bispectrum measures aswell as the compressed measures is shown in Table 1.In contrast to nulling, an unconditioned linear compressiondoes not eliminate the parameter bias, but increases or reducessome of them marginally. Regarding the parameter constraints,although the increase in the size of the ellipses is much less thanin the case of nulling, around one third of the information interms of FoM is lost through compression, which means thatthe amount of degradation in parameter constraints after com-pression is not negligible. This suggests that keeping only thefirst-order terms contributes to non-negligible information loss.To regain part of this information, one could add higher-orderweights to the nulling procedure. But the di ff erence between thenulled and the compressed FoM serves as an indication for theinevitable information loss through the nulling process, which isimposed by the nulling condition. Besides constraining cosmological parameters using three-point cosmic shear alone, we investigate the combined con-straints from both two-point and three-point cosmic shear mea-sures. The performance of the nulling technique on cosmic shearpower spectrum tomography alone and the resulting constraintson cosmological parameters were presented in JS08. For con-sistency, we use the same setting for the cosmic shear powerspectrum as described for the bispectrum in Sect. 3. In partic-ular, we neglect photometric redshift errors, use only a limitedrange and number of ℓ -bins, and adopt a power-law intrinsic-shear alignment model with a form described by (36) in JS08and a slope of 0.4. We have confirmed the consistency betweenour power spectrum and bispectrum codes with those used inBerg´e et al. (2010). Our power spectrum code agrees also withiCosmo (R´efr´egier et al. 2008).Figure 7 shows the resulting constraint ellipses after nullingfrom the cosmic shear power spectrum analysis, the bispectrumanalysis, and the two combined. To show how much informationis lost during the nulling process, we overplot the original two-and three-point combined constraints on top of the nulled con-straint ellipses in Fig. 7, but center them on the corresponding
12. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics nulled constraints by subtracting the bias di ff erence before andafter nulling. The information content in terms of FoM for eachparameter pair is presented in Table 2.One sees that the amount of information contained in bispec-trum measures and power spectrum measures are indeed compa-rable. With bispectrum information added, typically three timesbetter constraints in terms of FoM are achieved, both before andafter nulling. This factor is smaller than the result in TJ04, al-though the same angular frequency range and the same set of 7cosmological parameters are chosen for both studies. Howevera direct comparison is prohibited by di ff erent fiducial valuesadopted and di ff erent survey specification.Through the nulling procedure, around 15 % of the origi-nal information in terms of FoM is preserved in the two-pointcase, and around 20 % in the three-point case. It is a bit higherin the three-point case, in accordance to the fact that a roughly N z → N z compression is involved in the three-point case and a N z → N z one in the two-point case, while this fact is due to thesummation over one redshift bin index during the nulling proce-dure (the same trend is evident in Fig. 5). The information lossis considerable, but it is a price to pay for a model-independentmethod. As we have discussed in the previous subsection, thedi ff erence between the information loss through the nulling andthe unconditioned compression procedures represents the in-evitable loss of information through nulling. However, this dif-ference is less than 50 % in the considered three-point case. Theother information loss is due to the simplifications we adoptedin this study, including using only the first-order weights, anddiscarding the measures with two or three equal redshift bins. Afurther detailed consideration of these aspects can regain part ofthe lost information. Another simplification we have made in thethree-point case is to use only triangles with three di ff erent an-gular frequencies. This reduces both the original and the nulledinformation contained in the three-point measures. However, thissimplification can be easily removed with a careful distinction ofall cases.Also notice that, the dependence of number of possible bis-pectrum modes, i.e. triangles, on the maximum angular fre-quency ℓ max is roughly ℓ , while that of power spectrum modesis roughly ℓ . For this study ℓ max =
6. Conclusion
In this study we developed a method to control the intrinsic-shear alignment in three-point cosmic shear statistics by gener-alizing the nulling technique. We showed that the generalizationof the nulling technique to three-point statistics is quite natural,providing a model-independent method to reduce the intrinsic-shear alignment signals (GGI and GII) in comparison to the lens-ing GGG signal.To test the performance of the nulling technique, we as-sumed a fictitious survey with a setup typical of future multi-color imaging surveys, and applied the nulling technique to themodeled bispectra with intrinsic-shear alignment contamination.The lensing bispectra (GGG) was computed based on pertur-bation theory, while the GGI signal was modeled by a simplepower-law toy model. We focused on the reduction of the GGIcontaminant, since GII can be removed simply by not consider-ing tomographic bispectra with two or three equal redshift bins.The reduction of the intrinsic-shear alignment contaminationat the three-point level by the nulling technique was demon- strated both in terms of the GGI / GGG ratio, and in terms of bi-ases on cosmological parameters in the context of an extendedFisher matrix study. In terms of the GGI / GGG ratio, a factor of10 suppression is achieved after nulling over all angular scales.Correspondingly, the biases on cosmological parameters are re-duced to be less than or comparable to the original statistical er-rors. We studied the performance of the nulling technique when5, 10, 15, or 20 redshift bins are available, and found that the per-formance on bias reduction, rather than how much informationis preserved during the nulling procedure, depends more signifi-cantly on the number of redshift bins. In case one requires bettercontrol of intrinsic-shear alignment, more detailed redshift in-formation allowing more redshift bins is the most direct way togo. When dealing with real data, there is one further sourceof complication which we did not consider in this paper,that is the photometric redshift uncertainty. The photomet-ric redshift uncertainty can be characterized by a redshift-dependent photometric redshift scatter and catastrophic outliers.Joachimi & Schneider (2009) studied the influence of photomet-ric redshift uncertainty on the performance of the nulling tech-nique at the two-point level. They found that the photometricscatter places strong bounds on the remaining power to con-strain cosmological parameters after nulling. The existence ofcatastrophic outliers, on the other hand, can lead to an incom-plete removal of the intrinsic (II, III) alignments as well as theintrinsic-shear alignments (GI, GII, GGI). However, methods tocontrol the photometric redshift uncertainty have been proposed.For example, recent studies concerning the problem of catas-trophic outliers point to the solutions of either limiting the lens-ing analysis to z < .
13. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics
Acknowledgements.
We thank Joel Berg´e and Masahiro Takada for the helpin comparing power spectrum and bispectrum codes. We are also gratefulto the anonymous referee for the helpful comments and suggestions. BJ ac-knowledges support by the Deutsche Telekom Stiftung and the Bonn-CologneGraduate School of Physics and Astronomy. This work was supported bythe RTN-Network ‘DUEL’ of the European Commission, and the DeutscheForschungsgemeinschaft under the Transregional Research Center TR33 ‘TheDark Universe’.
References
Abdalla, F. B., Amara, A., Capak, P., et al. 2008, MNRAS, 387, 969Albrecht, A., Bernstein, G., Cahn, R., et al. 2006, arXiv:astro-ph / / / Appendix A: Counting of triangles
A triangle is specified by six indices, i.e. three redshift binindices { i , j , k } and three angular frequency bin indices ℓ , ℓ , ℓ .To ensure that we count each triangle configuration only once,we set the condition that ℓ ≤ ℓ ≤ ℓ . Moreover, we would likethe first index among { i , j , k } in (36) to have the lowest redshift,i.e. z i < z j and z i < z k , for the convenience of performing thenulling technique. The possible { i , j , k } combinations under theseconstraints in the case of N z = Fig. A.1.
List of possible triangles (redshift bin combinations)with condition z i < z j and z i < z k when 4 redshift bins are avail-able. An angular frequency combination satisfying ℓ ≤ ℓ ≤ ℓ is chosen. Note that the redshift indices and the angular frequen-cies are linked in pairs due to the definition of the tomographicbispectrum (14). In this paper a default of 10 redshift bins is as-sumed.However, setting both conditions is problematic. Inspectingthe definition of the tomographic bispectrum (14), one sees thatthe redshift indices and the angular frequencies are linked inpairs, e.g. convergence κ in redshift bin i has angular frequency ℓ , which is not desirable since the smallest angular scale doesnot necessarily correspond to the lowest redshift. To solve thisproblem, we perform nulling three times for each general an-gular frequency combination with ℓ < ℓ < ℓ , swapping theredshift-angular scale correspondence in-between, thus allowingeach redshift to be able to correspond to any angular frequency.Note that the situation complicates a bit when two of theangular frequencies are equal, since then the swapping may leadto exactly the same configuration. To avoid this, we will restrict
14. Shi et al.: Controlling intrinsic-shear alignment in three-point weak lensing statistics ourselves to three di ff erent angular frequencies. This can excludea high percentage of possible configurations. In our case, i.e. 20logarithmically spaced bins between ℓ min =
50 and ℓ max =
10 15 20 NZ ||
10 15 20 NZ || b f b i ||